Fixed-time adaptive composite fuzzy control via dynamic regressor extension and mixing

This paper addresses the problem of adaptive fuzzy fixed-time control for a class of uncertain nonlinear strict-feedback systems. Firstly, by utilizing the dynamic regressor extension and mixing (DREM) process along with the composite learning (CL) method, a novel weight estimation algorithm is proposed for approximating unknown functions in uncertain systems, which ensures that the elements in the weight vector can converge independently without requiring strict persistent excitation (PE) condition. Moreover, an adaptive practical fixed-time control approach is proposed by incorporating a new segmented continuous filter and fixed-time stability theory. Note that the information of the weight error and a tracking error term are included in the weight update law, which directly improves the estimation performance and indirectly enhances the control accuracy. Theoretical analysis substantiates that the proposed control method guarantees the boundedness of all signals in the closed-loop system while ensuring that tracking error approaches a compact neighborhood surrounding the origin within fixed-time. Finally, simulation results are presented to demonstrate both feasibility and effectiveness of the proposed method. • By integrating the dynamic regressor extension and mixing (DREM) technique and learning method, a novel enhanced weight estimator for FLSs is devised to accurately approximate the unknown terms in the strict-feedback nonlinear systems, where the strict PE condition is relaxed. • Different from existing results, due to the use of the DREM process, the proposed weight estimator can be designed in scalar form. It is guaranteed that each element of the weight estimation error will converge in fixed-time independently. • To overcome the issue of the ‘explosion of complexity’ that arises from computing the derivatives of virtual controllers repetitively, a fixed-time nonlinear filter is specifically designed. Meanwhile, a segmented differentiable function is used to ensure the continuity and differentiability of both the filter and the controller.