Novel flexible fixed-time stability theorem and its application to sliding mode control nonlinear systems

For the fixed-time nonlinear system control problem, a new fixed-time stability (FxTS) theorem and an integral sliding mode surface are proposed to balance the control speed and energy consumption. We discuss the existing fixed time inequalities and set up less conservative inequalities to study the FxTS theorem. The new inequality differs from other existing inequalities in that the parameter settings are more flexible. Under different parameter settings, the exact upper bound on settling time in four cases is discussed. Based on the stability theorem, a new integral sliding mode surface and sliding mode controller are proposed. The new control algorithm is successfully applied to the fixed-time control of chaotic four-dimensional Lorenz systems and permanent magnet synchronous motor systems. By comparing the numerical simulation results of this paper's method and traditional fixed-time sliding mode control (SMC), the flexibility and superiority of the theory proposed in this paper are demonstrated. Under the same parameter settings, compared to the traditional FxTS SMC, it reduces the convergence time by 18%, and the estimated upper bound of the fixed time reduction in waiting time is 41%. In addition, changing the variable parameters can improve the convergence velocity.