Model order reduction of port-Hamiltonian systems with inhomogeneous initial conditions via approximate finite-time Gramians

Authors

Abstract

Based on the approximate finite-time Gramians, this paper studies model order reduction method of port-Hamiltonian systems with inhomogeneous initial conditions. The approximate controllability and observability Gramians on the finite-time interval [ T 1 , T 2 ] ( 0 ≤ T 1 < T 2 < ∞ ) can be obtained by the shifted Legendre polynomials and the reduced port-Hamiltonian system is constructed by the union of dominant eigenspaces. Since the port-Hamiltonian system is square, the cross Gramian on the time interval [ T 1 , T 2 ] can also be approximated by using the shifted Legendre polynomials. Then, the truncated singular value decomposition of the approximate finite-time cross Gramian is carried out to obtain the projection matrix. Finally, the proposed methods are verified by two numerical examples.

Keywords

Model order reduction; Port-Hamiltonian systems; Structure-preserving; Inhomogeneous initial conditions; Finite-time Gramians; Shifted Legendre polynomials

Citation

Journal: Applied Mathematics and Computation
Year: 2022
Volume: 422
Issue:
Pages: 126959
Publisher: Elsevier BV
DOI: 10.1016/j.amc.2022.126959

BibTeX

@article{Li_2022,
  title={{Model order reduction of port-Hamiltonian systems with inhomogeneous initial conditions via approximate finite-time Gramians}},
  volume={422},
  ISSN={0096-3003},
  DOI={10.1016/j.amc.2022.126959},
  journal={Applied Mathematics and Computation},
  publisher={Elsevier BV},
  author={Li, Yanpeng and Jiang, Yaolin and Yang, Ping},
  year={2022},
  pages={126959}
}

