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Robust Control for MEMS Gyroscopes
Abstract
MEMS gyroscopes are composed of two perpendicular vibrating modes: the drive and the sense modes. The working principle is based on the transfer of energy between these modes caused by the Coriolis force, which is proportional to the angular rate. By controlling the drive mode oscillations with an excitation frequency and by estimating the Coriolis force, the angular rate can be recovered. Then, the better the drive mode oscillations are controlled and the Coriolis force is estimated, the better is the measure. The control architectures are usually optimized in terms of cost and simple implementation. Most of them are based on the complex envelope (amplitude and phase) of the signals, such that simple PI controllers can be used to independently regulate the amplitude and phase of the oscillations along each axis. To measure the complex envelope, nonlinear elements are introduced in the control loops. Moreover, the couplings between the drive and sense modes, as well as the dependence on environmental conditions, are not considered. The associated methods do not provide guarantees of stability or performance for the closed-loop system.An alternative approach is to consider the classical feedback control architecture, referred to as the direct control architecture, based on the signals themselves instead of their complex envelope. For this architecture, advanced control techniques have been developed for vibration control of mechanical systems. The potential interest is to explicitly consider the different couplings and the dependence on the environmental condition with formal guarantees of stability and performance. Nevertheless, their applicability to MEMS gyroscopes, including implementability, is still an open question. A possible reason is the controller complexity.In this thesis, we aim to propose design methods for both control architectures, guaranteeing stability and a certain performance level for the MEMS gyroscope, and to experimentally validate the obtained controllers. In the first part, we review the MEMS gyroscope literature and define the key performance indicators, which are not usually connected to the closed-loop specifications. Then, by using an input-output approach, we establish the relationships between the performance indicators and the closed-loop behavior. These relationships are a valuable tool for the control design. Based on these relationships, we propose design methods for the direct control architecture. First, we consider the case where the MEMS gyroscope works with a fixed operating condition and the excitation frequency is constant. In this context, the control objectives include the tracking of a sinusoidal signal and the standard H_∞ synthesis is applied for the controller design. However, the excitation frequency may vary over time. A control objective is then to track a “variable-frequency sinusoidal” signal. This particular problem is formulated as a weighted L_2 criterion with a new class of weighting functions modeling “variable-frequency sinusoidal” signals.We then revisit the theory of complex envelopes, which allows us to define a formal framework for the analysis of the envelope-based control architectures. If the complex envelope is ideally measured in real time, we establish links between the direct control approach and the envelope-based ones. These links reveal that the performances achieved with both strategies are equivalent. When the signal envelope is ideally measured, the same framework allows us to precisely model the nonidealities and to design controllers with formal guarantees of stability. The last part is dedicated to the controller design for digital implementation on two platforms: a flexible one, which can implement complex control architectures; and a platform designed for the electro-mechanical ΣΔ, which is a very particular control architecture. For both cases, the practical results validate the proposed methods.
... The control design step develops control strategies to achieve the required specifications. The PhD program carried out by Fabricio Saggin, and supervised by Gerard Scorletti, Xavier Bombois and Anton Korniienko [Sag21], provides systematic control design tools that allows to optimize the MEMS gyroscopes performance while offering formal guarantees of robustness. Next, adaptative control techniques can be necessary to compensate the effect of the temperature on the system behavior. ...
... 5 Then, considering the main control strategies that are studied and developed in the Next4MEMS project, and depending on the localization of the synchronous demodulation in the system, we can distinguish two main classes of Linear Harmonically Time-Varying (LHTV) systems to be analyzed, represented in Figure 1.2. The first configuration considers the so-called direct control strategies [DA09,Sag21], which consists in designing an LTI feedback control loop whose output is sent to the synchronous demodulator in an open-loop configuration. The second case considers the control strategies that require the implementation of the synchronous demodulation inside the closed-loop, such as the phasor-based control [SSK20]; the system is then represented by the interconnection of the LTI part of the system and a set of HTV parameters, which are the effect of the synchronous demodulator in the loop. ...
... Three main approaches, developed in this project, will be analyzed along this document. They are largely documented in the PhD manuscript of Fabricio Saggin [Sag21], and in our previous publications [ACSKS19, SACKS20, SSK20]. They are briefly listed below. ...
