Skip to main content
SpringerLink
Log in

Soft Computing Approach for Model Order Reduction of Linear Time Invariant Systems

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

Most of the physical systems can be represented by mathematical models. The mathematical procedure of system modeling often leads to a comprehensive description of a process in the form of higher-order differential equations which are difficult to use either for analysis or for controller synthesis. It is, therefore, useful and sometimes necessary to find the possibility of some equations of the same type but of lower order that may be considered to adequately reflect almost all essential characteristics of the system under consideration. This paper proposes a new method for order reduction of higher-order linear time invariant systems based on stability equation method and particle swarm optimization algorithm. Reduced-order model will definitely be stable if the original model is stable. The superiority of the proposed method is illustrated by numerical examples of single-input, single-output systems and multiple-input and multiple-output systems. The results are compared with well-known methods available in the literature.

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
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. D.A. Al-Nadi, O.M.K. Alsmadi, Z.S. Abo-Hammour, Reduced order modeling of linear MIMO systems using particle swarm optimization. In 7th International Conference on Autonomic and Autonomous Systems, Venice, Italy, pp. 62–66 (2011)

  2. M. Aoki, Control of large-scale dynamic systems. IEEE Trans. Autom. Control 3, 246–253 (1968)

    Article  Google Scholar 

  3. T.C. Chen, C.Y. Chang, K.W. Han, Reduction of transfer functions by the stability-equation method. J. Frankl. Inst. 8, 389–404 (1979)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  5. S.R. Desai, R. Prasad, A novel order diminution of LTI systems using big bang big crunch optimization and routh approximation. Appl. Math. Model. 37(16–17), 8016–8028 (2013). doi:10.1016/j.apm.2013.02.052

    Article  MathSciNet  Google Scholar 

  6. E. Eitelberg, Model reduction by minimizing the weighted equation error. Int. J. Control 34(6), 1113–1123 (1981)

    Article  MATH  Google Scholar 

  7. R.A. El-Attar, M. Vidyasagar, Order reduction by \(L_1\) and \({L_\infty }\) norm minimization. IEEE Trans. Autom. Control 23(4), 731–734 (1978)

    Article  MATH  Google Scholar 

  8. O.K. Erol, I. Eksin, A new optimization method: big bang–big crunch. Adv. Eng. Softw. 37, 106–111 (2006). doi:10.1016/j.advengsoft.2005.04.005

    Article  Google Scholar 

  9. K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their \({L_\infty }\)-error bounds. Int. J. Control 39, 1115–1193 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, 1989). doi:10.1007/s10589-009-9261-6

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

    Article  MATH  Google Scholar 

  12. M.F. Hutton, B. Friedland, Routh approximations for reducing order of linear, time-invariant systems. IEEE Trans. Autom. Control 20, 329–337 (1975). doi:10.1109/TAC.1975.1100953

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Kaveh, K. Laknejadi, A hybrid multi-objective particle swarm optimization and decision making procedure for optimal design of truss structures. Iran. J. Sci. Technol. 35(C2), 137–154 (2011)

    Google Scholar 

  14. A. Kaveh, S. Talatahari, A particle swarm ant colony optimization for truss structures with discrete variables. J. Constr. Steel Res. 65, 1558–1568 (2009). doi:10.1016/j.jcsr.2009.04.021

    Article  Google Scholar 

  15. A. Kaveh, S. Talatahari, Particle swarm optimizer, ant colony strategy and harmony search scheme hybridized for optimization of truss structures. Comput. Struct. 87, 267–283 (2009). doi:10.1016/j.compstruc.2009.01.003

    Article  Google Scholar 

  16. J. Kennedy, R. Eberhart, Particle swarm optimization. In IEEE International Conference on Neural Networks, vol. 4, IEEE, pp. 1942–1948 (1995). doi:10.1109/ICNN.1995.488968

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

    Article  Google Scholar 

  18. K.S. Lee, Z.W. Geem, A new structural optimization method based on the Harmony search algorithm. J. Comput. Struct. 82, 781–798 (2004)

    Article  Google Scholar 

  19. A.K. Mittal, R. Prasad, S.P. Sharma, Reduction of linear dynamic systems using an error minimization technique. J. Inst. Eng. IE (I) J. EL 84, 201–206 (2004)

    Google Scholar 

  20. B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Transa. Autom. Control 26(1), 17–32 (1981)

    Article  MATH  Google Scholar 

  21. S. Mukherjee, Satakshi, R.C. Mittal, Model order reduction using response matching technique. J. Franklin Inst. 342, 503–519 (2005)

