###
Scientific Journal Of King Faisal University

Basic and Applied Sciences

#### Scientific Journal of King Faisal University / Basic and Applied Sciences

### Model Order Reduction by Using Improved Approximation Techniques

(Santosh Kumar Suman and Awadhesh Kumar)### Abstract

A simplified approach for model order reduction (MOR) is presented in this article using the balanced singular perturbation approximation (BSPA) approach applicable to large-scale linear dynamical (LSLD) systems. The reduced system was so designed to preserve complete parameters of the original system with reasonable accuracy, employing MOR. The approach is based on the retention of the dominant states of the system and comparatively less important ones. The reduced system comes from the preservation of the dominant states (say ‘desirable states’) of the original system and, thus, from stability to preservation. The key disadvantage of the Balanced Truncation approach is that the ROM steady-state values do not correspond with the higher-order systems. This drawback has been eliminated in the proposed approach, which leads to hybridisation of balanced truncation and singular perturbation approximation into a novel reduction method without the loss of retaining its dynamic behaviour. The proposed approach has been tested on LSLD systems and the results obtained show the efficacy of the approach. The methodology presented has been tested on two typical numerical examples taken from the literature review to examine the performance, precision and comparison with other available standard order reduction methods.

KEYWORDS

Balanced truncation method, singular perturbation approximation, large scale linear dynamical system, steady state value, model order reduction

PDF

### References

Al-Saggaf, U.M. and Franklin, G.F. (1988). Model reduction via balanced realizations: An extension and frequency weighting techniques. IEEE Transactions on Automatic Control, 33(7), 687–92. https://doi.org/10.1109/9.1280

Antoulas, A.C., Sorensen, D.C. and Gugercin, S. (2012). A survey of model reduction methods for large-scale systems. Rice University, 1(12), 1-28. DOI: 10.1090/conm/280/04630

Antoulas, Athanasios C. (2005). Approximation of Large-scale Dynamical Systems (1st Edition). United States: Society for Industrial and Applied Mathematics. delphia, PA: SIAM Publications. DOI: 10.1137/1.9780898718713

Benner, P., Gugercin, S. and Willcox, K. (2015). A Survey of Model Reduction Methods for Parametric Systems. SIAM Review, 57(4), 483delphia, PA531. DOI: 10.1137/130932715

Benner, P. and Schneider, A. (2010). Balanced truncation model order reduction for LTI systems with many inputs or outputs.Theory of Networks and Systems–MTNS, 1(19), 1971–74.

Boley, D. and Datta, B.N. (1997). Numerical methods for linear control systems. In: Systems and Control in the Twenty-First Century, Birkhäuser, Boston, 22(1), 51–74. DOI: 10.1007/978-1-4612-4120-1_4

Cao, X., Saltik, M.B. and Weiland, S. (2019). Optimal Hankel norm model reduction for discrete-time descriptor systems. Journal of the Franklin Institute, 356(7), 4124–43, DOI: 10.1016/j.jfranklin.2018.11.047

Chen, T.C., Chang, C.Y. and Han, K.W. (1979). Reduction of Transfer Functions by the Stability-Equation Method. Journal of the Franklin Institute, 308(4), 389–404. DOI: 10.1016/0016-0032(79)90066-8

C. Huang, K. Zhang, X. Dai and W. Tang. (2013). A modified balanced truncation method and its application to model reduction of power system. In 2013 IEEE Power & Energy Society General Meeting , 21-25 July 2013, 1-5. IEEE. DOI: 10.1109/PESMG.2013.6672263

Clapperton, B., Crusca, F. and Aldeen, M. (1996). Bilinear transformation and generalized singular perturbation model reduction. IEEE Transactions on Automatic Control, 4(1), 3737–42. DOI: 10.1109/9.489281

Enns, D.F. (1984). Model reduction with balanced realizations: an error bound and a frequency weighted generalization. In 23rd IEEE Conference on Decision and Control, 12-14 Dec. 1984, 127–32. DOI: 10.1109/cdc.1984.272286

Fernando, K.V. and Nicholson, H. (1982). Singular perturbational model reduction of balanced systems. IEEE Transactions on Automatic Control, 27(2), 466–8. DOI: 10.1109/TAC.1982.1102932

Fernando, K.V. and Nicholson, H. (1983a). On the structure of balanced and other principal representations of siso systems. IEEE Transactions on Automatic Control, 28 (2), 228–31. DOI: 10.1109/TAC.1983.1103195

Gajic, Z. and Lelic, M. (2001). Improvement of system order reduction via balancing using the method of singular perturbations. Automatica, 37(11),1859–65. DOI: 10.1016/S0005-1098(01)00139-X

