On the Realization Theory of Polynomial Matrices and the Algebraic Structure of Pure Generalized State Space Systems A.I.G. Vardulakis, N.P. Karampetakis

On the Realization Theory of Polynomial Matrices and the Algebraic Structure of Pure Generalized State Space Systems

A.I.G. Vardulakis, N.P. Karampetakis and E.N. Antoniou

Department of Mathematics Faculty of SciencesAristotle University of ThessalonikiThessaloniki 54 006, Greece

Introduction Review of the Realization Theory of Polynomial Transfer

Function Matrices via “Pure" Generalized State Space Models

Study of associated concepts and features Comparison to results from the classical State Space

realization theory Key topics:

Generalized order of GSS realizations Cancellations of decoupling zeros at ∞ Irreducibility at infinity & Minimality Dynamic & Non-dynamic variables Isomorphism of spectral structures at ∞

A.I.G. Vardulakis, N.P. Karampetakis, E.N. Antoniou – Dept. of Mathematics - Aristotle University of Thessaloniki

Introduction

Given a polynomial transfer function matrix

As Aqsq Aq 1sq 1 A0 Rsp m , #

( ) ( ) ( )A x t x t B u t

yt C x t D ut #

One may obtain its generalized state space realization of the form:

C R p ,A R B R m ,D R p m and nilpotent, where

Such thatAs C I sA 1B D #


IntroductionRemark. A GSS realization of a polynomial matrix A(s) can be obtained from a state space realization of the strictly proper rational matrix


As : 1s A 1

s R prp m s

Example. Given

As

1 s3 s2

0 1 0

0 0 0

#

4 31 1 1

1( ) : 0 0 ( )

0 0 0

s s s

sA s C sI A B

0 1 0 0 1 0 01 0 0 0

0 0 1 0 0 0 00 0 0 1 , ,

0 0 0 1 0 0 10 0 0 0

0 0 0 0 0 1 0

C A B

where

then 1( )A s C I sA B

Generalized Order of a GSS Realization


Definition. Let be GSS realization of A(s). The generalized order fg of Σg is defined as follows:

g C ,A ,B ,D

: # of poles at of

# of zeros at of

g

M

f I sA

I sA

I sA

rankA

where δΜ(.) denotes the McMillan degree.

Example. Notice that the generalized order of the example in the previous slide is fg=3.

Irreducibility at infinity


Let and let be a GSS realization of A(s)

As Rsp m , g C ,A ,B ,D

Definition. The input decoupling zeros (i.d.z.) at s= ∞ of Σg are defined as the zeros at s= ∞ of the pencil:

I sA ,B Respectively, the output decoupling zeros (o.d.z.) at s= ∞ of Σg are defined as the zeros at s= ∞ of the pencil:

I sA

C

Finally, the input-output decoupling zeros (i.o.d.z.) at s= ∞ of Σg are the common zeros at s= ∞ of the pencils:

I sA

C

I sA ,B



Let 1 2 , , ,..., ,0J block diag J J J

J i

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

R i 1 i 1 #

where

SI sJ block diag sIfg , I , 1

s , , 1s 1

#

I sJ Then the Smith- McMillan form of the matrix pencil at s=∞ is



Remark. Candidates for (i.d.z.) and (o.d.z) at s= ∞ of a GSS realization of of a polynomial matrix A(s) are the zeros at s=∞ of A(s)

g C ,A ,B ,D

Example. Let

As

s 1 s2 0

0 s 1 0

0 0 1

, SAs

s2 0 0

0 1 0

0 0 1

i.e. A(s) has one pole at s=∞ of order q1=2 and no zeros at s= ∞

Continued…



…Continued

A GSS realization is with

7, 2, 1 2 q1 , 2 1, 2

The generalized order of Σg is fg rankRJ 1 2 3.Continued…

g C ,J ,B J∞3

0 1 0 0 0 0 0 1 0 0

0 0 1 0 0 0 0 1 0 0

1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0

0 1 0 0 1 1 0 , ,0 0 0 0 1 0 0 1 1 1

0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 0

0 0 0 0 0 0 0 0 1 1

C J B

C∞1 C∞2

C∞3

J∞1

J∞2

B∞1

B∞2

B∞2



…Continued

Continued…

It can be seen that both2

2

2

2 2

2

andI sJ

I sJ BC

have a zero of order 1 at s=∞, and thus Σg has an i.d.z., an o.d.z. and i.o.d.z.at s=∞. Now since

we may easily obtain a “smaller” GSS realization

of A(s), by simply eliminating the “middle” blocks from Σg, , with

2 2

0 11 1 1

0 1 00 0 0

0 1

C B

g C ,

J ,

B



…Continued

, ,I sJ

I sJ BC

It can be seen that both

have no zeros at s=∞, which leads to the following definition

0 1 0 0 0 1 0 0

1 0 0 0 0 0 0 1 0 0 1 0 0

0 1 0 1 0 , ,0 0 0 0 0 0 1 0

0 0 1 0 1 0 0 0 0 0 1 1 0

0 0 0 0 0 0 1 1

C J B



Let andAs Rsp m , C R p ,J R ,B R m ,D R p m

be a GSS realization of A(s) with J∞ in Jordan normal form. Then:

