Optimal adaptive control of linear systems with unknown measurement subsystems

Abstract

For the quadratic cost, adaptive stochastic control problem with linear plant and measurement models excited by white Gaussian noise, and with unknown time invariant model parameters in the measurement subsystem, the optimal stochastic control is obtained. It is shown that separation between estimation and control holds in this case. The controller is shown to be linear and deterministic, while the estimator is shown to consist of a bank of known nonlinear functions of model conditional, causal, mean-square state-vector estimates. The estimator can again be partitioned into a bank of linear, model conditional, optimal, nonadaptive estimates, one for each admissible value of unknown parameter vector θ, and a nonlinear bank of a posteriori model probabilities which incorporate the adaptive nature of the optimal adaptive control. The results are presented both for continuous-time and discrete-time systems. A special case of the above problem is also discussed. It is the case of joint detection and control.