List of figures

The red arrow represents the direction of actuation. The shaft is fixed by a bearing while the ring is free to rotate around it. One revolution of the contact point will cause the motor shaft to step one tooth [START_REF] Zsurzsan | Control and sensor techniques for PAD servo motor drive[END_REF]. 93

Résumé en français I. Introduction générale

Les moteurs électromagnétiques ont été largement utilisés dans de nombreuses applications d'ingénierie comme actionneurs. Les méthodes de conception, d'optimisation et de fabrication de ces moteurs ont été développées afin d'améliorer leurs performances. Toutefois, en raison des progrès rapides des technologies d'ingénierie, y compris les produits de haute technologie, l'aérospatiale, les équipements médicaux, l'automobile, les robots d'Imagerie par Résonance Magnétique (IRM), des nouvelles exigences des moteurs électriques sont apparues comme la taille réduite, la haute précision, un temps de réponse rapide, une forte densité massique et volumique, et l'insensibilité aux champs magnétiques.

Dans ce contexte où la réponse rapide et silencieuse, l'encombrement réduit, et les bonnes performances dynamiques des actionneurs sont appréciées, les moteurs piézoélectriques trouvent leurs places. Ces moteurs se caractérisent en effet par un couple massique important, des forts couples à basse vitesse sans réducteur, une discrétion acoustique dans le cas des moteurs ultrasonores, et ne sont pas affectés par les interférences magnétiques.

Cependant, les phénomènes non linéaires (hystérésis, creep, vibration, dérive thermique…) qui caractérisent leur fonctionnement et les exigences spécifiques de leurs circuits de commande, mettent un frein à leur utilisation et à leur développement. L'alimentation et le contrôle de ces moteurs soulèvent des problèmes liés au comportement capacitif de leur impédance et à la précision des modèles et aux techniques de contrôle qui leurs sont associées.

Pour faire face à ces problèmes de contrôle des moteurs piézoélectriques, deux démarches sont généralement abordées. La première consiste à utiliser des modèles simples tels que les circuits équivalents pour développer des techniques de contrôle non linéaires comme les réseaux de neurones et la logique floue. La deuxième repose sur la connaissance approfondie des comportements réels des moteurs et de leurs principes de fonctionnement, qui débouche sur l'obtention de modèles non linéaires reflétant leur comportement. Cependant la complexité des modèles ou des techniques de contrôle risque d'aboutir à l'élaboration de lois de commande non-implantables en temps réel.

Dans cette thèse, on traite le problème de contrôle en position des Moteurs Piézoélectriques (MPEs) dans des applications à haute précision, et on vise spécialement la robotique, dans le but d'évaluer la faisabilité de réalisation des systèmes de positionnement précis et robustes à base des MPEs dans des applications robotiques, et de dégager les limites de ces actionneurs dans tels applications. Les contributions de cette thèse peuvent être résumées de la manière suivante :

 L'établissement de modèles fidèles aux comportements des trois MPEs qui puissent être utilisés pour implémenter des lois de contrôle de position. Les trois moteurs étudiés sont :

 Un moteur rotatif ultrasonique à ondes progressives de type USR60 de chez Shinsei.Co.

 Un moteur rotatif quasi-statique de type PAD7220 de chez Noliac.

 Un moteur linéaire pas à pas de type N-310.13 de chez Physik Instrumente (PI)

GmbH.

 La synthèse des contrôleurs de position précis et robustes des PMEs qui répondent aux exigences d'applications spécifiques des trois moteurs étudiés. Deux contrôleurs sont proposés:

 Un contrôleur de type H-infini (H ∞ ) basé sur des filtres de pondérations.

 Un contrôleur en temps discret de type RST basé sur le placement des pôles.

 La garantie des critères de précision, stabilité, et robustesse dans un contrôleur efficace et simple à implémenter en temps réel quelle que soit la topologie du moteur.

 La validation des performances de précision et de robustesse des contrôleurs de position proposés sans faire appel ni à des systèmes de compensation supplémentaires des comportements non linéaires, et ni à des techniques d'adaptation des paramètres des contrôleurs pour surmonter le problème d'incertitude des modèles.

 La conception de bancs de positionnements expérimentaux basés sur les trois PMEs et l'implantation en temps réel des techniques de contrôle développées. Ces techniques sont comparées expérimentalement à un contrôleur PID conventionnel pour mettre en évidence les avantages des méthodes proposées.

 L'évaluation des atouts et des limites des trois moteurs par rapport aux applications robotiques. Les critères d'évaluation sont liés aux topologies des moteurs, aux caractéristiques électromécaniques, aux techniques d'entraînement et aux capacités de positionnement en boucle fermée.

Pour arriver à ces objectifs, une démarche de synthèse des contrôleurs de type H ∞ et RST est d'abord mise en place pour un système généralisé dans le second chapitre. Cette démarche est appliquée aux trois MPEs à travers leurs modèles et en respectant les performances des boucles fermées. Les travaux de modélisation et de contrôle en position des moteurs sont présentés dans les chapitres trois à cinq. Des conclusions générales ainsi que des recommandations liées à la modélisation, au contrôle/commande, et aux performances des moteurs étudiés sont présentées à la fin de cette thèse.

II. Modélisation et Contrôle en position de l'USR60

Le moteur ultrasonique à ondes progressives de type USR60 (Fig. 1) est commercialisé par le constructeur japonais Shinsei.Co. Son stator est constitué d'un disque de bronze sur lequel est collé un anneau de céramique piézoélectrique. L'anneau piézoélectrique est divisé en deux phases A et B. Chacune de ces phases est divisée en petits secteurs avec des polarisations opposées. Le principe de fonctionnement est basé sur la superposition des ondes stationnaires générées par chaque phase pour créer une onde progressive qui se propage tout le long de la surface du stator. Les vibrations du stator vont suivre des trajectoires elliptiques (Fig. 1.c). Ces micro-vibrations du stator sont transformées en mouvement rotatif du rotor par le biais du principe d'entrainement par frottement. Le modèle proposé de l'USR60 décrit le principe de génération de l'onde progressive, et la surface de contact entre stator et rotor. Celui-ci donne la possibilité d'évaluer la variation des grandeurs de sortie du moteur (vitesse, couple, position) en fonction des paramètres de commande, et les effets de variation des paramètres du stator et de la surface de contact sur les performances du moteur. Ce modèle permet aussi de simuler facilement des lois de contrôles (position, vitesse) en utilisant les amplitudes des tensions d'alimentation, la fréquence, et le déphasage comme grandeurs de contrôle.

Le modèle de l'USR60 est principalement composé des trois sous-systèmes montrés dans la Fig. 2 :

 Stator : Dans cette partie on modélise les processus de vibration de l'anneau et la génération de l'onde progressive (W, W max ).  Mouvement vertical de rotor : l'évolution temporelle de la position verticale du rotor est déterminée dans cette partie ainsi que la longueur de surface de contact entre le stator et rotor (x 0 ). Ce dernier paramètre va être utilisé pour le calcul du couple généré par le moteur (T).  Mouvement angulaire de rotor : Le mouvement angulaire ainsi que la vitesse de rotation (Ω) sont analysés dans ce sous-ensemble. Le modèle de l'USR60 est développé sur MATLAB/Simulink comme le montre la Fig. 3. La variation de la vitesse de rotation (Ω) en fonction des paramètres de commande (amplitude de tension, fréquence, déphasage) est simulée afin de choisir la meilleure grandeur de contrôle pour notre système de positionnement. Comme illustré dans la Fig. 4, le déphasage entre les deux tensions d'alimentation (φ) est le seul paramètre qui permette de faire tourner le moteur dans le deux sens. Cette grandeur (φ) va être donc utilisée comme signal de control de position de l'USR60 bien que la caractéristique vitessedéphasage présente une zone morte autour du déphasage nul quand un couple de charge est appliqué. Dans le modèle du PAD, on commence par la modélisation des actionneurs multicouches afin d'obtenir les déformations linéaires orthogonales dans les axes x, et y (Fig. 11). La surface de contact entre les actionneurs et l'anneau du stator est ensuite prise en compte, et la transmission de mouvement par engrenage est introduite en dernière étape. Ce modèle permet la simulation d'évolution des grandeurs de sortie du moteur en boucle ouverte (Fig. 12), et peut aussi servir pour tester des lois de contrôle en boucle fermée.

Un banc de positionnement expérimental (Fig. 13) est mis en place pour déterminer les caractéristiques de PAD7220 en boucle ouverte et pour valider les techniques de contrôle en boucle fermée. Une FT entre la position angulaire du moteur et la fréquence de commande est aussi identifiée expérimentalement : Un banc de positionnement expérimental pour le N-310.13 (Fig. 17.a) est mis en place pour identifier tout d'abord la FT entre le signal de contrôle (V c ) et la position de la tige (Y). Ensuite, ce banc va servir pour valider expérimentalement les lois de contrôle. La FT est identifiée à partir de la courbe de variation de vitesse en boucle ouverte suite à un échelon de fréquence (Fig. 17.b). Cette FT peut s'écrire sous la forme suivante :

( ) 2 1 ( ) ( ) 320 PAD rad G s f Hz s     (2)
(b) (a) (a) (b) (c) (a) (b) (c) ( ) (1 ) w WPZM c w k Y G s V s s     (3)
Avec k w = 1.2 mm/V, et τ w =0.15 s Une comparaison expérimentale entre H ∞ , RST, et PID est abordée pour le positionnement du moteur piézoélectrique linéaire. Les critères de comparaison sont, la précision en présence des diffèrents couples de charges, et à différentes vitesses de déplacements. La Fig. 18 montre les résultats de trois contrôleurs quand le moteur est soumis à des forces allant jusqu'à 7.5N. Le H ∞ garantit la haute précision quelle que soit la force appliquée, alors que pour les deux autres, on remarque que la précision et le temps de réponse se dégradent en augmentant la force exercée. Le H ∞ présente aussi une forte robustesse face au changement de vitesse de déplacement par rapport aux deux autres contrôleurs (Fig. 19).

V. Conclusion

Dans cette thèse, la problématique de contrôle en position des moteurs piézoélectriques est traitée, afin de valider la faisabilité de réalisation des applications robotiques à haute précision à base des MPEs. Pour arriver à ce but, trois moteurs avec différentes technologies sont testés.

Une démarche de modélisation de ces moteurs a été mise en place, ces modèles permettent à la fois la vérification des comportements des moteurs et l'implémentation des lois de contrôle.

Deux types de contrôles sont ensuite synthétisés, un en temps continu de type H ∞ avec des filtres de pondérations, et l'autre en temps discret de type RST.

Trois bancs expérimentaux à la base de ces trois moteurs sont mis en place afin d'implémenter en temps réel les lois de contrôle développées. Ces bancs ont tourné en boucle ouverte dans un premier temps pour évaluer les caractéristiques des moteurs et pour identifier les fonctions de transferts. Les contrôleurs synthétisés sont ensuite implantés en boucle fermée moyennent une carte dSPACE. La précision et la robustesse des systèmes de positionnements à base des MPEs ont été testées sous différentes conditions de fonctionnement.

Ces moteurs piézoélectriques contrôlés en position par le H ∞ ont montré une haute précision quelle que soit la charge appliquée (jusqu'à la valeur maximale) et la vitesse de mouvement, sans avoir besoin d'algorithmes d'adaptation des paramètres. La robustesse face au comportement non-linéaire des moteurs et aux bruits injectés est aussi confirmée sans faire appel à des systèmes de compensation supplémentaires.

Introduction

Traditional electrical motors based on electromagnetic principle have been widely used in many engineering applications as actuators and mechanical power sources. The design, optimization and manufacturing methods of these motors have been developed in order to improve their performances. However, due to the rapid progress of engineering technologies including the high-tech products, aerospace, medical equipment, automotive, semiconductor technologies, and smart robots, new requirements of electric motors appeared such as small size, light-weight, ultra-high precision, fast response, high torque density, and no electromagnetic interferences. Due to the structure and operating principle of electromagnetic motors, it becomes difficult to meet these requirements.

In parallel, the intensive worldwide research efforts in smart materials based actuators and related drive circuits promote the integration of new type motors. The piezoelectric motors are relatively mature among these innovative motor concepts. Where, since the discovery of the piezoelectric effect in 1880, the piezo-system technologies have made rapid progress in terms of reliability, efficiency, and compactness. The piezo-motors benefit from the direct drive principle using the converse piezoelectric effect to achieve interesting performances required in advanced engineering fields. They are well classified to actuate ultra-precision positioning system, where the quick response, small size, and selflocking criteria are important.

As an introduction, this chapter presents an initiation to the piezoelectric technology including the operating principles of piezo-actuators and motors, and the nonlinear behaviors of these devices. The piezo-motors are classified based on various criteria, and the motion generation principle of each category is explained. A survey of the industrial applications of piezo-motors is proposed in order to highlight the importance of these actuators in positioning applications. Finally, the thesis contributions and outlines are highlighted.

I.1 Objectives and scope of the research

In the recent decades, the piezoelectric motors (PZMs) are widely suggested for engineering applications where the high precision, fast response, and compact size requirements are very important. This is mainly due to the electromechanical characteristics of these motors such as the small size, fast response, theoretical unlimited resolution, self-locking, high torque density, and high torque at low speed without gear.

These features make the PZMs a very competitive candidate to the electromagnetic motors for industrial fields and especially robotic applications. In parallel to the precision and fast response potentials of PZMs, let us explain the advantage of high holding torque of PZM without supply. When actuating a robot arm by electromagnetic motor, to avoid the returning to initial position or laying down of the arm in case of off-power, a fail-safe brakes system is applied to stop and hold. These fail-safe brakes systems are generally based on permanent-magnetic and spring-set systems. This additional mechanism leads to an increase of system size and moment inertia, and in consequence slower response time. Due to the driving principles of PZMs, they benefit of their high holding torque without supply to maintain the final position without braking system. In other hand, the non-magnetic characteristic makes PZMs as a promising solution to actuate the Magnetic Resonant Imagery (MRI) robots for surgical operations.

Nevertheless, the insertion rate of PZMs into robotic applications do not reflect the real potentials of these motors. These integration difficulties are due to many factors that can be resumed in; the highly nonlinear behavior especially when operating close to the resonance, several sources of nonlinearities in different frequencies operations ranges (hysteresis and creep in low frequency, and vibration in high frequency), the requirement of specific drive system for high capacitive loads (PZMs), and the timevariant motor parameters due to the sensitivity to heating, load torque and electrical noises. Therefore, complexes, time consumption, and adaptive techniques are generally addressed to control the position of PZMs in closed loop operation mode, where an on-line parameters adaptation and additional compensation systems are required to overcome the motor nonlinear behaviors. The application of these methods in practical engineering applications remains difficult.

This thesis aims to study the feasibility of the design and implementation of robust position control of PZMs with different topologies. The control technique should meet the specific requirements of the PZMs robotic applications in terms of accuracy and robustness, and fulfill the simplicity criteria for practical implementation issues. This goal will be achieved through the establishment of electromechanical models of different topologies of PZMs for control purposes. Afterwards, the design and real time implementation of simple, precise, and robust position controllers of PZMs. The strengths and limitations of each piezo-positioning system in terms of control performance and the motor behaviors in open/closed loop operation conditions will be evaluated. This is in order to collect the most required data to design an "ideal" piezoelectric motor for robotic applications as perspective of this research work.

The piezoelectric motors that will be in the scope of this research are in the mechanical power range between 1 and 20 Watts. The output motion is in the range up to 100 mm for linear motors and bidirectional motion for rotary motors. These motors are mainly used to actuate MRI surgical robots, to adjust the driver's seat in automobiles, and for silent and smooth window operation of a car door. Therefore, in this thesis we will not be interested to the nano/micro-motors nor to the micro-robots based on PZMs.

I.2 Initiation to piezoelectric actuators and motors

I.2.1 Direct and converse piezoelectric effects

The origin of the world piezoelectric is derived from the Greek world "piezo" which means"press". The piezoelectric effect is the property of some materials to develop electric charge on their surface when a mechanical stress is exerted on them [START_REF] Ikeda | Fundamentals of Piezoelectricity[END_REF], and it was discovered in 1880 by the brother Pierre and Jacques Curie. There are many materials that exhibit a piezoelectric effect, and the lead zirconate titanate (PZT) is the most widely used material. The electrical response of these materials to the mechanicals effort is called direct piezoelectric effect and it is the principle of piezoelectric sensors. The same material can also generate a mechanical stress when an electric field is applied, and this mechanical response is called converse piezoelectric effect (Fig. I-1). The converse piezoelectric effect is at the origin of creation of the piezoelectric actuators.

I.2.2 Principle of piezoelectric actuator and motor

As aforementioned, the piezoelectric actuator (PZA) is based on the converse piezoelectric effect. The major drawbacks of PZAs is their small stroke, Amplified Piezoelectric Actuators (APA) [START_REF] Claeyssen | Amplified piezoelectric actuators: Static & dynamic applications[END_REF] and multilayer bending actuators [START_REF] Ballas | Piezoelectric multilayer beam bending actuators: Static and dynamic behavior and aspects of sensor integration[END_REF] are proposed to increase the actuators full stroke at the expense of stiffness and actuation force. However, the full stroke stills in the range of a few millimeters, even with such structures. The piezoelectric motor (PZM) consists on the conversion of the limited displacements of PZA into unlimited linear or rotational motion using different mechanical structures and drive principles. The PZMs can be classified into many categories based on the motion type, operation principle, and frequency operation range. The classification and the operation principle of PZM categories will be discussed in the section I.3.

I.2.3 Nonlinearities of piezoelectric actuators

As aforementioned, the piezoelectric motor is mainly composed by one ore many piezoelectric actuators emerged into a mechanical structure to produce unlimited motion. Thus, the nonlinear characteristics of the PZA outputs will generally affect the PZM performances. In this section, an overview of the common nonlinearities of PZAs will be discussed.

The strain of PZA is not only determined by the driving parameters (voltage amplitude, frequency, and phase shift), but it depends on many factors. These factors include the mechanical boundary conditions, especially on the pre-stress of the actuator, thermal drift, and self-heating [START_REF] Hunstig | Piezoelectric Inertia Motors-A Critical Review of History, Concepts, Design, Applications, and Perspectives[END_REF]. The PZA displacement shows also several nonlinear behaviors caused principally by the hysteresis, creep, and vibration phenomena.

I.2.3.1 Hysteresis

Hysteresis is a nonlinear behavior between the applied electric field (u) and the mechanical displacement of a piezoelectric actuator (x) as shown in Fig. I-2. This phenomenon is the cause of irreversible losses that occur when similarly oriented electric dipoles interact upon application of an electric field [START_REF] Jiles | Theory of ferromagnetic hysteresis[END_REF]. The hysteresis effect on the displacement of a piezoelectric actuator is more significant over large range motion. In general, the maximum error caused by the hysteresis can reach 15% of the travel range of the PZA. With the increase of driving frequency, the hysteresis caused errors may go beyond 35% [START_REF] Gu | Modeling and control of piezoactuated nanopositioning stages: a survey[END_REF].

Fig. I-2. Hysteresis between the applied voltage (u) and the actuator displacement (x)

To treat the hysteresis, many efforts have been made in the literature. The proposed solutions can be classified into feedforward and feedback control techniques. The feedforward techniques (also called inversion-based controls) consist on the modeling of the hysteresis nonlinear behavior, and in second time to apply the inverse model in order to compensate the hysteresis effect (Fig. I-3). Good tracking results can be achieved if the plant model and its frequency response are known with high accuracy [START_REF] Fleming | Design, modeling and control of nanopositioning systems[END_REF]. As a consequence several models are proposed to deal with the hysteresis phenomenon. Roughly speaking, the hysteresis model of piezo-actuated stages can be classified into physics-based and phenomenological models [START_REF] Gu | Modeling and control of piezoactuated nanopositioning stages: a survey[END_REF]. An overview of the hysteresis models of actuated piezo-stages is given in Fig. I-4, and the principle of each modeling approach can be found in [START_REF] Gu | Modeling and control of piezoactuated nanopositioning stages: a survey[END_REF][START_REF] Fleming | Design, modeling and control of nanopositioning systems[END_REF][START_REF] Ru | Nanopositioning Technologies: Fundamentals and Applications[END_REF][START_REF] Visintin | Differential models of hysteresis[END_REF]. However, it is demonstrated that, until now, there is no general reliable model to completely represent the hysteresis behaviors of piezo-actuated stages [START_REF] Gu | Modeling and control of piezoactuated nanopositioning stages: a survey[END_REF].

The second control approach of the hysteresis in piezo-stages is the feedback control method as illustrated in Fig. I-5. In this configuration, the hysteresis nonlinearity is treated as a nonlinear bounded disturbance or characterized by the aforementioned hysteresis models [START_REF] Gu | Modeling and control of piezoactuated nanopositioning stages: a survey[END_REF].Then, different feedback controllers can be applied to compensate the hysteresis effect [START_REF] Fleming | Design, modeling and control of nanopositioning systems[END_REF]. The major drawback of these techniques is the limitation of the closed loop bandwidth of the positioning system in presence of highly resonant modes.

The feedforward and feedback controllers can be combined to deal with the compensation of hysteresis nonlinearity in piezo-actuated stages as shown in Fig. I-6. With this methodology, the inverse compensators with different models are applied in the feedforward part and the feedback controllers are simultaneously designed in the feedback loop to mitigate the effect of the inversion errors and to handle the remaining dynamics of the systems [START_REF] Gu | Modeling and control of piezoactuated nanopositioning stages: a survey[END_REF]. There is also a hardware solution to reduce the hysteresis nonlinear behavior of piezoelectric actuators which is the use of charge amplifier rather than voltage amplifier. By using charge drivers and through the regulation of the piezoelectric current or charge, significant reduction of hysteresis effect can be achieved [START_REF] Ge | Tracking control of a piezoceramic actuator[END_REF][START_REF] Fleming | Charge drive with active dc stabilization for linearization of piezoelectric hysteresis[END_REF][START_REF] Fleming | Quantitative scanning probe microscope topographies by charge linearization of the vertical actuator[END_REF]. However, the capacitive behavior of piezoelectric actuators complicates the design of charge driver due to the finite output impedance and dielectric leakage [START_REF] Fleming | Design, modeling and control of nanopositioning systems[END_REF].

I.2.3.2 Creep

When a voltage step change is sent to drive a piezoelectric actuator, the response consists on a fast dynamic transient followed by low frequency drift (Δy) of the final position known as creep (Fig. I-7).

As shown in Fig The creep effect can be avoided by driving the piezo-stage at high speed, but this operation limits the use of piezoelectric positioning systems in slow and static applications. As done with the other nonlinearities sources, feedback [START_REF] Schitter | High performance feedback for fast scanning atomic force microscopes[END_REF][START_REF] Leang | Feedback-linearized inverse feedforward for creep, hysteresis, and vibration compensation in AFM piezoactuators[END_REF][START_REF] Gu | Motion control of piezoceramic actuators with creep, hysteresis and vibration compensation[END_REF] and feedforward [START_REF] Jung | Tracking control of piezoelectric actuators[END_REF][START_REF] Rakotondrabe | Complete open loop control of hysteretic, creeped, and oscillating piezoelectric cantilevers[END_REF] controllers are proposed to compensate the creep effect. To describe this phenomenon and to apply the compensation control methods, creep models are reported in the literature. One of them is usually called logarithmic model, and it can be expressed by the following equation [START_REF] Gu | Modeling and control of piezoactuated nanopositioning stages: a survey[END_REF]:

1 I-1
Where y(t) is the displacement of the piezo-actuator when subjected to a fixed input voltage, t 0 is the time at which the creep effect is obvious, y 0 is the displacement at the time of t 0 after applying the input voltage, and γ is a coefficient determining the rate of the logarithmic response. Generally, y 0, t 0, and γ are identified experimentally.

I.2.3.3 Vibrational dynamics

The dynamic vibration of piezoelectric devices is a relatively high frequency nonlinear behavior. The vibration effects are mainly caused by exciting the resonance modes of the piezo-actuated stage. Due to the high stiffness and low structural damping ratio of piezoelectric actuators [START_REF] Gu | Modeling and control of piezoactuated nanopositioning stages: a survey[END_REF], a sharp peak of the damped structure emerges in the frequency response of the piezo-actuated stages as shown in Fig. I-8.a.

The vibrational dynamics have a low-gain margin problem because of the rapid phase drop associated with the small structural damping ratio and hysteresis nonlinearity [START_REF] Clayton | A review of feedforward control approaches in nanopositioning for high-speed SPM[END_REF][START_REF] Gu | Motion control of piezoelectric positioning stages: modeling, controller design, and experimental evaluation[END_REF]. In consequence, the high frequency input signals can easily excite the motion vibration and tracking errors will be obtained as shown in Fig. I-8.c. Thus, the dynamic vibration is the main factor limiting the bandwidth of piezopositioning stages. To overcome this problem, several vibrational dynamic models are proposed. The concept consists of neglecting the hysteresis effect, and linear dynamic equations are identified using the input and output data of the piezoelectric system [START_REF] Gu | Modeling and control of piezoactuated nanopositioning stages: a survey[END_REF].

The common feedforward control of vibration is the model-based control method by inverting the linear dynamic model of the piezo-actuated stage and without considering the hysteresis effect [START_REF] Clayton | A review of feedforward control approaches in nanopositioning for high-speed SPM[END_REF][START_REF] Croft | Creep, hysteresis, and vibration compensation for piezoactuators: atomic force microscopy application[END_REF][START_REF] Croft | Vibration compensation for high speed scanning tunneling microscopy[END_REF].

The input shaping method is alternatively proposed to deal with the dynamical vibrations problems [START_REF] Schitter | Design and input-shaping control of a novel scanner for high-speed atomic force microscopy[END_REF][START_REF] Chang | Analysis and control of monolithic piezoelectric nano-actuator[END_REF]. The notch filter [START_REF] Gu | Motion control of piezoceramic actuators with creep, hysteresis and vibration compensation[END_REF], integral resonant [START_REF] Gu | Integral resonant damping for high-bandwidth control of piezoceramic stack actuators with asymmetric hysteresis nonlinearity[END_REF], and positive feedback [START_REF] Mahmood | Making a commercial atomic force microscope more accurate and faster using positive position feedback control[END_REF] controllers are also tested to compensate the dynamical vibration effects and to increase the piezo-stage bandwidth.

