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INTERNATIONAL SYMPOSIUM

ON

OPERATOR THEORY OF NETWORKS

AND SYSTEMS

Volume 2 EDITORS:

N. LEVAN R. SAEKS

AUGUST 17-19, 1977

SPONSOR ED BY

TEXAS TECH. UNIVERSITY U.S. AIR FORCE OFFICE

OF SCIENTIFIC RESEARCH UNDER GRANT 77·3382

Hilton Inn -lubbock, Texas

OPERATOR THEORY OF NETWORKS AND SYSTEMS. INTERNATIONAL SYMPOSIUM ON.

Vol. 1. Concordia University, Montreal, Canada. August 12-14, 1975. 153 pages. Paper ......... :....................................... $22.00

CONTENTS (partial):

• Structure Result for Nonlinear Passive Systems • Frequency Response Methods in Multivariable Infinite Dimensional

Linear Systems • A Walsh Operational Matrix for Solving Variational Problems • The Feedback Interconnection of Multivariable Systems: Simplifying

Theorems for Stability • Linear Hilbert Networks Containing Finitely Many Nonlinear Elements • Linear Network Synthesis Using Iteration Methods • A Note on the Nagy-Foias Lossy and Lossless Space • An Output Control Problems Containing Input Derivatives • Contractive Transfer Ratios of Operator Network

(Standing Orders Accepted)

Additional Copies Available From:

WESTERN PERIODICALS COMPANY 13000 RAYMER STREET

NORTH HOLLYWOOD, CALIFORNIA 91605

Copyright © 1977 by Western Periodicals Company 13000 Raymer Street North Hollywood, California 91605

ii

PRE F ACE

The International Symposium on the Operator Theory of Networks and Systems is a biannual

conference which is guided, ~rom year to year, by a Steering Committee composed of

representatives from a number of universities. The current symposium is the second in

the series, and the first which Texas Tech University has had the honor of hosting.

Although originally conceived as a forum for the presentation of research results in the

area of operator theoretic techniques applied to network and system theory, the scope

of the symposium has grown considerably over the years. The present volumn therefore

encompasses research spanning the entire field of Mathematical Ne~work and Systems

Theory. Mathematical tools include algebraic techniques, algebriac topology, and

differential geometry in addition to the operator theoretic and functional analytic

techniques to which the symposium was originally directed.

The proceedings clearly convey the international interests in the area of Mathematical

Network and Systems Theory. Contributions of Scientists from three continents - Asia,

Eroupe, and North America are presented.

The Symposium committee is please to acknowledge the support of the Department of

Electrical Engineering, the College of Engineering, and the Graduate School of Texas

Tech University. Moreover, the symposium could not have been conducted without the

financial support of the U. S. Air Force Office of Scientific Research.

III

SECOND INTERNATIONAL SYMPOSIUM ON THE OPERATOR THEORY OF

NETWORKS AND SYSTEMS

STEERING COMMITTEE

C. A. Desoer University of Calif. at Berkeley Berkeley, CA.

J. W. Helton University of Calif. at San Diego La Jolla, CA.

N. Levan University of Calif. at Los Angeles Los Angeles, CA.

W. A. Porter Louisiana State University Baton Rouge LA.

R. Saeks Texas Tech University Lubbock, TX.

A. H. Zemanian State university of New York Stony Brook, N.Y.

R. W. Newcomb (Chairman) University of Maryland College Park, MA.

SYMPOSIUM ORGANIZING COMMITTEE

R. M. DeSantis (Program Chairman) University of Montreal

N. Levan (Co-Chairman) University of Calif. at Los Angeles Los Angeles, CA. Montreal Quebec

Canada

R. Saeks (Co-Chairman) Texas Tech University

Lubbock, TX.

IV

TAB LEO F CON TEN T S

1: COLLOQUIUM LECTURE

"Input-Output Properties of Interconnected Systems: Part I"

C. A. Desoer, Univ. of Calif. at Berkeley ----------------------

2: GENERAL SESSION I

"Bernstein Systems for Approximation and Realization"

W. A. Porter, Louisiana State Univ. ----------------------

"A Linear Systems Theory in Multidimensional Time"

A. V. Balakrishnan, UCLA ---------

"Optimal Control in Hilbert Space" M. Steinberger, M. Schumitzky, and L. M. Silverman, Univ. of Southern Calif. ---------------

"Solvability and Linerzation of Monotone Hilbert Networks"

V. Dolezal, SUNY at Stony Brook ----------------------------

4: SESSION ON FUNCTION ANALYTIC TECHNIQUES

"Time-Varying Input-Output Systems Whose Signals are Banach-Space­Valued Distributions"

A. N. Zemanian, SUNY at Stony Brook ----------------------------

"Causality and C Operators" A. Feintuch, °Ben Gurion Univ. ----------------------------

"Wiener-Hopf Techniques in Resolution Space"

L. Tung, and R. Saeks, Texas Tech Univ. -----------------------

"Approximate Controllability and Weak State Stabilizability"

C. Benchimol, UCLA ---------------

"A Modified Discrete Convolution Operator for Simulation of Linear Continuous Systems"

H. B. Kekre and D. B. Phatak, Indian Inst. of Tech. at Bombay

5: SESSION ON DYNAMICAL SYSTEMS

"Differential Systems on Alterna­tive Algebras"

R. W. Newcomb, Univ. of

2

6

.7

13

17

21

28

34

35

__ Maryland -------------------------- 36

v

"Lagrangians with Integrals: An Approach to the Variational Theory of Dissipative Networks"

V. M. Fatic, Tri-State Univ., and W. A. Blackwell, Virginia Polytechnic Inst. and State 38 Univ. -----------------------------

"An Estimation of Parameters in Parabolic Equation with Spacially­Varying Coefficients"

Z. Jacyno, Univ. of Quebec in 56 Montreal --------------------------

"The Singularity Expansion Method _ in Electromagnetic Scattering" ~9'l

D. R. Wilton, Univ. of Miss. ------~.

"Variational Principels for Mechanical and Structural Systems with Applications to Optimality of Design"

V. Komkov, Texas Tech Univ. -------- 63

"A Numerical Calculation Method for Simultaneous Ordinary Dif­ferential Equation of High Order by the Momentary Diagonalized Modal Property"

S. Azuma, Ibaraki Univ., and B. F. Womack, Univ. of Texas at Austin -------------------------- 66

6: SESSION ON CONTROL

"On Structurally Stable Nonlinear Regulation with Step Inputs"

W. M. Wonham, Univ. of Toronto ----- 72

"Frequency Domain Stability for a Class of Partial Differential Equations"

D. Wexler, Univ. Notre Dame de la Paix ------------------------- 76

"Densensitizing Observer Design for Optimal Feedback Control"

M. M. Missaghie, Sentrol Systems Ltd. _______________________________ 81

"Grassman Manifolds and Global Properties of the Riccati Equation"

C. Martin, NASA/AMES Research Center _____________________________ 82

"Generalized Operator and Optimal Control"

S. M. Yousif, Calif. State 86 Univ. at Sacramento ----------------

"Absolute Invariant Compensators: Concepts, Properties and'Applications"

R. M. DeSantis, Univ. of Montreal --------------------------- 90

8: GENERAL SESSION II

"Stability Tests for One, Two, and Multidimensional Linear Systems"

E. I. Jury, Univ. of Calif. at Berkeley _______________________ 91

"A Darlington Realization Theory of Optimal Linear Predictors"

P. DeWilde, T.H. Delft.

"Operator Theory Techniques for Finite Dimensional Problems"

92

J. W •. Helton, Univ. of Calif. at San Diego ______________________ 96

"The Relation Between Network Theory, Vector Calculus and Theoretical Physics"

F. H. Branin, IBM Corp. ----------- 97

9: SESSION ON LINEAR NETWORKS AND SYSTEMS

"Detached Coefficients Represen­tation and Degree Functor of a Polynomial Matrix with Applica­tion to Linear Systems"

Y. S. Ho and P. H. Roe, Univ. of Waterloo _______________________ 103

"A Representation of Impedance Function in Terms of the Poles and Zeros for Transmission Lines"

F. Kato and M. Saito, Univ. of Tokyo --------------------------109

"Evaluation of Constituent Matrices of an Analytic Matrix Function"

F. C. Chang and S. R. Pulufani, Alabama A&M univ. -----------------113

"On the Losless Scattering Matrix Synthesis via State Space Techniques"

A. L. Dobruck and M. S. Piekarski, Wroclaw Technical Univ. -----------116

"Algebraic Characterizat:.ion of Matrices whose Multivariable Charac­teristic Polynomial is Hurwitzian"

M. S. Piekarski, Wroclaw Tech. Univ. -----------------------------121

10: SESSION ON OPERATOR METHODS

"Contraction Operator of Class C and the Structure of a class of 0 Infinite Dimensional System"

D. Hedberg, Hughes Aircraft Co., and N. Levan, UCLA -----------127

"Discrete-Time System Operators on Resolution Sets of Sequences"

R. J. Leake and B. Swanimathan, Univ. of Notre Dame ---------------128

VI

"Some Aspects in a Theory of General Linear Systems"

R. H. Foulker, Youngstown State Univ. -----------------------

"Certain Aspects of Inverse Filters"

V. P. Sinha, Indian Inst. of Tech. at Kanpui, and H. S. Sekhon, Punjab Argicultural Univ. -----------------------------

"Adaptive Antenna Polarization Schemes for Clutter Suppression and Target Identification"

G. Ioannids and D. Hammers, ITT Gilfillan Inc. ----------------

"On Limitations Based on Properties of the Strum-Liouville Operators in the Synthesis Procedures of Non­uniform Lines"

Z. Trazaska, Inst. of the Theory of Elec. Engrg. and Elec. Measure­ments of Warsaw -------------------

11: SESSION ON NONLINEAR SYSTEMS

"A Theory of Best Appro~imation of Nonlinear Functionals and Operators by Volterra Expansions"

133

134

135

'140

L. V. Zyla and R.J.P. de Figueiredo, Rice Univ. - ____________ 143

"Continued Fraction Describing Functions for Bilinear and Multiplicative Nonlinear Systems"

C. F. Chen, Univ. of Houston, and R. E. Yates, u.S. Army Research Lab. at Redstone Arsenal ____________________________ 144

"Nonlinear Analysis of Gyrator Networks: A Numerical Example"

M. B. Waldron and M. A. Smithers, Univ. of Houston ___________________ 148

"Lie Series and the Power System Stability Problems"

R. K. Bansal and R. Subramanina, Pubjab Argicultural Univ. ---------- 152

"A Study of Varying Efficiency Multiserver Queue Models"

H. B. Kekre, R. D. Kumar, and H. M. Srivastava, Indian Inst. of Tech. at Bombay ----------------- 156

"Modular Design of the Network which Realizes Original jProgram"

S. P. Kartashev,. Univ. of Nebraska, S. 1. Kartashev, DCA Assoc. ________ 161

INPUT-OUTPUT PROPERTIES OF INTERCONNECTED

SYSTEMS

C.A. Desoer Dept. of Electrical Engineering and Computer Science

University of California Berkeley, Ca. 94720

ABSTRACT

The purpose of this survey paper is to review some recent results on the input-output properties of both linear and nonlinear inter­connected systems. The results presented deal primarily with quali­tative system properties such as stability and parameter sensitivity with emphasis being placed on the robustness of these properties.

BERNSTEIN SYSTEMS FOR APPROXIMATION AND REALIZATION*

William A. Porter**

1. INTRODUCTION

This summary deals with the approximation of non­

linear systems by polynomic operators. For per­

spective it is helpful to consider the familiar

Volterra series expansion on L2 given by

p(x) ,= kO + S k~ (a)x(a)da +Hk2 (a,B)x(a)x(B)dadB + J~!k3(a,B,y)x(a)x(B)x(Y)dadBdY + ...

where the kernels kO' kl, ••. ,kn

satisfy properties

suitable to an operator on L2

. For the obvious

reasons we refer to each term on the right hand

side as a power function. If the number of terms

is finite then p is said to be a polynomic opera­

tor. Our interest in polynomic operators centers

on their use as approximates of the more general

nonlinear functions on L2

•

The relevant lit~rature may be grouped into two

subcategories. -First-there is the analytic theory

in which f is assumed to have derivatives (Frechet

or Gateau) of all orders and p arises as a power

series expansion on a bounded domain. This line of

devElopment was initiated by Volterra [1] and was

first e.pplied in a systems setting by Weiner and in

ensuing years several others including [2], [3],

and [0]. Mere recently [5], [6], [7], [8] have

investigated the Volterra expansion of solutions to

nonlinear differEntial equations with current

emphasis on computation and convergence problems.

The analytic theory identifies the power func­

tions with the deri vat:iv'~s of the system function.

f, to be approximated. The requisite differenti­

ability is a severe condition, however, a fringe

benefit accrued is that the causality of the power

functions is dictated by the causality of f.

In an independent line of development the polynomic

approximation problem has been approached as a

generalization of the classic Weierstrass result.

In this setting the function, f, to be approximated

need not be differentiable. Emphasis is placed on

uniformly approximating f, by the polynomial p,

over an arbitrary compact set. In this approach

the causality issue is less trivial. The computa­

tion of the power functions, which no longer repre­

sent derivatives, has here to fore been obsure. In

summary we introduce the concept of a Bernstein

differential system. This system provides one con­

structive realization of the Weierstrass approach.

An incidental bonus of the Bernstein system is a

pseudo sampling-theorem for systems. In short,

given an arbitrary system with a prescribed contin­

uity modulus, the density with which one must input­

output s~~ple in order to be able to approximately

reconstruct the system is established.

For efficiency of presentation we shall in many

cases be overly restrictive in the assumptions made,

for example we consider only Hilbert spaces. Also

the development is purposely c0nstrained so as to

ilse classical results on the Bernstein polynomials.

*Supported in par .. by the United States Air Force Office of Scientific Research, Grant No. 77-0352.

**Department of Electri_al Engineering, Louisiana State University, Baton Rouge, Louisiana.

2

In closing this introduction it is noted that the

admittance characteristic of the enhancement mode

MOSFET transistor provides an almost exact square

law (see [9]). It is easily shown that squaring

devices can be used to construct general polynomic

operators of the type necessary to realize the

Bernstein system. Thus it appears that the topic

of polynomic system approximation may have a ready

practicality in terms of microcircuit technology.

2. WEIERSTRASS APPROXIMATION

As the technical work of this study deals primar­

ily with the Weierstrass approximation it is use­

ful to comment in somewhat more detail on the

existing literature. In this regard we cite first

the original contribution of Weierstrass [10]

whose fundamental result in contemporary form

reads as follows.

Let f be an arbitrary continuous function on R,

the real line. Let D cR be an arbitrary compact

set. The~for every E > 0 there exists a finite

polynomial p, such that sup {If(x) - p(x)l:

XED} < E. The Weierstrass result, over the years,

has drawn the attention of several distinguished

mathematicians including Frechet [11], Bernstein

[12] and Stone [13] who investigated the relation­

ship to power series expansions, constructive

methods for finding the polynomial and extended

the result to Rn among other things.

