Optimal Control: An Introduction

Abstract

5R19. Optimal Control: An Introduction. - A Locatelli (Dept di Elettronica e Informazione, Politecnico di Milano, Piazza L da Vinci 32, Milano, 20133, Italy). Birkhauser Verlag AG, Basel, Switzerland. 2001. 294 pp. ISBN 3-7643-6408-4.Reviewed by S Sieniutycz (Dept of Chem Eng, Fac of Chem Eng, Warsaw Univ of Tech, 1 Warynskiego St, Warszawa, 00-645, Poland).Optimization is the collective process of finding the set of conditions required to achieve the best result from a given situation. Frequently its teaching is connected to an extent with control science as the field which has attained a high level of competence in advanced design of practical devices, complex industrial systems, robotics, and flying objects. Extensive research in dynamic optimization, especially with reference to robust control problems, has caused it to be regarded as valuable source of numerous useful, powerful, and flexible tools for both engineers and scientists. The link of optimal control with variational calculus, an older although still vital discipline, influences the opinion that many results of variational techniques are particularly suited when approaching nonlinear solutions of complex control problems. Therefore, even though first breaking discoveries in the optimal control were published more than 40 years ago, novel treatments of the fundamentals of classical optimal control theory are still important and of immediate interest. This book is a good example of accomplishing that task. The book is at a high scientific level and represents an integrated view of the discipline. Still its choice of topics, the relative weight given to them, and the nature of illustrative examples reflect the author’s commitment to effective teaching. The book is designed for both undergraduate and graduate students who have already been exposed to basic linear system and control theory and have the calculus background usually found in undergraduate curricula in engineering. Yet, graduates and teachers will also benefit from using the book. Almost any problem in the analysis, operation, and design of practical processes and industrial operations, and many associated problems, such as production planing, can be broken down in its final stage to the problem of determining the largest or smallest value of a function or a functional. Optimization of an arbitrary system requires knowledge of a model of a controlled system, the evaluation criterion called performance index, and constraints (not yet accounted for by the model). The constraints can be of different sort and may be given in diverse forms. When dynamical problems are in question, a typical optimization model consists of an integral functional, a set of constraining differential equations, and some local constraints imposed on control and/or state variables. That functional should be maximized or minimized. The task of optimization is then to find the best dynamics in terms of the best control functions and corresponding optimal trajectory. The book is composed of two parts: the first is devoted to global methods (sufficient optimality conditions), whereas the second deals with variational methods of the first order (necessary conditions) or second order (locally-sufficient conditions). The first part develops the basis of the Hamilton-Jacobi theory that provides sufficient conditions for global optimality along with its most essential accomplishments, namely the solution of the Linear Quadratic and Linear Quadratic Gaussian problems. Some attention is given to the Ricatti equations which play an essential role in these problems. The second part of the book begins with the presentation of the Pontryagin’s Maximum Principle, which represents the first-order variational approach and provides both powerful and elegant necessary conditions encompassing a wide class of complex problems. The second-order variational approach is displayed next and is shown to be capable of ensuring, under suitable conditions, the loc