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Advanced order diminution technique for linear time-invariant systems with applications in lag/lead compensators and PID controller design

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Abstract

This paper presents a new order diminution method for large-scale model simplification. The suggested strategy relies on the Mihailov stability method, which promises the stability of the approximated model under the condition that the intricate model is stable. In this approach, the Mihailov stability approach is applied to estimate the denominator coefficients of the original system, while the numerator coefficients are obtained using the Routh stability method. The proposed methodology is both straightforward and designed to guarantee the stability of the reduced-order system (ROS), provided that the higher-order system itself is stable. To ascertain the efficacy of the proposed methodology, a comparison is conducted between the step responses of the actual plant and those of the simplified ROS. The proposed approach undergoes a comparative analysis with various contemporary conventional reduced-order modelling methods, utilizing error indicators such as relative integral square error, integral square error, integral time absolute error, and integral absolute error. The abated system is then used to design controllers for the actual complex model using a Padé-type algorithm. The simulation findings demonstrate that the proposed methodology outperforms the most recent model diminution strategies reported in the literature.

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References

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  3. Prajapati AK, Prasad R (2018) 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

    Article  Google Scholar 

  4. Fortuna L (1992) Reduction techniques with applications in Electrical Engineering, 1st edn. Springer, London Limited, London

    Google Scholar 

  5. Snowden TJ, van der Graaf PH, Tindall MJ (2017) Methods of model reduction for large-scale biological systems: a survey of current methods and trends. Bull Math Biol 79(7):1449–1486

    Article  MathSciNet  Google Scholar 

  6. Javed A, Ahmad MI (2019) Projection-based model order reduction for biochemical systems. In: 2019 international conference on applied and engineering mathematics, ICAEM 2019, Taxila, Pakistan, 27–29 August 2019. IEEE Xplore, p. 133–138

  7. Vasu G, Sivakumar M, Ramalingaraju M (2019) Optimal model approximation of linear time-invariant systems using the enhanced DE algorithm and improved MPPA method. Circuits Syst Signal Process 39(9):2376–2411

    Google Scholar 

  8. Kumar MS, Anand NV, Rao RS (2016) Impulse energy approximation of higher-order interval systems using Kharitonov’s polynomials. Trans Inst Meas Control 38(10):1225–1235

    Article  Google Scholar 

  9. Gautam RK, Singh N, Choudhary NK, Narain A (2019) Model order reduction using factor division algorithm and fuzzy c-means clustering technique. Trans Inst Meas Control 41(2):468–475

    Article  Google Scholar 

  10. Ng B, Coman PT, Mustain WE, White RE (2020) Non-destructive parameter extraction for a reduced order lumped electrochemical-thermal model for simulating Li-ion full-cells. J Power Sources 445:1–12

    Article  Google Scholar 

  11. Celo D, Gunupudi PK, Khazaka R, Walkey DJ, Smy T, Nakhla MS (2005) Fast simulation of steady-state temperature distributions in electronic components using multidimensional model reduction. IEEE Trans Compon Packag Technol 28(1):70–79

    Article  Google Scholar 

  12. Lin YS, Nikravesh PE (2006) Deformable body model reduction with mean-axes. Mech Based Des Struct Mach 34(4):469–488

    Article  Google Scholar 

  13. Ischinger F, Bartel D, Brunk M, Solovyev S (2019) Non-linear model order reduction for elastohydrodynamic lubrication simulations of polymer seals. Tribol Int 140:1–9

    Article  Google Scholar 

  14. Veraszto Z, Ponsioen S, Haller G (2020) Explicit third-order model reduction formulas for general nonlinear mechanical systems. J Sound Vib 468:1–21

    Article  Google Scholar 

  15. Luo L, Dhople SV (2014) Spatiotemporal model reduction of inverter-based islanded microgrids. IEEE Trans Energy Convers 29(4):823–832

    Article  Google Scholar 

  16. Wang L, Long W (2020) Dynamic model reduction of power electronic interfaced generators based on singular perturbation. Electr Power Syst Res 178:1–13

    Article  Google Scholar 

  17. Al-Iedani I, Gajic Z (2020) Order reduction of a wind turbine energy system via the methods of system balancing and singular perturbations. Int J Electr Power Energy Syst 117:1–17

