Interconnection of port-hamiltonian systems via contact structures. An application to macro-economic systems

Authors

Abstract

This paper aims at extending a novel approach based on the port-Hamiltonian formalism and able to model macro-economic systems to include a more realistic formulation of the firm (or supplier) behaviour. The firm is a profit maximising entity, naturally described in terms of a contact structure, that interconnects the markets associated to the demand and to the input factors in order to create a profit. To determine the resulting dynamics, the paper shows how to interconnect the port-Hamiltonian systems associated to each market with contact structures, i.e. how to combine power conserving interconnecting structures, namely Dirac structures (related to the Walras’s Law of macro-economic), with contact structures, usually employed to describe irreversible phenomena. Beside the specific application to macro-economy, since a contact structure can be associated to a sort of “energy-based” maximisation/minimisation problem, it is also shown how to achieve such kind of interconnection via feedback control, i.e. how to shape a power-conserving interconnection to obtain a contact structure.

Citation

Journal: 53rd IEEE Conference on Decision and Control
Year: 2014
Volume:
Issue:
Pages: 6395–6400
Publisher: IEEE
DOI: 10.1109/cdc.2014.7040392

BibTeX

@inproceedings{Macchelli_2014,
  title={{Interconnection of port-hamiltonian systems via contact structures. An application to macro-economic systems}},
  DOI={10.1109/cdc.2014.7040392},
  booktitle={{53rd IEEE Conference on Decision and Control}},
  publisher={IEEE},
  author={Macchelli, Alessandro},
  year={2014},
  pages={6395--6400}
}

Download the bib file

References

Macchelli, A. Port-Hamiltonian formulation of simple macro-economic systems. 52nd IEEE Conference on Decision and Control 3888–3893 (2013) doi:10.1109/cdc.2013.6760483 – 10.1109/cdc.2013.6760483
lau, Production Economics A Dual Approach to Theory and Applications Volume 1 The Theory of Production Amsterdam North-Holland 1978 ch Applications of Profit Functions (0)
eberard, IFAC World Congress Proceeding of the 16th (2005)
Port contact systems for irreversible thermodynamical systems. Decision and Control and European Control Conference (CDC-ECC 2005) Proceedings of the 44th IEEE Conference on (2005)
Russell, T. Symplectic geometry: The natural geometry of economics? Economics Letters 112, 236–238 (2011) – 10.1016/j.econlet.2011.05.001
Favache, A., Dochain, D. & Maschke, B. An entropy-based formulation of irreversible processes based on contact structures. Chemical Engineering Science 65, 5204–5216 (2010) – 10.1016/j.ces.2010.06.019
gibbs, Method of geometric representation of the thermodynamic properties of substances by means of surfaces. Trans Connecticut Academy (1873)
Eberard, D., Maschke, B. M. & van der Schaft, A. J. An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics 60, 175–198 (2007) – 10.1016/s0034-4877(07)00024-9
Brayton, R. K. & Moser, J. K. A theory of nonlinear networks. I. Quart. Appl. Math. 22, 1–33 (1964) – 10.1090/qam/169746
Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM J. Control Optim. 37, 54–91 (1998) – 10.1137/s0363012996312039
paynter, Analysis and Design of Engineering Systems (1961)
sterman, Business Dynamics Systems Thinking and Modeling for a Complex World (2000)
Forrester, J. W. Counterintuitive behavior of social systems. Theor Decis 2, 109–140 (1971) – 10.1007/bf00148991
Brewer, J. W. & Craig, P. P. Bilinear, dynamic single-ports and bond graphs of economic systems. Journal of the Franklin Institute 313, 185–196 (1982) – 10.1016/0016-0032(82)90085-0
Structure and Cause and Effect Relations in Social System Simulations. IEEE Trans. Syst., Man, Cybern. 7, 468–474 (1977) – 10.1109/tsmc.1977.4309745
Duindam, V., Macchelli, A., Stramigioli, S. & Bruyninckx, H. Modeling and Control of Complex Physical Systems. (Springer Berlin Heidelberg, 2009). doi:10.1007/978-3-642-03196-0 – 10.1007/978-3-642-03196-0
maschke, Port controlled Hamiltonian systems: modeling origins and system theoretic properties. Nonlinear Control Systems (NOLCOS 1992) Proceedings of the 3rd IFAC Symposium on (1992)
Brewer, J. W. Progress in the bond graph representations of economics and population dynamics. Journal of the Franklin Institute 328, 675–696 (1991) – 10.1016/0016-0032(91)90048-8