Structure preserving spatial discretization of 1D convection-diffusion port-Hamiltonian systems

Abstract

Convection-diffusion is a physical phenomenon that appears in a multitude of dynamical systems, e.g. vibrating string with damping or chemical and thermal systems. This paper focuses on a structure preserving spatial discretization scheme of a general dynamical system with convection and diffusion in the port-Hamiltonian framework. The preservation of the port-Hamiltonian structure ensures that specific properties, such as passivity, of the infinite dimensional system are preserved. I. INTRODUCTIONSystems with spatial and temporal dynamics occur in many engineering applications where they are typically modeled by partial differential equations or as state space models over infinite dimensional state or phase spaces. The control of infinite dimensional systems is generally troublesome due to the fact that an infinite number of states needs to be controlled through a finite number of control variables. Currently, one can distinguish two common approaches for the control of infinite dimensional systems. The first one, late lumping [1], [2], amounts to designing an infinite dimensional control law which renders the infinite dimensional system stable while achieving some desired performance. The second approach, early lumping [3], amounts to first spatially discretizing the infinite dimensional system and, subsequently, designing a finite dimensional control law based on the dynamics of the spatially discretized system. All control design approaches based on finite element models belong to this category. This paper considers the first step in the early lumping control approach, namely the spatial discretization of a 1D convective and diffusive infinite dimensional port-Hamiltonian (pH) system [4]. There are compelling reasons to model convective and diffusive phenomena in infinite dimensional system in the port-Hamiltonian framework. The most important one is that pH systems have structural properties, such as passivity, that make them extremely suitable for control design, see [5], [6]. These properties are ensured by the special mathematical structure of a pH system and are typically lost by arbitrary spatial discretization schemes.The problem of structure preservation in discretization schemes of pH systems is of paramount importance. Indeed, from the general perspective of modeling, simulation, model approximation, and control system design, there is a need for specialized discretization schemes that preserve the pH structure in spatial discretizations. Classical spatial T. Voß and S. Weiland are with

Structure Preserving Discretization of 1D Nonlinear Port-Hamiltonian Distributed Parameter Systems

“…Using the expressions (33),(38) and substituting (31), (32), (34), (35), (36), (40) while setting σ = 0 (the system is conservative) yields that M 1 = 0 and…”

“…Partitioned finite element methods were considered in [29] and involve integration by parts on a subset of equations. Methods in [30,31,32,33,34,35,36] focus on the spatial discretization of boundary controlled systems and construct particular finite element modules from the direct approximation of the differential forms. This paper is in line with the latter approaches and addresses the direct spatial discretization of nonlinear distributed parameter Hamiltonian systems (with a decomposable Hamiltonian) controlled via ports at the boundary of a 1D spatial geometry.…”

“…The technical novelty to establish this, lies in the introduction of a fictitious point inside the mesh elements that partition the spatial geometry. This augments the degrees of freedom and remedies a number of limitations concerning compatibility conditions and choices of input-output variables [30], direct feedthrough terms that cause oscillations due to instantaneous power flows through the discretized network [32,33,34], non-sparsity or non feasibility to match nonlinear Hamiltonian functions [36]. As such, the proposed discretization scheme is numerically efficient, flexible towards choices of input, output and boundary variables, fully scalable to large networks and physically relevant on its preservation of passivity and energy distribution properties.…”

Energy-based modeling and control of interactive aerial robots

“…The work of Golo et al [2004] was extended to the case of a non-constant SDS in Pasumarthy and van der Schaft [2006a]; Pasumarthy et al [2012] for the shallow water equations, and extended to irreversible pH systems in Baaiu et al [2009b]; Voß and Weiland [2011]. Another method that was inspired by Golo et al [2004] is Bassi et al [2007] which was formulated for the functional analytic formulation of pH systems discussed in Sec.…”