Skip to main content
SpringerLink
Log in

A New Generalized Pole Clustering-Based Model Reduction Technique and Its Application for Design of Controllers

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

A new model diminution technique is proposed for the reduction of complexity of higher order linear dynamical systems. In this proposed method, a generalized pole clustering technique is used for obtaining the denominator polynomial of the lower order plant and the numerator polynomial is evaluated by applying the Padé approximation technique. The generalized pole clustering algorithm promises the preservation of stability and dominant poles of the actual system in the reduced order plant. The performance error indices such as integral square error (ISE), integral absolute error (IAE), integral time weighted absolute error (ITAE) and relative integral square error (RISE) are used to validate the proposed technique. By using the transfer function of the simplified order plant, the PID and lead/lag compensators are designed by using a moment matching algorithm. This controller is applied to the original large scale system and the response of the closed loop system is matching with the response of the desired reference model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data Availability

All data generated or analyzed during this study are included in this article.

References

  1. A.B.H. Adamou-Mitiche, L. Mitiche, Multivariable systems model reduction based on the dominant modes and Genetic algorithm. IEEE Trans. Ind. Electron. 64(2), 1617–1619 (2017)

    Google Scholar 

  2. H. Aridhi, M.H. Zaki, S. Tahar, Enhancing model Order Reduction for Nonlinear Analog Circuit Simulation. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 24(3), 1–14 (2016)

    Google Scholar 

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

    MATH  Google Scholar 

  4. J.C. Basilio, S.R. Matos, Design of PI and PID controllers with transient performance specification. IEEE Trans. Educ. 45(4), 364–370 (2002)

    Google Scholar 

  5. R. Bhatt, G. Parmar, R. Gupta, A. Sikander, Application of stochastic fractal search in approximation and control of LTI systems. Microsyst. Technol. 25(1), 105–114 (2019)

    Google Scholar 

  6. D. Binion, X. Chen, Coupled electrothermal–mechanical analysis for MEMS via model order reduction. Finite Elem. Anal. Des. 46(12), 1068–1076 (2010)

    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. Franklin Inst. 309(6), 473–490 (1980)

    MathSciNet  MATH  Google Scholar 

  8. T.C. Chen, C.Y. Chang, K.W. Han, Reduction of transfer functions by the stability-equation method. J. Franklin Inst. 308(4), 389–404 (1979)

    MathSciNet  MATH  Google Scholar 

  9. T.C. Chen, C.Y. Chang, K.W. Han, Model reduction using the stability-equation method and the continued fraction method. Int. J. Control 32(1), 81–94 (1980)

    MathSciNet  MATH  Google Scholar 

  10. A. Chu, H. Chiang, Constructing analytical energy functions for lossless network-reduction power system models; framework and new developments. Circuits Syst. Signal Process. 18(1), 1–16 (1999)

    MATH  Google Scholar 

  11. S.R. Desai, R. Prasad, A new approach to order reduction using stability equation and big bang big crunch optimization. Syst. Sci. Control Eng. 1(1), 20–27 (2013)

    Google Scholar 

  12. K.V. Fernando, H. Nicholson, Singular perturbational model reduction of balanced systems. IEEE Trans. Autom. Control 27(2), 466–468 (1982)

    MATH  Google Scholar 

  13. A. Ghafoor, M. Imran, Passivity preserving frequency weighted model order reduction technique. Circuits Syst. Signal Process. 36(11), 4388–4400 (2017)

    MathSciNet  MATH  Google Scholar 

  14. A.N. Gorban, Model reduction in chemical dynamics : slow invariant manifolds, singular perturbations, thermodynamic estimates, and analysis of reaction graph. Curr. Opin. Chem. Eng. 21, 48–59 (2018)

    Google Scholar 

  15. R. Goyal, P. Girish, A. Sikander, A new approach for simplification and control of linear time invariant systems. Microsyst. Technol. 25, 599–607 (2019)

    Google Scholar 

  16. A. Graham, R.C. Lathrop, The synthesis of ‘optimum’ transient response: Criteria and standard forms. Trans. Am. Inst. Electr. Eng. Part II: Appl. Ind. 72(5), 273–288 (1953)

    Google Scholar 

  17. G. Gu, All optimal Hankel-norm approximations and their L∞ error bounds in discrete-time. Int. J. Control 78(6), 408–423 (2005)

    MathSciNet  MATH  Google Scholar 

  18. Y. Gu, N. Bottrell, T.C. Green, Reduced-order models for representing converters in power system studies. IEEE Trans. Power Electron. 33(4), 3644–3654 (2017)

