Ph robust-and-optimal-control-kemin-zhou-john-c-doyle-keith-glover-603s[1]

ROBUST AND OPTIMAL
CONTROL
KEMIN ZHOU
with
JOHN C. DOYLE and KEITH GLOVER
PRENTICE HALL, Englewood Cli s, New Jersey 07632

Contents
Preface xiii
Notation and Symbols xvi
List of Acronyms xx
1 Introduction 1
1.1 Historical Perspective : : : : : : : : : : : : : : : : : : : : : : : : : : 1
1.2 How to Use This Book : : : : : : : : : : : : : : : : : : : : : : : : : 4
1.3 Highlights of The Book : : : : : : : : : : : : : : : : : : : : : : : : : 6
2 Linear Algebra 17
2.1 Linear Subspaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17
2.2 Eigenvalues and Eigenvectors : : : : : : : : : : : : : : : : : : : : : 20
2.3 Matrix Inversion Formulas : : : : : : : : : : : : : : : : : : : : : : : 22
2.4 Matrix Calculus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24
2.5 Kronecker Product and Kronecker Sum : : : : : : : : : : : : : : : : 25
2.6 Invariant Subspaces : : : : : : : : : : : : : : : : : : : : : : : : : : : 26
2.7 Vector Norms and Matrix Norms : : : : : : : : : : : : : : : : : : : 28
2.8 Singular Value Decomposition : : : : : : : : : : : : : : : : : : : : : 32
2.9 Generalized Inverses : : : : : : : : : : : : : : : : : : : : : : : : : : 36
2.10 Semide nite Matrices : : : : : : : : : : : : : : : : : : : : : : : : : : 36
2.11 Matrix Dilation Problems* : : : : : : : : : : : : : : : : : : : : : : : 38
2.12 Notes and References : : : : : : : : : : : : : : : : : : : : : : : : : : 44
3 Linear Dynamical Systems 45
3.1 Descriptions of Linear Dynamical Systems : : : : : : : : : : : : : : 45
vii

viii CONTENTS
3.2 Controllability and Observability : : : : : : : : : : : : : : : : : : : 47
3.3 Kalman Canonical Decomposition : : : : : : : : : : : : : : : : : : : 53
3.4 Pole Placement and Canonical Forms : : : : : : : : : : : : : : : : : 58
3.5 Observers and Observer-Based Controllers : : : : : : : : : : : : : : 63
3.6 Operations on Systems : : : : : : : : : : : : : : : : : : : : : : : : : 65
3.7 State Space Realizations for Transfer Matrices : : : : : : : : : : : : 68
3.8 Lyapunov Equations : : : : : : : : : : : : : : : : : : : : : : : : : : 71
3.9 Balanced Realizations : : : : : : : : : : : : : : : : : : : : : : : : : 72
3.10 Hidden Modes and Pole-Zero Cancelation : : : : : : : : : : : : : : 78
3.11 Multivariable System Poles and Zeros : : : : : : : : : : : : : : : : : 80
4 Performance Speci cations 91
4.1 Normed Spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 91
4.2 Hilbert Spaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93
4.3 Hardy Spaces H2 and H1 : : : : : : : : : : : : : : : : : : : : : : : 97
4.4 Power and Spectral Signals : : : : : : : : : : : : : : : : : : : : : : 102
4.5 Induced System Gains : : : : : : : : : : : : : : : : : : : : : : : : : 104
4.6 Computing L2 and H2 Norms : : : : : : : : : : : : : : : : : : : : : 112
4.7 Computing L1 and H1 Norms : : : : : : : : : : : : : : : : : : : : 114
5 Stability and Performance of Feedback Systems 117
5.1 Feedback Structure : : : : : : : : : : : : : : : : : : : : : : : : : : : 117
5.2 Well-Posedness of Feedback Loop : : : : : : : : : : : : : : : : : : : 119
5.3 Internal Stability : : : : : : : : : : : : : : : : : : : : : : : : : : : : 121
5.4 Coprime Factorization over RH1 : : : : : : : : : : : : : : : : : : : 126
5.5 Feedback Properties : : : : : : : : : : : : : : : : : : : : : : : : : : 130
5.6 The Concept of Loop Shaping : : : : : : : : : : : : : : : : : : : : : 134
5.7 Weighted H2 and H1 Performance : : : : : : : : : : : : : : : : : : 137
6 Performance Limitations 143
6.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 143
6.2 Integral Relations : : : : : : : : : : : : : : : : : : : : : : : : : : : : 145
6.3 Design Limitations and Sensitivity Bounds : : : : : : : : : : : : : : 149
6.4 Bode's Gain and Phase Relation : : : : : : : : : : : : : : : : : : : 151

CONTENTS ix
7 Model Reduction by Balanced Truncation 153
7.1 Model Reduction by Balanced Truncation : : : : : : : : : : : : : : 154
7.2 Frequency-Weighted Balanced Model Reduction : : : : : : : : : : : 160
7.3 Relative and Multiplicative Model Reductions : : : : : : : : : : : : 161
8 Hankel Norm Approximation 169
8.1 Hankel Operator : : : : : : : : : : : : : : : : : : : : : : : : : : : : 170
8.2 All-pass Dilations : : : : : : : : : : : : : : : : : : : : : : : : : : : : 175
8.3 Optimal Hankel Norm Approximation : : : : : : : : : : : : : : : : 184
8.4 L1 Bounds for Hankel Norm Approximation : : : : : : : : : : : : 188
8.5 Bounds for Balanced Truncation : : : : : : : : : : : : : : : : : : : 192
8.6 Toeplitz Operators : : : : : : : : : : : : : : : : : : : : : : : : : : : 194
8.7 Hankel and Toeplitz Operators on the Disk* : : : : : : : : : : : : : 195
8.8 Nehari's Theorem* : : : : : : : : : : : : : : : : : : : : : : : : : : : 200
9 Model Uncertainty and Robustness 207
9.1 Model Uncertainty : : : : : : : : : : : : : : : : : : : : : : : : : : : 207
9.2 Small Gain Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : 211
9.3 Stability under Stable Unstructured Uncertainties : : : : : : : : : : 214
9.4 Unstructured Robust Performance : : : : : : : : : : : : : : : : : : 222
9.5 Gain Margin and Phase Margin : : : : : : : : : : : : : : : : : : : : 229
9.6 De ciency of Classical Control for MIMO Systems : : : : : : : : : 232
10 Linear Fractional Transformation 239
10.1 Linear Fractional Transformations : : : : : : : : : : : : : : : : : : : 239
10.2 Examples of LFTs : : : : : : : : : : : : : : : : : : : : : : : : : : : 246
10.3 Basic Principle : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 256
10.4 Redhe er Star-Products : : : : : : : : : : : : : : : : : : : : : : : : 257
11 Structured Singular Value 261
11.1 General Framework for System Robustness : : : : : : : : : : : : : : 262
11.2 Structured Singular Value : : : : : : : : : : : : : : : : : : : : : : : 266
11.3 Structured Robust Stability and Performance : : : : : : : : : : : : 276
11.4 Overview on Synthesis : : : : : : : : : : : : : : : : : : : : : : : : 287

x CONTENTS
12 Parameterization of Stabilizing Controllers 291
12.1 Existence of Stabilizing Controllers : : : : : : : : : : : : : : : : : : 292
12.2 Duality and Special Problems : : : : : : : : : : : : : : : : : : : : : 295
12.3 Parameterization of All Stabilizing Controllers : : : : : : : : : : : : 301
12.4 Structure of Controller Parameterization : : : : : : : : : : : : : : : 312
12.5 Closed-Loop Transfer Matrix : : : : : : : : : : : : : : : : : : : : : 314
12.6 Youla Parameterization via Coprime Factorization* : : : : : : : : : 315
13 Algebraic Riccati Equations 319
13.1 All Solutions of A Riccati Equation : : : : : : : : : : : : : : : : : : 320
13.2 Stabilizing Solution and Riccati Operator : : : : : : : : : : : : : : 325
13.3 Extreme Solutions and Matrix Inequalities : : : : : : : : : : : : : : 333
13.4 Spectral Factorizations : : : : : : : : : : : : : : : : : : : : : : : : : 342
13.5 Positive Real Functions : : : : : : : : : : : : : : : : : : : : : : : : : 353
13.6 Inner Functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 357
13.7 Inner-Outer Factorizations : : : : : : : : : : : : : : : : : : : : : : : 358
13.8 Normalized Coprime Factorizations : : : : : : : : : : : : : : : : : : 362
14 H2 Optimal Control 365
14.1 Introduction to Regulator Problem : : : : : : : : : : : : : : : : : : 365
14.2 Standard LQR Problem : : : : : : : : : : : : : : : : : : : : : : : : 367
14.3 Extended LQR Problem : : : : : : : : : : : : : : : : : : : : : : : : 372
14.4 Guaranteed Stability Margins of LQR : : : : : : : : : : : : : : : : 373
14.5 Standard H2 Problem : : : : : : : : : : : : : : : : : : : : : : : : : 375
14.6 Optimal Controlled System : : : : : : : : : : : : : : : : : : : : : : 378
14.7 H2 Control with Direct Disturbance Feedforward* : : : : : : : : : 379
14.8 Special Problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : 381
14.9 Separation Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : 388
14.10 Stability Margins of H2 Controllers : : : : : : : : : : : : : : : : : : 390
15 Linear Quadratic Optimization 393
15.1 Hankel Operators : : : : : : : : : : : : : : : : : : : : : : : : : : : : 393
15.2 Toeplitz Operators : : : : : : : : : : : : : : : : : : : : : : : : : : : 395
15.3 Mixed Hankel-Toeplitz Operators : : : : : : : : : : : : : : : : : : : 397
15.4 Mixed Hankel-Toeplitz Operators: The General Case* : : : : : : : 399
15.5 Linear Quadratic Max-Min Problem : : : : : : : : : : : : : : : : : 401

CONTENTS xi
16 H1 Control: Simple Case 405
16.1 Problem Formulation : : : : : : : : : : : : : : : : : : : : : : : : : : 405
16.2 Output Feedback H1 Control : : : : : : : : : : : : : : : : : : : : : 406
16.3 Motivation for Special Problems : : : : : : : : : : : : : : : : : : : : 412
16.4 Full Information Control : : : : : : : : : : : : : : : : : : : : : : : : 416
16.5 Full Control : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 423
16.6 Disturbance Feedforward : : : : : : : : : : : : : : : : : : : : : : : 423
16.7 Output Estimation : : : : : : : : : : : : : : : : : : : : : : : : : : : 425
16.8 Separation Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : 426
16.9 Optimality and Limiting Behavior : : : : : : : : : : : : : : : : : : 431
16.10 Controller Interpretations : : : : : : : : : : : : : : : : : : : : : : : 436
16.11 An Optimal Controller : : : : : : : : : : : : : : : : : : : : : : : : : 438
17 H1 Control: General Case 441
17.1 General H1 Solutions : : : : : : : : : : : : : : : : : : : : : : : : : 442
17.2 Loop Shifting : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 445
17.3 Relaxing Assumptions : : : : : : : : : : : : : : : : : : : : : : : : : 449
17.4 H2 and H1 Integral Control : : : : : : : : : : : : : : : : : : : : : : 450
17.5 H1 Filtering : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 454
17.6 Youla Parameterization Approach* : : : : : : : : : : : : : : : : : : 456
17.7 Connections : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 460
17.8 State Feedback and Di erential Game : : : : : : : : : : : : : : : : 462
17.9 Parameterization of State Feedback H1 Controllers : : : : : : : : : 465
18 H1 Loop Shaping 469
18.1 Robust Stabilization of Coprime Factors : : : : : : : : : : : : : : : 469
18.2 Loop Shaping Using Normalized Coprime Stabilization : : : : : : : 478
18.3 Theoretical Justi cation for H1 Loop Shaping : : : : : : : : : : : 481
19 Controller Order Reduction 489
19.1 Controller Reduction with Stability Criteria : : : : : : : : : : : : : 490
19.2 H1 Controller Reductions : : : : : : : : : : : : : : : : : : : : : : : 497
19.3 Frequency-Weighted L1 Norm Approximations : : : : : : : : : : : 504
19.4 An Example : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 509

xii CONTENTS
20 Structure Fixed Controllers 515
20.1 Lagrange Multiplier Method : : : : : : : : : : : : : : : : : : : : : : 515
20.2 Fixed Order Controllers : : : : : : : : : : : : : : : : : : : : : : : : 520
21 Discrete Time Control 527
21.1 Discrete Lyapunov Equations : : : : : : : : : : : : : : : : : : : : : 527
21.2 Discrete Riccati Equations : : : : : : : : : : : : : : : : : : : : : : : 528
21.3 Bounded Real Functions : : : : : : : : : : : : : : : : : : : : : : : : 537
21.4 Matrix Factorizations : : : : : : : : : : : : : : : : : : : : : : : : : : 542
21.5 Discrete Time H2 Control : : : : : : : : : : : : : : : : : : : : : : : 548
21.6 Discrete Balanced Model Reduction : : : : : : : : : : : : : : : : : : 553
21.7 Model Reduction Using Coprime Factors : : : : : : : : : : : : : : : 558
Bibliography 563
Index 581

