Stability Analysis Of Fractional Singular System With Time Delay

The singular system with time delay is a generalized expression for real-world dynamic systems.However,these generalized features do not contain the order of the system.Therefore,studies on the performance promotion in fractional order systems are necessary.As the most important performance,the stability as well as the corresponding analysis methods are the key issue for fractional singular order systems with time delay.The common method for analyzing the stability of singular system with time delay is trying to obtain the stability preconditions through constructing a Lyapunov-Krasovskii function and analyzing the derivative.However,when considering the fractional order system,since the system forms and calculation are complex,this method is very difficult to apply.Therefore,how to solve the analytical expression of the system is motivated in the thesis.The main obstacle is that the accurate analytical expressions are unavailable because of time delays.By using related properties of the Mittag-Leffler function,the content of the Laplace transform and the inverse transform,the approximate stability expression of the system that can be obtained analytically.The stability conditions are thus obtained.Considering systems with nonlinear disturbances,the stability conditions of the system can also be obtained by using the relevant content of the matrix norm inequality and the nonlinear bounded premise.The stability conditions are used to find the stability domain of the system.The contributions are summarized as follows:Firstly,the stability of fractional singular system with time delay under zero input condition is considered when system order is 1<a<2.Because of the singularity,the system should be decomposed into two subsystems,and then the Laplace transform and the inverse transform are used to obtain the complex recursion of the system.Since the existence of the Mittag-Leffler function and its integral term,some simplification tricks are used to obtain the conditions for keeping the system stable.Accordingly,the feedback controller is designed.The relationship between the selection of feedback controller and system stability is obtained,and the stability theorem under feedback control is proposed.Secondly,considering nonlinear disturbances commonly exist in real-world systems,a nonlinear disturbance term is added to the fractional singular system with time delay.For analyzing the stability of this nonlinear system,the first step is also to decompose into two subsystems.The nonlinear bounded premise and the matrix norm inequality are then used for formal transformation.After that the approximate analytical expression of a system is obtained for deriving the stability conditions of the system.Correspondingly,the influence of feedback control on the nonlinear system is studied.The relevant stability theorem is proposed.Finally,the stability of the system under different nonlinear disturbances is considered,the system convergence ratio and the stability domain changes are also discussed.The corresponding simulation examples are included in each section.The experiments not only verify the correctness of the proposed conclusion,but also reflect the influence of system order,time delay constant and nonlinear disturbance parameters on the stability of the system.