Keywords: smart piezoelectric beam, finite element, active vibration control, velocity feedback, LQR and LQG controller.

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 one-dimensional 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 closed-loop 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 pole-zero representation of the system (as in the root-locus 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 trade-offs among various performance criteria. The relationship between cost function weights and performance criteria hold even for high-order 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 closed-loop 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 full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)