Glover, K. (1984). All optimal hankel-norm approximations of linear multivariable systems and their L,∞-error bounds†. International Journal of Control, 39(6), 1115–93. DOI: 10.1080/00207178408933239

Gugercin, S. (2008). An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems. Linear Algebra and Its Applications, 428(8), 1964–86. DOI: 10.1016/j.laa.2007.10.041

Gugercin, S. and Antoulas, A.C. (2004). A survey of model reduction by balanced truncation and some new results. In International Journal of Control, 77(8),748–66. DOI: 10.1080/00207170410001713448

Gugercin, S. and Antoulas, A.C. (2006). Model reduction of large-scale systems by least squares. Linear Algebra and Its Applications, 415(2), 290–321. DOI: 10.1016/j.laa.2004.12.022

Guillaume, P. and Huard, A. (2000). Multivariate Pade approximation. Journal of Computational and Applied Mathematics, 121(1),197–219. DOI: 10.1016/S0377-0427(00)00337-X

Guiver, C. (2019). The generalised singular perturbation approximation for bounded real and positive real control systems. Mathematical Control and Related Fields, 9(2), 313–50. DOI: 10.3934/MCRF.2019016

Gupta, A.K., Samuel, P. and Kumar, D. (2018). A mixed-method for order reduction of linear time invariant systems using big bang-big crunch and eigen spectrum algorithm. International Journal of Automation and Control, 13(2),158–75. DOI: 10.1504/ijaac.2019.10018127

Gutman, P., Mannerfelt, C.F. and Molander, P. (1982). Contributions to the model reduction problem. IEEE Transactions on Automatic Control, 27(2), 454–5. DOI: 10.1109/TAC.1982.1102930

Imran, M., Ghafoor, A. and Sreeram, V. (2014). A frequency weighted model order reduction technique and error bounds. Automatica, 50(12), 3304–9. DOI: 10.1016/j.automatica.2014.10.062

Kodra, K., Zhong, N. and Gajic, Z. (2016). Model order reduction of an islanded microgrid using singular perturbations. Proceedings of the American Control Conference, 6-8 July 2016, 3650–5. DOI: 10.1109/ACC.2016.7525480

Kokotovic, P.V., O’Malley, R.E. and Sannuti, P. (1976). Singular perturbations and order reduction in control theory - An overview. Automatica, 12(2), 123–32. DOI: 10.1016/0005-1098(76)90076-5

Krishnamurthy, V. and Seshadri, V. (1978). Model reduction using the routh stability criterion. IEEE Transactions on Automatic Control, 23(4), 729–31. DOI: 10.1109/TAC.1978.1101805

Kumar, D., Tiwari, J.P. and Nagar, S.K. (2012). Reducing order of large-scale systems by extended balanced singular perturbation approximation. International Journal of Automation and Control, 6(1),21–38. DOI: 10.1504/IJAAC.2012.045438

Lall, S., Marsden, J.E. and Glavaški, S. (2002). A subspace approach to balanced truncation for model reduction of nonlinear control systems. International Journal of Robust and Nonlinear Control, 12(6), 519–35. DOI: 10.1002/rnc.657

Lepschy, A. and Viaro, U. (1982). An improvement in the routh-pade approximation techniques. International Journal of Control, 36(4), 643–61. DOI: 10.1080/00207178208932921

Liu, Y. and Anderson, B.D.O. (1989a). Singular perturbation approximation of balanced systems. International Journal of Control, 50(4), 1379–405. DOI: 10.1080/00207178908953437

López-Caamal, F. and Marquez-Lago, T.T. (2014). Order reduction of the chemical master equation via balanced realisation. PLoS ONE, 9(8), e103521. DOI: 10.1371/journal.pone.0103521

Lucas, T.N. (1983). Factor division: a useful algorithm in model reduction. IEE Proceedings D: Control Theory and Applications, 130(6), 362–4. DOI: 10.1049/ip-d.1983.0060

Lucas, T.N. (1986). Linear system reduction by the modified factor division method. IEE Proceedings D: Control Theory and Applications, 133(6), 293-29. DOI: 10.1049/ip-d.1986.0047

Manohar, H. and Sambariya, D.K. (2016). Model order reduction of MIMO system using differentiation method. In 10th International Conference on Intelligent Systems and Control, 7-8 Jan. 2016. 1-5.

Mittal, A.K., Prasad, R. and Sharma, S. P. (2004). Reduction of linear dynamic systems using an error minimization technique. Journal of the Institution of Engineers (India): Electrical Engineering Division, 84(2), 201–6.

