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Lyapunov stability of a coupled ordinary differential system and a string equation with polytopic uncertainties. (English) Zbl 1511.93101

Valmorbida, Giorgio (ed.) et al., Accounting for constraints in delay systems. Based on the workshop, Gif-sur-Yvette, France; November 22–24, 2017. Cham: Springer. Adv. Delays Dyn. 12, 175-188 (2022).
The robust exponential stability of a coupled system composed of an uncertain polytopic ordinary differential system and a string equation is analysed. It mainly consists of an extension of a previous result based on a Lyapunov functional and which now requires a linear matrix inequalities (LMI) transformation to obtain the robust stability result. This is possible by considering less restrictive LMIs (through a relaxed positivity condition and the robust result taking advantage of the convex shape of the LMIs) and uncertainties on the ordinary differential part of the interconnected system. The transformation of LMIs to obtain the robust stability result is commonly the first extension proposed when using a Lyapunov functional that requires an LMI transformation to obtain the robust stability result. Finally, an example illustrates the effectiveness of the proposed method.
For the entire collection see [Zbl 1485.93019].

MSC:

93D09 Robust stability
93D23 Exponential stability
93C15 Control/observation systems governed by ordinary differential equations
93C41 Control/observation systems with incomplete information
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[1] M. Barreau, A. Seuret, F. Gouaisbaut, L. Baudouin, Lyapunov stability analysis of a string equation coupled with an ordinary differential system. IEEE Trans. Autom. Control 63(11), 3850-3857 (2018) · Zbl 1423.93287 · doi:10.1109/TAC.2018.2802495
[2] M. Barreau, F. Gouaisbaut, A. Seuret, R. Sipahi, Input/Output stability of a damped string equation coupled with an ordinary differential system. Int. J. Robust Non-linear Control 28, 6053-6069 (2018) · Zbl 1405.93203 · doi:10.1002/rnc.4357
[3] L. Baudouin, A. Seuret, F. Gouaisbaut, M. Dattas, Lyapunov stability analysis of a linear system coupled to a heat equation. IFAC-PapersOnLine, vol. 50, Issue 1, July 2017, pp. 11978-11983 (2017)
[4] J. Bontsema, R.F. Curtain, A note on spillover and robustness for flexible systems. IEEE Trans. Autom. Control 33(6), 567-569 (1988) · Zbl 0644.93043 · doi:10.1109/9.1253
[5] D. Bresch-Pietri, M. Krstic, Output-feedback adaptive control of a wave PDE with boundary anti-damping. Automatica 50(5), 1407-1415 (2014) · Zbl 1296.93081 · doi:10.1016/j.automatica.2014.02.040
[6] C. Briat, Convergence and equivalence results for the Jensens inequality-application to time-delay and sampled-data systems. IEEE Trans. Autom. Control 56(7), 1660-1665 (2011) · Zbl 1368.26020 · doi:10.1109/TAC.2011.2121410
[7] C. Briat, A. Seuret, Robust stability of impulsive systems: a functional-based approach, in 4th IFAC conference on Analysis and Design of Hybrid Systems (ADHS-2012), Eindhoven, Netherlands (2012) · Zbl 1270.93084
[8] N. Challamel, Rock destruction effect on the stability of a drilling structure. J. Sound Vib. 233(2), 235-254 (2000) · doi:10.1006/jsvi.1999.2811
[9] G. Chen, J. Zhou, The wave propagation method for the analysis of boundary stabilization in vibrating structures. SIAM J. Appl. Math. 50(5), 1254-1283 (1990) · Zbl 0712.73069 · doi:10.1137/0150076
[10] R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, 1989) · Zbl 0729.35001
[11] E. Fridman, S. Mondié, B. Saldivar, Bounds on the response of a drilling pipe model. IMA J. Math. Control Inf. 27(4), 513-526 (2010) · Zbl 1213.35150 · doi:10.1093/imamci/dnq024
[12] E. Fridman, Introduction to Time-Delay Systems: Analysis and Control (Springer, 2014) · Zbl 1303.93005
[13] K. Gu, J. Chen, V.L. Kharitonov, Stability of Time-Delay Systems (Springer, 2003) · Zbl 1039.34067
[14] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs. Voume 16 of Advances in Design and Control (SIAM, 2008) · Zbl 1149.93004
[15] M. Krstic, Delay Compensation for Nonlinear, Adaptive, and PDE Systems (Springer, 2009) · Zbl 1181.93003
[16] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differ. Equ. 50(2), 163-182 (2017) · Zbl 0536.35043 · doi:10.1016/0022-0396(83)90073-6
[17] Z.H. Luo, B.Z. Guo, Ö. Morgül, Stability and Stabilization of Infinite Dimensional Systems with Applications (Springer, 2012)
[18] L. Meirovitch, H. Baruh, On the problem of observation spillover in self-adjoint distributed-parameter systems. J. Optim. Theory Appl. 39(2), 269-291 (1983) · Zbl 0503.93062 · doi:10.1007/BF00934533
[19] M.B. Saldivar, I. Boussaada, H. Mounier, S.I. Niculescu, Analysis and Control of Oilwell Drilling Vibrations: A Time-Delay Systems Approach (Springer, 2015)
[20] A. Seuret, F. Gouaisbaut, Hierarchy of LMI conditions for the stability analysis of time delay systems. Syst. Control Lett. 81, 1-7 (2006) · Zbl 1330.93211 · doi:10.1016/j.sysconle.2015.03.007
[21] S. Tang, C. Xie, State and output feedback boundary control for a coupled PDE-ODE system. Syst. Control Lett. 60(8), 540-545 (2011) · Zbl 1236.93076 · doi:10.1016/j.sysconle.2011.04.011
[22] M. Tucsnak, G. Weiss, Observation and Control for Operator Semigroups (Springer, 2009) · Zbl 1188.93002
[23] J. Zhang, C.R. Knopse, P. Tsiotras, Stability of time-delay systems: equivalence between Lyapunov and scaled small-gain conditions. IEEE Trans. Autom. Control 46(3), 482-486 (2001) · Zbl 1056.93598 · doi:10.1109/9.911428
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