Preface
This book is inspired by the recent development in the robust and H1 control theory,
particularly the state-space H1 control theory developed in the paper by Doyle, Glover,
Khargonekar, and Francis 1989] (known as DGKF paper). We give a fairly compre-
hensive and step-by-step treatment of the state-space H1 control theory in the style
of DGKF. We also treat the robust control problems with unstructured and structured
uncertainties. The linear fractional transformation (LFT) and the structured singular
value (known as ) are introduced as the uni ed tools for robust stability and perfor-
mance analysis and synthesis. Chapter 1 contains a more detailed chapter-by-chapter
review of the topics and results presented in this book.
We would like to thank Professor Bruce A. Francis at University of Toronto for his
helpful comments and suggestions on early versions of the manuscript. As a matter
of fact, this manuscript was inspired by his lectures given at Caltech in 1987 and his
masterpiece { A Course in H1 Control Theory. We are grateful to Professor Andre Tits
at University of Maryland who has made numerous helpful comments and suggestions
that have greatly improved the quality of the manuscript. Professor Jakob Stoustrup,
Professor Hans Henrik Niemann, and their students at The Technical University of
Denmark have read various versions of this manuscript and have made many helpful
comments and suggestions. We are grateful to their help. Special thanks go to Professor
Andrew Packard at University of California-Berkeley for his help during the preparation
of the early versions of this manuscript. We are also grateful to Professor Jie Chen at
University of California-Riverside for providing material used in Chapter 6. We would
also like to thank Professor Kang-Zhi Liu at Chiba University (Japan) and Professor
Tongwen Chen at University of Calgary for their valuable comments and suggestions.
In addition, we would like to thank G. Balas, C. Beck, D. S. Bernstein, G. Gu, W.
Lu, J. Morris, M. Newlin, L. Qiu, H. P. Rotstein and many other people for their
comments and suggestions. The rst author is especially grateful to Professor Pramod
P. Khargonekar at The University of Michigan for introducing him to robust and H1
control and to Professor Tryphon Georgiou at University of Minnesota for encouraging
him to complete this work.
Kemin Zhou
xiii

xiv PREFACE
John C. Doyle
Keith Glover

xvi NOTATION AND SYMBOLS
Notation and Symbols
R eld of real numbers
C eld of complex numbers
F eld, either R or C
R+ nonnegative real numbers
C ; and C ; open and closed left-half plane
C + and C + open and closed right-half plane
C 0, jR imaginary axis
D unit disk
D closed unit disk
@D unit circle
2 belong to
subset
union
intersection
2 end of proof
3 end of example
~ end of remark
:= de ned as
' asymptotically greater than
/ asymptotically less than
much greater than
much less than
complex conjugate of 2 C
j j absolute value of 2 C
Re( ) real part of 2 C
(t) unit impulse
ij Kronecker delta function, ii = 1 and ij = 0 if i 6= j
1+(t) unit step function

NOTATION AND SYMBOLS xvii
In n n identity matrix
aij] a matrix with aij as its i-th row and j-th column element
diag(a1 ::: an) an n n diagonal matrix with ai as its i-th diagonal element
AT transpose
A adjoint operator of A or complex conjugate transpose of A
A;1 inverse of A
A+ pseudo inverse of A
A; shorthand for (A;1)
det(A) determinant of A
Trace(A) trace of A
(A) eigenvalue of A
(A) spectral radius of A
(A) the set of spectrum of A
(A) largest singular value of A
(A) smallest singular value of A
i(A) i-th singular value of A
(A) condition number of A
kAk spectral norm of A: kAk = (A)
Im(A), R(A) image (or range) space of A
Ker(A), N(A) kernel (or null) space of A
X;(A) stable invariant subspace of A
X+(A) antistable invariant subspace of A
Ric(H) the stabilizing solution of an ARE
g f convolution of g and f
Kronecker product
direct sum or Kronecker sum
angle
h i inner product
x ? y orthogonal, hx yi = 0
D? orthogonal complement of D, i.e., D D?
or D
D?
is unitary
S? orthogonal complement of subspace S, e.g., H?2
L2(;1 1) time domain Lebesgue space
L2 0 1) subspace of L2(;1 1)
L2(;1 0] subspace of L2(;1 1)
L2+ shorthand for L2 0 1)
L2; shorthand for L2(;1 0]
l2+ shorthand for l2 0 1)
l2; shorthand for l2(;1 0)
L2(jR) square integrable functions on C 0 including at 1
L2(@D ) square integrable functions on @D
H2(jR) subspace of L2(jR) with analytic extension to the rhp

xviii NOTATION AND SYMBOLS
H2(@D ) subspace of L2(@D ) with analytic extension to the inside of @D
H?2 (jR) subspace of L2(jR) with analytic extension to the lhp
H?2 (@D ) subspace of L2(@D ) with analytic extension to the outside of @D
L1(jR) functions bounded on Re(s) = 0 including at 1
L1(@D ) functions bounded on @D
H1(jR) the set of L1(jR) functions analytic in Re(s) > 0
H1(@D ) the set of L1(@D ) functions analytic in jzj < 1
H;1(jR) the set of L1(jR) functions analytic in Re(s) < 0
H;1(@D ) the set of L1(@D ) functions analytic in jzj > 1
pre x B or B closed unit ball, e.g. BH1 and B
pre x Bo open unit ball
pre x R real rational, e.g., RH1 and RH2, etc
R s] polynomial ring
Rp(s) rational proper transfer matrices
G (s) shorthand for GT(;s) (continuous time)
G (z) shorthand for GT(z;1) (discrete time)
A B
C D shorthand for state space realization C(sI ;A);1B + D
or C(zI ;A);1B + D
F`(M Q) lower LFT
Fu(M Q) upper LFT
S(M N) star product

xx LIST OF ACRONYMS
List of Acronyms
ARE algebraic Riccati equation
BR bounded real
CIF complementary inner factor
DF disturbance feedforward
FC full control
FDLTI nite dimensional linear time invariant
FI full information
HF high frequency
i if and only if
lcf left coprime factorization
LF low frequency
LFT linear fractional transformation
lhp or LHP left-half plane Re(s) < 0
LQG linear quadratic Gaussian
LQR linear quadratic regulator
LTI linear time invariant
LTR loop transfer recovery
MIMO multi-input multi-output
nlcf normalized left coprime factorization
NP nominal performance
nrcf normalized right coprime factorization
NS nominal stability
OE output estimation
OF output feedback
OI output injection
rcf right coprime factorization
rhp or RHP right-half plane Re(s) > 0
RP robust performance
RS robust stability
SF state feedback
SISO single-input single-output
SSV structured singular value ( )
SVD singular value decomposition

1
Introduction
1.1 Historical Perspective
This book gives a comprehensive treatment of optimal H2 and H1 control theory and
an introduction to the more general subject of robust control. Since the central subject
of this book is state-space H1 optimal control, in contrast to the approach adopted
in the famous book by Francis 1987]: A Course in H1 Control Theory, it may be
helpful to provide some historical perspective of the state-space H1 control theory to
be presented in this book. This section is not intended as a review of the literature
in H1 theory or robust control, but rather only an attempt to outline some of the
work that most closely touches on our approach to state-space H1. Hopefully our lack
of written historical material will be somewhat made up for by the pictorial history of
control shown in Figure 1.1. Here we see how the practical but classical methods yielded
to the more sophisticated modern theory. Robust control sought to blend the best of
both worlds. The strange creature that resulted is the main topic of this book.
The H1 optimal control theory was originally formulated by Zames 1981] in an
input-output setting. Most solution techniques available at that time involved analytic
functions (Nevanlinna-Pick interpolation) or operator-theoretic methods Sarason, 1967
Adamjan et al., 1978 Ball and Helton, 1983]. Indeed, H1 theory seemed to many to
signal the beginning of the end for the state-space methods which had dominated control
for the previous 20 years. Unfortunately, the standard frequency-domain approaches to
H1 started running into signi cant obstacles in dealing with multi-input multi-output
(MIMO) systems, both mathematically and computationally, much as the H2 (or LQG)
theory of the 1950's had.
1

2 INTRODUCTION
Figure 1.1: A picture history of control
Not surprisingly, the rst solution to a general rational MIMO H1 optimal control
problem, presented in Doyle 1984], relied heavily on state-space methods, although
more as a computational tool than in any essential way. The steps in this solution were as
follows: parameterize all internally-stabilizing controllers via Youla et al., 1976] obtain
realizations of the closed-loop transfer matrix convert the resulting model-matching
problem into the equivalent 2 2-block general distance or best approximation problem
involving mixed Hankel-Toeplitz operators reduce to the Nehari problem (Hankel only)
solve the Nehari problem by the procedure of Glover 1984]. Both Francis, 1987] and
Francis and Doyle, 1987] give expositions of this approach, which will be referred to as
the 1984" approach.
In a mathematical sense, the 1984 procedure solved" the general rational H1 op-
timal control problem and much of the subsequent work in H1 control theory focused
on the 2 2-block problems, either in the model-matching or general distance forms.
Unfortunately, the associated complexity of computation was substantial, involving sev-
eral Riccati equations of increasing dimension, and formulae for the resulting controllers
tended to be very complicated and have high state dimension. Encouragement came

1.1. Historical Perspective 3
from Limebeer and Hung 1987] and Limebeer and Halikias 1988] who showed, for
problems transformable to 2 1-block problems, that a subsequent minimal realization
of the controller has state dimension no greater than that of the generalized plant G.
This suggested the likely existence of similarly low dimension optimal controllers in the
general 2 2 case.
Additional progress on the 2 2-block problems came from Ball and Cohen 1987],
who gave a state-space solution involving 3 Riccati equations. Jonckheere and Juang
1987] showed a connection between the 2 1-block problem and previous work by
Jonckheere and Silverman 1978] on linear-quadratic control. Foias and Tannenbaum
1988] developed an interesting class of operators called skew Toeplitz to study the
2 2-block problem. Other approaches have been derived by Hung 1989] using an
interpolation theory approach, Kwakernaak 1986] using a polynomial approach, and
Kimura 1988] using a method based on conjugation.
The simple state space H1 controller formulae to be presented in this book were rst
derived in Glover and Doyle 1988] with the 1984 approach, but using a new 2 2-block
solution, together with a cumbersome back substitution. The very simplicity of the new
formulae and their similarity with the H2 ones suggested a more direct approach.
Independent encouragement for a simpler approach to the H1 problem came from
papers by Petersen 1987], Khargonekar, Petersen, and Zhou 1990], Zhou and Khar-
gonekar 1988], and Khargonekar, Petersen, and Rotea 1988]. They showed that for
the state-feedback H1 problem one can choose a constant gain as a (sub)optimal con-
troller. In addition, a formula for the state-feedback gain matrix was given in terms
of an algebraic Riccati equation. Also, these papers established connections between
H1-optimal control, quadratic stabilization, and linear-quadratic di erential games.
The landmark breakthrough came in the DGKF paper (Doyle, Glover, Khargonekar,
and Francis 1989]). In addition to providing controller formulae that are simple and
expressed in terms of plant data as in Glover and Doyle 1988], the methods in that
paper are a fundamental departure from the 1984 approach. In particular, the Youla
parameterization and the resulting 2 2-block model-matching problem of the 1984
solution are avoided entirely replaced by a more purely state-space approach involving
observer-based compensators, a pair of 2 1 block problems, and a separation argument.
The operator theory still plays a central role (as does Redhe er's work Redhe er, 1960]
on linear fractional transformations), but its use is more straightforward. The key
to this was a return to simple and familiar state-space tools, in the style of Willems
1971], such as completing the square, and the connection between frequency domain
inequalities (e.g kGk1 < 1), Riccati equations, and spectral factorizations. This book
in some sense can be regarded as an expansion of the DGKF paper.
The state-space theory of H1 can be carried much further, by generalizing time-
invariant to time-varying, in nite horizon to nite horizon, and nite dimensional to
in nite dimensional. A ourish of activity has begun on these problems since the publi-
cation of the DGKF paper and numerous results have been published in the literature,
not surprising, many results in DGKF paper generalize mutatis mutandis, to these cases,
which are beyond the scope of this book.

