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Reduced-Order Modelling of LTI Systems by Using Routh Approximation and Factor Division Methods

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Abstract

In this paper, a new model reduction technique for the large-scale continuous time systems is proposed. The proposed technique is a mixed method of Routh approximation and factor division techniques. In this technique, the Routh approximation method is applied for determining the denominator coefficients of the reduced model and the numerator coefficients are calculated by the factor division method. The proposed technique has two main advantages as it gives the stable reduced-order model if the original model is stable and ensures the retention of first “r” number of time moments of the actual system in the rth-order reduced system. This method is also applicable for those systems for which Routh approximation method fails. To illustrate the proposed method, a real-time system model is reduced where the reduced model retains the fundamental properties of the actual model. In order to examine the efficiency, accuracy and comparison to other existing standard model reduction methods, the presented technique has been verified on two standard numerical examples taken from the literature.

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References

  1. N. Ashoor, V. Singh, A note on lower order modeling. IEEE Trans. Autom. Control 27(5), 1124–1126 (1982)

    Article  MATH  Google Scholar 

  2. B. Bandyopadhyay, O. Ismail, R. Gorez, Routh–Padé approximation for interval systems. IEEE Trans. Autom. Control 39(12), 2454–2456 (1994)

    Article  MATH  Google Scholar 

  3. B. Bandyopadhyay, A. Upadhye, O. Ismail, γ–δ Routh approximation for interval systems. IEEE Trans. Autom. Control 42(8), 1127–1130 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Biradar, Y.V. Hote, S. Saxena, Reduced-order modeling of linear time invariant systems using big bang big crunch optimization and time moment matching method. Appl. Math. Model. 40(15–16), 7225–7244 (2016)

    Article  MathSciNet  Google Scholar 

  5. A. Birouche, B. Mourllion, M. Basset, Model order-reduction for discrete-time switched linear systems. Int. J. Syst. Sci. 43(9), 1753–1763 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. T.C. Chen, C.Y. Chang, Reduction of transfer functions by the stability-equation method. J. Frankl. Inst. 308(4), 389–404 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. T.C. Chen, C.Y. Chang, K.W. Han, Model reduction using the stability-equation method and the Padé approximation method. J. Frankl. Inst. 309(6), 473–490 (1980)

    Article  MATH  Google Scholar 

  8. T.C. Chen, C.Y. Chang, K.W. Han, Stable reduced-order Padé approximants using stability-equation method. Electron. Lett. 16(9), 345–346 (1980)

    Article  Google Scholar 

  9. W. Chen, X. Li, Model predictive control based on reduced order models applied to belt conveyor system. ISA Trans. 65, 350–360 (2016)

    Article  Google Scholar 

  10. S.R. Desai, R. Prasad, A novel order diminution of LTI systems using Big Bang Big Crunch optimization and Routh approximation. Appl. Math. Model. 37, 8016–8028 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Donida, F. Casella, G. Ferretti, Model order reduction for object-oriented models: a control systems perspective. Math. Comput. Model. Dyn. Syst. 16(3), 269–284 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Feng, J.G. Korvink, P. Benner, A fully adaptive scheme for model order reduction based on moment matching. IEEE Trans. Compon. Packag. Manuf. Technol. 5(12), 1872–1884 (2015)

    Article  Google Scholar 

  13. L. Fortuna, G. Nunnari, A. Gallo, Model Order Reduction Techniques with Applications in Electrical Engineering (Springer, London, 1992)

    Book  Google Scholar 

  14. S. Ghosh, N. Senroy, Balanced truncation approach to power system model order reduction. Electr. Power Compon. Syst. 41(8), 747–764 (2013)

    Article  Google Scholar 

  15. P. Gutman, C. Mannerfelt, P. Molander, Contributions to the model reduction problem. IEEE Trans. Autom. Control 27(2), 454–455 (1982)

    Article  MATH  Google Scholar 

  16. L.C. Hibbeler, M.M.C. See, J. Iwasaki, K.E. Swartz, R.J.O. Malley, B.G. Thomas, A reduced-order model of mould heat transfer in the continuous casting of steel. Appl. Math. Model. 40(19–20), 8530–8551 (2016)

