## Modeling and Reduction of Complex Systems

Module Number: MW0868

Duration: 1 Semester

Recurrence: Summer Semester

Language: English

Number of ECTS: 5

## Staff

Professor in charge: PD Dr.-Ing. habil Paul Kotyczka

## Amount of work

Class attendance: 45

Private study: 105

Total:150

## Description of achievement and assessment methods

The written exam (90 min) covers the two topics of the module in the corresponding ratio of approximately 2:1.

Based on theoretical questions, the students show their knowledge in the foundations A) of structured, port-Hamiltonian modeling of lumped and distributed parameter systems, energy-based control and B) model order reduction of high dimensional 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, to design energy-based controllers and to applying 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 handwritten (DIN A4, double-sided) sheets of paper (“cheat sheets”) - No calculators, computers and other electronic devices

## (Recommended) requirements

For the module, knowledge of linear state space methods is required, as taught e.g. in courses of linear systems' theory or "Advanced Control". Moreover, basics of linear algebra (vector spaces, linear systems of equations, etc.) are expected

## Contents

The module deals with modern methods for structured modeling, energy-based control and order reduction of highdimensional 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 systems and energy-based control

1. Port variables and interconnection: Finite-dimensional Dirac structures

2. Dynamics and constitutive equations: The port-Hamiltonian (PH) system representation

3. Structural invariants and energy shaping: Control by Interconnection (CbI) Module Description

4. PH model as target systems of state feedback control: IDA-PBC

5. Integration over density functions: Differential forms

6. Hyperbolic systems in PH representation: Wave and shallow water equations

7. Parabolic systems: Heat equation in structured representation

8. Beam models in PH representation

B) Model reduction

1. Introduction

2. Mathematic foundations from linear algebra

3. Projection-based model order reduction

4. Balancing and Truncation

5. Krylov subspace methods

## Study Goals

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 PH form. They know the basic mechanisms of energy-based control and are able to apply them, after having checked the conditions for their applicability. 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. The students are familiar with the variational derivative of an energy functional and can reproduce the common beam models according to Euler-Lagrange and Timoshenko in PH form. The students understand two approaches for model order 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.

## Teaching and learning methods

In the module 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. Lecture and the tutorial cover all topics relevant for the exam. Additionally, a revision course 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 lecture topics and tutorial problems, and

b) to support the preparation of the exam

## Media formats

Lectures with mainly blackboard notes, in parts presentation with beamer. Lecture and exersises are complemented by some Matlab and python examples. The lecture notes and supplementary material/slides as well as exercises with solutions are available for download under www.moodle.tum.de.

## Literature

Port-Hamiltonian systems and energy-based control

[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. Springer Science & Business Media.

[3] A. van der Schaft: L2-gain and passivity techniques in nonlinear control. Springer, 2017.

Model order reduction

[5] A. C. Antoulas. Approximation of Large Scale Dynamical Systems. Society for Industrial and Applied Mathematics, 2006.