Advertisement

Order Reduction in Linear Dynamical Systems by Using Improved Balanced Realization Technique

  • Arvind Kumar PrajapatiEmail author
  • Rajendra Prasad
Short Paper
  • 40 Downloads

Abstract

In this article, a new model order reduction scheme is proposed for the simplification of large-scale linear dynamical models. The proposed technique is based on the balanced realization method, in which the steady-state gain problem of the balanced truncation is circumvented. In this method, the denominator coefficients of the reduced system are evaluated by using the balanced realization, and the numerator coefficients are obtained by using a simple mathematical procedure as given in the literature. The proposed technique has been illustrated through some standard large-scale systems. This method gives the least performance error indices compared to some other existing system reduction methods. The time response of the approximated system, evaluated by the proposed method, is also shown which is the excellent matching of the response of the actual model when compared to the responses of other existing techniques.

Keywords

Controllability and observability Large-scale systems Lyapunov theorem Model order reduction Steady-state value 

Notes

References

  1. 1.
    M. Belhocine, M. Belhocine, A mix balanced-modal truncations for power systems model reduction, in 2014 European Control Conference (ECC) (Strasbourg, 2014), pp. 2721–2726Google Scholar
  2. 2.
    D. Casagrande, W. Krajewski, U. Viaro, On the asymptotic accuracy of reduced-order models. Int. J. Control Autom. Syst. 15(5), 2436–2442 (2017)CrossRefGoogle Scholar
  3. 3.
    C.F. Chen, L.S. Shieh, A novel approach to linear model simplification. Int. J. Control 8(6), 561–570 (1968)CrossRefGoogle Scholar
  4. 4.
    T.C. Chen, C.Y. Chang, K.W. Han, Reduction of transfer functions by the stability-equation method. J. Frankl. Inst. 308(4), 389–404 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    T.C. Chen, C.Y. Chang, K.W. Han, Model reduction using the stability-equation method and the Padé approximation method. J. Frankl. Inst. 309(6), 473–490 (1980)CrossRefzbMATHGoogle Scholar
  6. 6.
    B. Datta, Numerical Methods for Linear Control Systems, 1st edn. (Elsevier, Amsterdam, 2003)Google Scholar
  7. 7.
    M.F. Far, F. Martin, A. Belahcen, L. Montier, T. Henneron, Orthogonal interpolation method for order reduction of a synchronous machine model. IEEE Trans. Magn. 54(2), 1–6 (2018)CrossRefGoogle Scholar
  8. 8.
    K.V. Fernando, H. Nicholson, Singular perturbation model reduction in frequency domain. IEEE Trans. Autom. Control 27(4), 969–970 (1984)CrossRefzbMATHGoogle Scholar
  9. 9.
    Z. Gajic, M. Lelic, Improvement of system order reduction via balancing using the method of singular perturbations. Automatica 37, 1859–1865 (2001)CrossRefzbMATHGoogle Scholar
  10. 10.
    S. Ghosh, N. Senroy, Balanced truncation approach to power system model order reduction. Electr. Power Compon. Syst. 41(8), 747–764 (2013)CrossRefGoogle Scholar
  11. 11.
    K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L, -error bounds. Int. J. Control 39(6), 1115–1193 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    S. Gugercin, A.C. Antoulas, A survey of model reduction by balanced truncation and some new results. Int. J. Control 77(8), 748–766 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    P. Gutman, C. Mannerfelt, P. Molander, Contributions to the model reduction problem. IEEE Trans. Autom. Control 27(2), 454–455 (1982)CrossRefzbMATHGoogle Scholar
  14. 14.
    M.R. Hasan, L. Montier, T. Henneron, R.V. Sabariego, Matrix interpolation-based reduced-order modeling of a levitation device with eddy current effects. IEEE Trans. Magn. 54(6), 1–7 (2018)CrossRefGoogle Scholar
  15. 15.
    D. Huang, H.A. Khalik, C. Rabiti, F. Gleicher, Dimensionality reducibility for multi-physics reduced order modeling. Ann. Nuclear Energy 110, 526–540 (2017)CrossRefGoogle Scholar
  16. 16.
    M.F. Hutton, B. Friedland, Routh approximations for reducing order of linear, time-invariant systems. IEEE Trans. Autom. Control 20(3), 329–337 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    E. Jarlebring, T. Damm, W. Michiels, Model reduction of time-delay systems using position balancing and delay Lyapunov equations. Math. Control Signals Syst. 25, 147–166 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Y.L. Jiang, C.Y. Chen, P. Yang, Balanced truncation with ε-embedding for coupled dynamical systems. IET Circuits Devices Syst. 12(3), 271–279 (2018)CrossRefGoogle Scholar
