Adaptive integration control with satisfactory transients for uncertain linear system

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 sufficiently small. An existing adaptive version of the above result for the class of stable lumped systems, can achieve asymptotically tracking, but can not robustly achieve satisfactory transients for any system of this class. In this paper we derive an adaptive version with self-learning iteration of the above result for the class of stable lumped systems, so that asymptotically tracking and satisfactory transients are achieved for any system of this class.