I.3 Classification of piezoelectric motors and operating principles

The piezoelectric motors can be classified according to different criteria including the motion type (linear, rotational), frequency operation range, stator geometric, and operating principle [START_REF] Hunstig | Piezoelectric Inertia Motors-A Critical Review of History, Concepts, Design, Applications, and Perspectives[END_REF][START_REF] Ru | Nanopositioning Technologies: Fundamentals and Applications[END_REF][START_REF] Spanner | Piezoelectric motors, an overview[END_REF][START_REF] Prisacariu | A general view on the classification and operating principle of piezoelectric ultrasonic motors[END_REF][START_REF] Uchino | Piezoelectric ultrasonic motors: overview[END_REF]. In this report, the PZMs are classified in sub-categories in order to integrate the most used criteria as shown in The operating principle of the inertia motors is based on the stick-slip drive method, while the commercialized stepping PZMs can be devised into inchworm and walking principle motors. The ultrasonic motors (USM) are classified as function of the wave propagation method into traveling and standing waves USM.

I.3.1 Quasi-static piezoelectric motors

The quasi-static motors are the piezoelectric motors driven in a frequency range much lower than their mechanical resonance frequencies. These types of motors are principally characterized by the theoretically unlimited resolution at low speed without gear. The quasi-static PZMs are driven based on the inertia and steeping methods. In the following two sub-sections, the characteristics, and operating principles of the two PZM categories will be reviewed.

I.3.1.1 Inertia motors

Compared to the others PZMs, the inertia motors have a simply mechanical design, and are typically driven by one electrical voltage. These characteristics favorite the miniaturization of inertia motors for very compact positioning systems.

The inertia motors use the inertia of the driven part to drive it through an uninterrupted friction contact to generate motion [START_REF] Hunstig | Piezoelectric Inertia Motors-A Critical Review of History, Concepts, Design, Applications, and Perspectives[END_REF]. These motors are driven by asymmetric voltages to create the transition between the static (stick) and sliding (slip) friction, which explains the name "stick-slip" given to inertia PZMs. There are two basic functional principles of inertia motors, one with fixed actuator, and the second with moving actuator (called also impact drive motor). The operating principles of the two types of inertia motors are illustrated in Fig. I-10.

In the inertia drive PZM incorporating fixed actuator two movement stages can be distinguished. As shown in Fig. I-10.a, the actuator expands slowly and the slider follows this movement due to static friction. Next, the actuator contracts rapidly so that the slider is unable to follow this movement and lets the actuator slip back due to its own inertia and the not sufficiently high dynamic friction. The motion direction is determined by the mode of the slow movement (expansion or contraction). The inertia drive PZM based on moving actuator (impact drive motor) is typically composed by three main parts: the main body (m 2 ), the actuator, and the inertia weight (m 1 ). The main body is in frictional contact with a guiding surface, the actuator and the weight do not touch this surface (Fig. I-10.b). When the actuator expands (or contracts) slowly, the inertial weight moves with it, while the main body maintains its position due to the high friction force compared to the inertia force. This expansion is stopped suddenly to overpass the static friction, and immediately a rapid contraction of the actuator follows which causes impulsive inertia force to be exerted on the main body. As consequence, the main body follows the motion direction of the actuator. At the end of this cycle, one motor step motion is generated. In order to change the motion direction, the slope of the driving voltage will be inverted (rapid expansion and slow contraction). The first experimentally validated inertia PZMs are proposed firstly by Pohl in 1986 [START_REF] Pohl | Dynamic piezoelectric translation devices[END_REF] and by Higuchi et al. [START_REF] Higuchi | Apparatus for effecting fine movement by impact force produced by piezoelectric or electrostrictive element[END_REF]. There are others little different topologies of inertia motors derived from the two aforementioned principles, which are reported in the literature [START_REF] Hunstig | Piezoelectric Inertia Motors-A Critical Review of History, Concepts, Design, Applications, and Perspectives[END_REF][START_REF] Spanner | Piezoelectric motors, an overview[END_REF][START_REF] Spanner | Survey of the various operating principles of ultrasonic piezomotors[END_REF].

Piezoelectric motors

I.3.1.2 Stepping motors

The piezoelectric motors based on stepping mode are also called clamping-based motors. As the name reflects, these motors are based on the succession of clamping and unclamping movements. Depending on the operating principle and the employed moving parts, two categories of stepping PZMs can be distinguished, which are the inchworm and walking PZMs. This kind of motors are characterized by the unlimited resolution with high force capability, and it suffers from the low speeds.

 Inchworm PZMs

The first clamping-type PZM working with inchworm principle was developed and commercialized by Burleigh Instruments, Inc. in 1974 [START_REF] May | Piezoelectric electromechanical translation apparatus[END_REF]. The basic structure of inchworm motor consists of three piezoelectric actuators; two are used for clamping and one for extension [START_REF] Salisbury | Design, modeling, and closedloop control of a complementary clamp piezoworm stage[END_REF][START_REF] Tenzer | A systematic procedure for the design of piezoelectric inchworm precision positioners[END_REF]. In order to obtain continuous motion and higher forces, similar designs of clamping motors with five actuators are proposed [START_REF] Li | Design and development of a new piezoelectric linear Inchworm actuator[END_REF][START_REF] Frank | Design and performance of a high-force piezoelectric inchworm motor[END_REF]. The topology and the clamping sequences of the inchworm motor are illustrated in

Moving actuator type

The step size of the inchworm motor depends on the maximum strain of the extensional actuator, and the no-load speed is determined directly by the frequency of the clamping movements. The motion direction of the motor can be reversed by inverting the clamping sequences.

 Walking PZMs

The Walking Piezoelectric Motor (WPZM) is a modified clamping-type PZM. The WPZM was proposed for positioning in a large range of motion and with higher speeds than the inchworm motors. It consists of several piezoelectric drive elements (legs) engaging and disengaging in a frictional contact with a movable rod.

The motor proposed by Brisbane in 1965 [START_REF] Brisbane | Position control device[END_REF] (Fig. I-12) is one of the earlier PZMs working with the walk drive principle. It consists of two piezoelectric disc shaped plates (4 and 5) acting as clamping actuators and attached to a moving part which is a hollow cylinder [START_REF] Gu | Modeling and control of piezoactuated nanopositioning stages: a survey[END_REF]. Similar structure of WPZM employing multilayer piezoelectric actuators as clamping elements was presented by Galutva et al [START_REF] Galutva | Device for precision displacement of a solid body[END_REF]. A detailed description of the working principles of the two aforementioned motors is highlighted in [START_REF] Spanner | Piezoelectric motors, an overview[END_REF]. The design and the structure of the WPZMs are widely ameliorated through the research efforts. Actually, this kind of motors is commercialized by many piezoelectric devices companies, such as the Swedish company Piezo Motor Uppsala AB, and the German company Physik Instrumente (PI) GmbH.

A linear WPZM is studied in this thesis for modeling and position control purposes. Thus the operating principle and the motor topology will be explained in details in Chapter V.

I.3.2 Ultrasonic piezoelectric motors

The Ultrasonic Motor (USM) is a piezoelectric motor driven by a high frequency (above 20 kHz) voltage at one of its mechanical resonant frequencies. As the working principle is based on the converse piezoelectric effect, the applied voltages will be converted by the piezoelectric ceramics into resonant vibrations. These vibrations are transmitted to a movable part (rotor) often through frictional contact, leading to a rotational or linear motion. The working principle of USM can be described by the process given in Fig. I-13:  Step 1: The electrical energy is converted into ultrasonic vibrations by the piezoelectric materials mounted on the stator.  Step 2: The resonant vibrations will be amplified and transmitted to the rotor through friction coupling to create motion. The first USM was introduced by Lavrinenko et al in 1965 [START_REF] Lavrinenko | Electrical Motor[END_REF], and it consisted of piezoelectric plate excited at the ultrasonic range and pressed against a smooth rotor (Fig. I-14.a). This invention marked the beginning of research works related to ultrasonic motors, and similar concept to Lavrinenko motor was developed in [START_REF] Vishnevsky | Piezoelectric motor structures[END_REF]. In 1973, a practical USM was proposed by H.V. Barth of HBM [START_REF] Barth | IBM Tech[END_REF]. As shown in Fig. I-14.b, the rotor was pressed against two horns placed at different locations. By exciting one of the horns, the rotor was driven in one direction, and by exciting the other horn, the rotation direction was reversed [START_REF] Spanner | Survey of the various operating principles of ultrasonic piezomotors[END_REF]. The development of piezoelectric USM was significantly increased in the 1980s, due to the increase demand of more precise and sophisticated positioners which were not affected by magnetic fields, particularly by semiconductor industries. The USMs can be classified depending on many criteria, such as the wave propagation method (traveling wave, standing wave), motion type (rotational, linear), geometric shape of the stator (Disk, Ring, Bar…), contact state (contact, non-contact), and rotor directions (unidirectional, bidirectional) [START_REF] Ru | Nanopositioning Technologies: Fundamentals and Applications[END_REF][START_REF] Spanner | Piezoelectric motors, an overview[END_REF][START_REF] Uchino | Piezoelectric ultrasonic motors: overview[END_REF][START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF]. In this thesis report, the USMs will be classified based on the wave propagation method into traveling and standing waves.

I.3.2.1 Standing waves ultrasonic motors

The Standing Wave Ultrasonic Motor (SWUSM) is also called vibratory-coupler motor consists basically on a vibratory element driven close to its resonance frequency and generating elliptical motion of its tip portion. This motion is transferred to the rotor as a series of microscopic pushes. The SWUSM is generally a low cost motor (one vibration source) and has high efficiency, but its control in bidirectional motion is little complex [START_REF] Uchino | Piezoelectric ultrasonic motors: overview[END_REF]. In fact, the bidirectional motion of SWUSM can be obtained by employing two separate vibrating elements excited with a phase shift, or by superimposing two oscillations in a single resonator (bimodal motor). Among the earlier structures of SWUSM, the simple design of unidirectional motor proposed by Sashida (Fig. I-15) [START_REF] Sashida | Trial construction and operation of an ultrasonic vibration driven motor[END_REF]. It consists on a piezoelectric vibrator attached to a slider with a cant angle. When the piezoelectric element is excited, the vibrating piece will generate bending. If the bending deformation is sufficiently small compared to the piece length, elliptical motion of the piece tip will be created. This motion will be transmitted to the slider through friction contact. Endo et al [START_REF] Endo | Linear type ultrasonic motor using two-dimensionally positioned piezoelectric elements[END_REF] proposed a bidirectional SWUSM which employs two quadratic oscillations of piezoelectric elements in order to create elliptical motion of the common tip. The motion direction of the motor can be reversed by switching the driving voltages. An interesting design of bidirectional rotational SWUSM using two orthogonal bending modes of a hollow cylinder was developed by Uchino

I.3.2.2 Traveling waves ultrasonic motors

The Traveling Waves Ultrasonic Motor (TWUSM) was invented by Sashida in 1982 [START_REF] Sashida | Motor device utilizing ultrasonic oscillation[END_REF] and commercialized one year later by Shinsei Corporation [50] in a serie of USR motors. In this type of motor, the elliptical trajectory of the contact point between the stator and rotor results from a traveling wave. This Traveling Wave (TW) is formed by the superposition of two standing waves with a phase shift of 90° both in time and space. The TWUSM can be easily driven in bidirectional motion by switching the driving voltages phase shift between +/-90°. The original invention of TWUSM was a rotary motor and it has had a great success in the lenses drive of autofocus camera and watches. The structure of rotary TWUSM and its working principle will be explained in details in Chapter III, where an example of TWUSM motor will be studied for modeling and position control purposes.

Linear TWUSM was also proposed by Sashida in 1985 in two configurations; straight beam and ring beam types [START_REF] Sashida | Introduction to ultrasonic motors[END_REF]. The prototype of straight beam TWUSM is shown in Fig. I-18.a, it consists of two langevin vibrators fixed on both ends of the beam. When the vibrators are exited, one will act as vibrator and the second as absorber to prevent the reflection of the TW [START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF]. The generated TW will transmit through friction contact in order to move the slider pressed against the beam. TWUSM with ring-type stator was proposed by Hermann et al. [START_REF] Hermann | Piezoelectric travelling wave motors generating direct linear motion[END_REF] as shown in Fig. I-18.b. In this case, the TW is generated by the piezoelectric ceramics bonded inside the ring beam. The slider will perform linear motion when it is pressed against the linear portion of the beam. Many attempts to create linear TWUSM are also reported in [START_REF] Takano | Characteristics of the ultrasonic linear motor using radial and nonaxisymmetric vibration modes of an annular plate[END_REF][START_REF] Tominaga | An ultrasonic linear motor using ridge-mode traveling waves[END_REF].

I.4 Piezoelectric motor applications

Piezoelectric motors are increasingly used in various engineering applications where the accuracy, efficiency, miniaturization, and multi degrees of motion are required. In this section, a brief review of the PZMs applications in different engineering fields will be highlighted. Before starting this review, let us summarize the merits and demerits of PZMs in the table below: The compact size, fast response time, and self-locking at rest position make PZMs an attractive solution for focusing, zooming, and image stabilization in cameras. These features make also PZMs suitable candidate for the adjustment of lens mirrors of optical equipment. Actually, all major cameras, cellular phones, and lens manufacturers use PZMs especially the inertia and ultrasonic motors [START_REF] Spanner | Piezoelectric Motors and Their Industrial Applications[END_REF]. Canon realized the world's first practical application of an USM in auto focus lenses and fast zoom drive of digital camera [56] (Fig. I-19.a). Olympus is using multi-mode excitation type motor which employs multilayer vibrating element [START_REF] Spanner | Piezoelectric Motors and Their Industrial Applications[END_REF]. Samsung electro-mechanics in collaboration with the Penn State University developed in 2003 a zoom and focus mechanism for cellular phone using two rotary USMs [START_REF] Uchino | Piezoelectric motors for camera modules[END_REF]. Another camera model is also proposed by Samsung electro-mechanics employing linear multilayer USM. The Korean manufacturer designed also various miniature PZMs for lens positioning applications [START_REF] Spanner | Piezoelectric Motors and Their Industrial Applications[END_REF]. The New Scale Technology developed many solutions for optical and autofocus camera based on compact USMs (squiggle motor). This motor is employed also in an optical zoom module for higher resolution in smaller devices (Fig  Robotic applications

In addition to the precision and force capabilities, the high holding torque without supply and nonmagnetic features make PZMs to be a suitable candidate for robotic application and especially to actuate Magnetic Resonance Imagery (MRI) and surgical robots. The ultrasonic and walking piezoelectric motors are the major type of PZMs used in such applications. It should be mentioned that this survey do not focuses on the micro and nano-robots based PZMs.

The Piezo LEGS developed by the Swedish company Piezo Motor Uppsala AB, which works with the walking principle are integrated to actuate medical robots for MRI and surgical operations. The features of USMs make it suitable to meet the application requirements. The ultrasonic motors are also tested for the steering wheel adjustment mechanism. The proposition of USMs allows the avoiding of the slow response of the control system due to the considerable moment of inertia of motor associated to the gear system [START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF].  Aircraft and aerospace applications Compact, efficient, precise, and fast response actuators are generally needed to meet the aircraft and aerospace application requirements. These conditions facilitate the integration of PZMs into the space engineering fields. PZMs are mostly used for the flap actuation for noise cancellation, flight control surface actuation, and active flutter suppression, and scanning and image stabilization. The linear and rotary stepping piezo motors (LSPA and RSPA) developed by Cedrat Technologies [69] are integrated into space telescopes mainly to actuate high dynamic and high precise scan mechanism, and to control a laser fiber extension. In [START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF] two dimensional airfoil control system prototype was presented, in which the USM is used in wind tunnel model test.

The DARPA /AFRL/NASA Smart Wing program conducted by Northrop Grumman Corporation (NGC) team, aimed to develop novel and smart technologies and demonstration of relevant concepts to improve the aerodynamic performance of military aircraft [START_REF] Kudva | Overview of the DARPA/AFRL/NASA smart wing program[END_REF]. In one of the project phases, the team addressed a comparative study between smart materials based motors, USMs, and conventional electromagnetic motors, which are tested for an adaptive control of the aileron. The results show that the USMs have higher power density than the other motors, and enable simpler integration into the limited available space.

 Other applications

The ultra-precise PZMs positioning systems commercialized by the worldwide piezo-devices manufacturers [46, 47, 62, 67, 69, 71], are used in several applications where the precision , fast response, and small size criteria are important. These positioning systems are generally multi DOFs platforms. There are very interesting solutions for biotechnology and biomedical engineering applications, especially the cells manipulation and optical microscope actuation. The PZMs are highly requested to actuate medical equipment. Quasi-static and USM PZMs are used to drive high precision mechanism in ophthalmic surgery, adaptive diaphragm positioning, and compact piezoelectric micropumps.

Benefiting of the features mentioned above, the PZMs are integrated in many other applications including the semiconductor technology, surveillance camera platforms, and scientific instrumentation.

I.5 Thesis contributions

Despite the performances of piezoelectric motors listed above, their integration into the industrial fields does not reflecting the real motor potentials. This is due to many factors including the special structure and the specific driving circuit requirements, uncertain motor parameters, sensitivity to external perturbations (load, temperature, electrical noises, etc.), and the nonlinear variation of the motor output characteristics as function of the driving parameters. In consequence, the establishment of PZM models for control design and the synthesis and implementation of accurate and robust position controllers for practical engineering applications still in the heart of research efforts.

Thus, the contributions of this thesis can be summarized as following:

 The study of three different topologies of PZMs for modeling and position control purposes, the studied motors are:

 Rotary traveling wave ultrasonic motor from the USR series of Shinsei.Co (USR60) [50].

 Rotary Piezoelectric Actuator Drive (PAD) from Noliac [71].

 Linear walking piezoelectric motor from Physik Instrumente (PI) GmbH [67].

 The establishment of electromechanical models of the rotary PZMs (USR60 and PAD) which describe the motor motion generation process, reflect the output motor characteristics variation function of driving parameters, and can be used for closed loop control validation.

 The synthesis of precise position controllers of PZMs that fulfill the specific application requirements of the three studied PZMs. Two controllers are proposed:

 H-infinity control based on the mixed sensitivity approach.

 Discrete time RST control based on pole placement technique.

 The guarantee of precision, stability, and robustness criteria in one simple and effective controller whatever the motor topology.

 The validation of the precision and robustness performances of the proposed position controllers without additional compensation systems of the nonlinear behaviors (especially the speed-phase shift dead zone of the USR60). These techniques do not require parameters adaptation approach to overcome the uncertain motor parameters as reported in almost research works related to the position control of PZMs. This is in order to prove the feasibility of positioning engineering applications based on PZMs and with simple, precise, and robust position controllers.

 The design of compact experimental platforms based on the three PZMs, and the experimental validation of the proposed positioning systems. Comparison of experimental positioning results with conventional PID control method is carried out to highlight the advantages of the proposed methods.

 The evaluation of strengths and limitations of the three PZMs regarding to the robotic applications. The evaluation criteria are related to the motor topologies, electromechanical characteristics, drive techniques, and closed loop positioning capabilities.

I.6 Thesis outlines

The thesis is organized as follows:

Chapter II deals with the synthesis approaches of robust position controllers dedicated to piezoelectric motors. The requirement specifications of piezoelectric motor applications are firstly introduced to justify the choice of H-infinity and discrete time RST controllers. The two controllers are synthesized for a generalized plant in order to be applied later to each studied motor as function of each motor model.

Chapter III focuses on the modeling and design of robust closed loop position controllers for rotary traveling wave ultrasonic motor. This chapter starts with the illustration of the structure and working principle of rotary TWUSM. Afterwards, a literature review about the modeling and position control of TWUSM is proposed to clarify the contributions of the proposed work. Then, the modeling of USR60, position controller synthesis, and simulation and experimental validations of the precision and robustness of the proposed methods will be respectively detailed. A comparative study of experimental results between the H-infinity, RST, and PID controllers will be proposed at the end of this chapter to prove the performance of the proposed methods.

Chapter IV presents the modeling and design of robust closed loop position controllers for Piezoelectric Actuator Drive (PAD). Rotary quasi-static piezoelectric motor with innovative topology is studied in this chapter for position control purposes. The state of the art of the modeling and position control method of PAD is detailed in the introduction of this chapter. The working principle, and the specific features of the PAD are then illustrated. Similarly to the other motor, the modeling, simulation, and experimental results of the PAD positioning system based on robust control methods will be discussed.

Chapter V is dedicated to the modeling and design of robust closed loop position controllers for linear Walking Piezoelectric Motor (WPZM). In this chapter, linear quasi-static piezoelectric motor based on walking drive method is studied. A survey of the position control techniques of WPZM and the advantages of the proposed method are firstly introduced. The working principle of this motor and the positioning platform are described. The model experimental identification process and the synthesis procedure of position controllers are explained in details. The simulation and experimental positioning results of WPZM are presented and discussed at the end of this chapter.

The suggestions, recommendations, and general conclusions related to the studied motors will be addressed in Chapter VI.

II.1 Introduction

The position control of piezoelectric motors is in the heart of recent research topics related to these motors. This is principally due to the increase demand of high accurate positioning systems based on piezo-motors. In other hand, the special structure of the motor and the nonlinear behaviors of the used materials present a serious challenge for position control. To prove the performances of piezoelectric motors in closed loop positioning systems, the mostly developed techniques deal with the adaptive based model methods and the on-line learning methods which involve the fuzzy logics and neural networks.

The time consumption and the industrial integration efforts of such techniques into engineering applications are real locks.

Thus, the contribution of the proposed control methods is to guarantee the high precision performances and robustness against motor nonlinear behaviors and external disturbances using simple and easy implemented controllers. Moreover, the developed controllers will fulfill the positioning performances with different technologies (ultrasonic, quasi-static) and topologies of piezoelectric motors.

In this chapter, the synthesis approaches of two position controllers for piezoelectric motors will be detailed. The first one is a H-infinity controller integrating sensitivity functions to prove the robustness issues. The second one is a discrete time RST control based on pole placement technique to achieve high performances in a specified frequency operation range. The controller concepts for generalized plants is presented in this chapter, then it will be applied to control the different studied piezoelectric motors.

II.2 H-infinity position controller

The shortcomings of classic feedback control methods motivate researchers to shift to H ∞ optimization methods to achieve the desired performances with high robustness levels [START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF]. The significant development of the H ∞ control methods in an engineering context are started with Helton [START_REF] Helton | Operator theory and broadband matching[END_REF] and Zames [START_REF] Zames | Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses[END_REF]. This technique was then developed and applied to many process [START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF] . In this section, we will begin with a general control problem formulation into which we can cast the H ∞ optimization method. The selection of the controller parameters as function of the closed loop performances of generalized plant will be then explained. The generalized plant will be replaced by each motor plant to deal with the different studied systems. It is not our intention to explain in details the mathematical analysis of H ∞ optimization method since it can be easily found in the literature [START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF][START_REF] Chang | Robust output feedback H-infinity control and filtering for uncertain linear systems[END_REF][START_REF] Helton | Classical control using H∞ methods: theory, optimization, and design[END_REF][START_REF] Duc | Commande H∞ et µ-Analysis, des outils pour la robustesse[END_REF] and many softwares for solving such problems are available.

In the current research, the control system design is made as follow: defining the closed loop performances of the positioning system, then synthesis and parameters selection of the robust controller in order to achieve the desired performance. In comparison with the other feedback control approaches, the robust H ∞ control exhibits the advantage of combining the performance, robustness, and easy implementation requirements in one controller design procedure.

The H ∞ optimization problem can be posed in the standard formulation as shown in

P(s) K(s) w u e y

H ∞ optimization problem appears to be formulated as the task of designing a stabilizing controller K(s), which minimizes the H-infinity norm of the closed loop transfer matrix T ew from w to e for a given plant P(s) (see Fig. II-1), defined by state space equations [START_REF] Duc | Commande H∞ et µ-Analysis, des outils pour la robustesse[END_REF] :

II-1
Therefore, the state-space realization of the generalized plant P(s) can be written as:

II-2
The algorithms used to solve the H ∞ optimization problem are generally based on the state-space solutions [START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF]. To guarantee the existence of the solution of H ∞ problem, the following assumptions should be taken into account:

(A1): , , is stabilizable and detectable.

(A2): and have full rank.

(A3): has full column rank for all ω.

(A4): has full row rank for all ω.

Assumption A1 is required for the existence of stabilizing controllers K(s), and assumption A2 is sufficient to ensure that the controllers are proper and hence realizable, Assumptions A3 and A4 ensure that the optimal controller does not try to cancel poles or zeros on the imaginary axis which would result in closed loop instability [START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF]. Another assumption (A5) can simplify the resolution algorithm:

(A5): 11 0 and 22 0

The system presented in Fig. II-1 can be also described by the transfer matrix between the inputs and the outputs as:

II-3
The closed loop transfer function from w to e is given by the Linear Fractional Transformation (LFT):

, II-4 where ,

II-5

With reference to the generalized system plant (Fig. II-1), the optimal H ∞ control consists on finding all stabilizing controllers K(s) which minimize the following standard value [START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF]:

Therefore, the H ∞ control approach consists on finding all stabilizing controllers K which for a given γ , it satisfy the following inequality:

‖ , ‖ II-7
This problem can be solved using the resolution of Ricatti equations [START_REF] Duc | Commande H∞ et µ-Analysis, des outils pour la robustesse[END_REF][START_REF] Doyle | State-space solutions to standard H/sub 2/and H/sub infinity/control problems[END_REF], and by updating γ iteratively.

To deal with the motor position control, the motor plant is represented now by G(s) and controlled by K(s) as shown in Where S is the sensitivity function and it can be defined as the transfer function between the reference signal r and the tracking error ε. The transfer function SG represents the effect of the system input perturbation (b) on the error (ε). KS represents the effect of the reference (r) on the control signal (u). KSG represents the effect of the input perturbation on the control signal.

In II-9, as it will be done in most of the following work , the Laplace term "s" will be omitted for the sake of simplicity. Referring to the standard formulation (Fig. II-1), the equivalent LFT (F l (P(s), K(s))) can be identified for the system given in Fig. II-2.a. After that, finding γ>0 and the controller K which internally stabilize the closed loop and ensuring that its H ∞ norm is lower than γ:

‖ , ‖ II-9
In order to achieve the closed loop performances and to prove the robustness potentials, the mixed sensitivity H ∞ control method is often applied (Fig. weighting filter W 3 . In fact, the mixed sensitivity can be defined as the transfer function shaping problems in which the sensitivity function S is shaped along with one or more other closed loop transfer functions such as KS or the complementary sensitivity function T=1-S. For typical industrial process, the low order plant models (first or second order) are employed. As a consequence, applying this approach generates excessive overshoot in the time responses [START_REF] Ortega | Improved design of the weighting matrices for the S/KS/T mixed sensitivity problem-application to a multivariable thermodynamic system[END_REF]. The addition of weighting filters (Fig. II-2.b) to the mixed sensitivity approach will perform simultaneously the controller performances (response time, overshoot, and tracking) and robustness issues. Moreover, the weighting filter parameters will be selected in order to shape the closed loop transfer functions S and KS. The weighting filters are crucial to confirm the robustness of the H ∞ , for example the disturbance d can be successfully rejected if the maximum singular value of S is made small at low frequencies. To do this, we should select W 1 as low pass filter with a bandwidth equal to that of the disturbance. The size of KS is also important to improve the robustness to modeling uncertainty as additional perturbation [START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF].

