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Boundary output feedback control of a flexible string system with input saturation

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Abstract

In this paper, we consider the control problem of a flexible string system in the presence of input saturation. The dynamics of the string system is represented by the coupled partial differential equation and ordinary differential equations. Firstly, in order to suppress the transverse vibration of the flexible string system, full state feedback control is proposed by introducing a well-defined integral Lyapunov function. An auxiliary system is introduced to handle the effect of input saturation, and the proof for the existence and the uniqueness of the solution of the closed-loop system is presented. The exponential stability is achieved through rigorous analysis without any simplification of the dynamics. Subsequently, for the case that some of system states cannot be directly obtained, output feedback control is developed and uniform ultimate boundedness is guaranteed. Finally, the results are illustrated with numerical simulations for verifying the control performance.

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Abbreviations

\(L\) :

Length of the string

\(M\) :

Mass of the tip payload

\(\rho \) :

Mass per unit length of the string

\(T\) :

Tension of the string

\(w(x,t)\) :

Deflection of the string at the position \(x\) for time \(t\)

\(w(L,t)\) :

Boundary deflection of the string

\(\dot{w}(L,t)\) :

Velocity of the tip payload

\(w'(L,t)\) :

Boundary slope of the string

\(\dot{w}'(L,t)\) :

Time-varying rate of the boundary slope

\(u(t)\) :

Boundary control input applied on the tip payload

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Acknowledgments

The authors would like to thank the Editor-In-Chief, the Associate Editor and the anonymous reviewers for their constructive comments which helped improve the quality and presentation of this paper. This work was supported by the National Natural Science Foundation of China under Grant 61203057 and 61403063, the National Basic Research Program of China (973 Program) under Grant 2014CB744206, and the Fundamental Research Funds for the China Central Universities of UESTC under Grant ZYGX2013Z003.

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Authors and Affiliations

  1. School of Automation Engineering and Center for Robotics, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Wei He & Xiuyu He

  2. Department of Electrical and Computer Engineering, National University of Singapore, Singapore, 117576, Singapore

    Shuzhi Sam Ge

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  1. Wei He
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Correspondence to Wei He.

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He, W., He, X. & Ge, S.S. Boundary output feedback control of a flexible string system with input saturation. Nonlinear Dyn 80, 871–888 (2015). https://doi.org/10.1007/s11071-015-1913-8

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