5R18. Optimal Control of Singularly Perturbed Linear Systems and Applications: High-Accuracy Techniques. - Z Gajic (Elec and Comput Eng Dept, Rutgers Univ, Piscataway NJ) and Myo-Taeg Lim (Korea Univ, Seoul, Korea). Marcel Dekker, New York. 2001. 309 pp. ISBN 0-8247-8976-8. $150.00.
Reviewed by J Bentsman (Dept of Mech and Indust Eng, MC-244, Univ of Illinois, 1206 W Green St, 332e meb, mc 244, Urbana IL 61801).
This book is intended for researchers in control theory, graduate students, and control engineers. The book presents computational methods that permit high accuracy controller design of singularly perturbed systems. A number of applications are considered where the use of the methods is demonstrated.
The main theme of the book is the exact or highly accurate approximate decomposition of the numerically ill-conditioned optimal control and filtering problems for the original singularly perturbed system into two well-conditioned sub-problems separately governing fast and regular motion, with the latter being referred to as slow motion. The decomposition is accomplished via coordinate transformation that can be computed recursively. The decomposition is applied either to the original singularly perturbed system yielding two subsystems that govern pure-slow and pure-fast system motion or solution of the control problem for the original system, decomposing this solution into two easily computable pure-slow and pure-fast sub-solutions.
The book is organized into nine chapters. It starts with the detailed introduction, Chapter 1, that reviews the literature on the subject and presents the main technical milestones and approaches: Chang transformation that exactly decouples a linear singularly perturbed system into independent slow and fast subsystems, Hamiltonian approach that permits block-diagonalization of the Hamiltonian matrices for optimal control and filtering problems into pure-slow and pure-fast Hamiltonian matrices, and recursive numerical procedures for obtaining the necessary decompositions.
Chapters 2 and 3 consider continuous-time and discrete-time linear quadratic optimal control and filtering problems for singularly perturbed systems. They present the method for the exact decomposition of the algebraic Riccati equation corresponding to the original system into two reduced order algebraic Riccati equations corresponding to slow and fast time scales. Chapter 4 presents the methods for the exact decomposition of the Riccati equation for the singularly perturbed system with two fast motions with different motion rates into equation for slow subsystem and two equations for fast subsystems. Chapter 5 presents similar decompositions for the H-infinity control and filtering problems. Chapter 6 presents decompositions for high gain and related cheap control problems as well as the dual small measurement noise filtering problem. Chapter 7 presents Riccati equation decompositions using the numerically attractive eigenvector approach. Chapter 8 considers decomposition of nonstandard singularly perturbed systems, addresses finite horizon closed-loop optimization, reviews slow-fast manifold theory, and compares it with the Hamiltonian approach. Chapter 9 briefly gives concluding remarks and indicates open research problems in this area.
Optimal Control of Singularly Perturbed Linear Systems and Applications: High-Accuracy Techniques is well focused, considers application examples, and provides high quality computational procedures. It has detailed subject index. This reviewer would highly recommend this book to researchers in singular perturbation. The book is also a necessary addition to any good technical library.
