Neural Computing and Applications

, Volume 31, Issue 10, pp 6207–6221 | Cite as

A novel hybrid metaheuristic algorithm for model order reduction in the delta domain: a unified approach

  • Souvik GanguliEmail author
  • Gagandeep Kaur
  • Prasanta Sarkar
Original Article


Delta operator parameterization provides a unified framework in modeling, analysis and design of discrete-time systems, in which the resultant model converges to its continuous-time counterpart at high sampling limit. Capitalizing this unique property of delta operator, a new hybrid algorithm combining gray wolf optimizer and firefly algorithm has been proposed for model order reduction of high-dimensional linear discrete-time system. It has been shown that the reduced discrete-time model inherits all the dominant characteristics of the higher-order discrete-time model and with the increase in sampling frequency it converges to the continuous-time reduced model. The effectiveness of the proposed method is illustrated with the help of an example.


Model order reduction (MOR) Pseudo random binary sequence (PRBS) Delta operator modeling Hybrid gray wolf optimizer (HGWO) 



Artificial bee colony


Binary gray wolf optimizer


Differential evolution


Evolutionary population dynamics gray wolf optimizer


Firefly algorithm


Genetic algorithms


Gray wolf optimizer


Harmony search


Hybrid gray wolf optimizer


Integral of absolute error


Integral of time weighted absolute error


Integral of square error


Integral of time weighted square error


Invasive weed optimization


Pseudo random binary sequence


Particle swarm optimization


Micro-electro-mechanical system


Multi-input multi-output


Model order reduction


Single-input single-output


Sum of square error


  1. 1.
    Lucia DJ, Beran PS, Silva WA (2004) Reduced-order modeling: new approaches for computational physics. Prog Aerosp Sci 40(1):51–117CrossRefGoogle Scholar
  2. 2.
    Nayfeh AH, Younis MI, Abdel-Rahman EM (2005) Reduced-order models for MEMS applications. Nonlinear Dyn 41(1):211–236MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dorneanu B, Bildea CS, Grievink J (2009) On the application of model reduction to plantwide control. Comput Chem Eng 33(3):699–711CrossRefGoogle Scholar
  4. 4.
    Schilders WH, Van der Vorst HA, Rommes J (2008) Model order reduction: theory, research aspects and applications. Springer, Berlin. ISBN 978-3-540-78841-6CrossRefGoogle Scholar
  5. 5.
    Fortuna L, Nunnari G, Gallo A (2012) Model order reduction techniques with applications in electrical engineering. Springer, Berlin. ISBN 978-1-4471-3198-4Google Scholar
  6. 6.
    Mukherjee S, Mittal RC (2005) Order reduction of linear discrete systems using a genetic algorithm. Appl Math Model 29(6):565–578CrossRefGoogle Scholar
  7. 7.
    Desai SR, Prasad R (2013) A novel order diminution of LTI systems using Big Bang Big Crunch optimization and Routh Approximation. Appl Math Model 37(16):8016–8028MathSciNetCrossRefGoogle Scholar
  8. 8.
    Abu-Al-Nadi DI, Alsmadi OM, Abo-Hammour ZS, Hawa MF, Rahhal JS (2013) Invasive weed optimization for model order reduction of linear MIMO systems. Appl Math Model 37(6):4570–4577MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sikander AA, Prasad BR (2015) A novel order reduction method using cuckoo search algorithm. IETE J Res 61(2):83–90CrossRefGoogle Scholar
  10. 10.
    Sikander A, Prasad R (2015) Soft computing approach for model order reduction of linear time invariant systems. Circuits Syst Signal Process 34(11):3471–3487CrossRefGoogle Scholar
  11. 11.
    Biradar S, Hote YV, Saxena S (2016) Reduced-order modeling of linear time invariant systems using big bang big crunch optimization and time moment matching method. Appl Math Model 40(15):7225–7244MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sikander A, Prasad R (2017) New technique for system simplification using Cuckoo search and ESA. Sādhanā 42(9):1453–1458MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sikander A, Thakur P (2017) Reduced order modelling of linear time-invariant system using modified cuckoo search algorithm. Soft Comput. CrossRefGoogle Scholar
  14. 14.
    Mishra R, Das KN (2016) Chemo-inspired genetic algorithm and application to model order reduction problem. In: Proceedings of fifth international conference on soft computing for problem solving. Springer, Singapore, pp 31–41.
  15. 15.
