Zero dynamics modeling and boundary feedback design for parabolic systems

In this work the authors introduce a notion of zero dynamics for distributed parameter systems governed by linear parabolic equations on bounded domains with controls implemented through first order linear boundary conditions. The idea of zero dynamics presented here is motivated by classical root-locus constructs from finite dimensional linear systems theory. In particular for a scalar proportional output feedback trajectories of the closed loop system converge to those of the zero dynamics system as the gain tends to infinity. The main contribution of this work is a simple modeling and design mechanism for solving certain tracking and disturbance rejection problems. This mechanism provides an alternative to the geometric approach involving regulator equations. Our approach is based on the fact that for certain boundary control systems it is very easy to model the system’s zero dynamics, which, in turn, provides a simple systematic methodology for solving certain problems of output regulation. As special cases we describe dynamic and static controllers from the associated zero dynamics system for set-point and harmonic tracking. In order to provide a simple roadmap to using this methodology we present numerical examples for a one dimensional heat equation.