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Optimal and Robust Control, 2nd Edition

Name: Optimal and Robust Control, 2nd Edition
ISBN: 9781000486735

by Luigi Fortuna, Mattia Frasca, Arturo Buscarino

November 2021

Intermediate to advanced

322 pages

6h 47m

English

CRC Press

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Cover Page
Half-Title Page
Title Page
Copyright Page
Dedication Page
Contents
Preface
Symbol List
1 Modelling of Uncertain Systems and the Robust Control Problem
1.1 Uncertainty and Robust Control1.2 The Essential Chronology of Major Findings in Robust Control
2 Fundamentals of Stability
2.1 Lyapunov Criteria2.2 Positive Definite Matrices2.3 Lyapunov Theory for Linear Time-Invariant Systems2.4 Lyapunov Equations2.5 Stability with Uncertainty2.6 Further Results on the Lyapunov Theory2.6.1 Hystorical Notes2.6.2 Lyapunov Stability2.7 Exercises

3 Kalman Canonical Decomposition
3.1 Introduction3.2 Controllability Canonical Partition3.3 Observability Canonical Partition3.4 General Partition3.5 Remarks on Kalman Decomposition3.6 Exercises
4 Singular Value Decomposition
4.1 Singular Values of a Matrix4.2 Spectral Norm and Condition Number of a Matrix4.3 Exercises
5 Open-loop Balanced Realization
5.1 Controllability and Observability Gramians5.2 Principal Component Analysis5.3 Principal Component Analysis Applied to Linear Systems5.4 State Transformations of Gramians5.5 Singular Values of Linear Time-invariant Systems5.6 Computing the Open-loop Balanced Realization5.7 Balanced Realization for Discrete-time Linear Systems5.8 Exercises
6 Reduced Order Models and Symmetric Systems
6.1 Reduced Order Models Based on the Open-loop Balanced Realization6.1.1 Direct Truncation Method6.1.2 Singular Perturbation Method6.2 Reduced Order Model Exercises6.3 Symmetric Systems6.3.1 Reduced Order Models for SISO Systems6.3.2 Properties of Symmetric Systems6.3.3 The Cross-gramian Matrix6.3.4 Relations Between Wc2, Wo2 and Wco6.3.5 Open-loop Parameterization6.3.6 Relation Between the Cauchy Index and the Hankel Matrix6.3.7 Singular Values for a FIR Filter6.3.8 Singular Values of All-pass Systems6.4 Exercises
7 Variational Calculus and Linear Quadratic Optimal Control
7.1 Variational Calculus: An Introduction7.2 The Lagrange Method7.3 Towards Optimal Control7.4 LQR Optimal Control7.5 Hamiltonian Matrices7.6 Solving the Riccati Equation via the Hamiltonian Matrix7.7 The Control Algebraic Riccati Equation7.8 Optimal Control for SISO Systems7.9 Linear Quadratic Regulator with Cross-weighted Cost7.10 Finite-horizon Linear Quadratic Regulator7.11 Optimal Control for Discrete-time Linear Systems7.12 Exercises
8 Closed-loop Balanced Realization
8.1 Synthesis of a Compensator for High-Order Systems8.2 Filtering Algebraic Riccati Equation8.3 Computing the Closed-loop Balanced Realization8.4 Procedure for Closed-loop Balanced Realization8.5 Reduced Order Models Based on Closed-loop Balanced Realization8.6 Closed-loop Balanced Realization for Symmetric Systems8.7 Exercises
9 Positive-real, Bounded-real and Negative-imaginary Systems
9.1 Passive Systems9.1.1 Passivity in the Frequency Domain9.1.2 Passivity in the Time Domain9.1.3 Factorizing Positive-real Functions9.1.4 Passive Reduced Order Models9.1.5 Energy Considerations Connected to the Positive-real Lemma9.1.6 Closed-loop Stability and Positive-real Systems9.1.7 Optimal Gain for Loss-less Systems9.2 Circuit Implementation of Positive-real Systems9.3 Bounded-real Systems9.3.1 Properties of Bounded-real Systems9.3.2 Bounded-real Reduced Order Models9.4 Relationship Between Passive and Bounded-real Systems9.5 Negative-imaginary Systems9.5.1 Characterization of Negative-imaginary Systems in the Frequency Domain9.5.2 Characterization of Negative-imaginary Systems in the Time Domain9.5.3 Closed-loop Stability and Negative-imaginary Systems9.6 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 Model10.2 Passification10.3 Forward Action to make a System Negative-Imaginary10.3.1 The SISO Case10.3.2 The MIMO Case10.4 Exercises
