H∞ model reduction for port-controlled Hamiltonian systems

Authors

Dongbing Tong, Wuneng Zhou, Yan Gao, Chuan Ji, Hongye Su

Abstract

This paper is concerned with the problem of H ∞ model reduction for the linear port-controlled Hamiltonian systems. The development includes both the continuous- and discrete-time cases. Some sufficient conditions are obtained for the existence of solutions in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint set. In addition, an explicit parametrization of the desired reduced-order model can be constructed if these conditions are satisfied. Furthermore, the conditions based on the strict LMIs without rank constraint are derived for the zeroth-order H ∞ approximation problem. Finally, the effectiveness of the proposed model reduction method is illustrated via a practical example.

Keywords

Model reduction; Port-controlled Hamiltonian system; H ∞ performance; Continuous-time; Discrete-time

Citation

• Journal: Applied Mathematical Modelling
• Year: 2013
• Volume: 37
• Issue: 5
• Pages: 2727–2736
• Publisher: Elsevier BV
• DOI: 10.1016/j.apm.2012.06.031

BibTeX

@article{Tong_2013,
  title={{<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> model reduction for port-controlled Hamiltonian systems}},
  volume={37},
  ISSN={0307-904X},
  DOI={10.1016/j.apm.2012.06.031},
  number={5},
  journal={Applied Mathematical Modelling},
  publisher={Elsevier BV},
  author={Tong, Dongbing and Zhou, Wuneng and Gao, Yan and Ji, Chuan and Su, Hongye},
  year={2013},
  pages={2727--2736}
}

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References

• van der Schaft, (2000)
• Polyuga, R. V. & van der Schaft, A. Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity. Automatica 46, 665–672 (2010) – 10.1016/j.automatica.2010.01.018
• Polyuga, R. V. & van der Schaft, A. J. Structure Preserving Port-Hamiltonian Model Reduction of Electrical Circuits. Lecture Notes in Electrical Engineering 241–260 (2011) doi:10.1007/978-94-007-0089-5_14 – 10.1007/978-94-007-0089-5_14
• Ionutiu, R., Rommes, J. & Antoulas, A. C. Passivity-Preserving Model Reduction Using Dominant Spectral-Zero Interpolation. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 27, 2250–2263 (2008) – 10.1109/tcad.2008.2006160
• Moore, B. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Trans. Automat. Contr. 26, 17–32 (1981) – 10.1109/tac.1981.1102568
• Duindam, (2009)
• Zhou, W., Tong, D., Lu, H., Zhong, Q. & Fang, J. Time-Delay Dependent H ∞ Model Reduction for Uncertain Stochastic Systems: Continuous-Time Case. Circuits Syst Signal Process 30, 941–961 (2010) – 10.1007/s00034-010-9245-x
• Lu, H., Zhou, W., Xu, Y., Fang, J. & Tong, D. Time‐delay dependentH∞model simplification for singular systems with Markovian jumping parameters. Optim Control Appl Methods 32, 379–395 (2011) – 10.1002/oca.943
• Xu, S. & Lam, J. H∞ model reduction for discrete-time singular systems. Systems & Control Letters 48, 121–133 (2003) – 10.1016/s0167-6911(02)00279-7
• Gao, H., Lam, J. & Wang, C. Model simplification for switched hybrid systems. Systems & Control Letters 55, 1015–1021 (2006) – 10.1016/j.sysconle.2006.06.014
• Zhang, L., Huang, B. & Lam, J. H∞ model reduction of Markovian jump linear systems. Systems & Control Letters 50, 103–118 (2003) – 10.1016/s0167-6911(03)00133-6
• Zhang, L. & Lam, J. On H2 model reduction of bilinear systems. Automatica 38, 205–216 (2002) – 10.1016/s0005-1098(01)00204-7
• Alsmadi, O. M. K., Abo-Hammour, Zaer. S. & Al-Smadi, A. M. Artificial neural network for discrete model order reduction with substructure preservation. Applied Mathematical Modelling 35, 4620–4629 (2011) – 10.1016/j.apm.2011.03.028
• Gahinet, P. & Apkarian, P. A linear matrix inequality approach toH∞control. Intl J Robust & Nonlinear 4, 421–448 (1994) – 10.1002/rnc.4590040403
• Iwasaki, All controllers for the ganeral H∞ control problem: LMI existence conditions and state space formulas. Automatica (1994)
• de Souza, C. E. & Xie, L. On the discrete-time bounded real Lemma with application in the characterization of static state feedback controllers. Systems & Control Letters 18, 61–71 (1992) – 10.1016/0167-6911(92)90108-5
• Wang, Y., Xie, L. & de Souza, C. E. Robust control of a class of uncertain nonlinear systems. Systems & Control Letters 19, 139–149 (1992) – 10.1016/0167-6911(92)90097-c