The transfer matrix of the system plant shown in Fig. II-2.b is given by: II-10

Referring to II-9, the H ∞ problem consists now on finding γ >0 and the controller K which internally stabilize the closed loop and ensuring that:

II-11
If II-11 is satisfied, the following inequalities must be satisfied: The expression of the weighting filters is given by [START_REF] Ortega | Improved design of the weighting matrices for the S/KS/T mixed sensitivity problem-application to a multivariable thermodynamic system[END_REF]:

⁄ II-16
where ω 0 is the closed loop bandwidth, A 1 is the minimum steady state error, and M is the high frequency amplification gain.

The transfer 1/W 1 should have a low gain at low frequency to prove the controller precision and perturbation rejection capability. Whereas the filter cut-off frequency can be interpreted as the minimal system bandwidth.

The most important parameter of W 2 is the cut-off frequency that will limit the maximum system bandwidth to avoid very fast system response and therefore excessive control signal values. As W 2 is used to shape the transfer function KS, the primordial goal of W 2 is to perform the robustness to model parameters uncertainty.

The weighting filter W 3 is a supplementary degree of freedom to improve the perturbation rejection capability of other types of external perturbations.

This robust H ∞ control concept will be applied to each studied piezoelectric motor (in the next chapters) in order to synthesis ultra-precision robust controllers.

II.3 RST position controller

The RST controller presented in Fig. II-4 is a two degree of freedom digital controller based on a robust pole placement method allowing one to impose different specifications in terms of tracking and regulation performances. This digital control approach consists on finding the polynomials R, S, and T which satisfies the desired performances with respect to the disturbances and reference signal. Let us assume that a discrete-time plant, or a continuous time plant sampled at a given period t s can be described in terms of z-transforms as shown in in Fig. II-4. Where u is the control signal, and r and y are the reference and measured signals. The z-transform of the control signal u(z -1 ) can be written as:

II-17

The discrete time closed loop transfer function can be written as following:

.

II-18

where

. 1 ⋯ II-19
is the denominator of the closed loop system and it contains the system poles. It is also called as the characteristic polynomials of the closed loop system. This explain why the desired closed loop performances will be expressed in terms of closed loop poles and eventually in terms of desired zeros. The polynomial S(z -1 ) and R(z -1 ) perform respectively the regulation and tracking performances of the closed loop.

As can be seen in II-18, T(z -1 ) appears only in the numerator giving the second degree of freedom to this controller, which allows the distinction between tracking and regulation performance specifications. In the presence of disturbances (Fig. II-5), there are other important sensitivity functions linking the disturbances to the outputs and inputs of the plant that should be considered in the synthesis approach.

The transfer function between the disturbance v(t) (such as loads) and the output y(t) (output sensitivity function) can be written as: . .

. II-20

T (z -1 ) 1 / S (z -1 ) R (z -1 ) B (z -1 ) / A (z -1 ) r (z -1 ) + - Plant y (z -1 ) u (z -1 ) T (z -1 ) 1 / S (z -1 ) R (z -1 ) B (z -1 ) / A (z -1 ) r (z -1 ) + - + + + + d (z -1 ) v (z -1 ) + + y (z -1 ) b (z -1 ) u (z -1 )

Controller Plant

This sensitivity function allows the characterization of the perturbation rejection performances through the polynomial S(z -1 ) coefficients. In fact, to prove the disturbance rejection of the digital controller in a specified frequency, S(z -1 ) must contain a zero corresponding to this frequency [START_REF] Landau | Digital control systems: design, identification and implementation[END_REF].

The output measured signal can be noised (sensor + signal acquisition noises), the transfer function that relates the noises b(t) to the output signal y(t) can be used to improve the noises rejection potentials. This transfer function is given by:

.

II-21

The common noises in this side are generally a high frequency signals. Therefore, the idea is to achieve low gain of this transfer function in this frequency range in order to attenuate these noises.

The input sensitivity function is defined as the transfer function between u(t) and the disturbance v(t), and is given by:

.

II-22

This sensitivity function permits the evaluation of disturbance effects on the input signal. The control system robustness to such disturbances in a specific frequency range can be improved via the adjustment of polynomial R(z -1 ) coefficients [START_REF] Landau | Digital control systems: design, identification and implementation[END_REF].

The fourth sensitivity function is the transfer function that relates the disturbances d(t) to the output signal y(t) which describes the influence of d(t) on the output signal of the plant. It is also called the input disturbance-output sensitivity function and it is given by:

.

II-23

This sensitivity function serves to compensate the unstable poles of the plant by the zeros of R(z -1 ).

It should be mentioned that the closed loop system of the plant presented in Fig. II-5 is asymptotically stable if and only if all the four sensitivity functions (S yv , S yb , S uv , and S yd ) are asymptotically stable.

The first step of the digital RST pole placement method consists on defining the desired performances. These performances can be expressed by the reference transfer function model of the plant G m (z -1 ):

-

1 -1 -1 ( ) ( ) ( ) m m m B z G z A z  II-24
The idea is then to carefully select the system poles in order to make equal the reference closed loop model and the actual input-output closed loop transfer function y/r:

II-25

In other hand, the closed loop transfer function can be written as:

II-26

Where and contain all poles and zeros of that cannot be compensated. and contain all the other terms. As mentioned above that G m (z -1 ) was chosen in order to impose the desired closed loop poles, with the only constraint that its numerator had to contain as a factor the plant pure delay and its un-compensable zeros (zeros lying outside of the unit circle) and zeros that one decides not to compensate, all regrouped in the polynomial B -(z -1 ):

II-27

In order to ensure an unity gain in steady state 1 1, a constant factor ( ) is introduced in the expression of :

II-28

The partial transfer function from the reference r to the true error signal , is given according to II-25 as:

1 1 II-29
To cancel the steady state errors in response to polynomial references (r(t)=αt m ), which have a ztransform defined by:

1 II-30
Where r 1 is a polynomial in , it should be also mentioned that 0 in the case of step signal and equals to 1in the case of ramp. It is necessary and sufficient that 1 divides , according to the final limit theorem of the z-transform. This condition can be expressed also by:

1 II-31
Where L(z -1 ) is an unknown polynomial to be determined.

Fig. II-6. Flowchart of RST control synthesis approach

Referring to equations II-27 and II-31, the polynomial A m can be given by:

1 II-32
The discrete time RST pole placement method can be resumed in the following given in Fig. II-6.

In the case of sine-wave reference signal, the final limit theorem cannot be applied. Thus, the magnitude of the transfer function from the reference r to the true error signal can be considered [START_REF] Ostertag | RST-Controller design for sinewave references by means of an auxiliary Diophantine equation[END_REF]:

II-33
In order to cancel the error at a given angular frequency ω 0 , it is necessary to introduce a transmission zero into the transfer function (II-33) at this frequency. If we take the case of a complex conjugate pair of zeros, this is can be done by letting (1 ) (1 ) divide . Therefore, the equation (II-31) will be rewritten as:

1 1 1 2 cos II-34
And the auxiliary Diophantine equation which must be solved is: The synthesis approach for the sine-wave case can be applied to all polynomials reference by taking account the signal frequency. Moreover, the step signal can be tracked by using the auxiliary Diophantine equation.

-1 -1 -1 ( ) ( ) ( ) B z G z A z  -1 -1 -1 ( ) ( ) ( ) m m m B z G z A z  -1 -1 -1 -1 -1 -2 1 2 A A(z ).S(z ) + B(z )R(z ) = 1 + p z p z m   -1 -1 -2 1 2 S(z ) = 1 + s z z ... s   -1 -1 -2 1 2 R(z ) = 1 + r z z ... r   -1 1 -1 (1 z ) (z ) ' m m m A L B B      1 1 0 ( ) ' ( ) m T z A B z
The detailed digital poles placement concept based on RST method will be then applied to the studied piezoelectric motor plants. Therefore the discrete time plant model will identified in first time.

Afterwards, the poles of the closed loop will be fixed as function of the desired performances and the R, S, and T polynomial coefficients will be deduced.

II.4 Conclusions

In this chapter, the synthesis approaches of two position controllers for accurate positioning systems based on piezoelectric motors are explained in details. The first control method consists on robust mixed sensitivity H ∞ controller, and the second one is a digital RST controller based on pole placement method. Many precautions are taken into account in the synthesis concept, in particular the ultra-precision under system parameters uncertainties and the robustness to additional disturbances.

The constraints of piezoelectric motor positioning systems and the real time implementation in engineering applications issues are considered during the selection and the design of the proposed methods.

To deal with the different studied piezoelectric motors, the H ∞ and RST position controllers are synthesized for a generalized plant. Then, for each motor, the control methods can be easily applied by introducing each specified plant model.

The challenge of the position controllers is to validate the precision performances and robustness requirements in a simple controller without additional parameters adaptation or nonlinearity compensation systems

III.1 Introduction

Rotary traveling wave ultrasonic motor is presently the most typical and widely used ultrasonic motor [START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF]. TWUSM presents a good compromise between a compact, silent, and efficient motor. The technology used remains relatively simple and the performances are among the most interesting in the mechanical power range up to 20W. It combines two standing waves with a 90° of phase difference both in time and space. This operation mode makes the clockwise and counterclockwise motion of the motor very simple (by switching between +/-90° of phase shift). The TWUSM is an attractive solution for compact positioning systems and it was integrated in many engineering fields as detailed in Chapter I.

In order to improve the design efficiency of TWUSM and realize precise control, extensive research has been done on the models establishment and position controls synthesis.

In this chapter, the working principle of the rotary TWUSM will be explained, afterwards a literature review of the modeling and position control of such type of motors will be given. An electromechanical model of rotary TWUSM (USR60) is designed in the MATLAB/Simulink environment. It reflects the variation of the motor output characteristics as function of the driving parameters, and can be easily used for control purposes via the driving signals (voltage amplitude, frequency, and phase shift). The output characteristics of the USR60 will be verified via the open loop simulations of the proposed model. A compact experimental platform is setup in order to validate the USM characteristics, to identify the Transfer Function (TF) between the motor angular position and the control signal, and to implement the synthesized robust position controllers of USR60. The synthesized H-infinity and discrete time RST controllers are applied to the USR60 motor, and the controller parameters are determined based on the identified TF. The proposed positioning system must fulfill the accuracy and stability criteria needed for the most USR60 applications. Moreover, the challenge is to confirm the robustness of the closed loop system in case of loaded motor condition without additional compensation system, motor heating, and injected noises to the control signal or to the measured position. The proposed control methods are compared to a PID controller in order to confirm the advantage of the proposed methods. The closed loop positioning system is simulated and implemented in real time. The precision, stability and robustness of the proposed system are evaluated and compared through the simulation and experimental results.

III.2 Working principle of rotary TWUSM (USR60)

The studied motor is a commercialized rotary TWUSM (USR60) (Fig. The piezoelectric ring is divided into two semicircular sectors A and B, as shown in Fig. III-2.a. Each of these semicircular sectors is further divided into small sectors with opposite polarization. Therefore, when the motor is supplied by two sinusoidal voltages at a frequency range close to one of its mechanical resonance frequencies and with 90 degree of phase shift, one sector will contract while the other will expand. Two Standing Waves (SW) with equal amplitude and 90° of phase difference in time and space will be created based on the inverse piezoelectric effect. The traveling wave is formed by the superposition of the two SW. The surface points of the stator will follow an elliptical trajectory as shown in Fig. III-2.b. The micro-vibration of the piezoelectric ring will be transformed to rotary motion of the rotor through frictional forces. The traveling wave generation, the stator-rotor contact mechanism, and the motion of the rotor will be modeled in the USR60 modeling section.

III.3 Literature review

Since the invention of TWUSM, intensive research works have been done to establish an accurate model of such motor. Many control methods of TWUSM have been also developed based on the designed models. In this section, a literature review of the TWUSM models and position controllers will be proposed.

III.3.1 Modeling of rotary TWUSM

This section treats the modeling studies of TWUSM and its applications on the motor behavior analysis and parameters estimation. The TWUSM models can be classified into; Equivalent Circuit Models (ECM), Finite Element Method (FEM), and mathematical models.

A very good introduction to the construction and operating principle of TWUSM was undertaken by Sashida and Kenjo [START_REF] Sashida | An Introduction to Ultrasonic Motors[END_REF]. In this book, the motor is modeled by an equivalent electrical network of capacitors, resistors, inductors, and Zener diodes. This modeling approach is simple, however, the rotorstator interface which is the origin of torque generation is missing. The Equivalent Circuit Model (ECM) have been presented in [START_REF] Hirata | Characteristics estimation of a traveling wave type ultrasonic motor[END_REF] to estimate the TWUSM speed and load characteristics. Elghouti and Helbo [START_REF] Elghouti | Equivalent circuit modeling of a rotary piezoelectric motor[END_REF] highlight the importance of the electromechanical coupling factor in their proposed ECM, which is responsible for the electrical to mechanical energy conversion. The model is performed based on experimental approach combined with the electrical network method and some simplifying assumptions about the motor real behavior [START_REF] Bolborici | Modeling of the stator of piezoelectric traveling wave rotary ultrasonic motors[END_REF]. Similar ECM modeling approaches based on experimental tests have

(a) (b)
been also reported in [START_REF] Mojallali | Systematic modeling for free stators of rotary piezoelectric ultrasonic motors[END_REF][START_REF] Mojallali | A new scheme for experimentalbased modeling of a traveling wave ultrasonic motor[END_REF], to determine the stator parameters and vibrational behavior. When compared to other methods, the ECM has the advantages of simplicity and low computational efforts. However, the main drawback of this model is its inability to accurately model the contact mechanism between the motor stator and rotor.

The Finite Element Method (FEM) has been widely used for the analysis of the stator/rotor interaction and contact surface parameters. Maeno et al. [START_REF] Maeno | Finite-element analysis of the rotor/stator contact in a ring-type ultrasonic motor[END_REF] proposed FEM approach to model the TWUSM stator and the rotor/vibrator contact surface. Numerical calculation method is used to determine the load characteristics. FEM modeling of the TWUSM stator with comparison between closed and segmented ceramic ring has been presented by Krome and Wallaschek [START_REF] Krome | Finite element models for the piezoelectric actuation in ultrasonic traveling wave motors[END_REF]. Three-dimensional FEM have been reported in many works for modeling of TWUSM. It was applied by Kagawa et al. [START_REF] Kagawa | Finite element simulation of dynamic responses of piezoelectric actuators[END_REF] to simulate the dynamic behavior of USM including transient response. The performance of USM as a function of input parameters such as the driving frequency, voltage input and pre-load was evaluated using the threedimensional FEM in [START_REF] Frangi | Finite element modelling of a rotating piezoelectric ultrasonic motor[END_REF]. The stator is simulated as a fully deformable elastic body, the interaction through contact between the stator and the rotor is accounted by assuming that the rotor behaves as a rigid surface. A three dimensional finite element model with cohesive zone elements embedded between friction material and rotor is developed by Shen et al. [START_REF] Shen | Contact Modeling of Traveling Wave Ultrasonic Motors[END_REF] to model the TWUSM contact mechanism.

The traveling wave generation and the vibrational behavior of TWUSM have been also simulated through FEM in [START_REF] Iula | A high-power traveling wave ultrasonic motor[END_REF]. The time computation of FEM to analyze the motor parameters makes it not convenient for control implementation and fast evaluation.

Intensive research studies on the mathematical modeling of TWUSM have been reported in the literature. This modeling method is used to model the stator vibration, the rotor/stator interaction parameters, and the motor output characteristics. Working principle and mathematical modeling of the stator of TWUSM have been presented by Hagedorn and Wallaschek [START_REF] Hagedorn | Travelling wave ultrasonic motors, Part I: Working principle and mathematical modelling of the stator[END_REF]. In this paper, the stator is modeled using the theory of Mindlin and Reissner for circular plates of non-constant thickness, and the eigenvalue problem is solved using numerical methods. The Rayleigh-Ritz approach is used by Hagood and McFarland [START_REF] Hagood | Modeling of a piezoelectric rotary ultrasonic motor[END_REF] to model the distributed piezoceramics and the traveling wave dynamics of the stator. The purpose is to give a general framework for modeling TWUSM and predicting motor performances as a function of design parameters. Wallaschek [START_REF] Wallaschek | Contact mechanics of piezoelectric ultrasonic motors[END_REF] presented contact mechanics model of piezoelectric USMs to summarize the state of the art in the understanding of some fundamental processes governing the contact mechanics of piezoelectric USMs. Moregal et al. [START_REF] Morega | Numerical study of the stator motion in a piezoelectric ultrasonic motor[END_REF] presented mathematical modeling and numerical simulation of the TWUSM stator dynamics. In this study, modal analysis is provided for the structural Eigen frequencies of the stator. Mathematical model of USM for speed control is proposed by Senjyu et al. [START_REF] Senjyu | Mathematical model of ultrasonic motors for speed control[END_REF]. In this model the dead zone in control input effect is included, and the model parameters are derived by fitting the transfer function of dynamic speed response. Second order model for a rotary TWUSM was derived using Canudas-de-Witt method by Rochin et al. [START_REF] García-Rochín | Secondorder model for rotary traveling wave ultrasonic motors[END_REF]. The model parameters are identified through experimental tests under different frequencies and load torques. This model take into account the dependency of the motor velocity on the driving frequency. Another analytical modeling approach for the ultrasonic motor is presented by Elgotti [START_REF] Ghouti | Hybrid modeling of a traveling wave piezoelectric motor[END_REF]. The author presents the derivation of a general framework for a simplified analytical model using the energy method based on Hamilton's principle with simplifying assumptions. An interesting hybrid model is also proposed by Elgotti [START_REF] Ghouti | Hybrid modeling of a traveling wave piezoelectric motor[END_REF], which has the strengths of both an equivalent circuit model and an analytical model. The idea is to overwhelm the simplified analytical model of the unloaded stator with the equivalent circuit model. Therefore, the parameters of the ECM can be substituted in the framework of the simplified model.

III.3.2 Position control of rotary TWUSM

In parallel with the modeling and design of TWUSMs, a variety of position control methods of such motors have been developed. The speed/position of the TWUSM was controlled using the driving voltage frequency, phase shift, and dual control by combined the two control signals. The PID with fixed and variable gains, adaptive model based, fuzzy logic, neural networks, and mixed controllers are the commonly used methods for TWUSM position control. In the following section, an overview of the developed position controls methods (but not all) for rotary TWUSMs is given. The fixed gain PID controller was tested in [START_REF] Gencer | A new speed/position control technique for travelling wave ultrasonic motor under different load conditions[END_REF][START_REF] Sun | Control of multiple ultrasonic motors with robust parameter design[END_REF], where the equivalent circuit model of USM is used to determine the system parameters. It is simple and easy to be implemented but it is difficult to set the controller parameters and to meet the performance requirements of USM while the motor parameters are time varying. Tuning PID is proposed to overcome the shortcoming mentioned above [START_REF] Djoewahir | Adaptive PSO-based self-tuning PID controller for ultrasonic motor[END_REF]. The tuning parameters technique requires the setting of related parameters according to the USM operating state, load conditions, and temperature. Therefore, it is not easy to be implemented in practical applications [START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF].

The Adaptive methods are widely reported in the literature for USM position control. The main idea of these techniques is to design a Model Reference Adapting Control (MRAC) and self-correcting controller. On-line control parameters identification will be then executed by comparing the output with reference model in order to compensate the position error. The MRAC is mainly used to compensate the control input dead zone in case of load conditions [START_REF] Senjyu | Position control of ultrasonic motors with adaptive dead-zone compensation[END_REF][START_REF] Senjyu | Position control of ultrasonic motors using MRAC and dead-zone compensation with fuzzy inference[END_REF]. The generated position errors remain relatively important for precise application due to the uncertainty of the reference model. Senjyu et al.

[106] present a dual mode control by using both the driving frequency and voltages phase shift as control signals. This control scheme consists of a PI controller and an adaptive controller, which compensates the speed characteristic variations. However, as shown in the positioning experimental results of this paper, the oscillation of the final motor position increases significantly when operating close to the resonance frequency. Moreover, there is a weakness of sine-wave motion tracking for loaded motor.

Due to the fact that Fuzzy Logic Control (FLC) is preferred when the mathematical model of a control plant is difficult to obtain, this technique was proposed in several research works to achieve superior speed and position characteristics [START_REF] Bazrafshan | Fuzzy modeling and position control of a traveling wave ultrasonic motor[END_REF][START_REF] Bal | Fuzzy logic based DSP controlled servo position control for ultrasonic motor[END_REF][START_REF] Izuno | New fuzzy reasoning-based high-performance speed/position servo control schemes incorporating ultrasonic motor[END_REF][START_REF] Gencer | A comparative speed/position control technique based Fuzzy Logic control for travelling wave ultrasonic motor[END_REF]. It is based on the knowledge of the USM state and experimental data to make up the fuzzy inference rules. But under certain conditions, the motor parameters are variable, and the output motion can be outside the inference rules [START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF]. Moreover, this control method uses an empirically derived motor model, which does not reflect the real behavior of the motor [START_REF] Bolborici | Modeling of the stator of piezoelectric traveling wave rotary ultrasonic motors[END_REF].

The Neural Network (NN) control is also investigated in the position control system of USM [START_REF] Senjyu | Position control of ultrasonic motors using neural network[END_REF][START_REF] Lin | Identification and control of rotary traveling-wave type ultrasonic motor using neural networks[END_REF][START_REF] Yildirim | Design of adaptive robot control system using recurrent neural network[END_REF][START_REF] Ahmadi | Predictive Control of Traveling Wave Ultrasonic Motors using neural network[END_REF].

The output characteristics of the USM depend on the stator/rotor contact surface. The parameters of this contact surface are defined mainly by the amplitude of the traveling wave and the phase shift between the two standing waves. The dependency between the traveling wave and the motor output characteristics is at the origin of the design of NN control methods. The speed/torque-traveling wave characteristic curves are measured experimentally. Mapping function using neurons and connection weights are then investigated to model this dependency. With the NN better control accuracy can be obtained. However, USM behavior and parameter variation yields to the growth of the network's complexity for the NN controller which will increase the execution time and learning procedure [START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF].

The USM position control methods mentioned above have their advantages and disadvantages. In order to take advantage of each method, different types of mixed controllers have been developed such as fuzzy neural network (FNN) [START_REF] Lin | Fuzzy neural network position controller for ultrasonic motor drive using push-pull DC-DC converter[END_REF][START_REF] Yoshida | Position control of ultrasonic motors using dead-zone compensation with fuzzy neural network[END_REF][START_REF] Lin | An adaptive fuzzy-neural-network controller for ultrasonic motor drive using the LLCC resonant technique[END_REF], fuzzy PID [START_REF] Shi | SELF-TUNING FUZZY SPEED CONTROLLER OF TRAVELLING WAVE ULTRASONIC MOTOR[END_REF], NN-PID [START_REF] Mu | Intelligent control of USM using a modified NN with PSO[END_REF][START_REF] Tanaka | Precise position control of an ultrasonic motor using the PID controller combined with NN[END_REF], and fuzzy sliding mode control [START_REF] Leng | Fuzzy sliding mode control for pan-tilt-zoom system driven by ultrasonic motor[END_REF]. The key of the mixed controllers is to employ dual control via the voltages frequency and phase shift to handle system nonlinearities and parameters uncertainties. The FNN is adopted to determine a dead-zone compensation equivalent input and to construct a plant which does not have deadzone. Afterwards, PI controller will be applied to perform the accurate positioning. The mixed controllers can make the control precision with some improvements. However, the fuzzy rules and neurons connections are empirically designed. That means, a new parameters identification is required each time a new motor has to be controlled. Moreover, the implementation of such methods for engineering applications is still complicated [START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF].

Several robust position controllers are also developed based on USM model. Among them, the Behavior Model Control (BMC) proposed by F.Giraud et al. [START_REF] Giraud | Precise position control of a traveling-wave ultrasonic motor[END_REF]. Where the USM model is represented by a Causal Ordering Graph (COG), and the control law is then deduced by inversion of the COG. However, the model parameters are identified through the experimental characteristics of the studied motor. Therefore, a new identification is required for new motor or under other test conditions. The genetic algorithm method is also used for TWUSM position control [START_REF] Jahani | GA-neural network based position control of Traveling Wave Ultrasonic Motor[END_REF], the control effectiveness and prospective for engineering applications still need to be improved.

III.4 Proposed Simulink model of USR60

In this thesis, the aim of the TWUSM modeling approach is to establish a simplified model of the rotary USM which reflects the motor behavior and makes the position control implementation easier. In fact, the characterization, parameters identification, and complex contact mechanism of the USR60 motor will not be detailed here. Since the dynamic characteristics of the real motor are difficult to capture in analytical models, and the parameters of the motor are time varying and highly nonlinear [START_REF] Senjyu | Position control of ultrasonic motors using MRAC and dead-zone compensation with fuzzy inference[END_REF], then some assumption are required in order to design a suitable model for control purposes. The following section details the design, assembly, and simulation of electromechanical USR60 model in MATLAB Simulink. This model is mainly composed by three subsystems as shown in Fig. III-3, and the related parameters are illustrated in the equivalent stator/rotor contact mechanism in Fig. III-5.

 Stator: The piezo-ceramics vibration and traveling wave generation (w, w max ) based on the inverse piezoelectric effect and the stator/rotor interaction forces (F n,t ) are detailed in this subsystem.  Vertical motion of the rotor: The rotor vertical position, half contact length of the traveling wave(x 0 ), will be determined and used to calculate the output torque (T) and the angular motion of the rotor.  Angular motion of the rotor: The angular velocity (Ω) is analyzed to deduce the stick point (x s ) needed for the output torque calculation

III.4.1 Stator

The stator is composed by two set of piezoelectric ceramics with opposite polarization (A, B) which will be supplied by two sinusoidal voltages (V 1 (t), V 2 (t)) with a phase shift of 90°:

sin cos III-1
Where V max and ω are respectively the maximum voltage amplitude and angular frequency

In order to simplify the motor model, it is assumed that only two orthogonal modes and with the same amplitude can be excited within the piezo-ceramic plate. These modes are given by [START_REF] Ghouti | Hybrid modeling of a traveling wave piezoelectric motor[END_REF]:

sin cos III-2 Stator Rotor (vertical motion) Rotor (angular motion) w w max X 0 X s Ω,T F n,t
Input voltages where R r is the shape constant of the traveling wave in radial direction, and k is the wave number.

