Robustesse).

First of all, I would like to thank the director of doctorate school EEATS, M. Christian Commault, the x-director of Gipsa-lab, M. Jean-Marc Chassery, and the current director, M.

Jean-Marc Thiriet, for allowing me to perform my PhD research work in Gipsa-lab. I would also like to present my sincere regards to my PhD supervisors, Mme. Alina Voda and M.

Gildas Besançon, for their support and patience during all my stay in Gipsa-lab. I would also like to extend my appreciation to the committee members, M. Antoine Ferreira, M. Adrian Filipescu and M. Alireza Karimi, for the evaluation of my PhD research work, their encouragements and useful discussion. Indeed, I express my profound gratitude to the president of the committee, M. Jean-Marc Thiriet, for his time, kindness and always having confidence on the young generation.

I would like to sincerely thank the technical team of control system department, M. Didier, M.

Thierry and particularly M. Gabriel Buche for sharing their expertise and for their support and help in the experimental part of this PhD thesis. I will not forget M. Pascal for his work in order to make the presentation of this PhD thesis more attractive and understandable by videos using real-time experimental data. I sincerely appreciate the support of M. Sylvain Blanvillain who initiated this project for his PhD research work. Here, I will definitely express my profound regards to Marie-Thé and Patricia (the permanent staff members of control system department) who always encouraged me with their sweet words.

During this research work, I made many good friends and we all were like a family. The time I spent with them, within Gipsa-lab and also in our out-door activities, is really unforgettable. I list some of these friends here: Marouane, Lizeth, Amine, Joumana, Nesrine, Andi, Diana, Andra, Tony, Simona, Lam, Hieu, …. and indeed, there are many many others.

List of Figures

2.1 (a) Case of classical mechanics: a traveling electron of energy E 0 encountering a potential barrier of height V. In this case, the electron would be unable to pass through the potential barrier and would be reflected. (b) Case of quantum mechanics: the electron interacting with the potential barrier of height V. The electron's wave function acts as a travelling wave before (Region I) and after (Region III) the barrier, but like an exponential under the potential barrier (Region II). In this fashion, the electron has a non-zero probability of tunneling through a potential barrier. . . .

Schematic

2.9

The general approach to deal with cross-coupling effect, where vertical deflection z is affected by the z-axis input u Z and also the x-axis input

List of Notations

H R (z -1 ) Pre-specified fixed filter for R(z -1 )
H S (z -1 ) Pre-specified fixed filter for S(z -1 )

R ′ (z -1 ) Free filter for R(z -1 ) S ′ (z -1 )
Free filter for S(z -1 )

P D (z -1 ) Dominant closed loop poles P F (z -1 )
Auxiliary closed loop poles

List of Publications

Journal paper International conference proceedings

Introduction

General context

The recent years have witnessed the explosive growth of an emerging scientific field: nanosciences and nanotechnology, which is the study of manipulating matter with atomic and molecular scale. This explosive growth started around two and a half decades before when Gerd Binnig and Heinrich Rohrer from IBM laboratory, Zurich got Nobel Prize in physics by their invention of Scanning Tunneling Microscope (STM) with the ability to characterize any conductive matter morphology with atomic scale resolution [START_REF] Binnig | Scanning tunneling microscopy[END_REF]. Nanotechnology is very diverse and this has fundamentally changed the research in numerous areas, including physics, biology, chemistry, electronics and materials science. Nanopositioning is an important aspect of research in nanotechnology and it involves precision control and manipulation of devices and materials at a nanometer scale. The quality in terms of precision of the measured signal at nanometer scale is one of the key requirements where the interface (coupling) between physics, electronics, thermal and mechanical sciences play crucial role. One of the ways to achieve this key requirement is the use of some better control techniques.

This manuscript is written focusing on control design techniques for this emerging field of science.

Problems statement

Many new problems have emerged working at nanometer scale. For example, the small size and extreme fragility of the micro or nanoscale objects make them difficult to handle. Ultrahigh positioning precision is one of the pivotal requirements in many applications of nanotechnology. The precision of measured signal at nanometer scale is highly sensitive to the operating conditions which introduce the major difficulty in terms of repeatability. Besides this requirement of high precision, there is also an increasing need of high bandwidth for fast speed operations. To meet the dual goal of high precision at high bandwidth, many novel actuators and sensors have been studied

and developed. However, the precision is adversely affected by the sensor noise as well as actuator nonlinearities such as hysteresis, and drifts due to creep phenomenon.

Additional external environmental disturbances influence the precision of measured signal as well. Quantum mechanical effects are very important at this nanometer scale.

Indeed, micro and nano systems are complex to study and the techniques of automatic control become essential. A precision and accuracy of the measured signal at nanometer scale can not be achieved without a better control scheme. In addition, physical limitations and non-linearities in the control loop are limiting factors to be considered in order to get better performances. The control loop stability and robustness and its tradeoff with desired performances should not be neglected in order to achieve optimal solutions.

Thesis objectives and contribution

The objective of this thesis was to study, propose and experimentally validate modern control laws for high precision nanopositioning using tunneling current, while working at nanometer scale. The controller must indeed be robust and have a capability to reject the external disturbances in order to achieve the desired performances. The main contribution of this thesis is highlighted here:

1. Analysis of a system of tunneling current (quantum mechanical phenomenon which occurs at nanometer scale) measurement with proposition of a corresponding dynamic modeling and formulation of the related control problem with desired measurement performances.

2. Synthesis of modern robust control techniques for the considered system (in the vertical z-direction) in order to achieve the desired performances in terms of measurement precision, large closed loop bandwidth with robustness and stability.

3. Realtime experimental validation with time and frequency domain comparison between proposed and conventional control techniques for the considered system over an experimental setup (at ambient atmosphere) developed in Gipsa-lab, Grenoble.

4. For the application of scanning (in the horizontal x-direction), proposition of a dynamic modeling of the MIMO plant having horizontal, vertical and cross coupling dynamics and synthesis of a MIMO controller in order to reduce the couplingcaused positioning error for the tunneling current measurement system (in the vertical z-direction).

Manuscript organization

This thesis manuscript is organized as follows:

Chapter 2: This chapter presents very briefly the quantum mechanical phenomenon of tunneling current, its different applications in nanosciences and the associated control techniques. The last part of this chapter highlights the control issues for nanopositioning. Chapter 6: Finally, the general conclusion and perspectives of this work are presented in this last chapter of the manuscript.

Chapter 2

State of the Art

Nanotechnology is an emerging field of science, where nanopositioning is one of the fundamental requirement. Different control issues in terms of ultrahigh positioning accuracy with large closed-loop bandwidth, robustness, repeatability, presence of noise, physical limitations and non-linearities in the control loop etc., are the great limitations in order to enhance the performances of the nanopositioning systems. The study and development of new actuators and sensors for high performance nanopositioning systems are the recent topics in the research community. Among the phenomena of interest at nanoscale, one can find the tunneling current, which was practically experienced first time in 1980s and has vast applications in the domain of nanotechnology e.g.

surface imaging with atomic scale resolution, nanopositioning tunneling sensor and also tunneling accelerometer with high sensitivity. This thesis is focused on the control design issues in a tunneling current measurement system.

The tunneling phenomenon is briefly presented in Section 2.1 and its different applications are highlighted in Section 2.2.

The piezoelectric effect which is also of interest for nanosystems is described in Section 2.3.

The control issues associated with the the problem of nanopositioning in terms of performance compromises, dynamic nonlinearities, modeling errors and coupling effects are presented in Section 2.4.

Finally, Section 2.5 draws some conclusions.

In early 1980s, Gerd Binnig and Heinrich Rohrer first practically experienced the phenomenon of tunneling current when an extremely sharp metallic electrically charged tip is approached at the vicinity of a conductive sample surface (distance between tip apex and sample surface in the range of 0.1 -1 × 10 -9 m) [START_REF] Binnig | Scanning tunneling microscopy[END_REF]]. An important application of this tunneling current with the ability to scan the tip against the sample surface was the invention of Scanning Tunneling Microscope (STM). It was the first member of the family of Scanning Probe Microscopes (SPM) that can characterize the surface morphology with atomic resolution. The manipulation and interrogation at nanometer scale after the invention of STM necessitate the positioning systems with atomic scale resolution, where nanopositioning is needed to scan the probe (tip) over a sample surface. Tunneling current is also now widely used to measure the accelerations down to sub-micro-g [START_REF] Rockstad | A miniature, high-sensitivity, electron tunneling accelerometer[END_REF], [START_REF] Liu | Characterization of a high-sensitivity micromachined tunneling accelerometer with micro-g resolution[END_REF]], [START_REF] Liu | A high-precision, wide-bandwidth micromachined tunneling accelerometer[END_REF] and to sense sub-micrometer displacements [START_REF] Bocko | The scanning tunneling microscope as a high-gain, low-noise displacement sensor[END_REF], [START_REF] Ekinci | Electromechanical transducers at the nanoscale: Actuation and sensing of motion in nanoelectromechanical systems (NEMS)[END_REF], [START_REF] Blanvillain | The tunnel current as a subnanometer motion sensor[END_REF].

In the next section, a brief introduction of the phenomenon of tunneling current is presented.

Tunneling Phenomenon

According to classical mechanics, a particle (electron) that does not have enough energy to overcome a potential barrier, can not pass through it. On the contrary, in quantum mechanics [START_REF] Liboff | Introductory quantum mechanics[END_REF]], an electron is described by a wave function which can have a non-zero probability of tunneling through a potential barrier, even if its energy is lower than the barrier. When an electron moves through the barrier in this fashion, it is called tunneling effect. If the barrier is thin enough then there is always a probability of observing an electron on the other side of the region. Tunneling current falls in the category of quantum mechanics. In this section, we will describe very briefly the phenomenon of tunneling current, although the complete details can be found in [START_REF] Chen | Introduction to Scanning Tunneling Microscopy[END_REF].

Mathematically, in classical mechanics, an electron in a potential V (x) is described by:

p 2 2m + V (x) = E (2.1.1)
where m is the electron mass, E is its energy and p the momentum. In regions where E > V (x), the electron has a non-zero momentum p = [2m(E -V )] 1/2 . On the other hand, if E < V (x), the electron can not penetrate into any region or a potential barrier as mentioned above.

In case of quantum mechanics, the electron is described by a wave function ψ(x), which satisfies the Schrödinger equation [START_REF] Landau | Quantum Mechanics[END_REF]:

- 2 2m ∂ 2 ∂x 2 + V (x) ψ(x) = Eψ(x) (2.1.2)
where is the reduced Planck constant. The potential barrier can be generated by an empty space between two electrodes. For simplification, the potential between the points x = 0 and x = L (Region II) can be considered as constant, and also before x = 0 (Region I) and after x = L (Region III), the potential barrier is considered zero (see Fig. 2.1). Under such conditions, the potential barrier V (x) does not vary with time, so we can use time independent (one dimensional) Schrödinger equation for all three regions as follows:

Region I : ∂ 2 ψ I ∂x 2 + 2m 2 Eψ I = 0 (2.1.3) Region II : ∂ 2 ψ II ∂x 2 - 2m 2 (V -E)ψ II = 0 (2.1.4) Region III : ∂ 2 ψ III ∂x 2 + 2m 2 Eψ III = 0 (2.1.5)
The solutions ψ I , ψ II and ψ III of the equations (Eq. (2.1.3) to Eq. (2.1.5)) are given respectively by:

ψ I = Ae ik 1 x + Be -ik 1 x (2.1.6)
Figure 2.1: (a) Case of classical mechanics: a traveling electron of energy E 0 encountering a potential barrier of height V. In this case, the electron would be unable to pass through the potential barrier and would be reflected. (b) Case of quantum mechanics: the electron interacting with the potential barrier of height V. The electron's wave function acts as a travelling wave before (Region I) and after (Region III) the barrier, but like an exponential under the potential barrier (Region II). In this fashion, the electron has a non-zero probability of tunneling through a potential barrier.

ψ II = Ce -k 2 x + De k 2 x
(2.1.7)

ψ III = F e ik 1 x (2.1.8)
where, k 1 = √ 2mE/ is wave number outside the barrier (Region I and Region III) and

k 2 = 2m(V -E)/
|ψ II | 2 ∝ e -2 2m(V -E) 2 L
(2.1.9)

For two conductive electrodes (like Region I and Region III) of same nature with weekly polarized, the quantity (V -E) can be replaced with the work function (φ) of the conductive materials. Now, we can deduce the formula of the tunneling current (i t ),

where L can be interpreted as the distance between the two electrodes (so we change the notation from L to d):

i t ∝ e -2 √ 2m √ φd (2.1.10)
where, 2 √ 2m/ = 1.025 eV 1/2 Å -1 . The exponential characteristics of the tunneling current is first demonstrated in 1928 [START_REF] Fowler | Electron emission in intense electric fields[END_REF]. Later, in [START_REF] Bardeen | Tunnelling from a many-particle point of view[END_REF], the author formalized the description of the tunneling current using time-based Schrödinger equation which was used for many theoretical studies of tunneling current. Soon after, J. Simmons proposed a formula for tunneling current [START_REF] Simmons | Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film[END_REF] based on the difference of potential and the exponential of the distance between the two electrodes separated by vacuum. In continuation of this work, Tersoff and Hamann proposed an expression for the tunneling current [START_REF] Tersoff | Theory and application for the scanning tunneling microscope[END_REF] taking into account the geometry of the electrodes used for scanning tunneling microscope. In this manuscript, the exponential behavior of tunneling current proposed in [START_REF] Simmons | Generalized formula for the electric tunnel effect between similar electrodes separated by a thin insulating film[END_REF] is adopted, and this formula is:

i t (t) = g • v b • e -k•d(t) (2.1.11)
where, v b is a potential difference applied between the two electrodes and d is the distance between them, g is a constant and k = 2 √ 2m/ = 1.025 eV 1/2 Å -1 . In all applications, the tunneling current is generally obtained by approaching a sharp tip to a sample surface. These two are often with a same material in order to obtain a rectangular potential barrier. If the material is not similar, the work function (φ) is considered as an average value of the work functions of the two materials.

Finally, in nanopositioning applications, the exponential dependance of tunneling current on the distance (Eq. (2.1.10) and Eq. (2.1.11)) is used. This exponential dependance will be the origin of the main nonlinearity in the control system.

Applications of the Tunneling Current

Scanning tunneling microscopes

Scanning tunneling microscope (STM) was invented by G. Binnig and H. Rohrer in early 1980s [START_REF] Binnig | Surface studies by scanning tunneling microscopy[END_REF] in the IBM laboratories in Zurich and they got nobel prize for this great invention. This device was the first to be able to reconstruct the topography of a sample surface with an atomic scale resolution.

The STM works by scanning a very sharp metal wire tip over a sample surface. A very small voltage is applied between the tip and the sample surface, and then this tip is approached towards the sample surface until the tunneling current appears. In this process, the electrons must tunnel through the vacuum barrier between tunneling tip and sample surface, which represents a potential barrier. The tunnel effect allows an electron to tunnel through this potential barrier even though the electronŠs energy is lower than the barrier height (quantum mechanical phenomenon). The probability of such a process decreases exponentially with the geometrical distance between the tip and the sample surface and with increasing barrier height. The tunneling current appears when the distance between the tip and sample surface is less than 1 × 10 -9 m. After a tunneling current is established, the tip is moved over the surface by a piezoelectric scanning unit (see Fig. 2.2), whose mechanical extension can be controlled by applying appropriate voltages.

STM operation modes

The sample topography can be obtained by operating the STM in different modes. There are mainly two modes of operation for STM as shown in Fig. 2.3. The "constant height" mode keeps the position of STM tip constant within the tunneling region while scanning the surface. In this mode, the variation of tunneling current reflects the small atomic corrugation of the surface. However, the tip could be crashed if the surface corrugation is big. To avoid such a problem, generally another approach, called "constant current" mode is mostly used in STM imaging. This mode of operation is safe to use on rough surfaces since the distance between the tip and sample surface is adjusted and the tunneling current is kept constant by a feedback loop of STM.

Vertical control of STM

The vertical feedback loop monitors the tunneling current (i t ) and tries to keep it constant while scanning the sample surface. The current sensor measures the tunneling current which varies because of the distance variation between tip and sample surface while scanning. The control signal regulates the tunneling current by actuating the tip away from or towards the sample surface with the help of piezoelectric actuator. This compensating control signal is directly depends on the variations in the topography and therefore provides a measure of the sample topography. The general feedback loop of STM is presented in Fig. 2.4.

As tunneling current (i t ) is exponentially dependant on the distance (d) between tip and sample surface, a common approach for STM community is to use a logarithmic amplifier after the current sensor in order to deal with exponential nonlinearity in the feedback loop [START_REF] Oliva | Analysis of scanning tunneling microscopy feedback system[END_REF], [START_REF] Anguiano | Analysis of scanning tunneling microscopy feedback system : Experimental determination of parameters[END_REF], [START_REF] Anguiano | Optimal conditions for imaging in scanning tunneling microscopy : Theory[END_REF]]. This linearization approach is based on the different approximations that the dynamics of the current sensor, and also the presence of noise between current sensor and the logarithmic amplifier must be neglected. Based on this linearization approach, the linearized output given by:

v y (t) = log 10 (G amp • i t (t)) = log 10 (G amp • g • v b • e -k•d(t) ) = C 1 -k • log 10 (e) • d(t) (2.2.1)
where,

C 1 = log 10 (G amp • g • v b
) is a constant term and G amp is the current sensor gain.

From the point of view of controller design for the vertical motion of STM, not much efforts have been done till now. In most commercial equipments of STM, only simple classical proportional-integral (PI) or proportional-integral with derivative (PID) controllers are implemented [START_REF] Curtis | An ultrahigh vacuum high speed scanning tunneling microscope[END_REF], [START_REF] Sasaki | A stable scanning tunneling microscope designed for investigations of organic thin films in air[END_REF], [START_REF] Nakakura | A high-speed variable-temperature ultrahigh vacuum scanning tunneling microscope[END_REF]], [Bredekamp & Tapson, n.d.], [START_REF] Bredekamp | A scanning tunneling microscope control system with potentiometric capability[END_REF], where usually the parameters of such controllers are fixed manually by the operator. A feedback loop of STM in vertical z-direction with some stability conditions has been presented in [START_REF] Oliva | Analysis of scanning tunneling microscopy feedback system[END_REF], [START_REF] Anguiano | Analysis of scanning tunneling microscopy feedback system : Experimental determination of parameters[END_REF], [START_REF] Anguiano | Optimal conditions for imaging in scanning tunneling microscopy : Theory[END_REF]] and this analysis has been done with simple classical PI (PID) control technique and with a simplified version of the system model. A step variation in sample surface is studied in [START_REF] Bonnail | Analyse de données, modélisation et commande pour la microscopie en champ proche utilisant des actionneurs piézoélectriques[END_REF] and a variable structure control (VSC) design methodology in the presence of classical PI control is proposed [START_REF] Bonnail | Variable structure control of a piezoelectric actuator for a scanning tunneling microscope[END_REF] in order to avoid the tip collision with the sample surface. Still, there is a need to analyze the vertical system of STM with some modern control techniques in order to get better performances.

