Chapter 10 - Feedback Stabilization, Eigenvalue Assignment, and Optimal Control

Abstract

This chapter presents the problem of stabilizing a linear control system by appropriately choosing the control vector. Necessary and sufficient conditions are established for the existence of stabilizing feedback matrices, and Lyapunov-style methods for constructing such matrices are described in the chapter. A concept dual to stabilizability, called detectability, is then introduced and its connection with a Lyapunov matrix equation is established. In certain practical situations, stabilizing a system is not enough; a designer should be able to control the eigenvalues of matrix so that certain design constraints are met. This important result is established in the chapter. The proof of this result is constructive and leads to several well-known formulas, the most important of which is the Ackermann formula. Since there are no set guidelines as to where the poles (the eigenvalues) need to be placed, in practice, a compromise is very often made in which a feedback matrix is constructed in such a way that not only the system is stabilized but also a certain performance criterion is satisfied. This leads to the well-known Linear Quadratic Regulator (LQR) problem. Both continuous-time and discrete-time LQR problems are discussed in the chapter. The chapter also discusses the H∞-control problems. The H∞-control problems are concerned with stabilization of perturbed versions of a system, when certain bounds of perturbations are known. The solutions of the H∞-control problems also require solutions of certain algebraic Rieeati equations. Two algorithms are provided in the chapter for computing the H∞-norm.