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Publisher:

Princeton University Press

Publication Date:

2009

Number of Pages:

263

Format:

Hardcover

Price:

59.95

ISBN:

9780691140216

Category:

Textbook

[Reviewed by , on ]

John D. Cook

02/9/2010

*Linear Systems Theory* by João Hespanha is primarily concerned with systems of the form

x' =Ax+Bu;y=Cx+Du.

Here *x* = *x*(*t)* is a vector-valued function of time. *A*, *B*, *C*, and *D* are matrices, or possibly matrix-valued functions of time. In the continuous-time case, *x*' is the derivative of *x* with respect to time. An initial condition *x*(0) = *x*_{0} must be specified to uniquely determine the solution.

It would be easy to underestimate the scope and usefulness linear system theory. Some possible objections would be as follows.

- How useful is a theory that only considers linear systems?
- Why limit your attention to first order derivatives?
- Aren't linear differential equations very well understood?

First, many systems of practical interest are non-linear, even highly non-linear, and strictly speaking linear systems theory does not apply to such systems. However, the local behavior of non-linear systems is often described well by linear approximations. Also, devices designed to control linear systems often work surprisingly well when used to control non-linear systems.

Second, systems of first order differential equations are more general than they seem. A system of nth order differential equations can easily be reduced to a (higher dimensional) system of first order differential equations by introducing additional variables.

Finally, while linear differential equations are indeed well understood, there are open problems in the *control* of linear systems. A course on differential equations would consider the function *u*(*t*) as given. Also, such a course would consider the equation *y* = *Cx* + *Du* irrelevant since it does not impact the solution *x*. However, control theory is concerned with choosing *u* to regulate *y*. For example, one may be interested in determining the input *u* that drives *y* to zero most quickly, subject to energy constraints on *u*.

In addition to systems of differential equations, *Linear Systems Theory* also considers difference equations of the form

x^{+}=Ax+Bu;y=Cx+Du.

where *x*^{+}(*t*) = *x*(*t*+1). The theory of such discrete-time systems is surprisingly similar to that of continuous-time systems involving differential equations.

*Linear Systems Theory* developed from a set of lecture notes, and in some ways still reads like a set of lecture notes. The book is self-contained and polished, but the pace is brisk. Also, each chapter corresponds to the material that might be covered in one lecture.

The book makes extensive use of linear algebra. It assumes familiarity with basic linear algebra but reviews more advanced aspects of the subject that are not always covered in an introductory course: positive definite matricies, Jordan canonical form, pseudo-inverses, etc. Notes on using *Matlab*® are sprinkled throughout the text. Readers who use *Matlab*® will appreciate these notes but others can easy overlook these notes without loss of continuity.

*Linear Systems Theory* covers a great deal of material in around 250 pages. An instructor using the book as a text would need to help students unpack some of the condensed presentation. Also, an instructor would need to provide applications; the book does not contain many applications until near the end.

John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs at The Endeavour.

