Riemannian Geometric-Nonlinear Conjugate Gradient Model Order Reduction of Linear Port-Hamiltonian Systems on Finite Frequency Intervals

Authors

Kang-Li Xu, Yao-Lin Jiang

Abstract

In this article, we propose a new frequency-limited Riemannian geometric-nonlinear conjugate gradient model order reduction (MOR) method with the modified Armijo step-size control to solve the frequency-limited \( \mathcal {H}_{2} \) optimal MOR problem of linear port-Hamiltonian systems. This problem is formulated as a Riemannian optimization problem on the product of several manifolds. Based on the geometric properties of the product manifold, the Riemannian gradient of the cost function is derived. Further, by scaling the vector transport of the product manifold, we design a new Riemannian spectral conjugate gradient direction, which is always descent for the cost function and is independent of the line search used and the convexity of the cost function. Meanwhile, a new modified Armijo step-size control strategy is presented. The resulting reduced system can preserve the port-Hamiltonian structure and the passivity of the original system. Two numerical examples are simulated to demonstrate the efficiency of the proposed method.

Citation

• Journal: IEEE Transactions on Automatic Control
• Year: 2024
• Volume: 69
• Issue: 5
• Pages: 3317–3324
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tac.2023.3321902

BibTeX

@article{Xu_2024,
  title={{Riemannian Geometric-Nonlinear Conjugate Gradient Model Order Reduction of Linear Port-Hamiltonian Systems on Finite Frequency Intervals}},
  volume={69},
  ISSN={2334-3303},
  DOI={10.1109/tac.2023.3321902},
  number={5},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Xu, Kang-Li and Jiang, Yao-Lin},
  year={2024},
  pages={3317--3324}
}

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References

• 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
• 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
• 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
• Macchelli, A., Le Gorrec, Y. & Ramirez, H. Exponential Stabilization of Port-Hamiltonian Boundary Control Systems via Energy Shaping. IEEE Transactions on Automatic Control vol. 65 4440–4447 (2020) – 10.1109/tac.2020.3004798
• Jiang, Model Order Reduction Methods (2010)
• Model Reduction and Approximation. (2017) doi:10.1137/1.9781611974829 – 10.1137/1.9781611974829
• Nouri, B., Nakhla, M. & Deng, X. Stable Model-Order Reduction of Active Circuits. IEEE Transactions on Components, Packaging and Manufacturing Technology vol. 7 710–719 (2017) – 10.1109/tcpmt.2016.2635656
• 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
• Benner, P. & Breiten, T. Interpolation-Based ${\cal H}_2$-Model Reduction of Bilinear Control Systems. SIAM Journal on Matrix Analysis and Applications vol. 33 859–885 (2012) – 10.1137/110836742
• Li, P., Lam, J., Wang, Z. & Date, P. Positivity-preserving \(H_{\infty}\) model reduction for positive systems. Automatica vol. 47 1504–1511 (2011) – 10.1016/j.automatica.2011.02.032
• Wei-Yong Yan & Lam, J. An approximate approach to H/sup 2/ optimal model reduction. IEEE Transactions on Automatic Control vol. 44 1341–1358 (1999) – 10.1109/9.774107
• Jiang, Y.-L. & Xu, K.-L. Riemannian Modified Polak–Ribière–Polyak Conjugate Gradient Order Reduced Model by Tensor Techniques. SIAM Journal on Matrix Analysis and Applications vol. 41 432–463 (2020) – 10.1137/19m1257147
• 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
• Ionescu, T. C. & Astolfi, A. Families of moment matching based, structure preserving approximations for linear port Hamiltonian systems. Automatica vol. 49 2424–2434 (2013) – 10.1016/j.automatica.2013.05.006
• 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
• 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. Riemannian Optimal Control and Model Matching of Linear Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 62 6575–6581 (2017) – 10.1109/tac.2017.2712905
• Sato, K. Riemannian optimal model reduction of linear port-Hamiltonian systems. Automatica vol. 93 428–434 (2018) – 10.1016/j.automatica.2018.03.051
• Jiang, Y.-L. & Xu, K.-L. Model Order Reduction of Port-Hamiltonian Systems by Riemannian Modified Fletcher–Reeves Scheme. IEEE Transactions on Circuits and Systems II: Express Briefs vol. 66 1825–1829 (2019) – 10.1109/tcsii.2019.2895872
• Li, X., Yu, C. & Gao, H. Frequency-Limited \({\mmb H}_{\infty}\) Model Reduction for Positive Systems. IEEE Transactions on Automatic Control vol. 60 1093–1098 (2015) – 10.1109/tac.2014.2352751
• Ghafoor, A. & Sreeram, V. Model Reduction Via Limited Frequency Interval Gramians. IEEE Transactions on Circuits and Systems I: Regular Papers vol. 55 2806–2812 (2008) – 10.1109/tcsi.2008.920092
• Shaker, H. R. & Tahavori, M. Frequency-Interval Model Reduction of Bilinear Systems. IEEE Transactions on Automatic Control vol. 59 1948–1953 (2014) – 10.1109/tac.2013.2295661
• Petersson, D. & Löfberg, J. Model reduction using a frequency-limited -cost. Systems & Control Letters vol. 67 32–39 (2014) – 10.1016/j.sysconle.2014.02.004
• Jiang, Y.-L. & Xu, K.-L. Frequency-Limited Reduced Models for Linear and Bilinear Systems on the Riemannian Manifold. IEEE Transactions on Automatic Control vol. 66 3938–3951 (2021) – 10.1109/tac.2020.3027643
• Xu, K. & Jiang, Y. Structure‐preserving interval‐limited balanced truncation reduced models for port‐Hamiltonian systems. IET Control Theory & Applications vol. 14 405–414 (2020) – 10.1049/iet-cta.2019.0566
• Higham, N. J. Functions of Matrices. (2008) doi:10.1137/1.9780898717778 – 10.1137/1.9780898717778
• Vandereycken, B. & Vandewalle, S. A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations. SIAM Journal on Matrix Analysis and Applications vol. 31 2553–2579 (2010) – 10.1137/090764566
• Helmke, U. & Moore, J. Optimization and Dynamical Systems. Proceedings of the IEEE vol. 84 907 (1996) – 10.1109/jproc.1996.503147
• Jeuris, A survey and comparison of ceontemporary algorithms for computing the matrix geometric mean. Electron. Trans. Numer. Anal. (2012)
• Absil, P.-A., Mahony, R. & Sepulchre, R. Optimization Algorithms on Matrix Manifolds. (2008) doi:10.1515/9781400830244 – 10.1515/9781400830244
• Benner, Sparse-dense Sylvester equations in $\mathcal {H}_{2}$-model order reduction. (2011)
• Simoncini, V. Computational Methods for Linear Matrix Equations. SIAM Review vol. 58 377–441 (2016) – 10.1137/130912839
• Sato, K. Riemannian Optimal Model Reduction of Stable Linear Systems. IEEE Access vol. 7 14689–14698 (2019) – 10.1109/access.2019.2892071
• Kunisch, K. & Volkwein, S. Proper orthogonal decomposition for optimality systems. ESAIM: Mathematical Modelling and Numerical Analysis vol. 42 1–23 (2008) – 10.1051/m2an:2007054