# Linear Systems Theory

###### João P. Hespanha
Publisher:
Princeton University Press
Publication Date:
2009
Number of Pages:
263
Format:
Hardcover
Price:
59.95
ISBN:
9780691140216
Category:
Textbook
[Reviewed by
John D. Cook
, on
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) = x0 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.

1. How useful is a theory that only considers linear systems?
2. Why limit your attention to first order derivatives?
3. 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
3.8 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

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
20.8 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
22.8 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