PREAMBLE xiii

LINEAR SYSTEMS I - BASIC CONCEPTS

I: SYSTEM REPRESENTATION 3

Chapter 1: STATE-SPACE LINEAR SYSTEMS 5

1.1 State-Space Linear Systems 5

1.2 Block Diagrams 7

1.3 Exercises 10

Chapter 2: LINEARIZATION 11

2.1 State-Space Nonlinear Systems 11

2.2 Local Linearization around an Equilibrium Point 11

2.3 Local Linearization around a Trajectory 14

2.4 Feedback Linearization 15

2.5 Exercises 19

Chapter 3: CAUSALITY, TIME INVARIANCE, AND LINEARITY 22

3.1 Basic Properties of LTV/LTI Systems 22

3.2 Characterization of All Outputs to a Given Input 24

3.3 Impulse Response 25

3.4 Laplace Transform (review) 27

3.5 Transfer Function 27

3.6 Discrete-Time Case 28

3.7 Additional Notes 29

3.8 Exercise 30

Chapter 4: IMPULSE RESPONSE AND TRANSFER FUNCTION OF STATESPACE SYSTEMS 31

4.1 Impulse Response and Transfer Function for LTI Systems 31

4.2 Discrete-Time Case 32

4.3 Elementary Realization Theory 32

4.4 Equivalent State-Space Systems 36

4.5 LTI Systems in MATLABr 38

4.6 Exercises 39

Chapter 5: SOLUTIONS TO LTV SYSTEMS 41

5.1 Solution to Homogeneous Linear Systems 41

5.2 Solution to Nonhomogeneous Linear Systems 43

5.3 Discrete-Time Case 44

5.4 Exercises 45

Chapter 6: SOLUTIONS TO LTI SYSTEMS 46

6.1 Matrix Exponential 46

6.2 Properties of the Matrix Exponential 47

6.3 Computation of Matrix Exponentials Using Laplace Transforms 49

6.4 The Importance of the Characteristic Polynomial 50

6.5 Discrete-Time Case 50

6.6 Symbolic Computations in MATLABr_ 51

6.7 Exercises 53

Chapter 7: SOLUTIONS TO LTI SYSTEMS: THE JORDAN NORMAL FORM 55

7.1 Jordan Normal Form 55

7.2 Computation of Matrix Powers Using the Jordan Normal Form 57

7.3 Computation ofMatrix Exponentials Using the Jordan Normal Form 58

7.4 Eigenvalues with Multiplicity Larger than 1 59

7.5 Exercise 60

Part II: STABILITY 61

Chapter 8: INTERNAL OR LYAPUNOV STABILITY 63

8.1 Matrix Norms (review) 63

8.2 Lyapunov Stability 65

8.3 Eigenvalue Conditions for Lyapunov Stability 66

8.4 Positive-Definite Matrices (review) 67

8.5 Lyapunov Stability Theorem 67

8.6 Discrete-Time Case 70

8.7 Stability of Locally Linearized Systems 72

8.8 Stability Tests with MATLABr_ 77

8.9 Exercises 78

Chapter 9: INPUT-OUTPUT STABILITY 80

9.1 Bounded-Input, Bounded-Output Stability 80

9.2 Time Domain Conditions for BIBO Stability 81

9.3 Frequency Domain Conditions for BIBO Stability 84

9.4 BIBO versus Lyapunov Stability 85

9.5 Discrete-Time Case 85

9.6 Exercises 86

Chapter 10: PREVIEW OF OPTIMAL CONTROL 87

10.1 The Linear Quadratic Regulator Problem 87

10.2 Feedback Invariants 88

10.3 Feedback Invariants in Optimal Control 88

10.4 Optimal State Feedback 89

10.5 LQR with MATLABr_ 91

10.6 Exercises 91

Part III: CONTROLLABILITY AND STATE FEEDBACK 93

Chapter 11: CONTROLLABLE AND REACHABLE SUBSPACES 95

11.1 Controllable and Reachable Subspaces 95

11.2 Physical Examples and System Interconnections 96

11.3 Fundamental Theorem of Linear Equations (review) 99

11.4 Reachability and Controllability Gramians 100

11.5 Open-Loop Minimum-Energy Control 101

11.6 Controllability Matrix (LTI) 102

11.7 Discrete-Time Case 105

11.8 MATLABr Commands 109

11.9 Exercise 109

Chapter 12: CONTROLLABLE SYSTEMS 110

12.1 Controllable Systems 110

12.2 Eigenvector Test for Controllability 111

12.3 Lyapunov Test for Controllability 113

12.4 Feedback Stabilization Based on the Lyapunov Test 116

12.5 Exercises 117

Chapter 13: CONTROLLABLE DECOMPOSITIONS 118

13.1 Invariance with Respect to Similarity Transformations 118

13.2 Controllable Decomposition 119

13.3 Block Diagram Interpretation 120

13.4 Transfer Function 121

13.5 MATLABr Commands 122

13.6 Exercise 122

Chapter 14: STABILIZABILITY 123

14.1 Stabilizable System 123

14.2 Eigenvector Test for Stabilizability 124

14.3 Popov-Belevitch-Hautus (PBH) Test for Stabilizability 125

14.4 Lyapunov Test for Stabilizability 126

