1st Edition

Linear Control Theory Structure, Robustness, and Optimization

    930 Pages 367 B/W Illustrations
    by CRC Press

    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.




    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.


    Shankar P. Bhattacharyya, Aniruddha Datta, Lee H. Keel