et discipline ou spcialit

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Institut Suprieur de lAronautique et de lEspace (ISAE)

XuanWANG

lundi 16 dcembre 2013

Contrle de forme d'un miroir spatial par actionneurs pizolectriques

Shape control of a deformable spatial mirror with piezoelectric actuators

ED AA : Automatique, Gnie mcanique

quipe d'accueil ISAE-ONERA CSDV - Institut Clment Ader

M. Ayech BENJEDDOU - RapporteurM. Hui-Ji SHI - Rapporteur

M. Daniel GUYOMAR - ExaminateurM. Jean-Franois ROUCHON - ExaminateurM. Yves GOURINAT - Directeur de thse

Mme Valrie POMMIER-BUDINGER - Co-directrice de thse

M. Yves GOURINAT (directeur de thse)Mme Valrie POMMIER-BUDINGER (co-directeur de thse)

I

Abstract

The next generation of space-based observation systems will make use of larger primary mirrors to achieve higher image resolution. Large primary mirrors lead to the increase of structural flexibility and are more susceptible to distortions. Thus maintaining optical tolerances across the mirror surface becomes increasingly difficult. The techniques of active shape control may be required for spatial mirror surfaces in future space observation systems. Piezoelectric actuators are often studied as embedded elements for the active control of mirror structures due to their excellent properties. However, unwanted nonlinear effects in piezoelectric actuators, i.e., hysteresis and creep, severely limit the service performance. This thesis aims at developing open-loop control laws to compensate hysteresis and creep effects in piezoelectric actuators. The studies led during this thesis are applied to the shape control of spatial mirror surfaces.

An experimental setup with a small-scale mirror test structure involving multiple piezoelectric actuators is first developed and is used as support for all the measurements conducted during this thesis. Then the open-loop control methodologies of creep compensation, hysteresis compensation, and simultaneous compensation of both the nonlinear effects in a single piezoelectric actuator are respectively developed. To compensate creep, a nonlinear viscoelastic model is used to portray creep, and a new inverse model of creep based on the concept of voltage relaxation is proposed Regarding the hysteresis compensation, the classical Preisach model is modified by adding a derivative term in parallel to describe hysteresis more accurately with relatively few measurements, and the new inverse model is constructed in the similar way. For the simultaneous compensation of the two nonlinear effects, the hysteresis is first compensated and then, the creep of the hysteresis-compensated piezoelectric actuator is attenuated by open-loop control. The methodology is first developed for a single actuator. Finally, the shape control of a mirror surface with several piezoelectric actuators is achieved by actuating the points on the mirror surface in such a way as to reach the required displacements. The mirror test structure involving multiple piezoelectric actuators compensated in hysteresis and creep is considered as a linear system on which the superposition principle can be applied. The influence coefficients characterizing the coupling effect between the piezoelectric actuators are determined by measurements. The influence coefficient matrix is first constructed using the superposition principle, and is then inverted. By insertion of the inverse matrix in cascade with multiple piezoelectric actuators with hysteresis and creep compensation, a feed-forward control approach to actuate the multiple interesting points of the mirror surface is developed. A number of experimental results demonstrate that the developed control methodologies are effective and feasible in practice.

Keywords: spatial mirror surface, active control, shape control, piezoelectric actuator, open-loop control, hysteresis, creep, compensation, coupling effect, influence coefficient

III

Acknowledgements

During the past three years, I have got generous and continuous support and encouragement from my supervisors, colleagues, friends, and my family. I am really in a luxury situation because there are so many people around. I would like to express my deepest gratitude to them all.

Special thanks to my supervisor-Prof. Yves Gourinat! First of all, thanks to offer me the opportunity to start as a PhD student. Thanks for his guidance and support through my years here. Every discussion with him in PhD review gave me a lot of enlightenment and puts me on the right track when I got lost in the darkness. And, his immense and extensive knowledge, serious scientific attitude and genial personality deeply inflected and inspired me.

Special thanks to my supervisor-Associate Prof. Valrie Pommier-Budinger! It is her who brought me into such a big and interesting world of smart structure system and control. The investigation into a multi-disciplinary subject is not very easy, but her generosity and encouragement helped me to overcome all the obstacles. Thanks for her trust in my abilities! Thanks for providing whatever advice and resources necessary for me to accomplish my goals! Thanks for spending countless hours meeting with me to discuss aspects of my research! Thanks for patiently revising my various publications and this thesis! This work would not have been possible without her help. Also thanks for her care on my daily life.

Special thanks to Civil Aviation University of China (CAUC) and China Scholarship Council (CSC) for their jointly financial support throughout my work and daily life in France!

Deeply thanks to Aurlien Reysset for his preliminary work on classical Preisach model. The fourth chapter of this thesis is partially based on his work. Deeply thanks to Laurent Alloza for developing the real time control system under MATLAB and teaching me how to use it. Whenever I encountered problems on the real time control system, he would patiently and promptly help me to solve them. Deeply thanks to Matthieu Berranger for processing specimens and mounting fixtures. Whenever I demanded refit of experimental setup, he would unreservedly provide me his help he could. Thanks to Gilles Prusot and Philippe Henrich for their technical support during the testing processes. Thanks to my co-workers, Joseph Morlier, Miguel Charlotte, Guihem Michon, Yves Brire, Matthias Pointner and Jos-Maria Tamayo-Palau for their help during computing, analysis, testing, and writing.

Thanks to Franisco Pizarro-Torres, Franois Deudon, Sbastien Roche, Sammuel Rodriguez, Frdric Brigui, Jean-Philippe Goy, Jol Bordeneuve-Guibe, Stphanie Bidon, Romain Pascaud, Franois Vincent, Olivier Besson, Simon Foucaud, Irne Maillet and all the other labmates and friends for their advice, inspiration and friendship over the years. Thanks to Lunlong Zhong, Siyuan Zhang, Wei Li, Bin Sun, Qingyuan Mu, Hongying Wu, Jing Guan, Lu Wang, Ping Han, Weikun He, Chunzhi Du, Yajun Chen, Wei Zhang, Qiaoli Dong, Jingchang Zhuge, Jie Lin, Xuelan Hu, Huiyun Zhang, Jun Zhou, Ning Wang, Zhengxing Feng and Chaofan Zhang who studied or is studying in Toulouse coming from CAUC for their constant encouragement. Thanks to Cheng Chen, Zhongjie Tang, Zhongze Dong, Wen Peng, Lv Peng, Cheng Cheng, Zhongxun Liu, Mingxi Tan, Yao Chen, Qiaozhi Xu, Jing Li, Luhai Fan, Huijie Wang, Yi Zhang, Guodong Kang and other Chinese friends who I met with in Toulouse for their inspiration and support.

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Thanks to Prof. Jianxin Xu, Prof. Guanghui Qing, Prof. Yongsheng Shi, Prof. Zhenyu Feng, Associate Prof. Xiang Lu, M. Bo Zhang and other colleagues in CAUC for their long-term encouragement and support.

Finally, special thanks to my grandma, Mm. Ruzhen Gao, my parents, Zhenyu Wang and Wanmin Zheng, my aunt, Jing Wang, and other family members for their love and support in everything that I have done.

Toulouse, December 2013

Xuan WANG

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Contents

General Introduction ..................................................................................................................... 1

Chapter 1 A Survey of Shape Control of Spatial Mirror Surfaces ............................................ 5

1.1 Space-based Observation Systems ..................................................................................... 5

1.2 Wave-front Sensing and Control (WFSC) .......................................................................... 7

1.3 Deformable Mirrors ............................................................................................................ 9

1.3.1 Conventional deformable mirrors .......................................................................... 10

1.3.2 In-plane Actuated mechanisms .............................................................................. 11

1.4 Piezoelectric Actuators ..................................................................................................... 12

1.4.1 Piezoelectricity ...................................................................................................... 12

1.4.2 Linear Piezoelectric Theory .................................................................................. 14

1.5 Hysteresis and Creep Compensation ................................................................................ 15

1.5.1 Closed-loop and Charge Control ........................................................................... 16

1.5.2 Open-loop Control ................................................................................................. 16

1.5.3 Control of Multiple-Input-Multiple-Output (MIMO) system ................................ 19

1.6 Summary .......................................................................................................................... 19

Chapter 2 Experimental Set-up .................................................................................................. 21

2.1 The Mirror Test Structure ................................................................................................. 21

2.1.1 Introduction ........................................................................................................... 21

2.1.2 Design of Mirror Test Structure ............................................................................ 22

2.2 Experimental system ........................................................................................................ 27

2.2.1 Real Time Control System .................................................................................... 27

2.2.2 Capacitive Sensor .................................................................................................. 29

2.2.3 Power Amplifier for Piezoelectric Actuators ........................................................ 31

2.2.4 Integration of Experimental Systems .................................................................... 31

2.3 Summary .......................................................................................................................... 31

