Soft Computing

, Volume 22, Issue 10, pp 3449–3459 | Cite as

Reduced order modelling of linear time-invariant system using modified cuckoo search algorithm

  • Afzal Sikander
  • Padmanabh Thakur
Methodologies and Application


In this paper modified cuckoo search (MCS) algorithm is considered to develop reduced order model (ROM) of higher-order linear time-invariant systems. Firstly, the MCS algorithm has been employed to minimize the integral square error (ISE) between original and proposed ROM to obtain its unknown coefficients. Five systems of different order are considered to obtain their reduced order model. Finally, various performance indices, such as ISE, integral of absolute and integral of time multiplied by absolute error, have been estimated to reveal the efficacy of the proposed model. Also, time and frequency response characteristics of original higher-order model are compared with the proposed MCS-based and some of other existing techniques-based ROM available in the literature. Furthermore, the results are compared in terms of time response specifications such as rise time (\(t_\mathrm{r} \)) in second, settling time (\( t_\mathrm{s}\)) in second and maximum peak overshoot (\( M_\mathrm{p}\)) in percentage. It is revealed that the response of the proposed MCS-based ROM is much closer to the response of the original higher-order system.


Modified cuckoo search algorithm Optimization Order reduction Integral square error 


Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflicts of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Abu-Al-Nadi DI, Alsmadi OMK, Abo-Hammour ZS (2011) Reduced order modeling of linear MIMO systems using particle swarm optimization. In: 7th international conference on autonomic and autonomous systems, Venice, Italy, pp 62–66Google Scholar
  2. Alsmadi OMK, Abo-Hammour ZS, Al-Smadi AM, Abu-Al-Nadi DI (2011) Genetic algorithm approach with frequency selectivity for model order reduction of MIMO systems. Math Comput Model Dyn Syst 17(2):163–181. doi: 10.1080/13873954.2010.540806 MathSciNetCrossRefzbMATHGoogle Scholar
  3. Biradar S, Hote YV, Saxena S (2016) Reduced-order modelling of linear time invariant systems using big bang big crunch optimization and time moment matching method. Appl Math Model 40(15–16):7225–7244MathSciNetCrossRefGoogle Scholar
  4. Brown CT, Liebovitch LS, Glendon R (2007) Lévy flights in Dobe Ju/hoansi foraging patterns. Hum Ecol 35(1):129–138CrossRefGoogle Scholar
  5. Desai SR, Prasad R (2013a) A novel order diminution of LTI systems using big bang big crunch optimization and routh approximation. Appl Math Model 37(16–17):8016–8028. doi: 10.1016/j.apm.2013.02.052
  6. Desai SR, Prasad R (2013b) A new approach to order reduction using stability equation and big bang big crunch optimization. Syst Sci Control Eng Open Access J 1:20–27CrossRefGoogle Scholar
  7. Edgar TF (1975) Least squares model reduction using step response. Int J Control 22:261–270CrossRefzbMATHGoogle Scholar
  8. Eitelberg E (1981) Model reduction by minimizing the weighted equation error. Int J Control 34(6):1113–1123MathSciNetCrossRefzbMATHGoogle Scholar
  9. El-Attar RA, Vidyasagar M (1978) Order reduction by \({L_1}\) and \({L_\infty }\) norm minimization. IEEE Trans Autom Control 23(4):731–734CrossRefzbMATHGoogle Scholar
  10. Erol OK, Eksin I (2006) A new optimization method: big bang-big crunch. Adv Eng Softw 37:106–111. doi: 10.1016/j.advengsoft.2005.04.005 CrossRefGoogle Scholar
  11. Ghosh S, Senroy N (2013) Balanced truncation approach to power system model order reduction. Electr Power Compon Syst 41:747–764. doi: 10.1080/15325008.2013.769031 CrossRefGoogle Scholar
  12. Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Boston 10.1007/s10589-009-9261-6zbMATHGoogle Scholar
  13. Humphries NE, Weimerskirch H, Queiroz N, Southall EJ, Sims DW (2012) Foraging success of biological Lévy flights recorded in situ. Proc Natl Acad Sci 109(19):7169–7174CrossRefGoogle Scholar
  14. Hutton MF, Friedland B (1975) Routh approximations for reducing order of linear, time-invariant systems. IEEE Trans Autom Control 20:329–337. doi: 10.1109/TAC.1975.1100953 MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: IEEE International Conference on Neural Networks, Perth, WA, vol 4, pp 1942–1948. doi: 10.1109/ICNN.1995.488968