Mohamed, K.S. (2018). Machine Learning for Model Order Reduction. NY, NY: Springer Publishing Company .

Moore, B.C. (1981). Principal component analysis in linear systems: controllability, observability and model reduction. IEEE Transactions on Automatic Control, 26(1), 17–32. DOI: 10.1109/TAC.1981.1102568

Mukherjee, S. and Mishra, R.N. (1987). Order reduction of linear systems using an error minimization technique. Journal of the Franklin Institute, 323(1),23–32. DOI: 10.1016/0016-0032(87)90037-8

Pal, J. (1979). Stable reduced-order padé approximants using the routh-hurwitz array. Electronics Letters, 15(8), 225–6. DOI: 10.1049/el:19790159

Parmar, G., Mukherjee, S. and Prasad, R. (2007). System reduction using factor division algorithm and eigen spectrum analysis. Applied Mathematical Modelling, 31(11), 2542–552. DOI: 10.1016/j.apm.2006.10.004

Pernebo, L. and Silverman, L.M. (1982). Model reduction via balanced state space representations. IEEE Transactions on Automatic Control, 27(2), 382–7. DOI: 10.1109/TAC.1982.1102945

Prajapati, A.K. and Prasad, R. (2018a). Model order reduction by using the balanced truncation and factor division methods. IETE Journal of Research, 65(6), 827–42. DOI: 10.1080/03772063.2018.1464971

Prajapati, A.K. and Prasad, R. (2018b). Reduced order modelling of linear time invariant systems using the factor division method to allow retention of dominant modes. IETE Technical Review (Institution of Electronics and Telecommunication Engineers, India), 36(5), 449–62. DOI: 10.1080/02564602.2018.1503567

Prajapati, A.K. and Prasad, R. (2019a). Order reduction in linear dynamical systems by using improved balanced realization technique. Circuits, Systems and Signal Processing, 38(11), 5289–303. DOI: 10.1007/s00034-019-01109-x

Prajapati, A.K. and Prasad, R. (2019b). Order reduction of linear dynamic systems by improved routh approximation method. IETE Journal of Research, 65(5), 702–15. DOI: 10.1080/03772063.2018.1452645

Prajapati, A.K. and Prasad, R. (2019c). Reduced-order modelling of lti systems by using routh approximation and factor division methods. Circuits, Systems and Signal Processing, 38(7), 3340–55. DOI: 10.1007/s00034-018-1010-6

Prasad, R. (2000). Pade type model order reduction for multivariable systems using routh approximation. Computers and Electrical Engineering, 26(6), 445–59. DOI: 10.1016/S0045-7906(00)00002-1

Rao, A.S., Lamba, S.S. and Rao, S. V. (1979). Comments on model reduction using the routh stability criterion. IEEE Transactions on Automatic Control, 24(3), 518. DOI: 10.1109/TAC.1979.1102069

Rowley, C.W. (2005). Model reduction for fluids, using balanced proper orthogonal decomposition. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 15(3), 997–1013. DOI: 10.1142/S0218127405012429

Suman, S.K. (2019). Approximation of large-scale dynamical systems for bench-mark collection. Journal of Mechanics of Continua and Mathematical Sciences, 14(3), 196–215. DOI: 10.26782/jmcms.2019.06.00016

Safonov, M.G. and Chiang, R.Y. (1989). A schur method for balanced-truncation model reduction. IEEE Transactions on Automatic Control, 34(7), 729–33. DOI: 10.1109/9.29399

Saksena, V.R., O’Reilly, J. and Kokotovic, P.V. (1984). Singular perturbations and time-scale methods in control theory: Survey 1976-1983. Automatica, 20(3), 273–93. DOI: 10.1016/0005-1098(84)90044-X

Samar, R., Postlethwaite, I. and Gu, D.W. (1995). Model reduction with balanced realizations. International Journal of Control, 62(1), 33-64. https://doi.org/10.1080/00207179508921533

Sambariya, D. and Arvind, G. (2016). High order diminution of lti system using stability equation method. British Journal of Mathematics & Computer Science, 13(5), 1–15. DOI: 10.9734/bjmcs/2016/23243

Sambariya, D.K. and Prasad, R. (2013). Stable reduced model of a single machine infinite bus power system with power system stabilizer. 2013 International Conference on Advances in Technology and Engineering, ICATE 2013. 1-10.