Chapter 3 Creep Compensation in a Single Piezoelectric Actuator ........................................ 33

3.1 Introduction ...................................................................................................................... 33

3.2 Observed Behavior ........................................................................................................... 34

3.3 Creep Model ..................................................................................................................... 37

3.3.1 A Nonlinear Viscoelastic Model ........................................................................... 37

3.3.2 Parameters Identification ....................................................................................... 40

VI

3.4 Inversion of the Creep Model ........................................................................................... 45

3.5 Creep compensation in open-loop .................................................................................... 51

3.6 Summary .......................................................................................................................... 54

Chapter 4 Hysteresis Compensation in a Single Piezoelectric Actuator ................................. 57

4.1 Introduction ...................................................................................................................... 57

4.2 Classical Preisach Model .................................................................................................. 58

4.2.1 Model description and numerical implementation ................................................ 58

4.2.2 Limitations of the classical Preisach model .......................................................... 62

4.3 Modified Preisach Model ................................................................................................. 73

4.4 Inverse Modified Preisach Model..................................................................................... 82

4.5 Summary .......................................................................................................................... 95

Chapter 5 Simultaneous Compensation of Hysteresis and Creep in a Single Piezoelectric Actuator ......................................................................................................................................... 97

5.1 Introduction ...................................................................................................................... 97

5.2 Scheme of Simultaneous Compensation of Hysteresis and Creep ................................... 97

5.3 Low Pass Filter ................................................................................................................. 98

5.4 Creep Model and Its Inversion in Hysteresis Compensated Piezoelectric Actuator ...... 103

5.4.1 Creep Model ........................................................................................................ 103

5.4.2 Inversion of the Creep Model .............................................................................. 108

5.5 Simultaneous Compensation of Hysteresis and Creep ................................................... 114

5.6 Summary ........................................................................................................................ 124

Chapter 6 Shape Control of a Deformable Mirror with Piezoelectric Actuators ................. 125

6.1 Introduction .................................................................................................................... 125

6.2 Coupling Effects and Influence Coefficient Matrix ....................................................... 125

6.3 Control of Multiple Compensated Piezoelectric Actuators ............................................ 128

6.4 Experiments .................................................................................................................... 130

6.5 Summary ........................................................................................................................ 140

Chapter 7 Conclusions and Perspectives .................................................................................. 141

7.1 Overall Conclusions ....................................................................................................... 141

7.2 Perspectives for Future Work ......................................................................................... 143

Appendix Procedure X ............................................................................................................... 145

Bibliography ............................................................................................................................... 147

Publications ................................................................................................................................. 157

List of Figures ............................................................................................................................. 159

VII

List of Tables ............................................................................................................................... 165

Nomenclature .............................................................................................................................. 167

1

General Introduction

Background

The next generation of space-based observation systems will push the limits of current technologies, while achieving huge advances in technical and operational performance. To perform some of the most challenging observations to answer some of our most compelling questions, including Is there life elsewhere in the Galaxy? [1], the use of larger primary apertures is essential to obtain the desired improvement in optical resolution and sensitivity. However, larger primary apertures induce a number of technical challenges such as mass, volume, and flexibility. Package constraints due to the size and volume limits of the launch vehicle shroud must be considered for the mirrors larger than three meters in diameter. In addition, mass-to-orbit is limited and extremely expensive, requiring the areal density, or mass per unit area, of the mirrors to decrease as the diameter increases in order to maintain an acceptable launch mass. Furthermore, the larger size and the lower mass combine to significantly increase the flexibility of the mirrors, decreasing the flexible mode frequencies and making them more susceptible to quasi-static and dynamic distortions. Thus maintaining optical tolerances across the mirror surface becomes increasingly difficult. Active optics is used to correct quasi-static distortion while dynamic errors are cancelled out by adaptive optics. Both active and adaptive optics are needed in future space-based observation systems and are destined to require deformable mirror configuration involving active control elements. Piezoelectric actuators, which meet the requirements of greater amplitude, higher strength and faster response [2] can be employed as active control elements embedded in the mirror structure. However, piezoelectric actuators also show unwanted nonlinear behaviors in open-loop operation, i.e., hysteresis and creep, which may lead to undesirable inaccuracy and limit system performance [3]. Therefore, for the active deformable mirrors with piezoelectric actuators, one solution that arises is the compensation of the hysteresis and creep in piezoelectric actuators.

This thesis concerns the field of active optics and focuses on the active shape control of deformable mirrors surfaces actuated by piezoelectric actuators to correct the quasi-static errors in the mirrors, more particularly on the compensation the hysteresis and creep nonlinearities in piezoelectric actuators.

Problem Statement

The shape control of spatial mirror surfaces is a key technology to design future space telescopes. However, the nonlinear effects such as hysteresis and creep in piezoelectric actuators connected to the deformable mirrors affect directly and severely the precision of the shape control, and thus the capabilities of the space telescopes. Therefore, it is essential to study the hysteresis and creep in piezoelectric actuators and to develop an efficient compensation methodology to attenuate their effects on the mirror surfaces.

To compensate the effects of hysteresis and creep, closed-loop control techniques seem to be the best way to improve the positioning precision and to provide robustness against uncertainties and parameters variations. However, due to the very high number of piezoelectric actuators for a large spatial mirror and due to the space requirements of displacement sensors, the use of closed-loop control techniques is strongly limited in practice. The thesis is thus concerned with

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techniques of open-loop compensation. These compensation techniques require very accurate models and inverse models of the phenomena to control. In addition, the coupling effect between piezoelectric actuators on the mirror also needs to be taken into account when multiple piezoelectric actuators are used for the shape control.

Objectives

The objective of this thesis is to develop and validate an open-loop control approach that actuate points of the mirror so that they reach required displacements without hysteresis and creep effects in the piezoelectric actuators and so that the quasi-static distortions in the mirror are effectively corrected. As a result, the precise quasi-static shape control of spatial mirror surfaces is achieved. To achieve this goal, four efficient methodologies are proposed:

an efficient methodology to compensate the creep in a single piezoelectric actuator by open-loop control;

an efficient methodology to compensate the hysteresis in a single piezoelectric actuator by open-loop control;

an efficient methodology to simultaneously compensate hysteresis and creep in a single piezoelectric actuator by open-loop control;

an efficient methodology to attenuate the effects of hysteresis and creep in multiple piezoelectric actuators of the mirror and to precisely control the shape of the mirror surface.

These methodologies are validated through experiments conducted on an experimental set-up developed during the thesis. It consists of a rigid structure and of a glass circular plate supported by feet actuated by ring-type piezoelectric patch actuators. It is the small-scale reproduction of a real mirror with seven piezoelectric actuators.

Contributions

The primary contributions of this thesis are summarized below.

A new inverse model-based compensation strategy is proposed to reduce creep effect in a single piezoelectric actuator. The strategy uses a nonlinear viscoelastic model to precisely describe the creep behavior. The issue is to compute the inverse model. A new concept of voltage relaxation is then developed and a closed-loop control system with proportional-integral-derivative (PID) controller is implemented to compute the inverse model of creep. As a result, the difficulty of solving the nonlinear inverse model of creep mathematically is subtly circumvented, and the infinite voltage that may appear in the direct experimental method is avoided. By insertion of the inverse model of creep in open-loop operation, the methodology of creep compensation is developed.

A modified Preisach model that requires relatively few measurements and that describes the detachment between major and minor loops is proposed to precisely portray hysteresis in a single piezoelectric actuator. The modified model is obtained by adding a derivative term in parallel to the classical model. The new inverse model is similarly constructed. By insertion of the inverse modified Preisach model in open-loop operation, the methodology of hysteresis compensation is developed.

A new open-loop control strategy of simultaneous compensation of hysteresis and creep is presented, which cascades a first-order low pass filter, the inverse model of creep and

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the inverse modified Preisach model successively to a single piezoelectric actuator. Hysteresis is firstly compensated by using the proposed methodology of hysteresis compensation. The creep of hysteresis compensated actuator is then attenuated by extending the proposed methodology of creep compensation.

A feed-forward control methodology is proposed to actuate points on the mirror to reach the required displacements without hysteresis and creep effects when multiple piezoelectric actuators are supplied at the same time. The precise quasi-static shape control of spatial mirror surface is achieved.

Thesis Outline

This thesis is organized into seven chapters.

Chapter 1 provides the contextual background of the research area and an outline of current research in the field of shape control of the spatial mirror surfaces.

Chapter 2 introduces the experimental set-up developed in this thesis and presents the design procedures of the experimental set-up.

