Optimal and robust control : advanced topics with MATLAB.

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Cover Half Title Title Page Copyright Page Dedication Contents Preface Symbol List 1. Modelling of Uncertain Systems and the Robust Control Problem 1.1. Uncertainty and Robust Control 1.2. The Essential Chronology of Major Findings in Robust Control 2. Fundamentals of Stability 2.1. Lyapunov Criteria 2.2. Positive Definite Matrices 2.3. Lyapunov Theory for Linear Time-Invariant Systems 2.4. Lyapunov Equations 2.5. Stability with Uncertainty 2.6. Further Results on the Lyapunov Theory 2.6.1. Hystorical Notes 2.6.2. Lyapunov Stability 2.7. Exercises 3. Kalman Canonical Decomposition 3.1. Introduction 3.2. Controllability Canonical Partition 3.3. Observability Canonical Partition 3.4. General Partition 3.5. Remarks on Kalman Decomposition 3.6. Exercises 4. Singular Value Decomposition 4.1. Singular Values of a Matrix 4.2. Spectral Norm and Condition Number of a Matrix 4.3. Exercises 5. Open-loop Balanced Realization 5.1. Controllability and Observability Gramians 5.2. Principal Component Analysis 5.3. Principal Component Analysis Applied to Linear Systems 5.4. State Transformations of Gramians 5.5. Singular Values of Linear Time-invariant Systems 5.6. Computing the Open-loop Balanced Realization 5.7. Balanced Realization for Discrete-time Linear Systems 5.8. Exercises 6. Reduced Order Models and Symmetric Systems 6.1. Reduced Order Models Based on the Open-loop Balanced Realization 6.1.1. Direct Truncation Method 6.1.2. Singular Perturbation Method 6.2. Reduced Order Model Exercises 6.3. Symmetric Systems 6.3.1. Reduced Order Models for SISO Systems 6.3.2. Properties of Symmetric Systems 6.3.3. The Cross-gramian Matrix 6.3.4. Relations Between W2c, W2o and Wco 6.3.5. Open-loop Parameterization 6.3.6. Relation Between the Cauchy Index and the Hankel Matrix 6.3.7. Singular Values for a FIR Filter 6.3.8. Singular Values of All-pass Systems 6.4. Exercises 7. Variational Calculus and Linear Quadratic Optimal Control 7.1. Variational Calculus: An Introduction 7.2. The Lagrange Method 7.3. Towards Optimal Control 7.4. LQR Optimal Control 7.5. Hamiltonian Matrices 7.6. Solving the Riccati Equation via the Hamiltonian Matrix 7.7. The Control Algebraic Riccati Equation 7.8. Optimal Control for SISO Systems 7.9. Linear Quadratic Regulator with Cross-weighted Cost 7.10. Finite-horizon Linear Quadratic Regulator 7.11. Optimal Control for Discrete-time Linear Systems 7.12. Exercises 8. Closed-loop Balanced Realization 8.1. Synthesis of a Compensator for High-Order Systems 8.2. Filtering Algebraic Riccati Equation 8.3. Computing the Closed-loop Balanced Realization 8.4. Procedure for Closed-loop Balanced Realization 8.5. Reduced Order Models Based on Closed-loop Balanced Realization 8.6. Closed-loop Balanced Realization for Symmetric Systems 8.7. Exercises 9. Positive-real, Bounded-real and Negative-imaginary Systems 9.1. Passive Systems 9.1.1. Passivity in the Frequency Domain 9.1.2. Passivity in the Time Domain 9.1.3. Factorizing Positive-real Functions 9.1.4. Passive Reduced Order Models 9.1.5. Energy Considerations Connected to the Positive-real Lemma 9.1.6. Closed-loop Stability and Positive-real Systems 9.1.7. Optimal Gain for Loss-less Systems 9.2. Circuit Implementation of Positive-real Systems 9.3. Bounded-real Systems 9.3.1. Properties of Bounded-real Systems 9.3.2. Bounded-real Reduced Order Models 9.4. Relationship Between Passive and Bounded-real Systems 9.5. Negative-imaginary Systems 9.5.1. Characterization of Negative-imaginary Systems in the Frequency Domain 9.5.2. Characterization of Negative-imaginary Systems in the Time Domain 9.5.3. Closed-loop Stability and Negative-imaginary Systems 9.6. Exercises 10. Enforcing the Positive-real or the Negative-imaginary Property in a Linear Model 10.1. Why to Enforce the Positive-real and Negative-Imaginary Property in a Linear Model 10.2. Passification 10.3. Forward Action to make a System Negative-Imaginary 10.3.1. The SISO Case 10.3.2. The MIMO Case 10.4. Exercises 11. H∞ Linear Control 11.1. Introduction 11.2. Solution of the H∞ Linear Control Problem 11.3. The H∞ Linear Control and the Uncertainty Problem 11.4. Exercises 12. Linear Matrix Inequalities for Optimal and Robust Control 12.1. Definition and Properties of LMI 12.2. LMI Problems 12.2.1. Feasibility Problem 12.2.2. Linear Objective Minimization Problem 12.2.3. Generalized Eigenvalue Minimization Problem 12.3. Formulation of Control Problems in LMI Terms 12.3.1. Stability 12.3.2. Closed-loop Stability 12.3.3. Simultaneous Stabilizability 12.3.4. Positive-real Lemma 12.3.5. Bounded-real Lemma 12.3.6. Calculating the H∞ Norm Through LMI 12.4. Solving a LMI Problem 12.5. LMI Problem for Simultaneous Stabilizability 12.6. Solving Algebraic Riccati Equations Through LMI 12.7. Computation of Gramians Through LMI 12.8. Computation of the Hankel Norm Through LMI 12.9. H∞ Control 12.10. Multiobjective Control 12.11. Exercises 13. The Class of Stabilizing Controllers 13.1. Parameterization of Stabilizing Controllers for Processes 13.2. Parameterization of Stabilizing Controllers for Unstable Processes 13.3. Parameterization of Stable Controllers 13.4. Simultaneous Stabilizability of Two Systems 13.5. Coprime Factorizations for MIMO Systems and Unitary Factorization 13.6. Parameterization in Presence of Uncertainty 13.7. Exercises 14. Formulation and Solution of Matrix Algebraic Problems through Optimization Problems 14.1. Solutions of Matrix Algebra Problems Using Dynamical Systems 14.1.1. Problem 1: Inverse of a Matrix 14.1.2. Problem 2: Eigenvalues of a Matrix 14.1.3. Problem 3: Eigenvectors of a Symmetric Positive Definite Matrix 14.1.4. Problem 4: Observability and Controllability Gramian 14.2. Computation of the Open-loop Balanced Representation via the Dynamical System Approach 14.3. Concluding Remarks 14.4. Exercises 15. Time-delay Systems 15.1. Modeling Systems with Time-delays 15.2. Basic Principles of Time-delay Systems 15.3. Stability of Time-delay Systems 15.4. Stability of Time-delay Systems with q = 1 15.5. Direct Method 15.6. Exercises Recommended Essential References Appendix A. Norms Appendix B. Algebraic Riccati Equations Appendix C. Invariance Under Frequency Transformations Index