Direct adaptive control of parabolic systems: algorithm synthesis and convergence and stability analysis

This paper presents a model reference adaptive control of a class of distributed parameter systems described by linear, n-dimensional, parabolic partial differential equations. Unknown parameters appearing in the system equation are either constant or spatially-varying. Distributed sensing and actuation are assumed to be available. Adaptation laws are obtained by the Lyapunov redesign method. It is shown that the concept of persistency of excitation, which guarantees the parameter error convergence to zero in finite-dimensional adaptive systems, in infinite-dimensional adaptive systems should be investigated in relation to time variable, spatial variable, and also boundary conditions. Unlike the finite-dimensional case, in infinite-dimensional adaptive systems even a constant input is shown to be persistently exciting in the sense that it guarantees the convergence of parameter errors to zero. Averaging theorems for two-time scale systems which involve a finite dimensional slow system and an infinite dimensional fast system are developed. The exponential stability of the adaptive system, which is critical in finite dimensional adaptive control in terms of tolerating disturbances and unmodeled dynamics, is shown by applying averaging. A numerical example which demonstrates an averaged system and computer simulations are provided.