Chapter 3 develops a nonlinear viscoelastic model to describe the creep behavior in a single piezoelectric actuator. An inverse model is computed by the use of a PID closed-loop control system resulting from a new concept of voltage relaxation. Creep compensation is achieved by inserting the inverse model of creep in open-loop operation.

Chapter 4 modifies the classical Preisach model by adding a derivative term in parallel to precisely model the hysteresis behavior in a single piezoelectric actuator and inverts the Preisach model using the similar approach. Hysteresis compensation is achieved by inserting the inverse modified Preisach model in open-loop operation.

Chapter 5 presents an open-loop control strategy of simultaneous compensation of hysteresis and creep in a single piezoelectric actuator. The hysteresis compensation approach presented in Chapter 4 is implemented firstly and the creep of the new hysteresis compensated system is modeled and compensated.

Chapter 6 analyzes the coupling between multiple piezoelectric actuators, computes the influence coefficient matrix and its inversion, and presents a feed-forward methodology to precisely control the shape of the mirror surface.

Chapter 7 provides an overall conclusion of the thesis along with propositions for the future work.

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5

Chapter 1

A Survey of Shape Control of Spatial Mirror Surfaces

The work proposed herein is based on articles that can be assembled into a few major categories: space telescopes, wave-front sensing and control (WSFC), deformable mirrors, piezoelectric actuators and hysteresis and creep compensation. These areas encompass the bulk of the material that is pertinent to this thesis.

1.1 Space-based Observation Systems

The human knowledge of the universe is strongly based on observations from telescopes. The astronomical space telescope has the potential to greatly expand our knowledge of the universe. The main advantage of a space telescope is linked to the absence of the Earths atmospheric turbulence, which distorts astronomical images.

In recent years different projects of space telescopes have been developed and realized, the most famous one for astronomical observations being the Hubble Space Telescope (HST, Fig 1.1), launched in 1990. It allows taking images from ultraviolet (UV) to near-infrared (IR) wavelengths and has provided an immense amount of scientific data. Among numerous accomplishments HST has helped to determine the age of the universe, to improve understanding of planets formation, and to discover extra-solar organic matter [4]. HSTs successor, the James Webb Space Telescope (JWST, Fig. 1.2), will continue the tradition of large and orbiting observatories [5]. JWST is scheduled to launch in 2013 and will use infrared imaging to provide data that will help scientists understand the Big Bang theory. With its 6.5 m diameter segmented primary mirror, it provides greater mirror design challenges than Hubble, whose primary mirror measures 2.4 m in diameter.

Fig 1.1 Hubble space telescope [4]

Fig 1.2 James Webb space telescope [5]

HST has a monolithic mirror that produces a cleaner, more uniform and more stable point spread function (PSF) than a segmented aperture. Additionally, a monolithic aperture produces shorter wavelengths than a comparable segmented aperture because of the diffraction that limits

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the performance [6]. In general, a monolithic aperture is the geometry of choice for UV and optical-infrared (OIR) applications. By contrast, a segmented aperture is suitable for spectroscopic and for IR to Far-IR/Sub-mm applications. For very large apertures, segmented geometries are the only path forward. As mentioned in general introduction, the launch vehicles capabilities affect the path forward for both geometries, but in different ways. The monolithic aperture is constrained in volume by the launch vehicles fairing size, while the segmented geometry is constrained in mass by the lift capacity. The monolithic potential maximum (circular) aperture is 4 meters for the Evolved Expendable Launch Vehicle (EELV) and 8 meters for the Ares V launch vehicle [7]. Advanced Technology Large-Aperture Space Telescope (ATLAST) [1], a research program of the NASA on astrophysics strategic mission concepts, indicates that the next generation of UV/OIR space observatories will have a primary aperture diameter in the 8-m and 16-m range, because maximizing the chance of a successful search for life in the solar neighborhood requires space telescopes with an aperture size of at least 8 meters [1]. The ATLAST technology development plan is based on three point designs. Two of the concepts, the 8-m monolithic mirror telescope (hereafter ATLAST-8m) and the 16.8-m segmented mirror telescope (ATLAST-16m), span the range of UV/OIR observatories that are enabled by Ares-V launch vehicles. ATLAST-8m is studied because of the inherent advantages offered by a monolithic aperture telescope in terms of high-contrast imaging and wave-front error (WFE) control. ATLAST-16m is studied because it represents a pathway to truly large apertures in space and uses the largest extrapolation of a JWST-like chord-fold primary mirror packaging. The third concept, a 9.2-m segmented telescope (ATLAST-9.2m), is compatible with an Evolved Expendable Launch Vehicle (EELV) and also adopts JWST design heritage. All ATLAST concepts require many of the same key technologies. It is believed that these designs represent a robust set of options to achieve the next generation of UV/OIR space observatories in the 2020 era. Their technologies also will enable a new generation of more capable small and medium class space-based and balloon-borne observatories [1].

(a) (b) (c) Fig 1.3 (a) ATLAST-8m (cutaway view) (b) ATLAST-16m (c) ATLAST-9.2m after deployment

of sunshield [1]

Besides astronomical space telescopes, there are space-based Earth observation systems that can be used for Earth imaging, climate change, national security, and other applications. Kramer [8] presents a history of Earth observation. For example, the LandSat program [9], a resource for global change research in geology and agriculture fields, has been running since 1972. Recently,

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the European Space Agency (ESA) call for Advanced Lidar Concepts [10] has been met with a proposal in the form of a feasibility study for the development of advanced technologies of a light detection and ranging (Lidar) system, devoted mainly to the implementation of telescopes with large aperture and active control, in order to retrieve a sufficient signal from the laser transmitter. The Lidar telescope satellite is built with seven independent continuous mirrors, as shown in Fig 1.4.

Whatever astronomical space telescopes or Earth observation systems, the next generation of space-based observation systems requires increasing the aperture size to take advantages of the benefits of the increased resolution capabilities [11].

Fig 1.4 Lidar telescope with auxiliary optics layout (a) 2-D; (b) 3-D [10]

1.2 Wave-front Sensing and Control (WFSC)

The technologies that are necessary to enable the next generation of envisioned space telescopes and observatories are addressed in a capability roadmap [12] developed by the NASA.

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Amongst these technologies, new adaptive and active wave-front sensing and control (WFSC) is required for both monolithic and segmented apertures [12, 13]. To make a monolithic mirror (ex. ATLAST-8m) light enough to launch in current rockets, its stiffness maybe inadequate to passively maintain the required high quality of wave-fronts. In such cases, active WFSC can be used to limit on-orbit wave-front errors [14]. For segmented mirrors (ex. ATLAST-9.2m and ATLAST-16m), extremely high angular resolution is enabled by combining those smaller aperture telescopes with WSFC to obtain larger apertures than with monolithic mirrors [15]. All the mirror segments have to be actively aligned by WFSC using actuators built into the mirror support cell. In above examples, WFSC is a key component of an active optics system, which actively shapes the mirror surfaces to prevent deformations due to external influences such as optical manufacture errors, deployment errors, thermal deformations, shifts in alignments during launch and mechanical deformations in telescope structures. These influences that active optics should compensate are intrinsically slow (1 Hz [16], 0.05 Hz or less [17], quasi-static) and have large amplitudes in aberration. Fig 1.5 presents the VLT (very large telescope) active optics system of ESO (European Southern Observatory) [18]. The system provides complete control of the VLT optics and optimizes its performance in all telescope positions. This is achieved by changing the shape of the primary 8.2-m Zerodur mirror and also shifting the position of the secondary 1.1-m beryllium mirror at the top of the telescope structure. A stellar image is registered by the wave-front sensor and analyzed, where after corresponding correction signals are generated to move the mirror supports. Active optics should not be confused with adaptive optics, which also contains WFSC. Adaptive optics operates on a much shorter timescale to compensate atmospheric effects, rather than mirror deformations, which affect the image at 1001000 Hz (the Greenwood frequency [19], depending on wavelengths and weather conditions, very dynamic). These corrections need to be much faster, but also have smaller amplitude. Thus adaptive optics uses smaller, which used to be a separate mirror as [20, 21], third or fourth [22] mirror in a telescope. Astronomical space telescopes operate in an environment with no atmosphere and little gravity. The relevant timescales for WFSC are dictated by slow thermal drifts, which occur on day-to-week timescales. Therefore, only active optics is applied in astronomical space telescopes (ex. JWST). However, for space-based Earth observation systems and Earth-based telescopes, adaptive optics is necessary to achieve satisfactory image quality. The major components of an Earth-based adaptive optics system are presented in Figure 1.6 [23]. The light from a distant object, such as a star, is distorted by the atmosphere. As the light enters adaptive optics system, it is reflected by the deformable mirror and the beam splitter, and directed to the wave-front sensor (WFS). The WFS converts the wave-front of light into a measurement vector, e.g., a Shack-Hartmann WFS converts the local gradients of the wave-front to the spots displacement in the CCD/CMOS image sensor. The measurement is fed to the controller. The controller reconstructs the wave-front to a reference plane wave-front and computes the control signal to the deformable mirror, such that the deformable mirror deforms its surface to counter-act the wave-front aberration. Because the wave-front measurement, the control signal computation and the wave-front correction are implemented in closed-loop and in real-time, both the spatial and temporal variations in the wave-front aberration are tracked and corrected. Images with higher resolution are then achieved by science camera, when compared to those without adaptive optics system. A space-based Earth observation system needs the same components if resolution is increased to where real-time atmospheric corrections are required [24].

