Dynamic feedback control and exponential stabilization of a compound system. (English) Zbl 1297.93131
Summary: We study the exponential stabilization problem of a compounded system composed of a flow equation and an Euler-Bernoulli beam, which is equivalent to a cantilever Euler-Bernoulli beam with a delay controller. We design a dynamic feedback controller that stabilizes exponentially the system provided that the eigenvalues of the free system are not the zeros of controller. In this paper we describe the design detais of the dynamic feedback controller and prove its stabilization property.
MSC:
| 93D15 | Stabilization of systems by feedback |
| 93B52 | Feedback control |
| 70Q05 | Control of mechanical systems |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 35Q93 | PDEs in connection with control and optimization |
| 34H05 | Control problems involving ordinary differential equations |
Keywords:
compounded system; controller delay; Euler-Bernoulli beam; feedback control; exponential stabilizationReferences:
| [1] | Ait Benhassi, E. M.; Ammari, K.; Boulite, S.; Maniar, L., Feedback stabilization of a class of evolution equations with delay, J. Evol. Equ., 9, 103-121 (2009) · Zbl 1239.93092 |
| [2] | Avdonin, S. A.; Ivanov, S. A., Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems (1995), Cambridge University Press: Cambridge University Press New York · Zbl 0866.93001 |
| [3] | Datko, R., Not all feedback stabilized hyperbolic systems are robust with respect to small time delay in their feedbacks, SIAM J. Control Optim., 26, 697-713 (1988) · Zbl 0643.93050 |
| [4] | Datko, R., Two examples of ill-posedness with respect to small delays in stabilized elastic systems, IEEE Trans. Automat. Control, 38, 163-166 (1993) · Zbl 0775.93184 |
| [5] | Guo, B. Z.; Luo, Y. H., Controllability and stability of a second-order hyperbolic system with collocated sensor/actuator, Systems Control Lett., 46, 45-65 (2002) · Zbl 0994.93021 |
| [6] | Han, Z. J.; Xu, G. Q., Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks, ESAIM Control Optim. Calc. Var., 17, 552-574 (2010) · Zbl 1251.93106 |
| [7] | Han, Z. J.; Xu, G. Q., Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs, Netw. Heterog. Media, 6, 297-327 (2011) · Zbl 1258.93089 |
| [8] | Han, Z. J.; Xu, G. Q., Output-based stabilization of Euler-Bernoulli beam with time-delay in boundary input, IMA J. Math. Control Inform. (2013) |
| [9] | Kirane, M.; Said-Houari, B.; Anwar, M. N., Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Commun. Pure Appl. Anal., 10, 2, 667-686 (2011) · Zbl 1228.35242 |
| [10] | Komornik, V.; Loreti, P., Fourier Series in Control Theory, Springer Monogr. Math. (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1094.49002 |
| [11] | Liu, X. F.; Xu, G. Q., Exponential stabilization for Timoshenko beam with distributed delay in the boundary control, Abstr. Appl. Anal. (2013) · Zbl 1421.93120 |
| [12] | Lyubich, Y. I.; Phöng, V. Q., Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88, 34-37 (1988) · Zbl 0639.34050 |
| [13] | Nicaise, S.; Pignotti, C., Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45, 5, 1561-1585 (2006) · Zbl 1180.35095 |
| [14] | Nicaise, S.; Pignotti, C., Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21, 935-958 (2008) · Zbl 1224.35247 |
| [15] | Nicaise, S.; Pignotti, C., Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations, 2011, 41, 1-20 (2011) · Zbl 1215.35098 |
| [16] | Nicaise, S.; Valein, J., Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks, Netw. Heterog. Media, 2, 3, 425-479 (2007) · Zbl 1211.35050 |
| [17] | Nicaise, S.; Valein, J., Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim. Calc. Var., 16, 420-456 (2010) · Zbl 1217.93144 |
| [18] | Pazy, A., Semigroup of Linear Operator and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0516.47023 |
| [19] | Said-Houari, B.; Laskri, Y., A stability result of a Timoshenko system with a delay term in the internal feedback, Appl. Math. Comput., 217, 2857-2869 (2010) · Zbl 1342.74086 |
| [20] | Said-Houari, B.; Soufyane, A., Stability result of the Timoshenko system with delay and boundary feedback, IMA J. Math. Control Inform., 383-398 (2012) · Zbl 1252.93110 |
| [21] | Shang, Y. F.; Xu, G. Q., Stabilization of an Euler-Bernoulli beam with input delay in the boundary control, Systems Control Lett., 61, 11, 1069-1078 (2012) · Zbl 1252.93100 |
| [22] | Shang, Y. F.; Xu, G. Q.; Chen, Y. L., Stability analysis of Euler-Bernoulli beam with input delay in the boundary control, Asian J. Control, 14, 1, 186-196 (2012), online: 11 October 2010 · Zbl 1282.93199 |
| [23] | Wang, H.; Xu, G. Q., Exponential stabilization of 1-d wave equation with input delay, WSEAS Trans. Math., 12, 10, 1001-1013 (2013) |
| [24] | Xu, G. Q.; Guo, B. Z., Riesz basis property of evolution equations in Hilbert space and application to a coupled string equation, SIAM J. Control Optim., 42, 3, 966-984 (2003) · Zbl 1066.93028 |
| [25] | Xu, G. Q.; Wang, H. X., Stabilization of Timoshenko beam system with delay in the boundary control, Internat. J. Control, 86, 1165-1178 (2013) · Zbl 1278.93221 |
| [26] | Xu, G. Q.; Yung, S. P.; Li, L. K., Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12, 770-785 (2006) · Zbl 1105.35016 |
| [27] | Young, R. M., An Introduction to Nonharmonic Fourier Series (2001), Academic Press: Academic Press New York · Zbl 0981.42001 |
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