Output-Feedback Control of a Chaotic MEMS Resonator for Oscillation Amplitude Enhancement

The present work addresses the problem of chaos control in an electrostatic MEMS resonator by using an output-feedback control scheme. One of the unstable orbits immersed in the chaotic attractor is stabilized in order to produce a sustained oscillation of the movable plate composing the microstructure. The orbit is carefully chosen so as to produce a high amplitude oscillation. This approach allows the enhancement of oscillation amplitude of the resonator at a reduced control effort, since the unstable orbit already exists in the system and it is not necessary to spend energy to create it. Realistic operational conditions of the MEMS are considered including parametric uncertainties in the model and constraints due to the difficulty in measuring the speed of the plates of the microstructure. A control law is constructed recursively by using the technique of backstepping. Finally, numerical simulations are carried out to confirm the validity of the developed control scheme and to demonstrate the effect of controlling orbits immersed in the chaotic attractor

Forced large amplitude periodic vibrations of non-linear Mathieu resonators for microgyroscope applications

International audienceThis paper describes a comprehensive non-linear multiphysics model based on the Euler-Bernoulli beam equation that remains valid up to large displacements in the case of electrostatically actuated Mathieu resonators. This purely analytical model takes into account the fringing field effects and is used to track the periodic motions of the sensing parts in resonant microgyroscopes. Several parametric analyses are presented in order to investigate the effect of the proof mass frequency on the bifurcation topology. The model shows that the optimal sensitivity is reached for resonant microgyroscopes designed with sensing frequency four times faster than the actuation one

