Skip to main content
SpringerLink
Log in

Analytic hierarchy process based approximation of high-order continuous systems using TLBO algorithm

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

This paper presents an analytic hierarchy process based approach for approximation of stable high-order systems using teacher–learner-based-optimization (TLBO) algorithm. In this method, the stable approximant is derived by minimizing the errors of time moments and of Markov parameters of system and its approximant. Being free from algorithm-specific parameters, the TLBO algorithm is used for minimizing the objective function. The Hurwitz criterion is used to ensure the stability of approximant. The first time moment of the system is retained in approximant to guarantee the matching of steady states of system and approximant. The distinctive feature of this work is that the multi-objective problem of minimization of errors of time moments and of Markov parameters is converted into single objective problem by assigning some weights to different objectives using analytic hierarchy process. Also, the proposed method always produces stable approximant for stable high-order system. The results of proposed approach are compared with other existing techniques. To conclude the superiority of proposed approach, a comparative study is performed using the step responses and time domain analysis. The efficacy and systematic nature of proposed approach is shown with the help of two test systems.

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

Similar content being viewed by others

References

  1. Fortuna L et al (2012) Model order reduction techniques with applications in electrical engineering. Springer, Berlin

    Google Scholar 

  2. Quarteroni A, Rozza G (2014) Reduced order methods for modeling and computational reduction, vol 9. Springer, Berlin

    MATH  Google Scholar 

  3. Schilders WH et al (2008) Model order reduction: theory, research aspects and applications, vol 13. Springer, Berlin

    Book  MATH  Google Scholar 

  4. Singh J et al (2016) Biased reduction method by combining improved modified pole clustering and improved Pade approximations. Appl Math Model 40:1418–1426

    Article  MathSciNet  Google Scholar 

  5. Ionescu TC et al (2014) Families of moment matching based, low order approximations for linear systems. Syst Control Lett 64:47–56

    Article  MathSciNet  MATH  Google Scholar 

  6. Beattie C et al (2017) Model reduction for systems with inhomogeneous initial conditions. Syst Control Lett 99:99–106

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  8. Ganji V et al (2017) A novel model order reduction technique for linear continuous-time systems using PSO-DV algorithm. J Control Autom Electr Syst 28:68–77

    Article  Google Scholar 

  9. Bandyopadhyay B et al (1994) Routh-Padé approximation for interval systems. IEEE Trans Autom Control 39:2454–2456

    Article  MATH  Google Scholar 

  10. Singh V, Chandra D (2012) Model reduction of discrete interval system using clustering of poles. Int J Model Ident Control 17:116–123

    Article  Google Scholar 

  11. Singh V et al (2017) On time moments and Markov parameters of continuous interval systems. J Circuits Syst Comput 26:1750038

    Article  Google Scholar 

  12. Singh VP, Chandra D (2011) Model reduction of discrete interval system using dominant poles retention and direct series expansion method. In: 2011 5th International power engineering and optimization conference (PEOCO), pp 27–30

  13. Singh V et al (2004) Improved Routh-Padé approximants: a computer-aided approach. IEEE Trans Autom Control 49:292–296

    Article  MATH  Google Scholar 

  14. Singh V (2005) Obtaining Routh-Padé approximants using the Luus–Jaakola algorithm. IEE Proc Control Theory Appl 152:129–132

    Article  Google Scholar 

  15. Singh VP, Chandra D (2012) Reduction of discrete interval systems based on pole clustering and improved Padé approximation: a computer-aided approach. Adv Model Optim 14:45–56

    MathSciNet  MATH  Google Scholar 

  16. 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 21:2283–2293

    Article  Google Scholar 

  17. Chen T et al (1980) Stable reduced-order Pad? approximants using stability-equation method. Electron Lett 9:345–346

    Article  Google Scholar 

  18. Wan B-W (1981) Linear model reduction using Mihailov criterion and Pade approximation technique. Int J Control 33:1073–1089

    Article  MathSciNet  MATH  Google Scholar 

  19. Appiah R (1979) Padé methods of Hurwitz polynomial approximation with application to linear system reduction. Int J Control 29:39–48

    Article  MathSciNet  MATH  Google Scholar 

  20. Rao A et al (1978) Routh-approximant time-domain reduced-order models for single-input single-output systems. Electr Eng Proc Inst 125:1059–1063

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Saaty TL (1988) What is the analytic hierarchy process? In: Mathematical models for decision support, Ed Springer, pp 109–121

  24. Saaty TL (2004) Decision making—the analytic hierarchy and network processes (AHP/ANP). J Syst Sci Syst Eng 13:1–35

    Article  Google Scholar 

  25. Wasielewska K et al (2014) Applying Saaty’s multicriterial decision making approach in grid resource management. Inf Technol Control 43:73–87

    Google Scholar 

  26. Rao RV et al (2011) Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43:303–315

    Article  Google Scholar 

  27. Rao R et al (2012) Teaching–learning-based optimization: an optimization method for continuous non-linear large scale problems. Inf Sci 183:1–15

    Article  MathSciNet  Google Scholar 

  28. Gantmacher FR (1959) Matrix theory, vol 21. Chelsea, New York

    MATH  Google Scholar 

  29. Hsu C-C, Yu C-Y (2004) Design of optimal controller for interval plant from signal energy point of view via evolutionary approaches. IEEE Trans Syst Man Cybern Part B (Cybern) 34:1609–1617

    Article  Google Scholar 

  30. Saaty TL (2000) Fundamentals of decision making and priority theory with the analytic hierarchy process, vol 6. Rws Publications, Pittsburgh

    Google Scholar 

  31. Singh SP, Prakash Tapan, Singh VP, Ganesh Babu M (2017) Analytic hierarchy process based automatic generation control of multi-area interconnected power system using Jaya algorithm. Eng Appl Artif Intell 60:35–44

    Article  Google Scholar 

  32. Salim R, Bettayeb M (2011) H 2 and H\(\infty \) optimal model reduction using genetic algorithms. J Franklin Inst 348:1177–1191

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Indian Institute of Technology, Kanpur, India

    S. P. Singh

  2. Department of Electrical Engineering, National Institute of Technology, Raipur, India

    Varsha Singh & V. P. Singh

Authors
  1. S. P. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Varsha Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. P. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. P. Singh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, S.P., Singh, V. & Singh, V.P. Analytic hierarchy process based approximation of high-order continuous systems using TLBO algorithm. Int. J. Dynam. Control 7, 53–60 (2019). https://doi.org/10.1007/s40435-018-0436-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-018-0436-9

Keywords

Search

Navigation