Active vibration control of an elastic plate using multiple piezoelectric sensors and actuators
Introduction
Piezoelectric sensors and actuators are extensively employed in many practical applications involving smart technologies (e.g. in shape and vibration control of light structures and in radiated noise suppression) due to their lightness and their capability of coupling strain and electric fields.
In order to control structural vibrations, piezoelectric sensors and actuators can be easily bonded on the vibrating structure or embedded in it. Different kinds of control strategies can be designed for a piezoactuated structure such as a purely passive control, obtained by connecting the electrodes of the piezoelectric actuators to a suitable passive electrical network, a purely active control, obtained by directly applying an external control voltage to the piezoelectric actuator electrodes, and a hybrid control which consists of an active control chain together with a passive device; these three strategies are investigated in [2], [12].
This paper is focused on purely active control of piezoactuated structures. This topic has received much attention by researchers and different control techniques have been investigated in order to design suitable, high performance compensators; a simple technique is the state feedback control employed e.g. in [8], which is easy to design but requires the exact measure of the entire state of the system. In [3] an optimal control law for a piezoactuated beam is obtained by using a state observer and the Riccati equation. The LQG technique, adopted e.g. in [2], employs an observer able to estimate the state of a reduced order model. In [9] the μ-synthesis technique is used in order to build up a robust controller for the vibration control of a cantilever beam with a self-sensing piezoelectric actuator. In [10] an H∞ technique based on the μ-synthesis method is employed for the vibration suppression of a piezoactuated cylindrical shell.
In all these references, only a single piezoelectric actuator is employed in the control scheme; fewer contributions are available concerning the use of multiple piezoelectric sensors and actuators [4], [5]. As a matter of fact, the use of multiple sensors and actuators is important to effectively control complex structures with several vibration modes, where the use of a single sensor and actuator can result in very poor observability and controllability properties of some of the modes. These issues are here faced in the active vibration control of an elastic plate, clamped along one of its sides. The controlled structure, described in Section 2, is subject to an external disturbance consisting of an impulsive transversal force acting on one of its free corners, and the induced flexural vibrations are damped by using up to three piezoelectric sensors and actuators, as shown in Fig. 1.
The previous disturbance attenuation problem fits naturally in the framework of the H2 control technique [6], which is used here for the design of the controller. When a multiple input multiple output (MIMO) control approach is adopted, an accurate model for the coupled electromechanical behaviour of the piezoactuated structure is needed. To this end, the finite-element formulation developed in [1] is employed, and a modal reduction is applied in order to obtain a modal model of the coupled electromechanical structure, as described in Section 2.2.
Several compensators are considered, in order to make a comparison among the performance that can be achieved using different combinations of sensors and actuators, designed on the basis of increasingly accurate structural models. In particular, adjusting the weights characterizing the H2 index to be optimized, it is possible to select sensors and actuators to put in inactive state. Four archetypal cases are investigated: in the first case, just one sensor and one actuator are used, and the controller is designed on a model considering only the first structural eigenmode; then the use of three sensors and three actuators is explored still taking into account a single structural eigenmode; in the last two cases, compensators considering a single sensor and actuator and three sensors and actuators are designed on the basis of a modal model taking into account the first three structural eigenmodes. The selection of these four cases out of the many possible ones enables to show that, increasing the number of sensors and actuators, a better performance is achieved provided that a sufficiently accurate model of the controlled system has been employed in the design of the controller.
The proposed compensators are eventually tested in simulation by using a modal model taking into account the first five flexural structural eigenmodes and the mechanical damping of the plate. The simulation results, briefly reported in Section 3, confirm the expectations previously reported.
Section snippets
Controlled structure
The piezoactuated steel plate (E=210 GPa, ν=0.3, ρ=7850 kg/m3) shown in Fig. 1 is considered; its sides are 0.3 m × 0.2 m and its thickness is 0.005 m. Three couples of piezoelectric patches are bonded on the upper and lower surfaces of the plate symmetrically with respect to its mid-plane; the patches on the upper surface act like actuators whereas the patches on the lower surface are used as sensors, thus allowing a collocated vibration control. Each piezoelectric patch has dimensions 0.06 m ×
Design and evaluation of controllers
Using the modal model described in the preceding sections, a controller can be designed to achieve a high damping in the controlled structure. In this section, a controller will be designed to suppress the vibrations induced on the piezoactuated plate by an impulsive disturbance force. According to the block diagram in Fig. 3(a), the controller will use a measured output c(t) (the vector of the voltages c(t)=vs(t) measured by the piezoelectric sensors and proportional to the curvature of the
Conclusions
In this paper, the problem of damping flexural vibrations of an elastic plate by using multiple piezoelectric sensors and actuators has been addressed. To this end, the H2 control technique has been applied in the controller design. This technique, not explored in the cited literature, represents the optimal synthesis tool when the objective of the control is to minimize (in a suitable sense) the vibration energy due to impulsive or stochastic disturbances acting on the controlled structure.
Acknowledgements
This research was developed within the framework of Lagrange Laboratory, an European research group between CNRS, CNR, University of Rome “Tor Vergata”, University of Montpellier II, ENPC and LCPC.
The financial supports of MIUR and ASI are gratefully acknowledged.
Professor Franco Maceri is gratefully acknowledged for his valuable comments on this paper.
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