The works conducted in this thesis are constrained in the field of active optics. Only quasi-static actively shape control of mirror surface is studied, regardless of dynamic modes.

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Fig 1.5 ESO VLT Active optics system [18]

Fig 1.6 Earth-based adaptive optics system [23]

1.3 Deformable Mirrors

Both active and adaptive optics need deformable mirrors, which are mirrors whose surface can be deformable in order to achieve wave-front control and correction of optical aberration. Deformable mirrors are typical continuous surface mirrors adjusted by actuators to have peaks

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and valleys [25]. The space telescopes with a monolithic aperture such as ATLAST-8m have a single large deformable mirror, and the ones with segmented apertures such as JWST, ATLAST-16m, ATLAST-9.2m and Lidar have multiple deformable mirrors. The objective of the survey of deformable mirrors herein is to understand the mechanics to create large space mirrors that require actively shape control to maintain optical precision. Deformable mirrors are usually classified according to their actuation mechanisms: conventional, boundary, volume changing, and in-plane actuated [24]. The conventional and in-plane actuated deformable mirrors are usually found in the application of space-based observation systems, because they are appropriate to be used in large scale structures to correct distortions in the mirror surface due to disturbances.

1.3.1 Conventional deformable mirrors

Conventional mirrors refer to an actuation scheme where actuators act directly on the non-reflecting side of a mirror with a deformable face, and require a backing structure to which they are attached [24]. Conventional deformable mirrors are the type of deformable mirror pictured in the articles by Goodman [26] and Hecht [27]. A graphic showing a conventional deformable mirror is presented in Fig 1.7.

Fig 1.7 Side view of conventional deformable mirror

Menikoff [28] defines the term influence function as the characteristic shape corresponding to the mirror response to the action of a single actuator connected to the backing surface of a plate-like glass mirror. The mirror is modeled as a deformable plate without any tension and with actuators connected as linear springs. The resulting differential equations are solved using a Fourier series approach. Hiddleston, Lyman and Schafer [29] review deformable mirror surfaces and compare two different types of deformable mirror models. The first model creates the deformable mirror surface through the superposition of an influence function over the extended actuators. The second model uses a two-dimensional cubic spline fit over the extended actuators to create the deformable mirror surface. Lee, Uhm, Lee and Youn [30] derive an equation which estimates influence functions of thin-plate deformable mirrors based on the analytic calculation and finite element analysis and present the performance analysis for the case of equi-spaced actuators.

A number of conventional deformable mirrors with electrostatic actuators are proposed for use in a space telescope. Angel [31] studies the large aperture made of a stretched reflective membrane with weak concave curvature induced by electrostatic pressure. Gorinevsky, Hyde and Cabuz [32] discuss a distributed surface control approach for future gossamer structures with several thousand of electrostatic actuators. Stamper et al. [33] and Errico et al. [34] investigate stretched membranes with electrostatic curvature mirrors for extremely large space telescopes. Clafin and Bareket [35] configure an electrostatic membrane mirror by least-squares fitting with

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analytically derived influence functions. Wang and Hadaegh [36] compute static shapes and voltages for micro-machined deformable mirrors with nonlinear electrostatic actuators. Bush et al. [37] discuss the design and analysis of deformable mirrors with electrostatic membranes for wave-front control systems. Todovinin, Thomas and Vdovin [38] study the use of deformable mirrors with 50-mm electrostatic membranes in astronomical adaptive optics.

Besides electrostatic actuators, other types of actuators such as piezoelectric actuators are also employed for conventional deformable mirrors. Albertnetti [39] designs discrete, bimorph piezoelectric actuators for a deformable mirror, which has a dilatation sensitivity of 6 microns at 1.5 kV. Chellabi [25, 40] develops a new control design method for active shape control of dynamic flexible structures with piezoelectric inclusions, which is based on the Rayleigh-Ritz method, extended to both space and time dimensions in conjunction with variational work-energy principles that govern the physical system. Hardy, Lefebvre and Koliopoulos [41] develop an active optical imaging system capable of correcting optical wave-front errors in real time at frequencies in the kilohertz range. The system uses an ac shearing interferometer, a parallel analog data processor, and a monolithic piezoelectric active mirror arranged in a closed-loop configuration. Rousset et al. [42] introduce the use of stacked piezoelectric actuated mirrors in COME ON PLUS system. Manetti, Morandini and Mantegazza [43] presents a control scheme that provides precise active shape control of magnetically levitated deformable mirror shells using a very large number of control points. The proposed controller combines a low frequency centralized feed-forward and a high frequency fully decentralized feedback, with each a single actuator mated to a single sensor. Martin et al. [44] describe the active support system and optimize the support forces parallel to the optical axis by axial actuators for the 6.5 m primary mirror for the Multiple Mirror Telescope Conversion.

1.3.2 In-plane Actuated mechanisms

In-plane actuated deformable mirrors rely on piezoelectric (or other types of electro- or magnetostrictive actuators) regions to strain offset from and parallel to the structures neutral axis, thus imparting a surface curvature [24]. Tyson [45] uses firstly the term in-plane actuation to describe the bimorph deformable mirrors, due to the fact that the term bimorph is not appropriate to all cases. This class of deformable mirrors may be further subdivided into three types based on the actuation mechanism, as shown in Fig 1.8.

Unimorph: The unimorph deformable mirror uses piezoelectric actuators bonded to the backing face of the mirror [46]. A surface curvature is induced by the difference from the neutral axis of the composite structure caused by the expansion or contraction of piezoelectric actuators. Adelman [47] suggests using the term unimorph for this actuator configuration. Yang, Shcheglov and Trolier-McKinstry [48] present the concept, modeling and fabrication technologies for large-stroke piezoelectric unimorph deformable mirrors.

Bimorph: The bimorph deformable mirror use piezoelectric actuators in pairs at each actuator location [46]. The actions of the piezoelectric actuators in pair in opposite directions produce a local curvature. Rodrigues et al. [49] examines the possibility of constructing deformable mirrors for adaptive optics with a large number of degrees of freedom from silicon wafers with bimorph piezoelectric actuators.

Discrete: The discrete in-plane actuator uses fixed blocks affixed to the backing face of the mirror. A piezoelectric actuator pushes laterally on the blocks, which pivot on a

12

central axis. The surface therefore deforms in segments [24]. Yang, Dekany and Padin [50] use this type of in-plane actuated mirror in micro-electrical-mechanical systems.

The real mirror structure and its small scale reproduction investigated in this thesis employ the configuration of conventional deformable mirrors.

Fig 1.8 In-plane actuated deformable mirrors

1.4 Piezoelectric Actuators

As discussed in the previous section, an active deformable mirror in space-based observation system requires a number of compensating actuators to correct wave-front errors. The conventional active optic system used to compensate wave-front errors is limited to correct relatively small turbulence errors due to the physical limitations of the available correction devices. The relationship between spacing and stroke defines the spatial frequency and amplitude of the distorted wave-fronts that may be compensated by a closed loop system. The stroke range and the spatial frequencies of the correction device are inversely proportional [2]. Mulvihill and Ealey [2] compare the stroke performances and peak forces of a number of different types of actuators and indicate that the use of piezoelectric actuators allows compensating higher spatial frequency of the distorted wave-front. Additionally, piezoelectric actuators exhibit high resolution, high stiffness, no backlash, no friction and feature thermal stability in terms of low power dissipation and low thermal expansion [51, 52]. Piezoelectric actuators meet the greater amplitude and higher strength for the large mirror actuators in the future [1, 2]. The next generation of astronomical space telescope will employ piezoelectric stack actuators as active elements of deformable mirror [53]. It is also found in [53] that piezoelectric patch actuator is very promising to be used in future space telescopes. However, the research on compensating actuator of this type is little reported. A type of ring-type piezoelectric patch actuator detailed in next chapter is used to actively shape the deformable mirror surface, which is investigated in this thesis.

