Using Steady-State Response for Predicting Stability Boundaries in Switched Systems Under PWM with Linear and Bilinear Plants

  • A. El Aroudi
  • M. Al-Numay
  • K. Al Hosani
  • N. Al Sayari
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 168)


Switching systems under Pulse Width Modulation (PWM) are commonly utilized in many industrial applications. Due to their associated nonlinearities, such systems are prone to exhibit a large variety of complex dynamics and undesired behaviors. In general, slow dynamics in these systems can be predicted and analyzed by conventional averaging procedures. However, fast dynamics instabilities such as period doubling (PD) and saddle-node (SN) bifurcations cannot be detected by average models and analyzing them requires the use of additional sophisticated tools. In this chapter, closed-form conditions for predicting the boundary of these bifurcations in a class of PWM systems with linear and bilinear plants are obtained using a time-domain asymptotic approach. Previous studies have obtained similar boundaries by either solving the eigenvalue problem of the monodromy matrix of the Poincaré map or performing a Fourier series expansion of the feedback signal. While the former approach is general and can be applied to linear as well as bilinear plants, the later approach is applicable only to PWM systems with linear plants. The conditions for fast scale instability boundaries presented in this chapter are obtained from the steady-state analysis of the Poincaré map using an asymptotic approach without resorting to frequency-domain Fourier analysis and without using the monodromy matrix of the Poincaré map. The obtained expressions are simpler than the previously reported ones and allow to understand the effect of different system’s parameters on its stability. In PWM systems with linear plants, under certain practical conditions concerning these parameters, the matrix form expression can be approximated by standard polynomial functions expressed in terms of the operating duty cycle weighted by the Markov parameters of the linear part of the system.


Duty Cycle Pulse Width Modulation Period Doubling Monodromy Matrix Fourier Series Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by the Spanish ministerio de Economía y Competitividad under grant DPI2013-47437-R, the VPP of King Saud University, Riyadh, KSA and The Petroleum Institute, Abu Dhabi, UAE.


