Large-scale physical network systems arise in many application areas; from systems biology to power networks. In this talk we concentrate on the simplest, but paradigmatic, example of such systems: mass-spring-damper networks. The aim is to obtain reduced-order models which are still in the same class of systems. We discuss a method which is based on clustering. From a graph-theoretic perspective the novelty lies in the introduction of weights associated to the vertices of the graph; next to the weights associated to the edges. We extend the notion of equitable partitions to this class of graphs. Consequently, an explicit model reduction error expression in the sense of H2-norm is provided for clustering arising from such partitions (joint work with Nima Monshizadeh). We will discuss how other physical examples like hydraulic networks and single-species substrate/product chemical reaction networks, as well as consensus dynamics, fall into the same class. Finally, we mention the relationship with structure-preserving model reduction methods for power networks, and point towards relations with Kron reduction.
Arjan van der Schaft received the undergraduate (cum laude) and Ph.D. degrees in Mathematics from the University of Groningen, The Netherlands, under the guidance of Jan C. Willems. In 1982 he joined the Department of Applied Mathematics, University of Twente, Enschede, where he was appointed as a full professor in Mathematical Systems and Control Theory in 2000. In September 2005 he returned to his Alma Mater as a full professor in Mathematics. Arjan van der Schaft is Fellow of the Institute of Electrical and Electronics Engineers (IEEE). He was Invited Speaker at the International Congress of Mathematicians, Madrid, 2006, and delivered an invited plenary lecture at the 84th Annual Meeting of the Gesellschaft fur Angewandte Mathematik und Mechanik (GAMM), 2013. He was the 2013 recipient of the 3-yearly awarded Certificate of Excellent Achievements of the IFAC Technical Committee on Nonlinear Systems. He is (co-)author of the following books: System Theoretic Descriptions of Physical Systems (1984), Variational and Hamiltonian Control Systems (1987, with P.E. Crouch), Nonlinear Dynamical Control Systems (1990, with H. Nijmeijer), L_2-Gain and Passivity Techniques in Nonlinear Control (2000), An Introduction to Hybrid Dynamical Systems (2000, with J.M. Schumacher), Port-Hamiltonian Systems Theory: An Introductory Overview (2014, with D. Jeltsema). His research interests include nonlinear systems and control theory, systems modeling and analysis of multi-domain physical systems, network dynamics, geometric mechanics and hybrid systems. Present focus is on the network modeling and control of large-scale physical systems and their geometric formulation as port-Hamiltonian systems, and applications towards power systems, systems biology, and cyber-physical systems.