The Control And Synchronization Of Uncertain Chaotic Systems And Its Applications

This paper studies the problem of control and synchronization for uncertain chaoticsystems. As a result, this paper also considers the application of chaos synchronization insecure communication. Chaos is a special phenomenon of the nonlinear system. Beingsensitive to initial conditions, for a long time chaos is considered uncontrollable and un-predictable. So people treated the chaos with fear rather than interests. Since the end ofthe last century, more and more research results have shown that chaos is not only control-lable but also useful in many fields. The chaotic systems have many special characters.For example, chaotic attractors contain infinite unstable periodic trajectories. chaotic sys-tem is sensitive to initial conditions and its states are bounded. Just because of thesespecial characters, it is necessary to study chaotic systems in state space when research-ing on the chaotic systems. Chaos control and synchronization are two main aspects.There are two control problems for the chaotic systems. One is chaos control. Its aim isto stabilize the unstable periodic trajectory in the chaotic attractor or suppress the chaoticbehavior. The other is anti-control. Its aim is to strength the existing chaotic behavior ordrive the system which is not chaotic to be chaotic. Chaos synchronization is to drive theslave systems to follow up a chaotic system.In recent years, many researchers studied the chaotic systems. Specialists in physics,society, biologic, chemistry, information science and other many areas have done manyworks in chaotic systems based on the aspects of the investigation in their own fields.And they have got lots interesting results. Inspired by the theory and techniques that de-veloped recently from the control science, this paper studies the chaotic systems usingnonlinear design methods. It is true that a practical system always contains uncertainty.The uncertainty may come from the parameter varying, or from the modeling error. Soit is necessary to research the design of uncertain chaotic systems. CLF is an effectivedesign method for nonlinear systems. It's definition was proposed by Arstein and Sontagindependently when they researched the control of nonlinear systems in the 1980s. ThenLyapunov function is turned into a design tool from an analysis tool. Till now, few re-searcher has applied the CLF in the design of chaotic system to the best of the author's known. This paper introduces the CLF into the design of chaos control and synchroniza-tion. Integrating the finite-time stability theory, we firstly defined the idea of finite-timeControl Lyapunov Function(f-CLF). And we also designed the feedback control to realizefinite-time chaos control and synchronization by using the f-CLF.The main contributions of this thesis are as follows:1) Starting form the practical application aspect, we concern on simpler designmethod for control system. Based on cascade system stability theory and Input-to-State-Stability(ISS) theory, we propose a simple linear feedback for chaos control and synchro-nization. Compared with the existing chaos control and synchronization results, the linearcontroller given in this dissertation is advanced and it is one of the most simplest design.Furthermore, we take the lead in applying the CLF in chaotic system design. And weconstruct a simple and effective CLF for the unified chaotic system.2) To improve the transient response performancet, we consider finite time chaoscontrol and synchronization. Most control system design is to realize asymptotical sta-bility of the close-loop systems. This is to say that the system reach the control target ininfinite time. This dissertation studies realize such goals in finite time. Based on terminalsliding mode and CLF respectively, finite time chaos control and synchronization is real-ized. This makes the design of the control system more effective.3) This thesis proposes a step design technique. This design method suits to a classof semi-decouple system. Step by step, the system states can be driven to zero. For thestates that have been designed, they can be treated as zero in the next design. This designmethod greatly simplify the design of a class of nonlinear system. It has a wide applica-tion prospect.This dissertation consists of two parts. The first part is the design of chaos controland synchronization that contains Chapters Two, Three, Four and Five. Chapter Six isthe second part. It is about the secure communication based on descriptor model. Theconcrete contents and results of the thesis are as follows sincerely.Chapter One is a survey. It introduces the definition of chaos, chaotic characters, thedefinition of CLF, sliding mode control, finite-time stability and other items. It sums upthe research status of the chaotic systems. Different control methods that are introducedin chaotic system are also introduced especially.Based on the sliding mode control, Chapter Two presents chaos control for a class of uncertain chaotic systems. Both the parameters in the switching surface and the controllergains can be obtained through the solvability of the linear matrix inequality. Feedbackcontrol are designed for the Chua's circuit and Lorenz chaotic system respectively. At thesame time, this chapter proposes terminal sliding mode control to realize chaos control.A new nonsingular terminal sliding mode is presented to tackle the singularity problemof the controller. The finite-time chaos control is attained through such a design.Chapter Three studies chaos synchronization for uncertain chaotic systems based onthe Control Lyapunov Functions(CLFs). We have completed the task of construction ofthe CLF for the uncertainty unified chaotic system. Controllers are designed to realizeasymptotic synchronization. Then, by combining the finite-time stability theory with theCLF, we presente finite-time Control Lyapunov Function(f-CLF). Based on the f-CLF,robust and rapid controllers are proposed to realize finite-time chaos synchronization.By virtue of the stable theory of the cascade system and input-to-state stability(ISS)theory, we can get a semi-decouple form for the chaotic systems by feedback. Ultimately,we can design linear controller which is very simple to realize chaos control and synchro-nization.Based on the preceding results, Chapter Five extends the anterior results to the semi-decouple systems and puts forward a new stepping design method. This design methodis easy to be implemented and can tackle the design problem for a class of nonlinear sys-tems.Chapter Six considers secure communication which is an application of chaotic sys-tem. By making use of the descriptor system theory, we can design descriptor observerto realize chaos synchronization and secure communication. In the mean time, the gainmatrices in the observer can be chosen by solving of a linear matrix inequality. Takingthe transmitting signal as a new system state, we transform the chaotic system into adescriptor system.