Impulse Control Of Nonlinear Systems With Delay And Its Application

In recent years,many phenomena in the nature not only contain continuous control process,but also have the discrete dynamic characteristics.The impulsive control veritably reflects the discontinuous dynamic characteristics in the nature,engineering technology and modern social science,which breaks the limitation of continuous differential equation,provides more accurately description of the practical systems.The traditional control methods have continuously effects on the object,while the impulsive control instantaneously impacts on the object at impulsive time in discrete form.Therefore,the continuous control theory can not be directly applied to the impulsive control system.Moreover,the impulsive control theory is more complex than the continuous control theory.In practical engineering fields,impulsive control theory has been successfully applied in cancer chemotherapy,optimization control for the spacecraft fuel,chemical reaction and management of biological population dynamics,etc.However,due to the restriction of the theoretical analysis,most theories of the impulsive control consider the factors of model,and lack systematic and pragmatic results.In addition,with the limitation of some objective condition,some systems do not allow for the manipulation of multi-variables,which makes more difficult for the design of impulse controller for nonlinear systems with finite operational variable.In this paper,we focus on the problems above and study the impulsive control problems for the systems with time delay.We present the synchronization stability of complex systems with impulsive control and the complex dynamic behavior of the system caused by impulsive control.Some new conclusions and new algorithms are derived.The main research contents are given as follows:(1)With the Lyapunov theory and some analysis techniques,the synchronization method and the corresponding theoretical proof of synchronization stability in nonlinear system are proposed.In this scheme,a full variable impulse synchronization controller is designed for the two identical linear delay feedback chaotic systems,and the corresponding synchronization stability theorem is proposed.To assure the synchronization error system be asymptotically stable,the upper bound and the lower bound of the interval of the impulse controller are estimated by the Lyapunov theorem,and the synchronization states of the two systems are observed through both simulation and experiment,which verifies the correctness of the proposed theory.However,with the restricted condition in the engineering,the full variable impulse control cannot be applied to the systems whose partial states are limited due to the actual objective conditions.To solve this problem,the synchronization stability theorem of univariate impulse control is proposed in this paper,and the range of impulsive interval of univariate impulse controller is determined according to the proposed theorem.The synchronization states are observed through both simulation and experiment for the two identical delay feedback Chen systems,which verifies the correctness of the proposed theorem.Univariate impulse control and full variable impulse control have the characteristics of simple structure and easy realization.In application field,univariate impulse control can be applied to the chaotic systems with constrained manipulation of variables,so it is better than full variable impulse control method in practical engineering application.Moreover,univariate impulse control scheme reduces the number of controllers,but needs to further improve the response speed as compared with the fully variable impulse control.(2)For the complex network,the method of the local topology identification based on univariate data and synchronization with univariate impulse pinning control are proposed.The partial observation data from nodes of the complex network and an adaptive method are exploited to identify the local topological connection of the complex network,which solves the problem of accurately identify the network topology with less data.In the aspect of complex network control,the synchronization stability theorem of univariate impulse pinning control is proposed by using the Lyapunov theorem.Besides the pinning to a part number of the nodes in the network,the small perturbation for the nodes is also demanded.In this paper,the pinning nodes of the network and the controllability of the network is judged by the controllability rank condition.The univariate impulse synchronization controller is designed to the selected pinning nodes to implement the complete synchronization of the network.Both the correctness and effectiveness of the proposed method are verified by the simulation.The proposed univariate impulse synchronization controller not only realizes the synchronization of complex network nodes,but also reduces the control energy,which has a broad application prospect.(3)Chaos has been widely used in many fields to improve the performance,such as optimization,measurement and control,image processing and other engineering systems.The method of chaos generation in non-chaotic system is proposed by using univariate impulse control.The Chen system in the stable region is controlled by the univariate impulse control to generate chaotic attractors.This method has the same structure with the state feedback method,which not only generates but also eliminates chaos flexibly according to the requirement,and exceeds the limitations of continuous state feedback control.It has the characteristics of simple structure,easy implementation,and suitable for the systems that only allowing control at intermittent moments.The dynamic characteristics of chaotic attractors are studied.The power spectrum and the bifurcation diagram of the chaotic time series of the systems are analyzed qualitatively,and the Lyapunov exponents are estimated for the quantitative analysis.In order to clearly observe the chaotic phenomenon,the Chen circuit,delay circuit and impulsive control circuit are designed in the experiments.Using the designed analog circuit,four chaotic attractors are observed in the experiment.The experiment results consist with the simulations,which verify the effectiveness of the method.(4)Evidence for the chaotic attractor is not trivial.The detailed process for finding topological horseshoe in the impulse differential equation and time delay differential equation are more different and complex than that in ordinary differential equation.By using the Smale horseshoe lemma,the mechanism of chaos generated by impulse control is expounded,and the existence of chaos is theoretically proved in this chapter.The method based on the topological horseshoe theorem is proposed to find topological horseshoe on the Poincare section for impulse control chaotic system,which overcomes the problem of the initial value of the delay term when using the Poincare inverse mapping.The MATLAB analysis software of topology horseshoe for the impulse delay system is designed.The topological horseshoe in the impulsive control Chen system and the delayed feedback control Chen system are analyzed with the designed analysis software,respectively.(5)The drug treatment process is usually regarded as an impulse control system,which the concentration and frequency of drug can be considered as impulse gain and interval.It is of great significance to research tumor chemotherapy with impulse control theory.In this paper,the adaptive impulse control method for tumor chemotherapy model with uncertain parameters and the corresponding control stability are proposed.On the one hand,the differences between individuals or inaccurate measurement data lead to the uncertain parameters of the tumor chemotherapy model.On the other hand,the current chemotherapy only depends on the experience of the doctor,which needs theoretical optimization of chemotherapy project.Considering the two points above,a method of adaptive impulsive control for the system with uncertain parameters is proposed in this chapter.With the Lyapunov theory,we analyze the positivity and permanence of the solution of the tumor chemotherapy model,then the chemotherapy impulse interval is estimated.With this method,the dose of chemotherapy agent is adjusted by state feedback for each injection,which the parameters of the system are uncertain.Simulation results show that the proposed adaptive impulse control method can successfully eliminate tumor cells and maintain immune cells.