The orthogonal temporal amplitudes of the generated standing waves are denoted by w 1 (t) and w 2 (t) respectively sin cos III-3

where w max is the maximum temporal amplitude of the traveling wave (m),

The superposition of their corresponding standing waves (in case of 90° of phase shift) results in a traveling wave at the surface of the stator expressed by [START_REF] Murphy | Analysis of a Rotary Ultrasonic Motor for Application in Force-Feel Systems[END_REF]:

, , cos III-4 III-5
In the case of voltages phase shift different from 90°, the model will be developed based on two standing waves with phases different from 90°. The resulting simulated wave in this case is not a purely traveling wave.

The general state space model of the stator was also derived based on the hypothesis that it is composed by two orthogonal and symmetrical phases. It is possible to reduce the general state space model by decoupling the two orthogonal modes and thereby obtain a separate model for each phase of the stator [START_REF] Ghouti | Hybrid modeling of a traveling wave piezoelectric motor[END_REF]. The contact between the two orthogonal stator phases and the rotor is supposed symmetrical along the contact surface [START_REF] Ghouti | Hybrid modeling of a traveling wave piezoelectric motor[END_REF].

The The piezo-forces and F v2 are the forces generated by the stator piezoelectric ceramics when excited by two phase shifted sinusoidal voltages (V 1 , V 2 ). They can be written as following:

sin 2 cos 2 III-7
where is the piezoelectric force factor (N/V).

The feedback forces , are composed of normal and tangential components and they serves as the coupling terms between the stator and rotor. They depend mainly on the half-contact length x 0 (will be defined later) and the wave amplitudes (w 1 , w 2 ). The normal feedback forces can be calculated as in [START_REF] Maas | Simulation model for ultrasonic motors powered by resonant converters[END_REF]:

, sin 2 , sin 2
III-8
The tangential feedback forces for the ideal motor operating within its load carrying range are [START_REF] Maas | Simulation model for ultrasonic motors powered by resonant converters[END_REF]: where: is the coulomb friction coefficient, b is the inner radius of the stator (m), h is the half thickness of the stator (m), R 0 is the middle radius of the contact surface (m), and x r is the rotor stiffness per unit area (N.m -3 ).

For a non-ideal model, additional terms would be added to equation III-1 accounting for unequal stickpoint lengths (x s ) on either side of the traveling wave crest as done in [START_REF] Maas | Model-based control for ultrasonic motors[END_REF]. However, following the ideal wave assumption, the stick-point lengths to the right and left of the wave crest are of equal length and the additional terms cancel each other in the feedback calculation.

The total feedback force experienced by each stator phase is determined by combining equations III-8 and III-9:

, , , ,

III-10

where dir(R)=+/-1 for clockwise/counterclockwise rotor motion directions .

III.4.2 Contact surface

The contact mechanism refers to the friction between the stator and rotor surfaces. The main approach for the modeling of the contact mechanic is shown in Fig. III-5. The stator is assumed as undeformable while the rotor is deformable with some uniform stiffness. We will interested on two variables of this contact surface which are the half-contact length (x 0 ) and the stick points (x s ). The two variables will be used to determine the motor characteristics such as the no-load speed, feedback forces, and generated torque.

III.4.2.1 Half-contact length, x 0

The half-contact length is defined as half of the horizontal length of the traveling wave crest of the stator submerged within the rotor from its point of entry to its exit [START_REF] Murphy | Analysis of a Rotary Ultrasonic Motor for Application in Force-Feel Systems[END_REF]. The generated stator wave crest presses into the rotor contact material creating a normal force F N . As w max increases from zero, F N increases towards the preload between the rotor and the stator F ext . At a critical value of w max , with the contact length set at its maximum of λ / 4, F N becomes larger than F ext , and the rotor accelerates away from the stator surface to a height z(t). By monitoring the rotor height, the half-contact length can be calculated as in [START_REF] Maas | Simulation model for ultrasonic motors powered by resonant converters[END_REF]:

1 cos III-11
where z(t) is the vertical distance between the rotor's lowest surface and the top surface of the undeformed stator and h is the half-thickness of the stator.

The created normal force F N can be calculated as function of the half-contact length and the traveling wave amplitude w max as [START_REF] Ghouti | Hybrid modeling of a traveling wave piezoelectric motor[END_REF]:

2 sin cos III-12

III.4.2.2 Stick points, x s

The stick points, x s , are defined as the location where the horizontal velocity component of the stator surface point matches the horizontal velocity of the rotor, and their magnitude represents the length from the traveling wave crest to that location [START_REF] Murphy | Analysis of a Rotary Ultrasonic Motor for Application in Force-Feel Systems[END_REF]. If the rotor moves with an angular velocity Ω, then the horizontal velocity of the rotor is:

, 0 h rotor R    III-13
The horizontal velocity of the stator surface point is given by [START_REF] Ghouti | Hybrid modeling of a traveling wave piezoelectric motor[END_REF]:

, kωh III-14
The stick point condition is achieved when the following equation is fulfilled;

, , : acos III-15

III.4.2.3 Torque generation

The USM generated torque depends on the preload between the rotor and the stator, the applied external load, the stator-rotor friction coefficient, and the stator-rotor contact surface. The output torque of USM in steady state was derived in [START_REF] Ghouti | Hybrid modeling of a traveling wave piezoelectric motor[END_REF] using a linear spring contact model as:

2 ² 2ϕ ϕ III-16
where ϕ x sin kx cos III-17

In steady state, the stick-point length is less than or equal to the half-contact length, and the horizontal stator surface-point velocity at the stick-point is equal to the rotor horizontal velocity. For the USM, the transient condition is defined as the period in time that the traveling wave amplitude has not yet reached its steady state value. As the motor speed increases from zero to its steady state value, in the simulation model, it is possible for the stick-point to fall outside of the contact length ( > ). In the proposed motor model, if the stick point ( ) is calculated to be larger than the half-contact length (x 0 ), it will set equal to : III-18

III.4.3 Rotor (vertical motion)

The rotor of USR60 will be forced out of its initial vertical position (z 0 ) as shown in Fig. III-6. The vertical motion of the rotor will be described by the following equation:

Z III-19
where is the rotor mass (kg), and is the applied external force

III.4.4 Rotor (angular motion)

The angular motion of the rotor can be represented as shown in Fig. III-7 , where T USM and T L are respectively the motor and load torques, and J r is the rotor inertia. Then, the angular motion equation of the rotor is given by: Ω III-20 The TW amplitude plays a critical role in the operation mechanism of TWUSM, where it is used to determine the contact surface parameters (x 0 , x s ), feedback forces, and generated torque. It should be mentioned that a minimal TW amplitude value must be reached before the stator normal force (F N ) matches the external pressing force (F ext ). During the period where the TW is less than the minimal value, F N is less than F ext and the rotor remains pressed into the stator with a height of zero. For the TWUSM, the maximum amplitude of the TW is generated at the mechanical resonance frequency (f r ). The USR60 motor is driven at different frequencies in order to identify the motor resonance frequency. An electro-mechanical model of the rotary USR60 is successfully designed in Matlab/Simulink environment. The proposed model includes the TW generation process at the stator surface, the contact surfaces variables needed for the vertical rotor motion and torque generation, and the angular motion of the rotor. The simulation results reflect the behavior of the rotary TWUSM and the motor fast response is validated. The output characteristics variation as function of the driving parameters confirms that the model is suitable for closed loop control implementation via any control parameter (V rms , f, φ). Therefore the USR60 Simulink model will be used to validate the synthesized position controllers. In the open loop test configurations, the two phase shifted sinusoidal voltages (V 1 (t), V 2 (t)) are sent by the waveforms generator at the frequency range close to the mechanical resonance frequency of the USR60. The two voltages will be amplified using two switching power amplifiers type HSA4051 in order to obtain the adequate voltages range and power levels. The input voltage of the HSA4051 power amplifier is an analog signal between +/-10V (50 Ω/600 Ω) and its maximum output voltage is of 300 V p-p with a maximum RMS current of 1 A (2.83 A p-p ). The frequency response of the amplifier is of 500 kHz with a slew rate of 450 V/μs. The load test conditions of the USM are realized by applying a load torque at the motor shaft side using a powder brake type FAS21 (Fabricated by Merobel). The nominal torque of the brake is 2 N.m with a nominal current of 0. In the closed loop test configurations, the motor position is introduced to the control board via the incremental encoder interface of dSPACE microprocessor (type dS1104). The measured position is compared to a reference signal, and a position control algorithm will be executed in Matlab environment. The control signal parameter is then calculated and sent to the waveforms generator operating in external modulation mode. The driving voltages will be generated and connected to the two power amplifiers to drive the USR60. The synoptic of the driving diagram is illustrated in Fig. III-16.

III.6.2 Experimental characteristics of USR60

In this section, the open loop test results of the UR60 motor are carried out. These tests aim to validate the behavior of the USM discussed in section 3, and to analyze the motor output characteristics as function of the driving parameters in order to choose the suitable position control parameter. The motor is driven at first time in the frequency range between 30 and 45 kHz, which corresponds to the location of one its resonance frequencies. The RMS voltage amplitude is of 110 V and the phase shift is of 90°. By decreasing the driving frequency, the motor speed increases until the resonance frequency (f r =33 kHz). When the frequency decreases below f r a breakdown appears and the motor shutdown immediately. This breakdown is well known as the pull-out phenomenon of TWUSMs [START_REF] Ballas | Piezoelectric multilayer beam bending actuators: Static and dynamic behavior and aspects of sensor integration[END_REF] From the open loop experimental tests of the studied motor (Fig. , it can be seen that a large voltage amplitude variation (40 Vrms) generates a small speed variation. Thus, this drive technique permits only a smooth motor motion in one direction. The voltage phase shift (φ) can be also used to control the motor speed and position, and it is the only parameter that allows the inversion of motor rotation direction. This driving method also has the problem that the speed is not nicely linear with the voltage phase shift. Moreover, there is a dead zone around zero value of phase shift when load torque (T L ) is added (Fig. . Therefore, the phase shift is more suitable for a position control than a speed control. Regarding the intended use of the USM in robotic applications and the obligation of precision and bidirectional operation mode, φ is chosen as the position control parameter. For these purposes, a transfer function between φ and the angular motor position (θ) must be identified to be used for the synthesis of the position controllers. The TF between φ and Ω is estimated to a first order system using the Matlab identification toolbox from the introduced velocity and phase shift data with a sampling time of 100 μs. The TF between φ and θ is then deduced as a second order TF [9]:

s θ s φ s 1 s III-21
where the gain constant and the time constant are respectively equal to 11.5 and 4.25 ms.

As mentioned above, the dynamic behavior of the TWUSM is nonlinear. Firstly, because the mechanical resonance frequency of the TWUSM can change as function of the test conditions. The parameters of the motor TF are not the same if the open loop test was done at another driving frequency. For example the gain and time constants (k m , τ m ) are respectively equal to 6.5 (instead of 11.5) and 9.3 ms (instead of 4.25 ms) when the motor is driving at a frequency of 35 kHz. Moreover, these parameters are for an unloaded motor (T L =0 N.m), therefore when the motor is loaded the parameters will also change. The motor heating due to the contact mechanism between stator and rotor will affect the speed response (Fig.

III-21

) and therefore the TF parameters. For an operation temperature about 45°C the time constant (τ m ) reaches 6.5 ms due to the motor speed decrease. The minimum and maximum values of the TF parameters function of the drive frequency and the motor temperature are presented in table III.1. That is the reason why w rely on the robustness of the proposed controllers to overcome the model uncertainty problems and to ensure high precision levels at any operating temperature (<55°C) or under any loads (up to1 N.m).

Table III-1. Variation of the TF parameters

III.7 Synthesis of robust position controllers of USR60

The identified TF model G USM (s) (Eq. III-21), is used for the controller synthesis. As for the rest of studied PZMs in this manuscript, robust H-infinity (H ∞ ) and discrete time RST position controllers are designed, evaluated, and compared in order to select the accurate closed loop positioning system. The synthesis approach of H ∞ and RST controllers for general plant is explained in details in chapter.II. In this section, the USR60 plant will be studied and the position controllers will be found as function of the desired closed loop performances. A proportional, integral, and derivative (PID) controller will be designed and experimentally implemented and compared to the proposed robust position controllers to prove the advantage of the proposed techniques.

III.7.1 H-infinity position controller of USR60

As detailed in chapter II, the mixed sensitivity H ∞ optimization method is used to synthesize the feedback controller K(s). To deal with the USR60, the closed loop configuration of the H-infinity optimization process (Fig are respectively the reference and measured positions, and ε is the error between the reference and measured data. For the mixed sensitivity H ∞ optimization method, in order to shape respectively the sensitivity function S, the transfer G.S, and the complementary sensitivity function T (T=1-S), weighting filters (Wi) are introduced as shown in Fig. b. The W 1 filter parameters are chosen in order to reduce the error (ε) between the reference and measured position. W 2 is integrated to perform the control signal (u) quality which is the phase shift (φ) in this case, by overcoming the perturbation effects caused for example by the load torque. The input perturbation b is the output of the weighting filter W 3 which is addressed to reduce the effect of eventually electrical noises injected to the control signal.

As explained in chapter II, for the system configuration shown in Fig. III-26.b, the controller K(s) must satisfy the inequality given in II-11 by:

S SG KS KSG

For the USR60 plant, the desired closed loop performances are fixed in order to achieve a time response (t r ) lower than 10 ms with a gain and phase margins respectively of 10 dB and 45°. The steady state error ratio of the positioning system should be lower than 0.1% of the final position. The weighting filter parameters will be then determined to satisfy the performances listed above. The transfer 1/W 1 should have a low gain at low frequency to satisfy the steady state error criteria while its cut-of frequency represents the system bandwidth. The cut-off frequency of W 2 will be chosen much higher than that of W 1 to limit the maximum system bandwidth and therefore to avoid very fast system responses and high control signal values. The W 3 will be chosen as constant and can be readjusted as function of the introduced noises. The obtained weighting filters for the USR60 plant are: . ;

. .

; 0.1

III-22

Based on the filter parameters, the optimal controller and gamma constant can be found using the MATLAB function "hinfsyn":

K s . . . . . . . . . . . ² . . . ; γ 1.05 III-23
The controller order equal to the sum of system and filters order. Therefore, for the USR60 case, K(s) will be a fourth order controller. As given in Eq.III-23, the obtained controller contains zeros and poles outside the system operation frequency range. The useless zeros and poles can be reduced via the "reduce" MATLAB function. Finally, an optimal second order controller is obtained: K s 3.8. 10 s 1.12. 10 S² 2.6. 10 s 3.3. 10

III-24

As explained above, the H ∞ feedback controller and the associated weighting filters must satisfy the inequalities given in II-11. Therefore, the filters and system transfer functions behavior are verified As can be seen, that the desired performances of the closed loop can be easily introduced in the frequency domain via the H ∞ control principle with the minimum controller degree. The robustness of this controller is also ensured and will be confirmed later by simulation and experimental results. This plant is designed to be controlled so that the closed loop will have a second order type with a damping factor of 0.6 and a bandwidth of 500 rad/s. Thus, a discrete characteristic polynomial A m (z -1 ) is given by:

1 1.92 0.932 III-26
By the resolution of the primary Diophantine equation (ref), the polynomials S(z -1 ) and R(z -1 ) can be obtained:

1 0.02 ; 1.47 1.38 III-27
In order to track sine-wave position trajectories with angular frequency ω 0 =10rad/s, the auxilliary Diophantine equation (II-35) is rewritten as: The discrete time synthesis of the RST controller minimizes the delays introduced by the discretization stage of a continuous regulator. The robust pole placement method is effective for the tracking of polynomial reference signals. Moreover, the third degree of freedom which is the polynomial T, can be easily used to ameliorate the tracking performances.

1

III.7.3 Simulation of closed loop USR60 positionning system

The simulation results will be presented in this section to verify the performances of the proposed positioning system. The closed loop simulation results using the motor TF model (G USM (s)) will be firstly discussed to prove the stability, time response, precision, and robustness of the proposed system. Afterwards, the simulation results with the developed electro-mechanical model of USR60 (section III.4) will be presented to confirm that the proposed model is suitable for control purposes. In order to prove the advantage of the proposed controllers, PID controller is compared to the H ∞ and RST for the robustness issues.

III.7.3.1 Transfer function model simulations

The simulation results of the closed loop system are presented to verify the controller performances in terms of stability, time response, precision, and robustness. The fixed response time and the stability of the closed loop system are guaranteed with the two proposed controllers as illustrated in Table III It should be mentioned that the minimum margins to guarantee the stability of a closed loop system are around to 10 dB and 45° for the gain and the phase margins.

-2, T (z -1 ) 1 / S (z -1 ) R (z -1 ) B (z -1 ) / A (z -1 ) r (z -1 ) + - + + + + d (z -1 ) v (z -1 ) + + y (z -1 ) b (z -1 ) u (z -1 ) Controller USR60 plant
The phase shift (φ) between the two driven voltages of TWUSM is limited between +/-π/2 for practical use, because outside of this range the motor behavior will be highly nonlinear [START_REF] Visintin | Differential models of hysteresis[END_REF]. The control signal saturation between +/-π/2 will be taken into account in the simulation and experimental tests. To simulate the effect of electrical noises injected the generated control signal (φ) or to the measured position (θ), external Gaussian random noises are injected to the closed loop system to examine the perturbation rejection capability of the position controllers. b, the robustness of the H ∞ can be easily distinguished compared to the rest of controllers. In fact with the H ∞ controller, we keep a maximum position distortion of +/-0.007 rad for a step position of 1 rad. Moreover, the time responses of the positioning system based on PID and RST controllers is affected by these noises, and this is not the case with the H ∞ controller.

Table III-2: Stability margins

III.7.3.2 Simulation results based on electromechanical model

The

III.8 Real-time implementation of TWUSM closed loop positioning system

The compact experimental platform shown in Fig. III-17 is used for the real time implementation of the positioning system. As explained in section III.6.1, the proposed position controllers are implemented in the Matlab/Simulink environment and compiled via the dSPACE board (with a sampling time of 0.1 ms). In order to prove the advantage of the proposed method, the H ∞ , RST, and conventional PID controllers are experimentally implemented. A graphical interface (Annex A) is also built to configure the motor operation mode, and to change the system parameters in order to evaluate the performances of the closed loop system. Via the graphical interface one can also switch between the open and closed loop operation mode, and choose the motor motion trajectory.

III.8.1 H-infinity test results

The motor tests show that the mechanical resonance frequency and consequently the motor speed changes as functions of the motor temperature. Because of this, the closed loop tests are done with a driving frequencies higher than f r to avoid the instability around the resonance frequency. For this reason, the response time with the experimental tests is higher than obtained with the simulation results at the resonance frequency. The H ∞ controller is tested with different position trajectories, variable driving frequencies, and in the presence of load torque. In In this work, the challenge is to validate the precision and the robustness of the positioning system based on TWUSM even in the presence of dead zone. While the motor speed is load-depending, it is an expected result that the motor response time increases with load increasing as shown in It should be mentioned also that for the case of 0.35 N.m the dead zone reaches +/-30°. Due to the friction system between the stator and rotor of the TWUSM, the motor temperature will increase if no cooling system is used. The motor parameters and performances of TWUSM are temperature-depending [START_REF] Tavallaei | Robust motion control of ultrasonic motors under temperature disturbance[END_REF][START_REF] Lu | Analyses of the temperature field of traveling-wave rotary ultrasonic motors[END_REF][START_REF] Li | Temperature evaluation of traveling-wave ultrasonic motor considering interaction between temperature rise and motor parameters[END_REF]. In order to validate the performances of the proposed system under any motor temperature (up to 55°C), the H ∞ is tested under different motor temperatures (Fig. ). The precision is always guaranteed under temperature increasing, where for an operation temperature of 51°C the maximum static error is of 0.4° for a rotation of 90°.

III.8.2 RST test results

The RST position controller is synthesized in discrete time with a sampling period of 0. The robustness of the H∞ controller can be easily distinguished in these cases, where the controller gives a static error less than 0.1° in case of 90° rotation under 0.2 Nm. The resulting static error ratios of the three controllers in the case of loaded motor step response (90°) are summarized in Table III-3. From this table, it is clear that the very high precision levels can be achieved with the H∞ controller when the motor operates under load conditions.

Table III-3: Static error as function of load torques

III.9 Conclusions

This chapter deals with a high precision positioning closed loop system based on traveling wave ultrasonic motor. In order to achieve this goal, we started with the description of the working principle of TWUSM type USR60. A survey about the developed models of TWUSM and the related position controllers was presented in order to highlight the contributions of the proposed research work.

The problem with the TWUSM models for control issues is that; the Equivalent Circuit Models (ECM) which are generally used for TWUSM control purposes [START_REF] Mojallali | Systematic modeling for free stators of rotary piezoelectric ultrasonic motors[END_REF][START_REF] Mojallali | A new scheme for experimentalbased modeling of a traveling wave ultrasonic motor[END_REF] for the sake of simplicity, lack of the ability to capture the dynamic behavior of the contact zone between the stator and rotor which is at the origin of the torque generation. In other hand the developed analytical and numerical models which describe the highly nonlinear behavior of the contact mechanism [START_REF] Morega | Numerical study of the stator motion in a piezoelectric ultrasonic motor[END_REF][START_REF] Senjyu | Mathematical model of ultrasonic motors for speed control[END_REF][START_REF] Ghouti | Hybrid modeling of a traveling wave piezoelectric motor[END_REF][START_REF] Maas | Simulation model for ultrasonic motors powered by resonant converters[END_REF] are very complex for control design and implementation. A simplification effort of a full hybrid model was reported in [START_REF] Alsabbah | Mise en oeuvre de techniques issues de l'intelligence artificielle pour le contrôle de moteurs ultrasoniques à onde progressive[END_REF] for speed control implementation, but it leads to neglect the vertical motion of the rotor by imposing a constant length of the contact zone and a zero rotor vertical speed. Moreover, the proposed model can be controlled only via the driving frequency, and in consequence in only one motion direction.

The contribution of the designed model is that: Firstly, it is able to capture the dynamic of the motor including the traveling wave generation process at the stator surface, the contact mechanism leading to the torque generation, and the vertical and angular motions of the rotor. Secondly, it permits the implementation of motion control in the clock/counterclockwise directions via the supply voltages amplitude, frequency, and phase shift. Moreover, via the established USR60 model in the Matlab/Simulink environment, one can evaluate the effect of the stator (damping, stiffness, and coupling factor) and contact surface parameters on the output motor performances. The model simulation results reflect the behavior of the USR60 output characteristics variation published by the manufacturer. In second time it served to implement robust closed loop positioning controllers using the voltages phase shift as control signal.

Compact experimental positioning system based on USR60 is setup, and the motor characteristics are analyzed via the open loop tests. The experimental evaluation of the USR60 output characteristics shows that the frequency drive method is more suitable for speed control and not for position control. This is due principally to the "pull-out" phenomenon, where the motor shut-down immediately if the driving frequency decreases below the mechanical resonance frequency. Whereas, the resonance frequency is variable and depending on motor temperature and external loads. Moreover, the immediate switching between the on and off motion states of the USM is not feasible using the driving frequency, where a progressive increase of the frequency is needed to restart the motor. The USR60 was also driven in open loop using supply voltage amplitudes, where a minimum voltage of 80 V rms is needed to move the motor. The speed-V rms characteristic has a quasi-linear behavior, but a large variation of supply voltage (40 V rms ) produces a small speed variation and in consequence reduced motion range. In addition, the bidirectional motion feature is missing using this method.

The USR60 drive using the voltages phase shift is more suitable to control the shaft position. The motor can move in bidirectional motion by inverting the sign of the phase shift, and the switching between the on and off motion states can be done immediately by imposing a zero phase shift. However, the speedphase shift experimental characteristic shows dead zone when a load torque is added. The effect of this dead zone must be overcome to achieve high precision levels. In other hand, when the motor reaches the desired position, it has a high holding torque even without supply. The drawback of the drive technique using the voltages phase shift is that in "off" motion state, the motor continuous to consume power while the voltage amplitude is different to zero.

The voltage phase shift was chosen as position control signal in order to meet the specific requirements of the motor application. A Transfer Function (TF) between the control signal (phase shift) and the motor angular position is then experimentally identified. The H-infinity and RST position controllers are then synthesized on the basis of the TF model.

The advantage of the proposed control methods is that, they combine the precision performances and the robustness requirements in one simple design controller. The goals are to overcome the nonlinear behavior of the motor caused by the resonance structure, motor heating, and load torques leading to motor parameters variation, and to reject the external perturbation without supplementary compensation systems or complex adaptation methods.

The closed loop positioning systems based on H ∞ and RST controllers are simulated in closed loop using the designed electromechanical USR60 model. The simulation results confirm the accuracy of the proposed systems, and the robustness to model parameters uncertainties. The simulation comparison with PID control proves the advantage of the H ∞ in term of robustness to external disturbances. Afterwards, the positioning system is implemented in real time via a dSPACE microcontroller. A graphical interface is designed to configure the system parameters and test conditions in real time. The H ∞ and RST positioning accuracies in closed loop for the no-load conditions reached the maximum resolution of the position sensor (0.09°). While the motor temperature is a critical point for USM, the accuracy of the proposed system is also validated in case of motor heating. When load torques are applied to the motor shaft, the H ∞ and RST experimental positioning results show higher precision levels than the PID. This is because that in the synthesis approaches of the two first controllers the load torque is taken as disturbance and the related sensitivity functions are optimized in order to overcome the dead zone effect. To avoid the load torque effect on the PID positioning results, a supplementary dead zone compensation system is needed. Electrical noises are injected to the control signal (phase shift) and to the measured position data in order to evaluate the perturbation rejection capability of the tested controllers. The H ∞ control method shows higher robustness levels with respect to the external noises compared to the RST and PID controllers. This is due to the flexibility design of the H ∞ given by the three weighting filters, where one is selected to meet the precision and response time requirements, while the two others are designed to reject the external disturbances in a specific frequency range. The rejection capability of the RST controller can be improved by a rearrangement of the closed loop poles in discrete time, but at the expense of precision performances. For the PID controller, an on-line tuning parameters as function of the position and control signal quality is required to ameliorate the robustness to external disturbances.

Finally, rotary TWUSM positioning system based on H ∞ control method fulfills the requirements of high precision positioning applications including robotic engineering. The advantages of the proposed system are the accuracy, fast response, and robustness to system parameters uncertainties, load torques, and external perturbations without additional compensation systems or online parameters adaptation. These features make it suitable to be integrated into practical embedded system where the compact size, precision and robustness are important

IV.1 Introduction

The Piezoelectric Drive Actuator (PAD) is a novel structure of non-resonant piezoelectric motor which was introduced by Kappel et al. [START_REF] Kappel | PAD-Piezoelectric Actuator Drive[END_REF] in 2006. The working principle of PAD is based on the conversion of the linear actuator elongations into a precise rotation. The high torque (up to 5 N.m) at very low speed, high accuracy, low inertia, and high holding torque without supply characteristics make the principle use of PAD in robotic applications.

