Stochastic Control for Polynomial Systems
This chapter discusses the optimal control of stochastic polynomial systems. It presents discrete systems as the results will be useful in digital control schemes. The chapter shows that the results also apply to continuous systems. The solution to stochastic control problems depends on using the predictive formulation for the plant dynamics. The Kaiman filter requires the solution of a Riccati equation, while the polynomial system predictor requires the solution of a Diophantine equation. The chapter also discusses the closed-loop behavior under the influence of the control law. The importance of the predictive formulation is that it decomposes the cost index into a stochastic portion, and a deterministic portion that can be minimized by differentiation. The polynomial linear quadratic gaussian controller depends on the solution to the Diophantine equation. Its dynamics contain an implicit estimator for the portion of the internal state not appearing directly in the output.