More recently Prenter [14] considered a real sep­

arable Hilbert space H, and showed that if K c:: H

is compact, £ > 0 and f continuous on H then there

exists a finite polynomic operator p, such that

~~~llf(X) - p(x)11 < E

In a similar effort Prenter [15] and Ahmed [16]

were able to use normed linear spaces.

The Prenter result [14], for example, states that

if f is a continuous function on real L2

(a,b) then

on every compact subset there exists a finite

number of kernels kO' kl, ... ,kn such that p of

equation 1 is an E-approximat~on for f. Actually

we would suspect more, namely, that if f is causal

then each kernel ki is causal, tha: is for instance

3

0, T > t. More generally, can a caus-

ality structure be superimposed on the function

and its approximation? This question is answered

affirmatively in [17] and [18].

The setting for [17] and [18] is a Hilbert resolu­

tion space* {H,pt } where H is real and separable.

The set K c:: H is always compact. The sets: C, SC,

M, C(K), and P denote the causal, strictly causal,

memoryless, continuous on K c: H, and polynomic

functions, respectively, on {H,pt }. For brevity

we shall say that P is dense in C(K) in the sense

of Prenter's theorem.

The results of [17] include the following. The set

P" SC is dense in elK) n SC. In L2 the stronger

result that pO SC is dense in C(K) A C is also

established. This last result does not abstract.

In t2 it is known [18] that P n SC is not dense in

C(K) " c.

All of the above results are nonconstructive in

that they give no clue as to finding the polynomic

approximate of a given function. On the real line,

however, several constructive forms of the Weier­

strass result do exist and the Bernstein polyno­

mials constitute one of the more intriguing

approaches to such constructions. In the study we

develop a generalization of the Bernstein poly­

nomials to real Hilbert space. Using a causal data

interpolation scheme identification of the p E P~SC

that approximates f E C(K)~C results.

To be more explicit let {H,P}be any Hilbert resolu­

tion space. The set QCH is compact and E > 0 is

arbitrary, ~ denotes a tuplet of indicies. fC .!l

denotes a causal polynomial constructed explicitely.

We summarize our first result in the following

theorem

Theorem (1) If causal f is bounded on n then

for each X£Q and £>0 there exists .!l such that

I If(x) - fC(x)1 I < E at every continuity point n

of f. If f is continuous on n then .!l exists

such that ~~~I If(x) - f~(x)1 I < E.

The explicit nature of theorem (2) has answered the

central theoretical questions in rather complete

form. It is of interest, however, to give a real­

ization of the causal Bernstein function, fC

, in ~

state variable form. For this attention was fo-

cused on the real 12 [a,d] for O<d<oo e~uipped with

the usual inner product and the resolution of the

identity taken to be the familiar truncation

operators.

A second result which has been established is

summarized in the theorem

Theorem (2) In 12

, for every f E C(K)~ C and

E > a, there~ists a differential system

~(t) A(t)z(t) + B(t)u(t), z(O) = 0,

wet) ~(z(t),t)

where ~({·),t) is polynomic, such that the

map w = p(u) satisfies

s~PI If(u) _ p(u)1 I < E.

For obvious reasons the e~uations of this

theorem are called a Bernstein system.

4

REFERENCES

[1] Volterra, V., Theory of Functionals, Dover Publications Inc., New York, 1959.

[2] Bedrosian, E. and S. O. Rice, "The Output Properties of Volterra Systems Driven by Harmonic and Gaussian Inputs," Proc. IEEE, Vol. 59, pp. 1688-1707, 1971.

[3] Parente, R. B., "Nonlinear Differential Equations and Analytic Systems Theory," SIAM J. Appl. Math., Vol. 18, pp. 41-66,1970.

[4] Van Trees, H. L., "Functional Techniques for the Analysis of the Nonlinear Behavio~ of Phase-Locked Loops," Proc. IEEE, Vol. 52, pp. 894-911.

[5] Brockett, R. W., Volterra Series and Geo­metric Control Theory, Automatica, Vol. 12, 1976.

[6] Bruni, C., DiPillo, G., Koch, G., "On the Mathematical Models of Bilinear Systems," Ricerche di Automatica, Vol. 2, pp. 11-26, 1976.

[7] Krener, A. H., "Linearization and Bilinear­ization of Control Systems," Proc. 1974 Allerton Conf. on Cir. and Sys. Th., pp. 834-843, 1974.

[8] Gilbert, E. G., "Volterra Series and the Re­sponse of Nonlinear Differential Systems," Trans. Conf. on Inf. Sci. and Systems, Johns Hopkins University, March 1976.

[9] R. S. Cobbold, "Theory and Applications of Field-Effect Transistors," Wiley-Interscience, New York, 1970, Section 7.1.3.

[10] K. Weierstrass, "Uber die Analytische Dar­shell-bankeit Sogenannter Willkurlicher Funk­tionen Reeler Argumente," Math. Werke, III Bd., 1903.

[11] M. Frechet, "S~r les Fonctionelles Continues," Ann. de l'Ecole Normale Sup., Third Series, Vol. 27, 1910.

[12] S. Bernstein, Demonstrqtion du Theoreme de Weierstrass, fondee sur Ie calcul des prob­abilities, Com. Soc. Math., Kharkow, (2), 13, 1912-13.

5

[13] M. H. Stone, "The Generalized Weierstrass Approximation Theorem," Math. Mag., Vol. 21, pp. 167-183, 1948.

[14] P. M. Prenter, "A Weierstrass Theomem for real Separable Helbert Spaces," J. Approxi­mation Theory, Vol. 3, No.4, pp. 341-351, Dec. 1970.

[15] P. M. Prenter, "A Weierstrass Theorem for Real Normed Linear Space," Bull. American Math. Soc., Vol. 75, pp. 860-862,1969.

[16] N. W. Ahmed, "An Approximation Theorem for Continuous Functions on Lo spaces," Univ. of Ottawa, Ottawa, Can., TR-73-l8, Nov. 191 .

[17] W. A. Porter, T. M. Clark, and R. M. DeSantis, "Causality Structure and the Weierstrass Theorem," J. Math. Anal. Appli., Vol. 52, No.2, Nov. 1975.

[18] W. A. Porter, "The Common Causality Structure of Multilinear Maps and their Multipower Forms," J. Math, Anal. Appl., Vol.. 52, No.2, Nov. 1975.

A LINEAR SYSTEMS THEORY IN MULTIDIMENSIONAL TIME

A. V. Balakrishnan Department of System Science

university of California Los Angeles, Ca. 90024

ABSTRACT

A theory of linear systems which do not have a time-like parameter is formulated. The input-output properties of such systems are studied and a notion of state is developed. The resultant theory is applied to the study of Markov random fields.

6

I _

M.

OPTIMAL CONTROL IN HILBERT SPACE

* * * Steinberger, A. Schumitzky and L. Silverman * Department of Electrical Engineering ':'Department of Mathematics

University of Southern California Los Angeles, California 90007

Abstract

Using the key concepts of causal factorization and state space under Nerode equivalence, we show that for a bounded, linear, strictly causal operator on Hilbert resolution spaces with quadratic cost functional, the optimal con­trol may be expressed in memoryless state feedback form provided the forcing function is expressed as an initial state. We further conjecture that there is no causal feedback which realizes the optimal control for a larger subspace of forcing functions.

I. IN:TRODUCTION

In this paper will will examine the optimal con­trol of a linear system with respect to a quad­ratic performance criterion. In particular, we will be interested in the cases in which the optimal control can be put in causal state feed­back form. This problem was solved for the case of finite dimensional differential systems of the form

input-output point of view rather than from an explicit representation for the dynamics of the plant to be controlled. That is, we follow Porter [9] in studying systems of the form

y = Tu + f

where T is a linear, bounded, strictly causal operator from one Hilbert resolution space HI to another, HZ; and f is a given forcing func­tion. Using the concept of Nerode equivalence promoted by Kalman [10] and since adopted by many other authors, we construct a state space for T. We then go on to construct a Hilbert resolution space of state trajectories. Given this space of state trajectories, we then show that if the operator (I + T*T) admits a strictly causal factorization (in the manner of Gohberg and Krein (11]) and f is generated by an initial state, then the optimal control can be realized in memoryless state feedback form. Further­more, the optimal feedback operator can be extracted fairly directly from the factorization of (I+T*T).

x Ax + Bu

y Cx

by Kalman [1] with highly successful results. Subs equent efforts have been directed toward expanding Kalman's results to cover systems described by generalized versions of the above equations (see Delfour and Mitter [Z], Lions [3], Lukes and Russell [4]. Datko [5], Prit­chard [6], and Curtain and 1critchard [7]).

Recently, we have written a paper [8] in which we attack the optimal control problem from an

This work was supported in part by the National Science Foundation under Grants ENG 76-14379, GP - Z0130 and by JSEP through AFOSR/AFSC under Contract F446Z0-7l-C-0067.

7

In this paper we will review the pertinent de­tails of [8]. Section II will contain the neces­sary mathematical background and major supporting theorems, Section III will present the actual optimal control results, and Section IV will give some brief concluding remarks.

II. MA THEMA TICAL BACKGROUND

II. 1 Resolution of the Identity

A Hilbert space H is said to be equipped with a resolution of the identity if for every t e [to' t..,] a closed subset of IR, 3: pt:H"'H 3

to (i) p u 0 Vue H

(ii)

(iii)

(iv)

to:> p u u

ptp" = p" pt = pt

t t (u l , P uZ)H = (P u l ' uZ)H

Vue H

V ,. ~ t

Furthermore, we can define the complementary projection Pt: H'" H, Pt '" I - pt from wh ich we have I = pt + Pt' motivating the term "resolution

of the identity."

In the rest of this paper, we will associate with HI (HZ) the family of projections pt(i5

t).

A more complete treatment of this subject and the subject of causality can be found in the excellent article by DeSantis (13].

II. 2 Causality

In common usage, an operator is said to be causal if and only if past outputs are only effected by past inputs. In mathematical terms

we have:

Definition Z.l: An operator T is ~ iff

ptTPtu = ptTu , V t e [to' to>], u e HI

Similarly we have:

Definition 2. Z: An operator T is anticausal iff

An operator which is both causal and anticausal is called memoryless since its present output depends not on the past or the future but on the present input. Under reasonable conditions, any causal operator can be additively decom­posed into the sum of a memoryless operator and an operator which has no memoryless part. An operator which has no memoryless part is said to be strictly causal. We can define

strict causality more precisely using partitions

of the time set [to' t." ] •

Let 0 be a countable family of finite partitions of the time set [to, to>] with the following

properties:

8

(i) 11 cO VnelN n n+I

(ii) for every t e [to' to>] and e > 0 3: n, k :;I

o < tk

- t < e n

t k+l

For notational simplicity let t, k - P k P n n t

t k +l ~k ~ ~ n (:, - Pk P

n t n

causality:

n

Then we can define strict

Definition Z.3: An operator T is strictly causal

iff T is causal and

lim N(n) ~k k :0 [:; T (:,

n n o

k=O

for any family as defined above. This limit is

taken in the uniform topology.

The following results are immediate:

Pt T Pt = T Pt

pt T pt = T pt

(3) If S:Hr-+Hz is strictly causal and T: HZ'" H3 is causal and bounded, then ST is strictly causal.

(4) If T: HI ... H Z is causal and bounded, and S: HZ'" H3 is strictly causal, then TS is

strictly causal.

For convenience, we will hereafter call an operator T an LBSC operator if it is linear,

bounded, and strictly causal.

II. 3 The State Trajectory Space

In order to talk about state feedback in a Hil­bert resolution space setting, we first need a space of state trajectories which is a Hilbert resolution space. Such a space' is constructed

in [8].

The most basic concept underlying our construc­tion of a state space is the concept of Nerode equivalence, which, in our setting is defined as follows:

Definition 2.4: Two inputs U 1,u2 €H I are Nerode equivalent at time t with respect to operator T, written u I , T': t u 2 , iff

~ t ~ t P

t T P u

l P

t T P u 2

Clearly "T, t" is an equivalence relation over Hi' so that it is possible to form equivalence classes of HI under "T' t". This set of equi­valence classes, XT(t),' will be called the "state space of T at time t." Using a funda­mental theorem of modern algebra, Saeks [12] has shown that ESt T pt can be decomposed into the product of two mappings, kT(t) and gT(t) in such a way that the following diagram com-mutes:

H -H

lk~ ~)2 XT(t)

We then go on to form, for any LBSC operator T, a Hilbert resolution space of state trajec­tories which may be thought of informally as being of the form

x( .) where x(t) € XT(t) for each t € [to' tex> ]

We call this space XT

and define kT and gT es sentially as

(kT u)(t)

and

We prove that gT is memoryless (as has been stated by Saeks [12]), kT is strictly causal, and T = gT kT • The reader wishing a more rigorous treatment should refer directly to [8] as we have glossed over many important details here.

Having the required tools, we then go on to prove a theorem concerning the relationship of open loop control to state feedback control,

9

motivated by an analogous theorem by Hautus and Heymann [14] in the discrete time case. In order to quote this theorem, we will need a lemma due to DeSantis [13] which states that if W is LBSC, the (I+W)-I=I+V where V is LBSC. Then we have

Theor ern I: Given k T : HI -+ X T , kT an LBSC operator, and a dynamic input transformation (I+W):HI-+H I , where W is LBSC (therefore let (I+W)-I=I+V where V is LBSC).

Then there exists linear bounded memoryless state feedback F: X T -+ HI such that

v u + Fx

x and

v (I+W)u iff

The above condition may be interpreted as saying that the state of V may be found as a part of the state of T for any input at any time. The systems governed by the above equations are diagramed below.

u x

( a )

u x

( b )

Figure

What Theorem I says is that when the state of V can be found in the state of T, the open loop controller of Figure I b can be implemented as a state feedback controller of the type shown in Figure lao The solution to the optimal con­trol problem will es s entially consist of deriving the appropriate open. loop controller and then showing that Theorem I applies, thus giving us the desired state feedback control.

III. THE OPTIMAL CONTROL PROBLEM

With the background given in the previous sec-

p

tion, we are in a position to state in full the optimal control problem we wish to solve.

The t Problem: Given

(1) Two Hilbert resolution spaces HI and

HZ;

(Z) An LBSC operator

(3)

(4)

find

T : HI -t HZ

t~

A forcing function f = T P u;

Z A cost functional J(u, t, f} = I P t u I H +

IPt(Tu+f}l~ ; 1 Z

Uo e: Pt

HI :3 J(u, t, f} is minimized.

Note that the assumption f = T pt;: is equivalent to the assumption that the initial conditions of this problem are given as an initial state. In [8], we motivate the conjecture that all pro­blems of this type for which an optimal state feedback law exists can be expressed in this form by first noting that f can be decomposed

as

and that the optimal control for fZ' is identically 0, and then using Theorem I to suggest that a causal feedback solution only exists when f=Tpt;:.

The t-problem may be easily solved using the projection theorem (15] to yield

However, since T*T is neither causal nor anticausal, the causality properties of this solution are still very much in doubt. Much can be resolved, however, through the use of causal factorization theory.

We will now assume that (I+T*T) admits a strictly causal factorization; that is, we will as sume that there exists an LBSC operator V

such that

,-1+ T T

(therefore let (1 + V}-l = I + W where W is LBSC.) Then we have

-Pt(V + V* + V'''V} pt~ _p (V + V '''V) pt~

t

(V'" is anticausal)

t~

PT(I+V)UO

= -Pt YP u

(by multiplying by Pt(I+ W"-))

For notational Simplicity, we define

~

as the operator which maps the forcing input u

onto its optimal control uO •

Next we will prove that the (I + W) derived from factorization theory is exactly the dynamic input transformation needed to apply Theorem 1 to the optimal control problem.

10

Consider the feedback system of Figure Z

t~ P u

t~ u+P u

Figure Z

along with its associated equation

u t~

-Pt

V(u+ P u)

t~ (I + P

t V) u -P

t Y P u

but, since ue:PtH I , we have

t~ ~

P t u::: - (I + ~Nl P t V P u = Pt@(t) u

Thus the above system solves the t-problem for t~

f=TP u.

The following lemma will be found to be the central reason for the use of the notion of "state" in the solution of the optimal control

problem.

Lemma 1: Assume the strictly causal factori­zati';,n 1+ T*T = (1+ V'''}(I + V). Then V t e: [to' tee ];

ul T,t uZ-'ul V':t uZ·

Proof: t t

Pt

T P ul

= P t T P U z ~, t ," t

Pt

T Pt

T P u l = Pt

T P t T P Uz

P T*T pt u = P T'~T pt u tIt 2

(T* is anticausal)

'" * t * * t Pt(VtV tV V)P ul

= Pt(VtV tV V)Pu2

* t * t Pt(V tV V) P u l = Pt(VtV V) P u 2

(V* is anticausal)

~ t * t Pt(ItV~)PtVP u l = Pt(ItV )PtVP u 2

or, multiplying by Pt(I t W*),

t t P

t V P u

l = P

t V P u

2

Further insight can be gained by multi­plying the above equation by (I t W) to get the equation

Although it is not used in the theorem to follow, it is the heart of the concept. In words, this equation states that the initial state carries all the infor mation nec es sary for the derivation of the subsequent optimal control. This is exactly why state feedback optimal control is possible.

Note, ~ow, that by Lemma 1 and the fact that (I t W)-l = (I t V), Theorem 2 applies to the dynamic input transformation (I t W). Thus we have proved the following theorem:

Theorem 2: For any system of the form

y = Tu t f

wher e T: HI" H 2 is a bounded, linear, strictly causal operator on Hilbert resolution spaces, and I + T*T admits a strictly causal factoriza­tion, there exists a bounded· memoryless linear

t~ state feedback F: 'X T" H; such that V f = T P u, ~ e HI' t e [to,tCDj, the control Uo whic~ mini­mizes J(u, t, f) = \ :t(T P t u + f)\ H2 t \ P t U\H

l over

the clas s of admls sable controls u e P t H 1 satisfies

Review of Proof:

(1) P Uo minimizes J(u, t, f) over the class of admissable controls u e PtH l where f = T ptu:

iff PtUO = Pt@(t)U:.

(2) PtuO=!\@(t)\I: if£~PtUO=-PtV(PtUot pt \1:) where I t T~T = (I t V~)(I + V).

(3) Using Theorem 1 and Lemma I, there exists F linear, bounded and memoryless such that -V = FkT; 1. e., the following diagram com­mutes:

II

where F = -gv h •

IV. CONCLUSIONS

Using the key concepts of causal factorization and state space under Nerode equivalence, we showed that for a bounded, linear, strictly causal operator on Hilbert resolution spaces with quadratic cost functional, the optimal control may be expressed in memoryless state feedback form provided the forcing function is expressed as an initial state. We further conjecture that there is no causal feedback which realizes the optimal control for a larger subspace of forcing functions.

[1]

[2]

[3]

[4]

[5]

[6]

(7]

BIBLIOGRAPHY

R. E. Kalman, "Contributions to the Theory of Optimal Control:' Bol. Soc. Mat. Mexi­~, 5 (1960), pp. 102-119.

M. C. Delfour and S. K. Mitter, "Controlla­bility, Observability and Optimal Feedback Control of Affine Hereditary Differential Systems,"SIAM J. Contr., 10 (1972), pp. 298-327.

J. L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations," Springer-Verlag, Berlin, 1971.

D. L. Lukes and D. L. Russell, "The Quad­ratic Criterion for Distributed Systems," SIAM J. Contr., 7 (1969), pp. 101-121.

R. Datko, "A Linear Control Problem in Abstract Hilbert Space," J. Diff. Equ., 9 (1971), pp. 346-359.

A. J. Pritchard, "Stability and Control of Distributed Parameter Systems Governed by Wave Equations," IFAC Conference on Distributed Parameter Systems, Banff, Canada, 1971.

R. Curtain and A. J. Pritchard, "The In­finite-Dimensional Riccati Equation for Systems Defined by Evolution Operators," SIAM J. Contr., 14 (1976), pp. 951-983.

[8] M. Steinberger, A. Schurnitzky and L. M. Silverman," Optimal Causal Feedback Control of Linear Infinite Dimensional Systems," submitted to SIAM J. Contr.

[9] W.A. Porter, "A Basic Optimization Problem in Linear Systems," ~ Sys. Theory, 5 (1971), pp. 20-44.

(10] R. E. Kalman, "Lectures on Controllability and Observability," CIME Seminar on Controllability and Obs ervability, Bologna,

Italy (1968).

[11] I. C. Gohberg and M. G. Krein, "On the Factorization of Operators in Hilbert Space," Am. Math. Soc. Trans., Ser. 2,

51 (1966), pp. 155-188.

[12] R. Saeks, "Resolution Space Operators and Systems," Springer-Verlag, New York, 1973.

[13J R. M. DeSantis, "Causality Theory in Systems Analysis," IEEE Proc., 64 (1966), pp. 155-188.

[I4] M.l... J. Hautus and M. Heymann, "Linear Feedback - An Algebraic Approach," Center for Mathematical System Theory, UniverSity of Florida, 1976.

[15J D. G. Luenberger, Optimization by Vector Space Methods, Wiley, New York, 1969.

12

., - / I

SOLVABILITY AND LINEARIZATION OF MONOTONE HILBERT NETWORKS

Vaclav Dolezal State University of New York at Stony Brook

Stony Brook, New York

Abstract

Sufficient conditions for solvability of a class of monotone Hilbert networks are given. Moreover, a linearization of a nonlinear monotone Hilbert network in a neighborhood of an operating point is discussed.

1. INTRODUCTION

In the first part of the paper we give sufficient

conditions for solvability of a Hilbert network

some elements of which are described by monotone

operators defined only on subsets of the underly-

ing Hilbert space. As a special case we consider

a finite nonlinear LRC-network, whose inductors

are linear, time-varying.

In the second part we discuss a linearization of

a nonlinear monotone Hilbert network in a given

neighborhood of an operating point, which is sub-

optimal in a certain sense. An estimate for the

difference between the exact and approximate

current distribution is established.

2. SOLVABILITY

A Hilbert network ~ is called solvable, if for any

excitation by EMF and/or current sources there

exists in ~ a current distribution obeying

Kirchhoff laws. Known effective results concern-

ing solvability ([1] or ThEDrems 4, 5 in [2J) make

the assumption that the operators describing

\3

network elements are defined on the entire under-

lying Hilbert space H. Consequently, these re­

sults are unapplicable, if the network contains

d1fferentiators.

The theorem given below attempts to fill this gap.

It is based on properties of maximal monotone

operators [3J, [4J and has the following physical

interpretation:

If all elements in a (finite or infinite) monotone

network ~ which are described by operators defined

on the entire space are removed and a unit resis­

tor is inserted in every branch, and if the net-~

work ~l thus obtained is solvable, then ~ is also

solvable.

To avoid repetition of definitions, we will use

the notation and concepts introduced in the survey

paper [2J.

Theorem 2.1. Let H be a real Hilbert space, let G

be a locally finite oriented graph having

c2

:s; ~O branches, and let D CH c2 , N; n D f ¢.

A C2 zl:n-H is a monotone operator,

A C2 C2 Z2: H - H is a hemicontinuous operator

such that A A IP

(Z2Xl - Z2x2' Xl - x2> ~ c II xl - x21

for all x~,

and p >1,

x e HC2 with some fixed 2

(2.1 )

c >0

(iii) the Hilbert network 711 = (21

+I,G) possesses C2

a solution for any e e H

Then, for any e eHc2 , the network ~= (Zl +Z2,G)

possesses a unique solution in B corresponding to

e.

Using Theorem 2.1, we can prove a solvability re­

sult for a finite LRC-network with linear time-

varying inductors, whose underlying space is the

real space 12

[0,TJ (we will write 12 in the

sequel).

Definition. Let G be a finite oriented graph hav-

ing c2

branches, and let 1 S k S c2

•

(i) there exist loops sl, s2,

Assume that k

(ii )

•• , S such

that, for each m = 1, 2, ., k, the loop

Sm contains b and does not contain any m

other branch in the set tbl

, b2

, ••• , bk }, t(t) is a symmetric k x k matrix having a

continuous derivative t'(t) on [O,TJ such

that t(t) and t'(t) is positive definite

and positive semi-definite, respectively,

for each t e: [0, T J, A C c

(iii) T· 1 2 -1 2 is an operator • . 2 2A k c -k c

Let the operator 1 : K x 1 2 - 1 2 be defined by 2 2

(Lx)(t) = (1(t)x(t)} I, (2.2)

where 1(t) = Lr t(t) OJ is a c x c matrix, and K ° ° 2 2

is the space of all absolutely continuous functions

on [O,TJ. Then the Hilbert network 11 = (L+T,G)

will be called 1-proper.

Clearly, this definition requires that all induc­

tors (and possible mutual couplings) are confined

only to the branch~s bl

, b2

, ••• , bk

• Also note

that the operator T describes the behavior of both the resistors and the capacitors in the network.

Theorem 2.2. Let G be a finite oriented graph

14

having c2

branches, let lSk S C2

and let j , j , A c c l 2

••• , jk be real numbers. Let T: 1/-122 be a

hemicontinuous operator such that, for some c > 0

and p >1, we have

A A liP (TXl - TX2 , Xl - x2) ~ c Ilxl - x2 C2 A A A

for all Xl' x2

e:12

, and let 71=(1+T,G) be 1-c A

proper. Then, for any ee:12

2 , '1lpossesses a

unique solution i = [i l corresponding to e n-o k c2-k 0 () 0

1 e:K X12

and 1 0 =J for m=l, 2, •. m m

such 1h1:t

• , k.

Observe that Theorem 2.2 has the following

physical interpretation: if a network '1l satis­

fies the hypothesis, then for any EMF's el

, e2

,

••.• ' ec e: 12 and any values jl' j2' ••. , jk

there exi~ts in ~ a unique current distribution

il

, i2

, ••• , i e 12 such that, for m = 1, 2, c2

., k, im is absolutely continuous and satis-

fies the initial condition i (O)=j • m m

3. 1INEARIZATION

The problem dealt with in this part resembles the

small-signal analysis of nonlinear networks [5J,

but differs from it in several aspects. While in

the small-signal analysis a "strictly local" ap­

proximation (Frechet derivative) is used for ob-

taining an approximate solution, in the present

approach we use a certain "global" approximation.

Also, the Hilbert network setting is, of course,

more general.

To explain the underlying idea, consider a Hilbert

network ~= (i,G). Assume that we know the current

distribution iO in '1l corresponding to some EMF

vector eO (operating point), and that we seek dis­

tributions ie*' which correspond to excitations

eO +e*, where the e*'s satisfy the inequality

Ile*!1 Sr with some given r >0. To find approxima­

tions to the ie*'s, we linearize the network '1l

in a vicinity of iO

• More specifically, defining ,... I ,. I '" A

the operator Z by Z S = Z( iO +S) - ZiO' we assume

that we can find a linear operator Zo which satis­

fies the inequality liz 'e: -zosii sail e: lion a certain

ball centered at the origin, where a >0 is not

* t~~ large. If ie* is the s~lution of the linear

netw~rk ~o= (Zo,G) corresp~nding to e*, we will

take io + i* * as an appr~ximati~n to the solution A e

i * ~f 'TI. e

The overall relative error

A = sup II i *- (iO+i** )11·lle* i1-1

r lIe*ll~r e e e*f 0

(3.1)

depends, of course, ~n the choice ~f ZO' and con­

sequently, on the constant a. However, in speci­

fic cases ~f networks it is usually not difficult

to c~nstruct Zo s~ that, for a given r >0, a is

small en~ugh. In this context, let us emphasize

the fact that taking the Frechet derivative of

Z at iO f~r Zo (a counterpart of the small-signal

analysis) need not lead to the smallest values of

a.

It turns out that if Z is strongly monotone, we

can give a simple upper bound for Ar' which is

roughly prop~rti~nal to a for a small.

The theorems that follow use again the notation

introduced in [lJ or [2J. Without loss of gen­

erality we assume that the operating point eO and

the c~rresponding solution iO of ~ are zero. Also

note that Z is not required to be single-valued.

Theorem 3.1. Let H be a real Hilbert space, let

~ = (Z ,G) be a Hilbert network with Z being 'a set c

mapping defined on a nonempty subset D CH 2 such

that 0 € D, and let r >0. Furthermore, let A c c

F=X*(NAnD)CH 0, and let W :F ... e(H 0) be defined a A*A.A

by W = X zt.. Assume that

(i) there exists b >0 such that

(Yl -Y2' xl -x2)~b Ilxl - x211

2

for all xi €F, Yi €WXi ' i=l, 2,

(ii) there exists a linear bounded operator

(3.2)

c c ZO:H 2"' H 2 and a constant a with O<a<b

such that

15

I .... * .... ,. for all x €F nB and y eWx, where Wo =X ZOX,

c R -1 B~= (x:x€H 0, IIxll~R} and R=b r,

(iii) F nB~ is dense in B~.

If i €Hc

2 is the (unique) solution of ~ correspond-c A *

ing to some e* € H 2 with Ilx*e*1I ~r, and i is the

(unique) solution of "b = (Zo,G) corresponding to

e*, then

II i - i* II :s;ab-l(b -aflllx*e* 11, -1 -1

i.e., A ~ab (b-a) • r

(3.4)

Moreover, in (ii) the requirement that (3.3) holds

can be replaced by the stronger assumption that

r c2 for all v € D nBR

and u e Zv, where BR = tV : v € H ,

11 v II :S;R}.

A

The assumption concerning D and Zo made above can

be modified. Indeed, we have

Theorem 3.2. The Theorem 3.1 remains true, if its

conditions (ii) and (iii) are replaced by the

following assumptions:

(ii)* BRCD for R=b-lr, A C2 c2

(iii)* there exists a linear operator Zo : H "'H

and a constant a with O<a <b such that (3.3)

holds for all x € B~ and yeWx.

If, in addition, c

operator on H 0,

* c2 for any e e H ,

c D = H 2 and W is a hemicontinuous

A

then ~ possesses a unique solution c

i.e., (3.4) holds for any e*eH 2

Note that still another (but weaker) result like

Theorem 3.1 can be proved, provided it is assumed c

that D is a linear subspace of H 2 and ~ is

solvable. However, we omit the details.

References

[lJ V. Dolezal, Nonlinear Networks, Elsevier Scientific Publishing Co., 1977.

[2J , Basic Properties of Hilbert Networks, IEEE Trans. CAS, vol. CAS-23, #8, (1976), pp. 490-497.

[~l H. Brezis, Monotonicity Methods in Hilbert

Spaces and Some Applications to Nonlinear Partial Differential Equations, Contribu­tions to Nonlinear Functional Analysis, E. Zarantonello (ed.), Acad. Press, 1971, pp. 101-156.

[4J R. T. Rockafellar, On the Maximality of Suma of Nonlinear Monotone Operators, Tran& Amer. Math. Soc., 149 (1970), pp. 75-88.

[5J B. Feikari, Fundamentals of Network Analy­sis and Synthesis, Prentice-Hall, Inc., 1974.

16

j

TIME-VARYING INPUT-OUTPUT SYSTEMS WHOSE SIGNALS

ARE BANACH-SPACE-VALUED DISTRIBUTIONS

A. H. Zemanian State University of New York

Stony Brook , N. Y. 11794

Abstract

The composition of an operator-valued distribution and a Banach-space-valued distribution is established by extending Schwartz's kernel theorem in such a fashion that it has the form of the Cristescu-Marinescu composition of scalar distributions. This provides a representation for many linear continuous time­varying systems.

1. INTRODUCTION

The idea of a time-varying Banach system, which

was defined and analyzed in a prior work [10], led

to a study of composition operators acting on

spaces of distributions that take their values in

Banach spaces. Two types of composition were con­

sidered in [10]. The first, which was called

"composi tion • " makes use of Schwartz's kernel

theorem [8], [11] and its extension to Banach­

space-valued distributions. It provides an explic­

it representation n = f. for every continuous

linear mappingnof £leA) into [t); B] by means of

the composition product f-v, where .f is a distri­

bution on the real plane taking its values in

[A; B] and v~D(A). Here, A and B are complex

Banach spaces, [A; B] is the space of continuous

linear mappings of A into B,1)(A) is the space of

infinitely differentiable (i.e., smooth) A-valued

functions of compact support on the real line

supplied with its customary topology, 1:> = D(C), C

being the complex plane, and PD; B] is the space

of B-valued distributions equipped with the topo­

logy of uniform convergence on the bounded sets

of 1). A shortcoming of this representation for n is that it is defined only for certain suitably

restricted continuous functions v and not for

singular distributions v.

17

The second type of composition, which was called

"composition 0" in [10], is an extension to Banach­

space-valued distributions of a composition product

first introduced by Cristescu [2] and subsequently'

developed by Cristescu and Marinescu [3], Sabac

[7], Wexler [9], Cioranescu [1], Pondelicek [6],

and Dolezal [4]. In contrast to composition. ,­

not all continuous linear mappings ofi)(A) into

[D; B] can be represented by a compositiono opera­

tor. However, composition 0 has the virtue that,

when it does exist, it can be applied to singular

distributions.

The present work is aimed at this gap between com­

posi tion. and composition o· . A formula is given

for extending composition. , which we henceforth

refer to simply as "composition", onto singular

Banach-space-valued distributions by using

Cristescu's form of composition. The idea is as

follows. For the sake of simplicity let us con­

sider the case where f is a scalar distribution on

the real plane and v and ~ are members of~. Let

us also assume that f is of finite order so that

f(t,x) = (-D )i(_D )rh(t,x) where h is a continuous t x

function on the real plane and Dt

= J /d t. (These

assumptions will not be imposed subsequently.)

Then, by the composition arising from Schwartz's

£i£ ;is

kernel theorem, we formally write

<'f·v,q,> = Hh(t'X)D~cj>(t)D:V(x)dtdX

= S(D~V(x)Jh(t,X)D!q,(t)dtdX

= <V(X),(~Dx)rJh(t,X)D~q,(t)dt>

= (v(x) '(Yx(t) , Ht»>

(1.1)

where Yx is a mapping of!) into C and hopefully a

distribution depending on the parameter x. The

important thing is that the right~hand side of

(1.1) is precisely Cristescu's form of composition,

and therefore, if it turns out that (Yx,q,) is a

sufficiently smooth function of x, then (1.1) pro­

vides a means of extending the operator f· onto

some space of singular distributions v. This

paper gives a rigorous presentation of this idea

and overcomes the complications that arise when

the assumptions that f is of finite order and

that f and v are scalar-valued are dropped.

The present work extends the discussion in [10] in

still other ways. For instance, this discussion

encompasses the larger class of composition oper­

ators f - that are defined only on finite-order

distributions v [5], in contrast to the smaller

class considered in [10] of operators defined on

infinite-order distributions v. In addition, our

present results imply an estimate on the order of

the resulting composition product, something that

was not available in [10]. Another generalization

is that distributions on multidimensional euclid­

ean spaces are now allowed, whereas [10] was re­

stricted to distributions v on the real line.

It should also be pointed out that, although

Schwartz's extensive work [8] discusses the com~

position product f-v of vector-valued f and

scalar-valued v (see pages 124-126 of volume 7 in

[8]), it does not discuss the problem attacked

herein, namely, that of defining f.v, where f is

an operator-valued distribution and v is a Banach­

space-valued distribution, both of which may be

singular and of arbitrarily large order.

2. NOTATIONS

The notations used in this work are the same as

those of [11]. The reader should refer to that

18

work (especially to Section 1.2 and the Index of

Symbols) for the definitions of any symbols not

defined herein. There is one exception however.

In this work we will assign a more specialized

meaning to the symbol RS as follows. m will always e denote an ordered s-triple {ml ,m2,···,ms }' each

component of which is either a nonnegative integer

or 00 For example, {2,~,0} is such a 3-tuple

RS denotes the collection of such s-tuples e

Throughout this work we will always have m~R: and . n JERe'

pS is s-dimensional real euclidean space. K will

be a compact interval in RS , and K will be its in-

terior. If HRI or T=CO~ the notation [T] denotes

the s-tuple all of whose compontents are equal to

T. However, [0] is denoted simply by O. A and B

always denote complex Banach spaces with norms

V-itA and f{ '/IB

respectively. If U and V are

two topological linear spaces, the symbol [U; V] denotes the linear space of all continuous linear

mappings of U into V. Unless the opposite is

explicitly indicated, we always assign to [U; V] the topology of uniform convergence on the bound­

ed sets of U, which we call the "bounded topology".

Thus, for instance, [A; B] is assigned its opera­

tor-norm topology.

Let q, be a function from RS into some

Banach space. When we say that q, is continuous

or has a derivative, it will always be understood

that the continuity and derivative are with re­

spect to the norm topology of the Banach space.

Thus, for example, if the Banach space happens to

be [A; B], the said continuity is with respect to

the operator-norm topology of [A; B]. Let

k = {k ,"',k } be a nonnegative integer in RS

I s (i.e., every component kv is a nonnegative in-

teger.) Any {partial) derivative

J Ikl q, -k k-

d t 1 ,)t s s I k I = k 1 + ... +k s

. k Ikl of q, wlil be denoted by D q, = q, . We will also

write D~ q,(t) = q,(k) (t). We shall refer to k

(and not Ikl)as the order of the differential

operator Dk. The notation Ikl should not be

confused with the magnitude notation for the

members of RS

•

The support of any function or distribution f on

RS is denoted by supp f.

We will use the standard function spaces ~~lA), .DR~(A), c~~(a), and (?~(A), equipped with their

customary topologies. All of these are defined in

[10] and also in [11]. When m=oo, we drop the

superscript m. Similarly, we drop the subscript

RS to write 1J meA) = nm(A) and E. meA) ;: [meA) AS R'

whenever there is no need to specify the euclidean

space RS•

Schwartz's Kernel Theorem. n is a sequentially

continuous linear mapping of b,(A) into [DR"; B]

if and only if there exists an 'f € [.DR"+s (A); B] such that flv = f.v for every v~ D (A). Here, f is

R' uniquely determined byn, and conversely. The

composition product fov is defined by

<f.v,~> = <f(t,x), ~(t)v(x» (2.1 )

where ~fi 1) , Vii:!> •. (A), ttRn

, and xliRs

. R" R

A proof of this theorem is given in [11; Chapter

4] .

3, COMPOSITION OPERATORS ON [em ;A] INTO [~j ;B]. . RS R"

Lemma 1. Given any v£[em; A], define q by . v

(q , Fe) = F(v, e> v

(3.1)

where F~[A; B] and eED. Then, there exists a

unique Vf[em+[2] ([A; B]) whose restriction to the

elements of J)( [A; B]) of the form Fe, where

F€[A; B] and e~~ coincides with qv' Moreover,

supp v = supp v. In addi tion v ~V is a sequent­

ially continuous linear injection of [em; A] into

[Em+ [2] ([A; B]); B].

The proof of Lemma 1 is quite similar to that of

Lemma 4-2 of [10]. The assertion concerning the

supports is an easy consequence of the Hahn-Banach

theorem.

Next, let VE£m(A). Then, the same arguments as

those used for Lemma I show that there is a unique

V€[~m+[2]([A; Bl); B] which coincides with q on v

all elements of the form Fe. Furthermore, v gen-

erates a unique member of [~o; A], which we also

denote by v, through the definition:

(v, ~> : J v(x)~(x)dx ~EDo. It'

19

(Here, the superscript ° in ~O denotes the s-tuple

[0].) Moreover, v defines an [[A; B]; B]-valued

function v' on RS by the definition:

v'(x)F = Fv(x) Fe[A; B], x~Rs. By the Hahn-Banach theorem, supp v' supp v and,

for each x, v(x)~v'(x) is injective. Some

straight-forward arguments establish

Lemma 2. Let v€~(A). Then, for every integer

k€Rs with O~k~m, we have that Dkv' exists, is

continuous from RS into [[A: B]; B], and therefore

generates a regular member of [.Do ([A; B]); B].

Moreover,

k k D v'(x)F = FD vex)

for every F€[A; B]. Finally, v'=v in the sense of

equali ty in [1)[2] ([A; B]); BJ, and supp v' = supp v. = supp v'. Now, let fEr .DRn,,$(A); B]. Choose any two compact

intervals IeRn and VCRS. It can be shown that

there exist a jeRn not depending on I, an e

hfe~KL ([A; B]), and two nonnegative inte~ers itRn

and n;Rs with i::;j such that, for all ~Et>i and

ve.f)L (A) ,

<~.v,~> = <f(t,x), ~(t)v(x»

• SISLh(t,X).(i) (t)v(r) (x)dtdx.

1'.'e May now convert the integral on the right-hand

5i~e into a reneated inteRral:

S,[Srh(t,x)cl> (i) (t)dt]v(r) (x)dx.

Assume that the inner integral has continuous de­

rivatives on L of at least order r+[2]. Then, we

mar integrate by parts Irl times to obtain

Sr,fn:rrh(t,x)+(i) (t)dt]v(x)dx

jir ~ (-1) 'rl pr x x

" Let us now assume in addition that supp vC L. By

virtue of Lemma 2 and its notation, we may write

<f.v,,> = S"v' (x)n: SIh(t,X)cl>(i) (t)dtdx

= <~(x), n:Jrh(t.x),(i) (t)dt)

Tri s result mot ivates the following assumption

and definition, which provide a means of extend­

inp the definition (2.1) of the composition oper­

ator fo: vl+f.v onto the space [E~; A]. ~

u

Assumption. AssuMe tl·at c{)rresllondi nr to a .given n I m£P~ f"('r

f"t.{DR" .. CA); P.] tt..ere is a j€Pe am an e

Nhich the f"ollol-'inr condition!': arC' .<;atisfjecl.

Ct... 01' ce 04' the comnact ; nterva1 s TCpn ~r ~C~ n

1 Icns tj1ere exi st an rEer I (fA: TIl) aNl h:O anl " , .X . . r 1 pS • th .... h nonnepative inte?er~ lEP anr rE. ~1 • l.) suc

i trat, for all ¢E.flr and all vd\(,h),

(f(t ,x), Ht)vex» r') rr)

"STSyHt,xH·'·(t)v (x)rltdx

and, in ad,H tiol',

(n I'l+l+{'" Srh(t")<I>' (t)r1t€(\ .. ({.i\; nl).

nefinition.

Assumption. o

choose T nne! L ~UC1, t'·~t ,w'n ¢cT :m(~ SlInn veT,

Finn11", c"oose h, i, anel l' in accorchnc(' ",it"

tl,e ~s~l1fl'ntior. ,,'(' def"'ne (4'.v, <1» I,V

'A ~,.. J 'i) () > (r.v,¢> ~ (v(x), ''x T'·(t,xH t"t,

A "1+ r21 , .. here vis t.h:Jt !'1enh'r of rt <;' • (rA: nJ); l'\]

n' corre<;p{)n'~ ing to v i.n accordancp vi th LCT'lMa 1.

T1'coren. l'J1<'rT t "'(' ,\SSI1T'lnt ion a!'f1 nef"inition,

(f.v,¢) is inliepen,'ent {)I' t 1,I' C"nices 01' tl·c

paraneters T, L. 1~, i, and r amI is consistent

,dtl' (::.1). "orcover, tl,c onerator f.: \'~f.v is

a seouentially contimlous linear T'lanninp of"

[ & m. AJ into f.t).i; pl. cR' ' , R"

The proof of" this theoreM is nuite cxtensive an"

ldll appear dsel\'l'ere.

After maling annro"riate assUT'lntions on the SllP­

nort of" 4', "'e cnn refine comn~~ition onerators

f". that man [.oR;: /I] into r~1I.~: P], ~E~:: A' into • 1'1 1 m

{E '. "J Tf)" /11 into r t .: "1, and rf) ; '\) ~., . ! . ~.' . II" , -

into (.(t J ; Al. )forrover, convolution can he

shown to h~ a special ca~e of ollr cOl"no~ition,

Tris ~.'orl ,va~ s\lnnorte~ 'w t"e "'<tt;.onal Science

Foundation under r.rant "re:; 75-"52('8 .~n:;,

r 1] I. rioranescu, 'Tami 1 i i eompozahile de oper­

atori", St. r~TC. "at., vol. 1~ (1967),

nn. 44('\-454.

P] R. Cristescll, 'Tamille5 comnosaMe5 de dis­

triblltjon~ et ~vstemes rr.ysiques lineaires",

!lev. Pown. "ath, Pures ~t "pn1., vol. 9 (1%4)

1'p. 703-711.

f3] R. Cristescu and r. 'larinescu, "lInele

Aplicatii Ale Teori~i nistrilmtiilor",

Fclitllra Ikademiei "enuhl icii e:;ociali5te

Romania, Rucuresti, 1~66.

r4] V. f'olezal and .J. Sanborn, "A representation

ct linear continuous operators on testin~

functions and distributions", Siam .. J. !lath,

Anal., vol. I (1970), np. ~Ol-S"(.

[5] .J. I'orvath, Topological Vector Spaces and

Distrihutions, vol. T, Addison-"'esley,

l!eadinr" /lass., 191'i6.

P,. Pondel ice~:, "A contri hilt i on to the foun­

dation of networl' theory usinr distrihution

theorv", rzechos lova~' Hatr. ,~., vol. I!' r94)

(IQ69), nn. 6!'7-710.

.. C;ahac, ''ramilii COr:1nOzabi1e de distrihutii

si transformata Fourier:, "St, Cerc. ~!at.,

vol. 17 (1 0 ('5), ~p. 607-613.

rllJ L. .C;crwartz, "Tllforie des distributions a

valeurs vectod elIes", .~nnal es de 1 'T nst i tut

Fourier, tOT'll' 7 (1!'!i7) , nry. 1-130 , and tome 8

20

(IQ5!'), nn. 1-206.

[Q] l~. "'~xler, "Solutions periodioues et presoue­

np.riodic des systcl'1cS d'enuations differen­

tielles lineaires en di strihutions", ~

nifferent ial Fouat ions, vol. 2 (1966),

nn. 12-32.

nn) A. II. ?PT'lanian, "I\anach systems, llilhert

ports, and ""-norts", in :\etwork TheorY,

R. Roite (~d.), Gordon and Breac", ~ew York,

1972,

fIll A. II. Zemanian, Pealizahility Theorv for

rontinuous Linear Systems, Academic Press,

~!ew York., 1972,

Causality and Co Operators

Avraham Feintuch Ben Gurion University of the 'legev

Beersheva, Israel

Abstract It i~ shown that the class of C Contractions appear in a natural way in the algebra of causal

linear operators. Stability propertigs of such operators are studied.

1. Introduction:

The appearance of linear transformations 'J

which operate On infinite dimensional Hilbert

space has become quite common in systems theory

today. This is true both in the state space

theory and the input-output theory. In particular,

a significant number of authors ([1], [Il], [10],

[11], [14]) have noticed the relevance of the

Nagy-Foias model theory for contractions in

systems theory. An important suhclass of these

operators is the family of Co contractions. These

seem to be a quite natural ,generalization of

finite dimensional operators and such basic

finite dimensional concepts such as minimal

polynomials and .Jordan c,anonical forms have a

natural extension to the infinite dimensional

case.

The usefullness of these operators and the

unilateral shi ft operator to \~hich they are

closely related has been noticed by a number of

authors. Common to all of them is a state-space

approach.

My purpose here is to show how these

operators appear in a natural way in causality

structures and play an important role in the

theory of stability of linear feedhack systems.

~fost of the results presented here have appeared

in the literature in an operator theoretic setting.

However, this is, to my knOldedge, the first time

that they are presented from the point of view of

systems theory.

2. Causality:

Causality is usually descrihed on an complex

Hilhert space II in terms of a resolution space

21

structure. This is descrihed in detail in [17].

Here \~e present a slightly different formal ism

first descrihed in [17]. This will make it simpler

to translate operator theoretic results directly

into a systems theory framework.

'" Definition 2.1: A family:~ of suhspaces of II is a

nest if it is totally ordered hy inclusion. q is

complete if

(i)

(ii)

.A. {®}, fL'E N

forNoc~, n N :-.ltNo

and VN NtNo

arc in N.

for any 'I t ~l, ~I wi 11 denote the subspace

V (L : L t 'N and L c ~I).

with (0) = (0). If ~ #~, N is called the

predecessor of ~I.

1\ Definition 2.2: A nest N is maximal if

"\ (i) ~ is complete

(ii) for il E ~, dimel e '1)< 1

Definition 2.3: A nest space is a pair (H, ~) consisting of a lIilhert space II and a maximal

nest :-.l of suhspaces of 11.

PM will denote the orthogonal projection

with range '1.

The nest space structure allows us to define

causality in an ahstract setting.

" Definition 2.4: Let (II, 'J) he a nest space. A

bounded operator T on II is causal if 1':/ = 1'~ITP: I A

fOT all n E '\. T is ant i-causa 1 if Tl'q = 1':'ITP:'1

and memorvless if P.IT = TP~I for all :'f E ~'i.

rxaMple 2.1: The classical context of causality

is h'hen II = L2(-oo, co) and ~ = 1.2

(-<0, t) i'~' E iR). A

It is easily ch()cked that (H,N) is a nest space

and that the definition of causality given ahove

i;

coincides with the classical definition. It should /to

be noted that if H £ ~,~I ~I. ~ests which have

this property are called continuous. Throughout

this paper we will consider continuous nests

unless stated otherwise. This is done since the

corresponding results in the discrete case are

quite easily obtained.

3. Shift Invariant Subspaces:

Let m denote measure on the unit circle n

in the complex plane normalized so that men) 1.

L 2 will denote L 2 (1jI). Every f t L2 has a Fouries

f(e iO) = ! ina which converges to f in the n"-CO c e

2 2 n L -norm. H will denote the subspace.

112 = (f t L2 : f(e iO ) = ~o c eine

) . n- n

i. e. f t H2 if and only if its negative Fourier

coefficients are zero. The unilateral shift on 112

is the operator.

(Sf)(eiO) = eiof(eiS ).

. The basic facts about H2 and S can be found in

[12]. Any 112 function f has an analytic extrnsion

to the unit disc D = (z : Izl < 1) whose value. at

z will be denoted by fez).

The invariant subspaces of S were described in a

well-known theorem of Beurlinp,. For this we need

the notion of an inner function.

Definition 3.1: A non-constant function q analytic

in n is called inner if Iq(z)1 ~ 1 and Iq (eiS)I=1

almost everywhere on ~

Inner functions playa major role in the theory

of Co contractions and I will therefore briefly

discuss their structure. An elegant and complete

treatment can be found in [12].

If is inner then it can be factored as

rj>fil) = exB(A) s(A), Inl = 1

\~here B(A) is the Blaschke product determined by

the zeroes of rf> inside the unit disc and SeA) is

of the form _f2~eit+A

exp "t dUt , o e -1 + A S (A)

u being a non-negati.ve finite singular measure.

It is now possible tostate Beurling's Theorem

and to di.scuss its importance for cansality.

TZZT

Theorem 3.2: Let S be the unilateral shift on H2.

Then ~I c H2 is an invariant subspace for S if

and only if there exists an inner function rf> such

that ~I = rf>H2 = (rf>f : f t H2).

It will be more conventient to work with S* A

the adjoint of S. Every continuous chain N of

invariant suhspaces of S* has the form

~ft = (rf>t 1I2)"l where rf>t is inner. Moreover, each

rf>t is a singular inner function since the presence

of a Blaschke product would introduce a jump "-

discontinuity intoN.

The importance of such chains is that, up to

unitary equivalence, they are universal.

Theorem 3.2: [13]: Every continuous chain of

suhspaces (in a separable Hilbert space) is

unitarily equivalent to a continuous chain of

invariant subspaces of S*.

22

A

Thus given a nest space (H, N) with H

separable, infinite-dimensional we may as well 2 " 2 .l} assume that II = H and that N = {(rf>t II)' for

some chain {rf>t} of inner functions.

It is worth noting that the basic theory of

inner functions leads to the fact that

rf>l H2 c rf>2 H2 if and only if rf>2 divides 4>1' Thus

(rf>l 112)1;.:::> (4)2 112).l if and only if rf>2 divides 4>1.

It is then quite natural to use the expression

"a chain of inner functions". /'.

The identification of (H, N) with

(112, {(rf>t H2).l}) will allow as to show that the

algebra of causal operators contains many Co

operators.

4. Co Contractions:

In this section we present the relevant

material from the theory of Co contractions. No

attempt at completeness is made. A complete

treatment is given in [19, Ch. 3].

Let T be a contraction (IITII ~ 1) on H. Then

9[19], Ch. 1) there corresponds a decomposition

of II into an orthogonal sum of two subspaces

reducing T, say II = "0 III HI' such that the part

of T on "0 is unitary and the part of T on "1 is

completely non-unitary. This means that Ttlll has

no non-zero reducing subspace L for which T\L is

a unitary operator. This decomposition is unique

and 110 or HI may equal the trivial subspace (0).

We will concern ourselves with the case that

110 = (0); i. e. T is completely non-unitary.

By,,"" we denote that algebra of bounded

analytic functions on n with the usual norm (or 0;,

equivalenty, the subalr,ebra of L consisting of

those functions whose negative Fourier

coefficients vanish). Then if T is'c.n.u.

(completely non-unitary) it is possihle to define 00

the operator Ijl(T) for all ¢ t II

Oefinition: Co is the class of completely non­

unitary contractions T for which there exists a

non-zero function UtII'''' such that u(T) = o. u £ I~ can he factored into u u. with u an

e 1 e outer function and u. an inner function. It is

1

shown in [19] that if u(T) = n then Hi (T) = O.

\lso if T is of class COlT has a minimal function

mT

Ivhuch divides every flinction u such that

ueT) = O. This function is determined up to a

constant factor of modulus 1. In the sequel the

spectrum of T will play an important role. This

is completely determined hy mT

. Let

m,.().) = fl(A)S(X) be the factorization of mT

•

Theorem 4.1: [19, p. 126]: Let fTIT he the minimal

function and aCT) the spectrum of the contraction

T of class Cf)' Let 5r he the set consisting of

the zeroes of mT in the open unit disc n rl.nd of

the compliment, in the unit circle lT of the union

of nrc! of lTon which mT is analytic. Then

aCT) = ST' The simplest examples of Co operators are

given in the following theorem.

Theorem 4.2 [19, p. 124]: Let he a non-constant

inner function and let 14 = (dlH2).L in 112. Then

the operators p~lsl~1 and S*I~1 he long to Co and

their minimal function is

~low we return to callsa 1 i tv. A.s seen ahove, 1\ . •

if (II, ~) is a nest space, we Crl.n assume it is

(112 , {(¢t ,,2):-}).

the

2'" suppose ~I = (<Pt

II) for some t and consider

operator T .... = P~ISP~I' By Theorem 4.2 T is Co'

23

It is easily checked that T is causal and T* is

anti-causal. Thlls the algehra of causal operators

is rich in Co operators.

I I~ould like to end this section hy giving an

example of a Co contraction in a more traditional

contex for feedback systems.

Example 4.3:

2 Let II = L (_00, (0) Id th the usual nest of " 2 suhspaces N = {I. (_00, a) _00 < a < oo}. Let F be

the convolution operator on II with kernel

x < ()

2e -x x > 0 f(x) =~

Let P he t projection on L2(0, 1) and consider

the operator I - PF. This is the return difference

from the feedhack system with input-output map F

and feedhack P (which is just a truncation). It

is sho\Vn in ([181) that I - PF is unitarilly

equivalent to P~ISP~1 where M = (tj>1I2).L with

z+l ¢ = exp(z_l)'

S. Feedback Sta~ility

Let (H, N) be a nest space and consider the

feerlhack system described hy the equations

y Ke + d

e = ry + u (1)

"here K and F are causal operators on (H, N) IVhich

are, respectively, called the plant and feedback

operators.

Conceptually, u is the input signal, d is the

disturbance in the output of the system; u and d

are considered the independent variables of the

feedhack system. The dependent variables are the

output y and the plant input e.

Comhining the abOVe equations, we obtain

(I - KF) Y Ku + d

(I FK) e u + rd

\Vhich relate the dependent variables to the

independent ones. It is clear that for y and e to

be uniquelY determined hy u and d \Ve need that the

inverses of I - FK and I - FK to exist. KF is

called the open loop gain and I - KF is the return

difference of the system. It is easy to see that

(I - KF)-l exists if and only if (I - FK)-l exists

and one is causal if and only if the other is.

Stability of feedback system is usually

defined in terms of extension spaces (see [17],

p. 65). Fortunately an important theorem of

IHllems ([20]) allows us to avoid this approach.

IHllems result will be used as our definition of

stability. What is lost in the intuitiveness of

the extension spaces is more than made up for in

mathematical simplicity.

Definition 5.1:

rhe feedback system (1) is well posed and -1 -1 stable if (I - KF) (equivalently (I - FK) )

exists as a bounded operator on " and is causal on

(II, N).

Here we consider the case where the return

difference ( I - KF~ is a Co ·contraction. While

we do not have a complete solution the result to

be presented.

Definition 5.2:

A contraction r on a separable IIilhert space

is essentially unitary if hoth I - r*r and

I - TT* are compact.

I~ is worth making some remarks ahout the

operators I - r*r and I - TT* for a contraction T.

rhe square roots Dr = (I - r*T)1/2 and

Dr * = (I - TT*)1/2 are called the defect operators

for r, the closure of their ranges DT, Dr * are

called the defect spaces and dr = dim DT

,

~* = dim Dr * are called the defect indices of T.

It is easy to see that dr = 0 characterizes

the isompetric operators and dT = dr

* = 0

characterizes the unitary operators. Thus, the

defect indices measure, in a sense, the deviation

of the contraction T from heing unitary.

Another way of looking at this is the

following. The unilateral shift defined in 3 is in

a certain sense a universal operator. If T is a

contraction such that rn ~ 0 strongly then T is

unitarily equivalent to a compression of some

multiplicity (possihly infinite) unilateral shift

24

to some co-invariant suhspace. ([19]).

Our restriction means that T is related to

a unilateral shift of essentially multiplicity.

In the case of essentially unitary Co contractions we always have l~el1 posedness and

stability ([17J).

Theorem 5.3: I f (I - KF)-l is hounded and I - KF

is an essentially unitary Co contraction, then

the system (1) is well posed and stahle.

6. Strict Causality:

It turns out that for most desired properties

of feedhack system the assumption of causality of

K and F is not enough. Thus the concept of

causality was strenghed in various directions hy

a number of authors (see, for example [21J, [2J,

r33]). One particularly useful direction is the

concept of strict causality introduced by

De Santis, Porter and Saeks ([2J, [4J, [17J). One

of the rather surprising aspects of strict

causality is tlJat Nhile it is in some sense

natural for linear systems it turns out to be

important for non-linear systems as well ([12],

(5]) .

The "property of causality for an operator

means, in the finite dimensional case, that it has

a lONer triangular matrix representation. Strict

causality Nill reduce, in this case, to a

strictly lower triangular matrix repersentation.

We present the concept very hriefly and refer the

interested reader to [17J.

Let E he the set of projections onto the

memhers of N. By a partition P of E is meant a

finite suhset.

P = {Ei : 0 < i ~ n}

of E such that

... < E = I. n

Ai will denote the projection El - Ei _l • A

partition PI is a refinment. P ~ Pl' Note that the

partitions of E form a directed set under refinment.

Let T he a hounded operator on II and P any

partition of N. Form the sum.

Lp(T) =i~l Ei _l TAi

Theorem 6.1 ([6], [17]): T is causal if and only , ,

for any E > 0, there ,exists a partition P of such

that for any refinement PI of P

IIJJp (T) II < s: • 1

It is worth noting that this is wquivalent

to strong convergence of Lp(T) to zero; i.e.

causality = strong causality. This will not be

the case for strict causality which we define now.

Definition 6.2: Let T be a causal operator on

" n (H,N). Let Vp(T) = E ~. T~. i=l 1 1

T is strictly cousal if for any E > 0 there

exists a partition P of E such that for any

refinement PI of P,

IIVp (T) II < E. 1

Intuitively this means that the elements of

the diagonal of the matrix of T with respect to

any partition are zero.

It is of interest that the property of strict

causality defines the spectrum of T.

Theorem 6.3 [6]: If is strictly causal, then T is

quasinilpotent; i.e. the spectrum of T consists

of the point {oJ.

An immediate consequence of this is that if

the open loop gain of the system (1) is strictly

causal, then the system is well posed and stable.

We now return to Co contractions and consider

the problem of classifying the strictly causal

operators of this class. As mentioned above. the

spectral properties of such operators play an

important role. These were summarized in Theorem

4.1. The only quasinilpotent Co contractions are

the nilpotent ones. Of equal interest is the case

where T has a spectrum consisting of a single

point which is not necessarily O. An examination

of the spectrum of T allows us (up to a similarity)

to two possibilities:

(i) aCT) 0

(ii) aCT) 1.

In the first case the minimal function of T is

just u(t) = zn. In the second it is

. z+l u(z) = exp a(~).

Both situations are included in the next

theorem.

Theorem 6.4:(£7]) If T is caulal and I - T*T is

compact. then A - T is strictly causal if and

only if aCT) = A. At this point I would like to mention two

open prohlems in strict causality, the first in

the context of Co contractions and the second in

more general context.

(1) Can Theorem 6.4 he extended; i.e. Can the

condition I - T*T be dropped? I conjecture that

the answer in general is negative though I'm not

sure why.

(2) Suppose T is a nilpotent causal operator.

Is T strictly causal. The answer is yes ifAthe

nest fJ· is discrete (dim M 9 ~I = 1 for ~1 E N).

7. Strong Strict Causality:

Definition 7.1: Let T he a causal operator on

(H,~)

for a partition P. T is strongly strictly causal

if for any E > 0 and x E II there exists an

partition P of E such that for any refinemeT.t PI

of P,

IIVp (T)xll < E. 1

Clearly strict causality inplies strong

strict causality. The converse does no~ hold.

An example that shows this as well as the fact

that strong strict callsality is a more natural

concept will he given helow. First however, I

would like to mention where this concept prove<

useful in Systems Theory.

C Causality plays a role not only in stability

prohlems hut also in state decomposition for

non-time-invariant systems (see [17]). ,\s ~':lS

shown hy Saeks ([17], p. 130) in the case ~here

T is strongly strictly causal, a minimal state

decomposition gives a complete set of invariants

for T.

Example 7.2: Let II = [2(0.00) the space of

sequences {an}oon=o such that r la 12 < 00.

n=O n

For sequence (a O' aI' 8 2 , ... ) let s(ao' aI' a 2 ,

) = (0, aO' aI' ••• ). This is just the

previously mentioned unilateral shift in a

different context.

25

n Let N - {V 8 i - M } where e i is the i-th

i-O n

co-ordinate vector. Then S is causal. It is not

strictly causal since a (5) • h I I z I " 1l. lIowever

S is strongly strictly causal. For the matrix of S

with respect to {e } is n

/ 0 ..........

f 1 0

\ 0 0

L~t E ~ 0, and x = (aO

' al' ••• ,) • Then choose .. ~ .. uch that :.\a \2 < c.

~ n

Let P be the partition {o'. PI' ••• , PN, J} where 1

Pi is the projection on ~ ek •

n Then r ~.S~. has the matrix

i= 1 1 1

n ....... ..

o

o o

where the 0 in the upper left corner represents an ~ x 'i hlick.

If dim /I < 00 then aCT) • {O},

Note that for T a Co-contraction any

eigenvalue A of T mllst have IAI < 1.

This is not always the case. One need only

take 5* which is strongly strictly anti-causal

(n dual property) and has a large point spectrum

{z I I z 1 < I}. I'le wi 11 present what we hope is a

reasonable hypothesis.

Prohlems 7.3:(1) If T is strongly strictly causal

then the point spectrum of T has only one

component; Le. T has no more than one isolated

eihenvalue:

(2) If T is Co with I -T*T compact and

o(T) c {Izl I} then T is strongly strictly

cousal.

(3) Can the assumption I -T*T compact be

sropped?

8. Conclusion:

have tried to show that Co cOntractions

are an important class of operators for the study

of problems in linear feedback systems. Under the

assumption that the defect operators of such a

contraction are sufficiently small we have seen

that systems involving Co coptractions are well

behaved. I have also mentioned a numver of open

quest ions I~hich I feel are worth considering and

are basic to the understanding of the behavior of

linear feedback systems.

Bibliographr

[1) ? .J.S. Baras and R.W. Brockett, "H--functions

Thus n and infinite dimensional realizat ion theory", l: fli S fli x = (0, .... aN' aN+I , ... ).

i= I SIAl-I .J. Control. n

It follows that I I l: fl. S fl. xl I < E. This also ;=, 1 1

dearly holds for any refinement of P.

~e note that for strongly strictly causal

operators the spectrum can be quite large. 1I0liever

the propert), that we would hope carries over from

the finite-dimensional space is that the point s

spectrum can not consist of more than one point.

This is in fact the case for the class of Co

contractions that we considered ([S»).

Theorem 7.3: Suppose T is a Co-contraction which

is strongly strictly causal. and su~h that I-T*T

i~ compact. If dim H = "". then o(T" el' z I = 1 }.

(2) R.H. DeSantis, "Causality for nonliflear

systems in llilbert space". ~Iath. Sys. tho 7, No.4

(1974), 323-337.

[3] , "Causality, strict causality,

and invertibility for systems in lIilhert resolution

space, SIAII .J. Control 12. No.3 (1974),536-553.

[4] R.H. DeSantis and N.A. Porter. "On time­

related properties of non-linear system, SIAH .J.

App!. Hath. 24, :-lo. 2 (1973)'. 188-206.

[5) --------------, "On the analysis of feedhack systems with a polynomial

plant", Int . .J. Control 21. No.1 (1975), 159-175.

26

[6] A. Felntuch, "Causal alld strictly causal operators, to appear •

[7] ------, "on stability", to appear.

Co operators and feedbaCk1 I

[8] ------, "On strong strict causality for operators", to appear.

[9] P.A. Fuhrmann, "Realization theory in Hilbert

space for a class of transfer functions, J. Funct.

Anal. 18, No.4 (1975), 338-349.

[10] ------, "On realization of linear

systems and applications to some questions of

stability", Mat. Sys. Th .• 8, No.2 (1974), 132-141.

[11] J.W. IIilton, "Discrete time systems,

operator models and scattering theory" .J. Funct.

Anal. 16 (1974), 15-38.

[12] K. 1I0ffman, "Banach Spaces of Analytic

Functions", Prentice lIa11, Englewood Cliffs, N.J. 1962.

(13] T.L. Kriete, "Fourier transforms and chains

of inner functions", Duke Math. J. 40, No.1

(1973), 131-143.

(14] N. Levan, "The Nagy-Foias Operato~ ~iodels,

Networks, and Systems",

IEEE Trans. on Circuits and Systems, Vol. CAS-23,

No.6 (1976), 335-343.

[IS] W.A.Porter, "The conunon causality structure

of multilinear maps and their multipower forms",

J. Math. Analysis and Applic. 57 (1977).

[16] W.A. Porter and R.M. De Santis, "Linear

systems with mult iplicative control", Int. J.

Control 20, No.2 (1974), 257-266.

[17] R. Sacks, "Resolution Space Operators and

Systems", Lecture notes in Economics and Hath.

Systems 82, Springer-Verlag 1973.

[18] D.Sarason, "A remark on the Volterra operator,

J. Hath. Analysis and Applic. 12 (1965), 244-246.

[19] B. Sz.-Nagy and e. Foias, "lIarmonic Analysis

of Operators on Hilbert Space", North-Holland,

American Elsevier, New York 1970.

[20] J.e. l1i 11 ems , "Stability, instability,

invertibility and causality", SIAM J. Cont. 7

(1968), 645-671.

[21] , Analysis of Feedback Systemd, Ca,brodge, ~nT Press, 1971.

27

WIENER-HOPF TECHNIQUES IN RESOLUTION SPACE

L. Tung and R. Saeks Dept. of Electrical Engineering

Texas Tech University Lubbock, Texas 79409

I. INTRODUCTION

Wiener-Hopf filtering is a widely used

technique in certain kinds of optimization

problems. The purpose of this paper is to

formulate Wiener-Hopf filtering in abstract

spaces (reflexive Banach resolution spaces)

and to examine problems involved for the

formulation and the solving of the Wiener­

Hopf filter.

Referring to what has been done in the fre­

quency domain of the classical Wiener-Hopf

filteringl, we've found five major problems

for the formulation of Wiener-Hopf filter­

ing in abstract spaces. They are

i. Random variables in abstract spaces

ii. Causality

iii. Operator factorization

iv. Operator decomposition

v. Optimization.

These problems are briefly introduced as

follows:

i. Random process can be thought of

as a random variable which takes values in

a function space. In order to do so, we

need an adequate probability measure over

the space involved. Fortunately, this

kind of probability measure has been de-. 2 F fined over metr1c space or our pur-

poses, we assume that the space involved

is reflexive Banach space, not only be­

cause this kind of space possesses nice

properties but also because stochastic con­

cepts such as "mean" and "variance opera-

28

tion" can be defined therein. Random vari­

ables taking values in reflexive Banach

space is discussed in section II with pro­

bability measure assumed implicitly.

ii. Concepts of causality have been

introduced into Hilbert space-the so-called

Hilbert resolution space3 • In section III,

we extend the works done for Hilbert. space

to Banach space. Concepts of causality,

such as causal, anti-causal, miniphase and

maxiphase, are defined. Emphases are given

to reflexive Banach resolution space.

iii. Operators to be factorized in the

form of KK*, where K* denotes the adjoint

of K, have to be "positive"and "self-ad­

joint". These commonly-used properties

among operators on Hilbert space can be

extended to operators which map reflexive

Banach space to its dual space. Factoriza­

tion theorem is given in section IV. Fac­

tor operator K is required to be left-mini­

