Improved Bilinear Balanced Truncation for Order Reduction of the ‎High-Order Bilinear System Based on Linear Matrix Inequalities ‎

Nasiri Soloklo, H.; Bigdeli, N.

doi:10.22061/jecei.2022.8812.554

Document Type : Original Research Paper

Authors

Control Engineering Department, Faculty of Technical and Engineering, Imam Khomeini International University, Qazvin, Iran.

https://doi.org/10.22061/jecei.2022.8812.554

Abstract

Background and Objectives: This paper proposes a new Model Order ‎Reduction (MOR) method based on the Bilinear Balanced Truncation ‎‎(BBT) approach. In the BBT method, solving the generalized Lyapunov ‎equations is necessary to determine the bilinear system's controllability ‎and observability Gramians. Since the bilinear systems are generally of ‎high order, the computation of the Gramians of controllability and ‎observability have huge computational volumes. In addition, the accuracy ‎of reduced-order model obtained by BT is relatively low. In fact, the ‎balanced truncation method is only available for local energy bands due ‎to the use of type I Gramians. In this paper, BBT based on type II ‎controllability and observability Gramians would be considered to fix ‎these drawbacks.‎
Methods: At first, a new iterative method is proposed for determining ‎the proper order for the reduced-order bilinear model, which is related to ‎the number of Hankel singular values of the bilinear system whose real ‎parts are closest to origin and have the most significant amount of ‎energy. Then, the problem of determining of type II controllability and ‎observability Gramians of the high-order bilinear system have been ‎formulated as a constrained optimization problem with some Linear ‎Matrix Inequality (LMI) constraints for an intermediate middle-order ‎system. Then, the achieved Gramians are applied to the BBT method to ‎determine the reduced-order model of the bilinear system. Next, the ‎steady state accuracy of the reduced model would be improved via ‎employing a tuning factor.‎
Results: Using the concept of type II Gramians and via the proposed ‎method, the accuracy of the proposed bilinear BT method is increased. ‎For validation of the proposed method, three high-order bilinear models ‎are approximated. The achieved results are compared with some well-‎known MOR approaches such as bilinear BT, bilinear Proper Orthogonal ‎Decomposition (POD) and Bilinear Iterative Rational Krylov subspace ‎Algorithm (BIRKA) methods.‎
Conclusion: According to the obtained results, the proposed MOR ‎method is superior to classical bilinear MOR methods, but is almost ‎equivalent to BIRKA. It is out-performance respecting to BIRKA is its ‎guaranteed stability and convergence. ‎

Keywords

Main Subjects

Order Reduction

Open Access

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References

[1] S. A. Al-Baiyat, M. Bettayeb, U. M. Al-Saggaf, "New model reduction scheme for bilinear systems," Int. J. Syst. Sci., 25: 1631–1642, 1994.

[2] P. Benner, T. Damm, "Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems," SIAM J. Control Optim., 49(2): 686-711, 2011.

[3] P. Benner, P. Goyal, S. Gugercin, "H2-quasi-optimal model order reduction for quadratic-bilinear control systems," SIAM J. Matr Anal. Appl., 39(2): 983–1032, 2019.

[4] W. J. Rugh, Nonlinear System Theory, Johns Hopkins University Press, Baltimore, 1981.

[5] S. V. Dushin, A. N. Abramenkov, E. J. Kutyakov, A. B. Iskakov, A. M. Salnikov, "Developing a weakly nonlinear power system model using the carleman bilinearization procedure," 2020 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency (SUMMA): 963-967, 2020.

[6] U. Baur, P. Benner, L. Feng, "Model order reduction for linear and nonlinear systems: A system-theoretic perspective,” Arch. Comput. Methods. Eng., 21: 331–358, 2014.

[7] H. Nasiri Soloklo, N. Bigdeli, "A PFC-based hybrid approach for control of industrial heating furnace," J. Electr. Comput. Eng. Innovations (JECEI), 7(1): 83-94, 2018.

