A switchable exponent predefined time stability theory and adaptive sliding mode control for nonlinear systems

This paper proposes a novel flexible predefined-time stability theorem, and designs an adaptive fast sliding mode surface and control law based on this theorem to achieve synchronization control of chaotic systems with external disturbances and internal uncertain parameters. Specifically, a new predefined-time inequality is first constructed, which is characterized by a more flexible power term whose exponent is no longer limited to the conventional ranges of (0,1) and (1, +∞). Based on this inequality, a predefined-time stability theory is proposed, which significantly expands the existing research results. The designed adaptive fast predefined-time sliding mode surface and control law not only ensure that the system converges within the predefined time, but also can adaptively adjust the convergence rate according to the distance between the system state and the convergence point. This design, rooted in the flexible exponent, shows higher flexibility in parameter configuration compared with traditional methods. Finally, numerical simulation results verify the effectiveness and advancement of the proposed method.