Observer design for a class of parabolic PDE via sliding modes and backstepping

Observation problem for systems governed by Partial Differential Equations (PDE) has been a research field of its own for a long time. In this paper it is presented an observer design for a class or parabolic PDE's using sliding modes theory and bacstepping-like procedure in order to achieve exponential convergence. A Volterra-like integral transformation is used to change the coordinates of the error dynamics into exponentially stable target systems using the backstepping-like procedure. This gives as a result the output injection functions of the observer which are obtained by solving a hyperbolic PDE system. Sliding modes are used to find an explicit solution to the hyperbolic PDE system and to make the observer gains to be discontinuous which have well known advantages. Theoretical results were proved using the Lyapunov theory. A numerical example demonstrates the proposed method effectiveness.