Low - Gain Control of Uncertain Regular

It is well-known that closing the loop around an exponentially stable, nite-dimensional , linear, time-invariant plant with square transfer-function matrix G(s) compensated by a controller of the form (k=s)? 0 , where k 2 R and ? 0 2 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 G(0)? 0 have positive real parts and (ii) the gain parameter k is positive and suuciently small. In this paper we consider a rather general class of innnite-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 ? 0 can be tuned adaptively. The resulting adaptive tracking controllers are not based on system identiication or plant parameter algorithms, nor is the injection of probing signals required.