Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner
146 D. Mustafa, K. Glover
Minimum Entropy Hoo Control
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona
Series Editors M. Thoma • A. Wyner Advisory Board L. D. Davisson - A. G. J. MacFarlane • H. Kwakernaak J. L Massey • Ya Z. Tsypkin • A. J. Viterbi Authors Denis Mustafa Laboratory for Information and Decision Systems Massachusetts Institute for Technology Cambridge MA 02139, USA Keith Glover Department of Engineering Cambridge University Cambridge CB2 1PZ, UK
ISBN 3-540-52947-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52947-O Springer-Vedag NewYork Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its current version, and a copyright fee must always be paid. violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1990 Printed in Germany The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printing: Mercedes-Druck, Berlin Binding: B. Helm, BedJn 916113020-543210 Printed on acid-free paper.
To my parents D.M.
Preface Optimal control of linear dynamic systems by linear controllers generally reduces to minimizing some norm of a closed-loop transfer function over the class of all stabilizing controllers. Different norms can be motivated by different assumptions on the physical system and signals. Robust control then attempts to maintain good performance--such as attenuation of disturbances--in the presence of uncertainty in the plant. One realistic way of modelling plant uncertainty is to include perturbations with bounded 7/**-norm. In this context it is natural to require controllers to make particular closedloop transfer functions have 7/,o-norm less than some prescribed tolerance. The central question addressed in this monograph is to investigate which of these admissible closedloop transfer functions to choose. One possible choice is to take that which minimizes an entropy integral and the derivation and properties of this solution occupies most of this monograph. The approach taken here exploits recent progress in the state-space sohttion to the 7/00 control problem and explicit formulae can be obtained for the solution. At a mathematical level the entropy minimization problem considered here has close connections with maximum entropy spectral analysis. However the objective is more easily interpreted as a compromise between an 7/00 constraint and a Linear Quadratic Ganssian objective, rather than in terms of information theory or of one of the many other uses of the word entropy. Relations to other control methods are also exposed here. The appropriate background for this monograph is a first course in state-space methods, together with an elementary knowledge of 7too control and Linear Quadratic Gaussian control. The bulk of the research reported in this monograph was carried out at the Engineering Department at Cambridge University, UK, between October 1986 and May 1989. Financial support was provided by the Science and Engineering Research Council. Since September 1989 the first author has been a Harkness Fellow at the Laboratory for Information and Decision Systems at the Massachusetts Institute of Technology, USA, where the final stages of this research were carried out. Financial support was provided by the Air Force Office of Scientific Research, grant number AFOSR-89-0276. The first author would like to thank Dr Raimund Ober for his sound advice and for many useful discussions and comments; Glenn Vinnicombe for some influential discussions regarding Chapter 5; and Dr Duncan McFarlane, Pablo Iglesias, James Sefton, Frank van Diggelen, Richard Hyde and Paul McGinnie for their comments on earlier versions of the manuscript. Thanks too to Dr Basil Kouvaritakis, for sparking off the first author's interest in multivariable control. DENIS MUSTAFA, Cambridge, Massachusetts, USA KEITH GLOVER, Cambridge, UK
Contents Introduction
1
1.1 1.2 1.3
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outline of this Monograph . . . . . . . . . . . . . . . . . . . . . . . . . Background References . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 3 5
2
The 2.1 2.2 2.3 2.4 2.5
Entropy of a System Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of the Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . Some Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relations to the LQG Cost . . . . . . . . . . . . . . . . . . . . . . . . . The 7 ~ - N o r m Bound . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 8 8 9 11 12
3
The 3.1 3.2 3.3 3.4 3.5 3.6 3.7
M i n i m u m E n t r o p y 74~ C o n t r o l P r o b l e m Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution in the General Case . . . . . . . . . . . . . . . . . . . . . . . . Solution at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State-Space Formulae for the Minimum E n t r o p y . . . . . . . . . . . . . Upper Bounds and an 74oo/LQG Tradeoff . . . . . . . . . . . . . . . . . Recovery of the LQG Solution . . . . . . . . . . . . . . . . . . . . . . .
15
The 4.1 4.2 4.3 4.4 4.5
M i n i m u m E n t r o p y 7400 D i s t a n c e P r o b l e m
35
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From 7"/oo Control Problem to 740° Distance Problem . . . . . . . . . . Relations to the Band Extension Problem . . . . . . . . . . . . . . . . Solution in the General Case . . . . . . . . . . . . . . . . . . . . . . . . Solution at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36 36 38 38 42
4
R e l a t i o n s t o C o m b i n e d 74oo/LQG C o n t r o l 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Combined 74oo/LQGProblem . . . . . . . . . . . . . . . . . . . . 5.3 Equivalence with Minimum Entropy 7-/00 Control . . . . . . . . . . . . 5.4 Solution and Equivalence in State-Space . . . . . . . . . . . . . . . . .
16 16 19 22 25 30 31
49 50 50 52 54
VII
R e l a t i o n s t o R i s k - S e n s l t i v e LQG Control 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Risk-Sensltive LQG Control Problem . . . . . . . . . . . . . . . . 6.3 Equivalence with Minimum E n t r o p y 7~oo Control . . . . . . . . . . . .
59
T h e N o r m a l i z e d 7~o C o n t r o l P r o b l e m
65 66 66 68 72
7.1 7.2 7.3 7.4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Normalized LQG Problem . . . . . . . . . . . . . . . . . . . . . . The Normalized 7-/o~ Problem . . . . . . . . . . . . . . . . . . . . . . . A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . .
74oo-Characteristic Values 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 LQG-Balancing and LQG-Characteristic Values . . . . . . . . . . . . . 8.3 7~o~-Balancing and 7~oo-Characteristic Values . . . . . . . . . . . . . . . 8.4 Model Reduction by 7-/oo-BManced Truncation . . . . . . . . . . . . . . 8.4.1 7~oo-Balanced Truncation . . . . . . . . . . . . . . . . . . . . . . ........... 8.4.2 Relations to Balanced ~runeation . . . . . 8.4.3 Coprime Factorization via the Normalized ~ Problem . . . . . 8.4.4 Model Reduction via the Coprime Factors . . . . . . . . . . . . 8.4.5 Stability and Performance with the Reduced-Order Controller 8.4.6 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . .
LQG and 7~ Monotonieity 9.1 9.2 9.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The LQG Cost and its Derivative . . . . . . . . . . . . . . . . . . . . . On LQG Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . .
60 60 61
79 80 81 83 86 86 87 90 93 95 99 103 104 104 108
A P r o o f o f R e s u l t s N e e d e d in t h e T e x t A.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 A L e m m a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Proof of Theorem 2.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 State-Space Evaluation of the Entropy Integral . . . . . . . . . . . . . . A.5 Proof of L e m m a 8.4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Proof of Corollary 8.4.19 and Proposition 8.4.21 . . . . . . . . . . . . . A.7 Proof of Theorem 9.2.2(ii) . . . . . . . . . . . . . . . . . . . . . . . . .
111
B Entropy Formulae: Alternative Derivation
123 124 124 126 127 128
B.1 B.2 B.3 B.4 B.5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Full Information and Output Estimation Problems . . . . . . . . . Separation Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of Theorem 3.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . Proof of L e m m a 3.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112 112 112 113 116 118 121
VIII
C Notation C.1 Basic Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . C.2 Acrcnyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131 132 134
Bibliography
135
Index
141
List of Figures 2.1
Block diagram for the Small Gain T h e o r e m . . . . . . . . . . . . . . . .
14
3.1 3.2
T h e closed-loop system . . . . . . . . . . . . . . . . . . . . . . . . . . . All closed-loops satisfying the 7-/Go-norm bound
17 20
7.1 7.2 7.3 7.4 7.5 7.6 7.7
Block diagram for the Normalized Problems . . . . . . . . . . . . . . . Uncertainty types Aij covered by the Normalized 7"/oo Problem . . . . . IIHME**IIooand its upper bound 3', against 7 . . . . . . . . . . . . . . . 6'(HMEoo) and its u p p e r bound I(HMEoo; 7; CO), against 7 . . . . . . . HME® against frequency, for various 7 . . . . . . . . . . . . . . . . . . Components of HMEco against frequency, for various 7 . . . . . . . . . M a x i m u m allowable uncertainties Aij against frequency, for various 7 •
8.1 8.2 8.3
Reduced-order controller with full-order plant . . . . . . . . . . . . . . 7~oo-balanced truncation example: a priori numerical results . . . . . . 7/~o-balanced truncation example: a posteriori numerical results . . . .
.............
67 69 73 74 75 76 77 96 100 100
Chapter 1 Introduction
2
1.1
C H A P T E R 1. I N T R O D U C T I O N
Overview
In control system analysis and design the first step is to produce a mathematical model of the system being controlled. This model includes not only the dynamic equations of the system but also a characterization of any disturbances affecting the system. If the dynamic model is assumed to be exact and the disturbances are assumed to be Gaussian stochastic processes, then the control law that minimizes the expected value of a quadratic form in the errors is given by the Linear Quadratic Ganssian (LQG) controller [2, 38]. However there is no guaranteed robustness [15] for LQG-controllers, and a small perturbation of the plant dynamics may give rise to an unstable closed-loop system. The difficulty is that the design method concentrates on the gain of the closedloop system to the disturbance signals and does not explicitly consider uncertainty in the dynamic model. A reliable control synthesis procedure should therefore take account of the uncertainty inherent in a mathematical modal [18]. A suitable norm that can incorporate both signal gain and robustness to plant uncertainty is the 7~o-norm. The synthesis of controllers using 7-/oo-optimal control methods was initiated in [67] and a wide range of sensible robust controller synthesis problems may be phrased in the 7"~oo-optimal control framework (see for example [24, 40]). If the uncertainty has a particular structure or robust performance is required then structured singular values need to be considered as in [19]. However a central constituent of all these problems is to characterize controllers that satisfy a bound on the 7~o-norm of a (usually frequency weighted) closedloop transfer function. Advances in the algorithms [28, 17, 30] have now reduced the computational burden of these methods to one approaching that of the LQG-optimal control problem. The family of controllers that satisfy a closed-loop 7/oo-norm bound is characterized by a linear fractional transformation of a 'ball in 7~oo' and a natural question is which element of this ball to choose. One choice that has been considered in a closely related problem in mathematics is to choose that which minimizes an entropy integral [3, 4, 21, 22]. The full development of the minimum entropy solution to the present control problem, particularly exploiting state-space techniques, is the central problem considered in this monograph. This lninimum entropy solution is a precisely defined compromise between the LQG-optimal solution and the 7~oo-optimal solution, which are themselves obtained when the 7~oo-norm constraint is relaxed or given its minimum value.
1.2. OUTLINE OF THIS MONOGRAPH
1.2
3
Outline of this M o n o g r a p h
An outline of the contents of this monograph is now given. C h a p t e r 2: T h e E n t r o p y of a S y s t e m We introduce the entropy of a system satisfying an 71oo-norm bound, and give some properties. In particular, the entropy at infinity is shown to be an upper bound on the more familiar LQG cost. The 7too-norm bound on the system is given its usual robustness interpretation. The entropy itself and the properties and interpretations described above are vital components of the remaining chapters. C h a p t e r 3: T h e M i n i m u m E n t r o p y 71oo C o n t r o l P r o b l e m In this chapter the Minimum Entropy 7"[00 Control Problem is posed and solved. The problem is to determine a controller which minimizes the entropy of a system whilst stabilizing it and satisfying a closed-loop 71o0-norm bound. The problem is posed in the same general framework used in 71o0-optimal control, into which a wide range of useful control problems may be embedded. When entropy is evaluated at infinity, a particularly simple and explicit solution is obtained, and state-space formulae are derived. By exploiting the relationship between entropy and LQG cost, an inherent 71oo/LQG tradeoffis proved. Furthermore, if the 7"/~o constraint is relaxed entirely in the Minimum Entropy ?to0 Control Problem, then the corresponding LQG control problem is recovered. C h a p t e r 4: T h e M i n i m u m E n t r o p y 71o0 D i s t a n c e P r o b l e m The 71oo General Distance Problem of finding all causal extensions of a certain anticansal transfer function within a prescribed distance is art interesting problem in its own right, as well as being (until recently) a stepping-stone to the solution of 71o0-optimal control problems. We derive the minimum entropy solution. This forms an alternative solution to the Minimum Entropy 71oo Control Problem of the previous chapter. C h a p t e r 5: R e l a t i o n s to C o m b i n e d 7~O0/LQG C o n t r o l We have by now seen how the Minimum Entropy 71oo Control Problem combines both 71oo and LQG criteria. Other researchers have proposed apparently quite different methods for combining 7/00 and LQG criteria. One such Combined 71oo/LQG Problem minimizes an auziliary LQG cost. We prove that, under certain conditions, the auxiliary cost and the entropy are identical, so then the Combined ~O0/LQG Problem and the Minimum Entropy 71oo Control Problem coincide. This result has the bonus of allowing a simplification of the solution to the Combined 7fo0/LQG Problem, by using the state-space solution of the Minimum Entropy 74o0 Control Problem derived in Chapter 3. C h a p t e r 6: R e l a t i o n s to R i s k - S e n s i t i v e L Q G C o n t r o l For completeness, the equivalence between the Mini,hum Entropy 7too Control Problem and the Ris~-Sensitive LQG Control Problem is reviewed. Risk-sensitive LQG control is ~ generalization of standard LQG control to include an ezponen~ial-o.f-quadratie cost. We prove that, under certain conditions, the exponential-of-quadratic cost and the entropy are identical, so then the Risk-Sensitive LQG Control Problem and the Minimum Entropy 7400 Control Problem coincide.
4
CHAPTER 1. INTRODUCTION
C h a p t e r 7: T h e N o r m a l i z e d 7"/00 C o n t r o l P r o b l e m At the end of Chapter 3, it was shown that an LQG control problem may be recovered from each Minimum Entropy 7-/00 Control Problem by relaxing the ?/00 constraint entirely. Here this reasoning is reversed: beginning with the well-known Normalized LQG Problem, we invoke the minimum entropy/7/00 constraint to obtain the corresponding (minimum entropy) Normalized 7/00 Control Problem, and in the process gain robustness at the expense of the LQG cost bound. This Normalized 7/'/00 Control Problem is shown to be a sensible one, and a numerical example is given to illustrate the tradeoffs. C h a p t e r 8: 7/00-Characteristic Values Internally balanced state-space realizations are a well-known route to model reduction. We firstly review LQG-balancing of a system, where a given system is balanced in a closed-loop with a Normalized LQG Controller. Invoking the ideas of the previous chapter, we then provide a natural minimum entropy/7/oo generalization: 7/00-balancin9. Here the system is balanced in a closed-loop with a Normalized 7/-/00 Controller (from Chapter 7). This allows the definition of a new set of input-output invariants, called the 7"[00-characteristic values. Hence a new model reduction scheme--7/o0-balanced truncation--is proposed and its properties are explored. Some of the main features are highlighted in a short numerical example. Strong connections with eopr/me factorization are derived along the way. C h a p t e r 9: L Q G a n d 7-/oo M o n o t o n i e i t y In Chapter 3 an 7/oo/LQG tradeoff was proved: as the 7/°0 constraint is relaxed (hence as robustness guarantees loosen) so the entropy decreases. Here we explore two stronger properties displayed by many minimum entropy 7/00 controllers, that of LQG monotonieity and 7/'/00 monotonicity: as the 7/-/00 constraint is relaxed, the LQG cost decreases monotonically, whereas the 7/00-norm increases monotonically. We conjecture that any minimum entropy/7-/00 controller is both LQG monotonic and 7/00 monotonic. An expression for the derivative of the LQG cost is derived, based on the state-space solution to the Minimum Entropy 7too Distance Problem from Chapter 4. This is a promising direction for the construction of a proof of LQG monotonicity in general. A p p e n d i x A: P r o o f of R e s u l t s N e e d e d in t h e T e x t To preserve the flow of the main body of the text, some of the proofs are presented here. A p p e n d i x B: E n t r o p y F o r m u l a e : A l t e r n a t i v e D e r i v a t i o n For added insight, an alternative derivation of the entropy formulae and some related results from Chapter 3 is presented. A p p e n d i x C: N o t a t i o n Although notation is standard as far as possible, for clarity the main notational conventions are listed.
1.3. BA CKGRO UND REFERENCES
1.3
5
B a c k g r o u n d References
The reader is assumed to have a grounding in the fundamentals of the following three areas. * E s s e n t i a l s y s t e m t h e o r y , such as controllability, observability and related concepts. A comprehensive reference is [36, Chapters 1 & 2]. * L Q G c o n t r o l t h e o r y , as covered by [2] or [38], for example. * ? / ~ - o p t i m a l c o n t r o l t h e o r y , as covered by [24]. We shall assume a basic famillaxity with Hardy spaces, but not beyond that contained in [24, Chapter 2]. Each of these three areas is now sufficiently mature to have a vast literature. We shall therefore restrict review material to the appropriate parts of chapters and cite essential references only.
Chapter 2 T h e E n t r o p y of a S y s t e m
8
2.1
CHAPTER 2. THE E N T R O P Y OF A S Y S T E M
Introduction
Entropy has established itself as an important notion, with a wide applicability in a number of diverse subjects. For example, Shannon [54] introduced the concept of entropy as an information measure, whilst Burg [11] proposed a successful entropy method in spectral analysis. In this chapter we introduce the entropy of a system which satisfies an 740~-norm bound, and derive some important properties. We show that the entropy is an upper bound on the LQG cost, and interpret the ~foo-norm bound on the system as providing a prespecified level of robustness.
2.2
Definition
of the Entropy
Let us begin with a formal definition. Definition 2.2.1 ( E n t r o p y at s0 G (0,oo)) Let H E T~£oo and let 7 E IR be such that IIHIloo < 7. Let so E (0, oo). Then the entropy of H at so is defined by
l(H;~/; so):=
72
0o
2
-~"~/2,xIn Idet(l--'l-'H'(jlo)ll(j~))l[~]dl, ' [so+~oel "
Apart from an extra factor of so in the numerator, this definition is equivalent to the entropy integral defined in [4, 3] in the context of indeterminate extension problems. This extra factor allows us to extend the domain of definition to include the case when So ~ oo. In doing so, we obtain the entropy at infinity of the system, which is of great significance in what is to follow. Definition 2.2.2 ( E n t r o p y at infinity) Let H G 7~£0o and let 7 E IR be such that [IHlIoo < % Then the entropy of H at infinity is defined by
I(H; "f; co) := I 0llm {/(H; 7; so)}. ~oO R e m a r k 2.2.3 The entropy of H is a useful measure of how close H is to the upper bound 7 on ~rl{H(j~)}. This can be most clearly seen by rewriting the entropy as
¢
=
[,o'
l(H;ff;so) = - 2 ~ / ~ o o ~ . In ] i - ~f-ecr~{H(j~)}] [ ~ j
] d~.
(2.1)
If tr~{H(jw)} > 7' - 6 ' for some frequency range wl < w < wz, then I(H; 7; so) ~ oo as e --* 0. Also, it is clear front (2.1) that all the singular values el{H} of H are included, unlike the lf~o-norm which depends only on the largest singular value ¢1{H}. R e m a r k 2.2.4 By convention, in future we will use 'entropy' to mean 'entropy at infinity' unless otherwise stated.
2.3. SOME P R O P E R T I E S
9
R e m a r k 2.2.5 The domain of definition of the entropy at so can in fact be extended to the whole of the right-half plane by replacing the term sg/(S~o+W 2) with (Re{s0})z/(ls0jwl) 2, but we shall not need to do this. (To look ahead a little, the control systems derived in later chapters are physically realizable (real-rational) only if s0 is real.) As a glimpse of the importance of the entropy, we mention here some of its interpretations. We will see further on in this chapter precisely how to relate entropy to the LQG cost associated with a system. In Chapter 5, we outline the combined 2~¢o/LQG framework of [8, 9] in which an auziliary LQG cost is used. We prove that, under certain conditions, the auxiliary cost is exactly the entropy. In Chapter 6 we cover the equivalence between entropy and the exponential-of-quadratic cost of the Risk-Sensitive LQG Control Problem of [34, 57]. The entropy will also be interpreted in the context of 'band extension' problems [20} in Chapter 4.
2.3
Some
Properties
We now state various properties of the entropy, some of which will be needed later. P r o p o s i t i o n 2.3.1 Let H • 7~£o0 and let 7 • lit be such that IIHlloo < 7. So • (0, oo]. Then
(i) I(H;7; so) is
Let
well-defined.
(ii) I(H;7;So ) >_ O. (iii) I(H;7;so) < so for any So • (0, oo). (iv) I ( H ; 7 ; o o ) < so if and only if H(oo) = O.
(v) I(H; 7; so)
= 0 if and only if H = O.
(vi) I ( U H V ; 7 ; S o ) = I(H;7;So) for any U, V • 77.£oo such that U*U = I, VV* = I. (vii)
I(H*;7; so)
Proof
= I(HT; 7; 8o) =
I(H; 7; 8o).
For convenience, define 72 a(7; w) := -~-~ In I det(I - 7-2H*(jw)H(jw))l
(2.2)
~(~; so) . - ~ + ~
(2.3)
and
so that I(H;7;So ) =
F c~(7;w)/3(W;so) dw.
(2.4)
so
Begin with the fact that
JIHIIoo< 7
~=~
I >>I - 7-2H*(jw)H(jw) > 0,
Vw • lit U {so}.
(2.5)
C H A P T E R 2. THE E N T R O P Y OF A S Y S T E M
10
This implies that 0 < er(7; w) < b < 00,
Vw 6 ]R U {00} and for some b E IR.
(2.6)
Also, it is obvious that
0 < fl(w; so) < 1,
Vw 6 lit U {00} and Vs0 6 (0, ¢¢1.
Hence the integral is well-defined, which proves Part (i), and non-negative, which proves Part (ii). Now if So 6 (0, co), we also have (1 + w2)fl(w;
so) ~ s,~ <
00
as
w ~
4-00.
Part (iii) follows from this together with (2.6). For Part (iv) we note that, as so ~ ¢x), we have ~(w; So) ~ 1 monotonically from below for each w E ]1%. Now, because H is rational and proper, wch{H(jw)} is uniformly bounded if and only if H(00) = 0. It can then be shown, by using Lemlna A.2.1(i) of Appendix A, that
.'. :. (1+~)~(7;w)
H(00) = 0
wemu{00}
andforsomecem.
(2.7)
Clearly, when H satisfies (2.7), a(7; w) is integrable and I(H; 7; 00) is finite by dominated convergence, which proves Part (iv). Part (v) is a simple consequence of the fact that the integrand a(7;w)~(w; So) is non-negative. Parts (vi) and (vii) are immediate on using the well-known matrix identity d e t ( I - M ' M ) = det(I - MM*) = d e t ( I - (MT)*MT).
[] The next proposition shows that the entropy at so 6 (0, ¢~] of a fixed H 6 7~£~ is monotonically decreasing with 7" This will be useful in Section 3.6. P r o p o s i t i o n 2.3.2 Let H E gf-.oo and let 7 6 IFL be such that [IHII~ < 7. If So e (0,oo) then I ( H ; 7 ; so) is a monotonically decreasing function ofT. If H(00) = O then I ( H ; 7 ; 00) is a monotonically decreasing function OfT. P r o o f Firstly note that, by Proposition 2.3.1(iii) and (iv), I(H;'g; So) and I(H; 7; 00) are both finite under the stated assumptions on H. Define ~ := 7 -2 and let e 6 IR. It is easy to show that
- l n I d e t ( I - (5 + e ) H * ( j w ) H ( j w ) ) I -4- In ] det(! - ~ H ' ( j w ) H ( j w ) ) I = - In I d e t ( I - e(I - ~ H * ( j w ) H ( j ~ ) ) - I H * ( j w ) H ( j w ) ) ] =
e
trace[(/-
~H*(jw)H(jw))-lH*(jw)H(jw)] + O(e~),
2.4. RELATIONS TO THE LQG COST
11
using Lemma A.2.1(i) of Appendix A to get the second equality. It follows that
ff~{
~H*(joJ)H(jw))[}
- In [ det(I
= t r a c e [ ( / - ~H*(jw)H(jw)) -1H*(jw)H(jw)].
Therefore, continuing with the notation of equations (2.2), (2.3) and (2.4), we have
o~ -
1
- - In I d e t ( I - ~H*(jw)H(j~,)) I r~7 1
- - - t r a c e [ ( / - ~tt*(jw)ft(j,,))-leH*(j~)tt(j,,)]. ~r~7
Define ,~, := a,{~H*(j~)H(/~)}, and note that
0 < ~:i < 1 b e c a u s e
(2.8)
IIH]I® < 7. We
can then rewrite (2.8) as
Define f ( z ) := ln(1 - z) + (1 - x ) - l x for 0 _< z < 1. It is straightforward to show that df/dz = (1 - z)-~z _> 0 for 0 _< z < 1, so f ( z ) is a monotonically increasing function of z. But f(0) = 0 so f ( z ) > 0 for 0 < z < 1. Applying this to (2.9) gives
~7~(7; ~) _<0. Hence a ( 7 ; w ) is a monotonically decreasing function of 7 for each real w. The result follows on recalling (2.4) and noting that fl(w; So) is independent of 7. [3
2.4
Relations
to the
LQG
Cost
Our next aim is to relate entropy to the well-known concept of LQG cost [2, 38]. The LQG cost of a stable system is just the mean-square output of the system when it is driven by a white noise input. The formal definition is as follows. D e f i n i t i o n 2.4.1 ( L Q G cost) Let H E T~7-loo, and let z be the output of this system
when driven by a zero mean Gaussian white noise input w with spectrum equal to the identity. Then the associated LQG cost is defined by C(H) : : lira E{IcI~}, T--*oo
where
W r : = ~1
f_~ zr(t)z(t)dt,
and where E denotes expectation with respect to the noise.
CHAPTER 2. THE ENTROPY OF A SYSTEM
12
R e m a r k 2.4.2 The above definition is actually of the steady-state LQG cost. That is, C(H) is just the limit as T ~ oo of the finite-time LQG cost
Cr(H) := E{WT}. R e m a r k 2.4.3 By Parseval's Theorem (see e.g., [65]), we have an equivalent frequency domain characterization [55, p107],[53]: 1 r~°
C(H) = ~ ]_~ trace[H*(jw)H(jw)] dw = IIHIJ~,.
Clearly C(H) is finite if and only if g ( c o ) = 0 (using a similar argument to that used to prove Proposition 2.3.1(iv)). This corresponds to none of the noise w appearing instantaneously at the output z. For convenience, define
X(H; co; co) := 2m (Z(H; ~; co)}, that is, the entropy evaluated when the 7-/co-norm bound IIHIIoo < 7 is relaxed completely. We can now relate entropy and LQG cost, one of the main results of this chapter. T h e o r e m 2.4.4 Let H e nT-[oo with H(co) = O. Let 7 E lit be such that IIHHoo < 7.
Then (i) I ( H ; 7 ; c o ) > C(H). (ii) I ( H ; 7 ; co) = C(H) + 0(7-2). (iii) I(H; co; co) = C(H). Proof
Appendix A.3.
[]
The vital observation is: for 7 < co, the entropy is an upper bound on the LQG cost, but when the ://**-norm bound is relaxed completely (~/---* co), the entropy is equal to the LQG cost. Part (ii) shows how fast the entropy converges to the LQG cost as 7 increases.
2.5
The
~co-Norm
Bound
Here we review the interpretation of the 7/oo-norm bound I[H[[oo < % Since the ideas are now well-known, we will be brief and refer the interested reader to [24] for full details.
2.5. THE TI~-NORM BOUND
13
There axe two interpretations of the 7~o,-norm which are of interest in a control systems context. The first stems from the fact [24] that the 7/o,-norm is the induced norm from 7£~2 to 7~7"/2. In other words, if H E 7~74oo and w G 7~7"/~then Hw E 7~7~2, and furthermore,
II~rlloo = sup(llU~ll2
:
~o ~ ~ 2
and II~oll~ ~ 1}.
tO
From an input-output point of view, this means that the maximum energy gain, from bounded energy inputs w to bounded energy outputs z = ttw, is equal to IIHllo,. Hence [IHIIoo < 7 means that this energy gain is less than 7- If we interpret w as representing bounded energy disturbance signals, we see that HHII~o < 7 specifies a level of disturbance rejection. The second interpretation of [[hr[l~ < 7 is much more important, and is based on the well-known Small Gain Theorem. The version of the theorem of interest to us is stated below. T h e o r e m 2.5.1 ( T h e S m a l l G a i n T h e o r e m [66]) Let H 6 7~7-[oo. Suppose A 6 ~7-[~ is connected frora the output of H to the input of H, as illustrated in Figure ~.1. Then this closed-loop is (internally) stable if
~l{z~(j~)} ~I{H(j~)} < 1,
w ~ m u {~}.
R e m a r k 2.5.2 In our setup, we have H E :ggfo. and IIHIIo. < 7. Then by the Small Gain Theorem, A E 7~7-/~ and Ilall® ___7 -I is sufficient to guarantee stability of the system in Figure 2.1. We can interpret A as a perturbation to the system, representing system uncertainty. Thus the Small Gain Theorem tells us that an 7~o-norm bound on H implies stability in the presence of 7-~o-norm bounded system uncertainty. That is, the ~oo-norm bound implies a prespecified level of stability robustness. Notice that the size of tolerable uncertainty varies as 7-1: robustness increases as the ~**-norm bound on H decreases.
14
CHAPTER 2. THE E N T R O P Y OF A S Y S T E M
zx I. |l
w ['tt
Z
Figure 2.1: Block diagram for the Small Gain Theorem
Chapter 3 T h e M i n i m u m E n t r o p y 7/~ C o n t r o l Problem
16
3.1
C H A P T E R 3. T H E M I N I M U M E N T R O P Y Tloo C O N T R O L P R O B L E M
Introduction
In the previous chapter we saw how to define the entropy of an 7Coo-norm bounded system. We also saw that a level of robustness is given by the 7-/oo-norm bound, and that the entropy overbounds the LQG cost. This analysis prompts us to pose the synthesis problem of finding a feedback controller which stabilizes a system subject to an 7/oo-norm bound whilst simultaneously minimizing the entropy. We will solve this problem in this chapter. The solution exploits the parametrization of all stabilized closed-loop systems that meet the 7too-norm bound. The central member of this set is shown to minimize the entropy at infinity and for this case a particularly simple formula is derived for the minimum value of the entropy.
3.2
Problem Formulation
Consider a system P with a state-space description i: = A x + B l w + B~u Z ~- C1 x --~ D l l w + D 1 2 u
y = C2:e + D21w
where the signals are as follows: w E IR'm is the disturbance vector, u E ]R".2 is the control input vector, z E IRw is the error vector, t/ E IRv2 is the observation vector and a~ E IR'* is the state vector. As is usual in 7"/oo control problems we assume that ml _> 102, pl _> m2. The transfer function P from [w r u T ] r to [zr y r ] r is
P:=
P21 P2~
C1 C2
Dn D21
2
We connect a feedback controller K from y to u, as illustrated in Figure 3.1. This gives us a standard general framework into which a wide variety of control problems will fit (as explained in [24, Chapter 3]), so we shall adopt the usual convention of referring to P as a standard plant. This is to distinguish P from the actual plant, which is embedded in P together with the weighting functions and interconnections appropriate for the particular problem in hand. For example, a certain choice for P is made in Chapter 7--see equation (7.1). Until then, P is taken to be any standard plant, subject to the assumptions below. The closed-loop transfer function H from disturbance vector w to error vector z is given by the (lower) linear fractional map H = ,T(P, K ) := Pll + P t ~ K ( I - P 2 ~ K ) - I P n .