4 INTRODUCTION
1.2 How to Use This Book
This book is intended to be used either as a graduate textbook or as a reference for
control engineers. With the second objective in mind, we have tried to balance the
broadness and the depth of the material covered in the book. In particular, some
chapters have been written su ciently self-contained so that one may jump to those
special topics without going through all the preceding chapters, for example, Chapter
13 on algebraic Riccati equations. Some other topics may only require some basic linear
system theory, for instance, many readers may nd that it is not di cult to go directly
to Chapters 9 11. In some cases, we have tried to collect some most frequently used
formulas and results in one place for the convenience of reference although they may not
have any direct connection with the main results presented in the book. For example,
readers may nd that those matrix formulas collected in Chapter 2 on linear algebra
convenient in their research. On the other hand, if the book is used as a textbook, it
may be advisable to skip those topics like Chapter 2 on the regular lectures and leave
them for students to read. It is obvious that only some selected topics in this book can
be covered in an one or two semester course. The speci c choice of the topics depends
on the time allotted for the course and the preference of the instructor. The diagram
in Figure 1.2 shows roughly the relations among the chapters and should give the users
some idea for the selection of the topics. For example, the diagram shows that the only
prerequisite for Chapters 7 and 8 is Section 3.9 of Chapter 3 and, therefore, these two
chapters alone may be used as a short course on model reductions. Similarly, one only
needs the knowledge of Sections 13.2 and 13.6 of Chapter 13 to understand Chapter 14.
Hence one may only cover those related sections of Chapter 13 if time is the factor. The
book is separated roughly into the following subgroups:
Basic Linear System Theory: Chapters 2 3.
Stability and Performance: Chapters 4 6.
Model Reduction: Chapters 7 8.
Robustness: Chapters 9 11.
H2 and H1 Control: Chapters 12 19.
Lagrange Method: Chapter 20.
Discrete Time Systems: Chapter 21.
In view of the above classi cation, one possible choice for an one-semester course
on robust control would cover Chapters 4 5 9 11 or 4 11 and an one-semester
advanced course on H2 and H1 control would cover (parts of) Chapters 12 19.
Another possible choice for an one-semester course on H1 control may include Chapter
4, parts of Chapter 5 (5:1 5:3 5:5 5:7), Chapter 10, Chapter 12 (except Section 12.6),
parts of Chapter 13 (13:2 13:4 13:6), Chapter 15 and Chapter 16. Although Chapters
7 8 are very much independent of other topics and can, in principle, be studied at any

1.2. How to Use This Book 5
4
5 6 7 8
9
10 11
12
15 14
13
16
17 18
2 3
19 20 21
?
-
A
AU
13:2
13:6
3:9
?
-
?
@
@@R?
?
?
?
-
-
13:4
?
@
@@I
?
;
;;
?
?
Figure 1.2: Relations among the chapters

6 INTRODUCTION
stage with the background of Section 3.9, they may serve as an introduction to sources
of model uncertainties and hence to robustness problems.
Robust Control H1 Control Advanced Model & Controller
H1 Control Reductions
4 4 12 3.9
5 5.1 5.3,5.5,5.7 13.2,13.4,13.6 7
6* 10 14 8
7* 12 15 5.4,5.7
8* 13.2,13.4,13.6 16 10.1
9 15 17* 16.1,16.2
10 16 18* 17.1
11 19* 19
Table 1.1: Possible choices for an one-semester course (* chapters may be omitted)
Table 1.1 lists several possible choices of topics for an one-semester course. A course
on model and controller reductions may only include the concept of H1 control and
the H1 controller formulas with the detailed proofs omitted as suggested in the above
table.
1.3 Highlights of The Book
The key results in each chapter are highlighted below. Note that some of the state-
ments in this section are not precise, they are true under certain assumptions that are
not explicitly stated. Readers should consult the corresponding chapters for the exact
statements and conditions.
Chapter 2 reviews some basic linear algebra facts and treats a special class of matrix
dilation problems. In particular, we show
minX
X B
C A = max C A B
A
and characterize all optimal (suboptimal) X.
Chapter 3 reviews some system theoretical concepts: controllability, observability,
stabilizability, detectability, pole placement, observer theory, system poles and zeros,
and state space realizations. Particularly, the balanced state space realizations are
studied in some detail. We show that for a given stable transfer function G(s) there
is a state space realization G(s) = A B
C D such that the controllability Gramian P
and the observability Gramian Q de ned below are equal and diagonal: P = Q = =
diag( 1 2 ::: n) where
AP + PA + BB = 0

1.3. Highlights of The Book 7
A Q+ QA+ C C = 0:
Chapter 4 de nes several norms for signals and introduces the H2 spaces and the H1
spaces. The input/output gains of a stable linear system under various input signals
are characterized. We show that H2 and H1 norms come out naturally as measures of
the worst possible performance for many classes of input signals. For example, let
G(s) = A B
C 0 2 RH1 g(t) = CeAtB
Then kGk1 = sup
kg uk2
kuk2
and 1 kGk1
Z 1
0
kg(t)kdt 2
nX
i=1
i. Some state
space methods of computing real rational H2 and H1 transfer matrix norms are also
presented:
kGk2
2 = trace(B QB) = trace(CPC )
and
kGk1 = maxf : H has an eigenvalue on the imaginary axisg
where
H = A BB = 2
;C C ;A :
Chapter 5 introduces the feedback structure and discusses its stability and perfor-
mance properties.
e
e
+
+
++e2
e1
w2
w1
-
6
?
-
^K
P
We show that the above closed-loop system is internally stable if and only if
I ; ^K
;P I
;1
= I + ^K(I ;P ^K);1P ^K(I ;P ^K);1
(I ;P ^K);1P (I ;P ^K);1 2 RH1:
Alternative characterizations of internal stability using coprime factorizations are also
presented.
Chapter 6 introduces some multivariable versions of the Bode's sensitivity integral
relations and Poisson integral formula. The sensitivity integral relations are used to
study the design limitations imposed by bandwidth constraint and the open-loop un-
stable poles, while the Poisson integral formula is used to study the design constraints

8 INTRODUCTION
imposed by the non-minimum phase zeros. For example, let S(s) be a sensitivity func-
tion, and let pi be the right half plane poles of the open-loop system and i be the
corresponding pole directions. Then we show that
Z 1
0
ln (S(j!))d! = max
X
i
(Repi) i i
!
+ H1 H1 0:
This equality shows that the design limitations in multivariable systems are dependent
on the directionality properties of the sensitivity function as well as those of the poles
(and zeros), in addition to the dependence upon pole (and zero) locations which is
known in single-input single-output systems.
Chapter 7 considers the problem of reducing the order of a linear multivariable
dynamical system using the balanced truncation method. Suppose
G(s) =
2
4
A11 A12
A21 A22
B1
B2
C1 C2 D
3
5 2 RH1
is a balanced realization with controllability and observability Gramians P = Q = =
diag( 1 2)
1 = diag( 1Is1 2Is2 ::: rIsr)
2 = diag( r+1Isr+1 r+2Isr+2 ::: NIsN ):
Then the truncated system Gr(s) = A11 B1
C1 D is stable and satis es an additive
error bound:
kG(s) ;Gr(s)k1 2
NX
i=r+1
i:
On the other hand, if G;1 2 RH1, and P and Q satisfy
PA + AP + BB = 0
Q(A ;BD;1C) + (A ;BD;1C) Q+ C (D;1) D;1C = 0
such that P = Q = diag( 1 2) with G partitioned compatibly as before, then
Gr(s) = A11 B1
C1 D
is stable and minimum phase, and satis es respectively the following relative and mul-
tiplicative error bounds:
G;1(G ;Gr) 1
NY
i=r+1
1 + 2 i(
q
1 + 2
i + i) ;1
G;1
r (G ;Gr) 1
NY
i=r+1
1 + 2 i(
q
1 + 2
i + i) ;1:

Chapter 8 deals with the optimal Hankel norm approximation and its applications
in L1 norm model reduction. We show that for a given G(s) of McMillan degree n
there is a ^G(s) of McMillan degree r < n such that
G(s) ; ^G(s)
H
= inf G(s) ; ^G(s)
H
= r+1:
Moreover, there is a constant matrix D0 such that
G(s) ; ^G(s) ;D0
1
NX
i=r+1
i:
The well-known Nehari's theorem is also shown:
inf
Q2RH;1
kG ;Qk1 = kGkH = 1:
Chapter 9 derives robust stability tests for systems under various modeling assump-
tions through the use of a small gain theorem. In particular, we show that an uncertain
system described below with an unstructured uncertainty 2 RH1 with k k1 < 1 is
robustly stable if and only if the transfer function from w to z has H1 norm no greater
than 1.
∆
wz
nominal system
Chapter 10 introduces the linear fractional transformation (LFT). We show that
many control problems can be formulated and treated in the LFT framework. In par-
ticular, we show that every analysis problem can be put in an LFT form with some
structured (s) and some interconnection matrix M(s) and every synthesis problem
can be put in an LFT form with a generalized plant G(s) and a controller K(s) to be
designed.
wz
-
M
y u
wz
-
G
K

10 INTRODUCTION
Chapter 11 considers robust stability and performance for systems with multiple
sources of uncertainties. We show that an uncertain system is robustly stable for all
i 2 RH1 with k ik1 < 1 if and only if the structured singular value ( ) of the
corresponding interconnection model is no greater than 1.
nominal system
∆
∆
∆∆
1
2
3
4
Chapter 12 characterizes all controllers that stabilize a given dynamical system G(s)
using the state space approach. The construction of the controller parameterization is
done via separation theory and a sequence of special problems, which are so-called full
information (FI) problems, disturbance feedforward (DF) problems, full control (FC)
problems and output estimation (OE). The relations among these special problems are
established.
DF OE
FI FC
-
6
?
-
6
?
equivalent equivalent
dual
dual
For a given generalized plant
G(s) = G11(s) G12(s)
G21(s) G22(s) =
2
4
A B1 B2
C1 D11 D12
C2 D21 D22
3
5
we show that all stabilizing controllers can be parameterized as the transfer matrix from
y to u below where F and L are such that A + LC2 and A + B2F are stable.

cc c cc
66;
y1u1
z w
uy
- Q
?
G
?
-
6
-
-
D22
;L
F
A
B2
R
C2
Chapter 13 studies the Algebraic Riccati Equation and the related problems: the
properties of its solutions, the methods to obtain the solutions, and some applications.
In particular, we study in detail the so-called stabilizing solution and its applications in
matrix factorizations. A solution to the following ARE
A X + XA+ XRX + Q = 0
is said to be a stabilizing solution if A+ RX is stable. Now let
H := A R
;Q ;A
and let X;(H) be the stable H invariant subspace and
X;(H) = Im X1
X2
where X1 X2 2 C n n. If X1 is nonsingular, then X := X2X;1
1 is uniquely determined
by H, denoted by X = Ric(H).
A key result of this chapter is the relationship between the spectral factorization of a
transfer function and the solution of a correspondingARE. Suppose (A B) is stabilizable
and suppose either A has no eigenvalues on j!-axis or P is sign de nite (i.e., P 0 or
P 0) and (P A) has no unobservable modes on the j!-axis. De ne
(s) = B (;sI ;A );1 I P S
S R
(sI ;A);1B
I :

12 INTRODUCTION
Then
(j!) > 0 for all 0 ! 1
() 9 a stabilizing solution X to
(A ;BR;1S ) X + X(A;BR;1S ) ;XBR;1B X + P ;SR;1S = 0
() the Hamiltonian matrix
H = A;BR;1S ;BR;1B
;(P ;SR;1S ) ;(A ;BR;1S )
has no j!-axis eigenvalues.
Similarly,
(j!) 0 for all 0 ! 1
() 9 a solution X to
(A ;BR;1S ) X + X(A;BR;1S ) ;XBR;1B X + P ;SR;1S = 0
such that (A ;BR;1S ;BR;1B X) C ;.
Furthermore, there exists a M 2 Rp such that
= M RM:
with
M = A B
;F I F = ;R;1(S + B X):
Chapter 14 treats the optimal control of linear time-invariant systems with quadratic
performance criteria, i.e., LQR and H2 problems. We consider a dynamical system
described by an LFT with
G(s) =
2
4
A B1 B2
C1 0 D12
C2 D21 0
3
5 :
G
K
z
y
w
u
-

De ne
H2 := A 0
;C1C1 ;A ; B2
;C1D12
D12C1 B2
J2 := A 0
;B1B1 ;A ; C2
;B1D21
D21B1 C2
X2 := Ric(H2) 0 Y2 := Ric(J2) 0
F2 := ;(B2X2 + D12C1) L2 := ;(Y2C2 + B1D21):
Then the H2 optimal controller, i.e. the controller that minimizes kTzwk2, is given by
Kopt(s) := A+ B2F2 + L2C2 ;L2
F2 0 :
Chapter 15 solves a max-min problem, i.e., a full information (or state feedback)
H1 control problem, which is the key to the H1 theory considered in the next chapter.
Consider a dynamical system
_x = Ax + B1w + B2u
z = C1x + D12u D12 C1 D12 = 0 I :
Then we show that sup
w2BL2+
minu2L2+
kzk2 < if and only if H1 2 dom(Ric) and X1 =
Ric(H1) 0 where
H1 := A ;2B1B1 ;B2B2
;C1C1 ;A
Furthermore, u = F1x with F1 := ;B2X1 is an optimal control.
Chapter 16 considers a simpli ed H1 control problem with the generalized plant
G(s) as given in Chapter 14. We show that there exists an admissible controller such
that kTzwk1 < i the following three conditions hold:
(i) H1 2 dom(Ric) and X1 := Ric(H1) 0
(ii) J1 2 dom(Ric) and Y1 := Ric(J1) 0 where
J1 := A ;2C1C1 ;C2C2
;B1B1 ;A :
(iii) (X1Y1) < 2 .

14 INTRODUCTION
Moreover, the set of all admissible controllers such that kTzwk1 < equals the set of
all transfer matrices from y to u in
M1
Q
u y
-
M1(s) =
2
4
^A1 ;Z1L1 Z1B2
F1 0 I
;C2 I 0
3
5
where Q 2 RH1, kQk1 < and
^A1 := A + ;2B1B1X1 + B2F1 + Z1L1C2
F1 := ;B2X1 L1 := ;Y1C2 Z1 := (I ; ;2Y1X1);1:
Chapter 17 considers again the standard H1 control problem but with some as-
sumptions in the last chapter relaxed. We indicate how the assumptions can be relaxed
to accommodate other more complicated problems such as singular control problems.
We also consider the integral control in the H2 and H1 theory and show how the
general H1 solution can be used to solve the H1 ltering problem. The conventional
Youla parameterization approach to the H2 and H1 problems is also outlined. Finally,
the general state feedback H1 control problem and its relations with full information
control and di erential game problems are discussed.
Chapter 18 rst solves a gap metric minimization problem. Let P = ~M;1 ~N be a
normalized left coprime factorization. Then we show that
infK stabilizing
K
I (I + PK);1 I P
1
= infK stabilizing
K
I (I + PK);1 ~M;1
1
=
q
1 ; ~N ~M 2
H
;1
:
This implies that there is a robustly stabilizing controller for
P = ( ~M + ~M);1( ~N + ~N)
with
~N ~M 1 <
if and only if q
1 ; ~N ~M 2
H:
Using this stabilization result, a loop shaping design technique is proposed. The pro-
posed technique uses only the basic concept of loop shaping methods and then a robust

stabilization controller for the normalized coprime factor perturbed system is used to
construct the nal controller.
Chapter 19 considers the design of reduced order controllers by means of controller
reduction. Special attention is paid to the controllerreduction methods that preservethe
closed-loop stability and performance. In particular, two H1 performance preserving
reduction methods are proposed:
a) Let K0 be a stabilizing controller such that kF`(G K0)k1 < . Then ^K is also a
stabilizing controller such that F`(G ^K)
1
< if
W;1
2 ( ^K ;K0)W;1
1 1
< 1
where W1 and W2 are some stable, minimum phase and invertible transfer matri-
ces.
b) Let K0 = 12
;1
22 be a central H1 controller such that kF`(G K0)k1 < and
let Û ^V 2 RH1 be such that
;1I 0
0 I
;1 12
22
;
Û
^V 1
< 1=
p
2:
Then ^K = Û ^V ;1 is also a stabilizing controller such that kF`(G ^K)k1 < .
Thus the controller reduction problem is converted to weighted model reduction prob-
lems for which some numerical methods are suggested.
Chapter 20 brie y introduces the Lagrange multiplier method for the design of xed
order controllers.
Chapter 21 discusses discrete time Riccati equations and some of their applications in
discrete time control. Finally, the discrete time balanced model reduction is considered.

