Model order reduction using dual ν-gap metrics: A multi-objective optimization approach

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Abstract

This study investigated the problem of model order reduction by employing the ν-gap metric and introduced a metaheuristic optimization framework to address its inherent nonconvexity. The ν-gap metric quantifies the closed-loop distance between two systems, making it useful for generating low-order models that preserve closed-loop stability characteristics. However, minimizing the conventional ν-gap alone may result in poor dynamic fidelity. To address this limitation, a modified ν-gap metric based on frequency response matching was developed to explicitly capture frequency-wise discrepancies associated with time-domain behavior. By jointly considering the conventional and modified ν-gap metrics, a dual ν-gap–based multi-objective model reduction framework was formulated, in which stability-related and time-domain fidelity objectives are treated in a complementary manner. The resulting optimization problem is highly nonconvex and constrained due to the winding number condition. A multi-objective particle swarm optimization framework was therefore employed as a numerical tool to generate Pareto-optimal reduced-order models. Benchmark studies on large-scale systems demonstrated that the proposed framework enables a systematic exploration of trade-offs between stability preservation and time-domain fidelity, yielding practically meaningful reduced-order models with tunable performance characteristics.

Keywords

model order reduction / ν-gap metric / parameter identification / particle swarm optimization / multi-objective optimization

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References

[1] R. Kumar, D. Ezhilarasi, A state-of-the-art survey of model order reduction techniques for large-scale coupled dynamical systems, Int. J. Dyn. Control, 11 (2023), 900-916. https://doi.org/10.1007/s40435-022-00985-7

[2] S. K. Suman, A. Kumar, Investigation and implementation of model order reduction technique for large scale dynamical systems, Arch. Comput. Methods Eng., 29 (2022), 3087-3108. https://doi.org/10.1007/s11831-021-09690-8

[3] K. Lu, K. Zhang, H. Zhang, X. Gu, Y. Jin, S. Zhao, et al., A review of model order reduction methods for large-scale structure systems, Shock Vib., 2021 (2021), 6631180. https://doi.org/10.1155/2021/6631180

[4] O. M. Agudelo, C. Vermeersch, B. De Moor, Globally optimal H₂-norm model reduction: A numerical linear algebra approach , IFAC-PapersOnLine, 54 (2021), 564-571. https://doi.org/10.1016/j.ifacol.2021.06.117

[5] S. Lagauw, O. M. Agudelo, B. De Moor, Globally optimal SISO H₂-norm model reduction using Walsh’s theorem , IEEE Control Syst. Lett., 7 (2023), 1670-1675. https://doi.org/10.1109/LCSYS.2023.3276299

[6] L. Yu, J. Xiong, H₂ model reduction for negative imaginary systems , Int. J. Control, 93 (2020), 588-598. https://doi.org/10.1080/00207179.2018.1482501

[7] Y. Sakai, T. Wada, Y. Fujisaki, Mixed H₂/ H_∞ balanced truncations for discrete time linear systems , in 2019 12th Asian Control Conference (ASCC), (2019), 301-306.

[8] M. A. Hamadi, K. Jbilou, A. Ratnani, A data-driven Krylov model order reduction for large-scale dynamical systems, J. Sci. Comput., 95 (2023), 2. https://doi.org/10.1007/s10915-023-02122-8

[9] A. Bunse-Gerstner, D. Kubalińska, G. Vossen, D. Wilczek, H₂-norm optimal model reduction for large scale discrete dynamical MIMO systems , J. Comput. Appl. Math., 233 (2010), 1202-1216. https://doi.org/10.1016/j.cam.2008.12.029

[10] M. Vidyasagar, The graph metric for unstable plants and robustness estimates for feedback stability, IEEE Trans. Autom. Control, 29 (1984), 403-418. https://doi.org/10.1109/TAC.1984.1103547