    Article  MathSciNet  Google Scholar 

  17. C.S. Hsieh, C. Hwang, Reduced-order modeling of MIMO discrete systems using bilinear block Routh approximants. J. Chin. Inst. Eng. 12(4), 529–538 (1989)

    Article  MathSciNet  Google Scholar 

  18. C.S. Hsu, D. Hsu, Reducing unstable linear control systems via real Schur transformation. Electron. Lett. 27(11), 984–986 (1991)

    Article  Google Scholar 

  19. M.F. Hutton, B. Friedland, Routh approximations for reducing order of linear, time-invariant systems. IEEE Trans. Autom. Control 20(3), 329–337 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  20. C. Hwang, T.Y. Guo, L.S. Sheih, Model reduction using new optimal Routh approximant technique. Int. J. Control 55(4), 989–1007 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. C. Hwang, K.Y. Wang, Optimal Routh approximations for continuous time systems. Int. J. Syst. Sci. 15(3), 249–259 (1984)

    Article  MATH  Google Scholar 

  22. M. Imran, A. Ghafoor, V. Sreeram, A frequency weighted model order reduction technique and error bounds. Automatica 50, 3304–3309 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. O. Ismail, B. Bandyopadhyay, R. Gorez, Discrete interval system reduction using Padé approximation to allow retention of dominant poles. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 44(11), 1075–1078 (1997)

    Article  Google Scholar 

  24. C.Y. Jin, K.H. Ryu, S.W. Sung, J. Lee, I.B. Lee, PID auto-tuning using new model reduction method and explicit PID tuning rule for a fractional order plus time delay model. J. Process Control 24(1), 113–128 (2014)

    Article  Google Scholar 

  25. H.J. Kojakhmetov, J.M.A. Scherpen, Model order reduction and composite control for a class of slow-fast systems around a non-hyperbolic point. IEEE Control Syst. Lett. 1(1), 68–73 (2017)

    Article  Google Scholar 

  26. D. Kumar, S.K. Nagar, Reducing power system models by Hankel norm approximation technique. Int. J. Model. Simul. 33(3), 139–143 (2013)

    Google Scholar 

  27. D.K. Kumar, S.K. Nagar, J.P. Tiwari, A new algorithm for model order reduction of interval systems. Bonfring Int. J. Data Min. 3(1), 6–11 (2013)

    Article  Google Scholar 

  28. V. Krishnamurthy, V. Seshadri, Model reduction using the Routh stability criterion. IEEE Trans. Autom. Control 23(3), 729–731 (1978)

    Article  Google Scholar 

  29. B.C. Kuo, Automatic control systems, 7th edn. (Prentice Hall Inc., Upper Saddle River, 1995)

    Google Scholar 

  30. M. Lal, R. Mitra, Simplification of large system dynamics using a moment evaluation algorithm. IEEE Trans. Autom. Control 19(5), 602–603 (1974)

    Article  MathSciNet  Google Scholar 

  31. G. Langholz, D. Feinmesser, Model order reduction by Routh approximations. Int. J. Syst. Sci. 9(5), 493–496 (1978)

    Article  MATH  Google Scholar 

  32. A. Lepschy, U. Viaro, An improvement in the Routh–Padé approximation techniques. Int. J. Control 36(4), 643–661 (1982)

    Article  MATH  Google Scholar 

  33. T.N. Lucas, Factor division: a useful algorithm in model reduction. IEE Proc. D Control Theory Appl. 130(6), 362–364 (1983)

    Article  MATH  Google Scholar 

  34. B.C. Moore, Principal component analysis in control system: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26(1), 17–36 (1981)

    Article  MATH  Google Scholar 

  35. A. Narwal, R. Prasad, A novel order reduction approach for LTI systems using cuckoo search optimization and stability equation. IETE J. Res. 62(2), 154–163 (2015)

    Article  Google Scholar 

  36. A. Narwal, R. Prasad, Optimization of LTI systems using modified clustering algorithm. IETE Tech. Rev. 34(2), 201–213 (2016)

