1R6. Optimal Control Systems. - DS Naidu (Idaho State Univ, Pocatello ID). CRC Press LLC, Boca Raton FL. 2003. 433 pp. ISBN 0-8493-0892-5. $99.95.

Reviewed by I Kolmanovsky (Sci Res Lab, MD-2036, Ford Motor Co, 2101 Village Rd, Dearborn MI 48124).

The book, Optimal Control Systems, by DS Naidu provides an introduction to key concepts and methods of the optimal control theory for deterministic continuous-time and discrete-time finite-dimensional dynamic systems. The material of the book is based on the author’s experience teaching graduate level optimal control courses and has grown out of the lecture notes used by the author in these courses.

After an Introduction (the first chapter of the book), the second chapter describes in detail the main ideas in the calculus of variations (culminating in the study of Euler-Lagrange equations and second variation-based Legendre conditions) and seamlessly transitions to the treatment of the optimal control problems. The Hamiltonian function is introduced early on which subsequently features prominently in  
the treatment of the Pontryagin’s maximum principle. The end result of Chapter 2 is a solution to several types of optimal control problems (such as fixed end-time, with no control constraints) using the principles of the calculus of variations but stated in the formalism that will later be useful for more general optimal control problems. These developments are already sufficient for the in-depth treatment of Linear Quadratic Optimal Control problem for continuous-time systems, which is one of the central tools in the control theory. This treatment is a subject of Chapters 3 and 4 where the finite horizon and infinite horizon cases, set point and trajectory tracking, fixed-end point regulator system, Linear Quadratic Regulator with a specified degree of stability, and other topics are considered in detail. Frequency domain properties of the linear quadratic regulator are analyzed and the celebrated results on one half to infinity gain margin and 60 degrees phase margin are derived. Chapter 5 treats similar issues as in Chapters 2-4 but for discrete-time systems. In Chapter 6, the treatment of constrained control problems begins with the introduction of the Pontryagin maximum principle for these systems (using the notion of admissible variations) and dynamic programming techniques. The latter are also used to illustrate an alternative approach to deriving continuous-time and discrete-time linear quadratic regulator solutions. Time-optimal control problems are treated in Chapter 7 including the analysis of the number of switchings in the optimal “bang-bang” control laws for linear systems. Other special optimal control problems (energy and fuel-optimal control problems with integral costs) including the case of singular controls are touched upon as well. The developments are completed by considering the point wise-in-time state constraints and describing how they can be treated using either the penalty function method or the slack variable method. The appendices help make the book self-contained by reviewing some of the key results in linear algebra and state space analysis of linear systems, and by providing the listings of the relevant Matlab files (available electronically from the author).

The book is very readable and careful attention has certainly been given to provide the reader with a comprehensive set of tools without overburdening with the complex mathematical details and notations (although the key mathematical ideas are, indeed, well exposed and the logic underlying the developments is transparent and thorough). It should be accessible to a wide audience including graduate students and practitioners from engineering and other fields. Optimal Control Systems can be utilized both as a textbook in the introductory courses in the optimal control theory, or as a quick reference by practicing engineers. The use of Matlab/Simulink to treat the examples (including the snippets of the actual code), historical remarks about the scientists behind the major discoveries in the optimal control theory, recipe-style summaries of the key methodologies, exercise problems at the end of each of the chapters are undoubtedly the strong points of this book which significantly contribute to its pedagogical value.

On a somewhat more critical side, most of the examples in the book are, in fact, of relatively low order linear systems. This can actually be viewed as an advantage in that the in-depth treatment of these examples is possible so that the main ideas can be illustrated rather well. At the same time, the treatment of more of higher order nonlinear system examples in combination with a more extended introduction into the numerical methods of the optimal control could have been beneficial for the readers to gain an additional insight into how the underlying techniques can be applied.