The new theorem of stability and gain convergence in simple adaptive control

Abstract

Convergence of the adaptive control gains when the presence of sufficient excitation is not available has remained an open question for more than 30 years. The rather common opinion that in these cases the adaptive control gains do not actually converge and may continue wandering without reaching any limit at all, even when asymptotically perfect tracking is reached, is disturbing to practitioners and potential users of adaptive control. Recent publications have provided solutions to various aspects of this open question within the particular frame of so-called Simple Adaptive Control methodology. However, a thorough review of the problem shows that the solution to the ultimate behavior of the adaptive gains when sufficient excitation is not available may still be considered incomplete. The present paper revisits the issue in order to finally show that a new Theorem of Stability, which greatly simplifies stability analysis for nonautonomous nonlinear systems, in combination with Gronwall-Bellman Lemma provide the solution of the gain convergence problem. It is shown that the control gains do reach a constant value at the end of adaptation process, thus allowing the conclusion that simple and robust adaptive control systems can successfully be implemented in real-world systems.