Authors

Pablo Borja, Jacquelien M. A. Scherpen, Kenji Fujimoto

Abstract

In this article, we treat extended balancing for continuous-time linear time-invariant systems. We take a dissipativity perspective, thus, resulting in a characterization in terms of linear matrix inequalities. This perspective is useful for determining a priori error bounds. In addition, we address the problem of structure-preserving model reduction of the subclass of port-Hamiltonian systems. We establish sufficient conditions to ensure that the reduced-order model preserves a port-Hamiltonian structure. Moreover, we show that the use of extended Gramians can be exploited to get a small error bound and, possibly, to preserve a physical interpretation for the reduced-order model. We illustrate the results with a large-scale mechanical system example. Furthermore, we show how to interpret a reduced-order model of an electrical circuit again as a lower dimensional electrical circuit.

Citation

  • Journal: IEEE Transactions on Automatic Control
  • Year: 2023
  • Volume: 68
  • Issue: 1
  • Pages: 257–271
  • Publisher: Institute of Electrical and Electronics Engineers (IEEE)
  • DOI: 10.1109/tac.2021.3138645

BibTeX

@article{Borja_2023,
  title={{Extended Balancing of Continuous LTI Systems: A Structure-Preserving Approach}},
  volume={68},
  ISSN={2334-3303},
  DOI={10.1109/tac.2021.3138645},
  number={1},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Borja, Pablo and Scherpen, Jacquelien M. A. and Fujimoto, Kenji},
  year={2023},
  pages={257--271}
}

Download the bib file

References

  • Antoulas, A. C. Approximation of Large-Scale Dynamical Systems. (2005) doi:10.1137/1.9780898718713 – 10.1137/1.9780898718713
  • Borja, Data of the extended balanced truncation examples.
  • Caughey, T. K. Classical Normal Modes in Damped Linear Dynamic Systems. Journal of Applied Mechanics vol. 27 269–271 (1960) – 10.1115/1.3643949
  • Cheng, X., Scherpen, J. M. A. & Besselink, B. Balanced truncation of networked linear passive systems. Automatica vol. 104 17–25 (2019) – 10.1016/j.automatica.2019.02.045
  • de Oliveira, M. C., Bernussou, J. & Geromel, J. C. A new discrete-time robust stability condition. Systems & Control Letters vol. 37 261–265 (1999) – 10.1016/s0167-6911(99)00035-3
  • De Oliveira, M. C., Geromel, J. C. & Bernussou, J. Extended H 2 and H norm characterizations and controller parametrizations for discrete-time systems. International Journal of Control vol. 75 666–679 (2002) – 10.1080/00207170210140212
  • 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
  • Dullerud, G. E. & Paganini, F. A Course in Robust Control Theory. Texts in Applied Mathematics (Springer New York, 2000). doi:10.1007/978-1-4757-3290-0 – 10.1007/978-1-4757-3290-0
  • FUJIMOTO, K. Balanced Realization and Model Order Reduction for Port-Hamiltonian Systems. Journal of System Design and Dynamics vol. 2 694–702 (2008) – 10.1299/jsdd.2.694
  • Fujimoto, K. & Scherpen, J. M. A. Balanced Realization and Model Order Reduction for Nonlinear Systems Based on Singular Value Analysis. SIAM Journal on Control and Optimization vol. 48 4591–4623 (2010) – 10.1137/070695332
  • GLOVER, K. All optimal Hankel-norm approximations of linear multivariable systems and theirL,∞-error bounds†. International Journal of Control vol. 39 1115–1193 (1984) – 10.1080/00207178408933239
  • Hinrichsen, D. & Pritchard, A. J. An improved error estimate for reduced-order models of discrete-time systems. IEEE Transactions on Automatic Control vol. 35 317–320 (1990) – 10.1109/9.50345
  • Horn, R. A. & Johnson, C. R. Matrix Analysis. (1985) doi:10.1017/cbo9780511810817 – 10.1017/cbo9780511810817
  • Kawano, Y. & Scherpen, J. M. A. Structure Preserving Truncation of Nonlinear Port Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 63 4286–4293 (2018) – 10.1109/tac.2018.2811787
  • Kotsalis, G., Megretski, A. & Dahleh, M. A. Balanced Truncation for a Class of Stochastic Jump Linear Systems and Model Reduction for Hidden Markov Models. IEEE Transactions on Automatic Control vol. 53 2543–2557 (2008) – 10.1109/tac.2008.2006931
  • Kotsalis, G. & Rantzer, A. Balanced Truncation for Discrete Time Markov Jump Linear Systems. IEEE Transactions on Automatic Control vol. 55 2606–2611 (2010) – 10.1109/tac.2010.2060241
  • Moore, B. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control vol. 26 17–32 (1981) – 10.1109/tac.1981.1102568
  • Novikov, M. A. Simultaneous diagonalization of three real symmetric matrices. Russian Mathematics vol. 58 59–69 (2014) – 10.3103/s1066369x1412007x
  • Polyuga, Structure preserving model reduction of port-Hamiltonian systems. Proc. 18th Int. Symp. Math. Theory Netw. Syst. (2008)
  • Sandberg, H. Model reduction of linear systems using extended balanced truncation. 2008 American Control Conference 4654–4659 (2008) doi:10.1109/acc.2008.4587229 – 10.1109/acc.2008.4587229
  • Sandberg, H. An Extension to Balanced Truncation With Application to Structured Model Reduction. IEEE Transactions on Automatic Control vol. 55 1038–1043 (2010) – 10.1109/tac.2010.2041984
  • Scherpen, The Control Handbook: Control System Advanced Methods, Chapter Balanced Realizations, Model Order Reduction, and the Hankel Operator (2011)
  • Scherpen, J. M. A. & Fujimoto, K. Extended balanced truncation for continuous time LTI systems. 2018 European Control Conference (ECC) 2611–2615 (2018) doi:10.23919/ecc.2018.8550152 – 10.23919/ecc.2018.8550152
  • van der Schaft, A. L2-Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer International Publishing, 2017). doi:10.1007/978-3-319-49992-5 – 10.1007/978-3-319-49992-5
  • Willems, Model reduction by balancing.
  • Zhou, Robust and Optimal Control (1996)