A numerical approach to hybrid nonlinear optimal control

Abstract

This paper proposes a novel optimal control design framework for hybrid nonlinear dynamical systems involving an interacting combination of continuous-time and discrete-time dynamics. Two numerical algorithms are proposed to approximate the continuous-time and discrete-time portions of the hybrid Hamilton-Jacobi-Bellman (HJB) equation. Galerkin’s spectral method is utilised to approximate the value function involved in the continuous-time HJB equation, thereby computing the optimal control gains between impulsive events. Employing the spectral collocation method, the discrete-time HJB equation is then approximated to find the optimal control gain vector at impulsive instants. These two algorithms are ultimately combined to obtain the desired hybrid nonlinear optimal control law. Describing practical considerations for implementing the algorithms, some illustrative examples are presented to evaluate the functionality of the proposed hybrid nonlinear optimal controller.