Numerical Methods for Linear Control Systems

Abstract

This chapter presents a well-known procedure “Kalman filtering” developed by Kalman for optimal estimation of the states of a stochastic system, followed by a brief discussion on the Linear Quadratic Gaussian problem that deals with optimization of a performance measure for a stochastic system. The chapter discusses how the states of a continuous-time system can be estimated. The discussions apply equally to the discrete-time systems, possibly with some minor changes. So the main focus is on the continuous-time case. The chapter describes two common procedures for state estimation: one, via eigenvalue assignment and the other, via solution of the Sylvester-observer equation. The chapter also describes two other numerical methods, especially designed for Sylvester-observer equation; both are based on the reduction of the observable pair to the observer-Hessenberg pair and are recursive in nature. Both numerical methods seem to have good numerical properties.