The Applications Of Fractional Calculus In Adaptive Control

Fractional calculus is the generalization of the traditional calculus, and it was paid more and more attention to for its widely used in control, physics, signal processing and so on. In this paper, a survey of fractional calculus and its applications are introduced firstly, and the application in control is more explicit. And then, in chapter 2 the Riemann-Liouville and Caputo definitions and their properties of fractional calculus are introduced, and a system model is constructed which demonstrates that the fractional order system can describe the system characters effectivelly, and then, many important results of fractional order system are given in the end. In chapter 3, after introducing the history of the Lyapunov theory, the Lyapunov second method is extended based on the fractional calculus theory. The condition that the Lyapunov function 1 order partial derivative in t is negative is extended to that the k(0? k? 1) order partial derivative in t is negative. Then a general Lyapunov theorem is obtained. The Lyapunov stability of fractional system is defined, and the obtained theorem is extended to the fractional order dynamic system, and then a general Lyapunov theorem of fractional order dynamic system is obtained. In chapter 4, a novel fractional parameters law is designed. Using the general Lyapunov theorem in chapter 3, a new kind of stable fractional adaptive law is obtained which has the better dynamic characters such as less regulating time and less overshoot. Then the traditional adaptive control law is extended. In chapter 5 , using the backstepping method and the obtained theorem in chapter 3, a fractional control method of Lorenz system is given, followed with two chaos systems synchronization method which the slave system has the strict feed-back forms, then two important theorems are got. The simulation results show the effectiveness of the method.