On Gain Conditions and Convergence of Simple Adaptive Control

What is it about?

Recent publications have presented successful implementations of simple direct adaptive control techniques in various applications. Along with the successful applications, they also expose the fact that the convergence of the adaptive gains has remained uncertain until now. The gains may not necessarily converge to the ideal constant control gains that guarantee perfect tracking in linear time invariant systems. Although this conclusion may raise doubts about the robustness of the adaptive system, this paper shows that the adaptive control performs perfect tracking even when the linear time invariant solution does not exist. It is shown that the adaptation performs a “steepest descent” minimization of the errors, ultimately ending with the appropriate set of control gains that fit the particular input command and initial conditions. The adaptive gains do asymptotically reach an appropriate set of bounded ideal gain values that solve the problem at task.

Featured Image