Adaptive observer design for uncertain nonlinear ODE‐PDE cascades

Abstract

We are revisiting the problem of adaptive observer design for systems that are constituted of an Ordinary Differential Equation (ODE), containing a globally Lipschitz function of the state, and a linear Partial Differential Equation (PDE) of a diffusion-reaction heat type. The ODE and PDE are connected in series and both are subject to parametric uncertainties. In addition to nonlinearity and uncertainty, the system complexity also lies in the fact that no sensor can be implemented at the junction point between the ODE and the PDE. In the absence of parameter uncertainty, nonadaptive state observers are available featuring exponential convergence. However, convergence is guaranteed only under the condition that either the Lipschitz coefficient is sufficiently small or the PDE domain length is sufficiently small. To get around this limitation, and also to account for parameter uncertainty, we develop a design that involves two concatenated adaptive observers, covering the two subintervals of the PDE domain. The proposed design employs one extra sensor, providing the measurement of the PDE state at an inner position close to the ODE-PDE junction point.Both observers are shown to be exponentially convergent, under ad-hoc persistent excitation (PE) conditions, with no limitation on the Lipschitz coefficient and domain length.