Frequency Domain Methods in H-infinity Control - Applied

FREQUENCY DOMAIN METHODS

IN H∞ CONTROL

i

Gjerrit MeinsmaDepartment of Applied MathematicsUniversity of TwenteP. O. Box 2177500 AE EnschedeThe Netherlands

CIP – DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Meinsma, Gjerrit

Frequency Domain Methods inH∞ Control / Gjerrit Meinsma.– [S. l. : s. n.]. – Ill.Thesis Enschede. – With index, ref. – With summary in English,Dutch and Frisian.Subject headings: Linear system theory,H∞ control theory,Wiener-Hopf factorization.

ISBN 90-9006122-3

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FREQUENCY DOMAIN METHODS

IN H∞ CONTROL

PROEFSCHRIFT

ter verkrijging vande graad van doctor aan de Universiteit Twente,

op gezag van de rector magnificus,prof. dr. Th. J. A. Popma,

volgens besluit van het College van Dekanenin het openbaar te verdedigen

op vrijdag 21 mei 1993 te 16.45 uur

door

Gjerrit Meinsma,geboren op 29 januari 1965

te Opeinde.

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Dit proefschrift is goedgekeurd door de promotorProf.dr.ir. H. Kwakernaak

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Voorwoord

Net toen ik de wasmachine aangezet had met daarin al mijn nette kleren, werd ik opgebeld met devraag of ik de volgende dag op sollicitatiegesprek kon komen. In een vale spijkerbroek (geloofik) en met een slobbertrui die tot aan mijn knieen reikte (dat weet ik zeker) klopte ik de volgendedag op de deur van de kamer van professor Kwakernaak.

Zo begon een periode van vier jaar als AiO bij de Vakgroep Systeem- en Besturingstheorievan de Fakulteit der Toegepaste Wiskunde van de Universiteit Twente. Nu, vier jaar later, is hetalweer bijna voorbij en ga ik weer opnieuw solliciteren, zijhet deze keer gewapend met een heusproefschrift, een proefschrift dat er zonder hulp van anderen niet zou zijn.

De meeste dank ben ik verschuldigd aan mijn begeleider en promotor Huibert Kwakernaak.Dankzij zijn rechtlijnige no-nonsense kijk op zaken heb ik niet al mijn tijd verspild aan onzinnigewiskundige hersenspinsels. De beter leesbare zinnen in ditproefschrift zijn ongetwijfeld van zijnhand. (Ik vermoed dat hij tijdens het korrigeren van mijn stukjes tekst de afgelopen jaren zekereen dozijn rode pennen heeft versleten.)

Ik bedank het Netwerk Systeem- en Regeltheorie voor de mogelijkheid die zij aan AiOs geeftom een aantal hoogstaande kursussen te volgen. Ik behoor totde tweede generatie AiOs die aandeze kursussen deelgenomen heeft. Het heeft indirekt een grote invloed gehad op de uiteindelijkeinhoud van het proefschrift. Mijn kijk op de systeemtheorieis voor een groot deel het produktvan deze kursussen en Hoofdstuk5 van dit proefschrift zou zonder deze kursussen waarschijnlijkniet zijn geschreven.

Essentieel voor dit proefschrift is de bijdrage, direkt en indirekt, geleverd door Michael Green.Een artikel van hem was de reden voor mijn “dipje” in het tweede jaar van mijn AiO-schap.Hij had namelijk op een mooie manier zo ongeveer het hele probleem opgelost waar ik eenproefschrift over zou gaan schrijven. Tijdens zijn bezoek aan onze vakgroep bleek dat mee tevallen. We hebben toen afgesproken dat ik hem in Australie op zou gaan zoeken om daar deproblemen verder te bekijken. Daar onder (Australie bedoel ik) hebben hij en ik gewerkt aan datwat nu in Hoofdstuk5 staat. Het was heel spannend om ter plekke wiskunde te maken en ik weetzeker dat naast Hoofdstuk5 ook Hoofdstuk3 er heel anders uit had gezien als we toen niet metveel “wlogs” de ene na de andere spetterende konstruktie hadden bedacht (die natuurlijk de dagerna niet zo spetterend bleek te zijn als we eerst dachten). Bij een volgend bezoek aan Australiehoop ik echter wel beter weer mee te nemen.

De leden van de promotiekommissie, bestaande uit Okko Bosgra, Rien Kaashoek, David Lime-beer, Arun Bagchi, Ruth Curtain en Arjan van der Schaft bedank ik voor hun inspanningen. Metname bedank ik Ruth Curtain voor haar ongezouten uitvoerigekommentaar op het koncept vanhet proefschrift.

Ik heb met veel plezier gewerkt binnen de vakgroep SB, maar debeste herinneringen heb ik aande tijden dat we niet werkten en toch druk bezig waren: de pauzes. De laatste paar maanden wasik er misschien niet helemaal bij met mijn hoofd, en het vervelende is dat ik er straks helemaalniet meer bij zal zijn.

Het is af.

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Contents

1. Introduction 1

2. Systems 72.1. Systems described by ordinary linear differential equations . . . . . . . . . . . . 7

3. Frequency domain solution to suboptimal and optimal two- block H∞ problems 193.1. Preliminaries: Positive subspaces. . . . . . . . . . . . . . . . . . . . . . . . . . 213.2. Suboptimal solutions to a two-blockH∞ problem . . . . . . . . . . . . . . . . . 293.3. Optimal solutions to a two-blockH∞ problem . . . . . . . . . . . . . . . . . . . 323.4. Some state space formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4. The standard H∞ problem 434.1. The SSP2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2. The SSP1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3. On the computation of suboptimal and optimal solutionsto the SSP1. . . . . . . 594.4. Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.5. Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5. L2−-Systems and some further results on strict positivity 775.1. Three representations of systems. . . . . . . . . . . . . . . . . . . . . . . . . . 795.2. L2−-systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3. Strict positivity or strict passivity. . . . . . . . . . . . . . . . . . . . . . . . . . 905.4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6. Conclusions 105

A. Basics from H∞ Theory 107

B. Polynomial and Rational Matrices 115B.1. Polynomial matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115B.2. Real-rational matrices and fractions. . . . . . . . . . . . . . . . . . . . . . . . 119B.3. Wiener-Hopf factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

C. Proofs 131C.1. Proofs of Chapter 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

D. Bibliography 135

E. Notation 141

Index 143

vii

Contents

F. Summary 147

G. Samenvatting 149

H. Gearfetting 151

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1

Introduction

This thesis deals with a number of mathematical problems arising inH∞ control.The basic question in control theory is how to control a givenprocess (or system). Undoubtedly

the most successful and powerful technique in the control ofsystems isfeedback. Roughlyspeaking, feedback means correcting theinput signals of the system, based on the observedoutputsignals. The number of applications of feedback control areenormous, and probably thisthesis would have been written by hand—just the thought—if feedback control had not existed.

Most of the feedback controllers around in the “real world” are PID controllers—PID stands forproportional/integral/derivative—and this situation will probably not change in the near future.The positioning of the three knobs that make a PID controlleris in many cases conducted byengineering ingenuity rather than theory. This is not to saythat there is not a more systematicapproach for controller design. Far from it. Starting with the work of Bode (12) “classical”control theory has grown to a impressive theory.

Every method, despite its success, has its limitations. Forhard-core mathematicians the prob-lem with classical control techniques is the lack of an optimality criterion. More serious lim-itations of classical control theory are the difficulties indealing with multi-input-multi-outputsystems and poorly modelled systems.

Control theory received a new impetus at the beginning of theeighties when Zames introduceda control problem as an optimization problem in theH∞-norm (or∞-norm). In a famous pa-per (91) he considered minimizing the effect of disturbances acting on the feedback system. Itwas soon recognized that this∞-norm could be used to quantify, not only disturbance insensi-tivity, but also robustness against model uncertainties, and performance of the feedback system.Since then, hundreds of papers have appeared contributing to what nowadays is calledH∞ con-trol theory.H∞ control methods are systematic, work for multi-input-multi-output systems, andallow one to deal with model uncertainty far more directly than is possible with classical controland LQ control methods.

Equally lively has been the development of the mathematics that underlies theH∞ controlproblems. The beauty, and also the problem, of the∞-norm with its associated spaces is thatthey have an incredible amount of structure. (The∞-norm is an induced operator norm). As aresult, many different approaches toH∞ control problems work, each having its own merits andlimitations.

Some of theH∞ problems, like the minimum sensitivity problem of Zames (91), turned outto be a Nevanlinna-Pick interpolation problem. The historyof the Nevanlinna-Pick interpolationtheory dates back to beginning of the century (Nevanlinna (70) and Pick (71)). More complicatedproblems inH∞ theory turned out to be related to Nehari type problems, involving Hankel oper-ators and indefinite inner product spaces. These problems have a long history as well (Adamjan,Arov and Kreın (1), see also Francis (26)). Later, a connection was established with differentialgame theory (Khargonekar, Petersen and Rotea (41)).

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1. Introduction

We limit the discussion to systems governed by ordinary linear differential equations. Theseare the systems that we and the majority of researchers onH∞ control consider. The presentationdoes not give a full account on the diverse field ofH∞ control.

Most of the attempts to solveH∞ control problems focus on thestandardH∞ problemintro-duced in 1984 by Doyle (21) and treated in detail by Francis (26). The standardH∞ problem iscalled “standard” because it includes manyH∞ control problems, if not all, as special cases. Inthe first attempts at that time the idea was to rewrite the standardH∞ problem as a four-blockH∞problem via the Youla-Bongiorno-Jabr-Kuceraparameterization of all stabilizing controllers. Thestate space manipulations entered the scene in this basically frequency domain approach only asa computational tool (26). The results were not very satisfactory because of the endless chain ofreduction steps needed, leading to high degree solutions. Later, around 1988 in an epoch makingpaper (Doyle, Glover, Khargonekar and Francis (22)) the matter seemed to be decided in favourof a pure state space approach, though here and there some frequency domain arguments stillwere used. The by now famous pair of Riccati equations and thecoupling condition opened theway to compute efficiently solutions to a very large class of standardH∞ problems. The methodnaturally led to solutions of low degree.

The success of the state space approach overshadowed the originally frequency domain slantonH∞ control. Indeed an interesting feature of the state space set up is that it may be generalizedto time-varying systems (Limebeer et. al. (54)) and nonlinear systems (Van der Schaft (80)).However, since a paper by Green (35) appeared, we know that the unattractive chain of reductionsteps needed in those preliminary frequency domain attempts is not due to a possible impractica-bility of the frequency domain approach, but only to the way it was applied. In (35) it is shownthat the famous Riccati equations can be seen to arise from a pure frequency domain approachthrough what is calledJ-lossless factorization, in combination with the canonical factorizationtheorem (Bart, Gohberg and Kaashoek (11)). The J-lossless factorization approach has a con-ceptual advantage in that the overall algorithm, on a frequency domain level, is very compact.The state space manipulations in this approach, again, enter the scene only as a computationaltool.

Practically all research on the solution toH∞ control problems is directed to asuboptimal ver-sion of the problem. In suboptimal versions the aim is to find controllers that make the∞-normlessthan some prescribed bound, but donot necessarilyminimizethe∞-norm. The reason forlooking at these suboptimal problems lies in the inherent mathematical difficulties that come upwhen trying to solve the optimal case. For Nehari type problems it was known how to deriveoptimal solutions (Adamjan, Arov and Kreın (1), Ball and Ran (10) and Glover (29)). It wasto be expected that these results would carry over in some form to more generalH∞ problems.In Glover et. al. (31) the most systematic solution to the optimal case is reported. The methodrelies on all-pass embeddings. Other interesting approaches are Gahinet (27), and the polyno-mial approach, which we shall introduce soon. An operator theoretic approach allowing infinitedimensional systems is reported in Foias and Tannenbaum (25).

Independent of the mainstream of research onH∞ control problems is the polynomial ap-proach toH∞ control, reported in a series of papers (Kwakernaak (46; 47; 48; 49; 50)). In thisapproachsolutions toandcontrol aspects ofH∞ control theory are developed jointly. This ledto a slightly more general problem formulation and solution, allowing more practical problemsto be considered. It is basically a frequency domain approach, close in spirit to classical control.The polynomial solution method at present is capable of handling a large class ofoptimalH∞control problems. This is one of the results that we prove in this thesis.

In this thesis we develop a frequency domain solution methodto H∞ control problems. Thegoal is to develop a method that fits into the polynomial approach. To achieve this goal wedevelop a geometric interpretation of theJ-lossless factorization approach of Green (35). The

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idea behind it is that the geometry of spaces provides a deeper insight into the structure ofH∞control problems, certainly when theoptimal case is considered. With the help of geometricarguments we extend theJ-lossless solution method to the optimal case and, later, translate it toa polynomial method.

The material presented in this thesis is reasonably self-contained.An overview of the thesis:

Chapter 2 Systems The material of this chapter constitutes the background material needed insubsequent chapters. We give an introduction to systems whose signals are interrelated throughordinary linear differential equations. The language usedis that of Willems’ behavioral approachto mathematical systems theory (86). Closed-loop systems and inputs and outputs are defined.Stability of input/output systems is defined, and the notionof well-posedness of closed loopsis introduced. At this level of generality there is no such thing yet as “mapsfrom inputs tooutputs”; all we know—by definition—is that the signals satisfy a set of differentialequations.The “map” interpretation is then given using the concepts earlier introduced. The idea is this:If the input/output system is (part of) a stable loop, then the effect of initial conditions on thesignals eventually tends to zero. In this case one may therefore argue that there is no harm inomitting this part, leaving an output depending uniquely and linearly on the input. This leads towhat is known as convolution systems and convolution operators, or convolution maps. In ouropinion it makes sense to talk about convolution systems only after stabilizability of the system(not necessarily stability of the system itself) is ensured.

With this convolution map interpretation at our disposal wemake the step to the frequencydomain. The transfer matrix as introduced earlier is seen asthe Laplace transform of the convo-lution map, though also still as a differential operator in disguise. It is shown that under suitableminimality conditions, stability or instability of the input/output system may be decided on thebasis of properties of an input/output map representation of the system. This finally opens upthe possibility to identify an input/output system with itstransfer matrix and to see this transfermatrix as map from inputs to outputs. Only then we are at the point where most papers on controlstart. Finally we devote a few words to “plants”, “stabilizing compensators” andH∞ control.

Chapter 3 Frequency domain solution to suboptimal and optimal two-bl ock H∞ problemsIn this chapter we treat the two-blockH∞ problem in a frequency domain setting. As is quitecommon, we distinguish between suboptimal and optimal solutions. The reason for making thisdistinction mainly derives from the fact that the suboptimal case and the optimal case are quitedifferent in nature. The two-blockH∞ problem exhibits most of the essential features appearingin more generalH∞ problems that we wish to solve. On the other hand it has a structure that issimple enough to make it suited for an attempt to solve, besides the (easy) suboptimal case, alsothe (difficult) optimal case.

The optimal case is tricky and therefore we have to be particularly careful. The approach thatwe take is basically a geometric approach. The reason for doing so lies in the belief, adopted fromthe behavioral approach, that the “behavior”—the set of signals in the system—makes a systemand not so much the (arbitrary choice of) representation of the system. We start our analysisby identifying a subset of the closed-loop behavior that cannot be affected by the compensatorto be constructed. All that the compensator can do is to try tomould, so to say, the space itcan affect in the best possible way. One can feel right away all sorts of projection argumentsentering the scene. An important aspect of the approach, though not explicitly acknowledged inthis chapter, is the unusual choice of signal space. This space seems to work well. We return tothis in Chapter5. The main result proved in this chapter is about a one-to-onecorrespondence

3

1. Introduction

between strict positivity of a signal space andJ-losslessness of an associated transfer matrix. Wecall this result the strict positivity theorem.

Many preliminary results are required. These preliminary results are not really difficult andactually some of them are quite elegant. These results constitute a brief and incomplete surveyon positive and strictly positive subspaces as they arise inthe theory of indefinite inner productspaces or, more specifically, the theory of Kreın spaces (Bognar (14) and Azizov and Iokhvi-dov (3)). The suboptimal version of the two-blockH∞ problem is solved, and the optimal caseis solved under certain conditions.

Chapter 4 The standard H∞ problem The problem considered in this chapter is the standardH∞ problem in two versions. Only a vague remainder of the geometric approach is still apparentin this chapter. The reason is the difficulty of interpreting, in terms of signals, several dual resultsthat we need for the solution to the standardH∞ problem. We define a modified standardH∞problem, in line with the polynomial approach, which we needto solve more practical problems,like a mixed sensitivity problem with nonproper shaping filters. Some brute force mathematicsis needed to tackle this modified problem. An algorithm usingpolynomial matrix algebra is for-mulated that may be used to generate all optimal solutions tothe standardH∞ problem, providedsome assumptions hold. This algorithm is direct generalization of the procedure, developed inthe preceding chapter, for generating all optimal solutions to the two-blockH∞ problem. Thealgorithm is demonstrated on a mixed sensitivity example. At the end of this chapter we brieflysummarize how the results may be translated to the well-known state space formulas.

Chapter 5 L2−-systems and some further results on strict positivity Convolution systemsare extremely popular among system theoreticians, in particular whenH∞ problems are consid-ered. Of course, convolution systems as such do not exist; their success mainly owes to the factthat it provides the comfortable feeling of there being “causes” mapping in to “consequences”.

We do not want to give the impression that convolution systems are not worth considering, butone should realize that convolutions systems provide just one of the many possible frameworksfor studying systems. The aim of this chapter is to highlightsome properties of what we callL2−-systems. These are systems whose signals are assumed to have finite energy up to anyfinitetime. Roughly speaking,L2−-systems provide a means to focus the attention on the unstabledynamics, the unstable behavior of the system.

Some questions naturally arise. For example, when are two different representations ofL2−-systems equivalent, and when are such representations “minimal”? The results are not reallysurprising, though it may put some of the results of preceding chapters in a different perspective.

The two main theorems in this chapter are about a state space characterization of strictlypositive subspaces forL2−-systems. The results reduce to the well known Bounded Real Lemmawhen applied to the usual stable convolution systems. When applied to systems that have noinputs, the results reduce to results onfinite dimensional strictly positive subspaces, which maybe associated with one-sided Nevanlinna-Pick interpolation problems. It is argued thatL2−-systems make a suitable substitute of convolution systems,providing a easy framework for allsorts of problems inH∞ control theory. Actually we have already seen theL2−-systems in actionin Chapter3, though we did not call them that in that chapter. As exampleswe consider anH∞filtering problem and a one-sided Nevanlinna-Pick interpolation problem.

Chapter 6 Conclusions In this tiny chapter we recapitulate some of the points made in thisthesis. We dream up several of the inevitable “topics for future research”.

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Appendices A, B and C Of the three appendices in this thesis the first two contain importantbackground material that we use throughout this thesis without explicit mention. AppendixA isabout the basics ofH∞ theory and some of its connections with system theory. We expand onthe notion ofJ-losslessness, a notion that pervades the whole thesis. AppendixB summarizesproperties of polynomial and rational matrices. Some deep results on canonical and noncanonicalfactorization of rational matrices are discussed. The material presented is well known amongsystem theoreticians, except, perhaps, our definition of the McMillan degree of a rational matrix.Whenever possible we refer to proofs in the literature. The results on canonical and noncanonicalfactorization form an exception to this rule: The results on(non)canonical factorization are provedin detail. AppendixC contains some not very enlightening proofs of technical results needed inChapter4.

At the end of this thesis a general index list and a list of notation is added. The two lists areintended to be used as a look-up tables. Some conventions andoften used notations are listedbelow.

Notation The set of complex numbers is denoted asC. By C−, C0 andC+ we mean the subsetsof C that have nonzero negative, zero and nonzero positive real part, respectively. SimilarlyRdenotes the set of real numbers,R− represents the nonzero negative real numbers andR+ thenonzero positive real numbers. The norm‖ • ‖ on C andR and its subsets will always be theEuclidean norm:‖z‖ =

√z∗z. Herez∗ denotes the complex conjugate ofz. For the matrix case

the norm‖ • ‖ without exception is defined as‖M‖ =√

traceM∗ M, whereM∗ stands for thecomplex conjugate transpose ofM. The identity matrix is denoted asI , or In if we want to specifyits dimension. Very often we use so-calledJ-matricesor signature matrices. These are matricesof the form

Jq,p :=[

Iq 00 − I p

]

.

Bullets (•) denote either unspecified arguments or arguments that are not important for the prob-lem at hand. For example,f (•) = g(• + 1)meansf (s) = g(s+ 1) for all allowables.

A prominent role in this thesis is played by spaces of the form

L2(X;Y) := w : X 7→ Y | w is Lebesgue measurable,∫

X

w∗(t)w(t)dt< ∞ .

Here the domainX is an interval ofR or C0 of possibly infinite length and the image spaceY

is eitherRq or Cq for someq. Throughout this thesis the inner product and norm on the HilbertspaceL2((a,b);Y) is defined as

〈u, y〉 =∫ b

au∗(t)y(t)dt; ‖ f ‖2 =

√

〈 f, f 〉.

Here(a,b) is an interval ofR of possibly infinite length. For the Hilbert spaceL2(C0;Cq) andits subsets we use without exception

〈u, y〉 = 12π

∫ ∞

−∞u∗(iω)y(iω)dω; ‖ f ‖2 =

√

〈 f, f 〉.

It is convenient to absorb the constant factor12π in the above definition of inner product and norm

because then the two-sided Laplace transformL from L2(R;Cq) to L2(C0;C

q) defined by

L(w) (•) =∫ ∞

−∞e−•tw(t) dt

5

1. Introduction

preserves inner products. The set of locally square integrable functions fromR to Y is denotedby R L loc

2 (R;Y) and defined as

R L loc2 (R;Y) = w : R 7→ Y | for every−∞ < a< b<∞,

w restricted to(a,b) is in L2((a,b);Y) .

The Hardy spaces that we use frequently are

H2 := f : C+ 7→ C | f is analytic inC+ and supσ>0

∫ ∞

−∞‖ f (σ+ jω)‖2dω <∞ ,

H ⊥2 := f : C− 7→ C | f is analytic inC− and sup

σ<0

∫ ∞

−∞‖ f (σ+ jω)‖2dω <∞ ,

H∞ := f : C+ 7→ C | f is analytic inC+ and sups∈C+

‖ f (s)‖ < ∞ .

We take the convention to use hats on symbols to stress that itrepresents the Laplace transformof some function. For example,w = L(w) is a common expression in this thesis. Hats, likeoverbars, are sometimes also used to distinguish between related elements. Overbars are not usedto denote complex conjugates. Apart from the complex and real numbers we use the calligraphicfont for symbols that denote sets and spaces. Usually symbols in lower case denote scalars andupper case symbols denote matrices. For any setM , M n is the set ofn vectors with entries inMandM n×m is the set ofn × m matrices with entries inM . Sometimes when the dimensions areclear from the formulation, or the dimensions are unspecified, we useM rather thanM •×•. Forexample,P denotes the set of real polynomial matrices of unspecified dimension. That is,P isin P iff for some integersq and p andn, P(s) = P0 + sP1 + s2P2 + · · · + snPn with Pi ∈ Rq×p.By R we denote the set of real-rational matrices. In other words,R is in R iff qR∈ P for somescalarq ∈ P .

For matrix functionsG, we defineG∼ by G∼(s) = [G(−s∗)]∗. If G is a real-rational this re-duces toG∼(s) = [G(−s)]T, whereT denotes the transpose. If a rational matrixG ∈ R n×m

has no poles on the imaginary axis, including infinity, thenG may be seen as an operatorfrom L2(C0;Cm) to L2(C0;Cn) assigning to each elementu of L2(C0;Cm) the elementy ofL2(C0;Cn) defined pointwise asy(s) = G(s)u(s). Often rational matrices are identified withtheir associated operator, and it may be checked thatG∼ may be identified with the adjoint oper-ator:

〈u,Gy〉 = 〈G∼u, y〉.

By R H n×m∞ we mean the set of real-rational matrices inH n×m

∞ (that is, R H n×m∞ = R n×m ∩

H n×m∞ ).Functions that depend on “time” are often called signals. The number of components of a

vector-valued signalz is sometimes denoted asnz.

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2

Systems

In this chapter we give a general introduction to linear systems described by ordinary lineardifferential equations. It provides the background material that we need for subsequent chapters.We define what we mean by a system and what closed loops are, etcetera. The results in thischapter are mostly well known and whenever possible we referto proofs in the literature. Mostof the results are drawn from Kailath (39), Willems (86; 87; 88), Desoer and Vidyasagar (19) andCallier and Desoer (16).

2.1. Systems described by ordinary linear differentialequations

Following Willems (86) we define a dynamical systemΣ as a tripleΣ = (T, W, B ) with T ⊆ R

the time axis,W the signal space (the space where the time signals take on their value), andBthebehaviorof the system, that is, the set of all time signals that satisfy the laws of the system.We consider systems whose laws take the form of ordinary linear differential equations. In otherwords,

B = w : T 7→ Rq | R(d/dt)w(t)= 0 , (2.1)

whereR∈ P is some real polynomial matrix. We assume that the time signalsw are either definedon the whole real axis or for negative time only (T = R, or T = R−), and thatw(t) ∈ W = Rq

for someq. Elementsw of B are referred to asexternalsignals to distinguish them from othersignals we may like to introduce. Once in a while we allow complex-valued time signals andpolynomial matrices.

The differential equationR(d/dt)w(t) = 0 is unambiguously defined for signalsw that areinfinitely often differentiable. Most often, however, the classC ∞ of infinitely often differentialsignals is a class that is too restrictive to be suited for ourwants as it would rule out such commonsignals as the step1. Spaces that include practically all signals one is likely to consider are thespaces

Lp((a,b);Rq) := w : (a,b) 7→ R

q | (∫ b

a‖w(t)‖p dt)1/p <∞ ,

for a givenp ≥ 1 and a finite or infinite interval(a,b) in R. These spaces are complete normedspaces with norm

‖w‖p :=(∫ b

a‖w(t)‖p dt

)1/p

.

1step(t)= 0 if t < 0 and step(t)= 1 if t ≥ 0.

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2. Systems

The classC ∞ of infinitely often differentiable signals is dense in the normed spaces mentioned(see e.g. Treves (78, p. 159)). This fact allows us todefine2 solutions of a differential equationfor signals inLp((a,b);Rq). We say thatw ∈ Lp((a,b);Rq) is a (generalized) solution of thedifferential equationR(d/dt)w(t)= 0 if w is the limit of some sequencewi ⊂ Lp((a,b);Rq)∩C ∞ whose elements satisfyR(d/dt)wi(t) = 0. This will mean in practice that there is no harmin assuming that the signalsw we meet are infinitely often differentiable. For the rest of thisthesis we assume that the external signals are locally square integrable (that is, that they are inL loc

2 (R;Rq)).

Definition 2.1.1 (Auto regressive representations, ( 86)). Let Σ = (R, Rq, B ) be a systemwith external signalw. If B = w ∈ L loc

2 (R;Rq) | R(d/dt)w(t)= 0 for some polynomial matrixR∈ P , thenR(d/dt)w(t)= 0 is anauto regressive representation(or, AR representation) of Σ,andR is said to define an AR representation ofΣ.

AR representationsRw = 0 of a systemΣ are by definitionminimalwhen both the number ofrows ofRand the degreeδRof R is minimal amongst all AR representations ofΣ. The definitionof δRmay be found in AppendixB. A polynomial matrixU ∈ P is unimodularif U−1 ∈ P . Thisis the case iff detU is a nonzero constant.

Lemma 2.1.2 (Minimality and uniqueness of AR representatio ns, (88; 86)). Suppose thatΣis a dynamical system that admits an AR representation. Thenthere exist full row rank poly-nomial matrices R∈ P defining an AR representation ofΣ. Such representations are minimalAR representations. Moreover, two full row rank polynomialmatrices R andR define an ARrepresentation of the same system if and only if R= U R for some polynomial unimodular matrixU.

It may be very annoying to make distinction all the time between “systems”, “behaviors” and“representations”. We often identify a systems with its behavior; we say thatRw= 0 is a system,and so on.

Important: Because we consider only systems whose external signals arelocally square inte-grable and interrelated through ordinary linear differential equations, we incorporate this in thedefinition of system. That is, for the rest of this thesis a(dynamical) systemis a dynamical systemthat admits an AR representation as defined in Definition2.1.1.

2.1.1. I/O systems and stability

In this subsection we define what we mean by inputs and outputsof a dynamical system. Internalstability andL2 stability of systems with inputs and outputs is defined thereafter.

u y

Figure 2.1.: Inputs and outputs.

Diagrammatically an I/O system is often depicted as in Fig.2.1. The box indicates thatu andy are related. The arrows suggest thatu is not constrained by the system equations and thaty is

2A neater way to define solutions of such differential equations goes via a notion of “weak solutions”, see (89). For ourpurposes the present definition will do.

8

2.1. Systems described by ordinary linear differential equations

completely determined by it. This idea may be formalized to adefinition of inputs and outputs(see Willems (86) for the discrete time case) but for our purposes the following definition willdo.

Definition 2.1.3 (I/O systems). A system Σ = (R,Rm+p,B ) is an I/O systemΣI/O =(R,Rm,Rp,B ) with input uandoutput yif

B = [

uy

]

: R 7→[

L loc2 (R;Rm)

L loc2 (R;Rp)

]

| Nu= Dy

for some[

−N D]

∈ P p×(m+p) with D square nonsingular. The rational matrixD−1N is calledthetransfer matrixof the I/O system.

Note that an I/O system defined this way may have a nonproper transfer matrix3. The transfermatrix of an I/O system is unique because every system admitsa full row rank AR representation,and such representations

[

−N D]

are unique up to multiplication from the left by a unimodularmatrix, which cancels in the expressionD−1N. The converse is not true: An I/O system is notcompletely determined by its transfer matrix. Part of this section is devoted to the problem towhich extent an I/O systemis determined by its transfer matrix.

The transfer matrixD−1N may be nonproper for the systems considered in Definition2.1.3.Nonproperness — what’s in a name — is not what one generally wants. A mathematical expla-nation is:

Lemma 2.1.4 ( L loc2 stability). Suppose N∈P p×m and D∈P p×p are polynomial matrices with

D nonsingular. Then for every u∈ L loc2 (R;Rm) the solutions y of

Dy = Nu

are well defined inL loc2 (R;Rp) if and only if D−1N is proper.

In other words, an I/O system is “L loc2 -stable” iff its transfer matrix is proper. The proof of

Lemma2.1.4uses the following result.

Lemma 2.1.5 (I/S/O representations, ( 87; 88)). If an input/output systemΣI/O = (R,Rm,Rp,B )with input u and output y has a proper transfer matrix G, then there exist

[

A BC D

]

∈ R(n+p)×(n+m), (2.2)

with (C, A) observable4 such that

B = [

uy

]

∈ L loc2 (R;R

m+p) |[

xy

]

=[

A BC D

][

xu

]

for some x, (2.3)

or equivalently,

B = [

uy

]

∈ L loc2 (R;R

m+p) | for arbitrary t0 ∈ R there exists an x0 ∈ Rn

such that y(t) = CeA(t−t0)x0 +∫ t

t0

CeA(t−τ)Bu(τ) dτ+ Du(t) .

Moreover, in this case the transfer matrix G satisfies G(s) = C(sI − A)−1B+ D.

3A rational matrixG is proper if G(s) is bounded ats= ∞, it is nonproperif G(s) is unbounded ats= ∞.

4A pair (C, A) of constant matrices is observable if

[

A− sIC

]

has full column rank for alls∈ C.

9

2. Systems

In Willems (88) the above result is proved by construction, but it is not shown that the transfermatrix equalsC(sI − A)−1B + D. This is a standard result and may be proved in a number ofways (see Kailath (39)). We say that

[

xy

]

=[

A BC D

][

xu

]

is an input/state/output representation (I/S/O representation) of an I/O system with inputu andoutputy if its behavior equals (2.3) (see (87)).

Proof (Lemma2.1.4). If D−1N is proper then by Lemma2.1.5there exist constant matricesA,B, C andD such that

y(t) = CeA(t−t0)x0 +∫ t

t0

CeA(t−τ)Bu(τ) dτ+ Du(t).

This shows thaty is in L loc2 (R;R

p) wheneveru is in L loc2 (R;R

m).It remains to show that for someu ∈ L loc

2 (R;Rm) the solutionsy are not well defined inL loc

2 (R;Rp) if D−1N is nonproper. SupposeD−1N is nonproper and letP be a (noncon-stant) polynomial matrix such thatD−1N − P is proper. DefineM = N − DP. For everyu ∈L loc

2 (R;Rm) we have thatzdefined byDz= Mu is in L loc2 (R;Rp) becauseD−1M = D−1N − P

is proper. By linearity, we therefore have thaty is in L loc2 (R;Rp) iff y − z is in L loc

2 (R;Rp).Again by linearity, the signalsy − z andu are related asD(y− z) = (N − M)u = DPu. Fromthis equation it is clear that for manyu ∈ L loc

2 (R;Rm) the solutionsy− z are not well defined inL loc

2 (R;Rp).

Definition 2.1.6 (Internal stability). An I/O system with inputu and outputy is internallyasymptotically stable(or, internally stable, for brevity) if

limt→∞

‖y(t)‖ = 0

for every possible outputy and inputu ≡ 0.

The following lemma is immediate. Its proof is omitted.

Lemma 2.1.7 (Internal stability, ( 89)). LetΣI/O be an I/O system with input u and output y, andsuppose it has AR representation Dy= Nu with

[

−N D]

∈ P and D nonsingular. ThenΣI/O

is internally stable if and only if D isstrictly Hurwitz5.

Non-asymptotic stability (that is, when for zero input the output remains bounded but notnecessarily goes to zero) is not important for the purposes we have in mind.

Definition 2.1.8 ( L2 stability). SupposeΣI/O = (R,Rm,Rp,B ) is an I/O system with inputuand outputy. The system isL2-stableif for every

[ uy]

∈ B

u |(0,∞)

∈ L2((0,∞);Rm) =⇒ y |

(0,∞)∈ L2((0,∞);R

p).

Here,u |(0,∞)

stands foru with its domain restricted to(0,∞). Definition2.1.8formalizes theidea that in a stable system the output is “future-time-bounded” for future-time-bounded inputs,irrespective of the past. This definition ofL2 stability is a mix of internal stability and ofL2

stability as it is defined in Desoer and Vidyasagar (19).

5A square polynomial matrixD is strictly Hurwitz if det D has all its zeros inC−.

10


Lemma 2.1.9 ( L2 stability). An I/O system with input u and output y and AR representation

Dy = Nu, D nonsingular,

is L2-stable if and only if D is strictly Hurwitz and the transfer matrix D−1N is proper.

In other words: “L2 stability = internal stability +L loc2 stability.”

Proof . (Only if) D strictly Hurwitz andD−1N proper are necessary conditions according toLemma’s2.1.7and2.1.4, respectively. (If) Assume thatD is strictly Hurwitz and thatD−1Nis proper. By Lemma2.1.7the system is then internally stable, and by Lemma2.1.5it admitsan I/S/O representation. Soy(t) = CeAtx0 +

∫ t0 K(t − τ)u(τ) dτ+ Du(t), with K(s) = Ces AB.

By internal stability of the system it follows thatCet Ax0 tends to zero for allx0 ∈ R if t →∞. Therefore the elements of the matrixCet A are exponentially decaying time functions, and,hence, so are the elements ofK(t − τ) = Ce(t−τ)AB. This shows that

∫ ∞0 ‖K(s)‖ ds< ∞. Let

‖ • ‖2 denote the norm onL2((0,∞);R•). By the theorem on page 25 of (19), the expression‖∫ t

0 K(t − τ)u(τ) dτ‖2 exists and is finite foru |(0,∞)

∈ L2((0,∞);Rm). Since‖Ce•Ax0‖2 < ∞we therefore have that

y(•) = Ce•Ax0 +∫ •

0K(• − τ)u(τ) dτ+ Du(•)

restricted to(0,∞) is in L2((0,∞);Rp) wheneveru |(0,∞)

is in L2((0,∞);Rm).

It is common to identify an I/O system with its transfer matrix and to view this transfer matrixas a map from inputs to outputs. In general a transfer matrix does not determine the systemcompletely and it certainly is not a map (see, for instance, Example2.1.12). Nevertheless it isadvantageous to have this “I/O map” interpretation at our disposal, in particular as this interpre-tation lies at the basis ofH∞ control. The reason that this approach often works is that inpracticeone only encounters systems that are (part of an) internallystable system. In this case the effectof initial conditions on the output tends to zero, which, if discarded, leaves an output dependinguniquely and linearly on the input. We go through it in some detail for I/O systems that have aproper transfer matrix.

If an I/O systemΣI/O = (R,Rm,Rp,B ) has a proper transfer matrix, then by Lemma2.1.5there exist constant matricesA, B, C andD such that

B = [

uy

]

∈ L loc2 (R;R

m+p) | for arbitraryt0 ∈ R there exists anx0 ∈ Rn

such thaty(t) = CeA(t−t0)x0 +∫ t

t0

CeA(t−τ)Bu(τ) dτ+ Du(t) .

The effectCeA(t−t0)x0 of x0 tends to zero under the assumption that the system is part of astablesystem. OmittingCeA(t−t0)x0 leaves the desired linear map from inputs to outputs:

y(t) =∫ t

t0

CeA(t−τ)Bu(τ) dτ+ Du(t). (2.4)

Note that this map is not causal becauset is allowed to be less thant0. To overcome this technicalproblem it is often assumed thatt0 = −∞.

Definition 2.1.10 (Convolution systems). A convolution systemis a restricted I/O system wherefor some given constant real matricesA, B, C andD the outputy follows from the inputu as

y(t) =∫ t

−∞CeA(t−τ)Bu(τ) dτ+ Du(t). (2.5)

11

2. Systems

In this case equation (2.5) is said to represent the convolution system.

Lemma 2.1.11 (Convolution systems). LetΣ be a convolution system with input u and outputy. Denote its transfer matrix by G. LetL(w) denote the two-sided Laplace transform ofw.

1. The convolution system is completely determined by its transfer matrix G.

2. The system isL2-stable if and only if G has all its poles inC−. If the system isL2-stableand u∈ L2(R;R

m) then the two-sided Laplace transformsL(u) andL(y) of u and y aredefined almost everywhere on the imaginary axis, andL(y) = GL(u).

This is a standard result, see, for example, Desoer and Vidyasagar (19). Because of thislemma a rational matrix is calledstableif it is proper and its poles lie inC−. An expressionG(s) = C(sI − A)−1B + D with A, B, C and D constant matrices is called arealizationof G.Every proper real-rational matrixG has a realizationG(s) = C(sI − A)−1B + D with all fourmatricesA, B, C andD real (see e.g. Anderson and Vongpanitlerd (2)).

Example 2.1.12 (Indistinguishable convolution systems). In an obvious notation, the two I/Osystems

[

1 −1][

u1

y1

]

= 0, and[

ddt − d

dt

][

u2

y2

]

= 0

have the same transfer matrix. The solutions arey1 = u1 andy2 = u2 + c for some constantc,respectively. The second system is a convolution system iffc = 0, which shows that the twodifferential equations define the same convolution system.

2.1.2. Closed loops and well-posedness

Σ1

Σ2u

y

Figure 2.2.: A feedback interconnected system.

Fig 2.2shows the probably most common diagram in control theory. Itdepicts a closed-loopsystem in its simplest form. This closed loop represents an interconnection of two I/O systemsΣ1 andΣ2 with input/output pairs(u, y) and(y,u), respectively.

Suppose

N1u = D1y, N2y = D2u

are AR representations ofΣ1 andΣ2, respectively. In the closed loop in Fig.2.2 the signals(u, y) by definition have to satisfy the laws ofΣ1 and the laws ofΣ2 at the same time. In otherwords, an AR representation of the closed loop is

[

−N1 D1

D2 −N2

][

uy

]

= 0.

12


Such a closed-loop system is defined to beweakly well-posedif the closed-loop system de-picted in Fig.2.2 defines an I/O system with outputsu and y. In Vidyasagar (82) this is calledwell-posedness. More often well-posedness is reserved fora strengthened version of weak well-posedness (see e.g. Callier and Desoer (16)).

The notion of well-posedness as defined in Callier and Desoer(16), for example, may beintroduced as follows. WhenΣ1 andΣ2 represent two real physical systems connected as inFig.2.2via real physical signalsu andy, then there always will be a (presumably small) differencebetween the signals at the ports ofΣ1 and the ports ofΣ2. Diagrammatically this difference maybe visualized as disturbance signalsv1 andv2 (see Fig.2.3).

Σ1

Σ2 v1u

yv2

Figure 2.3.: A feedback interconnected system; setup for well-posedness.

Definition 2.1.13 (Well-posed closed loops). Let Σ1 andΣ2 be two I/O systems with in-put/output pairs(u, y) and (y,u), respectively. The closed loop in Fig.2.2 is well-posedifthe extended closed loop in Fig.2.3 has the property thaty andu are inL loc

2 (R;R•) wheneverv1 andv2 are inL loc

2 (R;R•).

Stated differently, the closed loop is well-posed if it isL loc2 -stable under perturbations(v1, v2).

Lemma 2.1.14 (Well-posedness). Consider the closed loop as depicted in Fig.2.3. LetΣ1 andΣ2 be I/O systems and suppose that they have AR representations

N1u = D1y, D1 nonsingular,

and

N2y = D2u, D2 nonsingular,

respectively. The closed loop as in Fig.2.3 with inputs(v2, v1) and outputs(u, y) then has theminimal AR representation

[

−N1 D1

D2 −N2

]

︸︷︷︸

D:=

[

uy

]

=[

N1 00 N2

]

︸︷︷︸

N:=

[

v2

v1

]

.

Let G := D−11 N1 and K := D−1

2 N2 denote the transfer matrices ofΣ1 andΣ2, respectively. Thefollowing statements hold.

1. The closed loop is weakly well-posed iff D is nonsingular,or equivalently, iff( I − GK) isa nonsingular rational matrix.

2. The closed-loop system is well-posed iff the transfer matrix

D−1N =[

−G II −K

]−1 [

G 00 K

]

(2.6)

from[v2v1

]

to[ u

y]

exists and is proper.

13

2. Systems

3. The closed loop is internally stable iff D is strictly Hurwitz.

4. The closed loop isL2-stable iff D is strictly Hurwitz and (2.6) is proper.

The proof is immediate from the results in the previous section. Note that well-posedness hasnothing to do with properness of the transfer matrices of thetwo subsystemsΣ1 andΣ2. Forinstance, we have a well-posed closed-loop system ifG(s) = 1 is the transfer matrix ofΣ1 andK(s) = −s is the transfer matrix ofΣ2. If both transfer matricesG andK are proper, then well-posedness is equivalent toI − G(s)K(s) being nonsingular ats= ∞. More complicated closedloops may be treated similarly.

Note also that whenever the system isL2-stable the transfer matrix (2.6) is stable (we call arational matrix stable if it is proper and all its poles lie inC−). In many cases the converse holdsas well:

Example 2.1.15 (Scalar transfer matrices). Let g ∈ R 1×1 be a scalar rational function and letn andd be a coprime pair of scalar polynomials6 such thatg = n

d . Suppose thatG is the transfermatrix of an I/O system with inputu and outputy. If p(d/dt)y(t) = q(d/dt)u(t) is an ARrepresentation ofΣ, then

qp

= G = nd.

This implies that[

−q p]

= f[

−n d]

for some nonzero polynomialf , sincen and d byassumption do not have common zeros. This factorf cancels in the expression of the transfermatrix and, therefore, is lost in a transfer matrix description of the system.

Definition 2.1.16 (Coprime polynomial matrices). Polynomial matricesN and D are left co-prime if F−1N andF−1D are polynomial for some polynomial matrixF only if F is unimodu-lar.

EveryG ∈ R may be written asG = D−1N with D andN left coprime polynomial matrices.The expressionG = D−1N is then called aleft coprime polynomial matrix fraction description(left coprime PMFD)of G. A pair N and D is left coprime iff

[

−N D]

has full row rankeverywhere in the complex plane. This shows that ifG = D−1N is a left coprime PMFD ofG,thenG(s) is unbounded ats= λ ∈ C iff D(λ) is nonsingular (consider the identity

[

−N D]

=[

−DG D]

). We callλ ∈ C a poleof G ∈ R if G(s) is unbounded arounds = λ. AppendixBcontains details and proofs on these algebraic properties of polynomial and rational matrices.

Definition 2.1.17 (Hidden modes). λ ∈ C is ahidden modeof a system with AR representationRw= 0 if R(s) drops below normal rank fors= λ. λ is anunstablehidden mode if it is a hiddenmode andλ ∈ C0 ∪ C+.

Lemma 2.1.18 (I/O systems with identical transfer matrices ). LetΣI/O be an I/O system withinput u and output y. Let G denote its transfer matrix and let D−1N = G be a left coprimePMFD of G. Then

Dy = Nu,[

−N D]

∈ P , D square nonsingular,

defines an AR representation ofΣI/O only if[

−N D]

= F[

−N D]

for some square nonsingular F∈ P .

6A pair of scalar polynomialsn andd is coprimeif n andd do not have common zeros.

14


A proof of this result is implicitly given by the results on PMFDs in AppendixB. The nextresult is important.

Lemma 2.1.19 (Stabilizability). Consider the closed-loop system in Fig.2.3with inputs(v1, v2)

and outputs(u, y). Suppose that neitherΣ1 nor Σ2 has hidden modes. Denote the transfermatrices ofΣ1 andΣ2 by G and K, respectively. Then the closed-loop system isL2-stable if andonly if the transfer matrix

[

−G II −K

]−1 [

G 00 K

]

from[v2v1

]

to[ u

y]

is stable.

Proof . Let D1y = N1u andD2u = N2y be AR representations of systemΣ1 andΣ2 respectively.The expressionsG = D−1

1 N1 andK = D−12 N2 are left coprime PMFDs because by assumption

there are no hidden modes. From Lemma2.1.14we know that the system isL2-stable iff D−1Nis proper andD strictly Hurwitz, whereN andD are defined by

D =[

−N1 D1

D2 −N2

]

, N =[

N1 00 N2

]

.

(Only if) By Lemma2.1.14L2-stability implies thatD−1N is stable.(If) SupposeD−1N is stable as a rational matrix. The idea is to show thatD and N are left

coprime, so that zeros ofD are poles ofD−1N. In this caseL2 stability is ensured wheneverG isproper and has all its poles inC−. Suppose, to obtain a contradiction, thatD andN are not leftcoprime. That is, suppose

[

−N(s) D(s)]

loses row rank at, say,s = λ ∈ C. Then there existsa nonzero constant row vectorc =

[

c1 c2]

such that

0 =[

c1 c2] [

−N(λ) D(λ)]

=[

c1 c2][

−N1(λ) 0 −N1(λ) D1(λ)

0 −N2(λ) D2(λ) −N2(λ)

]

.

From the above expression it follows that

c2N2(λ) = 0, c2D2(λ) = 0.

This is possible only ifc2 = 0, because by assumptionN2 andD2 are left coprime. Similarly wehave thatc1N1(λ) = 0 and thatc1D1(λ) = 0. Again by coprimeness this implies thatc1 = 0.This, however, contradicts the fact thatc =

[

c1 c2]

was taken nonzero. As a result[

−N D]

must have full row rank everywhere in the complex plane, and,hence,N andD are left coprime.The latter implies that all zeros ofD appear as poles ofD−1N and soL2-stability is implied bythe assumption thatG = D−1N is proper and has poles only inC−.

We implicitly assume from now on that there are no hidden modes in an I/O system if wechoose to represent it by its transfer matrix. This way the I/O system associated with the transfermatrix is unique. This is immediate from Lemmas2.1.18and2.1.2.