phase.

iv. The decomposition of operators over

Hilbert spaces is treaded in Ref. 3. For

operators over Banach spaces, this problem

is still under research. For our conveni­

ence, operators are restricted to those

which guarantee the decomposition.

v. As in the classical Wiener-Hopf

filtering, we would like to minimize the

variance of the error. However, when Wiener­

Hopf filtering is formulated in reflexive

Banach space, the variance of the error is

a positive and self-adjoint operator which

can only be minimized in the partial order­

ing of the positive operators.' This subject

is treaded in section V.

II. BANACH SPACE VALUED RANDOM

VARIABLES

The theory of Banach space valued random

variables has been studied in Ref. 2. For

our purpose, we discuss reflexive Banach

space valued random variables with proba­

bility measure over the space assumed im­

plicitly. The development follows that of

Parthasarathy (2) and Balakrishnan (4); the

reader is referred to these works for the

details.

Let p, TI denote finitely additive random

variables taking values in a reflexive

Banach space B. For such random variables,

we assume

* * * E{ I (p, x ) I} < CD , for all x £ B

* (2.1)

E{(p, x )} is continuous in x*

Here E{.} denotes the expected value of a

scalar valued random variable with respect

to the probability space underlying p. For

random variables satisfy condition (2.1),

there is a unique vector mp in B satisfy­

ing

* * * * E{(p, x )} = (mp, x ), for x £ B •

mp is termed as the mean of random variable

p. As in most stochastic processes, mean

is not our prime concern. Therefore, in

the sequel we only deal with zero-mean ran­

dom variables. For such random variables,

we further assume

E{ I (p , * * ) I } x ) (TI , Y < CD , * for all x , y* £ B*

(2.2) E{ (p , x*) (TI , y*) }

is continuous in x* and y*

It can be shown that condition (2.2) implies

condition (2.1). Now let's take a look at

E{(p,x*)

E{ (p, x*)

(TI, y*)}. If we fix y*, then

(TI, y*)} is a bounded linear

29

functional on B* (so an element of B**=B).

This means that there exists a unique Py*

in B such that E{ (p, x*) (TI, y*)}

= (Py*' x*) for x* £ B*.

QPTI

= B* + B, by QPTI

y*

Define a mapping

Py*.

Hence E{ (p, x*) (TI, y*)} (QPTI y*, x*).

It can be easily proved that Q is linear. PTI

Moreover, Q is PTI

bounded. Q is termed PTI

as the covariance operator of random vari-

abIes p and TI. Covariance operators satis­

fy following conditions:

i. Q(Lp) (MTI) = L QPTIM* , where Land are linear bounded operator on B.

ii. Define Qp Qpp then

iii.

iv.

QP

+TI = Qp + QPTI + QTIP + QTI •

Qp is called the variance operator

of p.

Q* in particular Q = Q * TIP , P P

Q is positive in the sense that p

(Qp y* , y*)

for all y* £ B*.

These conditions result from straight for­

ward manipulation of the defining equation

for the covariance operator. Using QPTI'

we say that p and TI are independent if

QPTI

= O.

III. BANACH RESOLUTION SPACE

By a Banach resolution space, we mean a

2-tuple, (B, BF), where B is a Banach space

and BF is the so-called resolution of iden­

tity in B, which is defined in the following:

(A) Resolution of identity

Definition 3.1. Let B be a Banach space.

By a resolution of identity, BF, in B, we

mean a family of linear bounded operators,

BF(~), on B defined for each Borel subset,

~, of the real number set R, satisfying

the followings:

i. BF (R) = IB-identity operator on B

ii. BF(~Il . BF(~2) = BF(~lM2)' for

all ~l' ~2 £ S (R)- the set of all

Bo el subsets of R.

iii. n n n

BF(?~i) = I BF(~l)' where {~l}l is

a finite set of disjoint Borel sub­

sets of R.

iv. IIBF(~) x II 2. Ilxll, for all

~ £ B (R) and x £ B (Equivalent

statement: Norm of BF(~) is either

o or 1)

The subscript on the left in the notation,

BF , is to notify that the resolution of

identity is defined over space B and will

be dropped if no ambiguity would result.

Working with a Banach resolution space,

(B'BF), it is natural and important to ask

whether we can define a resolution of

identity in B*, the dual space of B. The

following theorem gives us the answer.

Theorem 3.1. Let (B, BF) be a Banach re-

* solution space, then {BF (~) I~ £ B(6)}

is a resolution of identity in B*-termed

as the induced resolution space, (B*'BF*).

With the resolution of identity defined as

above, we'd like to point out that although

Hilbert space is a special case of Banach

space, Definition 3.1 does not lead to a

Hilbert resolution space3 • In Hilbert re­

solution space, the resolution of identity,

{E(~) I~ £ B(R)}, satisfies an additional

condition, i.e. E*(~) = E(~).

Example. Let p, q £ R, such that

lip + l/q = 1. Then Lp is a

reflexive Banach space with dual

space Lq . For each f £ Lp ' def-

ine (f ,q) = f""f (t) g (t) dt. Let

F(M f(t) = X (~) f(t) = 0 , tlb.

f(t),tF:~

It is easy to show that {F(~) ~£B(R)} is

a resolution of identity in Land F*(6) p

= X (~) for all ~ £ B (R).

(B) Concepts of causality

Definition 3.2. Let (X,xF), (Y'yF) be

30

Banach resolution spaces. T: X+Y, is a

linear bounded operator

(i) T is causal, if

Ft ~ Ft T Ft T X x 2 Y· Xl = Y x2 , t

where BF = BF(-w,t) , B = X, Y.

(ii) T is anti-causal, if

XFt Xl = xFt x 2 => yFt T Xl = yFt T x 2 ,

where BFt = BF (t,w), B = X, y.

(iii) T is memoryless, if T is causal and

anti-causal.

(iv) T is left-miniphase, if

(v) T is left-maxiphase, if

(vi) T is right-miniphase, if

(vii) T is right-maxiphase, if

According to above definitions, we've found

the following results:

(1) Miniphase, left-or right-, implies

causality.

(2) Maxiphase, left-or right-, implies

anti-causality.

(3) When X and Yare reflexive, we have

(a) T is causal <=> T* is anti-causal

(b) T is left-miniphase <=> T* is

right-maxiphase.

(4) When X and Yare reflexive and T is

invertable, we have

(a) Miniphases are equivalent, so

are maxiphases.

(b) is miniphase and -1

T => T T causal

is maxiphase and -1 . T => T T antl.-

causal.

Readers are referred to Ref. 5 for the de­

tails.

IV. OPERATOR FACTORIZATION

Not every operator over arbitrary Banach

spaces can be factorized in form desired.

For our purpose the desired form of factori­

zation is K K*. Operator to be factorized

in this form has to be positive and self­

adjoint. These two commonly-used proper­

ties for operators over Hilbert space can,

be extended to operators that map from re­

flexive Banach space to its dual space.

They are defined as follows:

Defini Hon 4.1.

(i) Let B be a reflexive Banach space.

(ii) Q B+B*, is linear and bounded.

Q is said to be positive if

(x, Qx) ~ 0, for each x E B.

Q is said to be self-adjoint if

Q* = Q.

Note that Q* : B**=B + B* 'so it

makes sense to compare Q with Q*.

For positive and self-adjoint operators, we

have the following theorem:

Theorem 4.1.

(i) Let B be a reflexive Banach space.

(ii) Q : B+B*, is linear, bounded,

positive and self-adjoint.

Then there exist a Hilbert space H

Theorem 4.2.

(i) Let (B,F) be a reflexive Banach re­

solution space. (B*, F*) denotes

the induced resolution space.

(ii) Q = (B,F) + (B*, F*), is linear,

bounded, positive and self-adjoint.

Then there exist a Hilbert resolution

space (H,E)

K = (H, E)

1. Q

and a linear bounded operator

+ (B*, F*) such that

K k*

2. K is a left-miniphase

3. The factorization is unique up to

a memoryless unitary transforma­

tion.

For the proof of this theory, please refer

to Ref. 5.

V. WIENER-HOPF FILTERING FORMULATED

IN REFLEXIVE BANACH SPACE

With the preparation of sections II, III

and IV, now we are ready for the formula­

tion of Wiener-Hopf filtering. The for­

mulation is done as follows:

Let X, n be random variables taking values

in a reflexive Banach resolution space

(B, F). X denotes the signal and n the

and a linear bounded operator K : H+B, noise. Both X and n satisfy condition

such that Q = K K*6. (2.2) in section II and they are assumed

When dealing with Banach resolution spaces,

the usefulness of operator factorization is

limited unless the factor operator possesses

certain causal properties. Referring to

factorization of the spectral density in

classical Wiener-Hopf filtering, we have

found what we need is a factorization theo­

rem which gives a causal operator and quar­

antees a causal inverse once the existance

of a inverse is granted, i.e. a theorem

that gives a miniphase factorization. Based

on Theorem 4.1, we construct the resolution

of identities in spaces involved and we come

up with the following theorem.

3J

to be zero-mean and independent. As such,

X and n have Qx and Qn

as their variance

operators respectively. The propbem we are

facing is to find a filter, T B+B, linear

and causal, to operate on X + n such

that the error, defined as x-y, where y is

the output of T, has a variance operator

that is minimal in the partial ordering of

the positive operators. We will describe

this ordering right after we find the vari­

ance of the error.

Let e deonte the error and Qe denote its

variance operator. Since

def e = X-Y = X - T(X+n) = (I~T)X + Tn

we have

* Qe = (I-T) Qx (I-T) * + T Qn

T , following

from the results in section II. Rearrang­

ing terms in Q , we get e

Qe = T(QX+Q~)T* - Qx T* - T Qx + Qx·

Q is dependent on T. e notify the dependence.

We write Qe(T) to

Q (T ) is said to e 0

be minimal for some filter To' if

Q (T) < Q (T ) => Q (T) = Q (T ) e - e 0 e e 0

(A ~ B if (B-A) is positive).

In the equation for Q , Q + Q represents e x n

the variance operator of X+n, hence is

positive and self-adjoint. Therefore, by

Theorem 4.2, there exists a resolution

space (H,E) and a linear bounded operator

K = (H,E) + (B,F), such that (a) Qe = K K*

(b) K is a left-miniphase. Without further

assumptions, the formulation would be stuck

right here. At this point, what we need

is an invertable factor operator K. The

invertability of K can be guaranteed by

the invertability of Qx+Qn. There are

several ways to secure the invertability

of Qx+Qn. One way is to assume that Qx+

Q in onto and Q is positive definite. n n With an invertable factor K, Q can be e rewritten an

Qe = T K K* T* - Qx T* - K Qx + Qx

[TK-Q (K~)-l] [K~T*-T-IQ ] + Qx X X

_ Q (K*)-l K- l Q X X

[TK-Q (K*)-l] [TK-Q (K*)-l]~+ Qx X X

_ Q (KK*)-lQ X X

The last two terms in the above equation, -1

Qx and Qx(KK*) Qx' are positive and in-

dependent of T. Hence, to find the mini­

mal of Q is the same as to find the mini-e -1 -1 *

mal of [TK-QX(K*) ] [TK-QX(K*) ] - de-

noted as Q(T) in the sequel. Right now,

we are facing the same kind of problem as

in classical Wiener-Hopf filtering. Mini-

32

-1 -1 mal Q(t) occurs when T = QX(K*) K , but "it

does not represent a causal system in gene­

ral. In order to get a possible optimal -1

causal filter, can we decompose QX(K*)

into "causal part" and "strictly anti­

causal" (a term to be generalized in Banach

resolution space) and under what conditions

can we do so? This subject has been,treated

in Ref. 3 and Ref. 7 when the reflexive

Banach space happens to be a Hilbert space.

However in reflexive Banach resolution

spaces, the subject is still under research.

While we follow the same pattern as that

of classical wiener-Hopf filtering in fre­

quency domain, we would like to ask whether

this decomposition would work and how it

would. The same question in classical

Wiener-Hopf filtering is not directly ans­

wered in frequency domain. In order to

find the answer, let's assume the decom­

position. Let

Q (K*)-l = C + A, where C is the X

-1 causal part of QX(K*) and A is the "stri-

ctly anti-causal part" (a term to be gener­

alized in Banach resolution space). Then

Q(T) [TK-C-A] [TK-C-A] * * * [TK-C] [TK-C] - A[TK-C]

- [TK-C]A* + AA*.

To claim TK-C=O is the condition for mini­

mal Q(T), we should demonstrate that those * cross terms, [TK-C]A* and A[TK-C] , have

no effect on the ordering of Q(T). Again

when the reflexive Banach resolution space

is Hilbert resolution space, we've found

two ways to achieve this. The first one

is to take the trace of Q{T). Surely, work

has to be done to guarantee Q{T) being

nuclear. The second one is to take the

memoryless part of Q{T). This is justified

once the decomposition is given. However

we've also found advantages and disadvant­

ages to each way. For the method of taking

trace, it gives a minimal variance operator

once the maximum of the trace is found, but

we have to restrict certain operators, and Systems, New York, Springer-Verlag,

such as Qx and T, to be Hilbert-Schmidt. 1973.

On the other hand, the method of taking 4. Balakrishnan, A-V., Introduction to Op-

memoryless part works for a broader class timization Theory in a Hilbert Space,

of operators-operators have decompositions, New York, Springer-Verlag, 1971.

but it does not give a minimal variance

operator. The best we can have is a vari­

ance operator that has a minimal memory­

less part. However, there is an important

aspect for taking the memory less part.

This method allows us to generalize the

idea in reflexive Banach space, while the

other method does not. The reason is quite

simple, for it does not make sense to talk

about the eigenvalue of an operator that

maps from Banach space to its dual, not

to mention the trace of such an operator,

while it does make sense to take the

memoryless part given the decomposition.

Readers are referred to Ref. S for the

destils of Wiener-Hopf filtering formulated

in Hilbert resolution space. When all the

problems mentioned above are solved, we

would come up with the optimal filter

T C K- l , a causal system o

VI. CONCLUS ION

Wiener-Hopf filtering has been formulated

and solved in Hilbert resolution spaceS.

In this paper, we outlined the formulation

in reflexive Banach resolution space and

the possible way of solving it. Generali­

zation would be accomplished once the

theory of operator decomposition in Banach

resolution space is completed.

REFERENCES

1. Cooper, G.R. and Mcgillem, C.D.,

Probabilistic Methods of Signal and

System Analysis, New York, Holf,

Rinehart and Winston, Inc., 1971.

2. Parthasarathy, K.R., Probability

Measures on Metric Spaces, Academic

Press, 1967.

3. Saeks, R., Resolution Space, Operators

33

S. Tung, L.J., "Random Variables, Wiener­

Hofp filtering and Control Formulated

in Abstract Spaces", Ph.D. Thesis,

Texas Tech University, to appear.

6. Masani, P., "An Explicit Treatment of

Dialation Theory", University of Pitts­

burgh, published notes.

7. DeSantis, R.M., "Causality Structure of

Engineering Systems", Ph.D. Thesis,

University of Michigan, Sept. 1971.

APPROXI}~TE CONTROLLABILITY

AND WEAK STABILIZABILITY

Claude D. Benchimol System Science Department

University of California at Los Angeles Los Angeles, California 90024

Abstract

First using Sz. Nagy, C. Foias approach for contraction operators, we prove the following theorem: Let H be a Hilbert space, and T(t} a C contraction semigroup in H. Then H can be dec~mposed into three or~hogonal subspaces Hcnu ' Wu and W , all reduc~ng T(t} and T*(t}, such that

Wu + W Hu

Wu + Wcnu W

such that, On H ,T(t} is completely non unitary, and weakly stable On Wc~uT(t} is unitary and weakly stable On WU , T(t) is unitary, and x w, T(t)x / 0 at t + • W is called the "weakly stable subspace" W is called the "weakly unstable subspace". Hu reduces T(t) to a unitary group. H reduces T(t} to a completely non unitary contraction sgb\Ygroup.

Next, we define the controllable subspace to be C = Range T(t)B

t 0 and the uncontrollable subspace

C N(B*T*(t)}. t 0

Or main result is: Let A be the infinitesimal generator of a C contraction semigroup T(t} in a Hilbert space H, and B a bounded Rperator mapping another Hilbert space H. into H. Then, the system x = Ax + Bu is weakly stabilizable if~and only if the "weakly unstable states" of T(t} are approximately controllable, or equivalently W C. Furthermore K = -B* is a stabilizing feedback gain. As a corollary, the above condition is necessary and sufficient for strong stabilizability, when A has a compact resolvent. These results generalize the well known finite dimensional condition (see Wonham) and considerably strengthen the results of Slemrod, where only a sufficient condition for weak stabilizability was given, assuming that the resolvent of A was compact.

34

A MODIFIED DISCRETE CONVOLUTION OPERATOR FOR

SIMULATION OF LINEAR CONTINUOUS SYSTEMS

H. B. Kekre and D. B. Phatak Indian Institute of Technology

Bombay, India

SU~Y

Linear continuous systems are characterized by the convolution integral which takes the form of convolution sum for discrete systems. When it is desired to obtain a discrete model for simulating a continuous system, the traditional time-domain me­thods make use of the sampling process as it can be most con­veniently implemented. However, there is a one to one corres­pondance between the convolution integral and the convolution sum only for band limited systems and signals, with sampling rate greater than the Nyquist rate.

Steiglitz has derived an isomorphism which preserves convolution in the above sense. The discrete sequences representing the con­tinuous signals are obtained as coefficients of signal expansion on a continuous domain basis {A (t)} ~, where A (t) are La-n n=-~ n guerre functions. Oppenheim has given a generalized form of such basis functions. But the coefficient computation requires analog filtering and hence is not very attractive from digital computer simulation point of view.

This paper presents a new relationship between the two domains which retains the sampling process and removes all approximations attached to the sampling interval T, for the so called "unband­limited" square integrable signals and linear time invariant causal systems. It is based on the expansion of functions on a real exponential basis. A modified discrete convolution operator is defined which is shown to correspond to the convolution integral.

* DIFFF,Rl~;NTIAL SYSTEHS ON ALmRNATIV~ ALG~BHg)

Robert W. Newcomb Electrical B~ngineering Department University of l'laryland College Park, 11aryland 20742

A 00 tract

Continuing a previous study, which gave a universal square - law canonical for~ for differential systems by embedding the description in an attached commutative nonassociative algebra, we show here that when the algebra is alternative the description can further be reduced to zero.

"It is my wish that the Treasury shall make a chronicle setting forth the g~neolo~."

[1, p.1J 1. Introduction

Previous1~ [2] we have shown that any system

g described in the state variable form x ~ f(x,t),

with f reasonably behaved, can be reduced to the

canonical form

x(O) .. x given - -0

where 2t(t) is a vector in an attached algebra

0(8) associated with g and ~ = dz/dt. This alge-

bra is commutative rut not necessarily associative.

Amo~g the more extensively studied nonassociative

algebras are the alternative algebras [3, Chap. 3].

Here we show that if O(g) is an alternative alge­

bra the canonical form can be further reduced to

the ultimate form of zero.

2. ~lain Result

We first recall that an algebra is a vector

space in which multiplication of any two vectors

is properly defined [4, p. 144J. An alternative

alge bra J, is an a1ge bra in which for all ~, X E a

o

(?a)

and (?b)

Since the attached algebras for (1) are assumerl

commutative, there is no difference between (2a)

and (2b) in the considerations here. Too, as is

known [5, p.3l9], we may assume that U has a unity

element~, since otherwise one may be adjoined

while preserving the alternative nature of the

algebra. Now, given an alternative algebra a(g)

associated with a system g to yield its repre­

sentation x = x.x, that is, representation by (1)

within the algebra, we can proceed somewhat as

with Riccati differential equations [6, p.12] by

introducing a new vector variable y through . r '" ~'l (J)

Then differentiation of (J), using (1), yields

.y = -x.y-x.y = -(x.x).y + x.(x.y). On usinF, (2a) -- ---- --- ---this gives our main result

'y = 0 - - (4)

* This work was supported in part by the US Natioal Science Foundation under Grant i:SF gNG 75-CJJ227 and in part by a ]i'ul bright - Hays Grant to 11alaysia.

36

In other words, any system Hhose att?ched al[iehra

is alternative can be reduced further Hithin the

a1ge bra to Y = .2.

3. ~;olution

:';quation (4) ca,n l:€ lnterr,ratod to yield a

solution: . y = a (5a)

y(t)"'at+b - - a /" b constant (Sb)

Sul:Etitution in (3) gives, -1 1 if y and then b-

exist,

«(,a)

( 6b)

As the initial conditions are

~(o)

He obtain

== x -0

-1 -~'t (6c)

Hhich ar;rees Hith [:, ::q. (JA)] previously obtained

for division a1r;ebras. :;ince only the ratio of .<::.

to b is important, in (5b) He may all"ays t.ako

a == ~, £ == ,£. In this case 12 is clearly nonr3in[;U­

lar, and, hence (as Z(!) =):2 = .§.), by continuity

~ t) [or sufficiently [;mall t. By analytic contin­

uation (7) is then sr,en to t:e the solution Hithin

the a,ttached al terna ti ve al;cc bra.

;!e have S(Jen hcre tht!.t ir thl) attached al,w bra

U(o,) of a SylltO[;l 3 h~ eJter112-tive, thc canonical

equations ~ = ~.~ take tho very simple linearized

form .i = 2 Hithin Lhe '>3.'1e 2.lso bra. This rurther

loacls to a sir:IJllc solution x( t) = x LO - tx rl ,.- ' 0 - . 0

r;iven the initial conoitions 2<0 ?:CU:), irri~r;nc,ctjve

of Hhether or not the alt8rnative alc;ebra. h; a

divbion a]i~r·L'Ta. 1t is of interest to Clotr" thai,

alternative cliv:!,[,ion illr:ebras then~~olves ilTe lTell­

studi()d and are either il,sGociativc or isonorphic to

an ei~ht - dinensional ~aylny - Dickson d~vision

algr; bra over thd,r center L'7, 'P. :;l!l:.

To H. :J. H. ,Tanalullail llith rene"lbrar.ce oT

!';alaysian developments.

37

"Hhf)n he hnc'.nl the HOrd of lib Highner;s, hr; tool-: t.he cO;;J'llanu lljlOll hb h"ad ami hib linbG Here boHed h)T)eath the Heirht 0[' it." .,

LI, PP.J-?:

l1ercre!lces:

L'll" 1 A I" (" " '1 ) (' , ' ,.a ay nna s "C Jarall ,'·0 ayll , '~. '. "rmrn,

L.3l

tran"lator, Oxrord Univerr;it.y rre~:.~. ;'lal'J

;,umpur, 1970.

:i. ;1. LeHcomb, ";.onlincar Li:~I'()rnnti8Jjyr;tel'l~~: II Canonic, i'iul tivariable Thecry," l'roceedinf's of the L';;~:;, Vol. (,5, [.0. (., June 1977.

il. ~). :,chai'er, "An Introul)ction to .,ona2.scci­a.tive l\l~:,p,bras." Acacierr,ic J-re~)f;, .,r.~; YorI", 19(/.

L. J. raic;e, " Jcrua.I: A13eb:r:'u.5," in ",)t~J.f!ies jn l:od ern Ale:'> bra," cd i ted by A. A. j,] lx,rt, 'i'hr· Lathe:latical Association of' A!1erica, 19(,3, r:m. JlfLf - 18G.

L51 j. •• A. Albert, liOn Rie;ht Alte~";rJ-LiV('; Alf':pr)}·;::l.~:, 11

Anm'.ls of Lathem2.tics, ·.jol. 5'~" ,D. 2, Anril 191f9, PP. 318 - 320.

L6l ::. '1'. i'lcid, "::lic',],t:i, ljE;'nronti~J ''111ation~~,'' AC8,(1'_8nic Pres~J, 1:(:H '{or~~, 1')72.

L.7J '. j(lfdnI'cld., "A C!hiJTact"rization 'l:' th;) ';ayl~y :·;urrbers,11 in "~)tHcJiGG in ;.odern t\ll/~l:r.a, II ed Hwl by A. A. Allxrt, 'l'he ,.a.thc'ni',t ir.iJJ ASGoci.3,tion or Ar~erica, 19('3, pp. J?:. - Ih3.

- r 'J, '-, '

~r ... • I ~) • (. ;;;.

LAGRANGIANS WITH INTEGRALS.

AN APPROACH TO THE VARIATIONAL THEORY OF DISSIPATIVE SYSTEMS

Vuk M. Fati6 Electrical Engineering Department

Tri-State University Angola, Indiana 46703

Abstract

William A. Blackwell Electrical Engineering Department Virginia Polytechnic Institute

and State University Blacksburg, Virginia 24061

Some fundamental results on Hamilton's principle with a class of Lagrangians which depend of path dependent integrals are derived. It is shown that dynamical equations of linear dissipative systems could never be identified with the sta­tionarity conditions for this class of action functionals. An equivalence, how­ever, is established between dynamical equations of purely dissipative linear systems and stationarity conditions for an action functional with a Langrangian containing integrals, and a Hamilton's principle for purely dissipative linear systems is thus established. Exponential Lagrangians for one-dimensional sys­tems are derived as a special case of the theory.

1. INTRODUCTION

Network and system theory is occasionally dressed up in the formalism of analytical mechanics, but no satisfactory method is known yet for incorpora­tion of dissipative forces in Lagrange-Hamilton's theory. Rayleigh's dissipative function does not resolve the problem, because it destroys the link between Lagrange's equations and Hamilton's prin­ciple. The difficulty lies in incompatibility of Lagrange's equations with dynamical equations of systems with dissipation. For linear systems

Rund's conditions reduce to: (a) ~ and ~ should be symmetric matrices, and (b) ~ should be a skew­symmetric matrix. These conditions are obviously met in the case of conservative (B = 0) and gyro­scopic (~= _~T) systems, but conditi~n (b) ex­cludes dissipative systems, which have symmetric ~ ~~. For the special case of systems with one degree of freedom it means that ~ - term cannot appear in Lagrangian equation.

H. Rund (1) has shown that equations

are identical with a system of Euler-Lagrange's equations:

(1)

(2)

for a L(t,gA) iff ~(t) is a symmetric matrix, B(t) is a skew-symmetric matrix, and B(t) = K(t) -~T(t). When coefficient matrices are-consta~t,

38

Rund's theorem, however, presumes the identity

which is not necessary to satisfy in order to es­tablish a variational principle of Hamilton's type:

tl 6 J L(t,.9..,g)dt = 0 (4)

to

Existence of Hamilton's principle requires only an equivalence of equations (1) and (2), i.e. identi­ty of their solutions. As a matter of fact, if a Lagrangian L(t,~,~) could generate

d aL aL dt ag: - a~ -

= !(t,~,g) .{ -It [~Jt)§.] + ~(t)§. + !(t)~ - Q.(t)}

with matrix! nonsingular in [to,tlJ ' equations (2) would be equivalent to (1), and the variation­al principle (4) would be demonstrated. This idea has already been explored: Havas(2), Rohrer(3), and Denman(4)-(6) examined its various aspects, and it is sUfficient to mention here that equation

Mq + Bq + ~(q) = 0 ( 5)

could be generated from Lagrangian:

~t [ ] L(t,q,q) = eM ~2 - ~~(X)dX qo

(6)

while the equation

Mq ± Bq2 + ~(q) = 0 (7)

follows from the Lagrangian:

M ±2~ lq L(t,q,q) = 2q 2e -

±2Bx e W ~(x)dx (8)

qo

However, Van der Vaart(7) proved that the multi­plier method cannot be generalized to systems with n>l degrees of freedom (unless some very special conditions are satisfied, or system is uncoupled). Lagranians of this type have been used in some quantum-mechanical problems (Buch-Denman(8), Kerner(9), Besieris-Fatic(lO~, and the multiplier method has been recentl) extended to some contin­uous systems (Denman(ll , Fatic-Blackwell (12».

The assumption (3) can also be circumvented, as it was done in the "image" method (Morse-Feshbach(13~ Leech(14», by adjoining an "image" system with negative dissipation (-~) and generalized coordin­ates ~ to the given system. The requirement (3) can then be replaced (for time-invariant systems) by

39

It it - ~~ = ~q + ~9. + ]Sq - Q.( t)

:4 aL - ~ = Mr - Br + Kr - R(t) ut ag: a~ -- -- -- -

which is satisfied by a Langrangian, bilinear in ~ and~. Applications in network theory and general­ization to time-varying systems are presented in (15)-(17)

Both of these methods have their shortcomings: the multiplier method is limited to systems with one variable (discrete or continuous), while the "image" method requires additional auxiliary var­iables; canonical formalism in both cases does not lead to physically meaningful results.

Another radically different approach was proposed by Huang and Blackwell(18): instead of modifying Lagrange's equations by Rayleigh function, dissipa­ted energy can be included in Lagrangian, together with other energy functions. But in contrast to other energy functions, which depend on the state of the system only, dissipated energy WD depends on the whole path 9.{t) in[to,tJ, so that

WD = J\_T(-r)~g{t)dt to

(9)

is a path-dependent integral. Presence of a path­dependent integral in Lagrangian raises a question of whether Lagrangian equations (2) can be used as stationarity conditions any more. In order to clarify the situation, we have developed some fun­damental results concerning variational principles with Lagrangians which contain one general class of path-dependent integrals. These results, interest­ing in themselves as a new variational problem, may eventually prove to be useful for variational theory of dissioative systems. At least, Rund's theorem does not apply to Lagrangians with inte­grals.

2. LAGRANGIANS WITH INTEGRALS

Lagrangians composed exclusively of integrals ap­pear i~ nonlinear system theory (White-Woodson(19), Stern(20), Meisel(21), MacFarlane(22),(23), Jones­Evans(24», where all relevant energy and co-energy functions are expressed as integrals. Lagrangians

-(6) and (8) also contain integrals. All these in­tegrals. however. are by their very nature path­independent; they represent state functions, which depend only on initial and final states of the system. In order to get suitable mathematical framework for variational theory of dissipative systems. integrals which are truly functionals need to be installed in Lagrangian.

Fairly general class of Lagrangians containing in­tegrals is represented by

L = L [t,g.<t), 9..(t), ~(t)] (10)

where

1 ~ t f CT. 9.(-r), 9..(T)] dt (11);

to

9.(t). 9..(t) and 9(t) are n X 1, n X 1 and m X 1 matrices (vectors) respectively:

T g(t) = {ql~t). q2(t)' .....• qn(t)}

g(t) ={ql(t), 42(t),·····'qn(t)}T

T 9(t) = {9l(t). 92(t)' ..... '~m(t)}

Dissipated energy belongs to this class (f=gT~). as well as Lagrangians (6) and (8): f = ~(q)q for

(6). and f = e±2l q ~(q) q for (8).

It is important to realize the difference be­tween classical Lagrangian

L = L [t, 9.(t) , 9..(t)] (12 )

and Lagrangian (10). L in (12) is a function of t. g(t). and g(t) in the ordinary sense, i.e. it depends on t. 9.(t), and 9..(t) pointwise. On the other hand. L in (10) contains terms q which de­pend on the path 9.(t) , taken by the system in the time interval [to.t]. Therefore, L in (10) con­tains integral functionals together with the point functions 9.(t) and 9.. (t). Lagrangians containing functionals ~(t), with functional nature of 9(t) spelled out - -

L = l{t, 9.(t) , 9..(t), 9 [9.(T): to$ t ~ t]} (13)

may nevertheless be point functions of its var­iables t. 9.(t) , 9..(t) , and 9(t). For convenience (which will be apparent soon) we shall restrict

40

consideration to this simpler class; Lagrangians (6) and (8) are typical examples of this type.

Lagranqians (13) should be distinguished from the case

L = l{t, 9.(t) , 9..(t), F[t, 9.(t), 9..(t)]} (14)

where L depends on t, 9.(t) and 9..(t) explicitly and implicitly by the way of a pOintwise function F: (14) is actually a classical Lagrangian of the same type as (12), only more complicated. The term 9(t) in (13) is not a pointwise function of t, 9.(t) and 9..(t). but a functional, which depends on the whole path ~(t) between to and t,i.e. on the history of the system.

Consideration of Lagrangians (13) leads to a fund­amental question: what is a necessary condition for stationarity of the action functional

J ~ Itl L[t, 9.(t) , 9..(t), q(t)] dt (15) to !

with fixed endpoints to and tl? The assumption taken in (5), (18) and (24) that

~ ltl u L[t, 9.(t), 9..(t), ~(t)J dt = 0

to

is equivalent to

(16)

(17)

is not apriori justified, because L is not of the type (12), which has been assumed to be the case in the derivation of Euler-Lagrange's equa­tions (17). Therefore, it becomes necessary to derive an appropriate equation from (16).

2.1. STATIONARITY OF THE ACTION FUNCTIONAL

In the class of functionals (15), with q given by (11), the simplest case is when -

( 18)

Introducing a new variable ~ ~ q(t) , ~ = 9.(t) and i = 9..(t) , and (10) becomes a fu~ction of r's: L = L [t, .t.<t) , i(t), .dt)]. For this ty;e of

Lagrangian Euler-Lagrange's equations are:

Returning to the variables ~, (19) reads

E.!:. - ~ E.!:. + ~ E..l;. = 0 (20) a9 dt a~ dt2 a~ -

This is the necessary condition for stationarity of J for Land q given by (15) and (18) respectiv­ely. In a more-general case former substitution procedure is not feasible, so that the derivation must be carried out directly calculating the var­iation of action functional in terms of ~~.

The first variation of J with fixed endpoints is:

t oJ = J 1 oL [t, ~(t), 9.(t), 9(t)J dt

to -(21)

The integrand, being a function of t, ~(t), ~(t), and q(t), can be expanded by Taylor's theorem into

By the well-known property of isochronous varia­

tions:

(23) ,

so that after integration by parts

tl eJ = J [(E.!:._ d.u. )\n(t) +(E.!:. )Teq]dt(24)

to a~ Of a~:L a~.:.

(the endpoints are fixed). Now oq(t) should also be expressed in terms of e~:

J tl T Jt . = (;~) 15 f[T, ~(T), 9.(T)]dT dt (24)

to .:. to

Using commutativity of variation and integration and Taylor's theorem again

tl t

eJ1 = J (;~)T J [~~ e~(T) + * e~(T)J dT dt (25) to .:. to

41

Integrating the last term in (25) by parts

t t (26) l ~i e9.(T)dT = (*)T=t oq(t) -l d~*Oq(T)dT because o~( to) = Q., and o~( t) 1 Q.. Therefore

It1 T

oJ 1 = t (~~) (~) t o~( t) dt + o .:.

t1 T t

+ I (~~) I ( ~~ -d~ * )e~(TldT dt (27)

to .:. to

Changing the order of integration in the last term (Fig. 1) it becomes

t

Fig. 1: Integration region in (27).

Interchanging dummy variables t and T the last ex­pression turns into

It1 tl T

[J (~~) dTJ( ~~ - a%*)o~(t)dt to t .:.

(28)

so that:

oJ = /tl{( ;~)T - a%( ~l + (~~r(*) + to .:.

(29)

Due to arbitrariness of e~(t), stationarity condi-

.. ,

tion oJ = 0 is equivalent to

(30)

Transposing this matrix equation we get:

This is the necessary condition for stationarity of J , which will be called EULER'S EQUATION in order to distinguish it from LAGRANGES EQUATIONS (17) .

When f = ~ /see (18)/ Euler's equation becomes

tl

!h.-.J!~+J!h.d =0 aB. dt aB. aq T t !

which· evidently reduces to (20) after differentia­tion with respect to t.

2.1.1. Discussion of Euler's Equation

Euler's equation (31) is a differential--integral equation, whose the most conspicuous feature is the presence of an integral from t to tl' The in­terval (t,t l ] is the "future" with respect to the moment t; hence Euler's equation apparently states that future behaviour of the trajectory (q(T) for t < T ~ t 1) influences its present state q(t). This is a feature typical of anticipative systems, not of dissipative ones. Therefore, Euler's equa­tion seems in principle unsuitable for dissipative systems.

However, the other terms in Euler's equation are also functionals in general: they involve q(t) terms (aL/aq also depends on q in general), 'i.e. they depend!on the past behavior of the system -of its history. Now we see that Euler's equation is a functional equation which relates the past and the future behaviour of the system via its

42

present state. This type of equation does not ne­cessarily violate causality; for instance

tl itt) 1 HT, !lh), itT)] dT = 0

t

is equivalent to

.9. + Ht, !l(t) , 9..(t)] = 0

which is just a system of ordinary differential equati ons.

To show that Euler's equation is not disqualified as a mathematical model for real systems, all we have to prove is that (31) can be reduced to an equation which does not contai1tl aL Indeed,

aq clT. t .

introducing the notation:

6 aL d aL + (af.)T aL ~(t) = a!l - dt.ag: 8! a~ ~xl

~xn

_C(t) 6 ~ = aq ~xl

for the sake of brevity, Euler's equation reads

(32)

For a coordinate qi (i = 1, 2, .... , n) this is an

algebraic equation in J(tl I(T) dT:

Differentiating (33) m times we get:

1.e.

k

+ L (~) d:: ,Q,=O

T dk-,Q, B. -;:-;;- I{ t) + -1 dt"'-,Q,

dk T Jl +-..B. C(r)dT=O dt~ -1 t -

(34)

(k = 1,2, ... ,m). This is a system of algebraic

Itl

equations in t ~(T) dT, which can be solved if

S.(t) is nonsingular (mxm matrix S.{t) is defined -1 k -1

by Skj ~ d!k Bji ) and the solutions substituted

in (33). The result is the eliminant

where ~i{t) is the vector mxl, whose components are the first two terms in (34) for k = 1,2, .. ,m. The determinant (35) does not contain (tl any more, although it is still a differential-i~egral equa­tion in ge~eral, because it may contain q. But they are all~t, so that Euler's equatio~ (3l) is

actually not l~mited to anticipative systems, but also suitable for hereditary processes (25) or

systems with memory.

2.1.2. Complementary Case of Euler's Equation

If q in (lO) is defined as !.

tl ~(t) ~f f[T,9JT),~{T)J dT

t

(36)

necessary condition for stationarity of J can be derived in the same fashion as (3l). The differ­ence begins in {26}, which reads now:

tl = - (!f.) 09.{t) -J ~ ll:. 09.(r) dT (37) a9. T=t dT a~

t

(because o~tl) = ~), so that (27) becomes

tl T oJ l = - tJ (~~) ( *)t oq{t) dt +

o

(38)

Double integral in the second term of (38) is now taken over the region shown in Fig. 2., so that changing the order of integration second term in

(38) changes into

43

t

Fig. 2: Integration region in (38).

i tl iT T [ (:~) dt] (~~ - i*)o9.{T) dT

to to !.

Interchanging the variables t and T it turns into

tl t T J [J (;~) dT] (;i -d~ *)Oq{t) dt to to &

Now evidently Euler's equation is t

aL d aL ( at)T aL + (U, d .!!.)T J aL a9. - dt ag: - ag: a~ a9. - dt ag, a~ dT = ~

- to - (39)

It can be proved in the same way as in 2.1.1. that~: may be eliminated from (39); the result is a differen­tial-integral equation with tftl only, so that (39) seems to be generally suitable for anticipative systems.

2.2. RELATIONS BETWEEN EULER'S AND LAGRANGIAN EQUATIONS

Euler's equation is obviously a generalization of Lagrangian equation: when f = ~, i.e. when Lagrangian does not contain integrals, (31) reduces to (17). There are several other situations in which Euler's equation is related to Lagrangian equation.

I

2.2.1. Transformation of Euler's Equation to Lagrangi an Form.

Lagrangians (10) depend on ~ and i explicitly and implicitly via q; one may expect, therefore, that necessary condition for stationarity of (15) is of the form

DL d DL _ ng:-Clt~-Q. (40)

where partial derivatives D/D~ and D/Di take into account both explicit and implicit dependence of L on ~ and i.

In order to verify this conjecture, we need expres­sions for Dl/D~ and Dl/D.9.; they can be obtained if one can separate contributions of increments of ~ and .9. to the increment of L. This separation can be achieved by avoiding elimination of 0.9. in the derivation of Euler's equation.

Changing the order of integration in (25) we get (Fig. 1)

t} t}

oJ} = f [f ( ~~ f dtJ (~i o~h) + * 0.9.( T) ) dT

to T

and interchanging dummy variables t and T

so that

(41)

The multiplying factors of o~ and oi (in square brackets) are measures of the rate of increment of L due to the increments of ~ and i. It seems rea­sonable, therefore, to define them as global par-tial derivatives: t

T } Dl/:,.£.!:.+(af.}J .£idT (42) D~ = a~ a~ aq

t .:.

Then (41) reads as

t}

oJ = J [( ~~)TOq(t) + ( ~)\gJt)J dt to

Integrating second terms by parts

tl

oJ = J (~~ -d~ ~rOq(t) dt to

Comparing (45) with (29) we see that

_ DL d DL = D~ - CIt og:

(43)

(44)

(45)

(46)

Therefore, with the global partial derivatives de­fined as in (42) - (43), Euler's equation can be put in the lagrangian form.

The artificial form in which DL/D~ and DL/Di appear in (42) - (43), however, points out that the reduc­tion of Euler's equation to an equation in Lagran­gian form is achieved by force. As a matter of fact, Euler's and lagrangian equations are two dif­ferent equations, which is in agreement with the well known fact that stationarity conditions for the action functional have additional terms, besides those in (17), if Lagrangian contains higher deriva­tives of q than the first.

2.2.2. Equivalence between Lagrangians containing Integral Functiona1s and Lagrangians with­out them.

Although Euler's equation cannot be reduced to La­granian equation with the same Lagrangian L, it does not prohibit the possibility that there is another Lagrangian L' such that left-hand side of

aL' d aL' a~ - CIt aq = Q. (47)

coincides with the left-hand side of Euler's equa-

tion (31). An insight into th~se equations makes the following guess plausible:

Theorem 1: If Lagrangian is of the form

L(t.9..9..~) = Ll (t.9. • .9.) + hl(t)9 (48).

Euler's equation (31) is identical to (47) with

Proof:

~ _ ~ ~ = ~ _ ~ ~ + (.£1 d af)T a9. dt a9. a9. dt a9. a9. - dt a9: X

tl tl

J h2{T) dT - (*)T d~J h2h) dr =

t t

=':':l_~aLl (af)T (af daf)T a9. dt ar- + aB: h2{t) + a9. - dt ~ X

The expression on the right is actually the Euler's expression for L given by (48) because ~~ = h2(t). Q.E.D. ..

Corollary: Instead of applying Euler's equation to Lagrangian (48). the same result can be ob­tained if Lagt'angian equation is applied to the Lagrangi an (49).

2.2.3. Path-independent Integrals and Euler's Equation

If Hamilton's principle is to be found from (16),

Euler's equation (31) ought to be equivalent to dynamical equations; the simplest form of equiva­lence is the identity:

tl

.£h _ ~ .u. + (~) T .£h + ( at. _ ~ g ) T J a L -a9. dt a9. a9. aq a9. dt a9. tqdr = . .

t -

(50)

In selecting a proper Lagrangian, f and aL/aq can be chosen independently of each other; in otfier

words. integrand of 9 and the functional dependence of L on 9 can be chosen separately. The most at­tractive-choice of f would be one which annihilates the last term in Euler's equation. one which has an­ticipative character. Such f is to satisfy the

equation af. d af _ a9. - dt aB: = Q.

(51)

identically, i.e. for every ~(t)

Theorem 2.: f is an identical solution of (51) iff

f = it f.{t.9.)

where f. is an arbitrary function of t and 9.. ~: If f is of the form (52), i.e.

(52)

(53)

for i = 1.2, ...• n (the summation convention implied),

then

so that (when mixed derivatives are continuous)

afi d afi _ aq-; - Of a"Cik = 0

~i ,k = 1.2, ...• n, which is tantamount to (51). If f satisfies (51) i.e. (54), then

2 2 2

(54)

af i a f1 a f i . a fi .. 3Cik - ataClk - aqja~k qj - a<ijaClk qj :: 0 (55)

The first three terms in (55) do not depend on q .• J hence the coefficients of q. must be zeroes:

J

so that

and

af.

2 a f.

---.-,--..l = 0 aqjaqk

a< = <Pik{t.9.)

~.j

fi = <Pi~(t,9.)q~ + ~i(t,9.)

(56)

(57)

(running ~).where <P'k and ~. are arbitrary differen-1 1

tiable functions of t and 9.. Inserting (56) and

45

F

II

(57) into (55) we get

a~i~. a~i a~ik a~ik. aqk q~ + aq; -at - aqj qj '= 0

i. e. ~--- q +----=0 ( a~i' a~ik). a~i a~ik aqk aqj j aqk at-

This identity is possible only if (~j = 1.2 •..• n)

a~ij a~ik _ a~i a~ik - (58) - -- = 0 and - - -- = 0 aqk aqj aqk at

Whenever the conditions (58) are satisfied. ~.dt + , + ~ikdqk is an exact differential. Then a function Fi(t.~) exists such that

aF. aFi ~i = ~ and ~ik = aqk

Substituting (59) in (57) we get

aF. aF. f ,. +-' i = aq- q~ at

~

( 59)

¥i = 1.2 •...• n. which is the same as (53). Q.E.D.

Therefore. every f satisfying (51) identically has to be of the form (52). But then

t (60)

q(t) J c& £[T.~(·r)] dT = £[t.~(t)]-E[to.~(to)] to

so that Lagrangian does not contain integral func­tiona1s any more. and reduces to the classical case (14). which - according to the Rund's Theorem - cannot serve the purpose.

t

+ J ( ~i -if * )c~(T)dT to t

c~(t) = * c~(t) + J( *- -if * )o~(T) dT to

(61)

because c~(to) = Q (see Fig. 3)

'I.

~--~t~o------~~~--~t----~~

Fig. 3. Variation of the path

The first term in (61) represents the contribution due to variation at T = t; it vanishes when o~(t) = Q (Fig. 4). Integral term in (61) collects all the contributions due to variations between to and t.

The path-independence of q(t) means that its value is the same no matter which path is followed be­tween two fixed points (Fig. 4). If

Equation (60) at the same time proves sufficiency t of the statement in

Theorem 3: q(t) is independent of the path (i.e. q is a state-function) iff f is given by (52).

Proof of necessity: If q(t) is independent of the path. its increment. oq.-corresponding to an arbitrary variation of the path between two fixed points. is zero: oq(t) = Q . The first variation of q(t) is: .:.

.:. t

Fig. 4. Independence of q of the path we set oq(t) = ~ and o~(t) = ~ in (61). path inde-

.:.

46

F

pendence is assured if

t .

J (~; -i if) 09.(T )dT = Q to

for arbitrary 69.(.). This can be true iff

al "' ~ ~ = a 39. dT 09. -

for every 1: E [to, tJ and every 9.(T), i.e. if f satisfies the equation (51). But then by Theorem 2. f is given by (52). Q.E.D.

From Theorems 2 and 3 follows Corollary 1: Integral (10, is path-independent iff f satisfies (51).

Corollary 2:The most general state function of the form (11) has f given by (52):

d [ ] aE 3E. f = at f. t,9.(t) = at + 39. 9. (62)

which is linear in~. (Of course, any function linear in 9.:

(63)

with arbitrary i and 1 does not always generate a state function). Theorems 2 and 3 show that one can get rid of the last (anticipative) term of Euler's equation iff integral q is path-independent. But then one can eliminate-integrals from ~agrangian, because f is given by (52), and q represents a function of t and 9.. as (60) show~. Euler's equation is then reduced to

and, since

~ _ ...!! a~ + (2.t.)T ~ = a a~ dt 39. 39. a~ -

u. = aE a9. a9.

Isee (62)/. eventually to:

~ _ d~ + (1E.}T ~ = a 3.9. at a 9. Cl9. a9 -

(64)

(65)

On the other hand. in this case Lagrangian is a classical one, of the type (14), so that Lagran­gian equations also apply. L depends on 9. both explicitly ~n~ implicitly, through f.(t,9.). Using the symbol a/a9. as partial derivative which takes

47

into account both explicit and implicit dependence on 9.. we get:

aL daL_aL+(<lE)ToL d3L 39. - dt IT - a9. 09. a9 - at aI

so that Lagrangian equation A A

aL d aL 39. - df N = Q. (66)

gives the same result as Euler's equation. This shows that in the case when lagrangian contains path-independent integrals. one can use Lagrangian equation (66) rather than Euler's equation. If any of the integrals is path-dependent. Lagrangian equa­tions are no more a)propriate, and one must use Euler's equation. (* Following examples will il­lustrate this pOint.

For dissipative systems Lagrangians with path-inde­pendent integrals are no better than Lagrangians without integrals; according to the Rund's theorem, they are both unable to generate identity (3).

2.2.4. Examples

~. Lagrangians (6) and (8) contain integrals with insegrand f = ~(q)~ in the first case, and

f = e±2~~(q)q in the second.They both belong to the

class f A G(q)q, which is a special case of (52) with q

F( t,q) = J G(x)dx qo

One can easily verify that they both satisfy (51).

Integrals from (6) and (8) are, thus, path-indepen­dent, which justifies the use of Lagrangian equa­tions in (5).

On the other hand, the integrals could be elimina­ted from (6) and (8). Theorem 1. generates an equivalent lagrangian

B tl B

L' = e~t ~2 + ~(q)q Jr (_eMr ) d. = t

* Another special case when one can use Lagran­gian equations, even if 9 is path-dependent, is the class of Lagrangians (48). but then, accord­ing to Theorem 1. L has to be replaced by an e­quivalent Lagrangian L'.

t ; j;

B t in the first case; the tenn ~ ~(q)q eP; 1 may be

omitted as an identical solution of Lagrangian e­quation, and finally

= /~[~q2 _ .p(q)q (t 1 - t)]

±~ • once again. the tenn e .p(q)q tl may be omitted, and finally

±2~ L' = e [ ~q2 + .p(q)q t] (68)

Lagranglans (67) and (68) are equivalent to (6) and (a) respectively, and at the same time they are sim­pler.

Ex. 2. If an external force is applied to the sys­tem, equations (5) and (7) are to be replaced by

Mij + 82q + .p(q) = Q(t)

and Mij + Blq2 + .p{q} = Q(t)

Appropriate lagrangians are (15)

q

(~q2 - / <l;(x)dx + Q(t)q] qo

B -2t

L I = eM [~ 42 + ;2 q.p(q) + Q(t)q]

for (69), and

(69)

( 70)

(71)

(72)

B q

L = e 2w1

q ~ q2 - f B B (73) ~lX 2~q

qo e 4>(x)dx + 2~1 e Q(t)

B

L I = e 2;{lq [~cj2 + q.p(q)t + 2~1 Q(t)] (74)

If both linear and quadratic dissipation are present:

Mq + Blq2 + B2q + .p(q) = Q(t) (75)

with Bl; 0 and B2 ; 0, it was shown in (15) that

B B B ~ +;t-~ . it X

L = e 2 q2 - e

B B B q 2p;l x ~q+i"t Je .p(x)dx + ~ Q(t)e

qo .

(76)

and B B

~q + -2t rl ] L' = e M l~q2 + ~2 q.p(q) + 2~1 Q(t) (77)

L' can be derived from L applying Theorem 1.:

B tl B .

+ e~ q 4>(q)q J _etf T dT t .

This is equal to the expression in (77) if one drops the term containing t l , which satisfies Lagrangian equation identically.

Ex. 3. With the multiplier method in mind, it seems reasonable to investigate the lagrangian:

L(t.q,q,~) = e~Ll(t,q.q) (78)

When substituted in Euler's equation (31) it gives

aL 1 d ( all) af eq - - -=- eq .,......-- + ~ eqL l + • aq ut . aq aq'

tl

+ { ~~ - d~ *)J e9 Ll dT = 0 t

(79)

for (70). Corresponding Lagrangians Land L' are ob. Now taking viously related by Theorem 1. f = ~ (80)

48

It is easy to check that Lagrangian equation with L'

Ll = ~ q2(t) - ~ q2(t) + Q(t)q (79) becomes:

(81) from (84) gives the same result (83) as Euler's equation with L from (82).

~t-to) r, B ] e LQ(t) - Kq - Mq - MMq = 0

which is equivalent to

Mq + Bq + Kq = Q(t)

With the choice (80) - (81) Lagrangian (78) is equal (except for the insignificant constant factor

B :M to e ) to the Lagrangian (71) with ~(q) = Kq and

qo = O.

Taking f = ~~i~ and the same Ll as in (81) the re­sult can obviously be extended to the case of time variable coefficients(3).

The multiplier method is thus incorporated in our theory.

Ex. 4. Lagrangian given in(18), apart from its topological complexities, is essentially of the form t t (82)

L l·T M' 1 T K 1 f· T • J T( ).( = 2S _9 - II _9 -"2 9 ~9 dT + Q. T 9.: T)dT to to

This Lagrangian contains two path-dependent inte-

The very fact that there is an equivalent Lagrangian without integrals is an evidence that Lagrangian (82) does not generate dynamical equations for lin­ear dissipative systems; the result only confirms thi s concl us i on.

The last term in (83) does not have any physical significance. Therefore, Lagrangian (82) does not fulfill the expectations of (15). At the same time, Lagrangian (84) has the same effect, but its struc­ture is simpler --- formally and in principle.

3. CAN EULER'S EQUATION GENERATE DYNAMICAL EQUATION OF LINEAR DISSIPATIVE SYSTEM?

Negative result in Ex. 4. is not incidental, i.e. due to an unfortunate choice of Lagrangian. As a matter of fact, it is a necessary consequence of

Theorem 4: There is no Lagrangian function of t, q, q, and ~, where ~ is defined by (11), which generates the identity tl d aL aL aL af + ( d af af )/ aL

Of aq - ail - ail aq Of aq - ail ail dT -. t .

= Mq + Bq + Kq - Q(t) (85) grals, because their integrands do not satisfy (51). Therefore, in using this Lagrangian one must apply Proof: Using handy notation Lq ~ ~~. etc., and taking

Euler's equation, rather than Lagrangian equation. total time derivatives:

In this case:

aL ail = H-] f=

so that

(*)T = [at (~T ~ g)! at (Q.T~)] = [2~g \ Q.(t)]

and Euler's equation gives:

- ~9 - Mg - B9 + g(t) + [~g - ~(t)](tl - t) = Q. (83)

It is possible to formulate for Lagrangian (82) an equivalent Lagrangian, which does not contain in­tegrals; by Theorem 1.:

(84)

49

d dt fq = fqqC:i + fqqq + fqt

(85) becomes~tl

(Lqq + fqq J LqdT) t .

+ Lq' t - L + L· f - L f· + (f. -q q~ ~ q qt

= Mq + Bq + Kq - Q(t) ( 86)

The only terms involving q are the first two terms on both sides of (86), so that we must have

tl

L.. + f.. J Lq dT = M (87) qq qq t

Differentiating (87) with respect to t we get:

d L. ·-M Lq = df~ (88)

. qq

when fqq 1- 0

Lemma 1: (88) is equivalent to (87) iff

Lqq = M at t = tl ( 89)

Proof: (8l) satisfies (89) and at the same time implies (88). On the other hand, from (88)

t

Lqq-M I -Lgg-M/ = f f.. f .. qq t=t qq t=t* t*

for any t*; especially for t=tl

tl L .• -M L· ·-M J ~f J - ~f I = L dT .. .. * q qq t=tl qq t=t t* .

Subtracting the last two equations, and taking (89)

into account we get:

t tl t tl L··-M / J ~:= L dT - L dT:= f.. q q qq t*· t*·

f Lq dT = -J Lq dT t 1 • t·

which is identical to (87). Q.E.D.

We can safely take the assumption fqq 1- 0, because the case fqq = 0 is of no interest, as Lemma 2 proves.

Lemma 2: If fqq = 0 the Euler's equation does not generate Sq - term.

Proof: If f·. = 0, then L .. = M ¥to In this case: qq qq

f q = A( t,q)

f = A(t,q)q + S(t,q)

Lq = Mq + F(t.q,~) _ 1 ·2 ( ). ) L - 2 Mq + F t,q,~ q + G(t,q,~

where A, S, F, and G are arbitrary functions of their respective variables. Euler's equation then

50

reduces to

None of these terms contain q, so that the case

fqq = 0 can be excluded from further considerations. (Especially when At-B = 0 q is path-independent, q • and then we know that Euler's equation reduces to Lagrangian equation, which cannot produce dissipa­tive term). Q.E.D.

The condition (87) eliminates the first terms in both sides of (86), and al.so helps to eliminate in­tegrals of Lq:

M-L .• (L. + f· ~) q + L· t - L + L. f-f.L + qq qq qq q q q~ q ~

M-L .. + (f· t - f ) ~ = Bq + Kq - Q(t) q q qq

From (88) we have:

L· .-M L =(2+·2+" -!+ f.L).=.gg-.:..:. q at q aq q aq aq f .. . . qq

But Lq does not depend on q, so that

and

From (91):

so that:

L· ·-M ~ = F (t,q,~)

qq

L·· = M + f·· F (t,q,q) qq qq .

(90)

(91)

(92)

L· := Mq + f· F(t,q,q),+ G(t,q,q) (93) qq. . and:

L = } Mq2 + f F + qG + H (94)

where f=f(t,q,q), while F, G, and H are arbitrary functions of t, q, and q.

r I

I !

These results simplifY (90):

(f. F + f·F + G - f· F) ~ + f·tF + f.F t + Gt -qq q q q qq q q

- f F - f F - ~G - H + (f·F + G ) f -q q q q q q q

- f· (Ft + F q + F f) - (f' t - f ) F = Bq + Kq-Q(t) q q q q q i.e. Gt - fF - H + G f = Bq + Kq - Q(t) q q q (95)

Taking derivative of L in (94) with respect to q, and comparing it with (92) we get:

f F + ~ G + H = F t + ~ F + f Fq 9 9 9 q

i.e. (G - F ) q + H - F

t = 0

9 q 9 which is possible only if

Gq = Fq , Hq Ft

But then (95) is reduced to

Gt - Hq = Bq + Kq - Q(t)

.

(96)

(97)

which is impossible because neither G nor H depend on~. Q.E.D.

Remark 1: There is no need to prove Theorem 4. for multidimensional case; one-dimensional case should not be an exception.

Remark 2: The same conclusion holds for the equa­tion

(99)

where ~, ~, y, and Q are arbitrary given functions of their variables, as can be seen glancing at the the former proof. This means that Van der Poh1 equation

q + y(q2_1} q + q = 0 ( 100)

for example, cannot be generated by Euler'S equa­tion.

The Theorem 4., therefore, shows that Euler's equa­tion, just like Lagrangian equation, cannot be identified with dynamical equations of linear (and some nonlinear) dissipative systems. This, however, does not mean that Hamilton's principle for dissi-pative systems cannot be found in the form (16),

type of equivalence - with a multiplier - was al­ready considered in Ex. 1-3. Different type of equivalence is based on observation that Euler's equation can generate the term B~, but accompanied by parasitic terms, like Bq(t1-t} in Ex. 4., which may not influence the solution. This idea is pur­sued in the next section.

4. HAMILTON'S PRINCIPLE AND MINIMUM THEOREMS IN NETWORK THEORY

For a network composed of capacitors and ideal volt­age sources ~(t) only, Lagrangian equations (2) with the Lagrangian

L = !- 9.T ~ 9. - ~T (t) 9. (101)

generate the correct equati ons ~ 9. = ~(t). (~i s the inverse capacitance matrix). Therefore, q's are distributed among capacitors so that

( 102)

has stationary value. In static equilibrium with ~ = const (102) reduces to the well-known Thomson's theorem.

Corresponding theorem for resistive network with ideal voltage sources ~(t) states (26), (27) that

the functi on

(103)

(~- loop currents, ~ - loop resistance matrix) has minimum for the actual distribution of currents. compared with all hypothetical distributions which satisfy Kirchoff's current law. When resistive net­work with ideal current sources let) is considered, corresponding function is

(104)

where ~ is the node conductance matrix, and ~ nodal voltages. Components of ~ are generalized velocit­ies, while generalized coordinates are the compon­ents of flux

t

t(t) ~ Jr ~ (T) dT because Hamilton's principle does not require the identity (85), but only a sort of equivalence. One so that ~=i.

51

The proof could be based on the modified form of Lagrangian equation

d -ll.- ~+ ~= n(t) Of c9. c9. c9. !t

with the Rayleigh's function

R~l9.·TB9.· - 2 -

which reduces for L = 0 to

i.e.

lB. = Q(t) c9. -

( 105)

(105) is the necessary condition for a minimum of R - !IT(t)~, which is only in notation different from (103) and (104).

Although close in spirit to Hamilton's principle, the former theorems for resistive networks have no direct relation to it. The connection with Hamilton's principle could be established if the Lagrangian

t

L = J [t ~T R ~ - IT(,:)~J dT (106) to

is considered. This is a Lagrangian with the inte­gral, actually composed of only one integral with

( • ) 1· T . T() . f T ,9.,9. = '2 9. R 9. - I T 9. ( 107}

Stationarity condition for the action functional

tl t

J J [1 • T . T()'J ( J = '2 9. R 9. - I T 9. dT dt 108) to to

is the Euler's equation (31), which reduces in this case to

(Ri - I) - ~ (R~ - I) (t l - t) = 0 (109)

This is not the equation

R ~ - I( t) = 0 (110 )

for the flow in resistive networks, but it can be shown that (109) and (110) are equivalent, i.e. they have the same solutions for the same initial

52,

conditions.

Proof: Equation (110) obviously implies (109). On the other hand, introducing

I(t) ~ R~ - I(t)

for brevi ty, (109) has the form

I - (~ - t) ~t = Q.

Now for the kth component of I

dXk Xk - (t1 - t) c.rt = 0

i.e. Xk (t) t

J d~~ = J Xk(tO) to

so that the solution of (109) is

i.e.

(111 )

( 112)

( 113)

Initial values i(to), common to (109) and (110), satisfy (110):

( 114)

so that I(to) = Q.. (113) then implies I(t) = Q. in [to,t l ], which means that (109) implies (110). Q.E.D.

Therefore, Hamilton's principle for resistive net­works with ideal voltage Sources is

tl t

IS f /[t p"T R ~ - IT(T)i] dT dt = 0 (115) to to

Similarly, for resistive networks with ideal current sources

tl t

IS J J [~yT R y - IT(T)y] dT dt = 0 (116) to to

It should be noted that the Lagrangian (106) is a special case of Lagrangian (82) with the conserva­tive part (first two terms) omitted.

Lagrangian (106) also belongs to the class (48), with Ll = 0 and h2= 1, and therefore can be replaced

by an equivalent Lagrangian without integrals:

When (117) is substituted in Lagrangian equation (47), it generates the equation

( 118)

which is identical to (109).

At the same time (118) supplies an additional proof of equivalence between (109) and (110): (118) im­

plies

(t1-t)(!!.9.. - g) = const

and the constant is zero, which follows from (114) for t=to. or directly for t=tl; but

(tl-t)(! 9.. - g)= Q.

can be true for every t in [to.tl ] only if

!!.9.. - Q(t) = Q. Q.E.D.

Hamilton's principles (115) and (116) can. there­

fore, be replaced by

iational principle with Lagrangians containing inte­grals exist. ensuing analytical formalism can bear only an indirect relationship to dynamical equations for dissipative systems.

However. all those difficulties might have been caused by an insufficiently general framework. In­tegrals considered here may not have been general

enough. Integrals 1 ike

t

~(t) = f ~ (t,T) f [T.!l(T).9..(·d] dT - to

( 121)

or even t

~(t) =J f [t.!l(t).gJt);T.!l(T),g.<-d] dT

to

(122)

may prove to be more suitable for dissipative systems. This seems to be the case indeed: recent developments (28)-(30) in variational operator theory (31)-(32) lead to Lagrangians with integrals as a result. The integrals which appear there are of more general na­ture than those admitted in Lagrangian h~re. Hence

tl the result presented here need to be generalized in () f (tl-t) [t 9..T !!. 9.. - IT(t) .9.] dt = 0 (119) order to make a meaningful comparison.

to On the other hand. the idea of equivalence between Euler'S equation and dynamical equations is worth pursuing further in order to generalize the result

of Section 4.

and tl () f (tl-t)

to

respectively (in (120) ~ = 1)· CONCLUSION

Apart from its possible use in the theory of dissi­pative systems, the theory of Lagrangians with inte-grals offers variational formulation for a class of

Although the framework of variational calculus was differential-integral equations (31) and (39). and extended by introduction of Lagrangians with inte- variational principles for systems described by such grals. Hamilton's principle was established only for equations. like some hereditary systems (25)

purely dissipative systems. Even in this case Lagrangians with integrals turned out to be unneces­sary: the same equations could be generated by Lagrangians without integrals. Furthermore, the result was achieved only indirectly. through an equivalence between dynamical equations and Euler's equation; it could not be otherwise, in the light of Theorem 4. and Rund's theorem. Variational form­ulation, simple and direct as one known for conser­vative systems, cannot be designed for dissipative systems using Lagrangians with integrals. If a var-

53

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17. Fatic. V. M. & Blackwell. W. A.: COMPOSITE VARIATIONAL PRINCIPLES FOR LINEAR TIME-VARYING LUMPED-PARAMETER SYSTEMS WITH DISSIPATION, "Proc. 8th Ann. SE Symp. on System Theory", Univ. of Tennessee, Knoxville, Tn., April 1976. pp. 95-101.

18. Huang, H. L. & Blackwell, W. A.: NEW LAGRANGIAN AND HAMILTONIAN FUNCTIONS FOR LINEAR DISSIPATIVE PHYSICAL SYSTEMS WITH IDEAL DRIVERS, "IEEE Trans. on Circuit Theory", Vol. CT-1B. 1971, No.4, pp. 461-463.

19. White, D. C. & Woodson, H. H.: ELECTROMECHAN­ICAL ENERGY CONVERSION, Wiley, 1959.

20. Stern, T. E.: THEORY OF NONLINEAR NETWORKS AND SYSTEMS, Addison-Wesley, 1965.

21. Meisel, J.: PRINCIPLES OF ELECTROMECHANICAL ENERGY CONVERSION, MCGraw-Hill, 1966.

22. MacFarlane, A. G. J.: DYNAMICAL SYSTEM MODELS. George G. Harrap & Co .• Ltd., 1970.

23. MacFarlane, A. G. J.: FORMULATION OF THE STATE­SPACE EQUATIONS FOR NONLINEAR NETWORKS, "Int. J.

Control", Vol. 5, 1967, No.2, pp. 145-161.

24. Jones, D. L. & Evans, F. J.: VARIATIONAL ANAL­YSIS OF ELECTRICAL NETWORKS, "J. Franklin Inst~

Vol. 293, March 1972. No.3, p. 9-23.

25. Volterra. V.: THEORY OF FUNCTIONALS AND OF INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS, Dover, 1959.

26. Jeans, J.: THE MATHEMATICAL THEORY OF ELECTRI­CITY AND MAGNETISM, 5th ed., Cambridge at the University Press, 1933.

27. Desoer, C. A. - Kuh, E. S.: BASIC CIRCUIT

···~--~~~·c.~ ______________ ~ __ ~_~

THEORY, McGraw-Hill, 1969.

28. Tonti, E.: A SYSTEMATIC APPROACH TO THE SEARCH FOR VARIATIONAL PRINCIPLES, An article

in the book: "International Conference on Var­iational Methods in Engineering", Vol. I-II, pp. 1/1 - 1/12. Ed. by Brebbia, C. A. &

Tottenham, H., Southampton University, 1972.

29. Magri, F.: VARIATIONAL FORMULATION FOR EVERY LINEAR EQUATION, "International Journal of Engineering Science", 12(1974), 6, 537-549.

30. Atherton, R. W. & Homsy, G. M.: ON THE EXIST­ENCE AND FORMULATION OF VARIATIONAL PRINCIPLES FOR NONLINEAR 01 FFERENTIAL EQUATIONS, "Studi es in App 1 i ed Mathemati cs," VoL LIV, March 1975, No.1, pp. 31-60.

31. Vainberg, M. M.: VARIATIONAL METHODS FOR THE STUDY OF NONLINEAR OPERATORS, Holden-Day, Inc.

1964.

32. Mikhlin, S. G.: VARIATIONAL METHODS IN MATH­EMATICAL PHYSICS, Pergamon Press, Oxford,

1964.

BIOGRAPHIES

VUK M. FATIC was born in Pancevo, Yugoslavia, on March 22, 1932. He received the Dip1. Ing. degree from Belgrade University, Belgrade, Yugoslavia, and the M.Sc. and Ph.D. degrees from Virginia Polytech­nic Institute & State University, Blacksburg, VA, in 1960. 1973. and 1976 respectively. all in elec­trical p.ngineering. He was a research engineer with "Elektroinstitut" and INTDI. Belgrade. from 1960 to 1968. and an assistant in control engin­eering at Novi Sad University. Yugoslavia. from 1968 to 1970. During the 1975-1976 academic year he was a Visiting Assistant Professor at Union College. Schenectady. N.Y., and since 1976 he has been an Assistant Professor of electrical engin­eering at Tri-State University. Angola. IN. He has worked in control, power, laser, and IR engineering. His special area of interest is application of variational calculus in system. network. and field theory. in which he has contributed six papers to­gether with W. A. Blackwell. He is a member of honor societies ~K~ and HKN. and a student member of IEEE.

WILLIAM A. BLACKWELL was born in Fort Worth. Texas, on May 17. 1920. He received the B.S. degree from Texas Technological College. Lubbock, in 1949. the M.Sc. degree from the University of Illinois. Urbana, in 1952. and the Ph.D. degree from the Michigan State University. East Lansing. in 1959. all in electrical engineering. He has been at Virginia Polytechnic Institute and State University since 1966. where he is presently Professor and

Head of the Electrical Engineering Department. He has also been a member of the electrical engineering faculty at five other universities. and has indus­trial experience with General Dynamics and General Electric corporations. He is a senior member 6f ASEE and NSPE. and is a registered professional engineer in the state of Texas. He has published two texts and numerous papers in his area of spec­ialty, network theory and control engineering.

55

iiji , ,;'11

r :\_

AN ESTIMATION OF PARAMETERS IN A PARABOLIC EQUATION

WITH SPATIALLY-VARYING COEFFICIENTS

Z. Jacyno Department of PhYSics

University of Quebec in Montreal Montreal, Canada

Abstract

An estimator is derived for distributed parameter systems described by a parabo­lic equation with spatially varying coefficients, based upon the variational approach with usual least-square criterion and the observation data accompanied by noise. As a result the iterative procedure follows with the adjoint non­homogeneous state equation. The'evaluation of probabilistic properties of the estimator shows the blased'results obtained even with the use of unbiased mea­surement data.

1. INTRODUCTION The distributed parameter system considered in this paper is described by a parabolic partial differen­tial equation with spatially-varying conductivity coefficient being a known function of space coor­dinate. Thus the estimator has to provide the values of unknown constants only. The considered estimating procedure uses variatio­nal approach, from which an iterative scheme re­sults. It is based upon the noisy 'input data gathered from the system. As a consequence the adjoint state equation becomes non-homogeneous with a "forcing" factor given by the measurements. The estimator provides the best estimation from each set of observations available, charged with noise and measurement errors. For the purpose of proba­bilistic optimization the procedure needs to be repeated for a statist; ca 11y representati ve number of input data sets. This produces a set of estimat­ed parameters, related in some way to probabilistic characteristics of input noise. Supposing the nor­mal distribution for input noise, the evaluation of the estimator from the probabilistic point of view is sought.

56

2. DETERMINISTIC ESTIMATOR The distributed parameter system under considera­tion is described by the parabolic partial diffe­rential equation Yt=(aYx)x+~(t,x); (x,t)€Qx]O,TC, Q=]O.l[cR, (l)

with a being a spatially varying coefficient, i.e. a=a(x) and the following initial

y{O.x) '" Yo(x); (x, t)€S1XO (2)

and boundary

Yxlx=o -aoh(t) , YxlX=l = 0 , h(t) H '" const.

conditions. This system can be interpreted as one­dimensional rod heated along x and at one boundary point and perfectly insulated at another, with y being its temperature. The parameters to be estimated are the spatially­varying thermal conductivity coefficient of the form

a '"

with al const., a2. = const.

(4)

(5)

(6)

and the heat exchange coefficient

ao = const. (7)

The estimator $.* = [a~ at a~]T (8)

uses the input data obtained from the k-th set of observations of the system

m(t,x) = y(t,x) t n(t,x), (9) n(t,x) - measurement error and noise

and the usual least-square criterion T 1

J(A) =jf(m-y)2 dt dx (10)

where o 0

A, Y - estimated parameter and state, to provide the best approximation A* of A from the information available at the k-th step, i.e. to minimize J

