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Robustness of Delayed Multistable Systems

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Delays and Interconnections: Methodology, Algorithms and Applications

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 10))

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Abstract

In this chapter sufficient conditions for input-to-state stability (ISS) of delayed systems are presented using Lyapunov-Razumikhin functions. It is shown that ISS multistable systems are robust with respect to delays in the feedback path. First, the approach is illustrated by establishing the ISS property for the model of a nonlinear pendulum, then delay-dependent robustness conditions are derived. Second, it is shown that, under certain assumptions, the problem of phase-locking analysis in droop-controlled inverter-based microgrids with delays can be reduced to the stability investigation of a nonlinear pendulum, and corresponding delay-dependent conditions for asymptotic phase-locking are derived for an exemplary microgrid consisting of two droop-controlled inverters.

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Notes

  1. 1.

    The overall delay reduces to \(\tau =1.5T_{S}\) if no moving average function for the measurement is used [29].

References

  1. Angeli, D.: An almost global notion of input-to-state stability. IEEE Trans. Autom. Control 49, 866–874 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Angeli, D. Efimov, On input-to-state stability with respect to decomposable invariant sets. In: Proceedings of 52nd IEEE Conference on Decision and Control, Florence (2013)

    Google Scholar 

  3. D. Angeli, D. Efimov, Characterizations of input-to-state stability for systems with multiple invariant sets. IEEE Trans. Autom. Control PP(99), 1–13 (2015)

    Google Scholar 

  4. Angeli, D., Ferrell, J., Sontag, E.: Detection of multistability, bifurcations and hysteresis in a large class of biological positive-feedback systems. Proc. Natl. Acad. Sci. USA 101, 1822–1827 (2004)

    Article  Google Scholar 

  5. Angeli, D., Praly, L.: Stability robustness in the presence of exponentially unstable isolated equilibria. IEEE Trans. Autom. Control 56, 1582–1592 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Chandorkar, D. Divan, and R. Adapa. Control of parallel connected inverters in stand-alone AC supply systems. IEEE Trans. Ind. Appl. 29(1), 136 –143 (1993)

    Article  Google Scholar 

  7. Chaves, M., Eissing, T., Allgower, F.: Bistable biological systems: a characterization through local compact input-to-state stability. IEEE Trans. Autom. Control 45, 87–100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dashkovskiy, S., Efimov, D., Sontag, E.: Input to state stability and allied system properties. Autom. Remote Control 72(8), 1579–1614 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Dashkovskiy, L. Naujok, Lyapunov-Razumikhin and Lyapunov-Krasovskii theorems for interconnected ISS time-delay systems. In: Proceedings of 19th International Symposium on Mathematical Theory of Networks and Systems(MTNS), pp. 1179–1184, Budapest (2010)

    Google Scholar 

  10. Efimov, D., Fradkov, A.: Oscillatority of nonlinear systems with static feedback. SIAM J. Optim. Control 48(2), 618–640 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Efimov, R. Ortega, J. Schiffer, ISS of multistable systems with delays: application to droop-controlled inverter-based microgrids. In: Proceedings of ACC 2015, Chicago (2015)

    Google Scholar 

  12. Efimov, D., Schiffer, J., Ortega, R.: Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids. Int. J. Control 89(5), 909–918 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fang, X., Misra, S., Xue, G., Yang, D.: Smart grid - the new and improved power grid: a survey. Commun. Surv. Tutor. IEEE 14(4), 944–980 (2012)

    Article  Google Scholar 

  14. Farhangi, H.: The path of the smart grid. IEEE Power Energy Mag. 8(1), 18–28 (2010)

    Article  MathSciNet  Google Scholar 

  15. Franci, A., Chaillet, A., Panteley, E., Lamnabhi-Lagarrigue, F.: Desynchronization and inhibition of Kuramoto oscillators by scalar mean-field feedback. Math. Control. Signals Syst. 24(1–2), 169–217 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fridman, E.: Tutorial on Lyapunov-based methods for time-delay systems. Eur. J. Control. 20, 271–283 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Green, T., Prodanovic, M.: Control of inverter-based micro-grids. Electr. Power Syst. Res. 77(9), 1204–1213 (2007). July

    Article  Google Scholar 

  18. Guckenheimer, J., Holmes, P.: Structurally stable heteroclinic cycles. Math. Proc. Camb. Phil. Soc. 103, 189–192 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guerrero, J., Loh, P., Chandorkar, M., Lee, T.: Advanced control architectures for intelligent microgrids - part I: decentralized and hierarchical control. IEEE Trans. Ind. Electron. 60(4), 1254–1262 (2013)

    Article  Google Scholar 

  20. Hatziargyriou, N., Asano, H., Iravani, R., Marnay, C.: Microgrids. IEEE Power Energy Mag. 5(4), 78–94 (2007)

    Article  Google Scholar 

  21. C.M. Kellett, F.R. Wirth, P.M. Dower, Input-to-state stability, integral input-to-state stability, and unbounded level sets. In: Proceedings of 9th IFAC Symposium on Nonlinear Control Systems, pp. 38–43, Toulouse (2013)

    Google Scholar 

  22. Kukrer, O.: Discrete-time current control of voltage-fed three-phase PWM inverters. IEEE Trans. Power Electron. 11(2), 260–269 (1996)

    Article  Google Scholar 

  23. R. Lasseter. Microgrids. In IEEE Power Engineering Society Winter Meeting, 2002, volume 1, pages 305 – 308 vol.1, 2002

