Robust periodic reference tracking by stable uncertain infinite-dimensional linear systems

A robust control scheme for tracking of periodic signals, consisting of a finite number of sinusoids, by uncertain exponentially stable infinite dimensional linear systems is presented. The scheme consists in constructing a cascade interconnection of the stable linear system and a partitioning filter and augmenting this cascade system with a simple internal model based filter. The stable system model is presumed to be unknown, but its transfer function gain at the frequencies to be tracked is assumed to be known and non-zero. A theorem guaranteeing the robust stability and performance of this scheme while tracking a sinusoidal reference is proved. The general theorem for tracking periodic signals is stated and can be established analogously. A discussion on quantitatively estimating the robustness of this scheme is presented. The efficacy of the scheme is demonstrated via simulation of an example. The simplicity of the proposed scheme, its quantitatively ascertainable robustness and a virtual lack of modeling requirements make it well suited for industrial applications.

I. Introduction

INTERNAL model principle [1] enables tracking of periodic signals with zero steady state error by embedding the generator of the signal into the closed-loop system. This approach has been used in [2] [6] for linear finite dimensional plants. In these works the plant model is assumed to be known. This paper addresses the tracking of periodic signals consisting of a finite number of sinusoids assuming no knowledge of the plant, other than that it is an exponentially stable regular linear system and its transfer function (TF) gain at the frequencies to be tracked, readily found by experiments, is known and non-zero. Davison [7] [9] provides a solution methodology for a similar problem in case of stable finite dimensional plants. This methodology has been extended to exponentially stable regular linear systems (a large subset of well-posed linear system) in [10] (only for step reference), to the class of stable plants with transfer function in Callier- Desoer algebra in [11] and to exponentially stable well-posed linear systems in [12].