Several Control Issues For Fractional And Degenerate Differential Systems With Delay

With the extensive use of mathematics and the development of interdisciplinary, in order to more accurately describe and simulate the practical phenomenon, the fractional integrals, derivatives and fractional differential equations are introduced. And it was found that the new fractional models are more suitable for modeling systems than the ordinary models. The advantage of fractional differentiation is more and more obvious. And in practical engineering, with the development of the modeling, design, analysis and application of practical systems, people attach more and more attention to the influence of delay phenomenon on systems, and make a systemic research. At the same time, the degenerate phenomenon was found to be common phenomenon of practical systems. The best-known systems, for example, Hopfield neural network model, Leontief dynamic input-output model, the power system model with nonlinear loads and so on are degenerate systems.In many practical applications, delay and degeneration are always ignored in order to facilitate future research. But that they actually exist causes the inaccuracy. And some systems described by fractional differential equations(FDEs) are more accurate than ordinary differential equations(ODEs). Though the optimal control and sliding mode control are not new field, but systems concerned are relatively novel and much more press close to actual systems. Therefore it has practical significance to research the control problems for systems with delay, degeneration or fractional order. Researching the optimal control or sliding mode control for separate system is the main work in this paper. The main contents are as following.The first chapter mainly introduces the reality backgrounds, the main work and the preliminary knowledge which is necessary in the paper.In chapter two, the optimal control of a kind of systems affected by external distur-bance with both state- and control input-delay is discussed. The optimal control is gained by introducing a sensitivity parameter. And a reduced-dimension observer for external disturbance is designed to realize the physical realization of optimal control.The chapter three is designed to research the optimal control of degenerate system affected by external disturbance with control input-delay. By constructing Hamilton func-tion and utilizing the necessary condition of the optimal control problem based on the maximum principle, the optimal control law can be obtained.Chapter four studies the optimal control for a class of fractional differential equations. Using Oustaloup's Recursive Approximation, the fractional derivative operator can be approximated in frequency-domain. And the optimal control law is gained.In the fifth chapter, the problem of sliding mode control for fractional differential systems with state-delay is considered, such that the state starting from any initial value will move toward the switching surface and reach the sliding surface in finite time and the state variables on the sliding surface will converge to equilibrium point. And the stability of the proposed system is discussed.