2
Linear Algebra
Some basic linear algebra facts will be reviewed in this chapter. The detailed treatment
of this topic can be found in the references listed at the end of the chapter. Hence we shall
omit most proofs and provide proofs only for those results that either cannot be easily
found in the standard linear algebra textbooks or are insightful to the understanding of
some related problems. We then treat a special class of matrix dilation problems which
will be used in Chapters 8 and 17 however, most of the results presented in this book
can be understood without the knowledge of the matrix dilation theory.
2.1 Linear Subspaces
Let R denote the real scalar eld and C the complex scalar eld. For the interest of this
chapter, let F be either R or C and let Fn be the vector space over F, i.e., Fn is either Rn
or C n. Now let x1 x2 ::: xk 2 Fn. Then an element of the form 1x1 +:::+ kxk with
i 2 F is a linear combination over F of x1 ::: xk. The set of all linear combinations
of x1 x2 ::: xk 2 Fn is a subspace called the span of x1 x2 ::: xk, denoted by
spanfx1 x2 ::: xkg := fx = 1x1 + :::+ kxk : i 2 Fg:
A set of vectors x1 x2 ::: xk 2 Fn are said to be linearly dependent over F if there
exists 1 ::: k 2 F not all zero such that 1x2 + ::: + kxk = 0 otherwise they are
said to be linearly independent.
Let S be a subspace of Fn, then a set of vectors fx1 x2 ::: xkg 2 S is called a basis
for S if x1 x2 ::: xk are linearly independent and S = spanfx1 x2 ::: xkg. However,
17

18 LINEAR ALGEBRA
such a basis for a subspace S is not unique but all bases for S have the same number
of elements. This number is called the dimension of S, denoted by dim(S).
A set of vectors fx1 x2 ::: xkg in Fn are mutually orthogonal if xi xj = 0 for all
i 6= j and orthonormal if xi xj = ij, where the superscript denotes complex conjugate
transpose and ij is the Kronecker delta function with ij = 1 for i = j and ij = 0
for i 6= j. More generally, a collection of subspaces S1 S2 ::: Sk of Fn are mutually
orthogonal if x y = 0 whenever x 2 Si and y 2 Sj for i 6= j.
The orthogonal complement of a subspace S Fn is de ned by
S? := fy 2 Fn : y x = 0 for all x 2 Sg:
We call a set of vectors fu1 u2 ::: ukg an orthonormal basis for a subspace S 2 Fn if
they form a basis of S and are orthonormal. It is always possible to extend such a basis
to a full orthonormal basis fu1 u2 ::: ung for Fn. Note that in this case
S? = spanfuk+1 ::: ung
and fuk+1 ::: ung is called an orthonormal completion of fu1 u2 ::: ukg.
Let A 2 Fm n be a linear transformation from Fn to Fm, i.e.,
A : Fn 7;! Fm:
(Note that a vector x 2 Fm can also be viewed as a linear transformation from F to Fm,
hence anything said for the general matrix case is also true for the vector case.) Then
the kernel or null space of the linear transformation A is de ned by
KerA = N(A) := fx 2 Fn : Ax = 0g
and the image or range of A is
ImA = R(A) := fy 2 Fm : y = Ax x 2 Fng:
It is clear that KerA is a subspace of Fn and ImA is a subspace of Fm. Moreover, it can
be easily seen that dim(KerA) + dim(ImA) = n and dim(ImA) = dim(KerA)?. Note
that (KerA)? is a subspace of Fn.
Let ai i = 1 2 ::: n denote the columns of a matrix A 2 Fm n, then
ImA = spanfa1 a2 ::: ang:
The rank of a matrix A is de ned by
rank(A) = dim(ImA):
It is a fact that rank(A) = rank(A ), and thus the rank of a matrix equals the maximal
number of independent rows or columns. A matrix A 2 Fm n is said to have full row
rank if m n and rank(A) = m. Dually, it is said to have full column rank if n m
and rank(A) = n. A full rank square matrix is called a nonsingular matrix. It is easy

2.1. Linear Subspaces 19
to see that rank(A) = rank(AT) = rank(PA) if T and P are nonsingular matrices with
appropriate dimensions.
A square matrix U 2 Fn n whose columns form an orthonormal basis for Fn is called
an unitary matrix (or orthogonal matrix if F = R), and it satis es U U = I = UU .
The following lemma is useful.
Lemma 2.1 Let D = d1 ::: dk 2 Fn k (n > k) be such that D D = I, so
di i = 1 2 ::: k are orthonormal. Then there exists a matrix D? 2 Fn (n;k) such that
D D? is a unitary matrix. Furthermore, the columns of D?, di i = k + 1 ::: n,
form an orthonormal completion of fd1 d2 ::: dkg.
The following results are standard:
Lemma 2.2 Consider the linear equation
AX = B
where A 2 Fn l and B 2 Fn m are given matrices. Then the following statements are
equivalent:
(i) there exists a solution X 2 Fl m.
(ii) the columns of B 2 ImA.
(iii) rank A B =rank(A).
(iv) Ker(A ) Ker(B ).
Furthermore, the solution, if it exists, is unique if and only if A has full column rank.
The following lemma concerns the rank of the product of two matrices.
Lemma 2.3 (Sylvester's inequality) Let A 2 Fm n and B 2 Fn k. Then
rank (A) + rank(B) ;n rank(AB) minfrank (A) rank(B)g:
For simplicity, a matrix M with mij as its i-th row and j-th column's element will
sometimes be denoted as M = mij] in this book. We will mostly use I as above to
denote an identity matrix with compatible dimensions, but from time to time, we will
use In to emphasis that it is an n n identity matrix.
Now let A = aij] 2 C n n, then the trace of A is de ned as
Trace(A) :=
nX
i=1
aii:
Trace has the following properties:
Trace( A) = Trace(A) 8 2 C A 2 C n n
Trace(A + B) = Trace(A) + Trace(B) 8A B 2 C n n
Trace(AB) = Trace(BA) 8A 2 C n m B 2 C m n:

20 LINEAR ALGEBRA
2.2 Eigenvalues and Eigenvectors
Let A 2 C n n then the eigenvalues of A are the n roots of its characteristic polynomial
p( ) = det( I;A). This set of roots is called the spectrum of A and is denoted by (A)
(not to be confused with singular values de ned later). That is, (A) := f 1 2 ::: ng
if i is a root of p( ). The maximal modulus of the eigenvalues is called the spectral
radius, denoted by
(A) := max1 i n
j ij
where, as usual, j j denotes the magnitude.
If 2 (A) then any nonzero vector x 2 C n that satis es
Ax = x
is referred to as a right eigenvector of A. Dually, a nonzero vector y is called a left
eigenvector of A if
y A = y :
It is a well known (but nontrivial) fact in linear algebra that any complex matrix admits
a Jordan Canonical representation:
Theorem 2.4 For any square complex matrix A 2 C n n, there exists a nonsingular
matrix T such that
A = TJT;1
where
J = diagfJ1 J2 ::: Jlg
Ji = diagfJi1 Ji2 ::: Jimig
Jij =
2
6
6
6
6
6
4
i 1
i 1
... ...
i 1
i
3
7
7
7
7
7
5
2 C nij nij
with
Pl
i=1
Pmi
j=1 nij = n, and with f i : i = 1 ::: lg as the distinct eigenvalues of A.
The transformation T has the following form:
T = T1 T2 ::: Tl
Ti = Ti1 Ti2 ::: Timi
Tij = tij1 tij2 ::: tijnij
where tij1 are the eigenvectors of A,
Atij1 = itij1

2.2. Eigenvalues and Eigenvectors 21
and tijk 6= 0 de ned by the following linear equations for k 2
(A ; iI)tijk = tij(k;1)
are called the generalized eigenvectors of A. For a given integer q nij, the generalized
eigenvectors tijl 8l < q, are called the lower rank generalized eigenvectors of tijq.
De nition 2.1 A square matrix A 2 Rn n is called cyclic if the Jordan canonical form
of A has one and only one Jordan block associated with each distinct eigenvalue.
More speci cally, a matrix A is cyclic if its Jordan form has mi = 1 i = 1 ::: l. Clearly,
a square matrix A with all distinct eigenvalues is cyclic and can be diagonalized:
A x1 x2 xn = x1 x2 xn
2
6
6
6
4
1
2
...
n
3
7
7
7
5
:
In this case, A has the following spectral decomposition:
A =
nX
i=1
ixiyi
where yi 2 C n is given by
2
6
6
6
4
y1
y2
...
yn
3
7
7
7
5
= x1 x2 xn
;1
:
In general, eigenvalues need not be real, and neither do their corresponding eigenvec-
tors. However, if A is real and is a real eigenvalue of A, then there is a real eigenvector
corresponding to . In the case that all eigenvalues of a matrix A are real1, we will
denote max(A) for the largest eigenvalue of A and min(A) for the smallest eigenvalue.
In particular, if A is a Hermitian matrix, then there exist a unitary matrix U and a
real diagonal matrix such that A = U U , where the diagonal elements of are the
eigenvalues of A and the columns of U are the eigenvectors of A.
The following theorem is useful in linear system theory.
Theorem 2.5 (Cayley-Hamilton) Let A 2 C n n and denote
det( I ;A) = n + a1
n;1 + + an:
Then
An + a1An;1 + + anI = 0:
1For example, this is the case if A is Hermitian, i.e., A = A .

22 LINEAR ALGEBRA
This is obvious if A has distinct eigenvalues. Since
An + a1An;1 + + anI = T;1 diag f::: n
i + a1
n;1
i + + an :::g T = 0
and i is an eigenvalue of A. The proof for the general case follows from the following
lemma.
Lemma 2.6 Let A 2 C n n. Then
( I ;A);1 = 1
det( I ;A)(R1
n;1 + R2
n;2 + + Rn)
and
det( I ;A) = n + a1
n;1 + + an
where ai and Ri can be computed from the following recursive formulas:
a1 = ;TraceA R1 = I
a2 = ;1
2 Trace(R2A) R2 = R1A + a1I
...
...
an;1 = ; 1
n;1 Trace(Rn;1A) Rn = Rn;1A + an;1I
an = ;1
n Trace(RnA) 0 = RnA + anI:
The proof is left to the reader as an exercise. Note that the Cayley-Hamilton Theorem
follows from the fact that
0 = RnA + anI = An + a1An;1 + + anI:
2.3 Matrix Inversion Formulas
Let A be a square matrix partitioned as follows
A := A11 A12
A21 A22
where A11 and A22 are also square matrices. Now suppose A11 is nonsingular, then A
has the following decomposition:
A11 A12
A21 A22
= I 0
A21A;1
11 I
A11 0
0
I A;1
11 A12
0 I
with := A22 ;A21A;1
11 A12, and A is nonsingular i is nonsingular.
Dually, if A22 is nonsingular, then
A11 A12
A21 A22
= I A12A;1
22
0 I
0
0 A22
I 0
A;1
22 A21 I

2.3. Matrix Inversion Formulas 23
with := A11 ;A12A;1
22 A21, and A is nonsingular i is nonsingular. The matrix
( ) is called the Schur complement of A11 (A22) in A.
Moreover, if A is nonsingular, then
A11 A12
A21 A22
;1
= A;1
11 + A;1
11 A12 ;1A21A;1
11 ;A;1
11 A12 ;1
; ;1A21A;1
11
;1
and
A11 A12
A21 A22
;1
=
"
;1
; ;1
A12A;1
22
;A;1
22 A21
;1
A;1
22 + A;1
22 A21
;1
A12A;1
22
#
:
The above matrix inversion formulas are particularly simple if A is block triangular:
A11 0
A21 A22
;1
= A;1
11 0
;A;1
22 A21A;1
11 A;1
22
A11 A12
0 A22
;1
= A;1
11 ;A;1
11 A12A;1
22
0 A;1
22
:
The following identity is also very useful. Suppose A11 and A22 are both nonsingular
matrices, then
(A11 ;A12A;1
22 A21);1 = A;1
11 + A;1
11 A12(A22 ;A21A;1
11 A12);1A21A;1
11 :
As a consequence of the matrix decomposition formulas mentioned above, we can
calculate the determinant of a matrix by using its sub-matrices. Suppose A11 is non-
singular, then
detA = detA11 det(A22 ;A21A;1
11 A12):
On the other hand, if A22 is nonsingular, then
detA = detA22 det(A11 ;A12A;1
22 A21):
In particular, for any B 2 C m n and C 2 C n m, we have
det Im B
;C In
= det(In + CB) = det(Im + BC)
and for x y 2 C n
det(In + xy ) = 1 + y x:

24 LINEAR ALGEBRA
2.4 Matrix Calculus
Let X = xij] 2 C m n be a real or complex matrix and F(X) 2 C be a scalar real or
complex function of X then the derivative of F(X) with respect to X is de ned as
@
@XF(X) := @
@xij
F(X) :
Let A and B be constant complex matrices with compatible dimensions. Then the
following is a list of formulas for the derivatives2:
@
@X TracefAXBg = ATBT
@
@X TracefAXTBg = BA
@
@X TracefAXBXg = ATXTBT + BTXTAT
@
@X TracefAXBXTg = ATXBT + AXB
@
@X TracefXkg = k(Xk;1)T
@
@X TracefAXkg =
Pk;1
i=0 XiAXk;i;1 T
@
@X TracefAX;1Bg = ;(X;1BAX;1)T
@
@X logdetX = (XT);1
@
@X detXT = @
@X detX = (detX)(XT);1
@
@X detfXkg = k(detXk)(XT);1:
And nally, the derivative of a matrix A( ) 2 C m n with respect to a scalar 2 C is
de ned as
dA
d := daij
d
so that all the rules applicable to a scalar function also apply here. In particular, we
have
d(AB)
d = dA
d B + AdB
d
dA;1
d = ;A;1 dA
d A;1:
2Note that transpose rather than complex conjugate transpose should be used in the list even if the
involved matrices are complex matrices.

