Trajectory Tracking For A Diffusin-Reaction Equation

As we know, there are a large number of diffusion-reaction problems in physics, chemistry, biology and other engineering area. The diffusion-reaction equation is the mathematical model to describe these diffusion-reaction problems. The trajectory tracking control objective for the diffusion-reaction equation, such as the temperature tracking control problem for the heat conduction process, the concentration tracking control problem for the fiber molding process, the rate tracking control problem for the enzyme diffusion process, is to make the output trajectory asymptotically tracking an external reference output.Because the trajectory tracking problem of diffusion-reaction equation has strong engineering background, so the research has some theoretical and practical value of such problems.The BACKSTEPPING control algorithm cooperated with boundaryapproach is proposed for distributed parameter systems modeled in partial differential equation. The algorithm based on the Volterra transformation, which establishes the mapping between unstable originalsystem and the exponentially stable target system to get the state feedback boundary controller to make the original system stable by solving the kernel function, is robust, inverse optimal, and potential for explicit control laws and exact solutions of closed system. The BACKSTEPPING control algorithm is novel and can be combined with the achievements of observer theory and the adaptive control to extend its application fields.In this paper, a trajectory tracking control design for the diffusion-reaction equation is introduced based on BACKSTEPPING control algorithm. Firstly, the heat equation which is the mathematical model for the heat conduction process is established then selects the heat equation as the target system by using the Lyapunov method to show its exponential stability. Secondly, the controllability of diffusion-reaction equation is studied.Based on the BACKSTEPPING control algorithm, the he state feedback boundary controller is obtained by using the Volterra transformation to establish the mapping between diffusion-reaction equation and heat equation after solving the kernel function. With the BACKSTEPPING boundary controller the diffusion-reaction can always reach the equilibrium for any initial condition. Finally, the BACKSTEPPING control algorithm is extended to the peculiar tracking for the diffusion-reaction equation. Based on the controllability of diffusion-reaction equation and using the method of Ta series expansion of signal model, a state feedback boundary controller is obtained to make the output trajectory asymptotically tracking an external reference output for any initial condition.The novel of this control algorithm is that using BACKSTEPPING control algorithm to solve trajectory tracking problem of diffusion-reaction equation. The control design is simple understanding and simulation study shows the control effects have good tracking performance of diffusion-equation.