1.4.1 Piezoelectricity

Piezoelectricity is an electromechanical phenomenon coupling the elastic field and the electric field [54] (note that piezo means press in Greek.). Ever since the discovery of piezoelectric behavior by the Curies brothers (Jacques and Pierre) in 1880, this coupled electromechanical characteristic has brought a new dimension in transducer (sensor and actuator) applications. The direct piezoelectric effect corresponds to a piezoelectric material that generates an electric

13

charge/voltage in response to a mechanical force/pressure. Conversely, an electric charge/field applied to the material induces mechanical stresses or strains, and this is called the converse piezoelectric effect [55]. The direct effect is usually the basis in sensor and measurement applications; the converse effect is for precision actuation and manipulation in control applications [56].

Piezoelectric materials may be ceramic- or polymer-based, along with naturally occurring quartz or other crystals. Ceramic-based piezoelectric material is generally directional, due to a process called poling, where the piezoelectric properties are strengthened by applying an electric field at high temperatures, leaving a residual polarization [57]. The type of ceramic-based piezoelectric material generally used is the Lead Zirconate Titanate (Pb(Zr,Ti)O3) crystal [58]; commonly called the PZT. At this point, it is worth mentioning that a piezoelectric actuator is partially polarized after being manufactured. In the absence of an applied electric field or when operated at temperature conditions that exceed the Curie temperature, the individual electric dipoles in the ceramic material are randomly oriented (Fig 1.9 (a)). In this state the ceramic material becomes unpolarized and paraelectric in nature, i.e., it loses the property of piezoelectricity. The individual crystals in the unpolarized ceramic material become symmetric in structure. A symmetrical crystal structure means that the external electric dipole moment is equal to zero. Therefore, the ceramic does not expand/contract [58]. Below the Curie temperature the ceramic material undergoes a phase change and becomes ferroelectric in nature, i.e., the property of piezoelectricity is regained [58]. This phase change makes the individual crystals in the ceramic materials, asymmetric in structure. When the ceramic material is subjected to large electric fields, the electric dipoles align themselves in a direction close to the applied electric field. This causes the asymmetric axis of an individual crystal and all neighboring crystals to expand in a direction close to that of the applied electric field. This process is called polarization (Fig 1.9 (b)). When the electric field is removed, the electric dipoles do not entirely return to their original position. This phase of polarization is called remanent polarization in which the ceramic remains partially polarized. In this phase the external electric dipole moment is not equal to zero. As shown in (Fig 1.9 (b)), the actuator will have some remanent displacement. When an electric field is re-applied, the ceramic material elongates thereby elongating the actuator (Fig 1.9 (c)) [59].

PZT as an actuator is commonly used in shape control of flexible structure [60]. Hishinuma and Yang [61] uses a PZT actuated deformable mirror for space coronagraph. In general, however, PZT is too stiff for membrane optics and reflector applications, whereas piezoelectric polymers offer a much more suitable compliance [3].

The most common piezoelectric polymer in membrane optics and reflector applications is polyvinylidene fluoride (PVDF). Patterson, Pellegrino and Breckinridge [3] investigate the use of PVDF as an actuator for a reconfigurable modular space telescope. Chen et al. [62] study the development of PVDF based piezopolymer actuators to control the surface of a membrane reflector. Fang, Pattom and Wang [63] develop an analytical model of a high precision membrane reflector shape control system with PVDF actuators to investigate the systems functionality on a 35-m diameter membrane reflector.

Piezoelectric actuators are also used in shape control of other types of structure, such as beam [64, 65], plate [66-79], shell [66, 67 and 80] and space truss [81, 82]. Besides the field of active structural shape control, piezoelectric actuators are employed in the active vibration control, ultra precision machines and MEMS systems applications [51, 83-86].

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Fig 1.9 Polarization process for a piezoelectric actuator [59] (a) unpolarized (b) polarized (c) after polarization

1.4.2 Linear Piezoelectric Theory

Most current research on the shape control of piezo-actuated structures is based on the linear piezoelectric theory. The linear piezoelectric theory indicates that the coupling between the electric field and mechanical field is a first-order effect, implying that an induced strain is proportional to the electric field and that the direction of the displacement is dependent on the sign of the electric field. In this section, the linear piezoelectric theory is reviewed, and the relations among various elastic and electric constants are defined [54].

The linear piezoelectric theory is based on a quasi-static assumption in which the electric field is balanced with the elastic field so that these two fields can be decoupled at a given time instant. The direct and converse piezoelectric effects are written as [87]:

EeSD S+= (1.1) EeScT E t= (1.2)

where D is the electric displacement vector; T is the stress tensor (second-order); S is the strain tensor (second-order); E is the electric field vector; e is a tensor of piezoelectric stress coefficients (third-order); et is the transposed tensor of e; S is the dielectric tensor (second-order) evaluated at constant strain; and cE is the elasticity tensor (fourth-order) evaluated at constant electric field. If the piezoelectric tensor e is set to zero, Eq (2.1) and Eq (2.2) become the conventional dielectric and Hookes equations. Note that the stress T and strain S are represented in column vectors:

[ ]t654321 TTTTTTT = (1.3) [ ]t654321 SSSSSSS = (1.4)

where the superscript t denotes the vector or matrix transpose. Subscripts 1-3 denote the normal components and 4-6 denote the shear components. The electric field E and the electric displacement (flux density) D are also written in column vectors:

[ ]t321 EEEE = (1.5) [ ]t321 DDDD = (1.6)

There are a number of ways to write the elastic, piezoelectric, and dielectric governing equations in which (1) stress and electric field, (2) strain and electric displacement, (3) strain and electric field are respectively used as dependent and/or independent variables. Table 1.1 summarizes the four sets of standard representations (according to IEEE standard on piezoelectricity [88]), including the equations defined above.

15

sD is the compliance matrix defined at constant dielectric displacement (note s=c-1); d is the piezoelectric strain constant matrix; g matrix relates the open-circuit voltage at a given stress; e is the piezoelectric stress constant matrix; h matrix relates the open-circuit voltage at a given strain;

T is a free dielectric impermeability matrix evaluated at constant stress and T can be obtained from the inverse of the dielectric matrix . The superscript T (inside the matrix) denotes the properties measured at constant stress T. (Recall that t denotes a vector or matrix transpose and -1 denotes the matrix inverse.). Table 1.2 shows that the four piezoelectric coefficient matrices d, e, g, and h are all related. Note that an adiabatic condition ensures that no heat is added or removed from a given space or volume. The elastic constants are defined at the adiabatic conditions. The piezoelectric and dielectric constants of a piezoelectric material are defined at isothermal and adiabatic conditions. The piezoelectric material is assumed to be nonpyroelectric in this case. Again, the superscripts T, S, E, and D denote the matrix/tensor defined at constant stress, strain, electric field, and dielectric displacement, respectively.

Table 1.1 Four sets of piezoelectric equations (IEEE standard [88])

No. Elastic relationship Electric relationship

1 DhScT D t= DhSE S+= 2 EeScT E t= EeSD S+= 3 DgTsS D t+= DgTE T+= 4 EdTsS t+= EdTD T+=

Table 1.2 Relations among elastic, piezoelectric, and dielectric constants

No. Elastic Piezoelectric Dielectric

1 gdss ED t= dg T= 1= TT 2 1= EE sc Edce = tedTS = 3 hecc ED t+= eh S= thgTS = 4 1= DD sc Dgch = 1= SS

However, piezoelectric actuators also show unwanted nonlinear behaviors in open-loop operation, i.e., hysteresis and creep [89]. The linear constitutive equations fail to explicitly describe the nonlinearities that are present in piezoelectric actuators and that may lead to undesirable inaccuracy and limit system performance [89]. For the active deformable mirror with piezoelectric actuators, one issue that arises is the compensation approach to attenuate the effects of hysteresis and creep in piezoelectric actuators on the deformable mirror.

1.5 Hysteresis and Creep Compensation

Creep, a slow drift of the displacement of piezoelectric actuator responding to a constant applied voltage as shown in Fig 1.10 [90], is the effect of the remanent polarization which continues to change over time [59]. Creep can cause unaffordable positioning errors in slow and static applications [91]. Hysteresis, which is observed between the voltages applied to the actuator and the obtained displacement as seen as in Fig 1.11, is a non-linearity such that the obtained displacement depends on present and previous input voltages. Hysteresis severely limits system performance, since it leads to a positioning uncertainty of up to 15% of the motion range [92].

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1.5.1 Closed-loop and Charge Control

To compensate hysteresis and creep effects, closed-loop control techniques seem to be the best way to improve positioning precision and provide robustness to modeling uncertainties and parameter variation [93, 94]. However, due to the high cost and space requirements of displacement sensors, the use of closed-loop control techniques is strongly limited in practice [95, 96].

