Research On The Stability Of Fractional Order Nonlinear System With Time Delay

Fractional calculus is the extension of integer order calculus in orders which describes the objects in nature more accurate.In recent years,with the development of fractional calculus theories,it has attracted the attention of mounting scholars both at home and abroad,and it has been widely applied in many fields such as physics,chemistry,biology and engineering.It has been a hot research topic in the area of nonlinear systems.Owing to the feedback delay and signal transmission delay,time-delay is commonly encountered.The existence of time-delay is also the reason of instability poor performance.The distributed delay can be applied in various models and it is more realistic and more accurate than instant time delay.Considered the above problems,the paper mainly considers on the stability analysis of the fractional-order system with discrete delay and distributed delay.A series of sufficient conditions for ensuring stability are proposed in this paper.The research in this article focuses on the following aspects:Firstly,the paper presents the basic theory of fractional calculus and the stability theorem.Combined with frequency distributed model of Riemann-Liouville integration,we propose the asymptotical stability of fractional-order nonlinear discrete delay system with Lyapunov indirect method and Lyapunov-Krasovskii function.Then according to Jensen inequalities,the asymptotical stability of fractional-order nonlinear system with distributed delay is proposed.On the basis of the sufficient condition of stability,the linear feedback controllers are designed.The feasibility of the sufficient conditions is proved by simulations.Secondly,we study the asymptotical stability of fractional-order nonlinear system with mixed delays based on the definition of Riemann-Liouville.We propose a simple and feasible stability criterion by adopting a novel algebraic manipulation with Lyapunov indirect method.Then we transform the system to a time-varying delay system,the asymptotical stability theorem of fractional-order nonlinear system with time-varying mixed delays is proposed.Thirdly,based on Caputo definition,we study the asymptotical stability of fractional-order nonlinear network system with mixed delays according to definition.The sufficient conditions are given to guarantee the asymptotically stable combined with generalized Gronwall inequality and Mittag-Leffler function.Finally,we study the finite-time stability of fractional-order nonlinear network system with mixed delays.At first,we convert the fractional differential equation into equivalent Volterra fractional integral equation with memory.And then,the sufficient condition of the finite-time stability is presented by using Mittag-Leffler function.At last,the feasibility is proved by the simuliations.