2.5. Kronecker Product and Kronecker Sum 25
2.5 Kronecker Product and Kronecker Sum
Let A 2 C m n and B 2 C p q, then the Kronecker product of A and B is de ned as
A B :=
2
6
6
6
4
a11B a12B a1nB
a21B a22B a2nB
...
...
...
am1B am2B amnB
3
7
7
7
5
2 C mp nq:
Furthermore, if the matrices A and B are square and A 2 C n n and B 2 C m m then
the Kronecker sum of A and B is de ned as
A B := (A Im) + (In B) 2 C nm nm:
Let X 2 C m n and let vec(X) denote the vector formed by stacking the columns of X
into one long vector:
vec(X) :=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
x11
x21
...
xm1
x12
x22
...
x1n
x2n
...
xmn
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
:
Then for any matrices A 2 C k m, B 2 C n l, and X 2 C m n, we have
vec(AXB) = (BT A)vec(X):
Consequently, if k = m and l = n, then
vec(AX + XB) = (BT A)vec(X):
Let A 2 C n n and B 2 C m m, and let f i i = 1 ::: ng be the eigenvalues of A
and f j j = 1 ::: mg be the eigenvalues of B. Then we have the following properties:
The eigenvalues of A B are the mn numbers i j, i = 1 2 ::: n, j = 1 2 ::: m.
The eigenvalues of A B = (A Im) + (In B) are the mn numbers i + j,
i = 1 2 ::: n, j = 1 2 ::: m.

26 LINEAR ALGEBRA
Let fxi i = 1 ::: ng be the eigenvectors of A and let fyj j = 1 ::: mg be the
eigenvectors of B. Then the eigenvectors of A B and A B correspond to the
eigenvalues i j and i + j are xi yj.
Using these properties, we can show the following Lemma.
Lemma 2.7 Consider the Sylvester equation
AX + XB = C (2:1)
where A 2 Fn n, B 2 Fm m, and C 2 Fn m are given matrices. There exists a unique
solution X 2 Fn m if and only if i(A)+ j(B) 6= 0 8i = 1 2 ::: n and j = 1 2 ::: m.
In particular, if B = A , (2.1) is called the Lyapunov Equation" and the necessary
and su cient condition for the existence of a unique solution is that i(A) + j(A) 6=
0 8i j = 1 2 ::: n
Proof. Equation (2.1) can be written as a linear matrix equation by using the Kro-
necker product:
(BT A)vec(X) = vec(C):
Now this equation has a unique solution i BT A is nonsingular. Since the eigenvalues
of BT A have the form of i(A)+ j(BT) = i(A)+ j(B), the conclusion follows. 2
The properties of the Lyapunov equations will be studied in more detail in the next
chapter.
2.6 Invariant Subspaces
Let A : C n 7;! C n be a linear transformation, be an eigenvalue of A, and x be a
corresponding eigenvector, respectively. Then Ax = x and A( x) = ( x) for any
2 C . Clearly, the eigenvector x de nes an one-dimensional subspace that is invariant
with respect to pre-multiplication by A since Akx = kx 8k. In general, a subspace
S C n is called invariant for the transformation A, or A-invariant, if Ax 2 S for every
x 2 S. In other words, that S is invariant for A means that the image of S under A
is contained in S: AS S. For example, f0g, C n, KerA, and ImA are all A-invariant
subspaces.
As a generalization of the one dimensional invariant subspace induced by an eigen-
vector, let 1 ::: k be eigenvalues of A (not necessarily distinct), and let xi be the cor-
responding eigenvectors and the generalized eigenvectors. Then S = spanfx1 ::: xkg
is an A-invariant subspace provided that all the lower rank generalized eigenvectors
are included. More speci cally, let 1 = 2 = = l be eigenvalues of A, and

2.6. Invariant Subspaces 27
let x1 x2 ::: xl be the corresponding eigenvector and the generalized eigenvectors ob-
tained through the following equations:
(A ; 1I)x1 = 0
(A ; 1I)x2 = x1
...
(A ; 1I)xl = xl;1:
Then a subspace S with xt 2 S for some t l is an A-invariant subspace only if all lower
rank eigenvectors and generalized eigenvectors of xt are in S, i.e., xi 2 S 81 i t.
This will be further illustrated in Example 2.1.
On the other hand, if S is a nontrivial subspace3 and is A-invariant, then there is
x 2 S and such that Ax = x.
An A-invariant subspace S C n is called a stable invariant subspace if all the
eigenvalues of A constrained to S have negative real parts. Stable invariant subspaces
will play an important role in computing the stabilizing solutions to the algebraic Riccati
equations in Chapter 13.
Example 2.1 Suppose a matrix A has the following Jordan canonical form
A x1 x2 x3 x4 = x1 x2 x3 x4
2
6
6
4
1 1
1
3
4
3
7
7
5
with Re 1 < 0, 3 < 0, and 4 > 0. Then it is easy to verify that
S1 = spanfx1g S12 = spanfx1 x2g S123 = spanfx1 x2 x3g
S3 = spanfx3g S13 = spanfx1 x3g S124 = spanfx1 x2 x4g
S4 = spanfx4g S14 = spanfx1 x4g S34 = spanfx3 x4g
are all A-invariant subspaces. Moreover, S1 S3 S12 S13, and S123 are stable A-invariant
subspaces. However, the subspacesS2 = spanfx2g, S23 = spanfx2 x3g, S24 = spanfx2 x4g,
and S234 = spanfx2 x3 x4g are not A-invariant subspaces since the lower rank general-
ized eigenvector x1 of x2 is not in these subspaces. To illustrate, consider the subspace
S23. Then by de nition, Ax2 2 S23 if it is an A-invariant subspace. Since
Ax2 = x2 + x1
Ax2 2 S23 would require that x1 be a linear combination of x2 and x3, but this is
impossible since x1 is independent of x2 and x3. 3
3We will say subspace S is trivial if S = f0g.

28 LINEAR ALGEBRA
2.7 Vector Norms and Matrix Norms
In this section, we will de ne vector and matrix norms. Let X be a vector space, a real-
valued function k k de ned on X is said to be a norm on X if it satis es the following
properties:
(i) kxk 0 (positivity)
(ii) kxk = 0 if and only if x = 0 (positive de niteness)
(iii) k xk = j jkxk, for any scalar (homogeneity)
(iv) kx + yk kxk+ kyk (triangle inequality)
for any x 2 X and y 2 X. A function is said to be a semi-norm if it satis es (i), (iii),
and (iv) but not necessarily (ii).
Let x 2 C n. Then we de ne the vector p-norm of x as
kxkp :=
nX
i=1
jxijp
!1=p
for 1 p 1:
In particular, when p = 1 2 1 we have
kxk1 :=
nX
i=1
jxij
kxk2 :=
v
u
u
t
nX
i=1
jxij2
kxk1 := max1 i n
jxij:
Clearly, norm is an abstraction and extension of our usual concept of length in 3-
dimensional Euclidean space. So a norm of a vector is a measure of the vector length",
for example kxk2 is the Euclidean distance of the vector x from the origin. Similarly,
we can introduce some kind of measure for a matrix.
Let A = aij] 2 C m n, then the matrix norm induced by a vector p-norm is de ned
as
kAkp := sup
x6=0
kAxkp
kxkp
:
In particular, for p = 1 2 1, the corresponding induced matrix norm can be computed
as
kAk1 = max1 j n
mX
i=1
jaijj (column sum)

2.7. Vector Norms and Matrix Norms 29
kAk2 =
p
max(A A)
kAk1 = max1 i m
nX
j=1
jaijj (row sum) :
The matrix norms induced by vector p-norms are sometimes called induced p-norms.
This is because kAkp is de ned by or induced from a vector p-norm. In fact, A can
be viewed as a mapping from a vector space C n equipped with a vector norm k kp to
another vector space C m equipped with a vector norm k kp. So from a system theoretical
point of view, the induced norms have the interpretation of input/output ampli cation
gains.
We shall adopt the following convention throughout the book for the vector and matrix
norms unless speci ed otherwise: let x 2 C n and A 2 C m n, then we shall denote the
Euclidean 2-norm of x simply by
kxk := kxk2
and the induced 2-norm of A by
kAk := kAk2 :
The Euclidean 2-norm has some very nice properties:
Lemma 2.8 Let x 2 Fn and y 2 Fm.
1. Suppose n m. Then kxk = kyk i there is a matrix U 2 Fn m such that x = Uy
and U U = I.
2. Suppose n = m. Then jx yj kxkkyk. Moreover, the equality holds i x = y
for some 2 F or y = 0.
3. kxk kyk i there is a matrix 2 Fn m with k k 1 such that x = y.
Furthermore, kxk < kyk i k k < 1.
4. kUxk = kxk for any appropriately dimensioned unitary matrices U.
Another often used matrix norm is the so called Frobenius norm. It is de ned as
kAkF :=
p
Trace(A A) =
v
u
u
t
mX
i=1
nX
j=1
jaijj2 :
However, the Frobenius norm is not an induced norm.
The following properties of matrix norms are easy to show:
Lemma 2.9 Let A and B be any matrices with appropriate dimensions. Then

30 LINEAR ALGEBRA
1. (A) kAk (This is also true for F norm and any induced matrix norm).
2. kABk kAkkBk. In particular, this gives A;1 kAk;1 if A is invertible.
(This is also true for any induced matrix norm.)
3. kUAV k = kAk, and kUAV kF = kAkF, for any appropriately dimensioned unitary
matrices U and V .
4. kABkF kAkkBkF and kABkF kBkkAkF.
Note that although pre-multiplication or post-multiplication of a unitary matrix on
a matrix does not change its induced 2-norm and F-norm, it does change its eigenvalues.
For example, let
A = 1 0
1 0 :
Then 1(A) = 1 2(A) = 0. Now let
U =
" 1p2
1p2
; 1p2
1p2
#
then U is a unitary matrix and
UA =
p
2 0
0 0
with 1(UA) =
p
2, 2(UA) = 0. This property is useful in some matrix perturbation
problems, particularly, in the computation of bounds for structured singular values
which will be studied in Chapter 10.
Lemma 2.10 Let A be a block partitioned matrix with
A =
2
6664
A11 A12 A1q
A21 A22 A2q
...
...
...
Am1 Am2 Amq
3
7775 =: Aij]
and let each Aij be an appropriately dimensioned matrix. Then for any induced matrix
p-norm
kAkp
2
6664
kA11kp kA12kp kA1qkp
...
...
...
kAm1kp kAm2kp kAmqkp
3
7775
p
: (2:2)
Further, the inequality becomes an equality if the F-norm is used.

