A modified invariance principle and gain convergence in adaptive control

In spite of successful implementations of simple adaptive control systems, the convergence of the adaptive control gains has remained an open question for more that 30 years. Moreover, the customary opinion is that the control gains do not actually converge and instead they may continue wandering without reaching any limit at all. Recently, this open question, that may give pause to practitioners and potential users of adaptive control, has recently been solved. The paper shows how a modified LaSalles invariance principle in combination with Gromwall-Bellman Lemma have finally allowed solving the gain convergence problem. It is shown that the control gains do reach a constant value at the end of a process of steepest descent error minimization, thus allowing the conclusion that simple and robust adaptive control systems can successfully be implemented in real-world systems.