Study On Stable Theorem And Control Of Fractional Nonlinear Systems With Delay

In view of the fractional calculus’s outstanding advantage than integer or-der’s, since Mandelbort point out that there are vast fractional phenomena in the nature and the domain of science and technology, the fractional calculus has been widely studied involving all fields, and became the research hotspot. Nonlinear always exists in real system, nonlinear is an essential attribute of real system. Meanwhile, owing to feedback delay and siginal transmission delay, time-delay is commonly encountered. Time-delay is frequently a source of instability poor performance, such as network congestion, data packet dropout, poor flow the ve-hicles, traffic jams, etc. However, how to predicate the stability on the fractional nonlinear system with delay have not been solved effectively, let alone realize control. So the scientific significance of the subject lies not only in developing the stability theorem and realizing control about the fractional nonlinear system, promoting the application of the control theorem and control method, but also in enriching the nonlinear system control theorem, pushing the development of control theory and control engineer.Firstly, the stability theories of the fractional nonlinear system with or with-out delay are proposed. Then, the Matrix configuration controller design methods are presented. Finally, control and synchronization of fractional nonlinear sys-tems with and without delay, as well as identification of unknown parameters of the systems are realized.The main achievements and innovations of this dissertation are as follows:1. Stability and control study of fractional nonlinear system. Based on stability theorems of integer order nonlinear systems and fractional linear systems, the stability theorem and control of fractional nonlinear system with-out delay are studied from Jacobian matrix eigenvalues point of view. In the meantime, by expanding integer order Lyapunov stability theory to fractional systems, a new stability theorem on fractional nonlinear system is proposed via constructing positive definite function V and taking integer derivatives of V. The superiority of the proposed theorem is reflected in not only avoiding difficultly taking fractional derivatives of V, but also avoiding losing stable efficacy while parameters perturbing. And on this basis we study impulsive control theorem on fractional nonlinear system, establish the general relationship between pulse strength and pulse period, and realize fractional impulsive control. Then we conduct intermittent control theorem study, create general intermittent stability condition, realize fractional intermittent control. Finally, simulation examples are given to illustrate the effectiveness of the developed theoretical results.2. stability and control study of fractional nonlinear system with time-delay. Based on the proposed stability theorem of the fractional nonlinear system and the stability theorem of integer time-delay nonlinear system, we s-tudy fractional Lyapunov-krasovskii theory by building positive definite function V and taking integer derivatives of V instead of taking fractional derivatives, and propose general constructing method of expanding function. Using this theory, most control method of integer delayed system can be expanded to fractional delayed system. Based on the eigenvalue stability theory, combined inequali-ty and reduction to absurdum, the computing methods of limit divergence and convergence are presented. Subsequently, intermittent control theorem on frac-tional nonlinear system with discrete time-delay is set up and the intermittent control on delayed fractional nonlinear system is realized. Finally, numerical ex-amples are provided to demonstrate the effectiveness and the applicability of the proposed method.3. Controller design of fractional nonlinear system via matrix con-figuration method. Although there are a large number of reports on nonlinear control and synchronization, but how to simply design controller is seldom report-ed so far. Stimulated by this, the matrix configuration controller design method is put forward. For non-delay fractional nonlinear system with or without unknown parameters, controller can be designed by matrix configuration method to realize system control and parameter identification. Based on the proposed stability theorem on the delayed fractional nonlinear system, by separating time-delay terms, the matrix configuration controller design method of fractional nonlinear system with discrete time-delay is presented also, and for delayed fractional non-linear system with or without unknown parameters, controller can be designed by matrix configuration method to realize system control and parameter identi-fication. We as well as expand this method to the integer order nonlinear system with distributed delay and achieve control. All the works show that the matrix configuration controller design method has good versatility and effectiveness.Finally, the main results of the dissertation are concluded, and the issues of future investigation are proposed.