Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay

Yang, Kun-Yi; Wang, Jun-Min

ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 4.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

This paper considers a one-dimensional Euler-Bernoulli beam equation where two collocated actuators/sensors are presented at the internal point with pointwise feedback shear force and angle velocity at the arbitrary position ξ in the bounded domain (0,1). The boundary x = 0 is simply supported and at the other boundary x = 1 there is a shear hinge end. Both of the observation signals are subjected to a given time delay τ ( >0). Well-posedness of the open-loop system is shown to illustrate availability of the observer. An observer is then designed to estimate the state at the time interval when the observation is available, while a predictor is designed to predict the state at the time interval when the observation is not available. Pointwise output feedback controllers are introduced to guarantee the closed-loop system to be exponentially stable for the smooth initial values when ξ ∈ (0, 1) is a rational number satisfying ξ ≠ 2l∕(2m − 1) for any integers l, m. Simulation results demonstrate that the proposed feedback design effectively stabilizes the performance of the pointwise control system with time delay.

[1] K. Ammari and M. Tucsnak, Stabilization of Euler-Bernoulli beams by means of a pointwise feedback force. SIAM J. Control Optim. 39 (2000) 1160–1181. | DOI | MR | Zbl

[2] K. Ammari, Z.Y. Liu and M. Tucsnak, Decay rates for a beam with pointwise force and moment feedback. Math. Control Signals Syst. 15 (2002) 229–255. | DOI | MR | Zbl

[3] Z. Artstein, Linear systems with delayed controls: a reduction. IEEE Trans. Autom. Control 27 (1982) 869–879. | DOI | MR | Zbl

[4] S.A. Avdonin and W. Moran, Simultaneous control problems for systems of elastic strings and beams. Syst. Control Lett. 44 (2001) 147–155. | DOI | MR | Zbl

[5] C. Castro and E. Zuazua, A hybrid system consisting of two flexible beams connected by a point mass: spectral analysis and well-posedness in asymmetric spaces. ESAIM: PROCS 2 (1997) 17–53. | DOI | MR | Zbl

[6] C. Castro and E. Zuazua, Boundary controllability of a hybrid system consisting in two flexible beams connected by a joint mass. SIAM J. Control and Optim. 36 (1998) 1576–1595. | DOI | MR | Zbl

[7] C. Castro and E. Zuazua, Exact boundary controllability of two Euler-Bernoulli beams connected by a point mass. Math. Comput. Model. 32 (2000) 955–969. | DOI | MR | Zbl

[8] G. Chen, M.C. Delfour, A.M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams. SIAM J. Control Optim. 25 (1987) 526–546. | DOI | MR | Zbl

[9] G. Chen, S.G. Krantz, D.L. Russell, C.E. Wayne, H.H. West and M.P. Coleman, Analysis, designs, and behavior of dissipative joints for coupled beams. SIAM J. Control Optim. 49 (1989) 1665–1693. | MR | Zbl

[10] F. Conrad, Stabilization of beams by pointwise feedback control. SIAM J. Control Optim. 28 (1990) 423–437. | DOI | MR | Zbl

[11] R.F. Curtain, The Salamon-Weiss class of well-posed infinite dimensional linear systems: a survey. IMA J. Math. Control Inf. 14 (1997) 207–223. | DOI | MR | Zbl

[12] R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim. 26 (1988) 697–713. | DOI | MR | Zbl

[13] R. Datko, Two examples of ill-posed with respect to time delays in stabilized elastic systems. IEEE Trans. Autom. Control 38 (1993) 163–166. | DOI | MR | Zbl

[14] R. Datko, Two examples of ill-posedness with respect to time delays revisited. IEEE Trans. Autom. Control 42 (1997) 511–515. | DOI | MR | Zbl

[15] R. Datko, J. Lagnese and M.P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24 (1986) 152–156. | DOI | MR | Zbl

[16] W.H. Fleming, Future Directions in Control Theory. SIAM, Philadelphia (1988).

[17] E. Fridman and Y. Orlov, Exponential stability of linear distributed parameter systems with time-varying delays. Automatica 45 (2009) 194–201. | DOI | MR | Zbl

[18] B.Z. Guo, On the boundary control of a hybrid system with variable coefficients. J. Optim. Theory Appl. 114 (2002) 373–395. | DOI | MR | Zbl

[19] B.Z. Guo, J.M. Wang and C.L. Zhou, On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwiseoutput feedback. ESAIM: COCV 14 (2016) 632–656. | Numdam | MR | Zbl

[20] B.Z. Guo, C.Z. Xu and H. Hammouri, Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation. ESAIM: COCV 18 (2012) 22–35. | Numdam | MR | Zbl

[21] B.Z. Guo and K.Y. Yang, Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation. Automatica 45 (2009) 1468–1475. | DOI | MR | Zbl

[22] B.Z. Guo and K.Y. Chan, Riesz basis generation, eigenvalues distribution, and exponential stability for a Euler-Bernoulli beam with joint feedback contol. Rev. Math. Complut. 14 (2001) 205–229. | MR | Zbl

[23] M. Krstic, Lyapunov tools for predictor feedbacks for delay systems: inverse optimality and robustness to delay mismatch. Automatica 44 (2008) 2930–2935. | DOI | MR | Zbl

[24] M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic pdes and application to systems with actuator and sensor delays. Syst. Control Lett. 57 (2008) 750–758. | DOI | MR | Zbl

[25] R. Rebarber, Exponential stability of coupled beam with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33 (1995) 1–28. | DOI | MR | Zbl

[26] M. Tucsnak, On the pointwise stabilization of a string. Int. Ser. Numer. Math. 126 (1988) 287–295. | MR | Zbl

[27] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups. Birkhäuser, Basel (2009). | DOI | MR | Zbl

[28] V. Komornik and P. Loreti, Fourier Series in Control Theory. Springer-Verlag, New York (2005). | DOI | MR | Zbl

[29] J.M. Wang, B.Z. Guo and M. Krstic, Wave equation stabilization by delays equal to even multiples of the wave propagation time. SIAM J. Control Optim. 49 (2011) 517–554. | DOI | MR | Zbl

[30] G. Weiss, Admissibility of unbounded control operators. SIAM J. Control Optim. 27 (1989) 527–545. | DOI | MR | Zbl

[31] G. Weiss and M. Tucsnak, How to get a conservative well-posed linear system out of thin air, Part I. Well-posedness and energy balance. ESAIM: Control, Optim. Calc. Var. 9 (2003) 247–273. | Numdam | MR | Zbl

[32] G. Weiss and R.F. Curtain, Exponential stabilization of a Rayleigh beam using collocated control. IEEE Trans. Autom. Control 53 (2008) 643–654. | DOI | MR | Zbl

[33] G.Q. Xu and B.Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation. SIAM J. Control Optim. 42 (2003) 966–984. | DOI | MR | Zbl

[34] G.Q. Xu and S.P. Yung, Stabilization of Timoshenko beam by means of pointwise controls. ESAIM: COCV 9 (2003) 579–600. | Numdam | MR | Zbl

[35] C.Z. Yao and B.Z. Guo, Pointwise measure, control and stabilization of elastic beams. Control Theory Appl. 20 (2003) 351–360. | MR