The main advantage of the “transfer matrix approach” is thatthe language needed to definesystems and to characterize their properties is somewhat easier, though of course we should keepin mind that the transfer matrix is nothing but a differential operator in disguise—at least, this isthe way we see it. Note also that in stable closed loops the effect of initial conditions tends tozero. Hence in the long run such systems behave like their related convolution systems, which inturn by Lemma2.1.9are determined completely by their transfer matrices anyway.

15

2. Systems

2.1.3. I/O systems in frequency domain

Transfer matrices play an important role in a frequency domain approach to problems in systemtheory. Roughly speaking the frequency domain is the Laplace transformed version of the timedomain. The main theorem that enables the translation is a result by Paley and Wiener (seeRudin (76)). Combined with a little Hardy space theory and the Parseval and Plancherel resultswe get Theorem2.1.20.

Theorem 2.1.20 (Paley-Wiener). Define the Hardy spacesH ⊥2 andH2:

H ⊥2 := f : C− 7→ C | f is analytic inC− and sup

σ<0

∫ ∞

−∞‖ f (σ + jω)‖2

2 dω <∞ ,

H2 := f : C+ 7→ C | f is analytic inC+ and supσ>0

∫ ∞

−∞‖ f (σ+ jω)‖2

2 dω <∞ .

LetL(w) denote the two-sided Laplace transform ofw.

1. The spacesH ⊥2 andH2 may be identified with the subspaces of elements ofL2(C0;C) that

have an analytic extension inC− andC+ respectively. Moreover, under this identificationH ⊥

2 andH2 are Hilbert spaces in the usual inner product onL2(C0;C)

〈 f, g〉 = 12π

∫ ∞

−∞f ∗(iω)g(iω) dω

and with itH ⊥2 andH2 are orthogonal subspaces ofL2(C0;C) andL2(C0;C)= H ⊥

2 ⊕H2.

2. H ⊥2 (H2) is isomorphic toL2(R−;C) ( L2(R+;C)) under the Laplace transform:

H ⊥2 = L(L2(R−;C)), H2 = L(L2(R+;C)),

and the transformation is bijective and preserves inner products. (Here elements ofL2(R−;C) or L2(R+;C) are identified with their embedding inL2(R;C).)

Example 2.1.21 (Paley-Wiener). Supposew(t) = et for t < 0 andw(t) = e−2t for t ≥ 0, thenw := L(w) equals 1

1−s + 12+s. This may be written asw = f− + f+, with f− = 1

1−s in H ⊥2 and

f− = 12+s in H2. The decomposition ofw as the sum of an element ofH ⊥

2 and of an element ofH2 is unique. Obviouslyf− = L(π−w) and f+ = L(π+w), whereπ− andπ+ are the orthogonalprojections fromL2(R;C) to L2(R−;C) andL2(R+;C), respectively.

We usually identify an elementw of L2(R;C) with its Laplace transform; for a time signaland its Laplace transform we use one symbol as long as no confusion can arise. Note that weworked with complex-valued time signals in the above theorem. It is possible to formulate asimilar result for real-valued signals but then things are bit messy.

We have already encountered the Laplace transform when we discussed convolution systems.Under the Laplace transform the convolution operator transforms in to a multiplication operator,which has some obvious conceptual advantages (see Lemma2.1.11). In the frequency domainit, therefore, is easier to characterize theL2 induced operator norm of a convolution operator.Concretely, SupposeG is the transfer matrix of a convolution system with inputu and outputy.Let y andu denote the Laplace transforms ofu andy, respectively. Then

y = Gu

16


by Lemma2.1.11. Therefore, by the Paley-Wiener results,

supu∈L2(R;C)

‖y‖2

‖u‖2= sup

u∈L2(C0;C)

‖Gu‖2

‖u‖2.

The latter expression is nothing but

‖G‖∞ := supω∈R

√

λmax(G∗(iω)G(iω)).

Here,λmax(•) denotes the largest eigenvalue of its argument. TheL2 induced operator norm ofa convolution operator equals the so calledinfinity norm (∞-norm)‖ • ‖∞ of the correspondingtransfer matrix.

The∞-norm of rational matrices is finite for rational matrices that are proper and have no poleson C0. However, it has the interpretation of theL2 induced operator norm of the correspondingtime domain convolution operator only if the transfer matrix is stable. The set of real-rationalstablep × m matrices is denoted asR H

p×m∞ . This is a subset of a more general Hardy space.

More details on the∞-norm and associated Hardy spaces may be found in AppendixA.

2.1.4. Stabilizing compensators and H∞ control

Consider the system depicted in Fig.2.4. The subsystems are represented by their transfer ma-

r y u zK P

Figure 2.4.: A feedback system.

trices P and K. A standard problem in controller design is to find a transfermatrix K, given atransfer matrixP, such that the closed-loop system with inputr in Fig. 2.4behaves well in somesense. The systemP is often called theplant, and the systemK that is to be constructed by thedesigner is usually referred to as the(feedback) compensatoror thecontroller. Obviously, un-stable closed loops do not “behave well”. Hence the primary task in controller design is to find acompensator that internally stabilizes the closed loop. Such compensators are called(internally)stabilizing compensators.

It turns out there are many compensators that stabilize a given plant. This freedom in thechoice of stabilizing compensator may be exploited to meet other objectives. Extremely popularnowadays areH∞ controller design problems. Practically all these problems consist of finding astabilizing compensator that makes an associated transfermatrix H satisfy

‖H‖∞ < γ,

for some prescribed boundγ (see, to name but a few, (91; 20; 42; 48; 49; 50; 56)). The ideabehind theseH∞ control problems is that the associated transfer matrixH may be constructedin such a way that it contains information on how well the closed loop behaves, in the sense thatthe smaller its∞-norm is the better the closed loop behaves. It is not our intention to give anoverview of what type of objectives may be translated in terms of ∞-norm bounds, and neitherwill we discuss and compare the various ways of translating these objectives in terms infinitynorm bounds. An excellent paper on this part of theH∞ control problem is Kwakernaak (50).We confine ourselves to the solution to a set of those problems.

17

2. Systems

Definition 2.1.22 (Mixed sensitivity problem, ( 48; 49; 50)). Consider the closed loop inFig. 2.4. Let rational matricesV, W1 and W2 of appropriate sizes be given. Thesuboptimalmixed sensitivity problemis to find compensatorsK, given a plantP, that internally stabilize thesystem in Fig.2.4 (with u, y andz considered as the outputs andr as the input) and that makethe∞-norm of the transfer matrix

H :=[

W1( I + PK)−1VW2K( I + PK)−1V

]

from w to[ z1

z2

]

as in Fig.2.5 less than some given boundγ. The optimal mixed sensitivityproblem is to find compensatorsK that minimize‖H‖∞ over all compensators that internallystabilize the system in Fig.2.4.

K P

V

W1

W2

w

z1

z2

uy

Figure 2.5.: The mixed sensitivity system configuration.

In Kwakernaak (48; 49; 50) this problem is treated in detail. In (49) it is argued that the rationalmatricesV, W1 andW2 need not all be proper. In particular,W2 will generally be nonproper. Therational matricesV, W1 andW2 are usually called(shaping) filters(see (48)). Note that in ourdefinition of the mixed sensitivity problem it is not assumedthat the allowable compensators areproper, and neither do we assume that the allowable compensator makes the closed loop well-posed. The idea is that properness of the compensator and well-posedness of the closed loopare two of the objectives that may be translated in terms of∞-norm bounds. In other words,for suitably chosen shaping filters, an internally stabilizing compensator that solves the mixedsensitivity problem is automatically proper and makes the closed loop well posed. (In fact, thisis the reason why it is useful or even necessary to takeW2 nonproper.) We return to the mixedsensitivity problem in Chapter4.

18

3

Frequency domain solution to suboptimaland optimal two-block H∞ problems

In this chapter we solve a certain class ofH∞ problems, known astwo-blockH∞ problems.

r e

y

z

u

A

B Q

Figure 3.1.: The two-block system configuration.

Consider the configuration in Fig.3.1. The two-blockH∞ problem (TBP) is to find stableQ,given stableA and B, such that the∞-norm of the closed-loop transfer matrixQB+ A from rto e is minimized.

The TBP in this chapter is handled by considering two relatedproblems. When combinedthey solve the TBP. The two related problems are thesuboptimal TBPand theoptimal TBP. Thesuboptimal TBP (STBP, from now on) with boundγ is to find stableQ, given stableA and B,such that

‖QB+ A‖∞ < γ.

In Section3.2 necessary and sufficient conditions are derived for the STBPto have a solution,and, provided solutions exist, a generator of all suboptimal solutionsQ is presented.

The optimal TBP (OTBP) is treated in Section3.3. The OTBP is to find stableQ, given stableA andB, such that

‖QB+ A‖∞ = γopt,

whereγopt is defined as

γopt := infstableQ

‖QB+ A‖∞.

The difference between the OTBP and the TBP is that in the OTBPthe value ofγopt is assumedgiven. Of course,γopt may be delimited using a line search that at each step invokesthe solutionmethod to the STBP. Thus, together the STBP and the OTBP solvethe TBP.

19

3. Frequency domain solution to suboptimal and optimal two-blockH∞ problems

The key idea that proves to be very useful originates from Khargonekar (40). In (40) it is usedin a state space setting for the suboptimal case. We use it in afrequency domain setting andextend it to the optimal case. The idea in time domain is this:Consider the TBP configurationin Fig. 3.2with, again,A andB given stable transfer matrices andQ a stable transfer matrix to

r e

y

z

u

A

B Q

Figure 3.2.: The two-block system configuration.

be determined. Letz− andz+ denote the signalz restricted toR− andR+, respectively. Supposethat the inputr is in L2(R−;C) 1 and suppose in addition that the resulting signaly in Fig. 3.2isin L2(R+;C) 1. The interesting thing to note is that such signalsr provide necessary conditionsfor the STBP to have a solution. To see this we argue as follows: If r is in L2(R−;C) and yis in L2(R+;C), then for negative timee(t) equalsz(t) = z−(t) since Q represents a causalconvolution system. The STBP with boundγ, therefore, has a solution only if

∫ 0−∞ z∗(t)z(t) dt

∫ 0−∞ r∗(t)r (t) dt

≤ γ2

for all r ∈ L2(R−;C) for which y lies inL2(R+;C), since

‖QB+ A‖∞ = supr∈L2(R;C)

‖e‖2

‖r‖2.

Stated in frequency domain terms: The STBP with boundγ has a solution only if

‖z−‖2

‖r‖2≤ γ (3.1)

for all elements of

B[B 0A −I

] :=[

rz−

]

∈[

H ⊥2

H ⊥2

]

|[

yz+

]

:=[

B 0A − I

][

rz−

]

∈[

R H 2R H 2

]

. (3.2)

For some reasons it is more convenient to replace (3.1) with the equivalent condition

γ2‖r‖22 − ‖z−‖2

2 ≥ 0. (3.3)

In the theory of indefinite inner product spaces the subspaces (3.2) whose elements satisfy (3.3)are calledpositivesubspaces ((3; 14)).

In view of this observation it will be no surprise that sets ofthe form

BG := w ∈ H ⊥m2 | Gw ∈ H g

2 1We identify such elements with their embedding inL2(R;C).

20

3.1. Preliminaries: Positive subspaces

play an important role in this chapter. In Section3.1 elementary properties of such sets arediscussed. The main result in Section3.1is Theorem3.1.12, which we think is the “basic buildingblock” of frequency domain solution methods toH∞ control problems. This theorem givesnecessary and sufficient conditions for a given setBG to be strictly positive. The conditioninvolves a canonical factorization problem, a problem which by now is well understood (seeAppendixB). Section3.2shows how all solutions to the STBP may be generated. In Section3.3the OTBP is solved provided one assumption is made. Technicalities are unavoidable whenhandling the OTBP. The “elaborate” theory that we need for the STBP and OTBP is the theoryof indefinite inner product spaces, or more specifically, thetheory of of Kreın spaces (see Azizovand Iokhvidov (3) and Bognar (14)).

In various books and papers (6; 7; 9; 10; 26; 34; 36) the suboptimal two-blockH∞ problem istreated. Most notable are a series of papers by Ball and Helton, in particular (7). The paper (7)contains several connections between various forms of positivity of signal spaces and propertiesof transfer matrices defining the system. In some respect theresults in (7) are more general thanours, in another respect they are more restricted than ours.(Their spaceχ+ is in one-to-onecorrespondence with the spaces we encounter, though theirsare finite dimensional, whereas wetreat the infinite dimensional case.) There is also a connection between our approach and thenotion of “null pair” as defined in Ball, Gohberg and Rodman (4). Like in (7), in (4) only thefinite dimensional case is considered. The results of Kimura, Lu and Kawatani (43) are linked toour results. They apply a “J-orthogonal complement approach” to solve a class ofH∞ controlproblems. The work (43) has in common with (4) that the transfer matrices defining the systemsform the basis of research. In our approach it is not necessary to restrict the attention to I/Osystems as is done in (43), and more or less by consequence our results are more general than theresults in (43).

The generator of optimal solutions as we derive it for the OTBP is of the same form as thegenerator of optimal Hankel norm approximations as given inBall and Ran (10). In Glover,Limebeer, Doyle, Kasenally and Safonov (31) optimal solutions are derived, using all-pass em-beddings, for the more general four-blockH∞ problem. The main difference between our ap-proach and other approaches, except (6; 7), is that our approach is based on signals, while otherapproaches tend to emphasize more the role of the transfer matrix relating the signals. The mainreason to approach the problem the way we do it is that it allows an extension to the optimal caseand that it may be used (in a slightly modified version) to solve H∞ control problems involvingnonproper transfer matrices, like mixed sensitivity problems. The computational aspects are dis-cussed only very briefly at the end of this chapter. In Appendix B an algorithm is formulatedfor the computation of noncanonical factorization. This algorithm in principle may be used tocompute the solutions based on the results we obtain in this chapter.


The solution of both the STBP and OTBP using frequency domainmethods require several resultsfrom operator theory. In this section we review some of thesepreliminary results, culminating ina theorem that connects strict positivity and co-J-losslessness.

The type of sets that we study in this section are sets of the form

BG := w ∈ R H⊥m2 | Gw ∈ R H

g2

depending on some stable rational matrixG ∈ R Hg×m∞ . We say thatBG is generatedby G. In

particular we are interested in whether or not they are “strictly positive”. This will be defined

21


later. Throughout this chapterJ and J denote matrices of the form[

Iq 00 − I p

]

.

This matrix is abbreviated toJq,p. The spaceBG is a subset of the Hilbert spaceH ⊥m2 with inner

product

〈 f, g〉 = 12π

∫ ∞

−∞f ∗(iω)g(iω) dω

(see AppendixA). For every proper real-rationalG that has no poles onC0 we have that

〈 f,Gg〉 = 〈G∼ f, g〉, with G∼(s) := [G(−s∗)]∗ = [G(−s)]T.

The set of real-rational stable matrices is denoted byR Hg×m∞ . The set of real-rational stable

matrices whose inverse is stable as well is denoted byGR Hm×m∞ . Note thatWH m

2 ⊂ H g2 if

W ∈ R Hg×m∞ and thatWH m

2 = H m2 if W ∈ GR H

m×m∞ .

Corollary 3.1.1 (Equivalent sets BG). Let G be an element ofR Hg×m∞ . Then

BG = BWG = BW−1G

if W ∈ GR Hg×g∞ .

Proof . WGBG ⊂ WH g2 = H g

2 . ThereforeBG ⊂ BWG. The converse may be proved similarly.

In fact a much stronger result may be proved (see Chapter5, Corollary5.2.11). Corollary3.1.1,however, will do for the solution of two-blockH∞ problems.

Corollary 3.1.2 (Closed subspace). If Z ∈ R g×m is proper, has no poles onC0 and has fullcolumn rank onC0 ∪ ∞, then ZH ⊥m

2 is a closed subspace ofL2(C0;Cg).

Proof . We show thatZH ⊥m2 is complete as a space in the norm induced byL2(C0;Cg). This

then implies that it is closed as a subspace ofL2(C0;Cg). Consider a Cauchy sequencezi inZH ⊥m

2 and definepi ∈ H ⊥m2 by zi = Zpi. We have that

‖zi − zj‖22 = 〈Z(pi − p j), Z(pi − p j)〉 = 〈(pi − p j), Z∼ Z(pi − p j)〉 ≥ ε‖pi − p j‖2

2

whereε is just any nonzero positive number for whichZ∼ Z − ε I > 0 everywhere onC0 ∪ ∞.(Such anε exists becauseZ∼ Z is positive definite everywhere onC0 ∪ ∞.) We therefore havelim i, j→∞ ‖pi − p j‖2

2 ≤ 1ε

lim i, j→∞ ‖zi − zj‖22 = 0. In other words,pi is also a Cauchy sequence.

SinceH ⊥m2 is closed, we have thatp∞ := lim i→∞ pi exists inH ⊥m

2 . Definez∞ := Zp∞. Thenobviously limi→∞ ‖zi − z∞‖2

2 = 0 and, thereforeZH ⊥m2 is closed.

Definition 3.1.3 (Positivity, ( 3)). A subspaceB of a q + p vector valued Hilbert spaceM withinner product〈•, •〉 is positive (P)with respect to theJq,p inner product

[ f, g] := 〈 f, Jq,pg〉,

if for everyw ∈ B

〈w, Jq,pw〉 ≥ 0.

22


It is strictly positive (SP)with respect to theJq,p inner product if there exists anε > 0 such thateveryw ∈ B satisfies

〈w, Jq,pw〉 ≥ ε〈w,w〉. (3.4)

Inequality3.4 is referred to as the SP inequality.

Lemma 3.1.4 (Connection between stability and positivity) . Suppose that G=[

H1 H2]

has full row rank onC0 ∪∞ with H1 ∈ R Hp×q∞ and H2 ∈ R H

p×p∞ . The following two statements

are equivalent.

1. ‖H−12 H1‖∞ ≤ 1; H2 ∈ GR H

p×p∞ .

2. B[

H1 H2

] ⊂ H ⊥q+p2 is positive with respect to the Jq,p inner product.

The following two statements are equivalent

3. ‖H−12 H1‖∞ < 1; H2 ∈ GR H

p×p∞ .

4. B[

H1 H2

] ⊂ H ⊥q+p2 is strictly positive with respect to the Jq,p inner product.

We say thatH is acontractionif ‖H‖∞ ≤ 1. It is astrict contractionif ‖H‖∞ < 1.

Proof . (1 ⇒ 2) BecauseH2 is GR Hp×p∞ we have by Corollary3.1.1thatB[

H1 H2

] = B[

H I],

with H := H−12 H1. Let w ∈ B[

H I] be partitioned asw =

[w1w2

]

, compatibly with the

partitioning ofG. Since

[

w1

w2

]

∈ H ⊥q+p2 ,

[

H I][

w1

w2

]

∈ H p2 ⇐⇒ w2 = −π−Hw1, w1 ∈ H ⊥q

2

we have〈w, Jq,pw〉 = ‖w1‖22 − ‖π− Hw1‖2

2 ≥ ‖w1‖22(1 − ‖H‖∞) ≥ 0. I.e. B[

H1 H2

] is

positive. (Hereπ− denotes the orthogonal projection fromL2(C0;Cp) on toH ⊥ p2 .)

(2 ⇒ 1) Suppose, to obtain a contradiction, thatH2 is singular for somes= s∈ C+. Then thereexists a nonzerov ∈ Cp such thatH2(s)v = 0. Definez as

z(s) :=[

01

s−sv

]

.

This z is in H ⊥q+p2 , z ∈ B[

H1 H2

] and〈z, Jq,pz〉 < 0. This contradicts the positivity of

B[

H1 H2

], and, hence,H2 is nonsingular inC+ and, in particular,H−12 exists. Next we

show that‖H−12 H1‖∞ ≤ 1. Let D andN be antistable rational matrices2 such that

[

Iq

−H−12 H1

]

= DN−1.

2A matrix M is antistable ifM∼ is stable.

23


We have that[

H1 H2]

D = 0, and as a consequenceDH ⊥q2 is a subset of the positive

subspaceB[

H1 H2

]. This implies thatD∼ Jq,pD ≥ 0 on the imaginary axis. Finally it

follows from nonsingularity ofN and

N∼( I − H∼ H)N = D∼ Jq,pD ≥ 0 (whereH := H−12 H1)

that‖H‖∞ ≤ 1.

The fact that‖H−12 H1‖∞ ≤ 1 also shows thatH2 is nonsingular onC0 ∪ ∞ because

[

H1 H2]

by assumption has full row rank onC0 ∪ ∞. ThereforeH2 is nonsingularin C0 ∪ C+ ∪ ∞, and, hence,H2 ∈ GR H

p×p∞ .

The equivalence of Items 3 and 4 may be proved similarly.

Lemma 3.1.5 (The inertia Lemma). Given a full row rank constant matrix G∈ C(r+p)×(q+p)

and a full column rank constant matrix P∈ C(q+p)×(q−r ). The following two statements are

equivalent if GP= 0.

1. P∗ Jq,pP> 0.

2. GJq,pG∗ is nonsingular, has p negative eigenvalues and r positive eigenvalues.

Proof . (1 ⇒ 2) SupposeP∗ Jq,pP> 0. It follows from[

GP∗

][

G∗ Jq,pP]

=[

GG∗ GJq,pP0 P∗ Jq,pP

]

that[

G∗ Jq,pP]

is nonsingular. As a result also

[

GP∗ Jq,p

]

Jq,p[

G∗ Jq,pP]

=[

GJq,pG∗ 00 P∗ Jq,pP

]

(3.5)

is nonsingular. In particular this implies that the matrixGJq,pG∗ is nonsingular. By Sylvester’sinertia law (see, for example, Lancaster (52, pp. 89-90)), the identity (3.5) implies that thenumberp of negative eigenvalues ofJq,p equals that of the right hand side of (3.5). The resultthen follows sinceP∗ Jq,pP by assumption is positive definite. That Item 2 implies Item 1maybe proved in a similar manner.

We refer to this result as theinertia lemma.

Definition 3.1.6 ( J-orthogonal complements). Given a subspaceY ⊂ R H⊥q+p2 and with J

defined asJ := Jq,p, the J-orthogonal complement ofY is denoted asY ⊥J and defined by

Y ⊥J := w ∈ R H⊥q+p2 | 〈w, Jy〉 = 0 for all y ∈ Y .

The ordinary orthogonal complement ofY is denoted asY ⊥.

A closed subspaceY and its ordinary orthogonal complementY ⊥ span the whole space. Thisis a well known fact. Unfortunately a closed subspaceY and itsJ-orthogonal complementY ⊥J

do not necessarily span the whole space. In many cases they do, however:

Lemma 3.1.7 ( J-orthogonal complements). Suppose G∈ R Hm×(q+p)∞ has full row rank on

C0 ∪ ∞ and define J= Jq,p. The following statements hold.

24


1. BG, G∼H ⊥m2 and JG∼H ⊥m

2 are closed subspaces ofH ⊥q+p2 , and

B ⊥JG = JG∼H ⊥m

2 .

2. If BG is strictly positive then

BG ⊕ B ⊥JG = R H

⊥q+p2 .

SubspacesM of H ⊥q+p2 that satisfyM ⊕ M ⊥J = H ⊥q+p

2 are so-calledregular subspacesof H ⊥q+p

2 under theJ inner product (see (3; 7)). Here,M ⊕ M ⊥J = H ⊥q+p2 means that every

element ofH ⊥q+p2 may uniquely be written as a sum of an element ofM and an element of

M ⊥J.

Proof .

1. The closedness ofBG is shown later implicitly by showing that it is the orthogonal comple-ment ofG∼H ⊥m

2 . The closedness ofJG∼H ⊥m2 andG∼H ⊥m

2 follows from Corollary3.1.2.

Since JG∼R H⊥m2 is closed, the claim thatB ⊥J

G = JG∼H ⊥m2 is equivalent to the claim

that [JG∼H m2 ]⊥J = BG. The latter claim is easy: TheJ-orthogonal complement of

JG∼R H⊥m2 is z ∈ R H

⊥q+p2 | 〈z, J JG∼g〉 = 0 for all g ∈ R H

⊥m2 = z ∈ R H

⊥q+p2 |

〈Gz, g〉 = 0 for all g ∈ R H⊥m2 = BG.

2. We will show that the ordinary orthogonal complement (inR H⊥q+p2 ) of BG +

JG∼R H⊥m2 is 0. The closedness ofBG + JG∼R H

⊥m2 , which is shown thereafter,

then gives thatBG + B ⊥JG = H ⊥q+p

2 . Finally we show that the intersectionBG ∩ B ⊥JG is

the zero element, and thus it follows thatBG ⊕ B ⊥JG = R H

⊥q+p2 .

Let v be an element of the orthogonal complement ofBG + JG∼R H⊥m2 . That is, letv be

an element of

v ∈ R H⊥q+p2 | v ∈ B ⊥

G = G∼H ⊥m2 , v ∈ [ JG∼R H

⊥m2 ]⊥ = BGJ .

v ∈ B ⊥G = G∼R H

⊥m2 implies thatv = G∼z for somez ∈ R H

⊥m2 . In additionv ∈ BGJ

impliesGJG∼z∈ R Hm2 . Our aim is to show thatz≡ 0, from which follows thatv≡ 0. To

obtain a contradiction, suppose that there exists a nonzeroz∈ R H⊥m2 such thatGJG∼z∈

R Hm2 . Thent defined ast = JG∼z is nonzero as well and thist ∈ H ⊥q+p

2 is an element

of BG sinceGt = GJG∼z∈ R Hm2 . This contradicts the strict positivity ofBG, because

〈t, Jt〉 = 〈JG∼z,G∼z〉 = 〈GJG∼z, z〉

is zero becauseGJG∼z ∈ R Hm2 andz ∈ R H

⊥m2 are perpendicular. Hence,z ≡ 0 and

therefore so isv.

Next we show thatZ := BG + JG∼R H⊥m2 is closed. Consider a Cauchy sequencezi =

bi + gi in Z with bi ∈ BG andgi ∈ JG∼R H⊥m2 . So limi, j→∞ ‖zi − zj‖ = 0. SinceBG is

SP there exists anε > 0 such that

|〈bi − b j, J(bi − b j)〉| > ε‖bi − b j‖2

25


for all bi,b j ∈ BG. Consider now the following two inequalities:

|〈zi − zj , J(bi − b j)〉| = |〈bi − b j, J(bi − b j)〉| ≥ ε‖bi − b j‖2,

(becausebi −b j andgi −g j areJ-orthogonal)

|〈zi − zj , J(bi − b j)〉| ≤ ‖zi − zj‖‖bi − b j‖ (Schwarz inequality).

It follows that‖bi − b j‖ ≤ 1ε‖zi − zj‖. In other words,bi is itself a Cauchy sequence,

and, hence, so isgi = zi − bi. Since the spacesBG andJG∼R H⊥m2 are closed it follows

thatb∞ := lim i→∞ bi andg∞ := lim i→∞ gi are well defined inBG and JG∼R H⊥m2 , re-

spectively. Thenz∞ defined asz∞ := b∞ + g∞ is in Z and limi→∞ ‖zi − z∞‖ = 0, hence,Z is a closed subset ofR H

⊥q+p2 .

It remains to show thatBG ∩ B ⊥JG = 0. Let m be an element ofBG ∩ B ⊥J

G . Then appar-ently m is J-orthogonal to itself, that is,〈m, Jm〉 = 0. On the other hand it follows fromstrict positivity ofBG and the fact thatm is an element ofBG that〈m, Jm〉 ≥ ε〈m,m〉 forsomeε > 0. Thereforem= 0, i.e.,BG ∩ B ⊥J

G = 0.

Example 3.1.8 ( J-orthogonal complements). SupposeG =[

1 −γ]

for someγ ∈ R, and takeJ = J1,1. We then have

BG =[

γ

1

]

H ⊥2 , B ⊥J

G =[

1γ

]

R H⊥2 , BG + B ⊥J

G =[

γ 11 γ

]

R H⊥22 .

It is obvious from the above expressions thatBG ⊕ B ⊥JG = R H

⊥22 iff γ2 6= 1. This is definitely

the case whenBG is strictly positive (|γ| > 1).

For the following result we need to know what canonical and noncanonical factorizations are.In AppendixB many results on factorizations are given and proved. We refer to this appendix foran overview of the definitions used and results involving factorization of rational matrices. Oneresult which is stated in subsectionB.3.2and proved constructively in the subsection thereafter,is:

Theorem 3.1.9 (Cofactorization). The following two statements are equivalent.

1. Z = Z∼, Z ∈ R m×m and Z and Z−1 have no poles or zeros onC0 ∪ ∞.

2.

Z = W

0 0 0 D+0 Ir−l 0 00 0 − I p−l 0

D∼+ 0 0 0

W∼, D+(s) =

0 0 ( s−1s+1)

kl

0 . ..

0( s−1

s+1)k1 0 0

(3.6)

for some nonnegative integer p, r:= m− p, l, strictly positive integers kj, j ∈ 1, . . . , land real-rational W∈ GR H

m×m∞ .

The expression (3.6) in Item 2 is called anoncanonical cofactorizationof Z if D+ is non-void.If D+ is void we call it acanonical cofactorizationof Z.

26


Lemma 3.1.10 (strict positivity, necessity results). Suppose G∈ R H(r+p)×(q+p)∞ has full row

rank onC0 ∪ ∞. ThenBG is SP with respect to the indefinite inner product[u, y] = 〈u, Jq,py〉only if GJq,pG∼ admits a canonical cofactorization

GJq,pG∼ = W Jr,pW∼; W ∈ GR H

(r+p)×(r+p)∞ .

Proof . Define J = Jq,p. Let P ∈ R (q+p)×(q−r ) be an antistable3 rational matrix of full columnrank such thatGP = 0. Without loss of generality we may assume thatP has full column rankon C0 ∪ ∞. By CorollaryB.3.5 there then existF ∈ GR H

(q−r )×(q−r )∞ such thatP∼ P = FF∼.

RedefineP as P := PF−∼. This way P is still an antistable rational matrix of maximal fullcolumn rank such thatGP= 0, but now in addition we have thatP∼ P = I . The spacePH ⊥q−r

2

is a subset ofBG as is easily seen. SinceBG is strictly positive by assumption,PH ⊥q−r2 is also

strictly positive. SinceP∼ P = I we must have that

P∼ J P≥ ε I on C0 ∪ ∞ (3.7)

because only then the strict positivity inequality〈w, Jw〉 ≥ ε‖w‖22 holds onPH ⊥q−r

2 . In par-ticular it follows that P∼ J P is nonsingular onC0 ∪ ∞. By application of the inertia lemmathis implies thatGJG∼ has no zeros onC0 ∪ ∞ and thatGJG∼ hasr positive andp negativeeigenvalues everywhere onC0 ∪ ∞. By Theorem3.1.9this in turn implies thatGJG∼ admits acanonical or noncanonical cofactorization:

GJG∼ = W

0 0 0 D+0 Ir−l 0 00 0 − I p−l 0

D∼+ 0 0 0

W∼, D+(s) =

0 0 ( s−1s+1)

kl

0 ...

0( s−1

s+1)k1 0 0

,

with W ∈ GR H(r+p)×(r+p)∞ and somek j > 0. Suppose the cofactorization is noncanonical. In

this case the vector

t := JG∼W−∼

0...0

1/(• − 1)

is an element ofBG (Gt ∈ H r+p2 ) and〈t, Jt〉 = 0 becauset∼ Jt = 0. This contradicts the strict

positivity of BG and, hence,GJG∼ admits a canonical cofactorization.

Now we have enough material to prove the main result of this section. The theorem links strictpositivity with co-J-losslessness.

Definition 3.1.11 (Stable co- Jq,p-lossless matrices). An M ∈ R H(r+p)×(q+p)∞ is co-Jq,p-

losslessif M Jq,pM∼ = Jr,p andM(s)Jq,p[M(s)]∗ ≤ Jr,p for all s∈ C+.

In AppendixA more results concerningJ-losslessness may be found. There it is proved thatM ∈ R H

(r+p)×(q+p)∞ is co-Jq,p-lossless iffM Jq,pM∼ = Jr,p and the lower rightp × p block

element ofM is in GR Hp×p∞ .

3A matrix P is antistableif P∼ is stable. Note thatP defined this way is void ifq = r .

27


Theorem 3.1.12 (The SP theorem). Let G∈ R H(r+p)×(q+p)∞ be a stable, real-rational matrix

that has full row rank onC0 ∪ ∞. The spaceBG is SP with respect to the Jq,p-inner product

[ f, g] := 〈 f, Jq,pg〉 if and only if GJq,pG∼ = W Jr,pW∼ has solutions W∈ GR H(r+p)×(r+p)∞ ,

and W−1G is co-Jq,p-lossless for one (and then all) such solutions W.

Proof . (Sufficiency) SupposeW as in Theorem3.1.12exists. With it defineM asM = W−1G.Hence,M stable co-Jq,p-lossless and, by Corollary3.1.1, BG = BM. Partition M compatiblywith the matrixJq,p as

M =[

M11 M12

M21 M22

]

.

SinceM is co-Jq,p-lossless, we have that

M Jq,pM∗ ≤ Jr,p in the closed right-half complex plane.

The lower rightp× p block element of the above inequality equals

M21M∗21 − M22M∗

22 ≤ − I p in the closed right-half complex plane.

As a resultM−122 is stable andH := M−1

22 M21 is stable and strictly contractive. Consequently[

H I]

generates a SP subspace (in the obvious indefinite inner product). So by Lemma3.1.4

the strict positivity inequality holds onB[

H I], and therefore it also holds on any subset of

B[

H I]. In particular, the SP inequality holds onBM ⊂ B[

M21 M22

] = B[

H I]. ThusBG = BM is

SP.(Necessity) The necessity part of the proof uses Lemmas3.1.7and3.1.10. By Lemma3.1.10

there exists aW ∈ GR H(r+p)×(r+p)∞ such thatGJq,pG∼ = W Jr,pW∼. Let W be one such solution

and defineM := W−1G. This M obviously satisfiesM Jq,pM∼ = Jr,p. We show that the lowerright p × p block elementM22 of M is in GR H

p×p∞ . By CorollaryA.0.13, this is equivalent to

M being co-J-lossless, which then completes the proof.To obtain a contradiction, suppose thatM22 has a zero inC+ ∪ C0 ∪ ∞. The matrixM22

cannot be singular onC0 ∪ ∞ becauseM Jq,pM∼ = Jr,p implies thatM22M∼22 = M21M∼

21 + I p

is positive definite onC0 ∪ ∞. ThereforeM22 is singular inC+, that is, there exists a nonzerov ∈ Cp ands∈ C+ such thatM22

1s−sv is stable. Define

z(s) =[

01

s−sv

]

given suchsandv. By Lemma3.1.7we know thatBM + Jq,pM∼R H⊥ r+p2 = R H

⊥q+p2 . There-

fore we may writez∈ H ⊥q+p2 as

z= b + g, b ∈ BM andg ∈ Jq,pM∼R H⊥ r+p2 .

Multiplying both sides from the left by M reveals that

π−Mz= Mg

sinceMb ∈ H r+p2 and Mg ∈ M Jq,pM∼H ⊥ r+p

2 = H ⊥ r+p2 . Let t ∈ H ⊥ r+p

2 be that element forwhich g = Jq,pM∼t (this is possible since by assumptiong is in Jq,pM∼H ⊥ r+p

2 ). We then havethat

π−Mz= M Jq,pM∼t = Jr,pt.

28

3.2. Suboptimal solutions to a two-blockH∞ problem

We have constructedz in such a way that[

M21 M22]

z is in H p2 . Hencet = Jr,pπ−Mz is of the

form

t =[

α

0

]

.

This gives rise to a contradiction. Namely, on the one hand wehave (with slight abuse of notation)

〈z, Jq,pz〉 = 〈[

01

s−sv

]

,

[

0− 1

s−sv

]

〉 = −‖ 1s− s

v‖22 < 0,

while on the other hand we have

〈z, Jq,pz〉 = 〈b, Jq,pb〉 + 2〈b, Jq,pg〉 + 〈g, Jq,pg〉= 〈b, Jq,pb〉 + 〈g, Jq,pg〉 (becauseb andg areJq,p-orthogonal)

= 〈b, Jq,pb〉 + 〈Jq,pM∼t, Jq,pJq,pM∼t〉= 〈b, Jq,pb〉 + 〈t,M Jq,pM∼t〉 = 〈b, Jq,pb〉 + 〈t, Jr,pt〉≥ 〈t, Jr,pt〉 = ‖α‖2

2 (becauseb is an element of a SP subspace)

≥ 0.

This is a contradiction, and, henceM22 is in GR Hp×p∞ and thereforeM is co-Jq,p-lossless.

3.2. Suboptimal solutions to a two-block H∞ problem

In this section we treat the suboptimal two-block problem.

Definition 3.2.1 (Suboptimal two-block H∞ problem). Let G ∈ R H(r+p)×(q+p)∞ be given and

suppose that it has full row rank onC0 ∪ ∞. DefineH1 andH2 by[

H1 H2]

= TG; H2 ∈ R Hp×p∞

depending on some stableT ∈ R Hp×(r+p)∞ . Thesuboptimal two-blockH∞ problem(STBP) with

boundγ is to findT ∈ R Hp×(r+p)∞ such thatH2 ∈ GR H

p×p∞ and‖H−1

2 H1‖∞ < γ.

This is not the most general version of the two-block problem. More general forms are possibleby dropping the assumptions thatT andG be rational and thatG be inH •×•

∞ . We consider in thispaper only two-block problems as defined in Definition3.2.1with boundγ = 1.

If G is of the form

G =[

B 0A − I p

]

(3.8)

and if T is partitioned compatibly withG as T =[

T1 T2]

, then[

H1 H2]

:= TG =[

T1B+ T2A −T2]

. SettingQ = −T−12 T1 gives H−1

2 H1 = QB+ A, which leads to the bet-ter known version of the two-block problem as defined at the beginning of this chapter.

Note that the STBP is nothing but the problem to determine stable T that have full row rank onC0 ∪ ∞, given stableG, such thatBTG is strictly positive. The next lemma is very important. Itshows that it is possible to separate the STBP in to two simpler problems of finding strictly pos-itivity subspaces. The arguments involveJ-orthogonal projections. One of the two problems isindependent of the parameterT, and, hence, provides necessity results. In the theorem followingthis lemma it is shown that this necessity result is sufficient as well. A generator of all solutionsto the STBP is given, and the method is demonstrated on an example. The next lemma we alsouse for the solution to the OTBP.

29


Lemma 3.2.2 (Positivity based on J-orthogonal projections). Suppose that G∈ R H(r+p)×(q+p)∞

is a given matrix that has full row rank onC0 ∪ ∞. Denote J= Jq,p and suppose that

T ∈ R Hp×(r+p)∞ is a matrix such thatBTG ⊂ BG + B ⊥J

G . Then

1. BTG has a J-orthogonal decomposition of the form

BTG = BG + [BTG ∩ B ⊥JG ]

︸︷︷︸

M :=

.

2. BTG is a positive subspace (in the J-inner product) iff bothBG andM are positive sub-spaces.

3. BTG is a strictly positive subspace (in the J-inner product) iffbothBG andM are strictlypositive subspaces.

Proof . 1. Obviously bothBG andM are subsets ofBTG, hence,BTG ⊃ BG + M . We nextshow thatBTG ⊂ BG + M . Since by assumptionBTG ⊂ BG + B ⊥J

G , we have that every

elementz of BTG may be written asz = b + g with b ∈ BG andg ∈ B ⊥JG = JG∼H ⊥ r+p

2 .The vectorg defined this way is an element ofM = BTG ∩ B ⊥J

G , becauseTGg= TG(z−b) ∈ H p

2 (sog ∈ BTG), and thereforez∈ BTG impliesz∈ BG + M .

2. If BTG is positive, then so areBG andM , because the latter two spaces are subspaces ofBTG. Conversely, if bothBTG andM are positive, then so isBTG because everyz ∈ BTG

may be written asz = b+ g with b ∈ BG andg ∈ M , and with it we see that

〈z, Jz〉 = 〈b + g, J(b+ g)〉= 〈b, Jb〉+ 〈g, Jg〉 (becauseb andg areJ-orthogonal)

≥ 0.

3. If BTG is SP, then so areBG andM , because the latter two spaces are subspaces ofBTG.Conversely, suppose that bothBTG andM are SP. So there exists anε > 0 such that for allb ∈ BG andg ∈ M the inequalities

〈b, Jb〉 ≥ ε‖b‖22, 〈g, Jg〉 ≥ ε‖g‖2

2

hold. Let z be an arbitrary element ofBTG and writez as z = b + g, with b ∈ BG andg ∈ M . Then

〈z, Jz〉 = 〈b + g, J(b+ g)〉= 〈b, Jb〉+ 〈g, Jg〉 (becauseb andg areJ-orthogonal)

≥ ε(‖b‖22 + ‖g‖2

2) = ε

2(‖b+ g‖2

2 + ‖b− g‖22)

≥ ε

2‖b + g‖2

2 = ε

2‖z‖2

2.

This shows thatBTG is SP.

30

3.2. Suboptimal solutions to a two-blockH∞ problem

Theorem 3.2.3 (Solution to the STBP). Suppose that G∈ R H(r+p)×(q+p)∞ has full row rank on

C0 ∪ ∞. The STBP with boundγ = 1 has a solution if and only ifBG is strictly positive. In thiscase T∈ R H

p×(r+p)∞ has the property that

[

H1 H2]

:= TG; H2 ∈ GR Hp×p∞ ; ‖H−1

2 H1‖∞ ≤ 1

if and only if T is of the form

T = A(U I p)W−1; A ∈ GR H

p×p∞ ; U ∈ R H

p×r∞ ; ‖U‖∞ ≤ 1,

where W is any solution to the canonical cofactorization problem

GJq,pG∼ = W Jr,pW∼; W ∈ GR H

(r+p)×(r+p)∞ .

Moreover,‖H−12 H1‖∞ < 1 if and only if‖U‖∞ < 1.

Proof . If the STBP has a solution then by assumption there exists aT such thatBTG = B[

H1 H2

]

is strictly positive (Lemma3.1.4). SinceTGBG ⊂ TH r+p2 ⊂ H p

2 we have thatBG ⊂ BTG, i.e.,

BG is a subset of a strictly positive subspace, hence,BG is strictly positive itself.Define for convenienceJ = Jq,p and J = Jr,p. Suppose from now on thatBG is strictly

positive. By Lemma3.1.7we know thatBG + JG∼H ⊥ r+p2 = H ⊥q+p

2 and thatGJG∼ admitsa canonical cofactorizationGJG∼ = WJW∼. (In fact, by Theorem3.1.12we know thatM :=W−1G is co-Jq,p-lossless, but we will no use this here).

Try T of the formT =[

B A]

W−1, with[

B A]

∈ R Hp×(r+p)∞ . This implies no loss of

generality sinceW is in GR H(r+p)×(r+p)∞ . Without loss of generality we assume that

[

B A]

has full row rank onC0 ∪ ∞—if it does not, then neither doesH2. Then

‖H−12 H1‖∞ (<) ≤ 1; H2 ∈ GR H

p×p∞ ,

by Lemma3.1.4, is equivalent toBTG being (strictly) positive. We work with (strictly) positivespaces rather than the conditions in the above displayed formula. SinceBG + B ⊥J

G = H ⊥q+p2

(Lemma3.1.7), the conditions under which Lemma3.2.2applies are satisfied. ThereforeBTG isa (strictly) positive subspace iff

M := BTG ∩ B ⊥JG = w | w ∈ BTG andw ∈ JG∼H ⊥ r+p

2

is a (strictly) positive subspace. With the help of the cofactorizationGJG∼ = WJW∼ we maymake the setM more explicit:

M = w | w ∈ B[

B A]

W−1Gandw ∈ JG∼W−∼H ⊥ r+p

2

= JG∼W−∼t |[

B A]

W−1GJG∼W−∼t ∈ H p2 andt ∈ H ⊥ r+p

2 = JG∼W−∼t |

[

B A]

Jt ∈ H p2 andt ∈ H ⊥ r+p

2 = JG∼W−∼t | t ∈ B[

B −A].

Let g be an element ofM and lett be that element ofH ⊥ r+p2 for which g = JG∼W−∼t. Then

〈g, Jg〉 = 〈JG∼W−∼t,G∼W−∼t〉 = 〈t, Jt〉.

31


This equality shows thatM is (strictly) positive iff B[

B −A] is (strictly) positive in theJ-

inner product. By Lemma3.1.4, B[

B −A] is positive iff ‖A−1B‖∞ ≤ 1 and A ∈ GR H

p×p∞ .

By Lemma3.1.4 B[

B −A] is SP iff ‖A−1B‖∞ < 1 and A ∈ GR H

p×p∞ . With U defined as

U := −A−1B we get the desired result.

Example 3.2.4 (STBP). Suppose

G =[

1 −θs−1s+2 0

]

.

Then

BG = 1s− 1

[

θ

1

]

C.

Obviously this space is strictly positive with respect to the J1,1 inner product iff|θ|> 1. Accord-ing to Theorem3.2.3the STBP with boundγ = 1 therefore has a solution iff|θ|> 1. To generateall solutionsT we need to factor the matrix

G

[

1 00 −1

]

G∼ =[1− θ2 −s−1

−s+2s−1s+2

1−s2

4−s2

]

= W

[

1 00 −1

]

W∼.

One solutionW ∈ GR H2×2∞ is

W =[

1 θ

(s− 1+θ2

1−θ2 )1

s+2−2θ1−θ2

1s+2

]

, W−1 =[

−2(θ2−1)(s+1)

s+2s+1

1θ(s+1) (s+ θ2+1

θ2−1 )s+2

−θ(s+1)

]

.

All solutions T look like T = A[

U 1]

W−1 with A ∈ GR H ∞ andU stable contractive. ForA = 1 andU = 0 the solution is

T =[

1θ(s+1) (s+ θ2+1

θ2−1 )s+2

−θ(s+1)

]

,[

H1 H2]

=[

1θ(s+1)

21−θ2

1s+1(s+ θ2+1

θ2−1 )

]

and forA = 1 andU = 1/θ < 1 the solution is

T =[

1θ

0]

,[

H1 H2]

=[

1θ

−1]

.

With the help of Theorem3.2.3we can only generate suboptimal solutions. The optimal caseis quite different. In this example, for instance,γ = 1 is the optimal bound ifθ = 1. But asθapproaches 1 (from above) the spectral factorW blows up and atθ = 1 it is not defined.

In this example optimal solutions do exist, however. Simplytake T =[

1 0]

. Are theremore solutionsT, and what about a general solution method? The optimal case is the subjectof the next section. It is shown there that for this particular example all optimal solutionsT areT = A

[

1 0]

, whereA is an arbitrary function inGR H ∞.

3.3. Optimal solutions to a two-block H∞ problem

In this section we treat the optimal case. The style of this section is similar to the style of theprevious section.

32

3.3. Optimal solutions to a two-blockH∞ problem

Definition 3.3.1 (Optimal two-block H∞ problem). SupposeG ∈ R H(r+p)×(q+p)∞ is given and

suppose that it has full row rank onC0 ∪ ∞. Let H1 andH2 be defined by[

H1 H2]

= TG; H2 ∈ R Hp×p∞

depending on some stableT ∈ R Hp×(r+p)∞ . Defineγopt as the infimal value of‖H−1

2 H1‖∞over all stableT ∈ R H

p×(r+p)∞ for which H2 ∈ GR H

p×p∞ . Theoptimal two-blockH∞ problem

(OTBP) is to find rationalT ∈ H p×(r+p)∞ such thatH2 ∈ GR H

p×p∞ and‖H−1

2 H1‖∞ = γopt.

Definition 3.3.2 (Parrott lower bound). Given a matrixG ∈ R H(r+p)×(q+p)∞ that has full row

rank defineγParrottas

γParrott= infT∈R H

p×(r+p)∞

‖H−12 H1‖∞ |

[

H1 H2]

:= TG∈ R Hp×(q+p)∞ .

