Research On Fractional-order Control Theory In Nonlinear System

The chaos, a complicated motion which is similar to random, has often been found in the determined nonlinear dynamic systems. Fractional calculus is the theory which researches on arbitrary order calculus, and it is the extension of integer order calculus to arbitrary order calculus. Compare with integer-order system, fractional-order system has a better reaction to engineering physics. As the fractional order can be adjusted, the fractional-order chaotic system has a wide application to cryptography, secure communications and other fields. In recent years, the chaotic synchronization in fractional-order nonlinear system has become a researching hotspot.Sliding mode control can make the system state move on the sliding surface by switching the control variable, so that the system can maintain invariant under parametric perturbation and external disturbances. Sliding mode control is widely used in motion control because of its good robustness, simple algorithm and high reliability.The research contents in this paper are as follows:Firstly, the chaotic synchronization in fractional order Coullet system with unmatched parameters using sliding mode control has been studied. The fractional order derivative operator has been introduced into exponential approach law of sliding mode control, which forms the fractional order exponential approach law. Finally, the control effort was increased and the jarring was eliminated with the design of sliding mode surface and the variable structure controller through the fractional order exponential approach law and the adjustment of parameters.Secondly, a new fractional piecewise linear system has been studied in depth. Classified discussion has been done by changing the system parameters according to the number of equilibrium points, then the system behavior with different initial values is discussed on the basis of characteristics that system expression differs as the different initial position of piecewise systems. Therefore, a large number of figures of phase trajectory have been given. From these figures, the phenomenon that fractional piecewise linear system presents a complex dynamic behavior can be seen, and some motion laws of this system can also be found.Finally, the chaotic synchronization of fractional-order piecewise linear system has been studied. Combined with the characteristics of the system itself, two controllers have been designed, one is fractional-order sliding mode controller, and the other is a simple feedback controller. Ultimately, two controllers realized chaos synchronization between the drive system and response system. In order to verify the synchronization effect, some chaotic synchronization diagrams have been given with different parameters and different initial values. By adjusting the order a and coefficientλof the fractional-order approach law, the flash burr caused by switching in system has been eliminated largely, which also verifies the flexibility of the control method.