Control Design For The ODE-PDE Cascade System

With control theory and computer technology development,many complex systems and control theories are gradually applied to the actual production in our life,the traditional dynamic system modeling can be divided into two categories:The lumped parameter system and the distributed parameter system.Combining the advantages of the two types of models,the model established by the cascade system of ordinary differential equation and partial differential equation is more beneficial to the practical application in production.Therefore,the research on a series of properties of ODE-PDE cascade system has aroused the interest of experts and scholars.Because of the coupling of boundary states,the controller format is more complex than the traditional model.In this paper,the properties and control methods of ODE-PDE cascade systems with boundary coupling are studied.The structure of this paper is mainly divided into the following three aspects.In the first part,the observer of ODE-heat equation and ODE-wave equation cascade system is considered respectively.The controller is located in the ODE system,and the coupling points of ODE and PDE and the measuring points of sensor installation are located at different ends of the PDE system.In this chapter,error system between original system and observer are mapped to a kind of homeomorphic system by using the design idea of Backstepping control method and coordinate transformation of function space.Lyapunov method is used to prove that the transformed system is asymptotically stable,and then the constraint conditions of transformation are solved.Finally,numerical simulation results are given to verify the correctness of theoretical derivation.In the second part,a class of output tracking control algorithm is designed for a class of ODE-heat equation cascade system described in the previous chapter.Considering the operating conditions of the actual industrial production model,the boundary disturbances have been considered in this chapter.The design of output tracking controller can also reject the interference signal,using Fourier decomposition and ignoring the high frequency term tracking signal and disturbance signal,Both references and disturbances can be generated by an exo-system.Then,a state feedback controller has been designed to stable the controlled system,and its stability and well-posedness are proved by Lyapunov method and operator semigroup method.Subsequently,the output feedback controller based on the state observer is designed.Finally,numerical simulation proves that the tracking effect meets the requirements.In the last part,a class of first-order ODE systems coupled with the heat equation has been considered as a single agent,the synchronization protocol between each agent has been given.This chapter assumes that the networked multi-agent system is connected in a fully connected directed topology.Then a continuous synchronization protocol for networked multi-agent systems is given.In addition,for the existing digital controllers,a kind of event-based consistency protocol is designed,and the design of the triggering time format of the protocol is given,as well as the proof process that the system can still maintain strict stability under the event triggering condition.Finally,simulation results are given on a computer platform to support the above conclusions.