(3.1)
We need the following standard assumption which is necessary and sufficient for the existence of a controller K which stabilizes _P.
17
3.2. P R O B L E M F O R M U L A T I O N
W
r
Ur--q
•
P
Z
hY
Figure 3.1: The closed-loop system A s s u m p t i o n 3.2.1 (A, B2) is stabilizable and (C2, A) is detectable. R e m a r k 3.2.2 Note that by ~stabilizes' we mean qnternally stabilizes'. Also, K stabilizes P if and only if K stabilizes P22. See [24, Chapter 4] for details. The following definitions will be convenient in the sequel. Definition 3.2.3 ((P,-/)-admissible controller) Let P be a standard plant. A controller K is said to be (P,7)-admissible i l K stabilizes P and [[.T'(P, K)11¢¢ < 3'. Definition 3.2.4 ((P, 7)-admissible closed-loop) A transfer function matrix H E TtTlco is said to be a (P, 7)-admissible closed-loop if H = :F(P, K ) and K is a (P, 7)admissible controller. From the discussion in Section 2.5, the 7-(co-norm bound 7 provides robustness which increases as ,y decreases. This motivates the usual 7"/oo-optimal control problem [24] of maximizing such robustness by minimizing the T/co-norm bound 7.
Problem 3.2.5 (The ?/~-Optimal Control Problem) Let P be a standard plant. Find a controller K , and % E IR which satisfy
7o = i~f{7 : K is a (P, 7)-admissible controller} = II~'(P, Ko)II~. Pick 7 E ]R such that 3' > %. Then the set of (P, 7)-admissible controllers is non-empty. We can now impose our minimum entropy constraint.
18
C H A P T E R 3. THE M I N I M U M E N T R O P Y 7-Io0 C O N T R O L P R O B L E M
P r o b l e m 3.2.6 ( T h e M i n i m u m E n t r o p y 7/00 C o n t r o l P r o b l e m ) Let P be a standard plant, let 7 > 7o, and let so E (0, ov]. Minimize the closed-loop entropy I(It; 7; so) over all ( P, 7)-admissible closed-loops 11. We will solve the Minimum Entropy 7/o0 Control Problem by firstly parametrizing all (P, 7)-admissible closed-loops. This will make use of the parametrization of all (P, 7)" admissible controllers given in [28]. We shall therefore require the following assumptions to go with Assumption 3.2.1. Assumptions 3.2.7 (i) D1, =
r]r and
Cx
Da~
=
- m2 = rank
A - C2j w I
B1 D~I
-P2,
VwEIR.
Assumption 3.2.7(i) ensures the existence of proper controllers, whilst Assumption 3.2.7(ii) ensures [2] that the LQG problem associated with the plant P has an asymptotically stable closed-loop. For completeness, we also state the LQG problem [2, 38] associated with the standard plant P. From Definition 2.4.1 and Remark 2.4.3 we see that the LQG cost is just C ( H ) = II1111Lwhere H = Y:(P, K) is a stabilized closed-loop. P r o b l e m 3.2.8 ( T h e L Q G C o n t r o l P r o b l e m ) Let P be a standard plant. mize the LOG cost C ( 3:( P, K ) ) over all stabilized closed-loops ~'(P, K).
Mini-
The Minimum Entropy 7-(~o Control Problem will be solved in Chapter 4, via the solution of the associated Minimum Entropy 7-/00 Distance Problem, by exploiting recent state-space results of [30] on the latter problem. It is then possible to back-substitute to find the solution of the original Minimum Entropy 7/00 Control Problem. However, in this chapter, by exploiting the new results of [28, 29, 17], the Minimum Entropy 7/00 Control Problem is completely solved directly in terms of the original standard plant P without the intermediate step of a general distance problem. Furthermore, with entropy evaluated at infinity and assuming Dll = 0 (which guarantees that the minimum value of the entropy evaluated at infinity is finite) simple and explicit statespace formulae are given for the minimum entropy controller and the minimum value of the entropy. These are in terms of the state-space realization of P and the solutions to the two algebraic Riccati equations involved in parametrizing all (P,7)-admissible controllers. The remainder of this chapter is structured as follows. In the next section we derive the solution to the Minimmn Entropy 7-/00 Control Problem for arbitrary so E (0, vo), after firstly parametrizing all (P,7)-admissible closed-loops. Section 3.4 deals with entropy at infinity; the minimum entropy solution is characterized in a particularly simple way in this case--explicit state-space formulae are derived in Section 3.5. Section 3.6 provides upper bounds on the 7/oo-norm and LQG cost, and their inherent tradeoff, whilst Section 3.7 shows how to recover the LQG problem and its solution from the Minimum Entropy 71oo Control Problem and its solution.
3.3. SOLUTION IN THE GENERAL CASE
3.3
19
S o l u t i o n in t h e G e n e r a l C a s e
We solve in this section the Minimum Entropy 7/00 Control Problem at an arbitrary So E (0,c~). To begin, all (P,7)-admissible closed-loop transfer functions H are parametrized using the recent results of [28]. All (P,7)-admissible closed-loops H are given as the linear fractional map of a stable all-pass transfer function matrix Ja and an arbitrary stable contraction @ (i.e., @ E B7~7"l~). Then, by adapting the arguments of [4, 3], the choice of @ which minimizes the entropy is derived, together with an expression for the resulting minimum value of the entropy. This problem has also been examined in [39], for the 'one-block' case when PI~ and P21 are square. To be specific, from [28], all (P,7)-admissible controllers K can be written K = 75r(K,, ~), where @ E B7~7~00, and K, can be calculated from the given data (i.e., P and 7). Throughout this chapter we find it convenient to keep the factor 7 explicit in a slightly different way to [28]. Instead of using K = ~'(K~, @), with @ E ~ 0 0 and II~IW < -r, we use K = "r~(Ka, ¢) with @ E B~7-/oo, and redefine Ka accordingly. It follows that all (P, 7)-admissible closed-loops are given by H = }'(P,7~(K,,@)) =: 7~'(J, @). It is then convenient to partition J into [ Jli J12 ] 321 J~2
J=
compatibly with @ in yr(j, @). The definition of J is made clear in Figure 3.2. Now J is asymptotically stable and contractive. It can therefore be thought of as a sub-block of an asymptotically stable and all-pass matrix J~. The matrix Ja is an all-pass dilation of J. By performing the dilation of J into Jo we can establish the following lemma; details are given in [29]. L e m m a 3.3.1 (All (P,7)-admlssible closed-loops) All ( P, 7)-admissible closed-loops H are given by H = 7Y:(Jo, O)
where P3
0 :----
m l --P2
,n2~
,
@ 6 BR7/00.
p l --rrl2
Furthermore, the matriz m 1
ml J, = rn| --P2
has the .following properties:
J~ J31 J~2 J33
20
CHAPTER 3. THE MINIMUM E N T R O P Y ~
W
Z
CONTROL PROBLEM
W
Z
J
,
I
|
l |
I i
II ¢ I1.. < I
ENN .................... K
Figure 3.2: All closed-loops satisfying the 7-/oo-norm bound
(i) J~ e nT~oo. (ii) d~ is all-pass. (iii) J2=(oo) = 0. It is now possible to exploit the structure of this parametrization in order to derive the choice of @ E BRT"(oo which minimizes the closed-loop entropy at So E (0, oo). T h e o r e m 8.8.2 Consider the class of all (P, 7)-admissible closed-loops H, as parametrized in Lemma 3.3.1. Then I(H;7;so), the closed-loop entropy at So G (0, oo), is minimized over this class by the unique choice
= J;2(s0). P r o o f Here we adapt the method of [4, 3]. From Lemma 3.3.1, we seek a @ E BRT-/~o which minimizes I(H; 7; so), where all (P, "/)-admissible closed-loops H are H = 73v(ja, [ •
00
0]
).
Now J~ is all-pass so J~J~ = I, which can be expressed in terms of its block partitions as
I = J~lJn +
Jzl
[,2,] d31
3.3. SOLUTION IN THE GENERAL CASE
o = < [ J,2 J,3 ]+
s,,
21
(33,
s,, s .
By exploiting (3.2), (3.3), (3.3)*, and (3.4), it may be shown by straightforward manipulations that
o
,]-[',
0
I 0
I - q - ' H * H = [J2~] *Jal J ' [ I o
/t
x[I-0"0
d,s
o 0
0
]}-
I.
Upon substituting this into the definition of I(H; 7; so) (see Definition 2.2.1) it follows that I(H;7;s0) = 72I(O; 1; so) 72 [oo
I.
+~
,~) ] &, &(s.,) ]'[ s'~('~') 4~(j,,) ]1[ , [d +.,~J
ln]det[Jzx(ri°)
In I det(I - J~(j~,,)O(i~))l I.*g + ,o'] &'
lr
=" 72I(0; 1; so)
+
(~)
+
(b).
Equation (3.2) allows us to write (a) = -r2/(&; 1;,o).
By Lemma 3.3.1, both • and Jm are in RT-/oo. Furthermore, I]OHoo< 1 and IJ&21loo~ 1 (since Jm is part of the all-pass matrix J~). This is enough to guarantee that det(I J226) ~'1 E 7Z7-/oo, allowing the use of Poisson's Integral Theorem [52, Theorem 17.16] to evaluate (b): (b) = 7'80 In I det(I - 22(*o)0(*o))1. Therefore
I(H; 7; so) = 72I(0; 1; So) + 72I(Jn; 1; so) + 7280 ha [ det(I - Jm(so)O(so))[.
(3.5)
To proceed, we reparametrize as follows. Define a constant Julia matrix [65, p148] by L=
L21 Lm
:=
(I-Jm(so)J~:(So)p/2
Jm(8o)
"
It is easy to verify that L is unitary. Then define a new arbitrary stable parameter E BTZT/oo, by = S=(L, 0), • ~ BTZn®.
C H A P T E R 3. THE MINIMUM E N T R O P Y TIoo CONTROL P R O B L E M
22
Note that this maps BT~oo onto itself, and in particular = 0 ~
¢ = J;~(So).
(3.6)
By applying an identical argument to that used to obtain (3.5) above, it can be shown that I ( ~ ; 1; so) = I ( ~ ; 1; so) + I(S;2(So); 1; So) +
8ola I det(/-
Jz2(so)~(so))l.
(3.7)
I(H; 7; So) : 7zI(~; 1; ao) + 7zI(J,,; 1; so) - 72I(d;2(so); 1; so).
(3.8)
Eliminating ~ between (3.5) and (3.7) gives
It is immediate from (3.8) that I(//;7;So) is milfimized by the unique choice ~ = 0. To complete the proof, it is only necessary to recall (3.6). [] Denote 'minimum entropy at s0' quantities by ( • )ME,0. The following corollary is then immediate. C o r o l l a r y 3.3.3 The minimum value of I(H;7;So), admissible closed-loops, is
as H varies over all (P,7)"
I(HMEoo; q; SO) = q2I(Jtl; 1; So) - 72I(J~2(So); 1; so). P r o o f Put ~ = 0 in (3.8).
3.4
[]
S o l u t i o n at I n f i n i t y
In this section we concentrate on the special case of the Minimum Entropy 7-/0o Control Problem based on I ( / / ; 7 ; oo), the entropy at infinity. As remarked in Chapter 2, this is the most interesting and important case because of the strong connections with other control problems. See [20] for entropy at infinity in the context of Nehari interpolation. R e m a r k 3.4.1 From Proposition 2.3.1(iv), I(tf; 7; co) is finite if and only i f / / ( c o ) = 0. Since ~~(co) = Dll + D~zK(oo)Dzl, a necessary and sufficient condition f o r / / ( c o ) = 0 is that lip,_,, 2 0(pl_,,~)×,~:]D~l = 0 and D~x[I,,,,_~ 0(m,_~)xp2] r = 0. In tMs case an equivalent problem with D u = 0 is easily derived by making the change of control T T input variable from u to v, given by v = u + Dl:DI1D21y. It is convenient to introduce the notation
//{~} :__7.~(do, [ •0 0 o] )
3.4. SOLUTION A T I N F I N I T Y
23
to denote a (P, 7)-admissible closed-loop produced via Lemma 3.3.1 by a ~I, E BT~Tfo,. Also define g'MEoo := $Olira {OM~oo} --4,OO = lira {J;2(So)} ° 0 -..-4~0
= O, where the second equality is taken from Theorem 3.3.2 and the third equality is from Lemma 3.3.1(iii). Then n { ¢ M ~ o } = H{0} = ~Jxx. (3.9) The minimum entropy at infinity solution is obtained by taking the limit, as So ~ oo, of the minimum entropy solution at finite 8o. A tittle care is needed when the minimum value of the entropy at infinity is not finite. This is all formalized in the next theorem. Theorem 3.4.2
(i) The choice = ~MEoo = 0
in the parametrization of Lemma 3.8.1 of all ( P,7)-admissible closed-loops H, is the unique choice of • which satisfies:
w ~ BnT~oo (~ # o) : 3s(q) < co such that Yso > s ( ~ ) :
I(H{fMEoo};7;So) < I(H{~2};7;So).
(ii) If Jn(co) = 0 then ~ = qMEoo = 0 is the unique choice of ~ which minimizes I(H{~}; 7; co) o~e, • e 13nn®. P r o o f Part (i) In view of (3.9), put H{OMEoo} = 7 J l l into equation (3.5). Then for any ~ E BTgT-/oo,
I(H(¢M~};
7; s0) -- X(H{~}; 7; So) = -7~X(~; 1; s0) - 7 % In l a e t ( 1 - s,~(,o)~,(so))[
=:
a(so)
+
b(so).
Consider • E BTZT-/oo such that q # 0. Then I ( q ; 1; So) # 0, and it is clear that
a(so) < 0 and b(so) >_ 0 for all as E (0, col. If q is proper, but not strictly proper, then a(ao) --* - c o (monotonically for each fixed q, by the argument in the proof of Proposition 2.3.1(iv)) as s0 --* co. An application of Lemma A.2.1(i) of Appendix A, shows that b(So) ~ b, a constant, as s0 ~ co, This guarantees the existence of a positive number s ( ~ ) < o% such that a(so)+b(s0) < 0 for all so > s(q). Alternatively, if q is strictly proper, then a(so) --~ a < 0, a constant, whereas (by Lemma A.2.1(i) again) b(s0) --* 0 as so --* co, and once more there exists a positive number s ( q ) < co,
24
C H A P T E R 3. T H E M I N I M U M E N T R O P Y ~oo C O N T R O L P R O B L E M
such that a(so) + b(so) < 0 for all so > s(e2). This establishes that @MEoo ---- 0 does have the property stated in the Part (i) of the theorem. Note that the above also proves that ¢MSoo = 0 gives a strictly smaller entropy than any other ~ E BT£7-/oo for sufficiently large s0, hence this ~Mgco = 0 is unique, completing the proof of Part (i). Part (ii) By hypothesis, Jlt(co) = 0, which implies that the minimum entropy closed-loop, H{~MEoo} = 7,/11, is strictly proper. By Proposition 2.3.1(iv), the minimum value of I ( H ; 7; co) is then finite. Suppose the claim of the Part (ii) is false, i.e., that there exists a ~ E B:R~/~o, call it ~1, such that I(H{ff~MSoo}; 7; Co) -- I ( H { ~ I } ; 7; co) = c, for some c > 0.
(3.10)
Obviously this is only possible if H ( ~ I } is strictly proper, so that I ( H ( ¢ ~ } ; 7 ; co) is finite, and there exists sl E (0, co), such that ] I ( H { @ l } ; 7 ; c o ) - I ( H { ~ , } ; 7 ; s o ) l < c/6,
Vs0 > sl.
(3.11)
Similarly, there exists s2 E (0, co), such that IX(H{~MEoJ; 7; So) - I(H{*~Eoo}; 7; oo)l < c/6,
Vso > s,.
(3.12)
The entropy evaluation in (3.5), together with ~MSoo = 0 and ~ME.o = J~(so), gives I(H{'~ME.o}; 7; So) -- I(H{#MEo~}; 7; So) = --72s0 In I det(I - J~2(s0)J;2(So))l ---~ 0 a s s 0 ----4 o o
by Lemma A.2.1(i) of Appendix A, since J~2(So) = O(so~). Thus, there exists s3 E (0, co), such that
II(H{@ME,o}; 7; So) -- I(H{'~MSoo}; 7; so)l < e/6, ¥so > ss.
(3.13)
Using (3.10)-(3.13) in conjunction with the identity
I(H{~.o};
7; so) - I ( H { ~ } ;
~; So)= I ( / t { ~ . 0 }; ~; So) - X ( H { ¢ ~ o o } ; ~; so) + r(H{~®}; 7; So) - I ( H { ~ M E ~ } ; 7; co) + I ( g { * M S o o } ; 7 ; co) - I(H{~1};7; co) + X ( H { ¢ , } ; 7 ; 0o) - Z(H{~I};7; So)
gives, for all so > max{sl,s2,ss}, I(g{~MF..o};7;So) -- I ( H { ~ , } ; 7 ; s o ) > - c / 6 + c - c/6 - c/6 = c/2 > O. So for all sufficiently large so,
I(H{¢~E.o};7; so) - I ( H { ~ } ; 7 ; so) > 0,
3.5. STATE-SPACE FORMULAE FOR THE MINIMUM E N T R O P Y
25
which contradicts the fact that ~ME,, is the minimum entropy solution at So. Therefore, (3.10) cannot hold, which establishes that '~Mg00 = 0 does indeed solve the Minimum Entropy 71oo Control Problem at infinity. The proof is completed by noting that uniqueness of this ffMgoo is implied by the proof of uniqueness of ~ME00in Part (i). [] The minimum entropy at infinity solution therefore has the simple and appealing characterization of q~ = 0: it is the 'central solution' from the ball of admissible closedloops. The following corollary arises by using this important fact in Lemma 3.3.1, Corollary 3.3.3 and in the controller given in [28]. C o r o l l a r y 3.4.3 With entropy evaluated at infinity, the unique controller which solves the Minimum Entropy 71oo Control Problem (Problem 3.2.6) is
KME00 = 7Jr(If~, 0) = 7(K.)11, that is, the central ( P, 7)-admissible controller. The minimum entropy closed-loop is HMEo~ = ~r(e, KMF~00) = 7~'(J~, O) = 7 J r , that is, the central (P,7)-admissible closed-loop. The minimum value of the closed-loop entropy is I(HME00;7; oo) = 72I(Jn; 1; oo).
3.5
S t a t e - S p a c e F o r m u l a e for t h e M i n i m u m tropy
En-
In this section we derive an explicit state-space solution to the Minimum Entropy 7100 Control Problem with entropy evaluated at infinity. In particular, we shall derive simple formulae for the minimum value of the entropy. For the remainder of this section we assume i)11 = 0. Then the minimum value of the entropy at infinity is finite (Remark 3.4.1), and is given by the state-space formulae of the following theorem. T h e o r e m 3.5.1 Assume that Dlx = O. Recall the standing assumptions that D12 : [0,~:×0,:_.,:) I~,] T and that D,: = [O~×(.,,_p:) Ip:]. Let D_t make [D,z D_L] square and orthogonal, and let D_t make [D~: DT] T square and orthogonal. Let X00 = X ~ > 0 be the solution to the algebraic Riccati equation 0 = X00(A - BzD Tf'~12~J1]+ ( a - B 2 D T 2 C 1 ) T x c o + '-'I"'TnL'-L~'-L'-'ImTf' + Xoo(7-2B1Br - B2BT)Xoo such that A - B2Drl,C1 + (7-2BIB~ - B~Br)Xoo
is asymptotically stable,
(3.14)
C H A P T E R 3. THE MINIMUM E N T R O P Y 7"loo CONTROL P R O B L E M
26
and let Yoo = y r >_ 0 be the solution to the algebraic Riccati equation 0 = Yoo(A - B~D~zC2) T + (A - B~Dr, C2)Y~o + B , D ~ b ± B ~ + Yoo(7-2CTc1 - C2TV2)Yoo
(3.15)
such that A - B, DraC2 + Yoo(7-=CrzCz - C~C2)
is asymptotically stable.
Define Zoo := (I - 7-2YooXoo) -1. Then the minimum value of the closed-loop entropy at infinity is I(HMEOO; 7; co) = trace[CaYooC T] + trace[(D2aBT + C2Yoo)XooZoo(BzDT1 + YooCT)] = trace[B~XooBx] + trace[(DT2C1 + arxoo)ZooYoo(CTD12 + XooB2)].
(3.16) (3.17)
R e m a r k 3.5.2 An alternative proof to the one given in the remainder of this section is given in Appendix B. The derivation there is based on the approaches developed in [17, 28, 29]. It has the advantage of providing an interpretation of the separate terms making up the total entropy. To use the terminology from [17, 29], it is shown in Appendix B that the total entropy in (3.17) is composed of the 'Full Information' entropy plus the entropy of the derived 'Output Estimation' problem. Alternatively, the total entropy in (3.16) is composed of the 'Full Control' entropy plus the entropy of the derived 'Disturbance Feedforward' problem. R e m a r k 3.5.3 Note that from [28] we know that there exists Xoo _> 0 and Yoo > 0 as in the theorem, with p(XooYoo) < 72, if and only if 7 > %. R e m a r k 3.5.4 It is valid to refer to Xoo (or Yoo) as the stabilizing solution to its Riccati equation because [36, Lemma 3.4-1] if a Riccati equation has a stabilizing solution then it is unique.
P r o o f of Theorem 3.5.1 The remainder of this section will be devoted to proving Theorem 3.5.1. We will prove formula (3.16) in detail. The dual formula (3.17) may be proved in a similar way, based on the fact (Proposition 2.3.1(vii)) that the entropy of H is the same as the entropy of H r. Since the proof is quite long, it is split into a number of intermediate le,nmas. The next lemma gives state-space formulae for the all-pass matrix d, needed in Lemma 3.3.1 to generate all (P,7)-admissible closed-loops.
3.5. STATE-SPACE FORMULAE FOR THE MINIMUM ENTROPY
27
L e m m a 3.~.5 With assumptions and definitions as in Theorem 27.5.1, the all-pass ma-
tri~. J. of Lemma 8.3.1 has a state-space realization
A =
.r-,/2B,
.rl/2Bz .r*/2B3
7-,/2/~1D21
3,,/2/~2 7*/2Bs
B2~1
7-112C1 7-11~Dn~1 .rl/2G~ .r~/~02 ,.),1/203
0 Dn bi
..t,1/2C3
D12 0 0
D± 0 0
where
O, = - ( D S C , + B~X®)Z® [~z = B~ +,7-~Y+CTDIz
C, = -(C, + 7-2Dz, B~X~)Zoo
ih = -Z~* X L C[ D.
O. = b , B~, r £
-X~C~Dj_
Bs =
c. = -A~ BfY£
and = Z~I(A - B2D~2C~ + (7-2B, B~ - B2B~)X=)Z~ + B, C2. Proof
See Appendix B.
[]
R e m a r k 3.5.6 Having already established that ~ M / ~ = 0, we immediately have from the above lemma together with Corollary 3.4.3 and [28] that, in the case of entropy evaluated at infinity and D n = 0, the unique minimum entropy 7-/~ controller has a realization
[ =
A -
IB1D~+Y~Cr2 ]
(D~C, + B~X~)Z=
0
"
This is an n-state realization, as is the given realization of P. The minimum entropy dosed-loop has a realization
HME~
=
=
"rJ::
A -B2(DT2C, + B~X~)Zo~ (B, DT~ + y+CT)C2 ft C~ -Dn(DT:C, + BTX+)Z+
B1 (B~DTI + l%C'[)D2, 0
Note that the above formulae for the nlinimum entropy "]-(¢~controller, closed-loop
and the resulting entropy only require the solutions of two independent algebraic Riccati equations, one for X,. and one for Y,,. We now perform a state-transformation which will make later calculations clearer.
28
C H A P T E R 3. THE M I N I M U M E N T R O P Y ~
CONTROL PROBLEM
L e m m a 3.5.7 There ezists a nonsingular state-transformation S which, when applied to the standard plant P, leads to 0
o]
0
'
YI>O.
~hen for
N:=A
T - BaD21C~,
partitioned compatibly with Yoo as follows, N=
Nn
Nm
'
C~
=
C21 C n
,
BI=
Bm
'
we have that
(i)
Bmb~= O.
(ii) N n = O.
(iii) Y~N~ + NlXY~ + Y~(',/-'C~Cn - C~C2~)~ + B n D~iTD- . c B nT = O. (iv) N,1 + Y~(7-2CTC,, -- C~C~,) is asymptotically stable. (v) Nm is asymptotically stable. P r o o f Since under a nonsingular state-transformation S, Yoo ~ SY¢oS r, and Yoo > 0, such an S clearly exists. Upon rewriting the Y¢o Riccati equation (3.15) in the implied block form, Part (i) is obtained immediately from the (2,2) block; Part (ii) follows from the (2,1) block and Part (i); Part (iii) is just the (1,1) block. Furthermore, it follows that A - B , D ~ C 2 + Y~(7-~Cr~C1 - Cr2C2)
]
[N,,+
=
N22
0
and since from Lemma 3.5.5, A - BtDr, C2 + Y o ~ ( 7 - ' c r c , - C[C~) is asymptotically stable, we conclude that both Nn + Y~(7-2CrCn - CTC2~) and Nm axe too, which proves Parts (iv) and (v). [] At this point we recall from Corollary 3.4.3 that the minimum value of the entropy at infinity is just
I(HM~;7; oo) = ~ir(J.; 1; oo) =
lim
,0-oo
°
-~-~,,,_ooln[det x
[ Jzl(jw) ] J3,(j~)
[J,1(jw) [ S2o Ida}
(3.18)
3.5. STATE-SPACE FORMULAE FOR THE MINIMUM E N T R O P Y
29
where (3.2) has been used to get (3.18). From Lemma 3.5.5, B2C1
B1
B1C~ C2
f4 C~
B, D2, Dn
c.
0.
b±
A
[J2,]= ']31
[*']
Note that
J31
ACA~DB[ ] =:
(3.19)
"
E :R7/¢~
because it is part of the asymptotically stable all-pass matrix Ja, but its inverse is not in 7~7-/oo. Lemma A.4.2 (Appendix A) shows us how to evaluate the integral in (3.18) for such a transfer function matrix: we find that I(HMEoo;7;oo) = --72 trace[D-IC/~] - 2 7 z ~
Ai{A - / ~ D - 1 C }
(3.20)
Re{AI}>O
To evaluate the second term we need to evaluate the unstable eigenvalues of A - / ~ D - 1 0 ,
J31
the 'A' matrix of
.
This is done in the next lemma.
L e m m a 3.5.8 With notation as above, 2~
A , { A - B D - I ¢ } = t r a c e [ B ~ b T b ± B r y t - Yoo(7-2cTc1 - C~C2)].
P r o o f Denoting a 'don't care' element of a matrix by (*), it is straightforward to exploit Lemmas 3.5.5 and 3.5.7 to show that A - B1D~IC2 - B1D±C8
=
0
(*)
Z[¢1MZ~.
[ -Y1NI~YI-1 - Y1(7-2C1~Cll - C~C21) (*) o 0
=
N,, 0
(*) (,)
]
Z g 1MZ~
where M "= A - B,D~C~ + ('~-'B,B~ - B,B~)X®. Therefore,
{~i{.~__ j~j~--l~*)} ___~{__~/{./~rll+ yl(,~-'C~lCll- cTc, I))} U {Ai{Nm}} U {AI{M}}. By Lemma 3.5.7(v), Nm is asymptotically stable, and by Lemma 3.5.5, so is M. Furthermore, by Lemma 3.5.7(iv), - ( N n + ]~(7-~CTCn - - C216~1)) ,T , is antistable. Hence
.~ -2 rT Re{)q}>O
i = trace[YiNTY~-1 + B n D ~ b ± B T y ~ -']
30
C H A P T E R 3. THE M I N I M U M E N T R O P Y 7-loo C O N T R O L P R O B L E M
(using Lemma 3.5.7(iii)Y~ -~) and therefore, using well-known properties of trace[ • ], 2~
A i { a - B/)-xO} = 2 trace[Y1NrY1-1 + B l l D- ±T D" z B nTY 1 - 1 ]
Re{Ai}>O =
trace[Y1/¥Tyl-1
+ NllYlYl-1
"T " T -1 ] + 2B,1D±Dj.BxlY1
"r " r -a + B l l b ~ b ± B T y l - , ] + B1,D±D±Bll)Y1
= trace[-Y~(7-2C~C~, - C~C2x) + B l l D- Tz D~± B l Tl Y , - I ] = trace[B, D- Ti-D z B ,T Y'~l - Yoo(7-2C~Cx - C~C,)]
as claimed, where we have used Lemma 3.5.7(iii)Y1-1 again to obtain the fourth equality, and the fact that Y~t = [ Y1-10 00] t ° ° b t a i n t h e f i f t h "
[]
It is now a simple matter to piece together the intermediate results to establish the entropy formula (3.16). Taking (3.20) with Lemma 3.5.8 and (3.19), gives I(HMEco;7; co) = -72 trace[D-1CB] - 272 ~ A~{A- BD-1C} Re{Ai}>0 = -~
trace[C2B1D~l + 02B1 + C a B l b ~ + OaJB1D21b r] - 7 2 trace[Blb~/9± BT y £ - Yoo( 7 - z c T c1 - c T c , ) ]
= -72 traceiC2B1Drl - (C~ + 7-'D~IBTXoo)Z=(B1D T, + YooCr)] t -T + D- ± B 1T Y~B1D± ! "r -- 72 trace[-D±B 1r Y~B1D± - Y o o ( 7 - ' C ~ C l - C~C~)]
= trace[C,Y~C~]
+'? tracet(C,Zoo- C. + 7-'V,~B3X.oZ®)(B,D~ + r.oC~)] = traceIC,Y=C~] + trace[(DzlB~ + C2Y~)X~Z~(B1D~I + YooC~)] as claimed, and the proof of the first entropy formula of Theorem 3.5.1 is complete. []
3.6
U p p e r B o u n d s and an 7-/oo/LQG T r a d e o f f
In Section 2.4 we saw how the entropy is related to the LQG cost. Here we shall expand on this. We will also show that the minimum value of the entropy is a monotonically decreasing function of 7, and hence obtain an, important 7f~/LQG tradeoff. To begin with, we shall for convenience collect together the upper bounds implied by the Minimum Entropy 7foo Control Problem and Theorem 2.4.4.
3.7. R E C O V E R Y OF THE LQG SOLUTION
31
Proposition 3.6.1 (Upper bounds on 7/oo-norm and LQG cost) Suppose KME~ solves the Minimum Entropy 7-[~ Control Problem (Problem 3.2.6), in the case of entropy evaluated at infinity, for a #iven standard plant P and 7 > %. Let HME~ = .T( P~ K M ~ ) be the corresponding minimum entropy closed-loop. Then the 7-[~-norm of ttME~ and the LQG cost of HME~ satisfy
(i) IIHMsoolloo< 7. (ii) C(HMEoo) < I(HMEo~;'r; oo).