J(A*) s J(A), A E Aadm (11 )

The estimating procedure G is based on a varia­tional approach [1]. The variation of the state equation (1)

0Yt = o[(aYxh] , (12) after multiplication by an adjoint state vector v and following substitutions

vOYt = (voY)t - vtoy , vo[(ayx)x] = [vo(ayx)]x - vxo(ayx) , avxoyx = (avxoy)x - (avx)xoy ,

becomes (voY)t = [Vtt(avx)x] Oy-vxyxoat(avoyx-avxoyt

tvyxoa)x • (13) The variations of the initial and boundary condi­tions are

~YI t= 0 = 0 ,

oYxlx= 0 = -h(t)oao

(14 )

(15 )

Green's theorem applied to (13) over the domain of independant variables gives

f (voy) I i~b d\"l= I![vtt(avx)x]oy dt dn -non J

- ~fvxyxoa dt dntJ (avoyx-avxoy t o n IX-l 0 t vyxoa ) x:O dt . (16 )

Thus the variation of augmented criterion with the initial condition

v(T) = 0 ; XEn, (17)

imposed upon the adjoint state vector and the ini­tial and boundary conditions for y taken into

account leads to

OJ~vtt(aVx)x-2(m-y)]oy dt d\"l~~Vxyxoa dt dn+

ilavxoYI X= 1 t[avxoyt(aovh-vyx)oa] I x=O } dt. (18) o

57

This allows to find the adjoint state equation Vt = -(avx)x + 2(m-y) , (19)

with initial condition (17) and boundary conditions

vx Ix=o = 0 Vx IX=l = 0

(x, t)Eo\"lX]O, n, (20)

T bringin

f18) to

~J=- jvxYxoa o n

dt dntj(aovh-vyx)oalx=o dt .(21) o

Introducing the function

1

1, x=O ~(O) ~

0, Xo'O (22)

(21) takes the final form T

oJ=~~-VXyx+~(O)(aoVh-VYx)J oA dt dn . (23) o n

The estimator G consists of the following steps: 1) initialize the procedure by making the first estimation of A, 2) find the state vector from (1)-(3), 3) compute the criterion (10) using the first set of observation data available, 4) find the adjoint state vector from (19), (20), (17) ,

5) from (23), find oA applying one of the known methods (steepest descent, conjugate gradient) to assure the decrease of (10), 6) repeat step 2 until the decrease (23) has attein­ed the value lesser than that initially imposed. It provides the best estimation A*(k) based upon a set of observation data m(k). The procedures then continues for m(ktl). As a result, the es­timation produces a set of the best estimates . {A*(k)} , being a random variable with a certain

probability distribution PG ,related to the cha­racteristics of input random variable n(t,x), true value of the coefficient A and the nature of the estimator G.

3. PROBABILISTIC PROPERTIES OF THE ESTIMATOR

iii

I'

\ I: I'

The noise n(t,x) represents a local random variable at given XEQ. We suppose that it has the normal probability distribution with both the local mean

Il(X) = E[n(x)J = 0, Y xdl (24)

and variance a2 (x) = E[n(x) nT(x)J. Y XEQ (25)

being known. The total values of these parameters

over all the domain Q are

II = !1l(X)dX = 0 ,

a2 = [a2(X)dX

The estimator defined in the previous chapter, symbolically denoted here as

A*(k) :: G [m(k)J ,

(26)

(27)

(28)

can then be characterized, [2J, by its mean value

A ~ E {A*(k) f :: jA*PG(A*)dA*=fi(m)p(m (29)

here as j(A) = (m-y)2 ,

can be also given as j(A) :: (y-y)2+n2+2n(y-y) ,

where jd(A) = (y_y)2

(34)

(34a)

(35) represents (34) for the deterministic estimator. The expectation

E[j(A)J :: (y_y)2+E(n2) , (36) extended over all the domain of independant varia-

bles ;r E[J(A)J = Jd(A)+ ~ a2 dt , (37)

o with

Jd(A) - deterministic least-square criterion, shows that the criterion used is biased with the bias

E[J(A)J - Jd(A) = o2T . (38) and covariance matrix

V~ E (A*-A)(A*-A) T:: j[G(m)-AJ Thus at the ~-th iteration, based upon m(k)

[G(m)-A]Tp(mIA)dm.(30) J~[A(k)]:: J~_l[A(k)] + 6J~ (39) the estimated parameters are biased because of (38).

The condition (24) imposed upon (9) means that the observations are unbiased

E[m(t,x)] :: y(t,x). (31)

To evaluate the estimator G in this regard, each of its steps needs to be considered. The state equation, rewritten expressly in terms

Ofy:h~ [::]f'r:;:Jx . f , (32)

shows the expected value of the state estimation 9 at each iteration linearly dependant upon the expected value of A. The characteristics of m do not intervene here explicitly. The same conclusion holds for the adjoint state

equation vt= - ra1]T [ v] + 2(m-y) , (33)

La2 4> vx x when (24) is taken into consideration. The step 5 uses the results provided by (32) and (33). The statistical characteristics of the ob­servations react here through the adjoint state, appearing as linear function. Consequently it is not affected by m. Finally let us consider the step 3 having an essen­tial importance in providing the best estimation of $.* through (11). The criterion function,denoted

58

The bias can be evaluated through the sensitivity function (23). Consequently the estimator G pro­vides, as a whole, biased results for A. The same approach can be used for the evaluation of the variance.

ACKNOWLEDGMENT The author wishes to acknowledge partial support obtained from the Canadian National Research Coun­cil through grant No. A 9516.

REFERENCES [1] Goodson R.E., Polis M. (editors). "Identifica­

tion of Parameters in Distributed Systems", The American Society of Mechanical Engineers, New York, 1974.

[2] Bard Y .• "Nonlinear Parameter Estimation", Academic Press, New York and London, 1974.

!

THE SINGULARITY EXPANSION METHOD IN ELECTROMAGNETIC SCATTERING

D. R. Wilton Depa~tment of Electrical Engineering

'Uni~ersity of Mississippi University, MS 38677

Abstract

1. INTRODUCTION

The advent of the high speed, large storage digital

computer has made possible the numerical solution

of electromagnetic scattering and radiation pro­

blems which were previously intractable. Most suc­

cessful of the various alternative numerical ap­

proaches for conducting scatterers has been to first

formulate the problem as an integral equation for

the induced current on the scatterer, and then to

solve the integral equation by the so-called method

of moments (MoM).(l) Formulation is usually in the

frequency domain so that frequency appears as a pa­

rameter in the integral operator. If time domain

results are required, they are generally obtained

by standard Fourier transform methods. More recent­

ly, however, it has been realized that a more com­

pact representation of the solution for both fre­

quency and time domain applications can be obtained

by determining the complex frequency-domain singu­

larities of the resolvent kernel. This procedure

has been termed the singularity expansion method

(SEM).(2) In the following sections we outline the

method for conducting scatterers, present two sim­

ple examples, and summarize some pertinent questions

related to the method and some of its extensions.

2. FORMULATION OF THE SINGULARITY EXPANSION METHOD

2.1 INTEGRAL EQUATION FOR CONDUCTING SCATTERERS

Assuming an exp(st) time dependence, the scattered

59

field at a point r due to a conducting scatterer S

whose induced current is J(r) is given by

-~ Ir-r' 1 -s - -1 (S2 ~ E (r) =-- - - 'V'V.

41TSE 2 e c -----dS' (1)

c S

where c is the speed of light in a medium with per­

mittivity E. If the incident field is denoted by -i -E (r), then the application of the boundary condi-

tion, which requires the total tangential electric

field to vanish on the scatterer surface, yields

the operator equation

(2)

-i -i where Et is the component of E tangential to Sand

3 is a linear operator defined by

= r~ (S2 - w'UJ(r~ 41TSE 2 c S

s 1- - 1 - - r-r'

~ c dS'J ,r on S. Ir-r' 1 t

(3) A numerical treatment of (2) amounts to discretiz-

ing the surface current J on.the object and enforc­

ing the equality in (2) at discrete points on the

object. A set of linear equations results which (1)

are then solved for the current on the scatterer.

2.2 SPECTRAL REPRESENTATION OF THE OPERATOR AND ITS INVERSE

By the electromagnetic reciprocity theorem,~ is a

complex symmetric operator. The spectral represen-

f filii

','I

"

'I Ii

i: i I

tat ion of the kernel r can be shown to be

1'(I.,r' ,s) ::: ~ A (s) C (r,s) C (r' ,s) L.. n n n n

where {A } and {C } are sets of eigenvalues and n n

(4)

~igenvectors, respectively, defined by the eigen-

value problem

~ C ::: A C . (5) iT n n n The inverse or resolvent kernel is then given by

n Hence, the inverse operator is

J ::: ~-l E;

<r-l(r,r'), E;(r' ~ (.7)

2.3 SINGULARITY EXPANSION OF THE INVERSE OPERATOR

It has been shown that r is an entire function of

the complex frequency s and, hence, that r 1 is

h · (3) I 1 h meromorp 1C. ts po es occur at t e zeros of

An(s), all of which must be in the left half plane.

Thus if s . denotes the ith zero of An(s), then n1

A(s.):::O. n n1

(8)

For three dimensional conducting scatterers, it

appears that a partial fraction expansion of the

form

1 I"TsT n I

i

[A'(s .)]-1 n1

s-s . n1 (9)

is always possible, where the prime denotes differ-

ention with respect to s and the sum is over all

the zeros (possibly infinite in number) of An(s).

The residues in (9) can alternatively be expressed

as

where C . n1

A'(s.) =<C., r,(s.)· C.) n n1 n1 n1 n1

_ C (s .). Thus (6) becomes n n1

~l - -r (r,r' ,s) L C (r) C (r')

::: -/;~C -,1'-:-' ",::,,~-c-.->-=a_(S-_S-) \: a a a a

a

(10 )

(11)

where the two indices on the natural frequencies

have been replaced by a single index a. The nor­

malized quantities Ca(r) are called the "modal

currents" and are source-free solutions at the

60

frequency sa'

In a numerical solution, the various quantities ap­

pearing in (11) are found as follows. First, since

the determinant of the matrix approximation to the

operator must also vanish at a pole, the complex

resonant frequencies are found by searching the

complex frequency plane for the zeros of the deter­

minant. The corresponding modal current is then

readily found as a homogeneous solution of the

matrix equation. Finally, the scale factors

(Ca.r~ • Ca )-1 are computed for each pole.

2.4 TIME AND FREQUENCY DOMAIN REPRESENTATIONS OF THE SOLUTION

From (7) and (11) the complex frequency domain solu­

tion for the current is

J(r,s)

a

n(s) c (I.) a

(s-s ) a

(12)

where the "coupling coefficient" n(s) is defined as

n(s) (13)

Eq. (12) constitutes an expansion of the current in

terms of singularities of the integral operator. If

the response due to time harmonic excitationexp(iwt)

is desired, one need only substitute s = iw.

The surface current in the time domain is found from

the Laplace inversion integral applied to (7); atiOO

)(r,t) =2;i J <r-l(r,r' ,s), E!(r,s» estds

a-ioo (14)

The integration along the Bromwich contour in (14)

is along a line to the right of and parallel to the

iw axis in the s-plane. If one assumes that

~l _ _ _ -sTL(r,r') r (r,r',s) ~ D(s) e Re s ~ _00 (15)

where D(s) is an algebraic dyadic function of s,

then Jordan's lemma may be used to close the con­

tour. Where the contour is closed in the right

half of the s plane, there will be no contribution

to the integral because there can be no poles in

the right half plane for a passive scatterer. If

the contour can be closed in the left half plane,

then, by the residue theorem,

s t ]"er,t) = I r\/r,t) Cae a

a -i -+ pole terms from E (r,s) (16)

The factor 110. in (16) might be termed a "general-. d" l' ., (4) I . . 11 ~ze coup ~ng coeff~c~ent. t ~s essent~a y an

inner product of the modal current C with Ei and :;.:.:.:.;.;;.;;.-~-...... ..;......;.;.....;;.;;;.;;.".; __ .. ~_ __ a a a gating function in the integrand which depends on

t, TL(r,r') and the form of the incident field,

i.e., all the terms determining the exponential

behavior of the integrand in (14). For sufficiently

long times, the integration will be over the whole

object, resulting in

-i -+ pole terms due to E (r,s) (17)

where, in contrast to (16), the coefficients of

the (damped) exponential terms exp(sat) are time­

independent. Obviously, the determination of the

gating behavior of the integrand, which is known

once TL(r,r') is known, is crucial to the SEM.

This determination has not been satisfactorily

solved to date.

3. EXAMPLES

3.1 THE WIRE LOOP

A conducting wire loop is a simple example of a

scatterer or radiator which can be reduced to a

scalar problem. For thin wires, the total current

flowing parallel to the wire axis can be shown to

satisfy the integral equation

1T

J H¢' )K(¢-¢' )d¢' (18a)

-1T

where 110 is the impedance of free space '" 1201T

ohms, and ¢ is the angle between the center and two

points on the loop. In operator notation, (18a)

can be succinctly written as

(l8b)

where E! is the component of Ei in the direction

of increasing ¢ at the wire surface.(5) Because

61

of the rotational symmetry of the structure, the

eigenvectors of the operator defined by (18) (and

hence the modal currents, since they are inde\Jend­

ent of s) are proportional to exp(-in¢); i.e., the

current is representable as the Fourier series

00

I(¢ ) I I (s) e- in</> n

(19)

n=-oo

where

-i < in</> i ? I

e ,E¢e¢,S)

n 11 1T a (s) o n (20)

Note that the term illo1T an(s) is just the eigen­

value of the operator in (18) corresponding to the

modal current exp(-in¢). an(s) has been defined in

terms of integrals of Anger-Weber functions and its

zeros have been determined for a wide range of loop . (5,6). 1 fl' h s~zes. Dom~nant po es or a oop w~t

n = 2~n(21Tb/a) = 15.0, where b is the loop radius

and a is the wire radius, are shown in Figure 1.

Time domain currents have been computed for a'pulse­

excited loop antenna and a plane wave excited loop ( 5)

scatterer by the SEM.

3.2 THE THIN LINEAR DIPOLE

For a thin linear dipole oriented along the z-axis,

the integral equation for the total wire current

has the form

~I = E! where Ei is the component of incident electric field

z along the wire axis. Tesche has computed poles and

associated modal currents for the dipole and some

results are given in Figures 2 and 3.( 7) The

principal difference here as compared with the loop

is that the modal currents (i.e., the operator's

eigenvectors) are not known ~ priori so that the

operator cannot be diagonalized before numerically

searching for the poles. Tesche also illustrates

time domain current calculations.

4. CONCLUSIONS AND RECOMMENDATIONS

The SEM is a useful technique in electromagnetic

scattering primarily because of the compactness of

the representation. The factorization of the.repre­

sentation into time-varying and spatially-varying

parts and the explicit exhibition of the manner in

I rni~ Ii"

which the object couples to the incident field are

also important in conceptually understanding the

scattering mechanism. It also appears that the

SEM will lead to the development of analysis and

synthesis techniques for loaded scatterers as well . .' . f (8) as ~n obtaining equ~valent Clrcu~ts or antennas.

It has also been shown that pole information for

antennas or scatterers in free space also complete­

ly determines the response of objects immmersed in

d. (5)

homogeneous lossy me lao

Further developments in SEM and extensions such as

the above are hampered by a lack of knowledge in

the following areas:

(1) Further information is needed on the

possible orders and the distribution of

the poles in the complex plane. The

(2)

influence of object topology on the kinds

and distribution of poles is also needed.

An asymptotic form =-1 - -of r (1'.1" ,s) is

needed in order to determine the correct

form of the coupling coefficient in

Eq. (16). The necessity for and the form

of the coupling coefficient is a subject

much in debate at present.

(3) For synthesis and equivalent circuit

problems, it is desirable to know. for

example, if eigenvalues of the integral

operator are positive real. Other prop­

erties must be known if synthesis is to

b d • . 1 d' (6) e one us~ng passlve oa lng,

It is hoped that new results pertaining to some of

these problems will be forthcoming in the near

future.

REFERENCES

1. R.F. Harrington, Field Computation by Moment Hethods, Macmillan. N.Y., 1968.

2.

3.

4.

5.

L.B. Felsen, Ed .• Transient Electromagnetic Fields, Ch. 3. C.E. Baum, "The Singularity Expansion Method," Springer-Verlag, N.Y., 1976.

L. Marin and R.W. Latham, "Natural-mode repre­sentation of transient scattered fields," IEEE Trans. Ant. and Prop., AP-21, pp. 809-818,---­Nov., 1973.

L.W. Pearson, "The Singularity Expansion Repre­sentation of the Transient Electromagnetic Cou­pling through a Rectangular Aperture," Ph.D. Thesis, Univ. of Ill., 1976.

K.R. Umashankar. "The Calculation of Electro­magnetic Transient Currents on Thin Perfectly Conducting Bodies Using Singularity Expansion

62

Method," Ph.D. Thesis, Univ. of Miss., 1974.

6. R.F. Blackburn, "Analysis and Synthesis of an Impedance-Loaded Loop Antenna Using the Singu­lari~y Expansion Method," Ph.D. Thesis, Univ. of M1SS., 1976.

7. F.M. Tesche, "On the analysis of scattering and antenna problems using the singularity expan­sion technique," IEEE Trans, Ant. and Prop., AP-21, pp. 52-63, Jan. 1973.

8. C.E. Bauro, "Emerging technology for transient and broad-band analysis and synthesis of anten­nas and scatterers," Proc. of IEEE. Vol. 64, No. 11, pp. 1598-1616, Nov. 1976.

BIOGRAPHY

Donald R. Wilton received his B.S •• M.S. and Ph.D.

degrees in 1964, 1966. and 1970, respectively, all

from the University of Illinois. Since 1970 he has

been with the University of Mississippi where he

holds the rank of associate professor.

iwb r--r--r--~-'---r--..... -r 2l a r '''r/j~ ::r·,,)., "'"i .

t/', ',I, : "!. /.0 60~ 'j', /'" : ,

f /., "1~' 30[!, " r " : I "! , Figure 2, Poles in s-plane

. 0.0 I ' /' 'L' ~;J!)j for linear dipole. wire diam-Lcb -6.0 -~.O 0.0 eter / leugth " 0.01 (from

, (7) . Tesche Figure 1. Poles ln

s-plane for wire loop.

- - n*2;.,L.1 (b)

(a) "3,101

Figure 3. Modal currents for wire of Figure 2.

l

VARIATIONAL PRINCIPLES FOR MECHANICAL AND STRUCTURAL

SYSTEMS WITH APPLICATIONS TO OPTIMALITY OF DESIGN

V. Kornkov Dept. of Mathematics Texas Tech University Lubbock, Texas 79409

ABSTRACT

Traditionally design of engineering systems was the prerogative of "a design engineer" who drawing on his experience produced some kind of an acceptable design. Then some changes or improvements would be made, after an analysis of the performance of the preliminary design disclosed some undesirable features, or hon-complience with perfor­mance criteria.

Only in the last decade were some efforts made to optimize the de­sign by introducing some mathematical theory and defining the con­cept of optimality. On the surface the problem is similar to some classical problems of control theory.

The behavior of the system is governed by a set of differential equa­tions of the form

L(z) = f

subject to some initial and boundary conditions.

Here z denotes the state vector, f is a given forcing term, L is a differential or integral operator.

For the time being the domain of L, and the spaces of functions, or generalized functions to which z and f belong, will be left delibera­tely vague.

There is of course a perfectly good explanation. All of the above must be determined by physical considerations rather then mathema­tical convenience.

Since the design is not fixed, the design vector ~ is introduced, u £ U, where U is the set of admissible designs, 1.e. designs which ~re physically feasible and satisfy a priori imposed constraints.

Unlike control theory problems, not only ~ and possible ~ depend on ~, but also the differential operator L depends on~. Tnis is easily 1llustrated by offering a very simple structural design problem. Suppose we wish to design a beam of minimum weight, supporting a given distributed weight over a span of length 1.

Maximum permissible stress, and maximum deflection are given. For simplicity of discussion we restrict the design to the I-beam shape welded from strips of constant thickness h. The design vector,

63

contains three components u l ' u 2 ' u 3 as shown.

L~'777':~

TI-~2~~1 ul

~ ~ T~3~

The cross-sectional area A (u), the moment of inertia about the neutral axis I(u), the deflection, ~(~), and the prescribed constraints all clearly~depend on the design vector ~ = (u

l' u 2 ' u

3). The state

variable w satisfies the differentia! equa~ion

2 2 ~2(EI(u)d w2 ) = f(x) dx dw (A)

If the elastic support is assumed, the inhomogeneous term f also be­comes a function of the design vector~. The optimal design problem can now be formulated.

Minimize J(~) = p J~ A(~) dx, subject to constraints ~w) = 4>0 '

o(u, f(u) < 00

•

The state equation (a) has to be regarded as a constraint condition imposed on w (~) .

The existence of Fieclet derivatives for the extended problem of minimization of the functional

10J(U) + 11(4) - 4>0) + 12(0 - 00

) = ~(~, w(~), a"(~),~.l

has been shown in [2] (Haug and Ko~k~v), under the hypothesis that w is an element of Sobolev space W2 '. The stability (i.e. sensiti­vity) analysis is much harder but partial results were also obtained in [2] for the structural case. For basic theory concerning suffici­ency conditions see [5].

The results of [2] are somewhat restricted by specific assumptions made on purely physical grounds, which are applicable to structural analysis. So far no general stability theory exists.

A much more serious defect is the absence of existence and "un"iqueness theorems. While reasonable copying of parallel results in control theory would offer some existence results, and theorems of this type are quoted in this paper, they are not immediately applicable to mechanical or structural systems. In fact a simple example of non­existence was offered by Komkov and Coleman in [1], and for the sake of completeness it is well known to be non-unique, but certain func­tions of such designs are unique in analogy with the control problems. (see Komkov [3], or [4], chapter 2.)

Taking a completely new approach to problems of optimal design, it is shown here that in the physically unrealistic case of relaxed de­signs an optimal solution always exists, and consquently approximately optimal admissible designs exist. This is illustrated on a specific example of structural design with a dynamic load. Some recent re­sults considering variational principles for dissipative systems are included in the derivation of sufficiency conditions for optimality of design (See [6]).

64

REFERENCES

[1] V. Komkov and N.P. Coleman, "Optimality of design and sensiti­vity analysis of beam theory", International J. Control, Vol. 17, #3, (1972) p. 455-463.

[2] V. Komkov, "Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems", Springer Verlag, Berlin (1973), Lecture Notes in Math. #253.

[4] Russian translation, revised and corrected, foreward by Se1ezov, Mir, Moscow, 1976.

[5] V. Komkov and N.P. Coleman, "An analytic approach to some pro­blems of optimal engineering design", Archives of Mechanics, Vol. 27 #4, (1975) p. 565-575.

[6] V. Komkov, "Another look at dual variational principles", to appear in J. Math. Anal. App1. in 1978.

6S

" ':

A NUMERICAL CALCULATION METHOD FOR SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATION

OF HIGHER ORDER BY THE MOMENTARY DIAGONALIZED MODAL PROPERTY*

Baxter F. Womack Department of Electrical Engineering

and Electronics Research Center The University of Texas at Austin

Austin, Texas 78712

Sadao Azuma Ibaraki University

4-12-1 Nakanarusawa-cho Hitachi-shi, Ibaraki-ken

JAPAN 316

Abstract A new numerical calculation method is treated for solving simultaneous ordinary differential equations of higher order as initial value problems. In this method of numerical calculation, the concept of a diagonalized modal property is introduced to achieve the simplification.

1. INTRODUCTION

For the transient analysis of a control system,

the analog-hybrid computer is generally used. It

has drawbacks, however, including limited accuracy

of calculation, the determination of many con­

stants, and the settings of nonlinear elements

that are necessary to achieve proper scale factor.

This requires much work and skill on the side of a

user. The digital computer, on the other hand,

has its own advantages, even though it is not so

suitable for calculating dynamic behavior of a

system. If the program of simulating the dynamic

process of control elements is incorporated, a

solution in the required analysis can be readily

obtained simply by giving as the input the infor­

mation constituting the control system, not

troubled by such problems as scale factor. Digi­

tal simulation is then suited to the analysis of

a complicated control system, and, for example,

can be useful in the design of an optimum

controller.

*This research was supported in part by the DoD Joint Services Electronics Program through the Air Force Office of Scientific Research (AFSC) Contract F49620-77-C-OlOl, and the Japanese Ministry of Education.

66

A control system in general is made up by several

combinations of simple elements described by trans­

fer functions or by simultaneous linear ordinary

differential equations of higher order. Several

types of digital simulators are now available for

carrying out the integrations and numerical calcu­

lations. Which to use is thus optional.

Numerical solutions for the transient analysis of

a given system are of two types, however, as

follows: First, in the case of an extremely gra­

dual change of the functions and the requirement

for high accuracy, for example, in cosmic space

dynamics, the usage of ultrahigh accuracy formula

is advantageous. Second, in the cases of an

extremely rapid change of functions, a large scale

with high complexity, or the simulation of a com­

plex control system, the employment of perfect and

tough formula, though of lower order, is useful.

In the present study, from the viewpoint above, an

attempt is made to develop an improved numerical

calculation method in digital simulation for simul­

taneous linear (or nonlinear) differential equations

of order n which describe a control system with

and without delay time.

2. FORMULATION OF NUXERICAL CALCULATION METHOD

2.1 DIFFERENCE FORM OF THE STATE SPACE EQUATION

The nonlinear. nonautonomous concentrated-parameter

continuous system may be given as

x = f(X.U. t) (1)

where on the right-hand side. fi(X.U.t) is a vector

function with i=1.2 •...• n. in columns. X is an

n x 1 vector; U is m x 1.

The hyper-curved surface fi(X.U.t) intersecting a

representative point of the (n+m+l) dimensional

space at the existing point of time tN' that is.

the static relation of nonlinear multi-variables is

schemed to momentary linearization as follows:

i 1.2 ••••• n. j = 1.2 •...• n. k

Equatid., (1) can be described as

it -i

n n

L: aij~j + L: bi~ j=l k=l

i 1.2 ••..• n

Rewrite Eq. (3) as

it = -i

m

l3 i = k bi~k

(2)

1.2 •...• m

(3)

(4)

Accordingly. Eq. (4) can be denoted by the follo­

wing momentary linear state space equation.

X A*X + 'U

where x -1 '\ 0

~2 0 Ct2

~3 0 0

X = ~ A* 0 0

~ 0 0

['~ 132.!! u= 13k

S U k = 1.2 ••••• m

0

0

~ 0

0

0

0

0

Ct4

0

U =

(5)

o o o o

a n

[J 67

Assume that each output. Yi is a momentary function

of the state vector X and the input vector U and is

determined by Eq. (6) including t explicitly.

y = F(X.U.t)

The momentary output equation corresponding to

Eq. (5) is given as

y = CX + DU

where C is the ~ x n matrix and D is the ~ x m

coefficient matrix.

(6)

(7)

To substitute a difference form for the momentary

linear state space equation given as Eq. (5). with

the initial value X(tN)=~' we use the solution as

i "'t X("'t) = exp(A*"'t)~ + 0 exp{A*("'t-T)}k(T)dT (8)

Assume the inputU(T) to be constant U in the N

integral section and let X("'t) = ~+l be the state

quantity "'t time afterwards; the solution then

takes the difference form:

-1 X 1 = exp(A*"'t)X + {exp(A*"'t)-I}A* U N+ -~ N

Define G* and H* as follows:

G*

H*

exp(A*"'t) }

(G-I)A*-l

The difference equation becomes

~+l G*~ + H*UN where

gl 0 0 0 hI 0

0 g2 0 0 0 h2

0 0 g3 0 0 0 G*= H*=

0 0 0 g4 0 0 0

0 0 0 0 g 0 0

i 1,2, •.• ,n

0 0

0 0

h3 0

0 h4

0 0

-1 Cti

(9)

(10)

(11)

0

0

0

0

h n

In Eq. (11). since matrix A* is diagonal, G* and H*

take the same diagonal form •. Equation (11), there­

fore, is finally separated into n-scalar linear

difference equations.

(12)

i = 1,2, ••. ,n

I

i i

2.2 DIAGONALIZED MODAL ELEMENTS

In the process described of introducing the dif­

ference equation, each element ~i of the diagonal

matrix ~* which determines the gi and hi values

has been given only as

(13)

i = 1,2, ••• ,n

Ordinarily, however, the values of elements ~i are

calculated independently of the modal matrix for

diagonalization. The ~i values themselves contain

information concerning the direction of an eigen­

vector and also the mode of determining exp(~i~t)

an instant after. It thus varies successively

with the transition of each state quantity.

The value of coefficient ~ .. in the infinitesimal ~J

interval ~t is taken to be constant due to the

assumption in Eq. (3). There is then the need that

the respective state quantities in Eq. (13), deter­

mining the direction of state trajectory, be

expressed as ~xi' in the same time interval.

In consequence, the relation is,

(14)

i 1,2, .•. ,n ;

the infinitesimal change of each state variable,

though it is an unknown quantity, satisfies

approximately the relation ~Xj/~Xi ~ Xj/xi'

The 'relation (14) can thus take the form:

a .. x. ~J J

i 1,2, ... ,n

~i' formed as in Eq. (15), will be called the

diagonalized modal elements" or "diagonalized

modal coefficients," and A*, the "diagonalized

modal matrix."

2.3 INTEGRAL MODAL COEFFICIENTS

(15)

Consider that the diagonalized modal elements ~ i

are all determined by the value of X, and join the

values in the difference equation for convenience

68

in numerical calculation~ then Eq. (11) can be

rewritten as

~+l = ~ + H*~ (16)

The difference form in Eq. (16) will be compared

with the existing numerical calculation formulae

from the simplest Euler's method generally used

for the solution of an ordinary differential equa­

tion of first order to the predictor-corrector type.

In the difference equation (16) derived here, H* in

the second term on the right-hand side is given as

a quantity successively changing. In Euler's

method, however, it is introduced throughout as a

constant in ~t. And then, in most other integral

formulae, ingenuity is exerted in the part corres­

ponding to the second term on the right-hand side

of Eq. (16).

When the difference equation takes the form of

Eq. (11), G* gives the rate change for ~+l calcu­

lated on the basis of ~, as follows:

exp(A*~t) (17)

Assume UN to be constant in ~t, then H* can be a

coefficient for UN as a result of the integral in

Eq. (8).

In Eq. (16) as rewritten, the essential charac­

teristic of diagonal matrix G* which is determined

by the diagonalized modal matrix A* is expressed

in the second term on the right-hand side. Then,

H* determines the integrated value in relation

with ~ and also the increment of ~ in an infinite­

simal time interval ~t. It is thus the most

important element.

H* gives directly the change of quantity XN in

relation with XN

• In this sense, it will be called

the "integral modal matrix," and the diagonal ele­

ments of H*, hi' the "integral modal elements" or

"integral modal coefficients."

3. MODIFIED DIFFERENCE EQUATION

The numerical calculation is such that when the

state trajectory is described along a hyper-curved

surface in state space, the hyper-plane passing

through a given representative point at a point in

time is first determined. And then, the next

representative point is obtained in this hyper­

plane. Therefore, in using Eq. (16) as the ori­

ginal form for the successive calculation, when

the state trajectory spins starting at the origin

in state space, the error of calculation tends to

increase in a region of large value of the

varying elements ai in the diagonalized modal

matrix A*. In the following, a method will be

introduced to calculate the diagonalized modal

coefficient with high accuracy, in the whole

region of calculation even when the coefficient is

almost infinitely large.

3.1 APPLICATION OF A CORRECTOR-TYPE FORMULA

The difference equation (16) given initially can

be expressed in scalar form as

(xi)N+l = (xi)N + hi(xi)N

hi {exp(aillt) - l}/ai

i = 1,2, ... ,n

(18)

If the state space equation is linear and non­

oscillatory, the solution can be very close to

the true value, with little error, owing to

successive calculations with Eq. (18). If then

the state space equation is oscillatory, the dis­

crete error can be held to 0(lIt3).

For example, when the numerical solution is desired

to be within an overall averaged error in the ana­

lysis of an oscillatory system, a corrector-type

formula such as Eq. (19) is applied only once.

n m

(xi)N+l = ~ aij(xi)N+l + ~ bik(uk)N+l j=l k=l

(xi)N+l = (xi)N + II; {(xi)N+(xi)N+l}

i = 1,2, ... ,n

3.2 INTRODUCTION OF THE APPROXIMATE CALCULATION FORMULAE FOR h.

1

(19)

If the system is stable and the calculated diago­

nalized modal coefficients all take a negative

sign, the integral modal coefficient hi in Eq. (18)

is expressed as

hi' = {I - exp ( - '1. t. t) } I ai

Equation (20) can be transformed as follows:

(20)

69

(21)

When the value of t.t is assumed to be sufficiently

small, it is possible that p(aillt) "'t.t/2.

In this case, the portion in { } on the right-hand

side of Eq. (21) corresponds with the trapezoidal

formula. It can be considered, therefore, that the

scalar difference equation (21) is A-stable and at

the same time a formula by the implicit method has

the least error of discretization. In [41, several

refinements of (21) are given but are omitted here

because of space limitations.

4. PROCEDURE OF THE NUMERICAL CALCULATION

The numerical calculation formula obtained thus,

in advance solution process, can be termed an

implicit formula where the value at the next point

of time is expressed as an implicit function of

the already obtained value. The implicit formula,

in general, is considered to be advantageous in

stability and accuracy. It is necessary, however,

to solve a numerical equation at each step. If

the differential equation is nonlinear, the corres­

ponding numerical equation also becomes nonlinear.

It is generally rare to solve a single differential

equation in an operation with a computer. Since

simultaneous differential equations are treated, the

numerical equations also take the simultaneous form.

In this case, to solve the numerical equations the

Gauss' or Crout's elimination method is used for a

linear problem; for a nonlinear problem, the

Newton's method is used.

In the present method of numerical calculation, the

concept of a diagonalized modal property is intro­

duced to achieve simplification. The tremendous

amount of elimination for the numerical equations,

necessary in a single calculation of the simul­

taneous ordinary differential equations, is thus

eliminated, so the treatment can be highly efficient

regradless of the linear or nonlinear problem.

Detailed computation steps are given in [41.

....

5. NUMERICAL SOLUTION OF THE GENERAL, SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS OF HIGHER ORDER

In a transient solution of the oscillatory system

equation, successive calculations are possible by

choosing the degree in the approximate formulae

according to the purpose. To make the numerical

solution applicable to higher order ordinary dif­

ferential equations in a general for~ the numerical

calculation formulae introduced thus far will be

rearranged as follows:

When the system state space equation is linear

simultaneous with order n and there exist extremely

large differences among the n values of ai' for

the purpose of an optimal numerical calculation

the degree of ~ as approximation in the function

of (ai~t) should be higher. With the degree of

about 10, however, the computing time for exponen­

tial function as approximation becomes nearly the

same as that for the exponential function as an

intrinsic function of the computer. In this case,

by utilizing the intrinsic function, a reasonable

value of exponential function can be calculated,

and the calculated value of hi can also approach

the theoretical one from the relation of this ex­

ponential value and the approximate function ~.

6. NUMERICAL CALCULATION METHOD OF THE ORDINARY DIFFERENTIAL EQUATION WITH DEAD TIME

The numerical calculation method can be applied

easily to a given system which also includes dead

time besides being in a nonlinear form.

7. APPLICATIONS AND RESULTS

In performing the calculation in a transient ana­

lysis, the usage of an explicit formula such as the

Runge-Kutta method is problematic. This is because,

due to the undominant eigen-value (for a large

absolute value), the time interval ~t in successive

calculation is limited, so that there occurs the

need for a tremendous number of steps to observe

the overall dynamic behavior of the system.

That the difference equation is A-stable means that

the solution always converges for any ~t regardless

of the eigen-value if only the sign is "minus."

The implicit formula has the advantage that its

calculation can be done easily with the ~t

determined in consideration of the dominant eigen­

value even when there are extreme differences be­

tween some eigen-values of the state space equatio~

Several examples are given in [4].

Whether it is nonlinear or time-varying, the state

space equation can be considered locally as a linear

state space equation with constant coefficients

when the variation is momentary in an infinitesimal

time interval. Then in order to advance one step

in the infinitesimal time interval ~t, it is pos­

sible to apply a conventional, numeric calculation

formula.

To obtain a calculation approaching the rigorous

solution in the course of successive calculations,

an attempt was made to improve the numerical calcu­

lation formulae. When the state space equation is

linear, the value of exp(A~t) plays an important

role in the rigorous solution, so the matrix calcu­

lation is generally required. To eliminate the need

of matrix calculation for a linear equation or to

facilitate solution of the nonlinear state space

equation, the system state equation is treated as a

momentary state space equation in an extremely small

time interval. And further, by introducing the

concept of a momentary diagonalized modal matrix

into the momentary state space equation, the dif­

ference equation necessary for the solution is given

in a simple form of a scalar equation.

As this results, it is understood that the numerical

calculation formula (the difference equation) becomes

one of the implicit methods which is A-stable with

small discretization error. Simplification of the

numerical calculation itself is thus possible by

switching over to the simple technique of calculating

the diagonalized modal coefficient from the complex

process rather than having to solve the numerical

equation in each step for an implicit formula.

And furthermore, in order to achieve the numerical

calculation formula suitable for the originally

oscillatory solution, the implicit formulae are

modified and expanded.

70

The programming of this numerical calculation method

is simple, and the method is designed for digital

simulation. The desired solution can be attained

by giving as the data the order of a state space

equation, the number of input variables, coef­

ficients a ij and bik (time-varying, the form in

the case of nonlinearity), and the initial value

X(O+) of a state variable. The calculating con­

ditions needed are the initial time, final time,

and time interval 6t.

As shown in [4], the concept of a diagonalized

modal property is incorporated into the group

of solutions based on difference, which is an

effective numerical solution of the ordinary

differential equation. Due to this introduction,

the procedure of calculations is more systematic

and purely numerical. And, formation and exten­

sion of the basic numerical-calculation algorithm

should lead to the design of a numerical calcu­

lation method with higher accuracy.

71

REFERENCES

1. Richard S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., New Jersey, U.S.A., 1963.

2. Baxter F. Womack and Troy F. Henson, "Time Domain Desensitized Specific Optimal System Design," Texas Biannual of Electronics Research, No. 20, 1974.

3. Hayato Togawa, Numerical Calculation for Differential Equation (Finite Element Method and Difference Method), Ohm Inc., Tokyo, Japan, 1974.

4. B. F. Womack and S. Azuma, "A Numerical Calculation 11ethod for Simultaneous Ordinary Differential Equation of Higher Order by the Momentary Diagonalized Modal Property," Texas Biannual of Electronics Research, No. 23, May 1976.

Ii

II' I

ON STRUCTURALLY STABLE NONLINEAR

REGULATION \HTH STEP INPUTS

w. M. Nonharnl

Systems Control Group Dept. of Electrical Engineering

University of Toronto Toronto, Canada M5S lA4

ABSTRACT

The internal model principle (IMP) is developed for nonlinear regulators with step reference and disturbance signals. The regu­lator structure is derived from a requirement of structural stabi­lity and a corresponding transversality property.

1. 2

Regulation: a general framework

In this note a system is a pair of

form (X,~) where X is a real, finite-dimen­

sional, COO differential manifold and

~:X + TX is a vector field on X. The no­

tation follows Lang [1]. We call (X,~) the

regulator (system). Next we bring in an

auxiliary system (V,n) called the exosystem.

The exosystem is envisaged as a factor sy­

stem of the regulator that drives it,

namely there is a submersion ~:X + V such

that the diagram below commutes.

X __ Z.~_~) TX

V __ ...:.n'--_-)o> TV

The regulator (X,~) is assumed to

1 This research was partially supported by The National Research Council of Canada, Grant No. A-7399.

2 For the system-theoretic background and motivation the reader is referred to [4] and [5].

72

be amenable to (V,n) in the following sense.

There exists a unique embedding v:V + X

such that

'lToV = id, (1)

Thus we have the diagram below. The amen­

ability property can be thought of as evi­

dence of willingness on the part of the

regulator to respond faithfully to the

exosystem: namely (V,v) is an 'induced'

submanifold of X to which the vector field

~ is tangent and on which the restriction

~lv(V) copies the action of n on V.

V n ) TV I

VI I : v*

id ~

~ '" X -4 TX id

~l !l

1 ~* V ~ TV",---

In the application to actual regulator

systems, v(V) would be required to have the

property of being a stable attractor ~or

the flow a of ~; namely with ~ complete,

v(V) is Liapunov stable and

a(t,x) ~ v(V) as t ~ 00 (2)

for all XEX. We do not explicitly need

such a property, although without it the

setup would probably be of little interest.

Let (K,K) be a closed submanifold3 of

X thought of as good, namely a desirable

place for the state x of the regulator to

reside. For instance, K(K)cX might be -1

given as 0 (zo) for some output map

o:X ~ Z, the point ZOEZ representing zero

error, i.e. ideal regulation. In view of

(2) the condition of output regulation

that we shall impose is

v(V)cK(K) • (3)

Thus, a(t,x) would tend to become good as

t~oo, this property being the geometric

counterpart of the usual control engineer­

ing requirement of zero steady-state error.

Since (3) is true we know (by Warner [2],

Th. 1.32) that v factors smoothly through

K, namely there exists a morphism l:V ~ K,

in the present case necessarily an immer­

sion, such that

Kol = V • (4)

So we have the diagram:

This diagram might Nell have ~erved

as our starting point, as n plays no fur­

ther essential role. From the diagram

there results

(5)

We think of (5) as a version of the output

regulation condition.

Let us suppose now that TX is given

the structure

-3 All submanifolds are declared to be embedded.

73

(6)

where the subbundles Eo ~ X, Ec ~ X are

identified with an object to be controlled

and a controller respectively, and E ~ X e

is identified with the exosystem; in par-

ticular

(7)

Accordingly we have a unique decomposition

~ = (~o'~c'~e) := ~o e ~c e ~e (8)

and the relation non = n*o~e .

From now on ~c' ~e' nand K are as­

sumed to be fixed; on the other hand, we

think of the section ~ :X ~ E as variable 00

00

in the space of smooth sections r (Eo)' on

the realistic assumption that our control­

led object depends on physical parameters

that need not be susceptible to direct

measurement or control. Of course if ~o

is varied to ~o' ~say, then v will in

general vary to v, in accordance with (1).

Following [3] we adopt the Whitney COO

topology for spaces of morphisms of mani­

folds. The output regulation condition

will be said to be structurally stable at

~ Eroo(E ) provided o 0

v = Kol , l:V ~ K

(~ ,~ ,~ ) with f: in some o ceo

(9)

for all f: nbhd of ~o. The general problem before us

is to explore the implications for the

control section ~ :X ~ E of the require-c c -ment of structural stability. Before pro-

ceeding it should be remarked that the

suitability of the topology chosen in our

definition is open to some doubt, as it is

not clear in general that interesting re­

gulators exist that are structurally stable

in the stated, strong sense. However, in

the special case to be treated next this

problem does not arise.

I l'

2. Regulation: step inputs

We denote the zero section of a

subbundle E.7 X by s. and write Z = s. (E.).

By regulation under the action of step

inputs we mean the special case of the

foregoing setup obtained when

n (v) '" 0v E Tv (V)

for all VEV. Thus

n*~ (x) On (x)' XEX.

(10)

(11)

In this case our condition of structural

stability reduces to

~OKO~(V) '" O\)(v) , VEV, (12)

for all ~ = (~ ,~ ,~ ) with ~ in some nbhd o ceo

N of ~ . o 0 There follows, in the notation

of zero sections,

~OKO~{V)CZX , (13)

which yields

~ OKo~(V)CZ o 0

(l4a)

and

~ OKo~(V)eZ c c

(14b)

be

Let the dimensions of our manifolds

dim X n, dim V

dim K

m, dim Ec

n-q.

Also, we impose the harmless technical re­

quirement4 that, when restricted to K(K),

~c is transverse to Zc:

~ OK iii Z c c (15)

Furthermore, by the results of ([3], Sec.

II. 4) we know that there exists in No a

section ~ such that ~ oK iii Z ; indeed, o 0 000

such sections are dense in r (E). It o follows that the inverse images of Zc' Zo

under ~coK,

folds of K.

~ oK respectively are submani­o With ~c fixed we may even

4 The requirement is harmless as long as dim K + dim Zc ~ dim Ec, namely (n-q)+n ~ n+nc ' or q s n-nc = no+m ; but this will always be true in the application to regulators.

74

choose ~o such that

~ -1 -1 (~OOK) (Zo) ili (~coK) (Zc)' (16)

so the intersection is again a submanifold

of K. Then (14) yields

dim [(~ OK)-l(Z ) n (~ OK)-l{Z )] o 0 c c

~ dim ~(V)

dim V

m (17)

Write dim (~ OK)-l(Z ) '" n. In view of 000

(14) and the transversality relations,

there follows

n dim K - dim ~ OK (K) 0 0

dim K - dim K /I dim E 0

0 v (dim K - dim E 0)

0 v [n - q - (n - n - m)] c

0 v (nc

+ m - q) . (18)

Writing dim -1

also (~COK) (Zc) = r we

have, by transversality,

dim [( ~ 0 K) -1 (Z ) n (~ 0 K) -1 (Z )] o 0 c c

o v [no + r - (n - q)]

o v [0 v{nc + m - q) +

+ r - (n - q)]

and by (17) there follows

o v (nc + m - q) + r - n + q ~ m .

(19)

Now if nc+m-q s 0 this gives r ~ n-q+m,

which is impossible since (~ OK)-l{Z )eK. c c

Therefore nc+m-q > 0 and we get finally

r ~ n - nco (20)

Since r s dim K n-q this shows in addit-

ion that

n ~ q (21) c

Our results may be paraphrased by

saying that the control section reduces

to zero on some submanifold of K, say K , c

of dimension at least r ~ n-nc

. By (21)

j

at least q 'components' of control vanish

on Kc' and this fact has the engineeering

interpretation that the inputs to at least

q 'integrators' in the control loop reduce

to zero when the 'error' is zero. So our

result is precisely analogous to that for

the linear case, and globalizes the local

structure theory developed in [4]. Con­

structing a tubular neighborhood of K(K ) c

in X, we may parametrize it as usual by

the fibers, and the n-r ~ q 'components'

of the fiber vectors provide a specific

realization of the required feedback vari­

ables.

3. References

[1] S. Lang. Differential Manifolds. Addison-Wesley: Reading, Mass., 1972.

[2] F.W. Warner. Foundations of Differ­entiable Manifolds and Lie Groups. Scott, Foresman: Glenview, Illinois, 1971.

[3] M. Golubitsky, V. Guillemin. Stable Mappings and Their Singularities. Springer-Verlag; New York, 1973.

[4] B.A. Francis,W.M. Wonham. The inter­nal model principle of control theory. Automatica 12(5), 1976, pp. 457-465.

[5] t"l.M. Wonham. Towards an abstract internal model principle. IEEE Trans. SMC ~(ll), 1976, pp. 735-740.

75

FREQUENCY DOMAIN STABILITY FOR A

CLASS OF PARTIAL DIFFERENTIAL EQUATIONS

D. Wexler Facultes Universitaires N.D. de la Paix,

Namur, Belgium

1. I NTRODUCTl ON

In recent years, an increasing interest has been taken in the system of integro-differential equa­tions

( 1.1)

ddt T ( t, C;) = ddC; l? 1 ( c;) ddC; T ( t, E; J +

+ P2(E;) T(t,E;) + ¢(a(t» a(E;)

d Y2 at a(t) = f b(E;) T(t,E;) dE; Yl

for all tE]O,+""[ and almost all E;E)Yl'Y2[ ,

subject to boundary conditions

( 1.2) tE]O,+oo[,

and"to initial conditions

a(O) = ao (1. 3)

where the real constants OJ verify

the nonlinear function ¢: R + R is continuous with r ¢(r) > 0 for all r E R , r ~ 0 and a , b are elements in the space L2(Yl,Y2) of real­valued square-integrable on ]Yl'Y2[ (classes of) functions. Also conditions on 8. and on the

J

76

real functions PI, P2 were required in order to insure the associated to (1.1) , (1.2) Sturm­Liouville operator L ,

( d d 1.5) L x(C;) = ~ [PI (C;) ~ x(E;)] + P2(E;) x(E;)

to be self-adjoint and negative in L2(Yl'Y2) .

Systems of this type arise as dynamic models of one-dimensional continuous medium nuclear reactors and one is interested in the asymptotic behaviour of the solutions as t + +"" , mainly in L iapunov type stability. The above problem has been studied by means of Volterra integral equations, energy functions,transform methods, Galerkin methods, semi group theory; for these approaches and more comp 1 ete references, we refer the reader to [1, 2 chapter 4 and It) .

In this contribution, we transform (1.1) , (1.2) into an abstract problem which we discuss by means of Popov type frequency domain methodS. We consi­der the system of differentiel equations

(1.6 ) du at = A u + ¢(o) a do at = <b ,u>

where A is a linear (possibly unbounded) operator with domain D(A) and range R(A) in a real Hilbert space H , ¢: R + R is a given nonlinear locally Lipschitz function verifying r ¢(r) > 0 , for all r E R , r F 0, a, b are given elements in Hand <.,. > denotes the inner product in H. System (1.6) is viewed in the Hilbert space J( = H x lR. It will also be assumed that A gene­rates a differentiable exponentially stable

Co-semi group on H .

System (1.6) may serve as an abstract version for a large class of significant partial differential equations, among which (1.1) , (1.2) is only a special example ( H = L2(Y1,Y2) , A = L). In our opinion, the advantage of this abst~act setting when discussing frequency domain stability for par­tial differential equations does not consist only in the generality we gain, but mainly in the fact that, in this way, we are somewhat guided by the methods and results which are now classical for the case when H is finite-dimensional (i.e. (1.6) is a system of ordinary differential equations).

The approach we choose here to study system (1.6) is an extension to the Hilbert setting of the approach used previously by Corduneanu for ordinary differential equations in finite-dimensional spaces 12, chapter 3]. In this approach, an essential point is the use of the frequency domain criteria for Volterra integral or integro-differential equa­tions established in 12 chapter 3,6, 7]. The additional technical difficulties which arise in the infinite-dimensional case may be overcome for a large class of unbounded operators A including significant partial differential operators.

In section 2, we state our main results and make a few comments; proofs will apprear elsewhere. In section 3, we apply the abstract results to problem

(1.1) , (1.2).

2. MAIN RESULTS

In this section, we assume that A generates a Co-semi group S on H which satisfies the follo­wing conditions:

(2.1) S is differentiable (i.e. S(t) He D(A)

for all t > 0 )

and there exist M> 1 , a > 0 such that

I I -at (2.2) S(t) ~(H) ~ M e for all t ~ 0

where ~(H) denotes the Banach space of bounded 1 inear operators from H to H; for the theory of semi groups of linear operators, we refer the reader

It is usefull to consider also the complexification HC of H ; the elements of HC will be written as x + i y , X E H , Y E H and the inner product of HC will be denoted by <','>HC' We denote by I the identity operator on H ,by IC the identity operator on HC and by AC the linear operator

defined as

AC (x + i y) = A x + A y

with domain

D(AC) = D(A) + i D(A)

An important case when our assumptions on A hold is the case in which, for some a> 0 , the opera­tor AC + a IC generates a bounded holomorphic semigroup (which is equivalent to saying that A is densely defined, there exists w > 0 such that the resolvent set P(AC) of AC contains the sector larg()..+w)1 < 2-131 +w and, for each £ E ]O,wl there exists ~ ~ 0 such that

c c -1 -1 I ().. I - A) It ( HC) ~ K£ I).. + a I .

for all )..EQ; with larg(Ha)1 ~ 2-1:n- + £ ) •

This condition is in turn satisfied when, for some a > 0 , the operator -(AC + a IC) is m-sectorial with vertex 0 15, pp. 490-491] (note that signi­ficant differential operators are m-sectorial 15, p. 280] ). The latter condition holds in particular when A is selfadjoint and there exists a> 0 such that A + a I is negative.

The function (u,o) from 10,+001 to jf is said to be a solution of (1.6) with initial data (uo,oo) E jf if it satisfies the following conditions:

(i) u is of class C1 on 10,+001, continuous at 0 , u(O) = Uo and

u(t) E D(A) , ¥t(t) = A u(t) + </>(o(t» a

for all t ~ 0

(ii) ° is of class C1 on 10,+001, 0(0) =00 and

~(t) = <b,u(t» for all t ~ 0

to 15, chapter IX and 8, chapter IX] . The zero solution of (1.6) is sais to be stable in

77

, !

the 1 arge if : (i) for each (uo,ao) EX, there exists a uni­

que solution of (1.6) with initial data (uo,ao) ; and

(ii) there exists a continuous strictly increa­sing function IT from [O,+oo[ to [O,+oo[ with IT(O) = 0 such that, for any solution (u,a) with initial data (uo,ao) and any

r> OJ

I (uo,ao) Ix';;; r implies

I (u(t) ,a(t)) Ix';;; IT(r) for all t;;;. 0

The zero solution is said to be uniformly asympto­tically stable in the large if it is stable in the large and if, for any bounded set S in X , the solution (u,a) with initial data (uo,ao) tends to 0 as t -->- + 00, uniformly with respect to (uo,ao) EtS .

We now state our main result.

Theorem 1. Assume that

and there exists q;;;. 0

(2.3) such that Re(l- isq)<b,(isl c - AC)-lA-la>Hc -

- q<b,A- 1a> .;;; 0 for all s;;;' 0 .

Then, the zero solution of (1.6) is uniformly asymptotically stable in the large.

Exponential estimates may be obtained by 1ineari­sation :

Corollary 1. Assume the conditions (2.3) hold and denote by l' ( . )( Uo ,a 0) the so 1 uti on of (1. 6) with initial data (Uo,ao) . Assume moreover that ¢ is differentiable at 0 with d ~(O)/d r> 0 . Then there exists S > 0 such that, for each bounded se t IS in}{, we may fi nd C;;;' 1 with

I T(t)(uo,oo)1 .;;; C e-St \(uo,ao)1 x x

for all (uo,ao) E <B and t;;;. 0

A question which seems of importance when (1.6) serves as a mathematical model of a physical sys­tem is the sensitivity of its stability properties with respect to small perturbations in parameters.

78

Theorem 2 below furnishes a partial answer to this question. For each y > 0 , denote by Gy the set of couples (a,b) E H x H satisfying

-1 <b,A a> > 0 and there exists q;;;. 0 (2.4)

such that Re(l - isq)<b,(isl c - Ac)-lA-1a>HC -

- q<b,A-1a> < y-1 forall s;;;' 0 ,

and denote by f'(" the class of locally Lipschitz real function ~ satisfying 0 < r ~(r) < y r2 , for all r E ~ , r f: 0 .

Theorem 2. Let y be a strictly positive number. Then Gy is open and, for each (a,b) E Gy , and each ~ E~y , the zero solution of (1.6) is uni­formly asymptotically stable in the large.

Consider now the important case when A is self­adjoint, A + a I is negative for some a> 0 and A-I is compact (these conditions hold for some elliptic operators A). There exists then a Hilbert basis (en)nE IN of H such that, for each n E IN, en is an eigenvector of A and the sequence (>.'n)nE IN of associated eigenvalues is decreasing; moreover, Ao"": 0 and lim An = _00

[3]. Denote the Fourier coefficients of a and b with respect to en by an and bn respecti­vely. Then the frequency domain conditions (2.3)

may be written r A-I an bn > 0 and there n=O n exists q;;;' 0 such that

n!O (1 +q An)(s +A~f1 an bn ;;;' 0 for all s;;;. 0 .

These conditions hold with q = \Ao\-l if, for all n EN, we have an bn .;;; 0 and there exists m ElN such that am bm < 0 .

To check the frequency domain conditions (2.3) requires the knowledge of the resolvent of AC

which, in the infinite-dimensional case, is far from being an easy matter. When A = L is defined by (1.5), it amounts to solving a Sturm-Liouville problem for an ordinary second order differential operator. That is why we state below some suffi­cient and easy to check conditions for (2.3) to hold.

Proposition 1. Assume that, for some a> 0 and

k < 0 • the operator _(Ac + a IC) is m-sectorial with vertex 0 and b = k a '1 0 . Then. conditions (2.3) hold.

It may be shown that the asymptotic stability re­sult established previously in [41 for system (1.6) with selfadjoint operator A by using the theory of nonlinear semi groups and some Liapunov functions is closely related to a specialization of Corollary 1.

Finally. we note that frequency domain methods may be applied to discuss also some other control systems in infinite-dimensional spaceS. as. for instance. system (1.6) under the assumption that o is a simple. isolated eigenvalue of AC • which is also of interest in reactor dynamics.

3. APPLICATION TO PROBLEM (1.1) • (1.2)

We assume that the parameters of system (1.1) • (1.2) satisfy the following conditions: PI E

Cl([Yl'Y21) and PI> 0 on [Y1'Y21; P2 E C([Yl'Y2 1) and P2":;0 on [Yl'Y2]; 15 1 c2~0

and 15 3 154>0; either P2$0 .or 11511+11531>0. Put H = L2(Yl'Y2) and define L by (1.5) with domain D(L) consisting of functions x E C1([ Yl'Y2 1) such that d2 x / d E;2E L2(Yl'Y2) and

dx . dx v °1 x(Yl) +15 2 ~(Yl) = 0 , 15 3 x('("2) +15 4 ~(.2) = O.

It is well-known that, under the above assumptions. L is selfadjoint. L-1 is compact and L - AO I is negative. where AO < 0 is the first eigenva-1 ue of L [31 .

Clearly, system (2.1) with A = L may then be viewed as a1\ L2-version of problem (1.1) • (1.2) and. since L generates a Co-semi group which satisfies (2.1) and (2.2). we may apply to (2.1) the results obtained in section 2.

Theorems 1 and 2 and Corollary 1 furnish stability results for the L2- vers ion of problem (1.1) • (1.2) under the norm of L2(Y1'Y2) . Most of the previ­ously established stability results for problem (1.1) • (1.2) are in terms of classical solutions and stronger norms. That is why it seems of inte­rest to show that, in fact, the solutions of the L2-version verify (1.1) • (1.2) in some classical

79

sense and stability results under a stronger norm hold. Given (To.oo) in JC = L2(Yl'Y2) x IR ,we say that the couple of functions (T.o) :

-T: ]0.+oo[x[Yl'Y21+1R • 0: [O.+oo[+lR.

is a classical solution of (1.1) • (1.2) with ini­tial data (To.oo) if the following conditions hold: TEC100.+oo[X[Y}'Y2]) and OECl([O.+oo[); for each t > 0 • one has a2 T ( t •. ) / a E;2 E

L2(Y1'Y2) ; (T.o) verifies (1.1) • (1.2) and (1.3).

We limit ourselves to state a "classical" version for Theorem 1, but such versions may be obtained also for Theorem 2 and Corollary 1.

Theorem 3. Assume conditions (2.3) with A = L hold. Let (To.oo) be an arbitrary element in JC

and denote by (u.o) the solution of the L2-version (1.6) with initial data (To.oo) . Then, (T.o) • where T(t.·) = u(t) • is the classical solution of problem (1.1) • (1.2) with initial data (To.oo). Moreover. for any bounded set ~ c JC. the solution (T ,0) of (1.1) • (1.2) ~Iith initial data (To.oo) satisfies :

do d2a ott) + 0 • Qf(t) + 0 • ~(t) + 0 , dt

sup I T(t.E;) I + O. sup I ~i(t,O I + 0 E; E [y l'Y 21 f,; E [ y l'Y 21

. sup 1~~(t.f,;)1 + 0 and <21 Ig(t.OI dE; + 0 E;E[Yl'Y2] af,;

as t + +00 • uniformly with respect to (To.oo) E6.

REFERENCES

[1] T.A. Bronikowski. J.E. Hall. J.A. Nohel, Quan­titative Estimates for a Nonlinear System of Integrodifferential Equations Arising in Reactor Dynamics. SIAM J. Math. Anal., 3 (1972), pp. 567-588.

[ 2] C. Corduneanu, Integral Equations and Stability of Feedback Systems. Academic Press, 1973.

[ 3] R. Courant and D. Hilbert, Methods of Mathema­tical Physics, vol. 1. Interscience. 1953.

(4) E.F. Infante and J.A. Walker. On the Stability

, I

Properties of an Equation Arising in Reactor

Dynamics,J.Math.Anal. Appl. 55(1976),p~ 112-124.

[5) T. Kato, Perturbation Theory for Linear Opera­tors, Springer Verlag, 1966.

[6) J.A. Nohel and D.F.. Shea, Frequency Domain Methods for Volterra Equations, Advances in Mathematics 22 (1976) 3, pp. 278-304.

[7) O. Staffans, Positive Definite Measures with Applications to a Volterra Equation, Trans. Amer. Math. Soc., 218 (1976), pp. 219-237.

[8) K. Yosida, Functional Analysis, Springer Verlag 1974.

aD

DESENSITIZING OBSERVER DESIGN

FOR

OPTIMAL FEEDBACK CONTROL

Manuchehr M. Missaghie Sentro1 Systems Ltd.

4401 Steeles Avenue West Downsview, Ontario

Canada

SUMMARY

This paper considers the problem of designing the observer in an observer based optimal feedback control scheme so as to offset the effect of plant parameter error. The optimal observer gain is determined as the solution of an equation which is formed by equating to zero the first variation of a linear combination of the usual performance index and the norms of the sensitivity matrices of that performance index with respect to the plant para­meter matrices. This equation is solved iteratively by the method of successive approximation. The computations involved in each iteration are linear and require the solution of four matrix Lyapunov equations. The results are tested on a single input, single output second order system and a number of observations regarding the effect of the desensitizing observer on the feed­back control system is noted.

81

GRASSMAN MANIFOLDS AND GLOBAL PROPERTIES OF THE RICCATI EQUATION

Clyde Martin* Ames Research Center (NASA) Moffett Field, California

Abstract

Riccati differential equations are studied as vector fields on a Grassman manifold. Such manifolds serve as a compactifi~ation for the Riccati equations.

1. INTRODUCTI ON

The Ricoati equations arise in many areas

of systems theory, the most familiar be­

ing in the linear quadratic optimization

problem. This particular class of Riccati

equations has received a great deal of

study and there is little additional that

needs to be said for the purposes of op­

timization. However, there are other

classes of Riccati differential equations

and other classes of algebraic Riccati

equations. The purpose of this paper is

to describe a framework which encompasses

all finite dimensional Riccati equations and gives a means of systematically study­

ing them. The basic idea is to compactify

the domain of the Riccati equation in such a way that the operator becomes a COO vec­

tor field and hence has an integral curve

for all t £ m. The resulting manifold is a Grassman manifold and as such has sev­

eral useful representations. The proper­ties of the manifold will be exploited to

obtain several properties of the Riccati

equation. A more detailed and more Lie

theoretic account will appear in a later

publication.

2. GRASSMAN MANIFOLDS

Let V be an n dimensional complex vector

space and let p be an integer less than or

equal to n. The set GP(V) will be the set

of all p-dimensional subspaces of V. The object of this section is to describe a

manifold topology for GP(V).

Let U and W be fixed subspaces of V with the following properties:

W £ GP(V), U £ Gn-p(V); (2.1) W 81 U = V. (2.2)

Let L(X,Y) denote the set of al1 linear maps from a vector space X to a vector

space Y. Now for A £ L(W,D) define UA £ GP(V) by

UA ='{w + Aw:A £ L(W,U), w £ W}.(2.3)

UA is clearly a wel1 defined element of

GP(V) and also note that UA = UB implies

*National Research Council Senior Research Associate

82

that A = B. Define E L(W,U)} (Z.4)

We see

r (U) =' {U A: A

that r(U) is in one to one corre-

spondence with L(W,U). Choosing a basis in Wand U we can then define a one to

one and onto function ~U from r(U) to a:(n-p)p by

(z. 5)

where A is the matrix representation of A E L(W,U). We can give r(U) the topology of a:(n-p)p. Now let S E GP(V) and let

Uo

E Gn-p(V) be such that S ti Uo = V and

W @ Uo = V. (The existence of such a Uo is an elementary exercise in linear alge­bra.) Let'{si} be a basis for S. There is a unique w. and u. such that s. = III

W. + u. with u. E U. Defin A E L(W U ) 1 1 1 0 ' 0

by Aw. = u .. It follows that S E r(U ) 1 1 0 •

Thus we have that GP (V) = U r (U) • (Z .6)

UEGn - p (V) It is actually easy to verify that GP(V) is the union of a finite set of r(U). With the topology induced by the r(U) GP(V) is called a Grassman manifold.

Thus we have that (r(U),~u) is a candi­date for a chart and chart map for a dif­ferentiable structure for GP(V). The analyticity of the map

-1 ~u 0 ~u' (Z.7)

needs to be verified.

Let S E r(U') n r(U) and suppose that S = UA. We need to determine a T such that S = UTe Each element of u' £ U' can be written uniquely as u' = u + w with u E U and w E W. Define Al and AZ by Al(u') = u and AZ(u') = w. The unique­ness of the decomposition determines that Al E L(U',U) and AZ £ L(U' ,W). In fact it is easy to show that Al is invertible. (Show it is one to one and note that dimU = dimU'). Now we have that UA =' {w + Aw : w £ W, A £ L(W,U')}

='{w + AlA\\' + AZAw : w E IV, A£L(\.;,II')}

83

='{(I + AZA)w + AIAw : (I + AZA) E L(W,W)AIA E L(W,U)}

=' {w + AIA(I + AZA)-lw : AIA(I + AZA)-l

E L(w,u)} U 1 (Z.8)

AIA(I+AZA) -

Thus the map Z.7 is such that

~u 0 ~~~ (A) = AIA(I + AZA)-l

and is analytic on ~u' (r (U' )) n ~u (r (U)) . Thus we have an atlas on GP(V). That it is Hausdorf follows from the fact that given two elements of GP(V) there is a

chart r(U') that contains both.

That GP(V) is a compact manifold follows from the fact that it can be represented as a set of left cosets of the group of orthogonal complex matrices--a compact Lie group. The proof is elementary but slightly involved and can be found in [IJ.

3. RICCATI EQUATIONS AS VECTOR FIELDS ON GP(V)

Let G(n,a:) be the Lie group of complex invertible matrices. For v E GP(V) and for a E G~(n,a:) define an action of G~(n,a:) on GP(V) by a(V) =' {ax:x E v}-­the natural action of G~(n,a:) on GP(V). We will consider this action in local coordinates on r(U). Partition a as

[::: :::] where all E L(W,W), a lZ £ L(U,W), a ZI £ L(W,U) and a ZZ E L(U,U). Then v E r(U) implies that v = U A'

UA ='{w + Aw:A £ L(W,U)}

and

aUA =' {allw + alZAw + aZlw + aZZAw A E L(W,U)}

='{(a ll + alZA)w + (a ZI + aZZA)w

(3.1)

A E L(W,U)} (3.Z)

If aUA £ r(U) then all + alZA is invert­ible and we have that

(3.3)

Thus a acts on the parameter space as-

a : A + (a Zl + azZA) (all + a A)-l lZ (3.4)

a generalized linear fractional transfor-mation.

Let aCt) be a one parameter subgroup of

G~(n,[) with infinitesimal generator B, da -1 ~ = B (3.5)

and assume B is a constant matrix. For UA € r(U) we have that a(t)(UA) = U

(aZl(t)+a2Z(t)A)(all(t) -1 +a lZ (t)A)

UX(t) (3.6)

We will show that X(t) is the solution of a Riccati differential equation. The

derivatives of a .. (t) can be obtained 1J

from 3.5 in terms of the partitioned B matrix. Calculating d at X(t) X' (t)

(aZl(t)+a2Z(t)A)(all(t) -1

+a1Z(t)A) - (aZl(t)+aZZ(t)A)

