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Active Vibration Control and Stability Analysis of Flexible Beam Systems pp 113–125Cite as

Distributed Control of a Flexible Beam

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Abstract

Since the 1980s and 1990s, plenty of control methods and techniques have been introduced to deal with vibration problem of the flexible mechanical systems on several papers and books. In [1], by introducing feedback variables through a regulator, a state-feedback controller that contains a dynamic compensator is designed to stabilize the vibration of the flexible system. In [2], various control schemes for a single flexible robotic arm are considered and the improved control performance can be obtained by using a linear optimal controller, which verifies the efficiency of LQR optimal method. According to [3], the authors investigate the problem of synthesizing PID controllers for robust performance for a given single-input-single-output system.

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References

  1. Y. Sakawa, F. Matsuno, S. Fukushima, Modeling and feedback control of a flexible arm. J. Robot. Syst. 2(4), 453–472 (1985)

    Article  Google Scholar 

  2. Y. Aoustin, C. Chevallereau, A. Glumineau, C. Moog, Experimental results for the end-effector control of a single flexible robotic arm. IEEE Trans. Control Syst. Technol. 2(4), 371–381 (1994)

    Article  Google Scholar 

  3. M.T. Ho, C.Y. Lin, PID controller design for robust performance. IEEE Trans. Autom. Control 48(8), 1404–1409 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. B.-Z. Guo, F.-F. Jin, The active disturbance rejection and sliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbance. Automatica 49(9), 2911–2918 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. B.-Z. Guo, H.-C. Zhou, The active disturbance rejection control to stabilization for multi-dimensional wave equation with boundary control matched disturbance. IEEE Trans. Autom. Control 60(1), 143–157 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. B.-Z. Guo, F.-F. Jin, Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance. IEEE Trans. Autom. Control 60(3), 824–830 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. G.Q. Xu, Z.J. Han, S.P. Yung, Riesz basis property of serially connected Timoshenko beams. Int. J. Control 80(3), 470–485 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. H.K. Khalil, Nonlinear Systems (Prentice Hall, New Jersey, 2002)

    MATH  Google Scholar 

  9. M. Krstic, I. Kanellakopoulos, P. Kokotovic, Nonlinear and Adaptive Control Design (Wiley, New York, 1995)

    MATH  Google Scholar 

  10. S.S. Ge, C.C. Hang, T.H. Lee, T. Zhang, Stable Adaptive Neural Network Control (Kluwer Academic, Boston, 2001)

    MATH  Google Scholar 

  11. J.D. Logan, Applied Mathematics, 3rd edn. (Wiley, New York, 2006)

    MATH  Google Scholar 

  12. J.-M. Wang, J.-J. Liu, B. Ren, J. Chen, Sliding mode control to stabilization of cascaded heat PDE-ODE systems subject to boundary control matched disturbance. Automatica 52, 23–34 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. J.M. Wang, B. Ren, M. Krstic, Stabilization and gevrey regularity of a Schrödinger equation in boundary feedback with a heat equation. IEEE Trans. Autom. Control 57(1), 179–185 (2012)

    Article  MATH  Google Scholar 

  14. B. Ren, J.-M. Wang, M. Krstic, Stabilization of an ODE-Schrödinger cascade. Syst. Control Lett. 62(6), 503–510 (2013)

    Article  MATH  Google Scholar 

  15. S.S. Ge, T.H. Lee, G. Zhu, A nonlinear feedback controller for a single-link flexible manipulator based on a finite element model. J. Robot. Syst. 14(3), 165–178 (1997)

    Article  MATH  Google Scholar 

  16. S.S. Ge, T.H. Lee, G. Zhu, Non-model-based position control of a planar multi-link flexible robot. Mech. Syst. Signal Process. 11(5), 707–724 (1997)

    Article  Google Scholar 

  17. A. Armaou, P. Christofides, Wave suppression by nonlinear finite-dimensional control. Chem. Eng. Sci. 55(14), 2627–2640 (2000)

    Article  Google Scholar 

  18. P. Christofides, A. Armaou, Global stabilization of the Kuramoto-Sivashinsky equation via distributed output feedback control. Syst. Control Lett. 39(4), 283–294 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. H.-N. Wu, J.-W. Wang, H.-X. Li, Design of distributed \(H_\infty \) fuzzy controllers with constraint for nonlinear hyperbolic PDE systems. Automatica 48(10), 2535–2543 (2012)

    MathSciNet  MATH  Google Scholar 

  20. B. Luo, H.-N. Wu, Approximate optimal control design for nonlinear one-dimensional parabolic PDE systems using empirical eigenfunctions and neural network. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 42(6), 1538–1549 (2012)

    Article  Google Scholar 

  21. J. Wang, H.-N. Wu, H.-X. Li, Distributed proportional spatial derivative control of nonlinear parabolic systems via fuzzy PDE modeling approach. IEEE Trans. Syst. Man Cybern. Part B: Cybern. 42(3), 927–938 (2012)

    Article  Google Scholar 

  22. H.-N. Wu, H.-X. Li, Finite-dimensional constrained fuzzy control for a class of nonlinear distributed process systems. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 37(5), 1422–1430 (2007)

