Estimation Of Attraction Domain Of Fractional Linear Systems Under Saturation Control

Fractional calculus,as a generalization of integer calculus,extends the order of calculus operations from the traditional integer order to the fractional order.It has a history of more than 300 years.With the development of fractional calculus,more and more scholars have begun to study fractional systems.Fractional order system is not only an extension of the traditional integer order system theory,but it can better describe the actual process of the system.In recent years,with regard to the study of fractional order systems,scholars have paid more attention to the stability of the system.How to introduce fractional order calculus into the stability of nonlinear systems.The stability theory of integer order systems is generally obtained by constructing the Lyapunov function,but for fractional order systems,it is difficult to obtain an accurate stability theory with the help of the Lyapunov function.Therefore,the study on the stability of fractional nonlinear systems is very meaningful.The problem of estimation of the attraction domain is also very important in the theoretical study of saturated nonlinear systems.The phenomenon of actuator saturation widely exists in practical engineering systems.In recent decades,more and more scholars in the field of control have begun to pay attention to saturation-constrained control.They have studied the global stabilization,local stability,and stabilization of saturated systems,and have obtained very considerable research results.There are very few studies on the estimation domain of the fractional system.Therefore,it is also very important to expand the estimation domain of the fractional linear system under saturation control.The main work of the thesis includes the following aspects.First,a brief introduction to the basic concepts and related theories of fractional systems,related concepts and stability theories of fractional systems under saturation control,the method of attracting domain estimation,the second method of Lyapunov and the ellipsoid and polyhedron sets The related theorem provides a sufficient theoretical basis for the following work.Second,by using the second method of Lyapunov to analyze the global stability of the fractional linear system,the invariant set condition of the fractional linear system is obtained,and then the asymptotic stability condition of the saturated control fractional linear system is obtained through the correlation theorem.On the basis of obtaining the asymptotic stability conditions and invariant set conditions of the fractional linear system with actuator saturation,the optimization method is used to find the attraction domain,the maximum volume invariant ellipsoid is obtained,and then the phase with the ellipsoid is found.The tangent polyhedron set proves that the polyhedron set is a positive invariant set,and a larger attracting domain is obtained.Finally,a numerical example illustrates the superiority of this method.Third,according to the Lyapunov direct method and the related fractional inequalities,sufficient conditions for asymptotic stability are obtained.The linear matrix inequalities are used to obtain the inverse ellipsoid estimation method.On this basis,a symmetric polyhedron containing the largest volume ellipsoid is found as a new attraction domain estimate,and then the symmetric polyhedron set is proved to be a positive invariant set.Finally,a numerical example illustrates the superiority of this method.Finally,the main content of the article is summarized,and further research work is proposed.