    Google Scholar 

  24. Lopes, J., Moreira, C., Madureira, A.: Defining control strategies for microgrids islanded operation. IEEE Trans. Power Syst. 21(2), 916–924 (2006). May

    Article  Google Scholar 

  25. Maksimovic, D., Zane, R.: Small-signal discrete-time modeling of digitally controlled PWM converters. IEEE Trans. Power Electron. 22(6), 2552–2556 (2007)

    Article  Google Scholar 

  26. P. Monzón, R. Potrie, Local and global aspects of almost global stability. In: Proceedings of 45th IEEE Conference on Decision and Control, pp. 5120–5125, San Diego, USA (2006)

    Google Scholar 

  27. Münz, U., Romeres, D.: Region of attraction of power systems. Estim. Control. Netw. Syst. 4, 49–54 (2013)

    Google Scholar 

  28. Nitecki, Z., Shub, M.: Filtrations, decompositions, and explosions. Am. J. Math. 97(4), 1029–1047 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nussbaumer, T., Heldwein, M.L., Gong, G., Round, S.D., Kolar, J.W.: Comparison of prediction techniques to compensate time delays caused by digital control of a three-phase buck-type PWM rectifier system. IEEE Trans. Ind. Electron. 55(2), 791–799 (2008)

    Article  Google Scholar 

  30. Pepe, P., Karafyllis, I., Jiang, Z.-P.: On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations. Automatica 44(9), 2266–2273 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Rajaram, U. Vaidya, M. Fardad, Connection between almost everywhere stability of an ODE and advection PDE. In: Proceedings of the 46th IEEE Conference Decision and Control, pp. 5880–5885, New Orleans (2007)

    Google Scholar 

  32. Rantzer, A.: A dual to Lyapunov’s stability theorem. Syst. Control Lett. 42, 161–168 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Rocabert, J., Luna, A., Blaabjerg, F., Rodriguez, P.: Control of power converters in AC microgrids. IEEE Trans. Power Electron. 27(11), 4734–4749 (2012). Nov

    Article  Google Scholar 

  34. Rumyantsev, V., Oziraner, A.: Stability and stabilization of motion with respect to part of variables. Nauka, Moscow (1987). [in Russian]

    MATH  Google Scholar 

  35. J. Schiffer, R. Ortega, A. Astolfi, J. Raisch, T. Sezi, Conditions for stability of droop-controlled inverter-based microgrids. Automatica (2014). Accepted

    Google Scholar 

  36. J. Schiffer, R. Ortega, C.A. Hans, J. Raisch, Droop-controlled inverter-based microgrids are robust to clock drifts. In: American Control Conference (ACC), pp. 2341–2346. IEEE (2015)

    Google Scholar 

  37. Simpson-Porco, J.W., Dörfler, F., Bullo, F.: Synchronization and power sharing for droop-controlled inverters in islanded microgrids. Automatica 49(9), 2603–2611 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747–817 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  39. Sontag, E.: On the input-to-state stability property. Eur. J. Control 1, 24–36 (1995)

    Article  MATH  Google Scholar 

  40. Sontag, E., Wang, Y.: New characterizations of input-to-state stability. IEEE Trans. Autom. Control 41(9), 1283–1294 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sontag, E.D., Wang, Y.: On characterizations of input-to-state stability with respect to compact sets. In: Proceedings of IFAC Non-Linear Control Systems Design Symposium. NOLCOS ’95, pp. 226–231. Tahoe City, CA (1995)

    Google Scholar 

  42. Stan, G.-B., Sepulchre, R.: Analysis of interconnected oscillators by dissipativity theory. IEEE Trans. Autom. Control 52, 256–270 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. Teel, A.R.: Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Trans. Automat. Control 43(7), 960–964 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  44. van Handel, R.: Almost global stochastic stability. SIAM J. Control Optim. 45(4), 1297–1313 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Varaiya, P.P., Wu, F.F., Bialek, J.W.: Smart operation of smart grid: risk-limiting dispatch. Proc. IEEE 99(1), 40–57 (2011)

    Article  Google Scholar 

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Acknowledgements

This work was supported in part by the Government of Russian Federation (Grant 08-08) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).

Author information

Authors and Affiliations

  1. Inria, 40 avenue Halley, 59650, Villeneuve d’Ascq, France

    Denis Efimov

  2. CRIStAL (UMR-CNRS 9189), Ecole Centrale de Lille, Avenue Paul Langevin, 59651, Villeneuve d’Ascq, France

    Denis Efimov

  3. Department of Control Systems and Informatics, University ITMO, 49 avenue Kronverkskiy, 197101, Saint Petersburg, Russia

    Denis Efimov

  4. Control Systems and Network Control Technology, Brandenburg University of Technology (BTU), Cottbus, 03046, Germany

    Johannes Schiffer

  5. Laboratoire des Signaux et Systémes, École Supérieure d’Electricité (SUPELEC), Gif-sur-Yvette, 91192, France

    Romeo Ortega

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  1. Denis Efimov
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  2. Johannes Schiffer
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  3. Romeo Ortega
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Corresponding author

Correspondence to Denis Efimov .

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Editors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Giorgio Valmorbida

  2. Laboratoire d’Analyse et d’Architecture de Systèmes-CNRS, Toulouse, France

    Alexandre Seuret

  3. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Islam Boussaada

  4. Mechanical and Industrial Engineering, Northeastern University, Boston, MA, USA

    Rifat Sipahi

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Efimov, D., Schiffer, J., Ortega, R. (2019). Robustness of Delayed Multistable Systems. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_6

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