The Fast Control And Synchronization Of Chaotic Systems And Their Applications In The Power System Management

Chaotic system,which is one of the special nonlinear systems,possesses the complicated phase structure and unpredictable dynamic behavior.The study of chaos control and synchronization belongs to the category of system cybernetics,and has always been a hot topic in the field of nonlinear science.The power system,which is closely related to people’s life,has strong nonlinearity and complex structures.It has been found that chaotic oscillations occur in the power system will cause system instability or even system collapse,which will bring huge economic losses.Due to the complexity of chaotic dynamics,it is difficult to eliminate chaos in the power system by using traditional methods,so it is not easy to control and manage power system to make it run stably.Therefore,from the perspective of economics and control management,the study of chaos control and synchronization has an important practical significance.Thanks to the efforts of scholars,many stability theorems and control approaches have been proposed,which provides many effective control schemes for chaotic systems.But most of the existing results are based on the asymptotic stability theory,which shows in theory that the system stability can be attained only when time tends to infinity.It is well known that the system convergence rate is one of the key factors to evaluate the quality of control scheme,then asymptotic stability control may not be applied to the actual system with high convergence rate requirements to a certain extent.Therefore,this paper seeks some other schemes to make the controlled chaotic system realize fast control and synchronization,and then applies these methods to the management of power system.In this paper,several new stability theorems with limited convergence time regardless of the initial value are proposed,which provide a theoretical basis for the design of control schemes and synchronization strategies of chaotic systems,as well as provide ideas and methods for the control design of other nonlinear systems.The control schemes and synchronization strategies of chaotic systems obtained in this paper can effectively realize the fast stability and synchronization of the nonlinear power systems including interconnected power system,permanent magnet linear synchronous motor system,wind generator system and so forth,which provides some flexible method references for eliminating chaotic oscillation to control and manage the power systems better.The main research works of this thesis are as follows:1.The problem of fast convergence control for a class of chaotic systems with uncertain parameters and external disturbances is studied.Firstly,an auxiliary system for the original system is presented,then a new auxiliary controller is devised and a new Lyapunov function is constructed by using the analytic solution approach.By combining the auxiliary system and the new Lyapunov function,a novel fast adaptive control scheme is developed for suppressing the state trajectories of the 3D chaotic system to its origin.In our control scheme,the system’s convergence rate can be flexibly adjusted by the control parameters,and Lyapunov function can be easily constructed by the Gram determinant.The control scheme is applied to the chaos control of the permanent magnet linear synchronous motor(PMLSM)system.The numerical results indicate that the given scheme can eliminate the chaotic oscillations of PMLSM system,which shows the effectiveness of proposed control scheme.2.The problem of variable exponential control for a class of lower triangular systems with uncertainties is addressed.Firstly,for reducing the complexity of control design brought by higher-order system,the n-D system is transformed into the 2-D system through the variable transformations.Then,a new nonsingular sliding mode manifold and a new controller are devised to make the state trajectories of 2-D system finitetimely move to the origin.After that,the exponential stability of the original system is then derived by the state inverse transformations.In the given control scheme,the sliding mode surface has few parameters,simple structure and no singularity.In addition,the controller contains a variable parameter which determines the convergence rate of the system.The control scheme is introduced to the second-order interconnected power system.The numerical results show that the system states can be stabilized under the controller,which implies that the proposed control scheme is feasible for the control and management of power system.3.The problem of fixed time stability for the chaotic systems with uncertainties and external disturbances is investigated.On the basis of finite time stability theory,a differential inequality with higher power term is constructed,which solution can converge to the origin in a fixed time.Then,by combining the given differential inequality and sliding mode method,a robust fixed time control scheme to ensure the fast stability of perturbed chaotic system is derived.The upper bound of the convergence time in the control scheme is determined by a formula consisting of control parameters,which is independent of the initial states.Using the obtained theoretical results,a robust control approach is designed for the coupled dynamos system.The numerical results show that the chaotic oscillation of the coupled dynamos system is well suppressed,which illustrates the feasibility and effectiveness of the obtained control scheme.4.The problem of control and synchronization of perturbed chaotic systems is researched.Firstly,a new differential inequality is constructed and the corresponding fixed time stability theorem is obtained.Then,the control scheme and synchronization strategy are designed by combining the presented theorem with sliding mode technique.In the new stability theorem,the convergence time is more accurate owing to the use of the integral interval segmentation method and extreme value method.Moreover,the provided control scheme can not only offset the adverse effect of external interference,but also give an accurate calculation formula for the upper bound of convergence time.The theoretical results are used to design the control scheme and synchronization strategy for the interconnected power system with magnetic disturbances.The numerical results show that these two control design criteria are effective and feasible to ensure the stable and orderly operation of the power system.5.The problem of fixed time control and projective synchronization for a class of perturbed chaotic systems is concerned.Firstly,a new fixed time stability theorem,which provides a more accurate convergence time,is derived by using direct integration approach.Then,based on the system’s structure and sliding mode control theory,a new integral sliding surface is constructed,and a sliding mode control criterion is accordingly put forward to suppress the chaotic trajectories to the origin.Besides,combing the given theorem and backstepping technique,a robust projective synchronization scheme is designed to achieve the purpose of fixed-time chaos synchronization.In the synchronization criterion,owing to the existence of projective scale factors,the synchronization forms can be flexibly changed according to the practical needs,which enhance the security of information transformation to some degree.The obtained theoretical results are applied to the chaos control of the permanent magnet synchronous generator(PMSG)system.The numerical results show that the control design can not only suppress the chaotic oscillations,but also realize the projective synchronization of the generator systems.6.The problem of function projective synchronization for chaotic systems with time delays and external disturbances is considered.Firstly,a new differential system is designed by using intermittent control.A new fixed time stability theorem is then obtained by the mathematical induction method.Based on the proposed stability theorem,a new state feedback control scheme is developed to achieve function projective synchronization of chaotic system within fixed time.In the synchronization strategy,the projective scale factors are the functions of time rather than the fixed constant,which not only generalizes the existing results of projective synchronization but also improve the security of information transformation.In addition,the convergence rate of system can be freely adjusted by the control parameters,and the upper bound of convergence time is a function of control parameters regardless of the initial values.The synchronization scheme is applied to the interconnected power system with time-delay,and the numerical results show the effectiveness of the function projective synchronization scheme.