Riesz basis approach to feedback stabilization for a cantilever beam system

详细信息    查看官网全文

• 英文论文题名：Riesz basis approach to feedback stabilization for a cantilever beam system
• 论文作者：Jun-Min Wang ; Meng-Qing Xiong ; Chao Yang
• 英文论文作者：Jun-Min Wang ; Meng-Qing Xiong ; Chao Yang ; School of Mathematics and Statistics ; Beijing Institute of Technology
• 年：2017
• 作者机构：School of Mathematics and Statistics,Beijing Institute of Technology;
• 英文论文关键词：Rayleigh beam ; feedback control ; exponential stability
• 会议召开时间：2017-07-26
• 会议录名称：第36届中国控制会议论文集（A）
• 英文会议录名称：Proceedings of the 36th Chinese Control Conference（A）
• 语种：英文
• 分类号：O231
• 学会代码：KZLL
• 会议名称：第36届中国控制会议
• 会议地点：中国辽宁大连
• 主办单位：中国自动化学会控制理论专业委员会
• 学会名称：中国自动化学会控制理论专业委员会
• 页数：6
• 文件大小：218k
• 原文格式：D
• 会议级别：全国

摘要

In this paper, we study the stabilization of an elastic beam system with axial force and a tip mass. The system is modeled as a Rayleigh beam equation. We propose a boundary feedback control moment to stabilize the closed-loop system. We first present the asymptotic expressions for the eigenpairs of the system and then show that the generalized eigenfunctions form a Riesz basis in the state space. Finally, we prove the exponential stability of the closed-loop system.
In this paper, we study the stabilization of an elastic beam system with axial force and a tip mass. The system is modeled as a Rayleigh beam equation. We propose a boundary feedback control moment to stabilize the closed-loop system. We first present the asymptotic expressions for the eigenpairs of the system and then show that the generalized eigenfunctions form a Riesz basis in the state space. Finally, we prove the exponential stability of the closed-loop system.

引文

[1]A.Bokaian,Natural frequencies of beams under tensile axial loads,J.Sound Vib.,142(3):481–498,1990.
    [2]S.Naguleswaran,Transverse vibration of an uniform EulerBernoulli beam under linearly varying axial force,J.Sound Vib.,275:47–57,2004.
    [3]R.Fung,Y.Liu,and C.Wang,Dynamic model of an electromagnetic actuator for vibration control of a cantilever beam with a tip mass,J.Sound Vib.,288:957–980,2005.
    [4]B.Rao,Uniform stabilization of a hybrid system of elasticity,SIAM J.Control Optim.,33(2):440–454,1995.
    [5]F.Conrad and.Morgül,On the stabilization of a flexible beam with a tip mass,SIAM J.Control Optim.,36(6):1962–1986,1998.
    [6]B.Guo,Riesz basis approach to the stabilization of a flexible beam with a tip mass,SIAM J.Control Optim.,39(6):1736–1747,2001.
    [7]B.Chentouf and J.Wang,Optimal energy decay for a nonhomogeneous flexible beam with a tip mass,J.Dyn.Control Syst.,13(1):37–53,2007.
    [8]X.Chen,B.Chentouf,and J.Wang,Exponential stability of a non-homogeneous rotating disk-beam-mass system,J.Math.Anal.Appl.,423(2):1243–1261,2015.
    [9]X.Li,A.Tang,and L.Xi,Vibration of a Rayleigh cantilever beam with axial force and tip mass,J.Constr.Steel Res.,80:15–22,2013.
    [10]A.Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations.New York:Springer-Verlag,1983.
    [11]J.Wang,Q.Vu,and S.Yung,Asymptotic frequency distributions for variable coefficients Rayleigh beams under boundary feedback control,in Proceedings of the American Control Conference,2003:113–118.
    [12]V.Maz’ja,Sobolev Spaces.New York:Springer-Verlag,1985.
    [13]G.Birkhoff and R.Langer,The boundary problems and developments associated with a system of ordinary linear differential equations of the first order,Proc.Amer.Acad.Arts Sci.,58(2):51–128,1923.
    [14]C.Tretter,Spectral problems for systems of differential equations y+A0y=λA1y withλ-polynomial boundary conditions,Math.Nachr.,214:129–172,2000.
    [15]C.Tretter,Boundary eigenvalue problems for differential equations Nη=λPηwithλ-polynomial boundary conditions,J.Differ.Equations,170(2):408–471,2001.
    [16]J.Wang,G.Xu,and S.Yung,Exponential stabilization of laminated beams with structural damping and boundary feedback controls,SIAM J.Control Optim.,44(5):1575–1597,2005.
    [17]C.Yang and J.Wang,Exponential stability of an active constrained layer beam actuated by a voltage source without magnetic effects,J.Math.Anal.Appl.,448(2):1204–1227,2017.
    [18]R.Young,An Introduction to Nonharmonic Fourier Series,revised 1st ed.New York:Academic Press,2001.