An alternative way to minimize hysteresis and creep is to drive the piezoelectric actuators using charge input, instead of voltage input [59, 97]. When connected electrically, the actuator acts like a nonlinear capacitor [98] which changes its capacitance even when the input voltage is kept constant. The change in the capacitance leads to a change in the amount of the charge acting on the actuator. This causes the nonlinear creep and hysteresis. Regulating the current and hence the charge prevents the actuator from changing its capacitance [98] thereby leasing to a significant reduction in hysteresis and creep. Therefore, a charge input allows linearizing approximately a piezoelectric actuator [99-102]. Charge control can significantly reduce the effect of hysteresis and creep, but at the expense of reducing the effective displacement range of the actuator [93] and of increasing the electronic complexity required for effective charge control [59]. Hence, the open-loop control techniques attract more attention [103-106] and it is demonstrated in extensive studies that the open-loop techniques can markedly reduce the positioning uncertainty and linearize the actuator.

Fig 1.10 Creep in piezoelectric actuators

Fig 1.11 Hysteresis in piezoelectric actuators

1.5.2 Open-loop Control

In open-loop control techniques, the creep and hysteresis can be compensated by the use of a feed-forward inverse model. Hence, the direct model must be accurate and can be inverted. Many models have been developed to characterize the hysteresis and creep behaviors. All the models can roughly be classified into three different types. One class includes so-called phenomenological models, which are mainly focused on mathematical aspect of the problem without considering the underlying physics. The second type comprises the constitutive models based on physical principles and basic physical elements including springs, massless carts and Coulomb frictions are used to characterize hysteresis or creep. The last class consists of domain wall approaches modeling hysteresis and creep by the use of nonlinear constitutive equations through consideration of domain and domain wall mechanisms within piezoelectric materials.

Time (s)

Out

put d

ispl

acem

ent (

m)

Input voltage (V)

Out

put d

ispl

acem

ent (

m)

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However, the domain wall approaches have limited performance characteristics as the domain wall dynamics of hysteresis and creep phenomenon has not been completely understood. Thus the domain wall models are relatively less applied in practice.

Creep

o Phenomenological model Jung, Gweon [107] and Kuhnen [108] use the logarithmic model to describe creep behavior. El-Rifai and Youcef-Toumi [90] compare the logarithmic model with the linear time-invariant (LTI) model for creep and find that the LTI model is more appropriate to predict creep.

o Constitutive model Croft, Shed and Devasia [103], Gaul and Becker [109] and Devasia et al. [110] portray creep by the use of a linear viscoelastic model made of springs and voigt elements, which is equivalent to LTI models from the control viewpoint. Richter et al. [111] propose a nonlinear viscoelastic model for creep, which can more precisely describe the steady portion of creep at a fixed time-frame than the LTI and logarithmic models due to a specific term corresponding to the steady component of creep.

o Domain wall model Some researchers [112-115] create the creep model using nonlinear constitutive relations based on the observation of the domain walls bending and translation in piezoelectric actuators.

Among these creep models, except from the LTI model that can be directly inverted [90, 103, 109 and 110], the other models are difficult to invert mathematically for control purposes due to the nonlinear functions existing in the models. Thus some researchers construct the inverse model of creep experimentally. Koops, Scholte and de Koning [116] conduct an experimental study on the performance of a piezoelectric actuator and observe that relaxation signal can reduce creep. Jung, Shim and Gweon [117] observe voltage creep phenomenon that can reduce creep and model voltage creep using a logarithmic model that is the inverse model of creep. Changhai and Lining [118] propose an exponential model as the inverse model of creep and identify the parameters by experiments. However, the experiments carried out for constructing these logarithmic and exponential inverse models can lead to an infinite voltage at the step time that can degrade the performance of the piezoelectric actuators [119]. The creep model adopted in this thesis [120] is a nonlinear viscoelastic model and it has been selected due to its capability of precisely capturing the steady portion of creep at a fixed time-frame. Moreover, a new approach is proposed to invert the creep model. A PID closed-loop control system is implemented in MATLAB to construct the inverse model of creep while limiting the infinite voltage appearing at the step time [118, 119]. It also subtly circumvents the difficulties to mathematically invert the nonlinear creep model and can easily be implemented.

Hysteresis

o Phenomenological model The most well-known phenomenological approach, known as Preisach model [121- 127], has found widespread acceptance in the modeling of hysteresis in piezoelectric materials. Ge and Jouaneh [128-130] first discuss the adaptation of the Preisach model to describe the hysteresis in piezoelectric actuators, develop generalized Preisach model by relaxing the congruency requirement on the hysteresis loops of a piezoelectric actuator, and present a tracking control approach for a piezoelectric actuator based on a feed-forward loop and a PID feedback

18

controller. Song et al. [131] propose fast and efficient numerical implementations of Preisach model and its inversion and conduct the tracking control of a piezoelectric actuator. Zhou et al. [132] employ successfully the same method to track control of a thunder actuator. Tan, Venkataraman and Krishnaprasad [133] propose a constrained least-squares algorithm to identify a discrete Preisach model from measurements and discuss a nonlinear inversion algorithm for the resulting Preisach model. Natale, Velardi and Visone [134] identify the Preisach model by the adoption of both a fuzzy approximator and a feed-forward neural network. Prandtl-Ishlinskii (PI) model also attract much attention [135-138], it is formed by a weighed superposition of infinite elementary hysteresis operators as in the Preisach model. There are still many phenomenological models, such as the Duhem model [139], the bond graph models [140], the parallel-series Iwan model [106], and the quadrilateral model [141].

o Constitutive model Goldfarb and Celanovic [97] develop a nonlinear lumped parameter model of a piezoelectric stack actuator including some basic physical elements to describe the actuator behavior. In formulating this model, the authors propose a generalized Maxwell resistive capacitor as a lumped-parameter causal representation of rate-independent hysteresis. Yeh, Lu and Wu [142] propose a model consisting of basic physical elements and utilizes a Maxwell-slip structure to describe the hysteresis.

o Domain wall model Smith and Ounaies [143] create a domain wall model for hysteresis based on the quantification of the reversible and irreversible motion of domain walls pinned at inclusions in the materials, which is an ODE model having five parameters. This model is based on a domain wall theory for hysteresis in general ferroelectric materials [144]. Ru and Sun [145] derive a polarization model to explain hysteresis from the analysis of microscopic polarization mechanisms and according to the domain wall theory of piezoelectric actuators.

The Preisach model is adopted in this thesis due to the compact and simple formalism leading to fast algorithms and reliable results. However, the model accuracy is dependent on the number of measured first-order reversal curves of hysteresis loop, and due to the limitations of experimental conditions, it is possible that the number of obtained curves is insufficient to accurately model hysteresis. Moreover, the detachment between the major and minor loops is not taken into account in the classical Preisach model. To overcome these limitations, this thesis proposes a modified Preisach model to accurately characterize the hysteresis in piezoelectric actuators.

In order to simultaneously compensate hysteresis and creep, models of the two effects have to be properly combined and the input is computed based on the inverse of the combined model. However, probably because the interacting mechanism between creep and hysteresis is not fully understood, in the literature there is no definite way to merge the two effects [146]. For example, hysteresis and creep are combined in a parallel manner such that the displacements generated by respective effects are added together in [96]. Models for hysteresis and creep, together with a vibration model which describes the high-frequency dynamics of piezoelectric actuators, are cascaded to describe the overall behavior for piezoelectric actuators in [95, 103]. A more complicated way of combining hysteresis and creep can be found in [147] where a hysteresis model is first cascaded with a creep model and then the cascaded model is combined in parallel with another hysteresis model. Among these three models, except the combined model in [95, 103] that can be directly inverted by multiplying together the inverses of the creep and hysteresis, the

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inversions in [96, 147] have to be realized by iterative procedures whose convergence are not guaranteed.

In addition to the combined models mentioned, another interesting method is adopted in [146, 148-153], where an integrated model is developed to simultaneously portray creep and hysteresis and the compensation is carried out by inversion of the model. However, the integrated model can result in a complicated identification procedure due to too many involved parameters and in a complex algorithm to compute the inverse model in real-time applications.

This thesis proposed a combined approach inspired by literatures [95, 103], which multiplies together the inverse models of hysteresis and creep and demonstrates very good results. All the approaches to compensate creep and hysteresis presented above are used for a single piezoelectric actuator (SISO system).

1.5.3 Control of Multiple-Input-Multiple-Output (MIMO) system

For multiple-input-multiple-output (MIMO) cases, many closed-loop control scheme that have been established to control SISO system can be extended to control multiple piezoelectric actuators (MIMO system), e.g., PID control [154, 155], H2 control [155], and sliding mode control [154] since such methods can be easily extended to MIMO cases. Few researchers study the simultaneous compensation of hysteresis and creep in MIMO system by open-loop control. Rakotondrable [156] investigates the modeling and compensation of multivariable creep in multi-degree of freedom (DOF) piezoelectric actuators by open-loop. The approach only deals with the compensation of creep, regardless of hysteresis. Furthermore, the efficiency of this approach is only confirmed by experiments performed in a single piezoelectric actuator with several DOFs, but not in the system with multiple piezoelectric actuators. In this thesis, a MIMO controller that can simultaneously cancel by open-loop the hysteresis and creep in multiple piezoelectric actuators of the system under study will be developed.

