Study On Stability And Control Of The Interconnected Systems Based On Vector Bounded And Lyapunov Function Methods

With the development of wireless power transfer(WPT)technology,a single point-to-point energy transfer mode has been unable to meet the power supply needs of many industrial equipment and social systems.Meanwhile,due to the limitation of magnetic field energy transmission distance,large distance wireless power transmission is greatly limited.In order to improve the effective distance of energy transmission and realize multi-point wireless power transmission,multi-node cascade and one-to-many WPT systems are introduced.In these systems,since the large number of nodes,the complex coupling relationship,the dynamic random uncertainty of spatial energy distribution,the high-order nonlinearity and the hybrid characteristics cuased by the dynamic process of multi-mode organization structure,meantime,the WPT system will inevitably be subject to external disturbances,such as load disturbance,parameter perturbation,and time delay,which may seriously affect the system dynamic behavior,stability and anti-interference ability and further influence the stability and energy efficiency of the energy transmission.At present,the circuit theory is mainly used to study the stability of the WPT system by using the method of the existence of a unique zero for the primary impedence angle.However,the dynamic behavior of the system cannot be described and the stability of the system can also not be well analyzed.Meantime,for complex WPT systems such as one-to-many WPT(namely,multi-receiver WPT)systems,due to the higher-order feature of the system,which brings great difficulties to the analysis and calculation of the system.In addition,the fractional-order feature for the WPT system is often ignored in the past research,so it is impossible to well study it.Essentially,the WPT system is an interconnected system that wirelessly transmits energy from the transmitter to the receiver by means of magnetic coupling.The modeling idea and research theory of the interconnected system can address the above problems well.However,at present,there are many problems in the research of the interconnected system.On the one hand,there exist the large number of subsystems in the interconnected system,and the relationships between subsystems are complex.Besides,the influence of various disturbances from the external environment,the parameter uncertainties and the time delays are unavoidable,which may cause the instability of the system.All the time,study on the stability of the interconnected system is usually carried out without considering or only considering a single factor.As a matter of fact,the system is usually affected by many factors simultaneously.In the meantime,the past studies on the stability of the interconnected system focus on the integer-order system,and less on the fractional-order system.In addition,the obtained stability results are very conservative,which largely restricts the application of the results.Besides,the stability domain for the system reflects the ability of the system to resist disturbances,however,up to now,there are few investigaitons on the stability domain for the interconnected system.On the other hand,in order to restrain the influence of the disturbance,the parameter uncertainty and the time delay on the stability of the system and ensure the reliable and stable operation for the system,it is necessary to control it.In terms of the control for the system,the traditional control modes only focus on the effect of control,but ignores the cost of control(namely,number of the controlled variables).Nevertheless,for the interconnected system,the control cost is very high because of the large number of internal subsystems,which greatly improves the difficulty for control.These problems also appear in many similar systems,such as interconnected power system,distributed energy system,multi-agent system and so on.This paper concentrates on the stability of the fractional-order uncertain interconnected systems,estimation of the stability domain for the fractional-order interconnected systems,the stability and control for the fractional-order interconnected systems,and the stability and time-delay control of the interconnected systems.The research results of this paper further enrich the theoretical system of the interconnected systems and provide theoretical guidance for modeling,stability study and robustness evaluation of the complex WPT systems.Main contributions of this paper are summarized as follows:For the problem of the stability for the uncertain fractional-order interconnected systems.And this uncertainty is time-varying and norm bounded.The vector bounded technique is utilized to deal with the problems.The model of the uncertain fractional-order interconnected system is established,and a corresponding state feedback controller is designed.Based on the fractional calculus theory and combined with vector bounded method,the stability of the fractional-order uncertain interconnected system is analyzed from the perspective of the solution of system.Meanwhile,the relevant stability criteria are obtained.The problems of the model order reduction for the multi-receiver WPT system and the influence of the parameter uncertainty on the dynamic behavior and stability of the whole WPT system and each subsystem are analyzed,respectively.For the estimation problem of the stability domain of the interconnected systems,the Lyapunov function technique is employed to investigate the stability domain of a typical interconnected system,namely,the WPT system.The corresponding fractional-order model of the typical WPT system is built.Based on fractional calculus theory,the boundedness of WPT system is analyzed by the Lyapunov function method and the induction method.Meantime,the relevant boundedness criteria are derived and extended to piecewise reflex system.Moon inequality is used to improve the conservatism of the obtained criteria.For the problems of the stability and control for the fractional-order interconnected system(including the fractional-order nonlinear system with external bias),the vector bounded technique is utilized to study.For the fractional-order nonlinear system with external bias and the fractional-order interconnected system,the corresponding fractional-order model is established,respectively.Meanwhile,in order to reduce the control cost(namely,the number of controlled variables),one single-variable control method with measurable state is proposed.The fractional calculus theory and the relevant inequality technique are used to analyze the stability of the fractional-order nonlinear system with external bias and the fractional-order interconnected system,respectively,and the corresponding stability criteria are obtained.The dynamic behavior and stability of the fractional-order multiple-receiver WPT system are improved.For the problems of the stability and control for the interconnected systems with time-varying delays,Lyapunov functional techinique is used to investigate.The corresponding mathematical model of the interconnected delayed systems is established.In order to overcome the influence of the time delays and save the control cost,one control method based on partial variables with measurable states is proposed.Based on the theory of stability,by constructing the new Lyapunov functionals,the exponential boundness and the exponential stability of the interconnected system are researched,respectively.Meanwhile,the relevant boundedness criteria and stability criteria are obtained,respectively.The free weight matrix method is used to reduce the conservatism of the obtained criteria.The stability for the multi-receiver delayed WPT system is improved.