2.7. Vector Norms and Matrix Norms 31
Proof. It is obvious that if the F-norm is used, then the right hand side of inequal-
ity (2.2) equals the left hand side. Hence only the induced p-norm cases, 1 p 1,
will be shown. Let a vector x be partitioned consistently with A as
x =
2
6664
x1
x2
...
xq
3
7775
and note that
kxkp =
2
6664
kx1kp
kx2kp
...
kxqkp
3
7775
p
:
Then
k Aij]kp := sup
kxkp=1
k Aij]xkp = sup
kxkp=1
2
6664
Pq
j=1 A1jxjPq
j=1 A2jxj
...Pq
j=1 Amjxj
3
7775
p
= sup
kxkp=1
2
66666664
Pq
j=1 A1jxj pPq
j=1 A2jxj p
...
Pq
j=1 Amjxj p
3
77777775
p
sup
kxkp=1
2
6664
Pq
j=1 kA1jkp kxjkpPq
j=1 kA2jkp kxjkp
...Pq
j=1 kAmjkp kxjkp
3
7775
p
= sup
kxkp=1
2
6664
...
...
...
kAm1kp kAm2kp kA1qkp
3
7775
2
6664
kx1kp
kx2kp
...
kxqkp
3
7775
p
sup
kxkp=1
h
kAijkp
i
p
kxkp
=
h
kAijkp
i
p
:
2

32 LINEAR ALGEBRA
2.8 Singular Value Decomposition
A very useful tool in matrix analysis is Singular Value Decomposition (SVD). It will be
seen that singular values of a matrix are good measures of the size" of the matrix and
that the corresponding singular vectors are good indications of strong/weak input or
output directions.
Theorem 2.11 Let A 2 Fm n. There exist unitary matrices
U = u1 u2 ::: um] 2 Fm m
V = v1 v2 ::: vn] 2 Fn n
such that
A = U V = 1 0
0 0
where
1 =
2
6664
1 0 0
0 2 0
...
...
...
...
0 0 p
3
7775
and
1 2 p 0 p = minfm ng:
Proof. Let = kAk and without loss of generality assume m n. Then from the
de nition of kAk, there exists a z 2 Fn such that
kAzk = kzk:
By Lemma 2.8, there is a matrix ~U 2 Fm n such that ~U ~U = I and
Az = ~Uz:
Now let
x = z
kzk 2 Fn y =
~Uz
~Uz
2 Fm:
We have Ax = y. Let
V = x V1 2 Fn n
and
U = y U1 2 Fm m
be unitary.4 Consequently, U AV has the following structure:
A1 := U AV = w
0 B
4Recall that it is always possible to extend an orthonormal set of vectors to an orthonormal basis
for the whole space.

2.8. Singular Value Decomposition 33
where w 2 Fn;1 and B 2 F(m;1) (n;1).
Since
A1
2
6664
1
0
...
0
3
7775
2
2
= ( 2 + w w)
it follows that kA1k2 2 + w w. But since = kAk = kA1k, we must have w = 0.
An obvious induction argument gives
U AV = :
This completes the proof. 2
The i is the i-th singular value of A, and the vectors ui and vj are, respectively,
the i-th left singular vector and the j-th right singular vector. It is easy to verify that
Avi = iui
A ui = ivi:
The above equations can also be written as
A Avi = 2
i vi
AA ui = 2
i ui:
Hence 2
i is an eigenvalue of AA or A A, ui is an eigenvector of AA , and vi is an
eigenvector of A A.
The following notations for singular values are often adopted:
(A) = max(A) = 1 = the largest singular value of A
and
(A) = min(A) = p = the smallest singular value of A :
Geometrically, the singular values of a matrix A are precisely the lengths of the semi-
axes of the hyperellipsoid E de ned by
E = fy : y = Ax x 2 C n kxk = 1g:
Thus v1 is the direction in which kyk is largest for all kxk = 1 while vn is the direction
in which kyk is smallest for all kxk = 1. From the input/output point of view, v1 (vn)
is the highest (lowest) gain input direction, while u1 (um) is the highest (lowest) gain
observing direction. This can be illustrated by the following 2 2 matrix:
A = cos 1 ;sin 1
sin 1 cos 1
1
2
cos 2 ;sin 2
sin 2 cos 2
:

34 LINEAR ALGEBRA
It is easy to see that A maps a unit disk to an ellipsoid with semi-axes of 1 and 2.
Hence it is often convenient to introduce the following alternative de nitions for the
largest singular value :
(A) := maxkxk=1
kAxk
and for the smallest singular value of a tall matrix:
(A) := minkxk=1
kAxk:
Lemma 2.12 Suppose A and are square matrices. Then
(i) j (A + ) ; (A)j ( )
(ii) (A ) (A) ( )
(iii) (A;1) = 1
(A) if A is invertible.
Proof.
(i) By de nition
(A + ) := minkxk=1
k(A + )xk
min
kxk=1
fkAxk;k xkg
min
kxk=1
kAxk; max
kxk=1
k xk
= (A) ; ( ):
Hence ; ( ) (A+ ); (A). The other inequality (A+ ); (A) ( )
follows by replacing A by A + and by ; in the above proof.
(ii) This follows by noting that
(A ) := minkxk=1
kA xk
=
r
minkxk=1
x A A x
(A) minkxk=1
k xk
= (A) ( ):
(iii) Let the singular value decomposition of A be A = U V , then A;1 = V ;1U .
Hence (A;1) = ( ;1) = 1= ( ) = 1= (A).

2.8. Singular Value Decomposition 35
2
Some useful properties of SVD are collected in the following lemma.
Lemma 2.13 Let A 2 Fm n and
1 2 r > r+1 = = 0 r minfm ng:
Then
1. rank(A) = r
2. KerA = spanfvr+1 ::: vng and (KerA)? = spanfv1 ::: vrg
3. ImA = spanfu1 ::: urg and (ImA)? = spanfur+1 ::: umg
4. A 2 Fm n has a dyadic expansion:
A =
rX
i=1
iuivi = Ur rVr
where Ur = u1 ::: ur], Vr = v1 ::: vr], and r = diag( 1 ::: r)
5. kAk2
F = 2
1 + 2
2 + + 2
r
6. kAk = 1
7. i(U0AV0) = i(A) i = 1 ::: p for any appropriately dimensioned unitary ma-
trices U0 and V0
8. Let k < r = rank(A) and Ak :=
Pk
i=1 iuivi , then
min
rank(B) k
kA;Bk = kA ;Akk = k+1:
Proof. We shall only give a proof for part 8. It is easy to see that rank(Ak) k and
kA;Akk = k+1. Hence, we only need to show that min
rank(B) k
kA;Bk k+1. Let
B be any matrix such that rank(B) k. Then
kA ;Bk = kU V ;Bk = k ;U BV k
Ik+1 0 ( ;U BV ) Ik+1
0 = k+1 ; ^B
where ^B = Ik+1 0 U BV Ik+1
0 2 F(k+1) (k+1) and rank( ^B) k. Let x 2 Fk+1
be such that ^Bx = 0 and kxk = 1. Then
kA ;Bk k+1 ; ^B ( k+1 ; ^B)x = k k+1xk k+1:
Since B is arbitrary, the conclusion follows. 2

36 LINEAR ALGEBRA
2.9 Generalized Inverses
Let A 2 C m n. A matrix X 2 C n m is said to be a right inverse of A if AX = I.
Obviously, A has a right inverse i A has full row rank, and, in that case, one of the right
inverses is given by X = A (AA );1. Similarly, if Y A = I then Y is called a left inverse
of A. By duality, A has a left inverse i A has full column rank, and, furthermore, one of
the left inverses is Y = (A A);1A . Note that right (or left) inverses are not necessarily
unique. For example, any matrix in the form I
? is a right inverse of I 0 .
More generally, if a matrix A has neither full row rank nor full column rank, then all
the ordinary matrix inverses do not exist however, the so called pseudo-inverse, known
also as the Moore-Penrose inverse, is useful. This pseudo-inverse is denoted by A+,
which satis es the following conditions:
(i) AA+A = A
(ii) A+AA+ = A+
(iii) (AA+) = AA+
(iv) (A+A) = A+A.
It can be shown that pseudo-inverse is unique. One way of computing A+ is by writing
A = BC
so that B has full column rank and C has full row rank. Then
A+ = C (CC );1(B B);1B :
Another way to compute A+ is by using SVD. Suppose A has a singular value decom-
position
A = U V
with
= r 0
0 0 r > 0:
Then A+ = V +U with
+ =
;1
r 0
0 0 :
2.10 Semide nite Matrices
A squarehermitian matrix A = A is said to be positive de nite (semi-de nite) , denoted
by A > 0 ( 0), if x Ax > 0 ( 0) for all x 6= 0. Suppose A 2 Fn n and A = A 0,
then there exists a B 2 Fn r with r rank(A) such that A = BB .
Lemma 2.14 Let B 2 Fm n and C 2 Fk n. Suppose m k and B B = C C. Then
there exists a matrix U 2 Fm k such that U U = I and B = UC.

2.10. Semide nite Matrices 37
Proof. Let V1 and V2 be unitary matrices such that
B1 = V1
B1
0 C1 = V2
C1
0
where B1 and C1 are full row rank. Then B1 and C1 have the same number of rows
and V3 := B1C1(C1C1);1 satis es V3 V3 = I since B B = C C. Hence V3 is a unitary
matrix and V3 B1 = C1. Finally let
U = V1
V3 0
0 V4
V2
for any suitably dimensioned V4 such that V4 V4 = I. 2
We can de ne square root for a positive semi-de nite matrix A, A1=2 = (A1=2) 0,
by
A = A1=2A1=2:
Clearly, A1=2 can be computed by using spectral decomposition or SVD: let A = U U ,
then
A1=2 = U 1=2U
where
= diagf 1 ::: ng 1=2 = diagf
p
1 :::
p
ng:
Lemma 2.15 Suppose A = A > 0 and B = B 0. Then A > B i (BA;1) < 1.
Proof. Since A > 0, we have A > B i
0 < I ;A;1=2BA;1=2 = I ;A;1=2(BA;1)A1=2:
However, A;1=2BA;1=2 and BA;1 aresimilar, and hence i(BA;1) = i(A;1=2BA;1=2).
Therefore, the conclusion follows by the fact that
0 < I ;A;1=2BA;1=2
i (A;1=2BA;1=2) < 1 i (BA;1) < 1. 2
Lemma 2.16 Let X = X 0 be partitioned as
X = X11 X12
X12 X22
:
Then KerX22 KerX12. Consequently, if X+
22 is the pseudo-inverse of X22, then Y =
X12X+
22 solves
Y X22 = X12
and
X11 X12
X12 X22
= I X12X+
22
0 I
X11 ;X12X+
22X12 0
0 X22
I 0
X+
22X12 I :

38 LINEAR ALGEBRA
Proof. Without loss of generality, assume
X22 = U1 U2
1 0
0 0
U1
U2
with 1 = 1 > 0 and U = U1 U2 unitary. Then
Ker X22 = spanfcolumns of U2g
and
X+
22 = U1 U2
;1
1 0
0 0
U1
U2
:
Moreover
I 0
0 U
X11 X12
X12 X22
I 0
0 U 0
gives X12U2 = 0. Hence, Ker X22 Ker X12 and now
X12X+
22X22 = X12U1U1 = X12U1U1 + X12U2U2 = X12:
The factorization follows easily. 2
2.11 Matrix Dilation Problems*
In this section, we consider the following induced 2-norm optimization problem:
minX
X B
C A (2:3)
where X, B, C, and A are constant matrices of compatible dimensions.
The matrix X B
C A is a dilation of its sub-matrices as indicated in the following
diagram:
d
c
d
c
cdcd
C A A
B
A
X B
C A
?
6
?
6
-
-

2.11. Matrix Dilation Problems* 39
In this diagram, c" stands for the operation of compression and d" stands for dilation.
Compression is always norm non-increasing and dilation is always norm non-decreasing.
Sometimes dilation can be made to be norm preserving. Norm preserving dilations are
the focus of this section.
The simplest matrix dilation problem occurs when solving
minX
X
A : (2:4)
Although (2.4) is a much simpli ed version of (2.3), we will see that it contains all the
essential features of the general problem. Letting 0 denote the minimum norm in (2.4),
it is immediate that
0 = kAk:
The following theorem characterizes all solutions to (2.4).
Theorem 2.17 8 0,
X
A
i there is a Y with kY k 1 such that
X = Y( 2I ;A A)1=2:
Proof.
X
A
i
X X + A A 2I
i
X X ( 2I ;A A):
Now suppose X X ( 2I ;A A) and let
Y := X
h
( 2I ;A A)1=2
i+
then X = Y ( 2I ; A A)1=2 and Y Y I. Similarly if X = Y( 2I ; A A)1=2 and
Y Y I then X X ( 2I ;A A): 2
This theorem implies that, in general, (2.4) has more than one solution, which is
in contrast to the minimization in the Frobenius norm in which X = 0 is the unique
solution. The solution X = 0 is the central solution but others are possible unless
A A = 2
0I.

