Boundary Controller And Observer Design For Distributed Parameter Systems

Over the last few decades,with the development of control theory for distributed parameter system(DPS)in practical applications,the boundary control problems of DPS have become one of the current research hotspots.And there are more and more research results on boundary controller and observer design for distributed parameter systems.However,most of the results of boundary control problems are limited to linear DPS by now;The boundary control of nonlinear DPSs is still a tough and open problem.T-S fuzzy model is one of the most effective methods to study nonlinear systems in finite dimensional control theory.Therefore,in this paper,the T-S fuzzy model and Backstepping method are used to study the boundary controller and observer design for a class of nonlinear distributed parameter systems;The stabilization conditions of T-S fuzzy distributed parameter systems are given and the simulation results show the effectiveness of the proposed results,and the feasibility of combining T-S fuzzy model with infinite dimensional distributed parameter system is further demonstrated.The main works of this paper are as follows:Firstly,the boundary control problem of a class of semilinear hyperbolic partial differential equations(PDEs)is studied by using T-S fuzzy theory.The semilinear partial differential equations are transformed into T-S fuzzy partial differential equations,then a simple fuzzy boundary controller is designed,and By using Lyapunov stability theory,a sufficient condition for the closed-loop system to be exponentially stable is given in terms of a set of linear matrix inequalities(LMIs).Finally,an application example of chemical reaction is given to show the effectiveness of the proposed method.Secondly,the asymptotic stability of a class of cascaded systems consisting of nonlinear parabolic partial differential equations and ordinary differential equations(ODEs)is studied.Based on T-S fuzzy theory,the nonlinear cascade system is transformed into T-S fuzzy cascade system model,Then,a fuzzy controller is designed,which only relies on the variable of ODE affects the entire system through a boundary condition of the PDE.and based on Lyapunov stability theory,a sufficient condition of exponential stabilization for the closed-loop cascaded system are given in terms of a set of algebraic linear matrix inequalities(ALMIs)in space.Finally,simulation results on a numerical example are provided to illustrate the correctness and effectiveness of the obtained results.Thirdly,the consensus problem of multi-agent systems with linear parabolic partial differential equations is studied.The multi-agent systems composed of ODE is extended,we discuss the case of Neumann type boundary controller for a class of thermal diffusion equations.Then,we design two kinds of non-fragile feedback control algorithms,and the sufficient conditions for the average consensus of multi-agent systems are given.Finally,an application example of a micro electro mechanical system(MEMS)is presented to show the validity of the obtained results by numerical simulation.Fourthly,based on the design method of high-gain observer,the observer design for a class of nonlinear distributed parameter systems is studied.The distributed parameter systems consist of a nonlinear ordinary differential equation and a linear first-order hyperbolic partial differential equation.The connection information of the two subsystems is located on the boundary of the partial differential equation,which is used as the observation point,and a high-gain observer is designed by using the Backstepping method.By using Lyapunov stability theory,the sufficient conditions for the exponential convergence of the observation error are given.Finally,an numerical simulation example is presented to demonstrate the effectiveness of the designed high-gain observer.