I sJ

C

Corollary.

rankRJ

C

I sJ B rank R J B

(i) has no zeros at s=∞, iff

(ii) has no zeros at s=∞, iff

Definition. A GSS realization of a polynomial matrix A(s) is called irreducible at s=∞, iff has no input and no output decoupling zeros at s= ∞.

g C ,A ,B ,D

Minimal GSS realizations of a polynomial matrix


Definition. A GSS realization

of a polynomial matrix A(s) is called minimal, iff it has the least number of generalized states.

[ , , , ]p m p mg C A B D

Theorem. Let

be the Smith – McMillan form of A(s) at s=∞. A GSS realization

of A(s), with J∞ in Jordan canonical form is minimal iff:

SAs diag

v

sq 1 , . . . , sq k , Iv k, 1s

q v1

, . . . , 1s

q r

, 0p r,m r

g C ,

J ,

B ,D

MAs v k i 1

k

qi 1 k i 1

k

q i #



Remark. (i) A necessary condition for the minimality of a GSS realization is

• must be irreducible at s=∞ • must not have non-dynamic variables

(ii) The least dimension of a GSS realization is

Where qi are the non-zero orders of the poles at s=∞, of A(s).

1 1k

i iq

Corollary. A GSS realization of a polynomial matrix A(s) is a minimal GSS realization iff it is an irreducible at s=∞ GSS realization and has no non-dynamic variables.



Example. Continuing the previous example, an irreducible at infinity GSS realization of A(s), was

0 1 0 0 0 1 0 0

1 0 0 0 0 0 0 1 0 0 1 0 0

0 1 0 1 0 , ,0 0 0 0 0 0 1 0

0 0 1 0 1 0 0 0 0 0 1 1 0

0 0 0 0 0 0 1 1

C J B

Obviously this realization is not minimal, since the expected least dimension of a GSS realization of A(s) should be

1 1 2 1 3k

i iq

(Recall that )2( ) { ,1,1}A sS diag s

Continued…



This realization Σg can be further reduced by moving its non-dynamic variables, from the state vector to the D∞ matrix.

The resulting minimal GSS realization is then given by

…Continued

1 0 0 0 1 0

0 1 0 , 0 0 1 ,

0 0 1 0 0 0

1 0 0 0 0 0

1 0 0 , 1 1 0

0 1 0 0 1 1

C J

B D



Proposition. Let A(s) be a column proper polynomial matrix with column degrees where are the columns of A(s).Then an irreducible at s=∞ GSS realization of A(s) can be obtained by inspection and is given by:

v j degajs, ja sC ,J ,B

C a1v1 , ,a10 a2v2 , ,a20 amvm , ,am0 #

J block diagJ 1 ,J 2 , ,J m R #

1 1

0 1 0

0 1

0 0

i iv vjJ

B block diagb1 ,b2 , ,bm R m # 1 10,0, ,0,1 jT v

jb



Theorem. Let

be the Smith-McMillan form at s=∞ of A(s). If the GSS realization is irreducible at s=∞ then the zero structure at is isomorphic to the zero structure of the GSS Rosenbrock system matrix

and its Smith-McMillan form at s=∞ is given by

SAs diag

v

sq 1 , . . . , sq k , Iv k, 1s

q v1

, . . . , 1s

q r

, 0p r,m r #

g C ,A ,B ,D s

Ps :I sA B

C D

SPs diag sI , I v k , 1

sq k1

, , 1s

q r

, 0

Conclusions We have investigated the mechanism of

cancellations of zeros at s=∞. We discussed the concepts of irreducibility

and minimality of pure GSS realization. The role of dynamic and non-dynamic

variables has been examined. The isomorphism between zeros at s=∞ of

the "infinite pole pencil" and the Rosenbrock system matrix has been presented.


Documents

On the Realization Theory of Polynomial Matrices and the Algebraic Structure of Pure Generalized State Space Systems A.I.G. Vardulakis, N.P. Karampetakis