Stability Analysis Of Caputo Fractional-Order Switching System

As a kind of special hybrid system,switching system has been involved in many engineering fields with complex switching characteristics,such as the multi-loop network scheduling control system,wind power generation control system,etc.With the development of fractional calculus,it can be divided into integer-order and fractional-order switching systems.The theoretical system and application of integerorder switching systems have tended to be mature,but for fractional-order switching systems,its special properties make the relevant stability research results incomplete and need to be further developed.Based on integer-order switching system,this paper analyzes the essential problems of fractional-order switching system,and then studies the related stability problems for various types of Caputo fractional-order switching system,so as to enrich the theoretical system and application field of fractional-order switching system.The main research contents are as follows:(1)Solve the key problems in the fractional-order switching system: 1)Whether the lower bound of Caputo’s fractional derivative can be updated with the change of switching time;2)The fractional integral cannot be taken directly in any interval inconsistent with the lower bound time of the fractional derivative.Thereafter,the exponential stability conditions of Caputo fractional-order switching linear systems under the multi-Lyapunov functions and model-dependent average dwell time method are given.(2)The Caputo fractional-order Markov switching nonlinear systems with fixed and variable lower bounds at their initial time are discussed.The conditions for them to be almost sure stability are studied by means of multi-Lyapunov functions and probabilistic analysis.The latter introduces the concept of short memory,reconstructs a short-term memory model,and makes the initial time and state value of the lower bound of the system update synchronously with the change of the switching time,so as to reduce the influence of memory and non-locality.(3)For the traditional Caputo fractional-order dual switching linear system(with deterministic and random switching signals),that is,the initial time of fixed lower bound,and the system consists of the top-level switching determined by deterministic switching signal and the bottom-level switching controlled by the random switching signal.The minimum energy attenuation switching of deterministic switching signals is designed to make the almost sure stability switching system with random switching signals.Then a novel Caputo fractional-order switching nonlinear system based on short memory theory is proposed.The system is designed to deterministic switching signal to control the initial condition of the lower bound and the order of fractional derivative.During the dwell time of each deterministic switching signal,the random switching signal generated by Markov process is used to make the jump between each subsystem,and the almost sure stability problem is analyzed.Mainly used in multi-Lyapunov functions and probability analysis methods.