Finite-dimensional observers for port-Hamiltonian systems of conservation laws

We consider the port-Hamiltonian formulation of systems of two conservation laws with canonical interdomain coupling in one spatial dimension. Based on the structure-preserving discretization in space and time, we propose two directions for the estimation of the discrete states from boundary measurement. First, we design full state Luenberger observers for the linear case. To guarantee unconditional asymptotic stability of the discrete-time error system, special attention is paid to the implementation of the correction term in the sense of implicit damping injection. Second, we exploit the flatness of the considered class of possibly nonlinear hyperbolic systems, which is preserved under the applied geometric discretization schemes, to obtain a state estimation based on boundary measurement. Numerical experiments serve as a basis for the comparison and discussion of the two proposed discrete-time estimation schemes for hyperbolic conservation laws.