Exponential convergence and robustness margins in adaptive control

The paper presents general results on the stability of ordinary nonlinear differential equations in the presence of bounded disturbances. These are used to study the robustness of a simple model reference adaptive control algorithm. It is shown that the system is guaranteed to remain stable in the presence of disturbances (arising from input disturbances, plant parameter variation, output disturbances, unmodelled dynamics...) provided that the unperturbed system is exponentially stable. Moreover, the bounds on the level of disturbances that can be tolerated increase with the rate of convergence. In the present application (and also for most adaptive systems), exponential convergence follows from persistent excitation of the exogeneous reference signal. The paper concludes with remarks on consequences of the results on practical applications.