Error bounds for port-Hamiltonian model and controller reduction based on system balancing

Authors

Tobias Breiten, Riccardo Morandin, Philipp Schulze

Abstract

We study linear quadratic Gaussian (LQG) control design for linear port-Hamiltonian systems. To this end, we exploit the freedom in choosing the weighting matrices and propose a specific choice which leads to an LQG controller which is port-Hamiltonian and, thus, in particular stable and passive. Furthermore, we construct a reduced-order controller via balancing and subsequent truncation. This approach is closely related to classical LQG balanced truncation and shares a similar a priori error bound with respect to the gap metric. By exploiting the non-uniqueness of the Hamiltonian, we are able to determine an optimal pH representation of the full-order system in the sense that the error bound is minimized. In addition, we discuss consequences for pH-preserving balanced truncation model reduction which results in two different classical H ∞ -error bounds. Finally, we illustrate the theoretical findings by means of two numerical examples.

Keywords

Port-Hamiltonian systems; Model order reduction; LQG control design; Error bounds

Citation

• Journal: Computers & Mathematics with Applications
• Year: 2022
• Volume: 116
• Issue:
• Pages: 100–115
• Publisher: Elsevier BV
• DOI: 10.1016/j.camwa.2021.07.022
• Note: New trends in Computational Methods for PDEs

BibTeX

@article{Breiten_2022,
  title={{Error bounds for port-Hamiltonian model and controller reduction based on system balancing}},
  volume={116},
  ISSN={0898-1221},
  DOI={10.1016/j.camwa.2021.07.022},
  journal={Computers & Mathematics with Applications},
  publisher={Elsevier BV},
  author={Breiten, Tobias and Morandin, Riccardo and Schulze, Philipp},
  year={2022},
  pages={100--115}
}