Endoftheorem

Note that in the definition ofγParrott the stability condition (H2 ∈ GR Hp×p∞ ) is dropped com-

pared with the definition ofγopt.

Lemma 3.3.3 (Parrott lower bound). Let a full row rank G∈ R H(r+p)×(q+p)∞ be given. Then:

1. γParrott≤ γopt.

2. γParrott = inf γ | G[ Iq 0

0 −γ2 I p

]

G∼ is nonsingular and has p negative and r positive eigen-values everywhere onC0 ∪ ∞ .

Proof . Item 1 is trivial. Item 2 is easy; it is a different way of reading Parrott’s theorem (seeYoung (90)).

We consider in this section one type of optimality:

Assumptions 3.3.4. γParrott< γopt.

Example3.2.4is of this type. By Lemma3.3.3, Item 2 the assumption ensures that the matrixG

[ Iq 00 −γ2 I p

]

G∼ for γ = γopt has a cofactorization. Another interpretation of Assumption3.3.4isthat some subspaces that we would like to use for the OTBP areclosedsubspaces. At the end ofthis chapter we comment on this assumption. From now on we assume that the problem is scaledso thatγopt = 1 (this rules out only the nongeneric casesγopt = 0 andγopt = ∞).

Lemma 3.3.5 (The OTBP, necessity results). Suppose G∈ R H(r+p)×(q+p)∞ is given and has

full row rank onC0 ∪ ∞. Assume thatγopt = 1. Under the assumption thatγParrott< γopt thefollowing statements hold.

1. BG is positive but not strictly positive (in the Jq,p inner product).

2. GJq,pG∼ is nonsingular onC0 ∪ ∞ and admits a noncanonical cofactorization (with D+non-void)

GJq,pG∼ = W

0 0 0 D+0 Ir−l 0 00 0 − I p−l 0

D∼+ 0 0 0

W∼; D+(s) =

0 0 ( s−1s+1)

kl

0 . ..

0( s−1

s+1)k1 0 0

. (3.9)

33


3. BG + Jq,pG∼H ⊥ r+p2 is a closed subspace ofH ⊥q+p

2 and its orthogonal complement (inH ⊥q+p

2 ) is the finite dimensional space

V := G∼W−∼

000L

Cn; (3.10)

L(s) =

( 1s−1)

1 · · · ( 1s−1)

k1 0 0 0 0 0 0 00 0 0 ( 1

s−1)1 · · · ( 1

s−1)k2 0 0 0 0

0 0 0 0 0 0. . . 0 0 0

0 0 0 0 0 0 0 ( 1s−1)

1 · · · ( 1s−1)

kl

,

where n:=∑l

j=1 k j .

Proof . DefineJ = Jq,p and J = Jr,p.

1. If the OTBP has a solutionT then BTG is positive. SinceTGBG ⊂ TH r+p2 ⊂ H p

2 we

haveBG ⊂ BTG, i.e. BG is a subset of a positive subspace, hence,BG is positive. BG isnot strictly positive because then by Theorem3.2.3the STBP has a solution, contradictingoptimality.

2. By assumptionγParrott< γopt so by Lemma3.3.3GJG∼ is nonsingular onC0 ∪ ∞ and hasp positive andr negative eigenvalues onC0 ∪ ∞. Therefore it admits either a canonicalor a noncanonical factorization. Supposing it admits a canonical factorization leads to acontradiction: ifGJG∼ admits a canonical factorization thenBG + JG∼H ⊥ r+p

2 = H ⊥q+p2

and all steps in the proof of Theorem3.2.3remain valid—we only use positivity, not strictpositivity of BG in the proof of Theorem3.2.3. As a result there would exist solutions tothe STBP, contradicting optimality. HenceGJG∼ has a noncanonical cofactorization.

3. We show that the orthogonal complement inH ⊥q+p2 of BG + JG∼H ⊥ r+p

2 is (3.10). There-

after we show thatBG + JG∼H ⊥ r+p2 is closed, which then completes the proof.

Let v be an element of the orthogonal complementV of BG + JG∼H ⊥2 :

V := v ∈ H ⊥q+p2 | v ∈ B ⊥

G , v ∈ [ JG∼H ⊥ r+p2 ]⊥ = BGJ .

In particular,v ∈ B ⊥G = G∼H ⊥ r+p

2 , i.e., v = G∼z for somez ∈ H ⊥ r+p2 . The additional

conditionv ⊥ JG∼H ⊥ r+p2 implies GJv = GJG∼z ∈ H r+p

2 . In other words,z ∈ H ⊥ r+p2

satisfies

W

0 0 0 D+0 Ir−l 0 00 0 − I p−l 0

D∼+ 0 0 0

W∼z ∈ H r+p2 ; D+(s) =

0 0 ( s−1s+1)

kl

0 ...

0( s−1

s+1)k1 0 0

,

which is the case for somez∈ H ⊥ r+p2 iff v = G∼z is an element ofV as defined in (3.10).

It remains to show thatZ := BG + JG∼H ⊥ r+p2 is closed. This turns out to be a very

technical problem, and at this point the assumptionγParrott< γ = 1 comes in. We know

34


thatG has a rectangular Wiener-Hopf factorization

G = A+[

D 0]

A−; D(s) =

( 1−s1+s)

m1 0 0

0... 0

0 0 ( 1−s1+s)

mr+p

, k j ∈ Z,

with A+ ∈ GR H(r+p)×(r+p)∞ and withA∼

− ∈ GR H(q+p)×(q+p)∞ . This follows from the fact

that G is stable and has full row rank onC0 ∪ ∞ (see SubsectionB.3.2of AppendixB).From this expression we may deduce that

BG = A−1−

[

0Iq−r

]

H ⊥q−r2 + w | w = A−1

−[ r1

0

]

, r1 ∈ H ⊥ r+p2 , Dr1 ∈ H r+p

2 ︸︷︷︸

M :=

.

M is finite dimensional becauseD is square nonsingular. DefineP asP = A−1−

[ 0Iq−r

]

. Wethus far have

BG + JG∼H ⊥ r+p2 =

[

P JG∼]

H ⊥q+p2 + M . (3.11)

The idea is to show that[

P JG∼]

is nonsingular onC0 ∪ ∞. Then by Lemma3.1.2the

space[

P JG∼]

H ⊥q+p2 is closed. Then also (3.11) is closed because the sum of a closed

subspace and a finite dimensional closed subspace is closed.So it remains to show that[

P JG∼]

is nonsingular onC0 ∪ ∞. Consider

[

P∼

G

][

P JG∼]

=[

P∼ P P∼ JG0 GJG∼

]

.

Since bothP∼ P andGJG∼ are nonsingular onC0 ∪ ∞ we have that also[

P JG∼]

isnonsingular onC0 ∪ ∞.

Central in the proof of Theorem3.2.3(the Theorem in which the STBP is solved) is the factthat BTG ⊂ BG + JG∼R H

⊥ r+p2 . Actually it follows trivially in the suboptimal case from the

identityBG ⊕ JG∼R H⊥ r+p2 = R H

⊥q+p2 , butBTG ⊂ BG + JG∼H ⊥ r+p

2 is what we really use inthe proof of Theorem3.2.3. In the optimal case it is no longer a triviality but it does hold:

Lemma 3.3.6 (The OTBP, necessity results). Let G∈ R H(r+p)×(q+p)∞ be a given matrix that

has full row rank onC0 ∪ ∞. Assume thatγParrott< γopt = 1. The matrix T∈ R Hp×(q+p)∞ then

is a solution to the OTBP only if

BTG ⊂ BG + Jq,pG∼H ⊥ r+p2 .

Proof . For convenience defineJ = Jq,p. We prove the equivalent inclusionB ⊥TG ⊃ V :=

[BG + Jq,pG∼H ⊥ r+p2 ]⊥ (see Lemma3.3.5, Item 3). First note thatB ⊥

TG = G∼T∼H ⊥ p2 (see

for comparison Lemma3.1.7, Item 1). Under the assumption thatT solves the OTBP we have

B ⊥TG = G∼T∼H ⊥ p

2 =[

H∼1

H∼2

]

H ⊥ p2 =

[

H∼

I

]

H ⊥ p2 ,

35


with H := H−12 H1 stable and contractive. ThereforeB ⊥

TG is a negative subspace in theJ innerproduct4. Also the space

Z := B ⊥TG + V

is negative as can be seen as follows. Writez∈ Z asz= b + v, with b ∈ B ⊥TG andv ∈ V. Then

〈z, Jz〉 = 〈b, Jb〉+ 〈v, Jb〉 + 〈b, Jv〉 + 〈v, Jv〉= 〈b, Jb〉 ≤ 0.

Here we used the fact that

〈v, Jb〉 = 0, 〈v, Jv〉 = 0,

which is easily checked.To obtain a contradiction, suppose thatB ⊥

TG 6⊃ V. Then there exists av ∈ V such thatv 6∈ B ⊥TG.

Take one suchv, partitionv compatibly with theJ matrix asv =[v1v2

]

and definew as

w :=[

w1

0

]

:=[

v1

v2

]

−[

H∼

I

]

v2

︸︷︷︸

∈B⊥TG

.

This vectorw is nonzero becausev by assumption is not inB ⊥TG, andw is an element ofZ. But

then〈w, Jw〉 > 0—that obviously holds—contradicts negativeness ofZ, hence,V is containedin B ⊥

TG.

Corollary 3.3.7. 1s+1 ∈ ZH2 holds for some Z∈ R H

1×1∞ iff Z is an element ofGR H ∞.

Proof . The “if” part is trivial. (Only if) Let p be that element ofH2 such that 1s+1 = Zp. Since

this p ∈ H2 is rational, we must have thatp is strictly proper, and, hence, thatZ−1 = (s+ 1)p isstable. In other wordsZ is in GR H ∞.

Theorem 3.3.8 (Solution to the OTBP). Let G∈ R H(r+p)×(q+p)∞ be a given matrix that has full

row rank onC0 ∪ ∞. Then the OTBP has solutions under the assumption thatγParrott< γopt.Assume thatγParrott< γopt = 1. Then there exists a unique integer l and a (nonunique) W∈GR H

(r+p)×(r+p)∞ such that

GJq,pG∼ = W

0 0 0 D+0 Ir−l 0 00 0 − I p−l 0

D∼+ 0 0 0

W∼, D+ =

0 0 ( s−1s+1 )

kl

0 . ..

0( s−1

s+1)k1 0 0

with kj > 0. Furthermore, given such a W, T solves the OTBP if and only if it is of the form

T = A

(

0 U I p−l 00 0 0 I l

)

W−1,

where A is inGR Hp×p∞ and U is a real-rational contractive matrix inR H

(p−l )×(r−l )∞ .

4A subspaceE is a negative subspace if〈z, Jz〉 ≤ 0 for all z∈ E .

36


Proof . SupposeγParrott< γopt = 1 and defineJ = Jq,p. Lemma3.3.5applies, so we know that

BG is positive, thatGJG∼ admits a noncanonical cofactorization as in (3.9) and thatV⊥ =BG + JG∼H ⊥ r+p

2 as in (3.10).

We takeT of the formT = XW−1, with X ∈ R Hp×(r+p)∞ . This implies no loss of generality

sinceW is in GR H(r+p)×(r+p)∞ . Without loss of generality we takeX to have full row rank on

C0 ∪ ∞ because if not, then neitherH2 has full rank onC0 ∪ ∞. By Lemma3.3.6T solvesthe OTBP only ifBTG ⊂ BG + B ⊥J

G . Therefore candidate solutionsT have the property thatthe conditions are satisfied under which Lemma3.2.2applies. ThereforeT solves the OTBP iffBTG ⊂ BG + B ⊥J

G and

M := BTG ∩ B ⊥JG = w | w ∈ BTG andw ∈ JG∼H ⊥ r+p

2

is a positive subspace. With the help of the noncanonical cofactorization ofGJG∼ we may makethe setM more explicit, but before we do this, we “shape”T a little further.

By Lemma3.3.6and Lemma3.3.5, Item 3 a matrixT = XW−1 solves the OTBP only if

BTG ⊂ BG + JG∼H ⊥ r+p2 = V⊥ (V = G∼W−∼

000L

Cn)

with L as in Lemma3.3.5, Item 3. The above inclusion we may rewrite as

G∼W−∼

000L

Cn ⊂ B ⊥

TG = G∼W−∼ X∼H ⊥ p2 ,

which is the case iff

000L

Cn ⊂ X∼H ⊥ p

2 ,

becauseG∼W−∼ is an injective map. By application of a matrix version of Lemma 3.3.7theabove inclusion holds iff

X∼ =

E∼ 0B∼ 0A∼ 0C∼ I l

A∼

for an A ∈ GR Hp×p∞ , and some stable real-rational matricesE, A, B andC to be determined

later. Without loss of generality we may assume thatC = 0. We thus far have

T = XW−1 = A

[

E B A 00 0 0 I l

]

W−1.

The next step is to show thatE = 0. Look at

TGJG∼T∼ = A

[

BB∼ − AA∼ ED+D∼

+ E∼ 0

]

A∼. (3.12)

37


As T is supposed to solve the OTBP we must have that (3.12) is negative semidefinite onC0 ∪∞.SinceD+ is nonsingular as a rational matrix we therefore must have that E = 0.

Summarizing the results obtained so far:T solves the OTBP only if it is of the form

T = A

[

0 B A 00 0 0 I l

]

W−1; A ∈ GR Hp×p∞ ;

[

B A]

∈ R H(p−l )×(r−l+p−l )∞ has full row rank onC0 ∪ ∞ .

(3.13)

We now use this expression forT to makeM more explicit. Recall thatT solves the OTBP iff itis of the above form andM is a positive subspace.

M = w | w ∈ BTG andw ∈ JG∼W−∼H ⊥ r+p2

= w | A

[

0 B A 00 0 0 I l

]

W−1Gw ∈ H p2 andw ∈ JG∼W−∼H ⊥ r+p

2

= JG∼W−∼t | A

[

0 B A 00 0 0 I l

]

W−1GJG∼W−∼t ∈ H p2 andt ∈ H ⊥ r+p

2

= JG∼W−∼t | A

[

0 B A 00 0 0 I l

]

0 0 0 D+0 Ir−l 0 00 0 − I p−l 0

D∼+ 0 0 0

t ∈ H p2 andt ∈ H ⊥ r+p

2

= JG∼W−∼t | t =

0t2t3•

∈ H ⊥ r+p2 and

[

t2t3

]

∈ B[

B −A].

Let g be an element ofM and lett be that element ofH ⊥ r+p2 for which g = JG∼W−∼t. Then

〈g, Jg〉 = 〈JG∼W−∼t,G∼W−∼t〉 = 〈[

t2t3

]

, J(r−l ),(p−l )

[

t2t3

]

〉.

This equality shows thatM is positive iffB[

B −A] is positive in theJ(r−l ),(p−l ) inner product. By

Lemma3.1.4the latter is the case iff‖A−1B‖∞ ≤ 1 andA ∈ GR H(p−l )×(p−l )∞ . With U defined

asU := −A−1B we get the desired result.

Example 3.3.9 (OTBP). We reconsider example3.2.4and now look at the optimal case. Thematrix G is

G =[

1 −θs−1s+2 0

]

.

As shown in example3.2.4the spaceBG is positive iff |θ| ≥ 1. Therefore we are in the optimalcase withγopt = 1 if |θ| = 1. From now on we assumeθ = 1. Note thatγParrott= 0 becauseG issquare nonsingular. Assumption3.3.4is satisfied and, consequently, Lemma3.3.6and Theorem3.3.8apply.

We need aW ∈ GR H2×2∞ such that

G

[

1 00 −1

]

G∼ =[

0 −s−1−s+2

s−1s+2

1−s2

4−s2

]

= W

[

0 s−1s+1

s+1s−1 0

]

W∼.

38


One solution is

W =[

0 1s+1s+2 − 1

2−s+1s+2

]

, W−1 =[ 1

2−s+1s+1

s+2s+1

1 0

]

.

Hence, by Theorem3.3.8all solutionsT to the OTBP are of the form

T = A[

0 1]

W−1 = A[

1 0]

,[

H1 H2]

=[

A −A]

,

whereA is an arbitrary function inGR H ∞.

Remark 3.3.10 (Alternative proof of Lemma 3.3.6). An equivalent formulation of Lemma3.3.6is thatT solves the OTBP only if

H 1×p2 T ⊃ Q := t | tGJG∼ ∈ H ⊥1×(r+p)

2 , t ∈ H 1×(r+p)2 .

In the special case that

G =[

B 0A Ip

]

this is particularly easy to see. First note that withG as defined above, the OTBP is equivalentto finding T =

[

Q Ip]

∈ R Hp×(r+p)∞ such that‖QB+ A‖∞ = γopt. We again assume that

γopt = 1.Let

[

t1 t2]

be an arbitrary nonzero element ofQ , partitioned compatibly withG. As Q is inR H

p×r∞ the following defines at in H 1×r

2 , depending onQ:

t2[

Q I]

=[

t1 + t t2]

.

Now defineH := QB+ A and look at the following equations.

〈t2(H H∼ − I ), t2〉 = 〈t2[

Q I]

GJG∼[

Q∼

I

]

, t2〉

= 〈[

t1 t2]

GJG∼︸︷︷︸

∈H ⊥1×(r+p)2

, t2[

Q I]

︸︷︷︸

∈H 1×(r+p)2

〉 + 〈[

t 0]

GJG∼, t2[

Q I]

〉

= 〈[

t 0]

GJG∼, t2[

Q I]

〉= 〈

[

t 0]

, t2[

Q I]

GJG∼〉= 〈

[

t 0]

,[

t 0]

GJG∼〉 + 〈[

t 0]

︸︷︷︸

∈H 1×(r+p)2

,[

t1 t2]

GJG∼︸︷︷︸

∈H ⊥1×(r+p)2

〉

= 〈[

t 0]

,[

t 0]

GJG∼〉 = 〈t, tBB∼〉= ‖tB‖2

2 ≥ 0.

That is,‖H‖∞ ≥ 1, and‖H‖∞ = 1 only if tB = 0. (It may be checked thatt2 is nonzero.) Bythe usual full rank assumptions onB, tB = 0 iff t = 0, or in other words, iff

H 1×p2

[

Q I]

⊃ Q := t | tGJG∼ ∈ H ⊥1×(r+p)2 , t ∈ H 1×(r+p)

2 .

Actually the arguments work also for the case whenA is an unstable rational matrix, as longas‖A‖∞ <∞. For instance, ifB = I andA an antistable, proper rational matrix (i.e., the Nehari

39


problem) thenQ ∈ R Hp×r∞ is such that‖Q+ A‖∞ = γopt = 1 only if H 1×p

2

[

Q I]

⊃ Q . It iseasily verified that withB = I andA proper and antistable, the row vector

[

t1 t2]

is an elementof Q iff the pair (t2, t2A+ t1) is a Schmidt pair of the Hankel operator

ΓA : H 1×p2 −→ H ⊥1×r

2 ; ΓA(t) = π−(tA),

corresponding to a Hankel singular value equal to 1:

ΓA(t2) = t2A+ t1, Γ∗A(t2A+ t1) = t2.

HereΓ ∗A denotes the adjoint operator ofΓA induced by the inner product onL2(C0;C•) and its

subsets:

Γ ∗A : H ⊥1×r

2 −→ H 1×p2 ; Γ ∗

A(t) = π+(tA∼).

3.4. Some state space formulas

We end this chapter with a few comments on how the results derived for the OTBP and STBPtranslate into state space manipulations. For the discussion we introduce a convenient notationwhich is fairly standard by now (see, for example, Doyle et. al. (22)). By

Gs=

[

A BC D

]

we mean thatG has a realizationG(s)= C(sI − A)−1B+ D. It is easily checked that ifG has arealization as above, thenGJq,pG∼ has a realization

GJq,pG∼ s=

[

H BC D

]

:=

A −BJq,pB∗ BJq,pD∗

0 −A∗ C∗

C −DJq,pB∗ DJq,pD∗

and that, ifGJq,pG∼ is biproper, a realization of its inverse is

(GJq,pG∼)−1 s=

[

H× BD−1

−D−1C D−1

]

;

H× := H − BD−1C

=[

A −BJq,pB∗

0 −A∗

]

−[

BJq,pD∗

C∗

]

[ DJq,pD∗]−1[

C −DJq,pB∗]

The matrixH× is a Hamiltonian matrix, that is,H×[ 0 In−In 0

]

+[ 0 In

−In 0

]

[ H×]∗ = 0.

Corollary 3.4.1 (State space manipulations). Suppose G∈ R H(r+p)×(q+p)∞ has full row rank

onC0 ∪∞. Let G(s)= C(sI− A)−1B+ D be a realization of G with A having all its eigenvaluesin C−, and suppose that the matrix

[

A− sI BC D

]

has no zeros onC0 ∪ ∞. Define J= Jq,p and J = Jr,p. Then:

40

3.4. Some state space formulas

1. GJG∼ has a canonical cofactorization GJG∼ = WJW∼ if and only if DJ D∗ = W∞ JW∗∞

for some nonsingular W∞; H× ∈ R2n×2n as defined above has no imaginary axis eigenval-

ues; and the antistable eigenspace of H× is of the formIm[ X1

X2

]

with X1, X2 ∈ Rn×n andX2 nonsingular.

2. BG is SP, or equivalently, W−1G is co-J-lossless, if and only if DJ D∗ = W∞ JW∗∞ for

some nonsingular W∞ and there exits a (unique) Q≥ 0 such that

AQ+ QA∗ − [ QC∗ + BJ D∗](DJ D∗)−1[CQ+ DJ B∗] + BJ B∗ = 0

with A− [ QC∗ + BJ D∗](DJ D∗)−1C having all its eigenvalues inC−. (Moreover, Q=−X1X−1

2 ).

3. Consider the OTBP as in Theorem3.3.8with this G. The assumptionγParrott< γopt = 1made in Theorem3.3.8is equivalent to

a) DJ D∗ = W∞ JW∗∞ for some nonsingular W∞ ∈ R(r+p)×(r+p);

b) H× has no imaginary axis eigenvalues (in other words, the antistable eigenspace ofH× is of the formIm

[ X1X2

]

with X1, X2 ∈ Rn×n);

c) The antistable eigenspace of H×, written asIm[ X1

X2

]

, exists and X2 is singular andX∗

2 X1 ≤ 0.

Proof . Item 1 is proved in AppendixB, TheoremB.3.7. (Item 2) In Green (35) it is provedthat there existW ∈ GR H

(r+p)×(r+p)∞ such thatGJG∼ = WJW∼ with W−1G co-J-lossless iff

Q ≥ 0. By Theorem3.1.12co-J-losslessness ofW−1G is equivalent toBG being SP. ThatQequalsQ = −X1X−1

2 is a matter of manipulation. (Item 3) Because the constant matrix A byassumption has no imaginary axis eigenvalues, all zeross ∈ C0 of H× − sI appear as zeros ofGJG∼ (see AppendixB, TheoremB.3.7). The rest follows from a continuity argument. Notethat in the suboptimal caseX2 is nonsingular, and thatQ = −X1X−1

2 ≥ 0. So certainly in thesuboptimal case we haveX∗

2 X1 = −X∗2 QX2 ≤ 0.

GivenQ := −X1X−12 as in Corollary3.4.1we may construct realizations of canonical cofactors

W ∈ GR H(r+p)×(r+p)∞ and of W−1G very easily: A realization of a canonical cofactorW of

GJG∼, combined withG, is

[

G W] s=

[

A B [ BJ D∗ + QC∗]W−∗∞ J

C D W∞

]

.

Presented this way may make clear thatW−1G has a realization

W−1Gs=

[

I [ BJ D∗ + QC∗]W−∗∞ J

0 W∞

]−1 [A B

C D

]

s=[

I −[ BJ D∗ + QC∗](DJ D∗)−1

0 W−1∞

][A B

C D

]

.

The trick we used here is that regular output injection transformation5 applied to a realization of[

G W]

does not affect the quotientW−1G. The construction of a realization of anoncanonical

5A transformation from[

A BC D

]

to[

I H0 W

][

A BC D

]

for someH and nonsingularW is called a regular output

injection transformation. See Chapter5.

41


cofactor ofGJG∼ is much more complicated. It can be done, however, (see (32)). We do notgo into the details. In AppendixB an algorithm is formulated that may be used to construct non-canonical factors. It uses polynomial matrix manipulations instead of state space manipulations,and it is unfortunately more of theoretical than of practical value.

42

4

The standard H∞ problem

G

K yu

w z

Figure 4.1.: The standard system configuration.

The subject of study in this chapter is the standardH∞ problem (Francis (26)). In a few words,the standardH∞ problem is to find compensatorsK that stabilize the closed loop in Fig.4.1andminimize the∞-norm of the closed-loop transfer matrix fromw to z over all stabilizing com-pensators. As with the two-blockH∞ problem we distinguish optimal solutions and suboptimalsolutions to this problem. It is called “standard” because it encompasses many, more practicalH∞ control problems as a special case.

The suboptimal standardH∞ problem (SSP) has been the subject of hundreds of papers sinceits introduction in 1984 (Doyle (21)). The derivation presented in this chapter is for a large partbased on the results of the previous chapter. It is a mix of results obtained by Kwakernaak (48)and Green (35), but in our language. It is fair to say that (35) is the first paper where the standardH∞ problem is solved in frequency domain terms in a satisfactory manner. The polynomialapproach by Kwakernaak on the other hand has the advantage ofbeing slightly more general.The SSP1, which we soon introduce, is in line with the approach taken by Kwakernaak; theSSP2 defined thereafter is practically equivalent to the oneconsidered by Green in (35). Othernoteworthy papers are the papers by Ball and Helton (7), Helton (36), and Ball and Cohen (5)and Ball, Helton and Verma (8).

We treat a class of optimalH∞ problems. On the theoretical side not much new is addedin comparison with the results on the OTBP. Since we want to beconcrete, we have addedan elementary algorithm for the construction of optimal solutions, related to the constructionof noncanonical factors. The algorithm is adjusted for use of the standardH∞ problem in thesense that only that part of the noncanonical factor is constructed that is really necessary for thegenerator of all optimal solutions to theH∞ problem. (As it turns out, a complete noncanonical

43

4. The standardH∞ problem

factorization reveals more structure than is necessary forthe construction of optimal solutions.)The derivation of optimal solutions to standardH∞ problems is a complicated problem. In Gloveret. al. (31) the optimal four-blockH∞ problem is treated in full detail. In Gahinet (27) a statespace method is proposed which is a variation of the well known state space formulas for thesuboptimal case (Doyle, Glover, Khargonekar and Francis (22)) but has the advantage that theparameterization behaves continuously around the optimalvalue, and, thus, also parameterizesoptimal solutions.

The essential difference between the“polynomial approach” and other approaches is that in thepolynomial approach nonproper plantsG can be handled. This, for instance, makes it possibleto deal with mixed sensitivity problems with nonproper shaping filters directly as standardH∞problems. Actually it is the other way around: Since we definitely want to have a method tosolve mixed sensitivity problems with nonproper filters, weadjust the definition of the standardH∞ problem in such a way that it includes the mixed sensitivity problem with nonproper filtersas a special case. The extension of the standardH∞ problem with proper plants to the ones withnonproper plants unfortunately requires some technical results. Nevertheless it can be done.

The discussion might give the impression that “nonproper” problems can not be translated into “proper” problems. This is not the case, but the existing trick (Krause (45)) that fixes thisproblem gives rise to a degree inflation of the compensator. This is shown in an example inSection4.4. The example gives a explanation of why polynomial methods are useful. Somegeneral comments are collected in Section4.5. For completeness—and to please the fans ofRiccati equations—we summarize in Section4.5 briefly how the results translate to the famousstate space formulas ((22)).

We writeG andK as left coprime PMFDs:

G =[

D1 D2]−1 [

N1 N2]

, K = X−1Y.

With these fractions we get a differential equation that completely describes the standard closedloop in Fig.4.1:

[

−N1 D1 D2 −N2

0 0 −Y X

]

w

zyu

= 0. (4.1)

The upper row block defines the plant, the lower row block defines the compensator and combinedthey define the closed loop. This is a very convenient and compact way to characterize the closedloop. We say that the closed-loop transfer matrix fromw to z is inducedby the above equations.In the closed loop we considerz, y and u as the outputs. Hence, the closed-loop system isinternally stable iff

Ω :=[

D1 D2 −N2

0 −Y X

]

is strictly Hurwitz. Thefirst suboptimal standardH∞ problem is:

Definition 4.0.2 (SSP1). Given a matrix[

−N1 D1 D2 −N2]

∈ P (p+r )×(q+p+r+t) the firstsuboptimal standardH∞ problem (SSP1)with boundγ is to find aK ∈ R t×r with a left coprimePMFD K = X−1Y, such that the transfer matrixH fromw to z induced by

(

−N1

0

∣∣∣∣

D1 D2 −N2

0 −Y X︸︷︷︸

Ω:=

)

w

zyu

= 0

44

satisfies‖H‖∞ < γ and such thatΩ is strictly Hurwitz.

SolutionsK to the SSP1 do not necessarily make the closed loop well-posed. As in Chapter2we define well-posedness with help of fictitious disturbancesignals: Consider the extended closedloop in Fig.4.2. The signals in the extended closed loop satisfy

G

Kyu v1

v2

w z

Figure 4.2.: The standard system configuration; setup forL2 stability.

[

D1 D2 −N2

0 −Y X

]

︸︷︷︸

Ω:=

zyu

=[

N1 D2 N2

0 0 0

]

︸︷︷︸

Ψ :=

w

v1

v2

.

Therefore the closed loop isL2 stable (internally stable and well-posed) iffΩ is strictly Hur-witz andΩ−1Ψ is proper. L2-stability is easily characterized if we switch from time do-main/polynomial fractions to frequency domain/stable fractions. Suppose that

G =[

D1 D2]−1 [

N1 N2]

, K = X−1Y

are left coprime fractions ofG and K overR H ∞. Then the system in Fig.4.2 is L2-stable iffΩ :=

[D1 D2 −N20 −Y X

]

is in GR H(p+r+t)×(p+r+t)∞ .

Definition 4.0.3 (SSP2). Given a[

−N1 D1 D2 −N2]

∈ R H(p+r )×(q+p+r+t)∞ the second

suboptimal standardH∞ problem (SSP2)with boundγ is to find a K ∈ R t×r with a left co-prime fractionK = X−1Y overR H ∞, such that the transfer matrixH from w to z induced bythe frequency domain equation

(

−N1

0

∣∣∣∣

D1 D2 −N2

0 −Y X︸︷︷︸

Ω:=

)

w

zyu

= 0 (4.2)

satisfies‖H‖∞ < γ and such thatΩ is in GR H(p+r+t)×(p+r+t)∞ .

And then there is of course the optimal version of both standard H∞ problems. The SSP2 isthe one that is treated extensively in the literature. Our formulation of the problems SSP1 andSSP2 is not standard. The way the problems are formulated anticipate the way they are solved.It is important to recognize that in neither definition is it assumed that

[

D1 D2]

is nonsingularor biproper or whatever. This may seem unimportant, but as wewill see later on, it does have

45


an advantage. It is also interesting to see that as far as the plant is concerned we do not assumecoprimeness, or to say it in a different way, we do not a prioriassume that there are no hiddenmodes. The reason for not imposing coprimeness is purely pragmatic, again anticipating theway the problems are solved. As it turns out coprimeness is sometimes more a curse than ablessing. In the formulations of the SSP2 and SSP1 we implicitly assume that the numeratorXis nonsingular. Our solution to the problem, however, does not address this part of the problem.It is not a very interesting problem. We see it as the task of the “engineer” to come up with asensibleH∞ problem. And sensibleH∞ problems presumably do not lead to a singularX. (Wecomment on this in Section4.4.)

CompensatorsK that solve the SSP1 or SSP2 for a given boundγ are sometimes referred to asadmissiblecompensators. A compensator isoptimalif it is admissible and in addition minimizesthe ∞-norm of H over all admissible compensators. From a mathematical point of view theSSP2 is much more transparent than the SSP1. The SSP1—the onethat we are really interestedin—may be derived using the solution to the SSP2. It is for this reason that we summarize firstthe more elegant solution to the SSP2.

4.1. The SSP2

In this section we review a frequency domain solution to the conventional SSP2. It is practi-cally a copy of the results obtained by Green (35), with a minor difference in that we here andthere use the results obtained in the previous chapter. Throughout

[

−N1 D1 D2 −N2]

∈R H

(p+r )×(q+p+r+t)∞ denotes the given “plant”. The assumptions that we impose are:

Assumptions 4.1.1 (Regularity assumptions).

1.[

−N1 D1]

∈ R H(r+p)×(q+p)∞ has full row rank onC0 ∪ ∞.

2.[

D2 −N2]

∈ R H(r+p)×(r+t)∞ has full column rank onC0 ∪ ∞.

In Section4.5these assumptions are translated in terms of state space data. As in the previouschapter we now derive necessary conditions for the SSP2 to have a solution, based on signals inthe closed loop that do not depend on the compensator. To get the idea, consider the closed loopas in Fig.4.3. Suppose thatw is a time signal that up to time 0 does not activate the outputy.Since the compensatorK is supposed to be a causal map—this is very vague, but don’t mind,it’s only to get the idea—necessarily alsou(t) is zero for negative time. If we assume thatGrepresents a causal system we then have that the control error z for negative time is completelydetermined byw. This provides necessary conditions. The characterization of all suchw as wellas the resultingz for negative time actually is very easy given a stable fraction of the generalizedplantG.

Lemma 4.1.2 (Necessity results based on compensator indepe ndent signals). The SSP2with data

[

−N1 D1 D2 −N2]

∈ R H(p+r )×(q+p+r+t)∞ and boundγ = 1 has a solution K only

if

B[

−N1 D1

] := [wz−

]

∈ H ⊥q+p2 |

[

−N1 D1][

w

z−

]

∈ H r+p2

is strictly positive in the Jq,p-inner product.

46

4.1. The SSP2

G

K

yu

w z

Figure 4.3.: The standard system configuration; necessity results.

Proof . Let H denote the transfer matrix fromw to z and suppose thatK solves the SSP2 withboundγ = 1. Let K = X−1Y be a left coprime fraction overR H ∞ of K, and let

[wz−

]

be an

arbitrary element ofB[

−N1 D1

]. We take thisw as the input to the closed loop described by (4.2).

We may rewrite (4.2) as

[

D1 D2 −N2

0 −Y X

]

︸︷︷︸

=Ω

z− z−yu

= −[

−N1 D1

0 0

][

w

z−

]

.

Note thatΩ is in GR H(p+r+t)×(p+r+t)∞ and that the right-hand side of the above equality is in

H p+r+t2 . Hencez− z−, y andu are inH •

2 . As a result we have

‖w‖22 − ‖z−‖2

2 ≥ ‖w‖22 − ‖z−‖2

2 − ‖z− z−‖22

= ‖w‖22 − ‖z‖2

2 (z− ∈ H ⊥ p2 andz− z− ∈ H p

2 are perpendicular)

≥ (1− ‖H‖2∞)‖w‖2

2 (becausez= Hw)

≥ 12(1− ‖H‖2

∞) (‖w‖22 + ‖z−‖2

2) (because‖z−‖22 ≤ ‖z‖2

2 ≤ ‖w‖22).

Since this holds for any[wz−

]

∈ B[

−N1 D1

] we have thatB[

−N1 D1

] is SP in theJq,p inner product.

It should be clear thatB[

−N1 D1

] is the frequency domain analog of the set of time signalsw

(andz) that, so to say, do not active the outputy. If we assume that[

−N1 D1]

has full row rank

onC0 ∪ ∞, then we know from Theorem3.1.12thatB[

−N1 D1

] is SP iff

W Jr,pW∼ =[

−N1 D1]

Jq,p

[

−N∼1

D∼1

]

has solutionsW ∈ GR H(r+p)×(r+p)∞ with W−1

[

−N1 D1]

co-Jq,p-lossless for one (and then all)suchW. So we have proved the following result:

Lemma 4.1.3 (Necessity results, cf. ( 35)). Let the matrix[

−N1 D1 D2 −N2]

∈R H

(p+t)×(q+p+r+t)∞ be given and assume that

[

−N1 D1]

has full row rank onC0 ∪ ∞. The

47


SSP2 with this data and boundγ = 1 has a solution only if there exist W∈ GR H(r+p)×(r+p)∞

such that

W Jr,pW∼ =[

−N1 D1]

Jq,p

[

−N∼1

D∼1

]

,

and one (and then all) such W makes W−1[

−N1 D1]

co-Jq,p-lossless.

Before handling the general SSP2 we examine a simplified problem. We consider plants whose[

−N1 D1]

block equals identity. The SSP2 in this case turns out to be a two-blockH∞ problem:

Lemma 4.1.4 (A two-block H∞ problem). Let[

D2 −N2]

∈ R H(r+p)×(r+t)∞ be given. In

what follows K= X−1Y and K= YX−1 are left and right coprime fractions overR H ∞ of K,respectively. Let H be the transfer matrix fromw to z induced by

( [Ir0

]

0

∣∣∣∣

[ 0I p

]

D2 −N2

0 −Y X︸︷︷︸

Ω:=

)

w

zyu

= 0. (4.3)

Then H= BA−1, and A is inGR Hr×r∞ if and only ifΩ is in GR H

(p+r+t)×(p+r+t)∞ , whereB and

A are defined as[

AB

]

:=[

D2 −N2][

XY

]

.

Proof . We first derive an alternative expression forΩ being inGR H(p+r+t)×(p+r+t)∞ . Let U an

element ofGR H(r+t)×(r+t)∞ depending onX andY such that

[

−Y X]

U =[

0 I t]

.

(SuchU exists becauseY andX are left coprime overR H ∞.) PartitionU compatibly as

U =[

X •Y •

]

.

ThenK = YX−1 is a right coprime fraction overR H ∞. DefineA and B as[

AB

]

:=[

D2 −N2][

XY

]

.

Then

[[ 0I p

]

D2 −N2

0 −Y X

]

︸︷︷︸

=Ω

[

I p 00 U

]

︸︷︷︸

∈GR H(p+r+t)×(p+r+t)∞

=

0 A •I p B •0 0 I t

.

We may infer from this expression thatΩ is in GR H(p+r+t)×(p+r+t)∞ iff A is in GR H

r×r∞ . Next

define the auxiliary signalsl1 andl2 as[

l1l2

]

:= U−1

[

yu

]

.

48

4.1. The SSP2

Then we may rewrite (4.3) as

0 A •I p B •0 0 I t

zl1l2

=

− Ir

00

w;[

yu

]

= U

[

l1l2

]

. (4.4)

As an immediate result we see thatl2 ≡ 0, that l1 = − A−1w and thatz = −Bl1 = BA−1w.ThereforeH = BA−1.

The problem to determine stable[

XY

]

such that

[

AB

]

:=[

D2 −N2][

XY

]

; ‖BA−1‖∞ < 1; A ∈ GR Hr×r∞

is an STBP. Under the assumption that[

D2 −N2]

has full column rank onC0 ∪ ∞ we maycopy from Chapter3, in a transposed version, that there exist such solutions

[XY

]

iff

Γ∼ Jr,tΓ =[

D∼2

−N∼2

]

Jr,p[

D2 −N2]

(4.5)

has a solutionΓ ∈ GR H(r+t)×(r+t)∞ and one (and then all) suchΓ have the property that

[

0 I p

Ir 0

][

D2 −N2]

Γ−1

[

0 Ir

I t 0

]

is Jp,r-lossless1. In the case that these conditions are satisfied, all solutions[

XY

]

are generated by

[

XY

]

= Γ−1

[

IU

]

A; U ∈ R Ht×r∞ ; ‖U‖∞ < 1; A ∈ GR H

r×r∞ .

Note that any such pairX, Y is right coprime overR H ∞ because[

A−1 0]

Γ is a stable left in-verse of

[XY

]

. Note also that the factorA in the above displayed formula cancels in the expression

K = YX−1 for the compensator. So without loss of generality we may take A = I if it is only thecompensators we are interested in.

The solution to the SSP2 for general plants consists of a reduction step that transforms theSSP2 to an equivalent SSP2 of the two-block type as considered just now.

Lemma 4.1.5 (A reduction to a TBP). Let the matrix[

−N1 D1 D2 −N2]

∈R H

(p+r )×(q+p+r+t)∞ be given. Assume that

[

−N1 D1]

has full row rank onC0 ∪ ∞ andthat

[

D2 −N2]

has full column rank onC0 ∪ ∞. In what follows K= X−1Y and K= YX−1

are left and right coprime fractions overR H ∞ of K, respectively. Let H be the transfer matrixfromw to z induced by

(

−N1

0

∣∣∣∣

D1 D2 −N2

0 −Y X︸︷︷︸

Ω:=

)

w

zyu

= 0. (4.6)

1The matrices[

0 II 0

]

are there to swap some blocks of[

D2 −N2

]

Γ−1. This is inevitable. Another possibility would beswap some of the signal blocks.

49


1. There exist K such that‖H‖∞ < 1 andΩ ∈ GR H(p+r+t)×(p+r+t)∞ only if there exist W∈

GR H(r+p)×(r+p)∞ such that

W Jr,pW∼ =[

−N1 D1]

Jq,p

[

−N∼1

D∼1

]

,

and one (and then all) such W have the property that the lower right p× p block elementof W−1

[

−N1 D1]

is in GR Hp×p∞ .

2. Assume a W as in Item 1 exists. Define[

−N1 D1 D2 −N2]

:= W−1[

−N1 D1 D2 −N2]

,

and use this to define H′ as the transfer matrix fromw′ to z′ induced by

( [Ir0

]

0

∣∣∣∣

[ 0I p

]

D2 −N2

0 −Y X︸︷︷︸

Ω′:=

)

w′

z′

yu

= 0. (4.7)

Then

‖H‖∞ ≤ 1, Ω ∈ GR H(p+r+t)×(p+r+t)∞

⇐⇒ ‖H ′‖∞ ≤ 1, Ω′ ∈ GR H(p+r+t)×(p+r+t)∞ .

Moreover,‖H‖∞ < 1 iff ‖H ′‖∞ < 1.

Proof . 1. See Theorem4.1.3. In AppendixA, Corollary A.0.13 it is proved that stableW−1

[

−N1 D1]

is co-Jq,p-lossless iff

W−1 [

−N1 D1]

Jq,p

[

−N∼1

D∼1

]

W−∼ = Jr,p

and the lower rightp× p block of W−1[

−N1 D1]

is in GR Hp×p∞ .

2. First we simplify the expression for the transfer matrixH by eliminating in a few steps thesignalsu andy in

[

−N1 D1 D2 −N2

0 0 −Y X

]

w

zyu

= 0. (4.8)

Let U be a matrix inGR H(r+t)×(r+t)∞ such that

[

−Y X]

U =[

0 I t]

,

and partitionU compatibly as

U =[

X •Y •

]

.

50

4.1. The SSP2

Definel1 andl2 through[

yu

]

= U

[

l1l2

]

,

then (4.8) is equivalent to

[

−N1 D1 D2X − NY •0 0 0 I t

]

w

zl1l2

= 0;[

yu

]

= U

[

l1l2

]

.

It follows immediately from this expression thatl2 ≡ 0, and therefore (4.8) is equivalent to

[

D1 D2X − N2Y][

zl1

]

= N1w;[

yu

]

=[

XY

]

l1. (4.9)

Moreover, Ω is in GR H(p+r+t)×(p+r+t)∞ iff the matrix

[

D1 D2X − N2Y]

is in

GR H(p+r )×(p+r )∞ because

[

D1 D2 −N2

0 −Y X

] [

I p 00 U

]

︸︷︷︸

∈GR H(p+r+t)×(p+r+t)∞

=[

D1 D2X − N2Y •0 0 I t

]

.

In the proof of Lemma4.1.4it is shown thatΩ′ is in GR H(p+r+t)×(p+r+t)∞ iff A defined

by[

AB

]

:=[

D2 −N2][

XY

]

= W−1[

D2 −N2][

XY

]

(4.10)

is in GR Hr×r∞ , and thatH ′ = BA−1. It therefore remains to show that

‖H‖∞ ≤ (<)1;[

D1 D2X − N2Y]

∈ GR H(p+r )×(p+r )∞

⇐⇒ ‖BA−1‖∞ ≤ (<)1; A ∈ GR Hr×r∞ .

(Only if) Now assume that‖H‖∞ ≤ 1 and thatΩ ∈ GR H(p+r+t)×(p+r+t)∞ . Define E in

GR H(r+p)×(r+p)∞ as E =

[

D1 D2X − N2Y]−1

and partitionE compatibly asE =[

TV

]

.Then

[

TV

][

D1 D2X − N2Y]

= E[

D1 D2X − N2Y]

=[

I p 00 Ir

]

. (4.11)

Multiplying both sides in (4.9) from the left byE =[

TV

]

yields

[

T D1 0V D1 •

][

zl1

]

=[

T N1

•

]

w (note thatT D1 = I ) .

ThereforeH = (T D1)−1T N1 = T N1. In other words, we have

‖ − H−12 H1‖∞ ≤ 1;

[

H1 H2]

:= T[

−N1 D1]

; H2 is in GR Hp×p∞ .

51


SinceW−1[

−N1 D1]

is co-Jq,p-lossless we know that this is the case only ifT is of theform

T =[

T1 T2]

W−1; T2 ∈ GR Hp×p∞ ; ‖T−1

2 T1‖∞ ≤ 1.

(See Theorem3.2.3.) It follows from

[

T1 T2][

AB

]

= TW(D2X − N2Y)

= TW W−1(D2X − N2Y) (from (4.10))

= 0 (from (4.11))

that BA−1 = −T−12 T1. Hence‖H ′‖∞ = ‖BA−1‖∞ ≤ 1. We also have that

[

T1 T2

I 0

]

W−1 [

D1 D2X − N2Y]

=[

I 0• A

]

.

ThereforeA is in GR Hr×r∞ .

(If) SupposeA is in GR Hr×r∞ and that‖BA−1‖∞ ≤ 1. We need to show that‖H‖∞ ≤ 1

and that[

D1 D2X − N2Y]

is in GR H(r+p)×(r+p)∞ . Because‖H ′‖∞ ≤ 1 by assumption,

we have thatH1 andH2 defined by

[

H1 H2]

:=[

−H ′ I]

W−1 [

−N1 D1]

satisfies‖H−12 H1‖∞ ≤ 1 and thatH2 ∈ GR H

p×p∞ . It is easily checked thatH = −H−1

2 H1.Finally

[

−H ′ II 0

]

W−1[

D1 D2X − N2Y]

=[

H2 0• A

]

shows that[

D1 D2X − N2Y]

is in GR H(r+p)×(r+p)∞ .

In a similar way it may be shown that‖H‖∞ < 1 ⇔ ‖H ′‖∞ < 1.

Lemma’s4.1.4, 4.1.3and4.1.5combined solve the SSP2 with boundγ = 1. The caseγ 6= 1may of course be reduced to theγ = 1 case by simply scaling some of the matrices:

Algoritm 4.1.6 (The SSP2 algorithm). [Given:[

−N1 D1 D2 −N2]

∈ R H(p+r )×(q+p+r+t)∞ .

Assumptions:[

−N1 D1]

has full row rank onC0 ∪ ∞ and[

D2 −N2]

has full column rankonC0 ∪ ∞. Definitions: K = X−1Y is a left coprime fraction overR H ∞ of K; H is the transfermatrix fromw to z induced by

(

−N1

0

∣∣∣∣

D1 D2 −N2

0 −Y X︸︷︷︸

Ω:=

)

w

zyu

= 0

depending onK. Out: All solutions K such thatΩ ∈ GR H(p+r+t)×(p+r+t)∞ and‖H‖∞ ≤ γ, for

some given boundγ, provided any suchK exist that makes‖H‖∞ < γ.]

