Abstract
In this chapter we propose a control scheme where a pre-designed linear controller in feedback with a time-delay plant is augmented with suitable jump rules that are activated in certain subsets of the state space to ensure closed-loop asymptotic stability. Under suitable feasibility conditions on the data of the linear time-delay plant, we show that the proposed scheme induces delay-independent stability of the closed loop with controller state jumps. Due to the hybrid nature of the proposed scheme, we address stability by proposing a hybrid version of the classical Lyapunov-Krasovskii theorem, relying on a dwell-time condition and on a Lyapunov-Krasovskii function that does not increase accross jumps. The results in the chapter can be seen as preliminary results in the direction of hybrid time-delay dynamical systems, which still remains largely unexplored. A simulation example shows the effectiveness of the proposed hybrid scheme.
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References
Baños, A., Barreiro, A.: Delay-independent stability of reset systems. IEEE Trans. Autom. Control 54(2), 341–346 (2009)
Baños, A., Barreiro, A.: Reset Control Systems. AIC Series, Springer, London (2012)
Baños, A., Pérez Rubio, F., Tarbouriech, S., Zaccarian, L.: Delay-independent stability via reset loops. LAAS-CNRS technical report (2013)
Barreiro, A., Baños, A.: Delay-dependent stability of reset systems. Automatica 46(1), 216–221 (2010)
Clegg, J.C.: A nonlinear integrator for servomechnisms. Trans. A.I.E.E.m Part II 77, 41–42 (1958)
Fichera, F., Prieur, C., Tarbouriech, S., Zaccarian, L.: Using Luenberger observers and dwell-time logic for feedback hybrid loops in continuous-time control systems. Int. J. Robust Nonlinear Control 23, 1065–1086 (2013)
Goebel, R., Sanfelice, R.G., Teel, A.R.: Hybrid dynamical systems. IEEE Control Syst. Mag. 29(2), 28–93 (2009)
Goebel, R., Sanfelice, R.G., Teel, A.R.: Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press, Princeton (2012)
Guo, Y., Xie, L.: Quadratic stability of reset control systems with delays. In: World Congress on Intelligent Control and Automation, pp. 2268–2273 (2012)
Haddad, W.M., Chellaboina, V., Nersesov, S.G.: Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton University Press, Princeton (2006)
Hale, J.K., Lunel, S.M.V.: Introduction to functional differential equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993)
Horowitz, I.M., Rosenbaum, P.: Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty. Int. J. Control 24(6), 977–1001 (1975)
Liu, J., Teel, A.R.: Generalized solutions to hybrid systems with delays. In: Conference on Decision and Control, pp. 6169–6174. Maui, HI, USA (2012)
Nešić, D., Teel, A.R., Zaccarian, L.: Stability and performance of SISO control systems with first order reset elements. IEEE Trans. Autom. Control 56(11), 2567–2582 (2011)
Nešić, D., Zaccarian, L., Teel, A.R.: Stability properties of reset systems. Automatica 44, 2019–2026 (2008)
Prieur, C., Tarbouriech, S., Zaccarian, L.: Lyapunov-based hybrid loops for stability and performance of continuous-time control systems. Automatica 49(2), 577–584 (2013)
Sipahi, R., Niculescu, S.-I., Abadallah, C.T., Michiels, W., Gu, K.: Stability and stabilization of systems with time delay. IEEE Control Syst. Mag. 31(1), 38–65 (2011)
Sipahi, R., Vyhlídal, T., Niculescu, S.-I., Pierdomenico, P. (eds.): Time Delay Systems: Method, Applications and New Trends. Springer, New York (2012)
Zaccarian, L., Nešić, D., Teel, A.R.: Analytical and numerical Lyapunov functions for SISO linear control systems with first order reset elements. Int. J. Robust Nonlinear Control 21(10), 1134–1158 (2011)
Acknowledgments
This work was supported by ANR project LimICoS contract number 12 BS03 005 01, by HYCON2 Network of Excellence “Highly- Complex and Networked Control Systems”, grant agreement 257462 and by MICINN under the project DPI 2007-66455-C02-01.
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Baños, A., Pérez Rubio, F., Tarbouriech, S., Zaccarian, L. (2014). Delay-Independent Stability Via Reset Loops. In: Seuret, A., Özbay, H., Bonnet, C., Mounier, H. (eds) Low-Complexity Controllers for Time-Delay Systems. Advances in Delays and Dynamics, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-05576-3_8
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DOI: https://doi.org/10.1007/978-3-319-05576-3_8
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