Finite-time observer for a time-varying cascade of an ODE in a system of balance laws

In this paper, we design a finite-time convergent observer for linear time-varying ODE-hyperbolic balance law cascade systems. This work extends the study of output regulation for linear time-varying hyperbolic systems. In Bai et al. (2025), a state feedback regulator was developed to achieve finite-time output regulation under the assumption that all disturbances and system states are known. However, this assumption is unrealistic in practical applications. Therefore, we design a finite-time observer under the condition that only the to-be-tracked reference signal and boundary values of the system state are the available measurement. The key idea is to use two ODE observers to achieve finite-time observation of the ODE subsystem and to exploit the finite-time stability structure of the hyperbolic subsystem to estimate its state. In this work, we utilize results on the observability and estimability of linear time-varying ODE systems to establish the existence of the proposed observer.