Backstepping Controller Design And Simulation Study For A Class Of Parabolic Distributed Parameter Systems

In this paper, a problem of stabilization of a distributed parameter system described by a class of parabolic partial differential equations (PDE) with boundary conditions is considered. Based on the structure of the PDE and integrator Backstepping design idea, for an unstable heat equation, a new kind of Backstepping boundary states feedback controller is designed.This new kind controller contains an integral operator called kernel function. The kernel function is required to satisfy a Klein-Gordon-type hyperbolic PDE. Then the hyperbolic PDE is converted into an equivalent integral equation. By applying the method of successive approximations, the kernel function's well posedness is proved. In many conditions, the analytical solution of the kernel function is received. Therefore boundary controller is constructed in closed form. Compared with the existing boundary control arithmetic, it doesn't involve much deep and complicated mathematical knowledge in this method. Operator Riccati equation is avoided, instead, the boundary stabilization problem is converted to a problem of solving a specific Klein-Gordon-type hyperbolic PDE. Due to which, the numerical calculation effort is reduced enormously. This is a novelty of this method. For a broad range of constant and non-constant parameters, the Backstepping boundary states feedback states feedback controller are constructed in this paper. Using Lyapunove theorem, the stabilizations of the control systems are approved.With this new method, the closed-loop solutions of the control system are found explicitly, which means the analytical solutions of the parameter PDEs are received. That results in a new direction for analytical solution of PDEs. This is another novelty of this method. For a family of parameters, the analytical solutions of parabolic PDEs are given using the new method in this paper, the result schemes are posed. The schemes show that the results are exact as the numerical results using finite difference method. This new boundary control arithmetic is deduced and proved completely in this paper. A thin staff system with heat conduction is modeled as a simulation example, and an unstable heat equation is constructed. For this model, the Backstepping boundary states feedback controller is designed in continuous closed form, and analytical solution of the closed loop system state is also received. At the end, the simulation curves are posed, the curves show that the effect of the controller is perfect, the simulation system is stabilized.