Polynomial chaos expansions for optimal control of nonlinear random oscillators

The polynomial chaos decomposition of stochastic variables and processes is implemented in conjunction with optimal polynomial control of nonlinear dynamical systems. The procedure is demonstrated on a base-excited system whereby ground motion is modeled as a stochastic process with a specified correlation function and is approximated by its Karhunen–Loeve expansion. An adaptive scheme for stochastic approximation with polynomial chaos bases is proposed which is based on a displacement–velocity norm and is applied to the identification of phase orbits of nonlinear oscillators. This approximation is then integrated in the design of an optimal polynomial controller, allowing for the efficient estimation of statistics and probability density functions of quantities of interest. Numerical investigations are carried out employing the polynomial chaos expansions and the Lyapunov asymptotic stability condition based control policy. The results reveal that the performance, as gaged by probabilistic quantities of interest, of the controlled oscillators is greatly improved. A comparative study is also presented against the classical stochastic optimal control, whereby statistical linearization based LQG is employed to design the optimal controller. It is remarked that the proposed polynomial chaos expansion is a preferred approach to the optimal control of nonlinear random oscillators.