Download the bib file

References

• Willems, J. Least squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on Automatic Control vol. 16 621–634 (1971) – 10.1109/tac.1971.1099831
• Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
• Willems, J. C. Dissipative dynamical systems Part II: Linear systems with quadratic supply rates. Archive for Rational Mechanics and Analysis vol. 45 352–393 (1972) – 10.1007/bf00276494
• van der Schaft, (1996)
• van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
• Duindam, (2009)
• van der Schaft, A. J. & Maschke, B. M. Port-Hamiltonian Systems on Graphs. SIAM Journal on Control and Optimization vol. 51 906–937 (2013) – 10.1137/110840091
• Beattie, C. A., Mehrmann, V. & Van Dooren, P. Robust port-Hamiltonian representations of passive systems. Automatica vol. 100 182–186 (2019) – 10.1016/j.automatica.2018.11.013
• Mehl, C., Mehrmann, V. & Sharma, P. Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations. SIAM Journal on Matrix Analysis and Applications vol. 37 1625–1654 (2016) – 10.1137/16m1067330
• Brugnoli, Port-Hamiltonian flexible multibody dynamics. Multibody Syst. Dyn. (2020)
• Celledoni,
• Kotyczka, P. & Lefèvre, L. Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters vol. 133 104530 (2019) – 10.1016/j.sysconle.2019.104530
• Mehrmann, Structure-preserving discretization for port-Hamiltonian descriptor systems. (2019)
• Serhani, A partitioned finite element method for the structure-preserving discretization of damped infinite-dimensional port-Hamiltonian systems with boundary control. (2019)
• Halevi, Y. Stable LQG controllers. IEEE Transactions on Automatic Control vol. 39 2104–2106 (1994) – 10.1109/9.328801
• Lozano-Leal, On the design of the dissipative LQG-type controllers. (1988)
• Wu, (2016)
• Wu, Structure-preserving reduction of port Hamiltonian systems using a modified LQG method. (2014)
• Wu, Y., Hamroun, B., Le Gorrec, Y. & Maschke, B. Reduced order LQG control design for port Hamiltonian systems. Automatica vol. 95 86–92 (2018) – 10.1016/j.automatica.2018.05.003
• Jacob, (2012)
• Antoulas, (2005)
• Egger, H., Kugler, T., Liljegren-Sailer, B., Marheineke, N. & Mehrmann, V. On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks. SIAM Journal on Scientific Computing vol. 40 A331–A365 (2018) – 10.1137/17m1125303
• Gugercin, S., Polyuga, R. V., Beattie, C. & van der Schaft, A. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica vol. 48 1963–1974 (2012) – 10.1016/j.automatica.2012.05.052
• Polyuga, R. V. & van der Schaft, A. J. Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems. Systems & Control Letters vol. 61 412–421 (2012) – 10.1016/j.sysconle.2011.12.008
• 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
• Liljegren-Sailer, (2020)
• Guiver, C. & Opmeer, M. R. Error bounds in the gap metric for dissipative balanced approximations. Linear Algebra and its Applications vol. 439 3659–3698 (2013) – 10.1016/j.laa.2013.09.032
• McFarlane, (1990)
• Curtain, Model reduction for control design for distributed parameter systems. (2003)
• Meyer, D. G. Fractional balanced reduction: model reduction via fractional representation. IEEE Transactions on Automatic Control vol. 35 1341–1345 (1990) – 10.1109/9.61011
• Chahlaoui, Y., Lemonnier, D., Vandendorpe, A. & Van Dooren, P. Second-order balanced truncation. Linear Algebra and its Applications vol. 415 373–384 (2006) – 10.1016/j.laa.2004.03.032
• Reis, T. & Stykel, T. Balanced truncation model reduction of second-order systems. Mathematical and Computer Modelling of Dynamical Systems vol. 14 391–406 (2008) – 10.1080/13873950701844170
• Gugercin, S. & Antoulas, A. C. A Survey of Model Reduction by Balanced Truncation and Some New Results. International Journal of Control vol. 77 748–766 (2004) – 10.1080/00207170410001713448
• 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
• Mullis, C. & Roberts, R. Synthesis of minimum roundoff noise fixed point digital filters. IEEE Transactions on Circuits and Systems vol. 23 551–562 (1976) – 10.1109/tcs.1976.1084254
• Harshavardhana, Stochastic balancing and approximation - stability and minimality. (1983)
• JOHNSON, C. D. State-variable design methods may produce unstable feedback controllers. International Journal of Control vol. 29 607–619 (1979) – 10.1080/00207177908922723
• Mehrmann, (1991)
• Damm, Balanced truncation for stochastic linear systems with guaranteed error bound. (2014)
• Sandberg, H. & Rantzer, A. Balanced Truncation of Linear Time-Varying Systems. IEEE Transactions on Automatic Control vol. 49 217–229 (2004) – 10.1109/tac.2003.822862
• Möckel, J., Reis, T. & Stykel, T. Linear-quadratic Gaussian balancing for model reduction of differential-algebraic systems. International Journal of Control vol. 84 1627–1643 (2011) – 10.1080/00207179.2011.622791
• CURTAIN, R. F. Robust stabilizability of normalized coprime factors: the infinite-dimensional case. International Journal of Control vol. 51 1173–1190 (1990) – 10.1080/00207179008934125
• Sefton, J. A. & Ober, R. J. On the gap metric and coprime factor perturbations. Automatica vol. 29 723–734 (1993) – 10.1016/0005-1098(93)90066-3
• Vidyasagar, M. The graph metric for unstable plants and robustness estimates for feedback stability. IEEE Transactions on Automatic Control vol. 29 403–418 (1984) – 10.1109/tac.1984.1103547
• Zhou, (1996)
• Golub, (2013)
• Unneland, A novel scheme for positive real balanced truncation. (2007)
• TOMBS, M. S. & POSTLETHWAITE, I. Truncated balanced realization of a stable non-minimal state-space system. International Journal of Control vol. 46 1319–1330 (1987) – 10.1080/00207178708933971
• Penzl, T. Algorithms for model reduction of large dynamical systems. Linear Algebra and its Applications vol. 415 322–343 (2006) – 10.1016/j.laa.2006.01.007
• Benner, P. & Quintana‐Ortí, E. S. Numerical Algorithms vol. 20 75–100 (1999) – 10.1023/a:1019191431273
• Opdenacker, P. C. & Jonckheere, E. A. A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds. IEEE Transactions on Circuits and Systems vol. 35 184–189 (1988) – 10.1109/31.1720