[8] H. Hooshmand, M. M. Fateh, "Voltage control of flexible-joint robot manipulators using singular perturbation technique for model order reduction," J. Electr. Comput. Eng. Innovations (JECEI), 10(1): 123-142, 2022.

[9] V. B. Nguyen, M. Buffoni, K. Willcox, B. C. Khoo, "Model reduction for reacting flow applications," Int. J. Comput. Fluid Dyn., 28(3-4): 91-105, 2014.

[10] N. Stahl, B. Liljegren-Sailer, N. Marheineke, "Moment matching based model order reduction for quadratic-bilinear systems," in Proc. Faragó I., Izsák F., Simon P. (eds) Progress in Industrial Mathematics at ECMI 2018. Mathematics in Industry, (2019) 30, Springer, Cham.

[11] O. Agbaje, D. L. Shona, O. Haas, "Multimoment matching analysis of one-sided Krylov subspace model order reduction for nonlinear and bilinear systems," in Proc. 2018 European Control Conference (ECC): 2599-2604, 2018.

[12] Y. Lin, L. Bao, Y. Wei, "A model-order reduction method based on Krylov subspace for MIMO bilinear dynamical systems," J. Appl. Math. Comput., 25: 293-304, 2007.

[13] Y. Lin, L. Bao, Y. Wei, "Order reduction of bilinear MIMO dynamical systems using new block Krylov subspace," Comput. Math. Appl., 58: 1093-1102, 2009.

[14] T. Breiten, T. Damm, "Krylov subspace methods for model order reduction of bilinear control systems," Syst. Control Lett., 59: 443-450, 2010.

[15] J. R. Philips, "Projection frameworks for model reduction of weakly nonlinear systems," in Proc. 37th Design Automation Conference, 184-189, 2000.

[16] Z. Bai, D. Skoogh, "A projection method for model reduction of bilinear dynamical systems," Linear Algebra Appl., 415: 406-425, 2006.

[17] M. Redmann, I. P. Duff, "Full state approximation by Galerkin projection reduced order models for stochastic and bilinear systems," Appl. Math. Comput., 420: 126561, 2022.

[18] L. Dai, Z. H. Xiao, R. Z. Zhang, "Laguerre-Gramian-based model order reduction of bilinear systems," in Proc. 2021 40th Chinese Control Conference (CCC): 1195-1200, 2021.

[19] X. Wang, Y. Jiang, "Model reduction of discrete-time bilinear systems by a Laguerre expansion technique," Appl. Math. Model., 40(13–14): 6650-6662, 2016.

[20] Y. Li, Y. Jiang, P. Yang, “Time domain model order reduction of discrete-time bilinear systems with Charlier polynomials,” Math. Comput. Simul., 190: 905-920, 2021.

[21] L. Zhang, J. Lam, "On H2 model reduction of bilinear systems," Automatica, 38(2): 205-216, 2002.

[22] P. Benner, T. Breiten, "Interpolation-based H2 model reduction of bilinear control systems," SIAM J. Matr. Anal. Appl., 33(3): 859-885, 2012.

[23] P. Benner, X. Cao, W. Schilders, "A bilinear H2 model order reduction approach to linear parameter-varying systems," Adv. Comput. Math., 45: 2241–2271, 2019.

[24] X. L. Wang, Y. L. Jiang, "On model reduction of K-power bilinear systems," Int. J. Syst. Sci., 45(9): 1978-1990, 2014.

[25] K. L. Xu, Y. L. Jiang, Z. X. Yang, "H2 order-reduction for bilinear systems based on Grassmann manifold," J. Franklin Inst., 352: 4467–4479, 2015.

[26] K. L. Xu, Y. L. Jiang, Z. X. Yang, "H2 optimal model order reduction by two-sided technique on Grassmann manifold via the cross-gramian of bilinear systems," Int. J. Control., 90(3): 616-626, 2017.

