Study On Some Issues Of Chaotic Control And Synchronization

Since E. N. Lorenz found the first chaotic attractor in 1960's, chaos phenomenon was observed one after the other in many systems such as electrical devices, chemical reactions, biological systems, mechanical systems and neural networks, among others. The discovery of chaos was honored as the third revolution in physics following relative theory and quantum mechanics. Chaotic control and synchronization has now attracted much attention due to its dramatic potential applications. It is of great benefit for human beings to analyze and control chaos and coupling synchronization in natural and man-made systems.This Dissertation focuses on the research on chaotic control, chaotic synchronization and complete synchronization based parameter estimation. Its main contributions include following seven points.1. This dissertation points out the drawbacks of the autosynchronization approach since it was established. A new method is suggested to ensure parameter estimation by the properties of chaotic dynamics, merged into the conventional Lyapunov's direct method. Then, the new approach is further extended to several useful cases: 1) when matrix M is semi-positive definite, 2) when output derivative is used for reducing the dimension of output, 3) in the present of noise. When a general system is considered, the scope applicable and some limitations of the new approach are also discussed.2. To design a response system from a scalar output of the driving system, a matrix match method is proposed, which firstly uses the differential embedding theory for achieving the static reconstruction model and then constructs a response model synchronized with the driving system by some rules. The method presented cannot only remove the potential singular problem of the static reconstruction model but also reduce the dimension of differential embedding (i.e. the order of derivative estimator in the static reconstruction model) and even remove completely the output differential.3. To estimate parameters from a scalar output, a guidance is presented for the design of parameter estimation law, namely parameter estimation is designed such that the coefficient of the lowest order of the local linearization around the synchronization manifold is positive almost everywhere. The Lorenz model and a unified chaotic system are illustrated to validate this method and to demonstrate also that it is possible to estimate all parameters from a scalar output time-series.4. Trying to solve the high-frequency chattering of conventional variable structure control, an adaptive chattering-free variable structure control method is proposed for controlling a class of uncertain time-variant chaotic systems by three steps: 1) constructing an augmented model of the selected system, 2) designing a conventional variable structure control law for this augmented model and 3) using some mathematical operation. The method not only eliminates the chattering phenomenon and tracks arbitrarily desired trajectory asymptotically, but also removes the prior knowledge about the upper bound of the system uncertainty. However this approach exhibits some drawbacks in practice such as the ill-pose problem in parameter estimation law happening in the present of noise and external disturbances, as well as imperfect avoidance of high-frequency chattering. An improved method is suggested to solve the drawbacks, which introduces a parameter adaptive estimator with dead-zone and nonlinearly learning rate, and in addition replaces the signum function by the saturation one as well. Theoretical analysis and numerical simulation validate that the improved method not only avoids the high-frequency chattering completely but also obtains high-accuracy tracking as desired in a finite time.5. A new approach is suggested to introduce an integral action in the sliding mode control, which uses the same sliding variable in the conventional sliding mode but adds a nonlinearly integral action of the sliding variable in the controller. Due to the nonlinear integral action applied in the new sliding mode controller, the system controlled can achieve good dynamical and static performances.6. A simple full-state asymptotic trajectory control method is suggested to asymptotically drive full states of a unified chaotic system to arbitrarily desired trajectories from only a scalar state. Theoretic analysis and numerical simulation validate that the method proposed can ensure the asymptotic trajectory control.7. An approach to reduced-order synchronization is suggested with a finite-time uncertainty estimation which evaluates correctly the system uncertainty in a finite time. In addition, the reduced-order synchronization criterions are discussed, and the design of complete and general reduced-order synchronizations is then investigated. Theoretical analysis and numerical simulation validate the effectiveness of the method proposed.