Optimal adaptive control: A non-linear separation theorem†

Abstract For the quadratic cost, non-linear, adaptive stochastic control problem with linear plant and measurement models excited by white gaussian noise, and unknown time-invariant model parameters, the optimal stochastic control is obtained and shown to separate (‘non-linear separation theorem’) into a bank of model-conditional deterministic controller gains and a corresponding bank of known non-linear functional of the model-conditional, causal, mean-square state-vector estimates. This separation may also be viewed as a decomposition of the optimal, non-linear adaptive control into a bank of model-conditional, optimal, non-adaptive linear controls, one for each admissible value fof the unknown parameter θ and a nonlinear part, namely the bank of a posteriori model probabilities, which incorporate the adaptive nature, of the optimal adaptive control. Results are given for a special case of the above problem— namely, uncertainty in the measurement matrix—that exhibit drastically reduced computational req...