Adaptive Stabilization of Mechanical Systems With Underactuation Degree One and Uncertainties Based on Controlled Lagrangians

This paper addresses the stabilization control problem for mechanical systems with underactuation degree one and inertia parameter uncertainties, based on the controlled Lagrangian (CL) method. A novel adaptive control strategy is first proposed by integrating the CL method with adaptive control theory, which effectively extends the applicability of the CL framework to systems with parameter uncertainties. The adaptive control strategy consists of three steps. First, a desired‐though not directly implementable‐controller that ensures closed‐loop stability is derived using the CL method. Then, an implementable nominal controller is constructed, along with an adaptive compensator to correct the error between the nominal and desired control laws. Finally, an adaptive parameter update law is designed to guarantee the stability of the overall closed‐loop system. As a case study, the proposed method is applied to stabilize a two degree of freedom Pendubot with inertia parameter uncertainties around its equilibrium point. Numerical simulations confirm the effectiveness and correctness of the designed controller.