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CivilComp Proceedings
ISSN 17593433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 8
Classical and Optimal Active Vibration Control of Smart Piezoelectric Beams C.M.A. Vasques and J.D. Rodrigues
Department of Mechanical Engineering and Industrial Management, Faculty of Engineering, University of Porto, Portugal C.M.A. Vasques, J.D. Rodrigues, "Classical and Optimal Active Vibration Control of Smart Piezoelectric Beams", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 8, 2004. doi:10.4203/ccp.79.8
Keywords: smart piezoelectric beam, finite element, active vibration control, velocity feedback, LQR and LQG controller.
Summary
The aims of this work are the analysis and comparison of the
classical and optimal feedback control strategies on the active
control of vibrations of smart piezoelectric beams. The
mathematical model utilized is succinctly presented and a brief
description of the onedimensional finite element (FE) previously
developed by the authors is presented. The kinematic and electric
potential assumptions of the theoretical and FE models are
presented and the resultant FE equations of motion and electric
charge equilibrium are recast into a state variable representation
in terms of the physical modes of the beam. Furthermore, some
theoretical considerations about the classical control strategies,
with two distinct velocity feedback control algorithms
being considered, constant amplitude and constant
gain velocity feedback (CAVF and CGVF), and optimal control strategies, linear quadratic regulator (LQR) and linear quadratic
Gaussian (LQG) controller, are presented. The control models
assume that one of the piezoelectric layers acts as a distributed
sensor and the other one as a distributed actuator. The sensor
signal is used as a feedback reference in the closedloop control
systems previously referred.
The classical techniques can avoid the necessity of digital control reducing the time delays in the control system. However, the CGVF and CAVF strategies require the differentiation of the sensor voltage, which for noisy measurements can become troublesome. The gain selection and control system design of such feedback controllers can be done using either a polezero representation of the system (as in the rootlocus method or pole placement, for example) or a frequency response representation (as, for example, in the Nyquist method). If one looks for adaptability in design, the optimal techniques can be more interesting but with the disadvantage of requiring a model of the system. The quantification of the control system's performance by a quadratic cost function provides the designer with lots of flexibility to perform tradeoffs among various performance criteria. The relationship between cost function weights and performance criteria hold even for highorder and multiple input systems, where classical control becomes cumbersome. A major limitation of the LQR is that the entire state must be measured when generating the control. The LQG controller overcomes the need to measure the entire state by estimating the sate using a Kalman filter, which utilizes noisy partial output information, and some modifications in its performance, which sometimes can improve the efficiency, often occur. However, observability and controllability spillover problems related with the reduced (truncated) model dynamics may compromise stability. The analyzed case studies concern the vibration reduction of a cantilever aluminum beam with a pair of collocated piezoelectric patches mounted on the surface. The displacement time history, for an initial displacement field and white noise force disturbance, and point receptance at the free end are evaluated with the open and closedloop classical and optimal control systems. The case studies allow to compare the performances of the classical and optimal strategies. It is shown that for an initial displacement field the CAVF and LQG with the first mode weighted are the most interesting solutions. However, for a white noise disturbance, the CAVF only manages to reduce the mode under control, and the others are destabilized. Moreover, the LQG with all the modes weighted presents a better control in bandwidth with a lower control voltage. The output weighting LQG can also be an interesting strategy in situations where attenuation of the outputs is of interest. In the present analysis a resemblance in performance between the LQG controller with output weighting and the CGVF is also shown. purchase the fulltext of this paper (price £20)
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