# Tensor Systems - Multilinear Modeling and Applications

### G. Pangalos, A. Eichler, G. Lichtenberg

#### Abstract

Tensor systems are a framework for modeling of multilinear hybrid systems with discrete and continuous valued signals. Two examples from building services engineering and multi-agent systems show applications of this framework. A tensor model of a heating system is derived and approximated by tensor decomposition methods first. Second, a tensor model of a multi-agent system with a structure already given in a decomposed form is reduced further by the same decomposition methods. The validity of these low rank approximations is shown for both examples.

#### References

- Bader, B. and Kolda, T. (2012). MATLAB Tensor Toolbox Version 2.5. Available online.
- Balaji, P. and Srinivasan, D. (2010). An Introduction to Multi-Agent Systems, volume 310 of Studies in Computational Intelligence, chapter 1, pages 1-27. Springer.
- Cichocki, A., Zdunek, R., Phan, A., and Amari, S. (2009). Nonnegative Matrix and Tensor Factorizations. Wiley, Chichester.
- Hackbusch, W. (2012). Tensor Spaces and Numerical Tensor Calculus, volume 42 of Springer Series in Computational Mathematics. Springer-Verlag Berlin Heidelberg.
- Kolda, T. and Bader, B. (2009). Tensor Decompositions and Applications. SIAM Review, 51(3):455-500.
- Lichtenberg, G. (2010). Tensor Representation of Boolean Functions and Zhegalkin Polynomials. In International Workshop on Tensor Decompositions.
- Lichtenberg, G. (2011). Hybrid Tensor Systems. Habilitation, Hamburg University of Technology.
- Lichtenberg, G. and Eichler, A. (2011). Multilinear Algebraic Boolean Modelling with Tensor Decompositions Techniques. In 18th IFAC World Congress, page TuC01.2.
- Mesbahi, M. and Egerstedt, M. (2010). Graph Theoretic Methods for Multiagent Networks. Princeton University Press.
- Nouidui, T.S., Zuo, K.P.W., and Wetter, M. (2012). Validation and Application of the Room Model of the Modelica Buildings Library. 9th International Modelica Conference, pages 727-736.
- Pangalos, G. and Lichtenberg, G. (2012). Approach to Boolean Controller Design by Algebraic Relaxation for Heating Systems. In 4th IFAC Conference on Analysis and Design of Hybrid Systems.
- Ren, W. and Beard, R. (2007). Distributed Consensus in Multi-Vehicle Cooperative Control - Theory and Applications. Communications and Control Engineering. Springer Publishing Company, Incorporated.
- Ren, W. and Cao, Y. (2011). Distributed Coordination of Multi-agent Networks. Springer-Verlag London Limited.
- Srinivasan, D. and Choy, M. (2010). Hybrid Multi-Agent Systems, volume 310 of Studies in Computational Intelligence, chapter 2, pages 29-42. Springer.
- Wetter, M. (2006). Multizone Building Model for Thermal Building Simulation in Modelica. 5th International Modelica Conference, pages 517-526.

#### Paper Citation

#### in Harvard Style

Pangalos G., Eichler A. and Lichtenberg G. (2013). **Tensor Systems - Multilinear Modeling and Applications** . In *Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,* ISBN 978-989-8565-69-3, pages 275-285. DOI: 10.5220/0004475602750285

#### in Bibtex Style

@conference{simultech13,

author={G. Pangalos and A. Eichler and G. Lichtenberg},

title={Tensor Systems - Multilinear Modeling and Applications},

booktitle={Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,},

year={2013},

pages={275-285},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004475602750285},

isbn={978-989-8565-69-3},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,

TI - Tensor Systems - Multilinear Modeling and Applications

SN - 978-989-8565-69-3

AU - Pangalos G.

AU - Eichler A.

AU - Lichtenberg G.

PY - 2013

SP - 275

EP - 285

DO - 10.5220/0004475602750285