    Ganji V, Mangipudi S, Manyala R (2017) A novel model order reduction technique for linear continuous-time systems using PSO-DV algorithm. J Control Autom Electr Syst 28(1):68–77CrossRefGoogle Scholar
  16. 16.
    Narwal A, Prasad R (2017) Optimization of LTI systems using modified clustering algorithm. IETE Tech Rev 34(2):201–213CrossRefGoogle Scholar
  17. 17.
    Sikander A, Prasad R (2017) A new technique for reduced-order modelling of linear time-invariant system. IETE J Res 63(3):316–324CrossRefGoogle Scholar
  18. 18.
    Soloklo HN, Farsangi MM (2013) Multi-objective weighted sum approach model reduction by Routh-Pade approximation using harmony search. Turk J Electr Eng Comput Sci 21(Suppl 2):2283–2293. CrossRefGoogle Scholar
  19. 19.
    Khademi G, Mohammadi H, Dehghani M (2015) Order reduction of linear systems with keeping the minimum phase characteristic of the system: LMI based approach. IJST Trans Electr Eng 39(E2):217–227. CrossRefGoogle Scholar
  20. 20.
    Bansal JC, Sharma H (2012) Cognitive learning in differential evolution and its application to model order reduction problem for single-input single-output systems. Memet Comput. CrossRefGoogle Scholar
  21. 21.
    Sharma H, Bansal JC, Arya KV (2012) Fitness based differential evolution. Memet Comput 4(4):303–316CrossRefGoogle Scholar
  22. 22.
    Rosner B, Glynn RJ, Lee ML (2006) The Wilcoxon signed rank test for paired comparisons of clustered data. Biometrics 62(1):185–192MathSciNetCrossRefGoogle Scholar
  23. 23.
    Middleton RH, Goodwin GC (1990) Digital control and estimation: a unified approach (Prentice Hall information and system sciences series). Prentice Hall, Englewood CliffsGoogle Scholar
  24. 24.
    Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61CrossRefGoogle Scholar
  25. 25.
    Mirjalili S (2015) How effective is the Grey Wolf optimizer in training multi-layer perceptrons. Appl Intell 43(1):150–161CrossRefGoogle Scholar
  26. 26.
    Kamboj VK, Bath SK, Dhillon JS (2016) Solution of non-convex economic load dispatch problem using grey wolf optimizer. Neural Comput Appl 27(5):1301–1316CrossRefGoogle Scholar
  27. 27.
    Khairuzzaman AK, Chaudhury S (2017) Multilevel thresholding using grey wolf optimizer for image segmentation. Expert Syst Appl 86:64–76CrossRefGoogle Scholar
  28. 28.
    Saremi S, Mirjalili SZ, Mirjalili SM (2015) Evolutionary population dynamics and grey wolf optimizer. Neural Comput Appl 26(5):1257–1263CrossRefGoogle Scholar
  29. 29.
    Emary E, Zawbaa HM, Hassanien AE (2016) Binary grey wolf optimization approaches for feature selection. Neurocomputing 172:371–381CrossRefGoogle Scholar
  30. 30.
    Heidari AA, Pahlavani P (2017) An efficient modified grey wolf optimizer with Lévy flight for optimization tasks. Appl Soft Comput 60:115–134CrossRefGoogle Scholar
  31. 31.
    Jayabarathi T, Raghunathan T, Adarsh BR, Suganthan PN (2016) Economic dispatch using hybrid grey wolf optimizer. Energy 111:630–641CrossRefGoogle Scholar
  32. 32.
    Kamboj VK (2016) A novel hybrid PSO–GWO approach for unit commitment problem. Neural Comput Appl 27(6):1643–1655CrossRefGoogle Scholar
  33. 33.
    Lu C, Gao L, Li X, Xiao S (2017) A hybrid multi-objective grey wolf optimizer for dynamic scheduling in a real-world welding industry. Eng Appl Artif Intell 57:61–79CrossRefGoogle Scholar
  34. 34.
    Rizk-Allah RM, Zaki EM, El-Sawy AA (2013) Hybridizing ant colony optimization with firefly algorithm for unconstrained optimization problems. Appl Math Comput 224:473–483MathSciNetzbMATHGoogle Scholar
  35. 35.
    Yang XS (2010) Firefly algorithm, stochastic test functions and design optimisation. Int J Bio Inspired Comput 2(2):78–84CrossRefGoogle Scholar
  36. 36.
    Khajehzadeh M, Taha MR, Eslami M (2013) A new hybrid firefly algorithm for foundation optimization. Natl Acad Sci Lett 36(3):279–288MathSciNetCrossRefGoogle Scholar
  37. 37.
    Talbi EG (2002) A taxonomy of hybrid metaheuristics. J Heuristics 8(5):541–564CrossRefGoogle Scholar
  38. 38.
    Anagnost JJ, Desoer CA (1991) An elementary proof of the Routh–Hurwitz stability criterion. Circuits Syst Signal Process 10(1):101–114MathSciNetCrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2018

Authors and Affiliations

  1. 1.Department of Electrical and Instrumentation EngineeringThapar Institute of Engineering and TechnologyPatialaIndia
  2. 2.Department of Electrical EngineeringNational Institute of Technical Teachers’ Training and ResearchKolkataIndia

Personalised recommendations