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 t.imeinvariant model parameters, the optimal stochastic control is obtained and shown to separate ('non-linear separation theorem') into a bank of model.conditionel deterministic controller gains and a corresponding bank of known non-linear functionals 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 8 and a non-linear 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 problemnamely, 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 8ho\\-'11 that only one detenninistic controller is required to be used with the non-linear, adaptive, mean-square state-vector estimate.

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