Stabilization Radius Of Fractional-Order System Under Disturbance And Its Application In Economic System

The theory of fractional calculus has been established for more than three hundred years.It is not a fractional calculus,nor a part of traditional calculus(differential,integral and variational).In fact,it is to find arbitrary derivatives and integrals a discipline.The number of applications of fractional calculus has grown rapidly in a large number of studies that enable fractional-orders to more accurately describe and model practical problems than traditional "integer" methods.Therefore,the fractional-order model is more suitable for describing the actual system than the integer-order model,and solves the corresponding problem,which reflects the superiority of the fractional-order system.Especially in the application of economics,it is very difficult to study the dynamic system of financial system dynamics and economics.However,by the non-limitation of fractional operators,it is convenient to construct long-sequence models.Models constructed using only integer order difference and integral have more memory effects,which is one reason why fractional differential equations have great advantages in describing long-term and large-volume economic phenomena,thus indicating that the fractional-order system is economic.There are more important positions and meanings in the study,so it is very necessary to study the stable radius of the fractional-order system in the study of economic systems.The main work of this paper includes the following aspects.First,the problem of the stable radius of the fractional-order linear discrete-time system under disturbance is studied.The form of the disturbance is structured disturbance,and the fractional-order constant-discrete time system under the structural disturbance described by the state equation is proposed.On the basis of determining the positive conditions of fractional-order discrete-time systems,the positiveness of fractional-order linear discrete-time systems with structural perturbations is derived,and the structural disturbances are derived according to the corresponding conditions for determining the stability of fractional-order discrete-time systems.The stable radius of the fractional-order linear discrete-time system is verified by numerical examples.Secondly,the problem of the stable radius of the fractional linear continuous time system under disturbance is studied.The same way as the stability radius of the discrete time system is studied.Firstly,the structure of the disturbance is specified,and then the positive of the fractional linear continuous time system with structural disturbance is introduced.Finally,the stability of the fractional linear continuous time system under structural disturbance is derived by using the condition of determining the stability of the fractional linear continuous time system.The validity of the stability radius formula is verified by numerical examples.Thirdly,the examples of fractional-order systems with structural disturbances in the economic field are given in discrete and continuous time cases.The application of fractional-order systems in economics is combined with the stability of fractional linear systems under the disturbance of three,four and two chapters.The conclusion of the radius,the problem is applied to the economic field,and the application conclusion of the stability radius in the economical fractional dynamic system is obtained.Finally,the main content of the article is summarized and further research work is proposed.