Advertisement

Participation Factors for Singular Systems of Differential Equations

  • Ioannis DassiosEmail author
  • Georgios Tzounas
  • Federico Milano
Article

Abstract

In this article, we provide a method to measure the participation of system eigenvalues in system states, and vice versa, for a class of singular linear systems of differential equations. This method deals with eigenvalue multiplicities and covers all cases by taking into account both the algebraic and geometric multiplicity of the eigenvalues of the system matrix pencil. A Möbius transform is applied to determine the relative contributions associated with the infinite eigenvalue that appears because of the singularity of the system. The paper is a generalization of the conventional participation analysis, which provides a measure for the coupling between the states and the eigenvalues of systems of ordinary differential equations with distinct eigenvalues. Numerical examples are given including a classical DC circuit and a 2-bus power system dynamic model.

Keywords

Participation factor Singularity Dynamical system Möbius transform Differential equations 

Notes

Acknowledgements

This work is supported by the Science Foundation Ireland (SFI), by funding Ioannis Dassios, Georgios Tzounas and Federico Milano, under Investigator Programme Grant No. SFI/15 /IA/3074.

References

  1. 1.
    S.L. Campbell, Singular Systems of Differential Equations, vol. 1 (Pitman, San Francisco, 1980)zbMATHGoogle Scholar
  2. 2.
    S.L. Campbell, Singular Systems of Differential Equations, vol. 2 (Pitman, San Francisco, 1982)zbMATHGoogle Scholar
  3. 3.
    L. Dai, in Singular Control Systems, Lecture Notes in Control and Information Sciences, ed. by M. Thoma, A. Wyner (Springer, Berlin, 1988)Google Scholar
  4. 4.
    J.H. Chow, Power System Coherency and Model Reduction, vol. 94, Power Electronics and Power Systems (Springer, New York, 2013)CrossRefGoogle Scholar
  5. 5.
    I.K. Dassios, On non-homogeneous linear generalized linear discrete time systems. Circuits Syst. Signal Process. 31(5), 1699 (2012).  https://doi.org/10.1007/s00034-012-9400-7 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    I.K. Dassios, G. Kalogeropoulos, On a non-homogeneous singular linear discrete time system with a singular matrix pencil. Circuits Syst. Signal Process. 32(4), 1615 (2013).  https://doi.org/10.1007/s00034-012-9541-8 MathSciNetCrossRefGoogle Scholar
  7. 7.
    I.K. Dassios, Optimal solutions for non-consistent singular linear systems of fractional nabla difference equations. Circuits Syst. Signal Process. 34(6), 1769–1797 (2015)CrossRefzbMATHGoogle Scholar
  8. 8.
    I. Dassios, Stability and robustness of singular systems of fractional nabla difference equations. Circuits Syst. Signal Process. 36(1), 49–64 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    I. Dassios, D. Baleanu, G. Kalogeropoulos, On non-homogeneous singular systems of fractional nabla difference equations. Appl. Math. Comput. 227, 112–131 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    I. Dassios, D. Baleanu, Optimal Solutions for Singular Linear Systems of Caputo Fractional Differential Equations, Mathematical Methods in the Applied Sciences (Wiley, London, 2019)Google Scholar
  11. 11.
    I.K. Dassios, A practical formula of solutions for a family of linear non-autonomous fractional nabla difference equations. J. Comput. Appl. Math. 339, 317–328 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    I.K. Dassios, D.I. Baleanu, Duality of singular linear systems of fractional nabla difference equations. Appl. Math. Model. 39(14), 4180–4195 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    I. Dassios, D. Baleanu, Caputo and related fractional derivatives in singular systems. Appl. Math. Comput. 337, 591–606 (2018).  https://doi.org/10.1016/j.amc.2018.05.005 MathSciNetGoogle Scholar
  14. 14.
    I. Dassios, G. Tzounas, F. Milano, The Mobius transform effect in singular systems of differential equations. Appl. Math. Comput. 361, 338–353 (2019).  https://doi.org/10.1016/j.amc.2019.05.047 MathSciNetGoogle Scholar
  15. 15.