(iii) I(IIMEoo; 7; oo) -- C(ItMEoo) = O(7-2). Equality is achieved in (ii} when 7 ~ oo. As described in Section 2.5, the bound on the ~oo-norm gives a robustness guarantee via the Small Gain Theorem; the robustness guarantee decreases with increasing 7. On the other hand, the other bound is an upper bound on the LQG cost. The next theorem tells us that these upper bounds trade off against each other: robustness can only be improved at the expense of the (upper bound on the) LQG cost.
Theorem 3.6.2 ( T h e 7 ~ / L Q G t r a d e o f f ) The minimum value of the closed-loop entropy, I( HME~; 7; oo), is a monotonically decreasing function o.t:7. Proof Suppose 72 > 71 > 70. If K1 is a (P,71)-admissible controller, then K1 stabilizes P and 117(P,K~)lloo < ~ ___ "r~, so K~ i8 ~aso (P,72)-admissible. So the set of (P,'rl)-admissible closed-loops is a subset of the set of (P,'r2)-admissible closedloops. In particular, if H/, i = 1, 2, minimizes I(H;Ti; oo) over all (P,7~)-admissible dosed-loops H, then //1 is a (P, 71)-admissible closed-loop and a (P,72)-admissible closed-loop. Therefore / ( g , ; ~ ; oo) __/(/x~; ~ ; oo). But from Proposition 2.3.2, I(H1;T1; oo) _> I(H1;72; ~ ) , SO
X(H1;~I;oo) >__I(H2;72; oo), []
and the result is proved.
3.7
Recovery
of the LQG
Solution
The previous section showed us that loosening the 7-(~o-norm bound (by increasing 7), brings about a monotonic decrease in the entropy. By Theorem 2.4.4, if the 7-(~o-norm bound is completely relaxed (7 ~ oo) then the entropy becomes the LQG cost, and the set of (P, 7)-admissible controllers becomes the set of all controllers which stabilize P. We have therefore established the following.
32
C H A P T E R 3. T H E M I N I M U M E N T R O P Y 7"l¢~ C O N T R O L P R O B L E M
T h e o r e m 3.7.1 The following two problems are equivalent: (i) The Minimum Entropy T"lo~ Control Problem (Problem 3.2.6), in the case of 7 ---, oo and entropy evaluated at infinity. (ii) The L Q G Control Problem (Problem 3.2.8). In allowing 7 --4 0% any robustness guaranteed by the Minimum Entropy 9/¢o Control Problem is forfeited to obtain LQG-optimality. As pointed out by [15], an LQGoptimal controller has no guaranteed robustness margins. But if we are prepared to sacrifice a tittle LQG performance, some robustness can be guaranteed by using a minimum entropy ~/~ controller. Of course, it is in any case of interest to know the LQG-optimal (resp. ?/~-optimal) solution, for this sets the achievable limit of LQO performance (resp. robustness). Suppose now that D n = 0, so that the minimum value of the entropy and the minimum value of the LQG cost are finite. Theorem 3.7.1 allows us to recover the statespace solution to the LQG problem as the limit as 7 --* oo of the state-space solution of the Minimum Entropy ~/~ Control Problem given in Remark 3.5.6 and Theorem 3.5.1. Although the solution to the LQG problem is well-known (see e.g., [2, 38]), it is worth checking that we do obtain it in the linfit. If X¢¢ and Y~ are non-singular then we can apply the main result of [62] to the stabilizing solutions X ~ I and Y~I of the algebraic Riccati equations X~l(3.14)X~ ' and Y~(3.15)Y~ 1. This shows that both X ~ and Y~ are monotonically decreasing with 7: for 7z > 71 > 70 and with an obvious notation X~(7~) _< X~(7~)
and
Y~(72) < Y~(71).
(If X ~ and Y¢~ are not both non-singular then the above argument can be applied after identifying the non-singular parts of X ~ and Y~ in the manner described in [29]. The same monotonic behaviour is obtained.) Allowing 7 --* c¢ in equations (3.14) and (3.15) gives X~(oo) = )(2
and
Y~(oo) = Y2,
where X2 = X~ > 0 is the solution to the LQG algebraic Pdccati equation 0 = X z ( A - BzDT, C,) + (A - B , Dr, Cx)TX, + C r D ± D r C , - XzB=BTX~ such that A - B2DTC~ - B 2 B r X 2
is asymptotically stable,
and Y2 = Y2T > 0 is the solution to the LQG algebraic Riccati equation 0 = Y , ( A --
J~lDT21c2)
T "~
(A
-
B1D21C2)Y~ 4- B1DT_D±Brl T ,
--
such that A - B1Dr21C2 - ~ c T c 2
is asymptotically stable.
Y 2 c '2T c '2 Y 2
3.7. RECOVERY OF THE LQG SOLUTION
33
The final step is to note that lim.-+~o{Zoo('r)} = I: then taking the linfit in Remark 3.5.6 we obtain the LQG controller
KLQa = [ A - B 1 D T C , - B, DT2Ct - y~cTC, - B,B~X~ I B1D~ + y2CT ] - DT2Ct - BTX2 0 ' and taking the limit in Theorem 3.5.1 we obtain the minimum LQG cost
C(HLQa) = trace[BrX~B1] ÷ trace[(D~,C1 H- BTx,)Y2(CTDI, ÷ X,B,)]. We shaU need these state-space formulae later.
Chapter 4 T h e M i n i m u m E n t r o p y 7~c~ Distance Problem
36
4.1
C H A P T E R 4. THE MINIMUM E N T R O P Y ~oo DISTANCE P R O B L E M
Introduction
In the previous chapter we solved the Minimum Entropy ?go0 Control Problem directly in terms of the given standard plant. Here we will provide an alternative solution via an associated Minimum Entropy 7-/o0 Distance Problem. The solution to this problem is interesting in its own right, not least because it gives us an appreciation of the underlying structure of the control problem. Indeed, it is only within the last year or two that it has been possible to solve ~o0 control problems without solving an associated 7-/o~ General Distance Problem. See [12] for details of the role of the General Distance Problem in 7-(o0-optimal control. Our solution of the Minimum Entropy ~o0 Distance Problem uses the parametrization of all error systems satisfying the £¢o-norm bound. The results are reminiscent of the solution of the Minimum Entropy ~/~ Control Problem, although there are some extra subtleties in the proofs. We find that the central member of the admissible class minimizes the entropy at infinity and an explicit state-space formula is derived for the minimum value of the entropy at infinity (when it is finite). A link with abstract 'band extension' problems is given.
4.2
F r o m T/oo C o n t r o l P r o b l e m Problem
to 7%0 D i s t a n c e
The transformation from an 7~o0 control problem to its equivalent 7-/00 General Distance Problem is by now well-known. The details may be found in [12]. We follow the same steps on our Minimum Entropy 7fo0 Control Problem to reduce it to an equivalent Minimum Entropy 7-/00 Distance Problem. We take the initial problem setup to be exactly as in Section 3.2, Figure 3.1, and Problem 3.2.6. That is, we are given s0 E (0, co], "7 > % and a standard plant
v, I
( ~q'~l )
(r'rt2)
P21
P22
;
mi R p2, Pl ~_ m2.
The aim is to minimize the closed-loop entropy, I ( ~ ( P , K);7; So), over all stabilizing controllers K which satisfy
[l~'(e, K)l]o0 < "r.
(4.1)
Use the parametrization of all stabilizing controllers of [64, 37] to reduce (4.1) to the equivalent model-matching problem of finding (~ E 7~7/o0 such that
lIT1 +
T:d2T3II+ <
"r,
(4.2)
and then exploit the unitary invariance of the £o0-norm to reduce (4.2) to the 7fo0 General Distance Problem of finding (~ E T~7~¢~ such that
[ [ Rll /]~12 ] I R21 R22 + (~ oo< 7,
4.2. FROM 7i00 CONTROL PROBLEM TO 7"[00 DISTANCE P R O B L E M
37
where rnt -P2
~
R21 R22
is anticausal (i.e., R* E ~ 0 0 ) and is known in terms of the standard plant P (see e.g., [16, Theorem 3.2.6]). Define the error" system E by
E:=
Rn R12 ] R~I R 2 2 + Q "
(4.3)
Then I(~'(P, K); 3'; so) = I(E; 7; so) because entropy is unitarily invariant (from Proposition 2.3.1(vi)). Hence, the closed-loop transfer function ~-(P, K) and the error system E have the same entropy, as well as the same £oo-norm. This allows us to solve our original Minimum Entropy 7"/00 Control Problem by solving the following equivalent error system Minimum Entropy 7-/00 Distance Problem. P r o b l e m 4.2.1 ( T h e M i n i m u m E n t r o p y 7~00 D i s t a n c e P r o b l e m ) Let so E (0, oo] and let
R =
p, I
Rn R22
be given, where R* C T~Tt00, r e x > p 2 ,
P l --> m s .
For Q E 7~7-[00, define the error system E by E:=
Rn R12 ] Rzl R 2 a + Q '
and let 70 = i~f(llEHoo :
0 E n7~00).
Then for ~/ > %, find Q c T~7"f00 such that the entropy I(E;7;so ) is minimized over those E which satisfy IIEII00 < "r. Of course, by construction % above is exactly the % defined in the ~00-optimal control problem (Problem 3.2.5).
38
4.3
CHAPTER 4. THE MINIMUM E N T R O P Y 7-loo DISTANCE PROBLEM
Relations to the Band Extension Problem
The Minimum Entropy 7f~o Distance Problem is related to signal processing via the problem of finding the positive definite 'band extension' of a given operator. To see the connection, use a well-known fact [32, Theorem 7.7.6] and the definition of the £oo-norm to show that I[ Rll
RI'
if and only if
M :~--
I 0 7-1R21 0 I 7-1Rn 7-1R~l 7-1R~, I 7-1((~ + Rz,)" 7-1R12 0
7-1(R22 + (~) 7-1R12 0 I
> 0,
Vs = fla.
(4.4)
Thus we seek a positive definite extension M of the 'band' data I 0 7-1R21 7-1R2z 0 I 7-1R11 7-1R~2 7-1R~I 7-1R~1 / 0 7-1Rh 7-1R~ 0 I This can be interpreted as 'band' data because only the anticansal component of Rzz is specified. In [21, 22] the band eztension of N is defined as M > 0 in (4.4) such that M -1 has the same banded structure as N. It is shown in [21, 22] that this unique band extension also minimizes the entropy. Although the very general results of [21, 22] coald be applied to our problem, we choose to adapt the method of [4, 3], as done in the previous chapter. This makes for a relatively short and self-contained derivation.
4.4
S o l u t i o n in t h e G e n e r a l C a s e
In this section we solve the Minimum Entropy 7"(oo Distance Problem at an arbitrary s0 E (0, oo). Solution proceeds by firstly parametrizing all error systems E which satisfy the bound HE[Ioo< 3'. Such a parametrization is given in [.5], in terms of a linear fractional map of a J-unitary matrix and an arbitrary stable contraction ~, but it is more convenient to use the parametrization of [30] in terms of the linear fractional map of an all-pass matrix and an arbitrary stable contraction ~. To begin, we parametrize the class of error systems E over which the entropy must be minimized.
4.4. SOLUTION IN THE GENERAL CASE
39
L e m m a 4.4.1 ([ao]) All aolutione m(t --P2 )
< P~ )
with R* E "R.$'I,~ and Q E T~Tt~,
mx >_p~ and p~ > m2,
to the Distance Problem IIEII,~ < 7, where 7 > %, are given by: E = 7Y:'(R.. + Q.., ~), where
Pl -- raat I
ra2
[°°] 0
4)
(4.5)
¢ E B~.7-/~.
'
Also, ( t'l'tI )
Ro. + Qo. :=
.,I
(
JR,,,, + Q.ohl
Pl
)
[n°o + Qo°],~ 1
[Roo + Qo.]~, [R.o + Qo.],, m l --P2 7-1 Rll
Pl --trl'2 I
,7-~R21
ra2
"(
, l'
(
)
7-Z R12 Rls 0 7-~(R22 + Q22) R2s + Q23 Q~4
Raa 0
rat--P:~ I
]
R3, + Q32 Q4,
R33 + Q33 Q34 Q4a Q.t
Furthermore, R~j E 7Z7-~oo, Qij E T~7-[oo, R~. + Q,o is all-pass and Q44(oo) = 0. State-space realizations of R/j and Qij are avMIable in [30], in terms of the realization of R and the solutions to two algebraic Riccati equations. These realizations will be stated and used in Section 4.5 The next lemma relates the entropy of the linear fractional map of an all-pass matrix V mad an arbitrary stable contraction • to the entropy of # itself. L e m m a 4.4.2 Suppose V=
V21 V~2
E'R£¢.
is all-pass, ~ E I3TLT-[oo and V22~ E ~7-/oo. Then
z(7~(v, ~);7; so) = 72I(*; 1; so) + 72xCV11;1; so) + 7~,0 m I det(Z - V~2(So)*Cso))l.
40
CHAPTER 4. THE MINIMUM E N T R O P Y Tl¢o DISTANCE PROBLEM
P r o o f Since V22 is part of an all-pass matrix, IWnll~ ~ 1. Therefore IIv22~11o0 < 1 because []~Hoo < 1. This, together with the assumption that V2z~ ~ R~/.o, implies that det(I - ½2~) *~ ~ "R,~/~o. As V is all-pass, we then have that [51]
b3=(v,))]*[7>'(v, ¢)] < ~ 1 , so the entropy I(7.T'(V, ~); 7; So) is well-defined. Using v*v = z
(4.6)
in block-partitioned form, it is straightforward to show that I -7-2[Tit(V, ~)]*[7~'(V, ~)] = V2][I- V m ~ ] - * [ I - ~*~/][I- V22~]-~ V2a. From this, and the fact that for any square real-rational transfer function matrix G ln[det(G*G)[ = In[detG[ + l n [ d e t G*[ = 21hider G[,
V8 = jw
we obtain, Vs = jw !n ] d e t ( / - 7-217E(V, ~)]*[7~'(V, ~)])[ = 111[ det(I - ~*~)1 + In I det(V2*xV2~)l- 2ha [ det(I - V~2~)[. Substituting this into the definition of I(7~'(V, ~2);7; so) and using the (1,1) block of equation (4.6) to write V;xVzx = I - V~*~V~ it follows that I(7:F( V, ~); 7; s0) = 72I(g2; 1; 80) + 5'zI(Vll; 1; s0) + ~ o-]~oIn [ det(I - ½2(jw)~(fio))[
+ O)2]
(4.7)
But, from above, det(I - V22~) .1 E Z¢~oo, which permits the. use of Poisson's Integral Theorem [52, Theorem 17.16] to evaluate the integral in (4.7), giving
/ ( T y ( r , ~); 7; so) = -?/(~; 1; 8°) + 72/(v~1; 1; s0) + 72801n [ det(I - V.(80)~(80))J as daimed.
I"1
We are now in a position to derive the unique stable contractive • in the parametrization of all error systems, which minimizes the entropy I(E; 7; 8o). T h e o r e m 4.4.3 Consider the class of error systems E which satisfy the condition IIEIIoo < 7 as parametrized in Lernma ~.~.1 by E=TgZ(R,,~,+Qa,,,[O0 ~O ] )'
~ e t~zzT-t~.
(4.8)
Then the entropy I(E; 7; so) is minimized over this class of E by the unique choice
~=Q~4(so).
4.4. S O L U T I O N IN T H E G E N E R A L CASE
41
P r o o f Here we adapt the approach of [4, 3] to the present setting. Lamina 4.4.1 gives all error systems in the form (4.8), where Pt.o + Q~a is all-pass. Also,
because 034, 044 and ~li are all in g ~ o o . Hence, we may apply Lemma 4.4.2 to E to obtain I ( E i T ; so) = 721(0; 1; *o) + 72I([R.,, + Q.~]n; 1; so) + 72So In [ det(I - Q44(so)O(ao))l.
(4.9)
Define a constant Julia matrix [65, p148] by u=
[ U n U12] u21 v n :=
[
-O~4(So)
(I-O~4(so)O44(80))1/2 ]
(1-Q44(,o)Q14(so)) ~/2
Q44(,0)
"
It is easy to verify that U is unitary. Also, u2 0(,) = Q , , ( , 0 ) ¢ ( , ) which is in 7Z~oo. Now map the unit ball in 7Z7-/oo onto itself by the linear fractional map * := y:(u, ¢), ¢ e s n u ® . Note that, under this mapping, il~ = Ql4(so)
,: :.
i~ = 0.
(4.10)
Lamina 4.4.2 is applicable: I ( f ; 1; *o) = I ( 0 ; 1; *o) + I(Q~4(so); 1; *o) + so ha I det(I - Q44(s0)O(so))l. Use this together with (4.9) to relate the entropy of E to the entropy of ~: I(E;'r;so) = 7 2 I ( ¢ ; 1 ; s o ) + 7 2 I ( [ R . . + Q ~ . l x t ; 1 ; s o )
• -- 7 2l(Q44(So); 1; So),
(4.11)
from which it is immediate that I(E; 7; 80) is minimized by the unique choice ~ = 0. But from (4.10), ~ = 0 -', ~- • = Q~4(s0), and the theorem is proved. [] An expression for the minimum value of the entropy now follows. Corollary 4.4.4 The minimum value of I ( E ; 7; So), over the class of all error systems E satisfying [IElloo < 7, is given by I(EMEoo; 7; so) = 72I([R~ + Q.~]u; 1; so) - 72I(Q44(so); 1; so) = 72So { - In I det R;l(S0)l - In I det Q42(so)l + (1/2) ln l d e t ( I - Q'44(so)Q44(so))l}.
(4.12) (4.13)
42
C H A P T E R 4. THE MINIMUM ENTROPY']to~ DISTANCE PROBLEM
Proof Equation (4.12) follows immediately from equation (4.11) on setting ~ = 0. To show (4.13), recall that Ra, + Q~a is all-pass i.e.,
+ Qoo)'(Ro. + Q.o) = x. The (1,1) block of this gives,
I - [R,,,, + Q,,,,]~ItR,,,~ + Q,,,,]u = [R,., + Q,,,,];I[R,~,, + Q,,,,]2~ so that, along the imaginary axis, In [ det(I - [/L. + Qo.]~x[/L. + Q..]x,)] = 21n ] det([R.~ + Q..]:x)I = 2]n [det R;x[ + 21n ]det Q42],
(4.14) (4.15)
where (4.15) follows from (4.14) on examination of the structure of Ra. + Q,, in Lemma 4.4.1. Substituting (4.15) into the definition of entropy, we see that 72 ~r f_'~¢o {ln [det R;x(jw)]
72I([R,,, + Q ~ ] u ; 1; s0) -
+ In [det Q42(jw)[} [,~ + w2 j Since R~* E ~9/oo and Q ~ E 7~9/oo [30], Poisson's Integral Theorem may be used to evaluate (4.16) as 72I([R.. + Q..]u; 1; so) = -TZso (In [ det R;x(So)l + In [det Q42(so)l) •
(4.17)
The second term in (4.12) is
-7'I(Q~4(so); 1; s0) =
In [ det(I - Q44(ao)O;4(so))l f-oo 72
= ~
In
Idet(I
and this with (4.17) gives (4.13).
4.5
Solution
-
Q44(s0)Q;4(s0))l(Trso) []
at Infinity
We turn our attention in this section to the special case of entropy at infinity, i.e., when *o --* oo. The results for the minimum entropy problem at infinity are particularly simple, and are similar to those derived in Chapter 3 for the Minimum Entropy ~o~ Control Problem. The minimum entropy solution is just the central solution, and an explicit formula for the minimum value of the entropy is derived in terms of the state-space realizations inherent in the solution of the Distance Problem of Lemma 4.4.1; these state-space realizations are stated in the next lemma. The state-space formulae assume that R(oo) = 0. This is a sufficient condition for I(EmEoo; 7; oo) to be finite. R(oo) = 0 is implied by the assumption that Plx(oo) = O, made in Section 3.4 to ensure that I(HMF.oo; 7; oo) is finite. See Remark 3.4.1.
4.5. SOL UTION A T I N F I N I T Y
43
L e m m a 4.5.1 ([30]) Suppose R has a realization
,,2rt . ,.I ~,,-~I •-, I
R=
'7,-r?
r A B, [C1 0 C2 0
P2 )
0 0
where R is anticausal i.e., - A is asymptotically stable. Then Ro, + Qa~, as in the parametrization of all solutions to the Distance Problem IIEII~. < ,,/ given in £emma ~.~.1, has a realization
R**+Q°.= A[C~-~DB"] where :-----
:=
A0] 0 A 7-1/2B~ 0
7-1/2B2 -7-s/2xc~ 7 - s / 2 y Z-a B2 _7-t/2C~
7-112C1 7-1/2C~
0:~___
0
0 0 I 0
0 0 0 I
I 0 0 0
o
7-*/2 Z - r Cr2
]
0 -7-s12C2X 7-1/2BTZr _7-1/2 BT2
0 I 0 0
and where X = X r solves
o = X A * + A X + ~-'XC~,C,X + B,B~ + B ~
(4.18)
such that - ( A + 7-2XC~Cx)
is asymptotically stable,
where Y = y r solves 0 = YA + ATY + 7-2YB, BTY +
CTtC, + Cz,C,
such that - ( A + 7-2B1B~Y)
is asymptotically stable,
and where Z := 7 - 2 X Y - I, /i := - A ~ - 7 - 2 Y Z-1B2BT2 - 7-~C~C1X.
(4.19)
44
CHAPTER 4. THE MINIMUM E N T R O P Y T-I¢~ DISTANCE PROBLEM
It is convenient to introduce the notation E{O} : = T J r ( / ~ . + O..,
[0 0] 0 • )
to denote an error system produced by Lemma 4.4.1 by a @ 6 BRT/oo. Also define ~ME~
lira { (I)ME~o} j0...~oo
:----
= .lilnoo{Q:,(s0) } = 0, where the second equality is from Theorem 4.4.3 and the third equality is from Lemma 4.4.1. Then E{{tME~} = E{0} = 7[R.. + V.,],,. (4.20) We can now apply the results of the previous section, using the state-space realization given in Lemma 4.5.1, to obtain the following important theorem. T h e o r e m 4.5.2
(i) The choice : ~MEoo : 0
in the parametrization of Lemma "~.4.1 of all error systems E satisfying IIE IIoo < 7 is the unique choice of ¢~ which satisfies:
V,~ •
BTCT-&o(~ ¢ o) :
3s(~) < oo such that Vso > s(!l?) :
I(E{@MEoo};7;so) < I ( E { ~ } ; 7 ; s o ).
Thus the minimum entropy error system is EME¢o = E{@MS~o} = E{O}.
(ii) If R(oo) = 0 then ~ = ffMEo~ = 0 is the unique choice of ~ which minimizes I(E{~};7; co) over ~ 6 B7~7"l~. (iil) If R(oo) = O, then the minimum entropy error system has a state-space realization EMEc¢ = 7[Raa +
Q..]n
A
0
o
A
BI 0
C~ 0 C2 - 7 - 1 C 2 X A
=
C1 C2
B1 0 0
h
B2 0 0
B2 7-1YZ-1B2 0 0 0 0 0
+ • ,
7-1YZ-1B2 0 0 0 0
0 -7 -IC2X Y
0
[0
°1
]
(4.21)
4.5. SOLUTION A T I N F I N I T Y
45
and the minimum value of the entropy is I(EMEoo;'r; co) = - trace[B~YB1] + trace[B~YZ-tB,] = - traceiO,XeT,] + trace[O~Z-'XC[].
(4.22) (4.23)
Proof Parts (i) and (ii) From Theorem 4.4.3, the nfinimum entropy error system, when entropy is evaluated at So 6 (0, co), is characterized by the unique choice # = Q~4(so). Letting s0 --~ oo gives the minimum entropy at infinity choice as = Q~4(oo) = 0, because Q44 is strictly proper from Lemma 4.4.1. When R(co) # 0, the minimum value of the entropy is not necessarily finite; the limiting argument then required is identical to that used in the proof of Theorem 3.4.2, and so it will not be repeated here. Part (iii) That the minimum entropy error system has a realization (4.21) follows easily by using (4.20) in Lemma 4.5.1. To obtain the minimum value of the entropy, we take the limit as so --~ co of the result of Corollary 4.4.4. That is,
I(EMEoo; 7; co) =
lira {7~s0{ - In I det R;l(so)] - In ]det Q42(s0)]
$0 -'¢OO
+ (1/2)In I det(I - Q*44(so)Q44(so))]}}.
(4.24)
From the proof of Lemma A.4.1 of Appendix A, we have, for a typical term from (4.24): ,o..,oo(--SolimIn I det(I + 0(SoI - d)-l/~)l } = - trace[0/~]. Apply this to the terms in (4.24) using R;l(so ) = Z + 7 - ~ e ~ ( s o I + A T ) - I Y B , Q4~(~o) = X - ~ - ~ B T ( s d -- A ) - I Y Z - ~ B 2 from Lemma 4.4.1, to get
I( EMB¢o; 7; co) = -- traceIBTy B*] + trace[B~Y Z-* B~] as claimed. Note that the third term in (4.24) is zero in the limit because Q44 is strictly proper. The dual expression (4.23) follows in an entirely similar fashion; one recalls from Proposition 2.3.1(vii) that ETMEoo has the same entropy as EMEoo, leading to
I(EMEoo; 7; co) = ,01imoo{72So{- In I det R~(so)l - In I det Q~4(so)] + (1/2)In J det(I - Q44(s0)Q;4(s0))J}}, which gives (4.23) in the limit.
[]
46
CHAPTER 4. THE MINIMUM ENTROPY 7"l~ DISTANCE PROBLEM
R e m a r k 4.5.3 Notice that the entropy formulae (4.22) and (4.23) depend only on the state-space realization of R and the solutions X and Y to the two algebraic Pdccati equations (4.18) and (4.19) which are inherent in the solution to the Distance Problem. Calculation of the minimum value of the entropy therefore imposes negligible extra computational problems. Furthermore, the minimum entropy error system (4.21), being the linear fractional map of • = 0, is simply 7 times the pl by ml (1,1) block of Ra~+Qa~ (from (4.20)), which is also available from the solution to the Distance Problem with no extra computation. R e m a r k 4.5.4 ( R e c o v e r y of t h e £2-optimal solution) Recall, from Theorem 2.4.4 and Remark 2.4.3, that I(E; co;oo) = IIEII~ for strictly proper E. Thus if w¢ let 7 --~ oo in our minimum entropy solution we should obtain exactly the/:z-optimal solution i.e., the error system with minimum £:~-norm. We show here that this is indeed the case. By using [62], it may be shown that the positive semidefinite matrices - X and - Y are monotonically decreasing as 7 increases. Taking 7 --~ co we obtain 0 = X ( o o ) A T + A X ( o o ) + B~B T + B2B T
0 = Y(oo)A + ATy(oo) + C~C~ + C[C2
(4.25) (4.26)
and
z(oo) = -I, (with an obvious notation). Equations (4.25) and (4.26) identify the matrices -X(oo) and - Y ( o o ) as the controllability and observability Gramians of R(-s), respectively. (Remember that R* 6 ~7-/oo so - A is asymptotically stable.) Using this fact, a simple calculation shows that (assuming R(c¢) = 0 to ensure finiteness)
I( EME ; oo; oo) = -- trace[BTr(oo)Bl] + trace[B[Y(oo)Z(oo)-lB2] = trace[tB1 B2IT[-y(oo)][B1 B21]
= IIRII]
(4.27)
Also, inspection of (4.21) leads to limx--,oo{QMsoo} = 0, hence liIn-c-.oo{EMEco} = R. It is well-known that the Q 6 7~7/¢0 which minimizes
IIEH2 =
[ Rn Rn
R~2 ] 2 Rn + d2
is Q = Q~2 = 0 (the £2-optimal solution) and in that case the/:2-optilnal error system is E~ 2 = R. We therefore have that linl~o~{(~ME¢o} = ( ~ ; that. lim~c~{EMEoo} = E~2 = R; and from (4.27) that I(EMEoo;oo;c~) = I[Ec~][22. This establishes the equivalence between the Minimum Entropy 7-/00 Distance Problem when 7, So --* o0 and the £2-optimal Distance Problem. This corresponds exactly to the recovery of the LQG problem from the Minimum Entropy 7-/00 Control Problem as in Section 3.7.
4.5. S O L U T I O N A T I N F I N I T Y
47
R e m a r k 4.5.5 Note that if 7 --' 70, then EMEoo ---+ Eo (£oo-optimal). Also (Remark 4.5.4), if 7 -" oo, then EME¢o ~ E£2 (/:2-optimal). Thus ~/can be used to move from £:oo-optimal to £:2-optlmal via the minimum entropy solutions for % < -y < oo. This, of course, corresponds exactly to the 7"/oo/LQG tradeoff of Section 3.6 for the original Minimum Entropy 7¢oo Control Problem.
Chapter 5 Relations to C o m b i n e d 7/c¢/LQG Control
50
5.1
CHAPTER 5. RELATIONS TO COMBINED 7-loo/LQG CONTROL
Introduction
In this chapter we provide further results on the interplay between minimum entropy 7/o~ control and LQG control. This will be done by proving the equivalence between the Minimum Entropy 7-/~ Control Problem of Chapter 3 and the Combined 7"[00/LQG Problem of [8, 9]. The Combined 7/~o/LQG Problem is approached in [8, 9] by constructing upper bounds on the LQG cost using the solution of algebraic Riccati equations. We will show that the procedure of [8, 9], in the case of full-order controllers and with 7"/~ and LQG criteria applied to the same closed-loop transfer function, is exactly the same as using minimum entropy 7"/oo control. This gives an exact interpretation of the bounds used in [8, 9]. By exploiting the simplicity of the solution to the Minimum Entropy 7"/00 Control Problem, we will also be able to obtain a useful simplification of the state-space solution to the Combined "H~/LQG Problem given in [8].
5.2
The Combined
T/oo/LQG Problem
We consider the same setup as for the Minimum Entropy 7-/~ Control Problem in Section 3.2. We assume that D n : 0. This assumption, as we saw in Section 3.4, ensures that the minimmn value of the entropy and the minimum value of the LQG cost arc both finite. It also ensures that the auziliary cost (defined later) has a finite minimum value. Thus we consider an n-state standard plant P with a state-space realization P=
P~I P~2
C1
0
C2 D n
i)12
,
0
connected with an n-state feedback controller K := as in Figure 3.1. It is easy to verify that the closed-loop transfer function H = ~'(P, K) has a state-space realization H = ~'(P, K) := where
and C': [ c.',
]
In order to motivate the Combined 7-/~/LQG Problem of [8, 9], recall the expression for the LQG cost associated with a stabilized closed-loop H, as given in Remark 2.4.3:
v ( H ) = IIHl]].
5.2. THE COMBINED 7-I~[LQG P R O B L E M
51
It is well-known that
C(H) = trace[00r0],
(5.1)
where Q = O r > 0 is the solution to the Lyapunov equation
0 = AQ + QA~ + BB ~.
(5.2)
In fact, Q is just the closed-loop controllability Gramian. Now suppose we require the stabilized closed-loop H to satisfy IIH[I~ < 7 (that is, suppose H is a (P, ~')-admissible dosed-loop). From [60], the Frequency Domain Inequality tells us that I]HIIoo < 7 if there exists a stabilizing solution Q = QT > 0 to the algebraic Riccati equation
0 = AQ + Qfi r + 7-2QOTOQ + [~[~T.
