Advertisement

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

  • S. P. Singh
  • Varsha Singh
  • V. P. Singh
Article
  • 34 Downloads

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.

Keywords

Analytic hierarchy process Model reduction Padé approximation Routh approximation Teacher–learner-based-optimization 

References

  1. 1.
    Fortuna L et al (2012) Model order reduction techniques with applications in electrical engineering. Springer, BerlinGoogle Scholar
  2. 2.
    Quarteroni A, Rozza G (2014) Reduced order methods for modeling and computational reduction, vol 9. Springer, BerlinzbMATHGoogle Scholar
  3. 3.
    Schilders WH et al (2008) Model order reduction: theory, research aspects and applications, vol 13. Springer, BerlinCrossRefzbMATHGoogle Scholar
  4. 4.
    Singh J et al (2016) Biased reduction method by combining improved modified pole clustering and improved Pade approximations. Appl Math Model 40:1418–1426MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ionescu TC et al (2014) Families of moment matching based, low order approximations for linear systems. Syst Control Lett 64:47–56MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beattie C et al (2017) Model reduction for systems with inhomogeneous initial conditions. Syst Control Lett 99:99–106MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Tiwari SK, Kaur G (2017) Model reduction by new clustering method and frequency response matching. J Control Autom Electr Syst 28:78–85CrossRefGoogle Scholar
  8. 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 28:68–77CrossRefGoogle Scholar
  9. 9.
    Bandyopadhyay B et al (1994) Routh-Padé approximation for interval systems. IEEE Trans Autom Control 39:2454–2456CrossRefzbMATHGoogle Scholar
  10. 10.
    Singh V, Chandra D (2012) Model reduction of discrete interval system using clustering of poles. Int J Model Ident Control 17:116–123CrossRefGoogle Scholar
  11. 11.
    Singh V et al (2017) On time moments and Markov parameters of continuous interval systems. J Circuits Syst Comput 26:1750038CrossRefGoogle Scholar
  12. 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 27–30Google Scholar
  13. 13.
    Singh V et al (2004) Improved Routh-Padé approximants: a computer-aided approach. IEEE Trans Autom Control 49:292–296CrossRefzbMATHGoogle Scholar
  14. 14.
    Singh V (2005) Obtaining Routh-Padé approximants using the Luus–Jaakola algorithm. IEE Proc Control Theory Appl 152:129–132CrossRefGoogle Scholar
  15. 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 14:45–56MathSciNetzbMATHGoogle Scholar
  16. 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 21:2283–2293CrossRefGoogle Scholar
  17. 17.
    Chen T et al (1980) Stable reduced-order Pad? approximants using stability-equation method. Electron Lett 9:345–346CrossRefGoogle Scholar
  18. 18.
    Wan B-W (1981) Linear model reduction using Mihailov criterion and Pade approximation technique. Int J Control 33:1073–1089MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Appiah R (1979) Padé methods of Hurwitz polynomial approximation with application to linear system reduction. Int J Control 29:39–48MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rao A et al (1978) Routh-approximant time-domain reduced-order models for single-input single-output systems. Electr Eng Proc Inst 125:1059–1063CrossRefGoogle Scholar
  21. 21.
    Shamash Y (1975) Model reduction using the Routh stability criterion and the Padé approximation technique. Int J Control 21:475–484CrossRefzbMATHGoogle Scholar
  22. 22.
    Shamash Y (1980) Stable biased reduced order models using the Routh method of reduction. Int J Syst Sci 11:641–654MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Saaty TL (1988) What is the analytic hierarchy process? In: Mathematical models for decision support, Ed Springer, pp 109–121Google Scholar
  24. 24.
    Saaty TL (2004) Decision making—the analytic hierarchy and network processes (AHP/ANP). J Syst Sci Syst Eng 13:1–35CrossRefGoogle Scholar
  25. 25.
    Wasielewska K et al (2014) Applying Saaty’s multicriterial decision making approach in grid resource management. Inf Technol Control 43:73–87Google Scholar
  26. 26.
    Rao RV et al (2011) Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43:303–315CrossRefGoogle Scholar
  27. 27.
    Rao R et al (2012) Teaching–learning-based optimization: an optimization method for continuous non-linear large scale problems. Inf Sci 183:1–15MathSciNetCrossRefGoogle Scholar
  28. 28.
    Gantmacher FR (1959) Matrix theory, vol 21. Chelsea, New YorkzbMATHGoogle Scholar
  29. 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) 34:1609–1617CrossRefGoogle Scholar
  30. 30.
    Saaty TL (2000) Fundamentals of decision making and priority theory with the analytic hierarchy process, vol 6. Rws Publications, PittsburghGoogle Scholar
  31. 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 60:35–44CrossRefGoogle Scholar
  32. 32.
    Salim R, Bettayeb M (2011) H 2 and H\(\infty \) optimal model reduction using genetic algorithms. J Franklin Inst 348:1177–1191MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical EngineeringIndian Institute of TechnologyKanpurIndia
  2. 2.Department of Electrical EngineeringNational Institute of TechnologyRaipurIndia

Personalised recommendations