Singular Problems in Linear Estimation and Control

Abstract

Publisher Summary This chapter discusses various problems in the estimation of the state of linear dynamic systems using measurements corrupted by noise, which is Markov or correlated in time, or more generally, by noise with noninvertible covariance. These problems are members of a broader class of linear optimization problems, that is, the problems in both estimation and deterministic control, in which certain critical system matrices or transformations are singular, thus, posing difficulties in solution via usual techniques. In the colored noise filtering problem, the inverse of the measurement noise covariance, which is otherwise part of the optimal gain calculations, does not exist. An analogous difficulty occurs in the linear quadratic optimal regulator problem when the quadratic form of the control in the cost functional is not positive definite but only nonnegative definite. In this event, the Euler equations cannot be solved directly for the optimal control because of the requirement for the nonexistent inverse of the particular matrix. The techniques used for solution of the colored noise filter might be applied to the singular control problem.