Analysis of the error in an iterative algorithm for solution of the regulator equations for nonlinear parabolic control system

Abstract

This work is based on the classical geometric method, which in turn is based on the classical geometric regulation theory, which involves asymptotic tracking and disturbance rejection for nonlinear parabolic control systems. The classical geometric method is based on the solution of a coupled pair of operator equations referred to as regulator equations. In general, solving the regulator equations or even obtaining accurate numerical solutions for the simple control problem is not an easy task. In fact, most of the time the classical geometric method gives the solvability conditions of the regulator problem, rather than the actual solution. We present a methodology for tracking and disturbance rejection, which is more general than the one based on the regulator equations, and can be applied to general smooth signals. This methodology is based on an iterative method known as the 􀀈- iterative method for obtaining approximate solution for the regulator problems for a class of infinite dimensional linear control systems. This work describes the error analysis for this iterative method regarding more general references and disturbances. In this work we consider bounded input and output operators. In particular, we obtain estimates showing geometric convergence of the error, controlled by the parameter 􀀈. In addition, we demonstrate our estimates on a variety of control problems in multi-physics applications by numerically solving the 􀀈-iterative algorithm by using the finite element solver “COMSOL”.