Les gyroscopes MEMS sont des micro-capteurs qui mesurent la vitesse angulaire d'un objet par rapport à un référentiel, en estimant la force de Coriolis. L'estimation est obtenue grâce à la commande en boucle fermée des oscillations mal amorties du système ressort-masse couplé à un démodulateur synchrone. En plus d'avantages intéressants (taille, poids, faible consommation d'énergie et faible coût), ils souffrent d'une dispersion de fabrication et d'une sensibilité importante aux changements de température. Les correcteurs sont conçus à partir de modèles fortement simplifiés, sans un niveau de performance certifié. Les performances réelles des MEMS sont ensuite évaluées par des expériences. Ce travail de thèse se concentre sur la validation pré-expérimentale des performances du système de commande conçu, en utilisant des modèles plus réalistes, c'est-à-dire, il s’agit d’un problème d'analyse des systèmes dynamiques. En raison de la démodulation synchrone, le système est modélisé comme un système linéaire avec des paramètres variant dans le temps de façon harmonique (HTV), c'est-à-dire des paramètres qui sont des fonctions sinusoïdales du temps. Nous abordons l'analyse des systèmes LHTV (linéaires et temps-variant harmonique) en adoptant une approche de type IQC (contraintes quadratiques intégrales). Une étape clé pour appliquer le cadre IQC est de caractériser les paramètres HTV par des IQC définies par un ensemble de fonctions appelées multiplieurs. Le choix approprié d'un ensemble de multiplieurs est crucial en ce qui concerne le conservatisme des résultats de l'analyse. Un cas bien documenté dans la littérature est celui des D-G scalings pour des incertitudes paramétriques. Le D scaling a été étendue au cas HTV. Dans cette thèse, nous étudions l'introduction du G scaling au cas HTV, car elle réduit considérablement le conservatisme dans le cas d'incertitudes paramétriques.Un gyroscope MEMS commercialisé doit vérifier des spécifications de précision et de bruit de sortie, définies par des normes. Nous proposons des critères de performance basés sur des modèles afin d'évaluer ces spécifications. La spécification de précision la plus importante est la non-linéarité du facteur d'échelle (SFNL), définie comme l'erreur maximale de mesure du gyroscope pour toutes les vitesses angulaires mesurées. En l'exprimant comme un problème d'optimisation robuste, le calcul de la SFNL est refondu en un problème d'optimisation convexe. L'approche proposée est validée par des résultats expérimentaux. La procédure standard pour évaluer le bruit de sortie des gyroscopes MEMS est la variance d'Allan qui est un outil statistique dans le domaine temporel calculé à partir de longues mesures de la sortie du gyroscope. Cette méthode permet de classer et de quantifier les différents processus stochastiques qui sont contenus dans le bruit de sortie du gyroscope. Afin de dériver un calcul de la variance d'Allan basé sur un modèle, nous adoptons une approche de filtre générateur, qui est à notre connaissance originale. Différents cas sont étudiés, des modèles LTI aux classes de modèles LHTV qui sont pertinents pour l'application aux gyroscopes MEMS, y compris les incertitudes. Les problèmes d'optimisation convexe sont obtenus en utilisant, par exemple, l'approche IQC développée pour les systèmes LHTV. L'approche proposée est validée par des résultats expérimentaux. Enfin, les outils d'analyse de systèmes proposés sont appliqués à la validation de stratégies de commande alternatives qui nécessitent des architectures plus complexes que la commande LTI classique.
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In this work, we study the dynamics of primary oscillations of a high-Q micromechanical resonator - a sensitive element of an RR-type MEMS gyroscope - under the action of various implementations of a phase-locked loop system operating in conjunction with an automatic gain control system for an electrostatic drive. The study of the dynamics of the object is carried out both numerically and analytically - using the averaging method. Conditions for the stability of a stationary regime in a linear approximation are obtained. The questions of accuracy of various methods of numerical solution of differential equations of the circuit of primary oscillations are considered. The influence of the mechanical nonlinearity of the resonator on the dynamics of the resonator and the control system has been studied. An implementation of a low-order PLL circuit that does not contain a double-frequency spurious signal at the output of the phase detector is proposed. The output characteristics of control systems (speed, capture bandwidth, etc.) are analyzed and qualitative conclusions are drawn about the features of the interaction between the dynamics of a mechanical oscillatory link and the PLL-ARC circuit.
The accuracy of micro-electro-mechanical systems (MEMS) gyroscopes is sensitive to variations of the drive mode resonance frequency ω0. To tackle this problem, we propose an H∞ control, which explicitly depends on ω0 and guarantees, with reduced conservatism, a specified level of performance of the drive mode. We consider the design of continuous- and discrete-time controllers. We then propose a method based on the µ-analysis to validate the performance of drive mode control even if the frequency ω0 is not perfectly measured. Numerical examples confirm the effectiveness of our approach.
