ℒ asso ℳ 𝒫 𝒞 -Based ℒ 1 Adaptive Control for Uncertain Euler–Lagrange Systems: Guaranteed Stability Robustness and Performance

Abstract

The challenge of assessing system states and considering the robot's physical limitations impedes the development of an ${ℒ}_1$ adaptive controller for robotic systems. To solve this challenge, this study proposes an ${ℒ}_1$ adaptive controller based on ℒasso $ℳ𝒫𝒞$ (${ℒ}_{1}Aℒℳ𝒫𝒞$) (for Euler–Lagrange systems) that combines the method by a Barrier Lyapunov Function($ℬℒℱ$) and an adaptive high-gain observer ($𝒜\mathscr{H}𝒢𝒪$). In the face of uncertainty, time delay, and inaccessibility of system states, the presented approach establishes a mechanism to compromise between fast adaptation and robustness. The $ℬℒℱ$ constrains the system's outputs and adjusts the observer gain to ensure the output estimation error stays within a predetermined range. Then, to increase the prediction accuracy, ℒasso $ℳ𝒫𝒞$ ($ℒℳ𝒫𝒞$) uses the estimated system parameter (obtained by $𝒜\mathscr{H}𝒢𝒪$). To manage additive and parametric uncertainties, the proposed approach adds a robustness constraint to the $ℒℳ𝒫𝒞$ optimization. Subsequently, the effectiveness of the proposed method is assessed using an uncertain six-degrees-of-freedom (6-𝒟𝒪ℱ) remotely operated underwater vehicle (ℛ𝒪𝒱), and the stability of the closed-loop system under input delay is evaluated. The extensive numerical results demonstrate that the system with the proposed controller achieves superior performance compared to other adaptive controllers, particularly in terms of integral absolute tracking and estimation errors for this specific application. Furthermore, the results validate the proposed approach's ability to reject uncertainties and disturbances that fluctuate over time, and they exhibit a satisfactory tracking performance even in cases where the system dynamics are uncertain.