Reduced CP Representation of Multilinear Models
Niklas J
¨
ores
1 a
, Christoph Kaufmann
1,2,4 b
, Leona Schnelle
2 c
, Carlos Cateriano Y
´
a
˜
nez
1,2,3 d
,
Georg Pangalos
1 e
and Gerwald Lichtenberg
2 f
1
Application Center for Integration of Local Energy Systems, Fraunhofer IWES, Hamburg, Germany
2
Faculty of Life Science, HAW Hamburg, Germany
3
Universitat Polit
`
ecnica de Val
`
encia, Instituto Universitario de Autom
´
atica e Inform
´
atica Industrial, Val
`
encia, Spain
4
Centre d’Innovaci
´
o Tecnol
`
ogica en Convertidors Est
`
atics i Accionaments (CITCEA),
Departament d’Enginyeria El
`
ectrica, Universitat Polit
`
ecnica de Catalunya (UPC), Barcelona, Spain
Keywords:
Multilinear Systems, Tensor Decomposition.
Abstract:
Large and highly complex systems can be found in various application areas. Modeling these systems requires
appropriate representation of the underlying phenomena. Furthermore, due to the large dimensions efficient
simulation and low memory requirements are needed for such models. Multilinear modeling is a promising
approach to address these challenges. In this paper, we introduce a reduced canonical polyadic (CP) repre-
sentation for implicit time-invariant multilinear (iMTI) models. This representation is capable of storing large
models with very low memory requirements. This is particularly useful for efficient analyses of large systems
with numerous inputs and states.
1 INTRODUCTION
Modeling and simulation of complex systems is an
active field of research. Currently, e.g., in the field
of modeling energy systems co-simulation method-
ology is used as one approach to address the high
complexity, while maintaining a realistic representa-
tion (L
´
opez et al., 2019; Farrokhseresht et al., 2021;
Wiens et al., 2021; Vogt et al., 2018). However, mod-
eling of such large and complex systems while cap-
turing the relevant dynamics results in large computa-
tional resources and simulation times with the exist-
ing modeling approaches. Therefore, more computa-
tional power, more efficient algorithms and new mod-
eling strategies are required (F. Milano et al., 2018).
Focusing on modeling strategies, a possibility is
to rethink the fundamental question: Which class of
models has the potential to cover all relevant non-
linear dynamics and at the same time enables effi-
a
https://orcid.org/0000-0003-2471-3892
b
https://orcid.org/0000-0002-0666-1104
c
https://orcid.org/0000-0002-2600-8110
d
https://orcid.org/0000-0001-5261-2568
e
https://orcid.org/0000-0001-5094-8033
f
https://orcid.org/0000-0001-6032-0733
cient simulations as well as analysis and design al-
gorithms? For some large scale complex application
domains with similar modeling problems, recent re-
search shows that multilinearity and tensor decom-
position methods could lead to breakthroughs (Ver-
straete et al., 2008).
In recent years the multilinear modeling frame-
work have been introduced first in an explicit form
by (Pangalos et al., 2013) and then in the more gen-
eral implicit form (Lichtenberg et al., 2022). The ad-
vantage of multilinear models is, that some nonlinear
phenomena can be modeled while still maintaining
an efficient and structured representation. Applica-
tion examples range from heating systems (Pangalos
et al., 2013) over chemical reactions (Kruppa et al.,
2014) to energy systems (Lichtenberg et al., 2022). In
addition, efficient simulation is possible, when using
decompositions. However, the multilinear model is
still an approximation and therefore, not as exact as
the nonlinear model. In addition, the tools for mul-
tilinear modeling are not yet standard and further de-
velopment is required.
Regarding controller synthesis, approaches to deal
with multilinear models in the application domain
of heating systems are given in (Pangalos, 2016;
Kruppa, 2018). The heating sector has the advantage
252
Jöres, N., Kaufmann, C., Schnelle, L., Yáñez, C., Pangalos, G. and Lichtenberg, G.
Reduced CP Representation of Multilinear Models.
DOI: 10.5220/0011273100003274
In Proceedings of the 12th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2022), pages 252-259
ISBN: 978-989-758-578-4; ISSN: 2184-2841
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c
2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved