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Backstepping boundary control: An application to rod temperature control with Neumann boundary condition

This article presents a backstepping boundary control technique to stabilize the temperature of a rod. The rod is modeled by a parabolic partial differential equation with a Neumann boundary condition at one end of rod and a Dirichlet boundary condition at the other end. The rod also includes an internal heat generator to emulate an unstable behavior. Based on backstepping boundary control, a partial differential equation gain kernel of the system is determined. It is calculated numerically by using a finite difference method and then used in a control law. Since the control law requires temperatures along the rod for feedback, a Luenberger-like observer is used to estimate the temperatures. A copper rod with a heater installed inside the rod is used as an experimental plant. The control setup is anticollocation. The temperature signal at one end is sent to the observer for estimation of the temperatures along the rod. The estimated values are then fed to the controller. Simulation results using a finite difference method and experimental results are compared and used to confirm the effectiveness of the control method.