    Article  Google Scholar 

  18. Tong D, Zhou W, Dai A, Wang H (2014) HN model reduction for the distillation column linear system. Circuits Syst Signal Process 33(10):3287–3297

    Article  Google Scholar 

  19. Nguyen VB, Buffoni M, Willcox K, Khoo BC (2014) Model reduction for reacting flow applications. Int J Comput Fluid Dyn 28(3–4):91–105

    Article  MathSciNet  Google Scholar 

  20. Nguyen VB, Tran SBQ, Khan SA, Rong J, Lou J (2020) POD-DEIM model order reduction technique for model predictive control in continuous chemical processing. Comput Chem Eng 133:1–43

    Article  Google Scholar 

  21. Panda S, Yadav J, Patidar N, Ardil C (2009) Evolutionary techniques for model order reduction of large scale linear systems. Int J Appl Sci Eng Technol 5:22

    Google Scholar 

  22. Sivanandam S, Deepa S (2009) A comparative study using genetic algorithm and particle swarm optimization for lower order system modelling. Int J Comput Internet Manag 17:1

    Google Scholar 

  23. Eker E, Kayri M, Ekinci S, Izci D (2021) A new fusion of ASO with SA algorithm and its applications to MLP training and DC motor speed control. Arab J Sci Eng 46(4):3889–3911. https://doi.org/10.1007/s13369-020-05228-5

    Article  Google Scholar 

  24. Izci D, Ekinci S, Eker E, Kayri M (2022) Augmented hunger games search algorithm using logarithmic spiral opposition-based learning for function optimization and controller design. J King Saud Univ Eng Sci. https://doi.org/10.1016/j.jksues.2022.03.001

    Article  Google Scholar 

  25. Izci D, Ekinci S, Eker E, Kayri M (2022) A novel modified opposition-based hunger games search algorithm to design fractional order proportional-integral-derivative controller for magnetic ball suspension system. Adv Control Appl Eng Ind Syst 4:e96. https://doi.org/10.1002/adc2.96

    Article  Google Scholar 

  26. Izci D, Ekinci S, Kayri M, Eker E (2022) A novel improved arithmetic optimization algorithm for optimal design of PID controlled and Bode’s ideal transfer function based automobile cruise control system. Evol Syst 13(3):453–468. https://doi.org/10.1007/s12530-021-09402-4

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  28. Chen TC, Chang CY, Han KW (1979) Reduction of transfer functions by the stability-equation method. J Frankl Inst 308(4):389–404

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  32. Prajapati AK, Prasad R (2018) Failure of padé approximation and time moment matching techniques in reduced order modelling. In: Proceedings of 3rd IEEE international conference for convergence in technology, Pune, India, 6–8 April 2018. IEEE Xplore, pp. 1–4

  33. Prajapati AK, Prasad R, Pal J (2018) Contribution of time moments and Markov parameters in reduced order modeling. In: Proceedings of 3rd IEEE international conference for convergence in technology, Pune, India, 6–8 April 2018. IEEE Xplore, p. 1–7

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

    Article  Google Scholar 

  35. Chen CF, Chang YC, Han WK (1980) Model reduction using the stability-equation method and the Padé approximation method. Int J Control 32(1):81–94

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  39. Wan BW (1981) Linear model reduction using Mihailov criterion and Padé approximation technique. Int J Control 33(6):1073–1089

    Article  Google Scholar 

  40. Singh V (1979) Nonuniquenes of model reduction using the Routh approach. IEEE Trans Automat Control 24(4):650–651

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  45. Prajapati AK, Prasad R (2018) Order reduction of linear dynamic systems with an improved Routh stability method. In: Proceedings of IEEE international conference on control, power, communication and computing technologies, Kannur, India, 23–24 March 2018. IEEE Xplore, p. 362–367

  46. Prajapati AK, Prasad R (2019) 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

    Article  Google Scholar 

  47. Sarasu J, Parthasarathy R (1979) System reduction by Routh approximation and modified Cauer continued fraction. Electron Lett 15(21):691–692

