Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos
Authors
Rostyslav V. Polyuga, Arjan van der Schaft
Abstract
Structure preserving model reduction of single-input single-output port-Hamiltonian systems is considered by employing the rational Krylov methods. The rational Arnoldi method is shown to preserve (for the reduced order model) not only a specific number of the moments at an arbitrary point in the complex plane but also the port-Hamiltonian structure. Furthermore, it is shown how the rational Lanczos method applied to a subclass of port-Hamiltonian systems, characterized by an algebraic condition, preserves the port-Hamiltonian structure. In fact, for the same subclass of port-Hamiltonian systems the rational Arnoldi method and the rational Lanczos method turn out to be equivalent in the sense of producing reduced order port-Hamiltonian models with the same transfer function.
Citation
- Journal: IEEE Transactions on Automatic Control
- Year: 2011
- Volume: 56
- Issue: 6
- Pages: 1458–1462
- Publisher: Institute of Electrical and Electronics Engineers (IEEE)
- DOI: 10.1109/tac.2011.2128650
BibTeX
@article{Polyuga_2011,
title={{Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos}},
volume={56},
ISSN={0018-9286},
DOI={10.1109/tac.2011.2128650},
number={6},
journal={IEEE Transactions on Automatic Control},
publisher={Institute of Electrical and Electronics Engineers (IEEE)},
author={Polyuga, Rostyslav V. and van der Schaft, Arjan},
year={2011},
pages={1458--1462}
}References
- Polyuga, R. V. & van der Schaft, A. Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity. Automatica vol. 46 665–672 (2010) – 10.1016/j.automatica.2010.01.018
- salimbahrami, Krylov subspace methods in linear model order reduction Introduction and invariance properties (2002)
- Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry (Springer Berlin Heidelberg, 2008). doi:10.1007/978-3-540-78841-6 – 10.1007/978-3-540-78841-6
- van der schaft, Port-controlled Hamiltonian systems: Towards a theory for control and design of nonlinear physical systems. J Soc Instrum Control Eng Jpn (SICE) (2000)
- Schaft, A. L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences (Springer Berlin Heidelberg, 1996). doi:10.1007/3-540-76074-1 – 10.1007/3-540-76074-1
- van der schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. Archiv fr Elektronik und bertragungstechnik (1995)
- van der Schaft, A. J. & Polyuga, R. V. Structure-preserving model reduction of complex physical systems. Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference 4322–4327 (2009) doi:10.1109/cdc.2009.5399669 – 10.1109/cdc.2009.5399669
- Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
- Wolf, T., Lohmann, B., Eid, R. & Kotyczka, P. Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspaces. European Journal of Control vol. 16 401–406 (2010) – 10.3166/ejc.16.401-406
- grimme, Krylov projection methods for model reduction (1997)
- golub, Matrix Computations (1996)
- gugercin, Interpolation-based
${\cal H}_{2}$ - grimme, ENUMATH 97 (1998)
- Polyuga, R. V. Discussion on: “Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspaces”. European Journal of Control vol. 16 407–409 (2010) – 10.1016/s0947-3580(10)70672-5
- Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
- Modeling and Control of Complex Physical Systems The Port-Hamiltonian Approach (2009)
- Antoulas, A. C. Approximation of Large-Scale Dynamical Systems. (2005) doi:10.1137/1.9780898718713 – 10.1137/1.9780898718713
- polyuga, Model Reduction of Port-Hamiltonian Systems (2010)