52

4.1. The SSP2

STEP (A) Chooseγ ∈ R+.

STEP (B) Compute, if possible, a canonical cofactorW ∈ GR H(r+p)×(r+p)∞ such that

W Jr,pW∼ =[

−N1 D1][

Iq

−γ2 I p

][

−N∼1

D∼1

]

.

If this solution exists and if the lower rightp × p block of W−1[

−N1 D1]

is inGR H

p×p∞ , then proceed to STEP (C). Otherwise, no admissible compensator exists;γ

needs to be increased and STEP (B) repeated.

STEP (C) Define[

D2 −N2]

as[

D2 −N2]

:= W−1[

D2 −N2]

.

STEP (D) Compute, if possible, a canonical factorΓ ∈ GR H(r+t)×(r+t)∞ such that

Γ∼ Jr,tΓ =[

D∼2

−N∼2

]

Jr,p[

D2 −N2]

=[

D∼2

−N∼2

]

[ N1N∼1 − γ2D1D∼

1 ]−1 [

D2 −N2]

︸︷︷︸

Π:=

. (4.12)

If this solution exists and if the upper leftr × r block of[

D2 N2]

Γ−1 is in GR Hr×r∞

then proceed to STEP (E). Otherwise, no admissible compensator exists;γ needs to beincreased and STEP (B-D) repeated.

STEP (E) There existK such thatΩ ∈ GR H(p+r+t)×(p+r+t)∞ and‖H‖∞ < γ, and all K that

makeΩ ∈ GR H(p+r+t)×(p+r+t)∞ and‖H‖∞ ≤ γ are generated by

K = YX−1;[

XY

]

= ΛΓ−1

[

IU

]

; U ∈ R Ht×r∞ ; ‖U‖∞ ≤ 1.

The computation of solutionsK is probably best done using state space manipulations. It iswell known how to obtain state space realizations of left coprime factors

[

D1 D2]

,[

N1 N2]

overR H ∞ of a givenG =[

D1 D2]−1 [

N1 N2]

, in terms of state space realization ofG (see,for example, Vidyasagar (82) and Section4.5). The computation ofJq,p-lossless matrices mayalso be performed using state space techniques involving Riccati equations (see Green (35) andAppendixB, TheoremB.3.7). The existence of aJ-lossless factorization is equivalent to theexistence of a nonnegative definite stabilizing solution toa related Riccati equation (see (35) andChapter3, Corollary3.4.1). The over all state space algorithm equals that of the famous H∞Riccati equations (See (22; 35) and Section4.5).

The problem how to obtain optimal solutions may now be answered partially. Lemma4.1.5shows that the general SSP2 may be transformed to a related two-block H∞ problem (seeLemma4.1.4). For two-blockH∞ problems we have derived a generator of all optimal solu-tions in Chapter3, provided the matrix to be factored is nonsingular onC0 ∪ ∞. In other words,if at optimality the “first” co-Jq,p-lossless matrix—W−1

[

−N1 D1]

in STEP (B)—exist, andif the matrixΠ in (4.12) is nonsingular everywhere onC0 ∪ ∞ at optimality, then we have agenerator of all optimal solutions to the problem:

53


Corollary 4.1.7 (Optimal solutions). Let the stable, partitioned matrix[

−N1 D1 D2 −N2]

∈R H

(p+r )×(q+p+r+t)∞ be given. Supposeγopt is the infimum overγ > 0 for which the SSP2 with

this data and boundγ has a solution K. Suppose that there exist W∈ GR H(r+p)×(r+p)∞ such

that

W Jr,pW∼ =[

−N1 D1][

Iq 00 −γ2

optI p

][

−N∼1

D∼1

]

,

with the lower right p× p block element of W−1[

−N1 D1]

in GR H(r+p)×(r+p)∞ . Suppose in

addition that

Π :=[

D∼2

−N∼2

]

W−∼ Jr,pW−1 [

D2 −N2]

=[

D∼2

−N∼2

]

[ N1N∼1 − γ2

optD1D∼1 ]−1

[

D2 −N2]

is nonsingular onC0 ∪ ∞. Then there exist unique integers kj ≥ k j+1 > 0 and l and (nonunique)

Γ ∈ GR H(r+t)×(r+t)∞ such that

Π = Γ∼

0 0 0 D+0 Ir−l 0 00 0 − I t−l 0

D∼+ 0 0 0

Γ ; D+(s) :=

0 0 ( s−1s+1)

kl

0 . ..

0( s−1

s+1)k1 0 0

.

Let Γ be one such solution. In this case optimal solutions K exist,and K is an optimal solutionif and only if it is of the form

K = YX−1;[

XY

]

= Γ−1

I l 00 Ir−l

0 U0 0

; U ∈ R H(t−l )×(r−l )∞ ; ‖U‖∞ ≤ 1.

Proof . Is immediate from a transposed version of Theorem3.3.8.

Many standardH∞ problems, but unfortunately not all of them, satisfy the conditions underwhich Corollary4.1.7applies. Our presentation starts with aleft coprime fraction of the gener-alized plantG. Similar “dual” results may be derived starting with a rightcoprime fraction. Itis worthwhile to note that the conditions under which Corollary4.1.7applies and the conditionsunder which its “dual” version applies are not equivalent. So if Corollary 4.1.7were to fail forplant G, then one might be better off with a dual version of the result. A lot of work on theoptimal standardH∞ problem still remains to be done. The optimality results obtained in theprevious chapter as it seems do not easily allow to handle a more general type of optimality.

4.2. The SSP1

In this section we solve the SSP1. The difference between theSSP1 and the SSP2 is essentiallynothing more than a condition at∞. This minor difference unfortunately gives rise to severalpeculiar complications. The proofs do not add anything to the theory nor do they provide fur-ther insight. All proofs are listed in AppendixC. In this section

[

−N1 D1 D2 −N2]

∈P (r+p)×(q+p+r+t). Throughout we assume the following:

54

4.2. The SSP1

Assumptions 4.2.1.

1.[

−N1 D1]

∈ P (r+p)×(q+p) has full row rank onC0.

2.[

D2 −N2]

∈ P (r+p)×(r+t) has full column rank onC0.

In terms of the corresponding generalized plantG =[

D1 D2]−1 [

N1 N2]

, the first assump-tion implies thatG21 has full row rank on the imaginary axis, though not necessarily at infinity.(For example, strictly properG21 and polynomialG21 are not excluded this way.) The secondassumption implies thatG12 has full column rank onC0. Note that the assumptions do not sayanything about properties of

[

−N1 D1]

and[

D2 −N2]

at infinity. To streamline the solutionto the SSP1 we introduce a notation that for lack of a better name we choose to call “internallystable matrix”:

Definition 4.2.2 (Internally stable matrix). A rational matrixG ∈ R is internally stableif all itsfinite poles (hence, excluding possible poles at∞) lie in C−.

Polynomial matrices are internally stable. The central result that we need is a polynomialversion of the two-blockH∞ problem.

Definition 4.2.3 (Polynomial two-block H∞ problem). Let F ∈ P (r+p)×(q+p) be a polynomialmatrix. Thepolynomial suboptimal two-blockH∞ problem (PSTBP)with boundγ ∈ R+, is tofind internally stableT ∈ R p×(r+p) such that

[

P R]

:= T F ∈ P p×(q+p); ‖R−1P‖∞ < γ; R−1 internally stable.

Lemma 4.2.4 (Polynomial two-block H∞ problem). Let F∈P (r+p)×(q+p) be a given polynomialmatrix that has full row rank onC0. Let γ ∈ R+ be given. The following two statements areequivalent.

1. There exist internally stable matrices T∈ R p×(r+p) such that[

P R]

:= T F ∈ P p×(q+p); ‖R−1P‖∞ < γ; R−1 internally stable.

2. There exist strictly Hurwitz solutions Q of the equation

F

[

Iq 00 −γ2 I p

]

F∼ = QJr,pQ∼,

with Q−1F proper, and one such Q (and then all such Q) has the property that Q−1F isco-Jq,p-lossless, or, equivalently, one such Q (and then all such Q)has the property thatthe matrix

[

Q1 F2]

consisting of the left r columns of Q and the right p columns ofF isstrictly Hurwitz.

Moreover, in the case that the conditions in Item 2 are satisfied, T has the property that[

P R]

:= T F; ‖R−1P‖∞ ≤ γ; R−1 internally stable,

if and only if T is of the form

T = A[

U I]

Q−1; U ∈ R Hp×r∞ ; ‖U‖∞ ≤ 1; A, A−1 internally stable.

Furthermore,‖PR−1‖∞ < γ if and only if‖U‖∞ < 1.

55


Strictly Hurwitz matricesQ that satisfyQJq,pQ∼ = Z for some givenZ are sometimes referredto as(Jq,p-spectral) cofactorsof Z.

As with the SSP2, we first consider a simplified problem, whichis equivalent to a polynomialversion of the two-blockH∞ problem:

Lemma 4.2.5 (A polynomial two-block H∞ problem). Let[

Q1 Q2 D2 −N2]

inP (r+p)×(r+p+r+t) be given and suppose that

[

D2 −N2]

has full column rank onC0 and thatQ :=

[

Q1 Q2]

is strictly Hurwitz. Let∆Λ−1 = Q−1[

D2 −N2]

be a right coprime PMFDof Q−1

[

D2 −N2]

. In what follows K= X−1Y and K= YX−1 are a left and a right coprimePMFD of K, respectively. Let H be the transfer matrix fromw to z induced by

(

Q1

0

∣∣∣∣

Q2 D2 −N2

0 −Y X︸︷︷︸

Ω:=

)

w

zyu

= 0. (4.13)

Then H= BA−1, and A−1 is internally stable if and only ifΩ is strictly Hurwitz, whereB andA are defined as

[

AB

]

:= Q−1[

D2 −N2][

XY

]

= ∆Λ−1

[

XY

]

.

The problem to determine internally stable[

XY

]

such that

[

AB

]

:= ∆Γ−1

[

XY

]

; ‖BA−1‖∞ < γ; A−1 internally stable

is a polynomial STBP. Under the assumption that[

D2 −N2]

has full column rank onC0 also∆ = Q−1

[

D2 −N2]

Λ has full column rank onC0, in which case Lemma4.2.4applies. Thislemma, in a transposed version, states that there then existsuch solutionsΛ−1

[XY

]

to the PSTBPiff

Γ∼ Jr,tΓ =[

D∼2

−N∼2

]

Jr,p[

D2 −N2]

(4.14)

has a strictly Hurwitz solutionΓ with ∆Γ−1 proper, and one (and then all) suchΓ have theproperty that the matrix

[

∆1

Γ2

]

consisting of the topr rows of∆ and the lowert rows ofΓ , is strictly Hurwitz. In the case thatthese conditions are satisfied, all solutions

[XY

]

are are of the form

[

XY

]

= ΛΓ−1

[

IU

]

A; U ∈ R Ht×r∞ ; ‖U‖∞ < 1; A, A−1 internally stable.

Note that the factorA in the above displayed formula cancels in the expressionK = YX−1 forthe compensator. So if it is only the compensators we are interested in, we may without loss ofgenerality takeA = I .

56

4.2. The SSP1

Lemma 4.2.6 (A reduction to a polynomial TBP). Let[

−N1 D1 D2 −N2]

in P (p+r )×(q+p+r+t)

be given. Assume that[

−N1 D1]

has full row rank onC0 and that[

D2 −N2]

has full columnrank onC0. In what follows K= X−1Y and K= YX−1 are a left and a right coprime PMFD ofK, respectively. Let H be the transfer matrix fromw to z induced by

(

−N1

0

∣∣∣∣

D1 D2 −N2

0 −Y X︸︷︷︸

Ω:=

)

w

zyu

= 0. (4.15)

1. There exist K such that‖H‖∞ < 1 andΩ is strictly Hurwitz only if there exist strictlyHurwitz Q such that

QJr,pQ∼ =[

−N1 D1]

Jq,p

[

−N∼1

D∼1

]

with Q−1[

−N1 D1]

proper, and one (and then all) such Q are such that[

Q1 D1]

isstrictly Hurwitz, where Q1 are the left r columns of Q.

2. Assume Q as in item 1 exist and let Q be one such solution. LetH ′ be the transfer matrixfromw′ to z′ induced by

(

Q1

0

∣∣∣∣

Q2 D2 −N2

0 −Y X︸︷︷︸

Ω′:=

)

w′

z′

yu

= 0. (4.16)

Then

‖H‖∞ ≤ 1, Ω strictly Hurwitz ⇔ ‖H ′‖∞ ≤ 1, Ω′ strictly Hurwitz.

Moreover,‖H‖∞ < 1 if and only if‖H ′‖∞ < 1 .

The proof follows the same lines as that of Lemma4.1.5, up to some extremely boring ma-nipulative arguments. It is good to be aware of the followingcomplication. In the discussion onthe SSP2 we used an argument based on compensator independent signals to prove that a certainco-J-lossless matrix necessarily must exist for the problem to have a solution. In the polynomialcase this elegant argument no longer works so easily. The reason is that the system is not neces-sarily L2-stable. We therefore have to take a different route to provethe polynomial equivalent(Item 1 in Lemma4.2.6).

Lemma4.2.6has a nice diagrammatical representation. Given the data inLemma4.2.6definethe transfer matrices

G :=[

D1 D2]−1 [

N1 N2]

, E =[

D1 −Q1]−1 [

N1 Q2]

,

G′ :=[

Q2 D2]−1 [

−Q1 N2]

.

With these and with the signals defined in Lemma4.2.6we may form the diagram of Fig.4.4. Itis clear that the system inside the dotted box hasG as its transfer matrix. The matrixE is co-inneras is readily established. Pretty as it may be, this diagrammatical presentation has its limitations.The numerator

[

Q2 D2]

of G′, for instance, may well be singular for the cases that we allow.The proofs do not rely on nonsingularity of

[

Q2 D2]

.Summarizing, reintroducingγ, we get the SSP1 algorithm.

57


E

G′

Ky u

w z

w′z′

Figure 4.4.: An associated system.

Algoritm 4.2.7 (The SSP1 algorithm). [Given:[

−N1 D1 D2 −N2]

∈ P (r+p)×(q+p+r+t).Assumptions:

[

−N1 D1]

has full row rank onC0 and[

D2 −N2]

has full column rank onC0. Definitions: K = X−1Y is a left coprime PMFD ofK; H is the transfer matrix fromw to zinduced by

(

−N1

0

∣∣∣∣

D1 D2 −N2

0 −Y X︸︷︷︸

Ω:=

)

w

zyu

= 0

depending onK. Out: All solutions K such thatΩ is strictly Hurwitz and‖H‖∞ ≤ γ for somegiven boundγ, provided any suchK exist that makes‖H‖∞ < γ.]


STEP (B) Compute, if possible, aJr,p-spectral cofactorQ such that

QJr,pQ∼ =[

−N1 D1][

Iq

−γ2 I p

][

−N∼1

D∼1

]

,

with Q−1[

−N1 D1]

proper. If this solution exists and if[

Q1 D1]

is strictly Hurwitz,with Q1 the left r columns ofQ, then proceed to STEP (C). Otherwise, no admissiblecompensator exists;γ needs to be increased and STEP (B) repeated.

STEP (C) Find right coprime polynomial matrices∆ andΛ such that∆Λ−1 = Q−1[

D2 −N2]

.

STEP (D) Compute, if possible, aJr,t-spectral factorΓ such that

Γ∼ Jr,tΓ = ∆∼ Jr,p∆,

with ∆Γ−1 proper. If this solution exists and if[

∆1

Γ2

]

58

4.3. On the computation of suboptimal and optimal solutionsto the SSP1

is strictly Hurwitz, withΓ2 the lowert rows ofΓ and∆1 the upperr rows of∆, then pro-ceed to STEP (E). Otherwise, no admissible compensator exists;γ needs to be increasedand STEP (B-D) repeated.

STEP (E) There existK such thatΩ is strictly Hurwitz and‖H‖∞ < γ, and all K that make‖H‖∞ ≤ γ andΩ strictly Hurwitz are generated by

K = YX−1;[

XY

]

= ΛΓ−1

[

IU

]

; U stable and‖U‖∞ ≤ 1. (4.17)


By far the most time consuming steps in the SSP1 Algorithm arethe two J-spectral(co)factorization problems. In this section we formulate an algorithm that may be used tocompute polynomialJ-spectral (co)factors. We have listed a “rational” versionof this algorithmin AppendixB, where it used to prove constructively the existence of canonical and noncanonicalfactors. For this reason that we do not prove the validity of the algorithm here. The algorithmis based on Callier’s method for ordinary polynomial spectral factorization by symmetric factorextraction (Callier (15)). For details we refer to Kwakernaak (48). By m we mean1,2, . . . ,m,the set of positive integers from 1 up to and includingm. Recall that Strictly Hurwitz matricesQ that satisfyQJq,pQ∼ = Z for some givenZ are referred to asJq,p-spectral cofactors ofZ. Astrictly HurwitzΓ is a Jr,t-spectral factor ofZ if Γ∼ Jr,tΓ = Z. By γ j(∆) we mean the columndegree of thejth column of∆ ∈ P .

Algoritm 4.3.1 (Symmetric factor extraction algorithm). [ Given Z = ∆∼ Jr,p∆, with ∆ tallcolumn reduced withm columns andZ nonsingular onC0, the algorithm determines a matrixJq,t and a strictly HurwitzΓ such thatΓ∼ Jq,tΓ = Z. Moreover, if possibleΓ is such that∆Γ−1

is proper.]

STEP (A) n := 12 graaddetZ. Compute alln zerosζ j ∈ C− of detZ. Set the virtual column

degreesd j to d j := γ j(∆) for j ∈ m. Seti := 0 andZ1 := Z.

STEP (B) i := i + 1. Compute a constant null vectore= (e1, . . . ,em)T such thatZi(ζi )e= 0.

STEP (C) Select a pivot indexk from the maximal active index set

Mi = j ∈ m | ej 6= 0 andd j ≥ dl for all l ∈ m for which el 6= 0 . (4.18)

STEP (D) Compute the polynomial matrixZi+1 = (T∼i )

−1Zi T−1i , whereTi is defined as

Ti(s) =

1 − e1ek

. . ....

1 − ek−1ek

s− ζi

− ek+1

ek1

......

− emek

1

. (4.19)

59


STEP (E) dk := dk − 1 (update of the virtual column degrees of∆T−11 · · · T−1

i ).

STEP (F) if i < n then goto STEP (B).

STEP (G) (Zn+1 is unimodular.) Compute aJq,t and unimodularW such that

W∼ Jq,tW = Zn+1, (4.20)

by whatever method (see for instance (38; 15)).

STEP (H) Γ = WTn · · · T1 is a Jq,t-spectral factor ofZ.

The matrixΓ generated this way may turn out to have complex valued coefficients. In caseZitself has only real valued coefficients, the extractions may be rearranged such thatΓ is also real((48; 49)). The role of the virtual column degreesd j is explained in AppendixB. Furthermore,copying from AppendixB, LemmaB.3.11there exists a solutionΓ such that∆Γ−1 is proper iffall virtual column degreesd j are zero on exit of the algorithm. In this caseZn+1 is constant, thesolutionW in STEP (G) may be taken constant and the result is a solutionΓ such that∆Γ−1 isproper.

Example 4.3.2 (Symmetric factor extraction algorithm). Consider∆ andZ for some fixedγas defined below.

∆ =

1 01 11 −s

; Z := ∆∼[

γ2

− I2

]

∆ =[

γ2 − 2 s− 1−s− 1 s2 − 1

]

For 1< γ 6=√

2 we go through the steps of the symmetric factor extraction algorithm.

(A) n = 1, ζ1 = −1, Z1 = Z, m= 1, and d1 = 0, d2 = 1.

(B) i = 1, Z(ζ1) = Z(−1) =[γ2−2 −2

0 0

]

, e=[ 2γ2−2

]

.

(C) As k is to be chosen from the setM1 = 2, we havek = 2.

(D) Z2 = (T∼1 )

−1Z1T−11 , with

T1 =[

1 − 2γ2−2

0 s+ 1

]

,

so that

T−11 =

[1 2

γ2−21

s+1

0 1s+1

]

and Z2 =[

γ2 − 1 11 −1

]

.

(E) d2 := 0.

(F) i = n = 1 and alld j are zero.

(G)

Z2 =[

γ2 − 2 11 −1

]

=[√

γ2 − 1 −10 1

][

1 00 −1

][√

γ2 − 1 0−1 1

]

= W∼ J1,1W.

60


(H)

Γ = WT1 =[√

γ2 − 1 0−1 1

][

1 − 2γ2−2

0 s+ 1

]

=[√

γ2 − 1 −2γ2−2

√

γ2 − 1

−1 γ2

γ2−2 + s

]

.

If γ =√

2 the pivot element can only bek = 1 because thene= (2 0)T andM1 = 1. This givesrise to a discontinuity of theJ-spectral factor as a function ofγ at γ =

√2.

We now show thatγopt =√

2 is the infimum over allγ for which the associated polynomialSTBP with boundγ has a solution. The PSTBP for this data is to find internally stableT ∈ R 2×1

such that[

RP

]

:= ∆T

satisfies‖PR−1‖∞ < γ and such thatR−1 is internally stable. By Lemma4.2.4suchT exist iff

[

∆1

Γ2

]

=[

γ 0

−1 γ2

γ2−2 + s

]

is strictly Hurwitz. This is obviously the case iffγ >√

2. One solutionT that then solves theproblem is

T := Γ−1

[

10

]

= 1s+ 1

[γ2

γ2−2 + s1

]

.

Like Γ , alsoΓ−1 andT as above blow up asγ approaches the optimal valueγopt =√

2. Thisexemplifies two properties that hold for practically allH∞ problems:

• Nearly optimal solutions have large coefficients.

• Optimal solutions do not follow straightforwardly from a continuity argument using thesuboptimal solutions.

It must be added that the factor extraction algorithm is not anumerically stable algorithm.The algorithm in principle allows to compute suboptimal solutions. In fact the factor extractionprocedure may be modified so that optimal solutions may be derived as well, provided someassumptions are satisfied. This is a technical procedure. The idea is this: For a given set ofdata

[

−N1 D1 D2 −N2]

we may check, using the SSP1 Algorithm in combination withthe symmetric factor extraction Algorithm, whether or not there exist solutions to the SSP1 withboundγ. A root finder may be employed to delimit the infimal value ofγ for which the SSP1has a solution. Call this infimal valueγopt. If for γ = γopt the SSP1 may still be reduced the two-block H∞ problem as in Lemma4.2.6then we may apply a polynomial version of the optimaltwo-blockH∞ problem to generate all optimal solutions to the reduced SSP1.

We now formulate an algorithm that solves the polynomial version of the OTBP and therebysolves a whole family of optimal standardH∞ problems.

Definition 4.3.3 (POTBP). Let∆ ∈ P (r+p)×(r+t) be given. DefineP ∈ R p×r andR∈ R r×r as[

RP

]

= ∆T

61


depending on some internally stableT ∈ R (r+t)×r. Defineγopt as the infimal value of‖PR−1‖∞over all internally stableT for which R−1 is internally stable. Thepolynomial OTBP (POTBP)isto find internally stableT ∈ R (r+t)×r such thatR−1 is internally stable and‖PR−1‖∞ = γopt.

Lemma 4.3.4 (Optimal solutions). Let∆ ∈ P (r+p)×(r+t) be given and suppose it has full columnrank onC0. Consider the POTBP with this∆ and letγopt be as defined in Definition4.3.3.Suppose∞ > γopt > 0, and that

Z := ∆∼[

γ2optIr 00 − I p

]

∆

is nonsingular onC0 and thatδZ = 2δ∆. In this case there exist solutions T to the POTBP, andall optimal solutions may be generated by following the procedure given below.

1. Apply the symmetric factor extraction algorithm with Z asinput.

2. Permute the columns of∆ in such a way that on exit of the symmetric factor extractionalgorithm the virtual column degrees as defined in this algorithm are ordered as

d1, . . . ,dl︸︷︷︸

d j<0

, dl+1, . . . ,dm−l︸︷︷︸

d j=0

, dm−l+1, . . . ,dm︸︷︷︸

d j>0

(m := r + t).

(In AppendixB it is proved that the number l of strictly negative indices dj equals thenumber of strictly positive indices dj and that l> 0.)

3. Then Zn+1 produced by the symmetric factor extraction algorithm, partitioned compatiblywith the ordering of the djs, is of the form

Zn+1 =

0 0 •0 C •• • •

,

with C ∈ C(m−2l )×(m−2l ), and C may be written as

C = W∼c Jr−l ,t−lWc

for some nonsingular Wc ∈ C(m−2l )×(m−2l ).

4. Let Ti , i ∈ n denote the elementary factors as produced by symmetric factor extractionalgorithm. Then T is a solution to the POTBP if and only if it isof the form

T = [T−11 · · · T−1

n ]

I l

W−1c

I l

I l 00 Ir−l

0 U0 0

A;

U ∈ R H(t−l )×(r−l )∞ ; ‖U‖∞ ≤ 1; A, A−1 internally stable.

5. If the last pivot index k in the symmetric factor extraction algorithm is less than l (in otherwords, if on exit dk < 0), then the generator of all optimal solutions as given in Item 4 is

62


also valid if Tn is replaced by the constant matrix

Tn :=

1 − e1ek

. . ....

1 − ek−1ek

1

− ek+1

ek1

.... . .

− emek

1

.

The basic idea of the proof is to rewrite the problem as a noncanonical factorization problemand then to solve the OTBP. We omit the proof.

A further examination of the factor extraction algorithm shows that it is very unlikely that thevirtual column degreesd j arenot all zero on exit. In other words, if thecomputedvalue ofγopt is not exactly equal to the real optimal value, then the procedure in Lemma4.3.4based onthe computed nonexact value ofγopt does not work. This seeming disadvantage can be madein to an advantage: Since weknow that for the exact value ofγopt at least one virtual columndegree is less than zero—in practice there will be exactly one that is less than zero—we may usethis knowledge to determineγopt as precisely that number for which some of the virtual columndegrees become negative. This way we have by construction that for the optimal value somevirtual column degrees are negative. We omit the precise details.

Example 4.3.5 (A POTBP; Example 4.3.2 continued). Consider the problem to find internallystableT such that

[

RP

]

:= ∆T

satisfies‖PR−1‖∞ ≤ γ and such thatR−1 is internally stable, with∆ as in Example4.3.2:

∆ =

1 01 −s1 1

.

We know from Example4.3.2that such solutions exist only ifγ ≥√

2. Furthermore, forγ >√2—the suboptimal case—all solutions may be derived from theresults in that example. In the

present example we consider the optimal caseγ = γopt :=√

2.

Takeγ =√

2. As may be checked, the symmetric factor extraction algorithm with input

Z := ∆∼[

γ2opt 00 − I2

]

∆ =[

0 s− 1−s− 1 s2 − 1

]

produces among other things

T1(s) =[

s+ 1 00 1

]

; Zn+1 = Z2(s) =[

0 −1−1 1− s2

]

; k = 1, d1 = −1, d2 = 1.

63


The virtual column degrees are already in the order as required in Item 2 of Lemma4.3.4. Thematrix C as defined in Item 3 of this lemma is void, and, therefore, all solutionsT according toItem 4 of Lemma4.3.4follow as

T = T−11

[

10

]

A =[

As+10

]

; A, A−1 internally stable.

The simplest solutionT that does it isT =[

10

]

. It is a reduced degree solution. A more systematicway to obtain reduced degree solutionsT is based on the generator of optimal solutions describedin Item 5 of that lemma. There it is stated thatT1 may be replaced with

T1 :=[

1 0−e2/e1 1

]

=[

1 00 1

]

,

provided for the last pivot indexk on exit dk < 0. In our example this is the case. Hence alloptimal solutionsT are of the form

T = T−11

[

10

]

A =[

A0

]


The most obvious choice forA is A = I , which leads toT =[

10

]

.

The observation made at the end of Example4.3.5holds in general. That is, if∆ has degreeδ∆ = n, then for constantA andU the optimal solutionsT generated by the construction inLemma4.3.4, Item 5, has McMillan degreeδMT at mostn− 1. This is easily checked.

r y u zK P

Figure 4.5.: A closed-loop system configuration.

4.4. Example

As stated earlier, the SSP1 and SSP2 are “standard” because manyH∞ control problems may berecast as an SSP1 or SSP2. In this section we examine one of these control problems in moredetail: the mixed sensitivity problem. The example of the mixed sensitivity problem that followsclarifies why we do not bother about properness and well-posedness much and, as a consequence,why we consider the SSP1 to be more useful than the SSP2, at least as far as real control problemsare concerned.

We consider the system depicted in Fig.4.5. The plantP is given and the compensatorKis to be determined such that it makes the closed-loop system“behave well”. The idea is that“behaving well” may adequately be translated in terms of∞-norm bounds. More precisely, if thefiltersV,W1 andW2 in the extended, artificial closed loop in Fig.4.6are designed “correctly” thenstabilizing compensators—we soon make precise what we meanby “stabilizing”—that make the∞-norm of the transfer matrixH fromw to (z1, z2) small, make the original closed-loop systemin Fig. 4.5 behave well. This is in a few words the goal of the mixed sensitivity problem. Howto translate “behaving well” in terms of these shaping filters is a problem on its own and we

64

4.4. Example

are not going to dwell on it here. For details, see Kwakernaak(48; 50). Properness ofK andwell-posedness of the closed-loop system (I + P(∞)K(∞) nonsingular) are usually essentialfor a closed-loop system to behave well. In other words, correctly designed shaping filters havethe property that (nearly) optimal compensatorsK are proper and that the resulting closed-loopsystem is well-posed. In many cases properness ofK is not enough;K should be strictly proper,or better, it should be small outside the closed-loop bandwidth. In terms of shaping filters thismeans thatW2 has to be chosen nonproper. We copy from Chapter2:

Definition 4.4.1 (Mixed sensitivity problem). Consider the closed loop in Fig.4.5. Let rationalmatricesV, W1 and W2 and P be given. Thesuboptimal mixed sensitivity problemis to findcompensatorsK that internally stabilize the system in Fig.4.5(with u, y andz considered as theoutputs andr as the input) and that make the∞-norm of the transfer matrix

H :=[

W1( I + PK)−1VW2K( I + PK)−1V

]

from w to[ z1

z2

]

as in Fig.4.6 less than some given boundγ. The optimal mixed sensitivityproblem is to find compensatorsK that minimize‖H‖∞ over all compensators that internallystabilize the system in Fig.4.5.

K P

V

W1

W2

w

z1

z2

uy

Figure 4.6.: A mixed sensitivity configuration.

Example 4.4.2 (A mixed sensitivity problem). Suppose that the given plant is

P(s) = 1s. (4.21)

It may be argued that

W1(s) = 1, V(s) = s+ 1s, W2(s) = c(1+ rs),

are correctly chosen shaping filters if 0≤ r ≤ 1 andc > 0 (See Kwakernaak (48)). With thesefilters, the smaller a stabilizing compensator makes

‖H‖∞ =∥∥∥∥

W1( I + PK)−1VW2K( I + PK)−1V

∥∥∥∥

∞=

∥∥∥∥

s+1s+K

c(1+ rs)K s+1s+K

∥∥∥∥

∞, (4.22)

the better it makes the closed-loop system behave. Expression (4.22) is finite only if K(s) isbounded at infinity and, hence, admissible compensators arealways proper and even strictlyproper if r 6= 0. It shows that admissible compensators always make the closed-loop systemwell-posed in this example (because 1+ P(∞)K(∞) = 1 is nonsingular).

65


The generalized plantG in the corresponding standard system is

G =

W1V W1P0 W2

−V −P

. (4.23)

In our exampleW2 is nonproper, which is typical for mixed sensitivity problems. As a result,Gis nonproper, too. This shows that in this example the standard systemneveris well-posed foradmissible compensators2. This is the reason for not insisting on well-posedness in the standardsystem. It also shows that the SSP2, which is the most commonH∞ control problem around, isnot suitable for dealing directly with such control problems. Krause (45) proposes to circumventthis problem by absorbing a stable factorF−1: If Ktmp is an admissible compensator for thestandard problem with generalized plant

Gtmp = G

[

I 00 F−1

]

=

W1V W1PF−1

0 W2F−1

−V −PF−1

,

thenK := F−1Ktmp is an admissible compensator for the original problem.F is chosen to makeGtmp proper. OftenF = W2 will do. This is an effective technique, but has the undesirableeffect that the compensatorsK computed this way have McMillan degree higher than necessary.Without cancellation (which, if at all possible, is numerically unattractive) we have for admissiblecompensatorsK computed this way that

δM(K) = δM(Gtmp)+ δM(F) = δM(P)+ δM(W1)+ δM(W2)+ δM(F). (4.24)

whereas for suboptimal compensators computed polynomially we have

δM(K) ≤ δM(G) ≤ δM (P)+ δM (W1)+ δM(W2). (4.25)

(See Remark4.5.2.) In (4.24) and (4.25) we use the assumption thatV and P have the samedenominators and thatV is proper (see Kwakernaak (48)). The mixed sensitivity problem is anSSP1 as we show next.

We write the data as polynomial fractions:

V = D−1M; P = D−1N; W1 = B−11 A1 W2 = B−1

2 A2,

and we describe theK to be constructed also as a polynomial fraction:

K = X−1Y.

Note that the denominator ofV equals that of the plant. (It may be argued that this makes sense((48; 50)).) Now we are back at our favorite form: the differential equation. The closed loop iscompletely characterized by the differential equations

M 0 0 D N0 B1 0 A1 00 0 B2 0 −A2

0 0 0 −Y X

w

z1

z2

yu

= 0.

2The transfer matrix fromv2 to z is W2( I + K P)−1, which behaves asW2 for high frequencies since for admissiblecompensatorsI + K P is biproper. See Fig.4.2on page 55.

66

4.4. Example

The SSP1 with this data is to findK = X−1Y such that the transfer matrixH from w to[ z1

z2

]

satisfies‖H‖∞ < γ and such that

Ω :=

0 0 D NB1 0 A1 00 B2 0 −A2

0 0 −Y X

is strictly Hurwitz. Obviously detΩ = ±detB1 detB2 det[

D −N−Y X

]

. So the SSP1 has solutionsonly if B1 andB2 are strictly Hurwitz. This gives rise to a problem only ifB1 or B2 has zeros onC0. If Bi has no zeros onC0 and is not strictly Hurwitz, then we may replaceBi by any strictlyHurwitz solutionBi,o of B∼

i,oBi,o = B∼i Bi . This has no effect on the∞-norm of the closed-loop

transfer matrixH. Once this has been done we have that the SSP1 with this data isto findK = X−1Y such that

[

D −NY X

]

is strictly Hurwitz and such that‖H‖∞ < γ for some given boundγ. This is precisely thesuboptimal version of the mixed sensitivity problem.

Example 4.4.3 (A mixed sensitivity problem, Example 4.4.2 continued). The polynomial so-lution to the suboptimal mixed sensitivity problem defined by (4.21-4.23) in Example4.4.2goesas follows. As input to the SSP1 Algorithm we define

[

−N1 D1 D2 −N2]

:=

M 0 0 D N0 B1 0 A1 00 0 B2 0 −A2

=

s+ 1 0 0 s 10 1 0 1 00 0 1 0 −c(1+ rs)

.

The matrix[

−N1 D1]

is square and strictly Hurwitz, so aJ-spectral cofactorQ in the SSP1Algorithm, STEP (B) is simply

Q =[

−N1 γD1]

=

s+ 1 0 00 γ 00 0 γ

.

The matrices∆ andΛ defined byQ−1[

−N1 D1]

= ∆Λ−1 follow from a left-to-right conver-sion.

Q−1 [

D2 −N2]

=

ss+1

1s+1

1γ

0

0 −c1+rsγ

=

γ −γs1 −1− s

−c(1+ rs) 0

[

γ −γ(1+ s)γ (s+ 1)γ

]−1

= ∆Λ−1.

The next step in the SSP1 Algorithm is the computation ofΓ . The(1 00 −1)-spectral factorΓ need

67


satisfy

Γ∼[

1 00 −1

]

Γ = ∆∼

1 0 00 −1 00 0 −1

∆ (4.26)

=[

γ2 − 1− c2(1− r2s2) −(γ2 − 1)s+ 1(γ2 − 1)s+ 1 −(γ2 − 1)s2 − 1

]

. (4.27)

For simplicity we taker = 0, in which case det(∆∼ J1,2∆) = c2 − γ2 + c2(−1 + γ2)s2. Fromthis we see that a(1 0

0 −1)-spectral factorΓ exists only ifγ > max(1, c). The stable zero of (4.27)then is

ζ1 = 1c

√

γ2 − c2

γ2 − 1.

The symmetric factor extraction algorithm may be applied and the result is that for max(1, c) <γ 6=

√1+ c2, a solution with the correct degree structure is

Γ =

√

γ2 − 1 −√

γ2 − 1(s+ 1c

√γ2−c2

γ2−1 − 1c

√(γ2−1)(γ2−c2)+cγ2−1−c2 )

c√(γ2−1)(γ2−c2)+cγ2−1−c2

.

The zero of

[

∆1

Γ2

]

=[

γ −γs

c√(γ2−1)(γ2−c2)+cγ2−1−c2

]

lies in the left-half plane iffγ >√

1+ c2 and, hence,γopt =√

1+ c2. Forγ > γopt all suboptimalcompensatorsK = YX−1 are given by

[

XY

]

= ΛΓ−1

[

1U

]

,

with U stable and‖U‖∞ ≤ 1. The central compensator (that is, the compensator forU = 0) is

K =√

(γ2 − 1)(γ2 − c2)+ c− c(1+ s)(γ2 − 1− c2)√

(γ2 − 1)(γ2 − c2)+ c+ c(1+ s)(γ2 − 1− c2).

This central compensator satisfiesδM(K) ≤ δM(G) = 1, which holds in general as we show inSection4.5. Note that the central compensator is not unique. Forγ >

√1+ c2 another solution

Γ with the correct degree structure is

Γ =

√

γ2 − 1− c2 −(γ2−1)s+1√γ2−1−c2

0√

c2(γ2−1)s+√γ2−c2√

γ2−1−c2

.

In this case the central compensator isK = 1, independent byγ andc. Exceptionally, in this caseδM(K) is strictly less thanδM(G). K = 1 turns out to be unique optimal solution as we shownext.

68

4.5. Remarks

We end this example with a discussion on how to obtain optimalsolutions to this mixed sensi-tivity problem. The manipulations are in fact very easy. (This is typical for low order systems.)We know already thatγopt :=

√1+ c2 is the best possible bound we can achieve by stabilizing

compensators. By Lemma4.3.4all optimal solutions may be derived provided that the matrix∆∼ J1,2∆ to be factored is nonsingular onC0 and satisfiesδ∆∼ J1,2∆ = 2δ∆. In our case this istrue, because forγ =

√1+ c2 we have

Z := ∆∼ J1,2∆ =[

0 −c2s+ 1c2s+ 1 −c2s2 − 1

]

which obviously is nonsingular and has degree 2= 2δ∆. With this Z as input to the symmetricfactor extraction algorithm, we may find on exit of this algorithm the data

T1 =[

s+ ζ1 00 1

]

; Z2 = Zn+1 =[

0 c2

c2 −c2s2 − 1

]

; k = 1; d1 = −1, d2 = 1.

By Lemma4.3.4, Item 5 we may simplifyT1 to T1 =[

1 00 1

]

if the last pivot indexk in the sym-metric factor extraction algorithm satisfiesdk < 0 on exit. In our case this holds, and thereforeby Lemma4.3.4all internally stable solutionsT follow as

T = T−11

[

10

]

A =[

A0

]


All compensators finally follow as

K = YX−1;[

XY

]

= ΛT =[

γopt −γopt(1+ s)γopt (s+ 1)γopt

][

A0

]

.

Therefore

K = γoptA

γoptA= 1

is the unique optimal solution to the mixed sensitivity problem.

Remark 4.4.4 (Two-block H∞ problem). For mixed sensitivity problemsSTEP(B) of the SSP1Algorithm may be performed symbolically: With input

[

−N1 D1 D2 −N2]

:=

M 0 0 D N0 B1 0 A1 00 0 B2 0 −A2

STEP (B) of the SSP1 Algorithm is always satisfied with

Q =[

−N1 γD1]

=

MγB1

γB2

.

∆ andΛ in STEP (C) of the SSP1 Algorithm follow from

∆Λ−1 = Q−1[

D2 −N2]

=

M−1D M−1N1γ

B−11 A1 00 − 1

γB−1

2 A2

=

V−1 V−1P1γW1 00 − 1

γW2

.

Stated differently, the mixed sensitivity problem is a polynomial two-blockH∞ problem.

69


G

K yu

w z

Figure 4.7.: The standard system configuration.

4.5. Remarks

The previous section more or less completes what we want to convey. A few comments are inorder, however. We consider in this section again the standard system and give some generalcomments concerning the SSP1 and SSP2. Firstly we briefly give some connections between theSSP2 Algorithm and the state space solution method to the conventional standardH∞ problem.This connection has been treated in Green (35) and is included in this thesis for completenessonly. We assume that the reader is familiar with some widely known concepts used in statespace approaches. Secondly we give a polynomial analog of the interesting result that for propergeneralized plants the McMillan degree of an admissible compensator may always be chosenless than or equal to that of generalized plantG. We show that this is the case also ifG isnonproper. This extension to the nonproper case is actuallynot at all trivial but it is worththe effort because this allows to give an upper bound for the McMillan degree of admissiblecompensators also for, for example, mixed sensitivity problems with nonproper shaping filters.As a result, it turns out that the technique proposed by Krause (45) to absorb a proper makingmatrix in the generalized plant, explained in the previous section, effective as it may be, leadsto higher degree admissible compensators than the ones computed polynomially with the SSP1Algorithm. Finally we comment on the dual version of the SSP1and SSP2.

4.5.1. State space manipulations

Consider the standard system in Fig.4.7and suppose thatG is proper. We briefly indicate how thesolution to the SSP2—the “elegant” one, not the SSP1—translates to state space manipulations.The state space manipulations are equivalent to the famous state space results ((22)).

As a first step we rewrite the map[ z

y]

= G[wu

]

as an equation

[

−G11 I 0 −G12

−G21 0 I −G22

]

w

zyu

= 0.

Now suppose thatG has a realization

Gs=

A B1 B2

C1 D11 D12

C2 D21 D22

70

4.5. Remarks

with (C2, A) detectable and(A, B2) stabilizable. Then an equivalent time domain description ofthe open loop is

000

=

A− s B1 0 0 B2

−C1 −D11 I 0 −D12

−C2 −D21 0 I −D22

xw

zyu

= 0.

Heres is identified with the differential operatord/dt. Let H be a constant matrix such thatA − HC2 is stable3. (SuchH exist because by assumption(C2, A) is detectable.) In Nett et.al. (69) it is proved that then

[

−N1 D1 D2 −N2] s=

I 0 H0 I 00 0 I

A B1 0 0 B2

−C1 −D11 I 0 −D12

−C2 −D21 0 I −D22

is a realization of a a left coprime fractionG =[

D1 D2]−1 [

N1 N2]

overR H ∞ of G. Thisrealization shows that the assumptions made for the SSP2:

•[

−N1 D1] s=

I 0 H0 I 00 0 I

A B1 0−C1 −D11 I−C2 −D21 0

has full row rank onC0 ∪ ∞;

•[

D2 −N2] s=

I 0 H0 I 00 0 I

A 0 B2

−C1 0 −D12

−C2 I −D22

has full column rank onC0 ∪ ∞,

are equivalent to

•[ A−sI B1

−C2 −D21

]

has full row rank for alls∈ C0 ∪ ∞;

•[ A−sI B2

−C1 −D12

]

has full column rank for alls∈ C0 ∪ ∞.

Suppose that these assumptions are satisfied. By Lemma4.1.3the SSP2 with boundγ = 1 hasa solution only ifB[

−N1 D1

] is strictly positive. We have shown in Chapter3 that this is the

case iff an associated Riccati equation has a stabilizing nonnegative definite solutionQ. It isno surprise that this Riccati equation, translated in termsof the realization of

[

−N1 D1]

thatwe constructed, is exactly theH∞ filter algebraic Riccati equation. The manipulations involvedbecome very messy if we do not assume that some normalizationhas been carried out first. Byway of example, assume the following quite common normalization assumptions hold:

• D11 = 0; D21[

B∗1 D∗

21

]

=[

0 I]

.

Then the assumption that[ A−sI B1

−C2 −D21

]

has full column rank onC0 ∪ ∞ reduces to(A, B1) not

having uncontrollable modes onC0. By Corollary 3.4.1, Item 2 the spaceB[

−N1 D1

] is then

strictly positive iff there existQ such that

HFARE :

AQ+ QA∗ + Q(C∗1C1 − C∗

2C2)Q+ B1B∗1 = 0

A+ Q[C∗1C1 − C∗

2C2] stable; Q ≥ 0. (4.28)

3A constant square matrix is stable if all its eigenvalues liein C−.

71


HFARE stands forH∞ filter algebraic Riccati equation. Assume from now on thatQ is one suchsolution. A realization of a solutionW ∈ GR H

(r+p)×(r+p)∞ to the canonical cofactorization prob-

lem W Jr,pW∼ =[

−N1 D1]

Jq,p[

−N1 D1]∼

, combined with a realization of[

−N1 D1]

,is

[

−N1 D1 W] s=

A B1 0 −QC∗2 QC∗

1

−C1 0 I 0 I−C2 −D21 0 I 0

A realization of[

−N1 D1 D2 −N2]

:= W−1[

−N1 D1 D2 −N2]

can now be obtainedby applying a regular output injection transformation similar to what we did in Chapter3 for theconstruction of a realization ofW−1G: It may be verified that

[

−N1 D1 D2 −N2] s=

I −QC∗1 QC∗

20 0 I0 I 0

A B1 0 0 B2

−C1 0 I 0 −D12

−C2 −D21 0 I −D22

.

According to the SSP2 Algorithm there then exist solutions to the SSP2 iff there exists a canonicalfactorΓ ∈ GR H

(r+t)×(r+t)∞ satisfying

Π := Γ∼ Jr,tΓ =[

D∼2

−N∼2

]

Jr,p[

D2 −N2]

such that[

0 II 0

][

D2 −N2

]

Γ−1[

0 II 0

]

is Jp,r-lossless. To avoid technicalities we assume that thefollowing quite common normalization assumptions are fulfilled:

• D22 = 0; D∗12

[

C1 D12]

=[

0 I]

.

Under these additional assumptions the matrix[

D2 −N2]

has full column rank onC0 ∪ ∞ iffthe pair(C1, A) has no unobservable modes on the imaginary axis. The given realization of[

D2 −N2]

may be now be simplified to

[

D2 −N2] s=

Atmp QC∗2 B2

−C2 I 0−C1 0 −D21

; Atmp := A+ Q(C∗1C1 − C∗

2C2).

With this realization of[

D2 −N2]

, a realization of the matrixΠ is not that complicated:

Π :=[

D∼2

−N∼2

]

Jn,p[

D2 −N2] s=

Atmp 0 QC∗2 B2

C∗1C1 − C∗

2C2 −Atmp C∗2 0

−C2 C2Q I 00 B∗

2 0 − I

.

The corresponding Hamiltonian matrix then is

H :=[

Atmp 0C∗

1C1 − C∗2C2 −A∗

tmp

]

−[

QC∗2 B2

C∗2 0

]

Jr,t

[

−C2 C2Q0 B∗

2

]

=[

A+ QC∗1C1 −QC∗

2C2Q+ B2B∗2

C∗1C1 −A∗ − C∗

1C1Q

]

=[

I Q0 I

][

A B2B∗2 − B1B∗

1C∗

1C1 −A∗

]

︸︷︷︸

H:=

[

I −Q0 I

]

. (4.29)

72

4.5. Remarks

The results in Corollary3.4.1, Item 2, say, in a transposed version, thatΠ admits a canonical fac-torizationΠ = Γ∼ Jr,tΓ iff the stable eigenspace ofH may be written as Im

[IZ

]

. The additionalJ-lossless property4 holds iff this Z ≤ 0.

