Control Theory for Linear Systems

Abstract

9R15. Control Theory for Linear Systems. - HL Trentelman (Dept of Math, Univ of Groningen, PO Box 800, Groningen, 9700 AV, Netherlands), AA Stoorvogel, M Hautus (Dept of Math and Comput Sci, Eindhoven Univ of Tech, PO Box 513, Eindhoven, 5600 MB, Netherlands). Springer-Verlag London Ltd, Surrey, UK. 2001. 389 pp. ISBN 1-85233-316-2. $84.95.Reviewed by L Dewell (Adv Tech Center, Lockheed Martin Advanced Tech Center, 3251 Hanover St, Palo Alto CA 94304).In the preface of this book, the authors stake out a strikingly broad boundary: “the theory of feedback control design for linear, finite-dimensional, time-invariant state space systems with inputs and outputs.” It is a testament to the recent growth of this field that had such a title appeared in the early 1970s, it would scarcely require 100 pages, let alone the nearly 400 pages of this book. Indeed, such a wide scope of a book today may either elicit suspicions of the authors of either being overly cursory in their treatment or overly parochial in their interpretation of the subject. In this book, however, the authors treat the subject with a coherence and an organization that made the large scope seem perfectly natural and pleasantly illuminating. The book appears to be adapted primarily to a reader interested in the theory of control for finite-dimensional linear systems, as opposed to applied engineers or scientists. The very restriction of the subject to such systems, however, allows for the applied control systems analyst to gain insights into recent developments in the field. The focus is certainly towards the development of a tight theoretical basis for the results, rather than applied examples, case studies or exercises. The authors assume that the reader is proficient in the linear algebra tools to analyze state space linear systems. In the opinion of this reviewer, it is probably best that the reader approach the book with an existing knowledge of the major results of the past few decades on the control of state space systems—most particularly linear quadratic regulators (LQR) and the linear quadratic Gaussian (LQG) problem. This book can quite readily stand on its own as an introduction to the recent results of H-infinity control theory from this foundation. The authors approach their subject from the point of view of geometric control theory, which requires a particular nomenclature and foundation with which the reader may not initially be familiar. The authors spend the first two chapters building such a foundation. In the next two chapters, the authors deal with the problem of disturbance decoupling and the notions of controlled and conditioned invariant subspaces. These are very fascinating chapters which alone make the book a worthwhile read. The following three chapters delve deeply into the structural properties of linear control systems, including system zeros and system invertibility. The next five chapters “deliver the goods” that the authors advertised in their preface—tracking and regulation, LQ optimal control, stochastic optimal control, and H-infinity control in finite- and infinite-time, with full-state and with measurement feedback. Each chapter was superbly organized into an introduction, a presentation of ideas, and a set of concluding notes. The notes were particularly pleasant to read, as they give the reader a clear connection between the material and the work of others over the last several decades. The index of the book is excellent (it includes a symbol index, which is frequently overlooked), and all results are painstakingly supported by proofs. If there is one area where the book seems inconsistent, it is in the paucity of numerical examples or case studies to illustrate the material of the book. This was doubtless a conscious decision of the authors, but the decision resulted in the work having a certain opacity and theoretical detachment that was distracting at times. The authors motivate the book in Chapter 1 by discussing, in detail, a practical problem of stationkeeping control of a geosynchronous satellite. After such a strong practical introduction, it would seem appropriate to apply the theory to a practical problem, perhaps even to the satellite problem suggested by the authors. Moreover, the authors stress in Chapters 4 and 6 that a disturbance decoupling controller can actually be numerically computed by algorithms they provide. An actual demonstration of such a controller synthesis would have been very powerful. Control Theory for Linear Systems is a fine contribution to the literature of control theory of linear systems. It offers a powerful organization and structure to the very broad subject of the control of finite-dimension linear systems and brings the torrent of activity of this field over the last several decades to a tight focus and perspective that has been lacking heretofore. This book is highly recommended for purchase by libraries and researchers in the field.