Research On Boundary Feedback Control And Stabilization Of Coupled Partial Differential Equation

In nature,many physical phenomena and most dynamic characteristics in actual engineering systems have not only time evolution characteristics,but also spatial distribution characteristics,such as material distribution in tubular chemical reactors,temperature distribution in large heating furnaces,spatial distribution of species,electromagnetic fields,string vibrations,etc.Compared with ordinary differential equation(ODE)that only contains time variables,partial differential equation(PDE)contains both time-space variables and have infinite-dimensional characteristics,which can more accurately describe the dynamics of actual systems.In particular,coupled systems composed of multiple partial differential equations or coupled systems composed of partial differential equations and ordinary differential equations can more comprehensively simulate various industrial application processes,such as road traffic,gas flow in pipelines,power converters connected to transmission lines,oil drilling,etc.Therefore,studying the control issues of such coupled systems has important theoretical value and practical significance.This thesis studies the state feedback/output feedback stabilization problem of several kinds of coupled systems with infinite-dimensional characteristics.By using infinitedimensional backstepping transformation,successive approximation method,adaptive parameter estimation method and Lyapunov stability theory,a variety of boundary control strategies and stability analysis methods are proposed.The main innovative work of this paper includes the following five parts:1.State feedback and output feedback boundary control strategies are proposed for a class of ODE-parabolic PDE-ODE coupled systems with nonlinear characteristics and spatially variable coefficients.Assuming that the system state information is measurable,the parabolic PDE-ODE coupled system is first transformed into an easily analyzed target system using infinite-dimensional backstepping transformation,and then the nonlinear ODE is addresses using finite-dimensional backstepping transformation.A state feedback controller is designed,a Lyapunov function is constructed,and the exponential stability of the state feedback closed-loop system in the H1-norm sense is analyzed.However,the PDE state information in the domain is usually difficult to measure directly or cannot be measured.Therefore,a state observer is designed using boundary measurable signals,an output feedback controller based on the observer is constructed,and the exponential stability of the output feedback closed-loop system in the H1-norm sense is analyzed.2.An adaptive state feedback control scheme is proposed for a class of ODE-parabolic PDE-ODE coupled systems with uncertain nonlinear characteristics.Firstly,an infinitedimensional backstepping transformation is constructed to transform the original parabolic PDE-ODE coupled system into a target PDE-ODE system thatis easy to analyze.An adaptive state feedback controller is designed by combining finite-dimensional backstepping transformation and adaptive parameter estimation method.Then,a Lyapunov function is constructed to analyze the boundedness of all signals in the closed-loop system and the asymptotic convergence of the closed-loop system.3.The adaptive stabilization problem of a class of heterodirectional 2 × 2 hyperbolic PDE with uncertain nonlinear dynamics is studied.Firstly,the heterodirectional 2 × 2 hyperbolic PDE with spatially varying coefficients is transformed into the target system by using infinite-dimensional backstepping transformation.Secondly,a new variable transformation is introduced to eliminate the spatially varying coefficients in the PDE.Then,an adaptive state feedback controller is designed by combining finite-dimensional backstepping transformation and adaptive parameter estimation method.The boundedness of all signals in the closed-loop system and the asymptotic convergence of the closed-loop system are analyzed.4.Motivated by the thermoelastic coupling effect in Micro Electro Mechanical Systems,the event-triggered control problem of a PDE-ODE system coupled at an intermediate point is studied.First,the dynamic triggering condition is constructed and the event-triggered controller is designed.Then,an auxiliary function is introduced to prove the existence of the minimum dwell-time and avoid the Zeno phenomenon.Finally,a Lyapunov function is constructed and the exponential stability of the closed-loop system is analyzed.5.The event-triggered state feedback/output feedback boundary control strategies are proposed for a class of underactuated coupled parabolic PDE with spatially varying coefficients.Assuming that the system state information is measurable,the internal dynamic variables are introduced to construct the dynamic triggering condition,and an event-triggered state feedback controller is designed.Secondly,the existence of the minimum dwell-time is proved by introducing the auxiliary function,and the exponential stability of the eventtriggered state feedback closed-loop system is analyzed.However,the PDE state information in-domain is usually difficult to measure directly or cannot be measured.Therefore,a collocated state observer is constructed using boundary measurable signals,and an observer-based event-triggered output feedback controller is designed.