Transient and Asymptotic Properties of Robust Adaptive Controllers in the Presence of Non-Coercive Lyapunov Functions

Adaptive control architectures often make use of Lyapunov functions to design adaptive laws. We are specifically interested in adaptive control methods, such as the well-known ${{\mathcal{L}}_1}$ adaptive architecture, which employ a parameter observer for this purpose. In such architectures, the observation error plays a critical role in determining analytical bounds on the tracking error as well as robustness. In this paper, we show how the non-existence of coercive Lyapunov operators can impact the analytical bounds, and with it the performance and the robustness of such adaptive systems.