Sparse approximation-based collocation scheme for nonlinear optimal feedback control design

This paper discusses the development of a computationally efficient approach to generate optimal feedback control laws for infinite time problems by solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The solution process consists of iteratively solving the linear generalized HJB (GHJB) equation starting with an admissible stable controller. The collocation methods are exploited to solve the GHJB equation in the specified domain of interest. Recently developed nonproduct quadrature method known as Conjugate Unscented Transformation is used to manage the curse of dimensionality associated with the growth of collocation points with increase in the state dimension. Furthermore, recent advances in sparse approximation are leveraged to automatically generate the appropriate polynomial basis function set for the collocation-based approximation from an overcomplete dictionary of basis functions. It is demonstrated that the solution process uses the basis function selection process to automatically identify a form for the feedback control law, which is frequently unknown. Several numerical examples demonstrate the efficacy of the proposed approach in accurately generating feedback control policies for nonlinear systems.