Later, the invention of Atomic Force Microscope (AFM) was a new significant step allowing to characterize any sample surface. Because of the related operation challenges, AFMs have also motivated various control studies over the last decade (see for instance [START_REF] Abramovitch | A tutorial on the mechanisms, dynamics, and control of Atomic Force Microscopes[END_REF] and references therein). In this context, many efforts have been dedicated to the improvement of scanning performances (horizontal control), while vertical control is mostly concerned with force control, either in contact mode, or in dynamic (oscillatory) mode (see also [START_REF] Besançon | Towards oscillation control in a vibrating cantilever nonlinear NEMS[END_REF] for a similar operation).

Some works have also been dedicated to direct force estimation in such AFMs as in [START_REF] Besançon | Force estimation in fundamental physics: an observer application[END_REF], [START_REF] Besançon | Kalman observer approach towards force reconstruction from experimental afm measurements[END_REF]. Because of the atomic resolution, these microscopes have vast applications in different domains of material science [START_REF] Yamanaka | Quantitative material characterization by ultrasonic AFM[END_REF], physics [START_REF] Tsukada | Physics of artificial nano-structures on surfaces[END_REF], chemistry [START_REF] Jandt | Aspects of the physical chemistry of polymers, biomaterials and mineralised tissues investigated with atomic force microscopy[END_REF] and biology [START_REF] Kassies | Combined AFM and confocal fluorescence microscope for applications in bio-nanotechnology[END_REF].

Tunneling accelerometers

The idea of using the tunneling current to measure the accelerations was first presented in [START_REF] Waltman | An electron tunneling sensor[END_REF]]. Because of the high sensitivity of the tunneling current, it is now widely used to measure the accelerations down to sub-micro-g [START_REF] Kenny | Wide-bandwidth electromechanical actuators for tunneling displacement transducers[END_REF], [START_REF] Rockstad | A miniature, high-sensitivity, electron tunneling accelerometer[END_REF], [START_REF] Liu | Characterization of a high-sensitivity micromachined tunneling accelerometer with micro-g resolution[END_REF]], [START_REF] Liu | A high-precision, wide-bandwidth micromachined tunneling accelerometer[END_REF]. The ability of tunneling accelerometers to detect nano-g magnitude accelerations makes them critical in high precision applications like micro-gravity measurements, acoustic measurements, and seismology.

The operating principle of such accelerometers is based on the position regulation of a proof mass by the tunneling current. When the mass is subject to an acceleration, the distance between the proof mass and the tunneling tip varies, thereby changing the tunneling current (as tunneling current exponentially depends on the distance). An electrostatic actuator is normally used which is located opposite to the proof mass. A feedback control circuit adjusts the electrostatic re-balance force on the proof mass in order to maintain a constant tunneling current. The external acceleration is measured by recording the feedback deflection voltage.

Control for tunneling accelerometers

For the control of tunneling accelerometers, [START_REF] Liu | Robust controller design via µ-synthesis for high-performance micromachined tunneling accelerometers[END_REF] has proposed a controller design via µ synthesis for robust performance with respect to parametric variations of the system model. This approach is successfully implemented by the same researchers [START_REF] Liu | Characterization of a high-sensitivity micromachined tunneling accelerometer with micro-g resolution[END_REF]], [START_REF] Liu | A high-precision, wide-bandwidth micromachined tunneling accelerometer[END_REF] in order to achieve a high resolution of 20 nano-g/ √ Hz and a 5 Hz-1.5 kHz bandwidth. In [START_REF] Wang | Synthesis of the modeling and control systems of a tunneling accelerometer using the matlab simulation[END_REF], the author has synthesized a simple control law by using the relation between open loop and closed loop transfer functions for the system of tunneling accelerometer.

Recently, in [START_REF] Oropeza-Ramos | Noise analysis of a tunneling accelerometer based on state space stochastic theory[END_REF], [Yie et al., n.d.], a complete state space model for the system of tunneling accelerometer has been proposed with a digital integral control design and the effect of different sources of noise (Brownian motion, Nyquist-Johnson and shot noise) are analyzed.

Piezoelectric Effect

The piezoelectric effect was discovered by Pierre Curie in 1880. The effect is created by squeezing the sides of certain crystals (quartz or barium titanate) and the result is the creation of opposite electrical charges on the sides. The effect can be reversed as well:

by applying a voltage across a piezoelectric crystal, it will elongate or compress. These materials are used to scan the tip over the surface in STM and in many other scanning probe techniques. A typical piezoelectric material used in STM is PZT (Lead Zirconium Titanate). Reference [START_REF] Katzir | The discovery of the piezoelectric effect[END_REF] can be consulted for a detailed understanding of piezoelectric effect.

Piezoelectric scanners are widely used in different scanning-probe microscopes such as in scanning tunneling microscopes (STM), atomic force microscopes (AFM). These piezoelectric scanners are used to move a probe (tip) over a sample surface, both parallel to the sample surface (x and y axes) and perpendicular to the sample surface (z axis). The piezoelectric actuators are now widely used for high positioning accuracy at nanometer and sub-nanometer resolution with high bandwidths [START_REF] Taylor | Dynamics of piezoelectric tube scanners for scanning probe microscopy[END_REF], [START_REF] Ohara | Dynamics and control of piezotube actuators for subnanometer precision applications[END_REF], [START_REF] Schitter | Identification and open-loop tracking control of a piezoelectric tube scanner for high-speed scanning probe microscopy[END_REF]. One of the advantages of using piezoelectric actuators is that under certain experimental conditions their dynamics can be well approximated by linear models [START_REF] Bhikkaji | Highperformance control of piezoelectric tube scanners[END_REF],

leading in general to a second order linear model of the classical form (in the Laplace domain):

G a (s) = γ 0 1 ω 2 0 s 2 + 1 Qω 0 s + 1 (2.3.1)
where γ 0 is sensitivity, ω 0 is the resonance frequency and Q is the quality factor of piezoelectric actuator model.

Control Issues

Performance compromises

For all nanopositioning applications, the control design has to consider the tradeoffs between precision, bandwidth, robustness and sufficient range of motion. There is always a compromise between these performance parameters [START_REF] Devasia | A survey of control issues in nanopositioning[END_REF] as shown in Fig. 2.5.

There is a fundamental tradeoff between the system bandwidth and positioning precision. The positioning precision inversely depends on the bandwidth: systems with larger bandwidth allow noise to affect the precision over a larger frequency range and hence provide poorer positioning precision. Moreover, precision also inversely depends on the positioning range because of the nonlinearities and also because of the quantization noise in digital controller implementations. The bandwidth tends to be inversely dependant on the range because the first vibrational resonance of the actuator tends to be higher for a smaller actuator with less range. The vibrations tend to degrade the positioning accuracy as the main frequency content of the input becomes close to the first resonance frequency of the system. Although, vibration-induced error can be reduced by limiting the input frequency content to a level well below the system resonance (low speed operation). Alternatively, to enable higher speed operation, the first resonance frequency of the system can be damped by some control techniques [START_REF] Ratnam | PPF control of a piezoelectric tube scanner[END_REF], or it can be increased by optimizing the actuators geometry (to make them stiffer) at the cost of small positioning range. With the proposed control schemes in [Bhikkaji, Ratnam & Moheimani, 2007], [START_REF] Bhikkaji | Highperformance control of piezoelectric tube scanners[END_REF], the authors were able to achieve a 30 dB damping of the resonant mode without much control efforts.

Although, the high speed or large bandwidth and even high precision can also induce robustness and stability issues because of water-bed effect. Because of the nonlinearities, large positioning range can cause instabilities as well.

A lot of studies are currently in progress and much more efforts required working at nanometer scale from control point of view to deal with all such performance compromises. In recent years, a lot of work has been dedicated to the control of piezoelectric actuators for nanopositioning [START_REF] Salapaka | High bandwidth nano-positioner: a robust control approach[END_REF], [START_REF] Schitter | Identification and open-loop tracking control of a piezoelectric tube scanner for high-speed scanning probe microscopy[END_REF], [START_REF] Sebastian | Design methodologies for robust nanopositioning[END_REF], [START_REF] Devasia | A survey of control issues in nanopositioning[END_REF], [START_REF] Bhikkaji | Highperformance control of piezoelectric tube scanners[END_REF], [Bhikkaji, Ratnam & Moheimani, 2007], [Aphale, Devasia & Mo-heimani, 2008], [START_REF] Aphale | Minimizing scanning errors in piezoelectric stack-actuated nanopositioning platforms[END_REF], but such efforts have not been done till now for the control of piezoelectric actuators for nanopositioning in the presence of tunneling phenomenon (e.g. the vertical control of STM).

Dynamic nonlinearities

Hysteresis: Piezoelectric actuators are widely used in SPM (scanning probe microscopy) [START_REF] Binnig | Single-tube three-dimensional scanner for scanning tunneling microscopy[END_REF]] and many other nano-positioning applications [START_REF] Gao | Construction and testing of a nanomachining instrument[END_REF], [START_REF] Shim | Piezo-driven metrological multiaxis nanopositioner[END_REF], [START_REF] Salapaka | Control of the nanopositioning devices[END_REF]. Such actuators have strong non-linear hysteresis effects which can cause inaccuracy and oscillations in the system response, and could also lead to instability of the closed loop system [START_REF] Brokate | Hysteresis and phase transitions[END_REF], [START_REF] Pare | Robust stability and performance analysis of systems with hysteresis nonlinearities[END_REF]]. That is why, it is very important to model the hysteresis. A number of hysteresis models have been developed [START_REF] Mayergoyz | Mathematical models of hysteresis and their applications[END_REF] in order to facilitate the design of controllers for compensating its effects. Mainly, there are differ-ential equation based and operator based hysteresis models. Differential equation-based hysteresis models [START_REF] Visintin | Differential models of hysteresis[END_REF] include Duhem model [Hodgdon, 1988a], [Hodgdon, 1988b], Bouc-Wen model [START_REF] Sain | Models for hysteresis and application to structural control[END_REF], Jiles-Atherton model [START_REF] Jiles | Theory of ferromagnetic hysteresis: determination of model parameters from experimental hysteresis loops[END_REF]], [START_REF] Salvini | Genetic algorithms and neural networks generalizing the jiles-atherton model of static hysteresis for dynamic loops[END_REF] etc. Operator based hysteresis models [START_REF] Macki | Mathematical models for hysteresis[END_REF] include Preisach model [START_REF] Mayergoyz | Generalized preisach model of hysteresis[END_REF], Krasnosel'skii-Pokrovskii model [START_REF] Krasnoskl'skii | Systems with hysteresis[END_REF], and Prandtl-Ishlinskii model [START_REF] Brokate | Hysteresis and phase transitions[END_REF] etc.. The most commonly used operator based hysteresis models are Preisach model and Prandtl-Ishlinskii model. In operator based hysteresis models, the elementary operator is a rateindependent backlash operator. It is commonly used in the modeling of backlash between gears with one degree of freedom. A backlash operator is defined by:

y(t) = H r [x, y 0 ](t) = max{x(t) -r, min{x(t) + r, y(t -T )}} (2.4.1)
where x is the control input, y is the actuator response, r is the control input threshold value or the magnitude of the backlash, and T is the sampling period. A detailed discussion about hysteresis operator can be found in [START_REF] Brokate | Hysteresis and phase transitions[END_REF].

The hysteresis phenomenon in piezoelectric actuators strongly depend on either the maximum value of input voltage being applied, or the frequency of the input signal, or both. Therefore, a dynamic or rate dependent hysteresis modeling is required. Many works related to rate-dependent hysteresis modeling can be found in the literature [START_REF] Hu | On the classical preisach model for hysteresis in piezoceramic actuators[END_REF], [START_REF] Janaideh | Development of the rate-dependent prandtl-ishlinskii model for smart actuators[END_REF], [START_REF] Janaideh | Experimental characterization and modeling of rate-dependent hysteresis of a piezoceramic actuator[END_REF] and references therein.

Based on the above formulated hysteresis models, two control approaches are commonly adopted in the literature to compensate hysteresis effects. The first approach is to design a controller which directly incorporates the actuator nonlinearities, and the second approach, which is very commonly used, is to construct the inverse of the hysteresis model and apply it as a feedforward compensator as shown in Fig. 5.2. For the first approach, many adaptive control approaches have been recently proposed for hysteresis compensation [START_REF] Su | Adaptive variable structure control of a class of nonlinear systems with unknown prandtl-ishlinskii hysteresis[END_REF], [START_REF] Wang | Robust adaptive control of a class of nonlinear systems including actuator hysteresis with prandtl-ishlinskii presentations[END_REF], [START_REF] Lin | Adaptive control with hysteresis estimation and compensation using rfnn for piezo-actuator[END_REF], [START_REF] Chen | Adaptive control for the systems preceded by hysteresis[END_REF], [START_REF] Liaw | Sliding-mode enhanced adaptive motion tracking control of piezoelectric actuation systems for micro/nano manipulation[END_REF]. A lot of research has already been done on the construction of inverse (rate-independent and rate-dependent) hysteresis models and to use them as a feedfor-ward compensator. Some latest work in this direction is cited here [START_REF] Song | Tracking control of a piezoceramic actuator with hysteresis compensation using inverse preisach model[END_REF],

[ [START_REF] Ang | Feedforward controller with inverse rate-dependent model for piezoelectric actuators in trajectory-tracking applications[END_REF], [START_REF] Janaideh | Hysteresis compensation for smart actuators using inverse generalized prandtl-ishlinskii model[END_REF], [START_REF] Iyer | Control of hysteretic systems through inverse compensation[END_REF].

Figure 2.7: Piezoelectric actuator linearization with feedforward inverse hysteresis controller. The input of the feedforward controller x d (t) is the desired displacement and output v(t) is the voltage to apply to the piezoelectric actuator after amplification.

Recently, it has been shown that the nonlinear hysteresis effects can also be compensated by actuating the piezoelectric actuators using charge amplifiers (rather than voltage amplifiers) [START_REF] Bhikkaji | Highperformance control of piezoelectric tube scanners[END_REF]. However, in spite of its benefits, the charge actuation has not been generally accepted due to the practical problems of voltage drift, poor low frequency response and also commercially unavailability of the charge sources.

However, the nonlinear hysteresis phenomenon can be neglected for very small displacements (in nanometers) of piezoelectric actuator (e.g. in case of vertical displacement of STM tip where objective is to keep the tunneling current constant, and this tunneling phenomenon occurs when the distance between the tip and sample surface is less than a nanometer) [START_REF] Oliva | Analysis of scanning tunneling microscopy feedback system[END_REF], [START_REF] Bonnail | Dynamic and static responses of a piezoelectric actuator at nanometer scale elongations[END_REF]. In this thesis, hysteresis is neglected in the vertical positioning but it is observed in practice in the lateral positioning.

Creep: Creep can be defined as the drift of the displacement of piezoelectric actuator for a constant applied electric field (see Fig. 2.8). Creep phenomenon has been investigated less frequently compared with hysteresis. Two creep models have been proposed in the literature. The first static nonlinear model [START_REF] Jung | New open-loop actuating method of piezoelectric actuators for removing hysteresis and creep[END_REF], based on a logarithmic behavior of the creep effect over time, is given by the following equation: where, x(t) is piezoelectric actuator's displacement for any fixed input voltage, t 0 represents the time at which creep effect is apparent, x 0 is the value of actuator displacement at time t 0 , and γ is the creep rate. This nonlinear model is difficult to use because of the strong dependance of the creep rate (γ) on the choice of the time parameter t 0 . Additionally, this model becomes unbounded for any fixed t 0 as time t becomes large, i.e., t → ∞ or even when t → 0. In order to solve this modeling difficulty, [START_REF] Croft | Creep, hysteresis, and vibration compensation for piezoactuators: Atomic force microscopy application[END_REF] has proposed a dynamic linear creep model as a series connection of springs and dampers, i.e.:

x(t) = x 0 1 + γlog t t 0 (2.4.2)
X(s) V (s) = 1 k 0 + N i=1 1 c i s + k i (2.4.3)
where X(s) is the measured response in the lateral direction (in the Laplace domain) and V (s) is the input voltage affecting the lateral movements. k 0 models the elastic behavior at low frequencies, k i is the spring constant and c i is the damping constant. It has been

shown in [START_REF] Croft | Creep, hysteresis, and vibration compensation for piezoactuators: Atomic force microscopy application[END_REF] that the model order N between 3 and 5 can model the creep effect with sufficient accuracy. This low frequency model can be appended to the linear model that describes the vibrational dynamics of the actuator at high frequencies (Eq.

(2.3.1)) and thus a controller can be designed for this augmented model to compensate for creep as well as other vibration effects associated with the actuator.

However, the creep phenomenon has been neglected in this thesis and can be included in future work (perspectives) since it has been observed in experimental results (see Chapter 4 and Chapter 5).

Modeling errors

The system performance should be robust in the presence of modeling errors due to parameter variations and unmodeled dynamics.

Even when the physical parameters of the system are known, they can change over relatively long time intervals because of aging effects (which can occur over short periods of time at nanoscale). Moreover, the parameters are very sensitive to variations in the temperature. Therefore, it is challenging to develop a priori accurate models, and experimental modeling and system identification are very important in that respect, but it is also crucial that the control be robust against such possible parameters error. In addition, when designing the controller, the high frequency modes are often neglected to obtain a simplified model. However, these high frequency modes can affect the stability of the closed loop system as well as impose limitations on the achievable performance of the closed loop system [START_REF] Balas | Active control of flexible systems[END_REF].

A control system must thus be robust to the differences between the actual system and the model of the system which was used to design the controller. This means beyond nominal stability and nominal performance, robust stability and robust performance conditions must also be verified for any considered system.

Coupling effects

Recently, many authors have addressed the cross-coupling problem that arises during high-speed nano-precision positioning using piezoelectric scanners. As mentioned before, these piezoelectric scanners are used to move a probe (tip) over a sample surface, both parallel to the sample surface (x and y axes) and perpendicular to the sample surface (z axis). The scanning movement of the probe is in a raster (back and forth) pattern. To achieve this specific movement of the probe, a slowly increasing ramp signal is applied to the y-electrode of a piezoelectric scanner while the x-electrode is driven by a fast triangular waveform. All three x, y and z actuation are done by a single piezoelectric tube actuator [START_REF] Binnig | Single-tube three-dimensional scanner for scanning tunneling microscopy[END_REF]. Normally, the lateral movement of the probe (x and y axes) is controlled by feed-forward compensator in open loop (because of the absence of position sensors in lateral direction) and the vertical movement of the probe (z-axis) is controlled by the feedback compensator.

A complete plant model P (ω) of such scanning systems can be represented by multiinput multi-output (MIMO) form [START_REF] Pao | Combined feedforward/feedback control of atomic force microscopes[END_REF], [START_REF] Butterworth | A comparison of control architectures for atomic force microscopes[END_REF]:

P (ω) =     P xx (ω) P xy (ω) P xz (ω) P yx (ω) P yy (ω) P yz (ω) P zx (ω) P zy (ω) P zz (ω)     (2.4.4)
where P xx (ω) represents the transfer function from the x-axis control input to the xposition, P xy (ω) represents the transfer function from the y-axis control input to the

x-position, P xz (ω) represents the transfer function from the z-axis control input to the x-position, and so forth. Although P (ω) is a full matrix because of the coupling effects, P xy (ω), P yx (ω), P zy (ω) and P yz (ω) are generally relatively small compared with the other entries [START_REF] Song | Tracking control of a piezoceramic actuator with hysteresis compensation using inverse preisach model[END_REF], [START_REF] Sebastian | Design methodologies for robust nanopositioning[END_REF], [START_REF] Butterworth | A comparison of control architectures for atomic force microscopes[END_REF]. The cross-coupling between the x and z-directions, however, can be significant and the scanning movement of the probe (tip) in the lateral x-direction can cause positioning error in the perpendicular z-direction [START_REF] Song | Tracking control of a piezoceramic actuator with hysteresis compensation using inverse preisach model[END_REF]. This error will become more significant when the sample surface is scanned at high speed or scan frequency becomes close to the piezo scannerŠs vibrational resonance frequency. This is an important limitation in order to perform fast scan speed which is highly required in different nano-fabrication and biological processes [START_REF] El Feninat | A tapping mode afm study of collapse and denaturation in dentinal collagen[END_REF].

