Research On Control Theory And Stability Of Fractional Differential Systems

H.H.Rosenbrock established the degenerated differential system in1974when he dis-cussed the complex grid problem. Then the degradation phenomenon has been spring up in reality systems, such as control systems, management systems, ecological systems, pow-er systems, industrial engineering systems etc. The experts and scholars both at home and abroad pay more and more attention to this branch, and make a systemic research. Mean-while, with the extensive use of mathematics and the development of interdisciplinary, it was found that the new fractional models are more suitable and accurate for modeling systems than the ordinary models.This thesis focuses on the problems of control theory and stability about these above systems. Although the study of control theory and stability has been relatively mature, most is for integer order systems, including with degeneration. However, the study of fractional differential systems, especially of singular fractional differential systcn is fewer. In this paper, we discuss about the problem of control theorems and stability of the singular fractional differential system as follows:The main content of this thesis is arranged as follows:In the first chapter, we first introduce briefly to the reality background and the significance of research of singular fractional differential systems, then the pre-knowledge which is useful in the paper are presented.In the second chapter, by the Laplace transform method, we first obtain the solution of state equation of the singular fractional differential system(1). Then based on it,we discuss the necessary and sufficient condition on the controllability for the system(1). Finally, we give two examples to illustrate the application.In the third chapter, by using a similar method as one in the previous chapter, we obtain a criterion on the observability for the system(1) which is necessary and sufficient.In the fourth chapter, by using analytical method, a novel stability criterion for a class of nonlinear fractional differential system is derived. Compared with the obvious results, our criterion is straightforward and suitable for application. Then a stability criterion for singular fractional linear differential system is also built. In the fifth chapter, we first make corrections of two existence theorems of state observers for a integer order differential system and a linear fractional differential systems. Then we discuss the existence of the state observer of the singular fractional differential systems.