Prescribed-time trajectory tracking control for a class of nonlinear system

Previous works have analyzed finite/fixed-time tracking control for nonlinear systems. In these works, achieving the accurate time convergence of errors must be under the premise of known initial values and careful design of control parameters. Then, how to break through the constraints of initial values and design parameters for this issue is an unsolved problem. Motivated by this, we successfully studied prescribed-time tracking control for single-input single-output nonlinear systems with uncertainties. Specifically, we designed a state feedback controller on $[0,T_p)$, based on the backstepping method, to make the tracking error (TE) tend to zero at $T_p$, in which $T_p$ is the arbitrarily selected prescribed-time. Furthermore, on $[T_p,\mathrm{\infty }),$ another controller, similarly to that on $[0,T_p)$, was designed to keep TE within a precision after $T_p$, while TE may not stay at zero. Therefore, on $[T_p,\mathrm{\infty })$, another new controller, based on sliding mode control, was built to ensure that TE stays at zero after $T_p.$