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Lyapunov and LMI analysis and feedback control of border collision bifurcations

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Abstract

Feedback control of piecewise smooth discrete-time systems that undergo border collision bifurcations is considered. These bifurcations occur when a fixed point or a periodic orbit of a piecewise smooth system crosses or collides with the border between two regions of smooth operation as a system parameter is quasistatically varied. The class of systems studied is piecewise smooth maps that depend on a parameter, where the system dimension n can take any value. The goal of the control effort in this work is to replace the bifurcation so that in the closed-loop system, the steady state remains locally attracting and locally unique (“nonbifurcation with persistent stability”). To achieve this, Lyapunov and linear matrix inequality (LMI) techniques are used to derive a sufficient condition for nonbifurcation with persistent stability. The derived condition is stated in terms of LMIs. This condition is then used as a basis for the design of feedback controls to eliminate border collision bifurcations in piecewise smooth maps and to produce the desirable behavior noted earlier. Numerical examples that demonstrate the effectiveness of the proposed control techniques are given.

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References

  1. Alvarez-Ramirez, J.: Dynamics of controlled-linear discrete time systems with a dead-zone nonlinearity. In: Proceedings of the American Control Conference, Baltimore, MD, Vol. 1, pp. 761–765 (1994)

  2. Banerjee, S., Grebogi, C.: Border collision bifurcations in two-dimensional piecewise smooth maps. Phys. Rev. E 59(4), 4052–4061 (1999)

    Article  Google Scholar 

  3. Banerjee, S., Karthik, M.S., Yuan, G.H., Yorke, J.A.: Bifurcations in one-dimensional piecewise smooth maps – theory and applications in switching systems. IEEE Trans. Circuits Syst. I 47(3), 389–394 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Banerjee, S., Ranjan, P., Grebogi, C.: Bifurcations in two-dimensional piecewise smooth maps – theory and applications in switching circuits. IEEE Trans. Circuits Syst. I 47(5), 633–643 (2000)

    Article  MATH  Google Scholar 

  5. Boyd, S., Ghaoui, L.El., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, Studies in Applied Mathematics, Vol. 15. SIAM, Philadelphia, PA (1994)

  6. Deane, J.H.B., Hamill, D.C.: Instability, subharmonics, and chaos in power electronic systems. IEEE Trans. Power Electron. 5(3), 260–268 (1990)

    Article  Google Scholar 

  7. Di Bernardo, M., Budd, C.J., Champneys, A.R.: Corner collision implies border-collision bifurcation. Physica D 154, 171–194 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Di Bernardo, M., Budd, C.J., Champneys, A.R.: Grazing and border-collision in piecewise-smooth systems: A unified analytical framework. Phys. Rev. Lett. 86(12), 2553–2556 (2001)

    Article  Google Scholar 

  9. Di Bernardo, M., Feigin, M.I., Hogan, S.J., Homer, M.E.: Local analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems. Chaos Solitons Fractals 10(11), 1881–1908 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dutta, M., Nusse, H.E., Ott, E., Yorke, J.A., Yuan, G.H.: Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems. Phys. Rev. Lett. 83(21), 4281–4284 (1999)

    Article  Google Scholar 

  11. Feigin, M.I.: Doubling of the oscillation period with C-bifurcations in piecewise continuous systems. Prikl. Mat. Mekh. 34, 861–869 (1970)

    MathSciNet  Google Scholar 

  12. Feigin, M.I.: The increasingly complex structure of the bifurcation tree of a piecewise-smooth system. J. Appl. Math. Mech. 59(6), 853–863 (1995)

    Article  MathSciNet  Google Scholar 

  13. Feng, G.: Stability analysis of piecewise discrete-time linear systems. IEEE Trans. Autom. Control 47(7), 1108–1112 (2002)

    Article  Google Scholar 

  14. Firoiu, V., Borden, M.: A study of active queue management for congestion control. In: Proceedings of the INFOCOM, Tel Aviv, Israel, Vol. 3, pp. 1435–1444 (2000)