Mathematical models have a crucial place in every engineering field. Models can be
used for several purposes such as the design of a controller, the prediction, the health
monitoring of a system, etc. Often, the performances of the controller, the accuracy of
the prediction and the ability to detect faults occurring in a system strongly depend
on the quality of the model. System Identification is the scientific field consisting in
the modeling of a system with experimental data collected on this system. Therefore,
it gives a more realistic modeling of the system since it can allow to identify some
dynamics that are not modeled by first principles. This type of modeling relies on three
ingredients: the model structure (set of candidate models), the experimental data and
the identification criterion. These three ingredients must be carefully chosen in order
to guarantee a good quality for the model.
In order to obtain an accurate identified model, it is essential to guarantee the consistency
of the estimate (identified model). This property directly depends on the three
aforementioned ingredients. It guarantees the convergence of the identified model to
the most accurate candidate model available in the model set when the number of data
increases endlessly. When considering the classical Prediction Error estimator (with a
least-square criterion), it has been shown that the consistency is achieved when two
properties are verified: the identifiability of the model structure and the informativity
of the data with respect to the model structure. For the identifiability, a lot of works
have been done to derive linear parametrized model structures having strong identifiability
properties such as, e.g., Box-Jenkins, Output-Error, Finite Impulse Response,
ARX and ARMAX model structures.
For the data informativity, a lot of works have been done for the identification of the
single-input single-output systems within the Prediction Error framework. Necessary
and sufficient conditions have been derived to verify the data informativity with respect
to the classical single-input single-output linear model structures (Box-Jenkins, Output-
Error, Finite Impulse Response, ARX and ARMAX model structures) for the open-loop
and the direct closed-loop identification. However, for multiple-inputs multiple-outputs
systems, the only results for the data informativity available in the literature are only sufficient and can be very restrictive with respect to the experiment design limitations
of the to-be-identified system. Therefore, in this dissertation, we consider the study of
data informativity for linear multiple-inputs and multiple-outputs systems in open-loop
and closed-loop and we develop less conservative conditions for the verification of this
property within the Prediction Error framework.
The second limitation of the data informativity studies available in the literature is
the lack of results for the identification of nonlinear systems which can be accurately
identified within the Prediction Error framework. This limitation is moreover very
restricting in practice since a majority of real-life systems have nonlinear behaviors.
Consequently, in this dissertation, we address the problem of the data informativity for a
particular type of nonlinear model structures with an input monomial nonlinearity. This
particular analysis is motivated in this dissertation by the problem of the identification
of a real-life system with an input monomial nonlinearity: the inertial gyroscope used
for the measurement of angular rate (rotation speed).
This type of gyroscopes presents a lot of advantageous properties such as its cheap
manufacturing price, its low energy consumption and its small size. However, it suffers
from several inaccuracies due to manufacturing imperfections and its instrumentation.
This problem can be tackled by putting the sensor in closed-loop with a controller.
In order to derive an optimal controller, we need an accurate model of the inertial
gyroscope and system identification advanced techniques such as Prediction Error can
solve this problem. However, in the literature, there is no complete model structure
and no identification method for the modeling of the main dynamics in an inertial
gyroscope. Therefore, the last problem addressed in this thesis is the development of
a complete model structure and an identification approach to accurately estimate the
main dynamics of the inertial gyroscope.
When dealing with nonlinear systems, regular notions of stability are not enough to ensure an appropriate behavior when dealing with problems such as tracking, synchronization and observer design. Incremental stability has been proposed as a tool to deal with such problems and ensure that the system presents relevant qualitative behavior. However, as it is often the case with nonlinear systems, the complexity of the analysis leads engineers to search for relaxations, which introduce conservatism. In this thesis, we focus on the incremental stability of a specific class of systems, namely piecewise-affine systems, which could provide a valuable tool for approaching the incremental stability of more general dynamical systems.Piecewise-affine systems have a partitioned state space, in each region of which the dynamics are governed by an affine differential equation. They can represent systems containing piecewise-affine nonlinearities, as well as serve as approximations of more general nonlinear systems. More importantly, their description is relatively close to that of linear systems, allowing us to obtain analysis conditions expressed as linear matrix inequalities that can be efficiently handled numerically by existing solvers.In the first part of this memoir, we review the literature on the analysis of piecewise-affine systems using Lyapunov and dissipativity techniques. We then propose new conditions for the analysis of incremental L2-gain and incremental asymptotic stability of piecewise-affine systems expressed as linear matrix inequalities. These conditions are shown to be less conservative than previous results and illustrated through numerical examples.In the second part, we consider the case of uncertain piecewise-affine systems represented as the interconnection between a nominal system and a structured uncertainty block. Using graph separation theory, we propose conditions that extend the framework of integral quadratic constraints to consider the case when the nominal system is piecewise affine, both in the non-incremental and incremental cases. Through dissipativity theory, these conditions are then expressed as linear matrix inequalities.Finally, the third part of this memoir is devoted to the analysis of uncertain Lur’e-type nonlinear systems. We develop a new approximation technique allowing to equivalently rewrite such systems as uncertain piecewise-affine systems connected with the approximation error. The proposed approach ensures that the approximation error is Lipschitz continuous with a guaranteed pre-specified upper bound on the Lipschitz constant. This enables us to use the aforementioned techniques to analyze more general classes of nonlinear systems.
