Recent results on pole placement for linear time-varying (LTV) systems are exploited here for the control of flexible structures. The infinite-dimensional system is approached as usual by a restricted number of modes of interest, according to the frequency range in which the system is exploited. The difference with the previous approaches (e.g. gain scheduling) is that only one finite dimension linear model is used and its parameters (frequency and damping of the modes) are varying according to the operating conditions. The control model is thus LTV and a regulator is analytically synthesised to ensure closed-loop stability and performances. This regulator is also LTV and thus automatically (without any special switching action) tracks the operating conditions of the system. The case of a flexible beam is studied in simulation as well as in the laboratory experimentation. Hence, a suitable application of the proposed LTV synthesis is done to find a compromise between the complexity of interpolation and its efficiency. Furthermore, the bad dissipation factor of flexible systems leads them to be good candidates to proof the efficiency of the proposed gain scheduling strategy. This work validates on a real system the LTV pole placement approach and opens the way to a new gain scheduling strategy for the control of LTV, non-linear or infinite-dimensional systems.

References

1. 1)
  - 25. Fliess, M.: ‘Une interprétation algébrique de la transformation de Laplace et des matrices de transfert’, Linear Algebr. Appl., 1990, 203–204, pp. 429–443 (doi: 10.1016/0024-3795(94)90212-7).
2. 2)
  - 23. Malgrange, B.: ‘Systèmes différentiels à coefficients constants’. Séminaire Bourbaki, vol. 246, 1962–1963, pp. 1–11.
3. 3)
  - 22. Hac, A., Lui, L.: ‘Sensor and actuator location in motion control of flexible structures’, J. Sound Vib., 1990, 167, (2), pp. 239–261 (doi: 10.1006/jsvi.1993.1333).
4. 4)
  - 6. Jeanneau, M., Alazard, D., Mouyon, P.: ‘A semi-adaptive frequency control law for flexible structures’. Proc. American Control Conf. (ACC)’, Portland, USA, 1999.
5. 5)
  - R. Holland , P. Young , C. Zhu . Robustness and performance trade-offs in control design for flexible structures. IEEE Trans. Control Syst. Technol. , 4 , 352 - 361
6. 6)
  - 17. Adami, T., Zhu, J. J.: ‘Control of a flexible, hypersonic scramjet vehicle using a differential algebraic approach’. Proc. AIAA Guidance’, Navigation and Control Conf., Honolulu, HI, August, 2008.
7. 7)
  - 3. Lasiecka, I.: ‘Galerkin approximations of infinite-dimensional compensators for flexible structures with unbounded control action’, Acta Appl. Math., 1992, (28), pp. 101–133.
8. 8)
  - 15. Gielen, R.H., Lazar, M.: ‘Stabilisation of linear delay difference inclusions via time-varying control Lyapunov functions’, IET Control Theory Appl., 2012, 6, (12), pp. 1958–1964 (doi: 10.1049/iet-cta.2012.0100).
9. 9)
  - 10. Bérard, C., Puyou, G.: ‘Gain scheduled flight control law for a flexible aircraft: a practical approach’, 17th IFAC Symp. on Automatic Control in Aerospace, 25–29 June 2007, Toulouse, France.
10. 10)
  - 1. Gerardin, M., Rixen, G.: ‘Théorie des vibrations, application à la dynamique des structures’ Masson, Physique fondamentale appliquée, 2nd edn., 1996.
11. 11)
  - 9. Leith, D.J., Leithead, W.E.: ‘Input-output linearization velocity-based gain scheduling’, Int. J. Control, 1999, 72, pp. 229–246 (doi: 10.1080/002071799221217).
12. 12)
  - 19. Bourlès, H., Marinescu, B.: ‘Linear time-varying systems, algebraic-analytic approach’ LNCIS, Springer–Verlag, 2011.
13. 13)
  - 13. Cavallo, A., D Maria, G., Pirozzi, S.: ‘Robust control of flexible structures with stable bandpass controllers’, Automatica, 2007, 44, pp. 1251–1260 (doi: 10.1016/j.automatica.2007.10.020).
14. 14)
  - H. Gu , G. Song . Active vibration suppression of a flexible beam with piezoceramic patches using robust model reference control. Smart Mater. Struct. , 1453 - 1459
15. 15)
  - 26. Lam, T.Y., Leroy, A.: ‘Wedderburn polynomials over division rings I’, J. Pure Appl. Algebr., 2004, 186, pp. 43–76 (doi: 10.1016/S0022-4049(03)00125-7).
16. 16)
  - K. Shojaei , A.M. Shahri . Adaptive robust time-varying control of uncertain non-holonomic robotic systems. IET Control Theory Appl. , 90 - 102
17. 17)
  - 18. Zhu J., J.: ‘Series and parallel d-spectra for multi-input–multi-output linear time-varying systems’. Proc. South-Eastern Symp. on Systems Theory, Baton Rouge (LA-US), 1996, pp. 125–129.
18. 18)
  - 20. Marinescu, B.: ‘Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems’, Automatica, (46), (4), 2010, pp. 1524–1530.
19. 19)
  - 27. Marinescu, B., Bourlès, H.: ‘An intrinsic algebraic setting for poles and zeros of linear time-varying systems’, Syst. Control Lett., 2009, 58, pp. 248–253 (doi: 10.1016/j.sysconle.2008.10.013).
20. 20)
  - 24. Cohn, P. M.: ‘Free rings and their relations’, Academic Press, London Mathematical Society Monograph No. 19, 2nd edn., 1985.
21. 21)
  - 12. Halim, D., Reza Moheiani, G. De M.: ‘Experimental implementation of spatial H∞ control on a Piezoelectric–Laminate beam’, IEEE/ASME Trans. Mechatron., 2002, 7, (3), pp. 346–356 (doi: 10.1109/TMECH.2002.802727).
22. 22)
  - 5. Schilders, W. H. A., Van der Vost, H. A., Rommes, J.: ‘Model order reduction: theory, research aspects and applications’ Springer–Verlag, 2008.
23. 23)
  - 21. Leleu, S.: ‘Amortissement actif des vibrations d’une structure flexible de type plaque à l’aide de transducteurs piézoélectriques’, PhD thesis, Ecole Normale Supérieure, Cachan, France, 2002.
24. 24)
  - 16. Adami, T., Sabala, R., Zhu, J. J.: ‘Time-varying notch filters for control of flexible structures and vehicles’. Proc. 22nd Digital Avionics Systems Conf.’, Indianapolis, Indiana, October, 2003.
25. 25)
  - 2. Preumont, A.: ‘Vibration control of active structures: an introduction’ in ‘Solid mechanics and its applications, vol. 179, 1997, 3rd edn., Springer–Verlag, 2011.
26. 26)
  - 4. Luo, Z. H., Guo, B. Z., Morgül, Ö.: ‘Stability and stabilization of infinite dimensional systems with applications’ Springer–Verlag, 1999.
27. 27)
  - 7. Mohammadpour, J., Scherer, C. W.: ‘Control of linear parameter varying systems with applications’, Springer–Verlag, 2012.

Linear time-varying control of the vibrations of flexible structures

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