Few approaches have been used until now in the literature to reduce the dynamics coupling-caused errors, particularly in SPM (scanning probe microscopy) applications where high positioning precision with fast scan speed is required. [START_REF] Tien | Control of dynamics-coupling effects in piezoactuator for high-speed afm operation[END_REF], [START_REF] Song | Tracking control of a piezoceramic actuator with hysteresis compensation using inverse preisach model[END_REF] have proposed iterative, inversion-based, feedforward control to compensate the dynamics coupling error (from x-axis to z-axis as shown in Fig. 2.9) in piezoelectric scanners during high-speed nano-positioning operations. In this work, the model of the x-to-z coupling dynamics P zx (ω) and the model of the z-axis dynamics P zz (ω) are found experimentally (recently, [START_REF] Maess | Simulation of dynamics-coupling in piezoelectric tube scanners by reduced order finite element analysis[END_REF] tried to find fully coupled piezoelectric scanner dynamics model by finite element (FE) analysis using the commercially available software package ANSYS). According to the authors, the proposed iterative control technique can significantly reduce the dynamics coupling-caused error.

For example, at the 13th iteration step of the proposed iterative control technique, the RMS error due to coupling was reduced by 81% (from 10.32 nm to 1.96 nm) and the maximum error was reduced by 86% (from 50.49 nm to 7.23 nm). Later, [START_REF] Wu | A control approach to cross-coupling compensation of piezotube scanners in tapping-mode atomic force microscope imaging[END_REF],

Figure 2.9: The general approach to deal with cross-coupling effect, where vertical deflection z is affected by the z-axis input u Z and also the x-axis input u X through coupling-effect dynamics.

[ [START_REF] Shi | A control approach to cross coupling compensation of piezotube scanners in tapping-mode imaging[END_REF] extended the inversion-based iterative control (IIC) algorithm to further reduce the dynamic coupling caused error and also to achieve better noise attenuation by avoiding potential input saturation during practical implementations. In their work, three external disturbances are considered: unknown sample surface variations, measurement noise and also the disturbance input caused by the x-to-z cross-coupling effect.

[ [START_REF] Mahmood | Making a commercial atomic force microscope more accurate and faster using positive position feedback control[END_REF] has highlighted the need of more detailed research in order to minimize the cross-coupling to the vertical axis of the scanner to achieve higher resolution images at faster scans. [START_REF] Pao | Combined feedforward/feedback control of atomic force microscopes[END_REF], [START_REF] Butterworth | A comparison of control architectures for atomic force microscopes[END_REF] have mentioned that the cross-coupling terms have not been explored extensively in the SPM control literature yet and pointed to analyze MIMO controllers for the full plant model (Eq. (2.4.4)) to determine the achievable performance gains.

Conclusions

In this chapter, the tunneling current phenomenon has been briefly presented and its two main applications (tunneling microscopes and tunneling accelerometers) have been highlighted. Some control related work considered by research community about these applications have been presented. From all this bibliographical work, it has been observed that no much effort has been done till now for the tunneling current measurement system in the the presence of piezoelectric actuator (like vertical control of STM), which requires very precise nanopositioning.

The control issues associated with the problem of nanopositioning have also been presented, including (performance compromises in terms of precision, bandwidth, robustness and range; dynamic nonlinearities; modeling error and coupling effects). Taking into account these issues, the present thesis focuses on the analysis of a specific system of tunneling current with modern robust control techniques and then its validation over an experimental platform developed in Gipsa-lab by control group.

This chapter is devoted to the control system design for very high performance measurement of tunneling current, where ultrahigh positioning accuracy together with high bandwidth are the big challenges. A common approach for such applications is to use conventional proportional integral (PI) control design methodology. In this chapter, modern robust control techniques are analyzed in order to obtain better performances in terms of precision, bandwidth, robustness and disturbance rejection. These control techniques are:

• Pole placement combined with sensitivity functions shaping

• Mixed sensitivity H ∞ control
The motivation for this work focusing on control for tunneling current measurement systems is presented in Section 3.1. Section 3.2 very briefly presents a system of tunneling current measurement and a corresponding dynamic modeling is proposed in Section 3.3.

The control problem is formulated with desired measurement performances in Section 3.4 and also the measurement requirements are translated into control requirements. Section 3.5 and Section 3.6 present the proposed control techniques applied to considered system of tunneling current measurement, also with simulation and comparison results with conventionally used classical PI controller.

Finally, Section 3.7 draws some conclusions.

Motivation

The phenomenon of tunneling current with its applications, is briefly explained in Chapter 2. The control scheme of all tunneling applications is mainly composed of a sensor for the tunneling current measurement and also a regulation feedback loop, having a piezoelectric actuator attached with the tip to move it precisely in appropriate direction.

Tunneling current is the only measurement in vertical z-direction of STM. During scanning, when the separation between the sample surface and the tip decreases or increases due to variations in the sample topography, the control signal regulates the tunneling current by actuating the tip away from or towards the sample surface. This compensating control signal depends on variations in the topography and therefore provides a measure of the sample topography. The quality of the surface image strongly depends on the precision of tunneling current measurement. Since the distance between tip apex and sample surface must be less than 1 × 10 -9 m to get the tunneling effect, ultrahigh positioning accuracy together with high bandwidth are here major challenges.

As tunneling current is of order of nano-amperes, the presence of different sources of noise [START_REF] Bordoni | Fundamental noise, electromechanical transduction and their role in resonant gravitational-wave detectors[END_REF] (thermal noise, shot noise, 1/f noise, quantization noise, etc.) and sample surface variations highly influence the precision of this measurement signal. In addition, non-linearities and physical limitations in the control loop are also limiting factors to be considered in order to get better performances.

As mentioned in the last chapter, not much efforts have been done till now for the vertical control of STM. The most commercial equipments of STM [START_REF] Curtis | An ultrahigh vacuum high speed scanning tunneling microscope[END_REF], [START_REF] Sasaki | A stable scanning tunneling microscope designed for investigations of organic thin films in air[END_REF], [START_REF] Nakakura | A high-speed variable-temperature ultrahigh vacuum scanning tunneling microscope[END_REF]], [Bredekamp & Tapson, n.d.], [START_REF] Bredekamp | A scanning tunneling microscope control system with potentiometric capability[END_REF] and also the analysis presented for vertical control of STM in [START_REF] Oliva | Analysis of scanning tunneling microscopy feedback system[END_REF], [START_REF] Anguiano | Analysis of scanning tunneling microscopy feedback system : Experimental determination of parameters[END_REF], [START_REF] Anguiano | Optimal conditions for imaging in scanning tunneling microscopy : Theory[END_REF], are based on classical PI (PID) controllers. Without proper analysis, the imaging process can not be optimum and the image of the surface does not correspond necessarily to the reality [START_REF] Anguiano | Optimal conditions for imaging in scanning tunneling microscopy : Theory[END_REF]]. The quality of the image strongly depends on the precision of tunneling current measurement. This chapter is devoted to the modern control system design for precise measurement of tunneling current, in the presence of piezoelectric actuator and a sensor for the tunneling current measurement. Desired high performance requirements are transformed into control requirement, which are expressed by means of constraints on the shape of closed-loop sensitivity functions. Then, first a robust control design methodology developed since several years in our laboratory [START_REF] Landau | Robust digital control using pole placement with sensitivity function shaping method[END_REF]] based on combined pole placement with sensitivity function shaping and then mixed sensitivity H ∞ control techniques [START_REF] Skogestad | Multivariable Feedback Control: Analysis and design[END_REF] are proposed and then analyzed for the considered system of tunneling current in order to obtain better performances in terms of precision, robustness and disturbance rejection.

For all numerical values in this chapter, we refer to the characteristics of an experimental platform developed in Gipsa-lab, and fully presented in Chapter 4.

System Description

The complete closed-loop control scheme which will be here considered is presented in Fig. 3.1. The working principle of the considered system is based on the measurement and control of the tunneling current (i t ) produced between the sharp metallic tip and a biased (v b ) sample surface, when the distance (d) between them is less than 1 × 10 -9 m. This tunneling current (i t ) depends exponentially on the distance (d) between tip and sample surface with following non-linear relation as mentioned earlier: procedure for tip approach mechanism is the part of next chapter. A feedback loop constantly monitors the tunneling current (i t ). The tunneling current sensor is a current to voltage convertor (CV C) with high gain which converts the small tunneling current into a voltage (v 3 ). This voltage is subject to measurement noise (n) resulting into the available voltage for feedback (v y ). On the other hand, a piezoelectric actuator is attached with the tip to move it in appropriate direction according to the applied voltage (v 2 ) in order to keep the distance (d) or in other words the tunneling current (i t )

i t (t) = g • v b • e -k•d(t) (3.
constant at its desired value (i des ). The nonlinear hysteresis phenomenon is expected to be negligible here for the piezoelectric actuator since the displacement is very small, less than a nanomemter for the vertical movement (z-direction) [START_REF] Bonnail | Dynamic and static responses of a piezoelectric actuator at nanometer scale elongations[END_REF], [START_REF] Oliva | Analysis of scanning tunneling microscopy feedback system[END_REF]. A voltage amplifier is used before piezoelectric actuator at the output of controller. Knowing the gain of piezoelectric actuator, its position effect (z) defines the distance (d(t) = d 0 + z S (t) -z(t)) between tip and sample surface, where d 0 is the initial distance between tip and sample surface when tunneling current (i t ) appears and z S the sample surface variations, considered here as an external unknown disturbance. One assumption is taken here which is commonly used in the literature [START_REF] Chen | Introduction to Scanning Tunneling Microscopy[END_REF]] that the tunneling current (i t ) appears from a distance (d) between tip and sample surface equal to one nanometer. Based on this assumption, it can be written:

i t (t) = g • v b • e -k•d(t) if 0 < d(t) ≤ 1 nm i t (t) = 0 if d(t) > 1 nm (3.2.2)

System Modeling

The purpose of this section is to model the considered system of tunneling current. The open loop system is shown in Fig. 3.2 where the overall system consists of:

• the voltage amplifier;

• the vertical z-piezoelectric actuator;

• the physical law which gives the relationship between tunneling current and the distance between tip and sample surface;

• the current sensor (CV C).

Global nonlinear system model

Each subsystem (as shown in Fig. 3.2) is modeled separately in order to obtain the overall global nonlinear system model.

The voltage amplifier used before piezoelectric actuator, has been modeled by a linear first order dynamics:

ẋ1 (t) = -ω v • x 1 (t) + v 1 (t) v 2 (t) = G v ω v • x 1 (t) (3.3.1)
where v 1 and v 2 are input and output of the voltage amplifier respectively, ω v is the bandwidth and G v the gain of the voltage amplifier.

The piezoelectric actuator attached to the tip to move it in vertical z-direction, has been modeled by a linear second order dynamics. The displacement of this piezoelectric actuator in vertical z-direction is less than one nanomemter (in order to remain within tunneling current region) so the hysteresis of the actuator can be neglected [START_REF] Bonnail | Dynamic and static responses of a piezoelectric actuator at nanometer scale elongations[END_REF], [START_REF] Oliva | Analysis of scanning tunneling microscopy feedback system[END_REF] and the dynamics can be well approximated by linear model [START_REF] Bhikkaji | Highperformance control of piezoelectric tube scanners[END_REF]. The considered second order dynamics is:

ẋ2 (t) = -2ζω a • x 2 (t) -ω 2 a • x 3 (t) + v 2 (t) ẋ3 (t) = x 2 (t) z(t) = G a ω 2 a • x 3 (t) (3.3.2)
where v 2 and z are input and output of the piezoelectric actuator respectively, ζ is the damping, ω a the bandwidth and G a the gain of the piezoelectric actuator.

The physical law between tunneling current (i t ) and distance (d) has been modeled by an expression as given in equation (3.2.2). This physical law introduces the exponential static nonlinearity in the model of the system.

The current sensor (CV C) to measure and convert the small tunneling current (i t ) into a voltage, has been modeled by a linear first order dynamics:

ẋ4 (t) = -ω c • x 4 (t) + i t (t) v 3 (t) = G c ω c • x 4 (t) (3.3.3)
where i t and v 3 are input and output of the current sensor (CV C) respectively, ω c is the bandwidth and G c the gain of the current sensor (CV C).

To summarize, the overall global nonlinear system (order 4) is given by:

                   ẋ1 (t) = -ω v • x 1 (t) + v 1 (t) ẋ2 (t) = G v ω v • x 1 (t) -2ζω a • x 2 (t) -ω 2 a • x 3 (t) ẋ3 (t) = x 2 (t) ẋ4 (t) = -ω c • x 4 (t) + gv b • e -k(d 0 +z S (t)-Gaω 2 a •x 3 (t)) v y (t) = G c ω c • x 4 (t) + n(t) (3.3.4)
For the purpose of a linear control design, this global nonlinear model Eq. (3.3.4) will be here transformed into an appropriate linear system.

Linearization

To deal with exponential static nonlinearity given by the physical law between tunneling current (i t ) and distance (d) between tip and sample surface (Fig. 3.3), two different approaches for linearization are discussed here.

Electronic linearization

A common approach for linearization by physicists in this specific situation is to use a logarithmic amplifier after current sensor (CV C) in order to deal with static exponential nonlinearity [START_REF] Oliva | Analysis of scanning tunneling microscopy feedback system[END_REF], [START_REF] Anguiano | Analysis of scanning tunneling microscopy feedback system : Experimental determination of parameters[END_REF], [START_REF] Anguiano | Optimal conditions for imaging in scanning tunneling microscopy : Theory[END_REF]]. This linearization approach is based on the following two assumptions:

• Neglecting dynamics of the current sensor (CV C), considered as constant;

• Neglecting presence of noise (n) between exponential and logarithmic nonlinearity. This linearization approach is represented in Fig. 3.4 resulting into the available voltage for feedback (v y ) is:

v y (t) = log 10 (v 3 (t)) = log 10 (G c • i t (t)) = log 10 (G c • g • v b • e -k•d(t) ) = log 10 (G c • g • v b ) -k • log 10 (e) • d(t) (3.3.5)
The output voltage (v y ) is linear with respect to the distance (d) between tip and sample surface. This approach has the advantage of working on a wide range of tunneling current (i t ).

Approximate linearization around an operating point

An alternative approach, which will be considered here, is to approximate the nonlinear dynamics around an operating point by a first order Taylor expansion. This allows to account for the current sensor dynamics (CV C) as well as the presence of noise (n)

and the model will be valid as long as current variations remain small enough. In this approach, the linearized equation corresponding to the nonlinear equation (3.2.1) is given by:

d(t) = - 1 k log( i eq g • v b ) + - 1 k • i eq × (i t (t) -i eq ) (3.3.6)
where i eq is the operating tunneling current. (Notice that a similar analysis could be done with the logarithmic amplifier taking into account CV C dynamics and the presence of noise).

Linear control design model

After linearization, the overall global linear system (order 4) between v 1 and v y is modeled as:

                   ẋ1 (t) = -ω v • x 1 (t) + v 1 (t) ẋ2 (t) = G v ω v • x 1 (t) -2ζω a • x 2 (t) -ω 2 a • x 3 (t) ẋ3 (t) = x 2 (t) ẋ4 (t) = -c 1 G a ω 2 a • x 3 (t) -ω c • x 4 (t) + c 1 • z S (t) + i eq v y (t) = G c ω c • x 4 (t) + n(t) (3.3.7)
where, c 1 = -k • i eq is a constant.

A linear control design model is required for the linear controller synthesis. For this purpose, the complete simulation model (Fig. 3

Noise considerations

For the system of tunneling current, one of the external disturbances considered here is the noise (n). It is important to list, characterize and then quantify the different types of noises effecting the tunneling current (i t ) and the performance in terms of precision of the whole system at the nanoscale. Each noise is quantified by the theoretical description of its power spectral density so that it can be integrated in the simulation.

Thermal noise

The considered system works at ambient temperature. The tunneling current (i t ) fluctuates because of the thermal noise, generated by the thermal agitation of the electrons crossing the potential barrier between the electrodes (tip and sample surface).

The concept of thermal noise has been first experimentally demonstrated by Johnson [START_REF] Johnson | Thermal agitation of electricity in conductors[END_REF] and mathematically calculated by Nyquist [START_REF] Nyquist | Thermal agitation of electric charge in conductors[END_REF]. Thermal noise is approximately a white noise and its power spectral density (S th ) is given by:

S th = 4k B T R t [A 2 /Hz] (3.3.8)
where, k B is Boltzmann's constant in joules per kelvin, T is the absolute temperature in kelvins and R t is the tunneling resistance in Ohms, which is the resistance across the two electrodes (tip and sample surface). For a typical resistance of 10 8 Ω of the tunneling current, the power spectral density of this thermal noise at 20 • C is approximately S th = 1.6 × 10 -28 A 2 /Hz. This fluctuation in tunneling current (i t ) can be observed in the measured signal at the output of the considered system, indicated as (n) in Fig. 3.2.

1/f noise

This noise is common in most electronic devices, its power spectral density (S(f )) is inversely proportional to the frequency (f ): S(f ) ∝ 1/f α , with α generally between 0.8 and 1.4. The first spectral density measurement of a 1/f noise was published by

Johnson [Johnson, 1925], although the origin of this noise is still unclear. In the case of tunneling current (i t ), different ideas about the origin and the influences of this 1/f noise are discussed in [START_REF] Lagoute | Images et fluctuations du courant sur surfaces métalliques et fils moléculaires adsorbés en microscopie à effet tunnel[END_REF], [START_REF] Sugita | Origin of low frequency noise and 1/f fluctuations of tunneling current in scanning tunneling microscopes[END_REF]. This noise is dominant at low frequencies. Beyond a certain frequency, it becomes negligible compared to the thermal noise. The noise intensity at a fixed tunneling current increases with increasing bias voltage between tip and sample surface. [START_REF] Hooge | 1/f noise is no surface effect[END_REF] has demonstrated a relationship for the spectral density of the fluctuations of the current measurements in homogeneous metallic samples with average current of I o .

S(f ) = cI o 2 f α [A 2 /Hz] (3.3.9)
where c is a constant term. Depending on the nature of the tip and the sample surface and also the environmental conditions, the average power spectral density S(f ) of this 1/f noise is around 10 -24 A 2 /Hz. This noise is also at the output of the considered system, indicated as (n) in Fig. 3.2.

Shot noise

Shot noise can also be one of the reasons of random fluctuations of the tunneling current, as the current being carried by discrete charges (electrons) between the electrodes (tip and sample surface), whose number per unit time fluctuates. Shot noise is often only a problem with small currents of few nano amperes and its theoretical concept was first described by Schottky [START_REF] Schottky | Small shot effect and flicker effect[END_REF]. Shot noise has a Poisson distribution and has white noise properties (flat power spectral density). The power spectral density (S sh ) of this noise for an average tunneling current (I t ) is given by:

S sh = 2eI t [A 2 /Hz] (3.3.10)
where, e is the electron charge in Coulombs. For an average tunneling current of 1 nA, the power spectral density of this shot noise is approximately S sh = 3.2 × 10 -28 A 2 /Hz, which is also considered at the output of the considered system, indicated as (n) in Fig. 3.2.

