Stability Analysis of Model Reference Adaptive Control with the New Theorem of Stability

Abstract

In the stability analysis of Model Reference Adaptive Control (MRAC), the Lyapunov's direct method is usually used and this method only results in stability of the adaptive system. Until now, Barbalat's Lemma, was commonly used to prove asymptotic convergence of the tracking error in these systems. One of the disadvantages of this Lemma is the imposition of strict condition of uniform continuity that restricts the use of this Lemma. In addition, this Lemma does not provide any result about convergence of adaptive gains. For this reason, until now it has been difficult and even mostly impossible to achieve the desired result of asymptotic stability for the MRAC by using methods based on the Barbalat's Lemma. The new Theorem of Stability is a direct extension of the Lyapunov method, which under very simple initial conditions and without the need for uniform continuity yields to more comprehensive results for all the states of the adaptive system and even in some cases results in asymptotic stability. In this paper, first, the new Theorem of Stability and its benefits are described and then, the stability analysis of the MRAC is performed by this new method and under simple conditions, the result of asymptotic stability for this adaptive system is obtained which is a very desirable result. Simulation of a single-link flexible-joint robot with MRAC has confirmed the accuracy of the performed stability analysis.