Feedback Control Design Maximizing the Region of Attraction of Stochastic Systems Using Polynomial Chaos Expansion

A feedback control design is proposed for stochastic systems with finite second moment which aims at maximising the region of attraction of the equilibrium point. Polynomial Chaos (PC) expansions are employed to represent the stochastic closed loop system by a higher dimensional set of deterministic equations. By using the PC expanded system representation, the available information on the uncertainty affecting the system explicitly enters the control design problem. Further, this allows Lyapunov methods for deterministic systems to be used to formulate the stability criteria certifying the region of attraction. These criteria are parametrized by the feedback gain and formulated in a polynomial optimization program which is solved using sum-of-squares methods. This approach offers flexibility in the choice of the stochastic feedback law and accounts for input constraints. The application is demonstrated by two numerical examples.