Backstepping observers for a class of parabolic PDEs

Abstract

In this paper 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 an invertible coordinate transformation of the observer error system into an 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. For several physically relevant problems the observer gains are obtained in closed form. The observer gains are then used for an output-feedback design in both collocated and anti-collocated setting of sensor and actuator. The order of the resulting compensator can be substantially lowered without affecting stability. Explicit solutions of a closed loop system are found in particular cases.