Fixed-Time Adaptive Stabilization of Underactuated Euler–Lagrange Systems with Certified Internal Dynamics

This paper addresses the fixed-time adaptive stabilization issue for a category of underactuated mechanical systems regulated by Euler–Lagrange dynamics, characterized by aligned parametric uncertainties. Unlike traditional adaptive control schemes that only guarantee asymptotic convergence and usually assume stable internal dynamics, a new framework is created that guarantees global fixed-time convergence and explicitly proves that the zero dynamics caused by underactuation are stable. The suggested controller combines partial feedback linearization with a recursive fixed-time backstepping design and an online parameter adaptation law that keeps the structural properties of Euler–Lagrange systems. A new composite Lyapunov function is presented to address the coupled dynamics of actuated and unactuated coordinates and to formulate a differential inequality of the following form, ?˙ ≤ −??^? − ??^?, where 0 < α < 1 and β > 1. This structure guarantees global fixed-time convergence with a clear upper limit on the settling time that is not affected by the starting conditions. A Lyapunov certificate for the internal (zero) dynamics is also created and shown to work with the adaptive outer-loop design. This means that minimum-phase assumptions are no longer needed. A thorough analysis shows that all closed-loop signals are globally bounded and that the system is robust against matched uncertainties. Numerical simulations of typical underactuated systems show the theoretical properties and show how fixed-time performance compares to asymptotic adaptive controllers. The results lay a systematic groundwork for the fixed-time adaptive control of underactuated mechanical systems and facilitate further advancements towards robustness and constrained control.