It is readily established that wheneverK is an admissible compensator for the plantG, thatthen K∗ is an admissible compensator for the plantG∗. The transposed version of the HFARE(4.28) is:

HCARE :

PA+ A∗ P+ P[ B1B∗1 − B2B∗

2] P− C∗1C1 = 0;

A+ [ B1B∗1 − B2B∗

2] P stable; P ≥ 0.(4.30)

It must have a solution if the SSP2 is solvable. HCARE stands for H∞ controller algebraicRiccati equation. The curious thing to note is that then

[I

−P

]

spans the stable eigenspace of thematrix H that we defined earlier in (4.29). Obviously the stable eigenspace ofH is also spannedby

[I −Q0 I

][IZ

]

. In other words,

Z = Z2Z−11 ;

[

Z1

Z2

]

=[

I Q0 I

][

I−P

]

.

That is,Z = −P( I − QP)−1. It is a standard result by now that bothQ andZ exist withQ≥ 0 andZ ≤ 0 iff both Q andP are nonnegative definite andλmax(QP) < 1 ((22)). The latter conditionis known as the coupling condition. Summarizing, the SSP2 with boundγ has a solution iffQandP satisfying (4.28) and (4.30), respectively, exist, andλmax(QP) < 1.

The decoupling that is possible on a state space level is not only pretty, it also has some practicaluse. For example, for certain control problems the two decoupled Riccati equations (withP andQ) turn out to be in a way independent of the boundγ (McFarlane and Glover (56)). To ourknowledge up to now there has not been a hint in the direction of a frequency domain analog ofthe coupling condition. This is a disadvantage of a pure frequency domain approach. The SSP1and SSP2 algorithms suffer also from this “lack of decoupling”. At the end of this section wegive a connection between the SSP1 (SSP2) solution method and its its dual version in terms ofa transfer matrix.

The optimal standardH∞ problems that we can handle are, in terms ofQ andP, exactly thoseproblems where at optimality the solutionQ ≥ 0 to the HFARE still exists and the HamiltonianH associated with the HCARE (P) still has no eigenvalues onC0. This fact illustrates that ourresults on optimal solutions do not coincide with that of itsdual version.

4.5.2. Some other remarks

Corollary 4.5.1 (McMillan degrees). If G =[

D1 D2]−1 [

N1 N2]

is a left coprime PMFD ofG, then every compensator K generated by the SSP1 Algorithm,STEP (E) with a constant Usatisfies

δM K ≤ δM G.

4The J-lossless property that we must have is that[

0 II 0

][

D2 −N2]

Γ−1[

0 II 0

]

is J-lossless. This block swapping is thereason thatZ ≤ 0 instead of the more commonZ ≥ 0.

73


Proof . As in AppendixB we defineρi(A) (γi(A)) as theith row (column) degree of its argumentA. Further we use the identity

δA = minunimodularW

∑

i

ρi (WA)

for full row rank A (see AppendixB). Recall thatQ−1[

−N1 D1]

constructed in the SSP1Algorithm is proper, and thatC := lims→∞ Q−1(s)

[

−N1(s) D1(s)]

has full row rank. Thisimplies that for every nonsingular polynomial matrixW and every integerk the following identityis satisfied.

lims→∞

0s−ρk(W Q)

0

W(s)Q(s)C = lims→∞

0s−ρk(W Q)

0

W(s)[

−N1 D1]

.

Hence for every nonsingular polynomial matrixW, ρk(W Q) = ρk(W[

−N1 D1]

). This, andits transposed version, we use to prove the result:

δM G = δ[

−N1 D1 D2 −N2]

(DefinitionB.2.4)

= minunimodularW

∑

ρi (W[

−N1 D1 D2 −N2]

)

= minunimodularW

∑

ρi (W[

Q D2 −N2]

)

= δ[

Q D2 −N2]

≥ δM Q−1[

D2 −N2]

= δM∆Λ−1 = δ

[

∆

Λ

]

= δ

[

Γ

Λ

]

≥ δM ΛΓ−1 ≥ δMΛΓ

−1

[

IU

]

.

Note that this result does not depend on whetherG is proper or not.

Remark 4.5.2 (McMillan degrees). As an immediate consequence of Corollary4.5.1we havethat the McMillan degreeδM K of the most obvious suboptimal solutionsK to the mixed sensi-tivity problem with plantP and filtersW1, W2 andV satisfies

δM K ≤ δM P+ δM W1 + δM W2 + δM V.

This follows directly from the construction of the associated generalized plantG. Actually, if Vis proper and has a PMFD of the formV = D−1M, whereD is the denominator polynomial of aleft coprime PMFDP = D−1N of the plant, then for the most obvious choices of admissibleKwe have

δM K ≤ δM P+ δM W1 + δM W2.

This follows directly from the fact that in this case (in the obvious notation)

δ[

−N1 D1 D2 −N2]

= δ

M 0 0 D N0 B1 0 A1 00 0 B2 0 −A2

≤ δ (M N D)+ δ (B1 A1)+ δ (B2 A2)

= δM (P)+ δM (W1)+ δM (W2).

74

4.5. Remarks

The present SSP1 algorithm is based on a left coprime fraction of the generalized plantG. Asimilar algorithm may be derived starting with aright coprime PMFD ofG. A simple proof usesthe fact thatK is admissible for the standard system with plantG iff KT is admissible for thestandard system with plantGT.

Algoritm 4.5.3 (Dual SSP1 Algorithm). [Given: A right coprime PMFD of the generalized plantG =

[ N1

N2

][ D1

D2

]−1∈ R (p+r )×(q+t). Assumptions:[ −N1

D1

]

∈ P (p+q)×(t+q) and[ D2

−N2

]

∈ P (t+r )×(t+q)

have full column rank and full row rank onC0, respectively. Out: All internally stabilizingK ∈ R t×r that make the closed-loop transfer matrixH satisfy‖H‖∞ ≤ γ, provided there existinternally stabilizingK that achieve‖H‖∞ < γ.]


STEP (B) Compute, if possible, a polynomialJt,q-spectral factorQ such that

Q∼[

I t 00 − Iq

]

Q =[

−N∼1 D∼

1

][

I p 00 −γ2 Iq

][

−N1

D1

]

,

with[ −N1

D1

]

Q−1 proper. If this solutionQ exists and if[ Q1

D1

]

is strictly Hurwitz, withQ1 the

top t rows of Q, then proceed to STEP (C). Otherwise, no admissible compensator exists;γ needs to be increased and STEP (B) repeated.

STEP (C) Find left coprime polynomial matrices∆ andΛ such that

Λ−1∆ =[

D2

−N2

]

Q−1.

STEP (D) Compute, if possible, aJt,r-spectral cofactorΓ such that

Γ Jt,rΓ∼ = ∆Jt,q∆

∼,

with ∆Γ−1 proper. If this solution exists and if[

∆1 Γ2]

is strictly Hurwitz, withΓ2 theright r columns ofΓ and∆1 the leftt columns of∆, then proceed to STEP (E). Otherwise,no admissible compensator exists;γ needs to be increased and STEPS(B-D) repeated.

STEP (E) There exist internally stabilizing compensators such that ‖H‖∞ < γ. All compen-satorsK that internally stabilize and make‖H‖∞ ≤ γ are generated by

K = X−1Y;[

X Y]

=[

I t U]

Γ−1Λ; U ∈ R Ht×r∞ ; ‖U‖∞ ≤ 1.

We refer to the above algorithm as thedual solution and to the SSP1 algorithm given in theSection4.2as theprimal solution. There must be a connection between the primal solution andthe dual solution since they both give a method to generate all admissible compensators. To makethings more compact we define two rational matricesΠ andΠ: Let

G =[

D1 D2]−1 [

N1 N2]

=[

N1

N2

][

D1

D2

]−1

∈ R (p+r )×(q+t)

be a polynomial left and right coprime fraction of the plantG, and define

Π =[

D∼2

−N∼2

]

(N1N∼1 − γ2D1D∼

1 )−1 [

D2 −N2]

,

75


Π =[

D2

−N2

]

(N∼1 N1 − γ2D∼

1 D1)−1 [

D∼2 −N∼

2

]

.

In terms of the data produced by the SSP1 algorithm and its dual version, we have that

Π = Λ−∼Γ∼ Jr,tΓΛ−1; Π = Λ−1Γ Jt,rΓ

∼Λ−∼.

The connection between the primal and dual solution is easy to formulate in terms ofΠ andΠ:

Π−1 = −[

0 Ir

− I t 0

]

Π

[

0 − I t

Ir 0

]

. (4.31)

The validity of this equation is shown in AppendixC. It is not clear whether this connection canbe exploited.

From STEP (B) of the SSP1 algorithm and its dual version it follows that the SSP1 with boundγ has a solution only if

[

−N1 D1][

Iq

−γ2I p

][

−N∼1

D∼1

]

, (4.32)

is nonsingular and hasr positive andp negative eigenvalues everywhere onC0, and

[

−N∼1 D∼

1

][

I p 00 −γ2 Iq

][

−N1

D1

]

, (4.33)

is nonsingular and hast positive andq negative eigenvalues everywhere onC0, respectively.Therefore the infimumγ1 over allγ > 0 for which (4.32) hasr positive andp negative eigenvaluesis a lower bound of the optimal value ofγ = γopt. Similarly the infimumγ2 over allγ > 0 forwhich (4.33) hast positive andq negative eigenvalues is also a lower bound ofγopt. The twolower boundsγ1 andγ2 may be seen as the Parrott lower bounds for the SSP1.

76

5

L2−-Systems and some further results onstrict positivity

In this chapter we take a closer look at the idea introduced inChapter3 to work with signals inthe closed loop that do no depend on the compensator. To refresh memory, consider the followingexample.

r (t) = et e(t) = et

y(t) = 0

z(t) = et

u(t) = 0

A = 1

B(s) = 1−s1+s Q

Figure 5.1.: a simple two-blockH∞ problem.

Example 5.0.4 (A simple two-block H∞ problem). Consider the system in Fig. 5.1, with trans-fer matricesA = 1 andB(s) = 1−s

1+s representing given convolution systems. If we take as inputr (t) = et for t < 0 andr (t) = 0 for t ≥ 0, theny(t) is zero for negative timet, and, hence, so isu. In this case, therefore,e(t) = z(t) = r (t) = et for t < 0. We see that this inputr gives rise toan output that for negative time is independent ofQ. In particular, this shows that for any causalhomogeneous mapQ

supr∈L2(R;R)

‖e‖2

‖r‖2≥ 1.

Obviously withQ = 0 we get equality in this inequality. Hence,Q = 0 minimizes the∞-normof the closed-loop transfer matrix, over all stableQ.

The arguments used in the example are very simple and elegant. The idea to look at“things” that can not be affected by the compensator, is not new, however (see Kimura, Luand Kawatani (43)). The difference with (43) lies in the implementation of the idea. Kimura et.al. apply the idea to transfer matrix descriptions of the system. Translated in terms of signalsfor Example5.0.4, their approach amounts to finding necessary conditions based on signalsr inL2(R;C) that do not activate the outputy overall time. However, in the example such signals

77

5. L2−-Systems and some further results on strict positivity

do not exist—except for the zero signal of course. The trick is to see that it suffices to considersignals that do not activate the output up to time zero.

The attempt described in (43) is a natural consequence of a common practice to identify I/Osystems right away with convolution systems and their transfer matrices, and to do the analysissolely in terms of transfer matrices.

In this chapter we develop a substitute for the convolution system, which we call theL2−-system.L2−-systems are systems whose signals by assumption are restricted to

L2−(R;F) := w : R 7→ F |∫ T

−∞w∗(t)w(t) dt<∞ for everyT ∈ R ,

in whichF = Rq or F = Cq. We think that the theory ofL2−-systems provide a more convenientbasis for studying a large class ofH∞ control problems than the theory of convolution systems:

Example 5.0.5 (Example 5.0.4, continued). Consider the system depicted in Fig. 5.1. In viewof the observation made in Example5.0.4, it is desirable to have a description of the subset ofsignals

[re

]

∈ L2(R−;R2) in the system for which the outputy is zero for negative time. Thissubset is

B := [

re

]

∈ L2(R−;R2) | e(t) = r (t) = cet for somec ∈ R .

A state space description of this set is:

B = [

re

]

∈ L2(R−;R2) | x = x; e= x; r = x .

There is no “input” in the above state space description. Nevertheless it is clear that the statespace description is “minimal” in some sense. The finite dimensional spaceB can not be seenas the behavior of a convolution system but can be seen as the behavior of anL2−-system. It isshown later that the state space description is “minimal” inthe sense ofL2−-systems.

The results onL2−-systems constitute a theory underlying some of the resultspresented inChapter3.

BecauseL2−-systems in general are not convolution systems, some problems have to be re-examined. For example, what does “minimality” mean forL2−-systems. The bulk of this chapteris devoted to time domain descriptions ofL2−-systems. In Section5.1 we define three repre-sentations of the systems we work with. In Section5.2 we consider elementary properties ofL2−-systems in the three representations. Problems concerning minimality and uniqueness ofrepresentations are discussed. The last subsection in Section 5.2 is about a frequency domainanalog ofL2−-systems. These are the systems we encounter in Chapter3, though we did notcall them like that in Chapter3. In Section5.3 we make use of the elementary properties ofL2−-systems introduced a little earlier, to characterize strict positivity of subspaces representedby state space descriptions. This is a direct analog of the SPTheorem proved in Chapter3. Theresults involve Riccati equations. The proofs of these results also serve as a proof for Corol-lary 3.4.1, Item 2. Some examples are given in Section5.4to illustrate the potential applications.We consider a Nevanlinna-Pick interpolation problem and anH∞ filtering problem.

Many of the results in this chapter are linked to the results on the so-calledL2-systems—notethat there is no minus sign here—as examined in Weiland (83).

In this chapter a constant square matrixA is calledstable(antistable) if its eigenvalues lie inC− (C+). Recall that for rational matrices stable means somethingelse. It is not very likely thatthe difference between the two definitions of “stable” will give rise to confusion.

78

5.1. Three representations of systems


This section contains a brief overview of elementary results on systems in various representations.The material is drawn from Weiland (83) and Willems (86). The AR representation as introducedin Chapter2 is considered again here for completeness.

One of the differences between convolution systems andL2−-systems is the favorite type ofrepresentation. The most natural time domain description of convolution systems are the so-calledinput/state/output representations (I/S/O representations):

x = Ax+ Buy = Cx+ Du

with u considered as the input andy considered as the output. When consideringL2−-systems itis more convenient to work with one of the following three representations.

Definition 5.1.1 (Representations of systems, ( 86; 83)). SupposeΣ= (R, Rq, B ) is a systemwith external signalw.

1. If B = w | w ∈ L loc2 (R;Rq), R(d/dt)w(t) = 0 for some polynomial matrixR∈ P g×q,

then R(d/dt)w(t) = 0 is anauto regressive representation(or, AR representation) of Σ,andR is said to define an AR representation ofΣ.

2. Let A, B, C and D be given constant real matrices. IfB is the set of signalsw inL loc

2 (R;Rq) for which there exist signalsx such that

x = Ax+ Bwz = Cx+ Dw, z≡ 0,

(5.1)

then the equations (5.1) form anoutput nulling representation(ONR) ofΣ, and the quadru-ple A, B,C, D is said to define an ONR ofΣ.

3. Let A, B, C and D be given constant real matrices. IfB is the set of signalsw inL loc

2 (R;Rq) for which there exist signalsx and signalsv ∈ L loc

2 (R;R•) such that

x = Ax+ Bvw = Cx+ Dv,

(5.2)

then the equations (5.2) form a driving variable representation(DVR) of Σ, and thequadruple A, B, C, D is said to define a DVR ofΣ. The signalv in (5.2) is called adriving variableandx in (5.1) and (5.2) is referred to as thestate.

Once in a while we allow complex valued signals.The three types of representations defined in Definition5.1.1are equivalent in the sense that

a system can either be described by all three types of equations or by none of them at all (seeWillems (87; 88), Weiland (83) and Lemma5.1.5). The I/S/O representation may readily betranslated into a DVR and an ONR:

Example 5.1.2 (I/S/O, DVR and ONR). Consider the system described by the I/S/O representa-tion

x = Ax+ Buy = Cx+ Du.

79


This system with external signalw =[ u

y]

has DVR

x = Ax+ Bv

w =[

0C

]

x+[

ID

]

v,

and ONR

x = Ax+[

B 0]

w

0 = Cx+[

D − I]

w.

A state transformation (x := Sx, Snonsingular) in DVR (5.2) does not affect the set of solutionsw. Also regular state feedback1 in DVR (5.2) has no affect on the solutionsw, since this is justa redefinition of the driving variable. Under suitable minimality conditions all “minimal” DVRsare generated this way. Similar arguments are worked out in Lemma5.1.4for ONRs. Minimalitywe define first.

Definition 5.1.3 (Minimality). AR representationsRw = 0 of a systemΣ areminimal whenboth the number of rows ofRand the degreeδRof R is minimal amongst all AR representationsof Σ. ONR (DVR) quadruplesA, B,C, D of a systemΣ areminimalwhen both theA andDmatrix have smallest dimension amongst all ONRs (DVRs) ofΣ.

In Chapter2 we stated that AR representationsRw = 0 are minimal iff R has full row rank.For ONRs and DVRs the following may be proved.

Lemma 5.1.4 (Minimality and uniqueness of representations , (83; 88; 86)). Sup- poseΣ isa system that admits an AR representation.

1. There exist quadruplesA, B,C, D defining an ONR ofΣ such that D has full row rankand(C, A) is observable2. Such representations are minimal ONRs. Moreover, two mini-mal ONR quadruplesA, B,C, D and A, B, C, D define an ONR of the same system ifand only if for some H∈ R•×• and nonsingular T∈ R•×• and S∈ R•×•

[

A BC D

]

=[

S−1 H0 T

][

A BC D

][

S 00 I

]

.

(That is, minimal ONRs are unique up to state transformationand regular output injectiontransformation3.)

2. There exist quadruples A, B, C, D defining a DVR ofΣ that are strongly observable (thatis, (C+ DF, A+ BF) is observable for all F) and such thatD has full column rank. Suchrepresentations are minimal DVRs. Moreover, two minimal DVR quadruples A, B, C, D

1 For systems of the form (5.2) the relationv = Fx+ Tv for some real matricesF andT is aregular (static) state feed-backif T is nonsingular. Ifv is eliminated in (5.2) using this feedback, we obtain another DVR with driving variablev, defined by the quadruple A+ BF, BT, C+ DF, DT. It is for this reason the transformation from A, B, C, Dto A+ BF, BT, C+ DF, DT is called aregular state feedback transformationwheneverT is nonsingular.

2A pair (C, A) is observable if[

A−sIC

]

has full column rank for alls∈ C.3A transformation from A, B,C, D to A, B, C, D is called a regular output injection transformationif

A, B, C, D = A+ HC, B+ H D, TC, T D for some real matrixH and real nonsingular matrixT. See footnote1on state feedback.

80


and A, B, C, D define a DVR of the same system if and only if for some F∈ R•×• andnonsingular T∈ R

•×• and S∈ R•×•

[

A BC D

]

=[

S−1 00 I

][

A BC D

][

S 0F T

]

.

(That is, minimal DVRs are unique up to state transformationand regular state feedbacktransformation.)

If D ∈ Rp×q has full row rank, then Ker(D) := w ∈ Rq | Dw= 0 equals Im(D⊥) := D⊥Rq−p

for some “orthogonal complement”D⊥ of D. HereD⊥ is a full column matrix whose columnsspan the Kernel ofD. The expressionDw = 0 is a simple ONR andw = D⊥v is an equivalentDVR. The following lemma explores the relationship betweenminimal ONRs and DVRs of asystemΣ. Its proof is a matter of manipulation and is omitted.

Lemma 5.1.5 (Transformation of ONRs into DVRs and back). In what follows P−R denotes aright inverse of a full row rank matrix P (PP−R = I), P−L denotes a left inverse of a full columnrank P (P−L P = I) and D⊥ is a maximal full column rank matrix constructed from a givenDsuch that DD⊥ = 0 or D⊥ D = 0, depending on whether D has more columns than rows or viceversa.

1. LetA, B,C, D define a minimal ONR of a systemΣ. Then A, B, C, D given by

[

A BC D

]

=[

A B0 I

][

I 0−D−RC D⊥

]

(5.3)

defines a minimal DVR ofΣ. Furthermore,λ ∈ C is a zero of[

A−λ I BC D

]

if and only if it isa zero of

[

A− λ I B]

.

2. Let A, B, C, D define a minimal DVR of a systemΣ. ThenA, B,C, D given by

[

A BC D

]

=[

I −BD−L

0 −D⊥

][

A 0C − I

]

(5.4)

defines a minimal ONR ofΣ. Furthermore,λ ∈ C is a zero of[

A−λ I BC D

]

if and only if it isa zero of

[

A− λ I B]

.

The McMillan degree of an ONR or a DVR may be defined as the dimension of the statespace, that is, as the dimension of the “A matrix”. The minimal ONRs of a given system have byLemma5.1.4the same state space dimensions, and by the above lemma this equals the state spacedimension of any minimal DVR of the system. In other words, the McMillan degree defined asthe dimension of the “A matrix” of either a minimal ONR or a minimal DVR, is a quantityofthe system independent of the choice of representation. Furthermore, it may be shown that thisquantity equalsδR, if Rw = 0 defines an AR representation of the system (see (86; 57) andAppendixB, CorollaryB.2.4).

Example 5.1.6 (Uniqueness of ONRs). Let Σ be a system with AR representation(d/dt −1)w(t) = 0.

81


1. Σ has ONR

[

x0

]

=[

1 0−1 1

][

xw

]

.

Multiplying both sides from the left by[

1 H0 T

]

∈ R2×2 with T nonzero—that is, applying a

regular output injection transformation—reveals anotherand equivalent ONR:

[

1 H0 T

][

x0

]

=[

1 H0 T

][

1 0−1 1

][

xw

]

⇐⇒[

x0

]

=[

1− H H−T T

][

xw

]

.

2. A minimal DVR ofΣ is

[

xw

]

=[

11

]

x.

Note that there is no driving variable here.

Example 5.1.7 (Minimality does not imply controllability) . Controllability plays no role in theminimality of an ONR or DVR. Consider for example the system whose signals are of the form

w(t) = CeAtx0; x0 ∈ Rn.

This system has ONR quadrupleA,0,−C, I and DVR quadrupleA, ,C, (the driving vari-able has null dimension). If(C, A) is observable, both the ONR and the DVR are minimal.

Note that it isnot true in general that a DVR A, B, C, D can be recovered from its associatedtransfer matrixC(sI − A)−1 B + D. In the example given here theB and D matrices have nulldimension, as does the associated transfer matrix. For ONRsthe situation is less dramatic. ForexampleA− HC, H,−C, I also defines an ONR of the system. If(C, A) is observable andHis chosen such that(A, H) is controllable then an ONR of the systemcanbe recovered from thetransfer matrix−C(sI − (A− HC))−1H + I .

5.2. L2−-systems

In this section we review properties ofL2−-systems in terms of the representations introducedearlier. TheL2−-systems have an equivalent frequency domain counterpart,which is examined insubsection5.2.3. It will be no surprise that the frequency domain counterpart of L2−-systems areexactly the systems with behaviors of the formBG, the spaces that we use throughout Chapter3.

Formally, anL2−-system is a tripleΣ = (R,Rq,B ∩ L2−(R;Rq)). Since we consider timeinvariant systems only, we may restrict our attention to behaviors of the formB ∩ L2(R−;Rq).An obvious advantage ofL2(R−;R

q) over L2−(R;Rq) is thatL2(R−;R

q) is a Hilbert space.Sometimes we allow complex behaviors of the formB ∩ L2(R−;Cq).

Definition 5.2.1 ( L2−-systems). An L2−-systemis a system of the formΣ = (R−,F,B ∩L2(R−;F)) with eitherF = Rq or F = Cq for some integerq.

82

5.2. L2−-systems

5.2.1. AR representations of L2−-systems

In this subsection we considerL2−-systems defined through an AR description. The behaviorswe look at are of the form

BR := w ∈ L2(R−;Rq) | R(d/dt)w(t)= 0 . (5.5)

We say thatR∈ P generatesB if B = BR.If R(s) = s+ 1 thenR(d/dt)w(t) = 0 iff w(t) = ce−t. Such a signalw is not bounded in

L2(R−;R). This shows thatR(s) = s+ 1 and, for instance,R = 1 generate the same spaceBR = BR = 0. In general we have:

Lemma 5.2.2 (Uniqueness of generators of BR). Suppose R andR are two full row rank poly-nomial matrices inP g×q. Then R andR generate the same space (5.5) if and only if R= AR forsome nonsingular rational matrix A∈ R g×g that has all its zeros and poles in the closed left-halfcomplex plane.

Proof . (Only if) SupposeR, R∈ P g×q are two full row rank polynomial matrices that generatethe same space. LetP ∈ P q×(q−g) be a polynomial matrix of full column rank such thatRP= 0.Thenw(t) = P(d/dt)l (t) satisfies the differential equationRw = 0 for all time signalsl ∈ C ∞.HenceRP= 0 as well, which implies thatR = AR for some rational matrixA. Completelysimilar it may be shown thatR = AR. So R = A(AR) and, therefore,A = A−1 is squarenonsingular.

Write A as a right coprime polynomial matrix fractionA = N D−1. It follows from[

RR

]

=[

ND

]

D−1 R, (5.6)

that R′ := D−1R is a polynomial matrix. Now we have

R= N R′, R= DR′.

We need to prove thatA = N D−1 has no poles and zeros inC+. Suppose, to obtain a contradic-tion, thatN(s) is singular fors= ζ ∈ C+. Let v be a constant vector such thatN(ζ)v = 0. If wedefineu(t) = eζtv then

N(d/dt)u(t)= 0, D(d/dt)u(t) = eζt D(ζ)v 6= 0.

Later we will show thatu = R′w has a solutionw in L2(R;Rq) if ζ ∈ C+, in which case we havethat

Rw = 0, Rw 6= 0, w ∈ L2(R−;Rq).

This is a contradiction, and, hence,N has zeros only inC− ∪ C0. By changing the role ofR andR it may be shown in the same way that alsoD has all its zeros inC− ∪ C0.

Now we show thatu(t) = eζtv = R′(d/dt)w(t) has a solutionw ∈ L2(R−;Rq) if ζ ∈ C+.It is trivial if R′ is scalar: writeR′(s) as (s− ζ)k p(s) with p polynomial andp(ζ) 6= 0, thenw = 1

p(ζ)v/(s− ζ)k+1 ∈ H ⊥q2 is the Laplace transform of a solutionw ∈ L2(R−;Rq). The

matrix case follows by writingR= U(∆ 0)W in Smith form (U andW are unimodular and∆ isdiagonal). Definew = W−1

[ρ0

]

with ρ defined by∆ρ = U−1u. This is a set of scalar equationsin the components ofρ. Each component ofρ is an exponential with exponentζ ∈ C+, and soalso isw = W−1

[ρ0

]

. I.e. w is in L2(R−;R•). It is easily checked that thisw is the signal weneed.

83


(If) Now supposeR and R have full row rank and that all zeros ofN and D lie in C− ∪ C0.We will show thatRw = 0 for somew ∈ L2(R−;R

q) iff R′w = 0 and similarly thatRw = 0 forsomew ∈ L2(R−;Rq) iff R′w = 0. This then completes the proof.

If Nu = 0 for some nonzerou, then some zeros ofN must be poles ofu, i.e., u has poles inC− ∪C0. Solutionsw to u = R′wmust have poles whereu has poles, i.e.,w has poles inC0 ∪C−,and, hence,w 6∈ H ⊥q

2 , i.e.,w 6∈ L2(R−;Rq). As a resultRw= N R′w = 0 withw ∈ L2(R−;Rq)

is equivalent toR′w = 0 with w ∈ L2(R−;Rq). Similarly Rw = 0 with w ∈ L2(R−;Rq) isequivalent toR′w = 0. This completes the proof.

It is an elementary fact that every full row rank polynomial matrix R may be written as aproductR = F R with F a square polynomial matrix that has all its zeros inC− ∪ C0, and R awide polynomial matrix that has full row rank everywhere inC− ∪ C0 (see AppendixB). ByLemma5.2.2, in this caseBR = BR. It will be argued next that thisR is minimal in some sense.

Definition 5.2.3 ( L2−-minimality). We say that an AR (ONR) (DVR) representation of anL2−-systemΣ is L2−-minimalwhen amongst the set of AR (ONR) (DVR) representations ofΣ it isminimal in the usual sense.

Lemma 5.2.4 ( L2−-minimal generators of BR). LetΣ be anL2−-system. The matrix R∈ Pdefines anL2−-minimal AR representation ofΣ if and only if R has full row rank inC− ∪ C0.Moreover, anL2−-minimal R∈ P defining an AR representation ofΣ is unique up to multipli-cation from the left by a unimodular matrix inP .

Proof . The proof consists of three parts. Firstly we show thatL2−-minimal AR representationsexist. Secondly we show they are unique up to multiplicationfrom the left by unimodularU, andthirdly we showR defines anL2−-minimal representation iffR has full row rank inC− ∪ C0.

Obviously we have that possibleL2−-minimal generatorsR are full row rank polynomial ma-trices. From Lemma5.2.2we know that the set of full row rank generators ofΣ all have the samenumber of rows. Therefore ifR has full row rank but is notL2−-minimal, then, by definition ofL2−-minimality, a full row rank polynomialR′ exist such thatBR = BR′ with δR′ < δR. If this R′

is notL2−-minimal, the process may be repeated withR′ in place ofR. Obviously this processstops becauseδR′ ∈ Z+, and, hence,L2−-minimal AR representations exist.

SupposeRandRdefine twoL2−-minimal AR representations ofΣ. By Lemma5.2.2we thenhave for certain polynomialR′ and Hurwitz4 N andD that

[

RR

]

=[

ND

]

R′. (5.7)

By Lemma5.2.2we haveBR = BR′ = BR. From the above expression we see thatδR≥ δR′ ≤ δ R,with equality holding iffN andD are unimodular. By assumptionR and R areL2−-minimal soequality must hold, i.e.,N and D are unimodular. ConsequentlyR = U R for the unimodularU := N D−1.

If R loses rank inC− ∪ C0, thenR= FR′ for some HurwitzF and polynomialR′. By Lemma5.2.2we then haveBR = BR′ . This shows that wheneverR has a zero inC− ∪ C0 R can notbeL2−-minimal. Suppose now thatR has full row rank inC− ∪ C0, that R is L2−-minimal andthatBR = BR. The proof is complete if we can show thatR is L2−-minimal. We have that (5.7)holds for some HurwitzN andD and polynomialR′. BecauseR has full row rank inC− ∪ C0,we must have that the HurwitzN is unimodular. Finally it follows fromδR = δN + δR′ =δR′ ≤ δD + δR′ = δ R that δR≤ δ R. Because by assumptionR is L2−-minimal alsoR is L2−-minimal.

4A square polynomial matrixN is Hurwitz if det N has zeros only inC− ∪ C0.

84

5.2. L2−-systems

5.2.2. ONRs and DVRs of L2−-systems

For AR representations,L2−-minimality has much to do with some matrix having no zeros intheclosed left-half complex plane. Intuitively, the reason isthat with a zero inC+ an exponentialtime function inL2(R−;Rq) may be associated, and, therefore, such zeros can not be removedwithout affecting theL2−-behavior. On the other hand, with a zero inC− ∪ C0 no time functionin L2(R−;Rq) can be associated, and, therefore, removing such zeros doesnot affect theL2−-behavior. For ONRs and DVRs the situation is alike.

Definition 5.2.5 (Zeros of ONR and DVR quadruples). The zeros of a quadrupleA, B,C, Ddefining an ONR or a DVR of a system, is the set of valuess∈ C for which

[

A− sI BC D

]

drops below normal rank.

Lemma 5.2.6 (States in ONRs and DVRs of an L2−-system). Letw be the external signal of asystemΣ.

1. w ∈ L2(R−;Rq) implies that x∈ L2(R−;Rn) and lim t→−∞ x(t) = 0, in which x is thestate variable in any minimal ONR ofΣ.

2. w ∈ L2(R−;Rq) implies thatv ∈ L2(R−;Rp), x∈ L2(R−;Rn) and thatlim t→−∞ x(t)= 0,in which x is the state variable andv is the driving variable in any minimal DVR ofΣ.

Proof . 1. Consider any minimal ONR quadrupleA, B,C, D. Introduce an antistabiliz-ing output injectionH. Sincex = (A + HC)x + (B + H D)w in which A + HC is an-tistable andw ∈ L2(R−;Rq), we havex ∈ L2(R−;Rn). Also, x = Ax + Bw impliesx ∈ L2(R−;Rn). Thus limt→−∞ x(t) = 0.

2. Consider any minimal DVR quadruple A, B, C, D and let E be a nonsingular matrixsuch thatED = ( I

0). SoEw= ECx+ ( I0)v. Apply regular state feedbackv = −E1Cx+ v,

whereE1 is the upper row block ofE. Now we have (withE2 the lower row block ofE)that

xE1w

E2w

=

A− BE1C B0 I

E2C 0

[

xv

]

.

This shows thatv = E1w, hence, v is in L2(R−;Rp). By Strong observabil-ity of A, B, C, D, we have that(E2C, A − BE1C) is observable, so there ex-ists an H such that A − BE1C + H E2C is antistable. Rewrite the dynamics asx = ( A − BE1C + H E2C)x + Bv − H E2w and proceed as in Item (a). This showsthat x, x ∈ L2(R−;Rn), lim t→−∞ x(t) = 0, v ∈ L2(R−;Rp), and, hence, that also theoriginal driving variablev = −E1Cx+ v is in L2(R−;Rp).

As a result, a state in a DVR that can not be controlled by the driving variable is not al-lowed to blow up at minus infinity as this would implyw 6∈ L2(R−;Rq). Stated differently,w ∈ L2(R−;Rq) implies that states corresponding to uncontrollable stable modes in an observableDVR must be zero. Leaving out these modes does not affect the set of solutionsw ∈ L2(R−;R

q):

85


Lemma 5.2.7 ( L2−-minimal ONRs and DVRs). LetΣ be anL2−-system.

1. A, B,C, D defines anL2−-minimal ONR ofΣ if and only if it is minimal in the usualsense and in addition has no zeros inC− ∪ C0. Moreover,L2−-minimal ONRs are uniqueup to state transformation and regular output injection transformation.

2. A, B, C, D defines anL2−-minimal DVR ofΣ if and only if it is minimal in the usualsense and in addition has no uncontrollable modes inC− ∪ C0. Moreover,L2−-minimalDVRs are unique up to state transformation and regular statefeedback transformation.

Proof . The transformation of ONRs to DVRs and back as derived in Lemma 5.1.5transformsminimal ONRs into minimal DVRs and back. In this Lemma it is noted that zeros of the minimalONR quadruple appear as uncontrollable modes in the corresponding DVR. This shows thatItems 1 and 2 of Lemma5.2.7are equivalent. We prove Item 2, the DVR result.

Consider any minimal DVR A, B, C, D definingΣ. By Lemma5.2.6, Item 2 the statex andthe driving variablev are both inL2(R−;R

•) andx(−∞) = 0. The state corresponding to anyuncontrollable modeλ ∈ C− ∪ C0 of ( A, B) is therefore identically zero. Removing all suchmodes from the DVR leaves the desired antistabilizable DVR.That such DVRs areL2−-minimaland that all minimalL2−-minimal DVRs are related as given in Lemma5.2.7, follows fromsimilar arguments as used in Weiland (83, Theorems 3.10 and 3.12) for so-calledL2-systems.

The transformation from DVRs to ONRs and back as given in Lemma5.1.5, transformsL2−-minimal ONRs toL2−-minimal DVRs and vice versa. This follows trivially from the fact thatthis transformation from an ONR to a DVR make zeros of the ONR appear as uncontrollablemodes of the DVR.

Example 5.2.8 (An L2−-minimal DVR). Consider the DVR

[

x1

x2

]

=[

A11 00 A22

][

x1

x2

]

, w =[

C1 C2][

x1

x2

]

,

with A11 antistable andA22 stable. Suppose it is a minimal DVR (that is, suppose it is ob-servable). Then ifx2(0) is nonzero, the signalx2(t) blows up at minus infinity, and by ob-servability so willw. Restricting our attention tows in L2(R−;Rq) therefore impliesx2 ≡ 0.Hence, the considered DVR represents the sameL2−-system as defined by the DVR quadruple A, B, C, D := A11, , C1, :

x1 = A11x1, w = C1x1.

The “A matrix” A11 is antistable, so the pair( A11, B) is antistabilizable—note thatB has nulldimension—and, therefore, by Lemma5.2.7it is anL2−-minimal DVR.

5.2.3. L2−-systems in frequency domain

L2−-systems have a frequency domain analog. The behaviors thatwe examine in this section arethe spacesBG that we encounter in Chapter3. The step to the frequency domain is most easilymade through an ONR description of theL2−-system:

86

5.2. L2−-systems

Lemma 5.2.9 (Connection between ONRs and BG). SupposeA, B,C, D defines anL2−-minimal ONR. Then by minimality there exists an H such that A− HC is stable. Given such anH define an equivalent ONR:

x = Ax+ Bw0 = Cx+ Dw

,

[

A BC D

]

:=[

I −H0 I

][

A BC D

]

∈ R(n+g)×(n+q). (5.8)

Thenw ∈ L2(R−;Cq) satisfies the ONR equations for some state x if and only if the LaplacetransformL(w) is an element of

BG := w ∈ H ⊥q2 | Gw ∈ H g

2

in which G(s) := C(sI − A)−1 B+ D. Conversely, if G(s)= C(sI − A)−1 B+ D is a realizationof a given G∈ R H

g×q∞ with (C, A) observable and( A, B) controllable, thenw is an element of

BG if and only ifw = L(w) for somew ∈ L2(R−;Cq) that satisfies the ONR equations in (5.8)for some state x.

Proof . In this proof signals inL2(R−;Cq) are identified with their embedding inL2(R;Cq). (Ifpart) Assume thatw is in L2(R−;Cq) and definez := Cx+ Du. Then

z(t) = CeA(t−T)x(T)+∫ t

TCeA(t−τ) Bw(τ) dτ+ Dw(t).

As in Lemma5.2.6we have thatx ∈ L2(R−;Cn) and limt→−∞ x(t)= 0 if w ∈ L2(R−;Cq). Thisin combination with stability ofA shows that the limit ofT → −∞ exists and that thenz followsfromw as

z(t) =∫ t

−∞CeA(t−τ) Bw(τ) dτ+ Dw(t).

Note thatz(t) = 0 for t < 0 and that by stability ofA, z is in L2(R;Cg). It follows from thePaley-Wiener theorem thatL(z) = GL(w) is in H g

2 . In other words,L(w) is an element ofBG.(Only if part) The only if part follows directly from the Paley-Wiener theorem.

It is for this lemma that we call systems with behaviors of theform BG alsoL2−-systems. Wesay thatG ∈ R H

g×q∞ generatesB if B = BG. It is also possible to define a frequency domain

analog ofL2−-systems that have real-valued signals only. The difference with the complex-valuedcase is only technical and not very enlightening.

In line with the definition of minimality of AR representations and ONRs and DVRs, we saythat:

Definition 5.2.10 ( L2−-minimality). A matrix G ∈ R Hg×q∞ as a generator ofBG is L2−-minimal

if G has full row rank as a rational matrix and minimal McMillan degree amongst all generatorsG of BG = BG.

The following result is not very surprising. Item 1 of the following corollary5.2.11character-izes when a generatorG of B is L2−-minimal.

Corollary 5.2.11 (Uniqueness and L2−-minimality of generators G of BG).

1. Every spaceBG may be generated by a stable G′ that has full row rank onC0 ∪ ∞ andhas zeros only inC+. Such generators areL2−-minimal.

87


2. Suppose G andG are two stable real-rational matrices that both have full row rank onqqC0 ∪∞. Then G andG generate the same spaceBG = BG if and only if G= WG for someW ∈ GR H

g×g∞ .

Proof . 1. Similar to the proof of Lemma5.2.4. May also be proven using Lemma5.2.9.

2. The if part is trivial. (Only if) SupposeG ∈ R Hg×q∞ andG ∈ R H

m×q∞ generate the same

space, and suppose that they both have full row rank onC0 ∪ ∞. Let P ∈ R H(q−g)×q∞

be a matrix of full row rank such thatGP∼ = 0, and suppose, without loss of generality,that P∼ has full column rank onC0 ∪ ∞. Then obviouslyP∼H ⊥q

2 is a subset ofBG, and

therefore it is also a subset ofBG. HenceGP = 0 as well, which implies thatG = WGfor some rational matrixW ∈ R m×g. Completely similar it may be shown thatG = WG.HenceG = W(WG) and, therefore,m= g andW = W−1 is square and nonsingular. NotethatW is biproper and has no poles and zeros onC0 ∪ ∞, because bothG andG have fullrow rank onC0 ∪ ∞.

Write W as a right coprime fractionW = N D−1 overR H ∞. BecauseW is biproper, wehave thatD is biproper as well. There exists a stableL ∈ R H

g×2g∞ such thatL

[ND

]

= Ig,sinceN andD are right coprime. It follows from

[

GG

]

=[

ND

]

D−1G, (5.9)

that thenG′ := D−1G= L[

GG

]

is also stable. Now letV be a canonical cofactor ofG′G′∼ =VV∼, and with it define

[

ND

]

:=[

ND

]

V; G := V−1G′.

This way we have[

GG

]

=[

ND

]

G; GG∼ = I ; and W = ND−1.

Note thatN andD are right coprime becauseV is in GR Hg×g∞ .

Our aim is to show that bothN and D are inGR Hg×g∞ . Then alsoW = ND−1 is in

GR Hg×g∞ which is what we set out to prove. To obtain a contradiction, suppose thatN(s)

is singular fors= ζ ∈ C+ and supposev is a constant vector such thatN(ζ)v = 0. Defineu(s) = 1

s−ζv then

u ∈ H ⊥ g2 ; Nu∈ H g

2 ; Du 6∈ H g2

(The latter follows by coprimeness: ifDu were inH g2 , then by the stability ofL, so would

V−1L[

ND

]

u = Iu = u. This is not true.) Finally definew = G∼u, thenw is in BG because

Gw = NGG∼u = Nu ∈ H g2 , but w is not in BG, becauseGw = Du 6∈ H g

2 . This is acontradiction, henceN has no zeros inC+. Completely similar it follows that alsoD hasno zeros inC+. In other words,W = ND−1 has neither poles nor zeros inC+ and becauseW is biproper this implies thatW is in GR H

g×g∞ .

88

5.2. L2−-systems

If G has full row rank onC0 ∪ ∞ andG = GcoGci is a co-inner-outer factorization (see Ap-pendixA), thenGci is anL2−-minimal generator ofBG. Item 2 of Corollary5.2.11is particularlyinteresting. It suggests a connection with coprime factorization overR H ∞ of a given rationalmatrix (see AppendixB). In Vidyasagar (82) it is shown that a matrix

[

−N D]

coming from aleft coprime fractionD−1N = H overR H ∞ of a givenH ∈ R p×m is unique up to multiplicationfrom the left by an element ofGR H

p×p∞ . Coprime fractions of transfer matrices representing

convolution systems are sometimes used in association withthegraphof the convolution system(see, for example, (82).) More precisely, ifH is the transfer matrix of a convolution systemwith input u and outputy, and if D−1N = ND−1 = H is a left and right coprime fraction ofH over R H ∞, then the graph with respect toL2(R+;Rm+p) is the set of input/output pairs[ u

y]

∈ L2(R+;Rm+p) | y = Hu. This graph equals

[

DN

]

L2(R+;Rm) =

[

uy

]

∈ L2(R+;Rm+p) |

[

−N D][

uy

]

= 0.

(We are a bit sloppy here with the distinction between time and frequency domain.) These setsare similar to the ones that we consider with two main differences. SetsBG describe the set ofsignals that satisfy the laws of the system up to time zero andhave finite energy up to time zero,and the setsBG are not confined to input/output systems only:

Example 5.2.12 (A difference between graphs in L2(R+;Rm+p) and spaces BG). Supposewe have a scalar input/output system with with inputu, outputy and transfer matrixH(s)= s−3

s−4.A left and right coprime fraction ofH overR H ∞ is, for example,

D−1N = N D−1 = H;[

−N(s) D(s)]

=[

− s−3s+5

s−4s+5

]

.

The graph with respect toL2(R+;R2) of the associated convolution system equals[

DN

]

L2(R+;R) = [

uy

]

|[

−N D][

uy

]

= 0,

[

uy

]

∈ L2(R+;R2).

The set of signals that satisfy the laws of the system up to time zero and have finite energy up totime zero is (in frequency domain)

B[

−N D] =

[

uy

]

∈ H ⊥22 |

[

−N D][

uy

]

∈ H2 .

Roughly speaking, this set allows us to focus the attention on the unstable dynamics, the unstablepoles and zeros of the system. The unstablepolesof the system appear inB[

−N D] through the

subset “u ≡ 0”:

Bpoles := [

uy

]

∈ B[

−N D] | u ≡ 0 =

0

C

s−4

,

and the unstablezerosappear inB[

−N D] through the subset “y ≡ 0”:

Bzeros:= [

uy

]

∈ B[

−N D] | y ≡ 0 =

C

s−3

0

.

89


Both Bpoles andBzeros are sets that are not of the input/output system type; it is clear that nononzero input signalu exists in the graph of the convolution system with respect toL2(R+;R

2)

that produces an outputy that is zero over all time, and for graphs of convolution systemsu ≡ 0automatically implies alsoy ≡ 0.

The results derived for theL2−-systems may be easily translated toL2+-systems andL2-systems. ByL2+-systems we mean systems whose signals have finite energy from any finiteT to plus∞, and byL2-systems we refer to systems that have finite energy over all time. TheL2+ case follows trivially from theL2− case by changing the direction of time, and theL2 casefollows trivially from the identityL2(R;Cq) = L2−(R;Cq) ∩ L2+(R−;Cq). For example, ifΣ = (R,Cq,B ∩ L2(R;C

q)) is anL2-system, then it is immediate that:

• An AR representationRw = 0 of Σ is L2-minimal iff R(s) has full row rank for everys∈ C.

• An ONR quadrupleA, B,C, D of Σ is L2-minimal iff it has no zeros,(C, A) is observ-able andD has full row rank.

• A DVR quadruple A, B, C, D of Σ is L2-minimal iff it is controllable, strongly observ-able andD has full column rank (or equivalently, iff it has no zeros, itis controllable andD has full column rank).

Moreover, if A and A do not have imaginary eigenvalues, thenw is an element of the behavioriff its Laplace transformw is an element of

PL2(C0;Cq) = w ∈ L2(C0;C

q) | Gw = 0

in which G(s) := C(sI − A)−1B + D and P(s) := C(sI − A)−1C + D. L2-systems are exam-ined extensively in Weiland (83) and their behaviors are linked with the graph with respect toL2(R;R

q) of an associated convolution system.

5.3. Strict positivity or strict passivity

An important problem in system theory is whether or not a given convolution system with inputu and outputy satisfies

∫ T

−∞‖u(t)‖2 − ‖y(t)‖2 dt ≥ 0 (5.10)

for all inputsu and timeT. It is well known that this is the case iff the system is stableand theassociated transfer matrixH satisfies‖H‖∞ ≤ 1. It also well known how this may checked givenan I/S/O representation of the system, for example via the notion of dissipativeness (Willems (85))or via the Bounded Real Lemma (Anderson and Vongpanitlerd (2)).

