A note on polynomial chaos expansions for designing a linear feedback control for nonlinear systems

This paper presents a polynomial chaos-based framework for designing optimal linear feedback control laws for nonlinear systems with stochastic parametric uncertainty. The spectral decomposition of the original stochastic dynamical model in an orthogonal polynomial basis, prescribed by the Wiener–Askey scheme, provides a deterministic model from which the optimal linear control law is designed. Optimality of the proposed control law is proved by solving the Hamilton–Jacobi–Bellman equation, and asymptotic stability of the controlled nonlinear systems is guaranteed in the Lyapunov sense. We are especially interested in synchronization of chaotic systems. For this reason, the control strategy is applied in the trajectory tracking of periodic orbits for the Duffing oscillator and the Rössler system with uncertain stochastic parameters and initial conditions. The results are verified with Monte Carlo simulations.