Energy efficient control of electrostatically actuated MEMS

Plenty of Micro-electro-mechanical Systems (MEMS) devices are actuated using electrostatic forces, and specially, parallel-plate actuators are extensively used, due to the simplicity of their design. Nevertheless, parallel-plate actuators have some limitations due to the nonlinearity of the generated force. The dissertation analyzes the dynamics of the lumped electrostatically actuated nonlinear system, in order to obtain insight on its characteristics, define desired performance goals and implement a controller for energy efficient robustly stable actuation of MEMS resonators. In the first part of the dissertation, the modeling of the electromechanical lumped system is developed. From a complete distributed parameters model for MEMS devices which rely on electrostatic actuation, a concentrated parameters simplification is derived to be used for analysis and control design. Based on the model, energy analysis of the pull-in instability is performed. The classic approach is revisited to extend the results to models with a nonlinear springs. Analysis of the effect of dynamics is studied as an important factor for the stability of the system. From this study, the Resonant Pull-in Condition for parallel-plate electrostatically actuated MEMS resonators is defined and experimentally validated. Given the importance of the nonlinear dynamics and its richness in behaviors, Harmonic Balance is chosen as a tool to characterize the steady-state oscillation of the resonators. This characterization leads to the understanding of the key factors for large and stable oscillation of resonators. An important conclusion is reached, Harmonic Balance predicts that any oscillation amplitude is possible for any desired frequency if the appropriate voltage is applied to the resonator. And the energy consumption is dependent on this chosen oscillation frequency. Based on Harmonic Balance results, four main goals are defined for the control strategy: Stable oscillation with large amplitudes of motion; Robust oscillation independently of MEMS imperfections; Pure sinus-like oscillation for high-grade sensing; and Low energy consumption. The second part of the dissertation deals with the controller selection, design and verification. A survey of prior work on MEMS control confirms that existing control approaches cannot provide the desired performance. Consequently, a new three-stage controller is proposed to obtain the desired oscillation with the expected stability and energy efficiency. The controller has three different control loops. The first control loop includes a Robust controller designed using on µ-synthesis, to deal with MEMS resonators uncertainties. The second control loop includes an Internal-Model-Principle Resonant controller, to generate the desired control action to obtain the desired oscillation. And the third control loop handles the energy consumption minimization through an Extremum Seeking Controller, which selects the most efficient working frequency for the desired oscillation. The proposed controller is able to automatically generate the needed control voltage, as predicted by the Harmonic Balance analysis, to operate the parallel-plate electrostatically actuated MEMS resonator at the desired oscillation. Performance verification of stability, robustness, sinus-like oscillation and energy efficiency is carried out through simulation. Finally, the needed steps for a real implementation are analyzed. Independent two-sided actuation for full-range amplitude oscillation is introduced to overcome the limitations of one-sided actuation. And a modification of standard Electromechanical Amplitude Modulation is analyzed and validated for position feedback implementation. With these improvements, a MEMS resonator with the desired specifications for testing the proposed control is designed for fabrication. Based on this design, testing procedure is discussed as future work.Molts microsistemes (MEMS) són actuats amb forces electrostàtiques, i especialment, els actuadors electrostàtics de plaques paral.leles són molt usats, degut a la simplicitat del seu disseny. Tot i això, aquests actuadors tenen limitacions degut a la no-linealitat de les forces generades. La tesi analitza el sistema mecànic no-lineal actuat electrostàticament que forma el MEMS, per tal d'entendre'n les característiques, definir objectius de control de l'oscil.lació, i implementar un controlador robust, estable i eficient energèticament. A la primera part de la tesi es desenvolupa el modelat del sistema electromecànic complert. A partir de la formulació de paràmetres distribuïts aplicada a dispositius MEMS amb actuació electrostàtica, es deriva una formulació de paràmetres concentrats per a l'anàlisi i el disseny del control. Basat en aquest model, s'analitza energèticament la inestabilitat anomenada Pull-in, ampliant els resultats de l'enfocament clàssic al model amb motlles no-lineals. Dins de l'anàlisi, l'evolució dinàmica s'estudia per ser un factor important per a l'estabilitat. D'aquest estudi, la Resonant Pull-in Condition per a actuadors electrostàtics de plaques paral.leles es defineix i es valida experimentalment. Donada la importància de la dinàmica no-lineal del sistema i la seva riquesa de comportaments, s'utilitza Balanç d'Harmònics per tal de caracteritzar les oscil.lacions en estacionari. Aquesta caracterització permet entendre els factors claus per a obtenir oscil.lacions estables i d'amplitud elevada. El Balanç d'Harmònics dóna una conclusió important: qualsevol amplitud d'oscil.lació és possible per a qualsevol freqüència desitjada si s'aplica el voltatge adequat al ressonador. I el consum energètic associat a aquesta oscil.lació depèn de la freqüència triada. Llavors, basat en aquests resultats, quatre objectius es plantegen per a l'estratègia de control: oscil.lació estable amb amplituds elevades; robustesa de l'oscil.lació independentment de les imperfeccions dels MEMS; oscil.lació sinusoïdal sense harmònics per a aplicacions d'alta precisió; i baix consum energètic. La segona part de la tesi tracta la selecció, disseny i verificació dun controlador adequat per a aquests objectius. La revisió dels treballs existents en control de MEMS confirma que cap dels enfocaments actuals permet obtenir els objectius desitjats. En conseqüència, es proposa el disseny d'un nou controlador amb tres etapes per tal d'obtenir l'oscil.lació desitjada amb estabilitat i eficiència energètica. El controlador té tres llaços de control. Al primer llaç, un controlador robust dissenyat amb µ-síntesis gestiona les incertes es dels MEMS. El segon llaç inclou un controlador Ressonant, basat en el Principi del Model Intern, per a generar l'acció de control necessària per a obtenir l'oscil.lació desitjada. I el tercer llaç de control gestiona la minimització de l'energia consumida mitjançant un controlador basat en Extremum Seeking, el qual selecciona la freqüència de treball més eficient energèticament per a l'oscil.lació triada. El controlador proposat és capaç de generar automàticament el voltatge necessari, igual al previst pel Balanç d'Harmònics, per tal d'operar electrostàticament amb plaques paral.leles els ressonadors MEMS. Mitjançant simulació se'n verifica l'estabilitat, robustesa, inexistència d'harmònics i eficiència energètica de l'oscil.lació. Finalment, la implementació real és analitzada. En primer lloc, un nou esquema d'actuació per dos costats amb voltatges independents es proposa per aconseguir l'oscil.lació del ressonador en tot el rang d'amplituds. I en segon lloc, una modificació del sensat amb Modulació d'Amplitud Electromecànica s'utilitza per tancar el llaç de control de posició. Amb aquestes millores, un ressonador MEMS es dissenya per a ser fabricat i validar el control. Basat en aquest disseny, es proposa un procediment de test plantejat com a treball futur.Postprint (published version

Review of nonlinear vibration energy harvesting: Duffing, bistability, parametric, stochastic and others

Vibration energy harvesting typically involves a mechanical oscillatory mechanism to accumulate ambient kinetic energy, prior to the conversion to electrical energy through a transducer. The convention is to use a simple linear mass-spring-damper oscillator with its resonant frequency tuned towards that of the vibration source. In the past decade, there has been a rapid expansion in research of vibration energy harvesting into various nonlinear vibration principles such as Duffing nonlinearity, bistability, parametric oscillators, stochastic oscillators and other nonlinear mechanisms. The intended objectives for using nonlinearity include broadening of frequency bandwidth, enhancement of power amplitude and improvement in responsiveness to non-sinusoidal noisy excitations. However, nonlinear vibration energy harvesting also comes with its own challenges and some of the research pursuits have been less than fruitful. Previous reviews in the literature have either focussed on bandwidth enhancement strategies or converged on select few nonlinear mechanisms. This article reviews eight major types of nonlinear vibration energy harvesting reported over the past decade, covering underlying principles, advantages and disadvantages, and application-specific guidance for researchers and designers