  19. 19.
    T. Johnson, T. Bartol, T. Sejnowski, E. Mjolsness, Model reduction for stochastic CaMKII reaction kinetics in synapses by graph-constrained correlation dynamics. Phys. Biol. 12(4), 1–16 (2015)CrossRefGoogle Scholar
  20. 20.
    R. Komarasamy, N. Albhonso, G. Gurusamy, Order reduction of linear systems with an improved pole clustering. J. Vib. Control 18(12), 1876–1885 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    V. Krishnamurthy, V. Seshadri, Model reduction using the Routh stability criterion. IEEE Trans. Autom. Control 23(3), 729–731 (1978)CrossRefGoogle Scholar
  22. 22.
    D.K. Kumar, S.K. Nagar, J.P. Tiwari, A new algorithm for model order reduction of interval systems. Bonfring Int. J. Data Min. 3(1), 6–11 (2013)CrossRefGoogle Scholar
  23. 23.
    B.C. Kuo, Automatic Control Systems, 7th edn. (Prentice-Hall, Upper Saddle River, 1995)Google Scholar
  24. 24.
    P. Kurschner, Balanced truncation model order reduction in limited time intervals for large systems. Adv. Comput. Math. (2018).  https://doi.org/10.1007/s10444-018-9608-6 MathSciNetzbMATHGoogle Scholar
  25. 25.
    G. Langholz, D. Feinmesser, Model order reduction by Routh approximations. Int. J. Syst. Sci. 9(5), 493–496 (1978)CrossRefzbMATHGoogle Scholar
  26. 26.
    T.N. Lucas, Factor division: a useful algorithm in model reduction. IEE Proc. D Control Theory Appl. 130(6), 362–364 (1983)CrossRefzbMATHGoogle Scholar
  27. 27.
    S.S. Mohseni, M.J. Yazdanpanah, A.R. Noei, Model order reduction of nonlinear models based on decoupled multimodel via trajectory piecewise linearization. Int. J. Control Autom. Syst. 15(5), 2088–2098 (2017)CrossRefGoogle Scholar
  28. 28.
    B.C. Moore, Principal component analysis in control system: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26(1), 17–36 (1981)CrossRefzbMATHGoogle Scholar
  29. 29.
    A. Narwal, R. Prasad, A novel order reduction approach for LTI systems using cuckoo search optimization and stability equation. IETE J. Res. 62(2), 154–163 (2016)CrossRefGoogle Scholar
  30. 30.
    S.V. Ophem, A.V.D. Walle, E. Deckers, W. Desmet, Efficient vibro-acoustic identification of boundary conditions by low-rank parametric model order reduction. Mech. Syst. Signal Process. 111, 23–35 (2018)CrossRefGoogle Scholar
  31. 31.
    J. Pal, Stable reduced-order Padé approximants using the Routh–Hurwitz array. Electron. Lett. 15(8), 225–226 (1979)CrossRefGoogle Scholar
  32. 32.
    L. Pernebo, L.M. Silverman, Model reduction via balanced state space representations. IEEE Trans. Autom. Control 27(2), 382–387 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    A. Pierquin, T. Henneron, S. Clénet, Data-driven model-order reduction for magnetostatic problem coupled with circuit equation. IEEE Trans. Magn. 54(3), 1–4 (2018)CrossRefGoogle Scholar
  34. 34.
    A.K. Prajapati, R. Prasad, Padé approximation and its failure in reduced order modelling, in 1st International Conference on Recent Innovations in Electrical, Electronic and Communications Systems. (RIEECS-2017) (Dehradun, 2017)Google Scholar
  35. 35.
    A.K. Prajapati, R. Prasad, Failure of Padé approximation and time moment matching techniques in reduced order modelling, in IEEE 3rd International Conference for Convergence in Technology (I2CT-2018) (Pune, 2018)Google Scholar
  36. 36.
    A.K. Prajapati, R. Prasad, Order reduction of linear dynamic systems with an improved Routh stability method, in IEEE International Conference on Control, Power Communication and Computing Technologies (ICCPCCT-2018) (Kerala, 2018)Google Scholar
  37. 37.
    A.K. Prajapati, R. Prasad, Model order reduction by using the balanced truncation method and the factor division algorithm. IETE J. Res. (2018).  https://doi.org/10.1080/03772063.2018.1464971 Google Scholar
  38. 38.
    A.K. Prajapati, R. Prasad, Order reduction of linear dynamic systems by improved Routh approximation method. IETE J. Res. (2018).  https://doi.org/10.1080/03772063.2018.1452645 Google Scholar
  39. 39.
    A.K. Prajapati, R. Prasad, Reduced order modelling of LTI systems by using Routh approximation and factor division methods. Circuits Syst. Signal Process. (2018).  https://doi.org/10.1007/s00034-018-1010-6 Google Scholar
  40. 40.