14.5 Feedback Stabilization Based on the Lyapunov Test 127

14.6 Eigenvalue Assignment 128

14.7 MATLABr Commands 129

14.8 Exercises 129

Part IV: OBSERVABILITY AND OUTPUT FEEDBACK 133

Chapter 15: OBSERVABILITY 135

15.1 Motivation: Output Feedback 135

15.2 Unobservable Subspace 136

15.3 Unconstructible Subspace 137

15.4 Physical Examples 138

15.5 Observability and Constructibility Gramians 139

15.6 Gramian-based Reconstruction 140

15.7 Discrete-Time Case 141

15.8 Duality (LTI) 142

15.9 Observability Tests 144

15.10MATLABr_ Commands 145

15.11 Exercises 145

Chapter 16: OUTPUT FEEDBACK 148

16.1 Observable Decomposition 148

16.2 Kalman Decomposition Theorem 149

16.3 Detectability 152

16.4 Detectability Tests 152

16.5 State Estimation 153

16.6 Eigenvalue Assignment by Output Injection 154

16.7 Stabilization through Output Feedback 155

16.8 MATLABr_ Commands 156

16.9 Exercises 156

Chapter 17: MINIMAL REALIZATIONS 157

17.1 Minimal Realizations 157

17.2 Markov Parameters 158

17.3 Similarity of Minimal Realizations 160

17.4 Order of a Minimal SISO Realization 161

17.5 MATLABr_ Commands 163

17.6 Exercises 163

LINEAR SYSTEMS II-ADVANCED MATERIAL

Part V: POLES AND ZEROS OF MIMO SYSTEMS 167

Chapter 18: SMITH-MCMILLAN FORM 169

18.1 Informal Definition of Poles and Zeros 169

18.2 Polynomial Matrices: Smith Form 170

18.3 Rational Matrices: Smith-McMillan Form 172

18.4 McMillan Degree, Poles, and Zeros 173

18.5 Transmission-Blocking Property of Transmission Zeros 175

18.6 MATLABr_ Commands 176

18.7 Exercises 176

Chapter 19: STATE-SPACE ZEROS, MINIMALITY, AND SYSTEM INVERSES 177

19.1 Poles of Transfer Functions versus Eigenvalues of State-Space Realizations

177

19.2 Transmission Zeros of Transfer Functions versus Invariant Zeros of State-Space Realizations 178

19.3 Order of Minimal Realizations 180

19.4 System Inverse 182

19.5 Existence of an Inverse 183

19.6 Poles and Zeros of an Inverse 184

19.7 Feedback Control of Stable Systems with Stable Inverses 185

19.8 MATLABr Commands 186

19.9 Exercises 187

Part VI: LQR/LQG OPTIMAL CONTROL 189

Chapter 20: LINEAR QUADRATIC REGULATION (LQR) 191

20.1 Deterministic Linear Quadratic Regulation (LQR) 191

20.2 Optimal Regulation 192

20.3 Feedback Invariants 193

20.4 Feedback Invariants in Optimal Control 193

20.5 Optimal State Feedback 194

20.6 LQR in MATLABr 195

20.7 Additional Notes 196

20.8 Exercises 196

Chapter 21: THE ALGEBRAIC RICCATI EQUATION (ARE) 197

21.1 The Hamiltonian Matrix 197

21.2 Domain of the Riccati Operator 198

21.3 Stable Subspaces 199

21.4 Stable Subspace of the Hamiltonian Matrix 199

21.5 Exercises 203

Chapter 22: FREQUENCY DOMAIN AND ASYMPTOTIC PROPERTIES

OF LQR 204

22.1 Kalman's Equality 204

22.2 Frequency Domain Properties: Single-Input Case 205

22.3 Loop Shaping using LQR: Single-Input Case 207

22.4 LQR Design Example 210

22.5 Cheap Control Case 213

22.6 MATLABr Commands 216

22.7 Additional Notes 216

22.8 The Loop-shaping Design Method (review) 217

22.9 Exercises 222

Chapter 23: OUTPUT FEEDBACK 223

23.1 Certainty Equivalence 223

23.2 Deterministic Minimum-Energy Estimation (MEE) 223

23.3 Stochastic Linear Quadratic Gaussian (LQG) Estimation 228

23.4 LQR/LQG Output Feedback 229

23.5 Loop Transfer Recovery (LTR) 230

23.6 Optimal Set Point Control 231

23.7 LQR/LQG with MATLABr 235

23.8 LTR Design Example 235

23.9 Exercises 236

Chapter 24: LQG/LQR AND THE Q PARAMETERIZATION 238

24.1 Q-augmented LQG/LQR Controller 238

24.2 Properties 239

24.3 Q Parameterization 241

24.4 Exercise 242

Chapter 25: Q DESIGN 243

25.1 Control Specifications for Q Design 243

25.2 The Q Design Feasibility Problem 246

25.3 Finite-dimensional Optimization: Ritz Approximation 246

25.4 Q Design using MATLABr and CVX 248

25.5 Q Design Example 253

25.6 Exercise 255

BIBLIOGRAPHY 257

INDEX 259

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