(5.3)
But it is easy to prove [8, 9, 60, 62] that any Q solving (5.3) overbounds the closed-loop controllability Gramian Q i.e., Q > Q. In view of the evaluation of the LQG cost in (5.1), this motivates the definition of the auxiliary cost.
Definition 5.2.1 (Auxiliary cost [8, O]) The auxiliary cost associated with a ( P,7)-admissible closed-loop H is defined by J ( H ; 7 ) := trace[QoCTC],
(5.4)
where Q° = Qr >_ 0 is the stabilizing solution of the algebraic Riccati equation (5.3). It is immediate that J(H; 7) > C(H) because of the previously noted bound Q > (~. Minimizing J ( H ; 7 ) subject to H being (P,7)-admissible therefore gives us a control problem which combines both 7"/00 and LQG objectives. The similarities with the Minimum Entropy 7/0o Control Problem will be shown in the next section to be much more than superficial. P r o b l e m 5.2.2 ( T h e C o m b i n e d 7/oo/LQG C o n t r o l P r o b l e m [8, 9]) Let P be a standard plant and let 7 > %. Minimize the auziliary cost J ( H ; 7 ) over all ( P, 7)'admissible closed-loops H. In the terminology of [8, 9], Problem 5.2.2 is a full-order (controller) problem with equalized 7-/~ and LQG weights. Problems with reduced-order controller and/or nonequalized weights can also be treated within the framework of [8, 9]. However, the number and complexity of coupled Riccati equations needed for the solution then increases substantially--in that case the equivalence result of this chapter does not apply. A s s u m p t i o n s 5.2.3 A number of mild assumptions will be made. We, of course, inherit Assumption 3.2.1 and Assumptions 3.2.7 from the Minimum Entropy 7-/~ Control Problem. The assumption that Dll = 0 has already been mentioned as a simple way to guarantee that the minimum value of the entropy, minimum value of the auxiliary cost and minimum value of the LQG cost are all finite. Finally, in Section 5.4 we will assmne
52
C H A P T E R 5. RELATIONS TO COMBINED Tloo/LQG CONTROL
that DT2C1 = 0 and B1DT1 = 0, for convenience. This eliminates cross-weighting terms which would otherwise cloud the argument, and is also assumed in [8, 9]. DT2C1 = 0 corresponds to orthogonality of Clz, (where z is the state vector of P) and D12u, whereas B1Drl = 0 corresponds to orthogonality of plant disturbance Blw and sensor noise Dzlw. It is the purpose of this chapter to establish that under these conditions the Combined 7"/¢o/LQG Problem and the Minimum Entropy 7-/0o Control Problem are in fact identical. We will prove this in Section 5.3 by establishing the key result equating entropy and auxiliary cost: I ( H ; 7 ; o0) = J ( H ; 7 ) . This link is interesting because it draws together two apparently unconnected frameworks in a fairly deep way. In Section 5.4 state-space solutions to the two problems are given, using the results of Chapter 3 and [8], and it is shown explicitly that these solutions are identical. The minimum value of the entropy, by the results of Chapter 3, can be written in terms of the two Riccati equations needed to find the minimum entropy ~ o controller. But the minimum value of the auxiliary cost, given by [8], requires a third Riccati equation which is coupled to the other two. Our equivalence result allows us to dispense with this third Riccati equation entirely.
5.3
Equivalence with M i n i m u m Entropy 7-/o0 Control
We begin with our main equivalence result. T h e o r e m 5.3.1 With definitions and assumptions as in Section 5.2, for any (P,'y)-
admissible closed-loop H, the entropy equals the auziliary cost. That is, I(H;~/; o0) = J(H;~/).
Hence, the Minimum Entropy ~oo Control Problem (Problem 3.2.6) and the Combined 7"[~/LQG Problem (Problem 5.2.2) are equivalent. We defer the proof of Theorem 5.3.1 until after the next lemma, which gives an evaluation of the entropy of H in terms of its state-space matrices and the solution to an associated Riccati equation. L e m m a 5.3.2 Let H E T~7-[~ with IIH]loo < 7 and
5.3. EQUIVALENCE WITH MINIMUM ENTROPY'Hoo CONTROL
53
Then its entropy is given by I(H; 7; ~ ) = trace[Q,OrO]
(5.5)
where Q, = Qr >__0 is the stabilizing solution to the algebraic Riccati equation 0 = AQ + o f l r + 7 - 2 o O r O o + B B r. P r o o f The assumption that H E 7~7"/~, with
IIHIIo.
(5.6)
< 7, implies that
I -- 7-2H(jw)H*(jw) > O, Vw E ]R U {oo). This guarantees the existence of a spectral factor N such that I - 7 - 2 H H ° = NN*,
where N at E 7~T/~.
It is easily verified that a state-space realization of N is (see e.g., [16, 60])
where Q, = QT _> 0 is the stabilizing solution to the algebraic Riccati equation 0 = AQ + Qf4r + 7-2QOrCQ + BB r.
(5.7)
Using the fact that h I det(I - "r-~H(jw)H*(jw))l = h I det(N(j~,)N*Uo~))l = ha I det(N*(jw)N(jw)) I we find, from the definition of entropy in Definition 2.2.2 and from Proposition 2.3.1(vii), that Z(H;-r;
=
{
oo)
= so-colim --~w
ln ldet(N*(j~)N(fl°))l
d~
.
(5.8)
We may immediately apply Lemma A.4.1 of Appendix A to get I(H;7; oo) = trace[Q,6'rC],
(5.9)
as claimed.
[]
P r o o f of T h e o r e m 5.3.1 By definition (see (5.4) and (5.3)) the auxiliary cost is J ( H ; 7 ) = trace[QscYc],
(5.10)
where Qs = QT > 0 is the stabilizing solution to the algebraic Riccati equation 0 = AQ 4- QA T 4- 7-2QC'rd'Q + / 3 B r.
(5.11)
A comparison with the evaluation of I ( H ; 7 ; oo) in Lemma 5.3.2 (compare (5.10) and (5.11) with (5.5) and (5.6)) establishes the result. []
54
CHAPTER 5. RELATIONS TO COMBINED 7/00//LQG CONTROL
R e m a r k 5.3.3 In the definition of auxiliary cost given in [8], it is not mentioned that the stabilizing solution Q0 of the Riccati equation (5.3) should be used. However, it may be shown using the results of [60, 62] (taking care over the sign convention), that Q > Qo, where Q is any solution to (5.3). Therefore, no other solution to (5.3) gives a smaller auxiliary cost than Q, does, justifying our insistence on taking the stabilizing solution of (5.3) to evaluate J(H; 7). Section 3.7 states that recovery of the LQG problem associated with P is achieved by relaxing the 7/~o-constraint completely (by allowing 7 --* oo in the Minimum Entropy 7/00 Control Problem or equivalently in the Combined 7/oo/LQG Problem). Then the entropy and the auxiliary cost both become the LQG cost: I(H; oo; oo) = J(H; co) =
C(H).
5.4
Solution and Equivalence in S t a t e - S p a c e
In the previous section we proved the equivalence of the Minimum Entropy 7/00 Control Problem and the Combined 7/00/LQG Problem. In this section we expand on this by explicitly showing the equivalence of the state-space solutions of the two problems. The equivalence result also allows us to use the solution to the Minimum Entropy 7-/00 Control Problem to simplify the solution to the Combined 7/00/LQG Problem. We have, from Chapter 3, that the minimum entropy 7/00 controller and the minimum value of the entropy may be written in terms of the stabilizing solutions, denoted X00 and Y00, to two algebraic Pdccati equations. The solution of the Combined 7/00/LQG Problem in [8] expresses the controller in terms of X ~ and Y00 in the same way, but the minimum value of the auxiliary cost requires the solution, (~, to a third algebraic Riccati equation coupled to the other two. Our equivalence result allows us to reduce the expression for the minimum value of the auxiliary cost to its equivalent, and simpler, expression as the minimum value of the entropy. This makes Q unnecessary. Before showing this, we can give a state-space realization of the controller. P r o p o s i t i o n 5.4.1 The unique controller which solves the Minimum Entropy 7/00 Control Problem subject to the assumptions in Section 5.2 (equivalently, the con. troller which solves the Combined 7/00/LQG Problem as stated in Section 5.2) has a state-space realization
K . 00 = [ a +
B[X00Z00-C.) - ... X00Z00 ] YOC ].
where X00 = X ~ > O, Y00 = y T >__0 are the stabilizing solutions to X00A + A T x ~ + CT~Cx + Xoo(-r-2BxBr~ - B2BT)X00 = 0
(5.12)
Yc,,A r + AY00 + BaB~ + Y00(7-2C~C1 - crc2)Y00 = O,
(5.13)
and
5.4. SOLUTION AND EQUIVALENCE IN STATE-SPACE
55
respectively, and where
Zoo := ( I - 7-~YooXo.) -1. In saying the stabilizing solutions, we mean the solutions X~o and Yoo such that A + (7-2B1B r - B2Br2)Xoo
is asymptotically stable,
(5.14)
and A + ]~(7-2cTc1 - crc2)
is asymptotically stable.
(5.15)
P r o o f The proposition follows in a straightforward way from the formulae given in Chapter 3 (Remark 3.5.6 in particular), on making use of the assumptions of Section 5.2. Alternatively, the method in [8] involves assuming the controller structure (with n states) and treating the problem as one of constrained optimization, yielding coupled equations which must be satisfied by the controller state-space matrices. Some straightforward manipulations lead to the above controller. [] The resulting closed-loop is given as usual by HMEoo ----.T'(P, KMEoo). State-space formulae for the minimum value of the entropy and the minimum value of the auxiliary cost can now be stated. These formulae are taken directly from Theorem 3.5.1 and [8], with certain simplifications implied by the assumptions of Section 5.2. Proposition 5.4.2 The minimum value of the entropy is given by
I(HME~; 7; 00) = trace[X~B~B T 4- X ~ Z ~ Y ~ X ~ B 2 B r 2 ] .
(5.16)
Proposition 5.4.3 ([8]) The minimum value of the auxiliary cost is given by J(HM~o;7) = trace[Yc~CTC1 + O,(CrC~ + X ~ Z ~ B 2 B T Z T X.o)],
(5.17)
where (~° = O,r > 0 is the stabilizing solution to the algebraic Riccati equation 0 = AO + ~)~r + 7 - 2 0 / ~ ) + yoocTc2y.o,
(5.18)
and A := A - B2BrXooZ¢o + 7 - 2 y ® c r c 1 ,
/~ := cTc1 + X ~ Z ~ B 2 B 2T Z ~T X ~ .
(5.19)
In saying the stabilizing solution, we mean the solution Qo such that
A + 72Q.R is asymptoticaUy stable. R e m a r k 5.4.4 By Theorem 5.3.1, I(HMEoo;7;Oo) = J(HME~;7), which implies equality of (5.16) and (5.17). Therefore, we never need to solve equation (5.18) for (~o to find J(HMF~oo;7), as we can always use (5.16) instead, which only needs Xoo and Y¢~. In this way, Theorem 5.3.1 and Proposition 5.4.2 allow us to discard Proposition 5.4.3 as redundant.
CHAPTER 5. RELATIONS TO COMBINED 7-loo/LQG CONTROL
56
It is the purpose of the remainder of this section to clarify the nature of the redundancy of Proposition 5.4.3. We do this by firstly deriving some properties of the solutions to (5.18). We then use these properties to prove that (5.17) equals (5.16), and along the way see why it is the stabilizing solution to (5.18) which leads to the minimum value of the auxiliary cost. In the interests of brevity and clarity, we make two mild simplifying assumptions as follows: xoo > 0 (5.20) all d
(A T,7-1R ~/2) is controllable.
(5.21)
Assumption (5.20) is also made in [8]; sufficient conditions for Xoo > 0 may be found in [17]. Also, by [31], assumption (5.21) guarantees the existence of both a stabilizing solution Q, and an antistabilizing solution ( ~ to (5.18). Note that since the equivalence results of Section 5.3 and the final entropy formula (5.16) do not require either of these assumptions, it follows that these assumptions are not necessary--they merely streamline the presentation. L e m m a 5.4.5 Assuming (5.20) and (5.21), the antistabilizing solution Q, = Q~ > 0 to the algebraic Riccati equation (5.18) is
Q,, = 7 : Z £ 1 X £ 1. P r o o f It is a simple matter to substitute Q~ = 72ZjoIXjo 1 into equation (5.18) and use (5.12) and (5.13) to show that it is indeed a solution. That ¢~,~ > 0 follows from the assumption that Xoo > 0 and the definition of Zoo. It only remains to check that (~, = 72Zj~Xjo 1 is antistabilizing. In other words, we must show that A, := A + 7-"~?,R is antistable. Substituting for A, R and Q~,,
A,,
=
(A - B2B~XooZoo + 7-2YooC~C~) + Z ~ X ~ ( C ~ C t = A + x2C C = x : J ( X o o A + c c,) - X 2 ( A T + Xoo('r-"B1Bf - B.BT))xoo,
+
X
T OoZooB2B2r Z~oX~) (5.22)
=
where (5.12) has been used to obtain the final equality. Therefore A~ and - ( A + (7-2B1BT -- B2 B~)Xoo ) are similar matrices. But A + (7 -~ B1B[ - B2 B T )Xoo is asymptotically stable by (5.14). Hence A~ is antistable, as required. [] The solution 72Z~.lX~ 1 has also been obtained (independently) by Bernstein and Haddad. In order to deduce what we need to know about the stabilizing solution, based
5.4. SOLUTION AND EQUIVALENCE IN STATE-SPACE
57
on our knowledge of this =tistabilizing solution, we will need the following lemma. It is convenient to define the 'gap'
A := Q,, _ 0,.
(5.23)
L e m m a S.4.6 Assuming (5.CO) and (5.21), we have that (i) A is the stabilizing solution of the algebraic l:ticcati equation a,a
(ii)
+ A a r - ,r-2zx_~A = o.
(5.24)
A > 0.
P r o o f Part (i) That A satisfies (5.24) is easily verified using the definitions of .zI,,, A and/~, and equation (5.18). To see that A is stabilizing, we note that
= X + -y-~¢),/~, which is asymptotically stable by construction. Part (ii) It is a consequence of assumption (5.21) that (A~,7-1/21/z) is controllable. This, together with the stability of - A T and a well-known theorem on Lyapunov equations [26, Theorem 3.3], implies that
(5.25)
¢(-~L) + (-~Lr)¢ + -r-~ = o
has a unique solution # > 0. Comparing q~-~(5.25)~ -~ with (5.24), it follows that if/-1 = A, so A > 0 as claimed. [] R e m a r k 5.4.7 If Q is any solution to (5.18), then
0,, _>_¢ >_ O, __ o
(5.26)
follows easily from [60, 62]. This ordering of solutions shows dearly that only the stabilizing solution, (~, can lead to the minimum value of the auxiliary cost. A glance at (5.17), (5.19) and (5.23), together with (5.26) tells us that using the antistabilizing solution (~,,, rather than the stabilizing solution Q,, leads to a cost which is exactly traee[A/~] bigger than the minimum. T h e o r e m 5.4.8 The expression for the minimum value of the auxiliary cost in equals the expression for the minimum value of the entropy in (5.16). That is,
(5.17}
J(ttME~; 7) = trace[~C1TC1 + w~t :1 "1 = trace[XooB1Brl + X ~ Z ~ t ~~X ~-B 2 B 2 r ] = I(ttME~;7; ~ ) , and hence we may dispense with Proposition 5.4.3 in favour of Proposition 5.4.2
CHAPTER 5. RELATIONS TO COMBINED 7-I¢~/LQG CONTROL
58
P r o o f The conclusion of the theorem is just that of Theorem 5.3.1. What is of interest here is the direct proof of equality, by working only with the formulae and their associated Riccati equations. The LHS is just LHS
=
trace[r~ClrC~+ ~),R]
=
trace[Y~CfC1 + (~,,R] - trace[A/~]
=:
(~)
-
(~).
Using Lemma 5.4.5 to put (~, = 72Zj~*X~ 1, we get
( '0 = trace[ Y . Crl Cl + .~,X :J C rl c, _ r . o r c, + .y2B, B2TZ~ X,,, =
T . traceb'x:Jc[c, +.~2 B,B,T Z~X.].
From Lemma 5.4.6, (b) = ~ t ~ a c e [ i . +
Ai~rA -1]
= 272 trace[.4u] = 27' trace[A + X£1CTC,] = 72 trace[X~l(ATX~ + X ~ A + cT1c1) "~ X~IC1TC1] = - t r a c e t B i B ~ X ~ - ,~2B, B~,X~ - 72X:J C~ Cl], where we have used (5.22) to obtain the tkird equality, and Xjo~(5.12) to obtain the final equality. Hence,
LHS = (~) ---
-
(b)
trace[v2X~XC~C1 -t 7 2B ~ BT~,T~ 2 ~ a ~ j , + trace[B,B~X+ - 7'B2B~X+
- 7'X~'C~CI]
= trace[X~BiB~ + ~'(Z~ - I)X~B2B[]
= RHS. []
Chapter 6 R e l a t i o n s to Risk-Sensitive L Q G Control
60
6.1
CHAPTER 6. RELATIONS TO RISK-SENSITIVE LQG CONTROL
Introduction
In previous chapters we have seen the importance of the entropy at infinity and have identified it as an auxiliary LQG cost. The aim of this chapter is to further our understanding of the entropy. We shall do this by relating the Minimum Entropy 7-/oo Control Problem to an apparently unrelated problem, the Risk-Sensitive LQG Control Problem. Risk-sensitive LQG control is a generalization of usual LQG control to include an ezponential-of-quadratic cost (the LEQG cost). The equivalence of the minimum entropy 7/oo controller and the steady-state version of the risk-sensitive optimal controller was first established in [28], for discrete-time systems. It is interesting that two conceptually quite different control methodologies give the same results and this can be considered as a bonus to both approaches; each performing well with respect to a criterion that had not been explicitly considered. We hence have a stochastic interpretation of the minimum entropy 7~oo controller and an interpretation of the LEQG controller with respect to robust stability in the presence of plant uncertainty and signal gain to worst case disturbances. The purpose of the present chapter is to give the continuous-time version of the equivalence. Our analysis follows [27]. Note that, in keeping with the remainder of this monograph, the finite-dimensional case is considered. In [27], the result is extended to cover a class of infinite-dimensional systems, too.
6.2
The Risk-Sensitive LQG Control Problem
The study of the LEQG problem was begun in [34] where the solution for perfect state observation was derived. Then in [57], the imperfect output observation case was solved in discrete-time, by developing a risk-sensitive certainty equivalence principle. The demonstration in [28] of the equivalence between the LEQG and entropy criteria prompted [58], where it is shown that the Hamiltonian methods developed in [59] can be applied to give a direct derivation of the entropy minimizing property of the LEQG controller. Although this latter paper only considers the discrete-time problem with perfect state observations, the techniques could be adapted to other situations. The continuous-time case with imperfect output observation, finite time horizon and statespace, and possibly time-varying system model, was solved in [7]. The set-up will be our standard one, as shown in Figure 3.1; H = ~-(P, K) is taken to be a stabilized closed-loop. The input w(t) is taken to be zero mean Gaussian white noise with spectrum equal to the identity. Then the standard finite-time LQG control problem is to minimize the finite-time LQG cost (see Remark 2.4.2) Cr(H) =
over all stabilizing controllers, where 1
r
~VT = "~ J-/T zT (t )z( t )dt"
(6.1)
6.3. EQUIVALENCE WITH MINIMUM E N T R O P Y 7/o0 CONTROL
61
The Risk-Sensitive L QG Control Problem is to minimize the modulus of the exponentialof-quadratic cost 12T(O) := - ( OT) -1 In E{exp(-OTWr) } (6.2) over all stabilizing controllers. The scalar 0 is the risk.sensitivity parameter. The standard LQG control problem is recovered when 0 ---+0 (risk-neutral); 0 > 0 is risk-seeking and O < 0 is risk-averse. To help interpret this, following [57, p765-6], if OTVar{WT} is small then a power series expansion gives ~y(O) ~ E{WT} - (OT/2)Var{Wr}. If 0 > 0 then, for a given value of E{WT}, any statistical variation in Wr improves the value of the exponential-of-quadratic cost. This is an optimistic situation where incidence of favourable values of Wr (i.e., less than E{WT}) is taken to be more significant than the incidence of unfavourable values of Wr (i.e., greater than E{WT}). If 0 < 0 then the situation is reversed and the controller becomes pessimistic, in that incidence of unfavourable values of Wr is taken to be more significant than incidence of favourable values of WT. The risk-neutral case (0 --+ 0) lies at the border of optimism and pessimism: occurrences of favourable or unfavourable excursions of WT are assumed to be equally significant.
6.3
Equivalence trol
with Minimum
Entropy
7-/00 C o n -
We shall be concerned with the risk-averse case so we set 0 = - 7 -2 < 0. The LEQG cost becomes J~(7) := ~r(0) = (TZ/T)In E{exp(7-2TWr)}, (6.3) where W7 is as in (6.1) The equivalence between the Risk-Sensitive LQG Control Problem and the Minimum Entropy 7/0o Control Problem is then a consequence of the following result. Proposition 6.3.1 ([27]) Let H E 7~7/0o, and let z be the output of this system when driven by a Gaussian white noise input w with spectrum equal to the identity. Then the LEQG cost in the risk-averse case (0 = - 7 -2 < 0), and the entropy, satisfy jr(7)= ~ I(H;'r;oo)+O(1/T)
oo
if IIHII0o < ' r ,
if IIHIIo:> 7 and T is sufficiently large.
Thus in the steady-state case (T --+ c~),
ifllHII0o <'r. That is, for any (P,7)-admissible closed-loop H, the entropy I ( H ; 7 ; c ~ ) equals the steady-state LEQG cost J~o(7).
CHAPTER 6. RELATIONS TO RISK-SENSITIVE LQG CONTROL
62
Proof This proof follows [27] and is included for completeness. Assume that the dosed-loop system H satisfies the following stochastic differentia/equation,
dzt = Axt + Bdvt where vt is a normalized Brownian motion. To evaluate JT(~/) consider the value function conditioned on the state at time t,
f(t,x) = E[.¢=, {exp (ff-~--~If r Z~TZa ds) }. A standard calculation (see for example [13, page 195]) now gives
/(t,~) = exp {(~-~/2)~ ~C~a~at + o(at) }
{
fo,,
]o,
x J'(t,z)+-~Axdt+(1/2)trace [Oz" BBTdt + ~ dt+o(dt)
}
which implies that
--or +
A" + ('Y-'/2)~rC~a~I(t'~) + (1/2)trace [OVto~'BBr
= O.
(6.4)
Following [34], this is satisfied by
/(t, ~) = g(t)exp {(~-2/2)~S(t)~} where S and g satisfy
-S(t) = CTC + S(t)A + ArS(t) + 7-2S(t)BBTS(t), -~(t) = (7-2/2)g(t) trace[S(t)BBT], g(T) = 1.
S(T) = 0
This is verified to be a solution by noting that~ O~ = [~ + ( _ 2 / 2 ) g z ~ z ] exp{( _ , / 2 ) z r S z } 0t
O~
Substituting into (6.4) gives,
exp{--(7-2/2)zTSz} x left-hand side of (6.4) = ~ + (7-2/2)gxrSx + ~/-2gzrSAz + (7-2/2)gzTCTCx + (~-~/z)g ~race [(S + ~-2S~.~ ~"SW" ~] = ~ + (~-~/2)g trace [SB~ ~'] + (7-~/2)grr(:~ + CrC + SA + ATS + 7-2SBBTS)z = 0 by (6.5) and (6.6).
(6.5) (6.1})
6.3. EQUIVALENCE WITH MINIMUM ENTROPY 7"lo~CONTROL
63
In order to evaluate J r ( 7 ) in (6.3) under the assumption that the state x_r ~ N(0, P), consider E {exp{7-'TWr} } ~ exp{--(1/2)xTP-X~) / dx
L
J
= /.../g(-T)exp{(7-'/2)zTS(-T)z}
{ exp{--(l/2)zTP-*z}%/d(2~r)n et(P)
~/det(I
}do:
g(-T) - 7-'PS(-T))
and hence
JT(7) = (72/T)ln(g(-T))
- (72/2T)In d e t ( I -
7-2PS(-T)).
(6.7)
The steady state case is obtained by taking the limit as T --~ oo. Firstly observe that $(t) --~ Soo as T - t --4 oo where S¢o is the stabilizing solution to the algebraic Riccati equation corresponding to (6.5), i.e.,
c T c -~ScoA + ATs~ + 7-'S~BBTsoo -- 0,
Re{A~{A + 7-2BBTS¢~}} < 0
and SoD exists if and only if HH[I~o < 7. This is a standard result on the Riccati differential equation and can be found in Remark 21 in [60]. Now let h(t) := ln(g(t)) then £ = - ( 7 - ' / 2 )trace [S(t) BB T] and
lim {h(-T)/T} = 7-2trace[SooBB T]
T--+ oo
and hence on substituting into (6.7), the LEQG cost is given by JT(7) =
trace[SooBBr] + O(1/T).
Finally this expression can be related to the entropy integral by using Lemma 5.3.2 on H r. We get, after noting Proposition 2.3.1(vii), that
I(H; 7; oo) and Proposition 6.3.1 follows.
= I ( H r ; 7; oo) =
trace[S~BBr], []
As a final remark, note that as 0 = - 7 -2 is made more negative there comes a point where all controllers give infinite exponential-of-quadratic cost. This corresponds to maximal risk-aversion or equivalently, to ~oo-optimality (since I[Hl[oo is then minimized over all stabilizing controllers).
Chapter 7 T h e N o r m a l i z e d ~/c~ C o n t r o l Problem
66
7.1
C H A P T E R 7. THE N O R M A L I Z E D 7Coo C O N T R O L P R O B L E M
Introduction
In Section 3.7 we found that relaxing the 7-/oo-norm bound in any Minimum Entropy 7/00 Control Problem recovers an LQG control problem. Here we shall reverse this argument. We begin with the Normalized LQG Control Problem of [35] and obtain its associated Minimum Entropy 7/00 Control Problem, which we call the Normalized 7/o0 Control Problem. Proposition 3.6.1 implies that in imposing the minimum entropy/7"/o0 criterion to obtain the Normalized 7/oo Problem, we obtain robustness at the expense of the (upper bound on the) LQG cost. We give a simple numerical example to make the ideas concrete and to illustrate that the improvements in the robustness guarantees may be large relative to the degradation in LQG cost.
7.2
The Normalized LQG Problem
The Normalized LQG Problem of [35] is based on a system k = Ax + Bwl + Bu Z 1 ~- C x ,
z 2 -~- u
w2,
y = Cx +
where "tO1 and w2 are zero mean Gaussian white noise signals, each with a spectrum equal to the identity, and the given plant G -- (A, B, C) is stabilizable and detectable. Put z := [zT zT] T and w :--- [wT wT] T. The LQG cost (see Definition 2.4.1) is just (1/-:
xT
}
See Figure 7.1 for a block diagram of the closed-loop system. It is easily seen that the Normalized LQG Problem has a standard plant (in the sense of Section 3.2 and Figure 3.1) given by
A
P=
[B
0] [B]
[0 [: :] [0] [0]
(7.1)
Some simple manipulations show that the closed-loop transfer function from w to z is H = . F ( P , K ) = [ SG I KSG
SGK ] KS J
(7.2)
where S := (I - G K ) -1. We recognise this system to be the one used to analyse internal stability [56, pl01]. Note that because of the form of P and Remark 3.2.2, a controller K stabilizes P if and only if it stabilizes P m = G. The solution to the Normalized LQG Problem follows immediately from the statespace formulae of Section 3.7, or from standard references [2, 38, 35].
67
7.2. THE NORMALIZED LQG P R O B L E M
W1
II---Z2
K
w2
Figure 7.1: Block diagram for the Normalized Problems Proposition 7.2.1 (Solution to the Normalized LQG Problem) Let G = (A, B, C) be stabilizable and detectable. Then there exists a unique positive semidefinite stabilizing solution X2 = X ~ of the control algebraic Riccati equation
(CARE): ATx2 + X~A - X 2 B B T X 2 + CTC = 0
CARE
(7.3)
and there also exists a unique positive semidefinite stabilizing solution Y2 = Y2T of the
filter algebraic Riccati equation (FARE): AY2 + Y2A T - Y2cTcy2 + B B T = O.
FARE
(7.4)
The Normalized LQG Controller I(LQo = (A, B, C) takes the form of an optimal ohserver
= (A - Y2CTC - B B T X 2 ) ~ + Y2C T y
(7.5)
together with the optimal state-feedback u = --BTX~ :~. 0
(7.6)
The minimum value of the LQG cost is
c(g Qa) = trace[Br Z2B + Br X2Y X2B],
(.7.7)
where HLQa is the LQG-optimal closed-loop (i.e., H in equation (7.2) with KLQG).
68
CHAPTER 7. THE NORMALIZED 'H~ CONTROL PROBLEM
R e m a r k 7.2.2 If G (A, B, C) is actually minimal, rather than just stabilizable and detectable, the above proposition is unchanged except both X2 and Y~ are then positive definite rather than positive semidefinite. =
Note that the first and second terms in (7.7) are associated with the control and filtering, respectively, that is implicit in the LQG control system.
7.3
The Normalized
T/oo P r o b l e m
Now let us examine the Normalized T/o¢ Problem--that is, the Minimum Entropy ~ Control Problem implied by the Normalized LQG Problem. By assumption, the given plant G = (A, B, C) is stabilizable and detectable, hence Assumption 3.2.1 is fulfilled. Also, it is easy to see that all of Assumptions 3.2.7 are fulfilled, so P as in (7.1) is indeed a legitimate standard plant for the Minimum Entropy 7-/~ Control Problem. The block diagram is identical to that for the Normalized LQG Problem shown in Figure 7.1; the closed-loop transfer function H is given in (7.2). We noted in Section 7.2 that because of the form of P and Remark 3.2.2, a controUer K stabilizes P if and only if it stabilizes G. Hence a controller K is (P, 7)-adnfissible if and only if K stabilizes G and IIHII~ < 7- This allows us to use the results of Section 3.5 to state the solution to the Normalized 7-/¢~ Problem. But firstly some remarks on robustness. R e m a r k 7.8.1 Each element of H has a robustness interpretation: SG corresponds to 'additive' uncertainty All on the controller K; SGK corresponds to 'output multiplicative' uncertainty A12 on the plant G; KSG corresponds to 'input multiplicative' uncertainty A21 on G; and K S corresponds to additive uncertainty A~2 on G. Block diagrams illustrating these four uncertainty types Ai~, i,j = 1,2, are provided in Figure 7.2. (See [19] for a discussion of various uncertainty types). The ~¢o-norm bound on H implies robust stability guarantees for each of the four uncertainty types, via the Small Gain Theorem. This follows from the simple observation that
I1[ SG KSG
SGK]o ° KS < "y
IISGII= < I{SGK,loo<'r
IIKSalloo < -r
(7.8)
IIKSII,~ < "7 Using the Small Gain Theorem as in Remark 2.5.2 implies that closed-loop stability is maintained for any one Aij E T£7~ satisfying II ,Jll
-<
i , j = 1, 2.
