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Computationally efficient simultaneous policy update algorithm for nonlinear H ∞ state feedback control with Galerkin's method
Author(s) -
Luo Biao,
Wu HuaiNing
Publication year - 2013
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.2814
Subject(s) - nonlinear system , lyapunov function , mathematics , partial differential equation , convergence (economics) , sequence (biology) , bottleneck , galerkin method , iterative method , function (biology) , newton's method , simple (philosophy) , computer science , mathematical optimization , mathematical analysis , physics , quantum mechanics , evolutionary biology , biology , economics , genetics , embedded system , economic growth , philosophy , epistemology
SUMMARY The 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.