Finite-dimensional optimal controllers for nonlinear plants

Abstract

Optimal risk sensitive feedback controllers are now available for very general stochastic nonlinear plants and performance indices. They consist of nonlinear static feedback of so called information states from an information state filter. In general, these filters are linear, but infinite dimensional, and the information state feedback gains are derived from (doubly) infinite dimensional dynamic programming. The challenge is to achieve optimal finite dimensional controllers using finite dimensional calculations for practical implementation. This paper derives risk sensitive optimality results for finite-dimensional controllers. The controllers can be conveniently derived for ‘linearized’ (approximate) models (applied to nonlinear stochastic systems). Performance indices for which the controllers are optimal for the nonlinear plants are revealed. That is, inverse risk-sensitive optimal control results for nonlinear stochastic systems with finite dimensional linear controllers are generated. It is instructive to see from these results that as the nonlinear plants approach linearity, the risk sensitive finite dimensional controllers designed using linearized plant models and risk sensitive indices with quadratic cost kernels, are optimal for a risk sensitive cost index which approaches one with a quadratic cost kernel. Also even far from plant linearity, as the linearized model noise variance becomes suitably large, the index optimized is dominated by terms which can have an interesting and practical interpretation. Limiting versions of the results as the noise variances approach zero apply in a purely deterministic nonlinear H ∞ setting. Risk neutral and continuous-time results are summarized. More general indices than risk sensitive indices are introduced with the view to giving useful inverse optimal control results in non-Gaussian noise environments.