Environmental vibrations

Environmental vibrations can not be neglected while working at nanometer scale. There can be different sources of such vibrations which induce the distortion into the measurement. The vibrations induced by the building structure have a very low resonance frequency, typically between 10 Hz and 30 Hz with an amplitude of several nanometers. The human activity in the working place proximity is the origin of the mechanical vibrations. These mechanical vibrations are significant only at low frequencies and the use of anti-vibration table for experimental setup can be one possible solution to minimize these vibrations. The acoustic waves in the environment (because of running air conditioning, computer fans, moving and speaking people) induce distortion too, which can be minimized by using proper acoustic shielding.

The effect of all these noises mentioned above are included in the simulation for the validation of the proposed controllers.

Uncertainty model

Usually, a control system is considered to be robust if it is insensitive to differences between the actual system and the model which has been used to design the controller.

These differences are referred to as model uncertainties.

To account for model uncertainty, it can be assumed that the dynamic behavior of the plant is described not by a single linear time invariant model G(s), but by a set Π of possible linear time invariant models G ∆ (s) ∈ Π, denoted as "uncertainty set". From the nominal plant model G 0 (s), the set Π of possible linear time invariant models G ∆ (s) can be built by varying the model parameters inside a given range of values. For the considered system of tunneling current, we have considered the variations in the following parameters of the system model:

• the gain of the piezoelectric actuator (G a ), being the most sensitive element of the closed-loop;

• the bandwidth of the voltage amplifier (ω v ) which is of the order of desired closedloop bandwidth. All the other elements in the loop have very high bandwidths;

• the constant term (k) of physical tunneling current phenomenon.

The nominal values together with their percentage variations are shown in Table 3.1.

The parametric uncertainty is usually represented by complex perturbations. The uncertainty regions around a nominal plant G 0 (s) can be generated by these additive or multiplicative complex norm bounded perturbations (additive or multiplicative uncertainty) [START_REF] Skogestad | Multivariable Feedback Control: Analysis and design[END_REF]. We have chosen a multiplicative uncertainty model for the considered system of tunneling current as generally the numerical value of multiplicative weights is more informative in order to attain the conditions for robust stability and robust performance (see coming section). The multiplicative uncertainty model G ∆ (s) can be described by the following equation (Fig. 3.6):

G ∆ (s) = G 0 (s)(1 + W I (s)∆ I (s)) (3.3.11)
where |∆ I (jω)| ≤ 1, ∀ω. Here, ∆ I (s) is any stable transfer function which at each frequency is less than one in magnitude, representing the normalized complex perturbations and W I (s) is the uncertainty weight. In case of multiplicative uncertainty model, the relative error function can be computed as:

l I (ω) = max G ∆ ∈Π G ∆ (jω) -G 0 (jω) G 0 (jω) (3.3.12)
and the rational weight W I (jω) is chosen as follows:

| W I (jω) | ≥ l I (ω), ∀ω (3.3.13)
Relative errors l I (jω) together with the rational weight W I (jω) are plotted in Fig. 3.7.

The uncertainty weight W I (s) is chosen as a transfer function that satisfies condition (3.3.13):

W I (s) = (1/ω B )s + A (1/(ω B M))s + 1 (3.3.14)
where, ω B = 6 × 10 4 rad/sec, A = 0.22 and M = 0.34.

Control Problem Formulation with Desired Performances

In this section, all desired performances for the considered system of tunneling current are translated into control requirements. These control requirements are expressed in terms of constraints on the closed-loop sensitivity functions which will be used for controller synthesis and for performance analysis of the considered closed-loop system of tunneling current.

Closed loop sensitivity functions

The closed-loop system mostly operates under the influence of different types of disturbances and it is very important to analyze the effect of them on the input and output of the system. The effect of these disturbances can be obtained by analyzing the so-called closed loop sensitivity functions. These sensitivity functions play an important role in the performance and robustness analysis of the closed-loop system.

For the considered closed-loop system of tunneling current (Fig. 3.5), two types of external disturbances are considered: sample surface variations (z S ) and noise (n). The other input signal of closed-loop system is the reference voltage (v ref ) which represents the desired tunneling current (i des ). The influence of all these three external inputs

(v ref , z S , n) over the control signal (v 1
) and the controlled output (i t ) can be expressed by following relations:

V 1 (s) = 1 1 + G(s)K(s) [K(s)V ref (s) -c 1 G b (s)K(s)Z S (s) -K(s)N(s)]
(3.4.1)

I t (s) = 1 1 + G(s)K(s) [K(s)G f (s)V ref (s) + c 1 Z S (s) -G f (s)K(s)N(s)] (3.4.2)
where,

G(s) = G f (s)G b (s) and c 1 = -k • i eq is a constant.
These relations will help to define the control problem and the desired performances in terms of constraints over the closed loop sensitivity functions. The closed loop sensitivity functions are generally defined as:

S o (s) = 1 1 + G(s)K(s) ; Output sensitivity f unction (3.4.3) KS o (s) = K(s) 1 + G(s)K(s)
; Input sensitivity f unction (3.4.4)

T(s) = G(s)K(s) 1 + G(s)K(s)
; Complementary sensitivity f unction (3.4.5)

Desired performance constraints

All desired performance requirements can be expressed by means of constraints on the shape of closed-loop sensitivity functions [START_REF] Landau | Digital Control Systems: design, identification and implementation[END_REF]. All these constraints are discussed here.

Robustness and stability constraints

The stability of the closed-loop system requires that all the sensitivity functions be asymptotically stable. The modulus margin and delay margin are generally considered as robustness margins.

Modulus margin: ∆M is defined as the radius of the circle centered in (-1, j0) and tangent to the Nyquist plot of H OL (jω). From the definition of the vector connecting the critical point (-1, j0) with the Nyquist plot of H OL (jω):

∆M = |1 + H OL (jω)| min = ( S o ∞ ) -1 (3.4.7)
In other words, the reduction (or minimization) of S o ∞ will imply the increase (or maximization) of the modulus margin ∆M. Typical values for good modulus margin are ∆M ≥ 0.5 (-6 dB). A good modulus margin guarantees satisfactory values for the stability margins e.g. ∆M ≥ 0.5 implies a gain margin ∆G ≥ 2 (6 dB) and a phase margin ∆φ > 29 • .

Delay margin: ∆τ is defined as the maximum admissible increase of the delay of the open-loop system without making the closed-loop system unstable.

∆τ = ∆φ ω cr (3.4.8)
where ∆φ is the phase margin and ω cr is the crossover frequency where the Nyquist plot of H OL (jω) crosses the unit circle centered at origin. The typical value for the delay margin is ∆τ ≥ T S where T S is the sampling time.

For sufficient stability margins, the maximum of output sensitivity function (S o ) should be less than 6 dB which will ensure good robustness margin as well. It can be denoted as Constraint 1 for the synthesis of the controller.

S o ∞ ≤ 6 dB, ∀ω (3.4.9)

In the same way, the maximum of complementary sensitivity function (T) should be less than 3.5 dB, expressed as Constraint 2 for the shape of closed-loop sensitivity function.

T ∞ ≤ 3.5 dB, ∀ω (

Robust stability and robust performance constraints

In addition to nominal stability and performance, the objectives of a robust control system include:

Robust Stability (RS): The system is stable for all perturbed plants around the nominal model up to the worst-case model uncertainty.

Robust Performance (RP): The system satisfies the nominal stability (NS)

and the nominal performance (NP) conditions. It also guarantees that performance specifications are meet for all perturbed plants up to the worst-case model uncertainty.

In the presence of multiplicative uncertainty (Section 3.3.5), the condition for Robust Stability (RS) according to Nyquist stability condition is given by [START_REF] Skogestad | Multivariable Feedback Control: Analysis and design[END_REF]:

RS ⇐⇒ |W I H OL | < |1 + H OL | , ∀ω ⇐⇒ W I H OL 1 + H OL < 1, ∀ω ⇐⇒ |W I T | < 1, ∀ω ⇐⇒ W I T ∞ < 1
where W I is the uncertainty weighting function. Thus, the requirement of robust stability (RS) for the case with multiplicative uncertainty gives an upper bound on the complementary sensitivity function (T ). This upper bound is expressed as Constraint 4 for the closed loop sensitivity function:

RS ⇐⇒ |T| < 1 |W I | , ∀ω (3.4.12)
Before giving the condition for Robust Performance (RP), the condition for Nominal Performance (NP) is:

|W P S o | < 1, ∀ω (3.4.13)
where W P is the designed performance weighting filter and 1/|W P (s)| will impose the upper bound on the magnitude of output sensitivity function (S o ). The performance weighting filter is chosen as:

W P (s) = (1/M s ) s + ω s s + ω s • ǫ s (3.4.14)
The details about all performance weighting filters are discussed later in Section 3.6. Now, for Robust Performance (RP), we require the performance condition to be satisfied for all possible plants, that is, including the worst case uncertainty. The condition for Robust Performance (RP) is obtained as [START_REF] Skogestad | Multivariable Feedback Control: Analysis and design[END_REF]:

RP ⇐⇒ |W P | + |W I H OL | < |1 + H OL | , ∀ω ⇐⇒ W P 1 + H OL + W I H OL 1 + H OL < 1, ∀ω ⇐⇒ |W P S o | + |W I T | < 1, ∀ω
So, the RP constraint is expressed as Constraint 5 for the closed loop sensitivity functions:

RP ⇐⇒ max ω (| W P S o | + | W I T |) < 1, ∀ω (3.4.15)

Measurement accuracy constraints

The desired performance requirement is to keep the tunneling current (i t ) constant at desired value in the presence of all external disturbances. For measurement accuracy, a maximum of ±10% variations is allowed in the tunneling current (i t ). This performance criteria will define the maximum allowed error voltage (v e ), or the lower limit (-20 dB) for output sensitivity function (S o ) in the measurement bandwidth (ω M ), where ω M defines the maximum allowed variations 800 rad/sec (127.32 Hz) of sample surface (z S ) with an amplitude of 0.5 Å. To limit the influence of surface variations (z S ) on the controlled output (i t ) in order to achieve the desired measurement accuracy, the transfer function between sample surface (z S ) and controlled output (i t ) is considered which is given by c 1 • S o as shown in Fig. 3.5, where c 1 is a known constant. According to the desired accuracy in measurement bandwidth, this constraint (Constraint 6) can be formulated as:

|c 1 • S o | dB ≤ -20 dB, 0 ≤ ω ≤ ω M ⇒ |S o | dB ≤ -27.2 dB, 0 ≤ ω ≤ ω M (3.4.16)
In addition, the transfer function between noise (n) and controlled output (i t ) has to be considered and it is given by -G n T (see Fig. 3.5), where G n is a known transfer function. According to the desired performance in terms of measurement accuracy, this constraint (Constraint 7) can be written:

|G n T | dB < 0 dB, ω > ω M ⇒ |T | < 1 G n , ω > ω M (3.4.17)

Surface variations reconstruction constraint

In order to limit the influence of noise (n) on the system input (v 1 ), the transfer function between them (KS o ) should respect the following relation, expressed as Constraint 8

for the closed-loop sensitivity function:

|KS o | dB < 0 dB, ω > ω M (3.4.18)
Table 3.2: Desired performances and corresponding constraints for the synthesis of controller

Desired performances Controller design constraints

Stability and robustness

∆M ≥ 0.5 , ∆τ ≥ T S S o ∞ ≤ 6 dB, ∀ω ∆G ≥ 2 , ∆φ > 29 • T ∞ ≤ 3.5 dB, ∀ω avoid actuator saturations KS o ∞ ≤ 20 dB, ∀ω robust stability |T| < 1 |W I | , ∀ω robust performance max ω (| W P S o | + | W I T |) < 1, ∀ω Measurement accuracy ±10% variations of tunneling current |S o | ≤ -27.2 dB, 0 ≤ ω ≤ ω M noise attenuation at controlled output |T | < 1 Gn , ω > ω M

Surface variations reconstruction

Controller structure

With usual notations, the structure of a linear time invariant discrete time model of the plant (G) used for digital controller design is described by:

G(z -1 ) = z -d l B(z -1 ) A(z -1 ) (3.5.1)
where, d l = delay (in number of sampling periods)

B(z -1 ) = b 0 + b 1 z -1 + . . . + b n B z -n B A(z -1 ) = 1 + a 1 z -1 + . . . + a n A z -n A
As the desired closed-loop bandwidth is taken of the order of the voltage amplifier An RS-type polynomial controller (K) is proposed (according to control scheme of Fig. 3.5) with controller polynomials R(z -1 ) and S(z -1 ) as follows:

R(z -1 ) = r 0 + r 1 z -1 + . . . + r n R z -n R (3.5.2) S(z -1 ) = 1 + s 1 z -1 + . . . + s n S z -n S (3.5.3)
The sensitivity functions shaping is done by appropriate selection of the desired closed loop poles and the introduction of pre-specified filters in the controller. Those filters are introduced in R(z -1 ) and S(z -1 ) as follows:

R(z -1 ) = H R (z -1 )R ′ (z -1 ) (3.5.4) S(z -1 ) = H S (z -1 )S ′ (z -1 ) (3.5.5)
where H R (z -1 ) and H S (z -1 ) correspond to pre-specified fixed filters and R ′ (z -1 ) and S ′ (z -1 ) are "free" filters:

R ′ (z -1 ) = r ′ 0 + r ′ 1 z -1 + . . . + r ′ n R ′ z -n R ′ (3.5.6) S ′ (z -1 ) = 1 + s ′ 1 z -1 + . . . + s ′ n S ′ z -n S ′ (3.5.7)
The desired closed loop poles are defined from a polynomial P (z -1 ) of the form:

P (z -1 ) = P D (z -1 ) • P F (z -1 ) (3.5.8)
where, P D (z -1 ) and P F (z -1 ) correspond to the dominant and auxiliary closed loop poles respectively. In order to compute the controller polynomials, the following equation known as "Bezout Identity" must be solved, for the unknown filters R ′ (z -1 ) and S ′ (z -1 ):

P (z -1 ) = A(z -1 )S(z -1 ) + z -d l B(z -1 )R(z -1 ) (3.5.9)
with P (z -1 ) as in Eq. (3.5.8), R(z -1 ) and S(z -1 ) as in Eq. (3.5.4) and Eq. (3.5.5) respectively.

The controller polynomials R(z -1 ) and S(z -1 ) (more precisely R ′ (z -1 ) and S ′ (z -1 ))

are the minimal degree solutions of Eq.

(3.5.9). This polynomial equation Eq. (3.5.9) has a unique solution with minimal degree for n P = degP (z -1 ) ≤

n A + n H S + n B + n H R + d l -1; n S ′ = degS ′ (z -1 ) = n B + n H R + d l -1; n R ′ = degR ′ (z -1 ) = n A + n H S -1.
For the controller implementation, R(z -1 ) and S(z -1 ) will be given by Eq. (3.5.4) and Eq. (3.5.5) respectively.

Controller synthesis

Based on the above mentioned algorithm, the designed controller characteristics are:

Closed-loop dominant poles P D : These dominant poles are placed at 3 kHz with damping coefficient of 0.9 so that the closed-loop natural frequency remains almost the same as the open loop one.

Closed-loop auxiliary poles P F : Double high frequency real poles are added at 0.2 in order to improve the controller robustness.

Controller Fixed Part H S : An integrator is used for H S fixed part of the controller in order to achieve zero steady-state error.

Controller Fixed Part H S : A real zero has been introduced at 0.3f S in order to shape the output sensitivity function.

The obtained controller polynomials after solving Bezout equation (3.5.9) are:

The closed-loop performance is discussed here with simulation results after the design of digital classical PI controller for comparison purpose.

Digital classical PI control design

For the sake of comparison, a digital PI controller is designed by a loop shaping technique, using the second order model as in the case of pole placement approach, keeping in mind all the constraints which are mentioned above. The PI controller polynomials are:

R(z -1 ) = 0.322 × 10 -3 + (0.322 × 10 -3 )z -1 S(z -1 ) = 1 -z -1 (3.5.11)
The comparison of performance in simulation between the two designed controllers Eq.

(3.5.10) and Eq. (3.5.11) is performed in the next sub-section.

Simulation results

For the sake of first validation, the sensitivity functions of the designed controller are plotted in Fig. 3.8. These sensitivity functions are plotted with complete linear time invariant model as given in Eq. 3.3.7 to analyze the performance of considered system of tunneling current. The black dotted lines indicate the maximum limits (constraints) of sensitivity functions which correspond to the desired performances as mentioned earlier.

It can be observed that all constraints are fully met.

The controller performance is then validated through time responses with a simulation model (Eq. (3.3.4)), having actual non-linearities (exponential, saturations), measurement noise (n) and physical limitations in closed-loop, aiming at representing the real system as close as possible. The controller validation with such a simulation model is an important step before experimental validation. variations in tunneling current (i t ). The desired tunneling current value was 0.5 nA. It can be observed that the tunneling current variation remains within the desired limits with both designed controllers, although less variations can be noticed with the proposed RS controller designed by pole placement with sensitivity function shaping than with conventional PI controller. If a simulation is performed with a slightly higher frequency of 800 rad/sec (127.32 Hz) of surface variations (z S ) with an amplitude of 0.5 Å as presented in Fig. 3.10 (first graph), it can be observed that the variation in tunneling current (i t ) still remains within acceptable bounds with the proposed RS controller (third graph) but it becomes unacceptable with classical PI controller (second graph). These results show the possible higher speed STM operation with the proposed RS controller than with the conventional PI controller. All these simulations are performed in the presence of measurement noise (n) of 10 mV / √ Hz.

Finally, the controllers performances are analyzed with random surface variations

H ∞ Controller Design

A mixed sensitivity H ∞ control design methodology is adopted in this section in order to give a more complete approach to the robust control problem previously considered.

The purpose is to achieve the same desired performances in terms of high measurement precision with large closed-loop bandwidth, good robustness and stability margins in the presence of external disturbances as mentioned in detail in Section 3.4. A brief description about standard H ∞ control design methodology is first recalled here, details can be found in [START_REF] Skogestad | Multivariable Feedback Control: Analysis and design[END_REF]. This methodology is applied for the considered system of tunneling current and then, the control loop performance is analyzed in simulation in the presence of designed H ∞ controller, which will be validated with experimental results in the next Chapter.

General H ∞ algorithm

The general H ∞ control problem is formulated using the general control configuration (Fig. 3.12) where P(s) is the generalized plant model, composed by the actual system G(s) and the performance weighting functions, w is the exogenous input vector (such as external disturbances (z S , n) and reference signal (v ref )), u is control input signal (v 1 ), y is the controlled output vector and e is the error signal (v e ) which need to be minimized in some sense to meet the desired control objectives.

The system of Fig. 3.12 is described by: generalized plant P(s) for the considered system of tunneling current is given in the next section. The state-space realization of the generalized plant P is given by:

P =   A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22   (3.6.2)
The system closed-loop transfer function from w to y is given by following linear fractional transformation:

y = F l (P, K) • w (3.6.3)
where:

F l (P, K) = P 11 + P 12 K(I -P 22 K) -1 P 21 (3.6.4)

The H ∞ control involve the minimization of the H ∞ norms of F l (P, K). The following assumptions are typically made in H ∞ problems [START_REF] Skogestad | Multivariable Feedback Control: Analysis and design[END_REF]: 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.

