Linear Control Theory : Structure, Robustness, and Optimization book cover
1st Edition

Linear Control Theory
Structure, Robustness, and Optimization

ISBN 9780849340635
Published January 30, 2009 by CRC Press
924 Pages 367 B/W Illustrations

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Book Description

Successfully classroom-tested at the graduate level, Linear Control Theory: Structure, Robustness, and Optimization covers three major areas of control engineering (PID control, robust control, and optimal control). It provides balanced coverage of elegant mathematical theory and useful engineering-oriented results.

The first part of the book develops results relating to the design of PID and first-order controllers for continuous and discrete-time linear systems with possible delays. The second section deals with the robust stability and performance of systems under parametric and unstructured uncertainty. This section describes several elegant and sharp results, such as Kharitonov’s theorem and its extensions, the edge theorem, and the mapping theorem. Focusing on the optimal control of linear systems, the third part discusses the standard theories of the linear quadratic regulator, Hinfinity  and l1 optimal control, and associated results.

Written by recognized leaders in the field, this book explains how control theory can be applied to the design of real-world systems. It shows that the techniques of three term controllers, along with the results on robust and optimal control, are invaluable to developing and solving research problems in many areas of engineering.


Table of Contents



PID Controllers: An Overview of Classical Theory

Introduction to Control

The Magic of Integral Control

PID Controllers

Classical PID Controller Design

Integrator Windup

PID Controllers for Delay-Free LTI Systems


Stabilizing Set

Signature Formulas

Computation of the PID Stabilizing Set

PID Design with Performance Requirements

PID Controllers for Systems with Time Delay


Characteristic Equations for Delay Systems

The Padé Approximation and Its Limitations

The Hermite–Biehler Theorem for Quasipolynomials

Stability of Systems with a Single Delay

PID Stabilization of First-Order Systems with Time Delay

PID Stabilization of Arbitrary LTI Systems with a Single Time Delay

Proofs of Lemmas 3.3, 3.4, and 3.5

Proofs of Lemmas 3.7 and 3.9

An Example of Computing the Stabilizing Set

Digital PID Controller Design



Tchebyshev Representation and Root Clustering

Root Counting Formulas

Digital PI, PD, and PID Controllers

Computation of the Stabilizing Set

Stabilization with PID Controllers

First-Order Controllers for LTI Systems

Root Invariant Regions

An Example

Robust Stabilization by First-Order Controllers

Hinfinity Design with First-Order Controllers

First-Order Discrete-Time Controllers

Controller Synthesis Free of Analytical Models


Mathematical Preliminaries

Phase, Signature, Poles, Zeros, and Bode Plots

PID Synthesis for Delay-Free Continuous-Time Systems

PID Synthesis for Systems with Delay

PID Synthesis for Performance

An Illustrative Example: PID Synthesis

Model-Free Synthesis for First-Order Controllers

Model-Free Synthesis of First-Order Controllers for Performance

Data-Based Design vs. Model-Based Design

Data-Robust Design via Interval Linear Programming

Computer-Aided Design

Data-Driven Synthesis of Three Term Digital Controllers


Notation and Preliminaries

PID Controllers for Discrete-Time Systems

Data-Based Design: Impulse Response Data

First-Order Controllers for Discrete-Time Systems

Computer-Aided Design


Stability Theory for Polynomials


The Boundary Crossing Theorem

The Hermite–Biehler Theorem

Schur Stability Test

Hurwitz Stability Test

Stability of a Line Segment


Bounded Phase Conditions

Segment Lemma

Schur Segment Lemma via Tchebyshev Representation

Some Fundamental Phase Relations

Convex Directions

The Vertex Lemma

Stability Margin Computation


The Parametric Stability Margin

Stability Margin Computation

The Mapping Theorem

Stability Margins of Multilinear Interval Systems

Robust Stability of Interval Matrices

Robustness Using a Lyapunov Approach

Stability of a Polytope


Stability of Polytopic Families

The Edge Theorem

Stability of Interval Polynomials

Stability of Interval Systems

Polynomic Interval Families

Robust Control Design


Interval Control Systems

Frequency Domain Properties

Nyquist, Bode, and Nichols Envelopes

Extremal Stability Margins

Robust Parametric Classical Design

Robustness under Mixed Perturbations

Robust Small Gain Theorem

Robust Performance

The Absolute Stability Problem

Characterization of the SPR Property

The Robust Absolute Stability Problem


The Linear Quadratic Regulator

An Optimal Control Problem

The Finite Time LQR Problem

The Infinite Horizon LQR Problem

Solution of the Algebraic Riccati Equation

The LQR as an Output Zeroing Problem

Return Difference Relations

Guaranteed Stability Margins for the LQR

Eigenvalues of the Optimal Closed Loop System

Optimal Dynamic Compensators

Servomechanisms and Regulators



The Small Gain Theorem

L Stability and Robustness via the Small Gain Theorem

YJBK Parametrization of All Stabilizing Compensators (Scalar Case)

Control Problems in the Hinfinity Framework

Hinfinity Optimal Control: SISO Case

l1 Optimal Control: SISO Case

Hinfinity Optimal Multivariable Control

Hinfinity Optimal Control Using Hankel Theory

The State Space Solution of Hinfinity Optimal Control

Appendix A: Signal Spaces

Vector Spaces and Norms

Metric Spaces

Equivalent Norms and Convergence

Relations between Normed Spaces

Appendix B: Norms for Linear Systems

Induced Norms for Linear Maps

Properties of Fourier and Laplace Transforms

Lp/lp Norms of Convolutions of Signals

Induced Norms of Convolution Maps


Robustness and Fragility

Feedback, Robustness, and Fragility





Exercises, Notes, and References appear at the end of each chapter.

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