Micro-Resonators: The Quest for Superior Performance

Microelectromechanical resonators are no longer solely a subject of research in university and government labs; they have found a variety of applications at industrial scale, where their market is predicted to grow steadily. Nevertheless, many barriers to enhance their performance and further spread their application remain to be overcome. In this Special Issue, we will focus our attention to some of the persistent challenges of micro-/nano-resonators such as nonlinearity, temperature stability, acceleration sensitivity, limits of quality factor, and failure modes that require a more in-depth understanding of the physics of vibration at small scale. The goal is to seek innovative solutions that take advantage of unique material properties and original designs to push the performance of micro-resonators beyond what is conventionally achievable. Contributions from academia discussing less-known characteristics of micro-resonators and from industry depicting the challenges of large-scale implementation of resonators are encouraged with the hopes of further stimulating the growth of this field, which is rich with fascinating physics and challenging problems

Nonlinear mechanics and nonlinear material properties in micromechanical resonators

Microelectromechanical Systems are ubiquitous in modern technology, with applications ranging from accelerometers in smartphones to ultra-high precision motion stages used for atomically-precise positioning. With the appropriate selection of materials and device design, MEMS resonators with ultra-high quality factors can be fabricated at minimal cost. As the sizes of such resonators decrease, however, their mechanical, electrical, and material properties can no longer be treated as linear, as can be done for larger-scale devices. Unfortunately, adding nonlinear effects to a system changes its dynamics from exactly-solvable to only solvable in specific cases, if at all. Despite (and because of) these added complications, nonlinear effects open up an entirely new world of behaviors that can be measured or taken advantage of to create even more advanced technologies. In our resonators, oscillations are induced and measured using aluminum nitride transducers. I used this mechanism for several separate highly-sensitive experiments. In the first, I demonstrate the incredible sensitivity of these resonators by actuating a mechanical resonant mode using only the force generated by the radiation pressure of a laser at room temperature. In the following three experiments, which use similar mechanisms, I demonstrate information transfer and force measurements by taking advantage of the nonlinear behavior of the resonators. When nonlinear resonators are strongly driven, they exhibit sum and difference frequency generation, in which a large carrier signal can be mixed with a much smaller modulation to produce signals at sum and difference frequencies of the two signals. These sum and difference signals are used to detect information encoded in the modulation signal using optical radiation pressure and acoustic pressure waves. Finally, in my experiments, I probe the nonlinear nature of the piezoelectric material rather than take advantage of the nonlinear resonator behavior. The relative sizes of the linear and nonlinear portions of the piezoelectric constant can be determined because the force applied to the resonator by a transducer is independent of the dielectric constant. This method allowed me to quantify the nonlinear constants

A Large-Stroke Electrostatic Micro-Actuator

Parallel-plate electrostatic actuators driven by a voltage difference between two electrodes suffer from an operation range limited to 30% of the gap that has significantly restrained their applications in Microelectromechanical systems (MEMS). In this thesis, the travel range of an electrostatic actuator made of a micro-cantilever beam electrode above a fixed electrode is extended quasi-statically to 90% of the capacitor gap by introducing a voltage regulator (controller) circuit designed for low frequency actuation. The developed large-stroke actuator is valuable contribution to applications in optical filters, optical modulators, digital micro-mirrors and micro-probe based memory disk drives. To implement the low-frequency large-stroke actuator, the beam tip velocity is measured by a vibrometer, the corresponding signal is integrated in the regulator circuit to obtain the displacement feedback, which is used to modify the input voltage of the actuator to reach a target location. The voltage regulator reduces the total voltage, and therefore the electrostatic force, once the beam approaches the fixed electrode so that the balance is maintained between the mechanical restoring force and the electrostatic force that enables the actuator to achieve the desired large stroke. A mathematical model is developed for the actuator based on the mode shapes of the cantilever beam using experimentally identified parameters that yields good accuracy in predicting both the open loop and the closed loop responses. The low-frequency actuator also yields superharmonic resonances that are observed here for the first time in electrostatic actuators. The actuator can also be configured either as a bi-stable actuator using a low-frequency controller or as a chaotic resonator using a high-frequency controller. The high-frequency controller yields large and bounded chaotic attractors for a wide range of excitation magnitudes and frequencies making it suitable for sensor applications. Bifurcation diagrams reveal periodic motions, softening behavior, period doubling cascades, one-well and two-well chaos, superharmonic resonances and a reverse period doubling cascade. To verify the observed chaotic oscillations, Lyapunov exponents are calculated and found to be positive. Furthermore, a chaotic resonator with a quadratic controller is designed that not only requires less voltage, but also produces more robust and larger motions. Another metric of chaos, information entropy, is used to verify the chaotic attractors in this case. It is found that the attractors have a common information entropy of 0.732 independent of the excitation amplitude and frequency

Nonlinear dynamics and applications of MEMS and NEMS resonators.