A few works related to the modeling and closed loop position control of the PAD are reported in the literature. The proposed PAD models are used to characterize the motor in static and dynamic operation modes, and for parameters identification. In [START_REF] Mangeot | Operation of a quasi-static piezomotor in transitory frequency range up to resonance[END_REF], an equivalent mechanical dynamic model of the stator combined with a Masson model are proposed in order to identify the motor parameters through the frequency analysis. A dynamic model of the PAD is also proposed in [START_REF] Giraud | Dynamic model of a PAD actuator: dynamic operations and pull-off at high speed[END_REF] to investigate the dynamical behavior of the motor with a special emphasis on the pull off. A generalized model is presented in [START_REF] Sven | Aufbau und Verifikation eines dynamischen Modells für einen piezoelektrischen Stellantrieb (PAD)[END_REF] to analyze the PAD output characteristics as function of the load torque and the actuator damping factors. However, PAD models for control purposes are not discussed in the modeling related works.

Due to the special structure of the PAD, the motor has a good position resolution in open loop. Nevertheless, the nonlinear behavior of the used stack actuators [START_REF] Sven | Aufbau und Verifikation eines dynamischen Modells für einen piezoelektrischen Stellantrieb (PAD)[END_REF], and the maladjustment between the stator ring and the rotor shaft (see section IV.3) which was widely discussed in [START_REF] Sven | Aufbau und Verifikation eines dynamischen Modells für einen piezoelektrischen Stellantrieb (PAD)[END_REF][START_REF] Wallenhauer | Effiziente Ansteuerung und Nutzung des Sensoreffekts piezoelektrischer Aktoren[END_REF][START_REF] Zeichfussl | A method for auto-adjustment of a new piezoelectric drive[END_REF], can affect the position accuracy of the motor in ultra-precision applications. Therefore a closed loop control is needed to overcome these problems.

In this chapter, the working principle and features of the PAD will be firstly introduced. Electromechanical model of PAD7220 motor is presented for control purposes. The proposed model includes the orthogonal piezoelectric actuators and the motion of the stator ring, the micro-teeth mechanical system, and the angular motion of the shaft. Afterwards, the experimental positioning system based on PAD7220 is discussed. This setup is used firstly to evaluate the output characteristics in open loop as function of the driving parameters. Secondly, to implement in real time the synthesized robust closed position controllers. An experimental comparative study between the H ∞ , RST, and PID controllers will be presented at the end of this chapter to discuss the accuracy and robustness issues of the proposed strategies.

IV.2 PAD working principle and features

IV.2.1 PAD working principle

The working principle of the rotary PAD is mainly based on the conversion of periodic elongations of linear Piezoelectric Multilayer Actuators (PMAs) into continuous rotation of a motor shaft. The PAD is composed by four piezoelectric stack actuators orthogonally oriented and stiffly connected to a ring as shown in Fig. IV-1.a. The ring has a square exterior profile with a circular cutout in its center which houses the motor shaft [START_REF] Zsurzsan | Control and sensor techniques for PAD servo motor drive[END_REF]. The shaft is fixed to the motor housing through ball bearings and has a diameter d shaft slightly smaller than the internal diameter of the ring d ring . The inside surface of the ring and the circumference of the shaft are machined into micro-mechanical teeth with diametral pitch of 38 μm [71]. The ring and shaft have 313 and 312 teeth respectively. Powerful piezoelectric stack actuators (2080 N blocked force) are used to create the linear elongation with a large displacement (free displacement of 50 µm). The four piezoelectric actuators are driven by two sinusoidal voltages with a phase shift of 90° between them. Thus, the actuators expand and contract linearly and thereby push and pull against the motor ring. Thereby, the motor ring moves in a circular trajectory defined by Cartesian coordinate pair (r x , r y ) expressed by: sin 2 sin 2 IV-1 where x a and y a represent the elongations of orthogonal stacks (m) in the vertical and horizontal directions, f is the driving frequency (Hz), φ 1 and φ 2 are the driven voltages phase shift (rad) with the restriction that φ 1 -φ 2 = +/-90°.

The shaft rolls on the inner surface of the ring and thus putting it into rotation. Each full period of the applied sinusoidal voltage generates a full rotation of the ring around the shaft and this, in turn, will cause the shaft to step one tooth in the opposite direction Fig. IV-2 [START_REF] Zsurzsan | Control and sensor techniques for PAD servo motor drive[END_REF].

IV.2.2 PAD features

The PAD benefits from the conventional features of piezoelectric motors such as the high precision levels, high torque/weight ratio, high holding torque without supply, and no electromagnetic interferences. The combination of PMAs and rotary shaft using micro-teeth mechanical system offers the PAD additional interesting characteristics. The motor can easily be driven at very low speed (<1rpm) compared to other piezoelectric motors and up to 100 rpm. Moreover, this topology makes the PAD scalable in size and output power [START_REF] Kappel | PAD-Piezoelectric Actuator Drive[END_REF]. In fact, depending on the type, number, and size of the linear actuators, the dimension of the ring and the micro-teeth mechanical system can be readjusted. Therefore, the motor power and dimensions are adaptive as function of the application requirements. Fig. IV-3 gives an idea about the scalable technology of PAD proposed by Noliac [71]. In other hand, the special structure of the PAD enables a smart load sensing through the phase shift between the actuators charge and voltage as demonstrated in [START_REF] Wallenhauer | Accurate load detection based on a new piezoelectric drive principle employing phase-shift measurement[END_REF]. This phase shift is equal to zero in no-load conditions, and it is different to zero if a load torque is applied to the motor shaft. A relationship between the load torque and phase shift including the motor parameters is established in [START_REF] Wallenhauer | Accurate load detection based on a new piezoelectric drive principle employing phase-shift measurement[END_REF], and can be used for sensor-less torque control.

The PAD motor has also a special speed-load characteristic (Fig. IV-4). As the stack actuators permit a permanent contact between ring and shaft, the force transmission is independent of how fast the ring translates. Consequently, the rotational speed is not affected by the applied load [START_REF] Kappel | PAD-Piezoelectric Actuator Drive[END_REF]. The aforementioned features explain well the competitiveness of the PAD motor in compact and precise positioning systems, and especially the robotic applications.

IV.3 Modeling of Piezoelectric Actuator Drive (PAD7220)

The studied motor is a PAD7220 from Noliac [71] with rated speed and torque of 56.25 rpm and 4 Nm respectively (the motor datasheet is given in Annex B). In this section, Simulink model of the PAD7220 is proposed to evaluate the motor behavior as function of driving parameters, and in second time to simulate the proposed closed loop position controllers. The synoptic of the PAD model is given in Fig. IV-5. It consists of the supply part to generate the two phase shifted sinusoidal voltages (U x , U y ). The orthogonal linear piezoelectric actuators will produce the linear mechanical elongations and forces in x and y directions. The last model part represents the micro-teeth mechanical system between the ring and the shaft.

The behavior of the piezoelectric actuators can be described by the linear piezoelectric equations: ( = = ; = = ). The maladjustment between the ring and shaft position will be neglected (the shaft is in the centered position). In fact, the effect of the maladjustment on positioning performance must be compensated later by the closed loop control.

∆ 1 IV-2 IV-3
The center of the ring moves on a circular trajectory with a diameter of about a=19 μm [START_REF] Sven | Aufbau und Verifikation eines dynamischen Modells für einen piezoelektrischen Stellantrieb (PAD)[END_REF] (Fig. IV-7). The actuators generated torque is given by the dot product of the forces and displacement vectors of the PMAs as:

acos asin IV-7
The fundamental dynamic equation of the actuator-ring contact can be also written as:

IV-8
where, T s is the ring torque (N.m), and is the transmission inertia (kg.m 2 ).

In fact, for ideal transmission system between the motor ring and shaft (T r . Ω r = T s . Ω s ), the rotor generated torque (T r ) can be expressed by: IV-9

where; is the transmission ratio In other hand, the generated rotor torque can be written also as:

IV-10

where; is the load torque (N.m), is the shaft inertia (kg.m 2 ), and is the shaft damping factor (Nms/rad).

From the equations IV-10, IV-9, and IV-7, the acceleration of the motor shaft can be formulated as following:

. .

IV-11

The motor speed and position can then be deduced as function of the model parameters, which are given in details in the Annex B. It should be also mentioned that in case of maladjustment between the shaft and ring position (Fig. IV-9), the circular coordinates of the ring (Eq.IV-6) will be given as following:

acos asin IV-12
where and correspond to the shaft deviations in x and y directions respectively.

Moreover, if the identical behavior condition of the orthogonally actuators ( = = , and = = ) is not considered, the rotor acceleration will be written as:

. asin acos 1

IV-13

The model is implemented in the Matlab/Simulink environment, the driving voltage amplitudes are taken equal to 160 V pk-pk and the phase shift is fixed to 90°. The applied voltages to the PMAs and the equivalent charges at a frequency of 100 Hz are given in

IV.4.1 Experimental platform description

The experimental platform (Fig. IV-14) consists of a PAD7220 from Noliac coupled to an optical encoder with a resolution of 10000 p/r (Baumer, Go356). High resolution sensor is used because with this motor topology, high precision levels can be achieved. The load torque is generated using a powder brake from Merobel (FAS21). The torque is measured and conditioned via a mini-smart torque sensor with a control/monitoring board from Kistler. A dSPACE controller board (dS1104) is used for the real time closed loop control.

In the open loop configuration, the driving analog voltages (+/-10V) are programmed in the Matlab/Simulink environment and sent to the power amplifiers via the DAC interface (Fig.

IV.4.2 Frequency-position relationship identification

To apply the position controller synthesis approaches (explained in Chap.II), a Transfer Function (TF) model between the control signal which is the voltage frequency in this case and the motor angular position is needed. The TF will be identified through the integration of the experimental frequencyspeed relationship. Therefore, the motor is driven at different frequencies and the angular velocities are measured and collected. The nominal operation frequency of the PAD7220 is of 300Hz. With the tested prototype, the speed and position control signal (f) will be saturated in order to avoid high signal values. The saturation levels are variable (< 300Hz) to perform the operation at low and high rotation speeds.

As mentioned above, the PAD speed is load independent, and to experimentally confirm this feature, the motor is tested in case of no-load condition and under a load torque of 2 Nm. The evolution of the motor speed in the two cases under different frequencies is shown in

IV.5 Synthesis of robust position controllers of PAD

The synthesis approach of robust H-infinity (H ∞ ) and discrete time RST position controllers will be applied here to the PAD case. The performances of the positioning system based on PAD7220 will be fixed, and using the motor TF G PAD (s), the controller parameters will be determined. A proportional, integral, and derivative (PID) controller will be designed and experimentally implemented and compared to the proposed robust position controllers to prove the advantage of the proposed techniques.

IV.5.1 H-infinity position controller of PAD

The feedback H ∞ control method is applied to control the position of the PAD plant represented by G PAD (s) as shown in show that the motor can move at very low speed, which means it can be driven at very low frequency. Moreover, when the frequency signal is perturbed, the ring tooth vibrate inside two teeth of the shaft without passing to the next tooth. This phenomenon generates a lot of acoustic noises and decreases the motor precision. The frequency perturbation can be caused by the generation process of this signal inside Matlab Simulink with high sampling frequency and the DAC conversion stages, and by noises coming from the measured position data. Therefore, the weighting filters (W 2 and W3) of the closed loop positioning system must be carefully selected in order to avoid these problems. In other hand, the challenge is to achieve high bandwidth in closed loop with respect to the motor resonance frequency but without destabilizing the closed loop system. The stability criteria are performed via the weighting filter W 1 .

To guarantee these performances, a closed loop bandwidth of 300 rad/s with minimum gain and phase margins of 10dB and 45° respectively are fixed.

Based on the aforementioned performances criteria, and after checking of the first closed loop results, the obtained parameters of the weighting filters are:

. .

;

. .

; 0.01 IV-15

The resulting controller K(s) is a third order controller. This controller is reduced to simplify the implementation effort to a first order controller. Finamlly, the obtained H ∞ feedback controller and the optimal gamma value are then:

. .

. . ; 1.09 IV-16

IV.5.2 RST position controller of PAD

The RST control method is applied based on robust pole placement method in discrete time domain. Thereby, zero order hold discretization method is applied to the continuous time model with a sampling time of 0.1 ms, to obtain the discrete time PAD model G PAD (z -1 ) :

1.96. 10 1 IV-17
The PAD plant is designed to be controlled so that the closed loop will have a second order type with a damping factor of 0.6 and a bandwidth of 300 rad/s. Thus , a the reference denominator A m (z -1 ) is given by:

1 1.94 0.94 IV-18 1 0.94 ; 447 427 IV-19
The polynomials T(z -1 ) will be obtained by fixing an unity gain in steady state and by resolving the auxiliary Diophantine equations (II-28 and II-32):

IV.6 Simulation results

The stability of the closed loop system is guaranteed with both the H ∞ and RST controllers (Fig. IV-18) with a large margin phases equal to 88° and 58° respectively. For practical uses, the PAD must operate with maximum driven frequency of 300Hz as recommended by the manufacturer. The step responses of the PAD positioning system based on the H ∞ and RST controllers under frequency limitation show high accuracy without overshoots at different rotation speeds as can be seen in

IV.7 Real-time implementation of PAD closed loop positioning system

The experimental platform presented in Fig. IV-14 is used to experimentally validate the synthesized PAD position controllers. The experimental results of H ∞ and RST control methods will be discussed respectively. A comparative study with a PID controller for robustness issues will be presented at the end of this section.

The proposed position controllers are implemented in the Matlab/Simulink environment and compiled via the dSPACE board with a sampling time of 0.1 ms. The three controllers are inserted together in the same Simulink file, which makes the switching between them easier through the graphical interface. The goals of the proposed system is to achieve fast and ultra-precise positioning performances in closed loop under different working conditions, and to confirm the robustness to external disturbances. In fact, the motor behavior depends on the motion speed, on-line control parameters adaptation is mostly addressed to deal with this problem. So, the idea is to propose a high performance positioning system without additional parameters adaptation or compensation systems.

IV.7.1 H-infinity test results

The PAD positioning system based on H ∞ control is implemented in real time and the experimental results will be discussed here. The motor speed is adjusted by the variation of the maximum value of the generated control frequency. The step response of the H ∞ controller at low speed shows high precision without overshoot (Fig. . The steady state accuracy of the unloaded motor is of 0.009° (which is the maximum resolution of the sensor) for a motion of 90°. The high precision level of the closed loop system is confirmed in the two motion direction of the PAD as shown in The robust mixed sensitivity H ∞ approach gives a high precision level in static operation mode (very low speed) and until the quasi-static operation zone. Moreover, it proves the robustness to reject the external injected noises to control signal and to the measured position.

IV.7.2 RST test results

The RST control approach based on discrete time pole placement method is also implemented in real time. The step responses of the PAD positioning system based on RST controller show high precision levels for very low speeds (≈1.5rpm) and also for relatively high rotational speeds as illustrated in Fig.

IV

IV.7.3 Comparative study

A comparative study between the H-infinity, RST, and a PID controllers is carried out in this section. The comparison criteria are the accuracy of the positioning system in static and quasi-static operation modes, and the rejection capability of external disturbances. The H ∞ and RST controller parameters are the same given in section IV.5, while the PID parameters are tuned by trial-and-error through intensive tests in order to achieve similar performances with the other controllers. The resulting PID parameters are K p =2750, K i =0. The tracking errors and oscillations in the case of PID control, can be avoided by modifying the PID parameters (K p =3000). Thus an adaptive PID method as function of the operation mode is needed to achieve the required performances. The robustness of the positioning system to control signal and sensor perturbations is also evaluated. c. The comparative study confirms the advantages of the H ∞ position control of PAD system in terms of accuracy, stability, and robustness. The ultra-precision of the positioning system based on H ∞ control is confirmed at very low speed (low driving frequency) and at relatively high speeds. The weighting selection approach gives to this method high robustness levels to reject the external noises.

IV.8 Conclusions

This chapter deals with a high precision positioning closed loop system based on Piezoelectric Actuator Drive (PAD). The working principle and features of the novel piezoelectric motor structure are firstly introduced.

An overview of the proposed models of PAD was presented to highlight the contribution of the proposed model. In fact, the developed PAD models are addressed to capture the motor behaviors in static and dynamic operation modes, and to simulate the effect of the multilayer actuator parameters on the output motor characteristics. The contribution of the proposed PAD7220 model is that it describes the motor dynamic including the elongation of the multilayer stack actuators, the actuators/stator ring contact mechanism, and the motion transmission to the rotor via the micro-teeth system. Moreover, it permits a control techniques implementation in closed loop through the driving signals (voltage amplitude, frequency, and phase shift).

Compact rotary positioning system based on PAD7220 is designed. The experimental platform is used firstly to drive the PAD7220 in open loop to determine the output characteristics variation as function of the driving signals and external loads. The open loop test results show that the motor speed is constant whatever the load torque until the maximum torque. The independent speed-load characteristic gives it a success in industrial fields.

Due to the working principle of the PAD, it can be controlled in speed and position only via the driving frequency. In fact, the phase shift between the supply voltages must be equal to +/-90° in order to keep

(a) (b) (c)
the synchronization of the orthogonal piezo-actuators and to generate the circular trajectory of the stator ring. The amplitude of the driving voltages must be minimum equal to 160 V pk-pk , otherwise the stator ring tooth will stay blocked between two rotor teeth which generates an acoustic noise.

The control method using the driving frequency permits the smooth motion of the motor from very low speed (<1 rpm) and until 56.25 rpm, and an easy and fast switching between the on and off motion states of the motor. The two listed features are very important for pickup and place automation missions of industrial robots. In other hand, when reaching the desired position, the motor has a high holding torque even without supply. One demerit of the frequency-drive method, is that in "off" motor motion state, it continuous to consume power while the voltage amplitudes are different to zero.

The maladjustment between the stator ring and the rotor shaft positions is one of the open loop drive inconvenient. In fact, the stator ring position depends directly on the state of the multilayer actuators after operating cycles (contracted or expanded), and the rotor shaft will not be necessarily at the center of the ring. The solution was often an auto-adjustment technique for the stator/rotor position.

The speed-frequency characteristics are then used to identify the transfer function (G PAD (s)) between the angular position (θ) and the control frequency signal (f). The transfer function serves to synthesis the robust H ∞ and the RST control approaches based on the desired closed loop performance.

The advantage of the proposed controllers of the piezo-motor is that, it combines the high precision performance for a large bandwidth, and the robustness requirements to the uncertain motor parameters and external disturbances, in one simple design controller. The goal is to perform a high resolution in closed loop both in static and dynamic mode (high rotational speed), to overcome the maladjustment between the stator and rotor, and to show a high robustness with respect to the uncertain motor parameters and external electrical noises.

The precision, stability, and robustness of the proposed methods is firstly discussed through the simulation results of the closed loop system. Where the stability of the closed loop system was guaranteed for a large bandwidth. Afterwards, the position control techniques are implemented in real time via the experimental positioning platform. A PID controller is also experimentally implemented in order to establish a comparative study between the three controllers and to prove the advantages of the proposed methods. The three controllers show high precision levels in static mode (very low speed), however, when the rotational speed increases, the PID control fails to quickly track the reference position while the two other controllers keep the high precision levels reaching the maximum resolution of the sensor (0.009°) whatever the speed. The degrees of freedom of the H ∞ and the RST given by the weighting filters and the poles placement technique respectively, are at the origin of the high precision levels in a wide frequency band. Where, the controllers design in the frequency domains permits the adjustment of the amplification gains regarding to the frequency operation range, leading to tracking errors reduction. The precision and response time of the closed loop system can be improved using a PID controller, but a tuning parameters method as function of the operation conditions is mandatory. This online-learning and parameters adaptation complicates the real time implementation of this controller.

The robustness of the positioning system with respect to the control signal and measurement noises is evaluated. Electrical noises are injected respectively to the driving frequency and to the position sensor data. The advantages of the H ∞ in term of perturbations rejection can be clearly distinguished compared to the RST and PID controllers. The superiority of the H ∞ in term of perturbation rejection capability is explained by the design flexibility of the weighting filters (W i ). Where, the parameters of the filter W 1 are fixed to mainly improve the precision and response time performances. Whereas, the parameters of W 2 and W 3 are principally selected to reject the additional perturbations in a specific frequency range. The robustness of the RST controllers to external disturbances can be ameliorated by a rearrangement of the pole locations in the discrete time domain, but it is at the expense of precision and response time performances.

Finally, rotary positioning system based on quasi-static PAD positioning system can fulfill the performance requirements of robust ultra-precision applications using mixed sensitivity H ∞ control method. The accuracy and robustness of this system are guaranteed in the static operation mode and in relatively high PAD rotation speeds and under uncertain operation conditions.

V.1 Introduction

The Walking Piezoelectric Motor (WPZM) is a quasi-static piezoelectric motor. It consists of two or more sets of drive piezoelectric legs alternately in friction contact with a movable rod in order to enable the motion steps or walks (the working principle will be detailed in section II). It was the subject of many patents [START_REF] Johansson | Fine walking actuator[END_REF][START_REF] Johansson | Switched actuator control[END_REF][START_REF] Johansson | Fine control of electromechanical motors[END_REF], where the motor's principle of operation and construction methods are detailed.

The WPZM can generate a translational or rotational motion depending on the mechanical configuration and it overcomes the very low speed of inchworm motors. The high force and high stiffness of the WPZM with high resolution (<1nm in open loop), small size, high holding force without supply, and non-electromagnetic interferences [START_REF] Spanner | Piezoelectric motors, an overview[END_REF], make this motor useful in ultra-positioning system including zooming and optical image stabilization, microscopy and biotechnology research such as cell manipulation, actuation of robot arms, guiding of surgical MRI robots, microscopy and biotechnology research such as cell manipulation, semiconductor processing, actuation of hexapod robots, and many other applications [START_REF] Spanner | Piezoelectric motors, an overview[END_REF]. The positioning performances of WPZM are proved in open loop and in closed loop for reduced motion range (nano/micro scales). Nevertheless, the high precision level for large motor stroke in closed loop and under load conditions is not yet confirmed.

V.2 Literature Review of WPZM position controllers

Positioning in macro and micro scales requires an accurate and robust position control techniques. Since the PZMs have non-linear behavior and inherit hysteresis, creep, and vibration, a closed loop operation mode is often mandatory. In [START_REF] Merry | Using a walking piezo actuator to drive and control a high-precision stage[END_REF], a WPZM is used to drive a single DOF position stage. The system was PID feedback-controlled by prescribing the driving voltages frequency from the measured position.

The voltages amplitude and phase are controlled based on reference velocity and using feed-forward control strategy. The presented experimental results show significant overshoot and steady state error when changing the amplitude and phase of the voltage waveforms with overlapping trajectories. Similarly, for the same motor, relations between the motor step size and the voltage amplitude and phase are derived in [START_REF] Golubovic | Adaptive control of piezoelectric walker actuator[END_REF], and by applying a cascaded type adaptive controller of step size, the positioning precision and overshoots are improved. Feedback control with Gain Scheduling (GS) of PiezoLEGS [62] motor is proposed in [START_REF] Merry | Gain scheduling control of a walking piezo actuator[END_REF]. The overshoot and oscillations problems still exist in the proposed PID control based on Hadamard transform of the applied voltages for a WPZM working in bending mode [START_REF] Golubovic | Adaptive control of piezoelectric walker actuator[END_REF][START_REF] Zhakypov | High precision control of a walking piezoelectric motor in bending mode[END_REF]. Based on the x-y elongations of the piezo legs, coordinate transformation of driving voltages is presented in [START_REF] Uzunovic | Piezo LEGS driving principle based on coordinate transformation[END_REF]. A called virtual time controller is then designed for the modulation of the step frequency as function of the error amount. The motor behavior under load conditions was considered in positioning systems, and force controllers should be applied to handle loading tasks. In [START_REF] Szufnarowski | Compliant piezo-flexdrives for muscle-like, antagonistic actuation of robot joints[END_REF], forcemotor speed relations are identified for PiezoLEGS motor. Based on this knowledge, motor is linearized by dynamic compensation, and PI control is applied. Therefore, an additional force sensor is needed and the speed variation as function of the applied force must be approximated to a linear relationship. Polynomial control technique for adaptive positioning was proposed in [START_REF] Škorc | Polynomial control technique for adaptive positioning within sub-micrometer scale[END_REF]. It is implemented by changing speed and motor step size to decrease overshoot and achieve desired position. The speedovershoot dependence characteristics are determined through experimental tests under three different step positions. Therefore, the polynomial control feasibility is limited to the relationships established in the range of the tested nano-step position. Repetitive Control (RC) to compensate disturbances that are independent of time are also used in [START_REF] Merry | Repetitive control applied to a walking piezo actuator[END_REF][START_REF] Merry | Delay-varying repetitive control with application to a walking piezo actuator[END_REF]. The RC is designed based on a periodic and repetitive disturbance. However, it is not always the case for a piezoelectric motor. In [START_REF] Škorc | Position control with parameter adaptation for a nano-robotic cell[END_REF], position control with parameter adaptation of XYZ nano-robotic cell is discussed. The position stage is actuated by PiezoLEGS and the control strategy uses a simple bang-bang control with step size modulation to achieve very precise positioning without overshoot. However, for the last two methods, the positioning accuracy is not improved for the full motion range of the motor but only for the nano-metric scale.

V.3 Contributions and outlines

In this chapter, the effectiveness and reliability of long travel (up to 50 mm) positioning system based on linear walking piezoelectric motor (WPZM) are studied. The contribution is to prove the high accuracy of WPZM in closed loop at low and high speeds (up to 10 mm/s) and under external applied loads (up to 10 N). This is in order to prove the feasibility of using such motor for robotic applications demanding high precision levels with relatively high generated forces. Experimental single DOF positioning system is designed and the motor position is controlled in real-time via dSPACE board. A laser position sensor is used to ensure high position resolution. For the synthesis approach of position controllers, a Transfer Function (TF) between the control signal and the motor position is experimentally identified in open loop. Then, robust H-infinity (H ∞ ) and discrete time RST position controllers are synthesized based on the identified TF. The proposed controllers are implemented in real time and the positioning accuracy and robustness are evaluated and compared with conventional PID controller.

This chapter is organized as follows. First, the working principle of the WPZM is explained in section II. In section III, the experimental test bench of the single DOF positioning system, and the motor transfer function identification are detailed. The synthesis approach and the simulation results of Hinfinity and RST position controllers are illustrated in section IV. The real time implementation of the proposed controllers, the evaluation of motor positioning performances, and comparison of accuracy and robustness results are discussed in section V. Finally, conclusions and discussions will be drawn in section VI.

