Stability and Control of Linear Systems

Abstract

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-input–single-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-input–bounded-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, 18–20, 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.