(Note that because K stabilizes G, we have SG E 7Z7-l~,,,SGK E RTI~, KSG E R~,,o and K S E 7Z7"l~.) If we use the Small Gain Theorem on a frequency-by-frequency
69
7.3. THE NORMALIZED Tloo PROBLEM
Additive
uncertainty on K
Output multiplicative uncertainty on G
Input multiplicative uncertainty on G
Additive uncertainty on G
Figure 7.2: Uncertainty types A~i covered by the Normalized 7"/oo Problem basis (as in Theorem 2.5.1) then a frequency-wise description of the tolerable Aij is obtained" closed-loop stability is maintained for any one Aii E 7~7-/oo satisfying a l l A l l ( j w ) } al{S(jw)G(jw)} al{~,2(jw)} ax{S(jw)G(jw)K(jw)} al{A21(jw)} al{K(jw)S(jw)G(jw)} al{A22(jw)} al{K(jw)S(jw)}
< < < <
1 1 1 1
w e ~ u {oo}; vw • lit u {co}; v ~ e ]R u {oo}; v ~ • lit u {oo}.
R e m a r k 7.3.2 A bound on the sensitivity is obtained from the identity S = I + SGK,
by taking the 7"/oo-norm. Use the triangle inequality together with the appropriate bound from (7.8) to give IlSll~ < 1 + 7.
C H A P T E R 7. THE NORMALIZED T100 CONTROL PROBLEM
70
Proposition 7.3.3 (Solution to the Normalized T/~ Problem) Let G = ( A, B, C) be stabilizable and detectable, and let 7 > 7*. Then there exists a unique positive semidefinite stabilizing solution X ~ = X ~ of the 7-l~ control algebraic Riccati equation (HCARE): ATx00 -t- X00A - (1 - 7-2)X00BBT X00 + CTC = 0
HCARE
(7.9)
and there exists a unique positive semidefinite stabilizing solution Y~ = y T of the 7£0o filter algebraic Riccati equation (ttFARE): AYes + Y00Ar - (1 - 7 - 2 ) y ~ c T c y ~ + B B r = O.
HFARE
(7.10)
There exists a c o n t r o l l e r which stabilizes G and satisfies IIHII00 < ~ (where the closedloop H is defined in (7.~)) if and only if there exists Xoo >_ 0 and Yoo >>_0 as above, which satisfy p(X00Y00) < 7 2. Define zoo : = ( z -
7-~Yoox00)-x.
(7.11)
The Normalized 7i00 Controller KME~ ---- ( a, f~, O) takes the form of an observer
(7.12) together with the state-feedback u = _BTXooZ%$.
(7.13)
Y
The minimum value of the entropy is I( HME~; 7; oo) = trace[Br Xo~B + BT x00z00YcoXooB],
(7.14)
where HMF,00 is the minimum entropy closed-loop (i.e., It in equation (7.2) with
KME00). R e m a r k 7.3.4 If G = (A, B, C) is actually minimal, rather than just stabilizable and detectable, the above proposition is unchanged except both Xoo and Yo~ are then positive definite rather than positive semidefinite. To solve the 7-l~-optimal problem for this plant, we apply the above proposition: there exists a stabilizing controller such that [[Hli~ < 7, if and only if there exists Xoo > 0 and ~% > 0 as above, which satisfy p ( X ~ ) < 7 2. The optimal 7~00-norm 70 is the infimal value of 7- In general a closed-form solution for 7o is not available, but 7-iteration can be used to isolate 7o to an arbitrary accuracy. The following remark and lemma are interesting here: Corollary 8.4.6 should also be consulted.
7.3. THE NORMALIZED TI~ PROBLEM
71
R e m a r k 7.3.5 By definition 7o = inf{llHIl~ whereH=
SG KSG
SGK KS
and S
[t:=H+
K stabilizes G},
(/- GK) -1. Consider the closely related problem
% := i}f{llHlloo where
:
: K stabilizes G},
[0,] [+ ,] 0 0
=
KSG
KS
"
(7.1~)
This latter problem has an exact (non-iterative) solution [40, Remark 4.21] in terms of X~ and Y2, the stabilizing solutions to the CARE and FARE, respectively:
~o = ¢ I + AI{X2Y2} > I. But the triangle inequality applied to the 7-Loo-normof (7.15) gives
II/~II~-
1 _< IIHII® <_II//II~+ 1.
Taking the intlmum over all K which stabilize G gives 0 < ' ~ o - 1 ___% <_ ~,o+ 1,
which provides (non-iterative) upper and lower bounds on 7°. L e m m a 7.3.6 Consider the Normalized 7"[o0 Problem for a given stabilizable and detectable system G = (A, B, C)--that is, consider the Minimum Entropy 7-i~,, Control
Problem (Problem 3.2.6) .for a standard plant P as given in (7.1). Then "go < 1 only if the given plant G = ( A, B, C) is asymptotically stable. Conversely, if the given plant G = (A, B, C) is not asymptotically stable then 7o > 1. Proof
Suppose 7o < 7 -< I. Then (7.10) can be written as a Lyapunov equation: o = Y ~ A r + A Y ~ + [aY~C r B][c,Y~C T B] T,
where a 2 := 7 -2 - 1 > 0. The pair (A, [ctY~,Cr B]) is stabilizable because, by assumption, (A, B) is. This together with Y® > 0 implies that A is asymptotically stable, by a standard result on Lyapunov equations [63, Lemma 12.2]. [] For convenience, we state the upper bounds on 7~¢~-norm and LQG cost provided by Proposition 3.6.1.
Proposition 7.3.7 The ~oo-norm and the LQG cost of the minimum entropy closed loop HMEoo = ~'(P, KMEoo), satisfy II//Msooll~o < 7
(7.16)
C(HME~o) <_ trace[BTX~B + BTxc~z~Y~X¢cB] •
(7.17)
The tradeoff of Theorem 3.6.2 between the 7"/co bound and LQG cost bound is then apparent from the discussion in Section 3.7: as the RHS of (7.16) increases (resp. decreases) so the RHS of (7.17) decreases (resp. increases) monotonically.
72
7'.4
CHAPTER 7. THE NORMALIZED 7-[~ CONTROL PROBLEM
A Numerical Example
Let us demonstrate the results of the previous section on a simple numerical example, chosen for its ability to illustrate the tradeoffs rather than for physical interpretation. Take
This has a right-hag plane zero at 0.1 and two right-half plane poles at 10. Using 7-iteration gives 70 = 40.87 (the minimum value of [}HII® i.e., this corresponds to the 7~o-optimal solution). Then, solving the X2 and Y2 equations, we find from (7.7) that C(HLQG) = 64 x 100 (the minimum value of the LQG cost i.e., this corresponds to the LQG solution). For any 7 > 70, Xoo and Yoo are found from (7.9) and (7.10), and the minimum entropy controller, KMEoo, is found using (7.12) and (7.13). Then the minimum entropy closed-loop is HME~ = Yr(P,K~tF~oo) := (A,/~/, C), say. The minimum value of the entropy is evaluated using (7.14). To calculate G(HME~o), solve the Lyapunov equation for the controllability Gramian Q of the minimum entropy closed-loop; then G(H~E~) = trace[QOTC] as in (5.1). Calculation of ]]HME~][o~ can be performed to prespecified accuracy using the algorithm in [10]. These calculations were done for a number of values of % ranging over several orders of magnitude from close to %. The results are illustrated in Figures 7.3-7.7. Figure 7.3 is a plot of (7.16) as 7 varies i.e., of the upper bound ~/on IIHMEoolIoand the actual value of I[HME~olIo., against % Figure 7.4 is a plot of (7.17) as -y varies i.e., of the upper bound I(ItMB®;~/; c¢) on C(HMEo~) and the actual value of C(HMEoo), against 7. The graphs illustrate clearly the 7/oo/LQG tradeoff of Theorem 3.6.2: the upper curve in Figure 7.3 increases with increasing "y, whilst the upper curve in Figure 7.4 decreases with increasing % In fact, we notice that the tradeoff is even stronger than this--the achieved values (i.e., the lower curves in Figures 7.3 and 7.4) exhibit the same behaviour as their upper bounds. See Chapter 9 for more on such monotonicity. Notice that in Figure 7.3, as "r becomes large, [[HM~ooI]ootends fairly slowly towards liHnQai{oo ~ 80, a number slightly less than twice 70. The variation with 7 is more rapid in Figure 7.4. Both C(HMEoo) and its upper bound I(tlME~o;%oo) decrease quickly with 7 when "r is close to %. So large improvements in the LQG properties can be obtained with only modest degradation of ~ o properties. Although theory predicts that 3' must be arbitrarily large before the Minimum Entropy 7f~ Control Problem problem reduces to the LQG problem, we see in Figure 7.4 that for 7 as small as twice 7o, the LQG cost and its bound are very close to their minimum (LQG-optimal) values. Furthermore, the upper bound provided by the entropy is quite tight, as we would expect from Theorem 2.4.4(ii). Figure 7.5 shows the maximum singular value of the maximum entropy closed-loop as a function of frequency. Five curves are plotted, corresponding to five representative values of 7: "/ = 40.87 (Tfoo-optimal), 43, 45, 60 and c¢ (LQG-optimal). The rapid convergence (as 7 increases) towards the LQG curve is clear, as is the fairly slow
73
7.4. A N U M E R I C A L E X A M P L E
200, s s oS o~
180 sJ °
160
II~ru~®ll®
oo ~° s°
140
n~ o ° s~ S s~ s
120
oS
100 o o os ~
80 os ~ ss °
60 4C 40
I
I
60
80
Figure 7.3:
~o
IIgMEoollooand
1~o
,
!
1,o
1~o
I
~80
its upper bound 7, against "y
CHAPTER 7. THE NORMALIZED7-looCONTROLPROBLEM
74
xlO 5 14
12
10 ...........
8
I(HMs~; */; oo)
C(HMEoo)
2[I \,, 40
60
Figure 7.4:
80
100
C(HMEoo)and
120
its u p p e r bound
140
160
I(HMEoo;7; c~),
180
against 7
200
7.4. A NUMERICAL EXAMPLE
75
... ........
..~.,
~¢¢-optirnal oJ
,
tS no
.."
.."
101
..:
.•°•.'""
i
\
/ #.a / .w"
.... ""
iS
,.\
;
•
2" s"
- ...... ............ .........
"•:
\\
7-~-optimal -r=43 -~=47 ~=60 LQG-optimal
"L i
L
10 o
10-2
10-1
100
101
102
103
"~
,
.,,, '.
104
Frequency in rad/s Figure 7.5: HME¢o against frequency, for various 7
rise of the peak around 10 rad/s. Recalling from Remark 7.3.1 that the four transfer functions SG, SGK, KSG and KS which make up the closed-loop H all have robusthess interpretations, it is of interest to plot these too--see Figure 7.6--again for the five representative values of "7. From these, Remark 7.3.1 shows us how to obtain a frequency-wise description of the maximum allowable uncertainty, for each of the uncertainties A~i pictured in Figure 7.2. The maximum allowable uncertainties Aii are plotted against frequency in Figure 7.7. Note that all of the maximum allowable uncertainties are larger at higher frequencies--robustness to high frequency uncertainty is good. This is what we would need in practice. Only in a narrow band of frequencies around 10 rad/s does the 7"(¢~-optimal controller offer greater robustness than the minimum entropy/7-(oo controllers. We can conclude that the (minimum entropy) Normalized ~oo Controller with V -60 ~ 1.5% gives us nearly LQG-optimal performance, with robustness at least as good as the 7-(oo-optimal case except over a small band of frequencies around 10 rad/s. Examine Figures 7.3-7.7 for ~, = 60 and compare with the ~/oo-optimal and LQGoptimal cases.
76
CHAPTER 7. THE NORMALIZED 7-l~o CONTROL PROBLEM
10o
MinimumentropyS,G
i
i
i
i
llJlw
i
t
101
Ill
, M~,mum,,e,mrop~SG,K....
/--7=60
.~ 10-1
•~ 10-2
•~ 10-2 10-3
10-4 10-2
I
I
I
I
I
|1[|
t
i
101 Frequencyin rad/s
,u,m }, 1'nM i lOt ~m
i
i
i
t
104
,enI~op~ 0 KSG .... 1
10-5
10-2
m
i
i
,
i ¢11
I
I
[
101
i
i
i
ii
10 4
Frequencyin rad/s ~ 102 ~
Minimum, ,
10-2
101~0.2
I
I
, ....... 101 104 Frequencyin rad/s I
I
I llll
10-1 10-2
101 Frequencyin rad/s
Figure 7.6: Componentsof HMEoo against frequency, for various 7
104
77
7.4. A N U M E R I C A L E X A M P L E
104 -d
Max additive uncert on K
I
I
I
I
I
I I I I
103
I
I
I
I
I
105
I I
Max,,°utput,, ,,,,,mult uncen | i oni iG|Hi
~ 1 ~
10 2 ~ T q . ~ - o p t i
•~
-lO,
.~
, , LQG-optlmal 102
.#:'1 /,.'1
....... .'%
ds:'s
,:oo
LQO-optimal 10 0
10-2
los
'
,
'
, ,,,,,
,
i
i
i,~
10 i Frequency in rad/s
M ~ ~pu,t ,m,,4,1tun~t, on 9
104
. . . .
LQG-optlmal
lo2 _.:::,,:,%, "'%' 10-1 10-2
1N-I
10 i
100
/::y/
=,,
43
,
104
Max additive uncen on G
I
I
I
I
I
I I I I
I
I
i
I
I
i I ~
•LQG-optimal 4 3 ~ _ "" ':' '/--~7=60 ,' ..- ,. 3' ," * "
10"1 .'
104
....
10 i Frequency in rad/s
.......
""'"
................ 10 i Frequency in rad/s
. . . . . . . . . .
10-2
'"
". ",.
=47 =
lO-~r '"r,-o;'~ ,'~, ~ 10-2
• .': .. /'./'//'
" '
e,
...... :
10 i Frequency m rad]s
104
Figure 7.7: Maximum allowable uncertainties Aij against frequency, for various 7
Chapter 8 7/c~- C h a r a c t e r i s t i c Values
CHAPTER 8. 7-loo-CHARACTERISTIC VALUES
80
8.1
Introduction
The now well-established method of balancing the state-space description of a linear, asymptotically stable system was initiated in [441. By representing a system in balanced coordinates, where the observability and controllability Gramians are equal and diagonal, the contribution of each state to the input-output map of the system is exposed. The Hankel singular values of the system, which are just the diagonal elements trl of the balanced Gramian, form a set of system input-output invarlants. Each ai explicitly quantifies the relevance of its associated state, in that states corresponding to small tr~ are only weakly observable and weakly controllable. Deleting states corresponding to 'small' ai has a 'small' effect on the input-output map. Such 'balanced truncation' generically leads to a stable reduced-order model and the truncation error may be bounded in terms of the neglected ai (see [23, 26] and Lemma 8.4.4). Open-loop balancing, however, has the restriction that the original system must be asymptotically stable. Furthermore, by its very nature, model reduction using standard balanced truncation takes no account of any closed-loop control considerations. These restrictions were lifted in [35], using a closed-loop balancing approach. The open-loop system (which may be unstable) is first compensated with the associated Normalized LQG Controller (Chapter 7). Two algebraic Riccati equations are needed--one for filtering and one for control. Balancing the solutions to these two Riccati equations, so that they are equal and diagonal, exposes the difficulty of filtering and controlling each state. To be exact, the diagonal elements pl of the solution to the 'LQG-balanced' Riccati equations form a set of closed-loop input-output invariants, known as the LQGcharacteristic values. States corresponding to small pi are easy to filter and control; these states may be discarded with only a small effect on the closed-loop input-output map. For a full discussion of the properties of the #i, we refer the reader to [35]. An alternative (but by [46] equivalent) approach to closed-loop balancing and model reduction, based on coprime factorization, may be found in [41]; this will be discussed later. Given the results of this monograph, and in particular given the results of Chapter 7, it is natural to ask whether an '~oo-balancing' method is possible in the spirit of LQGbalancing. It is the purpose of this chapter to show that this is indeed the case. We will use the Normalized 7-/00 Problem of Chapter 7 to provide the natural generalization of LQG-balancing to the minimum entropy/T/o~ case. A different approach to obtaining reduced-order LQG or 7"/00 controllers may be found in [33] and [9], respectively. There a fixed (reduced-order) structure for the controller is assumed and optimization theory is applied to derive four coupledmatrix equations which the controller state-space matrices must satisfy (two Pdccati equations and two Lyapunov equations in the LQG case, four Riccati equations in the T/oo case). Our approach, in contrast, whilst not known to be optimal, only requires the pair of decoupled Riccati equations needed for the full-order Normalized 7-/00 Controller. The layout of the chapter is as follows. Section 8.2 summarises the LQG-balancing method of [35]. With Chapter 7 as background to the Normalized LQG Problem, we
8.2. LQG-BALANCING A N D L Q G - C H A R A C T E R I S T I C VALUES
81
state the LQG-balancing method formally, and define the LQG-characterlstic values, /z~. In Section 8.3 the Normalized 7"/00 Problem of Chapter 7 is used to introduce the new concept of 7-~oo-balaacing, and the 7-Qo-characteristic values, v~, are defined. Some important properties of the v¢ are given. A new model reduction scheme---Tfoo-balanced truncation--is presented and analysed in detail in Section 8.4: the vl play a major role.
8.2
L Q G - B a l a n c i n g and L Q G - C h a r a c t e r i s t i c Values
Consider the Normalized LQG Problem of Chapter 7 for an n-state minimal system G = (A, B, C). The n-state Normalized LQG Controller solving this problem was given in Proposition 7.2.1--we need to solve the Control Algebraic Riccati equation (the CARE: equation (7.3)) A r X , + X2A - X 2 B B T X 2 + c T c = 0
CARE
for the unique stabilizing solution X2 = X T, and we need to solve the Filter Algebraic Riccati equation (the FARE: equation (7.4)) AY2 + Y~A T - y 2 c T c y ~ + B B T = 0
FARE
for the unique stabilizing solution Yz = Y2T. Since G = (A, B, C) is minimal, X2 and Y2 are positive definite (see Remark 7.2.2). Under a nonsingular state transformation S, ( A , B , C ) s ( S A S _ I , S B , CS_I) =: (A,/3,C), so X2 ~ S - T X ~ S -~ =: X2 and Y2 s SY2Sr =: ~ . Since ~72~ = S - T x 2 Y 2 S r, it follows that X2Y2 and ) ( 2 ~ are similar matrices, and therefore the eigenvalues of X2Y2 are similarity invariants. These invariants are the squares of the LQG-eharacteristic values of the system G, as defined in [35]; a formal definition and some basic properties are given next.
Proposition 8.2.1 (LQG-balancing and LQG-characteristic values [35]) Let the system G = (A, B, C) be minimal with n-states and let .7(2 and Y2 be the unique positive definite stabilizin9 solutions of the C A R E and FARE respectively. Then the ei#envalues of X2Y2 are real, strictly positive similarity invariants, as are their positive square roots which are called the LQG-characteristic values of G. Let g3 > #5 >- "'" >--#~ > 0 denote the n eigenvalues of X2Y2 arranged in decreasin9 order, then there ezists a similarity transformation which transforms both X~ and Y2 to the form M := diag(#l, #2,..., p,). The system is then said to be in LQG-balanced coordinates, and M is the diagonal matriz of LQG-characteristic vahtes of G.
R e m a r k 8.2.2 ( U n i q u e n e s s of L Q G - b a l a n c e d realizations) In [35], it was shown that an LQG-balanced realization of a system is unique to within block diagonal similarity transformations. The block diagonal elements are: 4-1 for each distinct gi, and any l by l orthogonal matrix for each pl with multiplicity I.
82
CHAPTER 8. 7-lo~-CHARACTERISTIC VALUES
Ordering of the blocks is done compatibly with the ordering of the #~. In the generic case where the #i are all distinct, the LQG-balanced re~dization is therefore unique to within similarity transformation by a sign matrix. As argued in [35], small pi correspond to states which are easy to filter and control. This motivates the following model reduction schemes. Procedure 8.2.3 (Reduced-order plant by LQG-balanced truncation [35]) Let G = (A, B, C) be minimal with n-states and in LQG-balanced coordinates with LQG-characteristic values #1 >_p2 >_ "'" >- #n > O. That is, M = diag(#l,p2,...,#,) is the stabilizing solution of the CARE and FARE associated with G = (A, B, C). Pick k < n such that pk > pk+a and partition M accordingly into M=[
MaO M20 ]
where M1 = diag(pl,...,#h) and M~ = diag(#k+l,...,#,,). conformably with the partitioning of M: A=
[ a x l A12] Azl A22 '
B=
[Bx] B2
Partition A, B and C
and C = [ C 1
C~].
(8.1)
A k-state reduced-order plant is then G~ = (All,B1, C1). Procedure 8.2.4 (Reduced-order control by LQG-balanced truncation [35]) Let G = (A, B, C) be minimal with n-states and in LQG-balanced coordinates with LQG-characteristic values pl >_tz2 > "'" > pn > O. That is, M = diag(pl,p~,...,pn) is the stabilizing solution of the CARE and FARE associated with G = (A, B, C). Pick k < n such that #~ > pk+l and partition M accordingly into
M=[
Mr0 M2O]
where M1 = diag(m,...,f,~) and 2142 = diag(pk+x,...,~,,). Let Kzqo = (A,B,O) be the Normalized LQG Controller .for the plant G = (A, B, C) (as given in Proposition 7.2.1). Partition A, B and 0 conformably with the partitioning of M:
A k-state reduced-order controller is then h'r = (ft11, 131,C'1). Remark 8.2.5 The reduced-order controller /t'~ is the full-order Normalized LQG Controller for the reduced-order plant Gr. This is an immediate consequence of the easily seen fact that M1 is the stabilizing solution to the CARE and FARE for G, = (All, B1,C1).
8.3. 7-[0o-BALANCING A N D TI0o-CHARACTERISTIC VALUES
83
In the remainder of this chapter, we will be concerned with various ways of obtaining reduced-order plants and controllers. To keep notation simple, wc will use G, to denote a reduced-order plant and will state explicitlywhich model reduction method was used to obtain it. Similarly for a reduced-order controller K,. Of course, it is important to know what happens to the stabilityand performance of the closed-loop when a reduced-order Normalized L Q G Controller is connected to the full-order system (A, B, C). Sufficientconditions are derived in [35]; they confirm the intuitive notion that the reduced-order controller is likely to perform well if the discarded #i are sufficientlysmall.
8.3
7-/oo-Balancing and 7-/oo-Characteristic Values
In this section,we show how all the results of the previous section can be generalized to the minimum entropy/7~oo case. Consider the Normalized 7"/0oProblem of Chapter ? for an n-state minimal system G ---(A, B, C). The n-state Normalized 7/0o Controller solving this problem was given in Proposition 7.3.3--we need to solve the 7/0o Control Algebraic Riccati equation (the H C A R E : equation (7.9)) ArX0o + X0oA - (1 - 7-2)X0oBBrX0o + CTC = 0
HCARE
for the unique stabilizing solution X0o = X T, and we need to solve the 7-/.o Filter Algebraic mccati equation (the HFARE: equation (7.10)) AY0O + Y0OAr - (1 - 7-2)YooCTCy0O + B B T = 0
HFARE
for the unique stabilizing solution Y0o = Y~. Since G = ( A , B , C) is minimal, X0o and Y0o are positive definite (see Remark 7.3.4). Just as in the LQG case, under a nonsingular state transformation S, X0oY0o ~ S-TXooY0OS T, so the eigenvalues of X0oY0o are similarity invariants, the positive square roots of which we define to be the 7too-characteristic values. All the arguments of Section 8.2 carry across and we obtain the following minimum entropy/7/oo generalization of Proposition 8.2.1. Note that since "7 > % by assumption, Proposition 7.3.3 gives p(X~Y0o) < 72 so each of the 7/~o-characteristic values is strictly less than q,. P r o p o s i t i o n 8.3.1 (7-/0o-balancing and 7/0o-characteristie values) Let the system G --- (Am By C) be minimal with n-states, let 7 > %, and let X0o and Y0o be the unique positive definite stabilizing solutions of the H C A R E and HFARE respectively. Then the eigenvalues of X0oY0o are real, strictly positive similarity invariants, as are their positive square roots which are called the 7-(0o-characteristic values of G. Let v~ > v~ >_ ... > v~ > 0 denote the n eigenvalues of A'0oY0o arranged in decreasing order, then "7 > vl and there ezists a similarity transformation which transforms both X0o and Yoo to the form N := diag(vz,v2,...,vn). The system is then said to be in 7"/oo-balanced coordinates, and N is the diagonal matriz of ~oo-characteristic values
o.fG.
84
C H A P T E R 8. 7"[,,-CHARACTERISTIC VALUES
R e m a r k 8.3.2 ( U n i q u e n e s s of 74o.-balanced realizations) It is easy to see that an 7~oo-balanced realization of a system is unique to within similarity transformations of an analogous type to those described in Remark 8.2.2 for LQG-balanced realizations. In particular, we note that in the generic case where the vl are all distinct, the 7-/.~balanced realization is unique to within a similarity transformation by a sign matrix. Note that the 7~-characteristic values are functions of % Strictly speaking, we should write vi(3') and N(7). In the interests of notational simplicity, we will write v~ and N. The following proposition states some interesting properties of the el. P r o p o s i t i o n 8.3.3 For the •i as defined in Proposition 8.3.1 we have
(i) ~ > m. (ii) Each vl is a monotonically decreasing function of 7. (iii) If the u~ are distinct then du~/d7 <_ O. (iv) Each vi is a continuous function of 7. ( v ) lim~_.,o{vJ = #,.
P r o o f Note firstly that X ~ and Y~ are positive definite, hence non-singular, because (A, B, C) is minimal by assumption (Remark 7.3.4). Part (i) Apply the main result of [62] to X~I(HICARE)X~ ~ and X~-I(CARE)X~"I, and then to Y~I(HFARE)Y~ and Y2-~(FARE)Y2-~. It follows that
X.~ > X 2 and Y~. > Y~.
Hence [32, Observation 7.7.2] we have
x
roox
>
,
and
r l"xoor 1" > r
'x rf
which implies [32, CoroUary ~.7.4(c)1 that A~{XooY~} > A,{--YooY2}_~ A,{Xzt~}.
But, by detlnition, ~,~ = X,{XooY~}and ~ = A~{X2~}, and the result follows. Part (ii) Let 72 > "Yl > %. Apply the main result of [62] again, this time to X ; o ~ ( H C A R E ) X ; o ~ with 7 ; "r~ and "r ; "r~, and then to t ~ ( I ' I F A R E ) Y ~
and 7 -- 7~. It follows that x ~ ( - r ~ ) _< x ~ ( ~ )
~ with "r = "r~
8.3. 7"loo-BALANCING AND ~oo- CHARA CTERISTIC VAL UES
85
and Yoo(~) <_ Yoo(~), (with an obvious notation). Using a similar argument to that in the proof of Part (i), this implies that vi(~'2) < v~(fft), and the result is proved. In fact, from [14, 50], X~o and You are differentiable functions of 7, so we actually have
dXoo
- d~ -<0 -
and
dY. m<0. d~
-
Part (iii) From the proof of Part (ii), Xoo and Yoo are differentiable functions of 7. Hence so is XooYo.. By definition, v~ = A;{XooYo.} and, by assumption, the v~ are distinct. Hence each vl is a differentiable function of ft. But, from Part (ii), each v~ is a monotonically decreasing function of 7. Therefore, for each v~, there holds dvdd 7 < 0, as claimed. Part (iv) From the proof of Part (iii), XooYo, is a differentiable function ofT. Hence it is a continuous function of 7. The result then follows from the well-known fact that the eigenvalues of a matrix are continuous functions of the elements of the matrix. Part (v) From Parts (i), (ii) and (iv), each vl is a continuous, monotonically decreasing function of 7, bounded below by #i. Hence lim.r--.oo{vl} exists. It is easy to see that as 7 -~ oo, the HCARE tends to the CARE, and the HFARE tends to the FARE. So,
Hm ( X . ) = X2 and lira {Y~o} = Y~.
q,---¢Oo
Rec~tUns that v~ = ~,{X®roo} and ~ = ~,{X~Y~}, the reset foUows.
[]
R e m a r k 8.3.4 In [47], it was shown that the LQO-balancing approach is particularly applicable to symmetric passive systems, such as large space structures with colocated rate sensors and actuators. We would expect this to be the case for the 9foo-bManeing approach, too. However, as pointed out in [47, p83], for systems with no symmetry or passivity properties, a non-normallzed approach may be more suitable. One possible way round this would be to apply the normalized approach not to the actual plant, but to a %haped' plant which has been (open-loop) pre- and post-compensated. The success of such 'loop-shaping' ideas has been demonstrated in [40], using controller synthesis via the robust stabilization of the normalized coprime factors of a plant. R e m a r k 8.3.5 In [35], there is a parametrization, in the case of distinct t~i, of singleinput single-output systems in terms of the /ti and the elements of the B-vector in the realization of G. Using properties of this parametrization, necessary and sufficient
C H A P T E R 8. 7-I*0.CHARA C T E R I S T I C VALUES
86
conditions were derived for dissipativeness of both the open-loop and closed-loop, with either the full-order or reduced-order controller. Note that necessary and sufficient conditions for dissipativeness provide sufficient conditions for stability. These results all carry across to the minimum entropy/74*0 case in a straightforward way, by using the ul and making the appropriate changes.
8.4
M o d e l R e d u c t i o n by 7-/so-Balanced T r u n c a t i o n
Let us now turn to the model reduction aspects of the Normalized 74*0 Problem. Controller or plant order reduction may be carried out using 7-[*0-balanced truncation, an analogous approach to the LQG-balanced truncation of Procedure 8.2.4 mad Procedure 8.2.3: write the plant and its Normalized 74,0 Controller in 74~-balanced coordinates, and discard those states in either the plant or controller associated with 'sm~lP u~. We now state this new 7"los-balanced truncation method formally.
8.4.1
7~oo-Balanced Truncation
Procedure 8.4.1 (Reduced-order plant by 7~*0-balanced truncation) Let G = (A, B, C) be minimal with n-states and in 74*0-balanced coordinates for given 7 > 70, with 74*0-characteristic values 7 > ul > t'2 > ... > u, > O. That is, N = diag(ul,u2,...,us) is the stabilizing solution of the H C A R E and HFARE associated with G = (A, B, C) and 7. Pick k < n such that uk > uk+x and partition N accordingly into N=[
NIO N2O ]
where Nx = diag(ul,...,vk) and N2 = diag(uk+l,...,u,). conformably with the partitioning of N:
A,,
B:
Partition A, B and C
and C=[C
a].
A k-state reduced.order plant is then G, = (All,B1, Ca).