V.4 Working Principle of Walking Piezoelectric Motor

The motor under consideration in this study is a commercially available linear WPZM (N-310.

V-1

The legs elongate in y-direction when equal voltages are applied to the two piezo stacks of one leg. Applying different voltages on the two piezo stacks of one leg causes the leg to bend. According to the applied voltages, a leg will perform a periodic movement in the plane (Fig. V-1.b). During the operation, the legs will drive a preloaded rod due to the friction force formed at the contact surface. The PZLs are driven in pair of two and perform a stepping motion that causes a forwards and backwards feed of the rod. As can be seen in Fig. V-1.b, the first pair of legs p 1 consisting of legs A and C is driven by the voltages U 1 (t) and U 2 (t). The second pair p 2 , consisting of legs B and D is driven by the voltages U 3 (t) and U 4 (t). In the stepping mode, the voltage waveforms are chosen such that at all times only one pair of legs is in contact with the drive surface (phase shift of π rad) and such that the pairs of legs perform a movement in x-direction for the rod. Each pair of legs will perform the clamping or fed of the rod depending on the applied voltages as explained in Eq.V-2.

: 1 pair Leg  : 2 pair Leg  V-2
The shape of the elliptical trajectory performed by each leg is determined mainly by the voltage amplitude A and phase φ [START_REF] Jung | Tracking control of piezoelectric actuators[END_REF]. The effects of A and φ on the elliptical motion of legs are explained in Fig. V-2. The elliptical shape variation will then change the motor step size. The N-310.13 step size can be varied between 5 nm and 5 μm and it depends also on the applied external forces [67]. Whereas, the drive frequency determines the number of elliptical trajectories per second and therefore the motor velocity. The latter depends on the step size and the drive frequency i.e. Speed (m/s) = step size (m) ×frequency (Hz). The studied motor has a maximum input voltage of 45 V and a maximum driven frequency of 1.5 kHz. A dSPACE board 1103 is used for the real time position control of the WPZM. The position sensor data are processed by a G3001PV conditioner that will generate analog voltage (+/-10 Vmax) as function of the measured position. The position data is introduced to the control program via the ADC interface. Three output signals are sent from the dSPACE DAC interface to the E-862 driver: one of 5 V as supply of the electronic circuit, 0/5 V signal to enable/disable the motor motion (0: enable, 5: disable), and the analog control signal of +/-10 V. The synoptic of the closed loop diagram of the experimental setup is illustrated in Fig. V-3.

V.5.2 Motor transfer function identification

The transfer function (TF) identification approach consists on applying a control signal (Vc (V)) step change and measuring the corresponding motor velocity (dY/dt (mm/s) in open loop. Therefore, the TF between the motor position and Vc will be the integrator of the identified TF between dY/dt and Vc. A control signal step change is then applied, the motor velocity is derived from the position sensor data as illustrated in Fig. V-5. The analog control signal of the linear WPZM is between +10 and -10V, i.e. the maximum speed is for +/-10 V (in the two motion directions) and for 0V there is no motion. The motor velocity depends on the drive frequency and on the phase shift (for motion direction).Therefore, the control signal will act on the step frequency and the voltage phase shift (+/-90°). The experimental open loop tests show that the motor velocity is not always the same for the right and left motion direction under the identical drive voltage waveforms. This nonlinearity is also mentioned by the motor manufacturer (PI). The controller parameters will be determined using only the given TF (V-3), and we will rely on the robustness of the proposed controllers to ensure the high precision levels in the case of uncertain motor parameter uncertainties (due to speed variation or applied load).

V.6 Synthesis of robust position controllers of WPZM

The synthesis approach of robust H-infinity (H ∞ ) and discrete time RST position controllers will be applied here to the WPZM case. The performances criteria of the positioning system based on WPZM will be fixed, and using the motor TF G WPZM (s) (Eq.V-3), the controller parameters will be determined.

The control signal (u) which is the analog voltage V c in this case is limited between +/-10 Vmax (V.5.1) . The motion speed of the linear motor is performed via the value of the control signal, therefore, V c will be saturated at different levels to change the motor speed.

V.6.1 H-infinity position controller of WPZM

The classical closed loop system configuration (Fig The control signal distortion is caused by the motor parameters variation under different test conditions, and from the position sensor with analog output signal (+/-10 V of laser sensor conditioner) and the conversion stages. Therefore, the filter (W i ) parameters of the H ∞ controller must be carefully chosen in order to reject the introduced disturbances and to internally stabilize the closed loop system. As the synthesis method is explained in chapter II, the performances of the positioning system based on the N310.10 motor will be introduced to deal with the WPZM case. The first challenge of the designed controller is to achieve a fast response without instability with respect to the maximum operating frequency of the motor, the closed loop bandwidth is fixed to 100rad/s with a minimum phase and gain margins respectively of 15 dB and 50°. The motor will guarantee high precision levels in closed loop with a steady state error ratio lower than 0.1%. Based on the listed performances and after checking of the first closed loop test results, the obtained weighting filters for the N-310.10 plant are: . ;

. .

;

.

V-4

Based on the filter parameters, the optimal controller and gamma constant satisfying the H ∞ norm can be found using the MATLAB function "hinfsyn". Otherwise, the obtained fifth order controller contains zeros and poles outside the system operation frequency range. The useless zeros and poles can be reduced via the "reduce" MATLAB function. Finally, a second order controller and optimal gamma gain are obtained:

V.8 Real-time implementation of WPZM closed loop positioning system

The experimental platform shown in Fig. V-4 is used to experimentally validate the performance of the positioning system. The experimental results of the H ∞ and RST control methods are discussed respectively. A comparative study with a PID controller for robustness issues will be presented at the end of this section. The proposed position controllers are implemented in the Matlab/Simulink environment and compiled via the dSPACE board. The three controllers are inserted together in the same Simulink file, which makes the switching between them easier. A graphical interface is also built to configure the motor operation mode, and to change the system parameters in order to evaluate the performances of the closed loop system. Via the graphical interface one can also switch between the open and closed loop operation mode, and choose the motor trajectory and the rod full stroke. In order to verify the robustness of the proposed controllers in case of low and high motor speeds, the velocity of the motor in closed loop can be also selected and changed in real time via the graphical interface. The purposes of the proposed system is to design an ultra-precision positioning system based on WPZM for micro and macro-scales (up to 40 mm). The accuracy and robustness of the closed loop under the system parameter variation due to the motor speed variation and external applied forces must be validated.

V.8.1 H-infinity test results

The mixed sensitivity H ∞ controller is tested with the parameters given in section V. the maximum static error is of 5 µm for a displacement of 15 mm. The closed loop test results confirm the high precision levels of the proposed H ∞ at low and high motor speeds. The robustness of the controller in case of model uncertainties due to the nonlinear behavior of the motor or to the applied force is also confirmed.

V.8.2 RST test results

The discrete time pole placement method based on RST controller is also implemented and evaluated. The experimental and simulation results given in Fig. V-16 validate the model conformity and the high precision levels when the motor moves with 2 mm/s of speed with a static error of 5 μm. From motor speed of 5 mm/s and higher, the closed loop response of the RST position controller presents a lot of overshoot up to 0. However, from a load of 5 N, the response time is higher than the expected values with an oscillation in steady state, which complicates the tests with higher load forces.

The experimental test results show that the proposed discrete time position control based on RST method is not robust in case of motor parameter uncertainties due to the speed variation or the applied external forces.

V.8.3 Comparative study

In this section, the H ∞ and RST techniques are compared to a PID controller. The synthesis approach of the H ∞ and RST parameters was already detailed. The PID controller parameters are determined based in the motor TF (V-3) and tuned by trial -and-error through intensive tests in order to achieve similar performances with the other controllers in normal conditions. The resulting proportional, integrator, and derivative gains are respectively equals to 48, 0.01, and 1.4. The comparison criteria are the positioning accuracy at low and high motor speeds, and the robustness in case of model uncertainties due to the nonlinear behavior of the motor or to an external forces. The experimental results of the three proposed controllers in no-load condition and with a velocity of 2 mm/s is given in Fig. V-18.a. As can be seen, all controllers present fast responses without overshoot in normal operation mode. In order to evaluate the performances of the positioning system at higher motor velocity, the motor is driven with 8 mm/s speed, and the tracking capability of 10 mm position steps sequence is tested. Using the graphical interface, we switch in real time between the three controllers as shown in Fig For a load force up to 7.5 N, the steady state error of the H ∞ controller is always less than 10µm. However, the response time of the RST controller is much affected by the applied load, and the PID method shows a weak precision (150 µm of static error for 7.5 N) with overshoot. The effect of both load force and speed variation is tested when an external force of 4 N is applied to the motor rod and moving at different speeds as shown in Fig. V-21. In terms of overshoot and oscillation, the H ∞ controller shows high robustness levels compared to the RST and PID controllers. Similarly, for the steady state errors, as can be seen in the zoomed areas that the maximum static error is of 8 μm for the H ∞ , and of 40 μm and 25 μm for the RST and the PID controllers respectively.

From the briefly comparative study between the three proposed position controllers, it is clear that the precision and robustness of the H-infinity controller are very high compared to the RST and PID controllers. With the H ∞ controller, the motor model uncertainties and the effect of the speed variation are completely compensated. The high precision levels under load conditions is also guaranteed. Moreover, with this robust controller, the stability of the closed loop system with large bandwidth is guaranteed.

(a)

V.9 Conclusions

High precise closed loop positioning system based on linear WPZM is presented in this chapter. The studied motor (N-310.13) has a full stroke of 50 mm and generates force up to 10 N. The motor structure and working principle are explained in details.

An overview of the proposed positioning systems based on WPZM and the corresponding position control methods was presented at the beginning of this chapter to highlight the contributions of our proposed system. The reported position controllers of WPZMs are limited to the nano/micrometric scales which are useful for cells manipulation and microscopy applications. Furthermore, depending on the application requirements, the accuracy of the closed loop system was performed for slow motion to avoid the nonlinear behaviors when operating close to the dynamic operation range.

The contributions of our closed loop WPZM positioning system are; the high precision level is proved for the total motor stroke (50 mm) and whatever the motion speed (until the maximum speed), the robustness to uncertain motor parameters caused by the external forces and the nonlinear behavior in dynamic mode is confirmed. The advantage of the proposed method is that, the position control method combines the precision and robustness requirements in one simple design control.

The motor is driven firstly in open loop in order to evaluate the output characteristics as function of the control signal (V c ) and external forces, and to experimentally identify the equivalent second order Transfer Function (TF) model between the slider position (Y) and V c . Robust mixed sensitivity H ∞ and discrete time RST controllers are then synthesized based on the identified TF.

The experimental tests prove that the WPZM is an accurate actuator for high precise positioning (nanometer resolution) with relatively high generated forces, provided that the position control signal be carefully selected. In fact, the least perturbation of the control signal (electrical noise) generates acoustic noises and a lot of oscillations around the final position of the slider. The vibration of the slider is a critical point when the motor is used to guide a needle in surgery robot [START_REF] Su | High-field MRI-compatible needle placement robots for prostate interventions: pneumatic and piezoelectric approaches[END_REF] .

The experimental positioning platform is used also to validate the closed loop position control methods. The H ∞ and RST are compared to a PID controller in order to confirm the advantage of the proposed method. The motor position is measured with a laser sensor and the controllers are executed in real time via a dSPACE microcontroller. The output signal of the position sensor and the motor control signal are both analog voltages, therefore two ADC/DAC conversion stages are mandatory. These conversion stages in addition to the high sampling frequency of the real time control block into Matlab/Simulink affect the control signal quality. Thus, the weighting filters of the H ∞ and the pole locations of the RST control must be carefully selected in order to overcome these perturbations.

Due to the special structure of the motor and the nonlinear behavior of the used piezoelectric actuators to create motion, the motor TF is uncertain and depending on operation conditions. Therefore, the closed loop positioning system is simulated in case of motor parameter uncertainties. The proposed position controllers show high robustness to the motor gain and constant time variation with an amount up to 50%. The rejection capability of injected electrical noises of the proposed system was also discussed. The results prove that the H ∞ is more robust to external perturbations, especially when they are injected to the measured position.

The experimental results prove the ultra-precision of the positioning system for the micro and macroscales motion. The high precision of the closed loop system is validated for a large motor stroke (40 mm) with a steady state error ratio lower than 0.1%. The H ∞ control technique shows high robustness to motor parameters uncertainties due to the speed variation. Where, the H ∞ closed loop system keeps a high precision level without overshoot for low motor speed (1 mm/s) and up to the maximum speed (10 mm/s). The RST and PID controllers present a robustness weakness to motor parameters variation. In fact, the WPZM is identified at low motor speed (2 mm/s), and when the motor moves with higher speeds, the positioning system based on RST and PID controllers generate overshoots with oscillations in steady state. The robustness of the closed loop system to load torque is also evaluated. External forces are applied in the opposite direction of the motor motion, and the positioning responses are compared. The H ∞ control method gives high position resolution (<0.1%) also in load conditions (up to 7.5 N). However, it is not the case with the two other controllers, where the position static error increases significantly (3% for the PID with a force of 7.5 N).

The mixed sensitivity design approach of the H ∞ gives it the precision and robustness performances simultaneously, where one weighting filter was selected to perform the accuracy and stability requirements, while the two others are adjusted to improve the sensitivity function and consequently to reject the effect of additional perturbations (loads, noises, etc.).

Finally, an ultra-precision WPZM positioning system based on H ∞ control can perform high closed loop resolution for micro scale even using the total motor motion. The robustness of the proposed method to system parameters variation, motor nonlinear behavior, and external forces is confirmed without additional compensation system or online parameters adaptation. These features make it suitable to be integrated into practical embedded system where the compact size, precision and robustness are important. In order to confirm this, an embedded control board prototype (65*50 mm) based on dSpic microcontroller is designed, which can replace the dSPACE platform. The topology and architecture of this board are given in annex C.

This chapter provides an overview of the content of this thesis dealing with the piezoelectric motors for robotic applications: modeling and robust closed loop position control issues. The proposed control models of PZMs, synthesis of robust position controllers, and simulation and experimental investigations are summarized. The drive technique, positioning performance, and behaviors in open and closed loop of each studied motor will be discussed. Recommendations about the selection and drive methods of PZMs as function of the application requirements will be carried out. Finally, suggestions for future works will be presented.

VI.1 Summary and contributions of this thesis

VI.1.1 Summary

This thesis deals with the study of three PZMs with different topologies and operating principles for modeling and robust position control purposes. The nano/micro piezo-motors are not in the interest of this research, the concerned motors are in the range of 0.1-20W as output mechanical power and with a linear motion up to 100 mm.

The first studied motor is a bidirectional Traveling Wave Ultrasonic Motor (TWUSM) type USR60 with a nominal output torque of 0.5 Nm and rated speed of 100 rpm. The second is a bidirectional quasi-static rotary motor type Piezoelectric Actuator Drive (PAD) with an output torque of 4 Nm and rated speed of 56 rpm. The last one is a linear quasi-static motor type N-310.13, working with walking drive method, and has a total stroke of 50 mm with a maximum speed of 10 mm/s and 10 N as generated force.

This manuscript begins with an initiation to the piezoelectric field by introducing the principle of the piezoelectric actuators and motors, a kind of piezo-motors classification, and a description of the nonlinear behaviors of these devices. An overview of the industrial applications of PZMs was also presented. The thesis contributions and outlines are mentioned at the end of this chapter.

For the control purposes, H-infinity and RST position controllers are synthesized for a generalized plant. Afterwards, it will be applied to each motor plant regarding to the motor parameters and the desired closed loop performances.

The three chapters focusing on the modeling and design of robust closed position control of the three piezo-motors are addressed in a similar way. An illustration of the motor structure and working principle is given at the beginning of each chapter in order to make the reader familiar with this. Literature review of the modeling and position control research works is presented in second step to highlight the advantages of the proposed systems. Thereafter, the development approach consists of three main parts:

 Design of accurate Simulink model which describes the motor topology, reflects the dynamic behavior of the motor, and permits the closed loop controls implementation.

 Design of experimental positioning platform based on the corresponding motor, evaluation of the experimental output characteristics, and identification of the Transfer Function (TF) model between the control signal and the motor position.

 Synthesis of position control parameters, simulation validation, and real time implementation via the experimental positioning system: the precision and robustness results are discussed and compared to conventional PID controller.

VI.1.2 Contributions

The contributions of this thesis can be divided into three main parts; the modeling approaches, position controllers, and performance of real time positioning systems.

 Modeling approaches: The reported models of TWUSM employed for control purposes are Equivalent Circuit Model (ECM). The parameters of the ECM are generally identified through experimental tests. These models lack to capture the dynamic of the contact mechanism which is at the origin of torque generation. The second category of TWUSM models are of type analytical models which use numerical and mathematical methods to describe the highly nonlinear motor behaviors. These models are not suitable for control techniques implementation due to the time execution and so no adapted to fast prototyping systems. In our modeling approach, some assumptions which not affect the motor dynamic and motion are taken. The geometry and parameters of the stator (stiffness, damping, coupling factor…), the contact surface leading to the torque generation, and the vertical and angular motions of the rotor are integrated in the proposed Simulink USR60 model. This model serves for speed and position control implementation in the two directions via the supply voltages amplitude, frequency, and phase shift. The output characteristics of the USR60 model are similar to those given by the motor manufacturer. Moreover, it permits to evaluate the effect of the stator and contact surface parameters variation on the output motion and the generated torque. The Piezoelectric Actuators Drives (PAD) models focused only on the motor characterization, parameters identification, and sensing of load torque. The motor behavior in static and dynamic operation modes are analyzed through equivalent circuit models. The electromechanical models served to evaluate the effect of the parameters variation of the employed multilayer actuators on the output characteristics. In this thesis, hybrid model of PAD7220 motor is proposed for position control purposes. This model includes equivalent spring-mass-damper systems which describe the motion of the orthogonal piezoelectric actuators, the contact model between the actuators and the stator ring, and the micro-teeth mechanical system used to transmit the motion to the rotor. The Simulink model of PAD7220 motor permits both, the simulation of closed loop position controllers through the driving signals, the evaluation of motor parameters effect on the actuator displacements and the motor angular speed.

 Position control design: The construction, topology, and motion generation process of the PZMs (ultrasonic, inertia, stepping…) are widely different. In addition, the behavior of the piezoelectric ceramics is not the same when operating in quasi-static (low frequencies) and ultrasonic (>20 kHz) modes. In consequence, the solution to perform the precision of such actuators in closed loop, is to propose different control methods regarding to the motor topology specifications.

 The first contribution of the position control part is that the proposed H-infinity and RST robust controllers are synthesized for a generalized plant and can be used similarly for the three types of PZMs by introducing the specified motor requirements through a flexible design concept.

The position accuracy of PZMs is much affected by the nonlinearities between the deformations of the employed actuators (stators) and the driving signals, caused principally by the hysteresis, creep, and dynamical vibrations. The parameters of PZMs (stiffness, damping, and coupling factor) depend also on the materials wear and operation conditions of the motor. Furthermore, the USM output characteristics present a highly nonlinear behavior when operating very close to the mechanical resonance frequency, and are affected by the motor heating. To overcome the parameters uncertainty problems, complex control techniques with on-line parameters adaptation are proposed. The nonlinearity problems are often addressed by the applying of neural networks and fuzzy logic compensation based models. The accuracy of these models and the feasibility of such techniques for practical engineering applications are the common locks of this methodology.

 The second contribution of the proposed controllers is that, they combine the precision and the stability performances, and the robustness again nonlinear behaviors and motor parameter uncertainties, in one simple design controller without additional parameters adaptation or compensation systems.

VI.2 General conclusions

In this thesis, modeling, simulation, and experimental achievements had been presented. General conclusions and recommendations related to each studied PZM can be drawn as following:

VI.2.1 Traveling wave ultrasonic motor (USR60)

 Modeling: In the modeling approach of the TWUSM, the USR60 model complexities related especially to the contact mechanism are simplified, without neglecting the principal parameters allowing the torque generation (location of the half-contact and stick points), or the vertical ADC DAC 65mm 50mm μC32MZ2048 motion of the rotor. This is in order to capture the dynamic of the motor and to implement closed loop control techniques. The evolution of the output characteristics (speed, torque) as function of the driving signals are similar to those given by the manufacturer. The model is used to simulate robust closed loop position controllers in the two motion directions. Moreover, the synthesized controller simulations parameters are very close to those used in the experimental USR60 platform, which confirms the accuracy of the proposed model.

 Experimental investigations: From the experimental results of the USR60 positioning system, it could be concluded that:

 The USM can be controlled using the voltages amplitude, frequency, and phase shift.

The drive signal should be selected regarding to aims of the control. In fact, the speedfrequency characteristic of the USR60 presents an asymmetrical and nonlinear behavior around the mechanical resonance frequency (f r ), where the motor shutdown immediately when the driving frequency decreases bellow f r . Moreover, the resonance frequency depends on many factors like the temperature and the load torque. To restart the motor motion, the driving frequency must be increased progressively exceeding the resonance frequency range. Therefore, many precautions should be taken if the position of the USR60 is controlled via the driving frequency. Reminder also, the motor operation in the two motion directions is not possible using only the frequency.

Based on these features, the driving frequency is recommended for speed control and not for position control.

 The amplitude of the supply voltages of USR60 must be higher than 80 Vrms to create motion, and recommended to be lower than 130 Vrms. The speed-voltage characteristic presents a quasi-linear behavior in the frequency range above the resonance. However, the voltage amplitude increase of 40 Vrms produces a speed increase of about 20 rpm in the best case (resonance). Thus, this technique permits only a slow motor motion in one direction.

 To create ideal traveling wave, the phase shift between the excitation voltages (φ) should be equal to +/-90°. Nevertheless, this phase shift can be used as speed and position control signal when variating between +90° and -90°, the no motion state corresponds theoretically to 0°. In addition, this is the unique variable permitting the motor operation in two motion directions by inverting the sign of the phase shift. The driving method using φ permits also rapid and easy switching between the on and off motion states of the motor. In the other hand, the speed (Ω) variation function of φ is not nicely linear, and the Ω-φ characteristic presents a dead zone when a load torque is applied. These features make the phase shift more suitable for position control than speed control, and it is the selected variable for our closed loop positioning application.

 The feedback voltage generated by the actuation electrodes placed in the stator gives information about amplitude of the traveling wave and can be used to estimate the resonance frequency which corresponds to the maximum traveling wave amplitude.

 The Transfer Function (TF) between the motor angular position (θ) and φ was experimentally identified in open loop, and used to synthesize the position controllers.

 The USR60 motor has a high holding torque even without supply, i.e. the motor can maintain its final position in off power condition. One demerit of the phase shift drive method (and also frequency drive method) is that in "off" motion state, the motor continuous to consume power while the voltage amplitude is different to zero.

 The efficiency of the USR60 can be improved if a resonance frequency tracking is added, in order to keep the motor frequency very close to this resonance one.

 Real time positioning results: The simulation and experimental results of the USR60 closed positioning system show that the listed challenges in terms of precision and robustness are achieved with the H ∞ and RST control methods:

 The closed loop positioning system achieves the maximum resolution of the position sensor (0.09°).

 The uncertain motor parameters and the dead zone effects caused respectively by the nonlinear behaviors of the motor and the load torques are overcome without parameters adaptation or compensation techniques.

 The H ∞ present higher robustness to the load torque variation and the electrical noise effect than the RST, justified by the design flexibility of the H ∞ given by the weighting filters.

 The comparison of the two controllers with conventional PID controller highlights their advantages.

VI.2.2 Rotary quasi-static motor (PAD7220)

 Modeling: An accurate model of PAD7220 was designed. It describes the motor dynamic including the elongation of the multilayer stack actuators, the actuators/stator ring contact mechanism, and the motion transmission to the rotor via the micro-teeth system, and it permits a control techniques implementation in closed loop through the driving signals (voltage amplitude, frequency, and phase shift).  Experimental investigations: from the experimental investigations related to the PAD motor, it could be concluded that:

 The speed and position of the PAD can be controlled only via the driving voltage frequency, while the phase shift can be used to change the motion direction but it must be kept to +/-90°.  The structure of the motor makes it capable to operate at very low speed (<1rpm) by imposing low drive frequencies, and able to generate high torque. This is an advantage compared to the TWUSM.

 The quality of the driving frequency must be monitored, because the least electrical noise injected to this signal (which can occur in closed loop) generate acoustic noise at the output motion.

 The driving voltages amplitude must be minimum equal to 160 V pk-pk (and maximum 200 V pk-pk ), because with lower voltage values the stator tooth still blocked between two rotor teeth leading also to acoustic noise without moving the shaft.

 Load torques are applied to the motor shaft without affecting the rotation speed, which confirm the speed-load independent feature of the PAD.

 The transfer function between the control signal (f) and the motor angular position is experimentally identified through the open loop experimental tests.

 As the case of the other PZMs, the PAD has a high holding torque without supply, but by stopping the motor using the driving frequency, it continuous to consume power in no motion state.

 Real time positioning results: In the PAD positioning platform, the employed position sensor has a resolution of 0.009°. With the implemented position closed loop controllers, the maximum resolution is achieved:

 The closed loop system shows high stability margins for large bandwidth with respect to the resonance frequency.

 The precision of the closed loop system is guaranteed at very low and relatively high speeds without overshoot and oscillation.

 The electrical noises generated by the ADC/DAC conversion stages of the control board reflected on the measurement and control signals are completely rejected by the mixed sensitivity H ∞ approach. In fact, the cutoff frequency and amplification gains of the weighting filters are adjusted in order to prove the robustness of the controller to these perturbations.

 The comparison of the two controllers with PID control method highlights their advantages in term of precision under different operation conditions. Where, in the case of high rotation speed, the PID control lacks to quickly track the reference position and generates overshoot in steady state. This problem can be avoided by introducing parameters adaptation technique function of the operation conditions, at the expense of real time execution.

VI.2.3 Linear walking piezoelectric motor (N-310.13)

 Modeling: In the case of linear walking motor, only motor model based on Transfer Function (TF) is developed. The control signal of the motor is an analog voltage (+/-10 V) sent to the input of the voltage amplifier. The TF between the control signal (V c ) and the slider position (Y) is experimentally identified in open loop by introducing a step change of V c and measuring the corresponding speed variation. Afterwards, the TF is identified using the experimental data and the identification toolbox of MATLAB. The TF parameters change regarding to the moving speed. Therefore, the motor TF model for low speed (2 mm/s) was taken to synthesize the position controllers, and we rely on the robustness of the position controllers to overcome the model uncertain parameters problem.

