Abstract
The issue of fixed-time tracking control of nonlinear systems subject to time-varying uncertain parameters is investigated in this article. In contrast to previous adaptive approach-based fixed-time control results which focus on driving the tracking error to a bounded region, it is technically challenging yet highly desired to achieve zero-error trajectory tracking. The primary difficulty lies in how to construct and analyze adaptive estimation schemes to completely compensate for uncertain parameters within the fixed-time convergence setting. Furthermore, the presence of time-varying uncertainties renders the systems fundamentally different from those in existing works. To tackle this challenge, a new fixed-time stability lemma utilizing an exponential decay function is proposed. Then, we develop an adaptive fixed-time controller design framework, it is demonstrated that the tracking error ultimately converges to zero after converging to a small neighborhood around zero within a fixed time, with all the closed-loop signals remaining bounded. Besides, the singularity problem in fixed-time control is circumvented. Finally, simulation results substantiate the effectiveness of the proposed strategy. Note to Practitioners-Actual engineering systems are invariably characterized by uncertainties and external disturbances, which should be adequately accounted for in the control design. Furthermore, convergence speed and steady-state accuracy serve as crucial metrics for evaluating system performance. Compared with the fixed-time stabilization, fixed-time tracking is more prevalent in practical control systems and poses greater challenges due to the temporal variability of the reference signal. However, due to the inability to fully compensate for the impact of adaptive estimation errors on the system, existing adaptive fixed-time control algorithms can only obtain bounded tracking results, which is obviously undesirable in application scenarios requiring high-precision tracking. Therefore, this paper proposes a stability lemma based on the exponential decay function, providing a systematic solution for achieving zero-error tracking under the adaptive fixed-time framework. Besides, the designed nonsingular fixed-time control strategy demonstrates robustness against more general time-varying uncertainties. This approach is particularly suited for high-performance tracking applications requiring both rapid convergence and precision, such as in wheeled robots and uncrewed surface vessels.