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Keywords:
boundary control; disturbance; wave equation; anti-disturbance
boundary control; disturbance; wave equation; anti-disturbance
Summary:
We study the anti-disturbance problem of a 1-d wave equation with boundary control matched disturbance. In earlier literature, the authors always designed the controller such as the sliding mode control and the active disturbance rejection control to stabilize the system. However, most of the corresponding closed-loop systems are boundedly stable. In this paper we show that the linear feedback control also has a property of anti-disturbance, even if the disturbance includes some information of the system. By choosing suitable parameters introduced in the proof, we can ensure the solution of the closed-loop system is bounded in an admissible range. As an application, we discuss the control problem of a nonlinear system. As a result, it is shown that the bounded estimation of the solution is suitable.
References:
[1] Cox, S., Zuazua, E.: The rate at which energy decays in a damped string. Commun. Partial Differ. Equations 19 (1994), 213-143. DOI 10.1080/03605309408821015 | MR 1257004 | Zbl 0818.35072
[2] Fu, Q. H., Xu, G. Q.: Exponential stabilization of 1-d wave equation with distributed disturbance. WSEAS Trans. Math. 14 (2015), 192-201.
[3] Guo, W., Guo, B.-Z., Shao, Z.-C.: Parameter estimation and stabilization for a wave equation with boundary output harmonic disturbance and non-collocated control. Int. J. Robust Nonlinear Control 21 (2011), 1297-1321. DOI 10.1002/rnc.1650 | MR 2840009 | Zbl 1244.74038
[4] Guo, B.-Z., Jin, F.-F.: The active disturbance rejection and sliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbance. Automatica 49 (2013), 2911-2918. DOI 10.1016/j.automatica.2013.06.018 | MR 3084483 | Zbl 1364.93637
[5] Guo, B.-Z., Kang, W.: The Lyapunov approach to boundary stabilization of an anti-stable one-dimensional wave equation with boundary disturbance. Int. J. Robust Nonlinear Control 24 (2014), 54-69. DOI 10.1002/rnc.2874 | MR 3149286 | Zbl 1278.93199
[6] Guo, B.-Z., Liu, J.-J.: Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schrödinger equation subject to boundary control matched disturbance. Int. J. Robust. Nonlinear Control 24 (2014), 2194-2212. DOI 10.1002/rnc.2977 | MR 3271988 | Zbl 1302.93060
[7] Guo, B.-Z., Liu, J.-J., Al-Fhaid, A. S., Mahmood, M. Arshad, Younas, A. M. M., Asiri, A.: The active disturbance rejection control approach to stabilisation of coupled heat and ODE system subject to boundary control matched disturbance. Int. J. Control 88 (2015), 1554-1564. DOI 10.1080/00207179.2015.1010179 | MR 3371068 | Zbl 1337.93078
[8] Guo, B.-Z., Zhou, H.-C.: The active disturbance rejection control to stabilization for multi-dimensional wave equation with boundary control matched disturbance. IEEE Trans. Autom. Control 60 (2015), 143-157. DOI 10.1109/TAC.2014.2335511 | MR 3299420 | Zbl 1360.93545
[9] Immonen, E., Pohjolainen, S.: Feedback and feedforward output regulation of bounded uniformly continuous signals for infinite-dimensional systems. SIAM J. Control Optim. 45 (2006), 1714-1735. DOI 10.1137/050623000 | MR 2272163 | Zbl 1127.93029
[10] Jayawardhana, B., Weiss, G.: State convergence of passive nonlinear systems with an $L^2$ input. IEEE Trans. Autom. Control 54 (2009), 1723-1727. DOI 10.1109/TAC.2009.2020661 | MR 2535777 | Zbl 1367.93435
[11] Jin, F.-F., Guo, B.-Z.: Lyapunov approach to output feedback stabilization for the Euler-Bernoulli equation with boundary input disturbance. Automatica 52 (2015), 95-102. DOI 10.1016/j.automatica.2014.10.123 | MR 3310818 | Zbl 1309.93122
[12] Ke, Z., Logemann, H., Rebarber, R.: Approximate tracking and disturbance rejection for stable infinite-dimensional systems using sampled-data low-gain control. SIAM J. Control Optim. 48 (2009), 641-671. DOI 10.1137/080716517 | MR 2486087 | Zbl 1194.93043
[13] Krstic, M.: Adaptive control of an anti-stable wave PDE. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 17 (2010), 853-882. MR 2757916 | Zbl 1219.93055
[14] Nakao, M.: Decay of solutions of the wave equation with a local nonlinear dissipation. Math. Ann. 305 (1996), 403-417. DOI 10.1007/BF01444231 | MR 1397430 | Zbl 0856.35084
[15] Rebarber, R., Weiss, G.: Internal model based tracking and disturbance rejection for stable well-posed systems. Automatica 39 (2003), 1555-1569. DOI 10.1016/S0005-1098(03)00192-4 | MR 2143463 | Zbl 1028.93012
[16] Shang, Y., Xu, G.: Dynamic control of an Euler-Bernoulli equation with time-delay and disturbance in the boundary control. Int. J. Control 92 (2019), 27-41. DOI 10.1080/00207179.2017.1334264 | MR 3928477 | Zbl 1415.93117
[17] Weiss, G.: Admissibility of unbounded control operators. SIAM J. Control Optimization 27 (1989), 527-545. DOI 10.1137/0327028 | MR 0993285 | Zbl 0685.93043
[18] Xie, Y. R., Xu, G. Q.: Stabilization of a wave equation with a tip mass based on disturbance observer of time-varying gain. J. Dyn. Control Syst. 23 (2017), 667-677. DOI 10.1007/s10883-016-9349-0 | MR 3688888 | Zbl 1372.35175
[19] Xu, G. Q.: Exponential stabilization of conservation systems with interior disturbance. J. Math. Anal. Appl. 436 (2016), 764-781. DOI 10.1016/j.jmaa.2015.11.079 | MR 3446978 | Zbl 1330.93210
[20] Zhao, Z., Guo, B.: Active disturbance rejection control to stabilize one-dimensional wave equation with interior domain anti-damping and boundary disturbance. Control Theory Appl. 30 (2013), 1553-1563. DOI 10.7641/CTA.2013.30966 | Zbl 1299.93247
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