 Experimental investigations:

The walking motor is composed by four piezo-legs guiding a linear rod by friction contact. From the experimental results, it could be concluded:

 The N-310.13 motor can operate in analog or stepping mode depending on the configuration of the legs. In the analog mode, all piezo-legs are always in contact with the rod and the maximum feed is of 10 μm and corresponding to the deflection of the piezo-actuators. In stepping mode only one pair of legs is in contact with the rod, and the motor is moving for long distances by a succession of clamping and unclamping movements.

 The motor speed depends on the step size and the driving frequency. The phase shift between the driving voltages of the two pairs of legs must be equal to 180° in order to keep always one leg in contact with the slider. The phase shift between the driving voltages of one leg can be switched between +/-90° in order to reverse the motion direction.

 The step size of the N-310-13 motor version varies between 5 nm and 5 μm, and it depends on the applied external load  The rod position is affected by the quality of the control signal, where noised signal generates vibration of the rod at the final position and positioning error. The quality of the control signal depends of the output signal of the position sensor (laser sensor) which is an analog voltage +/-10 V, and is automatically perturbed by the ADC/DAC conversion stages of the dSPACE control board.

 Real time positioning results: The designed N-310-13 positioning system based on H ∞ and RST controllers has reached the fixed precision with static error of 0.1% for a displacement range between 100 μm and 40 mm.

 The H ∞ experimental results show high accuracy without overshoot whatever the moving speed, overcoming the model parameters variation problem. This is due to the large closed loop bandwidth introduced via the first weighting filter independently of the frequency of the injected noises.  The precision of the H ∞ control method was also tested under load conditions (up to 7.5 N) keeping high resolution (error of 5 μm for a displacement of 10 mm).

 The experimental test results using the RST and PID controllers presents overshoots with oscillations in steady state when the motor moves with speeds higher than 5 mm/s.

 The static errors in case of load conditions are relatively high with the RST and PID controllers, exceeding the permissible tolerances for high precision applications.

VI.2.4 Recommendations for future works

This work opens new perspectives which are related to the studied PZMs in terms of modeling, drive techniques, and control: TWUSM (USR60): The USM achieves its maximum effectiveness when operating at the resonance frequency. Therefore, a resonance frequency tracking technique can be implemented in parallel to the position controller to improve the effectiveness of the USM, and to increase its response time at a fixed voltage phase shift. This can be done through a dual control method, the motor position will be controlled via the voltages phase shift, and the resonance frequency will be tracked using the driving voltage frequency.

The experimental results of the TWUSM show that the resonance frequency (and in consequence the rotational speed) drifts function of the motor temperature. To perform the effectiveness and the time response of the TWUSM under heating, one solution is to insert coiling system to the motor plant.

Alternative solution is to identify the relationship between the motor temperature and the resonance frequency, and adjust the driving frequency regarding to the motor temperature through dual control (phase shift and frequency).

Linear walking piezoelectric motor (N-310.13): For the walking motor, electromechanical model can be designed to take into account the dynamic of the four piezo-legs, and the contact mechanism between the legs and the rod. This model can be used to evaluate the effect of piezo-leg and contact surface parameters in the output motor characteristics.

The drive of the walking motor can be also realized through four output voltage amplifiers which make possible the evaluation of control results using the voltages amplitude, frequency, and phase shift as was done with TWUSM.

Linear encoder with high resolution (5 nm) can be installed with the walking motor for higher precision requirement and to avoid the generated noise by the ADC conversion stage of the analog output signal of the laser sensor.

The designed H ∞ and RST controllers can be implemented using dSPIC microcontroller for embedded systems. Control card based dSpic microcontroller was designed for the walking motor (Fig. VI-1), this card has a dimension of 65*50 mm and replaces the dSPACE board. The same concept of control board can be realized for the two other motors.

As demonstration of the robotic application, prototype of robotic arm is designed to be actuated by the three studied motors (Fig. VI-2). As shown in Fig. VI-2, the base joint will be articulated by the PAD7220 motor and the second joint is articulated by the USR60. The gripper of the prototype will be actuated by the linear walking motor N-310.13. Mini-control cards based on dSPIC microcontroller will be designed to control the arm in closed loop.

Abstract:

The Piezoelectric motors present an interesting alternative to electromagnetics motors for precise positioning systems. This is mainly due to their high accuracy in the nanometer scale, and to their very low electromagnetic noise levels. In this context, the work presented in this thesis deals with the modeling, design, and real time implementation of position controllers for piezoelectric motors. The objective is to propose robust closed loop position controller of piezoelectric motors for robotic applications. Based on the application specification requirements, three motors with different topologies (USR60, PAD7220, N-310.10) are selected. Their electro mechanic models are developed in order to validate their operating principle and to analyze their dynamics.

These models are also used to simulate the controller algorithms in closed loop. Experimental platforms based on the three motors are designed, and the reduced models linking the motor positions to the corresponding control signals are experimentally identified. Afterwards, two position controllers of type Hinfinity (H ∞ ) and RST are synthesized and simulated. These controllers are implemented in real time via the experimental platforms equipped by dSPACE boards. The performances of each motor in closed loop associated to the position controllers are evaluated using the experimental results. Comparative study between the experimental results of two proposed controllers and conventional PID controller is also presented. Résumé : Pour des applications de positionnement, les moteurs piézoélectriques présentent aujourd'hui une alternative intéressante aux moteurs électromagnétiques classiques en raison de leur forte précision de l'ordre de quelques nanomètres ainsi que de leur faible niveau de bruit électromagnétique. Dans ce contexte, les travaux de cette thèse portent sur la modélisation, la conception, et l'implantation en temps réel de contrôleurs en position pour des moteurs piézoélectriques. L'objectif est de proposer un système de contrôle en position robuste pour des applications robotiques avec un cahier des charges prédéfini. Trois moteurs piézoélectriques avec des principes de fonctionnement différents (USR60, PAD7220, N-310.13) ont été choisis. Leurs modèles électromécaniques ont été développés afin de valider leurs principes de fonctionnement et d'analyser leurs comportements dynamiques face à des perturbations (variation de couple de charge, de température, etc…). Ensuite, ces modèles sont utilisés pour simuler et valider les performances des algorithmes de contrôle en boucle fermée notamment en termes de précision, de robustesse et de stabilité. Des bancs de test expérimentaux ont été mis en oeuvre pour les trois moteurs, et des modèles réduits reliant les positions des moteurs aux signaux de commande correspondants ont été identifiés expérimentalement. Deux contrôleurs de position de type H-infini (H ∞ ) et RST sont ensuite synthétisés et simulés. Ces contrôleurs sont implantés en temps réel sur les bancs de tests expérimentaux via un système dSPACE. Les performances de chaque moteur associé à sa commande sont évaluées. Une étude comparative entre les résultats expérimentaux de ces deux contrôleurs et ceux d'un contrôleur PID classique est aussi présentée.

Fig. I- 1 .Fig. I- 2 .Fig. I- 3 .Fig. I- 4 .Fig. I- 5 .Fig. I- 7 .Fig. I- 8 .Fig. I- 9 .Fig. I- 10 .Fig. I- 15 .Fig. I- 16 .Fig. I- 17 .Fig. I- 18 .Fig. I- 20 .Fig. I- 23 .Fig. II- 1 .Fig. II- 2 .Fig. II- 3 .Fig. II- 4 .Fig. II- 5 .Fig. II- 6 .Fig. III- 1 .Fig. III- 3 .

 123457891015161718202312345613 Fig. I-1. Direct and converse piezoelectric effects Fig. I-2. Hysteresis between the applied voltage (u) and the actuator displacement (x) Fig. I-3. Feedforward control of Hysteresis effect Fig. I-4. Classification of hysteresis models for piezo-actuated stages [6] Fig. I-5. Feedback control of Hysteresis effect Fig. I-6. Feedback-feedforward control of Hysteresis effect Fig. I-7. Effect of creep in piezoelectric actuator displacement Fig. I-8. Vibration effect: (a) Frequency response, (b) Tracking results at 1 rad/s, (c) Tracking results at 20 rad/s Fig. I-9. Classification of piezoelectric motors Fig. I-10. Description of inertia motor principles: (a) Fixed actuator type, (b) Moving actuator type Fig. I-11. Schematic description of actuation stages in an inchworm motor Fig. I-12. Walking piezoelectric motor proposed by Brisbane [37] Fig. I-13. Synoptic of USM working principle Fig. I-14. Earlier structures of USMs proposed by : (a) Lavrinenko [40], (b) Bath [41] Fig. I-15. Vibratory-coupler type motor proposed by Sashida [43] Fig. I-16. The squiggle motor from New Scale Technology [46] Fig. I-17. Concept of PIline ultrasonic motor series from Physik Instrumente GmbH [48] Fig. I-18. Prototype of straight beam TWUSM [42] , (b) Linear ring-type TWUSM [52] Fig. I-19 . (a) Ring-type USM used in camera [56], (b) Optical zoom module using squiggle motor [46]

Fig. III- 7 . 66 Fig. III- 8 .Fig. III- 9 . 68 Fig. III- 10 . 8 Fig. III- 11 . 68 Fig. III- 12 .Fig. III- 13 . 70 Fig. III- 14 . 70 Fig. III- 15 . 70 Fig. III- 16 . 72 Fig. III- 17 . Experimental platform 72 Fig. III- 18 . 72 Fig. III- 19 . 73 Fig. III- 20 . 73 Fig. III- 21 . 74 Fig. III- 23 . 75 Fig. III- 24 . 75 Fig. III- 25 . 76 Fig. III- 26 . 77 Fig. III- 27 . 78 Fig. III- 28 . 79 Fig. III- 29 . 80 Fig. III- 30 . 1 Fig. III- 31 . 81 Fig. III- 32 . 81 Fig. III- 33 . 82 Fig. III- 34 . 83 Fig. III- 35 . 83 Fig. III- 36 . 84 Fig. III- 37 . 84 Fig. III- 38 . 85 Fig. III- 39 .torques 86 Fig. III- 40 . 86 Fig. III- 42 . 87 Fig. III- 43 . 88 Fig. IV- 1 . 93 Fig. IV- 2 .

 766896810811681213701470157016721772187219732073217423752475257626772778287929803013181328133823483358336843784388539864086428743881932 Fig. III-7.Description of rotor angular motion 66 Fig. III-8. Block diagram of the Simulink model of USR60 67 Fig. III-9. (a) Generated stationary waves, (b) Traveling wave amplitude at 40kHz 68 Fig. III-10. (a) TW amplitude as function of driving frequency, (b) Rotor height 6 8 Fig. III-11. (a) Half-contact length and stick point locations, (b) Motor generated torque 68 Fig. III-12. (a) Time analysis of motor velocity, (b) Speed-load characteristics 69 Fig. III-13. Speed-frequency characteristic under different load torques 70 Fig. III-14. Speed-Vrms characteristic at different frequency (T L =0 Nm) 70 Fig. III-15. Speed variation as function of voltages phase shift (T L =0 Nm) 70 Fig. III-16. Block diagram of experimental setup 72 Fig. III-17. Experimental platform 72 Fig. III-18. (a) The electromechanical specifications of FAS 21[127] , 72 Fig. III-19. Experimental motor speed variation as function of the driving frequency (T L =0 Nm) 73 Fig. III-20. Experimental motor speed variation as function of the driving frequency under different load torques 73 Fig. III-21. Motor temperature effect on : (a) Resonance frequency, (b) Rotational speed 73 Fig. III-22. (a) V fb amplitude versus excitation frequency at various load torques, (b) The motor speed versus V fb amplitude at various load torques 74 Fig. III-23. Voltage amplitude effect on the motor speed (T L =0 Nm) 75 Fig. III-24. Experimental motor speed variation as function of the voltage phase shift under different load torques 75 Fig. III-25. (a) Synoptic of TF identification method, (b) Open loop speed response for π/2 phase shift step 76 Fig. III-26. (a) USR60 Closed loop configuration, (b) USR60 closed loop augmented by weighting filter 77 Fig. III-27. Frequency analysis of sensitivity transfer functions 78 Fig. III-28. Discrete time USR60 plant controlled by RST regulator 79 Fig. III-29. Bode diagrams of USR60 control systems 80 Fig. III-30. Comparison of closed loop position responses: (a) Step response , (b) Step response with limited phase shift, (c) Sine-wave motion,(d) Triangular motion 8 1 Fig. III-31. Controller performances under motor parameters variation: (a) 50% k m , (b) 65%τ m 81 Fig. III-32. Comparison of position step responses under: (a) Control signal noises, (b) Measurement noises 81 Fig. III-33. Block diagram of Simulink closed loop system 82 Fig. III-34. Simulation of closed loop responses: (a) Step, (b) Triangular motion, (c) Sine-wave motion 83 Fig. III-35. (a) Comparison of control signal noises rejection capability, (b) Injected noises,(c) Comparison of measurement noises rejection capability 83 Fig. III-36. Step response of H ∞ position controller (T L =0 Nm) 84 Fig. III-37. H ∞ response for periodic position trajectory (T L =0 Nm): (a) Triangular motion, (b) Sinewave motion 84 Fig. III-38. Step responses of H ∞ position controller : (a) With different load torques, (b) Under different motor temperatures (T L =0 Nm) 85

Fig. IV- 3 : 93 Fig. IV- 4 . 94 Fig. IV- 5 . 95 Fig. IV- 6 . 95 Fig. IV- 7 . 96 Fig. IV- 8 . 96 Fig. IV- 9 . 97 Fig. IV- 10 . 98 Fig. IV- 11 . 98 Fig. IV- 12 . 98 Fig. IV- 13 . 99 Fig. IV- 14 . Experimental setup 99 Fig. IV- 15 . 100 Fig. IV- 16 . 101 Fig. IV- 17 . 102 Fig. IV- 18 . 102 Fig. IV- 19 . 103 Fig. IV- 20 . 103 Fig. IV- 21 . 104 Fig. IV- 22 . 104 Fig. IV- 23 : 105 Fig. IV- 24 : 105 Fig. IV- 25 : 106 Fig. IV- 26 . 106 Fig. IV- 27 . 107 Fig. IV- 28 . 108 Fig. V- 1 . 114 Fig. V- 2 .tip 114 Fig. V- 3 . 115 Fig. V- 4 .platform 115 Fig. V- 5 . 116 Fig. V- 6 . 117 Fig. V- 7 . 118 Fig. V- 8 . 118 Fig. V- 9 . 119 Fig. V- 10 . 120 Fig. V- 11 . 120 Fig. V- 12 . 121 Fig. V- 13 . 121 Fig. V- 14 . 122 Fig. V- 15 . 122 Fig. V- 16 . 123 Fig. V- 17 . 123 Fig. V- 18 . 124 Fig. V- 19 . 125 Fig. V- 20 . 125 Fig. V- 21 . 126 Fig. VI- 1 . 132 Fig. VI- 2 .

 39349459569579689699710981198129813991499151001610117102181021910320103211042210423105241052510626106271072810811142114311541155116611771188118911910120111201212113121141221512216123171231812419125201252112611322 Fig. IV-3: Scalable technology of PAD [71] 93 Fig. IV-4. PAD speed-load characteristics [68] 94 Fig. IV-5. Synoptic of PAD model 95 Fig. IV-6. Equivalent dynamic model of the actuators and the kinematic of ring-shaft system 95 Fig. IV-7. Circular trajectory of the ring and forces on the ring 96 Fig. IV-8. Model of shaft angular motion 96 Fig. IV-9. Working area of the ring; (a) Centered position, (b) Maladjusted position 97 Fig. IV-10. (a): Driving voltages, (b): Actuator charges 98 Fig. IV-11. (a): Linear actuators displacement, (b): Linear actuators forces 98 Fig. IV-12. (a): Rotational speed at 100Hz, (b): Rotational speed at different frequencies 98 Fig. IV-13. Block diagram of experimental setup 99 Fig. IV-14. Experimental setup 99 Fig. IV-15. PAD speed-frequency characteristics in cases of : (a) no-load condition, (b) 2 Nm of load torque 100 Fig. IV-16. (a) PAD closed loop configuration, (b) PAD closed loop augmented by weighting filters 101 Fig. IV-17. Discrete time PAD plant controlled by RST regulator 102 Fig. IV-18. Bode diagrams of PAD control systems 102 Fig. IV-19. Comparison of step responses ; (a) High rotation speed, (b) low rotation speed 103 Fig. IV-20. Comparison of perturbation rejection capability; (a) Sensor noises (t=0.4s), (b) Control signal noises (t=0.4s) 103 Fig. IV-21. Experimental H ∞ step response (Ω=6 rpm) 104 Fig. IV-22. (a) H ∞ response for two motion directions (Ω=10 rpm), (b) H ∞ responses at different motor speeds 104 Fig. IV-23: H ∞ perturbation rejection capability, (a) Control signal noises (Ω=6 rpm), (b) Sensor noises (Ω=6 rpm) 105 Fig. IV-24: RST step responses, (a) Ω=6 rpm, (b) at different speeds 105 Fig. IV-25: RST perturbation rejection capability, (a) Control signal noises, (b) Sensor noises 106 Fig. IV-26. Comparison of controller step responses (a) Under 30Hz, (b) Under 50Hz 106 Fig. IV-27. Comparison of rejection capability of control signal perturbations (Ω=6 rpm): (a) H ∞ , (b) RST,(c) PID 107 Fig. IV-28. Comparison of rejection capability of measurement perturbations injected at t=5.8s (Ω=6 rpm): (a) H ∞ , (b) RST,(c) PID 108

Fig. 1 .

 1 Fig.1. (a) Structure de USR60, (b) USR60 en vue de coupe, (c) mécanisme de contacte stator-rotor

Fig. 2 .

 2 Fig.2. Synoptique du modèle de l'USR60

Fig. 3 .

 3 Fig.3. Architecture du modèle de l'USR60 sur MATLAB/Simulink

Fig. 4 .

 4 Fig. 4. (a) Caractéristique vitesse-fréquence, (b) Caractéristique vitesse-tension, (c) Caractéristique vitesse-déphasageUn banc de positionnement expérimental est mis en place pour déterminer les caractéristiques de l'USR60 en boucle ouverte et identifier la fonction de transfert entre la position angulaire du moteur (θ) et le déphasage (φ). Ce banc (Fig.5) va être utilisé aussi pour implémenter en temps réel les lois de contrôle développées à travers la maquette dSPACE.La Fonction de Transfert (FT) du moteur est déterminée en boucle ouverte en intégrant la relation entre la vitesse angulaire du moteur et le déphasage identifié à partir de la Fig.6. On peut l'écrire sous la forme :

Fig. 5 .Fig. 6 .

 56 Fig.5. (a) Synoptique de système de positionnement de l'USR60, (b) Plateforme expérimentale

Fig. 7 .Tableau. 1 :

 71 Fig.7. (a) USR60 contrôlé en position par un H ∞ , (b) USR60 contrôlé en position par un RST Les contrôleurs H ∞ et RST sont implémentés en temps réel, et leurs résultats sont comparés avec un correcteur PID classique. Les performances du H ∞ en terme d'erreur statiques, en présence d'un couple de charge sont meilleures que celles du RST et du PID comme indiqué dans le tableau.1 et Fig.8. La robustesse de ces trois contrôleurs face aux perturbations au niveau des mesures ou au niveau des signaux de contrôle est aussi testée expérimentalement (Fig.9a et b). Ces figurent montrent bien l'avantage de H ∞ en terme de rejection des bruits injectés.

Fig. 8 .Fig. 9 .

 89 Fig.8. Comparaison des résultats expérimentaux en présence de couple de charge : (a) Echelon de position, (b) Trajectoire sinusoïdale

Fig. 10 .

 10 Fig.10. (a) Principe de fonctionnement de PAD, (b) Prototype de PAD7220 (b) (a)

Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .Fig. 15 .

 1112131415 Fig.11. Topologie du modèle de PAD

Fig. 16 .

 16 Fig.16. (a) N-310.13, (b) Mouvement elliptique des pieds, (c) Principe de fonctionnement de moteur

Fig. 17 . 1 : 2 : 3 : 4 :Fig. 18 .Fig. 19 .

 1712341819 Fig.17. (a) Plateforme expérimentale, (b) Variation de vitesse en boucle ouverte pour un échelon de tension de contrôle

Fig. I- 1 .

 1 Fig. I-1. Direct and converse piezoelectric effects

Fig. I- 3 .Fig. I- 4 .

 34 Fig. I-3. Feedforward control of Hysteresis effect

Fig. I- 5 .Fig. I- 6 .

 56 Fig. I-5. Feedback control of Hysteresis effect

Fig. I- 7 .

 7 Fig. I-7. Effect of creep in piezoelectric actuator displacement

Fig. I- 8 .

 8 Fig. I-8. Vibration effect: (a) Frequency response, (b) Tracking results at 1 rad/s, (c) Tracking results at 20 rad/s

Fig. I- 9 .

 9 In the first level of Fig.I-9, the PZMs are classified based on the motion type into linear or rotary motors. These two types of PZMs can be classified according to the frequency operation range into quasi-static and ultrasonic motors. Two categories of quasi-static PZMs can be distinguished based on the driving method, which are the stepping and inertia motors.

Fig. I- 9 .

 9 Fig. I-9. Classification of piezoelectric motors

Fig. I- 10 .

 10 Fig. I-10. Description of inertia motor principles: (a) Fixed actuator type, (b) Moving actuator type

Fig. I- 11 .

 11 The slider is initially clamped by one of the clamping actuators. In step 1, the extensional actuator expands allowing the slider to move away from the open clamp. In steps 2 and 3, the switching between the closed and open clamps is subsequently realized. When the extensional actuator contracts in step 4, the distance between the clamps reduces and the slider moves further in the same direction. The clamping actuators change their roles again and a new cycle starts in steps 5 and 6 respectively.

Fig. I- 11 .

 11 Fig. I-11. Schematic description of actuation stages in an inchworm motor

Fig. I- 12 .

 12 Fig. I-12.Walking piezoelectric motor proposed by Brisbane[START_REF] Brisbane | Position control device[END_REF]

Fig. I- 13 .

 13 Fig. I-13. Synoptic of USM working principle

14 .

 14 Fig. I-14.Earlier structures of USMs proposed by : (a) Lavrinenko[START_REF] Vishnevsky | Piezoelectric motor structures[END_REF], (b) Bath[START_REF] Barth | IBM Tech[END_REF]

Fig. I- 15 .

 15 Fig. I-15.Vibratory-coupler type motor proposed by Sashida[START_REF] Sashida | Trial construction and operation of an ultrasonic vibration driven motor[END_REF]

  45]. This design is at the origin of the development of the smallest worldwide linear PZM by New Scale Technology, Inc [46] in a series of squiggle motors, where the smallest version has a size of 1.5*1.5*6 mm. As shown in Fig. I-16, the squiggle motor is mainly composed by a threaded nut, screw, and four piezoelectric plates. The piezoelectric plates are excited by two phase shifted voltages (+/-90°), allowing the nut to vibrate in a fixed resonance frequency. The vibration sequences of the nut will cause the screw to rotate and translate with nanometer resolution.

Fig. I- 16 .

 16 Fig. I-16. The squiggle motor from New Scale Technology [46] An example of bimodal piezoelectric SWUSM was commercialized by Nanomotion Ltd in a series of HR motors [47]. Similar structure is also developed and commercialized by Physik Instrumente GmbH in a series of PIline ultrasonic motors [48]. The operation of these motors is based upon resonant asymmetric excitation of a two standing waves in a piezo-ceramic plate element. The elliptical motion of the piezo-ceramic plate is transmitted into linear movement of the bar through friction (rotational version is also commercialized). The basic design of this motor and the motion sequences of the piezoelement are shown in Fig. I-17.

Fig. I- 17 .

 17 Fig. I-17. Concept of PIline ultrasonic motor series from Physik Instrumente GmbH[START_REF] Wischnewskiy | New ultrasonic piezoelectric actuator for nanopositioning[END_REF]

Fig. I- 18 .

 18 Fig. I-18.Prototype of straight beam TWUSM[START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF] , (b) Linear ring-type TWUSM[START_REF] Hermann | Piezoelectric travelling wave motors generating direct linear motion[END_REF]

  . I-19.b)[46]. The same company presents an Ultra-Thin Auto Focus (UTAF) for the new generation of phone camera[START_REF] Henderson | Continuous Auto Focus for Next Generation Phone Cameras[END_REF]. Konica Minolta is using inertia PZMs for autofocus, zooming, and image stabilizations [59].

Fig. I- 19 .

 19 Fig. I-19 . (a) Ring-type USM used in camera [56], (b) Optical zoom module using squiggle motor [46]

  This motor is used by the Automation and Interventional Medicine (AIM) laboratory of the Worcester Polytechnic Institute (WPI) to design a various types of medical application robots. The robot shown in Fig. I-20 is guided by piezoelectric motors to capture ultra-high resolution images of the tumor using MRI and to actuate ultra-precise surgical tools to remove the tumor [60]. It consists on one module with X, Y, Z translations and two rotational modules. Four Linear Piezo LEGS are used to generate the linear motion to move the robot in the three directions and to insert the needle, while two rotary Piezo LEGS motors serve to drive aluminum lead screws. The AIM is very invested on PZMs powered MRI robots, the applications include stereotactic neurosurgery for deep brain stimulation, lead placement for Parkinson's disease, and brachytherapy seed implantation for prostate cancer therapy [60, 61]. The linear Piezo (b) (a) LEGS with maximum stroke up to 80 mm and forces up to 40 N [62] are used for the linear motion of the robots, while the rotary versions of these motors are employed for rotational degrees of freedom.

Fig. I- 20 .

 20 Fig. I-20. Piezoelectric motor powered MRI robots [60] Robots based on USMs are reported in many research and industrial projects in the literature. One of the famous project of USM powered robot is the NeuroArm project, where a medical robot for neurological operations was developed. The latest version of the NeuroArm robot (Fig. I-21.a) employs sixteen ultrasonic motors from Nanomotion Ltd for multi-degrees of freedom motion [63]. Keio university of Japan team was developed a five-fingered robot hand (Fig. I-21.b) which has totally 20 Degrees Of Freedom (DOFs) [42]. Each DOF is driven by an USR30-B4 motor from Shinsei.Co, the hand has approximately the same size of human hand and weighs 853 g. Referring to the designer documents [64], the weight of the lightest hand driven by the traditional motor is 1400 g, and due to the fast response of USM, the hand response speed is faster than the human hand. The PZMs are also a suitable candidate to actuate the production and assembling robots and for the pick and place automation. An example of these robot types, the prototype of Delta-3 robot proposed by Noliac (Fig. I-22.a). It is based on Clavel 's Delta robot [65], but without rotational degree of freedom. The Delta-3 robot can handle up to 4 kg within an area with a diameter of 20 cm and a height of 14 cm [66]. Physik Instrumente (PI) GmbH developed hexapod driver for robotic applications in production and assembling (Fig. I-22.b) [67]. The hexapod is driven by NEXLINE motors which are quasi-static stepping PZMs.

Fig. I- 21 .