In Ball and Helton (7) convolution systems that satisfy (5.10) are calledpassive, a notionborrowed from electrical network theory. Roughly speaking, in electrical network theory a systemis calledpassiveif it absorbs more energy than that it supplies. The left-hand side of (5.10)represents the net flow of energy into the system if the systemis given by a so-called scatteringdescription, and, hence, in this case (5.10) means that the system is passive (see e.g. (2)). Themore conventional passivity inequality (see, for example,(85))

∫ T

−∞w(t)∗z(t) dt ≥ 0

90


is also of the form (5.10) by substitutingw = 12(u + y) andz= 1

2(u− y).In this section we extend the Bounded Real Lemma for convolution systems toL2−-systems.

More precisely, we give necessary and sufficient conditionsin terms of state space data underwhich anL2−-system with partitioned signal

[ uy]

satisfies (5.10) for all time T and[ u

y]

in thebehavior of the system. As stated earlier, withL2−-systems we are not confined to studyinginput/output systems only. For example, theL2−-system whose signal

[ uy]

satisfies[

d/dt− 1 0−1 2

][

u(t)y(t)

]

= 0, (5.11)

is not of the input/output system type. Nevertheless the problem whether or not the passivityinequality (5.10) holds is still well defined (and in fact it holds for this example, becausey = u/2).

It is tempting to callL2−-systemspassiveif they satisfy the passivity inequality (5.10), butthere are some problems with the interpretation of passivity for L2−-systems. For example, it isnot clear how to interpret passivity of theL2−-system given by the finite dimensional behaviordefined by (5.11). We therefore prefer not to use the notion passivity of systems as far asL2−-systems are concerned. It is good, however, to be aware of a connection between passivity andthe results derived in this section.

Because we consider only time invariant systems, the passivity inequality (5.10) holds iff∫ 0

−∞‖u(t)‖2 − ‖y(t)‖2 dt ≥ 0

holds for all[ u

y]

in the behavior of the system. This we may recognize as the problem ofpositivityof the behavior of the system with respect to some indefinite inner product onL2(R−;Rq+p).

For the most part we deal in this section with the problem ofstrict positivity (SP), that is inthe present context, whether there exists anε > 0 such that

∫ 0

−∞‖u(t)‖2 − ‖y(t)‖2 dt ≥ ε(

∫ 0

−∞‖u(t)‖2 + ‖y(t)‖2 dt)

holds for all signals in the system. Obviously this SP problem is well defined only ifu andy arein L2(R−;C•) or L2(R−;R•), or, in other words, only if we considerL2−-systems.

We recapitulate the notion of positivity and strict positivity in Subsection5.3.1and in Subsec-tion 5.3.2we make two comments on the SP theorem in frequency domain proved in Chapter3.In Subsection5.3.3the state space version of the results are given using only time domain argu-ments. We choose to not consider the AR representation. We have the feeling that AR represen-tations of continuous time systems do not provide a good basis as far as the analysis of (strict)positivity is concerned. (The frequency domain SP theorem proved in Chapter3 may readily betranslated into an equivalent result for systems with an AR representation. The result is howevera bit awkward and it seems unlikely that an easy and elegant proof may be derived based on timedomain arguments only.)

5.3.1. Strict Positivity for L2−-systems

In this subsection a recapitulate the notion of strict positivity as defined in Chapter3.From Chapter3 we copy:

Definition 5.3.1 (Strict positivity). A subspaceB of a q + p vector valued Hilbert spaceMwith inner product〈•, •〉 is is strictly positive (SP)with respect to theJq,p inner product if thereexists anε > 0 such that everyw ∈ B satisfies

〈w, Jq,pw〉 ≥ ε〈w,w〉. (5.12)

91


Inequality5.12is referred to as the SP inequality.

In this section the inner product is

〈w, z〉 :=∫ 0

−∞w(t)∗z(t) dt

on the Hilbert spaceL2(R−;F) with eitherF = Rq+p or F = Cq+p. If the signalw is partitionedcompatibly withJq,p asw =

[ uy]

then the SP inequality takes the form

∫ 0

−∞‖u(t)‖2 − ‖y(t)‖2 dt ≥ ε(

∫ 0

−∞‖u(t)‖2 + ‖y(t)‖2 dt).

The definition of strict positivity easily generalizes to the case that the SP inequality takes theform

〈w, Ew〉 ≥ ε‖w‖22, E = E∗ ∈ R

m×m, E nonsingular. (5.13)

This is direct from the fact that every nonsingular real symmetric E may be written asE =F∗ Jq,pF, for some integersp andq and nonsingularF ∈ Rm×m.

5.3.2. Two comments on strict positivity in frequency domai n

In subsection5.2.3we showed that for every subspaceBG there exist a matrixG∈ R H(r+p)×(q+p)∞

that has full row rank onC0 ∪ ∞ such thatBG = BG. In Chapter3 we proved that ifG ∈R H

(r+p)×(q+p)∞ has full row rank onC0 ∪ ∞ that thenBG is SP in theJq,p inner product iff

GJq,pG∼ = W Jr,pW∼ has a solutionW ∈ GR H(r+p)×(r+p)∞ such thatW−1G is co-Jq,p-lossless.

These two results we may combine to rephrase the SP theorem toa more elegant formulationgiven in Corollary5.3.2.

Corollary 5.3.2 (Strict positivity). For every given G∈ R H•×(q+p)∞ , the spaceBG is SP with

respect to indefinite inner product

[ f, g] := 〈 f, Jq,pg〉 := 12π

∫ ∞

−∞f (iω)∗ Jq,pg(iω) dω

if and only ifBG = BM for some co-Jq,p-lossless M∈ R H•×(q+p)∞ .

Lemma 5.3.3 (co- J-lossless generators are L2−-minimal generators). Stable co- Jq,p-

lossless generators M∈ R H(r+p)×(q+p)∞ of a given subspaceB ⊂ H ⊥q+p

2 are L2−-minimal

generators ofB .

Proof . Define J = Jq,p and J = Jr,p. SupposeM is co-J-lossless, and letM = D−1R be a leftcoprime polynomial fraction ofM. By co-J-losslessness ofM we have automatically thatM hasfull row rank onC0 ∪ ∞, and, hence, thatR has full row rank onC0. ThereforeL2−-minimalityof M as a generator ofBM is ensured if we can show that thatR has no zeros inC−. SupposeR does have a zero inC−. This implies thatR = FR′ for some polynomialR′ and nonconstantstrictly Hurwitz F. BecauseM JM∼ = J we have thatRJ R∼ = DJD∼. Multiplying both sidesfrom the left byF−1 shows thatF−1DJD∼ = F−1RJ R∼ is polynomial. By the fact thatF isstrictly Hurwitz andD∼ is nonsingular inC− we must therefore have thatF−1D is polynomial.This contradicts the assumption thatD−1R is a left coprime fraction. Hence no suchF exists,i.e., R has zeros only inC+, which completes the proof.

92


5.3.3. Strict positivity for ONRs and DVRs

We discuss in this subsection the results on strict positivity for systems given by an ONR or aDVR. We exploit the nonuniqueness ofL2−-minimal representations of the system to obtain aspecial representation which exists iff the system is strictly positive (SP). It will be no surprisethat the manipulations involve Riccati equations. The Riccati equation we derive in case thesystem is given by an ONR, is known as theH∞-filter Riccati equation. The reason for thisterminology is clarified by Example5.4.3where theH∞ filtering problem is solved using thecharacterization of strict positivity. We first consider the DVR case. Recall that applying regularstate feedback in a DVR does not alter the system defined by theDVR. Stated otherwise, we mayuse regular state feedback at any point without altering thesystem. It is good to be aware of thisfact when reading the proof of the next theorem.

Theorem 5.3.4 (Strict positivity for L2−-systems in DVR form). Let the quadruple of realvalued matrices A, B, C, D define anL2−-minimal DVR of anL2−-systemΣ= (R−,Cq+p,B ).Define for convenience J as J= Jq,p and J = Jq−m,p where m is the number of columns ofD.

1. ThenB is SP with respect to J inner product if and only if

a) D∗ JD > 0 and

b) A unique X exists such that

A∗ X + XA− C∗ JC + [ XB−C∗ JD](D∗ JD)−1[−D∗ JC+ B∗ X] = 0 (5.14)

with A+ B(D∗ JD)−1[−D∗ JC + B∗ X] is antistable and

c) X> 0.

2. Furthermore, given the solution X to (5.14) with A+ B(D∗ JD)−1[−D∗ JC + B∗ X] anti-stable as in Item 1(b) and a nonsingular solution W∈ R

m×m of W∗W = D∗ JD, define theequivalent DVR quadrupleA2, B2,C2, D2 as

[

A2 B2

C2 D2

]

=[

A BC D

][

I 0(D∗ JD)−1[−D∗ JC + B∗ X] W−1

]

. (5.15)

Let DM be a constant matrix such that DM D = 0 and DM J D∗M = J. Then M(s) :=

DM − DMC2X−1(sI + A∗2)

−1C∗2 J ∈ R H

(r+p)×(q+p)∞ is a stable, rational matrix such that

BM = L(B ), and M is co-J-lossless if and only if X> 0.

Proof . 1. a) SupposeΣ is SP. Consider driving variablev(t) =√δ(t)v0, with x(t) = 0

for t < 0 and withδ(t) representing the Dirac pulse. Then

v∗0D∗ JDv0 =

∫ 0

−∞w∗(t)Jw(t)dt

≥ ε

∫ 0

−∞w∗(t)w(t)dt

= εv∗0D∗ Dv0.

SinceD has full column rank, this impliesD∗ JD > 0. (This argument can be madeprecise by consideringL2(R−;R) approximations to the Dirac pulseδ(t)).

93


To simplify the algebra for Items 2 and 3, consider the following regular state feed-back. Let

[

A1 B1

C1 D1

]

:=[

A BC D

][

I 0−(D∗ JD)−1D∗ JC W−1

]

, (5.16)

in which W is a nonsingular solution toW∗W = D∗ JD. The quadrupleA1, B1,C1, D1 defines a DVR of the same system as the two DVRs differ by aregular state feedback. It suffices therefore to consider the DVR

x = A1x+ B1v1

w = C1x+ D1v1.

It is easily checked that

D∗1 J D1 = Im and D∗

1 JC1 = 0.

b) SupposeΣ is SP. LetH be the Hamiltonian matrix

H =[

A1 B1B∗1

C∗1 JC1 −A∗

1

]

. (5.17)

We claim thatH has no eigenvalues on the imaginary axis. To obtain a contradiction,suppose thatH does have an eigenvalue onC0. In that case there existsX andΛsuch that

H X = XΛ ; Λ := iω ∈ C0 ; X a nonzero vector. (5.18)

Let T = 2πω and letα(t) satisfy

α(t) = Λα(t); α(0) 6= 0.

Note thatα(t) is periodic, with periodT. Choose an integern > 0. Let X1 and X2

denote the upper and lower block ofX:

X =[

X1

X2

]

.

Consider the driving variable defined by

v1n(t) =

Fx(t) t < −nTB∗

1 X2α(t) t ≥ −nT,

in which F is any antistabilizing regular state feedback matrix andx(0) := X1α(0).Thenx(t) = X1α(t) on [−nT,0], since

ddt(X1α(t)) = X1Λα(t)

= (A1X1 + B1B∗1 X2)α(t)

= A1(X1α(t))+ B1v1n(t).

The external signalwn = C1x + D1v1n is not identically zero on [−nT,0]. To seethis, we argue as follows:wn ≡ 0 ⇒ v1n ≡ D∗

1 Jwn ≡ 0, givingC1x ≡ 0 andx ≡ A1x.Since(C1, A1) is observable, this implies 0≡ x = X1α. From (5.17) and (5.18), we

94


also have−A∗1X2α ≡ X2Λα = X2 iωα. Combining this with 0≡ v1n = B∗

1 X2α, weget

α(t)∗ X∗2

[

A1 − iω B1]

≡ 0 t ∈ [−nT,0],

giving X2α ≡ 0 since(A1, B1) is antistabilizable. We now have[

X1

X2

]

α(t) ≡ 0 on [−nT,0] and α(0) 6= 0,

which contradicts the full column rank property ofX. Sown is not identically zeroon [−nT,0].For t ∈ [−nT,0],

wn(t)∗ Jwn(t) = α(t)∗(X∗

1C∗1 JC1X1 + X∗

2 B1B∗1 X2)α(t)

= α(t)∗((X∗2 A1 +Λ∗ X∗

2 )X1 + X∗2 (X1Λ− A1X1))α(t)

= α(t)∗(Λ∗ X∗2 X1 + X∗

2 X1Λ)α(t)

= ddtα(t)∗ X∗

2 X1α(t).

Thus∫ 0

−nTwn(t)

∗ Jwn(t) dt = α(0)∗ X∗2 X1α(0)− α(−nT)∗X∗

2 X1α(−nT)

= 0

sinceα(t) has periodT. Therefore∫ 0

−∞wn(t)

∗ Jwn(t) dt =∫ −nT

−∞wn(t)

∗ Jwn(t) dt

= x(−nT)∗QFx(−nT)

= x(0)∗ QFx(0) (note thatx(−nT)= x(0)),

in which QF satisfiesQF(A1 + B1F)+ (A1 + B1F)∗QF = (C1 + D1F)∗ J(C1 +D1F). So

∫ 0−∞wn(t)∗ Jwn(t) dt is constant (as a function ofn). Now note that

‖wn‖22 grows linearly withn, since, on [−nT,0],wn is periodic with periodT and is

not identically zero. Thus, for anyε > 0, there is ann such that

ε

∫ 0

−∞wn(t)

∗wn(t) dt>∫ 0

−∞wn(t)

∗ Jwn(t) dt

which contradicts the SP property. Consequently, the Hamiltonian matrixH in (5.17)has no imaginary axis eigenvalue.Since(−A1, B1) is stabilizable,B1B∗

1 ≥ 0 and H in (5.17) has no imaginary axiseigenvalue, it follows from standard Hamiltonian matrix results (see, for instance,Francis (26), Chapter 7 or Doyle et. al. (22), Lemma 2) that there exists aY such that

Y(−A1)+ (−A∗1)Y + Y B1B∗

1Y − C∗1 JC1 = 0

and−A1 + B1B∗1Y is stable. SettingX = −Y, we get

X A1 + A∗1X + X B1B∗

1 X − C∗1 JC1 = 0, (5.19)

with A1 + B1B∗1 X antistable. A straightforward calculation, using equation (5.16),

shows thatX also satisfies (5.14) and that the antistableA1 + B1B∗1 X equalsA +

B(D∗ JD)−1[−D∗ JC + B∗ X].

95


c) SupposeΣ is SP. Let X be as in Item (b) and consider the DVR quadrupleA1, B1,C1, D1. The signalsw, x andv1 are inL2(R−;C

•) andx(−∞)= 0. Com-pleting the square gives∫ 0

−∞w(t)∗ Jw(t)dt = x(0)∗ Xx(0)− x(−∞)∗ Xx(−∞)+ ‖v1 − B∗

1 Xx‖22

= x(0)∗ Xx(0)+ ‖v1 − B∗1 Xx‖2

2. (5.20)

Here we used the fact thatx(−∞) exists and equals zero (see Lemma5.2.7, Item 2).Now consider the driving variablev1 = B∗

1 Xx, which is inL2(R−;C•) sinceA1 +B1B∗

1 X is antistable. Letw be the external signal resulting fromv1, with “initial”statex(0). Then

x(0)∗ Xx(0) =∫ 0

−∞w(t)∗ Jw(t) dt (by (5.20))

≥ ε

∫ 0

−∞w(t)∗w(t) dt (by SP the inequality)

= εx(0)∗ Qx(0)

in which Q(A1 + B1B∗1 X)+ (A1 + B1B∗

1 X)∗ Q = (C1 + D1B∗1 X)∗(C1 + D1B∗

1 X).Note that(C1 + D1B∗

1 X, A1 + B1B∗1 X) is observable (by strong observability of the

DVR), so Q> 0. HenceX ≥ εQ> 0.

Conversely, supposeX > 0. Write

x = (A1 + B1B∗1 X)x+ B1(v1 − vopt)

w = (C1 + D1B∗1 X)x+ D(v1 − vopt)

in which vopt = B∗1 Xx. SinceA1 + B1B∗

1 X is antistable, there exist constantsβ ≥ 0andγ > 0 such that

‖w‖22 ≤ γ‖v1 − vopt‖2

2 + β‖x(0)‖2 ; γ >β

λmin(X).

Hence∫ 0

−∞w(t)∗ Jw(t) dt = x(0)∗ Xx(0)+ ‖v1 − vopt‖2

2

≥ (λmin(X)−β

γ)‖x(0)‖2 + 1

γ‖w‖2

2

≥ 1γ

‖w‖22

which proves the system is SP.

2. Let AM := −A∗2, BM := C∗

2 J andCM := −DMC2X−1, so M(s) = CM(sI − AM )−1BM +

DM. The DVRA2, B2,C2, D2 is normalized in the sense that

A∗2X + X A2 − C∗

2 JC2 = 0, B∗2 X − D∗

2 JC2 = 0, D∗2 J D2 = I,

which may be used to show that

AM X + X A∗M + BM J B∗

M = 0, BM J D∗M + XC∗

M = 0. (5.21)

96


Let DM be an arbitrary matrix of maximal full row rank such thatDM D = 0. To see thatamong suchDM there exist those satisfyingDM J D∗

M = J we invoke the inertia lemma.By the inertia lemma (Lemma3.1.5), the fact thatD∗ JD > 0 implies thatDM J D∗

M isnonsingular and that its inertia equals that ofJ. ConsequentlyDM J D∗

M = PJ P∗ for somenonsingularP and with such aP we may redefineDM as DM := P−1DM which has thedesired properties. We apply Theorem 5.3 from Green (35) stating thatM is co-J-losslessiff X > 0.

Remains to show thatM generatesL(B ). By Lemma5.2.9 it suffices to prove thatAM, BM,CM, DM defines anL2−-minimal ONR of Σ. As a first step, we applyLemma5.1.5, which, applied to our case, states that the quadrupleA, B,C, D defined as

[

A BC D

]

=[

A2 − B2D−L2 C2 B2D−L

2−D2⊥C2 D2⊥

]

defines an ONR ofΣ if D−L2 is a left inverse ofD2 andD2⊥ is a left orthogonal complement

of D2. The thing to note here is we may takeD2⊥ := DM andD−L2 := D∗

2 J, in which casethe ONRA, B,C, D becomes

[

A BC D

]

=[

A2 − B2D∗2 JC2 B2D∗

2 J−DMC2 DM

]

.

This ONR in turn transforms under output injection (which for ONRs does not affect thesystem)H = X−1C∗

2 D∗M J into

[

A2 − X−1C∗2 JC2 X−1C∗

2 J−DMC2 DM

]

:= (5.22)[

I X−1C∗2 D∗

M J0 I

][

A2 − B2D∗2 JC2 B2D∗

2 J−DMC2 DM

]

(Here we used the fact thatJ D2D∗2 J − D∗

M JDM = J and thatB2 = X−1C∗2 J D2.) Using

(5.21), the matrix in (5.22) may be recognized to be[

X−1AM X X−1BM

CM X DM

]

. (5.23)

A state transformation gives the desired ONR. Because the transformation from ONRs toDVRs in Lemma5.1.5transformsL2−-minimal DVRs inL2−-minimal ONRs we have in(5.23) anL2−-minimal ONR ofΣ and thus by Lemma5.2.9and the fact thatAM is stable,M is such thatBM = L(B ).

Exactly the same result holds forL2−-systems of the formΣ = (R−,Rq+p,B ) whose signalsare restricted to real-valued signals. Theorem5.3.4can be translated into a corresponding ONRresult using Lemma5.1.5. It is frequently the case that we would like to determine whether or nota given ONR defines an SPL2−-system without having to find anL2−-minimal representationfirst. This may be done:

Theorem 5.3.5 (Strict passivity for L2−-systems in ONR form). Let the quadrupleA, B,C, D define an ONR of anL2−-systemΣ = (R−,Cq+p,B ). Suppose D∈ R(r+p)×(q+p)

has full row rank,(C, A) is detectable and that[

A− sI BC D

]

has full row rank for all s∈ C0. Define J= Jq,p and J = Jr,p.

97


1. ThenΣ is SP in the J inner product if and only if

a) There is a nonsingular matrix W∈ R(r+p)×(r+p) such that DJ D∗ = WJW∗ and

b) A unique Q exists such that

AQ+ QA∗ − [ QC∗ + BJ D∗](DJ D∗)−1[CQ+ DJ B∗] + BJ B∗ = 0 (5.24)

with A− [ QC∗ + BJ D∗](DJ D∗)−1C stable and

c) Q ≥ 0.

2. Furthermore, given the solution Q to (5.24) with A− [ QC∗ + BJ D∗](DJ D∗)−1C stableas in Item 1(b) and a nonsingular solution W∈ R(r+p)×(r+p) to WJW∗ = DJ D∗ as inItem 1(a), define the equivalent ONR quadrupleAM, BM,CM, DM as

[

AM BM

CM DM

]

=[

I −[ BJ D∗ + QC∗](DJ D∗)−1

0 W−1

][

A BC D

]

.

Then M(s) := CM(sI − AM)−1BM + DM ∈ R H

(r+p)×(q+p)∞ is a stable matrix such that

BM = L(B ), and M is co-J-lossless iff Q≥ 0.

Proof . 1. Suppose first that the ONR isL2−-minimal. The idea of the proof is transform theONR into an equivalentL2−-minimal DVR, to apply Theorem5.3.4to this DVR and thentranslate things back in terms of the ONR data.

(Only if) SupposeΣ is SP. TheD matrix in anyL2−-minimal DVR ofΣ satisfiesD∗ JD >

0 according to Theorem5.3.4. Since the columns ofD span KerD, we may conclude fromthe inertia lemma thatDJ D∗ = WJW∗ for some nonsingularW. In particular we see thatDJ D∗ is nonsingular. This comes in handy when trying to find a DVR equivalent to theONR. By Lemma5.1.5the quadruple A, B, C, D defined as

[

A BC D

]

:=[

A B0 I

][

I 0−J D∗(DJ D∗)−1C D⊥

]

defines anL2−-minimal DVR of Σ. (Note thatJ D∗(DJ D∗)−1 is a right inverse ofD,that D⊥ denotes an orthogonal complement ofD and that the DVR is nicely normalized:D∗ JC = 0.) We may apply Theorem5.3.4, which, rewritten in terms ofA, B, C and D,states that there exists anX > 0 such that

[ A− BJ D∗(DJ D∗)−1C]∗ X + X[ A− BJ D∗(DJ D∗)−1C]

−C∗(DJ D∗)−1C + [ X∗ BD⊥](D⊥ J D⊥)−1[ D∗

⊥B∗ X] = 0 (5.25)

with

AX := A− BJ D∗(DJ D∗)−1C + BD⊥(D⊥ J D⊥)−1[ D∗

⊥ B∗ X]

antistable. This definesAX. Using the equalityJ− J D∗(DJ D∗)−1DJ = D⊥(D⊥ J D⊥)−1D∗⊥

we may rearrange equation (5.25) and the formula forAX as

X A+ A∗ X − [C∗ + X BJ D∗](DJ D∗)−1[C + DJ B∗ X] + X BJ B∗X = 0, (5.26)

98

5.4. Examples

and

AX = A− BJ D∗(DJ D∗)−1[C + DJ B∗X] + BJ B∗X.

DefineQ = X−1 and AQ = A− [ QC∗ + BJ D∗](DJ D∗)−1C. Finally we may use this torewrite (5.26) as

XA− BJ D∗(DJ D∗)−1[C + DJ B∗X] + BJ B∗X︸︷︷︸

=AX

+

A∗ − C∗(DJ D∗)−1[CX−1 + DJ B∗]︸︷︷︸

=A∗Q

X = 0.

In other words,AQ = −[ X AX X−1]∗, and, hence, by antistability ofAX we have thatAQ isstable. Multiplying (5.26) both from the left and from the right byX−1, changing the signand identifyingQ with X−1 gives the desired Riccati equation.

(If) Simply reverse the arguments of the only-if part of the proof.

The proof of the non-L2−-minimal case can be built up from an associatedL2−-minimalONR in a similar way as is done in Green et. al. (34, Theorem 3.2).

2. ThatBM = L(B ) follows from Lemma5.2.9. The rest follows from Green (35, Theorem5.3).

5.4. Examples

In this section we give three examples. In the first example weshow thatL2−-systems of stableinput/output systems are automatically convolution systems and that in this case the results onthe strictly positive subspaces reduce to the Bounded Real Lemma ((2)). The other extreme,when there are no inputs at all, is considered thereafter in Example5.4.2. Example5.4.2is aboutNevanlinna-Pick interpolation and handles the presumablyonly occasion that it is desirable tohave the non-real version of the results at our disposal as well. We end this section with anexample onH∞ filtering.

Recall that a rational matrixH is contractiveif ‖H‖∞ ≤ 1 andstrictly contractiveif ‖H‖∞ < 1.

Example 5.4.1 (Bounded Real Lemma, cf. ( 2)). Suppose we are given anL2−-system withexternal signalw =

[ uy]

and suppose that it has the I/S/O representation

x = Ax+ Buy = Cx

.

Suppose in addition thatA is stable. We check under which condition there exists anε > 0 suchthat

∫ 0

−∞‖u(t)‖2 − ‖y(t)‖2 dt ≥ ε(

∫ 0

−∞‖u(t)‖2 + ‖y(t)‖2 dt) (5.27)

for all inputs and outputs inL2(R−;Rm). To do this we transform the I/S/O representation to anequivalent ONR:

x = Ax+[

B 0]

w

0 = Cx+[

0 − I]

w.

99


In order to be able to apply Theorem5.3.5we need to have that the ONR quadruple has nozeros onC0 ∪ ∞, that(C, A) is detectable and that the “D-matrix”

[

0 − I]

has full row rank.The latter is trivial and the other conditions are ensured bythe assumption thatA is stable.Theorem5.3.5states that the SP inequality (5.27) holds for someε > 0 iff

1. AQ+ QA∗ + QC∗CQ+ BB∗ = 0 has a solutionQ such thatA+ QC∗C is stable, and

2. Q ≥ 0.

These are the same conditions as for the Bounded Real Lemma, though in a dual version. (The“primal” version may be obtained by rewriting the I/S/O representation to an equivalent DVR,instead of an ONR.) The Bounded Real Lemma states that the associated transfer matrixH(s) :=C(sI − A)−1B satisfies‖H‖∞ < 1 iff the conditions in the above two items hold ((2)). That theseconditions are equivalent follows from the fact that the restriction to signals inL2(R−;Rm+p)

makes a stable I/O system automatically into a convolution system. To see this we assume that(C, A) is observable (the case that(C, A) is only detectable also works but is a bit messy). Inthis case we have by Lemma5.2.6that the statex in the ONR (and therefore also in the I/S/Orepresentation) satisfies limt→−∞ x(t) = 0. This, in combination with stability ofA implies thatfor everyt the expression

y(t) = CeA(t−T)x(T)+∫ t

TCeA(t−τ)Bu(τ) dτ

converges to

y(t) =∫ t

−∞CeA(t−τ)Bu(τ) dτ

as T goes to−∞. In other words, theL2−-system coincides with the associated convolutionsystem.

Example 5.4.2 (Nevanlinna Pick Interpolation). The aim of this example is to show a connec-tion between SP subspaces and the Nevanlinna Pick Interpolation problem (NPIP). We show thata certain NPIP has a solution iff a corresponding finite dimensional subspace ofH ⊥q+p

2 is SP.Consider the problem of findingH ∈ R H

p×q∞ such that‖H‖∞ < 1 and such that a set of given

interpolation conditions is satisfied:

H(ζi)ai = bi for i ∈ 1, . . . ,n.

We assume that allζi lie in C+ and thatζi 6= ζ j if i 6= j.This problem is known as the (one-sided) Nevanlinna-Pick interpolation problem (NPIP) (see

e.g. Ball, Gohberg and Rodman (4) and the references therein). We claim that this NPIP has asolution iff then dimensional space

B =[

a1 · · · an

b1 · · · bn

]

sI −

ζ1

. . .ζn

−1

Cn ⊂ H ⊥q+p

2 (5.28)

is SP in theJq,p-inner product. (Note thatai ∈ Cq andbi ∈ Cp.)Suppose the NPIP has a stable strictly contractive solutionH. The interpolation conditions on

H imply that(H − I p)w ∈ H p2 for everyw ∈ B . In other words,B is a subset ofB(H −I ). As H

100

5.4. Examples

is stable and strictly contractive, the spaceB(H −I ) is SP. As a result the SP inequality also holdson the subsetB . This proves that solvability of the NPIP implies thatB is SP.

Now suppose thatB is SP. Then, by Lemma5.3.2, B = BM for some stable co-Jq,p-lossless

M. (Note thatM is square becauseBM = B is finite dimensional.) Define[

H1 −H2]

:=[

U I]

M,

where the partitioning is such thatH2 is square andH1 andU have the same size as the solutionHwe are trying to find. TakeU stable and strictly contractive. ThenH1H∗

1 − H2H∗2 ≤ UU∗ − I < 0

in the closed right-half plane, by co-Jq,p-losslessness ofM. It follows that H2 is in GR Hp×p∞ ,

and thatH = H−12 H1 is stable and strictly contractive. The productMw is in H q+p

2 if w ∈ B , so[

H − I]

w = H−12

[

U I]

(Mw) ∈ H p2 if w ∈ B , becauseH−1

2 andU are stable. This holds inparticular for

w =[

a j

b j

]

1s− ζ j

∈ B .

Thus(H(s)a j − b j)/(s− ζ j) is in H p2 , which impliesH(s)a j − b j is zero ats = ζ j . In other

words,H constructed this way is stable strictly contractive and satisfies the interpolation condi-tions as well. That is, anH defined by

H = H−12 H1;

[

H1 −H2]

:=[

U I]

M; U ∈ R Hp×q∞ ; ‖U‖∞ < 1 (5.29)

is a solution to the NPIP.Note that there is a freedom in the construction ofH. Every stable strictly contractiveU in

(5.29) gives rise to a solutionH to the NPIP. In fact, as we will show now,all solutionsHto the NPIP are generated this way. The idea behind the proof is not new and is included forcompleteness only (see e.g. (4)). The proof relies on the well known small gain argument (SeeAppendixA, TheoremA.0.7).

Let H be a solution to the NPIP and letM be a co-Jq,p-lossless generator ofB . First we showthat the rational matricesA andB defined as

[

B A]

=[

H − I]

M−1 ∈ R p×(q+p), A square (5.30)

are stable matrices. This we do by showing that[

B A]

mapsH q+p2 to H p

2 . Let f be an arbitrary

element ofH q+p2 , and with it defineg− ∈ H ⊥q+p

2 andg+ ∈ H q+p2 by the equation

g− + g+ = M−1 f.

ThereforeMg− = f − Mg+ ∈ H q+p2 and, hence,g− ∈ BM = BG. BecauseH satisfies the interpo-

lation conditions, we have thatBM = B is a subset ofB[

H −I]. Consequently

[

H − I]

g− ∈ H p2

and, hence,[

B A]

f =[

H − I]

M−1 f =[

H − I]

(g− + g+) ∈ H p2 .

Apparently[

B A]

mapsH q+p2 to H p

2 and, hence,B andA are stable.By the J-lossless property ofM we have that

BB∼ − AA∼ =[

B A]

M Jq,pM∼[

B∼

A∼

]

= H H∼ − I < 0 onC0 ∪ ∞.

101


This shows thatA is nonsingular and thatA−1B is strictly contractive. If we partitionM com-patibly with Jq,p as

M =[

M11 M12

M21 M22

]

,

we see from (5.30) that

M−122 = BM12M−1

22 + A.

BecauseM22 is in GR Hp×p∞ and‖M12M−1

22 ‖∞ < 1 (by co-J-losslessness ofM, AppendixA)and because‖A−1B‖∞ < 1 we may apply the small gain theorem (see LemmaA.0.8, Item 2)saying thatA has as many zeros inC+ as M−1

22 . SinceM22 is in GR Hp×p∞ we therefore must

have thatA is in GR Hp×p∞ . Finally, defineU asU = A−1B and recognize that thisU is stable

and strictly contractive and thatH then satisfies

H = H−12 H1,

[

H1 H2]

:=[

U I]

M.

This completes the proof of our claim.For the construction of the co-Jq,p-lossless generatorM we temporarily switch to the time

domain. The time domain version ofB obviously is

B := w ∈ L2(R−;Cq+p) | w =

[

a1 · · · an

b1 · · · bn

]

x, x =

ζ1

. . .ζn

x,

which is of the DVR type and in fact it is anL2−-minimal DVR. We may therefore apply The-orem5.3.4, Item 1, which for this specific problem says that the corresponding system is SP iffthe solutionX to the Lyapunov equation

ζ∗1. . .

ζ∗n

X + X

ζ1

. . .ζn

=

a∗1 b∗

1...

...a∗

n b∗n

Jq,p

[

a1 · · · an

b1 · · · bn

]

is positive definite. This is a standard result andX is known as the Pick matrix. By applicationof Theorem5.3.4, Item 2, we deduce that the co-Jq,p-lossless generatorM of B may be chosento be

M(s) =[

a1 · · · an

b1 · · · bn

]

X−1

sI +

ζ∗1. . .

ζ∗n

−1

a∗1 −b∗

1...

...a∗

n −b∗n

− I .

In (4) much more material may be found on interpolation in general, and Nevanlinna-Pickinterpolation in particular. In (4, part five) it is proved that the NPIP has a solution iff the pair

(C, A) :=

[

a1 · · · an

b1 · · · bn

]

,

ζ1

. . .ζn

is a so-calledright null pair over C+ of some square co-J-lossless matrixM. A null pair ofa rational matrixM over C+ is a notion that describes the zero structure ofM in C+. Theresults in (4) applied to the NPIP gives rise to the same realization ofM and the same Lyapunovequation.

102

5.4. Examples

Σ

F

- - -

-

6f

−

+d z

y u

e

Figure 5.2.: TheH∞ filtering configuration.

Example 5.4.3 ( H∞ filtering, ( 40)). TheH∞ filtering problem is to find filtersF, given a systemΣ, such that the closed-loop system map fromd to e as in Fig.5.2 is stable and strictly con-tractive. (We estimatez with error‖e‖2 < ‖d‖2.) We allow causal, homogeneous filtersF. Byhomogeneous we mean thatF maps the zero signal into the zero signal. The given systemΣ isassumed to be of the usual type:

Σ :

x = Ax + Bdz = C1xy = C2x + Dd.

(5.31)

We assume thatD[

B∗ D∗] =[

0 I]

. (These are the standard assumptions, c.f. Doyle et.al. (22)).

The idea is to show that there exists a subset of the closed-loop behavior that does not dependon the filter. If theH∞ filtering problem has a solution, i.e., if there is a filter such that theclosed-loop system satisfies the SP inequality then certainly this SP inequality must hold onthis filter independent subset. This provides a necessary condition for theH∞ filtering problemto have a solution, which in fact turns out to be sufficient as well. This idea originates fromKhargonekar (40).

The L2− behavior that is important here is the set of external signals (d,e) in L2(R−;R•)that do not “activate” the outputy(t) for t < 0, so that there is nothing to filter fort < 0. Thenu(t) = 0, t < 0 ande= z, by causality and homogeneity ofF. Consider therefore the behavior

B = [

de

]

|x = Ax + Bd0 = C1x − Ie0 = C2x + Dd

,

[

de

]

∈ L2(R−;R•) .

The behavior is of the ONR type. In order to be able to apply Theorem5.3.5, we assume

1. (

[

C1

C2

]

, A) is detectable ;

2.

A− sI B 0C1 0 − IC2 D 0

has full row rank for alls∈ C0 ∪ ∞.

We assumed in the first place thatD[

B∗ D∗] =[

0 I]

and this implies that the second as-sumption is equivalent to(A, B) not having uncontrollable modes on the imaginary axis.

It follows from Theorem5.3.5that under these assumptions the system with behaviorB is SPiff

AQ+ QA∗ + Q(C∗1C1 − C∗

2C2)Q+ BB∗ = 0 (5.32)

103


has a nonnegative definite solutionQ such thatA − [C∗1C1 − C∗

2C2] Q is stable. (Such solutionsQ are often calledstabilizingsolutions to the Riccati equation (5.32).)

Conversely, given a stabilizing nonnegative definite solution Q to (5.32) a filter that solves theproblem can be constructed (see Doyle et. al. (22) and Limebeer and Shaked (55)):

F :

˙x = Ax+ QC∗2(y− C2x)

u = C1x.

104

6

Conclusions

We presented a frequency domain solution method to a class ofH∞ control problems. The sys-tems we worked with are—take a firm stand—finite dimensional linear time invariant continuoustime systems. Especially for such systems state space approaches have been very successful inobtaining solutions toH∞ control problems. The decision to choose nevertheless for the fre-quency domain was much influenced by the desire to have a method that fits in the polynomialapproach toH∞ control. Besides, it was believed, and indeed proved by Green (35), that thefamous state space results may be seen to arise from frequency domain results, by invoking thecanonical factorization theorem.

The derivation of our solution to the suboptimal two-blockH∞ problem differs from the solu-tions to this problem reported in, for example, Francis (26) and Green et. al. (34). We emphasizemore the role of thesignalsof the system, instead of thetransfer matrixrepresenting the sys-tem. The reason for doing so lies in the belief, adopted from the behavioral approach, that the“behavior”—the set of signals in the system—makes the system and not so much the (arbitrarychoice of) representation of the system. This belief led to the strict positivity theorem; a resultof intrinsic value, revealing a one-to-one correspondencebetween strict positivity of spaces andJ-losslessness of transfer matrices. This result appears tobe new. It is an extension to a resultconcerning positivity of a spaceχ+ as formulated in Ball and Helton (7).

The effort put in the development of a geometric approach pays in the optimal two-blockH∞problem. The frequency domain solution to the optimal version of the two-blockH∞ problemthat we give appears not to be reported elsewhere. Those familiar with the results of Adamjan,Arov and Kreın (1) might, however, guess the over-all algorithm that we derive. The algorithmsthat we and (1) give are identical but apply to different problems (see also Ball and Ran (10)).

The solution to the suboptimal standardH∞ problem is practically a copy of the results ofGreen (35). The two constant matrices that need to factored in (35) are demanded to have aspecific structure (some sub-blocks need to be invertible).This is only a technical requirement.We circumvent this technicality by reformulating the problem a little. We formulated an algorithmto solve a class of optimal standardH∞ problems. With the algorithm in principleall optimalsolutions may be generated. The algorithm is an extended version of the one formulated inKwakernaak (48) that cannot generate all optimal solutions in some cases. We have to add thatour algorithm is not numerically stable.

In AppendixB we included a proof of a variation of the canonical factorization theorem de-veloped in Bart, Gohberg and Kaashoek (11). Our proof stresses thenecessitypart, which isoften thought to be the most complicated part. Of the resultson polynomial and rational matricescollected in AppendixB also worth mentioning is our re-definition of the notion ofMcMillandegree. We propose a definition of McMillan degree that works for nonproper rational matricesas easy as it does for proper rational matrices. The usual definition of McMillan degree does not

105

6. Conclusions

work directly for nonproper rational matrices (see (39; 75; 81; 16)). The extension to the non-proper case is important for our derivation of the result that the McMillan degree of a suboptimalcompensator need not accede that of the generalized plant. (Remember that in the polynomialapproach the generalized plant need not be proper.)

In writing this thesis we had two main goals in mind:

• The approach should cover the polynomial approach toH∞ control theory.

• The approach should be able to generate optimal solutions, as opposed to the usual subop-timal solutions.

These goals have been met only up to certain extent. The link with the polynomial approach isnot complete. In recent work ((50)) it is argued that in some cases (such as for designing forintegral control) it is desirable to choose shaping filters that have poles on the imaginary axis.The results presented in this thesis work for nearly all filters, but not when some of the filtershave poles on the imaginary axis.

Several problems concerning optimality still are not solved satisfactory in our setup. On thetheoretical side the so-called Parrott optimality problemfor both the two-blockH∞ problem andstandardH∞ problem remains unsolved. In the other cases, where our results do apply, it is not yetcompletely clear how the solutions can be computed in a numerically stable way. In this respect itis good to be aware that the solution presented in this thesis, based on noncanonical factorization,has more structure than necessary for the solution to theH∞ problems. It is desirable to have amethod that circumvents the construction of a complete noncanonical factor.

At present a frequency domain analog of the coupling condition does not exist. It is not clear towhich extend the “lack of decoupling” in the polynomial solution method affects the numericalreliability of the polynomial software compared to the state space software that exploits thecoupling condition.

The results presented in this thesis may be translated without any difficulty to discrete timesystems, and it is to be expected that also some of the resultsof our approach, if not all, may beextended to a class of infinite dimensional systems.

106

A

Basics from H∞ Theory

The material that is dealt with in this appendix is drawn fromFrancis (26), Hoffman (37),Young (90), Vidyasagar (82), Redheffer (73), and Ball and Helton (7). Additional referencesare (76; 18; 23; 22; 74).

Definition A.0.4 (Hardy spaces).

H m2 := f : C+ 7→ C

m | f is analytic inC+ and supσ>0

∫ ∞

−∞‖ f (σ + jω)‖2dω <∞,

H ⊥m2 := f : C− 7→ C

m | f is analytic inC− and supσ<0

∫ ∞

−∞‖ f (σ + jω)‖2dω <∞,

H g×m∞ := f : C+ 7→ C

g×m | f is analytic inC+ and sups∈C+

σmax f (s) <∞.

Hereσmax M is the largest singular value of its argumentM ∈ Cg×m.

Definition A.0.5 ( L2 and L∞).

L m2 := L2(C0;C

m) := f : C0 7→ Cm |

∫ ∞−∞ ‖ f ( jω)‖2dω <∞ ,

L g×m∞ := L∞(C0;Cg×m) := f : C0 7→ Cg×m | ess sups∈C0

σmax f (s) <∞ .

It is assumed that its elements are Lebesgue measurable onC0. The essential supremum of afunctiong overC0 (ess sups∈C0

g(s)) is the minimal boundM ∈ R for whichs∈ C0 | ‖g(s)‖ >M has zero measure.

Sometimes we abbreviateH ⊥m2 , H m

2 andL m2 to H ⊥

2 , H2 andL2 respectively. Similar conven-tions are used for other spaces.

For elementsf of H2 the expression

F(iω) := limσ∈R+, σ→0

f (σ+ iω)

exists for almost allω ∈ R; f may be identified withF, andF is an element ofL2. Similarly,an element ofH ⊥

2 may be identified with an element ofL2. In other words,H2 andH ⊥2 may be

seen as subsets ofL2. The spacesH ⊥2 , H2 andL2 are Hilbert spaces under the inner product and

norm

〈 f, g〉 = 12π

∫ ∞

−∞f ∗(iω)g(iω) dω, ‖ f ‖2 :=

√

〈 f, f 〉.

With this inner product,H ⊥2 andH2 (as a subset ofL2) are orthogonal andL2 = H ⊥

2 ⊕ H2.

107

A. Basics fromH∞ Theory

The spacesL g×m∞ andH g×m

∞ are Banach spaces under the norm‖ · ‖∞ defined, respectively, as

‖G‖∞ := ess supω∈R σmaxG(iω), ‖G‖∞ := sups∈C+

σmaxG(s).

The spaceH g×m∞ constitutes a set of bounded linear operators fromH m

2 to H g2 —in fact H g×m

∞is the set of shift invariant bounded linear operators fromH m

2 to H g2 (see Rosenblum and

Rovnyak(74))—and the∞-norm coincides with the operator induced norm:

‖G‖∞ = supu∈H m

2

‖Gu‖2

‖u‖2.

Likewise, elements ofL g×m∞ are bounded linear operators fromL m

2 to L g2 , whose norm equals

that of the operator induced norm. For elementsG of H g×m∞ the expression

G(iω) := limσ∈R+, σ→0

G(σ+ iω) (A.1)

exists for almost allω ∈ R; G may be identified withG, G is an element ofL g×m∞ and‖G‖∞ =

‖G‖∞. Matrices inH g×m∞ are referred to asstablematrices.

GHm×m∞ denotes the set of invertible elements inH m×m

∞ (G ∈ GHm×m∞ ⇔ G,G−1 ∈ H m×m

∞ ).G in H g×m

∞ or L g×m∞ is a contractionif ‖G‖∞ ≤ 1 and astrict contractionif ‖G‖∞ < 1. It

may be checked that aG in H g×m∞ is a contraction iffG∗G − I ≤ 0 in C+, with G∗ defined by

G∗(s) = [G(s)]∗.R H

g×m∞ denotes the set of real-rational matrices inH g×m

∞ . More concretely,R Hg×m∞ is the

set ofg × m real-rational matrices that are proper and have all their poles in C−. R L g×m∞ are

the real-rational matrices inL g×m∞ , i.e., the rational matrices that are proper and have no poles on

the imaginary axis. The elements ofR Hm×m∞ whose inverse is inR H

m×m∞ as well, is denoted

by GR Hm×m∞ . The formal route taken earlier in order to see thatH g×m

∞ may be seen as a subsetof L g×m

∞ is not necessary for the restricted case of rational matrices. ElementsG ∈ R Hg×m∞ are

automatically defined almost everywhere on the complex plane and certainly on the imaginaryaxis: G ≡ G in (A.1) if G ∈ R H

g×m∞ .

Corollary A.0.6 (Multiplicative property). ‖AB‖∞ ≤ ‖A‖∞‖B‖∞ for every A inL g×m∞ and B

in L m×q∞ .

Theorem A.0.7 (Small Gain Theorem). If H ∈ H m×m∞ and‖H‖∞ < 1, then( I − H)−1 ∈ H m×m

∞ .

The proof is trivial:( I − H)−1 =∑∞

n=0 Hn is well defined inH m×m∞ becauseH g×m

∞ is a Banachspace and‖Hn‖∞ ≤ ‖H‖n

∞.

Lemma A.0.8 (An extended small gain argument, ( 82, pp. 274-275)).

1. If H ∈ R L m×m∞ and‖H‖∞ < 1, then H∈ R H

m×m∞ ⇔ ( I − H)−1 ∈ R H

m×m∞ .

2. Let H= N D−1 ∈ R L m×m∞ be a right coprime fraction of H overR H ∞. Suppose‖H‖∞ <

1. Then D− N has as many zeros inC+ ∪ C0 ∪ ∞ as D.

Proof . An elegant proof uses a winding number and a homotopy argument (see e.g. (82)). Herewe give a more concrete proof.

108

1. Let N D−1 = H be a right coprime fraction ofH over R Hm×m∞ . Then D(D − N)−1

is a right coprime fraction of( I − H)−1, and, hence,( I − H)−1 is stable iff D − N ∈GR H

m×m∞ . Proceed with Item 2.

2. D(D − εN)−1 is a right coprime fraction of( I − εH)−1 for everyε 6= 0. (D − εN) =( I − εH)D is nonsingular on the imaginary axis (including infinity) for everyε ∈ [0,1].So, using a continuity argument, forε = 1 the matrixD − εN has as many zeros inC+ ∪C0 ∪ ∞ as forε = 0 (that is,D − εN = D).

Definition A.0.9 (Inner and outer matrices). Gi ∈ R Hg×q∞ is inner if ‖Giu‖ = ‖u‖ for every

u ∈ H q2 . Go ∈ H q×m

∞ is outer if GH m2 is dense inH q

2 . G = GiGo is aninner-outer factorizationof G ∈ R H

g×m∞ if Gi is inner andGo is outer.

Corollary A.0.10 (Inner and outer matrices, ( 82)). Gi ∈ R Hg×q∞ is inner iff G∼G = Iq, and

Go ∈ R Hq×m∞ is outer iff Go(s) has full row rank for every s∈ C+.

