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.
Keywords: Beam equation, time delay, pointwise control, estimated state feedback, stability
Yang, Kun-Yi 1 ; Wang, Jun-Min 1
@article{COCV_2019__25__A4_0,
author = {Yang, Kun-Yi and Wang, Jun-Min},
title = {Pointwise feedback stabilization of an {Euler-Bernoulli} beam in observations with time delay},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2019},
publisher = {EDP Sciences},
volume = {25},
doi = {10.1051/cocv/2017080},
zbl = {1437.35198},
mrnumber = {3943356},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2017080/}
}
TY - JOUR AU - Yang, Kun-Yi AU - Wang, Jun-Min TI - Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2017080/ DO - 10.1051/cocv/2017080 LA - en ID - COCV_2019__25__A4_0 ER -
%0 Journal Article %A Yang, Kun-Yi %A Wang, Jun-Min %T Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2017080/ %R 10.1051/cocv/2017080 %G en %F COCV_2019__25__A4_0
Yang, Kun-Yi; Wang, Jun-Min. Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 4. doi: 10.1051/cocv/2017080
[1] and , Stabilization of Euler-Bernoulli beams by means of a pointwise feedback force. SIAM J. Control Optim. 39 (2000) 1160–1181. | Zbl | MR | DOI
[2] , and , Decay rates for a beam with pointwise force and moment feedback. Math. Control Signals Syst. 15 (2002) 229–255. | Zbl | MR | DOI
[3] , Linear systems with delayed controls: a reduction. IEEE Trans. Autom. Control 27 (1982) 869–879. | Zbl | MR | DOI
[4] and , Simultaneous control problems for systems of elastic strings and beams. Syst. Control Lett. 44 (2001) 147–155. | Zbl | MR | DOI
[5] and , 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. | Zbl | MR | DOI
[6] and , 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. | Zbl | MR | DOI
[7] and , Exact boundary controllability of two Euler-Bernoulli beams connected by a point mass. Math. Comput. Model. 32 (2000) 955–969. | Zbl | MR | DOI
[8] , , and , Modeling, stabilization and control of serially connected beams. SIAM J. Control Optim. 25 (1987) 526–546. | Zbl | MR | DOI
[9] , , , , and , Analysis, designs, and behavior of dissipative joints for coupled beams. SIAM J. Control Optim. 49 (1989) 1665–1693. | Zbl | MR
[10] , Stabilization of beams by pointwise feedback control. SIAM J. Control Optim. 28 (1990) 423–437. | Zbl | MR | DOI
[11] , The Salamon-Weiss class of well-posed infinite dimensional linear systems: a survey. IMA J. Math. Control Inf. 14 (1997) 207–223. | Zbl | MR | DOI
[12] , 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. | Zbl | MR | DOI
[13] , Two examples of ill-posed with respect to time delays in stabilized elastic systems. IEEE Trans. Autom. Control 38 (1993) 163–166. | Zbl | MR | DOI
[14] , Two examples of ill-posedness with respect to time delays revisited. IEEE Trans. Autom. Control 42 (1997) 511–515. | Zbl | MR | DOI
[15] , and , An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24 (1986) 152–156. | Zbl | MR | DOI
[16] , Future Directions in Control Theory. SIAM, Philadelphia (1988).
[17] and , Exponential stability of linear distributed parameter systems with time-varying delays. Automatica 45 (2009) 194–201. | Zbl | MR | DOI
[18] , On the boundary control of a hybrid system with variable coefficients. J. Optim. Theory Appl. 114 (2002) 373–395. | Zbl | MR | DOI
[19] , and , On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwiseoutput feedback. ESAIM: COCV 14 (2016) 632–656. | Zbl | MR | Numdam
[20] , and , Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation. ESAIM: COCV 18 (2012) 22–35. | Zbl | MR | Numdam
[21] and , Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation. Automatica 45 (2009) 1468–1475. | Zbl | MR | DOI
[22] and , Riesz basis generation, eigenvalues distribution, and exponential stability for a Euler-Bernoulli beam with joint feedback contol. Rev. Math. Complut. 14 (2001) 205–229. | Zbl | MR
[23] , Lyapunov tools for predictor feedbacks for delay systems: inverse optimality and robustness to delay mismatch. Automatica 44 (2008) 2930–2935. | Zbl | MR | DOI
[24] and , Backstepping boundary control for first-order hyperbolic pdes and application to systems with actuator and sensor delays. Syst. Control Lett. 57 (2008) 750–758. | Zbl | MR | DOI
[25] , Exponential stability of coupled beam with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33 (1995) 1–28. | Zbl | MR | DOI
[26] , On the pointwise stabilization of a string. Int. Ser. Numer. Math. 126 (1988) 287–295. | Zbl | MR
[27] and , Observation and Control for Operator Semigroups. Birkhäuser, Basel (2009). | Zbl | MR | DOI
[28] and , Fourier Series in Control Theory. Springer-Verlag, New York (2005). | Zbl | MR | DOI
[29] , and , Wave equation stabilization by delays equal to even multiples of the wave propagation time. SIAM J. Control Optim. 49 (2011) 517–554. | Zbl | MR | DOI
[30] , Admissibility of unbounded control operators. SIAM J. Control Optim. 27 (1989) 527–545. | Zbl | MR | DOI
[31] and , 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. | Zbl | MR | Numdam
[32] and , Exponential stabilization of a Rayleigh beam using collocated control. IEEE Trans. Autom. Control 53 (2008) 643–654. | Zbl | MR | DOI
[33] and , Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation. SIAM J. Control Optim. 42 (2003) 966–984. | Zbl | MR | DOI
[34] and , Stabilization of Timoshenko beam by means of pointwise controls. ESAIM: COCV 9 (2003) 579–600. | Zbl | MR | Numdam
[35] and , Pointwise measure, control and stabilization of elastic beams. Control Theory Appl. 20 (2003) 351–360. | MR
Cité par Sources :