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References

Jiang, (2010)
Ibrir, Model reduction of a class of discrete-time nonlinear systems. Appl. Math. Comput. (2015)
Grimme, (1997)
Bistritz, Y. & Langholz, G. Model reduction by Chebyshev polynomial techniques. IEEE Transactions on Automatic Control vol. 24 741–747 (1979) – 10.1109/tac.1979.1102155
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
Daraghmeh, Optimal control of linear systems with balanced reduced-order models: perturbation approximations. Appl. Math. Comput. (2018)
Antoulas, (2020)
Gugercin, S., Antoulas, A. C. & Beattie, C. $\mathcal{H}_2$ Model Reduction for Large-Scale Linear Dynamical Systems. SIAM Journal on Matrix Analysis and Applications vol. 30 609–638 (2008) – 10.1137/060666123
van der Schaft, (1999)
Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
van der Schaft, Port-Hamiltonian systems theory: An introductory overview. (2006)
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
Breedveld, P. C. Port-based modeling of mechatronic systems. Mathematics and Computers in Simulation vol. 66 99–128 (2004) – 10.1016/j.matcom.2003.11.002
Breedveld, Port-based modeling of dynamic systems. (2009)
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
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
Afkham, B. M. & Hesthaven, J. S. Structure Preserving Model Reduction of Parametric Hamiltonian Systems. SIAM Journal on Scientific Computing vol. 39 A2616–A2644 (2017) – 10.1137/17m1111991
Maboudi Afkham, B. & Hesthaven, J. S. Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems. Journal of Scientific Computing vol. 81 3–21 (2018) – 10.1007/s10915-018-0653-6
Chaturantabut, S., Beattie, C. & Gugercin, S. Structure-Preserving Model Reduction for Nonlinear Port-Hamiltonian Systems. SIAM Journal on Scientific Computing vol. 38 B837–B865 (2016) – 10.1137/15m1055085
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
Sato, K. & Sato, H. Structure-Preserving $H^2$ Optimal Model Reduction Based on the Riemannian Trust-Region Method. IEEE Transactions on Automatic Control vol. 63 505–512 (2018) – 10.1109/tac.2017.2723259
Schulze, P. & Unger, B. Model Reduction for Linear Systems with Low-Rank Switching. SIAM Journal on Control and Optimization vol. 56 4365–4384 (2018) – 10.1137/18m1167887
Hauschild, Model reduction techniques for linear constant coefficient port-Hamiltonian differential-algebraic systems. Control Cybernet. (2019)
Benner, P., Goyal, P. & Van Dooren, P. Identification of port-Hamiltonian systems from frequency response data. Systems & Control Letters vol. 143 104741 (2020) – 10.1016/j.sysconle.2020.104741
Schwerdtner, Structure preserving model order reduction by parameter optimization. arXiv e-print, arXiv: 2011.07567 (2020)
Zulfiqar, Time/frequency-limited positive-real truncated balanced realizations. IMA J. Math. Control Inf. (2020)
Goyal, Time-limited H2-optimal model order reduction. Appl. Math. Comput. (2019)
Haider, S., Ghafoor, A., Imran, M. & Malik, F. M. Time-limited Gramians-based model order reduction for second-order form systems. Transactions of the Institute of Measurement and Control vol. 41 2310–2318 (2018) – 10.1177/0142331218798893
GAWRONSKI, W. & JUANG, J.-N. Model reduction in limited time and frequency intervals. International Journal of Systems Science vol. 21 349–376 (1990) – 10.1080/00207729008910366
Jazlan, Cross Gramian based time interval model reduction. (2015)
Xiao, Z.-H., Jiang, Y.-L. & Qi, Z.-Z. Finite-time balanced truncation for linear systems via shifted Legendre polynomials. Systems & Control Letters vol. 126 48–57 (2019) – 10.1016/j.sysconle.2019.03.004
Shen, J. & Lam, J. H ∞ model reduction for discrete‐time positive systems with inhomogeneous initial conditions. International Journal of Robust and Nonlinear Control vol. 25 88–102 (2013) – 10.1002/rnc.3075
Heinkenschloss, M., Reis, T. & Antoulas, A. C. Balanced truncation model reduction for systems with inhomogeneous initial conditions. Automatica vol. 47 559–564 (2011) – 10.1016/j.automatica.2010.12.002
Daraghmeh, Balanced model reduction of linear systems with nonzero initial conditions: singular perturbation approximation. Appl. Math. Comput. (2019)
Schröder, Balanced truncation model reduction with a priori error bounds for LTI systems with nonzero initial value. arXiv e-print, arXiv: 2006.02495 (2020)
Beattie, C., Gugercin, S. & Mehrmann, V. Model reduction for systems with inhomogeneous initial conditions. Systems & Control Letters vol. 99 99–106 (2017) – 10.1016/j.sysconle.2016.11.007
Beattie, Linear port-Hamiltonian descriptor systems. Math. Control Signal Syst. (2017)
Brogliato, (2001)
Spiegel, Theory and Problems of Fourier Analysis. (1974)
Gugercin, A time-limited balanced reduction method. (2003)
Simoncini, V. Computational Methods for Linear Matrix Equations. SIAM Review vol. 58 377–441 (2016) – 10.1137/130912839
Lu, A. & Wachspress, E. L. Solution of lyapunov equations by alternating direction implicit iteration. Computers & Mathematics with Applications vol. 21 43–58 (1991) – 10.1016/0898-1221(91)90124-m
Bartels, R. H. & Stewart, G. W. Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4]. Communications of the ACM vol. 15 820–826 (1972) – 10.1145/361573.361582
Golub, G., Nash, S. & Van Loan, C. A Hessenberg-Schur method for the problem AX + XB= C. IEEE Transactions on Automatic Control vol. 24 909–913 (1979) – 10.1109/tac.1979.1102170
Li, J.-R. & White, J. Low Rank Solution of Lyapunov Equations. SIAM Journal on Matrix Analysis and Applications vol. 24 260–280 (2002) – 10.1137/s0895479801384937
Xiang, S. On Error Bounds for Orthogonal Polynomial Expansions and Gauss-Type Quadrature. SIAM Journal on Numerical Analysis vol. 50 1240–1263 (2012) – 10.1137/110820841
Duff, (2017)
Fernando, K. & Nicholson, H. On the structure of balanced and other principal representations of SISO systems. IEEE Transactions on Automatic Control vol. 28 228–231 (1983) – 10.1109/tac.1983.1103195
Himpe, C. & Ohlberger, M. Cross‐Gramian‐Based Combined State and Parameter Reduction for Large‐Scale Control Systems. Mathematical Problems in Engineering vol. 2014 (2014) – 10.1155/2014/843869
Himpe, C. & Ohlberger, M. A note on the cross Gramian for non-symmetric systems. Systems Science & Control Engineering vol. 4 199–208 (2016) – 10.1080/21642583.2016.1215273
Shishkin, S. L., Shalaginov, A. & Bopardikar, S. D. Fast approximate truncated SVD. Numerical Linear Algebra with Applications vol. 26 (2019) – 10.1002/nla.2246
Golub, (2013)
Rossi, T. & Toivanen, J. A Parallel Fast Direct Solver for Block Tridiagonal Systems with Separable Matrices of Arbitrary Dimension. SIAM Journal on Scientific Computing vol. 20 1778–1793 (1999) – 10.1137/s1064827597317016
Gutknecht, Block Krylov space methods for linear systems with multiple right-hand sides: an introduction. (2007)
Soodhalter, K. M., de Sturler, E. & Kilmer, M. E. A survey of subspace recycling iterative methods. GAMM-Mitteilungen vol. 43 (2020) – 10.1002/gamm.202000016