Max-Plus Eigenvector Methods for Nonlinear H$_\infty$ Problems: Error Analysis

Abstract

The H$_\infty$ problem for a nonlinear system is considered. The corresponding dynamic programming equation is a fully nonlinear, first-order, steady-state partial differential equation (PDE), possessing a term which is quadratic in the gradient. The solutions are typically nonsmooth, and further, there is nonuniqueness among the class of viscosity solutions. In the case where one tests a feedback control to see if it yields an H$_\infty$ controller, or where either the controller or disturbance sufficiently dominates, the PDE is a Hamilton--Jacobi--Bellman equation. The computation of the solution of a nonlinear, steady-state, first-order PDE is typically quite difficult. In a companion paper, we developed an entirely new class of methods for obtaining the "correct" solution of such PDEs. These methods are based on the linearity of the associated semigroup over the max-plus (or, in some cases, min-plus) algebra. In particular, solution of the PDE is reduced to solution of a max-plus (or min-plus) eigenvector problem for known unique eigenvalue 0 (the max-plus multiplicative identity). It was demonstrated that the eigenvector is unique and that the power method converges to it. In the companion paper, the basic method was laid out without discussion of errors and convergence. In this paper, we both approach the error analysis for such an algorithm, and demonstrate convergence. The errors are due to both the truncation of the basis expansion and computation of the matrix whose eigenvector one computes.