Balancing rigorous theory with practical applications, **Linear Systems: Optimal and Robust Control** explains the concepts behind linear systems, optimal control, and robust control and illustrates these concepts with concrete examples and problems.

Developed as a two-course book, this self-contained text first discusses linear systems, including controllability, observability, and matrix fraction description. Within this framework, the author develops the ideas of state feedback control and observers. He then examines optimal control, stochastic optimal control, and the lack of robustness of linear quadratic Gaussian (LQG) control. The book subsequently presents robust control techniques and derives *H*_{∞} control theory from the first principle, followed by a discussion of the sliding mode control of a linear system. In addition, it shows how a blend of sliding mode control and *H*_{∞} methods can enhance the robustness of a linear system.

By learning the theories and algorithms as well as exploring the examples in **Linear Systems: Optimal and Robust Control**, students will be able to better understand and ultimately better manage engineering processes and systems.

**Introduction **

Overview

Contents of the Book **State Space Description of a Linear System**

Transfer Function of a Single Input/Single Output (SISO) System

State Space Realizations of a SISO System

SISO Transfer Function from a State Space Realization

Solution of State Space Equations

Observability and Controllability of a SISO System

Some Important Similarity Transformations

Simultaneous Controllability and Observability

Multiinput/Multioutput (MIMO) Systems

State Space Realizations of a Transfer Function Matrix

Controllability and Observability of a MIMO System

Matrix-Fraction Description (MFD)

MFD of a Transfer Function Matrix for the Minimal Order of a State Space Realization

Controller Form Realization from a Right MFD

Poles and Zeros of a MIMO Transfer Function Matrix

Stability Analysis **State Feedback Control and Optimization**

State Variable Feedback for a Single Input System

Computation of State Feedback Gain Matrix for a Multiinput System

State Feedback Gain Matrix for a Multiinput System for Desired Eigenvalues and Eigenvectors

Fundamentals of Optimal Control Theory

Linear Quadratic Regulator (LQR) Problem

Solution of LQR Problem via Root Locus Plot: SISO Case

Linear Quadratic Trajectory Control

Frequency-Shaped LQ Control

Minimum-Time Control of a Linear Time-Invariant System **Control with Estimated States**

Open-Loop Observer

Closed-Loop Observer

Combined Observer–CONTROLLER

Reduced-Order Observer

Response of a Linear Continuous-Time System to White Noise

Kalman Filter: Optimal State Estimation

Stochastic Optimal Regulator in Steady State

Linear Quadratic Gaussian (LQG) Control

Impact of Modeling Errors on Observer-Based Control **Robust Control: Fundamental Concepts and H_{2}, H_{∞}, and **

Important Aspects of Singular Value Analysis

Robustness: Sensitivity and Complementary Sensitivity

Robustness of LQR and Kalman Filter (KF) Feedback Loops

LQG/LTR Control

Well-Posedness, Internal Stability, and Small Gain Theorem

Formulation of Some Robust Control Problems with Unstructured Uncertainties

Formulation of Robust Control Problems with Structured Uncertainties

Loop Shaping

Controller Based on μ Analysis

Basic Concepts of Sliding Modes

Sliding Mode Control of a Linear System with Full State Feedback

Sliding Mode Control of an Uncertain Linear System with Full State Feedback: Blending

Sliding Mode Control of a Linear System with Estimated States

Optimal Sliding Mode Gaussian (OSG) Control

System of Linear Algebraic Equations

Eigenvalues and Eigenvectors

Matrix Inversion Lemma

Quadratic Functions

Derivative of a Quadratic Function

Derivative of a Linear Function

Fourier Integrals and Parseval’s Theorem

Vector Norms

Matrix Norms

Singular Values of a Matrix

Singular Value Decomposition (SVD)

Properties of Singular Values

Supremum and Infinimum

Stationary Stochastic Process

Power Spectrum or Power Spectral Density (PSD)

White Noise: A Special Stationary Stochastic Process

Response of a SISO Linear and Time-Invariant System Subjected to a Stationary Stochastic Process

Vector Stationary Stochastic Processes