The Simplicity of Robust Direct Adaptive Control with Disturbance Rejection for Linear Infinite-Dimensional Systems: An Introduction

Abstract

Our purpose in this paper is to provide an introduction to direct adaptive control in infinite dimensions and to prove the remarkable simplicity of infinite dimensional adaptive control even in infinite dimensions. However, we will show that the mathematical foundations are much more stringent in infinite dimensional theory. Consequently, this paper will emphasize some important fundamentals and will not be comprehensive. Much more can be done, but here we hope to provide a starting point for further exploration. Given a linear continuous-time infinite-dimensional plant on a Hilbert space and disturbances of known and unknown waveform, we show that there exists a very simple stabilizing direct adaptive control law with certain disturbance rejection and robustness properties. The closed loop system is shown to be exponentially convergent to a neighborhood with radius proportional to bounds on the size of the disturbance. The plant is described by a closed densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states. To illustrate the use of the control law, we apply the result to a partial differential equation example of an unstable diffusion system, but other applications are possible.