  16. Lee KS, Geem ZW (2004) A new structural optimization method based on the Harmony search algorithm. J Comput Struct 82:781–798CrossRefGoogle Scholar
  17. Mukherjee S, Satakshi R, Mittal C (2005) Model order reduction using response-matching technique. J Frankl Inst 342:503–519MathSciNetCrossRefzbMATHGoogle Scholar
  18. Obinata G, Inooka H (1983) Authors reply to comments on model reduction by minimizing the equation error. IEEE Trans Autom Control 28:124–125CrossRefGoogle Scholar
  19. Panda S, Yadav JS, Padidar NP, Ardil C (2009) Evolutionary techniques for model order reduction of large scale linear systems. Int J Appl Sci Eng Technol 5:22–28Google Scholar
  20. Parmar G, Mukherjee S, Prasad R (2007a) Reduced order modeling of linear dynamic systems using particle swarm optimized eigen spectrum analysis. Int J Comput Math Sci 1(31):45–52Google Scholar
  21. Parmar G, Mukherjee S, Prasad R (2007b) System reduction using eigen spectrum analysis and pade approximation technique. Int J Comput Math 84(12):1871–1880MathSciNetCrossRefzbMATHGoogle Scholar
  22. Parmar G, Mukherjee S, Prasad R (2007c) System reduction using factor division algorithm and eigen spectrum analysis. Appl Math Model 31(11):2542–2552. doi: 10.1016/j.apm.2006.10.004 CrossRefzbMATHGoogle Scholar
  23. Parmar G, Prasad R, Mukherjee S (2007d) Order reduction of linear dynamic systems using stability equation method and GA. Int J Comput Inf Eng 1(1):26–32Google Scholar
  24. Parmar G, Pandey MK, Kumar V (2009) System order reduction using GA for unit impulse input and a comparative study using ISE and IRE. In: International conference on advances in computing, communications and control, Mumbai, India, pp 23–24Google Scholar
  25. Salim R, Bettayeb M (2011) \({H_2}\) and \({H_\infty }\) optimal model reduction using genetic algorithms. J Frankl Inst 348:1177–1191. doi: 10.1016/j.jfranklin.2009.10.016 MathSciNetCrossRefzbMATHGoogle Scholar
  26. Sambariya DK, Arvind G (2016) High order diminution of LTI system using stability equation method. Br J Math Comput Sci 13(5):1–15. doi: 10.9734/BJMCS/2016/23243 Google Scholar
  27. Sikander A, Prasad R (2015a) Soft computing approach for model order reduction of linear time invariant systems. Circuit Syst Signal Process. doi: 10.1007/s00034-015-0018-4 Google Scholar
  28. Sikander A, Prasad R (2015b) Time domain order reduction method using improved Hermite Normal Form. In: National conference on emerging trends in electrical and electronics engineering, JMI, New Delhi, India, pp 224–229Google Scholar
  29. Sikander A, Prasad R (2015c) Linear time-invariant system reduction using a mixed methods approach. Appl Math Model 39(16):4848–4858MathSciNetCrossRefGoogle Scholar
  30. Sikander A, Prasad R (2017) A new technique for reduced-order modelling of linear time-invariant system. IETE J Res 1–9. doi: 10.1080/03772063.2016.1272436
  31. Sikander A, Uniyal I, Thakur P (2016) Hybrid method of reduced order modelling for LTI system using evolutionary algorithm. In: IEEE international conference on next generation computing technologies, Dehradun, IndiaGoogle Scholar
  32. Viswanathan GM (2010) Fish in levy-flight foraging. Nature 465:1018–1019CrossRefGoogle Scholar
  33. Vishwakarma CB, Prasad R (2008) System reduction using modified pole clustering and pade approximation. In: XXXII national systems conference, NSC 2008, pp 592–596Google Scholar
  34. Vishwakarma CB, Prasad R (2009) MIMO system reduction using modified pole clustering and genetic algorithm. Model Simul Eng 2009:1–5CrossRefGoogle Scholar
  35. Walton S, Hassan O, Morgan K, Brown MR (2011) Modified cuckoo search: a new gradient free optimisation algorithm. Chaos Solitons Fractals 44:710–718. doi: 10.1016/j.chaos.2011.06.004 CrossRefGoogle Scholar
  36. Wilson DA (1970) Optimal solution of model reduction problem. Proc Inst Electr Eng 117(06):1161–1165Google Scholar
  37. Yang XS, Deb S (2008) Nature-inspired metaheuristic algorithms. Luniver Press, LondonGoogle Scholar
  38. Yang XS, Deb S (2010) Engineering optimisation by cuckoo search. Int J Math Model Numer Optim 1:330–343zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Electrical EngineeringGraphic Era UniversityDehradunIndia

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