## About this course

The module deals with modern methods for structured modeling and order reduction of high-dimensional system models. The topics are important areas of current research, among others at the Chair of Automatic Control. The presented methods allow for a control-oriented approach to complex systems and design problems: The port-Hamiltonian approach is based on structured modeling and puts an emphasis on the power flows. It is very appropriate for the representation of coupled multi-physics systems. Order reduction is necessary to cope efficiently with very high-dimensional models in simulation and computational control. High-order models result for example from the spatial discretization of multi-physics distributed parameter systems.

The attendance of the module prepares interested students for research internships and theses in the corresponding research areas of the Chair of Automatic Control.

The following topics are presented:

A) Port-Hamiltonian (PH) systems 1. Bond graphs for the graphical description of multiphysics 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

B) Model reduction 1. Introduction and fundamentals from linear algebra 2. Projection-based model order reduction 3. Modal Reduction 4. Balanced Truncation 5. Krylov subspace methods

## Expected learning outcomes

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-Lagrange 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 understand three approaches for model order reduction (Modal Reduction, Balanced Truncation and Krylov subspace methods). They can evaluate their applicability to given technical examples. They are able to apply the corresponding mathematical tools (e.g. projections and singular value decomposition) to problems in model order reduction.

By the closeness of the topics to current research, the students are enabled to understand and discuss current scientific publications in the fields of PH systems and model order reduction.

## Examination

The written exam (90 min) covers the two topics of the module in the corresponding ratio of approximately 70:30. Based on theoretical questions, the students show their knowledge in the foundations A) of structured, port-Hamiltonian modeling of lumped and distributed parameter systems and B) model order reduction of high dimensional linear systems. Moreover, the students are expected to solve problems in the style of the exercises. They show that they are able to establish structured system models from different physical domains, to master the necessary mathematical formalism, like exterior calculus with differential forms, and to apply methods of projection-based model order reduction. Up to 20% of the achievable credits can be obtained by the solution of “multiple choice” questions according to the examination rules.

Allowed material for the exam:

- 2 HAND WRITTEN "cheat sheets" in the format DIN A4 (double-sided). Printouts are NOT allowed.
- Calculators, computers and other electronic devices are NOT allowed.

## Course requirements

For the module, knowledge of linear state space methods is required, as taught e.g. in modules like "Systems Theory and Modeling" (or "Systemtheorie in der Mechatronik", in German) or "Advanced Control" (or "Moderne Methoden der Regelungstechnik 1"). Moreover, basics of linear algebra (vector spaces, linear systems of equations, etc.) are expected.

## Activities

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 formalism, like exterior calculus with differential forms. They furthermore learn how to apply methods of projection-based model order reduction.

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.

- Local course code
**MW0868** - Study load
**ECTS 5** - Level
**master** - Contact hours per week
**2** - Instructors
**Paul Kotyczka** - Mode of delivery
**Hybrid** - Course coordinator

### Start date

15 April 2024

- End date
**19 July 2024** - Main language
**English** - Apply between
**20 Oct and 24 Nov 2023** - Time info
**[unknown]**

Enrolment period closed