Adaptive observers for slowly time varying infinite dimensional systems

We consider a class of infinite dimensional systems with an unknown time varying perturbation in the input term. The goal here is twofold, namely to estimate the state and identify the unknown parameter in the input term using only input and output measurements. An adaptive observer along with a parameter adaptive law that is based on Lyapunov redesign is presented and, under certain conditions imposed on the plant, is shown to achieve state error convergence. Parameter convergence can be established by imposing the additional condition of persistence of excitation. Examples that illustrate the applicability of this approach to a parabolic partial differential equation and a delay system are included along with some numerical results