Finite-dimensional observers for port-Hamiltonian systems of conservation laws Paul Kotyczka, Henning Joos, Yongxin Wu and Yann Le Gorrec Abstract— 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 di- rections 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. I. INTRODUCTION In order to implement state feedback control, observer de- sign is necessary for the lack of complete state measurement in real physical applications. Deterministic observer design for linear finite dimensional systems has been established in the 60s and 70s by Luenberger [1]. However, in the nonlinear and infinite dimensional cases, observer design is still an open research problem. In last two decades, a powerful modeling and control approach, called port- Hamiltonian (PH) approach has been proposed to cope with nonlinear and distributed parameter systems. Based on the energy and a structured representation of the power flows and dissipation in the system, the PH framework is particularly suited to describe the complex behavior of multi-physical systems [2]. The PH approach has been generalized to infinite-dimensional systems described by partial differential equations (PDEs) in [3], [4]. Observer design for finite- dimensional PH systems has been investigated in the last ten years. It has been shown that the passivity of PH systems is very useful for the observer design [5]. The idea of Interconnection and Damping Assignment has been extended to the observer design for PH systems in [6], [7]. In the infinite-dimensional case, particular attention has to be paid to numerical issues associated with the design and implementation of finite dimensional observers. Recently, This work was supported by the Agence Nationale de la Recherche/Deutsche Forschungsgemeinschaft (ANR-DFG) project INFIDHEM, ID ANR-16-CE92-0028. P. Kotyczka and H. Joos are with the Department of Mechanical Engineering, Technical University of Munich, 85748 Garching, Germany kotyczka@tum.de, henning.joos@tum.de Y. Wu and Y. Le Gorrec are with FEMTO-ST Institute, AS2M Department, Univ. Bourgogne Franche-Comt´ e, Univ. de Franche-Comt´ e/CNRS/ENSMM, 25000 Besanc¸on, France yongxin.wu@femto-st.fr, legorrec@femto-st.fr progress has been made on the structure preserving spatial discretization of PH systems, applicable to arbitrary spatial dimension and complex geometries [8], [9]. A definition of discrete-time PH systems based on time discretization with collocation methods, which extends the notion of symplectic integration schemes to open systems, has been proposed in [10]. The simplest approach, which leads to such a discrete- time PH system is the symplectic Euler scheme, applied to partitioned systems. In [11], it is shown that beyond the preservation of the PH structure, also the flatness property of the corresponding outputs is preserved, which allows for the explicit computation of discrete-time feedforward controls. In this paper, we present two state estimation schemes based on the full structure preserving discretization of hy- perbolic systems of conservation laws. First, we present Lu- enberger type observers based on implicit damping injection with collocated and non-collocated measurements. Second, we exploit the flatness of the discrete-time finite-dimensional approximate models to construct an explicit scheme for state estimation. The paper is structured as follows. Section II gives an overview of 1D PH systems of conservation laws and their structure preserving discretization in space and time. The main results of the paper, the implicit damping injection based observer and the flatness-based state estimation, are introduced in Section III. In Section IV, we show the effectiveness of the proposed observers on the benchmark example of the 1D wave equation, for which the solution is exactly known. At last, we conclude this paper with final remarks and some future perspectives. II. PRELIMINARIES A. Port-Hamiltonian systems of conservation laws We consider 1D systems of two conservation laws in PH form, written in terms of exterior differential calculus 1 . According to [3], the PDE representation can be split into structure, dynamics and constitutive equations,  f p f q  =  0 d d 0  e p e q  , (Structure) (1a)  ˙ p ˙ q  =  -f p -f q  , (Dynamics) (1b)  e p e q  =  δ p H δ q H  . (Constit. Eq.) (1c) Considering an open domain Ω = (0,L), the state differ- ential forms p, q ∈ L 2 Λ 1 (Ω) ⊂ R represent the conserved 1 See [12] for an introduction to differential forms.