    Google Scholar 

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

    MATH  Google Scholar 

  20. C. Hsieh, P.C. Hwang, Model reduction of continuous-time systems using a modified Routh approximation method. IEE Proc. 136(1), 151–156 (1989)

    MATH  Google Scholar 

  21. C. Huang, K. Zhang, X. Dai, W. Tang, A modified balanced truncation method and its application to model reduction of power system, in Proceedings on IEEE Power and Energy Society General Meeting, Vancouver, BC, Canada, pp. 1–5 (2013)

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

    MathSciNet  MATH  Google Scholar 

  23. T.C. Ionescu, A. Astolfi, Nonlinear moment matching-based model order reduction. IEEE Trans. Autom. Control 61(10), 2837–2847 (2016)

    MathSciNet  MATH  Google Scholar 

  24. T. Ishizaki, H. Sandberg, K. Kashima, J. Imura, K. Aihara, Dissipativity-preserving model reduction for large-scale distributed control systems. IEEE Trans. Autom. Control 60(4), 1023–1037 (2015)

    MathSciNet  MATH  Google Scholar 

  25. J.P. Liu, X.B. Shu, H. Kanazawa, K. Imaoka, A.G. Mikkola, X. Ren, A model order reduction method for the simulation of gear contacts based on Arbitrary Lagrangian Eulerian formulation. Comput. Methods Appl. Mech. Eng. 338(1), 68–96 (2018)

    MathSciNet  MATH  Google Scholar 

  26. M. Jamshidi, Large-Scale Systems: Modeling, Control, and Fuzzy Logic, 1st edn. (Prentice Hall PTR, Upper Saddle River, 1998)

    MATH  Google Scholar 

  27. R. Komarasamy, N. Albhonso, G. Gurusamy, Order reduction of linear systems with an improved pole clustering. J. Vib. Control 18(12), 1876–1885 (2011)

    MathSciNet  Google Scholar 

  28. D.K. Kranthi, 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)

    Google Scholar 

  29. V. Krishnamurthy, V. Seshadri, Model reduction using the Routh stability criterion. IEEE Trans. Autom. Control 23(4), 729–731 (1978)

    Google Scholar 

  30. D. Kumar, S.K. Nagar, Model reduction by extended minimal degree optimal Hankel norm approximation. Appl. Math. Model. 38(11–12), 2922–2933 (2014)

    MathSciNet  MATH  Google Scholar 

  31. D. Kumar, J.P. Tiwari, S.K. Nagar, Reducing order of large-scale systems by extended balanced singular perturbation approximation. Int. J. Autom. Control 6(1), 21–38 (2012)

    Google Scholar 

  32. V. Kumar, J.P. Tiwari, Order reducing of linear system using clustering method factor division algorithm. Int. J. Appl. Inf. Syst. 3(5), 3–6 (2012)

    Google Scholar 

  33. D. Kun, E.L. Shengbo, L. Sisi, L. Zhaojian, Aggregation-based thermal model reduction, in Automotive Air Conditioning (Springer, Basel, 2016), pp. 29–49

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

    MATH  Google Scholar 

  35. B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26(1), 17–32 (1981)

    MathSciNet  MATH  Google Scholar 

  36. S. Mukherjee, R.C. Mittal, Model order reduction using response-matching technique. J. Franklin Inst. 342, 503–519 (2005)

    MathSciNet  MATH  Google Scholar 

  37. 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 (2016)

    Google Scholar 

  38. A. Narwal, R. Prasad, Optimization of LTI systems using modified clustering algorithm. IETE Tech. Rev. 34(2), 201–213 (2017)

    Google Scholar 

  39. A. Narwal, R. Prasad, Order reduction of LTI systems and their qualitative comparison. IETE Tech. Rev. 34(6), 655–663 (2017)

    Google Scholar 

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

    Google Scholar 

  41. Y. Paquay, O. Brüls, C. Geuzaine, Nonlinear interpolation on manifold of reduced-order models in magnetodynamic problems. IEEE Trans. Magn. 52(3), 18–21 (2016)

    Google Scholar 

  42. G. Parmar, R. Prasad, S. Mukherjee, Order reduction by least-squares methods about general point ‘a.’ Int. J. Electron. Commun. Eng. 1(2), 268–255 (2007)

    Google Scholar 

  43. G. Parmar, S. Mukherjee, R. Prasad, System reduction using eigen spectrum analysis and Padé approximation technique. Int. J. Comput. Math. 84(12), 1871–1880 (2007)

    MathSciNet  MATH  Google Scholar 

  44. G. Parmar, S. Mukherjee, R. Prasad, System reduction using factor division algorithm and eigen spectrum analysis. Appl. Math. Model. 31(11), 2542–2552 (2007)