• (all(t)+alZA)-l(ailCt)+aiz(t)A) -1 (all Ct) +a lZ (t)A)

= (aZl(t)+azz(t)A)(all(t) -1 +alZ(t)A)

- X(t) (ail (t)+aiZCt)A) -1 (all(t)+alZ(t)A) •

Substituting the values obtained from 3.5 one obtains:

£t X(t) = BZlall+BZZaZl + (BZla lZ -1 + BZZaZZ)X(t)(all+alZA)

-X(t)Bllall + BlZa lZ -1 +(BllalZ+B1ZaZZ)X(t) (all+alZA)

BZl + BZZ X(t) - X(t)

• (Bll +B12 X(t)) (3.7)

Thus the action of a one parameter group~ locally represented by a Riccati differ­

ential equation and each Riccati equation

84

corresponds to a vector field on GP(V).

For a slightly different development see [ZJ.

Consider some of the immediate conse­

quences of this linearization. Let S be

an equilibrium point of the equation 3.5.

Then S is a fixed point of aCt) for all t,

i.e. a(t)S = S. It is easy to show that

a(t)S = S iff BS ~ S. This result is

essentially what Potter observed in [3J

o obtain his solution of the algebraic

Riccati equation except that we obtain

slightly more in that we have determined

all solutions of algebraic Riccati equa­

tions, even those at "infinity." The

results of Section Z can be used to obtain

the results of [3J and the slightly more general results of [4J.

4. LINEARIZATION OF RICCATI DIFFERENTIAL EQUATIONS

Consider the Grassman manifold GP(V), the

one parameter group aCt) and its associ­

ated vector field B. If a and B are con­

sidered as matrices then their representa­

tion is with respect to a given basis of

a given decomposition of V, i.e. V = W ~ U

and a basis is chosen as' {wI"" ,wp,ul

' .•• u }. Thus a change of representation n-p of a and B is equivalent to a change of charts in GP(V).

Let V = W' ~ U' be another representation of V and let y be the appropriate change

of basis. Then the change of representa­tion of a is just the usual change of

-1 basis with a' = yay. As usual we have that

g, = -1 y Bya' (4. I)

and the vector field B changes as a change of basis. On the other hand, if U

A '{w+Aw:A € L(W,U)} then U

A =

U' -1 U'A' is the (YZl+YZZA) (Y1I+YlZA)

representation in terms of W' and U'.

Thus a change in charts is represented as

a similarity transformation in the global

representation of the vector field and as a linear fractional transforcation in the

local representation.

Let

and

Y = ~2l + ~22Y - Y(~ll + ~12Y) (4.3)

Define 4.Z to be equivalent to 4.3 iff -1

X = (YZl+YZZY)(Yll+YlZY) for some

y = (Yll Y12) £ GR.(V). YZI YZZ

That is, two Riccati equations are equi­

valent iff ~heir solution curves are

related by a generalized linear frac­

tional transformation. We have immed­

iately the following theorem.

Theorem 4.1. Every Riccati equation is

equivalent to a linear differential equa­

tion.

Proof: There exists a similarity trans­

formation that transforms B to its Jordan form. The resulting Riccati equation is

[:' :,] where the representation is with respect to the positive and negative eigenspaces

of 4.4. We can assume that BZ is stable

and Bl is unstable. Then letting W be the positive eigenspace and U the negative

eigenspace we hdve that u(t)UA = U and hence as t goes to

expBZ(t)Aexp-B1(t) infinity a(t)UA approaches Wand so W is the unique stable equilibrium point in the

chart r(U). Thus all of the other equili­

brium points of 4.4 lie in the complement

of a canonical chart. It is possible to show that this set is an algebraic subset

of GP(V) and further is an invariant sub­

manifold of GP(V) for 4.4. It can be further decomposed to obtain a nested set

of invariant submanifolds of decreasing

dimension. It appears that certain re­

sults of [5J on the partial ordering of solutions are related to these invariant

submanifolds. A detailed study of this is beyond the scope of this paper.

REFERENCES

linear. 1. Griffiths, P. and Adams, J., Topics

Note, though, that not every Riccati equa­

tion is equivalent to a homo1eneous lin-o C °d B 0 0] B 0 ear equatIon. onsl er = 1 O. IS

not similar to a diagonal matrix.

Consider the familiar LQG problem and

assume that it is properly posed. Let P = -Q - A'P - PA + PBR-IB'P (4.3)

be ~ Riccati equation with Ham-

iltonian vector field

B = fA -Q

(4.4)

Since the LQG problem was assumed to be well posed we have that B is equivalent

to a matrix

85

in Algebraic and Analytic Geometry, Princeton University Press, Princeton,

New Jersey, 1974. Z. Hermann, R., Interdisciplinary Mathe­

matics, Vol. III, 1973, Math Sci

Press, Brookline, MA. 3; Potter, J. E., Matrix Quadratic Solu­

tions, SIAM J. Appl. Math., Vol. 14,

pp. 496-501, 1966. 4. Martensson, K., On the Matrix Riccati

Equation, Inform. Sci., Vol. 3,

pp. 17 - 49, 1971. 5. Willems, J. C., Least Squares Station­

ary Optimal Control and the Algebraic

Riccati Equation, IEEE Trans. Aut.

Con. 16 (1971), 621-634.

• GENERALIZED OPERATOR AND OPTIMAL CotlTFClL

Salah M. Yousif California State University, Sacramento

ABSTRACT

This paper presents the optimal control problem in terms of linear operator theoretic foundations. The optimal control that minimizes a specified obj ecti ve functional is derived utili zing the concept of generali zed inverse of linear operator. The operators are assumed to be defined on Hilbert spaces. The representation of the control and the operator inverse in fini te dimensional minimal energy control problem is treated, and examp les for thi s case wi 11 be gi ven .

I. INTRODUCTIQ~

In many control problems, large number of systems can

be represented as operators on vector spaces. ~Iore

precisely gi\"en, a linear dynamic system

~lt) = :\(t1x(t) + B(t)u(t) (1)

the zero state response of this system is given by t

xlt) I ¢l(t, s)B(s)u(s)ds (Z) o

Idlere ~(t, s) the transition matrix. A general class

of control problems may be described by determining the

control \"ector u(t) that steers the state from the

origin at time to 0 to final state x(t f ) at the final

time t f and minimizes the performance index J(u) given

by

., IZ J(u) = Ilull~ + Ilx(t f ) - xl (3)

I,here x is some specified state in the state space.

The norms given in (3) are the usual Eucleadean norms.

If I,e let the space of the controls to be Hilbert space

HI and the other space another Hilbert space HZ' then

equation (2) represents a linear operator from HI into

H ~ and can be wri t ten in the form

(Lu) (t) = x(t) ( 4)

iie Id 11 assume L to be bounded linear operator from HI

into H2• The perfol:lllance index J (u) becomes

J(u) = lIulJZ + IILu - xli Z (5)

where the norms Ilull and IILu - xii are defined res­

pecti vely in terms of the inner products in HI and HZ'

If I,e require HI to be LZ[O, I], then the inner product

in this case is defined by

T * (xl' xZ) = IXI (t) Xz (t) dt (6)

o

The control function u(t) may be assumed to lie in a

convex set in HI' This .condition may be necessary for

existence of the optimal control. The class of control

86

problems that fits the above description includes mini­

mum energy control, regulator problems and tracking

control systems. For the last two problems, J(u) may

assume the well known quadratic performance index given

by

J(u) = lIull~ + IILu - x,,~ (6)

where R is positive definite, and Q is at least positive

semidefinite. Similar problems were treated in the

literature. Balakrishnan [3] obtained a solution of

similar problem by generating sequence of control

function un that converges to the optimal control of

minimal norm. He also computed the optimal control

function using steepest descent method. Porter treated

a general problem in linear system optimization" des­

cribed by

x = Lu + TZ ( 7)

where L: Hl~ HZ and T: HZ~ H3 and Land T are bounded

linear transformations between Hilbert spaces [12].

The performance index J(u) is given by

( 8)

and the problem is to find for a fixed element, Xl in

HZ a control Uo in HI such that J(u) is minimum. This

control Uo was given by

u = o -(I + L*L)-lL*TX

1

where L* denotes the adjoint of A.

(9)

In fact, the closed

form solution obtained in (9) is similar to one obtained

by Balakrishnan in terms of convergent sequence of

control functions in HI' Although these solutions are

of interest to us, our main interest is to treat ~pe­

cial problems in this general form and their represen­

tation in finite dimensional spaces. Therefore, we

will proceed in section two to treat and solve these

problems; and in section three, we will present solu­

tion for the finite dimensional discrete minimal energy

control problem. In our approach, the pseudoinverse

of linear operator plays a principal role in the solu­

tions and synthesis of the optimal control.

II. GENERALIZED CONTROL PROBLEM

The optimal control problem described previously can

be modified to the following generalized form. We

are given a control system as in equation (4), that

is,

tLu)(t) = x(t) (4)

It is required to find the control uO(t) of minimum

norm which lies in a convex set in Hilbert space HI

that steers the system (4) from the origin to a state

which lies as close as possible to a specified fixed

state Xl in Hilbert space H2 . We will assume L here

to be bounded linear operator from HI into H2 with

close range R(L). This formulation of the control

problem in hand becomes similar to the linear least

square problem in Hilbert space. The solution of the

above problem that yields the optimal control uO(t)

is given by the following theorem.

Theorem 2.1 Let L be as described above. The optimal

control uO(t) of minimum norm that minimizes

is given by

uO

(t)

where L+ is the pseudoinverse operator of L.

(10)

(11)

Proof. The proof we will provide here is similar to

the proof of theorem 7.4.2 in reference [6]. We let

Xl = Y + z where ye:R(L) and y e:R(L) . Since Uo

=

+ Luo= + +. + h L Xl' then LL Xl = LL y, Slnce LL = PR(L)w ere

PR(A) is the projection of H2 on R(A). Therefore,

IILuo - XIII = Ily - XIII ·llzll·

Let ul be any element in HI' Lule:R(L). Thence,

IILUI - xll12 = IILw - Yl12 + Ilzl12~IILuo - xll12 (12)

which proves that uO(t) minimizes I ILu - XII 12.

To prove uO(t) has minimum norm, we let u ' be any

solution that minimizes I ILu - XII 12. That is, u ' is

least square solution. Then, L*Lu ' = L*x l and

u2 = u ' - UO is in the null space, N(L*L) of L*L. +

Since N(L*L) = N(L) and L xle:R(L*) N(A). Therefore,

IIu l l12 = II u211

2 + Iluol12~lluol12 (13)

Consequently, uO(t) has mlnlmum norm, which means in

engineering terms: the optimal control uO(t) that

steers the system to a state closest, in the least

square sense, to the target state Xl it steers it

with minimum energy. Since all finite dimensional

spaces are complete and therefore Hilbert spaces,

every matrix is a bounded linear operator with closed

range. Therefore, the above generalized control pro­

blem in finite dimensional spaces becomes more tangible

for computation since the pseudoinverse of a matrix is

computable by a variety of computation methods [2].

In the finite dimensional case, L becomes an mxm matrix

from En into Em' the finite dimensional Eucleadean

spaces of dimensions n and m respectively. Consequently,

theorem 2.1 will have the following equivalent version

in finite dimensional spaces.

Theorem 2.2. Let L be an nxm continuous matrix from

,Em to En as described by (4). The minimum norm uO that

minimizes (10) is given by

87

o + u = L Xl (14)

where L+ is the pseudoinverse matrix of L.

Proof. The proof of this theorem is similar to that

of theorem 2.1 and is provided in reference [IS].

The above theorem, as it will be seen in section three,

will be used to derive the optimal minimum energy

control of linear finite dimensional time invariant

control system.

The least square version of theorem ~.2 is the follo­

wing theorem [2].

Theorem 2. 3. Let L be an nxm continuous matrix from

Em into En described by (4).

which

The vector uOe:Rm for

and

are minimi zed, is given by

o u = GXl

where G satisfies

LGL = L, (QLG) T = QLG, (RGL) T = RGL

(15)

(18)

Proof. The proof of this theorem is proyided in refe­

rence [2]. The basic ingredients for the proof of

theorem 2.3 are the following trans formations which

yield the equivalent unweighted least squares pr~blem

given by theorem 2.2.

The above transformations imply

II Lu - XIII ~ = II Lu - xll12

and

(20)

(21)

In writing (19), we use the fact that every positive

definite matrix 0 has a unique positive definite square

root matrix 01

/2 and the inverse of 0 1/ 2 is denoted by

D- 1I2•

III. MINIMUM ENERGY CONTROL

In this section. we solve the problem of minimum energy

control of finite-dimensional. time-invariant discrete

control systems. The method of solution will depend

mainly on the material provided in section two. The

discrete control system will be described by

x(k + 1) = Ax(k) + Bu(k). k = 0, 1. 2 •••• (22)

where x (k) e:En and u(k) e:Ern and A and B are matrices of

appropriate dimensions. For this problem. we will not

assume that the system is completely controllable.

although we assume that the uncontrollable states are

stable. These assumptions and the nature of the solu­

tion make the approach for obtaining the control func­

tion less restrictive and more general. It is desired

to find a control sequence {u(k)}N-l of minimum energy o

that steers the system (22) to a state as close as

possible to the origin, in finite time N, starting

from initial state x(O). That is, N-l .

o = x(N) = ANx(O) + L A{N-I-lJBu(i) (23) i=o

or

x(O) N-l .

- L A-1-IBu(i) i=o

u{O)

u(1)

u(N-l)

where the submatrices ~ are given by

~ = A-k-lB, k = 0, 1, ••• , N-l.

Equation (24) may be written in the compact form.

LU = xeO)

L -{LO' Ll ••••• ~-l]

is the n x (mxn) controllability matrix

UI = [u(O), u(l), ••• , u(N_I)]T

(24)

(25)

(26)

(27)

(28)

is an Nm-row vector. In general. we will not assume

the rank of L to equal n. Theorem 2.2 yields the

unique optimal control Uo. That is.

UO

= L+ x(O} (29)

UO is the control vector in ENxm that yields the

closest vector in the range of L to x(O). That is.

UO minimizes \\Lu - x(O)1 \2 over all Ue:ENxm

' Moreover,

88

UO

is of minimum Eucleadean norm. The pseudoinverse

matrix L+ may be computed by many of the efficient

computational methods that exist to compute the pseudo­

inverse [2. 7]. The optimal control UO is still an

open loop control. This control can be synthesized by

the following straightforward method. It is evident

that the minimum energy control that steers the state

x(k) to the origin in N steps. using 28, is given by

uO{k)

uO(k+l)

uO(N+k-l)

(30)

Equation (30) implies that the control UO(k) is given by

uO

(k) = F x(k) (31)

where the matrix F is the first (mxn) submatrix of the

(Nxm)xn matrix L+. That is. F consists exactly of the

m rows of the matrix L+ That is.

o ] L+ m-n (32)

where 1m is the mxm unity matrix and 0m_n is the mx

(m-n) zero matrix. It is clear that F is to be com­

puted only once in each control action. and the minimum

energy feedback control is given by (31). Since the

synthesis procedure depends on the computation of the

pseudoinverse L+ of L. the following iterative proce­

dure can be used to compute L+. The procedure consists of the fOllowing steps {7].

(i) Let i k denote the kth column of L and let Lk

denote the submatrix consisting of the first columns of L. That is,

(33)

k = 1. 2 •.••• M ; M = number of columns of L.

(ii) Compute the following two vectors

(34)

we have

o otherwise (35)

(iii) Compute the row vector hk

(36)

(i v)

(v)

+ Compute the matrix Lk by

c; [::-1 -,>hk 1

Finally,

+ + L = LM

(37)

(38)

The above procedure is easy to implement since it does

not require any matrix inversion.

IV. CONCLUSION

The optimal minimum energy control problem was solved

in terms of an operator and its pseudoinverse in

Hilbert spaces. The finite dimensional version was

solved in terms of the pseudoinverse of matrix in the

discrete control systems case. A procedure based on

computing the matrix pseudoinverse was used to synthe­

size the optimal control that led to feedback optimal

minimum energy control.

REFERENCES

1. Albert, Arthur. Regression on the Moore-Penrose Pseudo Inverse, Academic Press, New York, 1972.

2. Ben-Israel, Adi and Thomas N. E. Greville. Gene­ralized Inverses: Theory and Applications, John Wiley, New York, 1974.

3. Balakrishnan, A. V., "An operator theoretic for­mulation of a class of control problems and a steepest descent method of solution," J. SIAM, Control, Ser. A, Vol. 1, No.2, 1963.

4. Balakrishnan, A. V., "Linear systems with infinite dimensional state space," Symp. On System Theory Proc., Polytechnic Institute of Brooklyn, 1965.

5. Balakrishnan, A. V., "Optimal Control Problems in Banach Spaces," J. SIAM, Control, Ser. A, Vol. 3, No.1, 1965.

6. Blum, E. K. Numerical Analysis and Computation, Addison Wesley, New York, 1972.

7. Bollion, L. T. and L. P. Odell. Generalized Inverse Matrices, John Wiley-Interscience, New York, 1971.

8. Desoer, C. A. Notes for a Second Course on Linear Systems, Van Nostrand Reinhold, 1970.

9. Goldstein, A. A •. Constructive Real Analysis, Harper and Row, New York, 1967.

10. Kalman, R. E., Y. C. Ho and K. S. Narendra, "Controllability of Linear Dynamic Systems," Contribution to Differential Equations, Vol. 1, No.2, John Wiley, New York, 1961.

11. Luenberger, D. G. "A Generalized Maximum Princi­ple," Recent Advances in Optimization Techniques, 1968.

12. Porter, W. A., "A basic optimization problem in linear systems," Math. System Theory, pp. 20-44, 1971.

89

13.

14.

15.

Porter, W. A. Modern Foundation of System Engineering, Macmillan, New York, 1966.

Porter, W. A. and J. P. Williams, "Extension of the Minimum Effort Control Problem," J. Math. Anal. Appl., 13, 1966.

Yousif, S. M., "Generalized Minimum Energy Control of Discrete System," Eighth Asi lomar Conference Proc. on Circuits, Systems and Computers, 1974.

16. Yousif, S. M., "On approximate controllability of linear dynamic systems," 18th Midwest Symposium on Circuits and Systems Proc., 1975.

17. Zadeh, L. A. and C. A. Desoer, Linear System Theory, McGraw-Hill, New York, 1963.

I, Iii

ABSOLUTE INVARIANT COMPENSATORS : CONCEPTS, PROPERTIES

AND APPLICATIONS

Romano M. DeSantis Ecole Poly technique de Montreal

University· of Montreal Montreal, Canada

ABSTRACT

The available theory concerning the concepts of absolute invariant compensators is extended by studying the structural properties of these compensators with respect to stability, feedback and compara­tive sensitivity. It is shown that the interconnections between these properties and the Cruz-Perkins comparative sensitivity matrix provides a natural vehicle to extend the application of the classical Nyquist plot approach to the design of multivariable feedback compen­sators. The engineering significance of the development is illus­trated via the design of a multivariable frequency-voltage compensa-tor for an interconnected power system. .

STABILITY TESTS FOR ONE, TWO AND MULTIDIMENSIONAL LINEAR SYSTEMS

E. I. Jury Department of Electrical Engineering and Computer Sciences

and the Electronics Research Laboratory University of California, Berkeley, California 94720

Abstract

This talk will review analytical stability tests for one-dimensional linear systems since the early tests of E. J. Routh in his famous Adams Prize Essay of 1877. The historical background of Routh's stability test and criterion, as well as Fuller's conjecture on its simplification, will be mentioned. In this historical review, the works of Hermite, Sylvester, Maxwell and others as related to the stability problem will also be discussed. This review will provide the context for a discussion of recent stability tests obtained for two dimensional and multidimensional linear systems. These tests will be described and their computational complexity will be discussed in detail. In addition, the applications of stability testing to the study of two and multidimensional digital filters, numerical analysis of stiff-differential equations, realization of mixed lumped and distributed parameter systems, and the design of output feed­back systems will be briefly mentioned. Comments on future research in this area will conclude the lecture.

Research sponsored by the .National Science Foundation Grant ENG76-2l8l6.

91

A DARLINGTON REALIZATION THEORY

OF OPTIMAL LINEAR PREDICTORS.

P. Dewilde T. H. Delft

The Netherlands

Abstract

A systematic junction of Darlington Synthesis theory, optimal linear prediction and stochastic modelling is discussed in this paper. Key elements in the theory are J-unitary coprime factorization, and the theory of orthogonal polynomials on the unit circle.

1. INTRODUCTION

In recent years, the similarity between time in­variant prediction and transmission line theory both leading to Sturm-Liouville type problems, has been discussed by a number of authors (1,2,3). Also, the relation between Wiener-Hopf type equa­tions and scattering problems has been observed by a number of authors (for a survey, see 1). It is clear that (non-uniform) transmission line theory is but a special case of the more general Chandrasekhar-Sobolev theory for non-uniform scattering and transmission in an optical medium. On the other hand, (non-uniform) transmission line synthesis can be viewed as a special case of Darlington synthesis wherby the transmission ei­genstructure is located at infinity in the complex plane. The Darlington synthesis is, in another direction than the Chandrasekhar theory, more general also and has a definite bearing on pre­diction or modelling theory. The simplest instance of this circle of arguments is the predictive time discrete filtering using orthogonal polynomials -or an equivalent Levinson type algorithm (4,5,2). This case also is but a special case of Darlington synthesis, and we will start with this fact in the next paragraph. In the third paragraph we shall show how the Darlington synthesis generalizes the Levinson algorithm and produces a novel identifica­tion algorithm. In the fourth paragraph, we shall introduce the time-continuous case and show its analogy with the previous discussion.

2. THE LEVINSON ALGORITHM, ORI'HOGONAL POLYNOMIALS

AND DARLINGTON SYNTHESIS.

The relationship between orthogonal polynomials on the unit circle and Darlington synthesis is well know (see e.g. 7). Also, the relation between the Levinson algorithm and orthogonal polynomials

92

has been discussed in the literature (1). In this paragraph, we shall show in what manner the Levin­son algorithm and the theory of orthogonal polyno­mials appear as a special case of Darlington syn­thesis. The general Darlington synthesis will be used in the next paragraph to obtain a generaliza­tion. Let x(k) be a stationary Gaussian time discrete (k = 0,1, ... ) stochastic process with zero mean and c~ance function r (k), whereb~ r (k) = Ex(n)x(n-k) for all suitable n. Let x(n) be the optimal least square linear prediction of x(n) in function of th~ data x(k) (k = O,l, .• ,n-l). Writing E(n) x(n) - x(n)

= - ~ ank x(n-k) k=l

for the nth step innovation, and

4> (z) n

where

EE (n)E\ri1

n z + n n-k I: a kZ

k=l n (1)

(2)

we have the following well-known and easy to derive facts : (1) the 4>k form a set of orthogonal polynomials on the unit circle with respect to the spectral mea­sure dF derived from the covariance sequence r(k) by Bochner's theorem. (2) They obey a two-tier recursion formula, similar to the Levinson algorithm, and given by (5)

whereby

* 4>k(z) (4)

and £

~+1 (5)

00 -k (3) defining Zl (z) = reO) + 2 E r(k)z analytic

k=l outside the unit circle, and Wk(z) PZ 1 (z)~k(z)

where the operator P denotes projection on the analytic part inside the unit cir~le, we have that Z is approximated by W (z)/~ (z) in a weak sense.

1 k k The adjoint polynomials Wk satisfy a recurs~on similar to (3) but with uk replaced by - uk·

(4) ~ (z) and W (z) are polynomial with zeros insid~ the unitkcircle, and satisfying an "ener­gy conservation relationshipll(supposing reO) = 1 and ~*(z) = (~(lrz»-):

~kWk71- + Wk~k* = 2

It follows that Z2 l/~k satisfies

1 Z2* Z2 = 2" (Zl* + Zl)

(6)

(7)

which is also the relation between the input im­pedance and the transfer impedance of a lossless network terminated in unit resistors and pictured in fig. 1.

fig.l Network interpretation of eq. (7).

It should be remarked, as pointed out in (5), that jjl jjl

Z2~ (e )Z2(e )d e is a weak approximation of dF,

at least if certain Szeg6 type regularity condi­tions are satisfied.

Proposition 1 The recursion for 1> k+l and W k+l can ·be written in the following form :

z

where

and z

1(1-1 uk

+ 1 12) -uk+1 z

(9)

is a J-unitary passive chain scattering matrix (6) of the first degree with transmission zero at infi­nity (a Darlington section). Writing the recursion out one obtains :

.'nl -h: ::1- li k=n

o k

(10)

A lossless circuit realization for this is given in fig. 2 :

~B---=Gl=~ o 01 b !L n ___ . ~

93

fig. 2 Circuit realization of the product (10).

whereby

(n)

o [::1 ( III

A "running" innovations filter is then oJjtained by taking a

2=b

2=1 so that, in this case, fig 3 produ­

ces a "wfiitening" filter :

fig. 3 : The "running" whitening filter.

If, on the other hand, a modelling filter is de­sired, one can put a

1=1, and in that case the

filter of fig.3 with a 2=b2 will produce the trans­fer 1/~ and act as an approximate modelling filter. This fi£ter, connected as in fig. 4, has output impedanRe Zo = Wn/~n' and inverse transfer impedance Zit = z N n ,

fig. 4 : The modelling filter.

and all its transmission zeros are at zero.

4. SYNTHESIS PROCEDURES

Possibly the easiest way to convert (10) into a synthesis procedure, is the following:

Let us temporarily assume that the model is auto­regressive, i.e. that there is an n such that the

spectral density is given by : s(eie ) = 1/1~ (eiS

) 12 n

-1 and thus S(z) exists and S(z) = ~n(z)~n~z). From (10) we have

En An] = [1 oJ G(n) (12)

where G(n) is polynomial, J-unitary and passive. (12) can also be written as

~n E+zo 1-Z0J = G o ] G(n) (13)

It follows that e(n) can be obtained by extraction of the pole at infinity from ~n E+zo 1-Z0] at

the right. We proceed to show (1) that this extraction procedure is independent from ~n in such a way

that e.g.

zn E+zo (14)

would produce the same J-factor to the right, and (2) that the procedure is essentially unique and leads to the factored form (10). Indeed, the right hand side of (13) is easily seen to be a coprime factorization of the left hand side with factors analytic outside the unit circle. This fac!~rization is known to be uni~uT except for a z -unimodular factor. Since G n is J-unitary, this unimodular factor can only be constant J­unitary. Moreover the same is true for (14) . This completes the proof. It should be noted that (13) can be performed on the time series Zo

immediately without conversion to the frequency domain. The procedure is then not only a network synthesis bUt also an identification i.e. it iden­tifies the system from its time series.

Suppose next that it is not known whether the system producing Zo is indeed autoregressive. The Levinson procedure (or the equivalent procedure using orthogonal polynomials) then produces a filter with all its transmission zeros at infinity which gives (1) the innovations at each stage and (2) - when n is sufficiently large - an autore­gressive approximation to the modelling filter (5). Both can be obtained in exactly the same way as described previously : again Z is known, and a factorization of (14) with SUf~iciently large n will produce the predictive filter. More generally, in the multivariable case, we have that a facto-rization of

u(n) 11+zo 1-Z

0] Gll 6 1J G(n) (15)

where u(n) is a (judiciously chosen) unitary matrix

94

with poles at infinity of degree n and ern) J-unitary and passive, will produce a multi variable version of the innovations filter.

The previous procedure can now be generalized to generate innovations filters and modelling filters which are not restricted in their transmission zero . location. E.g. suppose that such a filter is desi­red with n transmission zeros at infinity and m at a given point a with lal>l, then a factoriza­tion of :

n m r,. z (z-a) [.+ZO 6 ] e(n+m)

12 (16)

. (n+m) . w~ll produce the answer where G ~s chosen as a J-unitary passive matrix having apole of order n at infinity and a pole of order m at a, and the factorization (16) is required to be coprime. 6

12 can be so determined that

-n -m. z (z-a) 6 12 (17)

where 6'12 side the

is suitably bounded and analytiC out­unit circle.

More generally, (15) can be used with any u(n) unitary and having singularities only outside the unit circle. (17) then generalizes to :

(18)

where 6'12 is analytic outside the unit circle.

Again, the factorization (15) can be performed on the time series and does not require any Fourier transformation. In this case, time invariant line­ar combinations of data in the input stream will be estimated and the "innovations" obtained will be orthogonal on such(cynbinations. The choice of unitary matrices U n is n~t)arbitrary. In fact, one must require that U n .. 0 in some weak sense. This is the case e.g. for zn due to the Lebesgue lemma. From a numerical point of view, the modelling and innova~iyns filter will be "best" if the zeros of U n are chosen as close as possible to' the zeros of the (unknown) system which generates the covariance. It is known that the identification problem of a covariance is generally ill-conditione~.)Introducing the in­formation contained in U n adds information on the probable transmission properties of the system and makes the problemt~ell-posed. In the Levinson filter of the n order, only the n last data are stored, and all other past infor­mation is lost. In the generalized Levinson filter obtained by factorization of (15) time-invariant linear combinations of past data are stored. These combinations may be much more efficient for pre­dictive purposes - especially if the filter gene­rating the data has finite zeros.

5. THE CONTINUOUS-TIME ANALOG

Let x(t) be a time-invariant Gaussian stochastic process with covariance function r(t) and spectral measure given by (Bochner's theorem) :

oCt) + K(t)

In this case the innovation-(8) is given by

E: (t) x(t) - It h(t,T) X(T) dT o

where h(t,T) obeys the Wiener-Hopf equation

h(t,T) + Ig h(t,ul K(u-T)du = K(t-T)

(19)

(20)

(21 )

The Fourier transform (with respect to T) of 6(t-T) h(t,T):

P(t,w) -iwt = e (22)

obeys the following set of differential equations (2) :

a at P(t,w) -iwP(t,w) P 1Ir (t,w)h(t,O) (23a)

() at plt- (t,w) = -P(t,w)h(t,O) (23b)

where P -IE (t,w) = e -iwt P"("t";Wj (24)

Eqs. (23) are analogous to eq. (3). They describe a non-reciprocal transmission line with J-skew transition matrix

. ~iW A(t) =

:h(t,O) (25)

and gen~rate a transfer matrix 0 t given by a pro­duct integral :

o t

t

I o

(26)

and analogous to (10). The transfer function (22) so obtained has a finite impulse response just as in the previous paragraph. The 0 theory there is equally valid here, except that a ne w type of section - consisting of transmission line stru­cture - has to be introduced. Again, a coprime factorization of type (15) provides for alternative approximations and a generalization of the method.

6. BIBLIOGRAPHY

1. T. Kailath, "A view of Three Decades of Linear Filtering theory", IEEE Trans. on Info. Theory, vol. IT-20, No.2, pp. 145-181, March 1974.

2. T. Kailath, A. Vieira, M. Morf, "Inverses of Toeplitz Operators, Innovations, and orthogonal polynomials", Informations System Lab. Stanford University.

95 , ' . ..1

3. M.G. Krein, "The Continuous analog of Theorems on Polynomials Orthogonal on the Unit Circle", Dokl. Akad. Nauk. SSSR, Vol. 104, pp. 637-640, 1955.

4. N. Levinson, "The Wiener RMS (Root Mean Square) Error Criterium in Filter Design and Prediction", Jour. of Math. and Phys., Vol. XXV, No.4, Jan.1947 pp. 261-278.

5. L.Ya.Geronimus, Orthogonal Polynomials, Transl. Consultants Bureau, New York 1961.

6. P. Dewilde, "Input-Output Description of Roomy Systems", SIAM J. on Control and Optimization, Vol. 14, No.4, July 1976.

7. J.D. Rhodes, P.C. Marston, D.C. Youla, "Expli­cit Solution for the Synthesis of Two-Variable Transmission Line Networks", IEEE Trans. on Circuit Theory, Vol. CT-20, No.5, Sept. 1973.

8. T. Kailath, "An Innovations Approach to Least­Square Estimation - Part I : Linear Filtering in Additive White Noise", IEEE Trans. on Antom. Con. Vol. AC-13, No.6, Dec. 1968.

BIOGRAPHY

Patrick Dewilde was born in Korbeek-Lo, Belgium, on Jan. 17th, 1943. He received the engineer de­gree from the University of Louvain, Belgium, in 1966, and the Ph.D. degree from Stanford Universi­ty, in 1970. He has held research and teaching positions with the University of Louvain, the University of California at Berkeley, and the University of Lagos, Nigeria. He is presently professor of Network Theory at the T.H. Delft, the Netherlands. His main interests are in applied algebra (systems an network synthesis, facto­rization properties for matrix functions, scatte­rin,g matrix theory, numerical analysis and fast algorithms) and in teaching Network Theory.

OPERATOR THEORY TECHNIQUES FOR FINITE DIMENSIONAL PROBLEMS

J. W. Helton Dept. of Mathematics

Univ. of Calif. at San Diego San Diego, Calif. 92110

ABSTRACT

Problems with matrix valued rational functions can become complicated very quickly (once one starts chasing various sub-spaces correspond­ing two different orders of vanshing). Operator Theory provides techniques for handling such problems which are conceptual but which still have considerable power. This talk describes how one can use operator theory on the problem of broad band impedance matching. The (rather powerful) commutant lifting theorem allows one to obtain some results for input where none were previously available. Furthermore, it helps to complete the classical theory for one ports.

96

THE RELATION BETWEEN NETWORK THEORY,

VECTOR CALCULUS, AND THEORETICAL PHYSICS

Franklin H. Branin, Jr. System Communications Division Laboratory

IBM Corporation Kingston, N.Y.

Abstract

This tutorial paper is a brief review of previously published work by Roth, Branin, and Tonti relating to the underlying nature of net­work analogies and their use in engineering and physics. Roth's al­gebraic topological characterization of the l-network (linear graph) problem laid the foundation for Branin's treatment of the 3-network problem and its relation to the vector calculus and Maxwell's equa­tions. Tonti has extended these ideas to encompass a significant portion of theoretical physics.

1. SUMMARY lem -- as well as many related network prob-

The purpose of this tutorial paper is to lems -- involves a l-network, or linear

point out the intimate relation that exists graph, consisting of a set of intercon­

between network theory, the vector calculus, nected O-dimensional objects [points, nodes,

and much of theoretical physics -- in short vertices, O-cells] and l-dimensional ob-

to show the mathematical basis for network

analogies and the implications thereof. The paper summarizes contributions made by Roth (1) , Branin(2,6) , and Tonti. (3,4,5,7)

jects [lines, branches, edges, or l-cells].

However, many physical systems require the

use of surface and volume elements [2-cells

and 3-cells] as well as points and lines to Most familiarly, network analogies have characterize them properly. Without a clear

evolved from a comparison of the basic equa- understanding of the related 3-network prob-

tions of some physical phenomenon with the

equations describing an electrical network.

This heuristic approach has been fruitful,

but it has failed to identify the underly­

ing justification for many of these anal­

ogies. The reason for this failure is the

fact that the electrical network problem is

of lower dimension than many of the physi­

cal phenomena for which network analogies

have been sought.

Specifically, the electrical network prob-

lem, then, the full implications of the net­

work analogies for such systems cannot be

appreciated. Indeed, some of the inherent

richness of the mathematical theory that

describes these phenomena is missed.

In a fundamental contribution to network theory, Roth(l) rigorously characterized

the l-network problem from a completely

abstract algebraic topological point of

view, devoid of any necessary physical

interpretation. In so doing, he laid the

97

foundation for understanding fully just

what conditions are necessary for the ex­

istence of any particular physical example

of a I-network problem.

It is well known that the I-network or lin­

ear graph can have "through" and "across"

variables associated with its nodes, its

branches, and its loops. As a consequence

of interconnecting the branches, these

through and across variables are subjected

to topological constraints that force the

through variables to sum to zero at the

nodes and the across variables to sum to

zero around the loops of the I-network.

Topological matrices (A, the branch-node

incidence matrix and C, the branch-loop

or circuit matrix) are used to express the

topological constraints. In addition, the

constitutive relations between the through

and across variables in each branch can

In this diagram, the through variables ap­

pear in the uppermost sequence of transform­

ations, the mesh (loop) variables i' and

branch variables i being responses. The

variables I associated with the branches

are quite arbitrary and correspond in an

electrical network to independent current

sources. The across variables, appearing

in the lower transformation sequence, in­

clude the node variables e' and branch vari­

ables e, both of which are responses; the

branch variables E are again arbitrary and

correspond electrically to independent volt­

age sources.

The topological constraints on the through

variables comprise the equations

and/or i = C i' (1)

and

(2)

be expressed in matrix form. The topologi- while those on the across variables are

cal constraints and constitutive relations

together completely describe the I-network

problem from an engineering or physical

point of view.

and

and/or e = A e' (3)

(4) Roth's abstract view of this problem is more The constitutive relations involve the

penetrating and is summarized by the trans- composite through and across variables as-

formation diagram of Fig. 1. socia ted with the branches, namely

c "ill Zlly

------ 0 At r:l ---•• L!J

[rJ .... -----Ct I:I ... -_A_- GJ o •• -----

(mesh) ( branch) (node)

Fig. 1 Roth's Transformatjon Diagram for the I-Network Problem

E+e = Z(I+i) and/or I+i = Y(E+e) (5)

(In Eqs. (1), (3), and (5), the "and/or"

signifies that the two alternative expres­

sions are mathematically equivalent.)

Rather than considering Eqs. (1) to (4) in

terms of topological matrix operations,

Roth regards them in terms of boundary and

coboundary operations since these have much

deeper significance topologically. For the

same reason, he treats the Z and Y matrices

as de~ining isomorphisms rather than simple

matrix operations.

There is much more substance in the detail

of Roth's paper than we can discuss here, but the main implication of his work is the fact that the through and across variables in a I-network problem can belong to any

98

vector space. Accordingly, any physical and "across" are no longer appropriate for

phenomenon in which through and across vari- describing variables associated with 2- and

abIes exist, obey the constitutive and topo- 3-cells, the terms 0-, 1-, 2-, 3-chains and

logical constraint laws, and belong to some 0-, 1-, 2-, 3-cochains will be used instead.

vector space, can be treated as an example

of a 1-network problem. In particular, this includes structural analysis(8)where the

force and displacement variables are 6-vec­

tors and mechanical systems where they are

3-vectors. In the spirit of Roth's approach, Kron(9)

derived a transformation diagram purporting

to explain Maxwell's equations. However,

the present author found Kron's results to

be unacceptable and sought an alternative

In the chain sequence, appearing at the top

of Fig. 2, the symbols Ml and M2 represent

responses while the symbols P1 and P2 stand

for independently assignable variables. In

the cochain sequence, Ml and M2 represent

responses while pI and p2 are arbitrary.

Moreover, the symbols COl' C12 , and C23 stand for "connection matrices" (incidence

matrices) between the 0- and I-cells, the

1- and 2-cells, and the 2- and 3-cells. Al­

ternatively, these matrices may be regarded explanation. By extending Roth's ideas, he as boundary operators and their transposes

developed an analogous transformation dia- as coboundary operators. (Incidentally,

gram that describes the algebraic structure C~l is identical with the A matrix of Fig. 1.).

of the 3-network problem, as shown in Fig.2. Finally, in place of the single Z/Y isomorph-

3- chains 2-chains .i-chains O-chains

Chain

~ C23

~ Sequence • • 0

P2 C12

~ • • 0

t 11 t I S211 T 2

Pi C 01 B (~3S2 C23' ( C23 S2 C23 '- •

511 f T 1 (COIT1 colf11 (CoiT 1 C~11 Cochain ~ ..

~ Sequence t

0 .. C23

~ ..

C;2 C61 E] 0 M1 .. .. 3-cochains 2- cochains 1. - cochains O-cochoins

Fig. 2 ism of the I-network problem, there are ~

in the 3-network problem, namely the SllTl Here, the underlying topological structure isomorphism between the I-chains and l-co­

is a set of interconnected 0-, 1-, 2-, and chains and the S2/T2 isomorphism between

3-cells [points, lines, surface, and volume the 2-chains and 2-cochains.

Branin's Transformation Diagram for the 3-Network Problem

elements] having two sets of variables assoc-Now the 3-network problem definable on this iated with them, just as in the I-network algebraic structure has the very peculiar

case. However, since the terms "through" feature that if the matrices (boundary and

99

'L lilll --"I

coboundary operators) COl' C12 ' and C23 and

the isomorphisms SllTl and S2/T2 are given, only Pl and p2 need be specified in order

1 for all the responses -- AND P and P2 --

to be determined. The original motivation for defining this

3-network problem, of course, was to find

an explanation for Maxwell's equations -­

which involve the vector calculus opera­

tors grad, curl, and div. These operators

turn out to be the exact counterparts in

the continuum of the coboundary operators t t d c t . th h . COl' C12 ' an 23 ~n e coc a~n sequence

of Fig. 2. However, there is no counter­

part of the chain sequence in vector cal­

culus. What is more, a full analogy be­

tween the structure of Fig. 2 and that of

the vector calculus requires two cochain

sequences.

In order to satisfy this requirement, the

concept of a dual 3-network was introduced.

Here, the 0-, 1-, 2-, and 3-cells of the

primal 3-network can be made isomorphic to

the 3-, 2-, 1-, and O-cells of the dual 3-

network. As a consequence, the primal

chain sequence in Fig. 2 can be replaced

by the (isomorphic) dual cochain sequence,

as shown in Fig. 3.

Dual O-cochoins l-cochoins

Cochoin C23 •

The 3-network problem associated with this

new transformation diagram has exactly the

same structure as before -- except that the

dual cochain sequence symbols (superscrip­

ted and underscored) replace the primal

chain sequence symbols (subscripted and not

underscored. )

At this stage, the state of affairs is pre­

cisely equivalent to that of the earliest

stage in the classical derivation of the

vector calculus operators grad, curl, and

div. In other words, both in the classical

treatment and in our 3-network, we have

established algebraic relations between

certain quantities associated with discrete

points in space, finite line segments, fin­

ite surface elements, and finite volume ele­

ments. Moreover, these algebraic relations

reflect the topological connectedness of

these 0-, 1-, 2-, and 3-cells in exactly

the same way in the classical treatment

and in our more formal 3-network approach.

If, now, we let these finite 1-, 2-, and

3-cells approach the limit of zero size,

we find -- not surprisingly, in view of

the results of this limiting process in the

classical derivation -- that the coboundary

operators in both cochain sequences of Fig. 3

2-cochoins

• 0 ca ~ Sequence Cl2

3-cochoins

~ el .. • 0

t H t )-1 COl ra Dual

(C23S2C23) (CZ3SZC23 S2HT2 p2 • 3-Complex

St 1fT! Primal

~ ~ t -11 r t Cochain • (C01TIC01) (COITI C01) Sequence t

C23 0 • •

~ 3-cochoins t C!2 t B Mt.

cO! Primal 0 • • 3-Complex Z-cochains

l-cochoins O-cochains Fig. 3 Transformation Diagram for the

Primal-Dual 3-Network Problem

100

become identical to the vector calculus

operators. In particular, C~l becomes equal to -grad, ci2 becomes equal to ~, and C~3 becomes equal to div. The net re­sult, then, is the transformation diagram of Fig. 4 which describes the vector cal­

culus relations in the continuum.

0- cochains

Dual Cochain Sequence

9 -CJrad

(- div 5 CJrad) (-div s CJrad)-l

Primal Cochain Sequence

3- cochains

dlv 4

1- cochains

u

s

w

2-cochains

Fig. 4 Transformation Diagram for the vector Calculus

curl

curl 4

ELECTROMAGNETIC FIELD

dual dual l-cochoins dual

On the basis of these vector calculus re­lationships, the corresponding transforma­tion diagram for Maxwell's equations can be

derived as show in Fig. 5. Here it is worth remarking that the dimensionalities of all the physical variables are fully

consistent with their topological roles.

2 - cochains

J

z y

v 4

1- cochains

3- cochains

div p Dual

3-complex

(-div Y CJrad)-l (-div Y CJrod)

- CJrad Primal 3-complex

0- cochains

O-cochoins r--, -grad

GJ 2-cochains

1 1 I 1 ----- ..... ------ ..... L_..J curl

• 1+ I I

-I' ~, II II I I +1

r--, I 1 1 I L_..J

o

r-' 1 I I I 1 I

I-al 4 .. ----

+-------div ..

primal curl

3- cochains o .... 1------primal 2-cochoins

Fig. 5 Transformation Diagram for Maxwell's Equations

0 dual 3-cochoins

ill -----.. 0

div ---0 11 ~+C :t

~ . -,,,' primal l-cochains

101

primal O-cochains

This diagram highlights the fact that two

different types of vector are involved,

namely line-density vectors such as E and

H, associated with I-cells, and surface­

density vectors such as B and I, associ­

ated with 2-cel1s. Clearly, it is topo­

logically possible to take the curl of

the first and the divergence of the second

type of vector, but not vice versa. This

point is apparently not well recognized.

The missing variable associated with the

primal 3-cells in Fig. 5 would be magnetic

charge density. But if the diagram is ro­

tated 1800 about its center, the corres­

ponding relations would, in effect, de­

scribe a fully consistent "magnetoelectric"

field. Page and Adams (10) point out that

this is quite possible, both mathematically

and physically.

Tonti(3,4,5,7)has greatly extended and re-

fined the ideas embodied in the 3-network.

He has treated in some detail the connect­

~vity properties of 0-, 1-, 2-, and 3-cells

2 • REFERENCES

1. J. P. Roth, "An Application of Algebra­

ic Topology to Numerical Analysis: On the

Existence of a SOlution to the Network Prob­

lem", Proc. Nat'l. Acad. Sci., vol. 41, pp.

599-600, 1955.

2. F. H. Branin, Jr., "The Algebraic~Topo­

logical Basis for Network Analogies and the

Vector Calculus", Proc. Symp. on Generalized

Networks, vol. 16, Microwave Res. Inst. Symp.

Ser., Polytechnic Inst. of Brooklyn, pp. 453-

491, 1966.

3. E. Tonti, "On the Mathematical Structure

of a Large Class of Physical Theories",

Rend. Acc. Lincei, vol. 52, pp. 48-56, 1972.

4. E. Tonti, "A Mathematical Model for

Physical Theories", Rend. Acc. Lincei, vol.

52, pp. 175-181 (Part I)~ pp. 350-356 (Part

II), 1972.

5. E. Tonti, "On the Formal Structure of

Physical Theories", Quad. dei Gruppi di

Ricerca Matematica, Istituto di Matematica

del Politechnico Milano, 1975.

and shown numerous examples of how the asso- 6. F. H. Branin, Jr., "The Network Concept

ciated variables relate to physical quanti- as a Unifying Principle in Engineering and

ties. He has introduced concepts from the

mu1tivector calculus and tensor calculus,

from the Grassman and Clifford algebras,

and from the theory of external different-

the Physical Sciences", in Problem Analysis

in Science and Engineering, F. H. Branin

and K. Huseyin, Eds., New York: Academic

Press, to be published.

ial forms to supplement the algebraic-topo- 7. E. Tonti, "The Reason of the Analogies

logical principles already involved. Indeed,in Physics", in Problem Analysis in Science

he has laid a very broad foundation for a and Engineering, F. H. Branin and K. Huseyin,

comprehensive approach to theoretical phys- Eds., New York: Academic Press, to be pub­

ics involving 1-, 2-, and 3-network models. lished.

A partial (but representative) list of the 8. S. J. Fenves and F. H. Branin, Jr., "A

systems treated by Tonti in this way is as Network-Topological Formulation of Struc-

follows: classical and relativistic part- tural Analysis", Journ. Struct. Div., Amer.

icle dynamics, vibrations of strings and Soc. Civil Engr., vol. 89, pp. 483-514,1963.

rods, acoustics and fluid dynamics, thermo- 9. G. Kron, Diakoptics: The Piecewise Solu-

statics and irreversible thermodynamics, tion of Large-Scale Systems, London: Macdon-

geometrical optics, the Klein-Gordon and aId, 1963.

Schroedinger equations, classical and rela- 10. L. Page and N. I. Adams, Electrodynam-

tivistic gravitational theory, the wave ics, pp. 210-211, New York: Van Nostrand,

equation, and the Dirac and Proca equations. 1940.

Tonti's work is indeed a remarkable and

comprehensive contribution.

102

DETACHED COEFFICIENTS REPRESENTATION AND DEGREE FUNCTOR

OF A POL YNOUIAL MATRIX \lITH APPLICATION TO LINEAR SYSTEMS

Y • S. Ho P • H • Roe Department of Systems Design

University of IJaterloo Waterloo, Ontario, Canada

Abstract

The zeros of a polynomial matrix are classified as explicit infinite zeros, impli­cit infinite zeros and zeros in the finite complex plane. Using detached coeffi­cients method and an operator called the degree functor of the matrix, explicit formulae to calculate the multiplicity of the explicit infinite zero and the num­ber of finite zero in the complex plane are derived.

1. INTRODUCTION number of the explicit zeros and the number of

In the analysis of linear systems, we encounter very zeros in the finite complex plane are derived.

often computation in the polynomial ring F[x] where These lead to the determination of relative prime-

F is either the field of real numbers R or complex ness and number of common zeros of two polynomial

numbers C, and x is a formal indeterminate [1,2].

An important question has been asked [3]: Is it

possible to bypass the machinery of polynomial al­

gebra and relate everything to standard matrix com­

putation, such as the determination of rank? This

question is of some interest from the viewpoint of

pure mathematics, since it concerns the representa­

tion of polynomial algebra via matrices in the sense

analogous to group representations [17]. Even more

interesting perhaps are the implications on numeri­

cal analysis and computing art in general, since

the computations in F[x] are rather awkward. In

this paper, we attempt to answer partially the

above mentioned questions.

The zeros of a rectangular polynomial matrix of full

rank are classified as explicit infinite zeros, im­

plicit infinite zeros and zeros in the finite com­

plex plane. Using the "detached coefficients

method" and an operator called the "degree functor"

of the matrix, explicit formulae to calculate the

matrices of different sizes. The results are used

to express unifying the conditions of existence,

controllability, observability and invertibility of

a linear time-invariant dynamical system in terms

of the number of zeros in the finite complex plane

of appropriate polynomial matrices derived from the

equations representing the system.

2. MATHEMATICAL PRELIMINARIES

We consider first two polynomials in R[s]:

laCs) ,b(s)] [a ,b ]sp + [a l,b l]sP-l + p p p- p-

+ [ao,bo] (2.1)

where ap

I 0, bp

= bp

_l = ... = bq+l = 0, bq I 0,

and p ~ q. We called p the degree of the matrix

[a(s),b(s)] and denoted it as d[a(s),b(s)].

The resultant matrix R[(a) ,(b) ] of [a(s),b(s)] is p p

defined as follows:

103

+-p columns -"I+- P columns ....

R{ (a) , (b) ] a b p p 0 0

a1 b

l

O a b 0 2 P 0 0 rows a a

1 bl

bo 0

-1 a b p P

a p-l a b b P P p-l

a 0 q b P P

The classical resultant matrix R[(a) ,(b) ] can be q p

obtained by deleting the first (p-q) columns and

the last (p-q) rows of R[(a) ,(b) J. The subre-p p

sultant matrix R[ (a) j' (b) .] of [a (s), b (s)] is p- P-]

defined as the matrix obtained from R[(a) ,(b) J by P p

deleting the first and last j columns, and the first

and last j rows. The subresultant matrix

R[(a) j,(b) j] can be similarly defined. q- p-

Lemma 2.1 [6]: The greatest common divisor of

(a(s),b(s)], denoted as G.C.D[a(s),b(s»), has

degree greater than zero if and only if the result­

ant R{(a) ,(b) } ~ det R[(a) ,(b) 1 is zero, q P q P

Lemma 2.2 [6,7,8]: Degree of G,C.D[a(s),b(s)]

d[C.C.D[a(s).b(s)l}, is j if and only if

R{(a) ,(b) }=R{{a) l,(b) l} q p q- p-

and

R{(a) j,(b) j} f O. q- p-

In order to facilitate and motivate the discussions

in this paper, we relabel the columns of R[(a) , p

(b) ] and denote the reSUlting matrix as: p

Rix2

(a,b) R;X2(a,h} Rlx~(a,b) p-

R1x2(a b) = ao b 0 0 p , 0

a l hI a b 0 0

a b 0 0

a h a p-l b

p-l P P

0 0 a b p p

a b p P

where the lx2 denotes the order of the matrix laCs) ,b(s) J.

(2.2)

With the definition of Rlx2 (a,b). we can generate p

b:2 Rj (a,b) as follows:

(a) If j<P. R~X2(a,b) is defined as the matrix ob­

tained by deleting the last (p-j) rows and the'.last lx2 2(p-j) columns vf R (a,b). p

lx2 (b) If j>p, R (a,b) is defined as the matrix ob­p tained by bordering (j-p) rows to the bottom and

2(j-p) columns to the right of R1x2(a,b) in the p

lx2 lx2 same fashion as R2 (a, b) is obtained from 1): (a, b).

Definition 2.1: Let A be an nxm matrix over R, the

Taking into account a + 0, we have

p ~[AJ respectively. We call n-rank[A]. the nullity Theorem 2.1: d[C.C.D[a(s),b(s)]] = j if and only if of the row rank of A and denote it as ~[AJr'

rank and ullity of A are denoted as rank[A] and

R{{a) ,(b) } = R{(a) l,(b) I} = .•• p p p- p-

and

R{(a) j,(b) .}fO. p- P-J

Theorem 2.2 (22):

then for j>l

104

r. lx2 1 ~LRi (a,b)j'

r

ThElorem 2.3 [22]: d[G.C.D.[a(s),b(s)]] = j, if and

only if there exists an i, such that with

T][R~X2(a,b)t ~ 1 x d[a(s),b(s)]

r, lx2( J [ lx2 ] T]~i+l a,b) = 11 Ri (a.b) = j r r

(2.3)

3. POLYNOMIAL MATRICES

Definition 3.1: Let pes) be an nxm matrix over

R[s] and let

_ p p-l pes) - P s + P IS + ••• + PIs + P (3.1)

p p- 0

where the coefficient matrices Pi : i=O.1,2, ••• ,p

are nxm matrices over R. pes) is assum~d to be of

full rank and n:::m unless otherwise stated.

(1) P is called the leading coefficient matrix p

of pes);

(2) pes) is of degree p, denoted as d[P(s)] = p,

if its leading coefficient matrix P is not a p

(3)

(4)

zero matrix;

pes) is called proper, if its leading coeffi-

cient matrix P p is of full rank;

The rank r of pes) is equal to the dimension

of the largest minor of pes) which is not a

zero polynomial. If r<n. pes) is called

identically singular; otherwise. it is

called full or pes) has full rank;

(5) Any matrix formed by the coefficient matrices

of pes) is called an associated matrix of

pes) ;

(6) pes) is called non-singular if and only if

pes) is full and m=n.

Let A(s) be an nxn non-singular matrix over R[s].

If A(s) is proper. det[A(s)] is a polynomial over

R of degree nxd[A(s)]. It has nxd[A(s)] zeros

(counting multiplicities) over C. If A(s) is im­

proper, d[detA(s)] is less than nxd[A(s). In fact

d[detA(s)] is the number of zero of de~(s~ in the

finite complex plane : C'''''.

Definition 3.2: (a) A number AEC ''''' is said to

a zero of pes) if and only if

rank[P(A)] over C < rank[P(s)] over R[s].

(b) Let 6r

(s) be the G.C.D. of all minors of order

r of pes). 6 (s) is called the characteristic n

polynomial of pes).

Definition 3.2 leads directly to

Theorem 3.1: AEC' "" is a zero of P (s) if and only

if A is a zero of 6n

(s).

Definition 3.3: Let K = max{ d [ai (s)] I ai (s) is a

minor of order n of pes) where i=1.2 •••.• C:}.*

(a) n",,(P) ~ n x d[P(s)] - d[6n (s)] is called the

multiplicity of the infinite zero of pes) in

C; (b) n~(P) ~ n x d[P(s)] - K is called the multi­

plicity of the explicit infinite zero of pes)

in C; (c) ni(p) ~ K - d[6 (s)] is called the multiplicity

"" n of the implicit infinite zero of pes) in C.

Theorem 3.2 [22]: If pes) is improper. there exists

a non-singular polynomial matrix T(s), which is the' ** product of elementary row transformation matrices.

such that T (s) x P (s) ~ p* (s) is proper and

* d[P(s)] = d[P (s)] if and only if pes) is of full

rank.

Corollary 3.1: det T(s) = as~

where a is real and ~ = n:(P) •

Remark 3.1: The explicit infinite zero of pes) has

* been normalized to zero via T(s). That is P (s) e will contain additional n",,(P) number of zeros at

the origin of C besides the zeros of pes) in C'''''.

D fi i i 3 4 L Mnxm(p) Hnxm(p) and Rnxm(p) e n t on .: et p+j , p+j j be associated with pes) as follows:

* m Cn

is the combination of m objects taken n at a time.

** We mean the usual constant'elementary row transformations together with Tii(s): whose off-diagonal elements are zero and whose diagonal elements are unity except for the ith. which is equal to s.

105

Mnxm(p) = p+j P

P P p-l PI Po ° I .

° P Pz PI P I p 0 I

I P Ip P

PI p-l 0

11 =

o

where ~xm(P) is of order (ixn) x [(p+i)xmJ,

H~xm(P) is of order (ixn) x (ixm) and it consists of

the first (ixm) columns of M~xm(P), and R~xm(P) is

of order [(p+j)xn] x (jxm). The ~xm(P), H~xm(P) nxm

and Ri (P) are defined only for i>O.

Theorem 3.3 [ZZJ: If pes) is of full rank, then

the associ~ted matrices M~xm(P) : i=l,2, •.. of pes)

are of full rank.

Theorem 3.4 [22J: PCs) is of full rank if and only

if there exists an i>O, such that with nxm J 11 rank[Ho {P} = 0

rank [H~:(P) J = rank [H~(P)] + n ••• (3.2)

Corollary 3.2:

(a) If p is the first index such that (3.2) is

satisfied, then

rank[H~(p)] rank[H~xm(P)J + nj; j>l

(b) D[T(s)] = p

(c) T][H~xm(p)l = T][H~(P)J= d[det T(s)]

(d) K = d[l1n (s)] + n:(P) = n[dIP(s)]-p)

+ rank [H~xm(P) J . m Definition 3.5: Let Qi : 1.2, •.•• r=Cn

be the sub-

matrices of order nxn of PCs) and let

aiCs) = det Qi(s). Let R~xe(A). ~xt(A) and

Hkxt (A) be associated with an kxt (k ~ t) matrix i

106

A(s} over R[sl as were defined in Definition 3.4. kxe kxt

Rl (A) and MI (A) are called the detached coeffi-

cient representations of A(s). An operator 6

called the degree functor of pes) is defined with

the following propert.ies:

if and only if there exists a 0 which is the smal­

lest index that exists, such that with

(3.3)

where A(s) and B(s} are respectively nxm and nxe

matrices over R[sl.

Theorem 3.5:

Proof: This is all but obvious if we note that the nxn nxn nxn( )] matrix [Rl (Ql).Rl (Q2}' .••• Rl Q

r and the

matrix [R~xm(P)] will give the same 0 and same j in

(3.3).

Theorems (Z.3).(3.5) and Definition 3.5 lead direct­

ly to:

Theorem 3.6: With [a(s).b(s)] as in (2.1). we have

(a) ':{RiXl(a)] d[a(s)]

(b) 6[RiXZ

(a,b)J 6[RixlCa).RiXI(b)]

[Theorem 3.51

deC.C.D. of [a(s) ,b(s) 1]

[Theorem 2.3]

The importance of Theorem (3.6) lies in the fact lxZ that when 0 is operated on R1 (a.b) of [a(s),b(s)],

it turns out to be the degree of the G.C.D. of

laCs) ,b(s)].

Theorem 3.7: Let a(s) [al (s),a2(s), ••.• a

r(s)],

where a i : i=1.2 •••• ,4 can be any r polynomials.

Then

d[G.C.D.[a(s)]] = O[Rixr(a)].

Proof: We prove the theorem by induction on r.

That the theorem is true for r = 1 follows from

Part (a) of Theorem (3.6). Suppose the theorem is

true for n<r. and let

then

0.4)

Consider

[Theorem 3.6(b)]

(3.4)

[Theorem 3.5]

[Theorem 3.5]

The induction is completed.

Theorem 3.8: If pes) is square; Le •• n=m. then

(3.5)

Proof: (a) If pes) is proper. Definition 3.5 shows

that 6[Rnxn (P)] = n x d[P(s)] and Part (c) of 1 e

Corollary (3.2) shows that n~(P) = o. (b) If pes) is improper. Theorem (3.4) shows that

fJ[R~xn(P)]r=n x d[P(s)]. and Parts (c) and Cd) of

Corollary (3.2) show that

d[det pes)] = n x d[P(s)] - fJ[H~xn(P)] e

n x d[P(s)] - n~(P)

Theorem (3.5) follows by invoking Definition 3.5.

Theorem 3.9: If pes) is an nxm matrix over R[5]

of full rank and n<m. then

= d [G.C.D. of [al (s) .a2 (s) •••• a/s) J [Definition 3.5]

6 [Rixr (a) ] [Theorem 3.7]

r, lxl lxl lxl 1 6LRl (al).Rl (a2).···.Rl (ar )

[Theorem 3.5]

r, nxn nxn nxn 1 6 LRI (Ql) .Rl (Q2)'··· .Rl (Qr)

_ min{n!(Qi) I i=1,2, ••• r} [Theorem 3.8]

[Theorem 3.5]

Corollary 3.3: Let pes) = [A(s) ,B(s). where A(s) and

B(s) are matrices over R[s] of orders nxm and nxt

respectively. Then the following propositions are

equivalent.

(a) A(s) and B(s) are relatively (left) prime

(b) 6 [R~X(mxt) (p) ] = n! (p)

(c) Rank[P(s)] = n for all s in C"~.

Definition 3.6: Let pes) = [A(s).B(s)], where A(s)

and B(s) are of orders nxm and nxt respectively and

rank[P(s)] =n.

(a) 6[R~X(m+t)(P)] - n!(p)

is called the common zero of A(s) and B(s)

(b) the zeros of pes) which are zeros of A(s) and

Bes) are called the explicit common zeros of

A(s) and h(s). otherwise they are called the

implicit common zeros of A(s) and B(s).

Let

rQ(s)

Lv(s)

4. LINEAR SYSTEMS

U(S)][x] [01 W(s) u y J

be the system matrix in polynomial form [I], where

x.u and yare the vectors of system variables, in­puts and outputs. The following results can be

derived using the results derived in Section 3.

1. The system is completely controlable if and

only if

Proof: J r. ] 6rRnlx (n+t) (Q.U) - n:(Q,U).

d [fln (8)] = d I.G.C .0. of [detQl (s), detQ2 (s) •.. ,detQr (s)] l -[Definition 3.5] 2. The system is completely observable if and only

107

•

if

oG~(n+m)(QT,VT}J = U:(QT,VT).

3. The system is completely observable and control­

able if and only if

8 [a~x(nxt} (Q,U)] + 0 [R~X(n+m) (QT, VT)]

- 8 [R~X(n+t+m) (Q,U, VT) J e e T e T = noo(Q,U) + noo(Q,V ) - uoo(Q,U,V ).

4.

5.

The system is invertible if and only if the

number of zeros of the system matrix is finite.

The system is solvable if and only if the

number of zeros of Q(s} is finite.

5. DISCUSSION

The scope of application of the theory developed in

this paper is not confined only to the topics just

mentioned. Due to lack of space, the application

to rational function matrices is omitted.

Interested readers can refer to [22J. Recent in­

terest in coprime factorization of a regular trans­

fer function matrix [21] and the stability studies

conducted using inners [23) pointed out that the

study of "functors" may open up a promising

frontier of research.

REFERENCES

1. H.H. Rosenbrock, State Space and Multivariab1e I~rory. New York : Wi~f!' ~~lO.

2. L.A. Zadeh and C.A. Desoer, Linear System Theory. New York: McGraw-Hill Book Co., 1963.

3. R.E. Kalman, "Some computational problems and methods related to invariant factors and con­trol theory", John Leech edited, Computational Problems in Abstract Algebra. Hungary: Perga­mon Press, 1970, pp.390.393.

4. R.E. Kalman, "Irreducible realizations and the degree of a matrix of rational functions". SIAM J. Appl. Math., Vol.13, No.2, June 1965, pp.520-544.

5. R.E. Kalman, P.L. Falb and M.A. Arbib, Topics in Mathematical System Theory. McGraw-Hill, 1969.

6. S. Barnett, Matrices in Control Theory with Application to Linear Programming. London: Van Nostrand Reinhold Company, 1971.

7. A.S. Householder, "Bigradients, and the problem of Routh and Hurwitz", SIAM Review, 10, 1968, pp.56-66.

108

8. J.M. Thmnas, Differential Systems. American Mathematical Society Colloquium Publications, Vo1.21. Baltimore, Md. : Waverly Press, 1937.

9. J.M. Thomas, Systems and Roots. Richmond, Va. The William Byrd Press, Inc., 1962, pp.2l-49.

10. Y.S. Ho and P.H. Roe, "Degree of polynomial ma­trix and explicit formula for the order of com­plexity of linear active networks". Proceedings, IEEE International Symposium on Circuits and Systems, San Fransisco, April 22-25, 1974.

11. W.A. Wolovich, "On determining the zeros of state-space systems", IEEE Trans. Aut. Control, Vol.AC-18, No.5, October, 1973, pp.542-544.

12. C.T. Chen, "Irreducibility of dynamic equation realizations of sets of differential equations", IEEE Trans. Aut. Control, Vo1.AC-13, 1970, p.13L

13. B. McMillan, "Introduction to formal realiz­ability theory", Bell System Tech. J., 31, 1952, pp.217-279,541-600.

14. F.R. Gantmakher, The Theory of Matrices and Its Applications, Vol.I & II. New York: Chelsea, 1959.

15. V. Belevitch, Classical Network Theory. San Fransisco : Holden-Day, 1968.

16. E.G. Gilbert, "Controllability and observability in multivariable control system.:", SIAM J. Con­trol, Ser.A, Vol.2, No.1, 1963, pp.128-151.

17. M. Newman, Matrix Representations of Groups. National Bureau of Standard Applied Mathematics, Ser.No.60. Washington, D.C. : U.S. Government Printing Office, July, 196B.

18. S.H. Wang and E.J. Davison, "A new invertibility criteria for linear multivariable systems", IEEE Trans. Aut. Control, Vo1.AC-IB, No.5, October, 1973, p.538.

19. M. Heymann, "The prime structure of linear dy­namic systems", SIAM J. Control, Vo1.l0, No.3, August, 1972, pp.460-469.

20. T. Ohtsuki and L.K. Cheung, "A matrix decompo­sition-reduction procedure for the pole-zero calculation of transfer functions", IEEE Trans. CT, Vol.CT-20, No.3, May, 1973, pp.262-271.

21. C.A. Desoer and J.D. Schulman, "Zeros and poles of matrix transfer functions and their dynamical interpretation", IEEE Trans. Circuits and Sys­tems, Vol.CAS-2I, No.1, January, 1974, pp.3-7.

22. Y.S. Ho, Detached Coefficients Representations and Degree Functor of a Polynomial Matrix with Application to Linear Systems, Ph.D. Thesis, Department of Systems Design, University of Waterloo, Ontario, Canada, March, 1974.

23. E.I. Jury, Inners and Stability of Dynamic Sys­tems. New York: John Wiley & Sons, 1974.

A REPRESENTATION OF THE IMPEDANCE FUNCTION

IN TERMS OF THE POLES AND ZEROS FOR TRANSMISSION LINES

F. Kato and M. Saito University of Tokyo

Tokyo, Japan

Abstract

An expression of the input impedance is derived in terms of the poles and zeros for a nonuniform lossless transmission line terminated in an inductance. From the expression, residues at the poles are determined and Gel'fand-Levitan's theory is shown to be applicable to realize a line with the given poles and zeros. Some relations among poles and zeros of the four-terminal parameters are also given.

1. INTRODUCTION

Consider a nonuniform lossless transmission

line of finite length terminated in an inductance.

When the poles and residues of the input impedance

are given, the synthesis problem is reduced to a

linear integral equation by means of Gel'fand­

Levitan's algorithm for the inverse Sturm-Liouville

problem [1,5,6]. Two sequences of poles corres­

ponding to two different terminating inductances

can be treated in a similar way [2,4]. On the

other hand, poles and zeros of the input impedance

may be sometimes specified in, e.g., the phase­

matching problem for optical circuits with symme-

trical structures. Although Marcenko's algorithm

[3,4] is available for this kind of problems, it

has a difficulty in that a nonlinear integral equa-

tion should be solved. It is the problem that is

investigated in this paper.

The organization of the paper is as follows.

Sections 2 and 3 are devoted to the preliminary

presentation of basic equations and some properties

of the solution, respectively. In Section 4, we

obtain the expression of the input impedance in

terms of the poles and zeros and the characteristic

impedance at the input terminal. In Section 5,

109

the residues of the impedance at the poles are ob­

tained from which we get the asymptotic behavior of

the residues. The result allows us to apply

Gel'fand-Levitan's theory to the problem. Finally,

in Section 6, some collateral formulas are derived

for the poles and zeros of the four-terminal para-

meters.

2. BASIC EQUATIONS

Let a loss less transmission line have nonuni­

form distributed inductance L(x»O and distributed