  22. G. Obinata, H. Inooka, Authors reply to comments on model reduction by minimizing the equation error. IEEE Trans. Autom. Control 28, 124–125 (1983)

    Article  Google Scholar 

  23. J. Pal, Stable reduced order Pade approximants using the Routh–Hurwitz array. Electron. Lett. 15(8), 225–226 (1979)

    Article  Google Scholar 

  24. S. Panda, J.S. Yadav, N.P. Padidar, C. Ardil, Evolutionary techniques for model order reduction of large scale linear systems. Int. J. Appl. Sci. Eng. Technol. 5, 22–28 (2009)

    Google Scholar 

  25. G. Parmar, S. Mukherjee, R. Prasad, Reduced order modeling of linear dynamic systems using particle swarm optimized eigen spectrum analysis. Int. J. Comp. Math. Sci. 1(31), 45–52 (2007)

    Google Scholar 

  26. G. Parmar, S. Mukherjee, R. Prasad, System reduction using factor division algorithm and eigen spectrum analysis. Appl. Math. Model. 31(11), 2542–2552 (2007). doi:10.1016/j.apm.2006.10.004

    Article  MATH  Google Scholar 

  27. G. Parmar, M.K. Pandey, V. Kumar, 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. 23–24 (2009)

  28. G. Parmar, R. Prasad, S. Mukherjee, Order reduction of linear dynamic systems using stability equation method and GA. Int. J. Comput. Inf. Eng. 1(1), 26–32 (2007)

    Google Scholar 

  29. R. Parthasarathy, K.N. Jayasimha, System reduction using stability equation method and modified Cauer continued fraction. Proc. IEEE 70(10), 1234–1236 (1982)

    Article  Google Scholar 

  30. R. Prasad, J. Pal, Stable reduction of linear systems by continued fractions. J. Inst. Eng. IE (I) J. EL 72, 113–116 (1991)

    Google Scholar 

  31. M.G. Safanov, R.Y. Chiang, Model reduction for robust control: a Schur relative error method. Int. J. Adapt. Control Signal 2, 259–272 (1988)

    Article  Google Scholar 

  32. R. Salim, M. Bettayeb, H2 and H optimal model reduction using genetic algorithms. J. Frankl. Inst. 348, 1177–1191 (2011). doi:10.1016/j.jfranklin.2009.10.016

    Article  MATH  MathSciNet  Google Scholar 

  33. Y. Shamash, Stable reduced-order models using Pade-type approximations. IEEE Trans. Autom. Control 19(5), 615–616 (1974)

    Article  MATH  Google Scholar 

  34. Y. Shamash, Linear system reduction using Pade approximation to allow retention of dominant modes. Int. J. Control 21(2), 257–272 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  35. L. Shieh, Y. Wei, A mixed method for multivariable system reduction. IEEE Trans. Autom. Control 20, 429–432 (1975). doi:10.1109/TAC.1975.1100964

    Article  MATH  Google Scholar 

  36. N.K. Sinha, B. Kuszta, Modeling and Identification of dynamic systems (Springer, Berlin, 1983)

    Google Scholar 

  37. C.B. Vishwakarma, R. Prasad, Clustering method for reducing order of linear system using Pade approximation (2008). doi:10.4103/0377-2063.48531

  38. C.B. Vishwakarma, R. Prasad, System reduction using modified pole clustering and Pade approximation. In: XXXII National systems conference, NSC, pp 592–596 (2008)

  39. C.B. Vishwakarma, R. Prasad, MIMO system reduction using modified pole clustering and genetic algorithm. Model. Simul. Eng. 2009, 1–5 (2009). doi:10.1155/2009/540895

    Article  Google Scholar 

  40. D.A. Wilson, Optimal solution of model reduction problem. Proc. Inst. Electr. Eng. 117(06), 1161–1165 (1970)

    Article  Google Scholar 

  41. D.A. Wilson, Model reduction for multivariable systems. Int. J. Control 20, 57–64 (1974)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Afzal Sikander & Rajendra Prasad

  2. Department of Electrical and Electronics Engineering, Graphic Era University, Dehradun, India

    Afzal Sikander

Authors
  1. Afzal Sikander
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajendra Prasad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Afzal Sikander.

Additional information

Based on a paper presented at the IEEE International Conference on Microelectronic, Circuits and Systems (Micro-2014) held in Kolkata, India.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sikander, A., Prasad, R. Soft Computing Approach for Model Order Reduction of Linear Time Invariant Systems. Circuits Syst Signal Process 34, 3471–3487 (2015). https://doi.org/10.1007/s00034-015-0018-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-015-0018-4

Keywords

Search

Navigation