    MATH  Google Scholar 

  45. W.C. Peterson, A.H. Nassar, On the synthesis of optimum linear feedback control systems. J. Franklin Inst. 306(3), 237–256 (1978)

    MATH  Google Scholar 

  46. B. Philip, J. Pal, An evolutionary computation based approach for reduced order modelling of linear systems, in IEEE International Conference on Computational Intelligence and Computing Research, 2010, Coimbatore, India, pp. 1–8 (2010)

  47. A. Pierquin, T. Henneron, S. Clenet, Data-driven model-order reduction for magnetostatic problem coupled with data-driven model order reduction for magnetostatic problem coupled with circuit equations. IEEE Trans. Magn. 54(3), 1–4 (2018)

    Google Scholar 

  48. A.K. Prajapati, R. Prasad, Failure of Padé approximation and time moment matching techniques in reduced order modelling, in Proceedings 3rd IEEE International Conference for Convergence in Technology, Pune, India, pp. 1–4 (2018)

  49. A.K. Prajapati, R. Prasad, Model order reduction by using the balanced truncation and factor division methods. IETE J. Res. 65(6), 827–842 (2018)

    Google Scholar 

  50. A.K. Prajapati, R. Prasad, Order reduction of linear dynamic systems by improved Routh approximation method. IETE J. Res. 65(5), 702–715 (2018)

    Google Scholar 

  51. A.K. Prajapati, R. Prasad, Order reduction of linear dynamic systems with an improved Routh stability method, in Proceedings IEEE International Conference on Control, Power, Communication and Computing Technologies, Kerala, India, pp. 362–367 (2018)

  52. A.K. Prajapati, R. Prasad, Reduced order modelling of linear time invariant systems using the factor division method to allow retention of dominant modes. IETE Tech. Rev. 36(5), 449–462 (2018)

    Google Scholar 

  53. A.K. Prajapati, R. Prasad, A new model order reduction method for the design of compensator by using moment matching algorithm. Trans. Inst. Meas. Control. 42(3), 472–484 (2019)

    Google Scholar 

  54. A.K. Prajapati, R. Prasad, A new model reduction method for the linear dynamic systems and its application for the design of compensator. Circuits Syst. Signal Process. 39(5), 2328–2348 (2019)

    MATH  Google Scholar 

  55. A.K. Prajapati, R. Prasad, Model reduction of linear systems by using improved Mihailov stability criterion, in Proceedings 11th IEEE International Conference on Computational Intelligence and Communication Networks (CICN), Hawaii, USA, pp. 1–6 (2019)

  56. A.K. Prajapati, R. Prasad, Order reduction in linear dynamical systems by using improved balanced realization technique. Circuits Syst. Signal Process. 38(11), 5298–5303 (2019)

    Google Scholar 

  57. A.K. Prajapati, R. Prasad, Reduced-order modelling of LTI systems by using Routh approximation and factor division methods. Circuits Syst. Signal Process. 38(7), 3340–3355 (2019)

    Google Scholar 

  58. A.K. Prajapati, R. Prasad, A novel order reduction method for linear dynamic systems and its application for designing of PID and lead / lag compensators. Trans. Inst. Meas. Control. 43(5), 1226–1238 (2021)

    Google Scholar 

  59. A.K. Prajapati, R. Prasad, Model reduction using the balanced truncation method and the Padé approximation method. IETE Tech. Rev. (2020). https://doi.org/10.1080/02564602.2020.1842257

    Article  Google Scholar 

  60. A.K. Prajapati, R. Prasad, J. Pal, Contribution of time moments and Markov parameters in reduced order modeling, in Proceedings 3rd IEEE International Conference for Convergence in Technology, Pune, India, pp. 1–7 (2018)

  61. A.K. Prajapati, V.G.D. Rayudu, A. Sikander, R. Prasad, A new technique for the reduced-order modelling of linear dynamic systems and design of controller. Circuits Syst. Signal Process. 39(10), 4849–4867 (2020)

    Google Scholar 

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

    MATH  Google Scholar 

  63. M. Rydel, R. Stanisławski, A new frequency weighted Fourier-based method for model order. Automatica 88, 107–112 (2018)

    MathSciNet  MATH  Google Scholar 

  64. M.G. Safonov, R.Y. Chiang, A Schur method for balanced-truncation model reduction. IEEE Trans. Autom. Control 34(7), 729–733 (1989)

    MathSciNet  MATH  Google Scholar 

  65. 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), 1–14 (2016)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  69. Y. Shamash, Truncation method of reduction: a viable alternative. Electron. Lett. 17(2), 97–99 (1981)

