Stability and control of linear systems (Q1632080)

This is a book about time-invariant finite-dimensional linear systems. This book provides an organized introduction to some basic system and control notions and methods (the time-domain approach and the frequency-domain approach) for these systems. The material in this book is divided into eight chapters: 1. Introduction; 2. Unforced Linear Systems; 3. Stability of Unforced Linear Systems; 4. Linear Systems with Forcing Term; 5. Controllability and Observability of Linear Systems; 6. External Stability; 7. Stabilization; 8. Frequency Domain Approach. Chapter 1 is a brief introduction of some very basic concepts and properties in system theory, such as the input-output operator, finite-dimensional systems, BIBO-stable, open/closed loop control, and the time-invariant property for impulse response systems and differential systems. Chapters 2 and 3 introduce the basic theory for the qualitative behavior and stability of the linear unforced differential systems: \(\dot{x}=Ax\) where \(A\) is a square \(n\times n\) matrix with real entries and \(x\in \mathbb{R}^n\). Chapter 4 considers the problem of representing the solutions of the linear nonhomogeneous system of differential equations: \(\dot{x}=Ax+b(t)\) where \(x\in \mathbb{R}^n\) and \(b(t)\in \mathcal{P}\mathcal{C}(I,\mathbb{R})\) is the forcing term. Chapter 5 discusses controllability and observability of the linear system with input and output: \(\dot{x}=Ax+Bu\) and \(y=Cx\) where \(x\in \mathbb{R}^n\), \(u\in \mathbb{R}^m\) is the input and \(y\in \mathbb{R}^p\) is the output. Chapter 6 continues with the analysis of the relationship between the external stability and the internal stability of the linear system with input and output. Chapter 7 addresses the problem of stabilizability by means of a static state feedback, or a static output feedback, or a dynamic output feedback. The last chapter concerns the relationship between two approaches: the time domain approach and the frequency domain approach based on the Laplace transform. From the Preface, we learn that this book is the outcome of a course the author taught for many years for students in aerospace engineering or in applied mathematics. I think it is an interesting book, and I would like to recommend it to students and researchers interested in control theory.