Keywords: piezoelectric actuator, sensor networks, collocation, active control, attenuation of vibrations.

This paper presents a self-sensing approach to the adaptive control of smart structures with piezoelectric actuators. The mechanical model of a Kirchhoff plate [1,2], with two patches is rewritten in the form of an infinite dimensional Port controlled hamiltonian system with dissipation (PCHD) model [3]. The advantage of this description is that one can apply several controller design methods, like potential, interconnection shaping, damping injection, etc. Our design is based on the method where the pairing of the input with the so called collocated output is exploited. One can achieve collocation by a special combination of actuators and with sensors, or by self-sensing like in this paper. The latter approach requires the robust separation of the electric current due to the direct piezoelectric effect from the measured electric current. Because of the unfavorable ratio of these two signals, the design of an approximate observer for the electric current due to the direct piezoelectric effect is proposed. The control design goal is the asymptotic suppression of harmonic disturbances with known frequency but unknown in amplitude and phase. The control law is derived for the plant augmented by an appropriate exosystem, which models the properties of the disturbance [4]. The novelty of this paper is that the control design methods are extended from the finite dimensional case to the infinite dimensional one. The stability analysis for the infinite dimensional system is based on the concept of L₂-stability and the small gain theorem [5]. In the course of our investigations, we consider a rectangular plate equipped with two piezoelectric actuators. The plate has two opposite edges either clamped or free. The test rig is designed such that one actuator induces disturbances in a controlled manner for testing the performance of the vibration attenuation system to be designed which uses the other piezoelectric actuator. Vibration attenuation at a dominant eigenfrequency is demonstrated and corresponding sensor and control signals are depicted along with actual plate vibrations which are measured with a Polytec PSV 400 laser scanning Doppler vibrometer.

1

C. Fuller, S.J. Elliott, P.A. Nelson, "Active Control of Vibration", Academic Press, London, 1993.

2

H. Ennsbrunner, K. Schlacher, "Modeling of Piezoelectric Structures - A Hamiltonian Approach", 5th Vienna Symposium on Mathematical Modeling, 2006.

3

K. Schlacher, "Mathematical modeling for nonlinear control: a Hamiltonian approach", Mathematics and Computers in simulation, 79, 829-849, 2008. doi:10.1016/j.matcom.2008.02.011

4

A. Isidori, L. Marconi, A. Serrani, "Robust Autonomous Guidance - an Internal Model Approach", Springer, London, 2003.

5

H.K. Khalil, "Nonlinear Systems", Prentice Hall Inc., New Jersey, 1996.

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