_{System-theoretic Analysis and Controller Design for PDEs - A formal Approach based on Differential Geometry}

_{FWF Project P29964: 05/2017-04/2021}

**Principal Investigator (PI):** Markus Schöberl

**Collaborators: **Bernd Kolar, Tobias Malzer

**Publications:**

B. Kolar, H. Rams, M. Schöberl: Application of Symmetry Groups to the Observability Analysis of Partial Differential Equations,* Proceedings of the 23rd International Symposium on Mathematical Theory of Networks and Systems (MTNS), *pp. 247-254, 2018, arXiv

T. Malzer, H. Rams, M. Schöberl: Energy-Based Control of Nonlinear Infinite-Dimensional Port-Hamiltonian Systems with Dissipation, *Proceedings of the 57th IEEE Conference on Decision and Control (CDC), *pp 3746-3751 2018, arXiv

B. Kolar, M. Schöberl: Symmetry Groups and the Observability of PDEs, *Proceedings Applied Mathematics and Mechanics (PAMM)* Vol. 18, Issue 1, 2018, pamm

T. Malzer, H. Rams, M. Schöberl: Energy-Based In-Domain Control of a Piezo-Actuated Euler-Bernoulli Beam, *3rd IFAC/IEEE CSS Workshop on Control of Systems Governed by Partial Differential Equations and XI Workshop Control of Distributed Parameter Systems,* pp. 147-152, 2019, arxiv

B. Kolar, M. Schöberl: System-theoretic Analysis of Nonlinear Infinite-dimensional Systems with Generalized Symmetries, *11th IFAC Symposium on Nonlinear Control Systems*, pp. 438-439, 2019, arxiv

**Publications (submitted):**

T. Malzer, H. Rams, M. Schöberl: On Structural Invariants in the Energy-Based In-Domain Control of Infinite-Dimensional Port-Hamiltonian Systems, 2019, preprint

**Related Publications:**

M. Schöberl, K. Schlacher: On the extraction of the boundary conditions and the boundary ports in second-order field theories, *Journal of Mathematical Physics* 59(10), 2018, JMP, preprint

B. Kolar, N. Gehring, M. Schöberl: On the Calculation of Differential Parametrizations for the Feedforward Control of an Euler-Bernoulli Beam, 2018, preprint

B. Kolar, M. Schöberl, J. Diwold: Differential-Geometric Decomposition of Flat Nonlinear Discrete-Time Systems, 2019, preprint

B. Kolar, J. Diwold, M. Schöberl: Necessary and Sufficient Conditions for Difference Flatness, 2019, preprint

**Talks:**

Anwendung von Symmetriegruppen zur Beobachtbarkeitsanalyse von PDEs, Feb. 2nd. 2018, *SVP Kolloquium*, Hall in Tirol, Austria (Bernd Kolar)

Symmetry Groups and the Observability of PDEs, March 22nd. 2018, *GAMM Annual Meeting*, Munich, Germany (Bernd Kolar)

Application of Symmetry Groups to the Observability Analysis of Partial Differential Equations,* *July 17th 2018,* 23rd International Symposium on Mathematical Theory of Networks and Systems, *Hong Kong (Bernd Kolar)

Energiebasierte Regelung von nichtlinearen verteilt-parametrischen Hamiltonschen Systemen mit Dissipation, Sept. 18th, 2018, *GMA 1.40 Fachausschuss*, Anif, Austria (Tobias Malzer)

Energy-Based Control of Nonlinear Infinite-Dimensional Port-Hamiltonian Systems with Dissipation, Dec. 18th, 2018,* 57th IEEE Conference on Decision and Control (CDC),* Miami, Florida (Tobias Malzer)

Energy-Based In-Domain Control of a Piezo-Actuated Euler-Bernoulli Beam, May 22nd, 2019,* 3rd IFAC/IEEE CSS Workshop on Control of Systems Governed by Partial Differential Equations and XI Workshop Control of Distributed Parameter Systems,* Oaxaca, Mexico (Tobias Malzer)

System-theoretic Analysis of Nonlinear Infinite-dimensional Systems with Generalized Symmetries, Sept. 04th, 2019, *11th IFAC Symposium on Nonlinear Control Systems*, *Vienna, Austria* (Bernd Kolar)

Differentialgeometrische Zerlegung von flachen nichtlinearen zeitdiskreten Systemen, Sept. 24th, 2019, *GMA 1.40 Fachausschuss, Anif, Austria* (Bernd Kolar, in cooperation with FWF-P32151)

Differential Parametrizations for the Feedforward Control of an Euler-Bernoulli Beam, Sept. 25th, 2019, *4th International Workshop on Advanced Dynamics and Model Based Control of Structures and Machines*, Linz, Austria (Markus Schöberl)

**Executive Summary:**

In this project we will apply the formal theory of partial differential equations (PDEs) that is based on jet-bundles for the system and control theoretic analysis of infinite-dimensional dynamical systems. We will identify dynamical systems described by PDEs as geometric objects in order to analyze system and control theoretic properties on a structural level, and furthermore a main goal is to design energy based control laws. Primarily, differential geometric methods shall be employed in this formal setting but we are also interested to consider methods from several other mathematical disciplines including functional analysis and homological algebra to complement our geometric theory. Based on this proposed mathematical framework, two main tasks shall be addressed.

Firstly, in the case of ordinary differential equations (ODEs) it is well known that several system properties (observability, controllability, ...) are connected to the existence of appropriate normal-forms, whereas in the PDE scenario no comparable general results are available. This raises the question under which conditions and for which system classes an analogous structural analysis is also possible for PDEs based on formal, geometric tools. In this context the concept of transformation groups will play an important role for the geometric analysis of structural system properties. In a functional analytic setting, system features are checked by properties of certain maps associated with a dynamical system or by proving the existence of certain a-priori inequalities. It is our intention to bring these pictures together, as for example it can be expected that criteria derived based on transformation groups can be linked with these inequalities, and we expect that a symbiosis of these techniques should be very promising.

Secondly, Lagrangian and Hamiltonian formulations that have been beneficially used in the ODE case for the system analysis and the controller design shall be further studied from a geometric point of view in the PDE scenario. The main intention of a port-Hamiltonian approach is to link the differential equations to a power balance relation together with possible energy/power ports, where in the PDE case an accurate handling of the non-trivial boundary conditions (e.g. used for boundary control) is crucial and challenging. The aim is to enhance existing port-Hamiltonian formulations in order to capture higher-order field theories in multi-physics applications, and to design energy based controllers, on one hand by using interconnection techniques, and on the other hand by using classical energy based feedback.