    Article  Google Scholar 

  48. Prajapati AK, Prasad R (2021) 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

    Article  Google Scholar 

  49. Prajapati AK (2019) Model reduction of linear systems by using improved Mihailov stability criterion. In: Proceedings of 11th IEEE international conference on computational intelligence and communication networks (CICN), Honolulu, USA, 3–4 January 2019. IEEE Xplore, p. 1–6

  50. Kumar DK, Nagar SK, Tiwari JP (2011) Model order reduction of interval systems using Mihailov criterion and factor division method. Int J Comput Appl 28(11):8–12

    Google Scholar 

  51. Prajapati AK, Rayudu VGD, Sikander A, Prasad R (2020) A new technique for the reduced-order modelling of linear dynamic systems and design of controller. Circuits Syst Signal Process 39(10):4849–4867

    Article  Google Scholar 

  52. Towill DR (1970) Transfer function techniques for control engineers, 1st edn. Iliffe Books Ltd., London

    Google Scholar 

  53. Peterson WC, Nassar AH (1978) On the synthesis of optimum linear feedback control systems. J Frankl Inst 306(3):237–256

    Article  Google Scholar 

  54. Basilio JC, Matos SR (2002) Design of PI and PID controllers with transient performance specification. IEEE Trans Autom Control 45(4):364–370

    Google Scholar 

  55. Zakian V (1973) Simplification of linear time-invariant systems by moment approximants. Int J Control 18(3):455–460

    Article  MathSciNet  Google Scholar 

  56. Lal M, Mitra R (1974) Simplification of large system dynamics using a moment evaluation algorithm. IEEE Trans Automat Contr 19(5):602–603

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  60. Mukherjee S, Mittal RC (2005) Model order reduction using response-matching technique. J Frankl Inst 342(5):503–519

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  62. Soloklo HN, Farsangi MM (2015) Model order reduction by using Legendre expansion and harmony search algorithm. Majlesi J Electr Eng 9(1):25–35

    Google Scholar 

  63. Philip B, Pal J (2010) 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, 28–29 December 2010. IEEE Xplore, p. 1–8

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

    Article  Google Scholar 

  65. Gu G (2005) All optimal Hankel-norm approximations and their LN error bounds in discrete-time. Int J Control 78(6):408–423

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  67. Kumar D, Tiwari JP, Nagar SK (2012) Reducing order of largescale systems by extended balanced singular perturbation approximation. Int J Autom Control 6(1):21–38

    Article  Google Scholar 

  68. Chen TC, Chang CY, Han KW (1980) Model reduction using the stability-equation method and the continued fraction method. Int J Control 32(1):81–94

    Article  MathSciNet  Google Scholar 

  69. Chen TC, Chang CY, Han KW (1980) Model reduction using the stability-equation method and the Padé approximation method. J Frankl Inst 309(6):473–490

    Article  Google Scholar 

  70. Gautam SK, Nema S, Nema RK (2023) A novel order abatement technique for linear dynamic systems and design of PID controller. IETE Tech Rev. https://doi.org/10.1080/02564602.2023.2268582

    Article  Google Scholar 

  71. Aguirre LA (1991) New algorithm for closed-loop model matching. Electron Lett 27(24):2260–2262

    Article  Google Scholar 

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

    Article  Google Scholar 

  73. Safonov MG, Chiang RY (1989) A Schur method for balanced truncation model reduction. IEEE Trans Autom Control 34(7):729–733

    Article  MathSciNet  Google Scholar 

  74. Glover K (1984) All optimal Hankel-norm approximations of linear multivariable systems and their LN error bounds. Int J Control 39(6):1115–1193

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh, 462003, India

    Sunil Kumar Gautam, Savita Nema & Rajesh Kumar Nema

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  1. Sunil Kumar Gautam
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  2. Savita Nema
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Contributions

SKG and SN contributed to conceptualization, methodology, validation, formal analysis, investigation, resources, data curation, writing—original draft preparation, writing–review and editing, and visualization; RKN was involved in supervision; and all authors have read and agreed to the published version of the manuscript.

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Correspondence to Sunil Kumar Gautam.

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Gautam, S.K., Nema, S. & Nema, R.K. Advanced order diminution technique for linear time-invariant systems with applications in lag/lead compensators and PID controller design. Electr Eng (2024). https://doi.org/10.1007/s00202-024-02400-0

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