 21 Fig. I-21. (a) NeuroArm robot based on USMs [63], (b) five-fingered robot hand based on USMs [42] (a) (b)

Fig. I- 23 .

 23 Fig. I-23. Automotive applications of USMs [42]: (a) rearview mirror, (b) headrest, (c) steering wheel

Fig. I- 24 .

 24 Fig. I-24. PAD application in car door[START_REF] Kappel | PAD-Piezoelectric Actuator Drive[END_REF]

  Fig. II-1 [77]. In this figure, P(s) is the generalized plant, K(s) is the controller, u and y are respectively the control and measured signals, e are the error signals, and w are the exogenous signals such as the reference signal and the external perturbations.

Fig. II- 1 .

 1 Fig. II-1.Generalized plant[START_REF] Duc | Commande H∞ et µ-Analysis, des outils pour la robustesse[END_REF]

  Fig. II-2. (a) Closed loop configuration [77], (b) Closed loop augmented by weighting filters[START_REF] Duc | Commande H∞ et µ-Analysis, des outils pour la robustesse[END_REF]

  -2.b)[START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF][START_REF] Duc | Commande H∞ et µ-Analysis, des outils pour la robustesse[END_REF]. In Fig.II-2.b, the error ε is weighted by the filter W 1 , the control signal u is weighted by W 2 , and the input perturbation b is the output of the

Fig. II- 3 .

 3 Fig. II-3. Flowchart of H∞ control synthesis approach

  Fig. II-4. Discrete time RST control system

Fig. II- 5 .

 5 Fig. II-5. Discrete time RST control system in presence of disturbances

  -1.a) from Shinsei.Co [50] .The exploded figure of the motor is shown in Fig. III-1.a.

Fig. III- 1 .

 1 Fig. III-1. Structure diagram of USR60[START_REF] Zhao | Ultrasonic Motors: Technologies and Applications[END_REF]

Fig. III- 3 .

 3 Fig. III-3. Block diagram of USR60 model subsystems

Fig. III- 4 .

 4 Fig. III-4. Spring-mass-damper representation of the USM stator

Fig. III- 5 .

 5 Fig. III-5. Drawing of stator rotor contact surface [100]

Fig. III- 6 .

 6 Fig. III-6. Diagram of the rotor vertical motion

Fig. III- 7 .

 7 Fig. III-7.Description of rotor angular motion

Fig. III- 8 .Fig. III- 12 .

 812 Fig. III-8. Block diagram of the Simulink model of USR60

Fig. III- 13 .Fig. III- 15 .III. 6 . 1

 131561 Fig. III-13. Speed-frequency characteristic under different load torques

  45 A (Fig. III-18 (a)). The chosen brake produces a torque proportional to the low DC supply current as shown in Fig. III-18 (b). The torque is measured and conditioned via a mini-smart torque sensor type 4502A with a control/monitoring board type 4700B (Fabricated by Kistler). The measurement range of the torque sensor is 2 N.m with a linearity error <0.2%. The angular position of the USR60 motor is measured by the incremental encoder (1000 p/r) mounted at the rotor shaft as shown in Fig. III-17.

  The steady state motor speed values are collected for different frequency values as shown in Fig. III-19. The arrows in Fig. III-19 indicate the frequency sweep direction. The motor rotates when the excitation frequency reaches 37.2 kHz.

Fig. III- 16 .FigFig. III- 18 .Fig. III- 19 .

 161819 Fig. III-16. Block diagram of experimental setup

  . The motor speed variation as function of the frequency is also evaluated in case of loaded motor (Fig. III-20), which confirms the behavior given by the model simulation results (Fig. III-13). The driving frequency variation gives a flexible control range but with a nonlinear relationship between the speed and frequency. Moreover, the resonance frequency depends on the motor temperature as shown in Fig. III-21.a. In consequence, the motor speed changes depending on the motor temperature range as (a) (b) illustrated in Fig. III-21.b. Therefore, avoiding the pull-out phenomenon is complicated when the motor speed/position is controlled through the excitation frequency. As it has been explained above in section III.2, the feedback electrode of the piezo-ceramic placed between the two excited phases provides a voltage signal (V fb ) according to the piezoelectric direct effect. The feedback voltage gives information about the traveling wave characteristics. Consequently, an intuitive understanding of the behavior of the motor is possible. From Fig. III-22 (a), it can be noticed that the feedback voltage characteristics exhibit a resonance behavior around the fundamental resonance frequency of the USR60. However, the speed-V fb characteristics are almost linear for each load torque (Fig. III-22 (b)). Since the amplitude of standing waves depends on the voltage amplitude, speed and position can be controlled through adjusting the driving voltage amplitude.

Fig. III- 22 .Fig. III- 23 .Fig. III- 24 .

 222324 Fig. III-22. (a) Vfb amplitude versus excitation frequency at various load torques, (b) The motor speed versus Vfb amplitude at various load torques (a) (b)

Fig. III- 25 .

 25 Fig. III-25. (a) Synoptic of TF identification method, (b) Open loop speed response for π/2 phase shift step

  . III-26.a) will be used to synthesize the position controller K(s). In Fig. III-26.a, the USR60 plant is represented by G USM (s), b represents the input system perturbation, r and y

Fig. III- 26 .

 26 Fig. III-26. (a) USR60 Closed loop configuration, (b) USR60 closed loop augmented by weighting filter

  analysis as shown in Fig. III-27. The behavior of S(jω) is fixed at low gain in low frequencies to ensure the precision requirements of the controller. The frequency ω 1 represents the desired bandwidth for the controller. The cut off frequency of the weighting filter W 2 is fixed at high value to perform the rejection of high frequency noises of the control signal.

Fig. III- 27 .

 27 Fig. III-27. Frequency analysis of sensitivity transfer functions

Fig. III- 29 ,

 29 Fig. III-29, and Fig. III-30.a.It should be mentioned that the minimum margins to guarantee the stability of a closed loop system are around to 10 dB and 45° for the gain and the phase margins.

  The closed loop simulation results without and with control signal (φ) saturation are presented in Fig. III-30.a and b respectively. Fig. III-30.a is presented in order to verify the fixed performance criteria in terms of time bandwidth and overshoot. The time analysis of the closed loop system controlled by H ∞ and RST for different position trajectories is shown in Fig. III-30.c and d. The two proposed controllers show high precision levels with zero steady state error and no overshoot. The motor speed depends on the driving frequency, motor temperature, and load torque. Therefore, the identified motor transfer function parameters (k m , τ m ) will change if the motor operation conditions change. The robustness of the position controllers in the case of motor parameter variation is also evaluated. The motor gain and time constant are changed without modifying the controller parameters in order to validate the system robustness. The two presented controllers are compared also to a PID controller to confirm the advantage of the proposed strategies. The PID parameters are determined based on the motor TF (Eq.III-21) and tuned by trialand-error through intensive tests. The resulting PID parameters are finally; K p =5, Ki=0.1, and K d =1.15. Fig. III-31 presents the position responses respectively in case of gain constant (k m ) decreases by a ratio of 50% (5.75 instead of 11.5), and time constant (τ m ) increases by a ratio of 65% (7 ms instead of 4.25 ms). This is the case when the motor is driven with a frequency of 36 kHz. The simulation results show that the response of the three designed controllers is not much affected by the gain constant variation (Fig. III-31.a). Whereas, the time constant variation generates an overshoot of respectively 0.5%, 0.9%, and 3% for the RST, H ∞ and PID controllers as illustrated in Fig. III-31.b.

Fig. III- 29 .Fig. III- 30 .Fig. III- 31 .Fig. III- 32 .

 29303132 Fig. III-29. Bode diagrams of USR60 control systems

  established Simulink model of USR60 presented in section III.4 is used here to validate the closed loop position system based on H ∞ and RST controllers. As shown in Fig. III-33, the measured and reference positions are introduced to the position controller subsystem, and one of the synthesized control techniques will be then executed and generates the position control signal (voltages phase shift). This signal will be sent to the motor supply subsystem in order to generate the driving voltages. The voltages amplitude and frequency are constant and equal respectively to 130 V rms and 40 kHz. The step position responses of the closed loop system shown in Fig. III-34.a confirm the accuracy and fast response of the positioning system without overshoot. The tracking of variable position trajectories is also validated with high precision levels as illustrated in Fig. III-34.b and c for respectively triangular and sine-wave motions. A brief comparison with PID controller in term of robustness to injected disturbances has been done in order to prove the advantage of the proposed method.

Fig. III- 33 .

 33 Fig. III-33. Block diagram of Simulink closed loop system Therefore, Gaussian random noises (Fig. III-35.b) are injected to the control signal and the measured position (at t= 0.1s). The noises rejection capability of each control method is then evaluated. Fig. III-35.a and c show the comparison between the simulation results of the closed loop system in case of control signal and measurement noises. As clarified in the zoomed area, the H ∞ and RST controllers have a better noises rejection capability than the PID, the position error remains less than 0.1% for a step position of 30°.

Fig. III- 35 .Fig. III- 34 .

 3534 Fig. III-35. (a) Comparison of control signal noises rejection capability, (b) Injected noises,(c) Comparison of measurement noises rejection capability

  Fig.III-36, the control method is tested with a rotor velocity of 10 rpm (f r =36 kHz) and without load. As can be seen, there is no overshoot with a static error of 0.09° (0.1% which is the maximum resolution of the encoder). The tracking performances of different position trajectories are tested at the driving frequency 35 kHz (30 rpm) showing similar results in terms of precision and response time (Fig.. When a load torque is applied to the shaft of rotary TWUSM, a dead zone appears at the phase-speed characteristic (Fig.. The reported solution is generally the addition of dead zone compensation system to the principal position controller[START_REF] Senjyu | Position control of ultrasonic motors using MRAC and dead-zone compensation with fuzzy inference[END_REF][START_REF] Chen | Motion control with deadzone estimation and compensation using GRNN for TWUSM drive system[END_REF][START_REF] Chenglin | QUICK AND PRECISE POSITION CONTROL OF ULTRASONIC MOTORS USING ADAPTIVE CONTROLLER WITH DEAD-ZONE COMPENSATION[END_REF].

Fig. III- 36 .

 36 Fig. III-36.Step response of H ∞ position controller (T L =0 Nm)

  Fig. III-38.a. However the H ∞ closed loop system keeps a high precision level in case of loaded motor, where for example the steady state error is of 0.36° for a position step of 90° when a load torque of 0.35 N.m is applied (Fig. III-38.a).

Fig. III- 38 .

 38 Fig. III-38. Step responses of H∞ position controller : (a) With different load torques, (b) Under different motor temperatures (T L =0 Nm)

Fig. III- 39 . 8 . 3 .

 3983 Fig. III-39. RST responses: (a) Step motion (T L =0 Nm),(b) Sine-wave motion (T L =0 Nm),(c) With different load torquesIII.8.3 . Comparative study of controller performancesA comparative study between the H-infinity, RST, and a PID controllers will be presented in this section in order to prove the advantage of the proposed methods. The comparison criteria are the precision and time response, load torque (dead-zone) sensitivity, adaptation with polynomial reference signals, and the rejection of control signal or measurement disturbances. The controller parameters are the same used in the simulation study presented in section III.7. The proposed controllers show good performances in terms of time response and precision in case of no-load tests and without motor heating. However, when load torque is added, the H ∞ controller gives higher precision levels and faster response times as shown in Fig. III-40 for load torques of 0.1 Nm and 0.2 N m. The degradation of the precision levels is explained by the dead zone around zero phase values in the speed-phase shift characteristics with load torques (Fig. III-24).

Fig. III- 40 .

 40 Fig. III-40. Comparison of H∞, RST and PID step position responses for loaded motor: (a) 0.1N.m, (b) 0.2N.m

  Fig. shows the comparison test results for sinus position response (120°) with a load torque of 0.2 Nm, giving a maximum static error of 0.65% (0.8°) for H∞, 1% (1.2°) for RST, and 1.8% (2.2°) for PID controller. The steady state error in load conditions is caused by the quality of the control signal and the resulting dead zone (Fig.. The better results of the H∞ controller due mainly to the weighting filter W2 which filters directly the control signal. The disturbances rejection capability of the three proposed controllers is also compared. Additional Gaussian random noises are injected to the closed loop system as a kind of system disturbances. Firstly, a high frequency noises is injected to the measured position data (encoder output) to anticipate the noises that can be generated by the position sensing and acquisition system. Secondly the control signal (φ) is perturbed by injecting additional oscillations. The proposed positioning system show high rejection capability of the control signal distortion as shown in Fig.III-42.a. Where for an injected perturbation of +/-5° of the control signal, the maximum position distortion is 0.3°. However, when the measured position is perturbed by high frequency signal (10 kHz) and an amplitude of +/-5°, the robust H∞ position controller shows much better rejection capability than the RST and PID controllers as illustrated in Fig.III-42.b. In the zoomed area of this figure, the resulting static error is respectively 0.3, 1 and 1.6° for the H∞, RST, and PID controllers. The advantage of the H∞ controller in case of injected measurement noises is justified by the fact that the error signal between the measured and reference positions is directly filtered by the weighting filter W1.

Fig. III- 41 .Fig. III- 42 .

 4142 Fig. III-41. Comparison of controller performances for sine-wave motion tracking (TL=0.2 Nm)

Fig. IV- 1 .Fig. IV- 2 .Fig. IV- 3 :

 123 Fig. IV-1. (a) Working principle and assembled parts of PAD [138] , (b) PAD prototype [71]

Fig. IV- 4 .

 4 Fig.IV-4. PAD speed-load characteristics[START_REF] Kappel | PAD-Piezoelectric Actuator Drive[END_REF]

Fig. IV- 7 .

 7 Fig. IV-7. Circular trajectory of the ring and forces on the ring Its circular coordinates can be written as function of the angular position in x-y plane (θ s ): acos asin IV-6

Fig. IV- 8 .

 8 Fig. IV-8. Model of shaft angular motion

  Fig. IV-10.a and b respectively. The linear displacements and the generated forces of the orthogonally piezoelectric actuators driven are shown in Fig. IV-11. The time evolution of the PAD angular speed at 100 Hz is given in Fig. IV-12.a, showing a steady state speed around 2rad/s which is very close to the values given by the motor manufacturer [71] . Due the special structure of the PAD, the motor speed can be only controlled through the driving frequency. The phase shift can be switched between +/-90° to inverse the motion direction. The simulation results of the PAD model under different driving frequencies are shown in Fig. IV-12.b.

Fig. IV- 9 .

 9 Fig. IV-9. Working area of the ring; (a) Centered position, (b) Maladjusted position (a) (b)

Fig. IV- 13 .

 13 Fig. IV-13. Block diagram of experimental setup

Fig. IV- 15 .

 15 Fig. IV-15. PAD speed-frequency characteristics in cases of : (a) no-load condition, (b) 2 Nm of load torque

  Fig. IV-16.a, where the reference and measured position are represented by r and y respectively, and b is the input perturbation. In order to prove the accuracy and robustness of this method, and to satisfy the H ∞ proprieties, weighting filters are introduced as shown in Fig. IV-16.b. In Fig. IV-16.b, the error ε and the control signal u are filtered by W 1 and W 2 respectively. The input perturbation b is the output of the weighting filter W 3 . The open loop experimental tests of the PAD7220

Fig. IV- 16 .Fig. IV- 17 .

 1617 Fig. IV-16. (a) PAD closed loop configuration, (b) PAD closed loop augmented by weighting filters (a) (b)

  Fig. IV-19. The robustness of the closed loop system to external perturbations is also analyzed through the simulation study. External electrical noises are injected to the control signal and the measured position, and the rejection capability of such disturbances is evaluated. As shown in Fig. IV-20.a, when the noises are injected (t=0.4 s), the RST step response shows a position ripple of +/-0.5% and the H ∞ control method rejects completely this perturbation. The two proposed controllers show high robustness to control signal noises. Where, the injected disturbance at the instant t=0.4 s, is rejected by both the RST and H ∞ controllers as shown in Fig. IV-20.a.

Fig. IV- 18 .Fig. IV- 20 .

 1820 Fig. IV-18. Bode diagrams of PAD control systems

  Fig. IV-22.a for a motion

Fig. IV- 19 .

 19 Fig. IV-19. Comparison of step responses ; (a) High rotation speed, (b) low rotation speed

Fig. IV- 21 .Fig. IV- 23 :

 2123 Fig. IV-21. Experimental H∞ step response (Ω=6 rpm)

  -24. The perturbation rejection capability of the discrete time controller is also evaluated as done for the H ∞ . As shown in Fig. IV-25.a, the electrical noises are injected to the driving frequency at the instant t=8 s. This perturbation generates a position distortion of about +/-0.08° illustrated in the zoomed area. Similarly, the noises are injected to the position sensor data at the instant t=7.2 s as shown in Fig. IV-25.b. The resulting positioning response shows an oscillation of about 0.4° when the position data are perturbed. The RST pole placement method achieves a high positioning performances as explained in the aforementioned paragraph. However, it shows a weak robustness to reject the external disturbances.

Fig. IV- 24 :Fig. IV- 25 :

 2425 Fig. IV-24: RST step responses, (a) Ω=6 rpm, (b) at different speeds

  01, and K D =0. The three controllers show high precision without overshoot for a rotation of 30° under a driving frequency of 30 Hz as shown in Fig. IV-26.a. However, when the motor operates close to the dynamic zone, the PID controller fails to prove the positioning performances in relatively high speed as shown in Fig. IV-26.b.

Fig. IV- 26 .Fig. IV- 27 .

 2627 Fig. IV-26. Comparison of controller step responses (a) Under 30Hz, (b) Under 50Hz

Fig. IV- 28 .

 28 Fig. IV-28. Comparison of rejection capability of measurement perturbations injected at t=5.8s (Ω=6 rpm): (a) H ∞ , (b) RST,(c) PID

  [START_REF] Schitter | High performance feedback for fast scanning atomic force microscopes[END_REF] from Physik Instrumente (PI) GmbH [62], (Fig.V-1.a). The travel range of the N-310.13 motor is of 50mm with a maximum generated force of 10 N and a linear velocity of 10 mm/s. The motor can perform stepping or analog operation mode (for micrometer stroke). To achieve a long travel, the motor must operate in stepping mode which will be on the focus of this work. The WPZM has four Piezoelectric drive Legs (PZL) (Fig.V-1.b), each leg consists of two electrically separated piezoelectric stacks. These stacks consist of shear piezoelectric actuators (acting in x-direction) that are responsible of microstepping attached at the tip of longitudinal actuators (acting in y-direction) responsible of clamping[START_REF] Spanner | Piezoelectric motors, an overview[END_REF]. The four legs are supplied by four separated voltage signals. The motor drive voltages can be written as shown in Eq. V-1 and the leg states in each step are described in Fig.V-1.c.

Fig. V- 1 .Fig. V- 2 .

 12 Fig. V-1. (a) N-310 motor series, (b) Elliptical motion of legs, (c) Working principle of WPZM

1 : 2 : 3 : 4 :V. 5 Fig. V- 3 .

 123453 Fig. V-3. Block diagram of the closed loop experimental setup

  Fig.V-6shows two different open loop responses for two different control signal steps. As mentioned above, the main applications of such motors is for precise positioning at low speed. Therefore, the transfer function for the low motor speed (Fig.V-6.a) will be taken as motor TF reference model. It will be approximated to a first order function to reduce the complexity of the controller and the implementation effort. The model uncertainty should be compensated by the robustness of the applied closed loop controllers. The control signal (Vc)-Position (Y) transfer function is then deduced as:1 V-3where the gain constant k w and the time constant τ w are respectively equals to 1.2 mm/v and 0.15 s. The TF parameters can significantly change as function of control signal values and external forces. For example the time and gain constants are respectively of 30 ms and 0.79 mm/v when the motor is moving with speed of 8 mm/s.

Fig. V- 5 .Fig. V- 6 .

 56 Fig. V-5. Block diagram of experimental open loop setup

  . V-7.a) is used to synthesis the robust feedback H ∞ controller for the WPZM plant. Afterwards, the weighting filters are introduced to apply the mixed sensitivity H ∞ optimization method as shown in Fig.V-7.b. The first closed loop tests of the N-310.13 motor show that the rod positioning accuracy is very sensitive to the control signal quality, i.e. the least control signal perturbation generates a lot of oscillations at the final rod position with acoustic noises.

Fig. V- 7 .

 7 Fig. V-7. (a) Closed loop configuration, (b) closed loop augmented by weighting filterThe filters and system transfer function behaviors are verified through the frequency analysis as shown in Fig.V-8. The criteria of the H ∞ in the frequency domain are satisfied, the accuracy and robustness of the proposed method will be evaluated via the simulation and experimental results.

Fig. V- 8 .Fig. V- 10 .

 810 Fig. V-8. Frequency analysis of Filters and transfer functions behavior

Fig. V- 11 .Fig. V- 12 .Fig. V- 13 .

 111213 Fig. V-11. Comparison of simulation step responses at different velocities: (a) 1 mm, (b) 10 mm (a) (b)

6 . 1 Fig. V- 14 .

 6114 Fig. V-14. H ∞ experimental and simulation step responses: (a) 0.1mm,(b) 5 mm, (c) 20 mm,(d) 40 mm

  2 mm for a position step of 5 mm as shown in Fig. V-17.a. The overshoot and oscillation in steady state are critical points for the motor positioning applications. The performances of the closed loop system based on RST controller under load forces are also evaluated as illustrated in Fig. V-17.b, the system time response and precision are acceptable for external forces until 4 N.

Fig. V- 17 .Fig. V- 18 .

 1718 Fig. V-17. RST experimental results: (a) at different speeds, (b) under different load forces

Fig. V- 19 .Fig. V- 20 .Fig. V- 21 .

 192021 Fig. V-19. Comparison of controller experimental step responses at different velocities: (a) H∞, (b) RST, (c) PID



  Real time positioning systems:This thesis deals also with the development of three compact positioning platforms based on PZMs. The two rotational positioning systems based on the USR60 and PAD7220 are installed and validated in the Group of electrical engineering of Paris (GeePs), while the linear one based on the N-310.13 motor was installed and validated in the Laboratory of Actuation Technology (LAT) in Saarbrucken (Germany). Theses platforms permit a rapid and flexible real time implementation of closed loop position controllers through dSPACE microcontroller board. The positioning platforms are equipped with graphical interfaces which serves to configure the operation conditions in real time (switching between open/closed loop, selection of control method, motion trajectory, etc.). The positioning performances of the PZMs under load condition can be also evaluated, where a smart torque sensor is installed. The features of the drive techniques and the flexible design of the robust position controllers make the proposed control suitable for integration into embedded systems. In fact, compact control board prototype (65*50 mm) for the linear piezo-motor is designed (Fig.VI-1), where the controllers are executed via a dSPIC microcontroller.

Fig. VI- 1 .

 1 Fig. VI-1. Prototype of walking motor control board

Fig. VI- 2 .A. 3 .C. 4 .

 234 Fig. VI-2. Prototype of robotic arm actuated by PZMs

  Modeling and Position Control of Piezoelectric MotorsKeywords: Piezoelectric motors, Modeling, Position control, H-infinity, RST, Robustness.

Université

  Paris-Saclay Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France Titre : Modélisation et Contrôle en Position des Moteurs Piézoélectriques Mots clés : Moteurs piézoélectriques, Modélisation, Contrôle en position, H-infini, RST, Robustesse.

Table I -1: Characteristics of PZMs

 I

	Merits of PZMs	Demerits of PZMs
	 Ultra-high precision (nanometer scale)	 Nonlinear variation of the output
	 Fast response  Compact and flexible design  Miniaturization  Silent drive  High torque/size ratio  High torque at low speed without gear	characteristics as function of driving parameters  Nonlinear behaviors when operating close to the resonance  Specific drive circuits  Less life time due to frictional drive
	 High holding torque without supply
	 No-backlash
	 Vacuum compatible
	 Insensitive to high magnetic fields
	 Optics and autofocus camera:

(b) (a) (c) (a) (b) (c)

(a) (b) (a) (b) (c)

| | | | II-12 | | | | II-13 | | | | II-14 | | | | II-15Therefore, the problem resolution topology can be resumed as given in the flowchart bellow:

(a) (b)

(a) (b)(c)

(b) (a) (a) (b)

B.4. Real time graphical interface for PAD7220 (ControlDesk)

Acknowledgments

List of tables

where; Δ x is the actuator elongation (m), d 33 is a piezoelectric constant (m/v), U is the voltage (V), k is the actuator stiffness (N/m), C is the equivalent capacitance (F),Q is the actuator charge (C), and F is the generated force (N).

The elongations of linear actuators will create a circular motion of the motor ring, and due to the gear system, a rotation of the shaft will be generated. The dynamic model of the actuators and the kinematic of the ring-shaft system can be represented by the equivalent model [START_REF] Sven | Aufbau und Verifikation eines dynamischen Modells für einen piezoelektrischen Stellantrieb (PAD)[END_REF] shown in Fig. IV-6. where; , are the actuators stiffness in x-y directions (N/m), , are the actuators damping in x-y directions (Ns/m), , are the actuators effective mass in x-y directions (kg), is the ring mass (kg), is the top cap mass (kg), is the tubular spring mass(kg), , and , are the generated and applied forces in x-y directions (N) respectively.

In order to build a simple PAD model for control purposes, some assumptions should be taken. Therefore, the damping and stiffness of the actuators in the two direction will be assumed to be equals

Modeling and Design of Robust Closed Loop Position Controllers for Linear Walking

Piezoelectric Motor

V.6.2 RST position controller of WPZM

The RST position control for the N-310.13 motor is synthesized in dicrete time based on pole placement method. Therefore, a discrete time motor plant (G(z -1 )=B(z -1 )/ A(z -1 ) is needed to synthesis the controller. It can be obtained by the discretization of the continuous model G WPZM (s) (Eq.V-3). A zeroorder hold method is used for the discretization aproach and a discrete WPZM plant is obtained: This plant is designed to be controlled so that the closed loop will have a second order type with a damping factor of 0.7 and a bandwidth of 100 rad/s. Thus, a discrete characteristic polynomial A m (z -1 ) is given by:

By the resolution of the primary Diophantine equation (II-19), the polynomials S(z -1 ) and R(z -1 ) can be obtained:

1 3.9. 10 ; 9.68 9.45 . 10 V-8

The polynomials T(z -1 ) will be obtained by fixing an unity gain in steady state and by resolving the auxiliary Diophantine equations (II-28 and II-32):

1.054 1.032 . 10 V-9

V.7 Simulation of WPZM closed loop system

The simulation results of the WPZM closed loop system are presented in this section in order to prove the performances of the proposed methods. The robustness of the positioning system to motor parameters variation and injected disturbances are evaluated and compared to a PID control method. The stability of the closed loop system is always guaranteed with the H ∞ and RST controllers as illustrated in Fig. V