The Research For Dynamic Characteristics Of Discrete Fractional Order State Space Systems

With the deepening of the research for fractional order calculus, researchers generally believe that the dynamic characteristics of fractional order system almost inherit all the features of the integer order system as its extension of integer order calculus. Fractional order calculus system can more accurately describe the objective world because of its historical memory effect, its dynamic feature relating to system's order and so on. In general, Fractional order systems have great application value in image processing, neural network, signal processing, robust control, and other fields.The control problem about fractional order system is regarded as a new branch in the field of control, because the system has the characteristics of non-integerthe order, it can't simply use the traditional classic control theory to research and analyse. It's a new research focus of the fractional order system to look for fractional order system analysis and control method by using the theory and method of fractional order calculus.It was a basic theoretical research in the field of control and signal processing to study on the dynamic characteristics of discrete fractional order system state space analysis.Mathematical model of discrete fractional order state space was set up according to the fractional order calculus definition. The suitable stability condition of discrete fractional order state space system is researched from the structural characteristics of the fractional order system. The controller of discrete fractional order chaotic systems synchronization was taken into account with the basis of the discrete system stability theory. In order to achieve that goals,we will do the following plans in our work.Firstly, the paper mainly introduced the two kinds of definition of fractional calculus:G-L and Caputo fractional calculus. According to the definition of fractional calculus to establish the state space model of discrete fractional order system, and discussed the significant problem of controllability and observability of the system model.Secondly,in this paper, the discrete fractional order system state space model was applied to the analysis of chaotic systems. With the discrete fractional theory of G-L and Caputo definition, we discussed the application of fractional order difference equation in discrete chaotic system, and we also get the bifurcation diagram of discrete fractional order chaotic systems. Through the analysis of the bifurcation diagram,we obtained that the chaotic state is not only related with the discrete system parameters, but also associated with the order of system.Finally, the paper researched the synchronization control problem of discrete fractional order chaotic systems. Application of discrete sliding mode control theory,we proposed a new discrete reaching law which contained multiple parameter function by analyzing the Gause's reaching law. A synchronization controller was designed based on the new discrete reaching law. The simulation results can be confirmed that using the new design of controller the different discrete fractional order chaotic systems can be achieved synchronization even though existing external bounded disturbance.