Special Functions-Based Fixed-Time Estimation and Stabilization for Dynamic Systems

Fixed-time stability (FXTS) and fixed-time control (FXTC) of dynamic systems are reconsidered in this article based on special functions from the view of improving the estimate accuracy of settling time (ST) and reducing the chattering caused by the sign function. First, by means of the idea of contradiction and variable transformations, some generic FXTS criteria are established and some upper bounds of ST are directly calculated and expressed by several special functions. It is further proved that these estimates are the most accurate compared with the existing results. Besides, to suppress the chattering caused by the sign function, some saturation functions are constructed to replace the sign function and the FXTS of the new system obtained by replacing is ensured by rigorous theoretical analysis. As applications, the problem of stabilization for chaotic systems in fixed or preassigned time is explored. Especially, an innovative saturation controller is developed to realize preassigned-time stabilization, where the convergence time is prescribed in advance according to actual requirement and the control gains are finite, the existing control methods with time-varying infinite gains are essentially improved. Lastly, three numerical examples are provided to verify the improved estimates of ST and the chattering reduction.