Adaptive Free-Will Arbitrary Time Tracking of a Class of Uncertain SISO Nonlinear Systems

Abstract

This paper investigates the free-will arbitrary time (FWAT) tracking problem for a class of single-input single-output (SISO) strict-feedback nonlinear systems without/with parametric uncertainties. First, we design an FWAT backstepping tracking controller for strict-feedback nonlinear systems without uncertainties in a recursive manner. It is shown that the proposed virtual controllers and their derivatives are continuously differentiable for the backstepping design, and the tracking error converges to zero within the pre-specified settling time independent of design parameters and initial conditions and remains at zero after the pre-specified settling time. Second, we address the adaptive FWAT tracking problem of SISO strict-feedback nonlinear systems with parametric uncertainties in the practical stability sense. The continuously differentiable and nonsingular time-varying adjustment functions for practical FWAT tracking are developed to transform error variables for the recursive backstepping design. Then, the adaptive FWAT backstepping tracking design and stability analysis strategy are established to ensure practical FWAT convergence within the pre-specified settling time independent of design parameters and initial conditions. The designed adaptive virtual and actual controllers are continuous for all $t\ge 0$. Furthermore, a practical adaptive prescribed-time tracking scheme is provided using the proposed practical adaptive FWAT tracking scheme. Finally, the effectiveness of the theoretical approaches is confirmed through a simulation example and an experimental result.