Adaptive observer design with heat PDE sensor

The problem of state and parameter estimation is addressed for systems with cascade structure including finite-dimensional dynamics followed in series with infinite-dimensional dynamics. The formers are captured through an ODE that is state- and parameter-affine. The latter, referred to as sensor dynamics, are represented by a diffusion parabolic PDE. Both equations are subject to parameter uncertainty. Furthermore, the connection point between the ODE and the PDE blocs is not accessible to measurements. The aim is to get online estimates of all inaccessible states and unknown parameters of both the ODE and the PDE subsystems. This observation problem is dealt with by combining the backstepping design method and the extended Kalman observer approach. The obtained adaptive observer is shown to be exponentially convergent under an ad-hoc persistent excitation condition.