Energy-Based Modeling of Complex Systems

The module introduces to a structured, network- and energy-based methodology for the modeling and coupling of multi-physical systems. Departing from the bond-graph formalism, a graphical language illustrating power flows and interconnections in complex systems, the port-Hamiltonian (PH) approach highlights the exchange, conversion, and storage of energy in control-oriented state space models. In its modularity, it is an ideal vehicle to set up complex system models. In the context of simulation and control design or implementation, preserving the physical structure in numerical approximations is of major interest. The course gives, based on the example of hyperbolic systems of conservation laws, insights in the structure-preserving discretization in space and time. Conditions to preserve the port-Hamiltonian structure in projective model order reduction are given. The attendance of the module prepares interested students for research internships and theses in the Energy-Based Modeling and Control group of the Chair of Automatic Control. The following topics are presented:

1. Bond graphs for the graphical description of multi-physics systems
2. Port-based modeling, Dirac structures, energy storage and other modules
3. PH systems and passivity
4. PH representation of spatially distributed systems in one dimension, beam models
5. Integration and calculus with differential forms
6. Conservation and balance laws in PH form on higher dimensional spatial domains
7. Structure-preserving spatial discretization of PH systems
8. Numerical time integration of (port-)Hamiltonian systems

Having attended the module, the students understand the concept of structured, port-based modeling, which is based on the separation of power structure, dynamics, and constitutive equations. The students are able to model systems with lumped energy storage elements under this paradigm and to derive the state representation in port-Hamiltonian (PH) form. Likewise, they are familiar with the graphical system description via bond graphs, and they can derive the PH representation therefrom, and vice-versa. They know power-preserving interconnections of passive systems and understand the concept of Dirac structures as a formalism to describe such interconnections. The students are familiar with the extension of the PH system representation to one-dimensional distributed parameter systems. They can compute variational derivatives of an energy functional, and they can reproduce the common beam models according to Euler-Bernoulli and Timoshenko in PH form, both in energy and deflection coordinates. The students know the most important rules for the calculus with differential forms. They can deal with them in the context of the presented examples of the wave and heat equation and understand the relations to vector calculus. They understand the variables and operations in the corresponding PH representation for systems of conservation laws on higher dimensional spatial domains. The students can explain the necessity of structure preservation in the context of numerical approximation. They can set up the weak form for the considered linear examples of distributed-parameter systems and explain how skew-symmetry of the system operator is achieved with integration by parts in a mixed finite element ansatz. They know examples for symplectic and energy-preserving integration schemes and can implement the corresponding integrators. By the closeness of the topics to current research, the students are enabled to understand and discuss current scientific publications in the field of PH systems.

The written exam (90 min) covers the topics of the module in the ratio of their treatment in the lecture. Based on theoretical questions, the students show their knowledge in the foundations of network- and port-based modeling of multi-physical systems, finite- and infinite dimensional port-Hamiltonian systems and their underlying geometric interconnection structures, as well as of aspects of structure-preserving discretization. Moreover, the students are expected to solve problems in the style of the exercises, in which they also show their proficiency with the used mathematical formalisms. Less than 20% of the achievable credits can be obtained by the solution of single choice questions according to the examination rules. Allowed material for the exam:

• 2 DIN A4 sheets of paper (“cheat sheets”)
• No calculators, computers, and other electronic devices.

Familiarity with state space modeling is recommended, as taught e.g. in "Systems Theory and Modeling". A basic understanding of partial differential equations and the finite element method is advantageous.

• The lecture notes are self-contained, with references to relevant books or papers. For a general introduction to port-Hamiltonian systems, the following books are recommended: [1] A. van der Schaft, D. Jeltsema: Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends in Systems and Control, vol. 1, no. 2-3, S. 173-378, 2014. http://www.math.rug.nl/arjan/DownloadVarious/PHbook.pdf [2] V. Duindam, A. Macchelli, S. Stramigioli & H. Bruyninckx (Hrsg.): Modeling and control of complex physical systems: the port-Hamiltonian approach.- Berlin: Springer Science & Business Media., 2009. – 423 S. ISBN 9783642031953. Available in the TUM library. [3] A. van der Schaft: L2-gain and passivity techniques in nonlinear control. 3´rd ed.- Cham: Springer, 2017.- XVIII, 321 S. - ISBN: 9783319499925. In the TUM library as E-Book under https://doi.org/10.1007/978-3-319-49992-5.

The module consists of a lecture and an exercise. In these, all presented methods are derived systematically and consecutively on the blackboard and illustrated by examples. Lecture notes are available for preparation and self-study, as well as supplementary material (download). Problems can be downloaded, and a part of the solutions will be presented in the exercise. Active participation of the students (questions, comments) is desired. Problems whose solution are not discussed in detail in the exercise are for self-study. Solutions to all problems are available for download. Thus, the students learn, among other things, to establish structured system models from different physical domains on their own, and they practice the necessary mathematical formalisms, like exterior calculus with differential forms. They furthermore learn in practical examples how to implement numerical schemes that preserve the desirable physical system properties. Lecture and exercise cover all topics relevant for the exam. Additionally, a revision exercise is offered on a voluntary basis. It is meant to be attended based on the individual needs and interests of the course participants. As a discussion group with only a small number of participants it serves a) to discuss and deepen the competences learned, and b) to support the preparation of the exam.