Rich nonlinear behaviours have been observed in microelectromechanical and nanoelectromechanical systems (MEMS and NEMS) resonators. This dissertation has performed a systematic study of nonlinear dynamics in various MEMS and NEMS resonators that appear to be single, two coupled, arrayed, parametric driven and coupled with multiple-fields, with the aim of exploring novel applications. New study on dynamic performance of a single carbon nanotube resonator taking account of the surface induced initial stress has been performed. It is found that the initial stress causes the jumping points, the whirling and chaotic motions to appear at higher driving forces. Chaotic synchronization of two identical MEMS resonators has been theoretically achieved using Open-Plus-Closed-Loop (OPCL) method, and the coupled resonating system is designed as a mass detector that is believed to possess high resistance to noise. The idea of chaotic synchronization is then popularized into wireless sensor networks for the purpose of achieving secure communication. The arising of intrinsic localised mode has been studied in microelectromechanical resonators array that is designed intentionally for an energy harvester, which could potentially be used to achieve high/concentrated energy output. Duffing resonators with negative and positive spring constants can exhibit chaotic behaviour. Systematic calculations have been performed for these two systems driven by parametric pumps to unveil the controllability of chaos. Based on the principle of nanomechanical transistor and quantum shuttle mechanism, a high sensitive mass sensor that consists of two mechanically coupled NEMS resonators has been postulated, and the mass sensor which can be realized in large-scale has also been investigated and verified. Furthermore, an novel transistor that couples three physical fields at the same time, i.e. mechanical, optical and electrical, has been designed, and the coupled opto-electro-mechanical simulation has been performed. It is shown from the dynamic analysis that the stable working range of the transistor is much wider than that of the optical wave inside the cavity

Dynamicanalysis of electrostatically exited micro cantilever beam

Microcantilever beams find applications in modern micro and nano devices, such as, microswitches, microvalves and atomic force microscopes (AFM). These are generally fabricated from single crystal silicon. Dynamics and pull-in analysis of these cantilever resonators (Beams) is of vital importance before their fabrication. Microbeams are often actuated by several actuation techniques such as piezoelectric, electromagnetic, thermal and electrostatic excitations. In electrostatic actuation a controlled dc voltage is applied together with a harmonic or suddenly applied ac component. The electrostatic force is in fact a non-linear function of displacement of the beam. In such beams when electrostatic force exceeds the elastic restoring force in the beam contact between the beam and supporting substrate occurs. This situation is referred to as pull-in instability which defines the life of a resonator. The study of pull-in is very important to design process of microsystems, such as, microgyroscopes to analyze the frequency response, the dynamic range and sensitivity. In this line, present work attempts to predict the static and dynamic characteristics of a cantilever microbeam resonator. A single-DOF model incorporating squeeze-film damping forces is first considered. Dynamic pull-in voltages are obtained from time-domain analysis and phase trajectories, for values of ac voltage amplitudes. Parametric studies are carried-out to find the effect of ac voltage amplitudes and air film pressure on the intermediate stability of system. For this bifurcation analysis and Poincare maps are employed. The harmonic voltage frequency effects are also illustrated in the form of frequency-response curves. As stiffness of test structure reduces due to applied voltage, the spring –softening behavior is observed in frequency-response curves. Two benchmark cantilever beam geometries available in terature are considered to illustrate the approach

Energy Transfer into Period-Tripled States in Coupled Electromechanical Modes at Internal Resonance

Efficient energy transfer often occurs between oscillation modes in a resonator when they are tuned to internal resonance. We design the eigenfrequencies of two vibrational modes of an electromechanical resonator to be close to a ratio of 3:1 and demonstrate that the energy supplied to the upper mode can be controllably transferred to the lower mode. With the lower mode vibrating with a period tripled that of the upper mode, the discrete time-translation symmetry imposed by the periodic drive is broken. The lower mode settles into one of three stable period-tripled states with different phases. This channel for energy transfer from the upper mode can be turned on or off without changing system parameters. When the upper mode itself becomes multistable under strong resonant or parametric drive, additional sets of coexisting period-tripled states emerge in the lower mode. In the latter case, we measure a total of 6 coexisting vibration states with identical amplitude but phases differing by $\pi$/3. Excitation of coexisting states with three different phases could open new opportunities in designing mechanical memory based on ternary logic. Coupled resonators with period-tripled states can also be used to model complex interacting systems with spin equals one