Lemma A.0.11 (Inner and outer matrices). Every G∈ R Hg×m∞ that has full column rank on

C0 ∪ ∞ has an inner-outer factorization G= GiGo. In this case the outer factor Go is anelement ofGR H

m×m∞ and it is unique up to multiplication from the left by a constant real unitary

matrix.

Proof . First note that ifG = GiGo, thenG∼G = G∼o Go. The expressionG∼G is positive definite

on C0 ∪ ∞ because by assumptionG has full column rank onC0 ∪ ∞. From Appendix B,Corollary B.3.5 it follows that there exist solutionsGo in GR H

m×m∞ of the equationG∼G =

G∼o Go and that they are unique up to multiplication from the left bya constant unitary matrix.Gi

defined asGi = GG−1o then is obviously inner.

Rational matrices are inner if they are stable and satisfy a condition on the imaginary axis.These two conditions are equivalent to a condition in the closed right-half plane: Rational matri-cesG are inner iff [G(s)]∗G(s) ≤ I for all s ∈ C+ with equality holding fors ∈ C0 ∪ ∞. Thisequivalent formulation of innerness may be generalized:

Definition A.0.12 (Stable J-lossless matrices). SupposeJq,p and Jr,p are two given matricesof the form

Jq,p =[

Iq 00 − I p

]

, Jr,p =[

Ir 00 − I p

]

.

ThenG ∈ R H(r+p)×(q+p)∞ is Jq,p-lossless(or, Jq,p-inner) if

[G(s)]∗ Jq,pG(s) ≤ Jr,p

for all s∈ C+ with equality holding fors∈ C0 ∪ ∞.

Note thatJr,p in Definition A.0.12 is determined completely byJq,p andG. A factorizationG = RW of G is a Jq,p-inner-outer factorization if R is Jq,p-lossless andW is outer. Theconditions under which there exist aJq,p-inner-outer factorization given matricesJq,p andG area bit subtle. The most tractable case is whenG has full column rank onC0 ∪ ∞. ThenG admitsa Jq,p-inner-outer factorization iffG∼ Jq,pG = W∼ Jr,pW has a solutionW ∈ GR H

(r+p)×(r+p)∞ .

This is a so-called canonical factorization problem, a problem we review in Appendix B. In

109


Appendix B it is shown that solutionsW ∈ GR H(r+p)×(r+p)∞ to G∼ Jq,pG = W∼ Jr,pW are unique

up to multiplication from the left by a constantJr,p-unitary matrix1.M is co-Jq,p-losslessor co-Jq,p-inner if MT is Jq,p-lossless. Concretely:M is co-Jq,p-lossless

if M Jq,pM∼ = Jr,p and M Jq,pM∗ ≤ Jr,p in C+. Here Jr,p is a J-matrix of appropriate sizewith as many negative eigenvalues asJq,p. For square matricesJq,p-losslessness and co-Jq,p-losslessness are equivalent. A factorization ofG = GcoGci is a co-inner-outer factorization ofGif GT

co is outer andGTci is inner.Gco is co-outerif GT

co is outer, andGci is co-innerif GTci is inner.

The connection between innerness andJq,p-losslessness is one of “swapping inputs and out-puts”:

Corollary A.0.13 (Connection between inner and Jq,p-inner). Suppose p and q are given pos-itive integers. In what followsw and y are elements ofH p

2 and z∈ H r2 and u∈ H q

2 .

1. If M =[ M11 M12

M21 M22

]

∈ R H(r+p)×(q+p)∞ is Jq,p-lossless (with Jq,p and M partitioned compat-

ibly), then

(z, w,u, y) |[

zw

]

= M

[

uy

]

,

[

uy

]

∈ R Hq+p2 (A.2)

equals

(z, w,u, y) |[

zy

]

= G

[

w

u

]

,

[

w

u

]

∈ R Hp+q2 (A.3)

where G is the matrix defined as

G =[

M12M−122 M11 − M12M−1

22 M21

M−122 −M−1

22 M21

]

∈ R H(r+p)×(p+q)∞ .

Moreover, G is inner.

Conversely, if G∈ R H(r+p)×(p+q)∞ is an inner matrix whose lower left p× p block element

is in GR Hp×p∞ , then there exists a unique stable Jq,p-lossless M for which (A.2) and (A.3)

coincide.

2. M ∈ R H(r+p)×(q+p)∞ is Jq,p-lossless iff M∼ Jq,pM = Jr,p and the lower right p× p block

element M22 of M is inGR Hp×p∞ .

Proof . Define J = Jq,p and J = Jr,p. (Item 1) It follows from M∗ JM ≤ J that M∗12M12 −

M∗22M22 ≤ − I in C+ ∪C0 ∪∞. ThereforeM22 is nonsingular in this region. Hence, being stable,

M22 is in GR Hp×p∞ . The rest follows from manipulations. Note thatG =

[ I −M120 −M22

]−1[ 0 M11−I M21

]

.

This shows thatG∼G = I iff M∼ JM = J and thatG∗G ≤ I in the right-half plane iffM∗ JM ≤ Jin the right-half plane. In other words,M is J-lossless iffG is inner.

(Item 2) According to the discussion aboveM is J-lossless iffG is inner. By construction wehave thatG in Item 1 satisfiesG∼G = I . Hence,G is inner iff G in addition is stable. ObviouslyG is stable iffM−1

22 is stable.

Recall thatM is co-Jq,p-lossless ifMT is Jq,p-lossless. Therefore CorollaryA.0.13, Item 2shows thatM is co-Jq,p-lossless iffM Jq,pM∼ = Jr,p and the lower rightp× p block element ofM is in GR H

p×p∞ .

J-lossless matrices correspond to systems that in a way are without loss:

1A matrix P is E-unitary if P∼ EP= E.

110

Corollary A.0.14 (Lossless convolution systems, ( 7)). Suppose that the matrix M∈R H

(r+p)×(q+p)∞ is a transfer matrix of a convolution system with input

[ uy]

and output[

zw

]

,where y andw are in L2(R;Rp), u ∈ l2(R;Rq) and z∈ L2(R;Rr ). Then M is Jq,p-lossless iffthe inequality

∫ T

−∞‖z(t)‖2 − ‖w(t)‖2 dt ≤

∫ T

−∞‖u(t)‖2 − ‖y(t)‖2 dt (A.4)

holds for all inputs inL2(R;Rq+p) and time T∈ R, with equality holding for T= ∞.

Proof . In this proof a time signal and its Laplace transform are identified and they are denotedby one symbol. For convenience defineJ = Jq,p and J = Jr,p.

(Only if ) Let M be J-lossless and suppose firstp = 0 (that is, thatw andy are void). Stateddifferently, suppose thatM is inner. From the Paley-Wiener results (Theorem2.1.20) it followsthat‖z‖2

2 = ‖Mu‖22 = ‖u‖2

2. Hence forT = ∞ indeed equality holds inA.4. If inequality (A.4)were to fail for some finite timeT and inputu = u, thenu redefined asu(t) = u(t) for t < T,andu(t) = 0 for t ≥ T would contradict theT = ∞ case. Hence, inequality (A.4) holds for allfinite time T. By CorollaryA.0.13 inner systems andJ-inner systems basically represent thesame system (just convert some inputs into outputs and vice versa). This proves the only if part.

(If ) ‖z‖22 − ‖w‖2

2 = 〈[ u

y]

,M∼ JM[ u

y]

〉. By assumption‖z‖22 − ‖w‖2

2 = ‖u‖22 − ‖y‖2

2, andhence,M∼ JM = J. It follows from CorollaryA.0.13, Item 2 that the proof is complete if wecan showM22 is in GR H

p×p∞ . Here M22 is the lower rightp × p block of M. We proof by

contradiction thatM22 is in GR Hp×p∞ . PartitionM as

M =[

M11 M12

M21 M22

]

, M22 ∈ R Hp×p∞ .

Writing out M∼ JM = J for the lower right block reveals thatM∼22M22 = M∼

12M12 + I . So M22

is nonsingular onC0 ∪ ∞. ThereforeM22 is in GR Hp×p∞ iff in addition it is nonsingular inC+.

Suppose, to obtain a contradiction, thatM22 is singular somewhere inC+. That is,M22(α)v = 0for someα ∈ C+ and constant vectorv. Definey(s) = 1

s−αv andu ≡ 0. Then[

zw

]

=[

M12

M22

]

1s− α

v.

w defined this way is inH p2 (the possible unstable poles = α cancels). This means that the

corresponding time signalsy(t) andw(t) are of the form

y(t) =

−eαtv for t < 00 for t ≥ 0

w(t) =

0 for t < 0• for t ≥ 0

.

So we have for thisu andy that∫ 0

−∞‖z(t)‖2 − ‖w(t)‖2 dt>

∫ 0

−∞‖u(t)‖2 − ‖y(t)‖2 dt.

This contradictsA.4, hence,M22 is in GR Hp×p∞ . We silently assumed thaty as defined above is

a real valued signal. Of course this is in general not the case. To solve this not very interestingproblem, remember that sinceM22 is real-rational, zeros ofM22 come in conjugate pairs. A littlecontemplation then reveals that the same arguments work forthe “real” signal

y(s) = 1s− α

v+ 1s− α∗ (v

T)∗

instead ofy(s) = 1s−αv.

111


J-lossless matrices come in naturally in the theory of indefinite inner product spaces (forinstance Kreın spaces, see (3; 14) and Chapter3), a subject not pursued in this appendix.

G

K

Ky

y

u

u

w

w z

z

M

System (A) System (B)

Figure A.1.:

Lemma A.0.15 (Redheffer’s lemma, ( 22; 73)). Consider the closed-loop systems (a) and (b) asin Fig. A.1and assume they depict convolution systems with inputw and output z, represented intransfer matrix form by

(a) :

[

zy

]

=[

G11 G12

G21 G22

][

w

u

]

u = Ky, (b) :

[

zw

]

=[

M11 M12

M21 M22

][

uy

]

u = Ky

for system (a) and (b) respectively. Partition M and G compatibly with the partitioning of thesignals as

M =[ M11 M12

M21 M22

]

∈ R H(r+p)×(q+p)∞ and G=

[ G11 G12G21 G22

]

∈ R H(r+p)×(p+q)∞ .

Then:

1. If G is an inner real-rational matrix whose lower left p× p block element G21 is inGR H

p×p∞ , then the closed-loop transfer matrix H:= G11 + G12K( I − G22K)−1G21 from

w to z in system (a) is contractive and the closed-loop systemL2-stable iff the rationaltransfer matrix K is a stable contraction (that is, K∈ R H

q×p∞ and‖K‖∞ ≤ 1). Further-

more,‖H‖∞ < 1 iff ‖K‖∞ < 1.

2. If M is a real-rational stable Jq,p-lossless matrix, then the transfer matrix H:= (M11K +M12)(M21K + M22)

−1 fromw to z in system (b) is contractive and the closed-loop systemL2-stable iff the rational transfer matrix K is a stable contraction (that is, K∈ R H

q×p∞

and‖K‖∞ ≤ 1). Furthermore,‖H‖∞ < 1 iff ‖K‖∞ < 1.

Proof . By Corollary A.0.13 the items are equivalent. We prove Item 2. First we discuss atechnicality. DefineJ = Jq,p and J = Jr,p. According to Sylvester’s Inertia law (see (52)) itfollows from

[

I 0−M∼ J I

][

J MM∼ J

][

I −JM0 I

]

=[

J 00 J − M∼ JM

]

=[

J 00 0

]

112

and[

I −M J0 I

][

J MM∼ J

][

I 0− JM∼ I

]

=[

J − M JM∼ 00 J

]

that J − M JM∼ ≥ 0 on the imaginary axis, including infinity. This implies that M21M∼21 −

M22M∼22 ≤ − I on the imaginary axis including∞. In particular this implies—and this is what

we need—that‖M−122 M21‖∞ < 1.

H ∈ R L r×p∞ exists (stable or unstable) iffA := (M21K + M22)

−1 exists. We then have

[

HI

]

= M

[

KI

]

A.

From this expression it follows that(H∼ H − I ) = A∼(K∼K − I )A. Since‖H‖∞ ≤ 1 iff(H∼ H − I ) ≤ 0 on the imaginary axis, we see that‖K‖∞ ≤ 1 is necessary and sufficient forH ∈ R L r×p

∞ to be contractive. Assume therefore from now on that‖K‖∞ ≤ 1.Write K = N D−1 as a right coprime fraction overR H ∞. Then

[

uy

]

=[

KI

]

(M21K + M22)−1w =

[

ND

]

(M21N + M22D)−1w.

This shows, by right coprimeness ofN and D, that (M21N + M22D)−1 must be stable forthe closed-loop to be stable. By LemmaA.0.8, M21N + M22D = M22(M−1

22 M21K + I )D hasas many unstable zeros asM22D because by the multiplicative property‖M−1

22 M21K‖∞ ≤‖M−1

22 M21‖∞‖K‖∞ ≤ ‖M−122 M21‖∞ < 1. Therefore for stability it is necessary that(M22D)−1

is in R Hp×p∞ . BecauseM22 ∈ GR H

p×p∞ this is equivalent toD−1 being inR H

p×p∞ . Hence

K = N D−1 is in R Hq×p∞ . It is easily checked that then the whole closed-loop is stable. Note

that A = (M21K + M22)−1 is in GR H

p×p∞ if K is stable and contractive. In particularA is then

nonsingular onC0 ∪ ∞, and so it follows from(H∼ H − I ) = A∼(K∼K − I )A that‖H‖∞ < 1iff ‖K‖∞ < 1.

The map fromK to G11 + G12K( I − G22K)−1G21 is called alinear fractional transformation(see, for example, McFarlane and Glover (56)). The same terminology may be used for the themap fromK to (M11K + M12)(M21K + M22)

−1. It is a matrix version of the Mobius map (seeConway (18)).

113


114

B

Polynomial and Rational Matrices

In this appendix we review elementary properties of polynomial (SectionB.1) and rational ma-trices (SectionB.2). SectionB.3 is on Wiener-Hopf factorization of rational matrices. Mostof the results presented in this appendix may be found in Kailath (39), Callier and Desoer (16)and Vidyasagar (82). Some references on Wiener-Hopf factorization are (11; 17; 32), Green et.al. (34) and Helton (36). An algorithm is listed at the end of SectionB.1 which may be used toprove some of results stated. An extension of this algorithmis given in SectionB.3 where it isused to prove results on Wiener-Hopf factorization of rational matrices.

B.1. Polynomial matrices

The symbolP denotes the set of all real polynomials in one variable and with coefficients inRp×q for some integersp andq. That is,P ∈ P means

P(s) = P0 + sP1 + s2P2 + · · · + snPn

for somen ∈ Z+ andPi ∈ Cp×q. P n×m denotes the set ofn× mmatrix valued polynomials inP .A polynomial matrix isnonsingularif it is square and its determinant is not the zero polynomial.

Definition B.1.1 (Submatrix). A matrix that remains after removing several rows and columnsfrom a givenP ∈ P is called asubmatrixof P. n × n submatrices ofP are submatrices inP n×n.

Definition B.1.2 (Rank). Therank (or, normal rank) of a polynomial matrixP ∈ P , denoted byrankP, is defined as

rankP = maxn | there is a nonsingularn× n submatrix ofP.

A matrix P ∈ P p×q hasfull row rank if rank P = p, it hasfull column rankif rank P = q.

Definition B.1.3 (Zero). s∈ C is azeroof P ∈ P if rank P(s) < rankP.

The degree of a scalar polynomialP(s) = p0 + sp1 + · · · + sn pn is by definition equal tomax j | pi = 0 if i > j. The degree of a scalar polynomialP ∈ P is denoted byδP. If P isthe zero polynomial we setδP = −∞. The matrix version of “degree” depends on what role thepolynomial matrix plays.

Example B.1.4 (Degree). TakeP(s) =[

1 s0 1

]

and P(s) =[

1 00 1

]

.

1. P andP are equivalent in the sense thatP(d/dt)w(t)= 0 if and only if P(d/dt)w(t)= 0.

115

B. Polynomial and Rational Matrices

2. P and P are different as transfer matrices, that is,y = Pu 6⇔ y = Pu.

A degree for transfer matrices (polynomial or rational) is the McMillan degree. This is de-fined in SectionB.2. Here we define a degree for polynomial matrices which is in line with theobservation made in Item 1 of ExampleB.1.4.

Definition B.1.5 (Degree, ( 81)). Thedegreeof a P ∈ P with rankq is denoted byδP and definedas

δP = maxq× q submatricesM of P

δdetM .

Example B.1.6 (Degree). δ[

1 s0 1

]

= δ[

1 00 1

]

= 0.

Definition B.1.7 (Unimodular). A U ∈ P is unimodular(in P ) if U−1 ∈ P .

ObviouslyU ∈ P is unimodular iff detU is a nonzero constant. From Lemma2.1.2, Item 1we know that ifP has full row rank, thenP(d/dt)w(t)= 0 if and only ifU P(d/dt)w(t)= 0 forsome unimodularU. Stated differently, if it is the set ofws we are interested in we may exploitthe freedom (the multiplication byU) to our advantage. For example, in ExampleB.1.4, Item 1the matrixP looks easier to work with thanP. This leads to the notion ofrow reducedness.

Definition B.1.8 (Row and column degrees). ρi(P) is by definition theith row degreeof aP ∈ P n×m: ρi(P) = maxd | d = δPik, k ∈ m. (HerePik is the element in theith row andkthcolumn ofP.) γk(P) := ρk(PT) is thekth column degreeof P. ρ(P) is defined as the sum of rowdegrees of a full row rank polynomial matrix.γ(P) is by definition the sum of column degreesof a full column rank matrixP.

Definition B.1.9 (Row reduced matrices). A P ∈ P n×m is row reducedif

lims→∞

s−ρ1(P)

. . .s−ρn(P)

P(s)

exists and has full row rank.P is column reducedif PT is row reduced.

Lemma B.1.10 (Row reducedness, ( 39)). For every P∈ P that has full row rank there existunimodular U such that U P is row reduced.

A full row rank P is row reduced iffδP =∑

ρi (P). If polynomial matrices are used forcomputational purposes it is often desirable to have them inrow reduced or column reducedform.

Theorem B.1.11 (Smith form, ( 39)). Every polynomial matrix P∈ P can be written as

P = U

[

D 00 0

]

V, D =

ε1

. . .εq

, (B.1)

with U and V unimodular matrices of appropriate sizes andεi scalar monic polynomials suchthat εi|εi+1. For a given P theεi defined this way are unique.

116

B.1. Polynomial matrices

By εi |εi+1 it is meant thatεi+1/εi is polynomial. The expression forP as given in TheoremB.1.11is referred to as aSmith formof P and theεi are called theinvariant polynomialsof P.The Smith form reveals much information. For example, the rank of P in (B.1) equalsq andthe zeros ofP are the zeros of the invariant polynomials ofP. The number of zerosof P isby definition

∑

δεi. For square nonsingular matricesP this number of zeros equalsδgraadP.CorollaryB.1.13is immediate from TheoremB.1.11.

Definition B.1.12 (Left and right prime). P ∈ P is left prime if there is anR ∈ P such thatPR= I . P is right prime if PT is left prime.

It follows from the Smith form thatP is left prime iff P(s) has full row rank for everys∈ C.

Corollary B.1.13 (Left factors). A full row rank P∈ P may be written as P= FR, where F∈ Pis square nonsingular and R∈ P is left prime.

We end this section with an algorithm that may be used to proveCorollaryB.1.13and LemmaB.1.10constructively. The algorithm is trivial.

Algoritm B.1.14 (Left-sided factor extraction). [Given a full row rank polynomial matrixP ∈P n×m and a zeroζ of P the algorithm produces anF ∈ P n×n andR ∈ P n×m such thatP = FR,with F square nonsingular withζ as its unique zero.]

STEP 1. Find a constant null vectore= (e1, . . . ,en) ∈ C1×n such thateP(ζ) = 0.

STEP 2. Select a pivot indexk from the active index setN = j | ej 6= 0, j ∈ n.

STEP 3. DefineF as

F(s) =

1...

1− e1

ek· · · − ek−1

eks− ζ − ek+1

ek· · · − en

ek

1...

1

.

STEP 4. SetR equal toP except for thekth row of R which is set to the polynomial row vector1ek

eP(s)/(s− ζ).

The extraction of a zero performed this way may have the undesirable effect thatR is not rowreduced even ifP is row reduced. This problem can be circumvented.

Corollary B.1.15 (Row reduced extractions). Given a row reduced P∈ P the factor R asproduced by AlgorithmB.1.14is row reduced if and only if the pivot index k as selected in step 2of AlgorithmB.1.14is chosen from the maximal active index set

M = j | ej 6= 0 andρ j(P) ≥ ρq(P) for all for which eq 6= 0 .

Moreover, in this caseρi (R) = ρi(P) if i 6= k, andρk(R) = ρk(P)− 1.

117


Proof . Without loss of generality assume pivot indexk = 1. Let r i andρi denote theith rowdegree ofR andP, respectively. Then

lims→∞

s−ρ1

. . .s−ρn

P(s) = lim

s→∞Ω(s)

s−r1

. . .s−rn

R(s)

with

Ω(s) =

(s− ζ)s−ρk+rk − e2ek

s−ρk+r2 · · · − enek

s−ρk+rn

s−ρ2+r2

. . .s−ρn+rn

R is row reduced given the fact thatP is row reduced iffΩ is biproper, that is, iffrk = ρk − 1,r i = ρi if i 6= k andk ∈ M .

Extraction of “zeros at infinity” may also be performed in this manner. By this we mean that,if P ∈ P has full row rank but is not row reduced then a modified versionof Algorithm B.1.14may be used to find a unimodularF and polynomialR such thatP = FR andρ(R) < ρ(P).This process may be repeated withR instead ofP and stops the moment there are no zeros leftat infinity, that is, it stops whenR is row reduced (see (51)).

Most of the results are stated here for full row rank polynomials. Similar results exist for fullcolumn rank polynomials. The extraction algorithm we borrowed from (15), where a symmetricversion is given. We have not found a proof of CorollaryB.1.15in the literature which is thereason we add one here. The results stated in this section arefor real polynomials. However theextraction of zeros in AlgorithmB.1.14may give rise to complex polynomials. A “real” versionof the factor extraction algorithm is a bit technical, though essentially not more complicated (see(48; 63)). This “real” version is based on the fact that complex zeros of real polynomials comein conjugate pairs.

B.1.1. Linear polynomial matrix equations

Lemma B.1.16 (Generalized Bezout identity, ( 39)). Given a P∈ P there exist polynomial ma-trices Q, R, T ∈ P of appropriate sizes such that

[

PT

][

R Q]

=[

I 00 I

]

(B.2)

if and only if P is left prime.

This lemma may be proved using the Smith form ofP. Note that[

PT

]

and[

R Q]

are eachother’s inverse and, hence, are both unimodular.

Corollary B.1.17 (Linear equations). Suppose P∈ P has full row rank.

1. There is a right prime Q∈ P such that PM= 0 for M ∈ P iff M = QF for some F∈ P .

2. There exists a unimodular U such that PU=[

F 0]

with F square nonsingular.

118

B.2. Real-rational matrices and fractions

Corollary B.1.18 (Degrees). If the generalized Bezout identity (B.2) holds for some P, Q, R, Tin P , thenδP = δQ andδR= δT.

Proof . There are many ways to prove this result. We give a proof basedon the fact thatδP =12δPP∼ = 1

2δP∼ P. From

δ

[

PQ∼

]

= δ

([

PQ∼

][

R Q])

= δ

[

I 0• Q∼ Q

]

,

and

2δ

[

PQ∼

]

= δ

([

PQ∼

][

P∼ Q])

= δ

[

PP∼ 00 Q∼ Q

]

we deduce thatδPP∼ = δQ∼ Q, i.e.,δP= δQ. Completely similarly it follows thatδR= δT.


Polynomial matrices and real-rational matrices have many features in common.R denotes theset of real-rational matrices, that is,R is in R iff qR is in P for someq ∈ P . A square rationalmatrix R is (non)singularif its determinant is (not) the zero function. A matrix that remainsafter removing several columns and rows fromR∈ R is called asubmatrixof R. Therank of anR∈ R is defined as

rankR := maxn | there is a nonsingularn× n submatrix ofR.

A rational matrixR∈ R g×m hasfull row rank if rank R= g, it hasfull column rankif RankR=m. If we say that a matrixR ∈ R g×m has full column rank on, say,C0, then we mean thatrankR(s) = m for all s∈ C0.

Theorem B.2.1 (Smith-McMillan form, ( 39)). Every real-rational matrix G∈ R may be writtenas

G = U

[

D 00 0

]

V; D =

ε1ψ1

. . .εq

ψq

, (B.3)

with U and V polynomial unimodular matrices of appropriate sizes,εi scalar monic polynomialssuch thatεi|εi+1 andψi scalar monic polynomials such thatψi+1|ψi . Given P theεi/ψi definedthis way are unique.

The expression forG as given in TheoremB.2.1is referred to as aSmith-McMillan formof Gand theεi/ψi are called theinvariant rational functionsof G.

A numbers ∈ C is azeroof G if it is a zero of one its invariant rational functions;s ∈ C is apoleof G if it is a pole of one its invariant rational functions. It is also useful to have a definitionof zeros and poles at infinity, but the Smith-McMillan form isnot suitable for this.W ∈ P is aleft factorof P ∈ P if P = W Rfor some polynomial matrixR.

Definition B.2.2 (Coprime polynomials). Polynomial matricesN andD areleft coprimeif theydo not have common left factors other than unimodular left factors. Polynomial matricesN andD areright coprimeif NT andDT are left coprime.

119


ObviouslyN andD are left coprime iff[

−N D]

is left prime. We say thatG = D−1N is aleft coprime polynomial matrix fraction description(left coprime PMFD) ofG if N and D areleft coprime andG = D−1N. Right coprime PMFDs are similarly defined.

Lemma B.2.3 (Polynomial fractions of rational matrices, ( 39)). Every G∈ R admits a leftcoprime PMFD and a right coprime PMFD. Left coprime PMFDs of agiven G∈ R are uniqueup to multiplication from the left by a polynomial unimodular matrix (that is, if Ni and Di are leftcoprime, then D−1

1 N1 = D−12 N2 iff

[

−N1 D1]

= U[

−N2 D2]

for some unimodular U.)

Definition B.2.4 (McMillan degree). TheMcMillan degreeof G ∈ R is denoted byδM G anddefined asδM G = δ

[

−N D]

, whereN and D are left coprime polynomials satisfyingG =D−1N.

If G is proper then theψi from a Smith-McMillan form (B.3) of G satisfy

δM G =∑

δψi (B.4)

(see (57)). The McMillan degree is usuallydefinedthrough (B.4) (see (39; 75; 81; 16)). Wedeliberately take another definition than the one given in (39; 75; 81; 16), though the usualdefinition is equivalent to ours (see (57; 81)). Definition B.2.4has an advantage over the usualdefinition of McMillan degree: DefinitionB.2.4 works also for the case whenG is nonproper.In (39; 75) the nonproper case is handled by mapping the point infinity via a Mobius map to theinterior of C. The definition in (39; 75; 81; 16) has the advantage of showing directly (more orless by definition) that the McMillan degree is a measure of the number of poles ofG. However,it is also common in system theory to view the McMillan degreeas the minimal number ofintegrators (the number of states) needed to build the system y = Gu (see (39; 86)). The numberof states can be read off from the underlying differential equationNu= Dy (assuming that thereare no hidden modes), and it equalsδ

[

−N D]

(see (86)).

Example B.2.5 (McMillan degree). A left coprime PMFD ofG(s) =[

1 1s+1

0 s

]

is

G = D−1N, with[

−N D]

=[

−(s+ 1) −1 s+ 1 00 −s 0 1

]

.

Therefore,

δM G = δ

[

−(s+ 1) −1 s+ 1 00 −s 0 1

]

= 2

G has one pole ats= −1 and one “pole at infinity”.

From the definition of McMillan degree it is immediate thatδM G = δM G−1 wheneverG isinvertible. If D−1N = ND−1 = G are a left- and right coprime polynomial fraction ofG, thenby CorollaryB.1.18

δ[

−N D]

= δ

[

DN

]

.

In particular this shows thatδM G = δM GT.

120


B.2.1. Fractions over R H ∞

We briefly discuss properties of fractions of rational matrices over the set of rational matricesin R H

•×•∞ . For a good reason this subject has gained much popularity over the last years. The

standard book on this subject is Vidyasagar (82). Recall thatR Hg×m∞ denotes the set of real-

rationalg × m matrices that are proper—and so their “poles” are already defined—and have alltheir poles inC−.

A rational matrix G ∈ R Hg×m∞ is by definition left prime overR H ∞ if there exist H ∈

R Hm×g∞ such thatGH = Ig. Two matricesN, D ∈ R H

•×•∞ are left coprimeover R H ∞ if

[

−N D]

is left prime overR H ∞. Two matricesD, N ∈ R H•×•∞ areright coprimeoverR H ∞

if NT andDT are left coprime. With help of the results on polynomial matrices it may be showedthatG ∈ R is left prime overR H ∞ iff G ∈ R H

•×•∞ andG has full row rank onC0 ∪ C+ ∪ ∞.

Definition B.2.6 (Stable fractions). G = D−1N ∈ R g×m is a left coprime fraction overR H ∞of G if N ∈ R H

g×m∞ , D ∈ R H

g×g∞ are left coprime overR H ∞.

Theorem B.2.7 (Stable fractions, ( 82)). Every G∈ R g×m admits a left coprime fraction overR H ∞. Moreover, D−1N = G andD−1N = G are two left coprime fractions overR H ∞ of thesame G iff

[

−N D]

= U[

−N D]

for some real-rational U∈ GR Hg×g∞ .

Similar statements hold for right coprime fractions. In general

δM[

−N D]

≥ δM G, (B.5)

if G = D−1N. Given G ∈ R there always existN and D left coprime overR H ∞ such thatequality holds in (B.5). This and TheoremB.2.7may be proved using the results on polynomialfractions. As an extension of the definition of a zero in the interior ofC, we say thats∈ C+ ∪C0 ∪∞ is a zero ofG if rank N(s) < rankN whereN comes from a left coprime fractionG = D−1NoverR H ∞ of G. With the same datas∈ C+ ∪ C0 ∪ ∞ is a pole ofG if it is zero of D. Note thatpoles and zeros defined this way do not depend on the choice of coprime fraction overR H ∞taken. Note also, however, that this definition does not workfor possible zeros and poles ofG inC−.

Lemma B.2.8 (Generalized Bezout identity, ( 82)). Given a P∈ R H•×•∞ there exist Q, R, T ∈

R H•×•∞ of appropriate sizes such that[

PT

][

R Q]

=[

I 00 I

]

(B.6)

if and only if P is left prime overR H ∞.

Corollary B.2.9 (Linear rational matrix equations). Let H ∈ R L g×m∞ and suppose it has full

row rank at infinity. Then there exist W∈ GR Hm×m∞ such that

HW =[

E 0]

with E ∈ R L g×g∞ square, nonsingular and biproper.

Proof . After possible permutation of columns ofH we may assume thatH =[

H1 H2]

withH2 is nonsingular. DefineG = H−1

2 H1 and letD−1N = G be a left coprime fraction ofG overR H ∞. Then H = H2D−1

[

N D]

. Since[

N D]

is left prime overR H ∞ there exists byLemmaB.2.8a W ∈ GR H

m×m∞ such that

[

N D]

W =[

I 0]

. DefineE by[

E 0]

:= HW.This E is biproper becauseH is proper and has full row rank at infinity.

121


B.3. Wiener-Hopf factorization

To introduce Wiener-Hopf factorization of rational matrices we consider the scalar case. Ifz is ascalar biproper1 real-rational function without poles and zeros onC0, then

z(s) = w∼1 (s)(

s− 1s+ 1

)kw2(s) for somew1, w2 ∈ GR H ∞ andk ∈ Z. (B.7)

To see this writez= nd wheren andd are coprime polynomials. Writen = n−n+ as the product

of n− andn+ whose zeros lie inC− andC+, respectively. Similarly, writed = d−d+ with d−andd∼

+ strictly Hurwitz. The obvious choice forw2 is n−d−

, however the latter expression need not

be biproper. This is fixed by lettingw2 = n−d−(s+ 1)k for that k ∈ Z for whichw2 is biproper.

Similarly, definew∼1 = n+

d+(s− 1)m for that uniquem for whichw1 is biproper. Obviously, since

z is biproper, we havem= −k and, hence, (B.7) holds. The factorization ofz in (B.7) is called aWiener-Hopf factorization ofz. The function( s−1

s+1)k in (B.7) is referred to as the winding function

of z, and the (unique) indexk is the factorization index ofz. We make distinction between thecasek = 0 andk 6= 0; (B.7) is acanonical factorizationof z if k = 0; if k 6= 0 then we refer to(B.7) as anoncanonical factorizationof z.

We focus on factorization of real-rational para-HermitianmatricesZ = Z∼ mainly. Most ofthe results presented here may be found in (11; 17; 32; 36; 34; 59). The results concerningfactorization of para-Hermitian matrices are proved in detail.

B.3.1. Canonical factorization of para-Hermitian matrice s

Throughout this sectionJ denotes asignature matrix, that is, a matrix of the form

J =[

Iq 00 − I p

]

for some integersq and p. The matrix above is abbreviated toJq,p. Z is calledpara-Hermitianif Z = Z∼.

Definition B.3.1 (Canonical factorization). Given aZ ∈ R m×m, the expressionZ = W∼ JW isa (symmetric) canonical (spectral) factorizationof Z if W is in GR H

m×m∞ . W in this case is a

canonical factorof Z. Z = W JW∼ is a canonical cofactorizationof Z if W ∈ GR Hm×m∞ . In

this caseW is acanonical cofactorof Z.

Necessary for the existence of such factorizations is thatZ = Z∼ and thatZ has no poles andzeros onC0 ∪ ∞. If W∼ JW = Z is a canonical factorization ofZ then by Sylvester’s inertialaw (see (52)) we have for allω ∈ R that Z(iω) has as many negative eigenvalues and positiveeigenvalues asJ.

Definition B.3.2 (Triple of inertia, ( 29)). The (triple of) inertia In(C) of a constant matrixC ∈ Cm×m is a triple of integers denoting the number of eigenvalues ofC in C+, C− andC0

respectively. IfZ ∈ R m×m is proper, then In(Z) = In(C) is an abbreviation of In(Z(s)) = In(C)for all s∈ C0 ∪ ∞.

The necessary conditions onZ derived just now are not always sufficient as the next exampleshows.

1z is biproper ifz andz−1 are both proper.

122


Example B.3.3 (Canonical factorization). Let Z be

Z(s) =[

0 s−1s+1

s+1s−1 0

]

.

This para-HermitianZ and its inverse are inR L 2×2∞ and In(Z) = In

[1 00 −1

]

. Note that with

u :=[ 1

s+10

]

∈ H 22 we have thaty := Zu is in H ⊥2

2 . This is the reason whyZ does not have acanonical factorization, for if it would have a factorization Z = W∼ JW, thenW−∼ y = JWuyields a contradiction because the left hand side is inH ⊥2

2 whereas the right hand side is inH 22

and nonzero.

Theorem B.3.4 (Canonical factorization). A Z = Z∼ with Z, Z−1 ∈ R L m×m∞ admits a canoni-

cal factorization if and only if there does not exist nonzerou ∈ H m2 such that Zu∈ H ⊥m

2 .

Proof . In ExampleB.3.3it is shown that suchu can not exist ifZ admits a canonical factoriza-tion. The converse may be proved constructively. WriteZ = A + A∼ with A ∈ R H

m×m∞ and

with it defineG = 12

[I+AI−A

]

. This wayZ = G∼ Jm,mG. In the proof of TheoremB.3.7a canonicalfactorization is constructed if there do not existu ∈ H m

2 such thatGJG∼u ∈ H ⊥m2 .

Nonzero vectorsu in H m2 for which Zu ∈ H ⊥m

2 for lack of a better name are referred to ascritical vectorsof Z.

Corollary B.3.5 (Ordinary spectral factorization). There is a W∈ GR Hm×m∞ such that Z=

W∼W for some given Z if and only if Z, Z−1 ∈ R L m×m∞ , Z = Z∼, and Z(iω) > 0 for all ω ∈

R.

Proof . If u is a critical vector ofZ, then 〈Zu,u〉 = 0 becauseu ∈ H m2 and Zu ∈ H ⊥m

2 areperpendicular. On the other hand〈Zu,u〉 = 1

2π

∫ ∞−∞ u∗(iω)Z(iω)u(iω) dω is positive because

Z(iω) > 0 by assumption. This is a contradiction, henceZ has no critical vectors and, therefore,admits a canonical factorizationZ = W∼ JW. In(J) = In(W∼ JW) = In(Z) = In( I ). ThereforeJ = I . The converse is trivial.

Lemma B.3.6 (Uniqueness of canonical factors, ( 34)). Canonical factors W in a canonicalfactorization Z= W∼ Jr,pW of a given Z∈ R (r+p)×(r+p) are unique up to multiplication fromthe left by a constant Jr,p-unitary2 matrix.

Proof . If rationalW ∈ GR H(r+p)×(r+p)∞ and rationalW ∈ GR H

(r+p)×(r+p)∞ satisfyW∼ Jr,pW =

W∼ Jr,pW, thenM := WW−1 is in GR H(r+p)×(r+p)∞ and M∼ Jr,pM = Jr,p. ThereforeM−1 =

Jr,pM∼ Jr,p. The left-hand side ofM−1 = Jr,pM∼ Jr,p is stable and the right-hand side as anti-stable, hence,M is constant.

The theorem presented next is a variation of the canonical factorization theorem developed in(11). For completeness we include a proof.

Theorem B.3.7 (Canonical factorization using state space m ethods, cf.( 34)). Sup- pose G∈R H

(q+p)×(r+t)∞ is given and suppose G has full column rank onC0 ∪ ∞. Let G(s) = C(sI −

A)−1B+ D be a realization of G withλ(A) ⊂ C− such that[

A− sI BC D

]

has full column rank for all s∈ C0. Let J= Jq,p and J = Jr,t and let n denote the dimension ofthe matrix A: A∈ Rn×n. The following statements are equivalent.

2U ∈ Cm×m is by definitionJr,p-unitary if U∼ Jr,pU = Jr,p.

123


1. Z := G∼ JG admits a canonical factorization Z= W∼ JW.

2. D∗ J D = W∗∞ JW∞ has a nonsingular solution W∞ and the stable eigenspace of

H× :=[

A 0−C∗ JC −A∗

]

−[

B−C∗ J D

]

(D∗ J D)−1[

D∗ JC B∗] ∈ R2n×2n

is of the formIm[ X1

X2

]

with X1, X2 ∈ Rn×n and X1 nonsingular.

3. D∗ J D = W∗∞ JW∞ has a nonsingular solution W∞ and there is a solution P of the Riccati

equation

PA+ A∗ P− [ PB+ C∗ J D](D∗ J D)−1[ D∗ JC+ B∗ P] + C∗ JC = 0

such that A− B(D∗ J D)−1[ D∗ JC+ B∗ P] has all its eigenvalues inC−.

In the case that the conditions above are satisfied we have that W(s) = JW−∗∞ [ D∗ JC +

B∗ P](sI − A)−1B+ W∞ is a canonical factor of G∼ JG.

(1) =⇒ (2). It follows from the Hamiltonian structure ofH× thatλ is eigenvalue ofH× iff −λis an eigenvalue ofH×. Therefore the dimension of the stable eigenspace ofH× is n iff H× hasno eigenvalues onC0. We next show thatH× cannot have imaginary eigenvalues. Note that

G∼ JGs=

[

H BC D

]

:=[

A 0 B−C∗ JC −A∗ −C∗ J D

]

D∗ JCB∗ D∗ J D, (B.8)

and

(G∼ JG)−1 s=[

H× BD−1

−D−1C D−1

]

; (H× = H − BD−1C).

BecauseH has no eigenvalues onC0, we conclude from[

H − sI BC D

] [

I −(H − sI)−1B0 I

]

=[

H − sI 0C G∼ JG

]

,[

I −BD−1

0 I

] [

H − sI BC D

]

=[

H× − sI 0C D

]

that eigenvalues ofH× on C0 are zeros ofG∼ JG. G∼ JG does not have zeros onC0 because itadmits a canonical factorization.

We next show—and this is the fun part—thatX1 is nonsingular. The idea is to construct acritical vectoru of G∼ JG under the assumption thatX1 is singular. By ExampleB.3.3, thiscontradicts the assumption thatG∼ JG admits a canonical factorization and, hence, completesthe proof. We first do our analysis in the time domain: We construct nonzerou, y, x ∈ L2(R;C•)with u(t) = 0 for t < 0 andy(t) = 0 for t ≥ 0 that satisfy[

xy

]

=[

H BC D

][

xu

]

,

[

H BC D

]

=[

A 0 B−C∗ JC −A∗ −C∗ J D

]

D∗ JCB∗ D∗ J D. (B.9)

First note that the antistable eigenspace ofH equals Im[

0I

]

. For brevity we denote the antistableeigenspace ofH asXantistab(H). Similarly, we denote byXstab(H×) the stable eigenspace ofH×.

Suppose thatX1 is singular, or in other words, suppose that

Xstab(H×) ∩ Xantistab(H) = Im

[

X1

X2

]

∩ Im

[

0I

]

124


is nonempty. Letx(0) be in this intersection. Settingu(t) = 0 for t < 0 implies

y(t) = Cx(t), x = Hx for t < 0,

and sincex(0) is in the antistable eigenspace ofH, we see thaty(t) andx(t) defined this waytend to zero exponentially ast → −∞. We invert system (B.9) so that we can see whatu(t), t ≥ 0is if y(t) = 0 for t ≥ 0:

[

xu

]

=[

H× BD−1

−D−1C D−1

][

xy

]

.

Settingy(t) = 0 for t ≥ 0, we find that

u(t) = −D−1Cx(t), x = H×x for t ≥ 0.

Recall thatx(0) is in the stable eigenspace ofH×. Henceu and x defined this way are inL2(R;C•). Define the two-sided Laplace transforms

y := L(y) = −C(sI − H)−1x(0), u := L(u) = D−1C(sI − H×)−1x(0),

then u is in H r+p2 , y is in H ⊥ r+p

2 and y = G∼ JGu. The only thing that remains to be shownis that u is not identically zero. To see thatu is indeed not the zero function, we argue asfollows. For t ≥ 0 we haveH×x = (H − BD−1C)x = Hx + Bu. Thereforeu is identicallyzero only if x = Hx = H×x for t > 0. The latter is impossible because by assumptionx is inXstab(H×) ∩ Xantistab(H). Consequently, a nonzero critical vectoru exists if X1 is singular. ByExampleB.3.3, this contradicts the assumption thatG∼ JGadmits a canonical factorization and,hence,X1 is nonsingular.

[(2) =⇒ (3)] Define P := X2X−11 , then

[IP

]

spans the stable eigenspace ofH×, so that

H×[

IP

]

=[

IP

]

Λ (B.10)

with λ(Λ) ⊂ C−. ObviouslyΛ = A − B(D∗ J D)−1[ D∗ JC+ B∗ P]. Multiplying (B.10) fromthe left by

[

−P I]

shows[

−P I]

H×[IP

]

= 0. This, expressed term by term, is the Riccatiequation.

[(3) =⇒ (1)] A matter of manipulation (see (34)): W(s) := JW−∗∞ [ D∗ JC + B∗ P](sI −

A)−1B + W∞ satisfies W∼ JW = G∼ JG. W−1(s) = −(D∗ J D)−1[ D∗ JC + B∗ P](sI −Λ)−1BW−1

∞ + W−1∞ , withΛ = A− B(D∗ J D)−1[ D∗ JC+ B∗ P].

B.3.2. Noncanonical factorization

Definition B.3.8 (Asymmetric cofactorization, ( 32)). Z = W1DW∼2 is a (Wiener-

Hopf) cofactorizationof Z ∈ R L m×m∞ , if W1,W2 ∈ GR H

m×m∞ andD is a matrix of the form

D(s) =

( s−1s+1)

k1 0 0

0... 0

0 0 ( s−1s+1)

km

, k j ≥ k j+1 ∈ Z. (B.11)

The matrixD is awindingmatrix of Z and thek j are the(Wiener-Hopf) cofactorization indicesof Z.

125


In (32) it is shown thatZ admits an asymmetric cofactorization iffZ and its inverse are inR L m×m

∞ . The cofactorization indices are unique ((32)). Note that we switched from factorizationto cofactorization (the rational stable matrix in a cofactorization appears on theleft of the windingmatrix).

Definition B.3.9 (Symmetric cofactorization, ( 32)). Z = W DW∼ is a(symmetric Wiener-Hopf)cofactorizationof Z ∈ R L (r+p)×(r+p)

∞ if W ∈ GR H(r+p)×(r+p)∞ andD is a winding matrix of the

form

D =

0 0 0 D+0 Ir−l 0 00 0 − I p−l 0

D∼+ 0 0 0

; D+(s) =

0 0 ( s−1s+1)

kl

0 ...

0( s−1

s+1)k1 0 0

,

k j ≥ k j+1 > 0. (B.12)

The indicesk j are thecofactorization indicesof Z. The cofactorization is called anoncanonicalcofactorizationof Z if D+ is non-void.

In (32) it is shown that a real rationalZ admits a symmetric cofactorization iffZ = Z∼ andZ and its inverse are inR L m×m

∞ . We prove this result constructively in the next subsection. Thecofactorization indices are unique ((32)). Actually, D is unique and In(D) = In(Jr,p) if D is asin (B.12).

If a rational matrixZ ∈ R L g×m∞ has full row rank and no poles onC0 ∪ ∞, then it has a

(rectangular Wiener-Hopf)cofactorizationZ = Z1(D 0)Z∼2 , with rationalZ1, Z2 ∈ GR H

•×•∞

and D diagonal as in (B.11). This result is perhaps less standard than the square case,but caneasily be deduced from the square case. Indeed, supposeP∼ ∈ R H

(m−g)×m∞ is a full row rank

matrix such thatZ P = 0, and suppose thatP has full column rank in the closed left-half plane.Then there exists a completion(V P)∼ ∈ GR H

m×m∞ . The square matrixZ′ defined by(Z′ 0) =

A(V P) admits a cofactorization, from which a cofactorization ofZ can be deduced. Though thereduction to the square case is not unique, the cofactorization indices ofZ are unique. Because,supposeZ = Z∼

1 (D 0)Z2 = Z′1∼(D′ 0)Z′

2 are two cofactorizations ofZ. It is easily seenthat thenD = T+ D′T− for someT+, T∼

− ∈ GR Hg×g∞ . Consequently,D and D′ have the same

cofactorization indices.

B.3.3. Polynomial algorithm for canonical and noncanonica lcofactorization

By m we mean1,2, . . . ,m, the set of positive integers from 1 up to and includingm.

Algoritm B.3.10 (Canonical cofactorization, ( 15; 48; 51; 63)). [ Given a signaturematrix J := Jq,p and aG ∈ R H

m×(q+p)∞ having full row rank onC0 ∪∞ the algorithm produces,

if a canonical cofactorization exists, a signature matrixJ and a canonical cofactorW of GJG∼

such thatGJG∼ = WJW∼.]

STEP (A) Write G as a polynomial left coprime fractionG = M−1N and takeN row reduced.SetA1 := N J N∼ and letm be the number of rows ofN.

STEP (B) n := 12 graaddetA1. Compute alln zerosζ j ∈ C− of detA1. Set the virtual row degrees

d j: d j := ρ j(N) for j ∈ m. Seti := 0.

STEP (C) i := i + 1. Compute a constant null vectore= (e1, . . . ,em) such thateAi(ζi) = 0.

