A simple observer scheme for a class of 1-D semi-linear parabolic distributed parameter systems

In this paper, a simple observer design scheme is proposed for a class of 1-D semi-linear parabolic distributed parameter systems with in-domain measurement. The main idea of the design resides in imposing the measurement on the observer dynamics in a similar way as it is done in the finite-dimensional reduced order observer design. For the distributed system this is achieved through an additional algebraic constraint in the domain instead of an output injection mechanism. Correspondingly, the observer convergence is not induced by the correction scheme, but by the sensor location and the system parameters, and thus amounts on the underlying detectability property rather than the system observability. Possible applications to the case of unknown inputs are discussed. The proposed estimation approach is illustrated and tested through simulations with two representative examples: an unstable diffusive linear system with Dirichlet boundary conditions, and a stable isothermal single-species nonlinear tubular reactor with non-monotonic kinetics and Robin-Newman (i.e., Danckwerts) boundary conditions.