Output feedback boundary control by backstepping and application to chemical tubular reactor

We design exponentially convergent observers for a class of parabolic partial integro-differential equations (P(I)DEs) with only boundary sensing available. The problem is posed as a problem of designing invertible coordinate transformation of the observer error system into exponentially stable target system. Observer gain (output injection function) is shown to satisfy a well-posed hyperbolic PDE that is closely related to the hyperbolic PDE governing backstepping control gain for the state-feedback problem. It is shown how observer gain can be obtained directly from the control gain. Backstepping controller and observer are then combined to obtain a solution for the boundary output-feedback problem. Collocated and anti-collocated positions of sensor and actuator are considered. Explicit solutions to the output-feedback problem are obtained for certain classes of PDEs. It is shown that the order of the compensator can be substantially lowered without affecting stability. Simulation study for the model of chemical tubular reactor is presented.