Fabrício Saggin defends his PhD on Sept. 13, 2021 at 2:00 PM.

**Place : ** Ecole Centrale de Lyon, bâtiment W1, Amphi 203 .

**Jury** :

M. Alexandre TROFINO NETO, Professeur, UFSC, Florianópolis, Brésil - Rapporteur

M. Jean-Marc BIANNIC, Directeur de Recherche, ONERA - Rapporteur

Mme. Suzanne LESECQ, Directrice de Recherche, CEA - Examinatrice

M. Jérôme JUILLARD, Professeur, CentraleSupélec - Examinateur

M. Christophe LE BLANC, Ingénieur R&D Électronique, Asygn - Invité

M. Guillaume PAPIN, System Architect, TDK Tronics - Invité

M. Anton KORNIIENKO, Maître de Conférence, École Centrale de Lyon - Encadrant

M. Xavier BOMBOIS, Directeur de Recherche, CNRS - Co-directeur de thèse

M. Gérard SCORLETTI, Professeur, École Centrale de Lyon - Directeur de thèse

**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.

**Keywords : **

Robust control, MEMS gyroscopes, H_∞ synthesis, LPV synthesis, weighting functions, complex envelope, digital control implementation, electro-mechanical ΣΔ.