Adaptive boundary estimation for uncertain cascaded hyperbolic PDE-ODEs

The paper is concerned with the adaptive boundary estimation problem for a class of hyperbolic partial differential equations (PDEs) cascaded with a set of ordinary differential equations (ODEs). The cascade system consists of $n\times n$ coupled hyperbolic PDEs, subject to uncertain parameters in domain and the ODEs entering the PDEs through the left boundary of the PDEs. The adaptive boundary observer is designed to estimate the system state and the unknown parameters. In order to facilitate the design of the observer, the observer error system is transformed into a target system by a Volterra integral transformation and a state transformation, and then a swapping filter design method is proposed by combining the method of least-squares with a variable forgetting factor. The stability analysis shows the exponential convergence of both the state estimation error and the parameter estimation error under the persistent excitation condition of the filter. Finally, the obtained theoretical results are applied to the linearized Aw–Rascle–Zhang (ARZ) traffic flow model involving the in-domain relaxation time and the boundary flux fluctuation uncertainties, which verifies the effectiveness of the proposed method.