Optimal and Robust Control. Advanced Topics with MATLAB

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