Title :
Truncation and approximation errors in the max-plus algorithm for H∞ value function computation
Author :
McEneaney, William M.
Author_Institution :
Dept. of Math., California Univ., San Diego, La Jolla, CA, USA
Abstract :
We consider the H∞. problem for a nonlinear system. The corresponding dynamic programming equation (DPE) 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 are multiple viscosity solutions. The computation of the solution of a nonlinear, steady-state, first-order PDE is typically quite difficult. However, the semi-group associated with such a PDE is linear over the max-plus algebra. Combining this with a basis for the space of semi-convex functions over the max-plus algebra leads to a new class of numerical methods for such problems. Here we examine the errors associated with such a numerical method, and obtain a (conservative) bound on the convergence rate.
Keywords :
H∞ control; approximation theory; convergence of numerical methods; dynamic programming; group theory; nonlinear control systems; nonlinear differential equations; partial differential equations; H∞ value function; approximation errors; convergence rate; dynamic programming equation; max-plus algorithm; nonlinear first-order steady-state partial differential equation; nonlinear system; nonsmooth solutions; semi-group; truncation errors; viscosity solutions; Algebra; Approximation algorithms; Approximation error; Differential equations; Dynamic programming; Nonlinear equations; Nonlinear systems; Partial differential equations; Steady-state; Viscosity;
Conference_Titel :
Decision and Control, 2002, Proceedings of the 41st IEEE Conference on
Print_ISBN :
0-7803-7516-5
DOI :
10.1109/CDC.2002.1184592