[27] P. Yang, Y. L. Jiang, K. L. Xu, "A trust-region method for H2 model reduction of bilinear systems on the Stiefel manifold," J. Franklin Inst., 356(4): 2258-2273, 2019.

[28] K. L. Xu, Y. L. Jiang, "An approach to model reduction on finite interval for bilinear systems," J. Franklin Inst., 354: 7429-7443, 2017.

[29] U. Zulfiqar, V. Sreeram, M. I. Ahmad, X. Du, "Time and frequency-limited H2-optimal model order reduction of bilinear control systems," Int. J. Syst. Sci., 52(10): 1953-1973, 2021.

[30] B. C. Moore, "Principal component analysis in linear systems: controllability, observability, and model reduction," IEEE Trans. Autom. Control., AC-26: 17-32, 1981.

[31] C. S. Hsu, U. B. Desai, C. A. Crawley, "Realization algorithms and approximation methods of bilinear systems," presented at the 22nd IEEE Conference on Decision and Control, San Antonio, Texas, 1983.

[32] I. P. Duff, P. Goyal, P. Benner, "Balanced truncation for a special class of bilinear descriptor systems," IEEE Control Syst. Lett., 3(3): 535-540, 2019.

[33] S. A. Al-Baiyat, A. S. Farag, M. Bettayeb, "Transient approximation of a bilinear two-area interconnected power system," Electr. Power Syst. Res., 26(1): 11–19, 1993.

[34] M. Redmann, “Type II balanced truncation for deterministic bilinear control systems,”arXiv: 1709.05655, 2017.

[35] P. Benner, P. Goyal, "Balanced truncation model order reduction for quadratic-bilinear control systems," Technical Report, 2017.

[36] P. Benner, P. Goyal, M. Redmann, Truncated Gramians for bilinear systems and their advantages in model order reduction, In Benner P, Ohlberger M, Patera T, Rozza G, Urban K, eds., Model reduction of parameterized systems. Springer International Publishing, 2017.

[37] S. K. Suman, A. Kumar, "Linear system of order reduction using a modified balanced truncation method," Circuits Syst. Signal Process, 40: 2741–2762, 2021.

[38] M. Redmann, “The missing link between the output and the H2-norm of bilinear systems.” arXiv, 1910.14427, 2019.

[39] R. Choudhary, K. Ahuja, "Inexact linear solves in model reduction of bilinear dynamical systems," IEEE Access, 7: 72297–72307, 2019.

[40] L. Q. Zhang, J. Lam, B. Huang, G. H. Yang, "On Gramians and balanced truncation of discrete-time bilinear systems," Int. J. Control., 76(4): 414–427, 2003.

[41] P. Benner, T. Damm, Y. R. Rodriguez Cruz, "Dual pairs of generalized lyapunov inequalities and balanced truncation of stochastic linear systems," IEEE Trans. Automat. Contr., 62(2): 782-791, 2017.

[42] K. Kerschen, J. Golinval, A. F. Vakakis, L. A. Bergman, "The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An Overview., Nonlinear Dyn., 41: 147–169, 2005.

[43] A. C. Antoulas, D. C. Sorensen, "Approximation of Large-Scale Dynamical Systems: An Overview," Int. J. Appl. Math. Comput. Sci. 11 (5), 1093-1121, 2001.

[44] P. Benner, T. Breiten, "Two-sided projection methods for nonlinear model order reduction," SIAM J. Sci. Comput., 37(2): B239–B260, 2015.

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Journal of Electrical and Computer Engineering Innovations (JECEI)

Improved Bilinear Balanced Truncation for Order Reduction of the ‎High-Order Bilinear System Based on Linear Matrix Inequalities ‎

References

References

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Volume 11, Issue 1
January 2023
Pages 129-140

Improved Bilinear Balanced Truncation for Order Reduction of the ‎High-Order Bilinear System Based on Linear Matrix Inequalities ‎

References

References

Send comment about this article

Volume 11, Issue 1January 2023Pages 129-140

Volume 11, Issue 1
January 2023
Pages 129-140