Sambariya, D.K. and Sharma, O. (2016). Routh approximation: an approach of model order reduction in siso and mimo systems. Indonesian Journal of Electrical Engineering and Computer Science, 2(3), 486–500. DOI: 10.11591/ijeecs.v2.i3.pp486-500

Sandberg, H. and Rantzer, A. (2004). Balanced truncation of linear time-varying systems. IEEE Transactions on Automatic Control, 49(2), 217–29. DOI: 10.1109/TAC.2003.822862

Saxena, S. and Hote, Y.V. (2013). Load frequency control in power systems via internal model control scheme and model-order reduction. IEEE Transactions on Power Systems, 28(3), 2749–57. DOI: 10.1109/TPWRS.2013.2245349

Schilders, W. (2008). Model Order Reduction: Theory, Research Aspects and Applications. NY, NY: Springer.

Segalman, D.J. (2007). Model reduction of systems with localized nonlinearities. Journal of Computational and Nonlinear Dynamics, 2(3), 249–66. DOI: 10.1115/1.2727495

Shamash, Y. (1975a). Linear system reduction using pade approximation to allow retention of dominant modes. International Journal of Control, 21(2), 257–72. DOI: 10.1080/00207177508921985

Shamash, Y. (1975b). Model reduction using the routh stability criterion and the pade approximation technique. International Journal of Control, 21(3), 475–84. DOI: 10.1080/00207177508922004

Sikander, A. and Prasad, R. (2015a). Linear time-invariant system reduction using a mixed methods approach. Applied Mathematical Modelling, 39(16), 4848–58. DOI: 10.1016/j.apm.2015.04.014

Sikander, A. and Prasad, R. (2015b). Soft computing approach for model order reduction of linear time invariant systems. Circuits, Systems and Signal Processing, 34(11), 3471–87. DOI: 10.1007/s00034-015-0018-4

Sikander, A. and Prasad, R. (2017). New technique for system simplification using cuckoo search and ESA. Sadhana - Academy Proceedings in Engineering Sciences, 42(9), 1453–8. DOI: 10.1007/s12046-017-0710-0

Singh, J., Vishwakarma, C.B. and Chattterjee, K. (2016). Biased reduction method by combining improved modified pole clustering and improved Pade approximations. Applied Mathematical Modelling, 40(2), 1418–26. DOI: 10.1016/j.apm.2015.07.014

Singh, N., Prasad, R. and Gupta, H.O. (2006). Reduction of linear dynamic systems using routh hurwitz array and factor division method. IETE Journal of Education, 47(1), 25–29. DOI: 10.1080/09747338.2006.11415859

Škatarić, D. and Kovačević, N.R. (2010). The system order reduction via balancing in view of the method of singular perturbation. FME Transactions, 38(4), 181-–7.

Smamash, Y. (2007). Truncation method of reduction: a viable alternative. Electronics Letters, 17(2), 97–9. DOI: 10.1049/el:19810070

Sonker, B., Kumar, D. and Samuel, P. (2017). A modified two-degree of freedom-internal model control configuration for load frequency control of a single area power system. Turkish Journal of Electrical Engineering and Computer Sciences, 25(6), 4624–35. DOI: 10.3906/elk-1701-225

Suman, S.K. and Kumar, A. (2019). Investigation and reduction of large-scale dynamical systems. Wseas Transactions on Systems, 18(1), 175–180.

Suman, S.K. and Kumar, A. (2020a). Higher-order reduction of linear system and design of controller. The Scientific Journal of King Faisal University. DOI: 10.37575/b/eng/2235.

Suman, S.K. and Kumar, A. (2020b). Model order reduction of transmission line model. Wseas Transactions on Circuits and Systems, 19(1), 62-68. DOI: 10.37394/23201.2020.19.7

Tan, W. (2009). Tuning of PID load frequency controller for power systems. Energy Conversion and Management, 50(6), 1465–72. DOI: 10.1016/j.enconman.2009.02.024

Tan, W. (2010). Unified tuning of PID load frequency controller for power systems via IMC. IEEE Transactions on Power Systems, 25(1), 341–35. DOI: 10.1109/TPWRS.2009.2036463

Willcox, K.E. and Peraire, J. (2002). Balanced model reduction via the proper introduction. AIAA Journal, 40(11), 2323–30. DOI: 10.2514/2.1570

Zakian, V. (1973). Simplification of linear time-invariant systems by moment approximants. International Journal of Control, 18(3), 455–60. DOI: 10.1080/00207177308932525

Zhou, K., Salomon, G. and Wu, E. (1999). Balanced realization and model reduction for unstable systems. International Journal of Robust and Nonlinear Control, 9(3), 183–198.