Structure-preserving discretization and control of a two-dimensional vibro-acoustic tube

Authors

Ning Liu, Yongxin Wu, Yann Le Gorrec, Hector Ramirez, Laurent Lefèvre

Abstract

This paper deals with the structure-preserving discretization and control of a two-dimensional vibro-acoustic tube using the port-Hamiltonian framework. A discretization scheme is proposed, and a set of precise basis functions are given in order to obtain a structure-preserving finite-dimensional port- Hamiltonian approximation of the two-dimensional vibro-acoustic system. Using the closed-loop structural invariants of the approximated system an energy-Casimir controller is derived. The performance of the proposed discretization scheme and the controller is shown by means of numerical simulations.

Citation

• Journal: IMA Journal of Mathematical Control and Information
• Year: 2021
• Volume: 38
• Issue: 2
• Pages: 417–439
• Publisher: Oxford University Press (OUP)
• DOI: 10.1093/imamci/dnaa028

BibTeX

@article{Liu_2020,
  title={{Structure-preserving discretization and control of a two-dimensional vibro-acoustic tube}},
  volume={38},
  ISSN={1471-6887},
  DOI={10.1093/imamci/dnaa028},
  number={2},
  journal={IMA Journal of Mathematical Control and Information},
  publisher={Oxford University Press (OUP)},
  author={Liu, Ning and Wu, Yongxin and Le Gorrec, Yann and Ramirez, Hector and Lefèvre, Laurent},
  year={2020},
  pages={417--439}
}

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References

• Baaiu, A., Couenne, F., Lefevre, L., Le Gorrec, Y. & Tayakout, M. Structure-preserving infinite dimensional model reduction: Application to adsorption processes. Journal of Process Control vol. 19 394–404 (2009) – 10.1016/j.jprocont.2008.07.002
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. Applied Mathematical Modelling vol. 75 940–960 (2019) – 10.1016/j.apm.2019.04.035
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates. Applied Mathematical Modelling vol. 75 961–981 (2019) – 10.1016/j.apm.2019.04.036
• Cardoso-Ribeiro, F. L., Matignon, D. & Lefèvre, L. A structure-preserving Partitioned Finite Element Method for the 2D wave equation. IFAC-PapersOnLine vol. 51 119–124 (2018) – 10.1016/j.ifacol.2018.06.033
• David, P., Collet, M. & Cote, J.-M. Experimental implementation of acoustic impedance control by a 2D network of distributed smart cells. Smart Materials and Structures vol. 19 035028 (2010) – 10.1088/0964-1726/19/3/035028
• 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
• Durand, J.-F., Soize, C. & Gagliardini, L. Structural-acoustic modeling of automotive vehicles in presence of uncertainties and experimental identification and validation. The Journal of the Acoustical Society of America vol. 124 1513–1525 (2008) – 10.1121/1.2953316
• Gardonio, P. Review of Active Techniques for Aerospace Vibro-Acoustic Control. Journal of Aircraft vol. 39 206–214 (2002) – 10.2514/2.2934
• Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017
• Kotyczka, P. Finite Volume Structure-Preserving Discretization of 1D Distributed-Parameter Port-Hamiltonian Systems. IFAC-PapersOnLine vol. 49 298–303 (2016) – 10.1016/j.ifacol.2016.07.457
• Kotyczka, P. & Maschke, B. Discrete port-Hamiltonian formulation and numerical approximation for systems of two conservation laws. at - Automatisierungstechnik vol. 65 308–322 (2017) – 10.1515/auto-2016-0098
• Kotyczka, P., Maschke, B. & Lefèvre, L. Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems. Journal of Computational Physics vol. 361 442–476 (2018) – 10.1016/j.jcp.2018.02.006
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• Macchelli, A., Le Gorrec, Y., Ramirez, H. & Zwart, H. On the Synthesis of Boundary Control Laws for Distributed Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 62 1700–1713 (2017) – 10.1109/tac.2016.2595263
• Moulla, R., Lefévre, L. & Maschke, B. Pseudo-spectral methods for the spatial symplectic reduction of open systems of conservation laws. Journal of Computational Physics vol. 231 1272–1292 (2012) – 10.1016/j.jcp.2011.10.008
• Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
• Serhani, A., Matignon, D. & Haine, G. Structure-Preserving Finite Volume Method for 2D Linear and Non-Linear Port-Hamiltonian Systems. IFAC-PapersOnLine vol. 51 131–136 (2018) – 10.1016/j.ifacol.2018.06.037
• Seslija, M., van der Schaft, A. & Scherpen, J. M. A. Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. Journal of Geometry and Physics vol. 62 1509–1531 (2012) – 10.1016/j.geomphys.2012.02.006
• Vu, N. M. T., Lefèvre, L. & Maschke, B. Geometric spatial reduction for port-Hamiltonian systems. Systems & Control Letters vol. 125 1–8 (2019) – 10.1016/j.sysconle.2019.01.002
• Vu, N. M. T., Lefèvre, L., Nouailletas, R. & Brémond, S. Symplectic spatial integration schemes for systems of balance equations. Journal of Process Control vol. 51 1–17 (2017) – 10.1016/j.jprocont.2016.12.005
• Trenchant, V., Fares, Y., Ramirez, H. & Le Gorrec, Y. A port-Hamiltonian formulation of a 2D boundary controlled acoustic system. IFAC-PapersOnLine vol. 48 235–240 (2015) – 10.1016/j.ifacol.2015.10.245
• Trenchant, V., Hu, W., Ramirez, H. & Gorrec, Y. L. Structure Preserving Finite Differences in Polar Coordinates for Heat and Wave Equations. IFAC-PapersOnLine vol. 51 571–576 (2018) – 10.1016/j.ifacol.2018.03.096
• Trenchant, V., Ramirez, H., Le Gorrec, Y. & Kotyczka, P. Structure preserving spatial discretization of 2D hyperbolic systems using staggered grids finite difference. 2017 American Control Conference (ACC) 2491–2496 (2017) doi:10.23919/acc.2017.7963327 – 10.23919/acc.2017.7963327
• Trenchant, V., Ramirez, H., Le Gorrec, Y. & Kotyczka, P. Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct. Journal of Computational Physics vol. 373 673–697 (2018) – 10.1016/j.jcp.2018.06.051
• Trenchant, V., Vu, T., Ramirez, H., Lefevre, L. & Le Gorrec, Y. On the use of structural invariants for the distributed control of infinite dimensional port-Hamitonian systems. 2017 IEEE 56th Annual Conference on Decision and Control (CDC) 47–52 (2017) doi:10.1109/cdc.2017.8263641 – 10.1109/cdc.2017.8263641
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Wu, Y., Hamroun, B., Gorrec, Y. L. & Maschke, B. Power preserving model reduction of 2D vibro-acoustic system: A port Hamiltonian approach. IFAC-PapersOnLine vol. 48 206–211 (2015) – 10.1016/j.ifacol.2015.10.240