Computationally efficient simultaneous policy update algorithm for nonlinear H∞ state feedback control with Galerkin's method

• Published in: International journal of robust and nonlinear control 2013-06, Vol.23 (9), p.991-1012
• Main Authors: Luo, Biao, Wu, Huai-Ning
• Format: Article
• Language: English
• Subjects: convergence; Galerkin's method; Hamilton-Jacobi-Isaacs equation; H∞state feedback control; simultaneous policy update algorithm
• ISSN: 1049-8923; 1099-1239
• DOI: 10.1002/rnc.2814

SUMMARYThe main bottleneck for the application of H∞ control theory on practical nonlinear systems is the need to solve the Hamilton–Jacobi–Isaacs (HJI) equation. The HJI equation is a nonlinear partial differential equation (PDE) that has proven to be impossible to solve analytically, even the approximate solution is still difficult to obtain. In this paper, we propose a simultaneous policy update algorithm (SPUA), in which the nonlinear HJI equation is solved by iteratively solving a sequence of Lyapunov function equations that are linear PDEs. By constructing a fixed point equation, the convergence of the SPUA is established rigorously by proving that it is essentially a Newton's iteration method for finding the fixed point. Subsequently, a computationally efficient SPUA (CESPUA) based on Galerkin's method, is developed to solve Lyapunov function equations in each iterative step of SPUA. The CESPUA is simple for implementation because only one iterative loop is included. Through the simulation studies on three examples, the results demonstrate that the proposed CESPUA is valid and efficient. Copyright © 2012 John Wiley & Sons, Ltd.