capacitance C(x»O, then the telegraphists' equa­

tions are given by dV(x2!:!)

dx -pL(x)I (x, p) (1)

dI(x1E) - pC (x)V(x, p) (2) dx

where p is the complex frequency variable, O<x< t

is the spatial variable, and I(x,p) and V(x,p) are

the current and voltage, respectively. The elimi­

nation of I(x,p) between (1) and (2) results in

_iJldV(x,E)}_p2C(x)V(x,P)=O. (3) dxlL dx

When output terminal is terminated in an inductance

Lt , boundary condition

V' (t,p) + Lit) V(t,p) = 0 t

(4)

is to be met. In addition, we impose the norma1i-

zing condition to V(x,p)

JL(t) (5) V{t,p) ~ 4 CU.) .

The input impedance is, in terms of the solution

V{x,p), expressed as Z () ~~~_ pL{O)V(O,p) in p I(O,p) V' (O,p) (6)

Suppose that the characteristic impedance

ZO(x) =JL(x)/C{x) (7)

is twice continuously differentiable. By the

Liouville substitution

~(x) '" \JL(X)C(X) dx (8)

A=~(l!,) .(9)

yO(O '" l/ZO(~) (10)

_......l-. d 2J'YOfIT q (0 - )10 (~) d~2 (11)

y(~,p) =JYO(~) V(~.p) (12) Eqs.(3)-(6) are transformed, respectively, to

y"(!;,p) -{ p2 +q(O}y(~,p) '" 0 (13)

where

y' (A,p) + H y(A,p) 0

y(A,p) =1 pZO(O)y(O,p)

Zin(P) "'-y'(O,p) -hy(O,p}

3. PROPERTIES OF THE SOLUTION

(14)

(IS)

(16)

(17) , (18)

Concerning the solution V(x,p) of (3)-(5),

V(O,p) and V'(O,p) are even entire functions of p

and can be written as w 2

V(O,p) C11T(1+~) k=O zk co 2

V'(O,p) =C2(p2+p~) TT(l+~)

(19)

(20) k=l P

where C1 and C2 are constants, and jz:and jpk are

a zero of V(O,p) and of V'(O,p), respectively.

Two sequences of non-negative numbers {zk} and {Pk}

separate each other and asymptotic formulas

Pk=ak+O(-k1). zk=a(k+.!.2)+O(-k1 ) (2) ) , 1 ~ (22

hold, where

a=rr/A. (23) For real p, we can write the solution y(~.p)

of Eqs.(13)-(15) as

Y(!;,p) = cosh p(A-I',;) + J; K(~, t) cosh p (A-t) dt (24)

110

where the kernel K(~,t) is independent of p.

4. THE INPUT IMPEDANCE

In this section, we derive an expression of

the input impedance in terms of the poles and zeros

for the transmission line. From (16) and (24), it

can be seen that

(25)

On the other hand, substitution of (19) and (20)

into (6) yields it ( 1 + p2/z~) k=:;:..O ___ _

(26) TT'" 2 2 •

( 1 + l' /Pk

) k=l

Evidently. {jpk} and {jzk} are poles and zeros,

respectively, of Zi (p). Making use of the formu-

la n TI{1+x2/( k+!)2} k=O 2

coth lTX =: co

Eq.(26) is rewritten as

the series

Thus

lim Zi (p) p-.oo n

1fX IT (1+x2/k2 ) k=l

(27)

(30)

(33)

(34) Consequently, the input impedance is, from (25),

(28) and (34), given by

(35)

where

2 2 2 • (36»)(37) p + a (k+ 1/2 )

The leading term ZO(O)coth pl'l in (35) coincides

with the input impedance of the uniform line with

the same electrical length 1'1, and the infinite

product represents the effects of the deviation of

poles and zeros from those of the corresponding

uniform line.

5. THE ASYMPTOTIC ESTIMATE OF RESIDUES

The residue of Zin(P) at the pole jPn: 2 2

h(n)= lim (p +Pn) Zin(P) P+jPn 2p

is calculated from (35) without difficulties.

The result is as follows.

(38)

1 Case 1: Pn=a(k+Z )' ak, for any non-negative

integer k. 2 2 2

( ) ZO(O) p - a n 2 2 h n = 2 n cotp 1'I/T<I>k(-P )TI'I'k(-P ).(39)

Pn n kfon n k n Case 2: Pn = an.

Z (0) 2 TT 2 h(n)= ~lir<l>k(-P) 'I'k(-Pn)·

Ll k+n n k (40)

Case 3: p = am. for non-negative integer m = n. n 2 2 2

(n)_ ZO(O) Pn- a n 2 2 h - 1'1 2 2 IT <l>k(-Pn)n'l'k(-P ). (41)

Pn- Pm k+m,n k n

Case 4: Pn=~(m~1~2), for non-negetive integer m. Z (0) 1I p - a n

h(n)=_O __ n (z2_ p2)JT<I> (_p2)lT'I' (_p2) 4 2 m n k+n k n k k n·

In any case, th~nresidue hen) is positive. (42)

Suppose that the poles {p } and the zeros n

{z } have the asymptotic behavior n

a O a l 1 Pn= an+n +3+ 0(4)

n n

1 bO b l 1 z =a(n+z ) +1+--1-3 +0(4)· n Il+-2 (n~) n

(43)

(44)

respectively, where aO' al , bO and bl are const­

ants. Applying similar computations as [2], we

obtain the asymptotic estimate of hen): 3

h(n)=Z (O)[.!+l:..t~A-tO+ l'Iao)l]+o(l:..) (45) o 1'1 2 2 21T 3 1 3

n 1T n where

00 00

A= }(p2_a2k2_2aa )_L:{z2_ a2(k+.!)2_2ab} ~k 0k=Ok 2 0

2 bO 1T2 2 7 +1T bo(ao-T) -Tao-zaaO. (46)

Equation (45) is compatible with the asymptotic re­

quirement imposed on the normalizing constants by

Gel'fand-Levitan's theory. Thus, there exists a

transmission line with a finite length whose input

impedance has the specified poles {jp } and {jz }. n n

Furthermore, ,the resulting q(~) is absolutely conti-

nuous [2].

6. FOUR-TERMINAL PARAMETERS

Let the output terminal electrically open. We

denote the poles and zeros of Zll(P) by {jPn} and

{jzn}' respectively, and the zeros of Z22(P) by

{jw}. In a similar manner as Section 4, we obtain n

the following expressions:

Z11 (p) = ZO(O) coth pl'l TT<I>k(p2) IT 'I'k(p2) k k

Z12(p) = 4Z0(O)ZO(I'I) cosech pl'l IT <l>k(p2) k

2 TT- 2 Z22(P) =ZO(l'I) cothPl'llT<I>k(P) 'I'k(P) k k

where 2 2

(47)

(48)

(49)

- 2 P +wk 'I' k (p ) = 2 2 1 2 (50)

p+a(k+Z ) From (48), the residue of Z12(P) at the pole jPn is

given by r:c--7"""::-J",,, 2 2 2

( ) JZO(O)ZO(I'I) p - a n 2 hl~ 2 n cosec p 1'1 TT <l>k(-P ) (51)

Pn n k+n n provided Pn = ak, for any non-negative integer k.

Here, we notice the relation zll (p)

zI2 (p) -=:--=---=-~ IT (l+p2/z~) . k

Invoking Eq.(39), it can be seen

2 2 2 2

(52)

( ) Zo(O) p -a n TT 2:TT zk h n ____ n __ cosec P (0, 4>k(-Pn) I I 2 I 2· (53)

12 2 Pn n k ka(k+l)

Comparison between (51) and (53) yields 2

ZO(l'I) = TT zk

ZO(O) k a 2( k+.!)2 Analogously, making use of 2

Z22(P) Z (p) =-=-=-~

12 rr(l+p2/w~) k

2 1 2 ZO(I'I) = na (k+ Z)

we obtain

(54)

(55)

(56) ZO(O) k w~

The residue of Zll(P) at the pole jPn is given by

III

(39) • On the other hand, it follows from 2 2

l+p /zk Zl1(p) =Z22(P)TI 2 2

k l+p /wk

in all cases cited in Section 4. The formulas

(57)

correspond to the fact that four-terminal characte­

ristics of a transmission line is completely deter­

mined by Zll(P) alone and that fwn} can not be

independent of {p } and {z }. n n

7. CONCLUSION

For a nonuniform lossless transmission line

terminated in an inductance, the input impedance

has been explicitly determined by the poles, zeros

and the characteristic impedance at the input ter­

minal. The residue at each pole can be determined

from the expression of the impedance. It has been

shown that the sequence of residues meets the asym­

ptotic requirement for the normalizing constants of

the Sturm-Liouville equation. The result of

Gel'fand and Levitan for the inverse Sturm-Liouville

operators can be applied to arrive at a nonuniform

line with the specified poles and zeros. Expre-

ssions for four-terminal network parameters Zll,Z12

and Z22 have also been obtained in a similar manner.

Corresponding to the fact that only one of Zll and

Z22 can arbitrarily be specified, a series of rela­

tions among poles and zeros of Zll~ and zeros of

Z22 has been derived.

REFERENCES

[1] LM.Gel 'fand and B.M.Levitan, "On the Determina­

tion of a Differential Equation from its Spec­

tral Function," Amer.Math.Soc.Transl., Ser.2,

Vol.l, pp.253-304(1955).

[2] B.M.Levitan and M.G.Gasymov,"Determination of a

Differential Equation by Two of its Spectra,"

Usp.Mat.Nauk., pp.1-63(1964).

112

[3] V.A.Marcenko,"Some Questions in the Theory of

One-Dimensional Linear Differential Operators

of the Second Order, I," Amer.Math.Soc.Transl.,

Ser.2, Vol.101, pp.l-104(1973).

[4J M.R.Wohlers,"A Realizability Theory for Smooth

Lossless Transmission Lines," IEEE Trans. on

Circuit Theory, Vol.CT-13, No.4, pp.356-363,

Dec. 1966.

[5] M.R.Wohlers,"A Realizability Theory for Smooth

Lossless Transmission Lines - Part II," IEEE

Trans. on Circuit Theory, Vol.CT-14, pp.442-444,

Dec. 1967.

[6] K. Horiuchi , K.Kawakita and H.Watanabe,"A Note

on the Synthesis of Terminated Nonuniform Trans­

mission Lines," CT Res. Group, lnst.Elect.Comm.

Eng.Jap., No.CT68-l6, July 1968.

BIOGRAPHIES

Fumio Kato was born in Fukushima, Japan, on Novem-

ber 13, 1950. He received the B.S. degree from

Yokohama National University, Yokohama, Japan, in

1973, and the M.S. degree from University of Tokyo,

Tokyo, Japan, in 1975, both in electrical engineer­

ing.

At present, he is studying toword the Ph.D.

degree in the Graduate School of University of

Tokyo, specializing distributed-constant networks.

Mr.Kato is a member of the Institute of Elec­

tronics and Communication Engineers of Japan.

Masao Saito (M 1962) was born in 1933. He obtain-

ed B. of Engng. in 1956, M. of Engng. in 1958 and

D. of Engng. in 1962 from University of Tokyo.

Working for some time in Faculty of Engineering in

University of Tokyo, he is now serving as a profes­

sor of medical engineering in Faculty of Medicine

in the same university.

His principal interest is in circuit and sys­

tems theory, especially its application to biologi­

cal and medical systems.

He is now a member of the board of Japan Soci­

ety of Medical Electronics and Biological Engineer­

ing, Vice-President of the International Federation

for Medical and Biological Engineering, Vice-Presi­

dent of the World Association of Medical Inform­

atics.

EVALUATION OF CONSTITUENT MATRICES

OF AN ANALYTIC MATRIX FUNCTION

Feng-cheng Chang Surender Pulusani

Alabama A&M University Normal. Alabama

ABSTRACT

An analytic matrix function of a given arbitrary square matrix is expressed in terms of constituent matrices when all eigenvalues with multicities are known. A simple and practicable approach for computing the constituent matrices are then formulated which requires only straightforward matrix multiplications.

In the theory of linear time-invariant systems,

solutions are often found to be expressible by

analytic function of a matrix A. such as eAt

and sin (A%t). If A is an n)( n arbitrary

constant matrix and f(s) is an analytic func­

tion of complex variable s. then the analytic

matrix function f(A) of A is given by the

fundamental formula [1] [2] [3]

The calculation of constituent matrices is gener­

ally very involved. especially for a large n.

However. if all the eigenvalues are known before-.

hand, a simple and practicable method for comput­

ing Zkh may be developed as follows.

f{A) (1)

where eigenvalues sk with multiplicities rk are obtained from the characteristic polynomial

of A.

c(s) - det (sI - A)

~ sn-p ~ cp

m-l

II k-O

and constituent matrices Zkh' depenent on A

but not on f. are to be determined.

(2)

Let

and

m-l

{s - s {·rr (8 k k'-O

kr,.tt

k .. O ..... m-l; .t - o ..... rk-l.

k-O ..... m-l; t.-O ..... rk-l.

(3)

(4)

By direct differentiation of (3) we obtain the fol­

lowing:

113

V.' ;;'f, ',rifl11

'lll (,' "

, ~ I

, h

k' * k, h' 0, " . , rk,l k' k, h' 0, ... , t -1

(5)

k' = k. h' = Q, , rk-l

where

1

j

(6)

are the coefficients of Taylor series expansion of

at

c(s) ckO(s) r

k (s - sk) n-rk

j

1

{ dkO

i 0

bki i-I d (10)

-L: ki-p

bkp ' i 1, 2,

p=O dkO

are found to be the coefficients of Taylor series expansion of l/ckO(s) at s = sk'

1

L (11) i=O

Since the matrices Cki

can be directly computed

from (4) and the scalars. bki can be easily eva­

luated by (10) and (6) or some other methods [2]

[3] when all the eigenvalues are known, Eq. (9)

seems to be a very convenient formula for comput­

ing constituent matrices of a given matrix with

mUltiple eigenvalues. It does require some ma-

2: dkj (s - sk) .

j=O

If we set successively in (1), f(s) = ck(s),

~ = 0, 1, .. , , rk-l, for every k, then from (5) we obtain

(7) trix multiplications, however, it does not need

any matrix inversions as done by some other ap­

proaches [4] [5] [6]

(8)

k = 0, '" , m-l; ~ = 0, ... , rk-l.

The constituent matrices Zkh are therefore found by solving the rk simultaneous linear algebric equations (8) for every k

(9)

k 0, .••• m-l;

where

114

If we substitute (4) into (9), we find

m-l

-IT k'= 0

k' * k

k = 0, ••• , m-l;

(12)

which is also another useful and direct formula

for computing constituent matrices [7].

A complete computer program for finding the con­

stituent matrices of a given arbitrary constant

matrix will be developed in the future contribu­tion.

REFERENCES

1. F. R. Gantmacer, The Theory of Matrices, ___ Vol. I, New York: Chelsea, 1959.

2. J. S. Frame, "Matrix Functions and Applica­tions --- Part IV: Matrix functions and constitu­ent matrices," IEEE Spectrum, vol. I, pp. 123-131, June 1964.

3. F. C. Chang and H. Mott, "On the matrix re­lated to the partial fraction expansion of a proper rational function," Proc. IEEE, vol. 62, pp. 1162-1163, Aug. 1974.

4. c. F. Chen and R. E. Yates, "A new approach to matrix Heaviside expansion," Int. J. Control, vol. 11, pp. 431-448, Mar 1970.

5. F. C. Chang, "Evaluation of an analytical function of a companion matrix with multiple ei­genvalues," Proc. IEEE, vol. 63, pp. 818-820, May 1975. 6. F. C. Chang, "Evaluation of an analytical function of an arbitrary matrix with multiple eigenvalues," Proc. IEEE, vol. 65, (accepted to be published).

7. F. C. Chang, "A direct approach to the con­stituent matrices of an arbitrary matrix with multiple eigenvalues," Proc. IEEE, (accepted to be published).

BIOGRAPHIES

Feng-cheng Chang

Assistant Professor, Alabama A&H University Ph. D., University of Alabama, 1972.

Surender Pulusani

Assistant Professor, Alabama A&M University.

115

ON THE LOSSLESS SCATTERING MATRIX

SYNTHESIS VIA STATE-SPACE TECHNIQUES

Andrzej L. Dobrucki and Marian S. Piekarski

Institute of Telecommunication and Acoustics WrocXaw Technical Uni­versity, Wroclaw, Poland

Abstract

In the paper a new direct algebraic method for a loss less scatter­ing matrix synthesis is developed. The method rest mainly on the minimum state-variable realization theory, but it is more efficient from a computational point of view than other proposed methods. The synthesis algorithm relies on a direct calculation one of the possi­ble scattering matrices describing 10ssless nondynamic part of rea­lized network.

1. INTRODUCTION

LNN Sa

L-______ ~----____ ~ f---<> )1 S(p)

Fig. 1. Synthesis model of S (p)

2. PRELIMINARIES

1')+1

0+1<:.

1

H 1

H

Recently the state-space approach to the

network synthesis has been considered qui­

te extensively in scientific literature

/see list of references in the book 2/. In

the field of integrated circuits, the me­

thod has some significant app1ications,be­

cause fundamental requirements of integra­

ted technology are satisfied4 . In the pa­

per3 ,4,5 a new direct procedure for 10ss­

less admittance matrix synthesis has been

proposed, which is more efficient from a

computational point of vie than other me­

thods. In this paper, by revising and ex­

tending of the above procedure, a synthe­

sis method of lumped linear finite statio­

nary 10ssless /LLFSL/ n-port networks de­

scribed by a scattering matrix S (p) is

presented. The method leads to a model

consisting of a 10ssless nondynamic net­

work /LNN/ with /n+k+1/ terminals and

It is well known 1 ,2,6 that the necessary

and sufficient conditions for the realiza­

tion of the nxn scattering matrix S(p) by

a LLFSL n-port network are

/n+k/ portS,which is terminated by groun­

ded capacitors at its last k terminals as

shown in Fig. 1. The number of used capa­

citor is minimal and equal to the McMil­

lan's degree 2 of the scattering matrix S(p)

116

a. s(p) is rational an real for real p,

b. S (p) is analytic in Re p>O,

c. s(p) st(_p) = In. (1)

An nxn matrix. S(p) possesing the above pro­

perties is said to be bounded real regular

and para-unitary and can be written in the

t r-i 1_0Bi P

form

. g(p) (2 a)

where the Bi are real constant matrices

and the least common denominator of the

entries in S (p)

g(p) =

is Hurwitzian6 .

The scattering matrix s(p) seen at the

first n ports of the networks shown in

Fig. 1 is given by

S(p)

where Sa' the scattering matrix of the LN

coupling network is given by

n k

(4)

The S(p) may be also regarded as a funct­

ion of the new complex variable s:

(5)

According to a well-known result2 , the ma­

trix (1) has a minimal realization IA,B,C,

01 and can be expanded in the form

where k = b [s (p)] is the MCMillan's de­

gree of S (p) .

Comparison (5) and (6) imply the following

relation

Now, the loss less condition of S (s S t = )

a a a = In+k can be expressed in the following

equivalent conditions

A + At = -ctc, (8 a)

B -cto, (sb)

~Ot = 1 . (8 c) n

Hence the problem is to chose a minimal rea­

lisation /A,B,C,O/ to satisfy equation (6) and at the same time guarantee that equa­

tion (7) describes the scattering matrix of

the LNN.

3. SYNTHESIS ALGORITHM

The scattering matrix S (~ given by (2) can be expanded in the neighborhood of

p = c:c as

S (p) = S_l + (9)

By equating the right sides of (2 a) and (9)

r CIJ L r-i g (p) (S_l S.

). (10) Bip = +L ~

1=0 1=0 pi+l

Equating coefficients of like powers of p

on both sides of (10) ,

B. ]

o, ... ,r (lla)

117

and

r

L: bkS i - k - l for i ~ r+l. k=O

(ll~

The equations (ll~ give a simple recurrent

method to calculating of S .• ~

Substi tuting (9) to (2) and identyfying co-

efficients of like powers of PI

(12a)

Thus the real rational matrix given by (2)

is the scattering matrix of a LLF5L n­

-port if and only if (12) is satisfied

and g (p) is Hurwitzian. Expanding the right side of equation (6)

in the neighborhood of p =00 as

cI) •

""' CA 1.B 8 (p) = D + L.. i + 1

i=OP

and equating with (9), we identify

(13)

Equation (14a) uniquely defines the ma­

trix D satysfying the lossless condition

(8c), but the equations Cl4b) do not have

a unique solution for A,BtC satysfying

(8a) and C8b). For the purpose of choos­

ing a proper matrices A, Band C, it is

convenient to define the block matrix

where

H e

is a Henkel

t -s -1

On S

Xe

On °

= H X e e

block

-8 t 0

t -1

n

matrix and

-5 t e-l

S t

e-2

(-1) e+1S_1

is a triangular block matrix.

(17)

t

T blocks of matrix Te may be written as ij

t;j ( )k+j-l t T .. = -1 Si+j-k-lSk-2 .

J.] =1 (18)

118

Combining (8), (14b) and (18),

Then

(20)

where

Equations (20) and (21) suggest that a way

of obtaining a pair A, C would be to form

the matrix Tel factor it in the form of

equation (20), and then try to identify A

and C from these factors. We do not know

in advance if the matrix Te formed from

the expansion coefficients of S(p) about

p (:lO can always be factored as indicated

in equation (20) ; hence we first study the

properties of Te' to see if it can be fac­

tored in the desired form. Equation ~2a)

implies that the matrix S_l

Hence the matrix Xe is also

is nonsingular.

nonsingular.

::::nih:h::n:O:ee=~r::~ He' It is well

Hence

Using

Tij

rank Hr _l '" k "" b[ s(p)). (22)

rank Te "" rank Tr - 1 = k. (12b) and (18) it can be t. T T t

Tji ' ~.e. r-l r-l'

shown that

The generalized companion matrix f2 has the form2 ;

n -b 1 ].

,1 n

The characteristic polynomial of f2 is gi­

ven by det(lnrs - n.) [g(s)]n, i.e. all eigenvalues of !1 lie in the left half pla­

ne (Res.<O). ~

Equations (23), (IS) and (12b) imply the

following relation

(24)

where

(25)

It suggests that Tr _l

is the solution of

(24). Since Res i <: 0, the equation (24)

has the unique solution2 , which is symme­

tric and positive semidefinite matrix.

Therefore Tr - l is symmetric positive se­

midefinite matrix of the rank k. Such a

matrix can always be factored in the

form 7

T = MMt r-l

where M is nrxk real matrix. M can be

partitioned into nxk blocks Mi

t [ t t t ] M = Mo ,M l , ... ,Mr

_l

. (27)

Now by comparison of (27) with (21) we can immediately identify a suitable C as

C M (2S) 0

and

P r-l = M. (29)

Then (Sb) gives

B -CtD t -Mo S_1" (30)

To find a suitable A, if we define Td as

(31)

from (15), (IS) , (19), (20) ,(21) and (23) we

see that A must satisfy

119

(32)

The equation (32) has the unique solution

given by

(33)

where M+ is the Moore-Pemrose pseudoinver­

se matrix to M7.

It is easy to prove that the matrix A gi­

ven by (3~ satisfies the loss less condi­

tion (Sa). From (27), (2S) and (30) it -fol­

lows that (14b) is satisfied for i = 0, 1,

, ... ,r-l. To see that (14b) is also satis­

fied for i ~r we can calculate

Then

r b.Ar - i L 1

1-0

r

L b.Ar - l 1

1=1

and hence (12b) gives

CArB

for i = r, r+l, ...

Concluding, if the minimal realisation lA,

B,C,DI is calculated from (2S) , (30) and

(3~ then the lossless conditions (S) are

satisfied and Sa given by (~ is the scat­

tering matrix of LNN.

4. CONCLUSIONS

The synthesis algorithm presented in this

paper relies on a direct calculation of a

minimal realisation /A,B,C,D/ of the sca­

ttering matrix S(p), where /A,B,C,D/ sa­

tisfies the lossless conditions. If the

loss less minimal realisation of s(p) is

found, it is very simple to calculate one

of the possible scattering matrices Sa de­

scribing LNN.

In the method presented here, in contrast

to methods proposed by Youla and Tissil

and also by Anderson and vongpanitlerd2 ,

it is not necessary to solve any matrix

equation. A factorization of a real con­

stant symmetric positive semidefinite ma­

trix Tr - l into a form MMt is only needed.

5. REFERENCES

1. D. C, Youla, P. Tissi, N-Port Synthe­

sis Via Reactance Extraction - Part I,

IEEE Intern. Conv. Rec., 1966, pp.

183-208.

2. B. D. Anderson, S. Vongpanitlerd, Net­

work Analysis and Synthesis - A Mo­

dern Systems Theory Approach, Prentice­

Hall, New York 1973.

120

3. M. S. Piekarski, Synthesis of Lossless

N-Port Networks with Reference to In­

tegrated Circuits lin Polish/, I Natio­

nal Symposium of Radio Science /URSI/,

Warsaw, Poland, 17-18 February 1975,

pp. 141-143.

4. M. S. Piekarski, Selected Synthesis Pro­

blems of linear Microelectronic Circu­

its lin Polish/, Pro Nauk. Inst. Tele­

kom. i Akustyki, Politechnika Wroclawska,

Nr 24, Seria: Monografie Nr 9, Wroclaw

1976.

5. M. S. Piekarski, Lossless Integrated

N-Port Synthesis Via State-Space Tech­

niques lin Polish/, Archiwum Elektro­

techniki, t. XXV, z. 2, 1976, pp. 343-

357.

6. R. W. Newcomb, Linear Multiport Syn­

thesis, McGraw-Hill, New York 1966.

7. F. R. Gantmacher, The Theory of Ma­

trices, Nauka, Moscow 1966 lin Rus­

sian edition/.

Q

ALGEBRAIC CHARACTERIZATION OF MATRICES WHOSE

MULTIVARIABLE CHARACTERISTIC POLYNOMIAL IS HURWITZ IAN

Marian S. Piekarski

Institute of Telecommunication and Acoustics, Wroclaw Technical University, poland

Abstract

Suppose g(Al

, A2

, ••. Ar} = det(A n - A } is multivariable charac­nr

teristic polynomial of an arbitrar§ nr x nr complex matrix An ,whe-

re A All + + ... + A I is an nr x nr diagona1 nr ml r mr

matrix with diagonal complex variables Al' In the paper the necessary and sufficient conditions are given for

the matrix Anr

to have Hurwitz multivariable characteristic polyno-

mial, i.e., to have g( Al

, A2 , ••• , A r} ~ 0 in the polydomain

Re Ai Q 0 (1 " i ~ r) . This is generalization of well-known Theorem given by Lyapunov to

the multivariable case.

1. INTRODUCTION

In the paper the following symbols will

be used:

W '7 0, W~O, W<..O, W~O - positive, nonne­

gative, nonpositive definite matrix, res­

pectively,

1m - identity matrix of order m,

+ - direct sum of matrices. A i' Ai - complex and complex conjugate

variable, respectively,

ReA i - real part of Ai' g( A

l, A 2' ••• , Ar} - a polynomial g in

>'1' A2 ,···, A r'

An - n x n complex matrix,

t -A , A, A* - transpose, complex conjugate

and complex conjugate trans­

pose of A, respectively,

det A - determinant of A,

x = [Xl,x2 , ... ,xn ]t - a column vector,

diag [al,a

2, ••• ,ak] - a diagonal matrix

with diagonal elements

W* m

a l , a 2 ,··· ,ak ,

Wm - m x m Hermitian matrix,

+

. .+ A rlm

r be a diagonal matrix with diago-

121

nal complex variables Ai (i = 1,2, ... ,r),

where ml + m2 + ... + m. + . .. + m ~ r

= nr (1' mi

( nr

, r (. n r ). Let A be an ar­nr

bitrary n x n complex matrix and consi­r r

der the multivariable characteristic poly-

nomial

In the case ml takes the form

(1 )

n this polynomial

det( An 1

det(A1

I -A ) n n

(2)

and its properties have been studied for

many years, especialy, with regard to the

stability problem of differential equat­

ions in normal form.

It is well-known that in a study of the

stability problem the following idea of

Hurwitz polynomials is important l ,2

Definition 1

A polynomial g(A 1 ) of one independent com­

plex variable is called a Hurwitz polyno­

mial if g(A1

) 1 0 in the closed domain

ReAl ~ 0.

The important algebraic characterization

of matrices whose one variable characte­

ristic polynomial is Hurwitzian has been

given by the following Theorem of Lyapu­

nov 3 :

Theorem 1 (Theorem of Lyapunov)

Let An be an n x n complex matrix. Then

the characteristic polynomial g(A l ) = = det(A I - A ) is Hurwitzian if and an­Inn ly if, for any positive definite n x n

Hermitian matrix K , there exists a posi­n tive n x n Hermitian matrix Wn which sa-

tisfies the matrix equation

W A + A* W n n n n -K

n (3)

As the name implies, this is originally

due to Lyapunov, but we shall use a sligh­

tly modified version that is more conve­

nient in further consideration than the

original Theorem.

Theorem 2

Let An be an n x n complex matrix. Then

the characteristic polynomial g"( A 1) = = det(A 1I - A ) is Hurwitzian if and an­n n ly if there exists a positive definite

n x n Hermitian matrix Wn such that

W A + A* W < 0, n n n n

(4)

122

A proof of the above Theorem follows direc­

tly from the Theorem 1 and will be omitted

here.

Recently, multivariab1e rational functions

and matrices have been finding increasing

applications in analysiS, synthesis and

stability problems of networks and systems . 1,2,3,4,5,7 I (see list of references 1n . n

a study of the above problems the follo­

wing idea of mu1tivariable Hurwitz poly­

nomial has important application.

Definition 2

A polynomial g(A 1 , A2 , ... , Ar) of r inde­

pendent complex variables is called a Hur­

witz polynomial in r variables or to be

Hurwitzian, if g(A l , A2 , ... , Ar) 10 in

the closed polydomain ReA i 1 ° (1~ i~ r).

This paper is devoted to the establish­

ment of the algebraic characterization of

matrices whose multivariable characteris­

tic polynomial is Hurwitzian.

2. PRINCIPAL RESULTS

Let A be an arbitrary n x n complex ma­n trix. Initially, we consider the multiva-

riable characteristic polynomial

(5)

where An = diag [AI' A2 ,···, An1' The above polynomial is a special case of

the polynomial given by (1) when r = nr =

= n, i.e., when ml = m2 = ... = mn = 1.

In order to establish a basis for charac­

terization of the general case, and for

simplicity, we shall prove the following

Theorem in terms of the above special case:

Theorem 3

The multivariable characteristic polyno­

mial g(A l , A2 ,.·., An) = det( An - An) is

Hurwitzian if and only if there exists

a positive definite diagonal matrix

Wn = diag [wl , w2 ' '" ,wn], (wi) 0;

i 1,2, ... ,n) such that

(6)

Proof:

Sufficiency. Let us suppose that there

exists a positive definite diagonal ma­

trix Wn = diag [wl ,w2 ' ••• , wn], (Wi '> o~ i = 1,2, ... ,n), such that Tn = WnAn +

+ A*nWn< 0, Le., X*TnX(O for any complex

n-vector x t 0, and det(An - An) is not

Hurwitzian.

By Definition 2 there exist n complex num­

bers Al , A2 , ... , An with nonnegative

real parts (~i + Ai)O~ i = 1,2, ... ,n),

so that the linear system

° (7)

possesses a nontrivial solution xo' From

(7) we have

a contradiction. Thus the condition of

the Theorem is sufficient.

Necessity. For n = 1 this is obvious. For

arbitrary n we proceed by induction, that

is to say we assume the result to be true

for n = k - 1 and show that this implies

its truth for n = k.

Partitioning Ak and A k as follows:

A k

[--~'!.~-~£-J k-l Ac I ad 1

I

k-l 1

k-l 1

(8)

123

it is easily seen that the polynomial det

(Ak - Ak ) can be factored in the following

forms:

1. if det( Ak _ l - Aa) t 0, then

Ab]det(Ak _ l - AaJ , (9a)

2. if (~k - ad) t 0, then

(A k - ad)det(Ak _ l - Aa

AbAc ) . (9b)

Let the polynomial det(Ak

- Ak

) be Hurwi­

tzian. Then, according to the Definition 2

and (9a), for each A k-l such that

det(A k-l - Aa) to and ReAi~ ° (l(,i~ k-l),

the equation

° (10)

is satisfied only for Ak such that ReAk<O.

In particular, when A i 00 (1 ~ i ~ k-l) ,

(10) gives Ak - ad ° for Ak such that

ReAk (0, Le.,

in the domain ReAk ~O. Furthermore, (9b)

and (11) imply

in the polydoma.:ln ReAi') ° (l 'i~ k). Hence,

the matrix

A + a

in the domain ReA k ) 0, and the matrix

Cl2a)

(12b)

generate characteristic polynomials which are hurwitzian.

If Dk_l = diag [dl'd2 ,··. ,dk_l] matrix, then

is Hermitian matrix.

is real

(13 )

Hk- l , as a Hermitian matrix, has only real eigenvalues 3 , It is also clear from (12a)

and (13) that for a fixed Ak the maximal

eigenvalue Pm of Hk- l and Hk_ l and Hk

-l

itself are functions of Ak and Dk- l •

Therefore, for each Dk_l , there exists

Ak = Ako = h(Dk_ l ), (ReAko ) 0), such that

Pm reaches a maximal value Prnrn in the do­main ReAk ) O. Hence,

in the domain ReAk'q O.

f(hCDk

_l

) ,Dk

_l

)

{l4}

Since Bk- l generates characteristic poly­

nomial which is Hurwitzian, by the induc­

tion hypothesis, there exists a matrix

-Dk _ 1 == Wk - 1 ) 0 such that

(11) implies

ad + ad < O.

(15)

(16)

According to a well-known Theorem that a

Hermitian matrix is negative definite if

and only if its maximal eigenvalue is ne­gative3 , (14) and (15) imply

and

124

in the domain Re ~k q O.

Summarizing, it is shown that there exists

a constant diagonal matrix Wk- l > 0 such that

A* A* c b)W (0

- - k-l Ak - ad

for any Ak in the domain ReAk

) O.

Putting

Ta Wk_lAa + A*aWk_l'

t2 = A*c'

it is seen that (17) takes the form

t2t*1 + ----"'-_- < 0

Ak - ad

in the domain ReAk

) O.

(17 )

( ISa)

(ISb)

(lSc)

(19)

In particular, when Ak - 00 then Bk

_l

= Aa' Hk_1 Ta and therefore

Ta (0. (20)

Let

Wk· [-~~=l-t--i;] :-' k-l I

where T is given by (18a) and a

anci

k-l

1

The object of course is to prove that the­

re exists dk = wk " a such that Tk < a. Be­

cuase T a < a, it is only necessary3 to es­

tablish that det Tk and det Ta have the

oposite signs at dk = wk ) a.

Since

det T = det[-:.:'l-l::~--J= k T* IT

bl d

= det [~~-L~::-~~~~~~~~-J T* IT -T* (T IT)

bl d b a b

= det [~~_l ____ ~_-----J T* I T T* -IT

b I d- bT b

(Td - T* T- l b a .

I a

it is only neccessary to establish that -1

Td - T*bTa Tb < a or equivalently that

(21)

at dk

= wk

.., a. Let us consider four following cases:

Case 1. tl = a

Then for any dk w

k from the interval

ad + ad

t* T -It 2 a 2

inequality (21) is satisfied.

Case 2. t2 a

Then for any dk

= wk from the interval

t* T -It 1 a 1 ( w

k <.00 a (

inequality (21) is satisfied.

Case 3. tl = t2 a.

This is a trivial case in which inequali­

ty (21) is satisfied for any dk= wk ) a.

Case 4. tl ~ a and t2 ~ a.

Let x be any complex (k-l) vector. Then,

from (19) it follows that

P x*Hk

_l

x x*T x + x*t l t*2x

+ a Ak - ad

+ x*t2t*lx

< a (22) A -k ad

for any x ~ a and Ak in the domain ReAk~a.

It is easy to show that, at a fixed x ~ a,p

reaches a maximal value Pmax in the domain ~--'= _

ReAk )/ a for

Re .:id1m y ) ,

Re y + IYI (23a)

where

y=x*tlt*2x • (23b)

125

Substituting (23) into (22) and using the

identity 2 (Rely) 2 = Re y + \y\' one can

obtain

_, i 2 (Re lX*tlt*2x)

P x *T x - 4 -----=---=::..-.-- (a (24 ) max a

ad + ad

for any x ~ a.

Because x*TaX < a for any x ~ a and ad +

+ ad < a (inequalities (16) and (20)), (24}

gives

for any x ~ a.

Let

x = T -let + bt2),

a 1

where

b t* T -It

1 a 1

t* T- l t2 2 a

(25)

(26)

'7 a , (27)

Using (26) and (27), (25) may be transfor­

med into the form (t* +t* b)T -let + t 2b) _-=:.1_.-:2=-----=a:..-.--.::.l--'~ < a •

b (28)

Multiplying both sides of (28) by b, (28)

takes the final form

(ad + ad)b - (t*l + t*2b )Ta - l (t l + t 2b)(0

(29)

Now, it is very easy to observe that

(21) and (29) are identical for dk = wk:

= b. It means that det Tk and det Ta have the opposite signs for wk ~ b. The proof

is now completed by induction from n ~ 1,

Q.E.D.

Now, we consider the general case of the

multivariable characteristic polynomial

r n L m. and r ~ n .

r i"1 1. r In this case, the following

- A ) n ' r

Theorem gi-

ves the algebraic characterization of ma­

trices whose multivariable characteristic

polynomial is Hurwitzian:

Theorem 4

The multivariable characteristic polyno-

mial g( Al , A2 , ... , A) :: det( A -A ) r nr nr

is Hurwitzian if and only if there

exists a positive definite Hermitian ma-

Wm + W .;. .. '+Wmr

(W . 1 m2 mi

W* ) OJ i = 1,2, .•. ,r) such that mi

(30)

A proof of the above Theorem follows from

the Theorems 2 and 3, but the considera­

tions are rather long and will not be

presented here6 •

3. REFERENCES

1. N. K. Bose, E. I. Jury, Positivity

and stability tests for multidimen­

sional filters (discrete-continuous),

126

IEEE Trans. on Acoustics, Speech,

and Signal, ASSP-22, 1974, pp. 174-180

2. N. K. Bose, R. W. Newcomb, Tellegen's

theorem and multivariable realizabili­

ty theory, Int. J. Electronics, 36,

1974, pp. 417-425

3. F. R. Gantmacher, The Theory of Matri­

ces, Nauka, Moscow, 1966 (in Russian

edition)

4. T. Koga, Synthesis of finite passive

n-ports with prescribed positive real

matrices of several variables, IEEE,

Trans. on Circuit Theory, CT-15, 1968,

pp. 2-22

5. M. S. Piekarski, Absolute stability of

linear n-ports, Proc. 1975 IEEE Inter­

national Symp. on Circuits and Systems

Boston, April 21-23, 1975, pp. 112-

115

6. M. S. Piekarski, Algebraic characteri­

zation of matrices whose multivariable

characteristic polynomial is Hurwi­

tzian(to be published)

7. H. L.Van Trees, Synthesis of Optimum

Non-linear Control Systems, The M.l.T.

Press, Cambridge, Massachussetts,1962

CONTRACTION OPERATORS OF CLASS C o

AND THE STRUCTURE OF

A CLASS OF INFINITE DIMENSIONAL SYSTEMS

David J. Hedberg Hughes Aircraft Company

and

N. Levan Department of System Science

University of California, Los Angeles

SUMMARY

In this paper we identify a class of infinite dimensional linear sys­tems which behave in a sense like finite dimensional ones. Specifi­cally, we consider two types; 1) discrete time systems for which the state operator A is a C contraction on a Hilbert space, and 2) "com­patible" continuous timg systems for which A is the infinitessimal generator of a contraction semigroup on a Hilbert space and whose co­generator is a C contraction. This class of systems includes finite dimensional systgms as a particular case. It is shown that control­lability and observabi1ity for these systems can be characterized in terms of the "cyclic vectors" of the decomposition of a Co contraction.

The task of finding the multiplicity and the cyclic vectors for a given infinite dimensional contraction operator is generally diffi­cult. We propose in this paper alternate criteria for contro11abi1ty and observabi1ity involving known "cyclic subspaces". The characteri­zation of these subspaces is a result of the Nagy-Poias theory of contraction operations. sufficient conditions for controllability and observabi1ity of the above systems are thus obtained.

Finally, we cite physical examples illustrating some aspects of this class of (nearly finite dimensional) systems.

127

J !

'I,

DISCRETE-TIME SYSTEM OPERATORS

ON RESOLUTION SETS OF SEQUENCES

R. J. Leake and B. Swaminathan Department of Electrical Engineering

University of Notre Dame Notre Dame, Indiana

Abstract

This paper introduces the concept of a resolution set of sequences as a natural consequence of the Hilbert Space resolution theory of Saeks [1] and others.

1. INTRODUCTION

Much of the theoretical work being carried out on the study of discrete-time systems adheres to the consideration of l spaces which seem to have

p all the "right" properties one needs to carry out system analysis. It is felt, however, that occa­sions often arise in which some special non-l

p class of sequences are of particular interest, and that the "right" properties should be carefully de­fined. In this paper we apply certain resolution space [I} ideas to sequences and give an outline of how these ideas can be carried forth in the study of discrete-time operators and systems. It is hoped that the reader will bear with us on the terminology. We should use "sequence" as a prefix to the term resolution but drop it for simplicity.

2. SHIFT INVARIANT RESOLUTION SETS

Let s denote the linear space of sequences of complex numbers with typical elements denoted by

(1)

We deal here with the truncated subspace s+ of consisting of sequences which are zero in value for negative arguments with typical elements de­noted by

x = ( ..• ,0,0,xO,xl

,x2 ' ... ) (2)

or, presuming that (2) is understood, by x = (xo'xl ,x2'···) (3)

s

128

Introducing the delay shift operator D and

the truncation operators PN

and pN for N = 0,1,2, ...

by

Dx =(0,xO,xl ,x2 ' .•. )

PNx =(0, ..• ,0,xN,xN+l ' ... ) (4)

+ +. we say that a subset of sequences a C:s ~s

(i) Shift invariant iff DNa+ C a + all N=0,1,2, ...

(ii) pN invariant iff N + +

PaC a all N=0,1,2, ...

(iii)PN

invariant iff PNa +

Ca + all N=0,1,2, •..

It is easy to construct examples to show that these conditions are generally unrelated in that any com­bination of the three conditions may hold without the others being true. Motivated+by [1] we define a resolution set as a subset of s which is both

N P and PN

invariant. Notice that

N PNx + P x = x

or (5)

I

and the operators thus form a resolution of the + identity on a .

The starting time of a subset 'a+ C s+ is the

least+integer K such that YK fO for some sequence YEa .

Notice that if a+ is a resolution set with starting time K then the "impulse" sequence

a(K) = (0, .•. ,0,1,0, ... ) (6)

Kth

is in a+ since if y E a+, YK

fO then

(7)

A shift invariant resolution set is one for which condition (i), (ii) and (iii) hold simultan­eously.

A resolution space is a resolution set a+ which is a complex subspace of the vector space s+. That is, for complex c and any x, y E a+ we have

(8)

Theorem 1. A shift invariant subspace a+ of -+ s with starting time K is a resolution

a(K) E a+. space if and only if

Proof: If a+ is a resolution space then + a(K) E a trivally from (6) and (7) above.

+ On the other hand, suppose a(K) Ea. Then + if yEa ,

and hence we also have

Corollary 1.

Corollary 2.

For vector spaces, pN invariance is equivalent to PN invariance.

Because o(K) can be shifted, multi­plied, and added over any finite range, all shift invariant resolu-

tion spaces a+ with the same start­ing time K are equivalent over any finite range. That is

a)

b) DK sc+ C a+ , where sc+ denotes sequences of compact support.

Examples of shift invariant resolution spaces are

+ s+ and 0 + sc , the ~ spaces. p

The theory can be enriched through the z-transform, defined here as a formal series

+ specified by each sequence YES by

-1 -2 y(z)=YO+ylz +Y2z + ... (9)

and it is easy to see that the delay shift of y

corresponds to multiplication of A(Z) by z-l. A partial hierarchy of shift invariant spaces and their isomorphic images with starting time K = 0 is indicated below, with most of the spacial des­ignations given in [2], [3], or [4].

Tr + c. C + C. A + C C + HI c P + C D + C F + I I I I I r I I

sc+ Co rd+ c: ll+' c + C + + + +

al

c loo c. sg c s

+ For example, F denotes the set of all formal

series, D+ denotes the set of (one sided) periodic + distributions, Tr is the set of polynomials in

z-l To illustrate the use of the z-transform we have

129

Theorem 2.' Any shift invariant resolution

subspace of l2+ which is closed in the l2+

topology is of the form DKl2+·

Proof: By Beurling's Theorem [5], every

closed shift invariant subspace of H2 is of

the form {F(z) I F(z) = g-KB(z)S(z)G(z), -K where G(z)EH

2}z B(z) is a Blaschke product

and S(z) is a singular inner function. But

Theorem 1 implies that

, for some G(z)EH2, so G(z) = l/B(z)S(z)

But l/B(z) is not in any H space, nor is p l/S(z) [3,p28] except in the trivial case

,B(z) = S(z) = 1. Hence the shift invariant

resolution subspace must be of the form z~~2 in the Z-domain or DKl

2+ in the time sequence

domain.

Many interesting problems concerning resolu­tion sets and spaces remain to be investigated, but let us turn now to investigate their interac­tion with operators and systems. Henceforth,

a+ and b+ will denote shift invariant resolution

sets, and if an operator T:a+ ~ b+ is designated

as being linear, it will be implicitly assumed

that a+ and b+ are vector spaces. We have the following designations for T.

Causal: pNx pNx' implies pNTx pNTx'

Anticausal: PNx PNx' implies PNTx PNTx'

Time invariant:DNT P TDN N

(10)

The presence of the PN in the time invariant

condition is, perhaps, bothersome. It is pres­ent because we are dealing with truncated se­quence spaces rather than uniform sequence se­quences. This will be discussed later.

3. MATRIX REPRESENTATION OF LINEAR OPERATORS

In this section, we consider various condi­tions under which linear operators can be repre­sented by matrices, and how matrices of certain forms characterize the operators they represent. First, we review some results on causality and time invariance of linear operators on resolution spaces. Proofs are omitted for the most part but can be found in [6J.

Theorem 3.

A linear operator G: a+ ~ b+ is causal iff anyone of the following equivalent conditions is satisfied for N = 0,1,2, ..•

a) pNx ~Nx' implies pNGx = pNGx'

b) N o implies pNGx = 0 (11) P x

c) GPN PNGPN

d) pNG pNGpN

Theorem 4.

A linear operator G: a+ ~ b+ is time in­variant and causal iff DG = GD.

Theorem 5.

An operator T: a + ~ b + is time invariant iff

DT (12)

Now, we say that a linear operator

G: a+ ~ b+ is represented by a (semi-infinite) matrix.

goo gal g02

glO gll g12 (gka)

g20 g2l g22 {13)

130

iffy Gx is equivalent for k 0,1,2, ... to

(14)

An example of an operator G : a+ ~ s+ which is not representable by a matrix is given by y = Gx with

lim N-><x>

N 1 N+l l:

a=O x

ct

Yk = 0 for k = 1,2,3, ...

(15)

Note: a+ is the shift invariant resolution space for which the limit in (15) exists.

In general, it is not known when linear maps

G : a+ ~ b+ are representable by matrices, but some partial results follow:

Theorem 6.

A semi-infinite matrix (gkct) represents a

linear map G : s+ ~ s+ iff each row of the matrix has only a finite number of non-zero elements.

Theorem 7.

E b d d 1 0+ ~ 0+ is very oun e inear operator G : ~l ~ ~l

representable by a matrix (gka)' Each such matrix satisfies

sup l: ct k=O

Ig 1< "" ka (16)

Furthermore, a matrix (gka) represents a

b d d I' G .,e.+ ~ ,e.+ iff (16) is satis-oun e 1near map : 1 1

fied.

Theorem 8.

A matrix (gka) represents a bounded linear

map G : ,e.: ~ ,e.: if

Theorem 9.

sup l:: k a=O

Ig I < <lO ka (17)

+ + Each causal linear map G : a ~ b determines, and is determined by a lower triangular matrix

goo o 0

glO (gkct)

g20 (18)

o

Theorem 10.

Suppose a linear map G : a+ + b+ is repre­

sented by a matrix (gka)' then G is time invar­

ient iff there exists a sequence g such that

go g-l g-2

(gka) (gk-a) gl go g-l

(19) g2 gl go

Matrices of this form are called Toeplitz matrices.

The Toeplitz form (gk-a) is a natural form

to associate with the notion of time invariant operators.

(gk-a)

and if x' then y' Hence, xk

Note that ifG is represented by

1 1 0 0 0

0 1 1 0 0

0 0 1 1 0 (20)

(1 1 0 0 0 .•. ).x = (0 1 100 •.• ) (2 1 000 ... ).y = (1 2 100 ..• ). xk_l for all k and Yk = Yk-l for all

k > 1 so that the definition of the invariance (12) is satisfied. It is not true that ~ = ~-l for all k implies Yk = Yk-l for all k. This

would be the usual requirement for spaces running from minus infinity to plus infinity. A universal definition for time invariance could be the re­quirement (17)

(21)

which corresponds to (lO)because DND*N = PN'

where D*. the adjoint of D is defined by the ad­vance shift relation

(22)

4. NORN. INVARIANCE. BOUNDEDNESS. CONTINUITY AND GAIN

Let a+ be a normal shift invariant resolu­tion space. With starting time k. we make the following definitions for an operator T:s+ + s+

Finite Norm: The a+ norm of T. IITII IITxl1

sup+TIxIT' x£x

xlO

T is of finite a+ norm (FN) iff I ITI I <

Invariant: T is a+ invariant (INV) if Tx £ a+ for + all x £ a

Bounded: T is a+ bounded (BD) if it is of a+ finite norm and if TO = O.

Continuous: T is a + continuous (CON) if it is

any sequence x(n) £ +

invariant and if for a +

converging to x £ a • that is I Ix(n)-xl I + O.

have II Tx (n) -Tx II + O.

+ a

we

Finite Gain: T is a+ finite gain if there exists

Y. 0 < Y < 00 such that IlpNTx11 2. Y IlpNxl1 for

N = 0.1.2 •.•. and all x £ s~

Comments: Note that T s+ + s+ is a+ bounded iff

there exists 0 < y < such that IITxl1 2. y Ilxll

for all x £ a+ I iTI I is the infimum of all such y.

Theorem 11.

+ Given a particular a spac e. the classes of

maps T : s+ + s+ are related as follows:

131

Goneral Causal Linear Col~&l

S. MEMORY AND SYSTEM CONCEPTS

In this section we consider operators

T: a+ + b+ where a+ and b+ are shift invariant resolution sets. and discrete time systems. As a preliminary definition we say that T is memory­less if it is both causal and anti-causal. First we show that memoryless operators are completely specified by ordinary functions of a complex (or real. in the real case) variable.

+ + Lemma 1. T: a + b is memoryless iff xk = ~ implies (Tx)k = (Tx')k for k = 0.1.2 ••••

Lemma 2. T: a+ + b+ is memoryless iff there exist

functions Gk

: C + C k = 0.1.2 •••• such that

, ,I I

,il

I, !

for all x £ a+

(Tx\ = Gk (xk)

Theorem 12.

and all k 0,1,2, ...

T: a+ ~ b+ is time invariant and memoryless

iff there exists a function G:C ~ C such that for

all x £ a+ and all k = 0,1,2, ... (Tx)k = G(xk).

Corollary. T: a+ 7 b+ is linear, time-invariant,

and memory less iff there exists a homogeneous function G: C ~ C, G(ab) = aG(b), such that for

+ all x £ a and k = 0,1,2, ... (Tx)k = G(xk).

In order to delve further into the concept of memory it is convenient to introduce a discrete-time-system defined by the state equations

(23)

where the impact xk

£ X, the state sk £ S, and

the output Yk £ Y. (Here assume X and Yare

subsets of C, the complex numbers.) Now for any given value of s the discrete-time system (23)

o + + + induces a causal map T: a 7 b where a is a + resolution set with sequence values in X and b is a resolution set with sequence values in Y. The induced system operator is said to be of finite memory N if there exists a function G such that

(TX)k = Yk = G(Yk_l'···'Yk_N,xk,··,xk_N,k) (24)

for k~N, and of finite input memory N if ( (Tx)k = Yk = G(xk,··,xk_N,k) (25)

for k~N. These definitions can be carried+ove¥ directly to arbitrary causal operators T:a ~b ' and we see now that a "memory less" operator T is simply any causal T with zero input memory. It is well known that linear time invariant finite dim­ensional discrete-time system with a rational transfer function induce operators T of finite memory, whereas, if the_iransfer function happens to be a polynomial in z ,T will be of finite in­put memory.

The connection between causal operators T and discrete-time systems as defined above is strengthened further if we note that every such 0

operator induces, and thus can be represented by a discrete-time system with So = (0,0,0, ... ) and

(xk,O,O, ..• ) (26)

+

132

6. ACKNOWLEDGEMENT

This work was supported by the United States Air Force Office of Scientific Research under grant number AFOSR 76 3036.

7. REFERENCES

[1] R. Saeks, "Resolution Space Operators and Systems," Springer-Verlag, 1973.

[2] R. E. Edwards,"Fourier Series," Vol. II, Holt, Rinehart, and Winston, 1967.

[3] P. 1. Duren, "Theory of Hp Spaces,"

Academic Press, 1970.

[4] N. Dunford, J. T. Schwartz,"Linear Operators," Interscience, 1958.

[5] W. Rudin, "Real and Complex Analysis," McGraw Hill, 1966.

[6] B. Swaminathan, "Characterization of System Operators on Truncated Sequence Spaces," M.S.E.E. Thesis, Electrical Engineering Dept., University of Notre Dame, May 1977.

[7] R. Saeks, R. J. Leake," On Semi-Uniform Resolution Space," Proc. Midwest Circuit Theory Symposium, 1971.

SOME ASPECTS IN A THEORY OF

GENERAL LINEAR SYSTEMS

Robert H. Foulkes, Jr. Electrical Engineering

Youngstown State University Youngstown, Ohio

Abstract

The work discussed here deals with a theory of general linear systems. In a previous work, an exiomatic approach to such a theory was pre­sented. Using set theory and abstract algebra, the usual notions of a linear state space, response separation, and superposition, as well as standard results about controllability and observability, were developed in an unusually abstract setting. The work discussed. here concerns an extension of the superpositon and response separation properties and some results concerning finite automata within the axiomatic framework.

133

CERTAIN ASPECTS OF INVERSE FILTERS

V. P. Sinha Electrical Engineering Dept.

Indian Institute of Technology Kanpur, India

H. s. Sekhon Electrical Engineering Dept.

Punjab Agricultural University Ludhiana 141004, India

Abstract

The problem of inverse filters is often encountered in instrumentation and geophysics. The existing techniques, being based on Laplace transform approach, lend into difficulty when the transfer function has zeros in the right half of the s-plane. Also, they do not adequately cover the practically more significant case of finite duration and/or discrete signals. The inverse filtering problem is re-formulated in terms of the oper­ator techniques so as to include finite duration and/or discrete signals.

-'

11

. , , '

./ /

ADAPTIVE ANTENNA POLARIZATION SCHEMES FOR

CLUTTER SUPPRESSION AND TARGET IDENTIFICATION

George loannidis and David E. Hammers ITT Gilfillan

Van Nuys, California

Abstract

Optimum transmitter and receiver antenna polarizations for target discrimination in the presence of background clutter are obtained. The analysis uses the concept of Stokes vectors and Stokes target operators in a constrained maximization of the ratio of two bilinear forms, represt:nting the signal to interference powt:r ratio.

INTRODUCTION

Polarization characteristics of targets art: described in terms of

the spacial orientation of tht: electric !kld vector in the

im:ident and retlected electromagnetic waves. In current radar

technology designers take advantage of a target's polarization

properties by selecting an antenna polarization (transmittt:d

electric field orientation), which maximizes tht: t:nergy in the

backscattered wave for some assumed target orientation.

Uowevt:r, most targets do not hold their orientation constant

so that the backscattered ent:rgy is not always what it could

be. Designers have also attempted to exploit the polarization

characteristics of background reflections (clutter) by affixing

the antenna polarization such that background contributions

to the retlected energy are cancelled. For example, by taking

advantage of spherical properties of raindrops, which upon

retlection reverse the polarization of an orthogonally polarized

wave, rain echo is cancelled by antennas employing circular

polarization I I I, (2], (3),

Recent antenna developments make it reasonable to control

the transit/receive polarization in a more dynamic manner.

That is, rather than just applying a fixed, horizontal, vertical,

or circular reft:rence, the antenna's polarization can be con­

tinuously adjusted to optimize target detectability in a partic­

ular target and interference environment. Furthermore, the

problem can be set-up in an operator theoretic manner so

that optimization theory can be applied. In what follows we

do this by use of the Stokes Polarization vectors and Stokes

Target Scattering Operators, defined in the next section, and

then proceed to derive an optimum algorithm for selection of

antenna polarizations, which can be adaptively applied to

the problem.

POLARIZATION DESCRIPTION IN TERMS OF STOKES VECTORS

The scattering of plane polarized electromagnetic waves by

complex body scatterers such as aircraft targets, ground

clutter, rain, or chaff is governed by a 2 X 2 complex

symmetric operator known as the target scattering matrix

U given by

(I)

where hxx ' hyy, hxy = hyx are complex variables describing

the relative amplitude and phase of the x-plane and y-plane

polarized components of the incident and scattered waves.

, In terms of the scattering matrix, H, the voltage induced in a

receiving antenna with a polarization described by the electric

field vector.

135

e/ = (Er' Ey) r is given by

V = e/ He! (2)

where et' = EX' Ey t is the electric field vector of the trans­

mitting antenna, the superscript' indicates matrix or vector

transposition and EX ' Ey are complex numbers representing the

relative amplitude and phase of the x and y components of the

electric field.

In radar applications where the target's aspect angle varies

continuously, the quantity of interest is the average received

power for given transmitter and receiver antenna polarizations,

which is given by

Ii \

1\

i

p (V·V), (3)

where the bar indicates time averaging and the superscript*

indicates complex conjugation. Substitution of Equation (2)

into (3) gives

(4)

By rearranging terms in the above equation, Kennaugh [II has

shown that the average received power can be expressed as

p = y' A X (5)

where X and Yare four dimensional vectors, whose elements

are the four Stokes parameters (Kennaugh [II. Poelman [21,

Born and Wolf [3)) describing the transmitter and receiver

antenna polarizations respectively, and A is known as the

average Stokes scattering operator for the target (see

Kennaugh [I]).

The elements of the Stokes vectors X and Y satisfy the

constraint

Xo 2 - x 12 - X22 - X32 = 0 xo>O

and

Y02_YI2_yl-Y32=0 Yo>O

(6)

(7)

The average Stokes scattering operator A is 4 X 4 real symmetric

matrix and its diagonal elements satisfy the equation.

(8)

Furthermore it can be shown that if the vector Z is given by

Z=AX (9)

where X is a Stokes vector then the elements of Z satisfy

(10)

If we introduce a matrix R given by

R = (:.;.::) o 0 0 ·1 (11 )

then Equations (6), (7), and (10) can be expressed as

(12)

We note that Equation (12) defines a convex subset of the

Euclidean 4-space which is invariant under the Stokes target

operator A.

From the above relations it follows that the bilinear form

p = y'A X

is always positive, when X and Yare Stokes vectors and A is a

Stokes operator, an expected result since P is an expression

for the average received power.

136

Optimum Transmitter-Receiver Polarization Combination for Target Detection in the Presence of Background Clutter

When the average Stokes scattering matrices A and e for the

target and clutter have been determined. transmitter and

receiver Stokes polarization vectors are selected so as to

maximize the signal to clutter power ratio.

Y'AX -- = r Y' ex 03)

with the restrictions

y'RY = 0

and

y'RX = 0 Xo >0

where the matrix R is given by Equation (II).

The bars denoting average Stokes matrices have been dropped

here for ease of notation.

Since the vectors X and Y in Equation (22) can be normalized

by appropriate constants that force Xo > 0 and Yo > 0 with­

out changing the value of the ratio of Equation (22). the only

significant constraints are

X' RX = 0 (14) and

Y' RY = 0 (15)

Since the bilinear forms Y' AX and y' ex are non-negative

under the restriction that X and Y be Stokes vectors. one could

consider two cases as follows: the first case applies to clutter

suppression by itself and considers possible existence of a pair

of Stokes vectors X and Y such that

y'ex = 0

Y' AX> 0

(i)

(ii)

The second case assumes that there are no Stokes vectors X and

Y that satisfy Equation (i) and proceeds to maximize the ratio

of Equation (13).

Here we present the solution to the second case only.

Maximization of SignaI-to-CIutter Power Ratio

When there are no Stokes vectors X and Y such that Y' CX = 0

and Y' AX> 0, then maximization of the ratio of Equation (13)

under the restrictions of Equations (14) and (IS) is

equivalent to the maximization of

y' AX

subject to the restrictions

y' ex = k (constant)

Y'RY=O,yo>O

and

x' RX = 0, Xo > 0

Introducing Lagrangian multipliers A, and J.I. we obtain that the

optimum Stokes polarization vectors X and Yare given by the

solution of the system of equations

AX = p CX + J.I. RY, (16)

and

A Y = p CY + A RX . (17)

The solution vectors of the above equations are given by the

eigenvectors of

(RA - PRC)2 X = s2 X

where

s2 = AJ.!.

and s2 is not necessarily a positive number.

(18)

After some algebraic manipulation the above equation can be

expressed in the form

where B has the form

f30

is a scalar, b is a three-dimensional vector and B I is a

symmetric 3 X3 matrix.

(19)

(20)

From Equation (18) we obtain that the eigenvalues s2 are

solutions of

(21)

If for some value of s2 = sl2 satisfying Equation (21) the

determinant

then application of the constraint X' RX = 0 to the resulting

solution gives

b' (81 + s2 11 )-2 b = I

where I I is the 3 X 3 identity matrix. If relation (22) is

satisfied one can show (see Brauer and Nohel (41) that

(23)

D(p,s2) = Is21-BI = [(po-s2) - b'(B I +s21lrlbllBI +s2111.

(24)

From the above we obtain that Equations (21), (22). (23), and

(24) imply that s2 must satisfy

and

D(p.s2) = 0

0D(p.s2) = 0

0(s2)

i.e., s2 must be a double root of Equation (21).

If

(25)

C~6)

(27)

137

and also

(28)

then one can show that the vector b defined in

Equation (20) is either equal to zero or it is orthogonal to at

least one vector in the null space of the operator (B I + s 12 I I)'

Therefore there is always an eigenvector of the form

Since this is not a Stokes vector, the existence of a Stokes

vector solution requires that there are at least two eigenvectors

corresponding to the same eigenvalue, i.e., the root s2 = s 12 of

Equation (25) must have at least multiplicity two.

From the preceding discussion we conclude that a necessary.

although not sufficient, condition for an eigenvector of the

matrix (RA-pRC)2 to be a Stokes vector is that the deter­

minant of Equation (21) have s2 as a double root. This can

happen if either s = sl is a double root of

-f(p,SI) = IR(A-pC) - Sill = 0 (29)

or if both s = sl and s = -sl are roots of the above equation.

Since the trace of (RA-pRC) is always zero as it follows from

Equation (8) we obtain that

o . (30)

This equation has one double root if

(31 )

and two equal and opposite roots if

(32)

Since the sum of the roots is zero. the above include the case

of two pairs of repeated roots. From the above discussion the

positive values of p satisfying the above equations include the

extrema of p under the constraint that X and Y be Stokes

vectors. a search among these values will yield the ones that

result in X and Y being Stokes vectors. Since s2 must be real.

only those resulting in real values of s2 need to be examined.

Additional criteria can be obtained but will not be discussed

here due to lack of space.

We also note that since Equation (18) can be written as

[R(A-pC) - 51 • [R(A-pC) + 51 X = 0 (33)

If s = sl and s = -sl are eigenvalues of (RA-pRn. then since

the two matrix operators in the brackets in Equation (33)

commute X is given by

x = kl UI + k2 U2 (34)

and from Equations (16) and (17) follows that

(35)

where k I and k2 are constants and U I and U2 are the eigen­

vectors of (RA-pRC) corresponding to the eigenvalues sl and

-s I' respectively.

NUMERICAL APPLICA nON

As an application of the above method of selecting optimum

antenna polarizations we consider the problem of detecting a

small (radius of the order of the wavelength), horizontally

oriented, cylindrical target in the presence of a chaff cloud

consisting of a collection of randomly oriented dipole scatterers.

The Stokes scattering operator C for the dipole cloud is given by

c",o' (~ ; : D where 0

02 is a constant depending on the wavelength and

number of dipoles. If one assumes that the dimensions of the

cylinder are such so that its reflection matrix is given by

H=oc (2 0) . 0 I

where' ° c depends on the size of the cylinder and the wave­

length, we obtain that the Stokes operator A for this

cylindrical target is given by

A=~~(: ~ ~ ~) 2 0 0 4 0

o 0 0-4

In maximizing the ratio of Equation (13) the constants 0 02

and 0c 2 can be neglected. From the determinant equation

f(p,s) = I s I -R(A-pC) I = 0

we obtain f(p,s) = s4 _ [3p2 _ 19p + 32) s2 + p2 [2p-lI) s

-4(p-4) (2p2 - ISp + 16) = 0 .

If we let dip) equal to the coefficient of s2, d I (p) equal to

the coefficient of s, and do(p) equal to the constant term in

the above polynomial in s we obtain that the values of p

satisfying d I (p) = 0, are

138

PI =0 and P2 = 5.5

The values of p satisfying

are

P3 = 2.13, P4 = 1.34

PS,6 9.2 ± j 0.9

and

P7,8 = S.l7 ± j 1.6 .

The maximum value of the above Pj's is P2 = 5.5. The

corresponding values of s are sl = 4, s2 = -4, s3 = 1.5, and

s4 = -1.5. The eigenvectors corresponding to s = ± 1.5 do not yield Stokes vectors. From the eigenvectors U l' U2 (see

Equations (34) and (35)) corresponding to s = ± 4, we obtain

that the optimum pair of Stokes vectors X and Yare given by

and

where k I and k2 are constants chosen to satisfy the constraints

X' RX = ° and Y 'R Y = 0 .

The resulting values of X and Yare given by

I

2 2 -3 3

X and Y 0 0

ys -0 3 3

which correspond to left-handed and right-handed elliptical

polarizations respectively with a 9-to-4 ratio of the horizontal

relative to the vertical field amplitudes.

CONCLUSIONS

From the above analysis and example it is obvious that the

concept of Stokes vectors and target operators is an extremely

powerful and useful tool in the problem of target discrimina­

tion in the presence of background clutter. In [5) Hammers

and MacKinnon developed an operator theoretic approach for

generating probing waveforms to identify target operators. The

Stokes operators and the above analysis can be used to extend

their analysis to include the selection of optimum antenna

polarization for target discrimination and identification.

REFERENCES

[I J Kennaugh, E.M., "Effects on the Type of Polarization on

Echo Characteristics". Final Engr. Rpt. Antenna Lab.,

Ohio State University, Vol. AF contract 28(099) -90

June 1951, AMC3.51, Griffiss AFB, New York.

[2 J Poeiman, AJ., Reconsideration of the target detection

criterioll based on adaptive antenna polarizations.

AGARD Symposium Proceedings, 1976.

[3 J Born, M., and E. Wolf, Principles of Optics. Pergamon

Press, New York, 1965.

[4 J Brauer, F., and J.A. Nohel, Qualitative Theory of

Ordinary Differelltial Equations. W.A. Benjamin, Inc.,

New York, 1968.

[5 J Hammers, D.E., and A.J. MacKinnon, Radar Target

R('wgnitioll. An Operator Theoretic Approach Presented

at Operator Theory of Networks and Systems International

Symposium, Montreal, Canada, August 1975.

BIOGRAPHIES

George A. Ioannidis was born in Athens, Greece on February 14,

1945. He received a B.Eng. degree from Stevens Institute of

139

Technology, Hoboken, N.J., in 1968; M.S. in Electrical Engi­

neering in 1971, and PhD. in Electrical Engineering in 1973,

both from Cornell University, Ithaca, N.Y. He has been a

research assistant at the Center of Radiophysics and Space

Research, Cornell University, and at the National Astronomy

and Ionosphere Center in Arecibo, Puerto Rico. He was a

research associate in the Department of Electrical Engineering, .

Cornell University, and the Institute of Geophysics and

Planetary Physics UCLA at Los Angeles. Currently he is senior

design engineer at ITT Gilfillan, Van Nuys, Califor?ia.

Dr. Hammers was born in Chicago, Illinois in 1937. He received

a BS in Engineering in 1960 from Loyola University of

Los Angeles, MSEE from the University of Southern California

in 1963, and PhD. in Engineering in 1973 from the University

of California at Los Angeles, majoring in Systems Optimization

Theory. He joined ITT Gilfillan in 1963 and participated in the

early design and development of a digital Automatic Target

Detection system for the SPS-48 radar. Since then he has

performed systems analysis and design for processing systems

in various 20 and 3D air defense radars, battlefield surveillance

radars including mortar and artillery detecting systems and

Precision Approach Control radars such as the TPN-22 currently

under development. His present position is Director of Systems

Analysis and Programming.

! Ii

ON LIMITATIONS BASED ON PROPERTIES OF THE STURH-LIOUVILLE

OPllliATOnS IN TIlli SYNTIlESIS PHOCEDUIlli OF NC)NeUNIFOml LINBS

Zdzisl.al" ii. Trzasl,a

1{arsal .. Technical. Uni versi ty

IfARSAIY, Pol.and

Abstract

The method for the evaluation of taper functions of a non-uniform l.ine is proposed in this paper. Appl.ication of the Sturl:l-Liouvil.le

op0rator theory permite to calculate the inductance distribution al.ong the llon-uniform lossless line lvi th the prescribed totul ca­pacitance.The desired expressions are obtained by thfJ usc only the N+l eigenvalues of the appropriate operator \dth given boundary condi tions •

1. INTRODUCTION

Recently, synthesis of discret and distri

huted parameter systems has been carried

on throu~h various methods. One of the

pOlverful.l tool.s for the network synthesis

is the differentia1 and inte(..Tal opera tor

theory.

The cenera1 aspects of the app1ications

of the operator theory to the network

synthesis are given in [7,11]. Conditions for the rea1ization a non-uni­

for,m parameter dis trihuted netw'ork with

short circuits at its ends have been C-i­

ved in the paper i~10]. This method of

synthesis is based on a prescribed imped~

nce function of the network and can be re

garded as an extension of the synthesis

procedure of l.adder networks.

A procedure for synthesizinG' very smooth

l.osslesS non-uniform lines has been

proposed in [9J. The application of this

method is usefu1l. only for 1ines whose

characteristic impedance has very sma11

variations in the propagation direction.

140

Distributions of voltac-es al.onc- a non­

uniform 10ssl.ess line using notations of

the Sturm-Liouville operator theory in

the frequency domain' can be expressed as

L V(x,s = s2V(x,s (1) d 2

where L = - di2 + q (x) and q(x) denote

the Sturm-Liouvil.l.e operator and the ta­

per function, respectivel.y. The space

coordina te x f (0, l.) is measured from the

beginning of the l.ine. Horeover we denote

by S (a, b), where - 00 <a, b <00, the q spectrum of the line in the case of the

fol.l.owinz boundary

V(O,s)

V (1, s)

condi tions

+ aV (O,s) = 0 x + bVx\l.,s) = 0

Ifhere subindex x denotes the derivative of

the voltage in respect to the space coor­

dinate x.

G. Borg has proved in [11 that for two

functions q(x) and q~x) and some values

of a1

and a2

and b the equal.ity Sq(a,b) = S +,b)invol.ves the equal.ity q(x~ :: q~x)

q for Xf(0,1). V.A. Marcenko has extended

in ~ 8] this reaul. ts for the dependence

of the function q x and the values of

ai

and b on the appropriate spectra

Sq(ai,b) for i = 1 and 2. It is well known that the values of diffe

rent spectra Sq(ai,b) for i = 1 and 2

for the same line interlace, e.g. they

form an infinite sequence as follows 2 ·,2 2,2 ( ) sl~sl ~s2 ~s2'" _ 3

As stated in H.Krein papers [5, 6J the

knowledge of two spectra S (a,b) and q Sq{a,b) is sufficient to evaluation the

taper function q (x) and the values a i and b.

However from the practical point of view

only finite number of the values of' the

line spectrum may be taken into account.

Thus a followin~ question appeares. Has

such approximation a great influence on

the obtained results? This problem we can

transforme into following one. How' much

di verge two taper ftUlctiollS ql (x) and

Q2\x) for the same only N+l values of the

appropria te line spectra. I-Ioreover it is

alsO interesting to ans,,,er to the follo­

wing question. Are there any limitations

in the procedure for evaluations of the

line parameters?

These problems focuse author's attention

in this paper. \

