Calculation of stable and unstable periodic orbits in a chopper-fed DC drive

2021;
: pp. 43–57
https://doi.org/10.23939/mmc2021.01.043
Received: October 29, 2020
Revised: November 30, 2020
Accepted: December 01, 2020
Authors:
1
Hetman Petro Sahaidachnyi National Army Academy

It is well known that electric drives demonstrate various nonlinear phenomena.  In particular, a chopper-fed analog DC drive system is characterized by the route to chaotic behavior though period-doubling cascade.  Besides, the considered system demonstrates coexistence of several stable periodic modes within the stability boundaries of the main period-1 orbit.  We discover the evolution of several periodic orbits utilizing the semi-analytical method based on the Filippov theory for the stability analysis of periodic orbits.  We analyze, in particular, stable and unstable period-1, 2, 3 and 4 orbits, as well as independent on stability they are significant for the organization of phase space.  We demonstrate, in particular, that the unstable periodic orbits undergo border collision bifurcations; those occur according to several scenarios related to the interaction of different orbits of the same period, including persistence border collision, when a periodic orbit is changed by a different orbit of the same period, and birth or disappearance of a couple of orbits of the same period characterized by different topology.

  1. Chau K. T., Zheng W.  Chaos in Electric Drive Systems: Analysis, Control and Application.  Wiley-IEEE Press (2011).
  2. Chau K. T., Chen J. H., Chan C. C., Pong J. K., Chan D. T.  Chaotic behavior in a simple DC drive.  Proceedings of Second International Conference on Power Electronics and Drive Systems. 1, 473–479 (1997).
  3. Okafor N., Zahawi B., Giaouris D., Banerjee S.  Chaos, coexisting attractors, and fractal basin boundaries in DC drives with full-bridge converter.  Proceedings of 2010 IEEE International Symposium on Circuits and Systems. 129–132 (2010).
  4. Okafor N.  Analysis and Control of Nonlinear Phenomena in Electrical Drives.  Ph.D. dissertation, Newcastle University, Newcastle, UK (2012).
  5. Zhang Y., Luo G.  Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems.  International Journal of Non-Linear Mechanics. 96, 12–21 (2017).
  6. Zakrzhevsky M.  New concepts of nonlinear dynamics: Complete bifurcation groups, protuberances, unstable periodic infinitiums and rare attractors.  J. Vibroeng. 10 (4), 421–441 (2008).
  7. Yevstignejev V., Klokov A., Smirnova R., Schukin I.  Rare attractors in typical nonlinear discrete dynamical models.  2012 IEEE 4th International Conference on Nonlinear Science and Complexity (NSC). 229–234 (2012).
  8. Pikulin D.  Rare phenomena and chaos in switching power converters.  In: Awrejcewicz J. (eds) Applied Non-Linear Dynamical Systems. Springer Proceedings in Mathematics & Statistics. 93, 203–211 (2014).
  9. Pikulins D., Tjukovs S., Eidaks J.  Effects of control non-idealities on the nonlinear dynamics of switching DC-DC converters.  In: Stavrinides S., Ozer M. (eds) Chaos and Complex Systems. Springer Proceedings in Complexity. 117–131 (2020).
  10. Filippov A.  Differential Equations with Discontinuous Righthand Sides.  Springer (1988).
  11. Giaouris D., Maity S., Banerjee S., Pickert V., Zahawi B.  Application of Filippov method for the analysis of subharmonic instability in DC-DC converters.  International Journal of Circuit Theory and Applications. 37 (8), 899–919 (2009).
  12. Baushev V., Zhusubaliyev Zh., Kolokolov Yu., Terekhin I.  Local stability of periodic solutions in sampled-data control systems.  Automation and Remote Control. 53 (6), 865–871 (1992).
  13. Mandal K., Chakraborty C., Abusorrah A., Al-Hindawi M. M., Al-Turki Y., Banerjee S.  An automated algorithm for stability analysis of hybrid dynamical systems.  Eur. Phys. J. Spec. Top. 222 (3–4), 757–768 (2013).
  14. Mandal K., Banerjee S., Chakraborty C.  A new algorithm for small-signal analysis of DC-DC converters.  IEEE Transactions on Industrial Informatics. 10 (1), 628–636 (2014).
  15. Hayes B., Condon M., Giaouris D.  Application of the Filippov Method to PV-fed DC-DC converters modeled as hybrid-DAEs.  Engineering Reports. 2 (9), e12237 (2020).
  16. Muppala K. K., Kavitha A, Duraisamy J. C. N.  Analysis of intermittent instabilities in switching power converters using Filippov's method.  COMPEL – The international journal for computation and mathematics in electrical and electronic engineering. 37 (6), 2025–2049 (2018).
  17. Mandal K., Abusorrah A., Al-Hindawi M. M., Al-Turki Y., El Aroudi A., Giaouris D., Banerjee S.  Control-oriented design guidelines to extend the stability margin of switching converters.  2017 IEEE International Symposium on Circuits and Systems (ISCAS). 1–4 (2017).
  18. Tahir F. R., Abdul-Hassan K. M., Abdullah M. A., Pham V.-T., Hoang T. M., Wang X.  Analysis and stabilization of chaos in permanent magnet DC motor driver.  International Journal of Bifurcation and Chaos. 27 (11), 1750173 (2017).
  19. Ma Y., Kawakami H., Tse C. K.  Bifurcation analysis of switched dynamical systems with periodically moving borders.  IEEE Transactions on Circuits and Systems I: Regular Papers. 51 (6), 1184–1193 (2004).
Mathematical Modeling and Computing, Vol. 8, No. 1, pp. 43–57 (2021)