[11] A. El-Sakkary, The gap metric: Robustness of stabilization of feedback systems, IEEE Trans. Autom. Control, 30 (1985), 240-247. https://doi.org/10.1109/TAC.1985.1103926

[12] T. T. Georgiou, On the computation of the gap metric, Syst. Control Lett., 11 (1988), 253-257. https://doi.org/10.1016/0167-6911(88)90067-9

[13] J. H. Steele, G. Vinnicombe, Closed-loop time-domain model validation in the nu-gap metric, in Proceedings of the 40th IEEE Conference on Decision and Control, (2001), 4332-4337. https://doi.org/10.1109/CDC.2001.980882

[14] S. Taamallah, Nu-gap metric a sum-of-squares and linear matrix inequality approach, in 2014 18th International Conference on System Theory, Control and Computing (ICSTCC), (2014), 405-411. https://doi.org/10.1109/ICSTCC.2014.6982450

[15] G. Buskes, M. Cantoni, Reduced order approximation in the ν-gap metric, in 2007 46th IEEE Conference on Decision and Control, (2007), 4367-4372. https://doi.org/10.1109/CDC.2007.4434557

[16] M. Cantoni, On model reduction in the ν-gap metric, in Proceedings of the 40th IEEE Conference on Decision and Control, (2001), 3665-3670. https://doi.org/10.1109/CDC.2001.980431

[17] H. Kondo, Y. Ochi, I-PD flight controller design based on integral-type optimal servomechanism, J. Control Meas. Syst. Integr., 3 (2010), 448-455. https://doi.org/10.9746/jcmsi.3.448

[18] H. Kondo, Y. Ochi, Plant-model reduction for PID controller design, in 49th IEEE Conference on Decision and Control (CDC), (2010), 6201-6206. https://doi.org/10.1109/CDC.2010.5717005

[19] A. Sootla, Nu-gap model reduction in the frequency domain, IEEE Trans. Autom. Control, 59 (2014), 228-233. https://doi.org/10.1109/TAC.2013.2290817

[20] S. Taamallah, Model order reduction in the L₂-gap metric , in 2016 American Control Conference (ACC), (2016), 2453-2459. https://doi.org/10.1109/ACC.2016.7525285

[21] Y. Hwang, Y. Ko, Y. Lee, T. Kim, Frequency-domain tuning of robust fixed-structure controllers via quantum-behaved particle swarm optimizer with cyclic neighborhood topology, Int. J. Control Autom. Syst., 16 (2018), 426-436. https://doi.org/10.1007/s12555-016-0766-3

[22] S. M. Djouadi, On robustness in the gap metric and coprime factor uncertainty for LTV systems, Syst. Control Lett., 80 (2015), 16-22. https://doi.org/10.1016/j.sysconle.2015.03.006

[23] A. Mondal, S. Ganguli, P. Sarkar, Reduced order modeling of delta operator systems by optimal frequency fitting approach, IFAC J. Syst. Control, 27 (2024), 100240. https://doi.org/10.1016/j.ifacsc.2024.100240

[24] E. H. Houssein, M. K. Saeed, G. Hu, M. M. Al-Sayed, Metaheuristics for solving global and engineering optimization problems: Review, applications, open issues and challenges, Arch. Comput. Methods Eng., 31 (2024), 4485-4519. https://doi.org/10.1007/s11831-024-10168-6

[25] D. Freitas, L. G. Lopes, F. Morgado-Dias, Particle swarm optimisation: A historical review up to the current developments, Entropy, 22 (2020), 362. https://doi.org/10.3390/e22030362

[26] G. Vinnicombe, Frequency domain uncertainty and the graph topology, IEEE Trans. Autom. Control, 38 (1993), 1371-1383. https://doi.org/10.1109/9.237648

[27] Y. Chahlaoui, P. Van Dooren, A collection of benchmark examples for model reduction of linear time invariant dynamical systems, 2002. Available from: https://eprints.maths.manchester.ac.uk/1040/.