Linear Systems Theory

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Linear systems theory is the cornerstone of control theory and a well-established discipline that focuses on linear differential equations from the perspective of control and estimation. In this textbook, Joao Hespanha covers the key topics of the field in a unique lecture-style format, making the book easy to use for instructors and students. He looks at system representation, stability, controllability and state feedback, observability and state estimation, and realization theory. He provides the background for advanced modern control design techniques and feedback linearization, and examines advanced foundational topics such as multivariable poles and zeros, and LQG/LQR. The textbook presents only the most essential mathematical derivations, and places comments, discussion, and terminology in sidebars so that readers can follow the core material easily and without distraction. Annotated proofs with sidebars explain the techniques of proof construction, including contradiction, contraposition, cycles of implications to prove equivalence, and the difference between necessity and sufficiency. Annotated theoretical developments also use sidebars to discuss relevant commands available in MATLAB, allowing students to understand these important tools. The balanced chapters can each be covered in approximately two hours of lecture time, simplifying course planning and student review. Solutions to the theoretical and computational exercises are also available for instructors. * Easy-to-use textbook in unique lecture-style format * Sidebars explain topics in further detail * Annotated proofs and discussions of MATLAB commands * Balanced chapters can each be taught in two hours of course lecture * Solutions to exercises available to instructors

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Linear Systems Theory gives a good presentation of the main topics on linear systems as well as more advanced topics related to controller design. The scholarship is sound and the book is very well written and readable. -- Ian Petersen, University of New South Wales This book provides a sound basis for an excellent course on linear systems theory. It covers a breadth of material in a fast-paced and mathematically focused way. It can be used by students wishing to specialize in this subject, as well as by those interested in this topic generally. -- Geir E. Dullerud, University of Illinois, Urbana-Champaign

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

Joao P. Hespanha is professor of electrical engineering at the University of California, Santa Barbara, where he is associate director of the Center for Control, Dynamical Systems and Computation.

"This is a splendidly written textbook; in fact, the next time I teach linear systems theory, I intend to use it. It covers the right amount of theory and presents the material at a perfect level for students. It has many exercises, most of which are well suited for beginning engineering graduate students."--Alan J. Laub, Elements of Computation Theory

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