Adaptive stabilization of infinite dimensional linear systems

For finite dimensional nonlinear systems, an enhancement of root-locus methods has recently been developed for the analysis and design of nonlinear feedback systems. Central to this approach was a dynamical systems interpretation of zeroes, suggested by geometric control theory. For flexible systems, an analogous physical interpretation of system zeroes has been used for some time to compute system transmission zeroes. Motivated by this method, by work of Curtain on disturbance decoupling and by the success of this philosophy for stabilizing nonlinear systems, we give a formal definition of zero dynamics for a broad class of infinite dimensional systems. We illustrate this for scalar systems evolving on a Banach space by rigorously proving two stability results. First, any relative degree one, minimum phase system can be stabilized by proportional error feedback. We next demonstrate that a simple adaptive stabilization scheme presented by Byrnes and Willems, which did not require an a priori bound on the system dimension, continues to function for a wide class of infinite dimensional system.