Asymptotic regulation for distributed parameter systems via zero dynamics inverse design

In this paper, we introduce a new approach, zero dynamics inverse (ZDI) design, for designing a feedback compensation scheme achieving asymptotic regulation for a linear or nonlinear distributed parameter system in the case when the value w(t) at time t of the signal w to be tracked or rejected is a measured variable. Following the nonequilibrium formulation of output regulation, we formulate the problem of asymptotic regulation by requiring zero steady‐state error together with ultimate boundedness of the state of the system and the controller(s), with a bound determined by bounds on the norms of the initial data and w. Because a controller solving this problem depends only on a bound on the norm of w not on the particular choice of w, this formulation is in sharp contrast to both exact tracking, asymptotic tracking or dynamic inversion of a completely known trajectory and to output regulation with a known exosystem. The ZDI design consists of the interconnection, via a memoryless filter, of a stabilizing feedback compensator and a cascade controller, designed in a simple, universal way from the zero dynamics of the closed‐loop feedback system. This design philosophy is illustrated with a problem of asymptotic regulation for a boundary controlled viscous Burgers' equation, for which we prove that the ZDI is input‐to‐state stable. In infinite dimensions, however, input‐to‐state stable compactness arguments are supplanted by smoothing arguments to accommodate crucial technical details, including the global existence, uniqueness, and regularity of solutions to the interconnected systems. Copyright © 2011 John Wiley & Sons, Ltd.