Pole-zero cancellation problem in the convergence of explicit LQG self-tuning controllers

Abstract

A problem which arises in the convergence of LQG explicit self-tuning controllers is considered, concerning the possibility of unstable pole-zero cancellation in the estimated system model. Since the underlying plant cannot contain such a term the problem is to stabilize this estimated system model knowing that the cancellation does not arise from the plant description itself. The assumption is therefore made that the cancellation arises due to unstable terms in the external stochastic signal inputs to the system. The problem is then to derive the LQG controller for such a system and to show that it stabilizes the assumed system model. The LQG stochastic optimal controller is obtained for systems which can include unstable disturbance, reference or measurement noise models. Although an infinite-time solution is required, this must be obtained by considering the limiting case of a finite-time problem. The necessary and sufficient conditions for optimality are therefore derived in the time-domain for a finite-time situation. The required solution is then obtained using a polynomial systems approach. Finally the resulting self-tuning algorithm, and the convergence and stability proof which avoids the unstable pole-zero cancellation problem, are discussed.