Adaptive Integral Control of Time-Delay Systems

It is well-known that if the steady-state gain G(0) of a stable lumped system, with transfer function G(s), is positive, then compensating the system by an integral controller k=s, where k is a gain parameter, leads to a stable closed-loop system which achieves tracking of arbitrary constant reference signals, provided that the gain parameter k is positive and su ciently small. It is also well-known that this result extends to certain classes of di erential-delay and distributed parameter systems. In this paper we derive an adaptive version of the above result for the class of stable lumped systems with output delay, i.e. we show that the gain parameter k can be tuned adaptively, so that tracking is achieved for any system of this class. The resulting adaptive tracking controller is not based on system identi cation or parameter estimation algorithms, nor is the injection of probing signals required.