    R.F. Gantmacher, The Theory of Matrices I, II (Chelsea, New York, 1959) zbMATHGoogle Scholar
  16. 16.
    F. Garofalo, L. Iannelli, F. Vasca, Participation factors and their connections to residues and relative gain array. The Proceedings of the IFAC World Congress 35(1), 125 (2002).  https://doi.org/10.3182/20020721-6-ES-1901.00182 Google Scholar
  17. 17.
    A.M.A. Hamdan, Coupling measures between modes and state variables in power-system dynamics. Int. J. Control 43(3), 1029 (1986).  https://doi.org/10.1080/00207178608933521 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    M. Liu, I. Dassios, F. Milano, On the stability analysis of systems of neutral delay differential equations. Circuits Syst. Signal Process. 38(4), 1639–1653 (2019)MathSciNetCrossRefGoogle Scholar
  19. 19.
    M. Liu, I. Dassios, G. Tzounas, F. Milano, Stability analysis of power systems with inclusion of realistic-modeling of WAMS delays. IEEE Trans. Power Syst. 34(1), 627–636 (2019)CrossRefGoogle Scholar
  20. 20.
    F. Milano, I.K. Dassios, Primal and dual generalized eigenvalue problems for power systems small-signal stability analysis. IEEE Trans. Power Syst. 32(6), 4626 (2017).  https://doi.org/10.1109/TPWRS.2017.2679128 CrossRefGoogle Scholar
  21. 21.
    F. Milano, I.K. Dassios, Small-signal stability analysis for non-index 1 Hessenberg form systems of delay differential-algebraic equations. IEEE Trans. Circuits Syst. I: Regul. Pap. 63(9), 1521 (2016).  https://doi.org/10.1109/TCSI.2016.2570944 MathSciNetCrossRefGoogle Scholar
  22. 22.
    F. Milano, Semi-implicit formulation of differential-algebraic equations for transient stability analysis. IEEE Trans. Power Syst. 31(6), 4534 (2016).  https://doi.org/10.1109/TPWRS.2016.2516646 CrossRefGoogle Scholar
  23. 23.
    M. Netto, Y. Susuki, L. Mili, Data-driven participation factors for nonlinear systems based on Koopman mode decomposition. IEEE Control Syst. Lett. 3(1), 198 (2019).  https://doi.org/10.1109/LCSYS.2018.2871887 CrossRefGoogle Scholar
  24. 24.
    F.L. Pagola, I.J. Perez-Arriaga, G.C. Verghese, On sensitivities, residues and participations: applications to oscillatory stability analysis and control. IEEE Trans. Power Syst. 4(1), 278 (1989).  https://doi.org/10.1109/59.32489 CrossRefGoogle Scholar
  25. 25.
    I.J. Perez-Arriaga, G.C. Verghese, F.C. Schweppe, Selective modal analysis with applications to electric power systems, part i: heuristic introduction. IEEE Trans. Power Appar. Syst. PAS–101(9), 3117 (1982).  https://doi.org/10.1109/TPAS.1982.317524 CrossRefGoogle Scholar
  26. 26.
    J. Qiu, K. Sun, T. Wang, H. Gao, Observer-based fuzzy adaptive event-triggered control for pure-feedback nonlinear systems with prescribed performance. IEEE Trans. Fuzzy Syst. (2019).  https://doi.org/10.1109/TFUZZ.2019.2895560 Google Scholar
  27. 27.
    W.J. Rugh, Linear System Theory (Prentice Hall International, London, 1996)zbMATHGoogle Scholar
  28. 28.
    K. Sun, S. Mou, J. Qiu, T. Wang, H. Gao, Adaptive fuzzy control for non-triangular structural stochastic switched nonlinear systems with full state constraints. IEEE Trans. Fuzzy Syst. (2018).  https://doi.org/10.1109/TFUZZ.2018.2883374
  29. 29.
    T. Tian, X. Kestelyn, O. Thomas, H. Amano, A.R. Messina, An accurate third-order normal form approximation for power system nonlinear analysis. IEEE Trans. Power Syst. 33(21), 2128 (2018).  https://doi.org/10.1109/TPWRS.2017.2737462 CrossRefGoogle Scholar
  30. 30.
    G.C. Verghese, I.J. Perez-Arriaga, F.C. Schweppe, Selective modal analysis with applications to electric power systems, part II: The dynamic stability problem. IEEE Trans. Power Appar. Syst. PAS–101(9), 3117 (1982).  https://doi.org/10.1109/TPAS.1982.317525 Google Scholar
  31. 31.
    L. Zhang, C. Gao, Y. Liu, New research advance of variable structure control singular systems with time delays (2018).  https://doi.org/10.12677/DSC.2018.74038

Copyright information

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

Authors and Affiliations

  • Ioannis Dassios
    • 1
    Email author
  • Georgios Tzounas
    • 1
  • Federico Milano
    • 1
  1. 1.AMPSASUniversity College DublinDublinIreland

Personalised recommendations