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. This notion is motivated by classical root-locus constructs from finite dimensional linear systems theory. In particular, for scalar proportional output feedback this notion of zero dynamics leads to the following natural result: the closed loop poles vary from the open loop poles to the open loop zeros (transmission zeros) and infinity as the gain varies from zero to infinity. A similar result holds for convergence of trajectories of the closed loop system to those of the zero dynamics system. This concept of zero dynamics is used to obtain a 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.