Abstract
An infinite-dimensional approach for the active vibration control of a multilayered straight composite piezoelectric beam is presented. In order to control the excited beam vibrations, distributed piezoelectric actuator and sensor layers are spatially shaped to achieve a sensor/actuator collocation which fits the control problem. In the sense of von Kármán a nonlinear formulation for the axial strain is used and a nonlinear initial boundary-value problem for the deflection is derived by means of the Hamilton formalism. Three different control strategies are proposed. The first one is an extension of the nonlinear H∞-design to the infinite-dimensional case. It will be shown that an exact solution of the corresponding Hamilton–Jacobi–Isaacs equation can be found for the beam under investigation and this leads to a control law with optimal damping properties. The second approach is a PD-controller for infinite-dimensional systems and the third strategy makes use of the disturbance compensation idea. Under certain observability assumptions of the free system, the closed loop is asymptotically stable in the sense of Lyapunov. In this way, flexural vibrations which are excited by an axial support motion or by different time varying lateral loadings, can be suppressed in an optimal manner. A numerical example serves both to illustrate the design process and to demonstrate the feasibility of the proposed methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, R., Marsden, J. E., and Ratiu, T., Manifolds, Tensor Analysis and Applications, Springer-Verlag, New York, 1988.
Bailey, T. and Hubbard, J. E., ‘Distributed piezoelectric-polymer active vibration control of a cantilever beam’, Journal of Guidance, Control and Dynamics 8(5), 1985, 605–611.
Balas, M. J., ‘Feedback control of flexible systems’, IEEE Transactions on Automatic Control 23(4), 1978, 673–679.
Banks, H. T., Smith, R. C., and Wang, Y., Smart Material Structures: Modeling, Estimation and Control, Wiley, New York, 1996.
Burke, W. L., Applied Differential Geometry, Cambridge University Press, New York, 1994.
Byrnes, Ch. I., Isidori, A., and Willems, J. C., ‘Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems’, IEEE Transactions on Automatic Control 36(11), 1991, 1228–1240.
Choquet-Bruhat, Y. and De Witt-Morette, C., Analysis, Manifolds and Physics, Part I: Basics, North-Holland, Amsterdam, 1982.
Ge, P. and Jouaneh, M., ‘Tracking control of a piezoceramic actuator’, IEEE Transactions on Control Systems Technology 4(3), 1996, 209–215.
Haas, W., Kugi, A., Schlacher, K., and Paster, M., ‘Experimental results of the control of structures with piezoelectric actuators and sensors’, in Proceedings of the Fourth International Conference on Motion and Vibration Control (MOVIC'98), Zurich, Switzerland, August 25–28, 1998, pp. 393–398.
Irschik, H., Belyaev, A. K., Krommer, M., and Schlacher, K., ‘Non-uniqueness of two inverse problems of thermally and force-loaded smart structures: Sensor shaping and actuator shaping problem’, in Proceedings of the 1997 ASME International Mechanical Engineering Congress and Exposition, Dallas, TX, November 16–21, 1997, pp. 119–125.
Kugi, A., Schlacher, K., and Irschik, H., ‘Optimal control of nonlinear parametrically excited beam vibrations by spatially distributed sensors and actors’, in Proceedings of the ASME Design Engineering Technical Conferences (DETC'97), Sacramento, CA, September 14–17, 1997, DETC97/VIB-4171.
Lee, C.-K. and Moon, F. C., ‘Modal sensors/actuators’, Journal of Applied Mechanics 57, 1990, 434–441.
Lee, C.-K., ‘Piezoelectric laminates: Theory and experiments for distributed sensors and actuators’, in Intelligent Structural Systems, H. S. Tzou and G. L. Anderson (eds.), Kluwer, Dordrecht, 1992, pp. 75–167.
Marsden, J. E. and Hughes, T. J. R., Mathematical Foundations of Elasticity, Dover, New York, 1994.
Marsden, J. E. and Ratiu, T. S., Introduction to Mechanics and Symmetry, Springer-Verlag, New York, 1994.
Meirovitch, L. and Baruh, H., ‘The implementation of modal filters for control of structures’, Journal of Guidance, Control and Dynamics 8(6), 1985, 707–716.
Miller, S. E., Oshman, Y., and Abramovich, H., ‘Modal control of piezolaminated anisotropic rectangular plates. Part 1: Modal transducer theory’, AIAA Journal 34(9), 1996, 1868–1875.
Miller, S. E., Oshman, Y., and Abramovich, H., ‘Modal control of piezolaminated anisotropic rectangular plates. Part 2: Control theory’, AIAA Journal 34(9), 1996, 1876–1884.
Nijmeijer, H. and Van der Schaft, A. J., Nonlinear Dynamical Control Systems, Springer-Verlag, Berlin, 1990.
Nowacki, W., Dynamic Problems of Thermoelasticity, Noordhoff, Leyden, 1975.
Schlacher, K., Irschik, H., and Kugi, A., ‘Control of nonlinear beam vibrations by multiple piezoelectric layers’, in IUTAM Symposium on Interaction between Dynamics and Control in Advanced Mechanical Systems, D. H. van Campen (ed.), Kluwer, Dordrecht, 1996, pp. 355–362.
Schlacher, K., Kugi, A., and Irschik, H., ‘H1-control of nonlinear beam vibrations’, in Proceedings of the Third International Conference on Motion and Vibration Control (MOVIC'96), Tokyo-Chiba, Japan, September 1–6, 1996, pp. 497–484.
Schlacher, K. and Kugi, A., ‘Regelung von Hamiltonsystemen’, Elektrotechnik und Informationstechnik e&i 114(7/8), 1997, 353–359.
Tauchert, T. R., ‘Piezothermoelastic behavior of a laminated plate’, Journal of Thermal Stresses 15, 1992, 25–37.
Tzou, H. S., ‘Active piezoelectric shell continua’, in Intelligent Structural Systems, H. S. Tzou and G. L. Anderson (eds.), Kluwer, Dordrecht, 1992, pp. 9–74.
Tzou, H. S., Zhong, J. P., and Hollkamp, J. J., ‘Spatially distributed orthogonal piezoelectric shell actuators: Theory and applications’, Journal of Sound and Vibration 177(3), 1994, 363–378.
Van der Schaft, A. J., ‘Nonlinear state space H1 control theory’, in Essays on Control: Perspectives in the Theory and Its Applications, H. L. Trentelman and J. C. Willems (eds.), Birkhäuser Verlag, Basel, 1993, pp. 153–190.
Ziegler, F., Mechanics of Solids and Fluids, Springer-Verlag, Wien, 1995.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kugi, A., Schlacher, K. & Irschik, H. Infinite-Dimensional Control of Nonlinear Beam Vibrations by Piezoelectric Actuator and Sensor Layers. Nonlinear Dynamics 19, 71–91 (1999). https://doi.org/10.1023/A:1008393904114
Issue Date:
DOI: https://doi.org/10.1023/A:1008393904114