Uniform Parametric Convergence in the Adaptive Control of Mechanical Systems

It is known in the practice of closed-loop identification of mechanical systems that, since the Lagrangian dynamic model is linear in the constant physical parameters, a sufficient condition for parameter convergence is that the system follows reference trajectories that render the regressor matrix persistently exciting. This was established many years ago for a particular robot adaptive controller by Slotine and Li. This result is based on adaptive linear systems theory and relies on the condition that the applied control law makes the robot asymptotically track the imposed rich reference trajectories. This paper revisits this problem and establishes a proof that this condition is not only sufficient but also necessary to establish the stronger property of uniform global asymptotic stability of the overall closed-loop system. Our proof is based on modern tools and a relaxed formulation of the condition of persistency of excitation, tailored for nonlinear systems. Hence, it is not limited to the controller of Slotine and Li but applies to many other reported controllers. Furthermore, we give sufficient conditions for exponential stability (hence, in particular for uniform exponential parameter convergence). Finally, we provide some experimental results, illustrative of our theoretical findings.