(A1) (A, B 2 , C 2 ) is
Assumption (A5) significantly simplifies the H ∞ algorithm formulas. For simplicity, it is also sometimes assumed that D 12 and D 21 are given by assumption (A6).

The standard H ∞ optimal control problem is to find all stabilizing controllers K which minimize the following quantity:

F l (P, K) ∞ = max ω σ(F l (P, K)(jω)) (3.6.5)
where σ represents the maximum singular value. In practice, it is usually not necessary to obtain an optimal controller for the H ∞ problem, and it is simpler to design a suboptimal one, which is close to the optimal controller, in the sense of the H ∞ norm. Let γ min be the minimum value of F l (P, K) ∞ over all stabilizing controllers K. Then the H ∞ sub-optimal control problem consists in finding, given a γ > γ min , all stabilizing controllers K such that:

F l (P, K) ∞ < γ (3.6.6)
For the general control configuration of Fig. 3.12 described by Eq. 3.6.4, with assumptions (A1) to (A6) listed above, there exist a stabilizing controller K(s) such that F l (P, K) ∞ < γ if and only if:

(i) X ∞ ≥ 0 is a solution to the algebraic Riccati equation:

A T X ∞ + X ∞ A + C T 1 C 1 + X ∞ (γ -2 B 1 B T 1 -B 2 B T 2 )X ∞ = 0 (3.6.7) such that Re λ i [A + (γ -2 B 1 B T 1 -B 2 B T 2 )X ∞ ] < 0
, ∀i where λ is the eigen value.

(ii) Y ∞ ≥ 0 is a solution to the algebraic Riccati equation:

AY ∞ + Y ∞ A T + B 1 B T 1 + Y ∞ (γ -2 C T 1 C 1 -C T 2 C 2 )Y ∞ = 0 (3.6.8) such that Re λ i [A + Y ∞ (γ -2 C T 1 C 1 -C T 2 C 2 )] < 0, ∀i (iii) ρ(X ∞ Y ∞ ) < γ 2
The family of all admissible controllers is given by K = F l (K c , Q), where:

K c =   A ∞ -Z ∞ L ∞ Z ∞ B 2 F ∞ 0 I -C 2 I 0   (3.6.9) F ∞ = -B T 2 X ∞ , L ∞ = -Y ∞ C T 2 , Z ∞ = (I -γ -2 Y ∞ X ∞ ) -1
(3.6.10)

A ∞ = A + γ -2 B 1 B T 1 X ∞ + B 2 F ∞ + Z ∞ L ∞ C 2 (3.6.11)
and Q(s) is any stable proper transfer function such that Q ∞ < γ. For Q(s) = 0, we get:

K(s) = -Z ∞ L ∞ (sI -A ∞ ) -1 F ∞ (3.6.12)
The controller K(s) has the same number of states as the generalized plant P (s).

An important part of H ∞ control design methodology is the selection of some weights on the controlled outputs which represent the performance specifications in the frequency domain. The generalized plant P thus includes the actual system and the considered weights (W p , W u and W t ) as shown in Fig. 3.13 and in Fig. 3.14. The H ∞ control problem is then referred to as a mixed-sensitivity problem, W p , W u and W t thus appearing in the Eq. (3.6.4) as weights on the sensitivity functions.

Mixed-sensitivity is the name given to transfer function shaping problems in which closed-loop sensitivity functions are shaped according to the designed weighting functions. The transfer function shaping approach uses H ∞ optimization to shape the singular values of specified closed-loop transfer functions over the frequency. The maximum singular values are easy to shape by forcing them to lie below user defined bounds, thereby ensuring desirable bandwidths and roll-off rates. We have chosen to use the loop shaping approach by mixed-sensitivity H ∞ control for our considered system of tunneling current.

Figure 3.13: Design model with weighting functions for the considered system of tunneling current.

Controller synthesis

The desired performances are imposed on the closed-loop sensitivity functions using appropriate weighting functions and then the mixed-sensitivity H ∞ control design methodology is adopted to fulfill the requirements. The functions W p , W u and W t weight the controlled outputs y 1 , y 2 and y 3 respectively (Fig. 3.13) and should be chosen according to the desired performance specifications (Section 3.4).

The generalized plant P (Fig. 3.14) (i.e. the interconnection of the actual system and the weighting functions) for our considered system of tunneling current is given by:

     y 1 y 2 y 3 v e      =      W p -c 1 G b W p -W p -G f G b W 1 0 0 0 W u 0 c 1 W t 0 G f W t I -c 1 G b I -G f G b      P      v ref z s n v 1     
where G f , G b and c 1 have been defined earlier in section 3.3.3. Thus, the H ∞ control Figure 3.14: Generalized design model for considered system of tunneling current.

problem is to find a stabilizing controller K(s) which minimizes γ such that:

  (W p )S (-c 1 G b W p )S (-W p )S (W u )KS (-c 1 G b W u )KS (-W u )KS (G -1 b W t )T (c 1 W t )S (-G -1 b W t )T   ∞ < γ (3.6.13)
The obtained controller K(s) has the same number of state variables as P . So, the choice of the weighting functions is an important issue in the H ∞ control problem, keeping in mind the real time implementation of the controller for the purpose of validation. The chosen weighting functions according to the desired performance requirements (Section 3.4 and Table . 3.2) are as follows:

(1) : W p is used to impose the desired performances on the closed-loop output sensitivity function S o in terms of robustness and measurement precision, that is:

W p (s) = (1/M s ) s + ω s s + ω s • ǫ s (3.6.14)
where M s = 2 to have a good robustness and stability margins (i.e. S ∞ ≤ 6 dB) for all frequency range, ω s = 1.2 × 10 4 rad/sec (1.9 kHz) to have a good attenuation of disturbances from low frequency up to ω s and ǫ s is chosen a very small value to induce an integral effect and to eliminate the steady-state error in the presence of maximum allowed variations in the sample surface (z S ).

(2) : W u is designed to avoid actuator saturations and also to attenuate the noise (n)

at system input for the considered system of tunneling current. It is chosen as follows:

W u (s) = s + (ω u /M u ) ǫ u • s + ω u (3.6.15)
where M u = 10 to impose limitation on the maximum value of controller output up to the frequency ω u which is chosen ω u = 1.88 × 10 4 rad/sec (3 kHz) and ǫ u = 0.5 to limit the effect of noise (n) at high frequencies on system input as mentioned in desired performances.

(3) : W t is designed to impose limitations on the complementary sensitivity function (T ) in order to achieve good robustness and to attenuate the noise (n) at controller output. It is chosen as:

W t (s) = s + (ω t /M t ) ǫ t • s + ω t (3.6.16)
where M t = 1.5 to have a good robustness margin (i.e. T ∞ ≤ 3.5 dB) for all frequency range, ω t = 1.88 × 10 4 rad/sec (3 kHz) to attenuate the noise (n) at high frequencies with ǫ t = 0.5.

As mentioned for the case of pole placement with sensitivity function shaping control (Section 3.5.1), again a simplified system model is considered for the controller synthesis.

As three weighting functions with first order transfer functions are designed so, an H ∞ controller with 4 th order transfer function is achieved. After computation, the minimal cost achieved for the considered system of tunneling current was γ = 0.95, which means that the obtained sensitivity functions match the desired loop shaping. For an actual digital implementation, the controller is discretized using a standard Tustin method and giving the following coefficients for RS polynomials of the controller (with sampling frequency of 30 kHz): r 0 = 6.33 × 10 -3 , r 1 = 0.85 × 10 -3 , r 2 = -3.97 × 10 -3 , r 3 = 1.38 × 10 -3 , r 4 = -0.14 × 10 -3 and s 1 = -466.1 × 10 -3 , s 2 = -933.9 × 10 -3 , s 3 = 451.7 × 10 -3 , s 4 = -51.6 × 10 -3 .

Simulation results

The obtained sensitivity functions of the designed controller are plotted in Fig. 3.15.

Again, these sensitivity functions are plotted with complete linear time invariant model as given in Eq. (3.3.7). It can be observed that all constraints are almost met.

The weighting functions (W p , W u and W t ) were designed considering the desired performance requirements (Section 3.4). After the control design, the performance of the considered system of tunneling current is validated with a complete simulation feedback loop (Fig. 3.1), having actual non-linearities and physical limitations in closed-loop. All these simulations are performed in the presence of noise (n) of 10 mV / √ Hz. Again, the performance comparison is performed with conventional PI controller as designed earlier (Eq. (3.5.11)). The desired tunneling current was considered as 0.5 nA. Less variations can be observed with the proposed controller than with the conventional PI controller. Again, if a simulation is performed with a slightly higher frequency of 800 rad/sec (127.32 Hz) of surface variations (z S ) with an amplitude of 0.5 Å, still less variations can be observed in Fig. 3.17 with the designed H ∞ controller (remains within acceptable bounds) but these variations become unacceptable with conventional PI controller (exceeded the acceptable bounds).

Time (sec) Tunneling current (A)

with "mixed sensitivity H-infinity control"

Acceptable Variations Acceptable Variations

Figure 3.17: Simulation results comparison between classical PI control and mixed sensitivity H ∞ control in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz), an amplitude of 0.5 Å and measurement noise (n) of 10 mV / √ Hz.

Finally, the performances of controllers are analyzed with random surface variations (z S ) (first graph in Fig. 3.18). As in the case of pole placement, less variations in tunneling current (i t ) can be observed with the proposed H ∞ controller (third graph) as compared to conventional PI controller (second graph).

As mentioned for the former robust control design, closed-loop sensitivity functions are helpful in justifying these results and they will also be identified with real-time experimental data in next Chapter.

Conclusion

In this chapter, after highlighting the motivation of this work, a system of tunneling current measurement has been briefly presented, a corresponding dynamic modeling has been proposed and a related control problem with desired measurement performance has been formulated. Then, measurement requirements have been translated into control requirements and modern robust control techniques have been proposed. Firstly RS control designed by combined pole placement with sensitivity function shaping method and then mixed-sensitivity H ∞ have been designed. A common approach for the system of tunneling current is to use classical PI control, so a comparison has been performed with conventional PI control design methodology.

Simulations results have shown better performances in terms of precision and disturbance rejection with the proposed controllers for the system of tunneling current measurement. Less variations of tunneling current have been observed with the proposed controllers. By comparing the output sensitivity function with both proposed controllers (Fig. 3.8 and Fig. 3.15), a little stronger attenuation of external disturbances for a large band of frequencies has been observed with the proposed H ∞ control, as compared with the proposed controller designed by pole placement with sensitivity function shaping. The results obtained in this Chapter still need to be validated with real time experimental results.

The experimental setup, explained in details in next Chapter, works at ambient temperature so additional external environmental disturbances can be present beyond the disturbances considered for simulations (measurement noise (n) and sample surface variations (z S )). More than time-domain results, the closed loop sensitivity functions also carry much information about all performance requirements, like measurement precision, bandwidth, robustness and stability, as highlighted in this Chapter. So, it will be interesting to analyze the performance of the system of tunneling current measurement with real time experimental data. The closed loop sensitivity functions, which are identified with real time experimental data in coming Chapter, and also the time domain experimental results will help us to perform a comparison between the two proposed control techniques.

An experimental setup has been developed in Gipsa-lab, Grenoble by the control group, in order to analyze the influence of different control techniques on tunneling current measurement. This setup works at ambiant temperature and is based on STM principles, although the purpose is not to take images of the surface. At nanometer scale, working over an experimental platform at ambient atmosphere in the presence of many unknown environmental disturbances was the real challenge. Many problems in terms of handling and repeatability have been encountered.

All necessary details including hardware characteristics, experimental constraints and the procedure proposed in order to bring the tip in the tunneling region (distance between tip and sample surface less than 1 ×10 -9 m) without collision is presented in Section 4.1.

System identification is performed in Section 4.2 and also experimental identification of closed loop sensitivity functions with conventional and proposed control techniques is presented in Section 4.3. Time domain experimental results of tunneling current with conventional and proposed control techniques for comparison are presented in Section 4.4 and Section 4.5 respectively. Some discussion on achieved results is done in Section 4.6 and finally Section 4.7 draws some conclusions.

Experimental Platform

In this section, all necessary details of the experimental setup (Fig. 4.1), developed in Gipsa-lab, Grenoble is presented [START_REF] Blanvillain | Contrôle nanoscopique du mouvement par courant tunnel : étude et réalisation[END_REF]. Experimental constraints and also the strategy in order to bring the tip close enough to the sample surface (distance between tip and sample surface less than 1 nm) to observe the tunneling current are given. can be prepared by electrochemical reactions and few relevant work can be found in [START_REF] Libioulle | Very sharp platinum tips for scanning tunneling microscopy[END_REF], [START_REF] Weinstein | A method for preparation of Pt-Ir tips for the scanning tunneling microscope[END_REF], [START_REF] Sorensen | Preparation of platinum/iridium scanning probe microscopy tips[END_REF], [START_REF] Rogers | A method for production of cheap, reliable Pt-Ir tips[END_REF]. The tip is fixed in a holder which is attached with piezoelectric actuator to move the tip in vertical z-direction. The piezoelectric actuator (Piezomechanics / PSt 150 ) has a resonance frequency of 120 kHz, a gain of 1.2 nm/V and an input voltage range from -20 V to 130 V . The output of the controller is between ±10 V and a voltage amplifier (Piezojena / ENV 300 ) having a gain of 15 V /V and a bandwidth of 4 kHz, provides appropriate signals to the piezoelectric actuator. The change in distance (d) between tip and sample surface modifies the tunneling current (i t ) and a tunneling current sensor (CV C) (home-made) of bandwidth 13 kHz and of gain 10 9 V /A, gives the signal to the controller. The CV C is fixed close enough to the tip to minimize the measurement noise (n). The sample surface is placed over a small bench which can be moved laterally very precisely in x-and y-directions with the help of two micrometer screws (Newport / HR-13 ) having travel range of 13 mm and sensitivity of 0.5 µm.

Experimental details

Three other micrometer screws (Newport / AJS-0.5 ) having the same travel range of 13 mm and sensitivity of 0.6 µm are attached with the piezoelectric actuator platform to move the tip manually in vertical z-direction. These sensitivities of the screws are based on a 1 • rotation of the adjustment micrometer screws. These micrometer screws and camera (µEye / CMOS UI-1550-C ) with telecentric zoom (VS-Technology / VS-TC-10-65 ) help the operator to bring the tip manually close to the sample surface so that the distance between them is in the range of few micro-meters. Further tip approach mechanism is done with the help of piezoelectric actuator until the tunneling current (i t ) is obtained. Tip approach strategy in order to obtain the tunneling current is described in the next sections. The whole experimental setup is placed over an anti-vibration table (Microworld ). The control scheme is implemented in a computer (Development PC, processor 2.5 GHz, Matlab) connected with another computer (Target PC, processor 4.1. They have been used for the design models considered in Chapter 3. The values of this table have been obtained as follows:

• Identification on the real set-up has been used to obtain φ P t/Ir and φ Au , based on the exponential behavior of the tunneling current recorded [START_REF] Blanvillain | Contrôle nanoscopique du mouvement par courant tunnel : étude et réalisation[END_REF].

• Data sheet information has been used for G v , ω v and ω a .

• Home-made current sensor (CV C) allows to have G c and ω c .

• Finally, G a has been adjusted to give a global gain equal to the one identified on the real-time experiment (see Fig. 4.5 in section 4.2).

Notice that some uncertainty remains in the value of the current sensor gain G c (which could be 20 or 30 times smaller than expected), this information being available only at the end of this manuscript writing. In future work, this value has to be thoroughly identified, as it can have a large impact on the simulation of disturbances effect on the output.

Experimental constraints

Before presenting the approach for obtaining the tunneling current, it is necessary to clarify the constraints imposed by the experimental platform.

G v = 15 V/V Bandwidth of voltage amplifier ω v ≈ 4 kHz Gain of piezoelectric actuator G a ≈ 1.2 nm/V Bandwidth of piezoelectric actuator ω a = 120 kHz Gain of current sensor G c = 10 9 V/A Bandwidth of current sensor ω c ≈ 13 kHz
• The measurement range of the current sensor output (CV C) is between 0 V to 10 V, which corresponds to a range of motion of the tip under few angstroms. To keep the tip within such small measurement range by working in open loop is really difficult and thus, working in closed loop is really essential.

• The major difficulty is the repeatability of good operating and environmental conditions to obtain the tunneling current. Since the platform works at ambient temperature, the composition of tunneling current is more complex and many external environmental disturbances can be present beyond the disturbances considered for simulations. The controller must be indeed robust and have a capability to reject the external disturbances in order to achieve the desired performances.

• The tunneling current is a quantum mechanical phenomenon, and its understanding, exploitation and implementation require knowledge in the areas related to physics. The complete analysis of the results and understanding by working in a department of controls was a real challenge.

• By bringing a very clean sharp tip close to a clean sample surface, one might expect a very low level of noise in measured signal. But, suddenly a situation quite complex might occur where measured signal is full of noise and it is very difficult to find the explanation of this observed noise. The only way is to change the tip or/and sample surface. We have performed our experiments on different samples and different tips.

• At atomic scale, the materials move slowly with temperature variations. These movements can be the source of contact (collision) between the elements (like tip and sample surface) during inactive periods. To avoid this contact, the platform allows with the help of micro meter screws to remove these elements from each other a few millimeters. The elements can be placed in the desired configuration (distance less than one nanometer) to perform a test. Since it is impossible to put the elements in exactly identical conditions to the tests performed previously with nanometer resolution, the experimental conditions are different in different tests.

Tip approach for obtaining tunneling current

Because of the current sensor (CVC) output voltage saturation (0 to 10 V), the measurement range of tunneling current is between few hundreds of picometers. The tip must reach this measurement range with a resolution of the order of picometers. This type of approach in real-time can be very slow. A strategy of approach-withdrawal has been proposed in order to achieve the tunneling current [START_REF] Blanvillain | Contrôle nanoscopique du mouvement par courant tunnel : étude et réalisation[END_REF].

