Stability Analysis Of Fractional State Space System And Its Application

Stability theory of fractional order chaotic systems is being developed in research, however its stability theory is not yet complete, such as integer-order stability theory. Therefore, the development of method of fractional chaos control and synchronization is not so rapid. Aiming at the present research status, the following works have been done in this paper.Firstly, the mathematical modeling of fractional order nonlinear system is introduced, using the model of fractional space state system. Besides we discuss the Lyapunov theory. Based on the above two theories, we present a new method to judge the stability of the fractional space state system, and the mathematical proof is given to ensure the accuracy and the general applicability of the new method. In addition, the new method avoids solving the fractional system's equilibrium points, and we can easily choose the control law, which can control the state space fractional order system effectively.Secondly, the fractional order unified chaotic system is chosen as the controlled object. The system is a collection of multiple chaotic systems, which has not only the general characteristics of a chaotic system, but also its own characteristics. Selectingn the fractional order unified chaotic system as the controlled object to proof the new mathod presented above is highly persuasive. Then we combine the new method with the Back-Stepping method to control the farctional order unifed chaotic system. The simulation results obtain the expected results, and it verifies the correctness and general applicability of the new mathod.Finally, because the synchronization the drive system and the response system is equivalent to the stability control of their error system. We introduce non-linear state variable feedback synchronization method to synchronise the fractional order unified once again, based on the new method. The simulation results demonstrate the new method is correct again.The establishment of the new mathod has much theoretical significance for the extension of the stability theory from integer orde to fractional order, which has laid a good foundation for the research of fractional order state space system.