Set theoretic performance verification of low-frequency learning adaptive controllers

Although adaptive control has been used in numerous applications, the ability to obtain a predictable transient and steady-state closed-loop performance is still a challenging problem from the verification and validation standpoint. To that end, we considered a recently developed robust adaptive control methodology called low-frequency learning adaptive control and utilized a set of theoretic analysis to show that the transitory performance of this approach can be expressed, analyzed, and optimized via a convex optimization problem based on linear matrix inequalities. This key feature of this design and analysis framework allows one to tune the adaptive control parameters rigorously so that the tracking error components of the closed-loop nonlinear system evolve in a priori specified region of the state space whose size can be minimized by selecting a suitable cost function. Simulation examples are provided to demonstrate the efficacy of the proposed verification and validation architecture showing the possibility of performing parametric studies to analyze the interplay between the size of the tracking error residual set and important design parameters such as the adaptation rate and the low-pass filters time constant of the weights adaptation algorithm