Direct-Adaptive Control of Relative-Degree-Two Systems with Certifiable Derivative Error Bound

In this work we propose a new direct adaptive control scheme for relative degree-two systems that allows one to establish prescribed bounds on the tracking-error and its derivative in a model-reference control setting. The proposed adaptive controller enjoys some remarkable features, namely it does not require neither error augmentation nor normalization of the parameter-adaptation law. The approach can be easily extended to systems with relative degree one or with direct feed-through. The tracking error and its derivative are shown to satisfy
$\mathcal{L}_{2}$
and
$\mathcal{L}_{\infty}$
bounds, which can be made arbitrarily small by increasing the adaptation gains. The possibility to guarantee certifiable and tunable bounds on the tracking error and its first derivative opens the door to the application of the proposed adaptive control algorithm to a wide class of real-world systems with stringent requirements on transient performance.