Contents Chapter 1 Introduction to Control Theory 1 1.1 Historical Review and Classical Control Theory 1 1.1.1 Historical Review of Automatic Control 1 1.1.2 Classical control theory 5 1.2 Modern Control Theory 7 1.3 Design of Control Systems 9 1.4 Outline of This Book 12 Chapter 2 Preliminary Mathematical Knowledge 14 2.1 Foundations of matrix algebra 14 2.1.1 Matrices 14 2.1.2 Algebraic operations with matrices 17 2.1.3 Matrix Operations 18 2.2 Vectors and Vector Spaces 20 2.2.1 Vector 20 2.2.2 Vector Spaces and Subspaces 23 2.3 Linear Algebra 26 2.3.1 Eigenvalue and Eigenvector of A Square Matrix 26 2.3.2 Linear Algebraic Equations 28 2.3.3 Similarity Transformation 29 2.3.4 Diagonal Form and Jordan Form 31 2.3.5 Cayley-Hamiton Theorem 36 2.3.6 Definiteness of A Scalar Function and Quadratic Form 38 Exercises 39 Chapter 3 State Variables and the State-Space Description of Dynamic Systems 41 3.1 State-Space Representation of Dynamic Systems 41 3.1.1 State and State Variables 41 3.1.2 State-Space Representation 44 3.1.3 Block Diagrams and Simulation Diagrams 50 3.2 Obtaining State Equations 52 3.2.1 From the Block Diagram 52 3.2.2 From Input-Output Representation 54 3.2.3 Equivalence Transformation of State-Space Representation 67 3.3 Transfer Function And Realizations 68 3.4 State-Space Representation of Linear Discrete-Time Systems 72 3.5 Summaries 76 Exercises 76 Problems 78 Chapter 4 Time Response of Linear Systems 80 4.1 Solution of LTI State Equations 80 4.1.1 Linear Homogeneous State Equations 80 4.1.2 The State Transition Matrix 83 4.1.3 Linear Nonhomogeous State Equations 87 4.2 Numerical Solution of State Equations 89 4.3 Solution of Linear Discrete-Time State Equations 92 4.4 Discretization of Continuous-Time Systems 94 Exercises 96 Problems 98 Chapter 5 Controllability and Observability 99 5.1 Corollaries of Cayley-Hamilton Theorem 100 5.2 Controllability and Observability of LTI Systems 101 5.2.1 Controllability Definition and Rank Criterion 101 5.2.2 Observability Definition and Rank Criterion 110 5.2.3 Controllable Canonical Form and Observable Canonical Form 115 5.2.4 Principle of Duality 118 5.3 Structural Decomposition of LTI systems 119 5.4 Controllability, Observability and Transfer Function 125 5.5 Controllability and Observability of Discrete-Time Systems 128 5.5.1 Controllability of Discrete-Time Systems 128 5.5.2 Observability of Discrete-Time Systems 131 5.5.3 Controllability and Observability After Sampling 133 Exercises 135 Problems 137 Chapter 6 Lyapunov Stability 139 6.1 Preliminary Examples 139 6.2 Stability Concepts 141 6.2.1 Equilibrium State 141 6.2.2 Stability Definitions 142 6.3 First Method of Lyapunov 144 6.3.1 Eigenvalue Criterion for Linear Time-invariant System 144 6.3.2 Eigenvalue Criterion for Linearized Time-invariant Systems 146 6.4 Second Method of Lyapunov 148 6.5 Lyapunov Equation 153 Exercise 155 Problems 156 Chapter 7 State Feedback and State Observer 157 7.1 State Feedback and Output Feedback 158 7.1.1 State Feedback 158 7.1.2 Output Feedback 160 7.2 Pole Placement using State feedback 161 7.3 Pole placement using Output Feedback 165 7.4 State Observer 167 7.4.1 Full-Dimensional Observer 167 7.4.2 Reduced-Dimensional State Observer 171 7.5 Feedback From Estimated States 175 7.6 The Engineering Applications of State Feedback and Observer 178 7.6.1 State Feedback Controller Design For The Inverted Pendulum[9] 178 7.6.2 Asymptotic Tracking And Disturbance Rejection 182 7.6.3 System Stabilization 188 7.6.4 System Decoupling 191 Exercises 197 Problems 199 Chapter 8 Optimal Control 201 8.1 Problem Formulation 201 8.2 Preliminaries��The extremum problem of functional 205 8.2.1 Functional and The Calculus of Variation 206 8.2.2 The Extremum Problem of Functional Without Constraints 210 8.3 The Variational Approach to Optimal Control Problems 221 8.4 Minimum Principle and Its Application 228 8.4.1 Pontryagin's Minimum Principle 228 8.4.2 Application of Minimum Principle 232 Exercises 243 Problems 245 Answers to Selected Exercises 248 Experiment 1: Modeling and Stability Analysis of the Inverted Pendulum System 252 Experiment 2: Design State Feedback Controller for the Inverted Pendulum System 255 References 257