    Article  Google Scholar 

  23. B. Bhikkaji, S.O.R. Moheimani, I. Petersen, A negative imaginary approach to modeling and control of a collocated structure. IEEE/ASME Trans. Mechatron. 17(4), 717–727 (2012)

    Article  Google Scholar 

  24. E. Pereira, S.S. Aphale, V. Feliu, S.R. Moheimani, Integral resonant control for vibration damping and precise tip-positioning of a single-link flexible manipulator. IEEE/ASME Trans. Mechatron. 16(2), 232–240 (2011)

    Article  Google Scholar 

  25. M.J. Balas, Active control of flexible systems. J. Optim. Theory Appl. 25, 415–436 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Meirovitch, H. Baruh, On the problem of observation spillover in self-adjoint distributed systems. J. Optim. Theory Appl. 39(2), 269–291 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Krstic, A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs (Society for Industrial and Applied Mathematics, Philadelphia, 2008)

    Book  MATH  Google Scholar 

  28. D. Huang, J.-X. Xu, X. Li, C. Xu, M. Yu, D-type anticipatory iterative learning control for a class of inhomogeneous heat equations. Automatica 49(8), 2397–2408 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. W. He, S.S. Ge, B.V.E. How, Y.S. Choo, Dynamics and Control of Mechanical Systems in Offshore Engineering (Springer, London, 2014)

    Book  Google Scholar 

  30. W. He, S.S. Ge, Robust adaptive boundary control of a vibrating string under unknown time-varying disturbance. IEEE Trans. Control Syst. Technol. 20(1), 48–58 (2012)

    Google Scholar 

  31. S.S. Ge, S. Zhang, W. He, Vibration control of an Euler-Bernoulli beam under unknown spatiotemporally varying disturbance. Int. J. Control 84(5), 947–960 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. G.Q. Xu, H. Wang, Stabilisation of Timoshenko beam system with delay in the boundary control. Int. J. Control 86(6), 1165–1178 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. F. Wu, Distributed control for interconnected linear parameter-dependent systems. IEE Proc. Control Theory Appl. 150(5), 518–527 (2003)

    Article  Google Scholar 

  34. B. Bamieh, F. Paganini, M.A. Dahleh, Distributed control of spatially invariant systems. IEEE Trans. Autom. Control 47(7), 1091–1107 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. R. D’Andrea, G.E. Dullerud, Distributed control design for spatially interconnected systems. IEEE Trans. Autom. Control 48(9), 1478–1495 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. K.P. Tee, B. Ren, S.S. Ge, Control of nonlinear systems with time-varying output constraints. Automatica 47(11), 2511–2516 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. W. He, S.S. Ge, Vibration control of a flexible beam with output constraint. IEEE Trans. Ind. Electron. 62(8), 5023–5030 (2015)

    Article  Google Scholar 

  38. W. He, Y. Chen, Z. Yin, Adaptive neural network control of an uncertain robot with full-state constraints. IEEE Trans. Cybern. 46(3), 620–629 (2016)

    Article  Google Scholar 

  39. L. Lu, Z. Chen, B. Yao, Q. Wang, A two-loop performance oriented tip tracking control of a linear motor driven flexible beam system with experiments. IEEE Trans. Ind. Electron. 60(3), 1011–1022 (2013)

    Article  Google Scholar 

  40. N.H. El Farra, A. Armaou, P.D. Christofides, Analysis and control of parabolic PDE systems with input constraints. Automatica 39(4), 715–725 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. S. Dubljevic, N.H. El Farra, P. Mhaskar, P.D. Christofides, Predictive control of parabolic PDEs with state and control constraints. Int. J. Robust Nonlinear Control 16(16), 749–772 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. K. Endo, F. Matsuno, H. Kawasaki, Simple boundary cooperative control of two one-link flexible arms for grasping. IEEE Trans. Autom. Control 54(10), 2470–2476 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. K.P. Tee, S.S. Ge, E. Tay, Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica 45(4), 918–927 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. W. He, C. Sun, S.S. Ge, Top tension control of a flexible marine riser by using integral-barrier Lyapunov function. IEEE/ASME Trans. Mechatron. 2(20), 497–505 (2015)

    Article  Google Scholar 

  45. B. Ren, S.S. Ge, K.P. Tee, T. Lee, Adaptive neural control for output feedback nonlinear systems using a barrier Lyapunov function. IEEE Trans. Neural Netw. 21(8), 1339–1345 (2010)

    Article  Google Scholar 

  46. W. He, S. Zhang, S.S. Ge, Adaptive control of a flexible crane system with the boundary output constraint. IEEE Trans. Ind. Electron. 61(8), 4126–4133 (2014)

    Article  Google Scholar 

  47. E. Hewitt, K. Stromberg, Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable (Springer, Berlin, 1929)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing, China

    Wei He

  2. School of Automation Science and Electrical Engineering, Beihang University, Beijing, China

    Jinkun Liu

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  1. Wei He
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Correspondence to Wei He .

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He, W., Liu, J. (2019). Distributed Control of a Flexible Beam. In: Active Vibration Control and Stability Analysis of Flexible Beam Systems. Springer, Singapore. https://doi.org/10.1007/978-981-10-7539-1_8

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