Andrea Bacciotti - Stability and Control of Linear Systems

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This book is the natural outcome of a course I taught for many years at the Technical University of Torino, first for students enrolled in the aerospace engineering curriculum, and later for students enrolled in the applied mathematics curriculum. The aim of the course was to provide an introduction to the main notions of system theory and automatic control, with a rigorous theoretical framework and a solid mathematical background.

Throughout the book, the reference model is a finite-dimensional, time-invariant, multivariable linear system. The exposition is basically concerned with the time-domain approach, but also the frequency-domain approach is taken into consideration. In fact, the relationship between the two approaches is discussed, especially for the case of single-inputsingle-output systems. Of course, there are many other excellent handbooks on the same subject (just to quote a few of them, [3, 6, 8, 11, 14, 23, 25, 27, 28, 32]). The distinguishing feature of the present book is the treatment of some specific topics which are rare to find elsewhere at a graduate level. For instance, bounded-inputbounded-output stability (including a characterization in terms of canonical decompositions), static output feedback stabilization (for which a simple criterion in terms of generalized inverse matrices is proposed), controllability under constrained controls.

The mathematical theories of stability and controllability of linear systems are essentially based on linear algebra, and it has reached today a high level of advancement. During the last three decades of the past century, a great effort was done, in order to develop an analogous theory for nonlinear systems, based on differential geometry (see [7] for a historical overview). For this development, usually referred to as geometric control theory , we have today a rich literature ([2, 5, 13, 1820, 26, 30]). However, I believe that the starting point for a successful approach to nonlinear systems is a wide and deep knowledge of the linear case. For this reason, while this book is limited to the linear context, in the presentation and organization of the material, as well as in the selection of topics, the final goal I had in mind is to prepare the reader for such a nonlinear extension.

Concerning the prerequisites, I assume that the reader is familiar with basic differential and integral calculus (for real functions of several real variables) and linear algebra. Some notions of complex analysis are required in the frequency-domain approach. The book can be used as a reference book for basic courses at a doctoral (or also upper undergraduate) level in mathematical control theory and in automatic control. More generally, parts of this book can be used in applied mathematics courses, where an introduction to the point of view of system theory and control philosophy is advisable. The perspective of control systems and the stability problem are indeed ubiquitous in applied sciences and witness a rapidly increasing importance in modern engineering. At a postdoctoral level, this book can be recommended for reading courses both for mathematician oriented to engineering applications and engineers with theoretical interests. To better focus on the main concepts and results, some more technical proofs are avoided or limited to special situations. However, in these cases, appropriate bibliographic references are supplied for the curious reader.

It follows a short description of the contents. The first chapter aims to introduce the reader to the point of view of system theory: In particular, the notions of inputoutput operator and external stability are given. The second chapter deals with systems without external forces which reduce, according to a more classical terminology, to homogeneous systems of linear differential equations. In view of the application, we are interested in, the representation of the general integral in terms of exponential matrix and Jordan form is crucial, and it is treated in detail. Chapter , we introduce the frequency-domain approach and study the relationship with the time-domain approach. Two appendices follow. In the first one, the notions of internal stability are introduced. These notions are formulated with respect to a system of nonlinear ordinary differential equations. In fact, only in this part of the book nonlinear systems came into play. The reason of this choice is that all the aspects of the stability notions became more evident in the nonlinear context. The second appendix is a short list of useful facts about Laplace transform.