Contents. 1 State-Space Linear Systems 5. 2 Linearization Causality, Time Invariance, and Linearity 31

Transcription

1 Contents Preamble xiii Linear Systems I Basic Concepts 1 I System Representation 3 1 State-Space Linear Systems State-Space Linear Systems Block Diagrams Exercises 11 2 Linearization State-Space Nonlinear Systems Local Linearization Around an Equilibrium Point Local Linearization Around a Trajectory Feedback Linearization Practice Exercises Exercises 27 3 Causality, Time Invariance, and Linearity Basic Properties of LTV/LTI Systems Characterization of All Outputs to a Given Input Impulse Response Laplace and Z Transforms (Review) Transfer Function Discrete-Time Case Additional Notes Exercises 42 4 Impulse Response and Transfer Function of State-Space Systems Impulse Response and Transfer Function for LTI Systems Discrete-Time Case Elementary Realization Theory Equivalent State-Space Systems LTI Systems in MATLAB R Practice Exercises Exercises 53 5 Solutions to LTV Systems Solution to Homogeneous Linear Systems Solution to Nonhomogeneous Linear Systems 58

2 viii 5.3 Discrete-Time Case Practice Exercises Exercises 62 6 Solutions to LTI Systems Matrix Exponential Properties of the Matrix Exponential Computation of Matrix Exponentials Using Laplace Transforms The Importance of the Characteristic Polynomial Discrete-Time Case Symbolic Computations in MATLAB R Practice Exercises Exercises 74 7 Solutions to LTI Systems: The Jordan Normal Form Jordan Normal Form Computation of Matrix Powers using the Jordan Normal Form Computation of Matrix Exponentials using the Jordan Normal Form Eigenvalues with Multiplicity Larger than Practice Exercise Exercises 83 II Stability 85 8 Internal or Lyapunov Stability Lyapunov Stability Vector and Matrix Norms (Review) Eigenvalue Conditions for Lyapunov Stability Positive-Definite Matrices (Review) Lyapunov Stability Theorem Discrete-Time Case Stability of Locally Linearized Systems Stability Tests with MATLAB R Practice Exercises Exercises Input-Output Stability Bounded-Input, Bounded-Output Stability Time Domain Conditions for BIBO Stability Frequency Domain Conditions for BIBO Stability BIBO versus Lyapunov Stability Discrete-Time Case Practice Exercises Exercises 118

3 ix 10 Preview of Optimal Control The Linear Quadratic Regulator Problem Feedback Invariants Feedback Invariants in Optimal Control Optimal State Feedback LQR with MATLAB R Practice Exercise Exercise 125 III Controllability and State Feedback Controllable and Reachable Subspaces Controllable and Reachable Subspaces Physical Examples and System Interconnections Fundamental Theorem of Linear Equations (Review) Reachability and Controllability Gramians Open-Loop Minimum-Energy Control Controllability Matrix (LTI) Discrete-Time Case MATLAB R Commands Practice Exercise Exercises Controllable Systems Controllable Systems Eigenvector Test for Controllability Lyapunov Test for Controllability Feedback Stabilization Based on the Lyapunov Test Eigenvalue Assignment Practice Exercises Exercises Controllable Decompositions Invariance with Respect to Similarity Transformations Controllable Decomposition Block Diagram Interpretation Transfer Function MATLAB R Commands Exercise Stabilizability Stabilizable System Eigenvector Test for Stabilizability Popov-Belevitch-Hautus (PBH) Test for Stabilizability Lyapunov Test for Stabilizability Feedback Stabilization Based on the Lyapunov Test MATLAB R Commands Exercises 174

4 x IV Observability and Output Feedback Observability Motivation: Output Feedback Unobservable Subspace Unconstructible Subspace Physical Examples Observability and Constructibility Gramians Gramian-Based Reconstruction Discrete-Time Case Duality for LTI Systems Observability Tests MATLAB R Commands Practice Exercises Exercises Output Feedback Observable Decomposition Kalman Decomposition Theorem Detectability Detectability Tests State Estimation Eigenvalue Assignment by Output Injection Stabilization through Output Feedback MATLAB R Commands Exercises Minimal Realizations Minimal Realizations Markov Parameters Similarity of Minimal Realizations Order of a Minimal SISO Realization MATLAB R Commands Practice Exercises Exercises 219 Linear Systems II Advanced Material 221 V Poles and Zeros of MIMO Systems Smith-McMillan Form Informal Definition of Poles and Zeros Polynomial Matrices: Smith Form Rational Matrices: Smith-McMillan Form McMillan Degree, Poles, and Zeros 230

5 xi 18.5 Blocking Property of Transmission Zeros MATLAB R Commands Exercises State-Space Poles, Zeros, and Minimality Poles of Transfer Functions versus Eigenvalues of State-Space Realizations Transmission Zeros of Transfer Functions versus Invariant Zeros of State-Space Realizations Order of Minimal Realizations Practice Exercises Exercise System Inverses System Inverse Existence of an Inverse Poles and Zeros of an Inverse Feedback Control of Invertible Stable Systems with Stable Inverses MATLAB R Commands Exercises 250 VI LQR/LQG Optimal Control Linear Quadratic Regulation (LQR) Deterministic Linear Quadratic Regulation (LQR) Optimal Regulation Feedback Invariants Feedback Invariants in Optimal Control Optimal State Feedback LQR in MATLAB R Additional Notes Exercises The Algebraic Riccati Equation (ARE) The Hamiltonian Matrix Domain of the Riccati Operator Stable Subspaces Stable Subspace of the Hamiltonian Matrix Exercises Frequency Domain and Asymptotic Properties of LQR Kalman s Equality Frequency Domain Properties: Single-Input Case Loop Shaping Using LQR: Single-Input Case LQR Design Example 275

6 xii 23.5 Cheap Control Case MATLAB R Commands Additional Notes The Loop-Shaping Design Method (Review) Exercises Output Feedback Certainty Equivalence Deterministic Minimum-Energy Estimation (MEE) Stochastic Linear Quadratic Gaussian (LQG) Estimation LQR/LQG Output Feedback Loop Transfer Recovery (LTR) Optimal Set-Point Control LQR/LQG with MATLAB R LTR Design Example Exercises LQG/LQR and the Q Parameterization Q-Augmented LQG/LQR Controller Properties Q Parameterization Exercise Q Design Control Specifications for Q Design The Q Design Feasibility Problem Finite-Dimensional Optimization: Ritz Approximation Q Design Using MATLAB R and CVX Q Design Example Exercise 323 Bibliography 325 Index 327