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Easy-to-Implement Energy-Based Control Design for Systems of Conservation Laws

Periodic Reporting for period 1 - EasyEBC (Easy-to-Implement Energy-Based Control Design for Systems of Conservation Laws)

Reporting period: 2015-09-01 to 2016-08-31

Today's industrial applications comprise a large number of different, highly interconnected, non-trivial subsystems. Their efficient overall operation is an important factor for economic growth and environmental protection. Conservation laws govern the dynamics of a large variety of phenomena in energy, production and process industry which evolve in space and time. They describe, for example, mass and heat transport, or the evolution of concentrations in chemical reactions. The efficient operation of such complex processes requires mathematical models that reproduce the dominant system properties.

The EasyEBC project is based on the port-Hamiltonian (PH) perspective with its exciting conceptual simplicity: The storage of energy (a “Hamiltonian”), dissipation and power exchange between subsystems and the environment via power interfaces (“ports”) are the key modeling paradigms. They make the PH approach suited for modeling and control of complex multi-physics systems. The development of transparent and easy-to-handle – as a prerequisite for industrial deployment – control design methods for systems of conservation laws based on the PH representation is the ultimate goal of EasyEBC.

A series of spatial discretization methods hat preserve the PH structure have been proposed in the last decade. Their properties and accuracy, their parametrization and the relations to classical schemes from numerical mathematics, pose, however, a series of open questions. These issues, together with the benchmark problem suggested by the host laboratory, namely the modelling of heat and mass transport in catalytic foams, motivated us to re-orientate the action. Instead of control design for 1D hyperbolic systems based on given models, we concentrated on the elaboration of discrete systems of conservation laws, augmented with boundary port variables in higher spatial dimensions, and the analysis and adaptation of classical numerical approximation methods for this purpose. This research on more fundamental questions paves the way to develop EasyEBC control methods for a much wider class of complex dynamical systems.

Besides the research project, the scientific training of the fellow, and the dissemination of the results, the consolidation of the scientific cooperation and the organization of a summer school on PH systems were the main objectives of the action.

Clarifications on structure-preserving discretization with respect to the state of the art, the extensions of existing and the development of new discretization methods for open and/or controlled physical systems, are the scientific results of the action. They serve as a solid basis for a control design methodology in the spirit of EasyEBC. The successful application for a French-German research grant (DFG-ANR project INFIDHEM, 2017-2020), impulsed by the fellow’s and the supervisor’s labs and four partner groups, is an outcome of the project. It founds the follow-up of the results towards geometrical model order reduction and control with the elaborated models and guarantees the continuity of our collaboration beyond the MSCA action. Research, training, scientific collaboration and the integration into an excellent research network, allowed the fellow to strengthen his academic qualification and to significantly advance in his Habilitation project.
The scientific training covered the topics:
- Linear operators on Hilbert spaces, C0-semigroups, linear distributed-parameter PH systems and boundary control systems.
- Hyperbolic conservation laws and their numerical solution.
- Numerical methods for PDEs: Finite differences, finite volumes on staggered grids, mixed finite elements.
- Graph-based discrete modeling of conservation laws on chain complexes.
- PH models in process engineering.
- Geometric methods for time integration.

Attended courses:
- French as Foreign Language
- Applied Analysis of PDEs

Regular visits at partner labs:
- LCIS Valence, Prof. Laurent Lefèvre.
- ISAE SUPAERO Toulouse, Prof. Denis Matignon.
- ENSMM Besançon, Prof. Yann Le Gorrec.

The fellow had the pleasure to participate as a committee member in the PhD defenses of Dr. Yongxin Wu (Lyon) and Dr. Flavio Cardoso Ribeiro (Toulouse).

Projects prepared during the action to consolidate the French-German collaboration:
- INFIDHEM (Interconnected Infinite-dimensional systems for heterogeneous media). DFG-ANR cofounded project (2017-2020) with 6 partner groups both countries.
- Summer school on PH systems in 2017. Letter of intention to the Franco-German University (Dec 2015). Postponement (due to the prolongation of the fellow’s stay) to 2019 with the support of the INFIDHEM partners.
- Visiting researcher at LCIS Valence / Grenoble INP (Prof. Laurent Lefèvre) in spring/summer 2017 (2 months). In this frame, co-supervision of a Master’s project on geometric time integration of PH systems.

The results of EasyEBC are:
- Structure-preserving finite volume discretization for 1D PH systems on staggered grids, with increased accuracy by generalized leapfrog stencils for the numerical fluxes.
- Extension of discrete PH modeling of conservation laws on k-complexes (roughly: oriented meshes) by defining boundary ports with different causality on the system boundary.
- Finite volume numerical approximation for discrete PH models with nonlinear constitutive equations.
- Clarification of relations between structure-preserving and classical methods for the numerical treatment of PDEs. As an example, consistency and structure-preservation can be conflicting goals at the system boundary in finite volume methods.
- The INFIDHEM project, as a follow-up of EasyEBC, allows to apply the obtained results in order to tackle modeling, order reduction and control of complex processes like the benchmark problem of heat and mass transport in catalytic foams.

As for now, the results have been published in one journal paper and three conference proceedings. The fellow gave talks at two international conferences.

Outreach activities comprised the presentation of the MSCA actions during the Euraxess Roadshow at Lyon, two seminar talks in Lyon and Munich, a tutorial for the graduate school in Lyon and the webpage.
We developed methods for the representation and finite-dimensional approximation of open systems of conservation laws in arbitrary spatial dimension, which preserve the PH structure, and have validated numerical properties. The results of EasyEBC contribute to connect the discretization of PH systems with the state of the art in numerical methods. We expect that the mindset of numerical mathematics enables us to develop new approaches for order reduction and control design, which are more versatile and constructive than those initially intended in EasyEBC. The benchmark of heat and mass transfer in catalytic foams, which we treat in the follow-up project INFIDHEM stands as a prototypical example for complex, heterogeneous systems in process industry. The results of EasyEBC, in particular the discrete, graph-based system representation, form the well-understood basis to optimize the operation of such processes from a close-to-physics, model-based point of view.
The dual chain complex with its boundary for the second conservation law
A primal chain complex to represent integration domains for the first conservation law
Paul Kotyczka