40 LINEAR ALGEBRA
Remark 2.1 The theorem still holds if ( 2I ; A A)1=2 is replaced by any matrix R
such that 2I ;A A = R R. ~
A more restricted version of the above theorem is shown in the following corollary.
Corollary 2.18 8 > 0,
X
A (< )
i
X( 2I ;A A);1=2 1(< 1):
The corresponding dual results are
Theorem 2.19 8 0
X A
i there is a Y , kYk 1, such that
X = ( 2I ;AA )1=2Y:
Corollary 2.20 8 > 0
X A (< )
i
( 2I ;AA );1=2X 1 (< 1):
Now, returning to the problem in (2.3), let
0 := minX
X B
C A : (2:5)
The following so called Parrott's theorem will play an important role in many control
related optimization problems. The proof is the straightforward application of Theo-
rem 2.17 and its dual, Theorem 2.19.
Theorem 2.21 (Parrott's Theorem) The minimum in (2.5) is given by
0 = max C A B
A : (2:6)

Proof. Denote by ^ the right hand side of the equation (2.6). Clearly, 0 ^ since
compressions are norm non-increasing, and that 0 ^ will be shown by using Theo-
rem 2.17 and Theorem 2.19.
Suppose A 2 C n m and n m (the case for m > n can be shown in the same
fashion). Then A has the following singular value decomposition:
A = U m
0n;m m
V U 2 C n n V 2 C m m:
Hence
^2I ;A A = V (^2I ; 2
m)V
and
^2I ;AA = U ^2I ; 2
m 0
0 ^2In;m
U :
Now let
(^2I ;A A)1=2 := V (^2I ; 2
m)1=2V
and
(^2I ;AA )1=2 := U (^2I ; 2
m)1=2 0
0 ^In;m
U :
Then it is easy to verify that
(^2I ;A A)1=2A = A (^2I ;AA )1=2:
Using this equality, we can show that
;A (^2I ;A A)1=2
(^2I ;AA )1=2 A
;A (^2I ;A A)1=2
(^2I ;AA )1=2 A
= ^2I 0
0 ^2I :
Now we are ready to show that 0 ^.
From Theorem 2.17 we have that B = Y(^2I ; A A)1=2 for some Y such that
kYk 1. Similarly, Theorem 2.19 yields C = (^2I ; AA )1=2Z for some Z with
kZk 1. Now let ^X = ;YA Z. Then
^X B
C A = ;YA Z Y(^2I ;A A)1=2
(^2I ;AA )1=2Z A
= Y
I
;A (^2I ;A A)1=2
(^2I ;AA )1=2 A
Z
I
;A (^2I ;A A)1=2
(^2I ;AA )1=2 A
= ^:

42 LINEAR ALGEBRA
Thus ^ 0, so ^ = 0. 2
This theorem gives one solution to (2.3) and an expression for 0. As in (2.4), there
may be more than one solution to (2.3), although the proof of theorem 2.21 exhibits
only one. Theorem 2.22 considers the problem of parameterizing all solutions. The
solution ^X = ;YA Z is the central" solution analogous to X = 0 in (2.4).
Theorem 2.22 Suppose 0. The solutions X such that
X B
C A (2:7)
are exactly those of the form
X = ;YA Z + (I ;YY )1=2W(I ;Z Z)1=2 (2:8)
where W is an arbitrary contraction (kWk 1), and Y with kY k 1 and Z with
kZk 1 solve the linear equations
B = Y( 2I ;A A)1=2 (2.9)
C = ( 2I ;AA )1=2Z: (2.10)
Proof. Since 0, again from Theorem 2.19 there exists a Z with kZk 1 such
that
C = ( 2I ;AA )1=2Z:
Note that using the above expression for C we have
2I ; C A C A
= (I ;Z Z)1=2 0
;A Z ( 2I ;A A)1=2
(I ;Z Z)1=2 0
;A Z ( 2I ;A A)1=2 :
Now apply Theorem 2.17 (Remark 2.1) to inequality (2.7) with respect to the partitioned
matrix X B
C A to get
X B = ^W (I ;Z Z)1=2 0
;A Z ( 2I ;A A)1=2
for some contraction ^W, ^W 1. Partition ^W as ^W = W1 Y to obtain the
expression for X and B:
X = ;YA Z + W1(I ;Z Z)1=2
B = Y( 2I ;A A)1=2:
Then kYk 1 and the theorem follows by noting that W1 Y 1 i there is a
W, kWk 1, such that W1 = (I ;YY )1=2W. 2
The following corollary gives an alternative version of Theorem 2.22 when > 0.

Corollary 2.23 For > 0.
X B
C A (< ) (2:11)
i
(I ;YY );1=2(X + YA Z)(I ;Z Z);1=2 (< ) (2:12)
where
Y = B( 2I ;A A);1=2 (2.13)
Z = ( 2I ;AA );1=2C: (2.14)
Note that in the case of > 0, I ; Y Y and I ; Z Z are invertible since Corol-
lary 2.18 and 2.20 clearly show that kY k < 1 and kZk < 1. There are many alternative
characterizations of solutions to (2.11), although the formulas given above seem to be
the simplest.
As a straightforward application of the dilation results obtained above, consider the
following matrix approximation problem:
0 = minQ
kR + UQVk (2:15)
where R, U, and V are constant matrices such that U U = I and V V = I.
Corollary 2.24 The minimum achievable norm is given by
0 = maxfkU?Rk kRV?kg
and the parameterization for all optimal solutions Q can be obtained from Theorem 2.22
with X = Q+ U RV , A = U?RV?, B = U RV?, and C = U?RV .
Proof. Let U? and V? be such that U U? and V
V?
are unitary matrices.
Then
0 = minQ
kR + UQVk
= minQ
U U? (R + UQV ) V
V?
= minQ
U RV + Q U RV?
U?RV U?RV?
:
The result follows after applying Theorem 2.21 and 2.22. 2
A similar problem arises in H1 control theory.

44 LINEAR ALGEBRA
2.12 Notes and References
A very extensive treatment of most topics in this chapter can be found in Brogan
1991], Horn and Johnson 1990,1991] and Lancaster and Tismenetsky 1985]. Golub
and Van Loan's book 1983] contains many numerical algorithms for solving most of the
problems in this chapter. The matrix dilation theory can be found in Davis, Kahan,
and Weinberger 1982].

3
Linear Dynamical Systems
This chapter reviews some basic system theoretical concepts. The notions of controlla-
bility, observability, stabilizability, and detectability are de ned and various algebraic
and geometric characterizations of these notions are summarized. Kalman canonical de-
composition, pole placement, and observer theory are then introduced. The solutions of
Lyapunov equations and their connections with system stability, controllability, and so
on, are discussed. System interconnections and realizations, in particular the balanced
realization, are studied in some detail. Finally, the concepts of system poles and zeros
are introduced.
3.1 Descriptions of Linear Dynamical Systems
Let a nite dimensional linear time invariant (FDLTI) dynamical system be described
by the following linear constant coe cient di erential equations:
_x = Ax + Bu x(t0) = x0 (3.1)
y = Cx + Du (3.2)
where x(t) 2 Rn is called the system state, x(t0) is called the initial condition of the
system, u(t) 2 Rm is called the system input, and y(t) 2 Rp is the system output.
The A B C and D are appropriately dimensioned real constant matrices. A dynamical
system with single input (m = 1) and single output (p = 1) is called a SISO (single input
and single output) system, otherwise it is called MIMO (multiple input and multiple
45

46 LINEAR DYNAMICAL SYSTEMS
output) system. The corresponding transfer matrix from u to y is de ned as
Y(s) = G(s)U(s)
where U(s) and Y (s) are the Laplace transform of u(t) and y(t) with zero initial condi-
tion (x(0) = 0). Hence, we have
G(s) = C(sI ;A);1B + D:
Note that the system equations (3.1) and (3.2) can be written in a more compact matrix
form:
_x
y = A B
C D
x
u :
To expedite calculations involving transfer matrices, the notation
A B
C D := C(sI ;A);1B + D
will be used. Other reasons for using this notation will be discussed in Chapter 10.
Note that
A B
C D
is a real block matrix, not a transfer function.
Now given the initial condition x(t0) and the input u(t), the dynamical system
response x(t) and y(t) for t t0 can be determined from the following formulas:
x(t) = eA(t;t0)x(t0) +
Z t
t0
eA(t; )Bu( )d (3.3)
y(t) = Cx(t) + Du(t): (3.4)
In the case of u(t) = 0 8t t0, it is easy to see from the solution that for any t1 t0
and t t0, we have
x(t) = eA(t;t1)x(t1):
Therefore, the matrix function (t t1) = eA(t;t1) acts as a transformation from one
state to another, and thus (t t1) is usually called the state transition matrix. Since
the state of a linear system at one time can be obtained from the state at another
through the transition matrix, we can assume without loss of generality that t0 = 0.
This will be assumed in the sequel.
The impulse matrix of the dynamical system is de ned as
g(t) = L;1 fG(s)g = CeAtB1+(t) + D (t)
where (t) is the unit impulse and 1+(t) is the unit step de ned as
1+(t) := 1 t 0
0 t < 0:

3.2. Controllability and Observability 47
The input/output relationship (i.e., with zero initial state: x0 = 0) can be described by
the convolution equation
y(t) = (g u)(t) :=
Z 1
;1
g(t; )u( )d =
Z t
;1
g(t ; )u( )d :
3.2 Controllability and Observability
We now turn to some very important concepts in linear system theory.
De nition 3.1 The dynamical system described by the equation (3.1) or the pair
(A B) is said to be controllable if, for any initial state x(0) = x0, t1 > 0 and nal
state x1, there exists a (piecewise continuous) input u( ) such that the solution of (3.1)
satis es x(t1) = x1. Otherwise, the system or the pair (A B) is said to be uncontrollable.
The controllability (or observability introduced next) of a system can be veri ed through
some algebraic or geometric criteria.
Theorem 3.1 The following are equivalent:
(i) (A B) is controllable.
(ii) The matrix
Wc(t) :=
Z t
0
eA BB eA d
is positive de nite for any t > 0.
(iii) The controllability matrix
C = B AB A2B ::: An;1B
has full row rank or, in other words, hAjImBi :=
Pn
i=1 Im(Ai;1B) = Rn.
(iv) The matrix A ; I B] has full row rank for all in C .
(v) Let and x be any eigenvalue and any corresponding left eigenvector of A, i.e.,
x A = x , then x B 6= 0.
(vi) The eigenvalues of A+BF can be freely assigned (with the restriction that complex
eigenvalues are in conjugate pairs) by a suitable choice of F.

Proof.
(i) , (ii): Suppose Wc(t1) > 0 for some t1 > 0, and let the input be de ned as
u( ) = ;B eA (t1; )Wc(t1);1(eAt1 x0 ;x1):
Then it is easy to verify using the formula in (3.3) that x(t1) = x1. Since x1 is
arbitrary, the pair (A B) is controllable.
To show that the controllability of (A B) implies that Wc(t) > 0 for any t > 0,
assume that (A B) is controllable but Wc(t1) is singular for some t1 > 0. Since
eAtBB eA t 0 for all t, there exists a real vector 0 6= v 2 Rn such that
v eAtB = 0 t 2 0 t1]:
Now let x(t1) = x1 = 0, and then from the solution (3.3), we have
0 = eAt1x(0) +
Z t1
0
eA(t1; )Bu( )d :
Pre-multiply the above equation by v to get
0 = v eAt1x(0):
If we chose the initial state x(0) = e;At1v, then v = 0, and this is a contradiction.
Hence, Wc(t) can not be singular for any t > 0.
(ii) , (iii): Suppose Wc(t) > 0 for all t > 0 (in fact, it can be shown that Wc(t) > 0
for all t > 0 i , for some t1, Wc(t1) > 0) but the controllability matrix C does not
have full row rank. Then there exists a v 2 Rn such that
v AiB = 0
for all 0 i n ; 1. In fact, this equality holds for all i 0 by the Cayley-
Hamilton Theorem. Hence,
v eAtB = 0
for all t or, equivalently, v Wc(t) = 0 for all t this is a contradiction, and hence,
the controllability matrix C must be full row rank. Conversely, suppose C has full
row rank but Wc(t) is singular for some t1. Then there exists a 0 6= v 2 Rn such
that v eAtB = 0 for all t 2 0 t1]. Therefore, set t = 0, and we have
v B = 0:
Next, evaluate the i-th derivative of v eAtB = 0 at t = 0 to get
v AiB = 0 i > 0:
Hence, we have
v B AB A2B ::: An;1B = 0
or, in other words, the controllability matrix C does not have full row rank. This
is again a contradiction.

(iii) ) (iv): Suppose, on the contrary, that the matrix
A ; I B
does not have full row rank for some 2 C . Then there exists a vector x 2 C n
such that
x A; I B = 0
i.e., x A = x and x B = 0. However, this will result in
x B AB ::: An;1B = x B x B ::: n;1x B = 0
i.e., the controllability matrix C does not have full row rank, and this is a contra-
diction.
(iv) ) (v): This is obvious from the proof of (iii) ) (iv).
(v) ) (iii): We will again prove this by contradiction. Assume that (v) holds but
rank C = k < n. Then in section 3.3, we will show that there is a transformation
T such that
TAT;1 = Ac A12
0 Ac
TB = Bc
0
with Ac 2 R(n;k) (n;k). Let 1 and xc be any eigenvalue and any corresponding
left eigenvector of Ac, i.e., xcAc = 1xc. Then x (TB) = 0 and
x = 0
xc
is an eigenvector of TAT;1 corresponding to the eigenvalue 1, which implies
that (TAT;1 TB) is not controllable. This is a contradiction since similarity
transformation does not change controllability. Hence, the proof is completed.
(vi) ) (i): This follows the same arguments as in the proof of (v) ) (iii): assume
that (vi) holds but (A B) is uncontrollable. Then, there is a decomposition so
that some subsystems are not a ected by the control, but this contradicts the
condition (vi).
(i) ) (vi): This will be clear in section 3.4. In that section, we will explicitly construct
a matrix F so that the eigenvalues of A + BF are in the desired locations.
2
De nition 3.2 An unforced dynamical system _x = Ax is said to be stable if all the
eigenvalues of A are in the open left half plane, i.e., Re (A) < 0. A matrix A with such
a property is said to be stable or Hurwitz.

De nition 3.3 The dynamical system (3.1), or the pair (A B), is said to be stabilizable
if there exists a state feedback u = Fx such that the system is stable, i.e., A + BF is
stable.
Therefore, it is more appropriate to call this stabilizability the state feedback stabiliz-
ability to di erentiate it from the output feedback stabilizability de ned later.
The following theorem is a consequence of Theorem 3.1.
(i) (A B) is stabilizable.
(ii) The matrix A ; I B] has full row rank for all Re 0.
(iii) For all and x such that x A = x and Re 0, x B 6= 0.
(iv) There exists a matrix F such that A + BF is Hurwitz.
We now consider the dual notions of observability and detectability of the system
described by equations (3.1) and (3.2).
De nition 3.4 The dynamical system described by the equations (3.1) and (3.2) or
by the pair (C A) is said to be observable if, for any t1 > 0, the initial state x(0) = x0
can be determined from the time history of the input u(t) and the output y(t) in the
interval of 0 t1]. Otherwise, the system, or (C A), is said to be unobservable.
(i) (C A) is observable.
(ii) The matrix
Wo(t) :=
Z t
0
eA C CeA d
is positive de nite for any t > 0.
(iii) The observability matrix
O =
2
666664
C
CA
CA2
...
CAn;1
3
777775
has full column rank or
Tn
i=1 Ker(CAi;1) = 0.
(iv) The matrix A ; I
C has full column rank for all in C .

