Disturbance-to-state stabilization by output feedback of nonlinear ODE cascaded with a reaction-diffusion equation

In this paper, we analyze the output stabilization problem for cascaded nonlinear ODE with $1-d$ heat diffusion equation affected by both in-domain and boundary perturbations. We assume that the only available part of states is the first components of the ODE-subsystem and one boundary of the heat-subsystem. The particularity of this system is two folds i) it contains a nonlinear additive term in the ODE-subsystem, and ii) it is affected by both boundary and in-domain perturbations signals. For such a system, and unlike the existing works, we succeeded to design an output observer-based feedback that guarantees not only asymptotic stabilization result but also a globally {\it disturbance-to-state stabilization} for our cascaded system. The output feedback is designed using an adequate backstepping transformation recently introduced for coupled ODE-heat equations combined with high-gain observer and high-gain controller.