Nonlinear H∞ Control Based on Successive Gaussian Process Regression

Abstract

Nonlinear <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> controllers can be designed based on the solution of the celebrated Hamilton-Jacobi-Isaacs (HJI) equation, which is a nonlinear partial differential equation (PDE) and generally hard to solve. In this brief, a concise but effective Gaussian process regression based method is proposed, and the solution of the HJI equation is obtained without explicitly discretizing the PDE. Based on the successive approximation technique, the nonlinear HJI PDE is transformed into a sequence of linear PDEs, and it is shown in this brief that solution to the linear PDE can be obtained by dealing with easily solvable initial value problems. Then a successive Gaussian process regression based method is designed, and the solution of the HJI equation is calculated by successively solving initial value problems and constructing regression models. Simulations are conducted and numerical results demonstrate its superior performance in accuracy and efficiency.