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Hybrid Dynamical Systems with Finite Memory

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Recent Results on Nonlinear Delay Control Systems

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 4))

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Abstract

Hybrid systems with memory are dynamical systems that exhibit both hybrid and delay phenomena, as seen in many physical and engineered applications. A prominent example is the use of delayed hybrid feedback in control systems. This chapter outlines a framework that allows studying hybrid systems with delays through generalized solutions and summarizes some recent results on basic existence and well-posedness of solutions and stability analysis using Lyapunov-based methods.

This work is supported, in part, by Royal Society grant IE130106, EU FP7 grant PCIG13-GA- 2013-617377, US AFOSR grant FA9550-12-1-0127, and US NSF grant ECCS-1232035.

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References

  1. Banos, A., Rubio, F., Tarbouriech, S., Zaccarian, L.: Delay-independent stability via reset loops. In: Seuret, A., Ozbay, H., Bonnet, C., Mounier, H. (eds.) Low-Complexity Controllers for Time-Delay Systems, pp. 111–125. Springer, New York (2014)

    Chapter  Google Scholar 

  2. Cloosterman, M., van de Wouw, N., Heemels, W., Nijmeijer, H.: Stability of networked control systems with uncertain time-varying delays. IEEE Trans. Autom. Control 54(7), 1575–1580 (2009)

    Article  Google Scholar 

  3. Goebel, R., Teel, A.: Solutions to hybrid inclusions via set and graphical convergence with stability theory applications. Automatica 42(4), 573–587 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Goebel, R., Hespanha, J., Teel, A., Cai, C., Sanfelice, R.: Hybrid systems: generalized solutions and robust stability. In: Proceedings of the 6th IFAC Symposium on Nonlinear Control Systems, pp. 1–12 (2004)

    Google Scholar 

  5. Goebel, R., Sanfelice, R., Teel, A.: Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press, Princeton (2012)

    Google Scholar 

  6. Halanay, A.: Differential Equations: Stability, Oscillations. Time Lags. Academic Press, New York (1966)

    Google Scholar 

  7. Hale, J., Verduyn Lunel, S.: Introduction to Functional Differential Equations. Springer, New York (1993)

    Google Scholar 

  8. Liu, J., Liu, X., Xie, W.: Input-to-state stability of impulsive and switching hybrid systems with time-delay. Automatica 47(5), 899–908 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Liu, J., Teel, A.: Generalized solutions to hybrid systems with delays. In: Proceedings of the IEEE Conference on Decision and Control, pp. 6169–6174 (2012)

    Google Scholar 

  10. Liu, J., Teel, A.: Hybrid systems with memory: modelling and stability analysis via generalized solutions. In: Proceedings of the 19th IFAC World Congress, pp. 6019–6024 (2014)

    Google Scholar 

  11. Liu, J., Teel, A.: Hybrid systems with memory: existence and well-posedness of generalized solutions. SIAM J. Control Optim. (2014) (submitted)

    Google Scholar 

  12. Liu, J., Teel, A.: Lyapunov-based sufficient conditions for stability of hybrid systems with memory. IEEE Trans. Autom. Control (2014) (submitted)

    Google Scholar 

  13. Liu, X., Shen, J.: Stability theory of hybrid dynamical systems with time delay. IEEE Trans. Autom. Control 51(4), 620–625 (2006)

    Article  MathSciNet  Google Scholar 

  14. Rockafellar, R., Wets, J.: Variational Analysis. Series Grundlehren der mathematischen Wissenschaften, vol. 317. Springer, New York (1998)

    Google Scholar 

  15. Sanfelice, R., Goebel, R., Teel, A.: Generalized solutions to hybrid dynamical systems. ESAIM Control, Optimisation, and Calculus of Variations 14(4), 699–724 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Sanfelice, R., Goebel, R., Teel, A.: Invariance principles for hybrid systems with connections to detectability and asymptotic stability. IEEE Transactions on Automatic ConTrol 52(12), 2282–2297 (2007)

    Article  MathSciNet  Google Scholar 

  17. Sipahi, R., Niculescu, S.-I., Abdallah, C., Michiels, W., Gu, K.: Stability and stabilization of systems with time delay. IEEE Control Systems Magazine 31(1), 38–65 (2011)

    Article  MathSciNet  Google Scholar 

  18. Yan, P., Ozbay, H.: Stability analysis of switched time delay systems. SIAM Journal on Control and Optimization 47(2), 936–949 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Yuan, R., Jing, Z., Chen, L.: Uniform asymptotic stability of hybrid dynamical systems with delay. IEEE Transactions on Automatic Control 48(2), 344–348 (2003)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, S1 3JD, UK

    Jun Liu

  2. Center for Control, Dynamical Systems and Computation and Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, 93106, USA

    Andrew R. Teel

Authors
  1. Jun Liu
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  2. Andrew R. Teel
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Corresponding author

Correspondence to Jun Liu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Iasson Karafyllis

  2. Department of Mathematics, Louisiana State University (LSU), Baton Rouge, Louisiana, USA

    Michael Malisoff

  3. CNRS-CentraleSupélec-Université Paris Sud, EPI Inria “DISCO”, Laboratory of Signals and Systems (L2S, UMR CNRS 8506), Gif-sur-Yvette, France

    Frederic Mazenc

  4. Department of Information Engineering, Computer Science, and Mathematics, University of L’Aquila, L’Aquila, Italy

    Pierdomenico Pepe

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Liu, J., Teel, A.R. (2016). Hybrid Dynamical Systems with Finite Memory. In: Karafyllis, I., Malisoff, M., Mazenc, F., Pepe, P. (eds) Recent Results on Nonlinear Delay Control Systems. Advances in Delays and Dynamics, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-18072-4_13

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  • DOI: https://doi.org/10.1007/978-3-319-18072-4_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-18071-7

  • Online ISBN: 978-3-319-18072-4

  • eBook Packages: EngineeringEngineering (R0)

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