Event-triggered Control For First-order Linear Hyperbolic Distributed Parameter Systems

The first-order linear hyperbolic distributed parameter systems(DPSs)represent the dynamics of chemical reaction processes,traffic flow behavior,population growth process,hydraulic pipeline transportation process and so on.Due to the probably existence of infinite eigenvalues with positive real parts in hyperbolic DPSs and the infinite dimensional characteristic of distributing in spatial and temporal,so that many classical control problems have already well researched in centralized parameter systems are still a challenge in hyperbolic DPSs.In the last decade,event-triggered control has attracted extensive attention for its advantages of saving resources and unnecessary waste.However,the control of the first-order linear hyperbolic DPSs focused on sampled-data control,sliding mode control,iterative learning control,event-triggered boundary control,etc.There are few research results on distributed event-triggered control.In this paper,a class of first-order linear hyperbolic distributed parameter systems based on event-triggered control have been studied.The main techniques used are Linear matrix inequality(LMI),Schur’s complement lemma,Matrix spectral radius theory,Spatial integral form of Jensen’s inequality and Lyapunov stability theory.Compared with existing results,this paper extends the applicable scope of event-triggered control and improves the control theory of DPSs.The main research contents include:(1)For the first-order linear hyperbolic distributed parameter system with two different kind of boundary conditions,the exponential stability based on event-triggered control is studied.Firstly,the distributed state feedback controller is designed by the preset event triggered conditions.Secondly,different Lyapunov functions are constructed according to the given boundary conditions and linear boundary conditions of the general linear hyperbolic system and the improved hyperbolic distributed parameter system,sufficient conditions have been obtained for the exponential stability of the systems in terms of LMI.In addition,the existence of Zeno phenomenon is excluded by calculating the derivative of measurement error with respect to time.Finally,a simulation example is given to verify the feasibility of the results and the superiority of event-triggered control.(2)For the first-order hyperbolic distributed parameter system with time-delay,the exponential stability problem under event-triggered control is studied.By constructing the Lyapunov-Krasovskii function,using the integral by parts formula and the matrix coupling law,the exponential stability of the varying-delays and multi-varying-delays system is proved rigorously.The correlation stability of a series of delay derivatives based on LMI is obtained.In the end of this chapter,a numerical example is given to illustrate the validity of the proposed theorem.(3)For the first-order linear hyperbolic distributed parameter multi-agent systems,the event-triggered control protocol with periodic sampling schemes is proposed to study the consensus problem.By setting up a virtual leader,the error system is constructed using the kronecker product.Then,the event-triggered consistency problem converts into the asymptotic stability problem of the error system.An appropriate Lyapunov-Krasovskii functional is selected,with the spatial integral form of Jensen’s inequality,matrix transformation is applied to obtain the sufficient conditions for the asymptotic stability of the error system based on the linear matrix inequality,and the expressions for the controller gain.Finally,simulations of the actual model are performed to verify the effectiveness of the control protocol.