1.6 Summary

This chapter provides the contextual background for research in the field of shape control of spatial mirror surface. A summary of current research and literature encompassing space telescopes, wave-front sensing and control (WSFC), deformable mirrors, piezoelectric actuators, and the compensation of creep and hysteresis is presented. From this survey, it is found that a compensation approach by open-loop control that can simultaneously reduce hysteresis and creep in multiple piezoelectric actuators for the shape control of spatial mirror surfaces is lacking. In addition, the novel approaches that can model the creep and hysteresis more accurately and compensate them more efficiently for a single piezoelectric actuator is required. The work proposed herein will combine and expand upon the aforementioned literature to advance the state-of-art in the shape control of spatial deformable mirrors actuated by piezoelectric actuators.

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Chapter 2

Experimental Set-up

This chapter describes the mirror surface control system for laboratory experiments performed during this thesis at ISAE. The design of a real active mirror is a very interesting topic and can be the subject of a whole thesis. It is not our purpose in this work and we have only developed a small-scale mirror structure where the problems of control are present and on which the control laws can be tested. The design has been inspired by another study led at ISAE before this thesis. The experimental set-up consists of a small-scale mirror test structure, a real time control system and other lab equipment. Firstly, the mirror test structure and its simplified design are presented. Then, the real time control system and other equipment are described.

2.1 The Mirror Test Structure

2.1.1 Introduction

To realize the shape control of a deformable mirror surface, a small-scale mirror test structure with piezoelectric actuators is developed as study object for testing purposes. The mirror test structure shown in Fig 2.1 comprises a circular plate of glass, seven mirror feet, a backing plate, seven ring-type piezoelectric patch actuators, upper and lower supporting structures with seven holes. Except from the glass plate, the other components are made in aluminum alloy. The piezoelectric actuators deform the backing plate, the feet fixed to the backing plate have a vertical movement and the mirror fixed to the feet is thus deformed. The supporting structure has two functions:

Give stiffness to the structure Make it possible for the piezoelectric actuators to generate a bending of the backing plate

in each hole of the structure

The connections between the components and between the structure and the experimental table are as follows:

Glass plate and feet: bonded Feet and backing plate: fixed Piezoelectric actuators and backing plate: glue bonded Backing plate, upper and lower supporting structures: fixed Lower supporting structure and experimental table: fixed

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Fig 2.1 The developed mirror test structure

2.1.2 Design of Mirror Test Structure

Most of the design parameters were already set by the previous study led at ISAE: the diameters of the holes, the size of the piezoelectric actuators and the kind of bonding. In this study, the only parameter that was not set is the backing plate thickness. The objective is to control displacements of a few tens of m. Thus, the design of the mirror test structure has been performed with this objective and a single design parameter. The displacement of the mirror surface can be measured by capacitive sensors (D510.101, PI Inc., detailed in section 2.2). The measurement range of this type of capacitive sensor is 0~100 m according to the technical data [157]. The ring-type piezoelectric patch actuator employed is made of NCE51 PZT (Noliac Inc.). The allowable maximum operating voltage for static operations is 400V [158]. To this end, finite element method is employed to predict the displacement response to a voltage applied to a piezoelectric actuator and to compute the dimensions of backing plate thickness that allows being in the good operating range. The ANSYS software is used as the design tool due to its advantage in multiphysics simulation [159]. The specific design procedures are as follows:

Step 1: create the geometric model of the mirror test structure as depicted in Fig 2.2. Step 2: define material type include glass, aluminum alloy, glue and NCE51 PZT [160]. Step 3: define material property for each component of the mirror test structure, presented

in section 2.1.1. Step 4: define the connecting conditions between the components as presented in section

2.1.1. The conditions for glue bonded are realized by adding a layer of thickness 30m and of Young modulus 3GPa in the model while the conditions for bonded and fixed are realized with the ANSYS connection totally bonded.

Step 5: create the mesh, using SOLID226 element [161] for piezoelectric actuators due to its capabilities to include piezoelectric effect, shown in Fig 2.3.

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Step 6: define the excitations. Two cases are tested: (1) applying 100 V to the piezoelectric actuator at the center; (2) applying 100 V to the piezoelectric actuator at the side (applying voltages to any one of the six piezoelectric actuators at the side which resulting in the same response due to the symmetric structure).

Step 7: carry out the static analysis and discuss the results. The final dimensions of each component listed in Table 2.1. Fig 2.4 shows the deformation of the mirror structure for the two studied cases and the displacements of seven interesting points located at seven actuators (Fig 2.5) are listed in Table 2.2. (seven piezoelectric actuators and their corresponding interesting points are same numbered (Fig 2.5))

From the predicted displacement responses, it can be seen that applying a 100 V voltage to piezoelectric actuator generates a maximum displacement of less than 14 m. So for the maximal voltage 400 V and with the assumption of a linear piezoelectric phenomenon, the maximal displacement should be less than 56 m, which is within the measurement range of the capacitive sensors with a large enough margin.

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(a)

(b)

Fig 2.2 The geometric model of the mirror test structure created in ANSYS (a) block diagram (b) side view

25

Fig 2.3 The mesh of developed model

Table 2.1 Dimensions of the components of the mirror test structure

Component Dimensions Glass plate Diameter: 300 mm, thickness: 3 mm

Connecting screws Diameter of head and nut: 15 mm, thickness: 8 mm, diameter

of stem: 6 mm, length: 60.5 mm

Backing plate Diameter: 300 mm, thickness: 0.5 mm, diameter of seven

holes: 6 mm Upper and lower supporting

structures Diameter: 300 mm, thickness: 10 mm, diameter of seven

holes: 80 mm

Piezoelectric actuators Outer diameter: 50 mm, inter diameter: 34.6 mm, thickness:

0.6 mm Glue (making actuators adhere to

backing plate) Outer diameter: 50 mm, inter diameter: 34.6 mm, thickness:

0.3 mm

26

(a)

(b)

Fig 2.4 The deformation of the mirror structure for the two studied cases: (a) case 1 (b) case 2

27

Fig 2.5 Seven interesting points on the mirror surface (points and their corresponding actuators with the same numbers)

Table 2.2 Displacements of the mirror points computed with ANSYS (m)

No. Case 1

(100 V to actuator 1) Case 2

(100 V to actuator 3) 1 12.50 6.29 2 7.39 8.11 3 8.46 13.89 4 7.25 8.09 5 6.62 3.83 6 6.40 2.31 7 6.58 -0.27

2.2 Experimental system

In addition to the mirror test structure, the set-up also comprises the real time control system and other lab equipment.

2.2.1 Real Time Control System

The real time control system used herein was developed at ISAE based on Simulink, Real-Time Workshop and xPC Target from MATLAB families [162]. The xPC Target from MATLAB is chosen to support the real-time environment [163]. This environment includes a host PC and a target PC (Fig 2.6). The host PC is a development platform which has Visual C++, MATLAB Simulink and Real-Time Workshop. xPC Target does not require DOS, Windows, Linux, or any another operating system on the target PC. Thus, the target PC is booted with a boot disk that includes the highly optimized xPC Target kernel. It is created on the host PC by setting up the xPC Target environment properties using xPC explorer (Fig 2.7).

1

2

3

4

5

6

7

28

To create a target application, a Simulink model (ex. Fig 2.8) is first created on the host PC. Real-Time Workshop provides the utilities to convert the Simulink model into C code and then with a third-party C/C++ compiler, compiles the code into a real-time executable. This executable is then converted to an image suitable for xPC Target and uploaded from the host PC to the target PC through network. While the application is running, the target PC communicates with the mirror test structure via two analog I/O boards (NI PCI-6713 DA Analog Output, NI PCI-6251 AD Analog Input, supplied by National Instruments Company and shown in Fig 2.9). Finally the target PC transfers information to the host PC through network. The retrieved data can be post-processed or plotted under MATLAB on the host PC.

Fig 2.6 Target PC

Fig 2.7 xPC Target explorer

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Fig 2.8 A Simulink model creating a target application

(a) (b)

Fig 2.9 Analogy I/O boards (a) NI PCI-6713 DA Analog Output (b) NI PCI-6251 AD Analog Input

2.2.2 Capacitive Sensor

The displacements of the points on the mirror surface are measured by using noncontact capacitive sensors, D510.101 (Fig 2.10), with a measurement range [0-100 m] provided by Physik Instrumente Company. The sensors require a signal conditioner with 3 channels: E509-03 (Fig 2.11) [164]. Three sensors are used and placed above the specific measurement points (point 1, point 3 and point 6 (Fig 2.5)) on the mirror surface as shown in Fig 2.12. These points of the glass mirror surface are covered by very small and thin aluminum plates because capacitive sensors require electrically conductive measuring target surfaces.