Procedure 8.4.2 (Reduced-order control by 74*0-balanced t r u n c a t i o n ) Let G = (A, B, C) be minimal with n-states and in 74*0-balanced coordinates for given 7 > 7o, with 74~-characteristie values 7 > ul >_ u~ > ... >_ u, > O. That is, N = diag(vl, v2,..., u,) is the stabilizing solution of the H C A R E and HFARE associated with G = (A, B, C) and 7. Pick k < n such that ua > ua+~ and partition N accordingly into
N=[
N10 N2O ]
8.4. MODEL REDUCTION B Y ?too.BALANCED TRUNCATION
87
where N1 = d i a g ( v l , . . . , v h ) and N~ = di~g(vk+l,...,u,,). Let KM~oo = ( A , / J , G ) be the Normalized ~ Controller for the plant G = (A, B, G) and 7 (as given in Proposition 7.3.3). Partition A, JB and C conformably with the partitioning of N:
A21 A22 '
B2
and 0 = [ 01 C2 ] •
A k-slate reduced-order controller is then K,. = (.,411, J~l, 01). R e m a r k 8.4.0 The reduced-order controller K, is the full-order Normalized 7"/~ Controller for the reduced-order plant Gr. This is an immediate consequence of the easily seen fact that Nx is the stabilizing solution to the t t C A R E and ttFARE for G, = (Alt, B1,Ct).
8.4.2
Relations to Balanced Truncation
Here we relate Hoo-balancing to ordinary balancing. This sets the scene for later sections, where a precise connection with balanced truncation of a scaled factorization of the plant is derived. Suppose we carry out ordinary balanced truncation [44] on the open-loop plant G. Assume G = (A, B , C ) is asymptotically stable and minimal and in balanced coordinates. That is, the controllability Gramian and the observability Gramian are both equal to the balanced Gramian ~,~ where ~.. := d i a g ( a l , a 2 , . . . ,an) > 0 solves AB + EA r + BB T = 0
AT~ + Y~A + c r c = O. Any asymptotically stable and minimal system may be balanced by a suitable similarity transform [44]. The matrix ~ is just the diagonal matrix of Hankel singular values; by convention we order them ~1 >_ a2 >_ .-. > ¢r,, > 0. Choose k < n and partition
0
~2
where ~1 = diag(al,...,crk) and ~z = d i a g ( a k + l . . . , ~ , ) . Partition the plant statespace matrices conformably in the obvious way (as in (8.1)). Then a reduced-order plant is G, = ( A l l , B1, C1), obtained by deleting the states associated with ~2. L e m m a 8.4.4 Let G be asymptotically stable and minimal with n-states. Let G be in balanced coordinates with balanced Gramian ~ = d i a g ( a x , a ~ , . . . , a , ) and Hankel singular values ¢71 >~ ~2 >_ "'" >_ a,~ > O. Let k < n and partition ~ = diag(~l,~2) where ~1 = diag(aa,... ,~k) and ~2 = diag(ak+l ... ,or,,). Let Gr be the k-state reducedorder model obtained by truncating the balanced realization of G to k-states. Then
88
C H A P T E R 8. 7"loo-CI-IARA C T E R I S T I C VALUES
(i) [49] G~ is in balanced coordinates with balanced Gramian ~1. (it) [40] / f qk > ak+t then G, is asymptotically stable and minimal. (iii) [26, 23] ]IG -- G, II~ < 2 trace[~2]. We can now relate the ~o-characteristic values of G to the H a n k d singular values of G. P r o p o s i t i o n 8.4.5 Suppose G = ( A, B , C) is asymptotically stable and minimal, with Hankel singular values (rl >_ ~r2 > " " >_ cr,~ > O. Let 7 > % and let G have 7"[oocharacteristic values 7 > t'l > v2 >__" " >_ t,,~ > O. Then (i) I f 7 > 1 then at > vl. (it) l f 7 = 1 then ai = vi. (iii) 117 < 1 then (r~ < v~. P r o o f In this proof, (A, B, C) is not necessarily a balanced realization of any type. The controllability Gramian P = p r > 0 and observability Gramian Q = Q r > 0 of G = (A, B, C) are given by
(8.2) (8.3)
AP + PA T + BB r = 0 ATQ + Q A + c T c = 0,
and then
a~ = Ai{QP}. Subtract the HFARE from (8.2), and subtract the HCARE from (8.3): A(P - Y~) + (P - Y~)Ar + (1 - 7 - 2 ) y ~ c T c y ~ = 0
AT(Q - xoo) + (Q - Xoo)A +
(1 -
"r-2)XooBBTXoo
=
(8.4)
0.
Part (i) By assumption, A is asymptotically stable and 1 - 7 -2 > 0. A standard result on Lyapunov equations (see [26, Theorem 3.3(7)]) applied to (8.4) and (8.5) then gives P > Yoo and Q > Xoo. Hence, by the argument in the proof of Proposition 8.3.3(i), A I { Q P } >_ A~{XooY~}. Recalling tr~ = A~{QP} and v~ = A~{XooYoo}, the result follows. Part (it) Put 7 = 1 in (8.4) and (8.5): A ( P - Yoo) + ( P - Yoo)A T = 0 A T ( Q - X ~ ) + (Q - Xoo)A = O.
Since A is asymptotically stable by assumption, it follows from [48] that P = Yoo and Q = Xoo. Hence the result. Part (iii) The same argument as used to prove Part (i) may be applied, except now 1 - 7 -2 < 0. It follows that P < Yoo and Q < Xoo, which implies the result. [] The following corollary is immediate on combining Parts (ii) and (iii) of the proposition. (Minimality of G = (A, B, C) is relaxed to stabilizability and detectability in the corollary because this does not affect %, at or vl.)
8.4. M O D E L R E D U C T I O N B Y 7-l~-BALANCED T R U N C A T I O N
89
Corollary 8.4.6 Suppose O = (A, 13, U) is asymptotically stable, stabilizable and detectable, with largest Hankel singular" value 0"1 > O. Then % < 1 only if o'1 < 1.
We recoguise 0"a as the Hankel norm [26] of the system G. Now suppose we carry out balanced truncation on the normalized coprime factors of the plant (see [41] and also [40, Chapter 5]). To be precise, let G have a normalized left-coprime factorization: G = ~-x~,
where h7$, N are asymptotically stable and left-coprime,/17/-1 exists, and
ti. * +
=/.
Let the ttankel singular values of [ti hT/] be ~1 _> ~2 _> "'" >__ 8. > 0 and define = diag(~l,~2,...,~,,). Since the factorization is normalized, we have [41] that I > ~, so 1 > &i. Pick k < n such that &k > ak+x. We can then perform balanced truncation in the usual way (i.e., as described above) on [N hT/] to obtain a reducedorder representation IN, hT/r]. It was proved in [41] that G, = /I;/~lNr is in fact a normalized left-eoprime factorization of the reduced-order plant Gr. The error bound in Lemma 8.4.4(iii) applies:
II[~r - ~ ,
~ - ~r,]ll~ _< 2 traee[~2],
this time in terms of ~2, the diagonal matrix of the neglected Hankel singular values of IN M]. It is well-known [42] that the solution to a suitable Linear Quadratic Regulator problem leads to the state-space realization of the normalized left-coprime factors. So it is to be expected that the ttankel singular values of [ti /17I]are closely related to the LQG-characteristie values. L e m m a 8.4.7 ([40, p80]) Let 1 > £q > ~2 >_ "'" >_ ~, > 0 be the Hankel singular values o f [ t i lf'I]. Then -
1+
From Proposition 8.3.3(i), vi > #i. The following result, which relates the 7 ~ characteristic values of G to the Hankel singular values of [ti hT/], is immediate from this fact together with Lemlna 8.4.7. Proposition 8.4.8 Definitions as above. Then
,? The above is a link between 7-/oo-balancing and coprime factorization of the plant G. In the next section we shall see a much stronger link with coprime factorization of a scaled plant.
CHAPTER 8. 74so-CHARACTERISTIC VALUES
90 8.4.3
Coprime
Factorization
via the Normalized
7"/00 P r o b l e m
Motivated by the relations of the previous section, we derive here a much more precise interpretation of the vi: we find that the Ha~kel singular values of the normalized left-coprime factors of a scaled plant can be written exactly in terms of the vi. In Proposition 8.4.8, the 7"/¢~-characteristic values were related to the Ha~kel singular values of the normalized left-coprime factors of the plant G. A key step was to use [42] to construct the normalized left-coprime factorization of G using one of the LQG algebraic P~ccati equations of the Normalized LQG Problem. The key step in this section is to use [42] to construct the normalized left-coprlme factorization of/3G using one of the 74~ algebraic Riccati equations of the Normalized 7400 Problem. Here /3 := v / l - ~-~ is a necessary plant scaling--we assume 7 > max{1,7o} (refer to Lemma 7.3.6 and Corollary 8.4.6), so that 0 ma~{1,7o}. Let~3 = (1 ~-2)1[~. Let Yoo be the unique positive definite stabilizing solution of the associated HFARE. Define N and fill by L6N /Q']= ['4C~-~ -- ] := [ A - " Y = C T C [ ~ BO -fl'Yo°Cr I
.
(8.6)
Then f G = IVI-1~ ~[ is a normalized lefl-coprime factorization o.f flG. P r o o f Appendix A.5.
[]
R e m a r k 8.4.10 From [46, 43], the realization (8.6) of [/3N /17/]is minimal if and only if the realization G = (A, B, U) is minimal. Minimality of G = (A, B, C) is a standing assumption in this chapter. In order to work out the Hankel singular values of ~ N .~7/]we need to calculate the controllability Gramian/5 of [/3N ~r] and the observabflity Gramian (~ of [/3N 11~]. Since the realization of ~ N /17/]given in (8.6) is asymptotically stable and minimal, both/5 and Q are positive definite. In fact, these Gramians can be written explicitly in terms of Xoo (the unique positive definite stabilizing solution of the HCARE) and Y~o (the unique positive definite stabilizing solution of the HFARE). L e m m a 8.4.11 Consider the slate-space realization of[fiN lfI] given in Lemma 8.4.9. Its controUability Gramian is P = ~Y~,
and its observabilily Gramian is 0 =" (XooI 7!-~(~2Yec,)-l.
8.4. MODEL R E D U C T I O N B Y 7"loo-BALANCED T R U N C A T I O N
P r o o f The controllability Gramian P to the Lyapunov equation /3(A
-
-
= pT
91
is the unique positive definite solution
f2YooCTC)T + (A - f2YooCTc)/3 + f 2 B B T "b f4YooCTcYoo = O.
It is easy to show, using the HFARE, t h a t / 3 =/~Yoo solves this equation. The observability Gramian 0 = 0 T is the unique positive definite solution to the Lyapunov equation Q(A - f2YooCrC) + (A - f2YooCTC)T 0 + c T c = {}.
Pre- and post-multiply by 0 -1 and substitute 0 -1 = X/o1 + f2Yoo. It is then straightforward to use the HFARE and the HCARE to show that the equation is indeed solved by this choice of 0. [] The Hankel singular values of [ f N M] are =
ordered 1 > O1 >_ ~2 >_ "'" -> ~,~ > 0. Lemma 8.4.11 immediately allows the ~ to be expressed in terms of the 7-/o~-characteristic vMues vi. P r o p o s i t i o n 8.4.12 Let BIG = ff/I-tflfi; be the normalized left-coprime faetorizafion given in Lemma 8.4.9. Let 1 > oa >_ 02 ~ " " >_ o,~ > 0 be the Hankel singular values of ~3fi[ 1FI]. Let 3" > vl >_ v2 >_ " " >_ vn > 0 be the ~oo-eharacteristic values of G, where 3' > max{i,%}. Then
P r o o f We firstly recall the definitions ~ = "\,{0/3} and t,? = a,{X**Y~}. Then, using Lemma 8.4.11 to express t3 and 0 in terms of Xo. and Y**, we have -2
= )~i{(X~ ~ + f ' Y o . ) - t f ' Y . o } f l 2 2 i { X , Yoo} 1 + 1+
[] R e m a r k 8.4.18 Suppose [fin /t~/]is in balanced coordinates. Then its controllability Gramian/5 and its observability Gramian 0 satisfy
P=O=
CHAPTER 8. 7"too-CHARACTERISTIC VALUES
92
where ~ = diag(el, ~'z,..., an) is the balanced Gramian. But then from Lemma 8.4.11,
y® = B-2~ and x~
= (~-1
-
~)-1,
wMch are both diagonal, with product xooYoo = f l - 2 ( ~ _ 1 _ ~ , ) - a ~
=
diag(fl-2~:(1_ ~)-,,fl-2,~(1 _ ~22)-1,...
,fl-2#~(1 _ a~)-l)
= diag(q,q,...,,,~) N 2,
using Lemma 8.4.12. Thus, G may be put into ~oo-balanced coordinates by applying the diagonal state similarity transform fl~12(I _
p2)-0/4),
(which was derived in a straightforward way by applying the method of [26, Section 4]). R e m a r k 8.4.14 Suppose G is in 7"too-balanced coordinates. Then
X~=Yoo=N where N = dinE(v1, v2,..., Vn). But then from Lemma 8.4.11,
P = flZN and
Q = (2V-1 + f122V')-l, w h i c h are b o t h diagonal, w i t h p r o d u c t
P 0 = fl2N(N-1 + fl2N)-I =diag(~2q(1
2 _ 1 ,~ 2~:(i 2 + p•2 v2j 2,_1 ,...,f12v2(1+fl%2)-1) + ~ 2vi)
= diag(O'~, 0"~,..., O'n 2)
using Lemma 8.4.12. Thus, [/3N ~I] may be put into balanced coordinates by applying the diagonal state similarity transform
3-~1/2)(z
+/32N2)-v/'),
(which was derived in a straightforward way by applying the method of [26, Section 4]).
8.4. MODEL REDUCTION B Y 7-I¢~-BALANCED TRUNCATION 8.4.4
Model Reduction
93
via the Coprime Factors
We are now in a position to carry out balanced truncation of the normalized leftcoprime factors [fin 217/]of the scaled plant ~G, to obtained a reduced-order plant. The analysis here and in the next subsection is reminiscent of [40, Chapter 5].
Procedure 8.4.15 (Reduced-order plant b y c o p r i m e factorization of 13G) Let G = ( A , B , C ) be minimal with n-states and let 7 > mac{I,%}. Let/3 = (1 - 7-2)1/L Let fiG = Jfl-l~N be the normalized lefl-coprime factorization of fig based on the HFARE--that is, let [fl~ 217/]= (.4, B , C , / ) ) be as given in Lemma 8.4.9. Let the state-space realization of [13N JVI] be in balanced coordinates with Hankel singular values 1 > ~1 > #z >_ "'" >_ 6~,~> O~that is, let the controllability Gramian P and the observability Gramian 0 of ~ff¢ if/l] satisfy P=O=
where ~. = diag(~'x,~2,...,6r,). accordingly into
Pick k < n such that #~ > #~+x and partition 0
~32
where ~ = diag(0"l,...,~'k) and ~ = a i a g ( e k + l , . . . , e , ) . conformably with the partitioning of ~:
D4ne
M,I :=
Partition A, B and 0
Then a k-state reduced-order
plant is 0, :=
Y/I~-XN, and [43] flO, = M~"flff¢, is a normalized lefl-coprime fuctorization. How does this reduced-order plant compase with the one obtained via ~ - b a l a r t c e d truncation in Procedure 8.4.17 The next result answers this question.
Theorem 8.4.16 The plant model reduction schemes described in Procedure 8.4.1 and Procedure 8.4.15 yield identical reduced-order plants. To be precise, let G~ be the k-state reduced-order plant obtained by performin 9 7"l~-balanced truncation (Procedure 8.4.1) on the full-order plant G for 7 > max{1,7o}. Let G, be the k-state reduced-order plant obtained by performing balanced truncation of the coprime factors [fiN Jfi] of fiG (Procedure 8.4.15) where fl = (1 - 7-2) 1/2. Then G, = 6',,.
Proof Suppose G = (A, B, C) is ~o~-balanced, and we form G, by ?~oo-balanced truncation. Thus, according to Procedure 8.4.1, we have the partitioning X°°=Y°~=N=[
N~O N~O]
CHAPTER 8. Hoo-CHARACTERISTIC VALUES
94
where/71 = diag(vl,..., v~) and N, = diag(v~+l,..., v,), together with
A =
An
A22
'
B2
The k-state reduced-order plant G, is then
G" = [ An lB* 0
(8.7)
"
Using Lemma 8.4.9, we can use N to construct the normalized left-coprime factorization f i g = _~r-lfN (with the partitioning corresponding to that of N made explicit):
An [ff¢ M] =
A21
-
f2N,CTC, An- f2NIcTc2 f"Z%CTCl C1
A.2 - f"IV2CTC. C2
fB1 -f2/%C1~ ], -f"N,.C[
lB.
O
(8.8)
I
with Gramians /5 = f2 N and (~ = (N -~ + f ~ N ) - L But from Remark 8.4.14, this may be put into balanced coordinates by a diagonal state-transformation. Denote this diagonal balancing transformation by L = diag(L,,L2), partitioned conformably with N -- diag(N1,/72). (An explicit expression for the balancing transformation L is given in Remark 8.4.14 but we shall not need it.) Applying this diagonal balancing transformation to (8.8) gives
~
~¢] = [ L1(An - f2NICTC1)L~ 1 L t ( A n - f2NICT, C2)L~* L2(A,, f~N2C~C1)L~ ~ L 2 ( A n - f"N2CTC~)L~ 1 C1L~ 1 C2L~ 1
fLIB~ fL2B2 0
-f2LIN1C~ -fl2L2N~C~ I
which is a balanced realization with (Remark 8.4.14) balanced Gramian ~. Applying Procedure 8.4.15, the k-state reduced-order plant G, has a normalized left-coprime factofization riO, = M71fN, where [fiN, 217/,]is obtained by balanced truncation of [fin if/]. Thus, truncating the above balanced realization to k-states,
C1Lf 1
0
1 C1
0
"
But this is precisely G, in (8.7). The following coronary is immediate on taking note of Remark 8.4.3.
[]
8.4. MODEL REDUCTION B Y ~oo-BALANCED TRUNCATION
95
Corollary 8.4.17 Let Kr be the k.sfate reduced-order controller obtained by performing 7-[oo-balanced truncation (Procedure 8.~.~) on KMEco, where It'ME~ is the full-order Normalized 7"[oo Controller for the full-order plant G. Let [f~ be the k-state Normalized 7-[oo Controller for the k-state reduced-order plant G, (Procedure 8.~.I or equivalently Procedure 8.4.15). Then K , = IC,.
8.4.5
Stability and Controller
Performance
with
the
Reduced-Order
We can now piece together results from the previous sections to consider the stability and performance of the closed-loop consisting of the reduced-order controller K, with the full-order plant G (as illustrated in Figure 8.1). It is an advantage of using ~f~obalanced truncation that the results may be expressed in terms of a priori quantities only (i.e., 7 and the neglected vs). This comes about because the model reduction error ~ defined by ~ := II[~t,~ A~]II®, (8.9)
where A s := f i r - fir,
A~ := K / - KL, may be bounded above using q and the neglected vl only. To be precise, the balanced truncation error bound of Lemma 8.4.4(iii) applied to (8.9) gives an upper bound on ~/~ in terms of the neglected Hankd singular values of [~N 217f]. Hence, from Proposition 8.4.12, ~ may be bounded in terms of the H.o-characteristic values of G: n
/3~ <_ 2 trace[]~2] = 2 y]~ Therefore,
/3vl
(8.1o)
~t
(811, and by inspection, a weaker but simpler bound is
_< 2 ~
~, = 2 trace{Nz].
(8.12)
i=k+l
This last upper bound is 'twice the sum of the tail,' just as in ordinary balanced truncation (Lemma 8.4.4(iii)). Using either of the above upper bounds in place of ~ in the following results gives a (conservative) statement of stability and performance using the reduced-order controller without having to calculate the reduced-order controller in advance.
96
CHAPTER 8. 7-I~-CHARACTERISTIC VALUES
G
Figure 8.1: Reduced-order controller with full-order plant P r o p o s i t i o n 8.4.18 Let Kr be the k-state reduced-order controller obtained by performing 7-[oo-balanced truncation (Procedure 8.,t.2) on the full-order Normalized 7"[¢o Controller for the full-order plant G with 7 > max{l, %}. Let ~G = J~I-lfl~[ be the normalized le~-coprime factorization of flU given in Lemma 8.~.9. Let flGr = l ~ - l flNr be the normalized left-coprime factorization (Procedure 8.~.15) of the k-state reducedorder plant Gr obtained by performing 7"[oo-balanced truncation (Procedure 8.,~.1). Let Kr = UrVr-1 be any right-coprime factorization of K~. Define R.r := ~ r r E - N~U~.
Let ~ be the model reduction error as defined in (8.9) above. Then the condition
ensures closed-loop stability of the system consisting of the reduced-order controller Kr and the full-order plant G. Proof The proof is basically an application of the Small Gain Theorem. Rather than repeat the argument here, we will for brevity quote a result [40, Corollary 3.7]
8.4. MODEL REDUCTION B Y 7-I~-BALANCED TRUNCATION
97
which, for our case, states that K, stabilizes G ff K~ stabilizes G, (which it does by construction) and
where S,, := ( I - G,K,) -1. The claim of the proposition follows easily on making the necessary substitutions to obtain s , , = V,R-2 M,
(8.~3)
and then K,S,,/~r,--* ] = [ U, ] R~-). [] Corollary 8.4.17 shows that K, is in fact the normalized 7-/oo controller for G,. The associated closed-loop transfer function is Hr, :=
K,S,,G,
K,S,r
where S,, := ( I - G,K,) -1. Define @ := IIH-II¢~
(8.14)
to be the actual 7-(~o-norm of H,r. Then
-~ < ~
(8.1s)
is immediate, and the following corollary to Proposition to 8.4.18 is obtained. Corollary 8.4.19 With definitions as above, we have that
so the condition
e(5 +'~) < 1 is sufficient to ensure close&loop stability of the system consisting of the reduce&order controller If, and the full-order plant G.
Proof Appendix A.6.
[]
CHAPTER 8. 7foo-CHARACTERISTIC VALUES
98
R e m a r k 8.4.20 An a priori test for the stability of the system consisting of K, with G is immediate. Just replace ~ with its upper bound 7 (equations (8.14) and (8.15)), and replace ~ with ~, where e is an upper bound on ~, such as that given in (8.11). It is simple to verify that
~(~+'r)
~
~(~+~r)
Hence to test if K, stabilizes G it suffices to check if e(~ + 7) < 1, a simple test which depends only on 7 and the truncated vl. Now assume that ~(~ + "~) < 1 so that, by Corollary 8.4.19, the dosed-loop consisting of the reduced-order controller K, and the full-order plant G is stable. From Remark 7.3.1 we know that the bound IlHM~ooll®< -y inherent in Normalized 7foo Problem gives robust stability guarantees, where
[ HMEoo =
SG SGKMEoo ] KMEooSG KMEooS '
is the closed-loop transfer function associated with G and its Normalized ~oo Controller KMgao, and where S = (I - GKMEoo) -1. The next proposition tells us how much the bound IIHMgoolloo < 7 is degraded (increased) by using the reduced-order controller If, in place of the full-order Normalized 7foo Controller KMgoo. For convenience, define the associated closed-loop H,:=
K,S,G
K,S,
where S, := (I - G K , ) - I ; that is, the dosed-loop transfer function when K~ is used in place of KMgoo. P r o p o s i t i o n 8.4.21 Definitions as above. Assume ~(~ + 5) < 1 so that the reduce# order controller If, stabilizes the full-order plant G. Then the associated closed-loop transfer function satisfies
IlH, II.o < ~ + ~(1 + ~)(1 + B + ~) (1 -- ~ ( / / + -~)) Proof
(8.17)
Appendix A.6.
[]
R e m a r k 8.4.22 An a priori upper bound on HH~lloo is immediate from (8.17). Just replace ~ with its upper bound ~/ (equations (8.14) and (8.15)), and replace ~ with ¢, where e is an upper bound on ~ such as that given in (8.11). Provided that ¢(/3+7) < 1, Remark 8.4.20 predicts that K, stabilizes G. Under this condition, it is simple to verify that the right-hand side of (8.17) can only increase under the transformation ~ --* ~, "Y--+ 3'. That is,
IIH, II,~
< ~ +
~(1 +
~/)(1 + / 3 + 7)
(1 - ~(~ + ~))
which depends only on 7 and the truncated vl.
'
(8.18)
8.4. M O D E L R E D U C T I O N B Y 7-[oo-BALANCED T R U N C A T I O N
99
R e m a r k 8.4.23 Note that as '7 increases, the a posteriori result of Proposition 8.4.21 remains viable, whereas the a priori result of Remark 8.4.22 becomes increasingly weak. In the limit as 7 ~ co (the LQG case), Proposition 8.4.21 is still viable, whereas Remark 8.4.22 becomes vacuous. R e m a r k 8.4.24 The upper bound (8.17) is tight, in that as the model reduction error ~ --~ 0, so the proposition recovers the definition "~ = IIH,,lloo. Similarly, the upper bound (8.18) is tight, in that as the model reduction error bound e --~ 0, so the full-order bound II M oolloo < "7 is recovered. 8.4.6
A
Numerical
Example
The results derived in the previous subsection are illustrated here using a numerical example. Consider the system
G=
A
B
=
-98/201 1
-98/201 -9999/200 1
I 1 ] 1 . 0
This system was constructed in [35, Section VIII to be in LQG-balanced coordinates with LQG-characteristic values #t = 2.0000 and/*2 = 0.0100. To verify this, one merely has to check that M = diag(#l,#2) is the stabilizing solution of the C A R E and F A R E . The system has a left-half plane zero at -24.1349 and poles at 0.7547 and -49.9997. Using q-iteration gives "7° = 2.3559. W e chose seven representative values of '7 > '7o at which to perform 7"/oo-balanccd truncation: '7 = 2.36, 3, 7, 10, 40, 100 and co. The results are tabulated in Figures 8.2 and 8.3: the 'predicted' results in Figure 8.2 and the 'actual' results in Figure 8.3. We know that when "7 -4 co, the 7"/oo-bMancing method recovers the LQG-balancing method. This is confirmed in Figures 8.2 and 8.3, where the last row (corresponding to 7 ~ co) of Columns (a), (b) and (i) agrees with the results of [35, Section VIII. For each of the chosen values of '7, the HCARE and HFARE are solved and the 7/00characteristic values are calculated using Proposition 8.3.1. This gives Columns (a) and (b) of Figure 8.2. For each choice of '7 we have vl >> v2, which prompts us to consider obtaining single-state reduced-order controllers by discarding v2 via 7~oo-balanced truncation (Procedure 8.4.2). But before calculating the reduced-order controllers, we can predict their success using the a priori data "7 and u2. Using (8.11), an upper bound on the model reduction error ~ may be calculated from the neglected v2, giving Column (c). As explained in Remark 8.4.20, this upper bound on ~ may then be used in place of ~ in Corollary 8.4.19 to predict stability of each reduced-order controller Kr with the full-order plant G, Column (d). Notice that stability of Kr with G is predicted for all but the two largest choices of 7, when nothing can be deduced. In the cases where stability of K, with G is predicted, the performance of that configuration can be bounded using Remark 8.4.22, as shown in Column (e).
100
CHAPTER 8. 7"Zoo-CHARACTERISTIC VALUES
2.3600 3.0000
0,9058
Ca)
(b)
(c) Upper bound on ~ from (8.11)
(d) Does Rem a r k 8.4.30 =¢* K v s t e b l l i s e s G7
2.3544
0.0100
0.0200
`yes
2.6667
0.0100
0,0200
Yes
3.4293
0.9428
2.2030
(e) Upper bound o n IIH~I[~ f r o m R e m a r k 8.4.22
7.0000
0.9897
2.0339
0.0100
0.0200
"Yes
8.7119
10.0000
0.9950
2.0107
0.0100
0.0200
Yes
13.3828 231.3253
40.0000
0.9997
2.0010
0.01O0
0.0200
Yen
100.0000
0.9999
2.0002
0.0100
0,0200
No
oo
1.OOO0
2.0000
O.OlO0
0.0200
No
Figure 8.2: 7-/oo-balanced truncation example: a priori numerical results
(r)
(g) Actual value of f r o m (8.9)
(h) Does Cot. 8.4.10 =~ K~ s t a b i l i z e s G7
(i) Does Kr stabilize GY
(k)
0)
Upper bound
on
IIH, II** from P r o p . 9.4.21
Actual value of
O) Actual value
2.3600
2.3600
0.0182
Yes
Yes
2.6373
2.4273
2,3600
3,0000
2.9076
0.0189
"Yes
Yes
3.2939
3.0249
2.9087
7.0000
4.1398
0.0198
"Yes
"Yes
4.8339
4.4268
4.1524
10.0000
4.3451
0.0199
"Yen
"Yes
5.0997
4.6688
4.3612
40,0000
4.5472
0.0200
Yes
Yes
5.3642
4.9092
4.5669
100.O000
4.5590
0.0200
Yes
Yes
5.3794
4.9233
4.5790
4.5612
0.0200
`yes
Yes
5.3823
4.9260
4.5812
co
of
IIR,II~
Figure 8.3: ~oo-balanced truncation example: exact and a posteriori numerical results
8.4. MODEL REDUCTION B Y 7"loo-BALANCED TRUNCATION
i01
Having predicted the satisfactory behaviour of the reduced-order controllers we can go ahead and calculate them using Procedure 8.4.2, from which Figure 8.3 is constructed. The model reduction error ~ may be calculated exactly from (8.9) using the algorithm in [10], to give Column (g). Stability of K~ with G may be tested for explicitly by calculating the closed-loop poles, to give Column (i). The actual 7-/oo-norm of the closed-loop of K~ with G is also calculated explicitly using the algorithm in [10], to give Column (k). Likewise, -~, the actual 7~o-norm of the closed-loop of K, with G, is calculated explicitly to give Column (f). For reference, in Column (1) we give the fuli-order results, that is, the actual 7foo-norm of the closed-loop ttMBoo consisting of the fuR-order controller KMl~oo with G. Using the a posteriori data ~/and ~, Corollary 8.4.19 may be used to check if stability of K~ with G is predicted, see Column (h). Proposition 8.4.21 then allows calculation of an upper bound on the 7-/oo-norm of the closed-loop of K, with G (Column (j)). Notice that it is predicted that K, stabilizes G for all the values of 7, including infinity. Also, the upper bound in Column (j) is tight (within 10% of the actual value in Column (k)), again even for large 7. Comparing the two tables of results it is clear that the 7"/o~-balanced truncation method can give a good reduced-order controller. Furthermore, when 7 is small (of the order of %), indicating a preference for robustness over LQG performance, the 7"/00balanced truncation method gives accurate a priori prediction of the performance of K~ with G. This is particularly clear on comparing the first three rows of the two tables. With the a posteriori data "~ and ~, even tighter results are possible; they indicate that a good reduced-order controller can be obtained even when 7 is large (in which case the a priori predictions may be weak, as indicated in Remark 8.4.23). This is particularly clear on comparing the final three rows of the two tables.