2. CALCTlLYl':fC,I"- ()F THE TAPBR FUNCTION

In this section a mean for calculations

of' a taper function on the base of N+l

eiGenvalues of an appropriate Sturm-Liou­

ville operator f'or the non-unifor'l1 lOSS!

eSS line is demonstrated.

We introduce the follo,V'ing notations.

Let c(x) and d(x) denote h"o continuous

non-decreasing functions of the sp::tce

coordinate x. By u(x)we denote a class of

all operators L for that follmdne; rela­

tions are fulfilled )iq(Y~dY ~e(x) , llq'(y)ldY (d(x) (4) o

for 0 ~x " 1. Now we assume operators L j ,,,ith taper

conditions functions qj(x) and boundary

V~(O,s} = 0, VJ(l,S) = 0

vj(o,s) = 0, Vj(l,s) = 0

for j = 1,2. }1oreover ,.e take into account N+ 1 equal

eigenvalues of such two operators L. ,~be­J

re 1 N ~ 711M, H = J[* d(l)+ iqo + 5c

2(1)j l

, ,71 qo = n?ax max q. (x)

J=l,~ xE(o,i) For these operators ",e have the follo,V'ing

estimation of the appropriate t::tper funct

( 8)

where

p(x)

and

141

+ i [2Mx + 2d 1 + Q(x) _ 2c(1) x -~s;tJ!I)l

+ t (x2 + lOx +_lO)C(lB +

+ (10M!. +8c) expl2d(X)] .

The proof of the above relation (S)can be

easy constructed on the base of the theory

H.Krein [5, ~ and given by, among others,

T.Kato [4J. The estimation (S)permits to ev::tluate the

desired taper function, for example, ql~) from the q2(x) and the N+l eic;envalues of

the appropriate sturm-Liouville operators.

To evaluate the desired function q(x) \1e

may use the follo",ine; procedure.

Let be given a sequence of the eigenvulues

uf t,vo operators as stated by relation (3~ Then lV'e may evaluate the so called central

function H (x) as follo,,,V ~l) V 12') H(X) = H(l) + COllst v

2'4")- V:R') (9)

for of (:x ,l)",here v1(x) alld V2(x) are

the solutions of the equations L.V(x) =0 J

(j = 1, 2) wi th the cOlleli tions (5) and (6). solutions of the above equations for v(x) e;i ve the follO\vinG

V (x) = COllS t ( -

Finally \,c obtain

formula), _ -21 cJ}.: x dx

2 e; (x) = V - (x) d vex)

dx2 ( 11)

The limitations in the above procedure

are followinG.

a/ Functions M(x) and their first and se­

cond derivatives must be absolutely

continuous.

b/ D " t" dN(x) erl va 1 ve ctX must not vanish.

The central flmction M(x; may be taken

as the inductance distribution alon!3" the

line "'hose capacitance distribution is

I:n01m.

There are also additional limitations for

the central ftmction ~l (x), namely. If for

the given ftmction q(x) = q,(xJconditions

for the Krein's equation are not fulfilled

then the centr::tl function H x Growth to

infinity ""hile the taper ftmction (So to

zero. In the second case the central. feme

tion H Lx) is limited l"hil.e the lenGth of

the the l.in0 tends to infinity.

J. CONCLUSIONS

The conditions for the evaluation of the

Part I. IEEE Trans. on CT, vol. CT-14

1967 , No 4, pp.394-408

4. T.Kato: Perturbation Theory fo~ Linear

Operators. Sprine;er, Berlin, 1976

5. M.G. Krein: Solution of the Inverse

sturm6Liouville Problem. Dokl. Akad.

Nauk SSSR, vol..76 1959, pp.21-24

6. M.G. Krein: On a method of effective

solution of an inverse botmdary prob­

lem. ibid., vol.94 1954 , pp.987-990

7. N. Levan: Operator Theory of Network

Synthesis. Proc. 1974 huropean Conf.

on CTD, London, 1974, pp. 234-238

8. V.A. Marcenko: Concerning the theory

of a differential operators of the

second order. Dokl. Akad. Nauk SSSR,

vol. 72 1950 , pp. 457-460

9. S. Ridella: On a variational synthe­

sis of lossless non-uniform lines. Al­

ta Frequenza, vol. 40 1971 , No 6, pp.

527-533

10. E.N. Protonotarios and O. Wing: Theory

of NOn-tmiform Lines. IEEE Trans. on

CT, vol. CT-14 1967 No 1, pp.2-20

~rbitrary t~per function for a non-tmiforln 11. R. Saeks: Synthesis of general linear

line are shown in the paper. The expre- Network. SIAM Journ. Appl.ted Mathem.,

ssions for the error estimation of tho ea- vol.16, 1968 , NoS, pp. 924-930

lculated taper ftmction on the base of

the N+1 eir,-cnvalues of the appropriate

Sturm-Liouville operntore are given. So­

I!!C kinrl of' lin:i te<. tions in the presented

~lrocudere ure demonstrated. The l!lore de-.

taill.ed investi(Sations are tmder develo­

ppmcnts.

1. G. Llorg: Bine Uml<hertmG dor Sturm­

Liovvilleschen 1!:icen"ertaufeabe .. \cta

Hath., vol. '78 19 1f6 , pp. 1-96

2. N. Dunford and J. T. Sch.,ar tz: Lincrrr

Oper::.tnrs. Part 2. Inter. Publ., New

forl~, 1963

3. D.S. Eeim and C.B. Sharpe: The synth~

sis of Non-uniform Lines of Finite Le n.;th.

142

12. Z.Trzaska: Use of the Sturm-Liouville

operators to the study transient phe­

nomena in a non-uniform long transmi­

ssion 1.ine. Proc. 1974 Inter. Confer.

CTD, London, 1974, pp. 199-204

13. !-t.R. ,vohlers: A Realizability Theory

for Smooth Lossles Transmission Lines.

IEEE Trans. on CT, vol. CT-1 J 1966, No

4, pp.J56-368

Zdzislaw TRZASKA 1YaS born in Nagnajow,

Poland, on Aue;ust 10,1939. He received

the dee;ree in Blectrical Engineering

in 1970 from the Technical University

of \warsaw. In 1963 he joined Technical.

Universi ty of War sa .. , 1vhere he is pre­

sently Director of the Institute of the TIIeory of Electrical Engineering.

A THEORY OF BEST APPROXIMATION OF NONLINEAR FUNCTIONALS

AND OPERATORS BY VOLTERRA EXPANSIONS

L. V. Zyla and R. P. de Figueiredo Dept. of Mathematical Sciences

Rice University Houston, Texas 77001

ABSTRACT

Let H be a real Hilbert space and Fp(H), p > 0, denote the Fock space of order p associated with H. Let T be a nonlinear operator from H into the Sobolev space Hl(i) of real functions on an inter­val I of the real line. T is such that the functional u+(Tu) (t) , u E H, is an element of Fp(H) for every t E I. Finally, assume that 1 IT(t) 1 1 Fp(H) is Lebesgue measurable as a function of t and

£1 IT(t) 1 12FP (H)dt < 00. The space of such operators is known as the

Bochner space B2 (I,Fp(H)) of Hilbert valued functions on I.

T is viewed as the input-output map of a general nonlinear dynamical system and a specific representation for T is in terms of a Volterra expansion. A particular class of operators of this type arise from differential equations of the form x(t)=f(x,t)+u(t)·g(x,t), x(O) = xO·

The approximation problem we consider is the reconstruction of the output x of T based on the set of observations {~l(u), ~2(u), ... ,

~ (u)} E Rn on the input u (x = TU).- The functional ~l' ... ,~ n n

are not necessarily linear but we do assume that ~lEFp(H), i = 1,

... , n. Using the Hilbert space structures of B2 (I;Fp(H)) and

Fp(H) we derive approximations of T based on ~l' .•. '~n which are

optimal with respect to various criteria, and obtain results for "best" choice of the reconstruction subspace for x in Hl(I) as well as the optimal choice of observation functionals ~l' .•. , ~ for a nonlinear T. n

Supported by the NSF Grant ENG 74-17955.

143

, , h

CONTINUED FRACTION DESCRIBING FUNCTIONS FOR

BILINEAR AND MULTIPLICATIVE NONLINEAR SYSTEMS

C.F. Chen Electrical Engineering Department University of Houston Houston, Texas 77004

R.E. Yates U.S. Army Missile Research and

Development Command Redstone Arsenal, Alabama 35809

Abstract

A harmonic balance method is developed for finding the describing functions of bilinear and multiplicative nonlinear systems. The results belong to the classeof continued fraction approximations which are flexible and easy to use. We can truncate the function as we like to obtain various appropriate describing func­tions from very crude to very accurate ones.

1. I NTRODUCTI ON

When a nonlinear element Is slngle-v.lued t the Input-output rela­tionship can be expressed an.lytlcal IV In the following ways.

(a) PolynomIal

(b) PIecewIse llne.r

(c) Transcendental functions.

To obtain the descrIbing function of such a nonl1nearlty. one needs to evaluate an Integral whIch determines the fundamental coefficients of a Fourier series. The. derivations of certain descrlbt[l9 functions 'rom the piecewise linear representation are wet I known [ IJ. Those derived from the polynomIal expressIon whIch represents the non I Inearlty Ire In the form of garrwna functlons[2]. When the nonlInearity Is expressed by .. transcendental function, for example, a segment of a sine wave for the saturatIon nonlInear ease, the correspondIng descr.blng function formula 15 In terms of Besset functlons[31.

There Is a class of nont Inearltles which Is encountered frequently, but has no known expllclt describing functIon shown In the literature. ThIs class Is bilinear In particular end multiplicative nonlinear In gen­eral.

We wJll develop a harmonic balance method for finding the describIng functions for bilinear and multlpl ieatlve nonl1neer elements.

2. DERIVATION OF THE GENERAL FORMULA

ConsIder the multiplicative system shown 'n FIgure 1.

FIGURE 1. EXAIIPLE IIULTlPL1CATlVE SYSTEM

144

It 1s characterIzed by the dIfferentIal .quotlon[~l:

(1)

where xO(t} and XI (t) are the output and Input of the system respec­

tIvely, 0 - ~ ,and fl (0) ~ I - " .••• 4 are polynomial or r.tlonal

functions of D. Assume that the system has dead InItIal states (zero

InItIal conditIons) or Dl xo(t) .0, for 1 • I, 2, 3, .... If F~(D) ~ 0,

Eq. (1) can be sImplIfied as

(2)

where

(3)

end

(~)

Assume that the Impulse responses of Fp(D) end FlfD) are contained In l2. Taking the Fourier transfom of Eq. (l) and applv1nq the frequency convolution theorem to the product term xO(t} x,(t}. we have

where

Xo("') • F[xoft))

XI ("') • Frx i (t)]

let us assume that the Input Signal Is given by

(5)

(6)

(7)

(8)

The 5teady state Input Is

XI(t) -"0 +"1 cos At

Assume that the output signal In the steady state Is

The corresponding Fourier transforms of Eqs. (9) and (10) are

and

From Eqs. (11) and (12). we evaluate the complex convolution term In Eq. (5):

-J+ , + e n{~[w + (n + I)A) + ~[w + ( n - I)A]}J

Substituting Eqs. (II), (12) and (13) Into Eq. (5), we obtain

where

c -. H* 6{w + A) + • ~ H 6 (w + nA) n n ~ n

and

Define

and from Eqs. (15), (18), (19) and (ZO), we have

(10)

(II)

(12)

(13)

•

(14)

(15)

(16)

(17)

(18)

(19)

(zo)

(zl)

(ZZ)

(23)

145

(Z4)

Eq. (24) can be rewri tten as

(ZS)

If Bn ~ 0, we have

(Z6)

Eq. (26) can be expressed In continued fraction form:

_8n __ 1 _ ~ F UnA) - a - _____ ..>..!...L_ ----,;---- (Z7)

[ (;4 1 8n a l 0 0 ( \ Z _. _ f., Fo[{J(n+I)A] 0 FO[j(n+Z)A] - a

O

. From Eq. (Z3) we have

81 Therefore, the describing function defined by N ., - 15 given by

"I

"I _ a _ r

o 81 S;

Substituting Eq. (ZO) for n - Z Into Eq. (30), we get

bO + FZ(jA) N---------~--~-______ _

G(jA) __________ ~h'_Z ________ _

hZ G(Zj.) - ---------'~-------

hZ

G(Jl·) - -----....!!..-----

where

(z8)

(Z9)

(30)

a l h -"2 (33)

Slnee the DC term bO is unknown and must be determined from Eqs. (22) and .(31). we separate N as follows:

where

and

N - N N " D

ND ----------~---------hZ

G(l') - --------"'--------

hZ G(ZjA) - -----!!------

hZ G(3l·) - -----....!!..----

(3~)

I3S)

Bl By definition N .. - , therefore

a l

Let uS def i ne

where FIR' FII

, NDR

, Npi are all real functions of L Ther. from Eqs.

(34) and (37) we have

and therefore

or

Eqs. (41), (1,2) and (43) are the general forms of the DC term and the describing functions of the muhlplicative system.

3. CDNSIDERATIONS OF SOME SPECIAL CASES

Ca) ',The Input signal h.5 no DC component or

ao - 0

then

and

(b) FO(O) - 0:

b __ 2aO FI(O) + a~[NOR FIR - HOI F II ]

o 2 2.0 + at NOR

(c) 80th.O - 0 .nd Fo(O) - 0:

(37)

(40)

(4t)

(42)

(43)

(44)

(45)

(46)

(48)

146

We note that:

(a) When FO(O) - 0 and F2(j),) Is frequency Independent (a con­stant or zero), then FlU).) = FI(O) - FIR and FII = O. From Eqs. (46) and (47) we have bO ... -F, and N .. 0 which is a trivial case.

(b) When Fj (j).) Is a real function of ). (for example, F, (0) _ 02) then aO and FO(O can not both be zero. Otherwise, from Eq. (49), N .. 0 which is also is trivial case,

(c) N is frequency dependent in general, as Is bOo For the case

where both aO and FO(O) are zero the phase angle of N Is always r [from

Eq. (49) J.

4. ILLUSTRATIVE EXAMPLE

Consider the servo-system with multiplicative feedback as shown In FIgure 2. let us evaluate the descrIbing functIon of the part of the system in the dashed line bloc.k tn FIgure 2.

FIGURE 2. SERVO-SYSTEM WITH MULTIPLICATIVE FEEDBACK

The input-output relation of that part of the system Inside the dashed 1 Tnes Is given by:

Assume that the Input signal has no DC part or aO - 0, then

FI (jw) - ~[w2 + 40 + tSJwJ w + 400

From Eqs. (4S) and (49), we have

10K NOI 2 bO - ---[tSw - - w - 40]

w2 + 400 NOR

.nd

where NO' NOR' and NOI are defined by Eqs. (36) and (39).

N T and bO

versus frequency ware shown In Figures 3 and 4 respec-

tIvely. Some selected data for the non 1 'near system and the corre­sponding values from Eqs. (53) and (54) are shown for compartson fn Table I. The results are quIte satTsfactory.

w

1.0

1.5

2.0

2.5

TABLE I

COt\PARISON OF DESCRIBING FUNCTION FREQUENCY RESPONSE AND NONLINEAR ANALYTICAL RESULTS FOR a l - 1.0

N[Eq. 54] N [Ana t yt I cal J bO[Eq. 53J bO[Analytlcal]

INI L!! INI L!! 0.37 -90" 0.37 -90" -0.74 -0.75

0.24 -90" 0.24 -90" -0.S9 -0.69

O. tS -90· O.IS -90· -1.09 -t .09

o. t~ -90· O.I~ -90· -1. 3~ -t .22

(50)

(51)

(52)

(53)

(54)

•

50

~O

-'00

1r

.

0.02 0.05 0.' 0.' 0.5 '.0

--"---

~

.; .; 0 0

· · · . • · • •

~ (

I-- (

\

.; 0 0 .; . · · .

• • · • I

FIGURE 3. DESCRIBING FUNCTION FREqUENCY RESPONSE FOR VAR I OUS VALUES OF I NruT SIGNAL

0.02 0,05 0.' 0.' 0.5

--"----5

t·· ~

J 0 0

.; .; . · . . • • • .-•

•

\ \ 0

" t-- ~ ~ -.

1\ -,

-J

-. .; .; 0

-)0 . · . . .- .,?

-)5 1'-

'.0

FIGURE~. DC COEFFICIENT FOR VARIOUS VALUES OF INPUT SIGNAL

REFERENCES

[1] J.E. Gibson, "Honllnear Control System,lI "'cGraw Hili Co., 1965.

[2] R. Sridhar. "A General Method for Deriving The Describing Functions for A Certain Class of Honllnearitles," IRE Trans. on A.C •• Vol. AC-5, 1960, pp. 135-141.

[3] L. A. Pipes, "Applied "'athematlcs for Engineers and Physiclsts,1I Hew York: McGraw Hill Book Co., 1946, p. 321.

[J.,] R. lanber, "A New Method to Derive The Describing Function by Certain Nonlinear Transfer Systems,1I IFAC, 1964, p. 14.

{51 R.J. Kochenburger, "A Frequency Response Method for Analyzing and Symbolizing Contactor Serovmechanlsms,I' Trans. AlEE, Vol. 69, Part I, 1950, pp. 270-284.

[6] L.C. Goldfarh, liOn Some Nonlfnear Phenomena in Regulating Systems," Automation and Remote Control, Vol. 8, 1947, pp. 349-J8)i English Translation, National Bureau of Standards, Washington, D.C., 1952.

[71 Chen, C.F. and I.J. Haas, tlAn Extension of Oppelt CrTterion,lI IEEE Transaction on Automatic Control. 1965.

[8] J.E. Gibson, "Nonlinear Control System," McGraw Hill Co. 1965.

[9] J."'. Leoh, "Frequency Response,lI Edited by R. Oldenburger, New York: Hacmlllan, 1956, pp. 260-268.

[10] N. Minorsky, "Theory of NonlInear Control Systems," New York: McGraw HI", 1969, pp. 64-70.

C.F. Chen has been Professor of Electrical

Engineering, University of Houston, Houston,

Texas since 1966. He receIved the Ph.D. Degree

from Cambridge UnIversIty. He co-authored a

147

book on Control Systems AnalysIs and has authored

more than 50 papers on IdentIficatIon, stabIlity

and model reductIon. He Is a consultant for the

U.S. Army MIssile Research and Development Com­

mand, Redstone Arsenal, Alabama.

Robert E. Yates was born In CecIlIa, Kentucky

In 1936. He receIved the 8.S.E.E. and M.S.E.E •

Degrees from the University of Tennessee In 1960

and 1963 respectIvely. He receIved the Ph.D •

Degree In Electrical EngineerIng from the

UnIversIty of Houston In 1972. He has worked In

the analysis and desIgn of control systems for

IndustrIal and mIlItary systems sInce 1960. He

Is presently a Research Aerospace EngIneer wIth

the U.S. Army MIssIle Research and Development

Command at Redstone Arsenal, Alabama where he

Is presently Involved In large scale simula­

tIons of termInally guIded tactIcal mIssIles.

NONLINEAR ANALYSIS OF GYRATOR NETWORKS:

A NUMERICAL EXAMPLE

Miles A. Smither and Manjula B. Waldron Electrical Engineering Department

University of Houston Houston. Texas 77004

Abstract Gyrator circuits have traditionally been analysed with linear transistor models. Such an analysis cannot uncover the bias constraints inherent in a given circuit topology. A numerical analysis based on the equations of canonical form nonlinear networks is described for a simple gyrator circuit which clearly shows the operating constraints imposed by the bias requirements.

1. INTRODUCTION

Several gyrator networks have been proposed (1). (2) which demonstrate gyrator properties when ana­lyzed with linear transistor models. The linear models are obtained at the expense of bias infor­mation and are. therefore. incapable of detecting the restrictions which transistor nonlinearities and bias requirements place on the range of useful port variables. This paper treats a simple gyrator network (3) using the nonlinear Ebers-Moll tran­sistor model (4) and establishes the allowable operating range for the circuit. As a result of the analysis. an alternative circuit which is useful over a wider range of port variables is proposed.

2 • BACKGROUND

Lin~ar Analysis. Integrable gyrator circuits have been studied in detail using various linear tran­sistor models (5).(6). Figure 1 shows a proposed gyrator circuit (3) which will be analysed in this paper. If the transistors are replaced with linear models. the following relationships are obtained (for Rl =R2):

il~vl/hfe2Rl + v2/Ra

i2~-vl/~ +v2(1/Rc + l/hfelRa )·

This can be written as R -1

a -1 (hfel R)

which. to the extent that Ra «hfe 2Rl •

~ «hfel Ra II Rc

148

and

Ra=~ describes the operation of a gyrator.

The linear analysis cannot show the restrictions which a particular circuit topology places on the port variables. These restrictions can be quite severe. For example. in the circuit of Figure 1 the collector-base junction of T requires that Vl~V2 at all times. Further restrictions are un­covered with the nonlinear analysis.

Nonlinear canonical form networks. Equations based on nonlinear canonical form networks have been developed recently (7).(8). These require that the circuit be treated in two separate parts. linear and nonlinear. The circuit in Figure 1 is in can­onical form where all of the nonlinear elements are to the left of the dotted line. Such networks can be represented in the form (9)

A F(v) + 1!. v = C. (1) In eq~atiO'n (1),- -

!!, ~ = -,Q,,!, + ~ T T (2) where ~=(vl.v2 •••.• v6) • .!. = (il .i2 •.•.• i 6) are the port variables. C describes the sources within the linear portion. and matrices!!, and ,Q, describe the linear network. The transistors are described in the !E(~) term. The port relations to the left of the dotted line are

i = I. F (v) (3) with I a blO'ck-diagonal matrix of 3 (one for each transistor) 2X2 blocks (which result from the cross

OOUPl1~fi:~:r' Eb.,,-.,11 mod.l)

where the a's are the reverse and forward o's in the transistor model and E(~) has the form

T F(y)=(fl(vl),f2(v2),···,f6(v6» .

The functions fl, f 2 , •.. ,f6 contain the nonline­

arities, the re ationship oetween the transistor terminal voltages and currents.

Combining equations (2) and (3) results in ~ y = -,Q, I I(Y) +.f

or equation (1) with h, = ,Q, I·

3. NUMERICAL ANALYSIS

Equation (1) can be solved for the v vector as -1

Y = ~ {.f - h,I (y)}

which allows an iterative solution to the network equations. The gyrator circuit operating point is found in this manner for each combination of port variables (Vl,VZ) studied. The following para­meters were used for the numerical analysis:

Collector saturation current ICBS

=5pA

<1 =.5 r

<1f =·004

V =15V cc Ra=~=lKn

Rc=Rl=R2=10Kn

Note that the emitter saturation current is ob­tained from (4)

IEBS=ICBS Clr/Cl f · Figure 2 shows the results obtained by setting V2=13 volts and sweeping VI from 5.5 to 10 volts. Tfie figure illustrates a severe bias constraint at V1",7.8 volts. At this level T2 saturates, changing tfie sign of the response of I Z as a function of VI and rendering the circuit useless as a gyrator. For Vl <7.8 volts the circuit is properly biased and behaves as expected. The slope of 12 indicates a gyration impedance of -ll06n, a reasonable result for the circuit values assumed. The response of 11 to VI (for Vl <7.8V) is very weak as expected.

Similar results were obtained by setting Vl =7 volts and sweeping V2 between 7 and 13 volts. Over this range the slope of 11 indicates a gyration imped­ance of 1014n. I as a function of V is nonlinear in the vicinity ot V =7 volts, a result of the zero collector-base Eias voltage for Tl . Over the remaining range of V2 ' 12 indicates an undesired input impedance at port I of 8.83K due, of course, mainly to Rc'

4. CONCLUSION

This paper has demonstrated that it is relatively straightforward to cast a given transistor circuit into a form which is amenable to numerical solution. The advantage of this type of analysis is the com­plete characterization of the circuit, the results relating to both bias considerations and small signal performance. The small signal parameters at the ports of the gyrator circuit can be directly calculated from the numerical results. Of greater

J49

practical importance, however, is the ready identi­fication of allowed operating regions for the cir­cuit. For the circuit of Figure 1 it is possible to identify bias constraints by inspection. For more complex circuits a complete nonlinear circuit evaluation is essential. The use of canonical form equations is one method of obtaining such a com­plete evaluation. By considering the bias con­straints of the circuit in Figure 1 a circuit with much wider operating margins can be obtained. The circuit of Figure 3 has several advantages as a straight gyrator circuit. The inputs can operate around ground allowing direct signal coupling if desired. The signal level at one port has no ef­fect on the level requirements at the other port. It is clear that for Rl=R2=R3/2 and for the usual approximations for the transIstors as current mirrors (T3-T

5, T4-T6) and current sources

(T7-T

8-T

9) the operafion of the circuit is that of

a gyrator over very wide ranges of e l and e2 . The numerical analysis of this circuit could be handled with the canonical form network equations or with a special purpose computer program such as SCEPTRE (10). The special purpose program has the obvious advantage of relative ease of use but cannot provide the insight into the circuit that comes from developing the equations as outlined above.

5 • REFERENCES

1. W. Heinlein and H. Holmes, Active Filters for Integrated Circuits, New York, New York:' Springer­Verlag, 1974, p. 306.

2. J. A. Miller and R. W. Newcomb, An Annotated Bibliography on Gyrators in Network Theory, R-72-01, University of Maryland.

3. W. New and R. W. Newcomb, "An Integrable Time­Variable Gyrator," Proceedings of the IEEE (Cor­respondence), Vol. 53, pp. 2161-2162, Dec. 1965.

4. W. Heinlein and H. Holmes, ~ cit., pp. 236-237.

5. B. D. O. Anderson, D. A. Spaulding and R. W. Newcomb, "Useful Time-Variable Circuit-Element Equivalences," Electronics Letters, Vol. 1, No.3, pp. 56-57, May 1965.

6. B. D. O. Anderson and R. W. Newcomb, "A Capa­citor-Transformer Gyrator Realization," Proceedings of the IEEE, Vol. 53, No. 10, p. 1640, Oct. 1965.

7. 1. W. Sandberg and A. N. Willson, Jr., "Some Theorems on Properties of DC Equations of Nonlinear Networks," Bell System Technical Journal, Vol. 48, pp. 1-34, Jan. 1969.

8. A. N. Willson, Jr., "New Theorems on the Equations of Nonlinear DC Transistor Networks," Bell System Technical Journal, Vol. 49, pp. 1713-1738, Oct. 1970.

9. A. N. Willson, Jr., "Some Aspects of the Theory of Nonlinear Networks," Proceedings of the IEEE, Vol. 61, pp. 1092-1113, Aug. 1973.

10. H. W. Mathers, S. R. Sedore, and J. R. Sents, "Automated Digital Computer Program for Determining Responses of Electronic Circuits to Transient Nuclear Radiation (SCEPTRE)," IBM Space Guidance Center, Oswego, N. Y., IBM File 66-928-611, Feb. 1967.

6. BIOGRAPHIES

Miles A. Smither received the BSEE (cum laude) and MSEE degress from the University of Houston in 1967 and 1968 respectively. From 1968 to 1975 he worked for General Electric and Geosource Incorporated. Since 1975 he has been an instructor in the Elec­trical Engineering Department at the University of Houston and a solid state circuit consultant to local industry. Mr. Smither is a PhD candidate at the University of Houston and is a member of the IEEE, Sigma Xi, Tau Beta Pi, Eta Kappa Nu, and Phi Kappa phi. His current research is in the area of electron beam-material interaction.

Manju1a B. Waldron received her B.Sc(Hous) in Physics and B.E. degrees in India and M.S. and Ph. D. degrees in Electrical Engineering from Stanford University, CA in 1962, 1965, 1968 and 1971 respectively. From 1970 to 1974 she was a lecturer in the School of Electrical Engineering at the University of New South Wales, Australia. She joined the faculty at the University of Houston in 1975. Dr. Waldron is a member of IEEE, Sigma Xi, and *SEE. Her research interests in electrical engineering are Nonlinear systems modeling, speech and psycholinguistic processes of the deaf, bio­engineering and engineering and engineering educa­tion for women and the hearing handicapped.

150

R2 ~+---~+~--1--------v~~+VCC

151

+Figure 1. A Simple Gyrator Circuit in Canonical Form.

<l 9 <l E E N ... ...

8 -1.0

7 -2.0

Vz= 13 VOLTS 6 L-__ -'--__ --'-__ -'-_..=......_---''------.J - 3.0

5 6 7 8 9 10 II VI (VOLTS)

Figure 2. Nonlinear Analysis Results from the Circuit of Figure 1.

Figure 3. Proposed Gyrator Circuit with Relaxed Bias Constraints.

,I

I'

i'

,I, ,i

LIE S:ERlE3 AND THE roH:ER SYSTEl·; SUmLITY FROBLOO

R. K. Bansal Assistant Professor Electrical Engineering Departnent LA.U., Ludhiana (India)

R. Subramanian Assistant Professor Elect. EnGg. Department I.r.T., Kanpur (Irxiia)

Abstract

The moethd of Lie-series and pattern recoCnition technique are used to estimate the stability domains for po./er system problems. Feasibility of the proposoo method is demonstrated by considering wo munerice.l exar,1pl.es.

1. urmODUCTION

Several approaches have been proposEd to determine

the stability domain for power system problems

during the last decade. Amonr; these, the method

aimed towards a solution of Zubov's partial

differential equation has been considera:l recen~

by Yu & Vongsurya[n and DeSarkar and DharllIl

Rao [2 J. Theoretica]ly the Zubov' s method always

yields a Iuapumv function and if a closed form

of solution for the Zubov's partial differential

Equation can be obtained, the stability domains

are exact. However, in general, it is not possible

to find a closoo form solution. Hence a truncated

series form of solution is us(xi which leads to an

approximate stability boundary. The technique

used for the solution, utilizes a power series

method which does not have uniform convergence

and a large munber of terms are generally

necessary before one obtains the final solution.

A recent work by furnand and Sarlos[3] suggects a

nSoi method of solving the Zubov' s partial

differential equation by the use of Lie-series.

In this paper, the application of this method to

power system stability problems has been

investigated. PO"Ter system problems consisting of

a single ~chine connectoo to an infinite rus

(Wi th governer and flux decay effects includa:l)

152

have been investigated am the results obtainoo

are quite encouraeing.

2. LIE SIDlES TH:lr.:1QUE APPLICATION

2.1 NA.TlIE.:ATICAL BAcJ:anomm Lie series solution for the differential Equations

are essentially the power-series solutions and the

expansion co-ei'ficimts are derived from alrebraic

recl:rsion relations.

The Lie series is definoo as

exlta) x = ~ ... (1) - k....-Q

where ~ is, in general, a vector variable and D

is a linear operator defined by

I' a D = E 9.(x1, Xz, ••• , x ) -,,---

j= 1 J r x J

... (2)

where Q _ are an~tic functions of the variahle x J j

The solution to the differnntial equation

~ = f(~) is then obtained as:

?!(t) = ex:/ tD) ~(O)

2.2 Z1.~OV' S PARTIAL DI:r'FEnE:rrIAL lllUATION SOLU'ITON

... (3)

• .. (1,.)

Zubov's partial differential equation is r;iven

hy:

Lie_series technique can be a:)rJlied to obtain the

solution of this equation as follows:

Define a function ~(~) such that

V(~) = 1 - e'vi(~) ••• (6)

which implies that

~(~) = ct (!) ••• (7)

Thus 0 ~ V(~) (1 correspoIrls to 0 ~ w(~) > - CD,

and t-J(~!) = - mat the stability boundary.

Equation (3) and (7) can now be considered

simultaneously and the sol.utions ~(t) and Vl(~(t»

can be obtained by recursive computation.

The solutions H(x(t» and x(t) are expresse:l as: tD - -

w(x( t» = e W(~(O» ••• (8)

••• (9)

The operator D for this system is

D = £(~)"'lx + + (~) : w ••• (10)

3. Bi. TI' Elm RECOGl;ITION CONCEPTS

Pattern recognition algorithms are use:l to

generate an analytical expression for the hyper­

surface that se~retes two sets of points in the

Iilase space. One set of points is obtainoo. f'rom

recursive apIllication of Li&-series technique

indice.ted earlier. This set reprfo.-Sents ( for

sufficien~ large negative W) a close approxi­

mation to the actual boundary of the recion of

attraction. A seconi set of points is obtained

by arbi trarily choosing a small constant radial

dis~canent from the first set towards the stable

equilibrium point (or towards interiol' of

stabili ty domain). The s eparcting surface, in

effect, acts as a decision sur.f£.ce and provides

a guaranteEd region of stability.

The decision surface can be approximated by an ~ degree polynomial F(~) as follows:

F(!)= CO+P1(~)+ P2(!)+ ••• + Pr(~) .•• (11)

ware P. (x) is the :i:th degree homot>"eneous

1. - '" polynomial of ~.

The problem involved is in ccmpu~inC the constants

fer the pol:rnomials, which depend ulJOn the ~ber

of pattern classes, di."'lensions of pcttern s~ce

ani the decree or polynomial F(~).

These constants are evaluated by usinc Ho-Y.ashyap

algori thIJ, details of which can be found in [4].

/+. l!XiI.1TLES

(a) A third order system repreJSentine a lll[,chine

with const.ant dat1pin[ and a ve10ci ty governor will

be inv6stigated. The suill[: tquation under post

fault conditions is givE-n by [0]:

d\ ib dt3 + .943 dt2 + (. 714+cos6 ) ~~ + .667 sin 6

-.196=0

The singularities of interest are

••• (12) ,

1) Stable(focus-node) b =17.1°, b= i""= 0

2) Unstable(focus-node) b =162.90 , b= b= 0

After shiftinc the stable Sin...,'·lUllrit:· to the

origin through the coordinate transformation

° x1 = b -17.1. The state variable form of

description is

*1 = x2 *2 = x3 13 = - .943 x3 -(.714 +cos(x1+ 17.1°» x2

-.667 sin (x1+'7.10

) + .196 ••• (13)

Dharma P.ao [7J had observed while studying a

second order SystOOl that the fault trajE:Ctory in

genaral is r(':stricted to the first two quadrants

of the ~-X:2 luane. It was accordinGly felt that

for the considera::! third order system, it may

prove sufficient to obtain that segment of the

decision surface in the x,-x.z-x3 f-h.:'.se space w'··ich

lies in hcl.f s~ce X:2 > 0, x1-~ arbitrary. This

feature allor-,s si~:nificant reduction in computation

time arxl is to be advoc.::.ted especially for higher

order systems. Initial values of xl'x2,x3 woro

chosen to be (1450~ 0,0). The Lie seri~~

integration routine was EllCecu ted and various data.

points obtdnoo. at intervals of At = 0.1. From

the various solution points, twenty equispaced

points were selected in the recion of interest and

153

thes e were considered r;!erhcrs of clc.ss '1'.

'·(;flbors of (!lass '2 1 were grneratErl by a

displ",:anent {) = .03. r

A second deerE·e polynomial ~TaS considerErl as a

rossiblc dt.'Cision surface and the r(JSc'.lting

decision surf£'.ce obtained is:

F(~) = -67.53+39.89 x1+05.C6 x2+52.1~ Xj-5.) x~

-17.42 Xzx1-12.13 x1x)-9.8) ~

-11 • 13 Xzx) -1 • 88 x~ = 0 ••• (14.)

:.'he critical clearing tiJTle obt..?inErl ~J

in~~ecratin.z the caul te:: systCL1 equation and

notinG time for first crossire F(~) = 0 provides

a value of 11 .55 unit:). 'rIns compe.res very well

with the exact vclu(l of 11.65 units [b].

(?) flux decay effects and variable danpin;:: in

the synchronous machine are considerod. ilie

machine e:rua tions are [2] :

2 E' V ,f. (x -xci) 1:.!i1> = F.- ....9.-Sin6 + q sin 26

dt2 ~ xd l 2xd' Xq

-DC 0) £.Q.. dt

dZ E E ("c - x ) allli -s t = ~ _ ~ + • d d 1 V cosb ••• ( 15)

. dt TdO I Td , xd TdO'

,.here v<:criables J';, Pi etc Imve usual meaning and

are definErl as given in [2J. liurnerical data arH

borrowt.'C! frov{2 J and the state variable fom of

eqllations are

Xl = Xz *z = 2r.b1-8/;.99CL18+x

3) sin(x'+'/'Z8)

+2'.5)Sin2(X1+.478)-Xz("O~ sin~(xl+.478) +.G~2cos (x

1+.478))

*3= .36-.621(1.18+X;) +./;21 cos(x1+.478) ••• (16)

Equations(16) have been obtainro after shifting

the stable sineulari ty locf'. tcrl at b =27 .L;.o ,6 = 0,

E , =1.1(; to the ori{!in. The unstable q

sin[.w..ari ty is loca too at c= 13)°, ~ = 0,

E =.115. '!he Lie series solution is 0 1

c~nerated using initial values of x1 'Xz,~ as

(104.2°, 0, -1.061). It was noticed, however,

that for this example, while iL1IiLementina the

TIo-Yaslwap algorithm, convereence faile:l to occur.

:. possible reason coulrl be attri buted w the

nature of the eenerated dI.. ta points. In viEM of

154

this dirficd t.y it was decided to obtain wo

decision ='faces, one in the x1-Xz plane and the

other in the Xl-X) plane. Second decree

polynomials were considered as decision stu'flces

and the dec:i.sion surfB.ce obtained in the x1-Xz

plane is

Fl(~)= -172.3-76.5 x1 + 1.02 Xz + 94.69 x;

2 + 5.94 xl~ + 1.1,15 Xz = 0 ••• (17)

ilie decision surface in the x,-x3

plane is

Fz(~)= 55.23-15.92 Xl + 207.12 x3+4.95 x~

-67.04 x,x3 + 43.48 ~ = 0 ••• (18)

futh Fl and F2 define decision surfaces that are

confinOO w a two dimensional plane and the

follOWinG procedure was adopted to determine the

cri ti cal cl earing time.

step 1 : The faultErl system is intecratoo and the

set of points (x1,x2,x

3) defirJ.ng the

faulte:d trc,jectoI"J are obtained.

step 2 : At rech tiro.c instant t=k ~t, the value

of F 1 and F2 {!iven by(17), (If:) are

comp.lt 00.

step 3 The faulted traj ector.r integeration is

terminated as soon as either of F1

" or

F2 becomes zero.

Step II': The particulnr time instant t = k Llt

obtained in step 3 furnishes an estimate

of cri tical clearine time.

The value of the cri tical clearing time thus

obtainOO Was 0.28 seconds. This is com~rable

wi th the exact value of 0.32 seconds [b J . 5. CONCV':-SIOH

~~oosibili ty of application of Lie series and

pattern recogni tion me~hods have been

deI!Dnstr: ted for some third ordEr power system

problems. The stability regions compare

favourably as evidenced ~J COl'lIXlrison of critical

clearine title wit:. the exact· ... ' values.

b. REFi-RBICES

Y.lI.Yu and E. VOllo""Surya, "1:onlina:r power system stability suty by J~lapunov i'clnction and Zubov's method", IEEE Trans. PAS,vol.<:6 1~.12, pp 1420-14£5.

1 I

2 De-Sarkar, AS. and Dharma Rao,n. "Zubov's r.le~;hod and trnlSient stability problems of power systems", Proc. lEE, 1971, 118(8). pp 1035-1040 •

.3 :Jurnand,G. and Sarloz , G., "Determination of the domain of stability" Journal. of Hath antlysis ond ap:)lication, 2.3, 71/v-722, 1968.

4 funsal, R.I.. and Subramanian,R. ,"stability analysis of power systcr:JS nsing Lie series and ~ttern recor,nition techniques" Prof.lEE.1974, 121, (7), pp b23-629.

5 Li,C.C. and [orD1an:i.~"J., "Decision surface estiwate of no~lincar system stability domain by Lie-series I!Icthod", l~:E TJans. AG-17 1972, ;:n 66&-608.

b Tl..'1rlSal, H.r.,"EBticw.tion of stability domains for for the transient stability investigation of ?JCJWCI' system", i'h.D. ThesiS, Etectrical EnGZ· Daptt, l.l.T. }:~mpur, Aucust 1975.

7 Dharma Hac, [/., "Generation of I,yapunov functiOns for the transient stability problem" tranS. ll1(:C. Instt. of Can da, 19l1S, C-.3 (11).

7. BIOGRI.FRY

Dr. R. K. j}.nsa.l cO!:i,lleted his S.E. and ; .. E.

e in £].ectrical Engineering fro!:\ Birla Institute

,)f Tcchnolo{!y and Science, Pilani, India in 1909

and 1971 respectively. COI:Jplctai his Ph.D.

in Electrical Engineer1ng fror.l Indian Institute

of TechnoloCY, ICanpur, India in 1977. Presently

workine as j,ssistant ?rofessor of Electrical

Eh&im:erine at P.A.U. Ludhiana, lIdia. l-ain

aren of interest is application of system theory

techniqulll to power systB':1 problems.

155

11

Ii I:

, I

:il ,

-- i·

tl.b.Kekre, li..I.:.t.umar and .Il.h.~rivastavR

Co:nputer Centre

Indian Instttute of ~echnology, .oombay, India

A varyinL efficnency rnultiberver queue model has b:en ana.ly::>ed un­der l'~rkovian absumptions for service time and arrIval pattern. 1quution for probability den~ity func~ion of the waiting ~ime has been obtained as a function of system demand rate and maXImum num­ber of servers. Average 0:; standard deviation of the v'tli ting time from this equation are - in agreement witt- simulation resul ts.

1. INThCDUC'l' IOl~

1'lul ti server queu£:; mO<.1els stud ied so far

(1) aSstune constant server

Lo~ever, in Some practical

t~e server efficiency is

efficiency.

si tuations,

d ep endan t on

the number of active servers in the sys­

OP'a. A mul tiprop;rammed computer system

m:.'y be one of the examples where this

5i tuatiO(l (;xists if each active slot in

thcl mul tiprobrammj.ng system is assumed

to be a server. In a multiprogra~med

computer system, more "than one j ODS sha­

re the resources of the syste"l so as to

ach ieve concurren t operat ion of CPU and

its peripheral devices; the number of

active jobs define the level of multipr­

ogramming. In the model studied, tt.e

eificiency of each server varies and is

gi ven by Ui/i where Ui is the Sum of the

efr'iciencies of i ~ctive oervers. This

notation has been used so that the appl­

ication to the multiprogrammed computer

system becomes straight forward. Ui

wou­

ld represent the overall efficiency at

multiprograllll!ling level 1. :,rhe mo,lel is

156

analysed under ~arkovinn assumptlons for

system demand rate and Rrri val mecr.,mi far ..

2. QUEUE MODJ:"l.J

The model is shown in l:'ig.l. Each serv­

er works at an efficjency ot' Ui/i wten i

:.;ervers are active &: U, is the cum of I

efficiencies of i Berver~. The Gervicp

time requirements of the custom0rs to

the system has ben sf:sumed to hnvc a

negative exponent ial d tstri but10n wi ttL

an average of t. Th'~ arri val of cUbtc:rn­

ers is aHsumed to b~ a Poi~son process

wi th an average of 1\ arri. val b per unt t

time. The queueing iiscipline 1s first

come first served (YCt~). The actual

servi ce time will v~ry rlepen'iing upon

the m.unt;er of active :;erven:l, tt,e maxim­

um number of serverb oeing i". Therefore,

a term 'system deman(i rate' (::lDh) hru;

been used in place of traffic inten~ity;

system demand rate halo been d(~finerl t?

be the fraction of the time one server

w 111 be kept busy if the "f fici eney of

the server is unity. ior the systHm to

be p.rgod ic, averaf:e i.)U.l:t I.3hould b e les~

t;, ~n aVf>rage b~rvice ti me &.vailable • .r'or

t\,") above merlti oned j·jarkcvian a~su'Ilptio­

lle;, average ;;)lJh (~=~) and tP-,rvice rate u i.s U . where lJ is thp SUll! of effici.enceo

lV. !V! 01 sp.rverG ... hen Rll ,., t;"'l'verll !ire bUf'Y·

'lherlHClre, for tr.t -..,ystcl:I to tJe E'rgodici

should be 1 .. S8 than U,,,. 1'1

',( hi;, monel cm' UP a.l'alY.Jed uj the well

k Hown td 1'1, 1 ar. a d ea t·; proce GS \ 1) llnd cr

thd ad umpti or, tha t C'H,tom'::l's not being

servea can fl'(,~1 queuE: 01' infi nj te length.

' .. h,,: stat., tl'ansi t<or; riiabI'am t'al: bE."n

shown ill .r i ,:;. ~~. jo'or all. k > 1'1, LJk==U~l &.b

then the :.lel;',,' ...:erver~J are N. 'lhe ste-

:~d'y iO .. 8 ;,:; :,uu,ysi G gi Vt-!S

1-' n = .t' n-1 0 < n '1-, , .. ( 1 )

~ 1" n> ji. U1'1 l1-i

[

~ ~c: ~l'1-1 1 + U 1 + u~ U 2 +... l.i 1 U 2 ' , • UM_ 1

~L, 1 -1 , . + 77"'"'r.,_._--.-. ---1 ' .. (C!)

0,d.., ... U" \ 1- !u!) c: H 'I

wf,ere'" repr, ;sO.tf: the rrol>abi.li,ty of 11

findJng n .ioliB in the- cystern.

~.1 ~L!-; nglime:

If a C1)~;tcm€r needing serv'l ce time wi th

probar;ili ty dellsi ty flJl1cti Oll \ I'df) of

U exp \-ut), on Hrrival finis there arE

alrdady n cu ... t,'mol's jn t.Le <:iYbte"'l then;

a) if n ('" where .(~ i8 'the max imU!Il number

iUlfneul",tely wtth an efficiellC:f of

~erved U

n1-1 ll+1'

And pd!, of tid: v.ait::llis time will ('e

uU n -rl uU 1 = exp \- --ll±- tj n:;r' n1-1

LJ) if n )J'I" t.he cUbtomer wi. 11 have 'to

wai t in tht-' qUf'ue for, (n-1'.+l) CU~)1,oro'~rl,

to li. ~Hrved uefore bPtn~ served. The

pdf of th~ waiting time in

Id be convolution of pdf's

spent in the queup and the

ring actually bein~ served.

tbiR

of

ti me

The

Cuse wou-

tbe ti[r;(-

~·.pent d Il-

n'H' of the t;me when the queue a,Jvance" by 1 i <;

uU1>'l expl-uUr'l tJ • And thUb pdf for (n-H-+1)

advances in qU8UR will be In-M+1 )th conv-

t n - h n-~+l olution Le. In_MTllu'Ur-l) eXPl-uU1,!t). The pdf of the tir:lt.' actually 't:;()ing served

uUJI'i . uUM will be M expl- -M t), so the pdf of wa-

itin~ in this C8Le wtll be

t 'rn-I1,

f (n-l'i)T ()

UU~ UU~I 1 T exp \ (- -f,j-) • ~ t-l ) } d~. .. (4)

If tr.e prouatJilitv of finding n cudomen;

in the ..,.y'Etl~n· is Pn

, t,HlCl the gt>ncr9.l pdf

of waiting time will De 1'1 uU

i uUi

\1'( t) :- L -1' exp\-i- tj .21 _ 1 i=1

t

+ ;f"P j +JIl f'rj~IlUl,)j+1exp(-uD • .T). "v 0 '1 - ,'I

157

uU .. j. uU1· -

r exp ~(- ~'l~)\'t-T){d~ y' 1" )

a ti on lfl. tel' and rp place

reducps to:

P j+i'j uy

P.q Wi t~. or:

... (5)

... (7)

The first mom~nt of the waiting time wi­

II be

1.1:' 1-1 PM ~ 1 -M1 ~+-. 2 u i MUl\'-UM+~1 \l-f/UM)

••• (8)

And the second moment of' the Waiting ti­

me Will be:

2 1\12 :; ~ [ w i. s )] .

de s=o

••• l9)

3. APP.LICAT 1011 TO MULT IPHO<.HtAMMING

~quation ,tl) ~ (9) wer~ used to compute

th~ values of average and standard devi­

ation of response time for a multlprogr­

ammed computer system. Ui would repres­

ent the overall efficiency of the system

at mul tiprcgramm1ng level i and 1>1 would

rp.prE:l:;,ent the maximum multiprogramming

level allowed. The values of Ui

were

computed as below (2)

U ::: i

w . ~1-W)1

1- i

i! L: (-L) j L j=O 1-w • j !

... (10)

'" hel'e w is IIG wait at uniprogramming and has Oeen assumed to be 0.65 ~3). The

valud3 of average and stR.ndard d~via tiO­

na of the re3pon:oe time for dtfff,orent

system detrand rate and maximum multipro­

gramming level have been SrJOWn in b'lg.3.

~his mojel ~~s al~o si~ulated on EC-l030 (;ompLl tel' to check the validi ty of the

ar.alytica,l result:;;. For 'a s~ple of 4000 jobti • .l:'oi ;.>80n arri valt: and negative

\58

exponential jobtimea were generated using

transform technique with equiprobable ra­

ndom number generator. These jobs were

run on the Simulated model and their res­

ponse times found out. The results obta­

ined thereof are portrayed in Fig.,.

4. CQbCLUaIONS

A varying efficiency multi server queue

model has been studied and equation for

pdf of waiting time has been developed.

The results of an application to multipr­

ogrammed computer are in agreement with

those obtained by Simulation, thus valid­

ating the analytical analysis. The vary­

ing efficiency multiserver model is quite

general and could be s'li table for all mu-

1 tj server systems where idle servers help

the busy servers thUd efl'ectively varying

the eff:l.ciency of' service.

rtEFErtENCES

~ 1) .L.JUienrock, 'i,jueueing i.lystem Vol. 1 :

'.cheory' John dley M: 1975

~ 2) S.R.fI'.a.dnic.lq J.J .Donovan, 'Operatj ng

oy Gt em s' i'lauraw hill 1974 (3) l·I.N.J.lehmanj J ..... i:tosenfeld, 'Performa­

nce of a Simula ted ftIul t iprogrammi ng

l::lystem' .Proc Ai<'IPd 1968,.l"J(;C. Vol.32.

pp 1431-1442.

bIOGRAPHIES

Lr. H.D • .rI,,;kre was born on April 4. 1935. He received his graduation in Tp.lecotrJllun­

lcatian Engineering from JaOalpur Univer­

Eity in 195d. de Jid his A.Tech. in Ind­

lutrial Elec troni Cf; in 1960 from Ind ian

Institute of 'l:echnology, J:;omoay. In 196;,

he wati awarded Conmonweal th i;}cholarsbip

by lfovt. of Canada and completed hi s .{'II. ~c

}:ngi.nee ri ng in Con trol clys t e~na from the

Univel's.l -;,y of uttawa in 1965. After com­

ing back to Inaia he compl€'ted hie Ph.D.

:In 1:)70 on .;lYbte'TI Identificat:lon. He is

on tr,f: teachin~ faculty of Indian InsUt­

ute of Technclogy, borr:oay from 1960. At

preHent he i::; guiding a large number of

re~earct Echolars for their ~h.D. on va­

riuuH fidrjs suct. ae ..Jysiem I'jentificat­

iOll, oJimulation, :"igitCil filter", dpeech

prrJcessing, l'icture procebsinb, !~online­

Kr ciyGtc~s, Computer communication syst­

eroL, etc. Two cr the reueBrch schclars

hbvp. already comr,leted their rh.Dls. he

h&; .. pUlil.ished over fifty re"earch papers

in vc;.rioub international journals.

hr. h.D. l.uoar "a" born on i.Jept.7, 1941. He gradua twi in Electrical Engineering

from Lanar~G rtindu Univer~ity in 1964

an~ ot.tained hi..: masters deeree in Appl­

ied Lleci,ronics in 1967. In 1973, he

wa,_ aloo:o1rded (,;oflffion;%altb Jc;,olqrship and

com[llrted his N. ;;,c. in Comp'-lter :';cience

fro~ ~lliversity of ~Hnchester in 1975.

QUEUE

ne is on the tcachine faculty of Indian

Insti tute of Technology, .dombay since

1968.

h .11. iJri vaBtava wah born on 12th Ivlarch

1939. He received the Jj.~C. degree from

AllahabFFi UniverlJity, .!:l.E. in :r.lectrical

Eng! neE" rio ng from hoorke e University,

l'I.'l'ecIJ. in Applied Electronicti ar.d Ph.L.

from Ll.?, bombay in 1957,1960, 1967

and 1976 respectively. ~rom 1960 to

1967 he worked ~s Lecturer and from 1967

to 1976 as tteatier in Govt. :r:ngg.(;ollege

Jabalp~r. ne worked as ~roject Engineer

in AltEii Project at L 1.'1'., ilomoay from·

~arch 1976 to June 1977 in a special ati­

sigr.mlmt. his subjects of interests are

~ueueing theory application to Comoutpr

bystems, otochastic process etc.

I I

I I I UI/I I L _____ -.J COM PUTER SERVICE

UI- OVERALL UTILISATION FACTOR OF CP 4 AT NUL TIPROGRAMNING LEVEL

FIG.1. QUEUEING MODEL

Adt Adt Adt Adt

FIG.2. STATE TRANSITION DIAGRAM

159

IIJ ::E ~

IIJ III Z 0 II. III IIJ II: IIJ o,!)

'" a: !oJ >

'"

z 0 ... ~ > IIJ 0

0 a::

" 0 z « ... III

100

!oJ ::E i= III III Z ~ 10 III III a:: III o,!)

" a:: IIJ >

'"

1~ __ ~ ____ ~ __ ~ ____ ~ __ ~ ____ ~

·7 ·75 ·8 ·85 ·9 '95 ',0 SYSTEM DEMAND RATE

FIG. 3 Q. RESPONSE TIME (ANALYTICAL)

100

1101 = 10

1101 = 8

~7~---. .I..75""-'--I,.8---~. 8""5----"""9--·.L..9-5-...J, • a

SYSTEM DEMAND RATE

FIG.3c.STANDARD DEVIATION OF RESPONSE TIME (ANALYTICAL)

160

~~7---'~7-5---'~8----~'8-5-~'9---~.9-5---J,,0 SYSTEM DEMAND RATE

FIG3b. RESPONSE TIME (SIMULATION)

10

z 0 j:

~ > M :10 ~ 10 1101:8 0 a:: C§ Z

:5 III

lL-__ -L ____ +-__ ~ ____ ~ __ ~_~ ·7 ·75 ·8 ·85 ·9 ·95 1.0

SYSTEM DEMAND RATE

FIG.3d STANDARD DEVIATION OF RESPONSE TIME (SIMULATED)

1

MODULAR DESIGN

OF THE NETVI0RK WHICH REALIZES ORIGINAL PROGRAM

Svetlana P. Kartashev University of Nebraska - Lincoln,

and Steven I. Kartashev

DCA Association, Lincoln, Nebraska

Abstract

The paper considers modular design of the network which is assembled from n unit modules, and realizes given program. The network con­tains given number p (1 < p < n) of the distinct module types, and is assembled with the use of only pin-to-pin connections.

1. INTRODUCTION

Current trend of large scale integration is to augment the size of one module (IC package). It is therefore expedient to give a broader interpretation of a unit module used as a basic building block =or modular logical realizations.

In the literature on modular logic design it is assumed that a unit module imple­ments either solely a logical function (1-4), or a logial function and th~ flip-

~f1op excited by the function (5-9). The next step in this approach is to assign each unit module with control and execu­tional abilities, i.e. to let it act as a computer module (CH) and contain a control sequencer (sequential machine) and the operations enabled by the sequencer. This means that one output of the sequencer has to enable execution of one operation im­plemented in the executional portion of the CH. It is then said that this C:I ex­ecutes an instruction.

This paper continues earlier research pub-

161

1ished in (7-10). Its difference from published works is that a CH is provided with control and executional abilities. Thus it functions as a computing module which executes the portion of an original program. This portion is implemented as the sequencer contained in the CU.

The following problem is solved: for an original program P, a decomposition al­gorithm is found which decomposes Pinto, n subprogram~. each of which is implemented inside one CM as the transition diagram of an asynchronous sequential machine. Of the total n sequential machine P is decom­~osed into. only p(l~p<n) are described by distinct transition diagrams. The con­sequence of this decomposition is that of the total n CH's which compute P" only p unit module types. CH(l). CH(2), ... , CU(p) are different. This means that CMel) is copied k1 times. CH(2) is copied k2 times, ...• and CH(p) is copied kp times. There­fore. the overall number n of CH's is:

n =:' kl + k2 + ... + kp .

I' I'

II I' ,

[ I

i!

Ii

To achieve computation of the P program,

n CM's are assembled into tLle parallel

network, by connecting input pins of one

CH to the output pins of ot:1er CH's. It

is silOwn that the network N realizes pro­

gram P if the input pins of every CII are

connected with the output pins of otDer

CM's in accordance with a so-called shift

register principle first studied in (9)

for modular logical realizations. Thus

the proposed decomposition algorithm

partitions an original proGram Pinto n

sequential machines working in parallel

and gives the rule of connections among

n CM's into a parallel network ~ w~icri

realizes P. The decomposition algorithm

was completely computerized. It was used

for computer-aided design of trie network

N whicri realized 8 programs containing

altogether 343 vertices. T~e network N

wa~ formed from 28 CH's of two types

CH(l) and CM(2), of whicri CM(l) was copied

12 times and CM(2) was copied 16 times.

2. DESCRIPTION OF TdE PROG&hl1 P

The program P realized by the network N

is presented by the program graph, where

each vertex of the grapri identifies one

conventional program instruction and

arrows identify sequencing among program

instructions. There exist two types of

transitions in the program grap~: uncon-

Figure 1.

Program directed graph.

162

ditional and conditional. Each uncondi­tional transition may be eit~er simple

or conplex. By a simple unconditional

transition a ~ b we mean execution of the

b instruction at the next clock period

after the a instruction. By a complex un­conditional transition a ~ b, c, d, we

mean concurrent executions of b, c, U in­

structions in distinct CH's at the next

clock period after the a instruction.

There exist two types of conditional tran­sitions: simple and complex. By a simple

conditional transition a ~ b; a i c, we mean execution of the b instruction at the

next clock period after the a instruction

if the x sienal is 1 (x = 1) and execution of c instruction at the next clock period after the a instruction if the x signal is o (x = 1).

By a complex conditional transition a ~ b, c, d·, a ~ e, f k n h f 11 , ,~, we mean teo ow-ing: if x signal is I, then execution of a instruction is succeeded at the next

clock period by concurrent execution of b, c, d instructions in pairwise distinct

Cl1' s. If x siBnal is 0 (x = 1), the ex­

ecution of a instruction is succeeded at the next clock period by concurrent execu­

t~on 0-: e, f, k, £, instructions in pair­wise distinct CH' s. [In the program di­rected graph (Fig. 1) the d + f transition from vertex 4 to vertex 2 is th~ simple

unconditional transition; the f + d, a *

transition from vertex 3 to vertices 4 and 1 is the complex unconditional transttion· the a'~ ~ f, a)~ & u transition from vertex'

6 to vertices 3 and 8 is the simple con­ditional transition; the b* ~ a b* ~ w W Of J ,

w transition from vertex 10 to vertices 28, 13, 24, 22 is the complex conditional transition. ]

3. DESCRIPTION OF THE NET\m~ N

By network N we mean a set of interconnect­

ed CM's which work in parallel over compu-

tation of given program P. Each CM is one IC package and is described by the hard­ware diagram of Fig. 2. It consists of

the control part, CP, and executional, part, EP. The CP contains the control sequencer C realized as an asynchronous

(Hoore) sequential machine, one identifi­

cation flip-flop v and the logical circuit Lv which switches v. The EP contains the processor, memory for data word storage

and input/output device.

Every CM works only if the respective

identification flip-flop v=l. If v=O, CM is idle. Hence parallel execution of k

program instructions in k CM's (k ~ n) is represented by an n-dimensional binary

vector ~ = (VI' v2 ' ... , vn ) having k ones in which each ... oordinate vi £ {O,l} iden­tifies the working state of CM.. Vector

l. ~ is called a network state. In the end of each clock period, network N performs ~ + ~I transition from the network state

~to the network state ~'. As a rule,

v I: ~'.

[Let N contain five CM's, i.e., N =

{CMl , CM2 , CM3 , CM4 , CM5 }. If N performs the 1 1 0 1 0 + 0 1 1 1 0 transition,' then

h .l . + 1 2 3 4 5 ~ h w 1. e 1n v l = 1 1 0 1 0, h as CMl , CM2 and CM4 working over execution of program instruction say a, band c, respectively, whereas CM

3 and CM5 are idle. At the next

clock period N establishes the new network·

state ~2 = 0 1 1 1 0, which means that CMl and CM5 are idle whereas CU2 ' CM3 , CM4 execute new program instructions, for

example d, e, f, respectively.]

Thus for each ~ + ~' transition, one of four types of transitions may occur in

every CMi :

(a) vi = 1 for both ~ and ~'; CHi executes one instruction in ~ and one instruction

+' in v. This type of transition is called

the 1 + 1 transition. + +' (b) Vi = 0 for both v and v; CMi works

163

output dynamic signals

I ,egls,e,l· ··1 ,egltter 1 EP CP , , ~ ..

1 ... r-...

lAdder (J I: T P,ocellor

I (Jcontrol dtcodein C [I 0

IVO

T V, ) Signals from

Identlficotlon fl • .. h F F ~,- ,.glsteFl · · Vn_1 flip-flops I r.glster ... il regl.ter I + Lv

(I next-llate IOftien .

} Rondomr oeceu C""o..

I

~Adl m."!ory , lJi\i \,.,J

• word, : warde

, ,

word, worde I doto regl.ter t •

( 1I0)devlce

I

1 · · ·

1j 12

~ "-/ 1

}

input dynamic signols

Figure 2.

Hardware diagram of one eM.

neither in ~ nor in ~~ i.e. it executes the 0+0 transition.

(c) Vi 0 for ~ and Vi = 1 for ~'; CMi executes nO instruction in ~ and one in-

+' struction in v, or it performs the 0+1

transition.

(d) ,Vi = 1 for ~ and Vi = 0 for ~'; CMi executes one instruction in ~ and no in-

+' atruction in v, or it performs the 1+0

transition.

Every 1+1 and 0+1 transitions in CHi are performed under a special signal x.

l.

called a dynamic signal. The Xi signal enables the C-sequencer transition into a new state (new code in the R. register)

1.

which corresponds to a new prop,ram in-

struction executed by CMi ·

Every 0+0 and 1+0 transitions in CMi mean

that identification flip-flop Vi = 0

164

blocks decoding of the state stored in the Ri register. Thus CMi executes no instructions during the clock periods

when Vi = O. Every CHi may generate two types of dynamic Signals: unconditional

and conditional.

The unconditional dynamic signals are generated by the control decoder of every

C-sequencer and they are used for execu­tion of unconditional transitions from one program instruction to another. The conditional dynamic signals are generated by the EP-circuit of the CM.. One may

l. regard as a conditional dynamic signal either an adder overflow, or a counter

overflow. or a comparison signal, etc. The conditional dynamic signals are used for execution of conditiona~ transitions from one program instruction to another.

The dynamic signals generated by CM are

used for switching the C-sequencer of the same CN. Furthermore, the same signals may be fed to other CH's where they switch the respective C-sequencers into new states. {Suppose, instruction a ex­ecuted in CH2 calls for comparison of two words A and B. The result of comparison is identified by signal 3 (A = B) or S (A! B). The 8 signal is produced by the comparator of CM2 , and conceived as a conditional dynamic signal. The 6 = 1, enables three concurrent program tran­sitions a ~ b, c, d, from the a instruc­tion to b, c, d instructions, executed in parallel in CM2 , CM3 , CM4 respectively. The B = I also enables three concurrent program transitions a ~. e, f, h, also executed in parallel in Cl12 , Clf3 , CH4 . Thus in CM2 which executes the 1+1 tran­sition, two alternative program ~ran­sitions are executed a ~ b or a ~ e. The B-signal (8 or S) is fed to CH3 and CM4 . The CU3 performs the 0-.. 1 transition. This means, that if 8=1, it executes c instruc­tion, if B=l, it executes f instruction. Since in previous clock period CM3 was idle, the program transitions executed by CH3 will be denoted as ~.c, and ~'f. The CM4 also executes the 0+1 transition in­terpreted as either ~.d or ~'h.l Thus it follows from our description, that when N assumes a network state V, each CM. having v. = 1 computes one program

J. J.

instruction. If v has t ones in positions 1, 2, ... t, N computes concurrently pro­

gram instructions aI' aZ ' ... , at in CMl , CM2 , ... CM~, respectively. Concurrent computation by N of the instructions aI' aZ' ... , a v during one clock period is thought of as an instruction vector A = (a l .a2 , ...• a t ). Thus the computation of program P by N is conceived as the sequence of executed instruction vectors

A. ft •... , f ..... n, etc.

Note: One distinguishing quality of N is

165

that during execution of the same program p. N can establish the same network state several times, each time computing a new instruction vector. For example, in the

beginning when the network state vI =

I I 0 I I is established first, N =

{CMI , CM2 • CM3 . CM4 • CM5} may compute the instruction vector A = (a.b.O.f.k); in another portion of the program p. estab­lishment of the same vIII 0 I 1 may lead to computing by N of another in­struction vector Bl = (c,d,O,m,t), etc. For all instruction vectors induced by the same system state vI' their zero positions

coincide with those of VI' + +,

For each network transition v -+ v. one must switch to the working mode(enable) only those CM's which correspond to the ones positions of the state v'. To do this

one must generate the set of dynamic signals, each of which enables one or several cH' s. A collection of dynamic signals which causes one network tran­sition v -+ v' is called a vector of dy­

namic signals. [Let VI = 1 1 0 1 0 and v2 = 0 III O. For I 1 ° 1 0 +0 1 1 1 0,

one has to enable CM2 • CM3 and CM4 using the vector of dynamic signals x = (O.x2' x3 ,x4 ,0). Note: A portion of dynamic signals x2 ,x3 ,x4 might be external with respect to some CM's.] Also, network N may perfom several times the same tran­sition v + v'each time generating a new vector of dynamic signals x. Let the same v + v'be executed at two distinct moments of time t and t'. We assume that at the moment t, it is enabled by the vec~r i 1 of dynamic signals, i.e., V ---- v', and at the moment t' it is enabled by another~vector x2 of dynamic signals. 1. e., v __ Z_ v~~ Then at the moment t, N. executes A __ 1_ ~ transi~on and at the moment t' it executes ~ ---- ~ transition, where A. ~, C, Dare instcuction vectors. In future, each

': I

,I

1

~ transition will be called a pro-gram vector transition.

Since working or nonworking state of each CHi is identified by the identification flip-flop v. (v. - IVO)

l l '

vided with the interface each Vi is pro­function F. im-

l

plemented with the logic Lv (Fig. 2). Any function Fi is the function of a dynamic signal Xi fed to or generated inside of CMi and the present system state; stored in the identification flip-flop; vl ,v2 '

... ,vn : Fi f (xi';)'

Each CM is synchronized by two synchroni­

zation sequences '1 and '2' New meaning of the flip-flop v is established at the

moment of Tl-pulse. A transition of the C sequencer to the next state is executed at the moment of T2 pulse, i.e., during the time when all identification flip­flops v l ,v2 ' ... ,vn assumed their new mean­ings, thus forming a new network state. Any program instruction executed by each CHi lasts one clock period TO which is measured by two consecutive pulses '2'

4. THE CYCLIC SHIFT REGISTER PRINCIPLE USED FOR MINIMIZATION OF THE NUMBER

OF PAIRWISE DISTINCT CM'S.

Since each CMi is placed on a single in­tegrated package, one has to minimize the

fill

.1Il

v,

number of distinct module types used for fabrication of N. This minimization is conceived as using only a limited number p of pairwise distinct computer modules CM(1),CM(2), ... ,CM(p) (otherwise called basic CM's) i.e., N uses kl replicas of one basic CM(l) , k2 replicas of another basic CM(2) , ... ,kp replicas of the pth CM(p). [Network N of Fig. 3 has 5 CM's, N

{CM(1)l,CM(1)2,CM(2)3,CM(2)4,CM(2)S} where CM(l)l and CM(1)2 are the replicas of basic

CM(l) , and CM(2)3' CM(2)4' CM(2)5 are the replicas of basic CM(2).] To fabricate k.

l

computer modules by replication of the same basic CM(i) , one has to achieve the total hardware identity in their control and executional circuits. Let us show how this may be accomplished.

The initial program P to be computed by the MCS must be modularly decomposed by a special modular decomposition algo'rithm. As a result, computation of initial program P is reduced to the parallel computation

of n subprograms PI'.·.' Pj , ... , P n' where each P. is computed by CM.. Since N con-

J J tains only p basic CM's, all hardware identical eM's must compute identical sub­programs. Thus the whole set {PI'" .,Pn } of subprograms must be partitioned into p subsets p. such that each p. subset con-

l l

Os CM(2)4 Os CM(2l5

0, CL[31

0, (1[41

09 (1[5:]

aDI 0.(41 Ci(5]

°'0 °'0 0'0

Oil gDI

°Il (41

0" V(51

·3 '. V5

Figure 3. Connection diagram of the network N.

166

tains only mutua~y identical subprograms. In further treatment, a Pi subset of identical subprograms (p. {P., P. , ... ,

~ ~l 1.2

P . . J) will be called the p:-Modular Pro-l.k· :!:

gram (Pi - MP).

Let us show that minimization of the num­

ber of distinct. Pi - MP's results in mini­mization of the number of basic CM's, i.e. minimization of distinct module types which one has to produce to fabricate N. Suppose that the realization of the

Pi - MP calls for the realization of w(Pi) distinct microoperations. This means that

execution of the same Pi - MP in CM's, say, CM I , CM2 , eM3 necessitates that the executional portions EP of these modules contain only w(Pi) micro-operations. Con­sequently, if CMI , CM2 , CM3 realize the same p. - HP their executional parts are

~

pairwise identical, i.e., EP I = EP2 = EP 3 .

We now find in what way one can organize the hardware identity of the C sequencers

of the CH' s. The mere fact that CHI' Ct1Z '

CM3 compute the same Pi - HP does not mean that their sequencers CI , CZ ' C3 are pairwise identical. This originates from the fact that at the same clock period CI •

CZ' C3 sequencers may assume distinct states, enablin~ CMI , CMZ' CH3 for execu­tion of different instructions. For the unconditional transition a ~ b, c, d,

executed in CHI' CM2 , CM3 let the sequen­cer of the CHI perform the a .... b transi­tion under the dynamic signal x generated by the same CMI , i.e., symbolically,

a[ll~ btl]. The x[ll dynamic signal is fed to CMZ and CM3 to enable execution of the c instruction in CM2 and d instruc­tion in CM3 . Thus the Cz sequencer of CMZ

performs x[II~C[2] and the C3

sequencer

of eM3

performs x[I) •• d/3J (Fig 4a).

167

Thus to implement a/l]~b[l), c[Z],

d[3], the CI , CZ' C3 sequencers call for distinct hardware. Thus to achieve the

hardware identity of CI ' eZ' C3 sequen­cers, it is necessary that these sequen­cers contain the same transitions. It may be achieved by supplementing PI' PZ' P3 subprograms by the needed program transi­tions. Indeed, for our case the Pz sub­program must be supplemented by the a~ b transition to be executed in CH2 but at another clock period, i.e.,

a[2] x[21, b[Z) (Fig 4b). The x[2) dynam­

ic signal generated by a[Z] must imitate the x[l] signal in what is seen as the transition of the CI sequencer into the d

instruction ( x[2] •• d[l]) and the tran-

sition of the C3 sequencer into the c

instruction ( x[2] hc[3)). Similarly,

one has to implement the a .... b transition

x(3) in the CH3

, 1. e., a [ 3] .. b [3). The x[3] dynamic signal generat~d by a[3] has

to perform x13] .'c[l] in GMI and

x [3 J II. d [2] in CHZ (Fig 4c). Therefore,

the hardware identity of the CI , C2 , C3 s~quencers can be achieved only if the

PI' P2' P3 subprograms contain ing transitions: for, say, ~l work transition, one performs

the follow-+

... v2 net-

a [1 J x [1 J • b [1], c [2 J, d [3 J for another

network transition, say, ~14 ~ VIS' one

performs a [ 2 J x [Z] .. b [Z J, c [ 3 J, d [ 1), and

for a third network transition, for in­

stance, v30 + v32 one performs

a[3] x[3] .. b[3], c[l], d[2].

The set Ei (a + b) = {a[ill + brill, ariZ] ~ b[i2l, ... , a[ikil ~ b[iki1} con­taining ki replicas of the a + b transi­

tion belonging to Pi-MP, where a[i1 ] +

brill belongs to Pil , a[i2l ~ b[ belongs

to'Pi2 , ",a[ikJ "+ b[ikJ belongs to Pik

, will

be called the complete modular expansion.

For the a "+ b, c, d transition considered

earlier and assigned to PI - MP {PI' PZ' P3}, the Pz and P3 subprograms are to be supplemented by the wanted program transitions. This results in obtaining the following modular expansion EI(a ~ b, c, d);

t [ I J ..2Ull. b [ I] ,c [ 2 J ,d [ .1]

El (a ~ b,c,d) == [2J~ b[2] ,c[3] ,del] (1) [3]2lll b[3] ,c[l] ,d[~J

It follows from the above that any modular expansion E. (a -+ b, c, d) contains k.; mem-1. . ~

bers 1f [i l ], 11 [i2],."., 11 [1.kil, where each 11[j] denotes the program transition a[j] x[j)b[g], crt], d[k] implemented in

the P. subprogram (which is executed by J

the CM.). The index j of the member 1T[j] s~ows the position of the CMj which executes the predecessor instruction a[j]. The a[j] instruction induces the dynamic signal x[j]. For every member 1T [j], each instruction f must be assigned with position ~(i.e. ,f [£]), which shows that the f instruction is executed in CM~.

f----C~ ___ + ___ C_M.::..2 __ -+ ___ C_M-=3:.....-_---1

I I ~ I I

(0)1 ~ I I ~ I I I I I ~--------~--------~----~

I I ~ ~ I

(b) :

I

I 1 I

( c)

Figure 4.

~ ~ ~

Steps of achieving the hardware identity in CMI , CM2 , CM3

",

168

Similar position numbers are assigned to

the dynamic signals.

Assignment of positions for each instruc­

tion and dynamic signal contained in the

member rr[j] £ E. (a ~ b.c.d) is governed ~

by a so-called cyclic shift operator ~.

For the modular expansion EI(a ~ b.c.d)

{ rr[l]. TI[2]. TIr3]} [given by (1)], one

can easily notice that the a instruction

contained in TI[l]. TI[2]. TI[3] is assigned

with positions described by the shifting

rule 1 ~ 2 + 3. The same rule is used

in assigning positions for x dynamic

signal and b instruction. For the c

instruction. the position assignment is

described by the rule 2 + 3 -+ 1 which is

one bit shifted with respect to 1 -+ 2 -+ 3.

The 2 + 3 + 1 rule instead of 1 -+ 2 -+ 3

originates from the fact that the c in­

struction is executed not in CMI but in

CM2

. For the d instruction the positions

are assigned under the rule 3 -+ 1 -+ 2.

two bit shifted with respect to 1 -+ 2 -+ 3;

since the d instruction is executed in

CM3 ·

Thus for the modular expansion E1(a -+ b,

c.d) the position assignment is described

by the cyclic shift operator ~l =

1 + 2 -+ 3. Each cyclic shift operator ~ ~i may assume k i distinct states: generic state ~.O [which identifies positions of --- ~ instructions and dynamic signals in gener-

ic member rr[i l ] of the modular expan-

sion E. (a + b. c. d)] and derivative ~ 1 2 ki-l hI. states ~ .• ~ .•...• ~. • w ere~; ~s