(v) Let and y be any eigenvalue and any corresponding right eigenvector of A, i.e.,
Ay = y, then Cy 6= 0.
(vi) The eigenvalues of A+LC can be freely assigned (with the restriction that complex
eigenvalues are in conjugate pairs) by a suitable choice of L.
(vii) (A C ) is controllable.
Proof. First, we will show the equivalence between conditions (i) and (iii). Once this
is done, the rest will follow by the duality or condition (vii).
(i) ( (iii): Note that given the input u(t) and the initial condition x0, the output in
the time interval 0 t1] is given by
y(t) = CeAtx(0) +
Z t
0
CeA(t; )Bu( )d + Du(t):
Since y(t) and u(t) are known, there is no loss of generality in assuming u(t) =
0 8t. Hence,
y(t) = CeAtx(0) t 2 0 t1]:
From this equation, we have
2
6664
y(0)
_y(0)
...
y(n;1)(0)
3
7775 =
2
666664
C
CA
CA2
...
CAn;1
3
777775
x(0)
where y(i) stands for the i-th derivative of y. Since the observability matrix O
has full column rank, there is a unique solution x(0) in the above equation. This
completes the proof.
(i) ) (iii): This will be proven by contradiction. Assume that (C A) is observable but
that the observability matrix does not have full column rank, i.e., there is a vector
x0 such that Ox0 = 0 or equivalently CAix0 = 0 8i 0 by the Cayley-Hamilton
Theorem. Now suppose the initial state x(0) = x0, then y(t) = CeAtx(0) = 0.
This implies that the system is not observable since x(0) cannot be determined
from y(t) 0.
2
De nition 3.5 The system, or the pair (C A), is detectable if A + LC is stable for
some L.

(i) (C A) is detectable.
(ii) The matrix A ; I
C has full column rank for all Re 0.
(iii) For all and x such that Ax = x and Re 0, Cx 6= 0.
(iv) There exists a matrix L such that A + LC is Hurwitz.
(v) (A C ) is stabilizable.
The conditions (iv) and (v) of Theorem 3.1 and Theorem 3.3 and the conditions
(ii) and (iii) of Theorem 3.2 and Theorem 3.4 are often called Popov-Belevitch-Hautus
(PBH) tests. In particular, the following de nitions of modal controllability and ob-
servability are often useful.
De nition 3.6 Let be an eigenvalue of A or, equivalently, a mode of the system.
Then the mode is said to be controllable (observable) if x B 6= 0 (Cx 6= 0) for all left
(right) eigenvectors of A associated with , i.e., x A = x (Ax = x) and 0 6= x 2 C n.
Otherwise, the mode is said to be uncontrollable (unobservable).
It follows that a system is controllable (observable) if and only if every mode is control-
lable (observable). Similarly, a system is stabilizable (detectable) if and only if every
unstable mode is controllable (observable).
For example, consider the following 4th order system:
A B
C D =
2
66664
1 1 0 0 0
0 1 1 0 1
0 0 1 0
0 0 0 2 1
1 0 0 0
3
77775
with 1 6= 2. Then, the mode 1 is not controllable if = 0, and 2 is not observable
if = 0. Note that if 1 = 2, the system is uncontrollable and unobservable for any
and since in that case, both
x1 =
2
664
0
0
1
0
3
775 and x2 =
2
664
0
0
0
1
3
775
are the left eigenvectors of A corresponding to 1. Hence any linear combination of x1
and x2 is still an eigenvector of A corresponding to 1. In particular, let x = x1 ; x2,
then x B = 0, and as a result, the system is not controllable. Similar arguments can be
applied to check observability. However, if the B matrix is changed into a 4 2 matrix

3.3. Kalman Canonical Decomposition 53
with the last two rows independent of each other, then the system is controllable even
if 1 = 2. For example, the reader may easily verify that the system with
B =
2
664
0 0
1 0
1
1 0
3
775
is controllable for any .
In general, for a system given in the Jordan canonical form, the controllability and
observability can be concluded by inspection. The interested reader may easily de-
rive some explicit conditions for the controllability and observability by using Jordan
canonical form and the tests (iv) and (v) of Theorem 3.1 and Theorem 3.3.
3.3 Kalman Canonical Decomposition
There are usually many di erent coordinate systems to describe a dynamical system.
For example, consider a simple pendulum, the motion of the pendulum can be uniquely
determined either in terms of the angle of the string attached to the pendulum or in
terms of the vertical displacement of the pendulum. However, in most cases, the angular
displacement is a more natural description than the vertical displacement in spite of the
fact that they both describe the same dynamical system. This is true for most physical
dynamical systems. On the other hand, although some coordinates may not be natural
descriptions of a physical dynamical system, they may make the system analysis and
synthesis much easier.
In general, let T 2 Rn n be a nonsingular matrix and de ne
x = Tx:
Then the original dynamical system equations (3.1) and (3.2) become
_x = TAT;1x + TBu
y = CT;1x + Du:
These equations represent the same dynamical system for any nonsingular matrix T,
and hence, we can regard these representations as equivalent. It is easy to see that the
input/output transfer matrix is not changed under the coordinate transformation, i.e.,
G(s) = C(sI ;A);1B + D = CT;1(sI ;TAT;1);1TB + D:
In this section, we will consider the system structure decomposition using coordinate
transformation if the system is not completely controllable and/or is not completely
observable. To begin with, let us consider further the dynamical systems related by a
similarity transformation:
A B
C D 7;!
"
A B
C D
#
= TAT;1 TB
CT;1 D :

The controllability and observability matrices are related by
C = TC O = OT;1:
Moreover, the following theorem follows easily from the PBH tests or from the above
relations.
Theorem 3.5 The controllability (or stabilizability) and observability (or detectability)
are invariant under similarity transformations.
Using this fact, we can now show the following theorem.
Theorem 3.6 If the controllability matrix C has rank k1 < n, then there exists a simi-
larity transformation
x = xc
xc
= Tx
such that
_xc
_xc
= Ac A12
0 Ac
xc
xc
+ Bc
0 u
y = Cc Cc
xc
xc
+ Du
with Ac 2 C k1 k1 and (Ac Bc) controllable. Moreover,
G(s) = C(sI ;A);1B + D = Cc(sI ;Ac);1Bc + D:
Proof. Since rankC = k1 < n, the pair (A B) is not controllable. Now let q1 q2 ::: qk1
be any linearly independent columns of C. Let qi i = k1 +1 ::: n be any n;k1 linearly
independent vectors such that the matrix
Q := q1 qk1 qk1+1 qn
is nonsingular. De ne
T := Q;1:
Then the transformation x = Tx will give the desired decomposition. To see that,
note that for each i = 1 2 ::: k1, Aqi can be written as a linear combination of qi i =
1 2 ::: k1 since Aqi is a linear combination of the columns of C by the Cayley-Hamilton
Theorem. Therefore, we have
AT;1 = Aq1 Aqk1 Aqk1+1 Aqn
= q1 qk1 qk1+1 qn
Ac A12
0 Ac
= T;1 Ac A12
0 Ac

for some k1 k1 matrix Ac. Similarly, each column of the matrix B is a linear combi-
nation of qi i = 1 2 ::: k1, hence
B = Q Bc
0 = T;1 Bc
0
for some Bc 2 C k1 m.
To show that (Ac Bc) is controllable, note that rank C = k1 and
C = T;1 Bc AcBc Ak1;1
c Bc An;1
c Bc
0 0 0 0 :
Since, for each j k1, Aj
c is a linear combination of Ai
c i = 0 1 ::: (k1 ;1) by Cayley-
Hamilton theorem, we have
rank Bc AcBc Ak1;1
c Bc = k1
i.e., (Ac Bc) is controllable. 2
A numerically reliable way to nd a such transformation T is to use QR factorization.
For example, if the controllability matrix C has the QR factorization QR = C, then
T = Q;1.
Corollary 3.7 If the system is stabilizable and the controllability matrix C has rank
k1 < n, then there exists a similarity transformation T such that
TAT;1 TB
CT;1 D =
2
64
Ac A12 Bc
0 Ac 0
Cc Cc D
3
75
with Ac 2 C k1 k1, (Ac Bc) controllable and with Ac stable.
Hence, the state space x is partitioned into two orthogonal subspaces
xc
0 and 0
xc
with the rst subspace controllable from the input and second completely uncontrollable
from the input (i.e., the state xc are not a ected by the control u). Write these subspaces
in terms of the original coordinates x, we have
x = T;1x = q1 qk1 qk1+1 qn
xc
xc
:
So the controllable subspace is the span of qi i = 1 ::: k1 or, equivalently, Im C. On the
other hand, the uncontrollable subspace is given by the complement of the controllable
subspace.
By duality, we have the following decomposition if the system is not completely
observable.

Theorem 3.8 If the observability matrix O has rank k2 < n, then there exists a simi-
larity transformation T such that
TAT;1 TB
CT;1 D =
2
64
Ao 0 Bo
A21 Ao Bo
Co 0 D
3
75
with Ao 2 C k2 k2 and (Co Ao) observable.
Corollary 3.9 If the system is detectable and the observability matrix C has rank k2 <
n, then there exists a similarity transformation T such that
TAT;1 TB
CT;1 D =
2
64
Ao 0 Bo
A21 Ao Bo
Co 0 D
3
75
with Ao 2 C k2 k2, (Co Ao) observable and with Ao stable.
Similarly, we have
G(s) = C(sI ;A);1B + D = Co(sI ;Ao);1Bo + D:
Carefully combining the above two theorems, we get the following Kalman Canonical
Decomposition. The proof is left to the reader as an exercise.
Theorem 3.10 Let an LTI dynamical system be described by the equations (3.1) and (3.2).
Then there exists a nonsingular coordinate transformation x = Tx such that
2
664
_xco
_xco
_xco
_xco
3
775 =
2
664
Aco 0 A13 0
A21 Aco A23 A24
0 0 Aco 0
0 0 A43 Aco
3
775
2
664
xco
xco
xco
xco
3
775+
2
664
Bco
Bco
0
0
3
775u
y = Cco 0 Cco 0
2
664
xco
xco
xco
xco
3
775+ Du
or equivalently
TAT;1 TB
CT;1 D =
2
66664
Aco 0 A13 0 Bco
A21 Aco A23 A24 Bco
0 0 Aco 0 0
0 0 A43 Aco 0
Cco 0 Cco 0 D
3
77775

where the vector xco is controllable and observable, xco is controllable but unobserv-
able, xco is observable but uncontrollable, and xco is uncontrollable and unobservable.
Moreover, the transfer matrix from u to y is given by
G(s) = Cco(sI ;Aco);1Bco + D:
One important issue is that although the transfer matrix of a dynamical system
A B
C D
is equal to its controllable and observable part
"
Aco Bco
Cco D
#
their internal behaviors are very di erent. In other words, while their input/output
behaviors are the same, their state space response with nonzero initial conditions are
very di erent. This can be illustrated by the state space response for the simple system
2
664
_xco
_xco
_xco
_xco
3
775 =
2
664
Aco 0 0 0
0 Aco 0 0
0 0 Aco 0
0 0 0 Aco
3
775
2
664
xco
xco
xco
xco
3
775+
2
664
Bco
Bco
0
0
3
775u
y = Cco 0 Cco 0
2
664
xco
xco
xco
xco
3
775
with xco controllable and observable, xco controllable but unobservable, xco observable
but uncontrollable, and xco uncontrollable and unobservable.
The solution to this system is given by
2
664
xco(t)
xco(t)
xco(t)
xco(t)
3
775 =
2
6664
eAcotxco(0) +
Rt
0 eAco(t; )Bcou( )d
eAcotxco(0) +
Rt
0 eAco(t; )Bcou( )d
eAcotxco(0)
eAcotxco(0)
3
7775
y(t) = Ccoxco(t) + Ccoxco
note that xco(t) and xco(t) are not a ected by the input u, while xco(t) and xco(t) do
not show up in the output y. Moreover, if the initial condition is zero, i.e., x(0) = 0,
then the output
y(t) =
Z t
0
CcoeAco(t; )Bcou( )d :

However, if the initial state is not zero, then the response xco(t) will show up in the
output. In particular, if Aco is not stable, then the output y(t) will grow without bound.
The problems with uncontrollable and/or unobservable unstable modes are even more
profound than what we have mentioned. Since the states are the internal signals of the
dynamical system, any unbounded internal signal will eventually destroy the system.
On the other hand, since it is impossible to make the initial states exactly zero, any
uncontrollable and/or unobservable unstable mode will result in unacceptable system
behavior. This issue will be exploited further in section 3.7. In the next section, we will
consider how to place the system poles to achieve desired closed-loop behavior if the
system is controllable.
3.4 Pole Placement and Canonical Forms
Consider a MIMO dynamical system described by
_x = Ax + Bu
y = Cx + Du
and let u be a state feedback control law given by
u = Fx + v:
This closed-loop system is as shown in Figure 3.1, and the closed-loop system equations
are given by
_x = (A + BF)x + Bv
y = (C + DF)x + Dv:
e ee
y ? u v
6
-
6
-
D
F
A
x
BC
R
Figure 3.1: State Feedback
Then we have the following lemma in which the proof is simple and is left as an
exercise to the reader.

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