This strategy is in two steps:

1. Manual approach 2. Fine approach.

Manual approach starts by bringing the tip, initially far from the sample surface, close to sample surface with the help of vertical micrometer screws and camera with telecentric zoom. One has to be very careful in order to avoid any contact between tip and sample surface. Once the tip is close enough to sample surface, the fine approach will start. According to this fine approach, the tip first approaches rapidly towards the sample surface with the help of piezoelectric actuator until the tunneling current saturates (approach phase). Tunneling current saturation implies that the tip has exceeded the measurement range and is now very close to the sample surface. Now, the input of piezoelectric actuator is changed and the tip is moved very slowly away from the sample surface (withdrawal-phase) until the tip arrives within the measurement range. The control loop is closed as soon as the tip arrives at the desired reference value of tunneling current. Fig. 4.3 shows the complete strategy of this fine approach for obtaining the tunneling current.

vector. The control signal u(t) delivered by the controller is given by:

u(t) = T (q -1 ) S(q -1 ) r(t) - R(q -1 ) S(q -1 ) y(t) (4.2.2)
where r(t) is the input excitation signal and R(q -1 ), S(q -1 ) and T (q -1 ) are controller parameters. The adjustable closed loop predictor uses a controller identical to the one used in the real time system and its output is described by:

ŷ• (t + 1) = -â 1 (t)ŷ(t) + b1 (t)û(t) = θT (t) φ(t) (4.2.3)
where ŷ• (t + 1) is the predicted output at the instant t based on the knowledge of the parameters estimated at time t. θT (t) = [â 1 (t), b1 (t)] is the vector of estimated parameters at time t and ŷ• (t + 1) is called the a priori prediction. A priori prediction error is given by:

ε • CL (t + 1) = y(t + 1) -ŷ• (t + 1) (4.2.4)
To evaluate the quality of the new estimated parameter vector θ(t + 1), which will be provided by the parameter adaptation algorithm, it is useful to define the a posteriori output of the adjustable predictor, which corresponds to re-computing Eq. (4.2.3) with the new values of the parameters estimated at t + 1. The a posteriori predictor output is defined by: ŷ(t + 1) = -â 1 (t + 1)y(t) + b1 (t + 1)û(t) = θT (t + 1) φ(t) (4.2.5)

and also an a posteriori prediction error is:

ε CL (t + 1) = y(t + 1) -ŷ(t + 1) (4.2.6)
A recursive parametric adaptation algorithm with memory is desired. The structure of such an algorithm is:

θ(t + 1) = θ(t) + f θ(t), φ(t), ε • CL (t + 1) (4.2.7)
where the correction term f must only depend upon the information available at instant t + 1. This correction term should allow to minimize at each step the a priori prediction error with respect to the criterion:

min θ(t) J(t + 1) = [ε • CL (t + 1)] 2 (4.2.8)
One can use the gradient parameter adaptation algorithm to minimize the above criterion, however such parametric adaptation algorithm based on a priori prediction error can introduce possible instabilities [START_REF] Landau | Digital Control Systems: design, identification and implementation[END_REF]. In order to avoid the possible instabilities, one can use the same gradient approach but with a different criterion based on minimization of the a posteriori prediction error at each step according to:

min θ(t+1) J(t + 1) = [ε CL (t + 1)] 2 (4.2.9)
By using the gradient algorithm, at each step ε 2 (t+1) is minimized but such minimization at each step does not necessarily lead to the minimization of

t i=1
ε 2 (i)on a t-steps time horizon. Therefore, we have chosen the least square recursive algorithm which minimizes the criterion:

min θ(t) J(t) = t i=1 y(i) -θT (t) φ(t -1) 2 = 1 t t i=1 ε 2 CL i, θ(t) (4.2.10)
The objective is therefore the minimization of the sum of the squares of the prediction errors. Thus, the objective of the identification in closed loop is to minimize the prediction error between the measured output of the real system and the predicted closed loop output by finding the best plant model close to real system.

Application:

The external excitation signal must have a rich frequency spectrum in order to cover the bandwidth of the plant to be identified. Thus, a PRBS (Pseudo-Random Binary Sequences) which is a sequence of rectangular pulses that approximates a discrete time white noise and has a spectral content rich in frequencies, can be superposed to the reference signal. In order to correctly identify the steady state gain of the plant dynamic model, the duration of, at least, one of the pulses must be greater than the rise time of the plant. The characteristics of PRBS signal are chosen as: ampliture = ±0.1 V, number of registers = 10, sampling frequency = 30 kHz and frequency divider = 2. The identification method CLOE (Closed Loop Output Error) is used to identify the best model of the system (plant).

The theoretical gain of the open loop system (plant) which includes voltage amplifier, piezoelectric actuator, the physical tunneling current phenomenon and the current sensor

(CV C), is 309 V/V

Sensitivity Functions Identification

In this section, the experimental results with comparison between conventional PI and proposed control techniques are presented for the considered system of tunneling current.

This comparison is performed with the experimentally identified closed loop sensitivity functions. The capability of the control system to reject all external disturbances can be analyzed through closed loop sensitivity functions. All the desired performances are already expressed in the previous chapter by means of constraints on the shape of the closed loop sensitivity functions (section 3.4) which helped us for the controllers synthesis.

It will be seen that those identified sensitivity functions maybe slightly different from those obtained with the simulation model in Chapter 3. This can be due to the differences between experiment and simulation conditions.

For the purpose of experimental identification of the closed loop sensitivity functions, the external excitation signal PRBS (Pseudo-Random Binary Sequences) is superposed to the reference, with the same characteristics as before: amplitude = ±0.1 V, number of registers = 10, sampling frequency = 30 kHz and frequency divider = 2. Based on the experimental data in the presence of the proposed control techniques (controller designed by pole placement with sensitivity function shaping and by mixed sensitivity H ∞ control)

and classical PI controller, the following closed loop sensitivity functions are identified.

Open loop identification method ELS (Extended Least Squares, details in [START_REF] Landau | Digital Control Systems: design, identification and implementation[END_REF]) has been used: Gn , ω > ω M and also T ∞ ≤ 3.5 dB, ∀ω in order to achieve good robustness. It can be observed that all controllers attenuate well measurement noise (n), while the proposed mixed sensitivity H ∞ control even provides a larger closed loop bandwidth, compared to the conventional PI control and the pole placement controller. Based on above experimentally identified closed loop sensitivity function, we can remark that stronger attenuation of external disturbances for a large band of frequencies is achieved with the prosed H ∞ control, even it provides a larger closed loop bandwidth as compared to the other control techniques. Now, these results need to be verified with the time domain experimental results which are presented in the next section.

Current reference tracking

Tunneling current (i t ) behavior was also observed with step variations of desired tunneling current. Again, after arriving in the tunneling region, a series of step variations of desired tunneling current is considered. Fig. 4.10 shows the measured tunneling current (i t ) variations with conventional PI controller together with the corresponding piezo control signal. Again, this result will be used for comparison between conventional and proposed control techniques.

Tunneling Current with Modern Control Techniques

In this section, the time domain experimental results with comparison between conventional and proposed control techniques are presented for the considered system of tunneling current. Here, we will observe the resulting variations in the measured tunneling current with the proposed controller designed by pole placement with sensitivity function shaping and by mixed sensitivity H ∞ control. Also the comparison with conventional PI control results as presented in previous section, is here illustrated.

Steady state current control

Reference tunneling current of 0.25 nA The standard deviation of measured tunneling current is 10.8 pA with the proposed controller designed by pole placement with sensitivity function shaping and 12.7 pA with the mixed sensitivity H ∞ control (the standard deviation with PI controller was 24.3 pA).

The tunneling current variations with proposed control techniques remain within more or less ±10% variations of desired value which is in good accordance wit simulation results. Reference tunneling current of 0.5 nA Next, the tunneling current (i t ) variations are examined with a slightly larger desired value which requires to move the tip precisely closer to the sample surface. Fig. 4.12 shows the behavior of tunneling current (i t ) with the proposed and with conventional PI controller when the desired value of tunneling current was 0.5 nA. It can be observed again that the tunneling current (i t ) variations are lower with proposed controllers then with conventional PI controller. The standard deviation of measured tunneling current is 16.7 pA with the proposed controller designed by pole placement with sensitivity function shaping and 14.8 pA with the mixed sensitivity H ∞ control (the standard deviation with PI controller was 36.1 pA). It can also be observed that with all controllers, the tunneling current (i t ) variations are a little larger than in the previous case. This can be explained by stronger mechanical noise. Reference tunneling current of 0.75 nA Again, the tunneling current (i t ) variations are examined with a more larger desired value of tunneling current (0.75 nA). Fig. 4.13 shows the behavior of tunneling current (i t ) with all the controllers. It can be observed again that the tunneling current (i t ) variations are lower with proposed controllers then with conventional PI controller. The standard deviation of measured tunneling current is 25.9 pA with the proposed controller designed by pole placement with sensitivity function shaping and 22.1 pA with the mixed sensitivity H ∞ control (the standard deviation with PI controller was 50.1 pA).

As shown in these results, the tunneling current variations are indeed lower with the proposed control techniques than with the conventional PI control technique, which is in good accordance with simulation results and also with the identified closed loop sensitivity functions.

Current reference tracking

Tunneling current (i t ) behavior was observed in the presence of proposed controllers with step variations of desired tunneling current. Fig. 4.14 and Fig. 4.15 show the tunneling current (i t ) variations with proposed controller designed by pole placement and with mixed sensitivity H ∞ control respectively. It can be noticed very clearly that at each step of desired tunneling current, the tunneling current (i t ) variations with proposed controllers is lower than the ones with the conventional PI controller. The step variations of tunneling current (i t ) with conventional PI controller was already shown in Fig. 4.10.

Discussion

In the previous section, we have observed the variations in the measured tunneling current with different control techniques and also we have experimentally identified the closed loop sensitivity functions which give us much information about desired performances in terms of disturbance rejection, robustness and closed loop bandwidth. In this section, we will further analyse the above achieved results which will help us finally to present a conclusion in the end of this Chapter. The power spectral densities of the measured tunneling current (i t ) with both proposed controllers can also be compared from Fig. 4.16 to Fig. 4.18. At all desired values of tunneling current (0.25 nA, 0.5 nA, 0.75 nA), a slightly stronger attenuation of disturbances at higher frequencies can be observed with the mixed sensitivity H ∞ control than with controller designed by pole placement with sensitivity function shaping e.g. around -5 dB more attenuation is achieved at high frequencies with the mixed mixed sensitivity H ∞ control than with pole placement with sensitivity function shaping control. Although, at lower frequencies, there is no much difference between the level of attenuation of disturbances. It can also be noticed that some peaks appeared at certain frequencies with all the controllers, e.g. at 5 kHz a small peak is observed with mixed sensitivity H ∞ control at all desired values of tunneling current. This peak can also be noticed in the output sensitivity function (S o (z -1 )) with H ∞ control (Fig. 4

Conclusion

In this chapter, an experimental setup developed in Gipsa-lab, Grenoble by control group in order to analyze the influence of different control techniques on tunneling current In the second section, system identification has been performed and the identified model of the system has been found very similar to the theoretical one, in terms of overall gain and the bandwidth of the open loop system (plant).

In the third section, the closed loop sensitivity functions have been identified experimentally with conventional PI control and the proposed control techniques (Chapter 3) for performance analysis of the tunneling current measurement system. Stronger attenuation of external disturbances for a large band of frequencies has been observed with the proposed H ∞ control, even it provides a larger closed loop bandwidth, as compared with the proposed controller designed by pole placement with sensitivity function shaping and the conventional PI control.

In the fourth section, time domain experimental results have been presented for tunneling current measurement system for a comparison between different control techniques. It has been observed that the tunneling current variations are indeed lower with the proposed control techniques, particularly with mixed sensitivity H ∞ control, than with conventional PI control technique.

Finally, a comparison between proposed and conventional control techniques have been performed with power spectral densities and the standard deviations of the measured tunneling current, which are indeed in good accordance with the frequency and time domain results discussed earlier.

This chapter is devoted to the analysis of the tunneling current measurement system while scanning the surface in the horizontal x-direction. Scanning for long range and large duration induces the problem of dynamic nonlinearities (hysteresis and creep respectively) of piezoelectric actuator. Cross-coupling problem of piezoelectric actuator is also an important issue for nanopositioning. Measurement precision of the tunneling current can be adversely affected by the coupling-caused positioning error in the vertical z-direction, while scanning the surface. In this chapter, a MIMO controller is analyzed in simulation in order to obtain the better performances for the considered system of tunneling current, in the presence of coupling-caused positioning error.

The problem of coupling-caused positioning error with some recent research work from control point of view is presented in Section 5.1.

A dynamic modeling of the overall MIMO system having horizontal, vertical and cross coupling dynamics is performed in Section 5.2.

The problem of dynamic nonlinearities of piezoelectric actuator is highlighted and

analyzed (in open-loop and also in closed-loop) in Section 5.3, while scanning for long range in horizontal x-direction over the experimental platform of Gipsa-lab.

In the first part of the Section 5.4, the measurement precision of the considered system of tunneling current in vertical z-direction is analyzed in simulations while scanning in the horizontal x-direction, in the presence of coupling dynamics. In the second part, LQR control for overall MIMO system is analyzed for the considered system.

Finally, Section 5.5 draws some conclusions.

Introduction

The cross-coupling problem that arises during high-speed nano-precision positioning using piezoelectric scanners is highlighted in Chapter 2. This problem can adversely influence the high measurement precision of the tunneling current in vertical z-direction.

Coupling-caused positioning errors in the vertical z-direction can arise due to the scanning movement in the x and y directions. However, as pointed out in different recent research articles [START_REF] Song | Tracking control of a piezoceramic actuator with hysteresis compensation using inverse preisach model[END_REF], [START_REF] Sebastian | Design methodologies for robust nanopositioning[END_REF], [START_REF] Butterworth | A comparison of control architectures for atomic force microscopes[END_REF], the effect of scanning movement in the horizontal x-direction on the vertical z-direction is more significant because the scan frequency in the x-direction is higher than the scan frequency in the y-direction (all three x, y and z directions can be seen in Fig. 2.3). Coupling-caused positioning error becomes more significant while scanning at high speed for long range. Long range scanning for some large duration in horizontal

x-direction also induces the problem of dynamic nonlinearities (hysteresis and creep) of piezoelectric actuator.

Some research work from control point of view in order to reduce the couplingcaused positioning errors is already mentioned in Chapter 2, e.g. [START_REF] Tien | Control of dynamics-coupling effects in piezoactuator for high-speed afm operation[END_REF], [START_REF] Song | Tracking control of a piezoceramic actuator with hysteresis compensation using inverse preisach model[END_REF], [START_REF] Wu | A control approach to cross-coupling compensation of piezotube scanners in tapping-mode atomic force microscope imaging[END_REF], [START_REF] Shi | A control approach to cross coupling compensation of piezotube scanners in tapping-mode imaging[END_REF]. However, some recent papers [START_REF] Pao | Combined feedforward/feedback control of atomic force microscopes[END_REF], [START_REF] Mahmood | Making a commercial atomic force microscope more accurate and faster using positive position feedback control[END_REF], [START_REF] Butterworth | A comparison of control architectures for atomic force microscopes[END_REF] have pointed out the interest to analyze MIMO controllers in order to analyze the performances of the overall positioning system (the x-y-z directions positioning). In this chapter, a MIMO controller is analyzed for tunneling current measurement system, in the presence of coupling dynamics (proposed from the experimental results presented in [START_REF] Tien | Control of dynamics-coupling effects in piezoactuator for high-speed afm operation[END_REF]).

System Modeling

The purpose of this section is to model the considered MIMO system. The complete open loop system which will be here considered is presented in Fig. 5.1, where the horizontal Figure 5.1: Complete MIMO plant having horizontal, vertical and cross-coupling dynamics system consists of:

• the voltage amplifier;

• the horizontal x-piezoelectric actuator;

• the capacitive sensor and the vertical system consists of:

• the voltage amplifier;

• the vertical z-piezoelectric actuator;

• the physical law which gives the relationship between tunneling current and the distance between tip and sample surface;

• the current sensor (CV C).

Here, in order to obtain a complete MIMO plant model, a simple linear model of horizontal and vertical systems with coupling dynamics are considered.

Horizontal dynamics

For horizontal positioning system, the dynamics of the piezoelectric actuator (bandwidth 630 Hz) is considered as a second order system, while the dynamics of voltage amplifier (bandwidth 4 kHz) and of the capacitive sensor (bandwidth 8.5 kHz) are considered as constant gains (closed-loop dominant poles are placed at 200 Hz). So, the output of the voltage amplifier (v 1h ) is:

v 1h (t) = G v • u h (t) (5.2.1)
where G v is the gain and u h is the input voltage of the voltage amplifier.

The dynamics of the piezoelectric actuator for horizontal x-direction has been modeled as:

   ẋ1 (t) = -2ζ ah ω ah • x 1 (t) -ω 2 ah • x 2 (t) + v 1h (t) ẋ2 (t) = x 1 (t) z h (t) = G ah ω 2 ah • x 2 (t) (5.2.2)
where v 1h and z h are input voltage and output displacement of the piezoelectric actuator respectively, ζ ah is the damping, ω ah the bandwidth and G ah the gain of the piezoelectric actuator.

The output voltage (v h ) of the capacitive sensor is:

v h (t) = G c • z h (t) (5.2.3)
where G c is the gain and z h is the input of the capacitive sensor.

So, the overall dynamics of the horizontal system is:

   ẋ1 (t) = -2ζ ah ω ah • x 1 (t) -ω 2 ah • x 2 (t) + G v • u h (t) ẋ2 (t) = x 1 (t) v h (t) = G c G ah ω 2 ah • x 2 (t)
(5.2.4)

Cross-coupling dynamics

First of all, the displacement of the piezoelectric actuator z h in horizontal direction is retrieved by using the gain of the capacitive sensor. As the bandwidth of the capacitive 123 5.3 Dynamic Nonlinearities for Horizontal Movement of Piezoelectric Actuator

Hysteresis

The piezoelectric actuator is driven by a fast triangular waveform in order to scan the sample surface in a raster (back and forth) pattern. The amplitude and frequency of this waveform defines the scanning range and speed respectively. Scanning for long range introduces the phenomenon of hysteresis, which has been observed for the horizontal motion of piezoelectric actuator in x-direction over the experimental platform of Gipsalab. A number of hysteresis models and a lot of research has already been done to compensate the hysteresis phenomenon. Some corresponding details have already been highlighted very briefly in Chapter 2. In next sections, we will just rely again on a direct RS-type design.

Creep

The drift of the displacement of piezoelectric actuator for a constant reference voltage is another undesirable phenomenon, commonly known as creep. This phenomenon has been observed (Fig. 5.6) for the horizontal motion of piezoelectric actuator in x-direction over the experimental platform of Gipsa-lab.

Two types of creep models (based on static logarithmic nonlinearity and a dynamic linear model) have been proposed in the literature, which have already been mentioned in Chapter 2. In the next section, we will show how this effect can be handled, together with hysteresis, via an RST controller.

Controller design for scanning (horizontal) movement of piezoelectric actuator

For the purpose of horizontal scanning control, an RST controller again based on pole placement with sensitivity functions shaping is designed, considering model Eq. ( 5

Cross-Coupling Effects and Compensation

In this section, the measurement precision of the tunneling current in vertical z-direction is analyzed while scanning in horizontal x-direction in the presence of cross-coupling phenomenon.

Cross-coupling effects on tunneling current measurement

The controller design for the considered system of tunneling current in the vertical zdirection is discussed in Chapter 3 and experimentally validated in Chapter 4. The controller design for horizontal x-direction (scanning) is discussed in previous section.

Here, the purpose is to analyze the performance of these controllers in the presence of the coupling phenomenon. The general feedback block diagram is shown in Fig. 5.9.

Some simulation results are presented with maximum scanning speed of 100 Hz and different scanning range in horizontal x-direction. Fig. 5.10 shows the simulation result with scanning range of 0.01 µm. The desired tunneling current value was 0.5 nA and the maximum ±10% variations was allowed in the tunneling current (i t ). It can be observed that the tunneling current variation remains within the desired limits. If a simulation is performed with a slightly larger scanning range 0.05 µm as presented in Fig. 5.11, it can be observed that the variation in tunneling current becomes unacceptable because of the coupling based positioning error in vertical z-direction. More increase in scanning range (e.g. 0.1 µm) causes more variation in the tunneling current as shown in Fig. 5.12.

One possible solution, proposed in the next section, is to design a MIMO controller by considering the dynamics of the system in the horizontal x-direction, the vertical z-direction and the cross-coupling effect, as depicted by Fig. 5.13.