    Article  Google Scholar 

  37. T.S. Nguyen, T.L. Duc, T.S. Tran, J.M. Guichon, O. Chadebec, G. Meunier, Adaptive multipoint model order reduction scheme for large-scale inductive PEEC circuits. IEEE Trans. Electromagn. Compat. 59(4), 1143–1151 (2017)

    Article  Google Scholar 

  38. Y. Ni, C. Li, Z. Du, G. Zhang, Model order reduction based dynamic equivalence of a wind farm. Int. J. Electr. Power Energy Syst. 83, 96–103 (2016)

    Article  Google Scholar 

  39. J. Pal, Stable reduced-order Padé approximants using the Routh-Hurwitz array. Electron. Lett. 15(8), 225–226 (1979)

    Article  Google Scholar 

  40. D.P. Papadopoulo, D.V. Bandekas, Routh approximation method applied to order reduction of linear MIMO systems. Int. J. Syst. Sci. 55(4), 203–210 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  41. G. Parmar, S. Mukherjee, R. Prasad, System reduction using factor division algorithm and Eigen spectrum analysis. Appl. Math. Model. 31, 2542–2552 (2007)

    Article  MATH  Google Scholar 

  42. A.K. Prajapati, R. Prasad, Model order reduction by using the balanced truncation method and the factor division algorithm. IETE J. Res. (2018). https://doi.org/10.1080/03772063.2018.1464971

    Article  Google Scholar 

  43. A.K. Prajapati, R. Prasad, Order reduction of linear dynamic systems by improved Routh approximation method. IETE J. Res. (2018). https://doi.org/10.1080/03772063.2018.1452645

    Article  Google Scholar 

  44. A.K. Prajapati, R. Prasad, Padé approximation and its failure in reduced order modelling, in 1st Int. Conf. on Recent Innovations in Elect. Electr. and Comm. Sys. (RIEECS-2017), Dehradun, India (2017)

  45. R. Prasad, Padé type model order reduction for multivariable systems using Routh approximation. Comput. Electr. Eng. 26(6), 445–459 (2000)

    Article  MATH  Google Scholar 

  46. A. Ramirez, A.M. Sani, D. Hussein, M. Matar, M.A. Rahman, J.J. Chavez, A. Davoudi, S. Kamalasadan, Application of balanced realizations for model-order reduction of dynamic power system equivalents. IEEE Trans. Power Deliv. 31(5), 2304–2312 (2016)

    Article  Google Scholar 

  47. E.R. Samuel, L. Knockaert, T. Dhaene, Matrix-interpolation-based parametric model order reduction for multiconductor transmission lines with delays. IEEE Trans. Circuits Syst. II Express Briefs 62(3), 276–280 (2015)

    Article  Google Scholar 

  48. G.V.K.R. Sastry, V. Krishnamurthy, Biased model reduction by simplified Routh approximation method. Electron. Lett. 23(20), 1045–1047 (1987)

    Article  Google Scholar 

  49. G.V.K.R. Sastry, V. Krishnamurthy, Relative stability using simplified Routh approximation method (SRAM). IETE J. Res. 33(3), 99–101 (1987)

    Article  Google Scholar 

  50. S.S. Sazhin, E. Shchepakina, V. Sobolev, Order reduction in models of spray ignition and combustion. Combust. Flame 187, 122–128 (2018)

    Article  Google Scholar 

  51. G. Scarciotti, A. Astolfi, Data-driven model reduction by moment matching for linear and nonlinear systems. Automatica 79, 340–351 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  52. G. Scarciotti, A. Astolfi, Model reduction of neutral linear and nonlinear time-invariant time-delay systems with discrete and distributed delays. IEEE Trans. Autom. Control 61(6), 1438–1451 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  53. Y. Shamash, Failure of the Routh-Hurwitz method of reduction. IEEE Trans. Autom. Control 25(2), 313–314 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  54. Y. Shamash, Linear system reduction using Padé approximation to allow retention of dominant modes. Int. J. Control 21(2), 257–272 (1975)

    Article  MATH  Google Scholar 

  55. Y. Shamash, Model reduction using the Routh stability criterion and the Padé approximation technique. Int. J. Control 21(3), 475–484 (1975)