  15. Hassouneh, M.A.: Feedback Control of Border Collision Bifurcations in Piecewise Smooth Systems. PhD Thesis, University of Maryland, College Park, MD (2003)

  16. Hassouneh, M.A., Abed, E.H., Nusse, H.E.: Robust dangerous border-collision bifurcations in piecewise smooth systems. Phys. Rev. Lett. 92(7), 070201 (2004)

    Article  Google Scholar 

  17. Hassouneh, M.A., Lee, H.-C., Abed, E.H.: Washout filters in feedback control: Benefits, limitations and extensions. In: Proceedings of the American Control Conference, Boston, MA, pp. 3950–3955, June/July (2004)

  18. Hommes, C.H., Nusse, H.E.: Period 3 to period 2 bifurcation for piecewise linear-models. J. Econ. 54(2), 157–169 (1991)

    MATH  MathSciNet  Google Scholar 

  19. Johansson, M., Rantzer, A.: Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Autom. Control 43, 555–559 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kantner, M.: Robust stability of piecewise linear discrete time systems. In: Proceedings of the American Control Conference, Albuquerque, NM, pp. 1241–1245, June (1997)

  21. Linsay, P.S.: Period doubling and chaotic behavior in a driven anharmonic-oscillator. Phys. Rev. Lett. 47(19), 1349–1352 (1981)

    Article  Google Scholar 

  22. Maistrenko, Y.L., Maistrenko, V.L., Vikul, S.I., Chua, L.O.: Bifurcations of attracting cycles from time-delayed Chua circuit. Int. J. Bifurcation Chaos 5, 653–671 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mignone, D., Ferrari-Trecate, G., Morari, M.: Stability and stabilization of piecewise affine and hybrid systems: An LMI approach. In: Proceedings of the Conference on Decision and Control, Sydney, Australia, Vol. 1, pp. 504–509, December (2000)

  24. Nordmark, A.B.: Non-periodic motion caused by grazing incidence in impact oscillators. J. Sound Vib. 2, 279–297 (1991)

    Article  Google Scholar 

  25. Nusse, H.E., Ott, E., Yorke, J.A.: Border-collision bifurcation: An explanation for an observed phenomena. Phys. Rev. E 201, 197–204 (1994)

    MathSciNet  Google Scholar 

  26. Nusse, H.E., Yorke, J.A.: Border-collision bifurcations including ‘period two to period three’ for piecewise smooth maps. Physica D 57, 39–57 (1992)

    Article  MathSciNet  Google Scholar 

  27. Nusse, H.E., Yorke, J.A.: Border-collision bifurcations for piecewise smooth one-dimensional maps. Int. J. Bifurcation Chaos 5, 189–207 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  28. Ranjan, P., Abed, E.H., La, R.J.: Nonlinear instabilities in TCP-RED. In: Proceedings of the INFOCOM, New York Vol. 1, pp. 249–258, (2002)

  29. Sun, J., Amellal, F., Glass, L., Billette, J.: Alternans and period-doubling bifurcations in atrioventricular nodal conduction. J. Theor. Biol. 173, 79–91 (1995)

    Article  Google Scholar 

  30. Tanaka, H., Ushio, T.: Analysis of border-collision bifurcations in a flow model of a switching system. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E85-A(4), 734–739 (2002)

    Google Scholar 

  31. Yuan, G.H.: Shipboard Crane Control, Simulated Data Generation and Border Collision Bifurcations. PhD Thesis, University of Maryland, College Park, MD (1997)

  32. Yuan, G.H., Banerjee, S., Ott, E., Yorke, J.A.: Border collision bifurcations in the buck converter. IEEE Trans. Circuits Syst. I 45(7), 707–716 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland, College Park, MD, 20742, USA

    Munther A. Hassouneh & Eyad H. Abed

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  1. Munther A. Hassouneh
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  2. Eyad H. Abed
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Correspondence to Eyad H. Abed.

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Hassouneh, M.A., Abed, E.H. Lyapunov and LMI analysis and feedback control of border collision bifurcations. Nonlinear Dyn 50, 373–386 (2007). https://doi.org/10.1007/s11071-006-9169-y

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