Creating a computational tool to simulate vibration control for piezoelectric devices

Piezoelectric materials have the unique ability to convert electrical energy to mechanical vibrations and vice versa. This project takes a stab to develop a reliable computational tool to simulate the vibration control of a novel “partial differential equation” model for a piezoelectric device, which is designed by integrating electric conducting piezoelectric layers constraining a viscoelastic layer to provide an active and lightweight intelligent structure. Controlling unwanted vibrations on piezoelectric devices (or harvesting energy from ambient vibrations) through piezoelectric layers has been the major focus in cutting-edge engineering applications such as ultrasonic welders and inchworms. The corresponding mathematical models for piezoelectric devices are either heuristic or mathematically oversimplified differential equations. Moreover, their “unjustified” approximated reductions consider only the first several vibrations on these devices. In this project, a novel partial differential equation model, accounting for all vibrational modes, is analyzed to provide new insights for a cost-efficient sensor feedback design. Therefore, the sensor feedback signals are not allowed to be contaminated by the residual modes. Our primary goal is to develop reproducible computational tools by an emerging stable approximation technique, so-called filtered Finite Difference Method, which is proved to provide faster and reliable computation. Filtering in the approximation is necessary since the spurious vibrations, due to the blind application of the Finite Difference Method, provide a false stability result. To see the efficiency of the algorithm, we compare the approximation to the one obtained by the Finite Element Method based on the Galerkin\u27s approximation, which is a common technique being used in the engineering literature. The mathematical techniques and computational tools developed in this project are essential to provide new insights into the active controlling of piezoelectric devices. Improving the efficiency of active controlling enables us to take better advantage of piezoelectric technology change since one-time design and fabrication may be unavoidable for many applications such as cardiac pacemakers or NASA/commercially-operated inflatable space antennas. Our state-of-the-art partial differential equation model and its stable approximations will be adaptable for a large class of piezoelectric devices

Modeling an elastic beam with piezoelectric patches by including magnetic effects

Models for piezoelectric beams using Euler-Bernoulli small displacement theory predict the dynamics of slender beams at the low frequency accurately but are insufficient for beams vibrating at high frequencies or beams with low length-to-width aspect ratios. A more thorough model that includes the effects of rotational inertia and shear strain, Mindlin-Timoshenko small displacement theory, is needed to predict the dynamics more accurately for these cases. Moreover, existing models ignore the magnetic effects since the magnetic effects are relatively small. However, it was shown recently \cite{O-M1} that these effects can substantially change the controllability and stabilizability properties of even a single piezoelectric beam. In this paper, we use a variational approach to derive models that include magnetic effects for an elastic beam with two piezoelectric patches actuated by different voltage sources. Both Euler-Bernoulli and Mindlin-Timoshenko small displacement theories are considered. Due to the magnetic effects, the equations are quite different from the standard equations.Comment: 3 figures. 2014 American Control Conference Proceeding

Sensor Choice for Minimum Error Variance Estimation

© 2016 IEEE, Morris, K. A., & Özer, A. ö. (2014). Modeling and Stabilizability of Voltage-Actuated Piezoelectric Beams with Magnetic Effects. SIAM Journal on Control and Optimization, 52(4), 2371–2398. https://doi.org/10.1137/130918319A Kalman filter is optimal in that the variance of the error is minimized by the estimator. It is shown here, in an infinite-dimensional context, that the solution to an operator Riccati equation minimizes the steady-state error variance. This extends a result previously known for lumped parameter systems to distributed parameter systems. It is shown then that minimizing the trace of the Riccati operator is a reasonable criterion for choosing sensor locations. It is then shown that multiple inaccurate sensors, that is, those with large noise variance, can provide as good an estimate as a single highly accurate (but probably more expensive) sensor. Optimal sensor location is then combined with estimator design. A framework for calculation of the best sensor locations using approximations is established and sensor location as well as choice is investigated with three examples. Simulations indicate that the sensor locations do affect the quality of the estimation and that multiple low quality sensors can lead to better estimation than a single high quality sensor.NSERC Discovery Grant US AFOSR grant || FA 9550-16-1-006

A Novel Passivity-Based Controller for a Piezoelectric Beam

This paper presents a new passivity property for distributed piezoelectric devices with integrable port-variables. We present two new control methodologies by exploiting the integrability property of the port-variables. The derived controllers have a Proportional-Integral (PI) like structure. Finally, we present the simulation results and an in-depth analysis on the tuning gains explaining their transient and the steady-state behaviors

Modeling and Stabilizability of Voltage-Actuated Piezoelectric Beams with Magnetic Effects

Models for piezoelectric beams and structures with piezoelectric patches generally ignore magnetic effects. This is because the magnetic energy has a relatively small effect on the overall dynamics. Piezoelectric beam models are known to be exactly observable and can be exponentially stabilized in the energy space by using a mechanical feedback controller. In this paper, a variational approach is used to derive a model for a piezoelectric beam that includes magnetic effects. It is proved that the partial differential equation model is well-posed. Magnetic effects have a strong effect on the stabilizability of the control system. For almost all system parameters the piezoelectric beam can be strongly stabilized, but it is not exponentially stabilizable in the energy space. Strong stabilization is achieved using only electrical feedback. Furthermore, using the same electrical feedback, an exponentially stable closed-loop system can be obtained for a set of system parameters of zero Lebesgue measure. These results are compared to those of a beam without magnetic effects

Modeling and Stabilizability of Voltage-Actuated Piezoelectric Beams with Magnetic Effects

Models for piezoelectric beams and structures with piezoelectric patches generally ignore magnetic effects. This is because the magnetic energy has a relatively small effect on the overall dynamics. Piezoelectric beam models are known to be exactly observable and can be exponentially stabilized in the energy space by using a mechanical feedback controller. In this paper, a variational approach is used to derive a model for a piezoelectric beam that includes magnetic effects. It is proved that the partial differential equation model is well-posed. Magnetic effects have a strong effect on the stabilizability of the control system. For almost all system parameters the piezoelectric beam can be strongly stabilized, but it is not exponentially stabilizable in the energy space. Strong stabilization is achieved using only electrical feedback. Furthermore, using the same electrical feedback, an exponentially stable closed-loop system can be obtained for a set of system parameters of zero Lebesgue measure. These results are compared to those of a beam without magnetic effects