11 H∞ Linear Control
11.1 Introduction11.2 Solution of the H∞ Linear Control Problem11.3 The H∞ Linear Control and the Uncertainty Problem11.4 Exercises
12 Linear Matrix Inequalities for Optimal and Robust Control
12.1 Definition and Properties of LMI12.2 LMI Problems12.2.1 Feasibility Problem12.2.2 Linear Objective Minimization Problem12.2.3 Generalized Eigenvalue Minimization Problem12.3 Formulation of Control Problems in LMI Terms12.3.1 Stability12.3.2 Closed-loop Stability12.3.3 Simultaneous Stabilizability12.3.4 Positive-real Lemma12.3.5 Bounded-real Lemma12.3.6 Calculating the H∞ Norm Through LMI12.4 Solving a LMI Problem12.5 LMI Problem for Simultaneous Stabilizability12.6 Solving Algebraic Riccati Equations Through LMI12.7 Computation of Gramians Through LMI12.8 Computation of the Hankel Norm Through LMI12.9 H∞ Control12.10 Multiobjective Control12.11 Exercises
13 The Class of Stabilizing Controllers
13.1 Parameterization of Stabilizing Controllers for Stable Processes13.2 Parameterization of Stabilizing Controllers for Unstable Processes13.3 Parameterization of Stable Controllers13.4 Simultaneous Stabilizability of Two Systems13.5 Coprime Factorizations for MIMO Systems and Unitary Factorization13.6 Parameterization in Presence of Uncertainty13.7 Exercises
14 Formulation and Solution of Matrix Algebraic Problems through Optimization Problems
14.1 Solutions of Matrix Algebra Problems Using Dynamical Systems14.1.1 Problem 1: Inverse of a Matrix14.1.2 Problem 2: Eigenvalues of a Matrix14.1.3 Problem 3: Eigenvectors of a Symmetric Positive Definite Matrix14.1.4 Problem 4: Observability and Controllability Gramian14.2 Computation of the Open-loop Balanced Representation via the Dynamical System Approach14.3 Concluding Remarks14.4 Exercises
15 Time-delay Systems
15.1 Modeling Systems with Time-delays15.2 Basic Principles of Time-delay Systems15.3 Stability of Time-delay Systems15.4 Stability of Time-delay Systems with q=115.5 Direct Method15.6 Exercises
Recommended Essential References
Appendix A. Norms
Appendix B. Algebraic Riccati Equations
Appendix C. Invariance Under Frequency Transformations
Index

Content preview from Optimal and Robust Control, 2nd Edition

Appendix C. Invariance Under Frequency Transformations

In this appendix, we briefly summarize some important properties of systems subject to a frequency transformation that is loss-less. In particular, we show that for these systems the singular values, the characteristic values and the shape of the Nyquist plot remain the same of the original (i.e., not transformed) system. The proofs and technical details can be found in the references reported at the end of this appendix.

Frequency Transformations

Given a linear time-invariant system, either SISO or MIMO, with transfer function matrix $G (s)$ , a frequency transformation replaces s with a SISO transfer function $F (s)$ . The frequency transformation is indicated as $s \leftarrow F (s)$ and the transformed system ...

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Optimal and Robust Control, 2nd Edition

by Luigi Fortuna, Mattia Frasca, Arturo Buscarino

Appendix C. Invariance Under Frequency Transformations

Frequency Transformations

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