Research On Some Problems Of Chaos Control And Chaos Anti-control

Chaos is one type of complex dynamic behaviors that possess some very special features, such as its extreme sensitivity to tiny variations of initial conditions and system parameters, with bounded orbits but possessing a positive leading Lyapunov exponent, with a finite Kolmogorov Sinai entropy, a continuous like power spectrum and a fractional topological dimension etc. Often, chaos coexists with some other complex dynamical phenomena such as bifurcations and fractals. So, it is important to do research work about chaos system.Chaos should be restrained or eliminated quickly when it is harmful.One of the research branches of this dissertation is about chaos control and its applications, emphasis is put on constrained control of Lorenz system family, the transfer control of chaos system with multiple strange chaos attractors and adaptive compensation control of chaos oscillation in power system;while the following aspects are followed:1 Survey is provided to explain the state of the art of the chaos control, where pro and con of the already existing chaos control means are reported. The promising trend of chaos control is also put forward to initiate the related dissertation work.2 Constrained control of Lorenz system family is realized through the Minimum Principle. The two equilibria affiliated with one of the certain member of Lorenz system family, which essentially unstable, can be stabilized by the constrained controller. In case of clarity and completeness, the all three different switching surfaces or curves which are depend on the imposed positions of the control signal are derived and the resulted two control laws are proved to be bang-bang ones, and the third control law is bang-bang control with logic switching. Simulation results of the transient process of the states of the closed control system are provided to demonstrate the effectiveness of the suggested scheme.3 Transfer control of chaos system with multiple chaos strange attractors is realized through passive equivalence. The final structure of this controller for equilibria stabilization has a simple feedback form. Using a passive method, we prove the stability of the closed loop system. Based on the controller derived from the passive principles, three different kinds of chaos control of the system are investigated , respectively: the origin control for which control aims to force thechaos motion settling down to the origin from the arbitrary position of the phase space;intra-attractor chaos control for which control aims to stabilize the equilibrium points only belong to the upper chaos attractor or the lower chaos attractor and the inter attractor control for which control aims to compel the chaos oscillation from one basin of the attractor to another one. Both theoretical results and simulation verify the validity of the suggested method.4 Adaptive compensation control of electrical power chaos oscillation system is realized. Chaos oscillating states and their differential signals are obtained through tracking differentiator, and on account of that, nonlinear influences induced by periodic load disturbance are compensated adaptively. And then, linear state feedback control law is designed to make the system stable and safe. At last simulation is given to verify the validity and lightness of the suggested method.With the thorough investigation of chaos, chaos is found with some special advantages which can't be compared with, so chaos anti-control which aims to generating new chaos attractors, enhance the already existing chaos attractors, or make the controlled non chaotic system tracking the reference chaos system pattern precisely is becoming more and more important.Another research branch of this dissertation is paid upon chaos anti-control of uncertain systems, where modern control theory and computer simulation are employed;while the following aspects are probed:1 Survey is given to describe the status of the chaos anti-control and the characteristics of the already existing chaos anti-control methods, and the controlled system tracking the chaos reference system in a precise way is a promising chaos anti-control method is put forward to initiate the related dissertation work.2 Robust chaos anti-control of the uncertain nonlinear system is realized by Nonlinear Proportional Integral Differential (NPID) control. Differential signals of the driving chaos system and the controlled system were obtained by tracking differentiator and numerical derivation, respectively. NPID controller was constructed according to the principle of the minimum of the integral of time-weighted absolute error (ITAE) based on the available differential signals to force the controlled system to follow the chaotic trajectory of the given driving system in an well arranged way. Chaos anti-control was not only extended from discrete system to continuous system but also from linear stable continuous system with clear structure and parameters to nonlinear systems with uncertainties of structure and parameters, which can be stable or not. It can be concluded from the analysis and simulation results that differential signals of the driving chaos system and the controlled system can be obtained precisely in virtue of tracking differentiator and numerical derivative, respectively;the suggested controller designmethod is free of the constraints of Lyapunov exponents allocation, which is often difficult to get and differential geometry control, which is too complicated to be employed.3 Robust chaos anti-control of the uncertain nonlinear system is realized by inverse system method. Outputs and their differential signals of the reference known chaos system and the controlled non-chaotic system are obtained in virtue of the given and output tracking differentiators respectively, the expected system is designed according to tracking performance, then certain order inverse of the controlled system is constructed based on inverse system theory to force the controlled system to follow the chaotic trajectory of the given reference system in an well arranged way;on account of that, chaos anti-control is extended from discrete system to continuous system, from linear stable continuous system with clear structure and parameters to nonlinear stable or unstable systems with uncertain structure and parameters;it can be concluded from the results that the dynamic uncertainties of the controlled system can be determined in real time by the output tracking differentiator, the suggested controller design method is free of the constraints of Lyapunov exponents allocation, which often being difficult to be achieved and unlike the other controller synthesis schemes which strongly depending on the precise model and parameters of the controlled system.4 Chaos anti-control of continuous high order nonlinear uncertain systems is realized through differential homeomorphism. States of reference chaos system are reconstructed by differential homeomorphism and the given high order tracking differentiator from one single scalar output signal. At the same time, states of the controlled non-chaotic system are obtained in virtue of output high order tracking differentiators from the single scalar output signal. The controller is designed according to the expected tracking performance. So, robust chaos anti-control of a class of high order nonlinear system with uncertainties is realized. On account of that, chaos anti-control is not only extended from discrete system to continuous system but also from low order system to high order system. The suggested controller design method is free of the constraints of Lyapunov exponents allocation, which is often difficult to get and the constraints of the other method strongly depends on accuracy of model and parameters of the controlled system. Both correctness and robustness of the suggested project are verified by computer simulation.