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. However, they also expose the fact that the convergence of the adaptive gains has remained uncertain. The gains may not converge to the ideal constant control gains predicted by the underlying linear time-invariant system considerations. As those prior conditions that were also needed for stability may not hold, this conclusion may raise doubts about the robustness of the adaptive system. This paper intends to show 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 constant ideal gain values that solve the problem at task.

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