~~~ ~ ~ ~ ~

the state of ~ .• which is 1 bit shifted ~ h . 0 \1/ 2 . with respect to t e gener~c ~i ; Ti ~s

the state of ~. which is two bit shifted ~

with respect to ~.O. etc. For the example ~

considered earlier. ~l = l~ forms the followin8 states: generic state

~l 0 = [(1). (2). (3)] -- because accord­

ing to (1). the generic TI[l] € El(a -+ b. c. d) enables instruc­

tions and dynamic signals in CMl • CM2 •

1 CM3 -- and two derivative states ~l and 2 1 - ~l • where ~l = [(1) ~ 2, (2) -+ 3, (3) -+

2 1] and ~l = [(1) ~ 3, (2) ~ 1, (3) -+ 2].

where each (p) ~ w shows that a position

p belonging to the generic state ~.O must ~

be replaced by the position w in the

derivitave state. Therefore for the modu­

lar expansion EI (a ~ b. c. d), containing

the members TI[l]. TI[2]. TI[3]. we obtain 1 2 that TI[2] = ~l (TI[l]). TI[3] = ~l (TI[l]).

Furthermore. the cyclic shift operator ~i

which describes a modular expansion E.(a ~ b) must be the same for any modular ~

169

expansion Ei(c ~ d. e, f) built for the

same Pi - MP. Thus each Pi - HP is described by its own cyclic shift operator

~i' Therefore. if the initial program P is described by I modular programs

PI - MP •...• Pi - MP..... Pe - MP. then for any Pi - ~W there exists the cyclic shift operator ~. which describes all mod-

]-

ular expansions of this Pi - HP.

Henceforth. for any modular expansion Ei

assignment of the positions to instruc­

tions and dynamic signals for every member

TI[j] (j = i l • i 2 •...• i ki ) must be govern­ed by the following rule: Closure of

Ej(a ~ c. d. e. f) with respect to o/i.

Each modular expansion Ei(a ~ c. d. e. f)=

{TI[i l ]. TI[i2 ].···. TI[iki ]} bu{~t for the

Pi - MP must be closed with respect toP i ,

i.e .• for any TI[j] € Ei(a -+ c, d, e, f), 0/ .(TI [j]) = TI [m], where TI [m] is also a mem-~

ber of the same Ei . [For El(a -+ b, c.d)=

{TI.[l], TI[2], TI[3]} given by (1), o/ll(TI[l]) = TI[2] and TI[2] € El(a -+ b. C;

d), ~/(TI[I]) = rr[3], 'l'13(rr[l]) = TI[1].]

Let us show that the Closure Rule can be

applied for organization of the pin-to­pin connection between k i identical CM's.

In our example for the modular expansion

El(a + b, c, d) built for the CM1 , CM2 , CM3 , respectively, and described by

~l l~, the Cl , C2 , C3 sequencers are transferred to the states c and d by

external dynamic signals. In CMl , CM2 , Cli3 , let the x dynamic signal fed through

the pin 0c transfer each C sequencer into the state c. Transition of the same C

sequencer into the d state is performed by

the dynamic signal fed through the pin 0d' Then, as follows from (1). in CMl the Cl sequencer assumes the d state under x[2]

signal fed to 0d; the same sequencer as­sumes the c state under x[3] signal fed

to 0d' etc. Therefore the following matrix describes feeding of dynamic sig­

nals x[l], x[Z], x[3] to the pins 0c and

°d'

2 f[21

ia ..... Id > .... S; Itt > i!l

..... -~ ,.....

7 5 ;::; 0

..... ::t

27 u[2]. zUJ

(0)

° °d c CMI x [3] x[2J

CMZ x[ 1J xl3]

CM3 x[Z] xlI]

Since for El(a ~ b, c, d) given by (1) the positions of dynamic signals are described

by ~l = 1~3, feeding of the dynamic signals to the pins 0c and 0d is estab­lished as follows.

Let the generic state ~O specify the posi­

tions of dynamic signals fed to the pins

0c and 0d in the CMl • i.e., ~O = «3),

(2». Then for CM2 , one has to feed, to

the pins 0c and 0d' the dynamic signals which positions are identified by ~l «3) + 1, (2) + 3), and for CM3 , the posi-

HI)

~ ...... tt > .... S; ..... Itt ~ > E2 -!!:!. @

8 4 '6

..... N ..... :::lI

9 uUl z[2]

(b)

Figure 5. Sub-programs computed by CM(l)l and CH(l)2'

170

tions of the dynamic signals· fed to Dc and 0d are identified by ~2 = «3) ~ 2, (2) ~ 1) .

4. MODULAR DECOMPOSITION

The modular decompostion algorithm con­

sists of the following stages:

Stage 1: Form sets of identical instruc­

tions otherwise called generic sets

[For program P given by Fie. l.the follow­

ing generic sets are formed: a*(1,6),

f(2,3), d(4,S), u(7,8), b*(19,20,lO), e(11,23,2l), c(lS,17,16), g(12,2S,28),

w(13,24,22), h(18,26,14), z(27,9»)

StaRe 2: Select n, p, and ki(i=l, ... ,p) where n is the overall number of CM's, p is the number of CM's types, k. - is the

~

number of copies of one CM(i) type.

Algorithm: Of all generic sets, form p blocks Bi , each block containing generic sets of the same size ki(i=l, ... ,p). Then:

n = kl + k2 + ... + kp ' P is the number of

g[41

d.[41v0L5J

(0) (b)

distinct blocks Bi . [For our case, all generic sets have sizes 2 or 3, therefore

p = 2, kl = 2, k2 = 3, n = S. Bl = {a* (1,6), f(2,3), d(4,S), u(7,8), z(27,9)1.

B2 = {b*(19,20,lO), e(11,23,2l), c(lS,17, 16), g(lS,2S,28), w(13,24,22), h(18,26,

14) }. )

Stage 3: (1) For every k., give arbitrar-~ .

ily the shift-register operator ~i : i l ~

i2 ~ ... ~ ~k. [For our case, kl = 2, ~ k2 = 3. Ther~fore, ~l shifts positions 1 and 2 (l~), ~2 shifts positions

3, 4, S (3~).)

(2) For each generic set a(i l , i 2 , ... ,

i k ) containing k. members, assign arbi~ . ~

tr~rily one element a(j) to one position j £ {i l , i 2 , ... , i k ) [For a*(1,6) =

{a*(l), a*(6)}, a*(I) is assigned to

CM(l)l' (a*(l) = a*[l)), a*(6) = a*[2); for f(2,3) = {f(2), f(3)}, f(2) = f[l),

f(3) = f[2). etc. Fig's S,6 show all the

remaining assignments made for Bl and B2)

9 (5] g[31

ii [5Jvc1[3J c1[31vG[41

(c)

Figure 6.

Subprograms computed by CM(2)3' CH(2)4' CI1(2)S·

171

I'

Stage 4:

(1) For each original program transition

a + b construct a complete modular expan­

sion E(a + b) which satisfies the closure

rule with respect to ~i. If E(a + b)

contains only original program transitions,

no modifications are necessary. If

E(a + b) contains supplemented program

transitions the modifications indicated

in p. 2 are to be performed:

(2a) For every CMi , introduce position

code i which binary value is one fewer

than i [For CM1 , i = 000, for CM2 ,

i = 001, ... , for CMS' i = 100.]

(2b) Equip every position code with de­

coder containing n outputs r l , r 2 ,···, r n ,

thereafter called recognizers. For every

CM., only r. [i] = 1, the remaining recog-1. 1.

nizers, rj[i] = O. [For CM1 , decoder

contains S outputs, rl-rS . Of those,

only rl[l] = 1, since it decodes i = 00

stored in CM1

. The remaining recognizers

r2[l], r

3[l], r 4 [l], rS[l] are zeros.

For CM2

, only r 2 [2] = 1, etc.]

(2c) For the modular expansion E(a + b)

containing, say, only one original pro­

gram transition, it is required that re­

maining program transitions (which war­

rant the closure rule) be supplemented.

[Let for E(a + b) = {a[3] + b[3],

a[4] + b[4], a[S] + b[S]}, described by

~ : 3 + 4 + S, only a[3] + b[3] be origi­

nal. Since it is executed in CM3

, enable

it with valid recognizer r3

[3] = 1:

a[3] r3[3] )b[3]. Enable a[4] + b[4]

with recognizer r 3 [4] = ~(r3[3]):

a[4] r3[4] ,b[4]. Enable a[S] + b[S]

with recognizer r3

[S] = ~2(r3[3]), namely,

a[S] r 3 (S] I b[S]. Since r3

[4] = r3

[S]

0, two supplemental program transitions

will never be executed.

Therefore, introduced procedure of supple­

mentation is such that no supplemented

172

transition is actually executed. Conse­

quently program P will be executed cor­

rectly. Fie. 's Sand 6 contain all five

subprograms, the original P is decomposed

into.

CONCLUSIONS

This paper presents new mathematical ideas

which allow one to perform a total formal­

ization in designing network U destined

to realize an original program P. It is

shown that the principle of shiftinp, op­

erator f = (~l' ~2'···' ~p) whic~ opera­

tion is equivalent to p concurrently

working shift-registers with total size in

n bits results in the network N containing

n CM' s, of which only p Ct1' s are distinct.

The contributed ideas affect the hardware

identity among modules at the expense of

equivalent supplementation of P.

REfoERE,ICES

1. F. P. Preparata and D. E. Muller,

"Generation of near-optimal universal

Boolean functions", J. Comput. Syst.

Sci. vol. 4, pp. 93-102, Apr. 1970.

2. F. P. Preparata, "On the design of

universal Boolean functions", IEEE

Transactions, Comp., vol. C-20,

pp. 418-423, Apr. 1971.

3. S. S. Yau, and C. K. Tang, "Universal

lop;ic modules and their applications",

IEEE Trans. Comput., vol. C-19, pp.

141-149, Feb. 1970.

4. T. F. Tabloski and F. J. Mowle, "A Nu­

merical Expansion Technique and its

Application to Minimal Multiplexer Lo­

gic Circuits" IEEE Trans. on Comput.,

vol. C-2S, pp. 684-703.

S. P. Heiner and J. E. Hopcroft, "Hodular

decomposition of synchronous sequential

machines", in Proc. 8th Annu. IEEE Symp.

Switching and Automata Theory, 1967,

~p. 233-239.

6. T. F. Arnold, C. J. Tau, and M. M. New­

born, "Iteratively realized sequential

circuits", IEEE Trans. Comput., vol.

C-19, pp. 54-66.

7. S. p. Kartashev, "Methods for realizing

sequential machines on identical inte­

grated circuits", Proc. Symp. Comput.

and Automata, Poly tech. Inst. of Brook­

lyn, April 13-15, 1971. 8. S. P. Kartashev, "State Assignment for

Realizin8 Modular Input-Free Sequential

Logical Networks without Invertors",

Journal of Comput. and System Sciences,

vol. 7, no. 5, 1973.

Svetlana P. Kartashev was born in Kiev,

USSR. She received the B.S. and M.S. de­

grees both in electrical engineering from

the Kiev Poly technical Institute, Kiev, in

1960 and 1961, respectively. In 1969 she

received the Ph.D. degree in computer sci­

ences from the Institute of Cybernetics,

Kiev.

From 1961 to 1969 she worked, respectively,

as a Research Engineer and a Research

Associate in the Institute of Cybernetics.

Becoming a permanent resident of the Unit­

ed States, from 1970 to 1972 she was a Re­

search Associate of the Department of

Computer Science, at The Johns Hopkins University. She is currently an Associate

Professor of Computer Science at the Univ­

ersity of Nebraska, Lincoln.

173

9. S. P. Kartashev, "Theory and Implemen­

tation of p-Multiple Sequential Ma­

chines", IEEE Transactions on Computers,

May 1974, pp. 500-523.

10. S. P. Kartashev, "Parallel Computation

in Modular Computing Systems", Proceed­

ings of International Symposium on Uni­

formly Structured Automata and Logic,

Japan, Aug. 21-23, 1975, pp. 184-191.

Steven I. Kartashev was born in 1934 in

the USSR. He received the B.S., the M.S.,

and Ph.D. degrees from the Institute of

Cybernetics, Kiev, all in computer sciences

(computer systems and architecture) in

1958, 1959, and 1966, respectively.

From 1961 to 1965 he was with the Insti­

tute of Cybernetics, where he developed

and computerized formal design techniques

for constructin8 the control units which

vIere widely used for large computer sys­

tems. From 1966 to 1969, he was an Associate Professor with the Computer Sci­

ence Department in the Kiev Civil Aviation

Institute. Since 1969 he has been a per­

manent resident of the U.S. From 1970 to

1972, he was a Research Scientist in the

Computer Science Department, The Johns Hop­

kins University, Baltimore, MD. Concurrent··

ly, he was a Consultant of the Defense,

Space, and Special Systems Groups, Bur­

roughs Corp., Paoli, PA. From 1973 to

1974, he was Chief Investigato: of the LSI multicomputer system project with the firm

Dynamic Computer Architecture, Lincoln,

NE. He is currently President of this

firm.

, I'

I'

i I I \

I I'

Azuma, S. ------------------------Balakrishnan, A. V. --------------Bansal, R. K. --------------------

_ Benchimol, C. -------------------­Blackwell, W. A. -----------------Branin, F. H. --------------------Chang, F. C. ---------------------de Figueriredo, R.J.P. ----------­DeDantis, R. M. ------------------Desoer, C. A. --------------------Dewilde, D. ----------------------Dobruck, A. L. -------------------Dolezal, V. ---------------------­Fatic, V. M. ---------------------

../ Feintuch, A. ---------------------Foulker, R. H. -------------------Haas, W. H. ---------------------­Hedberg, D. ----------------------Helton, J. W. --------------------Ho, Y. S. ------------------------Ioannids, G. ---------------------Jacyno, A. ----------------------­Jury, E. I. ----------------------Kartashev, S. I. ----------------­Kartashev, S. P. -----------------Kekre, H. B. --------------------­Komkov, V. ----------------------­Kumor, R. D. --------------------­Leake, R. J. --------------------­Levan, N. ------------------------Liberty, S. R. ------------------­Lindquist, C. S. -----------------

-Martin, C. -----------------------Mishra, K.L.P. ------------------­Missaghie, M. M. -----------------Naylor, A. -----------------------Newcomb, R. W. -------------------Phatak, D. B. --------------------Piekarski, M. S. -----------------Porter, W. A. --------------------Pu1ufani, S. R. ------------------Rajski, C. ----------------------­Ramarajan, S. --------------------Reichert, J. D. ------------------Roe, P. H. ----------------------­Saeks, R. -~---------------------­Saito, M. ------------------------Schumitzkum, M. ------------------Sekhon, H. S. --------------------

~Silverman, L. M. -----------------Sinha, V. P. ---------------------Smithers, M. A. -----------------­Srivastava, H. M. ---------------­Steinberger, M. -----------------­Subramanian, R. -----------------­Swanimathan, B. ------------------Trzaska, A. ---------------------­Tung, L. -------------------------Waldron, M. B. -------------------Wexler, D. -----------------------Wilton, D. R. --------------------Womack, B. F. -------------------­W~nham, W. M. --------------------

AUTHOR INDEX

5 2

11 4 5 8 9

11 6

1,7 8 9 2 5 4

10 9

10 8 9

10 5 8

11 11 11

5 11 10

1 3 9 6

11 6

10 2,5

4 9

2,7 9 5

11 3 9 4 9 2

10 2

10 11 11

2 11 10 10

4 11

6 5 5 6

Yates, R. E. ----------------------Yousif, S. M. ---------------------Zames, G. -------------------------Zemanian, A. H. -------------------Zyla, L. U. -----------------------

174

11 6 4 4

11