Indirect stochastic adaptive control using optimal joint parameter and state estimation

Abstract

The SISO and MIMO adaptive problems are approached via the theory of stochastic optimal control, using results in the literature and previously-unexploited properties of canonical singular-pencil dynamical models. For both the SISO and MIMO cases, it is shown how to choose the input to minimize the sample mean-square error between the filtered output and the filtered desired output.

In the SISO case, the proposed method, which is an indirect scheme, leads to control laws which not only use the estimates of the parameters and the states but also the variances of the estimation errors. In other words, the SISO cautious control problem is solved for the general delay-white noise case.

In the MIMO case, it is proved that the cautious control problem can be solved only if the interactor matrix of the system is known. However, if the interactor matrix is unknown, there will be no exact solution for the optimal problem. Hence, in this research the certainty equivalence principle is employed based on the result obtained by Das [17] and a priori knowledge of the degrees of the diagonal entries of the interactor matrix. It is shown that the conventional approaches discussed in the literature cannot be applied to a stochastic case. However, the indirect approach proposed in this research is capable of dealing with stochastic cases.

Simulation results show the performance of the proposed methods both in the SISO case and the MIMO case.