Optimal adaptive control: A non-linear separation theorem†

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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 requirements. In this special case, we have explicit separation between control and estimation, and it is shown that only one deterministic controller is required to be used with the non-linear, adaptive, mean-square state-vector estimate.

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