Research On Control And Synchronization Methods For Uncertain Chaotic Systems

In recent years, one can see some great developments in the area of chaos control and synchronization methods, not only some improvements in theoretical concern, but also many successful applications in the area of nonlinear circuits, security communications, laser, plasma, and bioscience, etc, which exhibits enormous and exciting application prospects. It is also necessary to consider the uncertainties caused by modeling a chaotic system in laboratory, where unknown control direction is one of the most difficult uncertainties that are hard to solve with common adaptive methods.Synchronization of chaotic systems with Nussbaum gain method is the main line of our research and it contains the following part. The first part is about the Nussbaum gain control problem for the situation that the controller with limit gain or limit control input. The second part is about the adaptive synchronization problem of uncertain driven-response system. The third part is about the nonlinear sliding mode synchronization of chaotic systems with limit Nussbaum gain method.The fourth part is about the limit Nussbaum gain synchronization of chaotic system with uncertain parameters.The main research is surrounded with the above context and it can be expended as follows:First, some basic Nussbaum gain theories are studied and a limit Nussbaum gain control method is proposed.(1)With common Nussbaum gain method, the initial control and the gain of the controller sometimes is too big and over the reasonable scope. The concepts of limit gain Nussbaum algorithm and limit input Nussbaum algorithm are proposed respectively to solve the problem.(2) For a simple one order system with unknown control direction, the Nussbaum gain method is studied for the special situations with limit gain or limit input, and two strategies are proposed to cope with the high gain and saturation threatens respectively.Second, a kind of adaptive synchronization for uncertain driven and response system with unknown parameters is studied and three adaptive strategies are proposed.(1) Based on the assumption that all uncertainties are bounded, adaptive method is used to estimate the bounds of uncertainties. Integrated with the advantage of PID control, a kind of adaptive PID algorithm is studied and applied in synchronization of uncertain chaotic systems.(2) With a further simplification of the above assumption, a kind of robust algorithm is used to compensate the uncertain nonlinear functions. With all unknown parameters estimated by adaptive method, a robust adaptive synchronization is proposed for uncertain chaotic systems.(3) Based on the concept of poles feedback in linear systems, a kind of synchronization method with adaptive poles is studied with the use of boundless characteristics of chaotic systems. It is unnecessary to known the accurate structure and forms of uncertainties with this design method. Also, some compare improvement for the above three methods are done.Third, the hybrid synchronization with nonlinear sliding mode method and Nussbaum gain method is researched for uncertain chaotic systems with limit gain.(1) Two kinds of construction methods for nonlinear sliding mode are proposed to improve the accuracy of synchronization such that it can satisfied the match condition of Nussbaum gain algorithm. Also, adaptive synchronization laws with constant coefficients or adaptive coefficients are designed respectively.(2) For some chaotic synchronization systems, if choose a small a gain, the response speed is very slow for the situation with a small error; but if choose a big gain, the gain is viewed too high for the situation with big errors. A kind of variety gain synchronization with nonlinear sliding mode is proposed to solve the problem. So with this method, the accuracy of synchronization can be improved and also the high gain feedback problem can be avoided.(3) Based on above research work, the match of limit Nussbaum gain method and nonlinear sliding mode synchronization is researched. And both the unknown control direction and limit gain problem are solved for the synchronization of uncertain chaotic systems.At last, the synchronization of chaotic systems with both unknown control direction and unknown parameters is researched with the restriction that the gain of the system is limit or the input of the system is limit. (1) For the simple situation that there is no unknown control direction, a kind of synchronization with adaptive algorithm for unknown parameters is studied and good performance is achieved.(2) For the complex situation with unknown control directions, especially with the requirement that the gain of control system is limit, a kind of limit gain Nussbaum synchronization method is proposed for chaotic systems with unknown parameters and unknown control directions. So the Nussbaum gain is guaranteed to be limit in theory. Also, simulation results show that the Nussbaum gain can satisfy to the desired limitation.(3) Consider the requirement that both the gain and the actuator of the system are limited, a kind of hybrid synchronization with limit input Nussbaum gain and adaptive algorithm, is proposed for chaotic systems with both unknown control directions and unknown parameters. Simulation results testify that the proposed strategy can guarantee the limitation for Nussbaum gain and input.Above all, synchronizations of uncertain chaotic systems with some complex situations are studied, such as unknown control direction situation; the available gain of the synchronization system is limit or actuator saturation problem. All of the proposed methods make the synchronization of chaotic systems have a strong stability under some bad conditions such as there exist big disturbances. It also provides a solid base for the application of chaotic systems in secure communication.