LQR control design for MIMO system

In this subsection, a Linear Quadratic Regulator (LQR) approach is considered as a first solution for the MIMO control of the plant described by Eq. (5.2.16) of the considered system of tunneling current, having horizontal, vertical and cross-coupling dynamics.

The general block diagram of the feedback loop considered here is shown in Fig. 5.13.

Complete details about LQR controller design can be found for instance in [START_REF] Zhou | Robust and optimal control[END_REF], the main features are briefly recalled here. Consider a general linear time- x-direction. For vertical z-direction, the desired tunneling current value was again 0.5 nA.

It can be observed that with this large scanning range in the horizontal x-direction, the tunneling current variation in the vertical z-direction remains within the desired limits (the maximum ±10% variation allowed). This simulation result shows the improvement in the performance as compared to (Fig. 5.12) individual controllers for each direction for the considered system of tunneling current.

The control could be of course improved by extending the approach to robust techniques.

Conclusion

In this chapter, the vertical system (tunneling current measurement) and horizontal system (scanning) have been considered together, in the presence of cross-coupling phenomenon. In the first section, the coupling-caused positioning error in vertical z-direction has been highlighted. Then, a corresponding dynamic modeling of MIMO plant has been proposed in the next section.

The dynamic nonlinearities (hysteresis and creep) of the piezoelectric actuator have been experimentally observed, while scanning in the horizontal x-direction. A controller based on pole placement with sensitivity function shaping has been designed and experimentally validated in order to achieve better tracking. The creep phenomenon has been eliminated by the proposed control design and the nonlinear hysteresis phenomenon has been reduced, although fast tracking in the horizontal x-direction still requires a better control scheme in the presence of the hysteresis phenomenon. A lot of research has already been done in this particular area, which has been highlighted in Chapter 2.

Finally, the coupling-caused positioning error for the tunneling current measurement system in the vertical z-direction has been analyzed in simulations in the presence of individual controllers and also with MIMO LQR controller for the horizontal and the vertical directions. The variation of tunneling current has been observed within the desired limits with the proposed MIMO LQR controller better than with the individual controllers, in the presence of fast speed with large scanning range in the horizontal Chapter 6

General Conclusion and Perspectives

General Conclusion

The main idea behind this work was to provide a contribution to an emerging domain of nanotechnology from control point of view. The tunneling current phenomenon, which occurs at nanometer scale and can be of strong interest for actual applications, was chosen for this purpose. Based on this phenomenon, an experimental platform has been developed in Gipsa-lab Grenoble by the control group allowing to validate the control techniques (at ambient atmosphere) proposed in the present work.

In the first part of this work, the tunneling current phenomenon has been explained and corresponding applications have been highlighted. Nanopositioning is required for the precise measurement of the tunneling current. The problems associated with the nanopositioning in terms of precision, bandwidth, robustness, actuator nonlinearities and cross-coupling issues have been presented, and then analyzed for the considered system of tunneling current in the other parts of the work.

A dynamic modeling corresponding to the considered system has been proposed and x-direction and vertical z-direction) for the considered system of tunneling current.

Perspectives

In view of the results obtained in this work, various perspectives can be thought of which are listed below:

Considering the dynamics of the piezoelectric actuator (120 kHz), modifications in the hardware can be made in order to achieve large closed-loop bandwidth by increasing the bandwidth of the voltage amplifier (4 kHz) (which was the limiting bandwidth in vertical z-direction) and also the tunneling current sensor (13 kHz) (at the cost of a lower gain). Another issue of the hardware was with the data acquisition cards which introduced the limitation of the sampling frequency (30 kHz) and didn't allow to work with high order controllers. Field-Programmable Gate Array (FPGA) technology could be exploited for high speed data acquisition.

During power spectral density analysis of the measured tunneling current, a peak at low frequency (around 350 Hz) was observed in all the tests with all type of controllers.

The amplitude of the peak varied in different tests. The reason of this peak seems to be experimental, although the exact reason is not determined yet which needs to be identified to further enhance the performance. A Notch filter [START_REF] Procházka | Pole placement with sensitivity function shaping using 2nd order digital notch filters[END_REF] can be introduced as the fixed part of the controller, at the cost of increasing the order of the controller by two.

Exact model of cross-coupling dynamics needs to be identified. Unfortunately, tunneling current is not a good sensor to accurately identify the coupling-caused positioning error in the vertical z-direction. By scanning in the horizontal x-direction, the variations in the tunneling current can be because of the coupling-caused positioning error and in the same time, because of the sample surface variations. The proposed MIMO controller needs to be validated experimentally, after the identification of cross-coupling dynamics. The control can of course be improved by extending the approach to robust MIMO control techniques.

This thesis was focused on the analysis of tunneling current measurement, although for scanning in horizontal x-direction, the experimentally observed hysteresis phenomenon can be modeled and then compensated (many related work is highlighted in Chapter 2) in order to enhance the tracking performance. The same holds for the creep phenomenon of the piezoelectric actuator.

The experimental platform allows for the situation with an oscillating sample surface, the analysis of the performance of the tunneling current measurement system in such a configuration is also a perspective of interest.

More generally, the impact of a better tunneling current measurement can be analyzed in various other applications, and for instance when using tunneling current as a displacement sensor in the presence of the electrostatic actuator as in [START_REF] Blanvillain | Contrôle nanoscopique du mouvement par courant tunnel : étude et réalisation[END_REF].

u

  X through coupling-effect dynamics. . . . . . . . . . . . . . . . . . . . . 3.1 Complete simulation model with block diagram representation. . . . . . . 3.2 Open loop system with block diagram representation. . . . . . . . . . . .3.3 Block diagram representation of tunneling current and current sensor (dashed block indicates the nonlinear element). . . . . . . . . . . . . . . 3.4 Physicists linearization approach with logarithmic amplifier. . . . . . . . 3.5 Control design model G and feedback loop. . . . . . . . . . . . . . . . . 3.6 Feedback system with multiplicative uncertainty. . . . . . . . . . . . . . . 3.7 Relative plant errors l I (jω) (dashed lines) and rational weight W I (jω) (solid line) for 27 possible combinations of the system parameters. . . . . 3.8 (top left) Output sensitivity function (S o ); (top right) Complementary sensitivity function (T); (bottom left) Input sensitivity function (KS o ); (bottom right) System robust performance test; with controller designed by pole placement with sensitivity functions shaping and associated constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Simulation results comparison between classical PI control and controller designed by pole placement with sensitivity functions shaping in the presence of sinusoidal surface variations (z S ) of frequency of 300 rad/sec (47.75 Hz), an amplitude of 0.5 Å and measurement noise (n) of 10 mV / √ Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Simulation results comparison between classical PI control and controller designed by pole placement with sensitivity functions shaping in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz), an amplitude of 0.5 Å and measurement noise (n) of 10 mV / √ Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Simulation results comparison between classical PI control and controller designed by pole placement with sensitivity functions shaping in the presence of random surface variations (z S ). . . . . . . . . 3.12 Generalized plant with controller. . . . . . . . . . . . . . . . . . . . . . . 3.13 Design model with weighting functions for the considered system of tunneling current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Generalized design model for considered system of tunneling current. . . 3.15 (top left) Output sensitivity function (S o ); (top right) Complementary sensitivity function (T); (bottom left) Input sensitivity function (KS o ); (bottom right) System robust performance test; with mixed sensitivity H ∞ control and associated constraints . . . . . . . . . . . . . . . . . . . 3.16 Simulation results comparison between classical PI control and mixed sensitivity H ∞ control in the presence of sinusoidal surface variations (z S ) of frequency of 300 rad/sec (47.75 Hz), an amplitude of 0.5 Å and measurement noise (n) of 10 mV / √ Hz. . . . . . . . . . . . . . . . . . . . 3.17 Simulation results comparison between classical PI control and mixed sensitivity H ∞ control in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz), an amplitude of 0.5 Å and measurement noise (n) of 10 mV / √ Hz. . . . . . . . . . . . . . . . . . . . 3.18 Simulation results comparison between classical PI control and mixed sensitivity H ∞ control in the presence of random surface variations (z S ). 4.1 Experimental platform developed in Gipsa-lab. . . . . . . . . . . . . . . . 4.2 Closed loop with all hardware elements of experimental platform. . . . . 4.3 Strategy of tip approach for obtaining tunneling current. . . . . . . . . . 4.4 Closed loop output error identification method. . . . . . . . . . . . . . . 4.5 Bode diagram of tunneling current measurement system plant (G) model. 4.6 Experimentally identified closed loop output sensitivity functions (S o ). . 4.7 Experimentally identified closed loop complementary sensitivity functions (T). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Experimentally identified closed loop input sensitivity functions (KS o ). . 4.9 Experimental result with classical PI control with reference tunneling current of 0.25 nA, 0.5 nA and 0.75 nA. . . . . . . . . . . . . . . . . . . 4.10 Experimental result with classical PI control having step variations of desired tunneling current. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Comparison of experimental results with reference tunneling current of 0.25 nA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Comparison of experimental results with reference tunneling current of 0.5 nA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Comparison of experimental results with reference tunneling current of 0.75 nA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Experimental result with controller designed by pole placement with sensitivity function shaping having step variations of desired tunneling current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Experimental result with mixed sensitivity H ∞ control having step variations of desired tunneling current. . . . . . . . . . . . . . . . . . . . 4.16 Comparison of power spectral densities with reference tunneling current of 0.25 nA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 Comparison of power spectral densities with reference tunneling current of 0.5 nA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 Comparison of power spectral densities with reference tunneling current of 0.75 nA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Comparison of robust stability test: |T| < 1 |W I | , ∀ω . . . . . . . . . . . . . 4.20 Comparison of robust performance test: max ω (| W PS o | + | W I T |) < 1, ∀ω5.1 Complete MIMO plant having horizontal, vertical and cross-coupling dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The hysteresis phenomenon observed over the experimental platform at different input voltage amplitudes for the horizontal movement of the piezoelectric actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The hysteresis phenomenon observed over the experimental platform for the horizontal movement of the piezoelectric actuator. . . . . . . . . . . . 5.4 The hysteresis phenomenon observed over the experimental platform at different rates of input voltage for the horizontal movement of the piezoelectric actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Experimental result of open-loop reference tracking and observed hysteresis loop for the horizontal movement of the piezoelectric actuator. . . . . 5.6 Experimental result of open-loop step response for the horizontal movement of the piezoelectric actuator with creep phenomenon. . . . . . . . . 5.7 Experimental result of closed-loop reference tracking and observed hysteresis loop for the horizontal movement of the piezoelectric actuator. . . 5.8 Experimental result of closed-loop step response for the horizontal movement of the piezoelectric actuator with compensation of creep phenomenon.133 5.9 Complete block diagram of MIMO plant with individual controllers for horizontal and vertical directions in the presence of cross-coupling dynamics.134 5.10 Simulation result with horizontal scanning speed of 100 Hz, scanning range of 0.01 µm and corresponding tunneling current variations in vertical z-direction in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz) and an amplitude of 0.5 Å. . . . . 135 5.11 Simulation result with horizontal scanning speed of 100 Hz, scanning range of 0.05 µm and corresponding tunneling current variations in vertical z-direction in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz) and an amplitude of 0.5 Å. . . . . 135 5.12 Simulation result with horizontal scanning speed of 100 Hz, scanning range of 0.1 µm and corresponding tunneling current variations in vertical z-direction in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz) and an amplitude of 0.5 Å. . . . . . . . . . . 136 5.13 Complete general block diagram of MIMO plant with MIMO controller for horizontal and vertical directions in the presence of cross-coupling dynamics.136 5.14 With MIMO control, the horizontal scanning speed of 100 Hz with scanning range of 0.1 µm and corresponding tunneling current variations in vertical z-direction in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz) and an amplitude of 0.5 Å. . . . . 139 6.1 Random peaks in the measurement signal . . . . . . . . . . . . . . . . . 145 6.2 Random peaks in the measurement signal . . . . . . . . . . . . . . . . . 146 6.3 Undesire-able response of tunneling current with random peaks, no tunneling current and large variations . . . . . . . . . . . . . . . . . . . . . . 146 List of Tables 3.1 System parameters together with their values and percentage variations . 3.2 Desired performances and corresponding constraints for the synthesis of controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Characteristics of devices used for the considered system of tunneling current 4.2 Comparison of standard deviations of tunneling current with different control techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q

  Wave function of electron ψ I and ψ III Wave function of electron outside potential barrier ψ II Wave function of electron within potential barrier k 1 Wave number outside the potential barrier k 2 Wave number within the potential barrier A Amplitude of the incoming wave function of electron B Amplitude of the reflected portion of wave function of electron C Amplitude of the wave function crossing within the potential barrier D Amplitude of the wave function reflecting within the potential barrier F Amplitude of the wave function transmitted through the potential barrier Constant term (C 1 = log 10 (G amp • g • v b )) Quality factor of piezoelectric actuator G a (s) Transfer function of piezoelectric actuator γ Creep rate of piezoelectric actuator x(t) Horizontal displacement of piezoelectric actuator x d (t) Horizontal desired displacement of piezoelectric actuator x 0 Horizontal displacement of piezoelectric actuator at time t 0 v(t) Applied voltage to piezoelectric actuator for horizontal displacement X(s) Horizontal displacement of piezoelectric actuator (in Laplace domain) V (s) Applied voltage to piezoelectric actuator (in Laplace domain) Matrix of complete MIMO plant model of piezoelectric actuator P xx (ω) Transfer function from x-axis control input to the x-position P xy (ω) Transfer function from y-axis control input to the x-position P xz (ω) Transfer function from z-axis control input to the x-position P yx (ω) Transfer function from x-axis control input to the y-position P yy (ω) Transfer function from y-axis control input to the y-position P yz (ω) Transfer function from z-axis control input to the y-position P zx (ω) Transfer function from x-axis control input to the z-position P zy (ω) Transfer function from y-axis control input to the z-position P zz (ω) Transfer function from z-axis control input to the z-position z Vertical deflection of piezoelectric actuator u Z Z-axis input of piezoelectric actuator u X X-axis input of piezoelectric actuator Chapter 3: Control Problem and Robust Design i t Tunneling current d Distance between tip and sample surface v b Potential difference applied between tip and sample surface g Proportionality constant of tunneling current k Constant of tunneling current (k = 2 √ 2m/ = 1.025 eV

Chapter 3 :

 3 This chapter is related to the first two points of the thesis contribution as mentioned above. It starts with the motivation of the work and brief description of the considered system of tunneling current measurement. Then, a corresponding system dynamic modeling is proposed and a related control problem with desired measurement performances are formulated. Measurement requirements are translated into control requirements and then modern robust controllers based on pole placement with sensitivity function shaping technique and mixed sensitivity H ∞ controller are proposed. Finally, the performance of the considered system of tunneling current is analyzed with the proposed controllers, and a comparison is performed by simulation results with the more classically used PI controller for this type of systems.Chapter 4: This chapter is related to the third point of the thesis contribution, in order to experimentally validate the proposed controllers of chapter 3 and to perform a comparison with the more classical PI controller for the considered system of tunneling current. It starts with all necessary details about the experimental setup developed in Gipsa-lab, Grenoble. Then, system identification is performed and closed loop sensitivity functions are identified experimentally with all control techniques of chapter 3. Time domain experimental results are subsequently presented, and finally, based on all time and frequency domain results, a comparison is performed between all control techniques for the considered system of tunneling current.Chapter 5: This chapter is related to the last point of the thesis contribution, where scanning related issues in the horizontal x-direction are considered. It starts by highlighting very briefly the coupling-caused positioning error and also the dynamic nonlinearities (hysteresis and creep) of the piezoelectric actuator. Then, corresponding dynamic modeling of the MIMO plant (having horizontal, vertical and cross coupling dynamics) is proposed and some experimental results related to the dynamic nonlinearities of the piezoelectric actuator (in open and in closed-loop) are presented. On this basis, a MIMO controller is designed and the measurement precision of the tunneling current is analyzed (in simulations) in the presence of coupling-caused positioning error in the vertical z-direction of the considered system.

Figure 2 . 2 :

 22 Figure 2.2: Schematic representation of the scanning tunneling microscope.

Figure 2

 2 Figure 2.3: "Constant height" and "Constant current" modes of operation.

Figure 2 . 4 :

 24 Figure 2.4: Feedback loop for scanning a sample surface (STM).

Figure 2 . 5 :

 25 Figure 2.5: Tradeoffs between different parameters.

Figure 2

 2 Figure2.6: Hysteresis loop example for lateral motion of piezoelectric actuator between input voltage of piezoelectric actuator through amplifier and the output displacement sensed by capacitive sensor. This hysteresis loop was observed from experimental platform of Gipsa-lab.

Figure 2

 2 Figure 2.8: Creep phenomenon of piezoelectric actuator observed from experimental platform of Gipsa-lab with a constant input voltage.

  Figure 3.1: Complete simulation model with block diagram representation.

Figure 3 . 2 :

 32 Figure 3.2: Open loop system with block diagram representation.

Figure 3

 3 Figure 3.3: Block diagram representation of tunneling current and current sensor (dashed block indicates the nonlinear element).

Figure 3

 3 Figure 3.4: Physicists linearization approach with logarithmic amplifier.

  Figure 3.5: Control design model G and feedback loop.

Figure 3

 3 Figure3.7: Relative plant errors l I (jω) (dashed lines) and rational weight W I (jω) (solid line) for 27 possible combinations of the system parameters.

  3.4.10) To avoid instability due to saturation effects in the electronic part, the maximum of input sensitivity function (KS o ) should be less than 20 dB which is expressed as Constraint 3 for the closed-loop sensitivity function. KS o ∞ ≤ 20 dB, ∀ω (3.4.11)

  noise attenuation at system input |KS o | < 0 dB, ω > ω M Closed-loop bandwidth Closed-loop dominant poles ≤ 4 kHz ∆M = Modulus margin, ∆τ = Delay margin, ∆G = Gain margin, ∆φ = Phase margin, T S = Sampling time, W I = Uncertainty weighting function, W P = Performance weighting filter

( 4

 4 kHz) (bandwidth constraint), a simple linear model is considered for the controller synthesis, where the voltage amplifier dynamics (bandwidth 4 kHz) and tunnel current sensor (CVC) (bandwidth 13 kHz) are taken into account and the dynamics of the piezoelectric actuator (bandwidth 120 kHz) is taken as constant gain. The coefficients of the model polynomials, used for the controller synthesis are: b 1 = 83.31, b 2 = 25.93, a 1 = -0.4984, a 2 = 0.0284.

Fig. 3 .Figure 3

 33 Fig. 3.9 shows the simulation result with the classical PI controller and with the proposed RS controller in the presence of surface variations (z S ) (first graph) with a frequency of 300 rad/sec (47.75 Hz) and an amplitude of 0.5 Å. The two horizontal

  Figure 3.9: Simulation results comparison between classical PI control and controller designed by pole placement with sensitivity functions shaping in the presence of sinusoidal surface variations (z S ) of frequency of 300 rad/sec (47.75 Hz), an amplitude of 0.5 Å and measurement noise (n) of 10 mV / √ Hz.

  (z S ) (first graph in Fig. 3.11). Again, less variations in tunneling current (i t ) can be observed with the proposed RS controller (third graph) as compared to classical PI controller (second graph). The reason can be investigated by comparing the closed loop sensitivity functions (particularly closed loop output sensitivity function (S o (z -1 ))) as they carry much information about the disturbance rejection. For proper analysis in a real-time environment, these closed loop sensitivity functions are also identified with real-time experimental data in next Chapter.