    MathSciNet  Google Scholar 

  70. T. Shimotani, Y. Sato, T. Sato, H. Igarashi, Fast finite-element analysis of motors using block model order reduction. IEEE Trans. Magn. 52(3), 3–6 (2016)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  72. A. Sikander, R. Prasad, A new technique for reduced-order modelling of linear time-invariant system. IETE J. Res. 63(3), 316–324 (2017)

    Google Scholar 

  73. A. Sikander, R. Prasad, New technique for system simplification using Cuckoo search and ESA. Sadhana – Acad. Proc. Eng. Sci. 42(9), 1453–1458 (2017)

    MathSciNet  MATH  Google Scholar 

  74. J. Singh, C.B. Vishwakarma, K. Chatterjee, Biased reduction method by combining improved modified pole clustering and improved Padé approximations. Appl. Math. Model. 40(2), 1418–1426 (2016)

    MathSciNet  MATH  Google Scholar 

  75. 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)

    Google Scholar 

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

    Google Scholar 

  77. H.N. Soloklo, M.M. Farsangi, Model order reduction by using Legendre expansion and harmony search algorithm. Majlesi J. Electr. Eng. 9(1), 25–35 (2015)

    Google Scholar 

  78. S.K. Tiwari, G. Kaur, Model reduction by new clustering method and frequency response matching. J. Control Autom. Electr. Syst. 28(1), 78–85 (2017)

    Google Scholar 

  79. S.K. Tiwari, G. Kaur, Improved reduced-order modeling using clustering method with dominant pole retention. IETE J. Res. 66(1), 42–52 (2018)

    Google Scholar 

  80. D. Tong, Q. Chen, Delay and its time-derivative-dependent model reduction for neutral-type control system. Circuits Syst. Signal Process. 36(6), 2542–2557 (2017)

    MathSciNet  MATH  Google Scholar 

  81. D. Tong, W. Zhou, A. Dai, H. Wang, X. Mou, Y. Xu, H∞ model reduction for the distillation column linear system. Circuits Syst. Signal Process. 33(10), 3287–3297 (2014)

    MathSciNet  MATH  Google Scholar 

  82. D.R. Towill, Transfer Function Techniques for Control Engineers (First Edit. IliffeBooks Ltd., London, 1970)

    Google Scholar 

  83. M.K. Transtrum, A.T. Sari, A.M. Stankovi, Measurement-directed reduction of dynamic models in power systems. IEEE Trans. Power Syst. 32(3), 2243–2253 (2017)

    Google Scholar 

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

    Google Scholar 

  85. C.B. Vishwakarma, R. Prasad, Order reduction using modified pole clustering and Padé approximations. Int. J. Electr. Comput. Eng. 5(8), 1003–1007 (2011)

    Google Scholar 

  86. C.B. Vishwakarma, R. Prasad, MIMO system reduction using modified pole clustering and Genetic Algorithm. Modell. Simul. Eng. 2009(1), 1–5 (2009)

    Google Scholar 

  87. C.B. Vishwakarma, R. Prasad, Time domain model order reduction using Hankel matrix approach. J. Franklin Inst. 351(6), 3445–3456 (2014)

    MathSciNet  MATH  Google Scholar 

  88. P. Vorobev, P. Huang, M.A. Hosani, J.L. Kirtley, K. Turitsyn, High-fidelity model order reduction for microgrids stability assessment. IEEE Trans. Power Syst. 33(1), 874–887 (2018)

    Google Scholar 

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

    MATH  Google Scholar 

  90. Q. Wang, Y. Wang, E.Y. Lam, N. Wong, Model order reduction for neutral systems by moment matching. Circuits Syst. Signal Process. 32(3), 1039–1063 (2013)

    MathSciNet  Google Scholar 

  91. X. Wang, M. Yu, C. Wang, Structure-preserving-based model-order reduction of parameterized interconnect systems. Circuits Syst. Signal Process. 37(1), 19–48 (2018)

    MathSciNet  MATH  Google Scholar 

  92. W. Zhou, D. Tong, H. Lu, Q. Zhong, J. Fang, Time-delay dependent H∞ model reduction for uncertain stochastic systems : continuous-time case. Circuits Syst. Signal Process. 30(5), 941–961 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electronics Engineering, Vellore Institute of Technology, Andhra Pradesh, Amaravati, 522237, India

    Arvind Kumar Prajapati

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

    Rajendra Prasad

Authors
  1. Arvind Kumar Prajapati
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajendra Prasad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arvind Kumar Prajapati.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Prajapati, A.K., Prasad, R. A New Generalized Pole Clustering-Based Model Reduction Technique and Its Application for Design of Controllers. Circuits Syst Signal Process 41, 1497–1529 (2022). https://doi.org/10.1007/s00034-021-01860-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-021-01860-0

Keywords

Search

Navigation