Chapter 9 LQG and 7%0 Monotonicity
CHAPTER 9. LQG AND 7/00 MONOTONICITY
104
9.1
Introduction
The aim of this chapter is to explore two properties of minimum entropy 7/00 controllers which we shall call LQG monotonicity and 7/00 monotonicity. Recall the Minimum Entropy 7-100Control Problem of Chapter 3.2.6; in particular, recall Section 3.6.1. There it was shown that the minimum entropy closed-loop HME00 satisfies (Proposition 3.6.1)
(i) IIHME II00 < "r. (a) C(HM~00) <_I(HMF~00;7; oo). Here C(HME00)is the LQG cost (Definition 2.4.1) of the minimum entropy closed-loop and I(HMEoo;'~;oo) is its entropy at infinity. Theorem 3.6.2 states that the upper bounds on the right-hand sides of (i) and (ii) above trade off: I(HMB00;7; oo) is a monotonically decreasing function of 3'. Computational evidence points towards a much stronger tradeoff: that the left-hand side of (i) is a monotonically increasing function of 7, and that the left-hand side of (ii) is a monotonically decreasing function of 3'. We state this formally as two conjectures. C o n j e c t u r e 9.1.1 ( L Q G m o n o t o n i c i t y ) Suppose KMF,00 solves the Minimum En-
tropy 7"[00 Control Problem at infinity (Problem 3.~.6) for a given standard plant P and 7 > 70. Then the minimum entropy closed-loop HME00 = br(P, KME00) exhibits LQG monotonicity, that is, the achieved LQG cost C(HME00) is a monotonically decreasing function of'y. C o n j e c t u r e 9.1.2 (7/00 m o n o t o n i c i t y ) S u p p o s e KME00 solves the Minimum En-
tropy 7/00 Control Problem at infinity (Problem 3.~.6) for a given standard plant P and 7 > 70" Then the minimum entropy closed-loop HME00 = ~'(P, KME00) e~hibits 7"/oo monotonicity, that is, the achieved 7~00-norm IIHME®II00 is a monotonically increasing function o f f . See Figures 7.3 and 7.4 for an example of LQG and 7/00 monotonicity. The proof, or otherwise, of the above conjectures in the general case is currently an open problem. In the remainder of this chapter we will focus on LQG monotonicity. Our approach will be to derive a formula for the derivative of the LQG cost with respect to 7. Proof of LQG monotonicity then reduces to proving that this derivative is non-posltive.
9.2
The LQG
Cost and its Derivative
The first step in deriving the derivative of the LQG cost with respect to 7 is to use the method of Section 4.2 to reduce the Minimum Entropy 7/00 Control Problem to the equivalent Minimum Entropy 7/00 Distance Problem (Problem 4.2.1), in both cases with entropy evaluated at infinity. Using the terminology of Chapter 3, let P be a given standard plant and let 3' > %; let KME00 be the controller which solves the associated
105
9.2. THE LQG COST AND ITS DERIVATIVE
Minimum Entropy 7-/~ Control Problem at infinity and let HMEzo = .~(I9, KMEzo) be the corresponding minimum entropy closed-loop. Then using the terminology of Chapter 4, let EME~ be the minimum entropy error system for the Minimum Entropy 7/oo Distance Problem at infinity associated with the above Minimum Entropy ~/~ Control Problem. The error system EME¢~ has the structure EME¢~ =
[Rll R,2 ] R21 R22 + OME~ '
where
m~
R21 R2~
'
R* E R'/-~¢o, ^
is determined by the standard plant P only, whilst Q M ~ ~ Rl"l,,~ and minimizes the entropy over the class of error systems satisfying the £:~-norm bound []EII~ < 7- Then we have
I(HME~;% c~) = I(EME~;7; ~ ) ,
These relations hold because (Section 4.2) ~'(P, KMEoo) = [fEMEooV for suitable transfer function matrices U and V satisfying U*U = I and VV* = I: the £~-norm, entropy, LQG cost and £2-norm are all unitarily invariant. We assume that R(oo) = 0--this is sufficient for a finite value for the minimum entropy and LQG cost, as noted in Section 4.5. We shall need the following state-space formula for EME~ from Theorem 4.5.2. Suppose
R =
pl-.~, I
C1
0
0
=:
is minimal and R* E 9 Z ~ i.e., - A is asymptotically stable. Then the minimum entropy error system EME~ is EME~ =
[Rll R12 ] R21 R22 "4- (~ME¢~
^
w h e r e Q M E o o E '~'~'~oo h a s a r e a l i z a t i o n
106
C H A P T E R 9. LQG A N D 7too M O N O T O N I C I T Y
and where X = X r < 0 solves XA r + AX + 7-zXCTCaX + BB T = 0
(9.2)
such that -
(A + 7 - ~ x c T c 1 )
is asymptotically stable,
(9.3)
where Y = y r < 0 solves YA + ATy + 7-2YB1BTy + cTc = 0
(9.4)
such that - (A + 7 - 2 B 1 B T y )
is asymptotically stable,
(9.5)
and where Z := 7 - 2 X Y - I.
Let QMEOO have controllability Gramian P and observability Gramian 6. Then (since QM~oo 6 7~7"/~o) P = pT >_ 0 is the unique solution of the Lyapunov equation P A T + A P +/~/~T = 0,
(9.6)
and Q = QT ___0 is the unique solution of the Lyapunov equation
QA + AT~ + ~ T 0 = 0.
(9.7)
Similarly, let R ( - s ) have controllability Gramian - X and observability Gramian - y . Then (since R ( - s ) 6 7ZTfoo and R is minimal) X = X T < 0 is the unique solution of the Lyapunov equation X A T + A X + B B T = O, and y = y~" < 0 is the unique solution of the Lyapunov equation Y A + A T y + c T c = O.
Note carefully that X and y are independent of 7 but P and Q are not. An expression for IIEMEOOlI~ (= C(H~EOO) the LQG cost) may now be stated. L e m m a 9.2.1 Consider the Minimum Entropy 7-[oo Distance Problem at infinity as described above. Let EM~oo be the minimum entropy error system. Then IIEMEOOlI] = -- trace[CXC r] + trace[CT$cT].
P r o o f Since R" 6 7~Tfoo and OMEoo 6 •71oo with R(oo) = 0 and QMEoo(c~) = 0, we have R* 6 7gTfz and QMEOO6 7ZTf2. Hence R and QMEOOare orthogonal in the Hilbert space 7~£~. It follows that
IIEM~+tl] -- IIRII] + IIOM~II~ = IIR'II] + IK)ME+II~ -'- IIR(-,)II~ + IlOMEooll~
= trace[O(-,-V)CT] + trace[OPOr],
9.2. THE LQG COST AND ITS DERIVATIVE
107
where the last llne is obtained using the standard evaluation of the ?/z-norm of a system in terms of its controllability Gramia~. [] The next result allows us to calculate the derivative of [IEMB¢~II] with respect to 7, without needing to differentiate ~b or 6. T h e o r e m 9.2.2 Definitions and assumptions as above. Then
(1) A
^
^
(ii) Define ff := 27-aX - dX/d7 and Y := 27-1Y - dY/dT. Then
dfi_ -7-2Z-r:YZ-XB2BT + 7-~cTcIff + 7-4YZ-~-~YZ-XB2Br2 , d7 d_~_B = 7-'Z-r~'Z-'B2 + 7-2YZ-XB2 _ 7-nyz-I~yZ-aB2, d7 d~ = .y_2C2X 1- dX
d--4
-7- c'2 .
P r o o f Part (i) We follow the method used in [61, Appendix 10.1] in a different context. Using Lemma 9.2.1 we have
d ( - trace[CXCr] + ~7 (HEM~=I]~) = "~7
trace{O$,0 .l)
A
^
(9.8) on substituting for O r C from (9.7). Next, differentiate (9.6), postmultiply by Q and take the trace to get A
.
Substitute this into (9.8) and use the easily verified fact that d
dr
]
CHAPTER 9. LQG AND 7"loo MONOTONICITY
108
The result follows on collecting terms. Part (ii) Appendix A.7.
[]
The following corollary provides the desired expression for the derivative of [[EMEoo[[ (which equals the derivative of the LQG cost of the minimum entropy 7-/¢~ control system).
Corollary 0.2.3
~
(IIEMEooH~) = 27 -2 trace[Z-T~Z-1B2BT(yz -1 _ 7~)~] _ 27 -4 trace[Z-Yy~-~Tyz-~B2Br(yz-1 - 75)Q] + 27 -2 trace[CTc, X ~ ]
+ 27 -2 t r a c e [ 7 5 x c T c ~ ] .
P r o o f Immediate on substitution of Part (ii) of Theorem 9.2.2 into Part (i): collect terms and use trace[75Cr0] = trace[/3BTQ] which follows from (9.6) and (9.7). []
9.3
On LQG Monotonicity
LQG monotonicity is equivalent to d(IIEME~II~)/d 7 <_ O. In order to deduce anything about the sign of the expression for d([IEMEoo[l~)/d7 in Corollary 9.2.3 above, we need to know more about the sign definiteness of its constituent matrices. We certainly know that X < 0 and Y < 0; that 75 > 0 and Q > 0; that ~ { Z } < O. The next lemma deals with the remaining matrices. L e m m a 9.3.1 Let X, Y, Z, 75 and Q be as defined in the previous section. Then (i) dX/dT >_O and dY/d 7>_0.
and ~ <_0. (iii) Y Z -1 - 75 > O. (ii).~_<0
Proof
Part (i) Differentiate equation (9.2) with respect to 7 (valid by [50]) to get JX .T Ax .--~-~-+27-3xcTc1x=o, dX --~TAx+
(9.9)
where
Ax := - ( A + y - 2 x c T c 1 ) is asymptotically stable from (9.3). Applying a standard result on Lyapunov equations [26, Theorem 3.3(7)] to (9.9) gives dX/d7 > 9, as claimed. A similar method based on differentiating (9.4) and using (9.5) leads to dY/d7 ~ O.
9.3. ON LQG M O N O T O N I C I T Y
109
Part (ii) Immediate from Part (i), the definition of )~ and ~" in Theorem 9.2.2 and the fact that X _< 0 and Y _< 0. Part (iii) It is first claimed that
yZ-t fiT + f~yz-X + ~ T + z-Tc[c,z-I + CZxCI =
O.
(9.10)
This is easily shown using (9.2), (9.4) and the expressions for i and /~ from (9.1). Subtract (9.6) from (9.10): (YZ-' -
+
i(YZ -I - 9 ) + z-Te
c2z-' +
e , = o.
This is a Lyapunov equation, and A is asymptotically stable. Hence, using [26, Theorem 3.3(7)] again, it follows that ( Y Z -1 - ~) ~_ O. [] If M
=
M T
> 0 and N = N T > 0 are arbitrary nonnegative definite symmetric
matrices then trace[MN] = trace[M1/2NM 1/2] >_ 0 follows. However, this result does not extend to three or more arbitrary nonnegative definite symmetric matrices, as the following example shows. Let
-10
1
'
1 1
0 100
"
Then L > 0, M > 0 and N > 0 but trace[LMN] = -708 ~ 0. Thus it is not clear how to deduce nonpositivity of d(liEM~ooi[~)/d 7 in Corollary 9.2.3 from knowledge of the definiteness of the constituent matrices alone. (See [6] for more on positivity of products of positive definite matrices.) We can only say with certainty that LQG monotonicity holds whenever R has only one stat~---for then all the terms in Corollary 9.2.3 are scalar and the nonpositivity is obvious by inspection.
Appendix A P r o o f of R e s u l t s N e e d e d in t h e Text
A P P E N D I X A. P R O O F OF RESULTS NEEDED IN THE T E X T
112
A.1
Outline
Proofs of various results are stated here, rather than in the main body of the text, to preserve continuity of exposition.
A.2
A Lemma
The following technical lemma will be needed in various places--for example, in the proof of Propositions 2.3.1 and 2.3.2 and in the proof of Theorems 2.4.4 and 3.4.2. L e m m a A . 2 . 1 Let M be a real square matriz, let N be a real or complez matriz, and let e G IR. Then (i)
- In d e t ( I - eM) = e trace[M] + O(e2).
(it) - In d e t ( I - c~N'N) >_ ~2 trace[N*N]. Proof
Part (i)
Use the Faddeev formula (see [25, Vol. 1, p88]) to obtain d e t ( l - eM) = 1 - ,
traceiM] + O(~ 2)
and expand the logarithm of this as a power series. Part (it) Using the well-known inequality - l n ( 1 - z 2) > z 2 (for Ix] < 1), we have - In d e t ( I - e ' N ' N ) = - E ln(1 - e'Ai{N*N}) i
> Z"a~{N'N) i
= c2 t r a c e [ N ' N ] , claimed.
A.3
[]
Proof of Theorem
2.4.4
We shall derive a relationship using entropy at So G (0, co), then take the limit as so ~ oo to obtain the result. Using Lemma A.2.1(i) we can write I ( / / ; 7 ; S o ) = ~1
'::
= 2--~
tra~e[~'(j~)/t(j~)l ~,
trace
{[ >, H(jw
So So + jw
d~, + O('y -~)
>1
+o,,-.,.
A.4. STATE-SPACE E V A L U A T I O N OF THE E N T R O P Y I N T E G R A L
113
Therefore, I ( H ; 7 ; S o ) = ][8--~801H(s)l]i + 0 ( 7 - ' ) "
(A.1)
The integrands in (A.1) are monotonically increasing with so, and are continuous. Henc% by dominated convergence, both sides of (A.1) tend to a limit as so -~ oo. Each side is finite because H(co) = 0 by assumption (see Proposition 2.3.1(iii) and (iv) and Remark 2.4.3). So, taking so --~ co mad noting that the 0 ( 7 -2) term may be bounded independently of So,
(A.2)
.r(R;.-r; oo) = It/Zll~, + o("r-'),
which proves Part (ii). Part (i) of the theorem follows from this on noting (using L e m m a A.2.1(ii))that the O(~/-2) terms are non-negative. Finally,Part (iii)is obtained by taking 7 "-* oo in (A.2). []
A.4
State-Space Evaluation of the Entropy Integral
The following lemmas are useful in establishing state-space formulae for entropy integrals. In particular, Lemma A.4.2 is used in the derivation of the minimum value of the entropy in Theorem 3.5.1. Before stating and proving Lemma A.4.2, it is convenient to state and prove a more restricted result, which we will also need in Section 5.3 and in the proof of the entropy formula in Theorem 4.5.2.
det D = 1 and G ~1 ~_ ~ o .
{ l j:
o0li~moo -2-4
Then
ln ldet
a*(jw)O(j~o)l
&o
}= -
trace[D-'OB].
P r o o f Using the fact that In [det G'(jw)G(j~)I = 21n[det G(jw)[ we can rewrite the integral as
I, := U m By assumption, G 4-I E ~ , Theorem 17.16]:
{ :: _I
l-ldetC0")I
~o
.
which permits the use of Poisson's Integral Theorem [52,
I1 = -- lim {soln I det G(so)l} = - $0lim {soln ] det(D 4- C(soI - i)-~/~1]} --~oo = - $0lim {soln [det b I + soln I det(I + D-~O(SoI -/i)-~/~)1}. =-~00
114
A P P E N D I X A. P R O O F OF RESULTS NEEDED IN THE T E X T
The first term in this last expression is zero because det/7) = 1 by assumption. A power series expansion of the second term in terms of s~ 1 gives Ix = - $0lim {soln I det(I + D-XO(s0I - A)-~/3)I} --40~ = - 80l~m {so h [det(I + soXD-xOB + --400
o(s~))l}
_- - l~m { trace[D-~OB] + O(sox)} $0 --¢00
= - trace[D-rOB} as required, having used Lemma A.2.1(i) to obtain the third equality. [] By relaxing the assumption that G -1 E 3Z7-/o0 we obtain the following lemma, which is the one we need in the proof of the entropy formula of Theorem 3.5.1. A similar result for scalar systems has been derived independently in a different context in [1]. L e m m a A.4.2 Suppose G = ]l". 4~1] ~ [/~
is a square transfer function matriz with
det b = 1 and G E 7~7"loo, but with no assumptions on the stability of G -1. Then ~m
$~-*00
{1-~ f_®hldetG'(j~)G(j~)l
Lso+ ~"J
= - trace[Z~-'0~]
-- 2 ~
~,{~
~) -- B b - l O } .
Re{Xi}>o Proof
It is convenient to define A
F:=
B
ICeD B ]
• _.!. G - 1
-o- o I?-:1 .
(A.3)
Clearly,
{ j:o $o " + ~
=
llm #o --~
-2-~
[ .:
so ln ] det G*(jw)G(j~)[ ~
{ 2-~f/-oolnldetF'(jwlF(j~')t[sg+w,J
and F -1 E ~ o o , but F itself is not necessarily in RT-/,o. Performing an appropriate state transformation on F allows us to write without loss of generality that F =
[a0 ] A21 A2 Ct C2
B2 D
A.4. STATE-SPACE EVALUATION OF THE E N T R O P Y INTEGRAL
115
where A1 is not asymptotically stable and A2 is asymptotically stable. Note that (A1, B1) is completely controllable because/I is asymptotically stable. Let W1 be the unique positive definite, symmetric, stabilizing solution to
0 = ATwI + W1A1 - WxBxBTW1 and define 0
0] 0
(A.4)
"
Then, because W1 is the stabilizing solution of (A.4), A - BBTW is easily seen to be asymptotically stable. In addition, it can be verified that
Brw
U :=
I
is all-pass, and that
F := FU =
A0 A -BB~W 1 BB ] BBTW C -DBTW
=
C - DBTW
D D
and removing uncontrollable states. Now 1~ E gT~o by construction, and also
F-l=[
A - BD-1C ] _ D - , C + B T w I BDD~'
is in 7~740~ because G E ~-.T/oo by assumption. Then, exploiting the all-pass nature of U, det F*F = det FF* = det FUU*F* = det 1~I~* --= det $'°F, and hence o0-~¢o ~ = lim
~
laldet F*(jw)F(jw)] lnldet P'(jw)[~(jw)l[
= trace[D-~(C - DBTW)B] = trace[D-1CB]- trace[BrWB], where the third equality is by Lemma A.4.1.
A P P E N D I X A. P R O O F OF R E S U L T S N E E D E D I N T H E T E X T
116
The second term may be further simplified in the following way:
trace[BTWB] = trace[BTW1B,] -_ trace[WiB1B T] = trace[A T + W ~ A , W [ ~] =
trace[A~ + At]
= 2 ~-~'.A,{Ax } i R¢{AI}>O
where the third equality is from (A.4)W1-x. Hence, using the definition of A, B, C and D in (A.3), it follows that
]2 = - trace[D-lOB] -- 2 ~
~,{a -- n D - l O }
Re{~tl}>0
as claimed.
A.5
Proof
[]
of Lemma
8.4.9
To show that f i g = ff4"-lflN is a normalized left-coprime factorization of f i g we need to show that:
(i) ~ r ~ ~z~® and K/~ g~o=.
(ii) M -1 exists. (iii) ~ r and 2f/are le£t-coprime. (iv) ~zNN* + M M " -- I.
(v) ~G
Proof because Part Part Part
=
M-~N.
Part (i) is immediate on noticing that A - fl2YooCrC is asymptotically stable Yoo is the stabilizing solution of the HFARE. (ii) is true because Jf/(oo) = I, which is nonsingular. (iii) follows from [45]. (iv) is verified by routine state-space calculation, as follows.
A.5.
PROOF OF LEMMA
117
8.4.9
[--(A--~2Y°°CTC)T] - c T ] BB r
x
A -
0
-/3~CYo~ I ~2Yc~CrC 132(BBr + fl2YooCrCYoo) --fl2YooCr -( A - ~2YooCrC)r -C r 0 C -fl2CYoo I
Apply a state transformation
] to get 0I -/32Y°° I
A -/3"Yo~CYC 0
~2/~r/~r* + 2~ 2~* =
o]
(,) - ( A - ~2YooCT c ) T
-C T
0
I
C
,
(A.5)
where
(*) = fl2(BBT -/32Y¢oCTCYoo + AY~o + YooAT)
= O,
from the HFARE. Hence all the states of (A.5) axe either unobservable or uncontrollable; these may all be deleted to leave/322~rN * + 217/~r" = I. P a r t (v) Put v = 13Nut + (2~'I - I)u2, say. If u2 = - v then v = / ~ N u , + (I - K/)v
==~
,~7/v =/3fi[u,
==~
v = ,~/-~j3fi/ul.
(A.6)
This gives a simple means of forming the state-space realization o f / ] 7 / - 1 / ~ in terms of the realization of LBN JlT/]. To do this, begin with the state-space equations for [#N ( M - I)] which are: =
(A
-
~YooCrC)x + ~Bu, - ~YooCru~,
V ~ 6f3%
Defining u2 = - v as above, gives = (A - ~2YooCrC)z + flBu~ + ~Yo, C T c x = Ax + flBul,
Hence,
v = C(sI - A)-l~Bul = ~Gul.
The result follows on comparing this with (A.fi).
[]
APPENDIX A. PROOF OF RESULTS NEEDED IN THE TEXT
118
A.6
P r o o f o f Corollary tion 8.4.21
8.4.19
and
Proposi-
Proof of Corollary 8.4.19 We only have to prove that (8.16) holds: the corollary follows immediately from that and Proposition 8.4.18. Now, K, is the Normalized 7~oo Controller for the plant G,: the closed loop transfer function is
H,,=[
S,,G, S,,¢,K,] K,S,,G, K,S,,
where S,, = (I - G , K , ) - ' , and we have IIH-II~ = ~ < 7. w e need to rewrite the closed-loop transfer function in terms of the coprime factorizations G, = 21~r--1~, and K,. = U,.Vj 1. We have
R,, = M,V, - ~r,U, by defitfition and
s,, = V,R;;M, from (8.13). Straightforward manipulations then give
~'" =
K,S,,G,
K,S,,
-
0 0
[ V~RT~r, V~RT~M, L
V,
Hence
0 I
,]-[o
o]
0 I
[~]R:E~ ~I=[ °o o ']+ H , u,
and
O
[ flV~
(A.7)
01H,,)
(A.S)
Similarly,
K
=
0 i
H,~
0
(A.9)
and also
/gv,
o o ~. ]~:c~ ~J= [~: ,]([o I
,
] (A.10)
A.6. PROOF OF COROLLARY8.4.19 AND PROPOSITION 8.4.21
119
Take the 7-/oo-norm of (A.7)-(A.10) and use the triangle inequality and the submultiplicative property of the ~ o - n o r m . Also use that [[H,,l[~o = ~ by definition and that 0 < 3 _
oo-and
(A.12) Also, v~
I[~ 1~ I~ ~ ~L~I[
_< 1 + -~,
(A.13)
and finally
1~ E~ ~ ~L~ I1[~ The first equality in (A.13) and in (A.14) arises because ~ N , ~7/,] is normalized. Equation (A.14) is (8.16) as claimed. The remaining equations (A.11)-(A.13) will be needed later. [3 P r o o f of P r o p o s i t i o n 8.4.21 We shall stick to the notation and definitions used in the above proof. Introduce / ~ := MY, - N U .
then
Noting that
an application of the weU-known matrix inversion lemma [32, p19] gives
(A.16) where
120
APPENDIX A. PROOF OF RESULTS NEEDED IN THE T E X T
Substitute (A.16) into the expression for H, in (A.15) and use the identity IN ~r] = [iV, + a ~ M, + A ~ l to obt~n H,=H.
+
+
+
u,
o ,ei
[ K ]RT)[A~ V,
-~-'A. ]
V, x T-'
0 I I 0
~V~ U,
R-I[
fl-'AM]
Taking ~/oo-norms, and using the triangle inequality and sub-multiplicative property of the ?'/oo-noml, leads to
x II[
A~ ~ - , A ~
] IloollT-~ll®
V,
+ I[ ,,. ] "~.'I1oI[ ~ To complete the derivation, use (A.11)-(A.14), and recall that IIH,,11oo- ~/by definition and that I][ AR ~-tAM ]I10==~ (A.17) by definition. Also, using the inequality [32, p301]
I[(I- A)-'II~ ~ (, - [IA[[~)-~
for IIAII~ < 1,
we get
<(1 [~v, ]R2 oo11[~x. ~-,z~, ]llo.) < (1 - (~ + -~)~)-',
(A.18)
A.7. PROOF OF T H E O R E M 9.2.2(II)
121
where the last inequality follows from (A.14) and (A.17). (Validity of the use of (A.18) and the existence of the necessary inverses is assured because (/3 + ~/)~ < 1 by assumption.) Collecting all of the above together, gives
Ilmll~ < ~' + (1 + ~)~ + (1 + ~)(fl + ~)(1 - (~ + ~)~)-'~ + (1 + ~,)(fl + ~)(1 - (fl + ~)~)_,~2 = - ~ + ~(1 + ~)(1 + fl + -~)
(1
-
~(~ + ~))
'
as claimed.
A.7
D
Proof
of Theorem
9.2.2(ii)
Differentiate Z = 7 - 2 X Y - I to obtain
dZ
-27-1(1 + Z) +
2dX-
2xdY
(Existence of the derivatives is assured by [501.) By differentiating (Z-t)Z = I we get _
d
1)= _ z - l d Z z 1 -
~-~(Z-l) ---~2~f-lZ-I ..it_~.~-Iz-2_~f-2z-ld_~_.Xyz-1 _,.~-2Z-Ixd_~Y_Yz-1 a7
a7
which will be of use later. The expressions for the derivatives of A, b and C are obtained by straightforward manipulations of the state-space fornmlae after using the product rule for differentiation, and substituting for d(Z-*)/d7 when necessary. Firstly,
dO ~7 1 dX d-~ = (-'r-lC~X) = "r-2OxX - 7- C2 dT" Then
db d7
~
(7-1YZ-1B~)
= -7-2yZ-1B2 +'7-1~TZ-1B2 +7-1Y(27-IZ-l+27-1Z-2-7-2Z-l~yz-l_7-2Z-1X~-TZ-')B = 7-2yZ-1B2 _ 7 - 1 ( 7 - 2 y z - I x _ I ) ? Z - 1 B 2
2
122
APPENDIX
+ 27-2YZ-ZB~ -
A. PROOF OF RESULTS NEEDED IN THE TEXT
"~-3yz-I
~TYZ-' B,
-_ 7-2yZ-IB2 _ 7-'Z-r~T Z-XB2 + 27-2z-TYZ-XB~ _ 7-ayz -I ~yZ-XB2 = 7-'Z-T~'Z-'B~ +-~-2YZ-IB~ _ 7-3yz-I~TyZ-aB ~. Finally, dA
./
~ ( - A v - 7-1hBT _
7-2cTcIx)
1 d/} -r + 2._SCrlC, X _
_2C,TC ~~x
= 7-3yZ-aB2B~_7-ayZ-1BzB~_7-2Z-rFZ-1BzB~
+ 7-4YZ-x~YZ-XB~BT+ .,/-2cTc,f( = _7-2Z-TFZ-'B~BT +7-2CTC,f( +7-4YZ-' ~TYZ-' B2BT2, which completes the proof.
[]
Appendix B E n t r o p y Formulae: A l t e r n a t i v e Derivation
124
APPENDIX B. E N T R O P Y FORMULAE: A L T E R N A T I V E DERIVATION
B.1
Introduction
Our purpose here is to provide an alternative derivation of the entropy formulae of Section 3.5, in the spirit of the recent work [17, 28] and its extensions in [29]. Done this way, the derivation is less manipulative and more intuitive. Proof of Theorem 3.5.1 is the main aim; proof of Lemma 3.5.5 is also included. Assumptions and notation is inherited from Chapter 3 and Section 3.5. In particular, P is a standard plant satisfying the Assumptions 3.2.1 and 3.2.7; also Dal = 0. In keeping with [17, 28, 29] we assume throughout this appendix that 7 = 1. This results in no loss of generality since it is achieved by the scalings 7-1/~B1, 7-1/2C1, 71/2B2, 71/2C2, and 7-XK. We will therefore talk of (P, 1)-admissible controllers and closedloops rather than (P,7)-admissible controllers and closed-loops. Note that there are some notational differences between this monograph and [17, 28, 29].
B.2
T h e Full I n f o r m a t i o n and O u t p u t E s t i m a t i o n Problems
In this section we are concerned with the Full Information and Output Estimation problems. These problems are used in [17, 29] as stepping stones to the parametrization of all (P, 1)-admissible controllers. The first lemma deals with the Full Information problem, where the controller has direct access to both the state and the disturbance input. L e m m a B.2.1 (Full I n f o r m a t i o n [29, T h e o r e m 3.1]) Consider the 'Full Inforrnation' standard plant A B1 B2 0 DI~ P F I ~-
satisfying Assumptions 3.~.1 and 3.~.7 as appropriate. Let Da_ make [Dx2 D±] square and orthogonal. Then (i) There ezists a ( PFt, 1)-admissible controller if and only if there ezists X¢o = X r > 0 solving the Full Information algebraic Riecati equation 0 = X , ( A - B2DT2C:) + (A - B2D~2C:)Tx¢o + Cr:D±DTCI + X , ( B : B T - B 2 B ~ ) X . such that A - B2DT2C1 + (BaB T - B2BT)Xo~
is asymptotically stable.
B.2. THE FULL INFORMATION AND OUTPUT ESTIMATION PROBLEMS 125
(ii) All ( Pez , 1)-admissible closed-loops Hrt are generated by HFI = ~'(R, ~)
where • 6 B7~7"l¢~,
where
A.~ C1F
R= [ R,1 R12] R2x R n
:=
-B x®
B1 B2 0 D12
x o
with
AF := A - B2(0 2Cl + B[X ) CaF : = Ci - Dn(DT2C~ + B[X®). (iii) The transfer function matriz R given in (ii} above has the following properties: R E 7~7-l¢,,, R*R = I and R ~ E 7~7"loo. In the general output feedback problem, if D~, = I and A - B1C2 is asymptotically stable then the state and the disturbance input can be reconstructed exactly by using an observer. Hence the achievable closed-loops would be identical to those of the Full Information problem. Taking the transpose of the above result leads to the following coroUary for the Output Estimation problem. Corollary B.2.2 ( O u t p u t E s t i m a t i o n [29]) Consider standard plant PoE=
C, Cz
0
the
'Output
Estimation'
I
D2, 0
satisfying Assumptions 3.2.1 and 3.~.7 as appropriate, together with A - B2Ca asymptotically stable. Let D± make [Dr2x br~]r square and orthogonal. Then
(i) There ezists a (POE, 1)-admissible controller if and only if there ezists Yo. = y T >__0 solving the Output Estimation algebraic Riccati equation 0
=
Y~(A -
B I D 2 1TC 2 )
r
+
( A - BxDTC2)Y~
+ B~br, b±Br~ + Yoo(7-2CrtC, - C[C2)Yoo such that r A - B1Dr2xCz + }~(7
-2
CaT Ct - c r c 2 )
is asymptotically stable.
(ii) All (POE, 1)-admissible closed-loops Hos ave generated by HOE = ~r( S, ~)
where q~ 6 BTZTtoo,
APPENDIX B. ENTROPY FORMULAE: ALTERNATIVE DERIVATION
126
where
Sit
St2 ]
S = $21 $22 :=
[ Az Cx
BtLD210- Y°°CrlIo ]
C2
with AL : = A - (B,D~I + Yo~C[)C,
B~r. := B, - (B,D~,, + Y ® C T ) D , .