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Fig 2.10 The capacitive sensor D510.101

Fig 2.11 Signal conditioner E509-03 for capacitive sensors D510.101

Fig 2.12 The capacitive sensors placed at specific measurement points

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2.2.3 Power Amplifier for Piezoelectric Actuators

The maximum working voltage for NI analog I/O boards is 10V. So the control signal coming from the NI analog output board must be amplified. A high power amplifier with three channels (Fig 2.13) is adopted in the set-up, for which the output voltages are 100 times greater than the input voltages.

Fig 2.13 The 3 channels high power amplifier

2.2.4 Integration of Experimental Systems

Fig 2.14 shows the functional diagram of the complete experimental set-up integrating the developed mirror test structure and the equipment for control and measurements.

Fig 2.14 The functional diagram of experimental set-up

2.3 Summary

This chapter describes the experimental set-up that will allow investigating hysteresis and creeping compensation in a single piezoelectric actuator as well as an active control approach to actuate the deformable mirror with multiple piezoelectric actuators to reach the required shape.

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Chapter 3

Creep Compensation in a Single Piezoelectric Actuator

3.1 Introduction

This chapter focuses on the compensation of creep nonlinearity in a single piezoelectric actuator by open-loop control. The approaches for modeling, identification and compensation of the creep are presented in detail and the creep is compensated in three actuators (actuator 1, actuator 3 and actuator 6 (Fig 2.1, Fig 2.5 and Fig 3.1) one after the other. To investigate the creep in a single actuator, the actuator is considered as a single-input and a single-output (SISO) system. In other words, when an actuator is actuated, other actuators are free of inputs. The input is the voltage applied to the actuator and the output is the displacement of the interesting point (point just above the actuator and on the mirror as shown in Fig 3.1).

Note: x represents the displacement of the interesting point. x1, x3 and x6 are respectively related to point 1, point 3 and point 6 (Fig 3.1). U represents the voltage applied to the piezoelectric actuator. U1, U3 and U6 respectively correspond to the input voltage applied to actuator 1, actuator 3 and actuator 6 (Fig 3.1). In this chapter, displacements x1, x3 and x6 corresponding respectively to the constant input voltages U1, U3 and U6 (three SISO systems) are of interest. The nonlinear relationships, x1=C11(U1,t), x3=C33(U3,t) and x6=C66(U6,t), are respectively employed to describe the creep behavior in actuator 1, actuator 3 and actuator 6.

(a) (b) (c)

Fig 3.1 The conjugate actuators and interesting points (a) actuator 1, point 1 (b) actuator 3, point 3 (c) actuator 6, point 6

In this chapter, a nonlinear viscoelastic model is above all used to construct the creep model. Then, the inverse model of creep is derived by using a new method. Finally, by insertion of the

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inverse model of creep in open-loop operation, the creep compensation is realized in a single piezoelectric actuator.

3.2 Observed Behavior

As mentioned in Chapter 1, creep corresponds to the slow drift of the displacement of the piezoelectric actuator responding to a constant applied voltage. Fig 3.2 presents the typical creep response of a piezoelectric actuator for a constant step input voltage. At the step time t0, instantaneous elastic response followed by damped oscillation appears. At td, damped oscillation ends and the response reaches a static gain. If the constant input is still kept, a slow drift begins to emerge from td. Physically, creep is caused by a gradual alignment of dipoles, and therefore the slow drift should converge to a steady value. Convergence is achieved, however, over a period of time that is much larger than the time constant of any practical control system. Hence, the real experimental creep response shows that a steady (or constant) rate of drift emerges from ts to the time-frame of the experiment. We divide the creep behavior into two parts: the transient portion and the steady portion. The moment ts is regarded as a watershed. The steady portion of creep presents a slow drift with a constant rate (after ts), while the instantaneous elastic response, the damped oscillations and the drift with changing rate (from td to ts) are together considered as the transient portion of creep (from t0 to ts).

Fig 3.2 Typical creep response

A series of step input voltages is applied to actuator 1, actuator 3 and actuator 6. Fig 3.3 shows the experimental creep responses for the three actuators under study. The sampling time is 0.01 s. For each input voltage, ranging from 20 to 200 V with 20 V increment, data are recorded for 10 seconds, i.e., the time-frame of experiment is 10 seconds. It is observed that the step time t0 is at 0 s, the transient portion last till 3.5 s, time for which the steady portion starts.

Time (s)

x (

m)

steady portion of creep

static gain

drift

final value

dampedoscillation

td

t0

ts

transient portion of creep

instantaneouselastic response

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The level of creep (= |drift/static gain| 100%) is used to evaluate the severity of the creep. Table 3.1 gives the levels of creep after 10 s (the time-frame of experiment) of actuator 1, actuator 3 and actuator 6.

(a)

(b)

Fig 3.3 Experimental creep response for (a) actuator 1, (b) actuator 3 and (c) actuator 6

0 1 2 3 4 5 6 7 8 9 10 110

1

2

3

4

5

6

7

8

9

10

Time (s)

x 1 (

m)

20V

200V

60V

40V

180V

160V

140V

120V

100V

80V

td

ts

0 1 2 3 4 5 6 7 8 9 10 110

2

4

6

8

10

12

14

Time (s)

x 3 (

m)

20V

200V

180V

160V

140V

80V

60V

40V

100V

120V

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(c)

Fig 3.3 Experimental creep response for (a) actuator 1, (b) actuator 3 and (c) actuator 6 (continued)

Table 3.1 Levels of creep of actuator 1, actuator 3 and actuator 6

Input voltage Level of creep (= |drift/static gain| 100%)

Actuator 1 Actuator 3 Actuator 6 20 V 2.71% 2.41% 4.09% 40 V 3.53% (max) 2.89% 4.18% (max) 60 V 3.13% 2.93% 4.13% 80 V 3.47% 3.01% 3.79%

100 V 2.90% 3.21% 3.79% 120 V 2.64% 3.43% (max) 3.74% 140 V 2.99% 3.23% 3.68% 160 V 3.45% 3.28% 3.76% 180 V 2.83% 3.37% 3.76% 200 V 2.89% 3.29% 3.79%

From Fig 3.3, we can also see that the experimental responses of three actuators when applying 100 V are different from the computed results by the use of ANSYS software in Chapter 2 and that the responses of actuator 3 and actuator 6, that are both on the periphery of the mirror, are unequal when applying the same constant voltages while they theoretically should have the same response due to the symmetric mirror structure. The main reason for these differences is that the practical connecting conditions between the components of the mirror structure are not perfectly the same.

0 1 2 3 4 5 6 7 8 9 10 110

5

10

15

20

Time (s)

x 6 (

m)

20V

40V

60V

80V

100V

120V

140V

160V

180V

200V

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3.3 Creep Model

3.3.1 A Nonlinear Viscoelastic Model

A large number of models have been proposed and tested over the last years to explain the observed phenomena with a certain degree of accuracy, but there is no all-embracing theory which is likely to accommodate all eventualities [165]. As in other fields, models can be constructed by empirical observation or by physical considerations. It is observed that creep leads to time-varying displacement in the mechanical domain, so it is postulated that creep is induced by the materials nonlinear viscoelastic behavior. Based on this, a nonlinear viscoelastic model is employed to model creep. The major advantage of the model of this type is that it can model steady creep portion with constant rate of drift for each input voltage as observed through the step responses experiments.

Fig 3.4(a) illustrates the classical nonlinear viscoelastic model to describe the creep behavior. The classical nonlinear viscoelastic model (Fig 3.4(a)) consists of:

a linear spring that corresponds to an instantaneous elastic response xe (Fig 3.4(b)) (EC represents the elastic coefficient between the input voltage U and the output displacement x for a piezoelectric actuator).

a single nonlinear dashpot that relates to the steady portion of creep xs (Fig 3.4(b)) for constant voltage. (It is considered that the constant rates of drift in the steady portion depend on voltage in a nonlinear fashion.)

a series of nonlinear Voigt elements (parallel arrangement of a nonlinear spring and a dashpot [166]) that represents the transient portion of the creep (Fig 3.4(b)).

As described in the preceding section, the instantaneous elastic response is followed by a damped oscillation. This is possibly a structural vibration of the piezoelectric actuator, generally not to be dealt with the creep model. Moreover, for piezoelectric actuators, it is considered that there is no instantaneous elastic component xe, as the subsequent drift is much slower than the initial fast response [111].

(a)

Fig 3.4 (a) The nonlinear