Piecewise fast multi-power reaching law: Basis for sliding mode control algorithm

A piecewise fast multi-power reaching law (PFMPRL) is proposed aiming at the problems of chattering and slow convergence in the reaching phase of sliding mode control (SMC). In this paper, the fast power reaching law and the double power reaching law are combined, and a nonlinear function is introduced to design the exponential term in PFMPRL. The proposed method ensures the characteristic of fast convergence of the system at all the phases of tendency. The characteristic of fixed-time convergence has also been satisfied. The study proves that the system state can converge to steady-state error bounds within a finite time in the presence of system uncertainty and bounded external disturbance. Compared with the existed methods, the proposed method has shorter convergence time and smaller steady-state error bound. To suppress the influence of model uncertainty and disturbance in system control, a non-linear disturbance observer (NDO) is introduced, and combined with the reaching law-based non-singular terminal sliding mode control (NTSMC), is applied to the cart inverted pendulum system. Simulation results and numerical analysis verify the effectiveness and superiority of this approach.