(iii)
B.3
The transfer function matrix S given in (ii) above has the following properties: S E 7 ~ o , SS* = I and S~ x ~ 7~7"[oo. Separation
Structure
Consider the problem of characterizing all (P, 1)-admissible controllers K. A key step in the solution of this problem as given by [17, 29] is to separate the problem into a Full Information problem and an Output Estimation problem. This approach will be useful for our purposes, so it will be outlined here. Define a new standard plant 15, in terms of the given standard plant P, as follows (as in equation (4.7) of [29])
A 5 =
01 02
B1 [~2 ] [ A+B1BT1X~' 0 D12 := DT~C1+ BTXo~ D21 0 C2 + D21BTX~
B1 B2 0 I
(B.1)
D:I 0
Then [29, Lemma 4.4] says that a controller K is (P, 1)-admissible if and only if it is (/5,1)-admissible. So we can shift our attention from P onto 15; moreover,/5 satisfies the assumptions for the Output Estimation problem. Furthermore, the feedback connection of K with P gives the same closed-loop transfer function as that given by the feedback connection of K with R with P, where R is as defined in Lemma B.2.1 for the Full Information problem. So ~'(e, K) = ~'(R, 2-'(/5, K)). (B.2) But now we can apply Corollary B.2.2 to 15: all (15, 1)-adnfissible closed-loops are generated by ~'(P, K) = 9v($, ¢) where ¢ e B7~7-/oo, (B.3) with an obvious notation. Combining (B.2) and (B.3), it follows that all (P, 1)-admissible closed-loops H are generated by H = .r(n, •($, ¢)) where¢ e Bnn~. (B.4)
B.4. PROOF OF THEOREM 3.5.1
B.4
Proof
of Theorem
127
3.5.1
The following lemma shows how the separation structure outlined in the previous section manifests itself in the entropy formulae. L e m m a B.4.1 Let R be as defined in Lemma B.2.1 for the Full Information problem associated with the given standard plant P. Let P be as defined in (B.1). Let S be as defined in Corollary B.L2 for the Output Estimation problem based on P. Let HMEo~ be the minimum entropy ( P, 1)-admissible closed-loop. Then I(HMsoo; 1; co) = I(Rll; 1; co) + I($11; 1; co). P r o o f From (B.4), all (P, 1)-admissible closed-loops H are given by
H=
where ¢ e
Setting ~ = OMEoo = 0 (Theorem 3.4.2) gives
HMEoo = Since R*R
= I,
~'(R, Sll).
it is easy to verify that
I - H~,f~ooHMF.o. = R ; , ( I - R~5',1)-*(I - S;xS, I)(I - R,~Six)-iR~I. Substitution into the definition of I(HMEoo; 1; CO) gives I(tIMEoo; 1; co) = I(R1,; 1; co) + I(.~,,; 1; co) + ao~, where a~o := ,olimoo{aoo} and as 0 := ~g-oo In I det(I - R~z(j~)Sn(j~))}
I
Now observe that R2~ and ~11 are strictly proper, so soRz2(so)Sll(so) ~ 0 as s0 ~ co. Also note that (I - Rzz,911):L1 E 7Z:Hoo. An application of Lemma A.4.1 then shows that in fact aoo = 0. [] The above h m m a expresses the total entropy as the sum of the 'Full Information entropy' I(Rxx; 1; co) and the 'Output Estimation entropy' I(5'xl; 1; oo). The former is easily evaluated using Lemma A.4.1 since I - R~lRll = R~1R21 and R~) E T~/oo. We find I(Rll; 1; oo) = trace[BTXooB1]. Similarly, the latter entropy is I($11; 1; co) = trace[Oll~ooOT].
128
APPENDIX B. E N T R O P Y FORMULAE: A L T E R N A T I V E DERIVATION
Here Yoo solves the Output Estimation algebraic Riccati equation associated with P; this is shown in the proof of Theorem 4.1 of [29] to be
f'® = zo~Y®.
(B.5)
To complete the proof, one only has to recall that C'x = D~zC1 + B2zX~o• Lemma B.4.1 together with the above expressions then gives
Z(HMso~;1;oo)~-- trace[B~X.B,] + trace[(DT,Cx + Br~X.)Z®Y~(Dr.C, + B~X.)~}, which is the second entropy formula of Theorem 3.5.1. The other formula given in Theorem 3.5.1 follows by duality. []
B.5
Proof
of Lemma
3.5.5
Complete derivations of the characterization of all (P, 1)-admissible controllers, as stated without proof in [28], have now appeared in [29] using techniques established in [17] (see also [30]). These papers give the characterization of all (P, 1)-admissible closed-loops but do not give the state-space formulae for the all-pass dilation that is required in Lemma 3.5.5. We fill in here the steps required for that calculation. As in the previous sections, we exploit the separation structure as in [17, 29] with the assumption that Dtx = 0. There is no difficulty in principle with the DI1 # 0; the expressions just become more involved. Recall the discussion of the separation structure given in Section B.3. We saw in equation (B.4) that all (P, 1)-admissible closed-hops are generated by
H =~=(R,Y(S,~))
where* ~ ~nno~.
To obtain the all-pass dilation needed in Lemma 3.5.5 we will first dilate R into an aU-pass transfer function matrix P~, then we will dilate ~' into an all-pass transfer matrix ~ . Augmenting 6 with zero rows and cohmns as appropriate, after combining R~ and S, into a single feedback dement, will give the desired result. Dilating R into an all-pass transfer function matrix R~ gives =
AF
B1
elF
0 D12 D± I 0 0
B2 B3
The observability Gramian of R. is X¢o (see the proof of Lemma 3.4 of [29]). It follows from [26, Theorem 5.1] that R~ is all-pass if
I
0
0
- B ~ X + +[BI B2 B3lrX+--O,
B.5. PROOF OF LEMMA 3.5.5
129
which is satisfied by
B~ = -X£C~D~.. Before dilating S, apply a state transformation of -ZooI to P so that
P=
[ Z~oX(A+BtB~X-)Z-(Dr, Ca + B [ X , ) Z . -(C2 + D,iBr~X = ) Z .
-Z~oaB1-ZgolB2 ] 0 I D2a
0
Then S is defined in Corollary B.2.2 based on the above P:
2=
01
02
o
I
•
D21 0
Before writing down expressions for IL, B1L and /32L we note that the solution to the Output Estimation algebraic Riccati equation associated with the above 15 is Y.Z;o r = Z~lYoo. (This comes from (B.5) after taking the -Z~o1 state transformation into account). Now we can apply Corollary B.2.2 to P to get
Az = Z~oa(A+ B, BTX=)Z,. - (-ZZ.1B, D~, + Y~Z~aO[)02 = .;i- JB,02- Z~olB2¢,- ((-I + Y.X.)BaDTa - Y.(C [ + X . B , DT,))O, = ~ - ZyaB20a, after substituting from the second expression for/~ and Zo:~TOT (from the statement of Lemma 3.5.5). Also,
= --Z~IBI + [31D21 and
~,L = - Y . Z : I O ~ = Y+(Cr, D12 + X . B : ) =B,-Z::B2. Further, the all-pass dilation of S is just
AL
£~L B2L
^
Ct
S° =
02 Os
0
I
D2a 0 b± 0
130
APPENDIX B. E N T R O P Y FORMULAE: ALTERNATIVE DERIVATION
The feedback combination of P~ with ,-qa will then give an all-pass system as follows
{ ~ = AFz + BlW + B2v + Baw3 z = CIFZ + Dlzv + D±wa r = -B[Xooz + w
v = 01~ + w 2
z2 = C2~ + D21r z3 = C3~ + b . r
Eliminating v and r gives
ie Z
z2
A + BzO1Z;o1 =
B201
B1
-B~LBr~X~ ~ - Z;2B~O~ CIF D1201 -D21B(X~,
C2
~3
X
w w2
0
w3
bz Finally a state transformation of
[,0] Z~ 1 I
B~
B1L B2L 0 0 D~2 D . 0 D~I 0 0
is easily shown to give the realization
for J~ given in the statement of the lemma. The equivalence of the two expressions for A given in the lemma is obtained by substitution from the two Riccati equations. []
Appendix C Notation
132
C.1
A P P E N D I X C. N O T A T I O N
Basic Notational Conventions
All systems are linear, multivariable, finite-dimensional and time-invariant, and possess reM-rational transfer function matrices. In general, we use capital letters to represent matrices and lower case letters to represent vectors. We shall not distinguish between a time domain signal and its Laplace transform. For example, w represents both the time domain signal w(t) and the Laplace domain signal w(s). The context will determine which domain is used. We work exclusively in continuous time. A (proper) transfer function matrix is represented in terms of state-space data by
or by (A, B, C, D), where A, B, C and D are real matrices of appropriate dimension and I is the identity matrix. If D = 0, the zero matrix, then the system is strictly proper and we shall write (A, B, C). The matrix A is asymptotically stable if and only if each of its eigenvalues has a strictly negative real part. In that case the system (A, B, C, D) is also called asymptotically stable (or causalin Chapter 4). A matrix A is called antistable if - A is asymptotically stable. In that case, in Chapter 4, the system (A, B, C, D) is called anticausal. As far as possible standard notation has been used. For convenience some of the global conventions are listed below. Notational conventions defined and used locally in the text in a single section are not listed here. Firstly the spaces of interest--see [24, Chapter 2] for more details.
7Z/;2
Lebesgue space of real-rational transfer function matrices squareintegrable on the imaginary axis.
R~2
Hardy space of real-rational transfer function matrices squareintegrable on the imaginary axis with analytic continuation into the right-half plane.
7Z/;~o Lebesgue space of real-rational transfer function matrices bounded on the imaginary axis. 7£~oo Hardy space of real-rational transfer function matrices bounded on the imaginary axis with analytic continuation into the right-half plane. Next some miscellaneous notation and the definition of the norms on the above spaces.
C.I. BASIC NOTATIONAL CONVENTIONS
¥ 3 X:=Y X=:Y m Re{(. )) 7¢ hi{M} p(M) MT M* Mt
M>0 M>0 M~/2
a*(,) 0",{a(s)) Ila(s)ll,
IIa(s)ll~
133
For each, for all. There exists. X is defined to equal Y. Y is defined to equal X. The real numbers. Real part of ( • ). (Prefix) Real-rational. The ith eigenvalue of a square matrix M. The spectral radius of a square matrix M. The transpose of a matrix M. The complex conjugate transpose of a matrix M. The Moore-Penrose generalized inverse of a matrix M. M = M* is positive definite. M = M* is positive semidefinite. For a matrix M > 0, any square matrix L such that M = L*L. := G ( - s ) r, the parahermitian conjugate of G(s). := A~/2{G*(s)G(s)}, the ith singular value of G(,); by convention ordered O"1 ~___0"2 ~_~ " ' " ~_~ 0"n ~_~ 0. X/2 := {(l/2~r)$y~o trace[G*(jw)G(jw)lcho} ,the £ , - a n d 74z-norm. := sup,~ o-x{G(j~)},the £®- and 7"/®-norm. Open unit ball in "~7"/0o i.e., an ,I, ~ "~7/oo such that II~l[oo < 1.
A square but not necessarily stable transfer function matrix G(s) is said to be all-pass if G*(s)G(s) = 1 for all s (equivalently, if G(s)G*(s) = I for all s). Finally some of the system-theoretic quantities.
")MEJo (.)o (.)~, (.)L~. Y(P,K) c( fir) CT( Y~) J(H;7)
/(fir; 7; so)
I(n;7;oo) nT(O) JT("?) E{(. )) Var{(. )) max{. }
Denotes minimum entropy at So. Denotes 7~oo-optimal. Denotes £2-optimal. Denotes LQG-optimal. Lower linear fractional map of P and K (Equation (3.1)). LQG cost of the system H (Definition 2.4.1). Finite-time LQG cost of H (Remark 2.4.2). Auxiliary LQG cost of fir (Definition 5.2.1). Entropy at 80 of fir (Definition 2.2.1). Entropy at infinity of H (Definition 2.2.2). Exponential-of-quadratic cost (Equation (6.2)). Exponential-of-quadratic cost (Equation (6.3)). Expected value of ( • ). Variance of ( - ) . Largest element of the set { • }.
134
C.2 CARE FARE HCARE HFARE LEQG LHS
LQG RHS
APPENDIX C. NOTATION
Acronyms Control Algebraic Riccati Equation (Equation (7.3)). Filter Algebraic Riccati Equation (Equation (7.4)). ~ o Control Algebraic Riccati Equation (Equation (7.9)). ~/~ FilterAlgebraic Riccati Equation (Equation (7.10)). Linear Exponential-of-Quadratic Gaussian. Left-hand side. Linear Quadratic Gaussian Right-hand side.
Bibliography [1] B. D. O. Anderson and D. L. Mingori. Use of frequency dependence in Linear Quadratic control problems to frequency-shape robustness. Journal of Guidance Control and Dynamics, 8(3):397-401, 1985. [2] B. D. O. Anderson and J. B. Moore. Linear Optimal Control. Prentice-Hall, 1971. [3] D.Z. Arov and M. G. Krein. Problem of search of the minimum entropy in indeterminate extension problems. Functional Analysis and its Applications, 15:123-126, 1981. [4] D.Z. Arov and M. G. Krein. On the evaluation of entropy functionals and their minima in generalized extension problems. Acts Scienta Mathematica, 45:33-50, 1983. (In Russian). [5] J. A. Ball and N. Cohen. Sensitivity minimization in ~¢,-norm. Parametrization of all suboptimal solutions. International Journal of Control, 46(3):785-816, 1987. [6] C. S. Ballantine. Products of positive definite matrices. III. Journal of Algebra, 10:174-182, 1968. [7] A. Bensoussan and J. W. van Schuppen. Optimal control of partially observable stochastic systems with an exponential-of-integral performance index. SIAM Journal on Control and Optimization, 23(4):599-613, 1985. [8] D. S. Bernstein and W. M. Haddad. LQG control with an 7"/00performance bound: a Riceati equation approach. In Proceedings of the American Control Conference, Atlanta, GA, 1988. [9] D. S. Bernstein and W. M. Haddad. LQG control with an 7-/oo performance bound: a Riccati equation approach. IEEE Transactions on Automatic Control, 34(3):293305, 1989. [10] S. Boyd, V. Balakrishnan, and P. Kabamba. On computing the ~ - n o r m of a transfer matrix. In Proceedings of the American Control Conference, Atlanta, CA, 1988.
136
BIBLIOGRAPHY
[11] J. P. Burg. Mazimum Entropy Spectral Analysis. PhD thesis, Stanford University, 1975. [12] C. C. Chu, J. C. Doyle, and E. B. Lee. The general distance problem in 7"/00 optimal control theory. International Journal of Control, 44(2):565-596, 1986. [13] M. H. A. Davis. Linear Estimation and Stochastic Control. Chapman and Hall, 1977. [14] D. F. Delchamps. A note on the analyticity of the Riccati metric. In C. I. Byrnes and C. F. Martin, editors, Lecture Notes in Applied Mathematics, pages 37-42. American Mathematical Society, 1980. [15] J. C. Doyle. Guaranteed margins for LQG regulators. IEEE Transactions on Automatic Control, 23(4):756-757, 1978. [16] J. C. Doyle and C. C. Chu. Robust Control of Multivariable and Large Scale Systems. Final Technical tteport, Honeywell Systems and Research Center, 1986. [17] J. C. Doyle, K. Clover, P. P. Khargonekar, mad B. A. Francis. State-space solutions to standard 7/2 and 7-/oo control problems. IEEE Transactions on Automatic Control, 34(8):831-847, 1989. [18] J. C. Doyle and G. Stein. Multivariable feedback design: concepts for a clasSical/modern synthesis. IEEE Transactions on Automatic Control, 26(1):4-16, 1981. [19] J. C. Doyle, J. E. Wall, and G. Stein. Performance and robustness analysis for structured uncertainty. In Proceedings of the Conference on Decision and Control, Orlando, FL, 1982. [20] H. Dym. J-Contractive Matriz Functions, Reproducing Kernel Hilbert Spaces and Interpolation, volume 71 of Regional Conference Series in Mathematics. American Mathematical Society, 1989. [21] H. Dym and I. Gohberg. A maximum entropy principle for contractive interpolants. Journal of Functional Analysis, 65:83-125, 1986. [22] H. Dym and I. Gohberg. A new class of contractive interpolants and maximum entropy principles. In I. Gohberg, editor, Topics in Operator Theory and Interpolation. Operator Theory: Advances and Applications, volume 29, pages 117-150. Birkhiiuser Verlag, 1988. [23] D. F. Enns. Model Reduction for Control System Design. PhD thesis, Stanford University, 1984. [24] B. A. Francis. A Course in Tloo Control Theory, volume 88 of Lecture Notes in Control and Information Sciences. Springer-Verlag, 1987.
BIBLIOGRAPHY
137
[25] F. R. Gantmacher. The Theory of Matrices. Chelsea Publishing Co., 1959. [26] K. Clover. All optimal Hankel-norm approximations of linear multivariable systems and their £oo-error bounds. International Journal of Control, 39(6):11151193, 1984. [27] K. Clover. Minimum entropy and risk-sensitive control: the continuous time case. In Proceedings of the Conference on Decision and Control, Tampa, F1, 1989. [28] K. Clover and J. C. Doyle. State-space formulae for all stabilizing controllers that satisfy an 7/oo-norm bound and relations to risk sensitivity. Systems and Control Letters, 11:167-172, 1988. [29] K. Clover and J. C. Doyle. A state-space approach to 7"/0o optimal control. In H. Nijmeijer and J. M. Schumacher, editors, Three Decades of Mathematical System Theory: A Collection of Surveys at the Occasion of the 50th Birthday of Jan C. WilIems, volume 135 of Lecture Notes in Control and Information Sciences. Springer-Verlag, 1989. [30] K. Glover, D. J. N. Limebeer, J. C. Doyle, E. M. Kasenally, and M. G. Safonov. A characterization of all solutions to the four block general distance problem. To appear in SIAM Journal on Control and Optimization, 19907 [31] I. Gohberg, P. Lancaster, and L. Rodman. On Hermitian solutions of the symmetric algebraic Riccati equation. SIAM Journal on Control and Optimization, 24(6):1323-1334, 1986. [32] R. A. Horn and C. R. Johnson. Matriz Analysis. Cambridge University Press, 1985. [33] D. C. Hyland and D. S. Bernstein. The optimal projection equations for fixed-order dynamic compensation. IEEE Transactions on Automatic Control, 29(11):10341037, 1984. [34] D. H. Jacobson. Optimal stochastic linear systems with exponential criteria and their relation to deterministic differential games. IEEE Transactions on Automatic Control, 18(2):124-131, 1973. [35] E. A. Jonckheere and L. M. Silverman. A new set of invariants for linear systems-application to reduced order compensator design. IEEE Transactions on Automatic Control, 28(10):953-964, 1983. [36] T. Kailath. Linear Systems. Prentice-Hall, 1980. [37] V. Ku~era. Discrete Linear Control: The Polynomial Equation Approach. Wiley, 1979.
138
BIBLIOGRAPHY
[38] H. Kwaltemaak and R. Sivan. Linear Optimal Control Systems. Wiley, 1972. [39] D. J. N. Limebeer and Y. S. Hung. An analysis of the pole-zero cancellations in ~o,-optimal control problems of the first kind. SIAM Journal on Control and Optimization, 25(6):1457-1493, 1987. [40] D. C. McFarlane and K. Glover. Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, volume 138 of Lecture Notes in Control and Information Sciences. Springer-Verlag, 1990. [41] D. G. Meyer. Model Reduction via Fractional Representation. PhD thesis, Stanford University, 1987. [42] D. G. Meyer and G. F. Franklin. A connection between Linear Quadratic Regulator theory and normalized coprime factorizations. IEEE Transactions on Automatic Control, 32(3):227-228, 1987. [43] D.G. Meyer. A fractional approach to model reduction. In Proceedings of the American Control Conference, Atlanta, GA, 1988. [44] B. C. Moore. Principal component analysis in linear systems: controllability, observability and model reduction. IEEE Transactions on Automatic Control,
26(1):17-32, 1981. [45] C. N. Nett, C. A. Jacobson, and M. J. Bolas. A connection between state-space and doubly coprime fractional representations. IEEE Transactions on Automatic Control, 29(9):831-832, 1984. [46] R. J. Ober and D. C. McFarlane. BManced canonical forms for minimal systems: a normalized coprime factor approach. Linear Algebra and Its Applications, 122124:23-64, 1989. [47] Ph. Opdenacker and E. A. Jonckheere. LQG balancing and reduced LQG compensation of symmetric passive systems. International Journal of Control, 41(1):73109, 1985. [48] A. Ostrowski and H. Schneider. Some theorems on the inertia of general matrices. Journal of Mathematical Analysis and Applications, 4:72-84, 1962. [49] L. Pernebo and L. M. Silverman. Model reduction via balanced state-space representations. IEEE Transactions on Automatic Control, 27(2):382-387, 1982. [50] A. C. Ran and L. Rodman. On parameter dependence of solutions of algebraic Riccati equations. Mathematics of Control, Signals, and Systems, 1(3):269-284~ 1988. [51] R. M. Redheffer. On a certain linear fractional transformation. Journal of Mathematics and Physics, 39:269-286, 1960.
BIBLIOGRAPHY
139
[52] W. Rudin. Real and Complez Analysis. Me Craw-Hill, third edition, 1986. [53] M. G. Safonov, A. J. Laub, and G. L. ttartman. Feedback properties of multivariable systems: the role and use of the return difference matrix. IEEE Transactions on Automatic Control, 26(1):47-65, 1981. [54] C. E. Shannon and W. Weaver. The Mathematical Theory of Communication. University of Illinois Press, 1949. [55] G. Stein and M. Athans. The LQG/LTR procedure for multivariable feedback control design. IEEE Transactions on Automatic Control, 32(2):105-113, 1987. [56] M. Vidyasagar. Control Systems Synthesis: A Factorization Approach. MIT Press, 1985. [57] P. Whittle. Risk-sensitive Linear/Quadratic/Ganssian control. Advances in Applied Probability~ 13:764--777, 1981. [58] P. Whittle. Entropy-minimizing and risk-sensitive control rules. Systems and Control Letters, 13:1-7, 1989. [59] P. Whittle and J. Kuhn. A Hamiltonian formulation of rlsk-sensitive Linear/Quadratic/Gaussian control. Internation Journal of Control, 43:1-12, 1986. [60] J. C. WiUems. Least-squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on Automatic Control, 16(6):621-634, 1971. [61] D. A. Wilson. Optimum solution of model-reduction problem. Proceedings of the IEE, 117(6):1161-1165, 1970. [62] H. K. Wimmer. Monotonicity of maximal solutions of algebraic Riccati equations. Systems and Control Letters~ 5:317-319, 1985. [63] W. M. Wonham. Linear Multivariable Control." A Geometric Approach. SpringerVerlag, 1979. [64] A. D. C. Youla, H. A. Jabr~ and J. J. Bonglorno. Modern Wiener-Hopf design of optimal controllers. Part II--the multivariable case. IEEE Transactions on Automatic Control, 21(3):319-338, 1976. [65] N. Young. An Introduction to Hilbert Space. Cambridge University Press, 1988. [66] G. Zames. On the input-output stability of time-varying nonlinear feedback systems, Part I. IEEE Transactions on Automatic Control, 11(2):228-238, 1966. [67] G. Zames. Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Transactions on Automatic Control, 26(2):301-320, 1981.
Index Acronyms, 134 admissible, see (P, 7)-admissible all-pass, 19, 20, 38, 128, 130 definition of, 133 anticausal, 37, 43, 132 antistable, 132 assumptions, 17, 18, 51 asymptotic stability, 71, 87-89, 132 auxiliary LQG cost and entropy, 52, 57 definition of, 51 minimum value of, 55, 57
Background reading, 5 balanced realization 7/oo-, 83, 92 LQG-, 81 open-loop, 80, 87, 88, 92 balanced truncation 7/oo-, 86 LQG-, 81 open-loop, 80, 87, 88 Band Extension Problem, 38 Causal, 132 central solution, 25, 44 characteristic values 7/oo-, 83 LQG-, 81 Combined 7/oo/LQG Control Problem, 49-58 assumptions, 51 definition of, 51 solution of, 54, 55 conjectures, 104 contraction, 19, 39
controller uncertainty, 68 convergence of entropy to LQG cost, 12, 32, 72 of 7/oo-characteristic values, 84 coprime factorization minimality of, 90 Normalized 7/oo Control and, 90-95 Normalized LQG Control and, 80, 89 Definition list, 133 derivative of 7/oo-charactefistic values, 84 of LQG cost, 107, 108 of solutions to Riccati equations, 85, 108 of state-space matrices, 107, 121, 122 detectable, 17, 66 dilation, 19, 128 dissipative system, 86 distance problem, see Minimum Entropy 7/~o Distance Problem Disturbance Feedforward Problem, 26 duality, 26 Entropy~ 7-14 and spectral factorization, 53 as a bound on LQG cost, 12, 31, 71, 104 at infinity (definition), 8 at So (definition), 8, 9 convergence to LQG cost, 12, 31, 72, 112, 113 evaluation of, 53, 113
142 minimum value of, 22, 26, 41, 45, 55, 70, 127 monotonicity of, 10, 31, 71, 72, 104 properties of, 9 separation structure and, 127 when finite, 22, 42, 46, 51 equivalence between auxiliary LQG cost and entropy, 52, 57 between LEQG cost and entropy, 61 error bound, 88, 89, 95 closed-loop, 98, 99, 121 error system definition of, 37 minimum entropy, 40, 44, 105 parametrization of, 39, 43 exponential-of-quadratic cost, 60, 61 F a d d e e v formula, 112 fixed-order control, 80 Frequency Domain Inequality, 51 Full Control Problem, 26 Full Information Problem, 26, 124, 127 if-iteration, 70 gap, 57 General Distance Problem, see Minimum Entropy 9/oo Distance Problem Gramian, 46, 51, 80, 87, 90, 106, 128 7~oo-btdanced t r u n c a t i o n of controller, 86 of plant, 86 7/oo-characteristic values, 79-101 and coprime factorization, 90-95 and Hankel singular values, 88 and LQG-characteristic values, 84 definition of, 83 properties of, 84 7-/oo/LQG tradeoff, 31, 47, 72, 104 ~/oo-monotonicity, 72, 104 ~~oo-norm and disturbance rejection, 13
INDEX
and Frequency Domain Inequality, 51 and robustness, 13 and spectral factorization, 53 as an induced norm, 13 bound, 12, 31, 71, 104 definition of, 133 numerical calculation of, 72, 101 ~oo-Optimal Control Problem, 17, 707 71 Hankel norm, 89 Hankel singular value, 80, 89, 91 definition of, 87, 88 Hardy space, 132 Hilbert space, 106 I n f o r m a t i o n theory, 8 input/output invariants, 807 81, 83 internal stability, 13, 17, 66 J - u n i t a r y , 38 Julia matrix, 21, 41 Laplace transform, 132 Lebesgue space, 132 LEQG cost, 60, 61 linear fractional map, 16 loop-shaping, 85 LQG-balanced truncation and coprime factorlzation, 89 of controller, 82 of plant, 82 LQG-characteristic values, 80-83 definition of, 81 LQG Control Problem definition of, 18 solution of, 32, 33 LQG cost and 7"~2-norm, 12, 50, 106 finite-time, 12, 60 minimum value of, 33, 67 steady-state~ 11, 66 when finite, 12 LQG monotonicity, 72, 103-109
INDEX
Lyapunov equation, 51, 57, 71, 106 Matrix inversion lemma, 119 minimal, 68, 70 minimMity of coprime factors, 90 Minimum Entropy 7/oo Control Problem, 15-33 assumptions, 17, 18 definition of, 18 solution at infinity, 23 solution at s0, 20 state-space solution 25-30, 54 Minimum Entropy 7/oo Distance Problem, 35-47 definition of, 37 solution at infinity, 44~ 45 solution at so, 40 state-space solution, 42-45 Model-Matching Problem, 36 ruonotonicity of entropy, 10, 31, 71, 72, 104 of 7/oo-characteristic values, 84 of 7/oo norm, 72, 104 of LQG cost, 72, 103-109 of solution to a Riccati equation, 32, 46 Nehari interpolation, 22 Normalized 7/oo Control Problem, 65-7? definition of, 68 solution of, 70 Normalized LQG Control Problem definition of, 66 sohtion of, 67 norms, definition of 7/oo / £oo , 133 7"/2/£2,133 Ha~akel, 89 notation, 131-134 numerical examples 7/oo-characteristic values, 99-101 Normalized 7ioo control, 72-77 One-block problem, 19
143 optimality, 2, 47, 70 Output Estimation Problem, 26, 125, 127 (P,7)-admissible closed-loop, and Full Information Problem, 125 and separation structure, 126, 128-130 central, 27 definition of, 17 parametrization of, 19, 27 controller, central, 27 definition of, 17 parametfization of, 19 parametrization of closed-loops, 19, 27, 126 of error systems, 39, 43 of (P, 7)-admissible controllers, 19 using 7/oo-characteristic values, 85, 86 using LQG-characteristic values, 86 Parseval's Theorem, 12 passive system, 85 Poisson's Integral Theorem, 21, 40, 42, 113 Problem Band Extension, 38 Combined 7/OO/LQG Control, 49-58 Disturbance Feedforward, 26 Full Control, 26 Full Information, 26, 124 7/oo-Optima] Control, 17 LQG Control, 18, 32, 33 Minimum Entropy 7/oo Control, 1533 Minimum Entropy 7/oo Distance, 35-47 Model-Matching, 46 Normalized 7/oo Control, 65-77 Normalized LQG Control 66, 67
144 Output Estimation, 26, 125 Risk-Sensitive LQG Control, 59-63 proper, 132 Real-rational, 132 recovery of £2-optimal solution,46 of LQG solution, 32, 33, 46 reduced-order Normalized ~ Control, 86 closed-loop error bound for, 98 error bound for, 95 stability criteria for, 96-98 via reduced-order copfime factorization, 93 reduced-order Normalized LQG Control, 82 via reduced-order coprime factorization, 80, 89 reduced-order plant via balanced truncation, 80, 87, 88 via balanced truncation of coprlme factors, 82, 89, 93 via 7~oo-balanced truncation, 86 via LQG-balanced truncation, 82 Riccati equation and Frequency Domain Inequality, 51 and spectral factorization, 53 antistabilizing solution of, 56 CARE, 67, 81 FARE, 67, 81 for X, 43, 106 for X¢o, 25, 124 for X2, 32 for Y, 43, 106 for Yoo, 26, 125 for Y2, 33 gap, 57 HCARE, 70, 83 HFARE, 70, 83 stabilizing solution of, 26, 54
INDEX
robustness o£ 7f¢o controllers, 2, 13, 17, 31 of LQG controUer~, 2, 32 of Normalized T/co Controller, 68, 69, 75, 98
Sensitivity, 69 separation structure, 126-128 singular values definition of, 133 Hankd (definition), 87, 88 Small Gain Theorem, 13, 68, 96 spectral analysis, 8 spectral factorization, 53 stabilizable, 17, 66 standard plant for Combined 7~¢o/LQG Problem, 5O definition of, 16 for Full Information Problem, 124 for Normalized Problems, 66 .,for Output Estimation Problem, 125 strictly proper, 132 sub-multiplicative, 119, 120 suboptimal, 2 system uncertainty, 68 Tradeoff, 31, 47, 72, 105 triangle inequality, 119, 120 Uniqueness of 7~oo-balanced realizations, 84 of LQG-balanced realizations, 81 of minimum entropy error system, 40, 44 of minimum entropy ~foo controller, 20, 23 unitary, 21, 41 unitary invariance, 36, 105 unit ball in 7~7/~, 41, 133