Dynamical behavior of a hybrid system of nonhomogeneous timoshenko beam with partial non-collocated inputs

The dynamical behavior of a kind of elastic hybrid system is considered. This system consists of a nonhomogeneous Timoshenko beam linked at its boundary to a rigid body. The non-collocated input terms are required in the boundary feedback controls. It is proved that this closed loop system is well-posed. By a complete eigenfrequency analysis, it is shown that this infinite-dimensional hybrid system satisfies spectrum-determined growth condition. The stability of this system is discussed and some simulations are given to show that this kind of hybrid system can be exponentially stable under certain condition.

[1]  J. Conway,et al.  Functions of a Complex Variable , 1964 .

[2]  Genqi Xu,et al.  Exponential Decay Rate for a Timoshenko Beam with Boundary Damping , 2004 .

[3]  Stephen W. Taylor,et al.  Boundary control of a rotating Timoshenko beam , 2008 .

[4]  Dexing Feng,et al.  Boundary feedback stabilization of Timoshenko beam with boundary dissipation , 1998 .

[5]  M. Krstić,et al.  Backstepping Boundary Controllers and Observers for the Slender Timoshenko Beam: Part I - Design , 2006, 2006 American Control Conference.

[6]  Shang-Teh Wu.  Virtual passive control of flexible arms with collocated and noncollocated feedback , 2001, J. Field Robotics.

[7]  Genqi Xu,et al.  The expansion of a semigroup and a Riesz basis criterion , 2005 .

[8]  Bao-Zhu Guo,et al.  Dynamic stabilization of an Euler-Bernoulli beam under boundary control and non-collocated observation , 2008, Syst. Control. Lett..

[9]  A. H. Von Flotow,et al.  Traveling wave control for large spacecraft structures , 1986 .

[10]  Bao-Zhu Guo,et al.  Riesz Basis Property and Exponential Stability of Controlled Euler--Bernoulli Beam Equations with Variable Coefficients , 2001, SIAM J. Control. Optim..

[11]  M.S. de Queiroz,et al.  Boundary control of the Timoshenko beam with free-end mass/inertial dynamics , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[12]  Thomas Thundat,et al.  Glucose biosensor based on the microcantilever. , 2004, Analytical chemistry.

[13]  Genqi Xu,et al.  Boundary feedback exponential stabilization of a Timoshenko beam with both ends free , 2005 .

[14]  Genqi Xu,et al.  Riesz basis property of serially connected Timoshenko beams , 2007, Int. J. Control.

[15]  Bao-Zhu Guo,et al.  The Stabilization of a One-Dimensional Wave Equation by Boundary Feedback With Noncollocated Observation , 2007, IEEE Transactions on Automatic Control.

[16]  Blake Hannaford,et al.  Control of a flexible manipulator with noncollocated feedback: time-domain passivity approach , 2004, IEEE Transactions on Robotics.

[17]  R. Mennicken,et al.  Non-Self-Adjoint Boundary Eigenvalue Problems , 2003 .

[18]  Wei Guo,et al.  The strong stabilization of a one-dimensional wave equation by non-collocated dynamic boundary feedback control , 2009, Autom..

[19]  Dexing Feng,et al.  Exponential decay of Timoshenko beam with locally distributed feedback , 2001 .

[20]  Jun-Min Wang,et al.  Spectral analysis and system of fundamental solutions for Timoshenko beams , 2005, Appl. Math. Lett..

[21]  Marianna A. Shubov,et al.  Riesz basis property of root vectors of non-self-adjoint operators generated by aircraft wing model in subsonic airflow , 2000 .

[22]  Ömer Morgül.  Dynamic boundary control of the timoshenko beam , 1992, Autom..

[23]  Ömer Morgül,et al.  Boundary control of a Timoshenko beam attached to a rigid body : planar motion , 1991 .

[24]  R S Pereira,et al.  Atomic force microscopy as a novel pharmacological tool. , 2001, Biochemical pharmacology.

[25]  Walter Littman,et al.  Exact boundary controllability of a hybrid system of elasticity , 1988 .

[26]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[27]  Genqi Xu,et al.  Riesz basis property of the generalized eigenvector system of a Timoshenko beam , 2004, IMA J. Math. Control. Inf..

[28]  M. Krstic,et al.  Backstepping Boundary Controllers and Observers for the Slender Timoshenko Beam: Part II---Stability and Simulations , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[29]  J. U. Kim,et al.  Boundary control of the Timoshenko beam , 1987 .

[30]  Genqi Xu,et al.  Stabilization of Timoshenko Beam by Means of Pointwise Controls , 2003 .

[31]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[32]  Stephen P. Timoshenko,et al.  Vibration problems in engineering , 1928 .