Introduction

This book represents a continuation of the work on parallel algorithms for optimal control of singularly perturbed linear deterministic and stochastic control problems of (Gajic and Shen, 1993). The book presents the most efficient methods for solving exactly (or with very high accuracy) optimal control and filtering problems of singularly perturbed linear systems by removing numerical ill-conditioning and obtaining well-conditioned, reducedorder, exact (or highly accurate) pure-slow and pure-fast subproblems. The class of problems solvable by the newly presented techniques are steady state linear-quadratic optimal control and filtering problems whose Hamiltonian matrices under appropriate scaling and permutation preserve singularly perturbed forms such that they can be block diagonalized into pure-slow and pure-fast Hamiltonian matrices. We call this method the Hamiltonian approach to singularly perturbed linear control systems. The problems presently solvable by the Hamiltonian method are: linear-quadratic optimal regulator and Kalman filter in continuous-and discrete-time domains, optimal open-loop control of continuous-and discrete-time linear systems, multimodeling estimation and control, H∞ optimal control and filtering of linear systems, linear-quadratic zero-sum differential games, linear-quadratic high gain, cheap control, small measurement noise problems, sampled data control systems, nonstandard linear singularly perturbed systems, and limited classes of finite horizon optimal linear control and filtering problems. Some other classes of linear-quadratic type optimal control problems that can be solved by the methodology considered in this book may emerge in the near future.