126


STEP (D) Select a pivot indexk from the maximal active index set

Mi = j ∈ m | ej 6= 0 andd j ≥ dl for all l ∈ m for whichel 6= 0 . (B.13)

STEP (E) Compute the polynomial matrixAi+1 = T−1i Ai T−∼

i , whereTi is defined as

Ti(s) =

1...

1− e1

ek· · · − ek−1

eks− ζi − ek+1

ek· · · − em

ek

1...

1

. (B.14)

STEP (F) dk := dk − 1.

STEP (G) If i < n then goto (c).

STEP (H) (An+1 is constant ifGJG∼ has a canonical cofactorization.) FactorAn+1 as

QJQ∼ = An+1, Q ∈ Cm×m. (B.15)

STEP (I) W = M−1T1 · · · TnQ is a canonical cofactor ofGJG∼ = WJW∼.

In this algorithm it is nowhere really used thatG(s) is real-rational (G rational suffices). IfGis real-rational then the extractions may be arranged such that alsoW is real-rational (see (48)).The only step in the algorithm that calls for an explanation is step (H). In Step (H) it is stated thatAn+1 is constant ifGJG∼ admits a canonical factorization. The converse holds as well:

Lemma B.3.11 (Degree structure). In the notation of AlgorithmB.3.10, assume that q of the nstable zeros have been extracted in AlgorithmB.3.10and that the virtual row degrees dj havebeen updated. Define the degree matrix∆q as

∆q =

sd1

. . .sdm

.

Then, independently of whether GJG∼ admits a (non)canonical cofactorization or not,

GJG∼ = M−1T1 · · · Tq∆q︸︷︷︸

=: Wq

∆−1q Aq+1∆

−∼q

︸︷︷︸

=: Dq

∆∼q T∼

q · · · T∼1 M−∼

︸︷︷︸

= W∼q

,

and both Wq and Dq are biproper. Moreover, GJG∼ admits a canonical cofactorization iff∆n = Im.

Proof . ThatWq andDq are biproper may be proved by induction, using a “symmetric”versionof CorollaryB.1.15(see (57)). ThatGJG∼ admits a canonical factorization iff∆n = I is provedin (57). An alternative proof is implicitly given in AlgorithmB.3.12presented next. In Algo-rithm B.3.12a noncanonical cofactorization is constructed given the fact that∆n 6= I . Obvi-ously a matrix cannot both have a noncanonical cofactorization anda canonical cofactorization.That∆n = I is sufficient for the existence of a canonical factorization, is immediate from theconstruction in AlgorithmB.3.10.

127


The role of the “degree” matrix∆q is to keep track of the degree structure during the successiveextractions of zeros fromN J N∼. Algorithm B.3.10completely solves the canonical case. NowsupposeGJG∼ does not admit a canonical cofactorization, but stillGJG∼ and its inverse are inR L m×m

∞ . Then by LemmaB.3.11∆n is nonconstant. The following algorithm is an extensionof Algorithm B.3.10. Assuming∆n is nonconstant it produces a noncanonical cofactorW and awinding matrixD of GJG∼. The algorithm is technical and is included for completeness only.In this algorithm several times we use expressions likeL = F∼(• + β). This means thatL = N∼,with N(s) = F(s+ β).

Algoritm B.3.12 (Noncanonical cofactorization). [Consider the data as in AlgorithmB.3.10and go through the steps of AlgorithmB.3.10until the point is reached that all zeros have beenextracted—that is, thatAn+1 is unimodular—and only STEPS(H) and (I) remain to be done.Assume GJG∼ and its inverse are inR L m×(q+p)

∞ . Proceed with STEPS(H’), ( I ’), ( J’), ( K ’) and (L’).]

STEP (H’) Permute the diagonal entries of∆n such that

∆n =

∆

I t−l−1

F−1

,

∆(s) =

sd1

. . .sdl

, F−1(s) =

sdt

. . .sdm

,

with d j > 0 if j ≤ l andd j < 0 if j ≥ t. PermuteAn+1 correspondingly. (That is, find anorthogonal permutation matrixK such that∆n := K−1∆nK is as above, and replaceAn+1

with An+1 := K−1An+1K−∼ andTn with Tn := TnK.)

Comment:Necessarily,F and ∆ are of the same size (l − 1 = m− t), and An+1 partitionedcompatibly with∆n is of the form

An+1 =

X Y∼ P∼

Y K 0P 0 0

, (B.16)

whereP is an l × l unimodular matrix,X andY are polynomial matrices andK is a constantnonsingular matrix.

STEP (I ’) Let K be as given in (B.16). Find Q ∈ C(m−2l )×(m−2l ) such that

K = QJQ∼.

STEP (J’) Define D andW as

D =

∆(• + α)

QF−1(• + β)

−1

An+1

∆(• + α)

QF−1(• + β)

−∼

,

W = M−1T1 · · · Tn

∆(• + α)

QF−1(• + β)

, α, β > 0,

128


Comment:ThenGJG∼ = WDW∼, W ∈ GR Hm×m∞ andD is biproper and is of the form

D =

A B∼ P∼+

B JP+

(B.17)

for some proper rational matricesA, B and biproperP+.

STEP (K ’) Write P∼+ defined in (B.17) as

P∼+ = ∆−1(• + α)P∼ F∼(• + β) = V1D+V∼

2 (B.18)

with

V1(s) = ∆−1(s+ α)∆(s+ 1)L−11

D+(s) = L1∆−1(s+ 1)∆(s− 1)L2 (B.19)

V∼2 = L−1

2 ∆−1(• − 1)P∼ F∼(• + β).

HereL1 andL2 are constant permutation matrices such thatD+ is of the form

D+(s) =

( s−1s+1)

kl

. ..

( s−1s+1)

k1

, k j ≥ k j+1 > 0.

STEP (L’) Set V ∈ GR Hm×m∞ to

V =

V112 AP−1

+I BP−1

+I

II

V2

,

and with it define

D = V−1DV−∼, W = WV.

ThenW is a noncanonical cofactor andD a symmetric winding matrix ofGJG∼ such thatW DW∼ = GJG∼.

Proof . Suppose STEP (H’) has been performed. From LemmaB.3.11 it follows that∆−1

n An+1∆−∼n is biproper. Owing to the specific structure of∆n and the fact thatAn+1 is

polynomial, the matrix∆−1n An+1∆

−∼n is biproper only if it is of the form

An+1 =

X Y∼ P∼

Y K 0P 0 0

, K constant.

The partitioning is compatible with the partitioning of∆n.Biproperness of∆−1

n An+1∆−∼n implies FP∆−∼ to be proper. The result is thatP can not be

strictly wide, for if it would be strictly wide, some minor ofFP∆−∼ would be nonproper (notethat F and∆ are polynomial with the same degree). HenceP is tall. P strictly tall is out of thequestion, because it would implyAn+1 to be singular. ThereforeP is square unimodular, andFand∆ are bothl × l polynomial matrices.W obviously is inGR H

m×m∞ .

AP−1+ = [∆−1(s+ α)X∆−∼(s+ α)][ F(s+ β)P∆−∼(s+ α)]−1 = ∆−1(s+ α)AP−1F−1(s+

β)which is stable. SimilarlyBP−1+ is stable and, hence,V is inGR H

m×m∞ . The rest is trivial.

129


An immediate consequence of AlgorithmB.3.12is:

Corollary B.3.13 (Cofactorization indices). On exit of AlgorithmB.3.12the virtual row degreesd j in AlgorithmB.3.10equal, up to ordering, the Wiener-Hopf cofactorization indices of GJG∼.

Proof . Equation (B.19) shows that the positive indicesd j, j ≤ l equal the cofactorization in-dicesk j of GJG∼. That the negative virtual row degreesd j, j ≥ t also equal the cofactorizationindices follows from another factorization ofP∼

+ :

P∼+ = ∆−1(• + α)P∼ F∼(• + β)

= ∆−1(• + α)P∼ F(• + 1)L−13

︸︷︷︸

=: V1

L3F−1(• + 1)F(• − 1)L4︸︷︷︸

=: D+

L−14 F−1(• − 1)F∼(• + a)

︸︷︷︸

=: V∼2

.

L3 andL4 are appropriate permutation matrices.

130

C

Proofs

C.1. Proofs of Chapter 4

Proof of Lemma4.2.4. We prove the caseγ = 1 because this simplifies some of the formulas.By CorollaryA.0.13M := Q−1F ∈ R H

(r+p)×(q+p)∞ is co-Jq,p-lossless iffF Jq,pF∼ = QJr,pQ∼

and the lower rightp× p block elementM22 of M is in GR Hp×p∞ . The identity

Q−1 [

Q1 F2]

=[

I •0 M22

]

shows that if a strictly Hurwitz matrixQ satisfies

F Jq,pF∼ = QJr,pQ∼; Q−1F proper (and, hence, stable),

that thenQ−1F is co-Jq,p-lossless iff[

Q1 F2]

is strictly Hurwitz.(If part) Let Q be one such solution. DefineT asT =

[

0 I p]

Q−1. Then[

P R]

= T F =[

0 I p]

Q−1F satisfy

‖R−1P‖∞ < 1; R∈ GR Hp×p∞

sinceQ−1F is co-Jq,p-lossless. In particularR−1 is internally stable, and therefore thisT solvesthe PSTBP.

(Only if part) Let T be one such solution and define[

P R]

:= T F. Let B be a strictlyHurwitz matrix such that

G := B−1F

is stable and has full row rank onC0 ∪ ∞. (SuchB exist becauseF has full row rank onC0.)Next defineT = R−1T B. Then

TG= [ R−1T B] B−1G =[

R−1P I]

,

hence,T solves the STBP as defined in Chapter3. Note that this displayed equation implies thatT is in R H

p×(r+p)∞ because the right hand side is proper andG has full row rank at infinity.)

By Theorem3.2.3and Theorem3.1.12this implies thatGJq,pG∼ = W Jr,pW∼ has a solution

W ∈ GR H(r+p)×(r+p)∞ and thatW−1G is co-Jq,p-lossless. Finally, defineQ = BW given such

a solutionW. Obviously Q = BW has full rank inC+ ∪ C0 and Q−1F = W−1G is co-Jq,p-lossless andQJr,pQ∼ = F Jq,pF∼. Therefore the proof is complete if in addition we can show

131

C. Proofs

that Q is polynomial. Q = BW has possible poles onlyC− becauseB is polynomial andW isstable. Using the identityQJr,pQ∼ = F Jq,pF∼ this then implies that also the right-hand side ofQ = F Jq,pF∼Q−∼ Jr,p has possible poles only inC−. The latter shows that there are no poles,becauseF J F∼—being polynomial—has no poles andQ−∼ Jr,p has possible poles only inC+.ConsequentlyQ is polynomial.

All solutions T to the STBP with dataG are of the formT = A[

U I]

W−1 with A ∈GR H

p×p∞ andU stable and strictly contractive. RewritingT = R−1T B shows thatT solves

the PSTBP iffT is as the Lemma states.

Proof of Lemma4.2.5. We first derive an alternative expression forΩ being strictly Hurwitz. LetU a unimodular matrix depending onX andY such that

[

−Y X]

U =[

0 I]

.

(SuchU exists becauseY andX are left coprime polynomials.) PartitionU compatibly as

U =[

X •Y •

]

.

ThenK = YX−1 is a right coprime polynomial fraction ofK. Consider the following equation,which definesA and B:

[

Q−1

I

][

Q2 D2 −N2

0 −Y X

]

︸︷︷︸

=Ω

[

I 00 U

]

=

0 A •I B •0 0 I

.

We may infer from this expression thatΩ is strictly Hurwitz iff A−1 is internally stable, becauseQ andU do not have unstable zeros. As in the proof of Lemma4.1.4we get thatH = BA−1.

Proof of Lemma4.2.6.

1. First we derive an expression for the transfer matrixH fromw to z induced by

(

−N1

0

∣∣∣∣

D1 D2 −N2

0 −Y X︸︷︷︸

Ω:=

)

w

zyu

= 0.

SinceK = YX−1 is by assumption a right coprime PMFD ofK, we have that[

yu

]

=[

XY

]

l1, (C.1)

and, hence, the closed-loop system is equivalently described by

[

D1 D2X − N2Y][

zl1

]

= N1w, (C.2)

in combination withC.1. Note that[

D1 D2X − N2Y]

is strictly Hurwitz iff Ω is strictlyHurwitz because there exist unimodularU12 andU22 such that

[

D1 D2 −N2

0 −Y X

]

︸︷︷︸

Ω

I 0 00 X U12

0 Y U22

︸︷︷︸

unimodular

=[

D1 D2X − N2Y •0 0 I

]

.

132

C.1. Proofs of Chapter 4

Now assume thatK is a solution to the SSP2. LetE be a unimodular matrix such that

E(D2X − N2Y) =[

0F

]

; F square nonsingular.

PartitionE compatibly asE =[

TV

]

. Multiplying (C.2) from the left byE yields[

T D1 0V D1 F

][

zl1

]

=[

T N1

•

]

w.

ThereforeH = (T D1)−1T N1. Note thatT D1 is strictly Hurwitz because its zeros are

closed-loop poles. In other words, we have

‖H = −R−1P‖∞ < 1;[

R P]

:= T[

−N1 D1]

; R−1 is internally stable.

ApparentlyT solves the PSTBP. So by Lemma4.2.4Q as in Item 1 exists.

2. Basically the same as the proof of Item 2 of Lemma4.1.5.

Proof of Equation4.31. Given are a left and right coprime fraction ofG:

G =[

D1 D2]−1 [

N1 N2]

=[

N1

N2

][

D1

D2

]−1

∈ R (p+r )×(q+t)

Define polynomial matricesA, A andB, B as

A =[

−N1 D1]

, B =[

D2 −N2]

,

A =[

−N1

D1

]

, B =[

D2

−N2

]

.

And for convenience of notation define

Jp,q =[

I p 00 − Iq

]

, Jq,p =[

Iq 00 − I p

]

, Lp,q =[

0 Iq

− I p 0

]

, Lt,r =[

0 Ir

− I t 0

]

.

With these definitions the connection between the left and right fraction ofG may be expressedas

ALp,qA = BLt,r B.

Furthermore, we have forΠ andΠ in this notation

Π = B∼(AJq,pA∼)−1B and Π = B( A∼ Jp,qA)−1 B∼.

We proof that

ΠL∼t,rΠLt,rΠ = −Π,

which is equivalent to equation (4.31).

ΠL∼t,rΠLt,rΠ = B( A∼ Jp,qA)−1B∼ L∼

t,r B∼(AJq,pA∼)−1BLt,r B( A∼ Jp,qA)−1 B∼

= B( A∼ Jp,qA)−1 A∼L∼p,q

︸︷︷︸

C

A∼(AJq,pA∼)−1 A︸︷︷︸

E

Lp,qA( A∼ Jp,qA)−1 B∼︸︷︷︸

C∼

.

133

C. Proofs

This definesC andE. Note thatL∼p,qJq,pLp,q = −Jp,q, and therefore,

ΠL∼t,rΠLt,rΠ = C Jq,pC

∼ + C(E− Jq,p)C∼ = −Π + C(E− Jq,p)C

∼.

Next we show thatC(E − Jq,p) = 0, which then completes the proof. Fixs and introduce twosubspacesV = Im(Jq,pLp,qA) andF = Ker( A∼L∼

p,q). The subspacesV andF are complemen-tary for almost alls because( A∼L∼

p,q)(Jq,pLp,qA) = − A∼ Jp,qA is nonsingular for almost alls.Let πV be the projection alongF ontoV and defineπF similarly. Restricted toF we have

C FπF (E − Jq,p) = 0.

This is immediate becauseC F = 0 F . Restricted toV we have

C VπV (E − Jq,p) = 0,

becauseAJq,p(E− Jq,p) = 0 andC V = −L−1t,r X AJq,p V , with X a left inverse ofB. ThatC V =

−L−1t,r X AJq,p V follows from

C(Jq,pLp,qA) = B( A∼ Jp,qA)−1 A∼L∼p,q(Jq,pLp,qA) = −B,

and

−L−1t,r X AJq,p(Jq,pLp,qA) = −L−1

t,r X(ALp,qA) = −L−1t,r X(BLt,r B) = −B.

A left inverse X of B exists becauseB =[

D2 −N2]

is tall and has full column rank byassumption. Summarizing we have thatC(E − Jq,p) = 0 for almost alls ∈ C0 and, hence,C(E − Jq,p) = 0 as a rational matrix.

134

D

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140

E

Notation

For any positive integersp andq and setM , the setM p×q denotes the set ofp× q matrices withentries inM . Explicit indication of dimensions is sometimes omitted ifno confusion can arise.

Jq,p[ Iq 0

0 −I p

]

Z−, Z+ negative nonzero integers, positive nonzero integersC ∞ w : R 7→ F | F = Rq or F = Cq for someq ∈

Z+, w is infinitely often differentiable onR C−, C+, C0, C open left half complex plane, open right half complex plane,

imaginary axis,C = C− ∪ C0 ∪ C+R−, R+, R negative nonzero real numbers, positive nonzero real num-

bers, real numbers:R = R− ∪ 0 ∪ R+P , R set of polynomial matrices with real matrix-valued coeffi-

cients,R∈ R iff qR∈ P for some polynomialq ∈ PAT, A∗ transpose ofA and complex conjugate transpose ofAtraceA

∑ni=1 Aii of A ∈ Cn×n

nz the number of components of a vector signalz‖z‖ Euclidean norm of a constant matrix or vector (‖z‖2 =

tracez∗z)D−L, D−R, D⊥ left inverse, right inverse and orthogonal complement of a

constant matrixD: D−L D = I , DD−R = I , D⊥ is a maximalfull rank matrix such thatDD⊥ = 0 or D⊥ D = 0, dependingon whetherD has more columns than rows or vice versa

H∼, H∗, H−∼ H∼(s) = [ H(−s∗)]∗, H∗(s) = (H(s))∗ and H−∼ =(H−1)∼ = (H∼)−1

A> B, A ≥ B A(s)− B(s) > 0, A(s)− B(s) ≥ 0 for all s∈ C0 ∪ ∞ess sup essential supremum

141

E. Notation

In(A) The triple of inertia ofA ∈ Cn×n, that is, a triple of integers

denoting the numbers of eigenvalues ofA in C+, C− andC0

δM G, δ P McMillan degree of rationalG, the degree of polynomialP‖H‖∞ essential supremum of the largest singular value ofH(s) over

all s∈ C0

L2(X;Y) w : X 7→ Y |∫

Xw∗(t)w(t)dt<∞ (X is always a subset of

R or C0, andY is eitherY = Rq or Y = Cq for some positiveintegerq)

L2−(R;Y) w : R 7→ Y |∫ T−∞w

∗(t)w(t)dt < ∞ for all T ∈ R (Y iseitherY = Rq or Y = Cq for some positive integerq)

L loc2 (R;Y) w : R 7→ Y |

∫ ba ‖w(t)‖2

2 dt < ∞ for all a,b ∈ R (Y iseitherY = Rq or Y = Cq for some positive integerq)

H m2 , H ⊥m

2 the set of functionsf : C+ 7→ Cm analytic inC+ such thatsupσ>0

∫ ∞−∞ ‖ f (σ + jω)‖2dω < ∞, the set of functionsf :

C− 7→ Cm such thatf ∼ ∈ H m2

RH2, RH ⊥2 f ∈ H2 | s∈ R+ ⇒ f (s) ∈ R, f ∈ H ⊥

2 | s∈ R− ⇒ f (s) ∈R

L q2 H ⊥q

2 ⊕ H q2 = L2(C0;C

q)

〈u, y〉 inner productH m×p

∞ the set ofm× p matrix valued functions that are analytic andbounded inC+

R Hm×p∞ , GR H

m×m∞ the set of real-rational matrices inH p×m

∞ , the subset of ele-ments ofR H

m×m∞ whose inverse is also inR H

m×m∞

M ⊥, M ⊥J orthogonal andJ-orthogonal complement ofMπ−, π+ orthogonal projection fromL q

2 to H ⊥q2 or from L2(R;Y) to

L2(R−;Y), π+ = 1− π−BG, BR w ∈ H ⊥m

2 | Gw ∈ H g2 for some givenG ∈ R H

g×m∞ , w ∈

L2(R−;Rm) | R(d/dt)w(t)= 0 for some givenR∈ P g×m

L(w) left or two-sided Laplace transform of a time signalw inL2(R−;Y) or L2(R;Y)

u(t) dudt (t)

λ(A), λmax(A) the set of eigenvalues of a constant square matrixA, thelargest eigenvalue ofA if the eigenvalues are real-valued

σ(A), σmax(A) the set of singular values ofA, the largest singular value ofA

W T the set of maps fromT to W:= is by definition equal toIm F, Ker F the image ofF: w |w= Fl, the kernel ofF: w | Fw= 0G

s=[

A BC D

]

G has a realizationG(s) = C(sI − A)−1B+ D

p|q p dividesq (in our case:q/p is polynomial for polynomialp andq)

Xstab(H), Xantistab(H) the stable eigenspace and antistable eigenspace ofX = Cn of

an operatorH, H : X 7→ X• unspecified parameterm the set of nonnegative integers1,2, . . . ,m⊕ direct sum

142

Index

J-inner-outer factorization,109J-lossless,109

co-,110J-matrices,5J-orthogonal complement,24, 26Jq,p-spectral (co)factor,56L2-stability,10H∞ filtering, 103L2− minimal,84L2−-minimal

generators ofBG, 87generators ofBG, 87ONR and DVR,86

L2−-systems,82L2

inner product,5

admissible,46antistable

constant matrix,78rational matrix,23

AR representation,8AR representation,79

L2−-minimality, 84hidden modes,14minimal,8, 80

AR representationsL2− minimal,84

behavior,7Bounded Real Lemma,99

closed loopwell-posed,13

closed loops,12co-J-inner,110co-J-lossless,110co-inner,110co-outer,110

cofactorization,26compensator,17

internally stabilizing,17contraction,108

strict,108convolution system,11, 12

lossless,111coprime,14coupling condition,73critical vectors,123

degree,115McMillan, 120

driving variable,79driving variable representation,79DVR, 79

L2−-minimality, 84minimal,80

dynamical system,7, 8

Euclidean norm,5

factorizationJ-inner-outer,109asymmetric Wiener-Hopf,125canonical,122, 123

polynomial algorithm,126state space formulas,123uniqueness of,123

indices,125noncanonical,125

polynomial algorithm,128noncanonical co-,126Wiener-Hopf,122

factorization indices,125factorization indices,126filters (shaping),18fractions overR H ∞, 121frequency domain,16

143

Index

generatorof BR, 83

generatorsco-J-lossless,92

generators ofBR

uniqueness,83

Hardy spaces,6, 16, 107HCARE,73HFARE,71hidden modes,14Hurwitz, 84

strictly, 10

I/O system,8, 9I/S/O representation,79I/S/O representation,10inertia lemma,24infinity norm,17inner,109

J, 109co-,110

inner product,5input,8, 9input/output systems,8internal stability,10internally stable matrix,55invariant polynomials,117invariant rational functions,119

Laplace transform,5Laplace transform,12left prime overR H ∞, 121left coprime overR H ∞, 121left coprime PMFD,14left factor

of a polynomial matrix,119linear fractional transformation,113lossless

J, 109co-J-, 110convolution system,111

McMillan degree,81, 120McMillan degrees,73, 74minimal

L2−, 84AR representation,8AR representation,8, 80AR representations,8

DVR, 80ONR,80

minimal representation,80mixed sensitivity problem,18, 65, 69, 74multiplicative property,108

Nevanlinna-Pick interpolation,100nonproper,9norm

2, 5∞, 17Euclidean,5

NPIP,100

observable,9ONR,79

L2−-minimality, 84minimal,80

optimal (compensator/solution),46OTBP,33outer,109

co-,110output,8, 9output nulling representation,79

Paley-Wiener theorem,16para-Hermitian,122Parrott lower bound,33Parseval,16passivity,90Plancherel,16plant,17PMFD,14, 120polynomial matrix

unimodular,8polynomial matrix,115

coprime,14degree,115, 116, 119generalized Bezout identity,118invariant polynomials,117left and right prime,117left coprime,14left coprime and right coprime,119left factor,119left factors,117linear equations,118rank,115row and column degrees,116row reducedness,116

144

Index

Smith form,116submatrix,115unimodular,116zero,115

positive subspace,21, 22strictly, 21strictly , 22

positivity, 22POTBP,61proper,9PSTBP,55

rational matrix(non)proper,9finite pole,119PMFD,120rank,119stable,12submatrix,119

rational matrix,119(non)singular,119antistable,23finite zero,119inner/outer,109internally stable,55invariant rational functions,119McMillan degree,120polynomial fraction,120realization,12Smith-McMillan form,119stable,17unstable zero,121

real-rational,6realization,12Redheffer’s lemma,112regular state feedback,80regular output injection,80representation

L2− minimal,84AR, 8, 79DVR, 79I/S/O,10, 79minimal,80ONR,79

representationsL2−-minimal AR,84

signals,6signature matrix,5

small gain Theorem,108extended,108

SP,22SP inequality,22SSP1,44, 54SSP1 Algorithm,58

dual,75SSP2,45, 46

algorithm,52state space formulas,70

stability,10L2, 10, 11L loc

2 , 9L2, 10internal,10

stabilizability,15stable

constant matrix,78constant matrix, anti-,78rational matrix,12, 17rational matrix, anti-,23

stable fraction,121stable fractions,121standardH∞ problem

first suboptimal,44standardH∞ problem,44standardH∞ problem,43

optimal,43suboptimal,43

standardH∞ problem,45second suboptimal,45

state,79state transformation,80state feedback,80STBP,29strict positivity,22

for DVRs,93for ONRs,97

strictly Hurwitz,10strictly positive subspace,22strongly observable,80symmetric factor extraction algorithm,59system,8

behavior of,7closed-loop,12convolution,11, 12dynamical,7I/O, 8, 9

L2 stable,10

145

Index

internally stable,10I/S/O,9stable closed-loop,15

systemsL2−, 82

transfer matrix,9transform

Laplace,5regular output injection,80regular state feedback,80state,80

two-blockH∞ problemsuboptimal,29

two-blockH∞ problemmixed sensitivity-,69optimal polynomial,61OTBP,33polynomial,55

two-blockH∞ problem,19optimal,19OTBP,19STBP,19suboptimal,19

unimodular,8uniqueness

generators ofBG, 87

virtual row degrees,126, 127

well-posed,13weakly,13

Wiener-Hopf(co)factorization indices,125

Wiener-Hopf factorization,122

zeroat infinity, 121finite

of polynomial matrix,115of rational matrix,119

unstable,121zeros

of ONR and DVR quadruples,85

146

F

Summary

Frequency Domain Methods in H∞ Control

This thesis deals with a number of problems arising inH∞ control theory.H∞ control theory is a reasonably new off-shoot of the theory ofcontrol, and may be seen as

a synthesis of “classical” and “modern” control theory. Roughly speaking, the control problemis how to design acontroller for a given system(theplant) such that theclosed loop(the plantconnected with the controller) behaves well in some sense. The theory ofH∞ control aims atproviding an answer to such questions by translating the problem into an optimization problem intheH∞-norm, based on a mathematical model of the given plant. Thisway the control problemis divided into three linked problems:

1. What is an adequate mathematical model of the given plant.

2. How can the question of “behaving well” of the closed loop be translated into anH∞optimization problem.

3. How canH∞ optimization problems be solved.

An important motivation forH∞ control theory is that it provides a means to designrobustcontrollers. More concretely, often knowledge of the nature of modeling errors may be useddirectly to formulate theH∞ optimization problem in such a way that controllers that solvethe H∞ optimization problem not only work for the mathematical model but also for the morecomplex real given plant.

In this thesis Item 3 is considered from a mathematical system theoretical angle. The systemsconsidered in this thesis are those whose signals are interrelated by a set of ordinary linear dif-ferential equations. TheH∞ optimization problem (or, for brevity, theH∞ problem) is to findstabilizing controllers that minimize theH∞-norm of some given matrixH that depends on thecontroller. The problem is handled by looking at two simplerproblems: ThesuboptimalH∞problem with boundγ is to find stabilizing controllers that make theH∞-norm of H strictly lessthanγ, and theoptimal H∞ problem is to find stabilizing controllers that make theH∞-normequal to the minimally achievable bound ofγ. It is possible to delimit the minimally achievablebound with arbitrary precision using the solution to the suboptimal version in combination witha line search. The precise minimal bound may then be pinpointed with the help of a rootfinderand then, finally, optimal controllers may be generated based on the solution to the optimalH∞problem.

In this thesis a frequency domain approach is used because itallows for very compact for-mulations and manipulations and because it provides a solidbasis for the polynomial approachto H∞ control. In this respect the approach deviates from the morecommonly used state space

147

F. Summary

approach. Another feature of this thesis is that the solution to theH∞ problems presented is notconfined to the suboptimal version. A large part of thesis is devoted to an analysis and solutionof optimalH∞ problems.

Chapter2 contains important background material from mathematicalsystem theory for linearsystems that is relevant to theH∞ problems.

In Chapter3 the two-blockH∞ problem is treated. This problem is a stylizedH∞ problemneeded to pave the way for more generalH∞ problems. It is argued that working with signalspaces instead of transfer matrices helps in studying two-block H∞ problems. With the helpof geometrical arguments it is shown that, under mild conditions, the (sub)optimal two-blockH∞ problem has a solution if and only if an energy inequality holds on some subspace that isindependent of the controller to be constructed. All solutions may be generated in the case thatthis energy inequality is satisfied. Along the way of provingthis result an interesting theoremof intrinsic value is obtained. This theorem reveals a one-to-one correspondence between strictpositivity of subspaces andJ-losslessness of their representations. The results in this chapterform the core of the thesis.

In Chapter4 thestandardH∞ problemis considered. This problem is called “standard” be-cause many, if not all,H∞ problems are examples of the standardH∞ problem. The frequencydomain solution method to this problem presented in this thesis is analogous to the extensivelyreported state space solution method, with the difference that the frequency domain solutionmethod may also directly be applied to standardH∞ problems withnonproper transfer matri-ces. This is important because many practicalH∞ design problems give rise to a standardH∞problem with nonproper transfer matrices. The optimal version is solved for a large class ofstandardH∞ problems. The suboptimal solution method as well as the optimal solution methodare demonstrated on a mixed sensitivity problem. For completeness the link with the state spacesformulas is given.

Chapter5 has a somewhat different style than the preceding chapters.Systems that may bedescribed by ordinary linear differentialequationsare often identified with convolution systemsand their associated transferfunctions. It is an assumption that this makes sense, an assumptionthat hardly ever is made explicit.

In the successful approach taken in Chapter3 the signals of the system, and not the transferfunctions, are the main object of study. In Chapter5 a theory underlying the geometric approachof Chapter3 is developed. It leads to what are calledL2−-systems. These are systems whosesignals by assumption have finite energy. The theory ofL2−-systems forms a basis for the geo-metric approach applied in Chapter3 and replaces the theory of convolution systems. It is arguedthatL2−-systems form a better basis for studying two-block typeH∞ problems than convolutionsystems.

The two main theorems in this chapter are about a state space characterization of strictlypositive subspaces forL2−-systems. The two theorems generalize the well known Bounded RealLemma for convolution systems. Two examples show that the Bounded Real Lemma and thesolution to the one-sided Nevanlinna-Pick interpolation problem follow as special cases. Thechapter ends with a solution to the problem ofH∞ filtering.

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Frekwentiedomeinmethoden in H∞-Regeling

In dit proefschrift wordt een aantal problemen uit deH∞-regeltheorie bestudeerd.De H∞-regeltheorie is een relatief jonge tak binnen de regeltheorie, en kan worden gezien

als een synthese van de “klassieke” en de “moderne” regeltheorie. Het regelprobleem is, inglobale termen, het probleem hoe voor eengegeven systeem(bijvoorbeeld een cv-installatie)eenregelaar(een thermostaat) gekonstrueerd kan worden zodanig dat degesloten-lus(de cv-installatie met thermostaat) zo goed mogelijk werkt. DeH∞-regeltheorie beoogt op dergelijkevragen een antwoord te geven door aan de hand van een wiskundig model van het gegevensysteem het probleem te vertalen in een optimalisatieprobleem in deH∞-norm. Op deze manierwordt het probleem in drie gekoppelde deelproblemen geknipt:

1. Wanneer is een wiskundig model van een gegeven systeem “adekwaat”.

2. Hoe kan aan de hand van het wiskundig model het “goed” werken van de gesloten-lusworden vertaald in eenH∞-optimalisatieprobleem.

3. Hoe kunnenH∞-optimalisatieproblemen worden opgelost.

Een belangrijke motivatie voorH∞-regeltheorie is dat het een manier geeft omrobuusterege-laars te ontwerpen. Meer konkreet, de kennis van de aard van modelleringsfouten kan vaak opeen direkte manier worden gebruikt bij het opstellen van hetH∞-optimalisatieprobleem op eendusdanige manier dat oplossingen van hetH∞-optimalisatieprobleem niet alleen “goed” werkenvoor het gebruikte wiskundige model, maar ook voor het werkelijke gegeven systeem.

In dit proefschrift wordt punt drie belicht vanuit een wiskundig systeemtheoretische inval-shoek voor systemen die kunnen worden beschreven door een stelsel van gewone lineaire dif-ferentiaalvergelijkingen. HetH∞-optimalisatieprobleem (of, iets korter, hetH∞-probleem) ishet probleem van het vinden van een stabiliserende regelaardie deH∞-norm van een gegevenoverdrachtsmatrixH minimaliseert met betrekking tot alle stabiliserende regelaars. Dit prob-leem wordt opgelost door naar twee eenvoudigerH∞-problemen te kijken. HetsuboptimaleH∞-probleem met grensγ is het probleem van het vinden van een stabiliserende regelaar zodanig datdeH∞-norm vanH strikt kleiner is danγ, en hetoptimaleH∞-probleem is het vinden van eenstabiliserende regelaar zodanig dat deH∞-norm vanH gelijk is aan de optimale waarde van degrensγ. Door gebruik te maken van een iteratieprocedure en de oplossing van het suboptimaleH∞-probleem kan de optimale waarde van de grensγ worden bepaald en kunnen vervolgensoptimale oplossingen worden gegenereerd met de oplossing van het optimaleH∞-probleem.

Er is gekozen voor een frekwentiedomeinbenadering omdat infrekwentiedomein-termenH∞-problemen en hun oplossingen kompakt te formuleren zijn, enomdat in het frekwentiedomein

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een solide basis voor de meer praktische “polynoombenadering” van H∞-theorie opgezet kanworden. Hierin wijkt de beschouwing in dit proefschrift af van de vaker gehanteerde toestand-sruimtebenadering. Een tweede essentieel punt is dat de beschouwing vanH∞-problemen in ditproefschrift zich niet beperkt tot de gebruikelijkesuboptimaleoplossingen. Een groot deel vandit proefschrift is gewijd aan een analyse en oplossing van de meer complexeoptimaleversie vanhet probleem.

Hoofdstuk2 is een algemeen hoofdstuk waarin enkele basisbegrippen uitde wiskundige theorievan lineaire systemen worden gegeven in zoverre deze relevant zijn voor het oplossen vanH∞-problemen.

In Hoofdstuk3 wordt het twee-blokH∞-probleembestudeerd. Dit is een gestileerdH∞-probleem dat fungeert als opstapje voor meer algemeneH∞-problemen. Beargumenteerd wordtdat een geometrische aanpak helpt bij het bestuderen van de suboptimale en optimale versie vanhet probleem. Gebruikmakend van geometrische argumenten en resultaten uit de theorie van in-definiete inproduktruimten, wordt aangetoond dat, onder milde voorwaarden, het (sub)optimaletwee-blokH∞-probleem met grensγ een oplossing heeft dan en slechts dan als een zekere en-ergiekonditie geldt op een regelaaronafhankelijk deel vande gesloten-lus. In het geval dat daaraan is voldaan kunnen alle (sub)optimale oplossingen gegenereerd worden. In dit hoofdstukwordt als tussenresultaat een stelling van intrinsieke waarde bewezen. In deze stelling wordt eendan-en-slechts-dan verband gelegd tussen strikt positieve deelruimten enJ-verliesloosheid vanhun representaties. De resultaten in dit hoofdstuk vormen de kern van dit proefschrift.

In Hoofdstuk4 wordt hetstandaardH∞-probleembestudeerd. Dit probleem heet “standaard”omdat vele, zo niet alle bestaandeH∞-problemen een voorbeeld zijn van het standaardH∞-probleem. Een frekwentiedomeinoplossing van de suboptimale versie wordt afgeleid. Dezeoplossing is analoog aan de bekende gedokumenteerde toestandsruimteoplossing met dit ver-schil dat de frekwentiedomeinoplossing ook bruikbaar is voor oneigenlijke overdrachtsmatri-ces. Dit is een belangrijk aspekt van de frekwentiedomeinbenadering omdat vele praktischeH∞-regelproblemen leiden totH∞-problemen met oneigenlijke overdrachtsmatrices. De opti-male versie wordt opgelost voor een grote klasse van standaard H∞-problemen. De oplosmeth-ode van zowel de suboptimale als de optimale versie wordt ge¨ıllustreerd aan de hand van eengemengd-gevoeligheidsprobleem. Voor de volledigheid wordt de koppeling met de bekende toe-standsruimteformules gegeven.

Hoofdstuk5 is anders van stijl dan de voorgaande hoofdstukken. Systemen die kunnen wordenbeschreven door gewone lineaire differentiaalvergelijkingenworden vaak geıdentificeerd metkonvolutiesystemen en de bijbehorende overdrachts-funktie. Het is in feite een veronderstellingdat dit kan en dat dit zinvol is, een veronderstelling die bijna nooit expliciet wordt gemaakt.

In de benadering in Hoofdstuk3 die zo suksesvol blijkt, worden de signalen in het systeem alsobjekt van studie gezien, en niet de bijbehorende overdrachtsfunkties. Een nadere beschouwingvan dit leidt tot wat in Hoofdstuk5 L2−-systemenworden genoemd. Dit zijn systemen waarvande signalen tot aan elk eindig tijdstip volgens veronderstelling eindige energie hebben. De theo-rie vanL2−-systemen vormt een basis voor de geometrische theorie gehanteerd in Hoofdstuk3en vervangt de theorie van konvolutiesystemen. Er wordt geargumenteerd datL2−-systemen eenbetere basis vormen voor het bestuderen vanH∞-problemen van het twee-blok type dan konvo-lutiesystemen.

De twee centrale stellingen in dit hoofdstuk betreffen een toestandsruimte-karakterisering vanstrikt positieve deelruimten voorL2−-systemen. Dit resultaat is een generalisatie van het bekende“Bounded Real Lemma” voor konvolutiesystemen. Aan de hand van twee voorbeelden wordtaangetoond dat met deze generalisatie het “Bounded Real Lemma” en de bekende oplossing vanhet enkelzijdige Nevanlinna-Pick-probleem als speciale gevallen volgen. Het hoofdstuk wordtbesloten met de oplossing van hetH∞-filterprobleem.

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Frekwinsjedomeinmetoaden yn H∞-Regeling

Yn dit proefskrift wurdt in tal problemen ut deH∞-regeling bestudearre.De H∞-regeling is in frij jonge tak fan de regelteory en kin sjoen wurde as in syntese fan de

“klassike” en de “moderne” regelteory. It regelprobleem isyn in pear wurden it probleem hoe’tfoar in beskaat systeem(bygelyks in c.v.-ynstallaasje) inregelaar (in termostaat) konstrueerdwurde kin sadat desletten-lus(de c.v-ynstallaasje mei termostaat) sa goed mooglik wurket. DeH∞-regelteory hat op’t each op soksoarte fragen in antwurd te jaan middels in wiskundich modelfan it beskaat systeem it probleem oer te setten yn in optimalisaasjeprobleem yn deH∞-noarm.Op dizze wize wurdt it probleem yn trije koppele dielproblemen ferdield:

1. Wannear is in wiskundich model fan it beskaat systeem “adekwaat”.

2. Hoe kin oan’e han fan in wiskundich model it “goed” wurkjen fan de sletten-lus oersetwurde yn inH∞-optimalisaasjeprobleem.

3. Hoe kinneH∞-optimalisaasjeproblemen oplost wurde.

In wichtige motifaasje foar deH∞-regelteory is dat it oanjowt hoe robuste regelaars te untwerpen.Meer konkreet, de kennis fan de aard fan modelleringsfoutenkin faak op in direkte wize meinaamwurde yn it opstellen fan itH∞-optimalisaasjeprobleem op sadanige wize dat de oplossingenfan it H∞-optimalisaasjeprobleem net allinne “goed” wurkje foar it brukte wiskundige model,mar ek foar it werklike beskaat systeem.

Yn dit proefskrift wurdt punt trije besjoen fan in wiskundich systeemteoretyske ynfalshoek utfoar systemen dy’t beskreaun wurde kinne troch in stelsel fan gewoane lineere differinsjaalfer-likens. ItH∞-optimalisaasjeprobleem (of koartsein, itH∞-probleem) is it probleem fan it finenfan in stabilisearjende regelaar dy’t deH∞-noarm fan in beskaat oerdrachtsmatriksH mini-malisearret oangeande alle stabilisearjende regelaars. Dit probleem wurdt oplost troch nei twaienfaldigerH∞-problemen te sjen. ItsuboptimaleH∞-probleem mei grinsγ is it probleem fanit finen fan in stabilisearjende regelaar sadat deH∞-noarm fanH strikt lytser is asγ, en it op-timaleH∞-probleem is it finen fan in stabilisearjende regelaar sadatdeH∞-noarm gelyk is oande optimale waarde fan de grinsγ. Middels it bruken fan in yteraasjeprosedeure en de oplossingfan it suboptimaleH∞-probleem kin de optimale grins utrekkene wurde en kinne dˆernei optimaleoplossingen genereard wurde mei de oplossing fan it optimaleH∞-probleem.

Der is keazen foar in frekwinsjedomeinoanpak omdat yn frekwinsjedomeintermenH∞-problemen en harren oplossingen kompakt te formulearjen binne, en omdat yn it frekwinsje-domein in deeglike basis foar de mear praktyske “polynoom”-oanpak fanH∞-teory opset wurdekin. Hjirin is de beskoging yn dit proefskrift oars as de faker hantearre tastansromte-oanpak. In

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twadde essinsjeel punt is dat de beskoging fanH∞-problemen yn dit proefskrift har net beheintta de suboptimale ferzy. In grut part fan dit proefskrift is wijd oan in analise en oplossing fan demear komplekse optimale ferzy fan it probleem.

Haadstik2 is in algemien haadstik weryn’t inkele basisbegrippen ut de wiskundige teory fanlineere systemen jown wurd foar safier dy fan belang binne foar it oplossen fanH∞-problemen.

Yn Haadstik3 wurdt it twa-blokH∞-probleembestudearre. Dit is in stylearreH∞-probleemdat fungeart as opstapke foar mear algemieneH∞-problemen. Der wurdt bearguminteard dat ingeometryske oanpak helpt bij it bestudearjen fan de suboptimale en optimale ferzy fan it prob-leem. Der wurdt gebruk makke fan geometryske arguminten enresultaten ut de teory dan yndefi-nite ynproduktromten. Der wurdt oantoand dat, under mildebetingsten, it (sub)optimale twa-blokH∞-probleem mei grinsγ in oplossing hat dan en alline dan as der in sekere enerzjykondysjejildich is op in regelaarunofhinklik part fan de sletten-lus. Mocht deroan foldien weze, dan kinnealle (sub)optimale oplossingen genereard wurde. Yn dit haadstik wurdt as tuskenresultaat instelling fan yntrinsike wearde bewiisd. Yn dizze stelling wurdt in dan-en-alline-dan ferban leintusken strikt positive dielromten enJ-ferliesleazens fan harren representaasjes. De resultaten yndit haadstik foarmje de kearn fan it proefskrift.

Yn Haadstik4 wurdt it standertH∞-probleembestudearre. Dit probleem wurdt “standert”neamt omdat in soad, en nei alle gedachten alle,H∞-problemen in foarbyld binne fan it standertH∞-probleem. De oplossing fan de suboptimale ferzy rint lyk opmei de bekende dokumentearretastansromte-oplossing mei dit ferskil dat de frekwinsjedomeinoplossing ek te bruken is foaruneigentlike oerdrachtsmatriksen. Dit is in wichtig aspekt fan dizze frekwinsjedomeinoan-pak omdat in soad praktyskeH∞-regelproblemen liede taH∞-problemen mei uneigentlikeoerdrachtsmatriksen. De optimale ferzy wurdt oplost foar in grutte klasse fan standertH∞-problemen. De oplosmetoade fan sawol de suboptimale as de optimale ferzy wurdt yllustreardoan de han dan in mingd-gefoelichheidsprobleem. Foar de folsleinens wurdt de koppeling meide tastansromteformules jown.

Haadstik 5 is oars fan styl as de foarofgeande haadstikken. Systemen dy’t beskreaunwurde kinne troch gewoane lineere differinsjaalferlikens wurde faak ydentiseard mei kon-volusjesystemen en de derbij hearrende oerdrachtsfunksje. It is yn feite in understelling dat ditkin en dat it sin hat, in understelling dy’t hast noait eksplisyt makke wurdt. Yn de wurkwize ynHaadstik3, dy’t suksesfol blykt te wezen, wurde de signalen fan it systeem as objekt fan studzjesjoen en net de oerdrachtsfunksje dy’t der bij heart. It neier besjen fan dit alles liedt ta wat ynHaadstik5 L2−-systemen neamd wurdt. Dit binne systemen der’t de signalen oant elk eindichtiidstip neffens understelling eindige enerzjy fan hawwe. De teory fanL2−-systemen foarmet inbasis foar de geometryske teory dy’t yn Haadstik3 hanteard wurdt en komt yn’t plak fan de teoryfan konvolusjesystemen. Der wurdt arguminteard datL2−-systemen in bettere basis foarmje foarit bestudearjen fanH∞-problemen fan it twa-blok type as konvolusjesystemen.

De twa sintrale stellingen yn dit haadstik slagge op intastansromte-karakterisearing fan strikt positive dielromten foarL2−-systemen. Dit resultaat is

in generalisaasje fan it bekende “Bounded Real Lemma” foar konvolusjesystemen. Oan de hanfan twa foarbylden wurdt oantoand dat mei dizze generalisaasje it “Bounded Real Lemma” en debekende oplossing fan it inkelsidige Nevanlinna-Pick ynterpolaasjeprobleem as spesiale gefallenneikomme. It Haadstik wurdt ofsletten mei de oplossing fanit H∞-filterprobleem.

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Curriculum Vitae

Gjerrit Meinsma was born on the 29th of January 1965 in Opeinde, a little village in the northernpart of The Netherlands. There he attended the nursery school “De Kindertuin” and elementaryschool. In the off-school hours he spent many hours with his friends making and throwingboomerangs, and in the weekends he used to play korfball. From the age of 12 till the age of 18 heattended the grammar-school “Het Drachtster Lyceum” in thenearby town Drachten. Somewherein the middle of this period the fascinating world of music caught his attention. Listening to musicand playing the piano have been his major hobbies since. At the same time Gjerrit developed akeen interest in mathematics, and he decided to pursue his career in this direction. In 1983 hewent to Enschede to study applied mathematics at the University of Twente. At the end of 1988he finished his master’s thesis entitled “Chebyshev approximation by free knot splines” and inMarch 1989 he received his master’s degree. After a refreshing break of five weeks he returnedto the faculty he graduated with, only this time as a researchassistant (AiO) with the Systemsand Control Group. Half a year later he bought himself a brandnew Klug & Sperl upright piano.After finishing his thesis Gjerrit wants to travel through Australia, New Zealand and several Asiancountries for about half a year, before continuing his career.

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Frequency Domain Methods in H-infinity Control - Applied