    Article  MATH  Google Scholar 

  56. Y. Shamash, Stable biased reduced order models using the Routh method of reduction. Int. J. Syst. Sci. 11(5), 641–654 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  57. Y. Shamash, Stable reduced-order models using Padé-type approximations. IEEE Trans. Autom. Control 19(5), 615–616 (1974)

    Article  MATH  Google Scholar 

  58. Y. Shamash, Truncation method of reduction: a viable alternative. Electron. Lett. 17(2), 79–98 (1981)

    Article  MathSciNet  Google Scholar 

  59. A. Sikander, R. Prasad, Linear time-invariant system reduction using a mixed methods approach. Appl. Math. Model. 39(16), 4848–4858 (2015)

    Article  MathSciNet  Google Scholar 

  60. A. Sikander, R. Prasad, Soft computing approach for model order reduction of linear time invariant systems. Circuits Syst. Signal Process. 34(11), 3471–3487 (2015)

    Article  Google Scholar 

  61. V. Singh, Non-uniqueness of model reduction using the Routh approach. IEEE Trans. Autom. Control 24(4), 650–651 (1979)

    Article  Google Scholar 

  62. N. Singh, R. Prasad, H.O. Gupta, Reduction of linear dynamic systems using Routh Hurwitz array and factor division method. IETE J. Educ. 47(1), 25–29 (2006)

    Article  Google Scholar 

  63. N. Singh, R. Prasad, H.O. Gupta, Reduction of power system model using balanced realization, Routh and Padé approximation methods. Int. J. Model. Simul. 28(1), 57–63 (2008)

    Article  Google Scholar 

  64. A.K. Sinha, J. Pal, Simulation based reduced order modelling using a clustering technique. Comput. Electr. Eng. 16(3), 159–169 (1990)

    Article  Google Scholar 

  65. S. Timme, K.J. Badcock, A.D. Ronch, Gust analysis using computational fluid dynamics derived reduced order models. J. Fluids Struct. 71, 116–125 (2017)

    Article  Google Scholar 

  66. H.F. Vanlandingham, B.Z. Yang, Model order reduction using a nonlinear model strategy. Int. J. Model. Simul. 12(2), 34–38 (2016)

    Article  Google Scholar 

  67. C.B. Vishwakarma, Order reduction using modified pole clustering and Padé approximations. Int. J. Electr. Comput. Energ. Electron. Commun. Eng. 5(8), 998–1002 (2011)

    MathSciNet  Google Scholar 

  68. C.B. Vishwakarma, R. Prasad, Clustering method for reducing order of linear system using Padé approximation. IETE J. Res. 54(5), 326–330 (2008)

    Article  Google Scholar 

  69. B.W. Wan, Linear model reduction using Mihailov criterion and Padé approximation technique. Int. J. Control 33(6), 1073–1089 (1981)

    Article  MATH  Google Scholar 

  70. Z.H. Xiao, Y.L. Jiang, Multi-order Arnoldi-based model order reduction of second-order time-delay systems. Int. J. Syst. Sci. 47(12), 2925–2934 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  71. J. Yang, C.S. Chen, J.A.D.A. Garcia, Y. Xu, Model reduction of unstable systems. Int. J. Syst. Sci. 24(12), 2407–2414 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  72. V. Zakian, Simplification of linear time invariant systems by moment approximations. Int. J. Control 18, 455–460 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  73. A. Zilochian, Balanced structures and model reduction of unstable systems, in IEEE SOUTHEASTCON ‘91, Williamsburg, VA, USA (1991), pp. 1198–1201

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  1. Electrical Engineering Department, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247667, India

    Arvind Kumar Prajapati & Rajendra Prasad

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  1. Arvind Kumar Prajapati
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  2. Rajendra Prasad
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Correspondence to Arvind Kumar Prajapati.

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Prajapati, A.K., Prasad, R. Reduced-Order Modelling of LTI Systems by Using Routh Approximation and Factor Division Methods. Circuits Syst Signal Process 38, 3340–3355 (2019). https://doi.org/10.1007/s00034-018-1010-6

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