Long-time average cost control of stochastic systems using sum of squares of polynomials

This paper presents a computationally attractive long-time average cost control approach for a class of nonlinear stochastic systems, where the deterministic dynamical part is of polynomial type. Instead of minimizing the time-averaged cost itself, we use its upper bound as the objective function for controller design. As such, under the framework of sum-of-squares-based optimization, the control law and a tunable function similar to the Lyapunov function are optimized simultaneously. The inherent non-convexity of the optimisation is resolved by assuming that the controller takes a small-feedback structure, which actually is a series in a small parameter with all the coefficients being finite-order polynomials of the system state. The effectiveness of the proposed controller is demonstrated by means of simulation of a simple cylinder flow model under persistent perturbation of random noise.