    A.K. Prajapati, R. Prasad, Reduced order modelling of linear time invariant systems using factor division method to allow retention of dominant modes. IETE Tech. Rev. (2018).  https://doi.org/10.1080/02564602.2018.1503567 Google Scholar
  41. 41.
    A.K. Prajapati, R. Prasad, Reduced order modelling of linear time invariant systems by using improved modal method. Int. J. Pure Appl. Math. 119(12), 13011–13023 (2018)Google Scholar
  42. 42.
    R. Prasad, Padé type model order reduction for multivariable systems using Routh approximation. Comput. Electr. Eng. 26(6), 445–459 (2000)CrossRefzbMATHGoogle Scholar
  43. 43.
    J. Qi, J. Wang, H. Liu, A.D. Dimitrovski, Nonlinear model reduction in power systems by balancing of empirical controllability and observability covariances. IEEE Trans. Power Syst. 32(1), 114–126 (2017)CrossRefGoogle Scholar
  44. 44.
    D. Qian, S. Tong, X. Liu, Load frequency control for micro hydro power plants by sliding mode and model order reduction. Autom. J. Control Meas. Electron. Comput. Commun. 56(3), 318–330 (2015)Google Scholar
  45. 45.
    Y. Shamash, Stable reduced-order models using Padé-type approximations. IEEE Trans. Autom. Control 19(5), 615–616 (1974)CrossRefzbMATHGoogle Scholar
  46. 46.
    Y. Shamash, Linear system reduction using Padé approximation to allow retention of dominant modes. Int. J. Control 21(2), 257–272 (1975)CrossRefzbMATHGoogle Scholar
  47. 47.
    Y. Shamash, Truncation method of reduction: a viable alternative. Electron. Lett. 17(2), 97–98 (1981)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Z. Shi, W.O. Brien, Building energy model reduction using model-cluster-reduce pipeline. J. Build. Perform. Simul. (2017).  https://doi.org/10.1080/19401493.2017.1410572 Google Scholar
  49. 49.
    A. Sikander, R. Prasad, Linear time-invariant system reduction using a mixed methods approach. Appl. Math. Model. 39(15–16), 4848–4858 (2015)MathSciNetCrossRefGoogle Scholar
  50. 50.
    A. Sikander, R. Prasad, Soft computing approach for model order reduction of linear time invariant systems. Circuits Syst. Signal Process. 34(11), 3471–3487 (2015)CrossRefGoogle Scholar
  51. 51.
    A. Sikander, R. Prasad, A new technique for reduced-order modelling of linear time-invariant system. IETE J. Res. 63(3), 316–324 (2017)CrossRefGoogle Scholar
  52. 52.
    V. Singh, Nonuniqueness of model reduction using the Routh approach. IEEE Trans. Autom. Control 24(4), 650–651 (1979)CrossRefGoogle Scholar
  53. 53.
    N. Singh, R. Prasad, H.O. Gupta, Reduction of linear dynamic systems using Routh–Hurwitz array and factor division method. IETE J. Educ. 47(1), 25–29 (2006)CrossRefGoogle Scholar
  54. 54.
    J. Singh, C.B. Vishwakarma, K. Chatterjee, Biased reduction method by combining improved modified pole clustering and improved Padé approximations. Appl. Math. Model. 40, 1418–1426 (2016)MathSciNetCrossRefGoogle Scholar
  55. 55.
    A.K. Sinha, J. Pal, Simulation based reduced order modelling using a clustering technique. Comput. Electr. Eng. 16(3), 159–169 (1990)CrossRefGoogle Scholar
  56. 56.
    A. Sootla, J. Anderson, On projection-based model reduction of biochemical networks part II: the stochastic case, in Proceedings of the 53rd IEEE Conference on Decision and Control (Los Angeles, 2014), pp. 3621–3626Google Scholar
  57. 57.
    F.A. Taie, H. Werner, Balanced truncation for temporal- and spatial-LPV interconnected systems based on the full block S-procedure. Int. J. Control (2018).  https://doi.org/10.1080/00207179.2018.1440087 Google Scholar
  58. 58.
    C.B. Vishwakarma, Order reduction using modified pole clustering and Padé approximations. Int. J. Electr. Comput. Energ. Electron. Commun. Eng. 5(8), 998–1002 (2011)MathSciNetGoogle Scholar
  59. 59.
    C.B. Vishwakarma, R. Prasad, Clustering method for reducing order of linear system using Padé approximation. IETE J. Res. 54(5), 326–330 (2008)CrossRefGoogle Scholar
  60. 60.
    Q. Wang, Y. Wang, E.Y. Lam, N. Wong, Model order reduction for neutral systems by moment matching. Circuits Syst. Signal Process. 32, 1039–1063 (2013)MathSciNetCrossRefGoogle Scholar
  61. 61.
    H. Zhang, L.W.P. Shi, Y. Zhao, Balanced truncation approach to model reduction of Markovian jump time-varying delay systems. J. Frankl. Inst. 352, 4205–4224 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical EngineeringIndian Institute of Technology RoorkeeHaridwarIndia

Personalised recommendations