A new adaptive optimal LQ regulator for linear systems based on two-point-multirate controllers

This article solves the adaptive LQ optimal regulation problem for continuous-time systems using a new class of multirate controllers called two-point-multirate controllers. In this type of controller, the control is constrained to a certain piecewise constant signal, and the controlled plant output is detected many times over a fundamental sampling period. The adaptive control strategy suggested here relies on solving the continuous LQ regulation problem. On the basis of this strategy, the original problem is reduced to an associate discrete-time LQ regulation problem for the performance index with cross-product terms, for which a fictitious static-state feedback controller needs to be computed. Thus, the present technique essentially resorts to the computation of simple gain controllers rather than to the computation of state observers, as compared to known techniques. As a consequence, the exogenous dynamics introduced in the control loop is of low order. The proposed adaptive scheme is readily applicable to non-minimum phase systems and to systems that do not possess the parity interlacing property (i.e., they are not strongly stabilizable). Persistency of excitation of the controlled system is assured without making any assumption about the existence of special convex sets in which the estimated system parameters belong, or about the coprimeness of the polynomials describing the ARMA model, as in known techniques. The a priori knowledge needed to implement the proposed adaptive algorithm is controllability and observability of the continuous and the discretized system under control and its order.