Stabilization Of An ODE-PDE Cascaded System

The boundary control problem of an ODE-PDE cascaded or coupled control systems has always been a hot topic in the control system research.In this paper,The stability of an cascade system of ordinary and partial differential equations is considered,and the control law is designed by backstepping method,the exponential stability of the closed-loop system with state feedback is also proved.In this thesis,we mainly use the backstepping method.The basic idea of this method is to decompose a complex nonlinear system into subsystems of lower system order,Then design Lyapunov functions and intermediate dummy variables for each subsystems,all the way back to the system and gets control law.The first chapter of this paper mainly introduces the background of the modern control theory,the background and current situation of ODE-PDE couple or cascade problem,as well as the main object of study and related preliminary knowledge.The second chapter is to design the controller of the studied system.Let's introduce an integral transformation,and the original system is transformed into the selected exponentially stable target system.The boundary control law is established.The kernel function is derived in this course,and solve the kernel function equations by mathematical method,and the specific analytical formula of control law is obtained.At last,it is primarily concerned with take the inverse transformation of the integral transformation and proved its boundedness.Through the exponential stability of the target system and the boundedness of the transformation and its inverse transformation,the exponential stability of the closed-loop system controlled by boundary control law is established.