In February 1851, Léon Foucault published in the Comptes rendus his famous pendulum experiment performed at the "Observatoire de Paris". This ended two centuries of quest for an experimental demonstration of Earth rotation. One month later, the experiment was reproduced at larger scale in the Panthéon and, as early as the summer of 1851, it was being repeated in many places across the world. The next year, Foucault invented the gyroscope to get a still more direct proof of Earth rotation. The theory relied on the masterpiece treatise of Laplace on celestial mechanics, published in 1805, which already contained the mathematical expression of the force that would be discovered by Gustave Coriolis 30 years later. The idea of a fictitious inertial force proposed by Coriolis prevailed by the end of 19th century, as it was conceptually simpler than Laplace's approach. The full theory of the Foucault pendulum, taking into account its unavoidable imperfections, was not obtained until three decades later by Kamerlingh Onnes, the future discoverer of liquid helium and superconductivity. Today, Foucault's exceptional creativity is still a source of inspiration for research and the promotion of science through experimental proofs widely available to the public.
In this work, we propose a systematic and flexible method for designing the electronic filter of electro-mechanical ΣΔ (EM-ΣΔ) feedbacks, widely used for the closed-loop operation of high-performance MEMS gyroscopes. We formulate the filter design as an optimization problem based on the 𝐻∞ norm of weighted closed-loop transfer functions with an appropriate 𝐻∞ criterion. The desired closed-loop system specifications are then expressed through weighting filters, which can be chosen by the system designer. Practical implementations demonstrate the effectiveness of our method. When compared to the results of an established filter, we obtain performance improvements of 30% for the scale factor nonlinearity, 40% for the RMS noise, 35% for the angle-random walk, to cite a few.
This research aims at solving a particular vibration control problem of smart structures. We aim at reducing the vibration in a specific zone of the smart structure under the disturbance that covers a wide frequency band. Moreover, at this specific zone, neither actuation nor sensing is possible.Here we face several main challenges. First, we need to control the vibration of a specific zone of the structure while we only have access to measurements at other zones. Second, the wide bandwidth of the disturbance implies that numerous modes should be controlled at the same time which requires the use of multiple actuators and sensors. This leads to a MIMO controller which is difficult to obtain using classical controller design methods. Third, the so-called spillover problem must be avoided which is to guarantee the closed-loop stability when the model-based controller is applied on the actual setup. To tackle these challenges, we investigate two control strategies: the centralized control and the distributed control.For centralized control, we propose a methodology that allows us to obtain a simple MIMO controller that accomplishes these challenges. First, several modeling and identification techniques are applied to obtain an accurate low-order model of the smart structure. Then, an H_∞ control based synthesis method with a particularly proposed H_∞ criterion is applied. This H_∞ criterion integrates multiple control objectives, including the main challenges. In particular, the spillover problem is transformed into a robust stability problem and will be guaranteed using this criterion. The obtained H_∞ controller is a standard solution of the H_∞ problem. The final controller is obtained by further simplifying this H_∞ controller without losing the closed-loop stability and degrading the performance. This methodology is validated on a beam structure with piezoelectric transducers and the central zone is where the vibration should be reduced. The effectiveness of the obtained controller is validated by simulations and experiments.For distributed control, we consider the same beam structure and the same control objectives. There exist methods aiming at designing distributed controllers of spatially interconnected system. This research proposes a FEM based method, combined with several model reduction techniques, that allows to spatially discretize the beam structure and deduce the state-space models of interconnected subsystems. The design of distributed controllers will not be tackled in this research.