  1. 1.
    Liberzon, D. Switching in Systems and Control, Springuer, 2003Google Scholar
  2. 2.
    El Aroudi, A., Debbat, M., Martinez-Salamero, L.: Poincaré maps modelling and local orbital stability analysis of discontinuous piecewise affine periodically driven systems. Nonlinear Dyn. 50(3), 431–445 (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    El Aroudi, A.: Prediction of subharmonic oscillation in switching converters under different control strategies. IEEE Trans. Circuits Syst. II: Express Briefs 62(11), 910–914 (2014)CrossRefGoogle Scholar
  4. 4.
    Fang, C.-C.: Using byquist or nyquist-like plot to predict three typical instabilities in DC-DC converters. J. Franklin Inst. 350(10), 3293–3312 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fang, C.-C.: Critical conditions for a class of switched linear systems based on harmonic balance: applications to DC-DC converters. Nonlinear Dyn. 70(3), 1767–1789 (2012)CrossRefGoogle Scholar
  6. 6.
    Giaouris, D., Maity, S., Banerjee, S., Pickert, V., Zahawi, B.: Application of Filippov method for the analysis of subharmonic instability in DC-DC converters. Int. J. Circuit Theory Appl. 37(8), 899–919 (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    El Aroudi, A., Calvente, J., Giral, R., Martinez-Salamero, L.: Effects of non-ideal current sensing on subharmonic oscillation boundary in DC-DC switching converters under CMC. In: Industrial Electronics Society, IECON 2013–39th Annual Conference of the IEEE, pp. 8367–8372, 10–13 Nov 2013Google Scholar
  8. 8.
    Fang, C.-C.: Critical conditions of saddle-node bifurcations in switching DC-DC converters. Int. J. Electron. 100(8), 1147–1174 (2013)CrossRefGoogle Scholar
  9. 9.
    Robert, B., Robert, C.: Border collision bifurcations in a one-dimensional piecewise smooth Map for a PWM current-programmed H-bridge Inverter. Int. J. Control 7(16), 1356–1367 (2002)CrossRefzbMATHGoogle Scholar
  10. 10.
    Miladi, Y., Feki, M., Derbel, N.: On the model identification of an incubator based on genetic algorithms. In: 9th International Multi-Conference on Systems. Signals and Devices, Chemnitz, Germany (2012)Google Scholar
  11. 11.
    Gardini, L., Tramontana, F., Banerjee, S.: Bifurcation analysis of an inductorless chaos generator using 1D piecewise smooth map. Math. Comput. Simul. 95, 137–145 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Huh J.-Y., Wennmacher, G.: A study on the stability analysis of a PWM controlled hydraulic equipment. KSME Int. J. 11(4), 397–407 (1997)Google Scholar
  13. 13.
    Liu, G., Xia, W., Qi, D., Hu, R.: Analysis of Dither in electro-hydraulic proportional control. Telekominika 11(11), 6808–6814 (2013)Google Scholar
  14. 14.
    Mandal, K., Chakraborty, C., Abusorrah, A., Al-Hindawi, M., Al-Turki, Y., Banerjee, S.: An automated algorithm for stability analysis of hybrid dynamical systems. Eur. Phys. J. Special Topics 222, 757–768 (2013)CrossRefADSGoogle Scholar
  15. 15.
    Mazumder, S.K., Nayfeh, A.H., Boroyevich, D.: Theoretical and experimental investigation of the fast- and slow-scale instabilities of a DC-DC converter. IEEE Trans. Power Electron. 16(2), 201–216 (2001)CrossRefGoogle Scholar
  16. 16.
    Leine, R.L., Nijemeijer, H.: Dynamics and Bifurcations of Non-smooth Mechanical Systems. Lecture Notes in Applied and Computational Mechanics, vol. 18. Springer, Heidelberg (2004)Google Scholar
  17. 17.
    Fang, C.-C., Abed, E.H.: Harmonic balance analysis and control of period doubling bifurcation in buck converters. IEEE Int. Symp. Circuits Syst. 3, 209–212 (2001)Google Scholar
  18. 18.
    Hiskens, I.A., Pai, M.A.: Trajectory sensitivity analysis of hybrid systems. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 47(2), 204–220, (2000)Google Scholar
  19. 19.
    Fossas, E., Olivar, G.: Study of Chaos in the buck converter. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 43(1), 13–25 (1996)CrossRefGoogle Scholar
  20. 20.
    El Aroudi, A.: A closed form expression for predicting fast scale instability in switching buck converters. In: The International Conference on Structural Nonlinear Dynamics and Diagnosis, Marrakech, Morocco (2012)Google Scholar
  21. 21.
    Tanitteerapan, T., Mori, S.: Fundamental frequency parabolic PWM controller for lossless soft-switching boost power factor correction. In: The 2001 IEEE International Symposium on Circuits and Systems, ISCAS 2001, vol. 3, pp. 57–60 (2001)Google Scholar
  22. 22.
    Fang, C.-C. Exact orbital stability analysis of static and dynamic ramp compensations in DC-DC converters. In: Proceedings IEEE International Symposium on Industrial Electronics, ISIE 2001, vol. 3, pp. 2124–2129 (2001)Google Scholar
  23. 23.
    Lehman, B., Bass, R.M.: Switching frequency dependent averaged models for PWM DC-DC converters. IEEE Trans. Power Electron. 11(1), 89–98 (1996)CrossRefGoogle Scholar
  24. 24.
    van der Woude, J.W., De Koning, W., Fuad, Y.: On the periodic behavior of PWM DC-DC converters. IEEE Trans. Power Electron. 17(4), 585–595 (2002)CrossRefGoogle Scholar
  25. 25.
    El Aroudi, A., Giaouris, D., Martinez-Salamero, L., Banerjee, S., Voutetakis, S., Papadopoulou, S.: Bifurcation behavior in switching converters driving other downstream converters in DC distributed power systems applications. In: MEDYNA’2013 Marrakech Morocco (2013)Google Scholar
  26. 26.
    El Aroudi, A.: A time-domain asymptotic approach to predict saddle-node and period doubling bifurcations in pulse width modulated piecewise linear systems. In: The International Conference on Structural Nonlinear Dynamics and Diagnosis, Agadir, Morocco (2014)Google Scholar
  27. 27.
    Corriou, J.-P.: Process control: theory and applications. Springer, London (2014). ISBN 978-1-4471-3848-8Google Scholar
  28. 28.
    Lewin, L.: Polylogarithms and Associated Functions. North-Holland, New York (1981)Google Scholar
  29. 29.
    Hamill, D.C., Deane, J.H.B., Jefferies, D.J.: Modelling of chaotic DC-DC converters by iterated nonlinear mapping. IEEE Trans. Power Electron. 7(1), 25–36 (1992)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • A. El Aroudi
    • 1
  • M. Al-Numay
    • 2
  • K. Al Hosani
    • 3
  • N. Al Sayari
    • 3
  1. 1.University Rovira i VirgiliTarragonaSpain
  2. 2.King Saud UniversityRiyadhKSA
  3. 3.The Petroleum InstituteAbu DhabiUAE

Personalised recommendations