Low-Gain Control of Uncertain Regular Linear Systems

It is well known that closing the loop around an exponentially stable, finite-dimensional, linear, time-invariant plant with square transfer-function matrix $\BG(s)$ compensated by a controller of the form $(k/s)\Gamma_0$, where $k\in {\Bbb R}$ and $\Gamma_0\in {\Bbb R}\mm$, will result in an exponentially stable closed-loop system which achieves tracking of arbitrary constant reference signals, provided that (i) all the eigenvalues of $\BG(0)\Gamma_0$ have positive real parts and (ii) the gain parameter $k$ is positive and sufficiently small. In this paper we consider a rather general class of infinite-dimensional linear systems, called regular systems, for which convenient representations are known to exist, both in time and in frequency domain. The purpose of the paper is twofold: (i) we extend the above result to the class of exponentially stable regular systems and (ii) we show how the parameters $k$ and $\Gamma_0$ can be tuned adaptively. The resulting adaptive tracking controllers are not based on system identification or parameter estimation algorithms, nor is the injection of probing signals required.