Figure 3

 3 Figure 3.10: Simulation results comparison between classical PI control and controller designed by pole placement with sensitivity functions shaping in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz), an amplitude of 0.5 Å and measurement noise (n) of 10 mV / √ Hz.

Figure 3

 3 Figure 3.11: Simulation results comparison between classical PI control and controller designed by pole placement with sensitivity functions shaping in the presence of random surface variations (z S ).

  Figure3.12: Generalized plant with controller.

  stabilizable and detectable. (A2) D 12 and D 21 have full rank. rank for all ω. (A5) D 11 = 0 and D 22 = 0. (A6) D 12 = 0 I and D 21 = 0 I .Assumption (A1) is required for the existence of a stabilizing controller K, and assumption (A2) is sufficient to ensure the controllers are proper and hence realizable.

Figure 3

 3 Figure 3.15: (top left) Output sensitivity function (S o ); (top right) Complementary sensitivity function (T); (bottom left) Input sensitivity function (KS o ); (bottom right) System robust performance test; with mixed sensitivity H ∞ control and associated constraints

Figure 3

 3 Figure 3.16: Simulation results comparison between classical PI control and mixed sensitivity H ∞ control in the presence of sinusoidal surface variations (z S ) of frequency of 300 rad/sec (47.75 Hz), an amplitude of 0.5 Å and measurement noise (n) of 10 mV / √ Hz.

Figure 3

 3 Figure 3.18: Simulation results comparison between classical PI control and mixed sensitivity H ∞ control in the presence of random surface variations (z S ).

Figure 4 . 1 :

 41 Figure 4.1: Experimental platform developed in Gipsa-lab.

  Figure 4.2: Closed loop with all hardware elements of experimental platform.

3. 2

 2 GHz, XPCTarget) through Ethernet. The Target PC has acquisition card (PCI DAS 1602/16, 8 differential inputs, 2 outputs, 16 bits resolution) connected with the experimental setup. The complete feedback loop with all signals and the hardware involved in shown in Fig. 4.2. The numerical values of the most important elements of this system are summarized in Table

  Figure 4.3: Strategy of tip approach for obtaining tunneling current.

Figure 4

 4 Figure 4.5: Bode diagram of tunneling current measurement system plant (G) model.

•

  Fig. 4.6 shows the closed loop output sensitivity function (S o (z -1)) with different controllers. This sensitivity function shows the relationship between the disturbance of surface variations (z S ) and the tunneling current (i t ) as mentioned in the desired performance constraints (section 3.4) and also it shows the influence of noise (n) on the measured signal (v y ). According to the desired performance constraints, (S o (z -1 )) must be |S o | ≤ -27.2 dB, 0 ≤ ω ≤ ω M in order to achieve desired measurement accuracy of tunneling current and also S o ∞ ≤ 6 dB, ∀ω in order to achieve good robustness (see Table3.2). A much stronger attenuation at low frequencies can be observed with

Figure 4

 4 Figure 4.6: Experimentally identified closed loop output sensitivity functions (S o ).

  Figure 4.7: Experimentally identified closed loop complementary sensitivity functions (T).

Figure 4

 4 Figure 4.9: Experimental result with classical PI control with reference tunneling current of 0.25 nA, 0.5 nA and 0.75 nA.

Figure 4

 4 Figure 4.10: Experimental result with classical PI control having step variations of desired tunneling current.

Fig. 4 .Figure 4

 44 Fig. 4.11 shows the measured tunneling current (i t ) with the proposed control techniques (controller designed by pole placement with sensitivity function shaping and mixed sen-

Figure 4

 4 Figure 4.12: Comparison of experimental results with reference tunneling current of 0.5 nA.

Figure 4

 4 Figure 4.13: Comparison of experimental results with reference tunneling current of 0.75 nA.

FirstlyFigure 4

 4 Figure 4.14: Experimental result with controller designed by pole placement with sensitivity function shaping having step variations of desired tunneling current

Figure 4 .Figure 4 Figure 4 Figure 4

 4444 Figure 4.15: Experimental result with mixed sensitivity H ∞ control having step variations of desired tunneling current.

Figure 4 Figure 4

 44 Figure 4.19: Comparison of robust stability test: |T| < 1 |W I | , ∀ω

Fig. 5 .

 5 Fig. 5.2 and Fig. 5.3 shows the hysteresis phenomenon dependance over the amplitude of the input voltage (scanning range) of piezoelectric actuator. It can be observed that at small amplitude of input voltage (0.1 V ) with horizontal displacement of piezoelectric actuator of around 300 nm, the hysteresis phenomenon is almost negligible. This result justifies our approximation of neglecting the hysteresis phenomenon for the vertical movement (in z-direction for tunneling current measurement) since the vertical displacement was very small (less than 1 nm). Fig. 5.4 shows the hysteresis phenomenon at different frequencies of input voltage for the horizontal movement of piezoelectric actuator. It can be observed that the hysteresis loop remains almost similar at all frequencies of input signal. Finally, an open loop reference tracking for the horizontal movement of the piezoelectric actuator and the corresponding hysteresis loop are presented in Fig. 5.5.

Figure 5 . 2 :Figure 5 Figure 5 . 4 :Figure 5

 525545 Figure 5.2: The hysteresis phenomenon observed over the experimental platform at different input voltage amplitudes for the horizontal movement of the piezoelectric actuator.

Figure 5 . 6 :

 56 Figure 5.6: Experimental result of open-loop step response for the horizontal movement of the piezoelectric actuator with creep phenomenon.

  Fig.5.7 shows a good closed-loop tracking of a reference scanning signal in horizontal x-direction and also it can be observed that the hysteresis phenomenon is dealt with the designed controller. However, a fast scanning speed in horizontal x-direction can not eliminate completely the hysteresis phenomenon as shown in the second part of Fig.5.7.

Fig. 5 .Figure 5

 55 Fig. 5.8 shows that the feedback control law also virtually eliminates the phenomenon of creep and the system tracks the constant reference signal.

Figure 5 . 8 :

 58 Figure 5.8: Experimental result of closed-loop step response for the horizontal movement of the piezoelectric actuator with compensation of creep phenomenon.

Figure 5 . 9 :

 59 Figure 5.9: Complete block diagram of MIMO plant with individual controllers for horizontal and vertical directions in the presence of cross-coupling dynamics.

Figure 5

 5 Figure5.10: Simulation result with horizontal scanning speed of 100 Hz, scanning range of 0.01 µm and corresponding tunneling current variations in vertical z-direction in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz) and an amplitude of 0.5 Å.

Figure 5

 5 Figure5.12: Simulation result with horizontal scanning speed of 100 Hz, scanning range of 0.1 µm and corresponding tunneling current variations in vertical z-direction in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz) and an amplitude of 0.5 Å.

Fig. 5 .

 5 Fig. 5.14 shows the simulation result with the proposed LQR controller in the presence of maximum scanning speed of 100 Hz and the scanning range of 0.1 µm in the horizontal

Figure 5

 5 Figure 5.14: With MIMO control, the horizontal scanning speed of 100 Hz with scanning range of 0.1 µm and corresponding tunneling current variations in vertical z-direction in the presence of sinusoidal surface variations (z S ) of frequency of 800 rad/sec (127.32 Hz) and an amplitude of 0.5 Å.

  desired performance requirements have been translated into control requirements. Two robust control techniques, firstly pole placement with sensitivity function shaping method and then mixed-sensitivity H ∞ approach, have been used and a comparison with the more conventionally used classical PI controller has been performed in simulations. The comparison results for the considered system of tunneling current have confirmed better performances in terms of precision and disturbance rejection with the proposed controllers than with the PI.The challenging part of this work was the experimental validation (at ambient atmosphere) of the proposed control schemes, while working at nanometer scale by keeping the distance between two electrodes (tip and sample surface) less than 1 × 10 -9 m without contact. This part of the work started with the system identification, went on with obtaining experimental results in time and frequency domains, finally allowing for an experimental comparison between the proposed and conventional control techniques. The experimental results have shown stronger attenuation of external disturbances for a large band of frequencies with the proposed H ∞ control, even providing a larger closed loop bandwidth, as compared with the proposed controller designed by pole placement with sensitivity function shaping and the conventional PI control.One of the great applications of tunneling current is to scan a sample surface with atomic scale resolution. This scanning (in horizontal x-direction) introduces the problem of actuator nonlinearities and also positioning error for the tunneling current measurement (in vertical z-direction) because of the actuator cross-coupling. These issues have been discussed in the last part of this work. Experimental results with the proposed control design based on pole placement with sensitivity function shaping method for horizontal x-direction have shown an attenuation of actuator nonlinearities effects. Finally, a Linear Quadratic Regulator (LQR) approach has been analyzed in simulation for the MIMO control of the plant having horizontal, vertical and cross coupling dynamics. Less variations in the tunneling current has been observed with the proposed MIMO controller as compared to the individual controllers for each direction (horizontal

Figure 6

 6 Figure 6.2: Random peaks in the measurement signal

  Initial distance between tip and sample surface v 1 Input voltage of the voltage amplifier v 2 Output voltage of the voltage amplifier ω v Bandwidth of the voltage amplifier G

	1/2-1 ) Output measured voltage of the vertical feedback loop Work function Measurement noise z S Sample surface variations φ n Damping of the piezoelectric actuator ω a Bandwidth of the piezoelectric actuator G a Gain of the piezoelectric actuator v 3 Output voltage of the current sensor (CVC) ω c Bandwidth of the current sensor (CVC) (v y ) i eq Tunneling current at operating point i des Desired value of tunneling current v ref Reference input voltage c 1 Constant for linearization of tunneling current (c 1 = -k • i eq ) (v e ) Error voltage G f (s) Plant model in feedforward path G b (s) Plant model in feedback path G n (s) Noise transfer function G(s) Plant transfer function K(s) Controller transfer function H Uncertainty weighting transfer function ω B Bandwidth of the uncertainty transfer function A Minimum value of uncertainty weighting transfer function M Maximum value of uncertainty weighting transfer function S th Power spectral density of thermal noise S(f ) Power spectral density of 1/f noise S sh Power spectral density of shot noise k B Boltzmann's constant R t Tunneling resistance in Ohms e Electron charge S o (s) Output sensitivity function KS o (s) Input sensitivity function T Complementary sensitivity function ∆M Modulus margin ∆τ Delay margin G(z -1 ) Discrete time plant model B(z -1 ) Numerator polynomial of the discrete time plant model A(z -1 ) Denominator polynomial of the discrete time plant model d l Delay (in number of sampling periods) R(z -1 ) Numerator polynomial of the discrete time controller d 0 ζ G c Gain of the current sensor (CVC) S(z -1 ) Denominator polynomial of the discrete time controller
	∆G	Gain margin
	∆φ	Phase margin
	T S	Sampling time
	f S	Sampling frequency
	ω cr	Cross-over frequency

v Gain of the voltage amplifier v 2 Input voltage of the piezoelectric actuator z Output displacement of the piezoelectric actuator in vertical z-direction OL (s) Open-loop transfer function G 0 (s) Nominal plant model G ∆ (s) Multiplicative uncertainty model W I (s)

Table 3 .

 3 1: System parameters together with their values and percentage variations

	Parameter description	Value	% variation
	Sensitivity of piezoelectric actuator (G a ) Bandwidth of voltage amplifier (ω v ) Constant term for tunneling phenomenon (k) 2.29 ± 0.229 [1/Å] 1.2 ± 0.12 [nm] 4000 ± 400 [Hz]	% % %
	Figure 3.6: Feedback system with multiplicative uncertainty.

Table 4 .

 4 1: Characteristics of devices used for the considered system of tunneling current

	Material	Characteristics
	Work function of tip	φ P t/Ir = 5.6 eV
	Work function of sample surface	φ Au = 5.4 eV
	Gain of voltage amplifier

  Table3.2). A much stronger attenuation at low frequencies can be observed with the proposed control techniques than with conventional PI controller, which was evident with the simulation results as well, where less variations were observed with proposed control techniques than with conventional PI control technique in the presence of surface variations (see Sections 3.5.3 and 3.6.3). For the comparison between the two proposed control techniques, a much stronger attenuation for large band of frequencies can be noticed with the proposed mixed sensitivity H ∞ control than with the controller designed by pole placement with sensitivity function shaping. This result in a better attenuation of external environmental disturbance, in order to achieve less fluctuations in the tunneling current (i t ). From the robustness point of view, the desired constraint is almost fulfilled with all the controllers.

Table 4 .

 4 2: Comparison of standard deviations of tunneling current with different control techniques

	Reference current PI control Sensitivity functions shaping H ∞ control
	0.25 nA	24.3 pA	10.8 pA	12.7 pA
	0.5 nA	36.1 pA	16.7 pA	14.8 pA
	0.75 nA	50.1 pA	25.9 pA	22.1 pA
	and 3 kHz. The reason is the high positive gain (peaks) of output sensitivity function
	and also the complementary sensitivity function within this frequency range (see Fig. 4.6
	and Fig. 4.7) with mixed sensitivity H ∞ control.

R(z-1 ) = 3.22 × 10 -3 -(2.34 × 10 -3 )z -1 + (0.47 × 10 -3 )z -2 -(0.01 × 10 -3 )z -3 S(z -1 ) = 1 -(1264 × 10 -3 )z -1 + (253 × 10 -3 )z -2 + (10.85 × 10 -3 )z -3(3.5.10)

Acknowledgements

The work presented in this PhD thesis is done in Gipsa-lab, department of control systems, Grenoble, France, with the affiliation of a research team SLR (Systèmes Linéaires et

Chapter 3

Control Problem and Robust Design

From equations Eq. (3.4.3) and Eq. (3.4.5), it is obvious that:

Before the controller synthesis, first it is needed to define the control problem and also the desired performances for the considered system of tunneling current.

Control problem formulation

For the feedback control system of Fig. 3.5, the control problem can be formulated as a disturbance rejection problem: variations in the sample surface (z S ) as well as measurement noise (n) are indeed considered as external disturbances, where the first one can be considered as a slowly varying disturbance and the latter one can be considered as a fastly varying disturbance. These disturbances must be rejected by moving the tip in appropriate direction with the help of piezoelectric actuator so that the tunneling current (i t ) should always remain constant at its desired value. In addition, as usual in control design, the control variable itself must not be too much affected by noises, and this is even more important for STM applications (since the control is also used for imaging purposes). So, the associated sensitivity functions must be shaped in order to obtain the best rejection of these disturbances.

The control objectives for the considered system of tunneling current are:

• Good robustness and stability margins;

• High measurement (or equivalently current control) accuracy;

• Noise attenuation at system input (STM application).

• Large closed-loop bandwidth;

All those control objectives will be tackled altogether by the proposed design approaches which are described in coming sections.

Closed-loop bandwidth constraint

The maximum sampling frequency (f S ) chosen for all signals must at least be 6 times the closed loop bandwidth [START_REF] Landau | Digital Control Systems: design, identification and implementation[END_REF]]. An important part of this thesis is the experimental validation of the proposed control schemes for the considered system of tunneling current (details in coming chapter), where the maximum sampling frequency (f S ) chosen for all signals is 30 kHz. A higher sampling frequency was not possible because of the hardware limitations. This sampling frequency imposes the limits on the closed-loop bandwidth which is taken of the order of the voltage amplifier (4 kHz) in order to avoid any aliasing phenomenon. Because of this limitation, a simple linear model is considered for synthesis of the controller, where dynamics of the piezoelectric actuator (bandwidth 120 kHz) is taken as constant gain. The obtained controller is validated in simulation with complete continuous non-linear time invariant model as given in Eq.

(3.3.4).

All these desired performances and constraints, listed in Table 3.2, are used in coming sections for controller design and for performance analysis of the closed-loop system.

Digital Control Design using Pole Placement with Sensitivity Function Shaping Technique

In order to meet with the specifications listed above, a robust digital RS controller is firstly designed by using pole placement combined with the shaping of sensitivity functions. This technique is a polynomial approach which allows to place the poles in order to justify the constraints on the shape of closed-loop sensitivity functions. Details on this controller design methodology can be found in [START_REF] Landau | Robust digital control using pole placement with sensitivity function shaping method[END_REF]], [START_REF] Landau | Digital Control Systems: design, identification and implementation[END_REF]. It is adopted here since it takes into account simultaneously robustness and performance specifications for the closed-loop. Notice that this methodology, aiming at a digital implementation, is based on a discrete-time model of the plant.

Chapter 4

Experimental Analysis and Control Results

System Identification

In this section, the open loop system (plant) between the output of the controller (v 1 )

and the measured voltage (v y ) is identified by the techniques of identification in closed loop. The objective of the identification in closed loop is to find the best plant model which minimizes the prediction error between the measured output of the true closed loop system and the predicted closed loop output. Here, we give a brief recall the closed loop identification principle and then apply it it to our considered system in order to identify the plant model. All details with algorithms about identification in closed loop can be found in [START_REF] Landau | Digital Control Systems: design, identification and implementation[END_REF].

Identification in closed-loop: The first order discrete time plant model is described by:

where θ T = [a 1 , b 1 ] is unknown parameter vector and φ T (t) = [-y(t), u(t)] measurement

Tunneling Current with Conventional PI Control Technique

In this section, the time domain experimental results with conventional PI control technique are presented for the considered system of tunneling current. After arriving in the tunneling region (distance (d) between tip apex and sample surface less than 1 ×10 -9 m)

with the strategy proposed earlier, different desired values of tunneling current (i t ) are

given and the resulting variations in the measured tunneling current with conventional PI control technique are examined. Chapter 5

Steady state current control

Towards MIMO Analysis and Control sensor (8.5 kHz) is very large compared to the bandwidth of the piezoelectric actuator (630 Hz) in horizontal direction, we can assume that: ẑh = z h . Then, the retrieved displacement of the piezoelectric actuator ẑh in horizontal direction is:

where G c is the gain of the capacitive sensor.

As mentioned in the previous sections, the scanning movement of the tip in the horizontal x-direction induces a positioning error in the vertical z-direction and this coupling-caused positioning error becomes more significant with high scan speed and also with large scanning range. A high pass filter is then proposed to model this crosscoupling effect and the gain of the filter dynamics will define the maximum positioning error (3% of the scanning displacement in horizontal x-direction at high scan speed of 100 Hz, taken from [START_REF] Tien | Control of dynamics-coupling effects in piezoactuator for high-speed afm operation[END_REF]), which will act as a disturbance for the vertical system.

The cross-coupling dynamics can thus be modeled as:

where z c is the output (positioning error) of the coupling dynamics, ω f c is the bandwidth (scan speed for maximum positioning error) and G f c the gain (maximum positioning error).

The overall cross-coupling dynamics between the output voltage of the capacitive sensor (v h ) and the positioning error (z e ) is:

Vertical dynamics

The modeling of the vertical system (nonlinear model) and then linearization of the static exponential nonlinearity (as tunneling current depends exponentially on the distance between tip and sample surface) in order to achieve the linearized model, is already discussed in Chapter 3. A simple linear model (dynamics of the voltage amplifier) of uncoupled vertical system was considered in Chapter 3 for the purpose of synthesis of the controller which was later validated in Chapter 4 with identification results performed with experimental data.

For the vertical positioning system, the voltage amplifier has been modeled as:

where u v and v 1v are input and output voltages of the voltage amplifier respectively, ω vv is the bandwidth and G v the gain of the voltage amplifier.

The output displacement (z v ) of the piezoelectric actuator is:

where G av is the gain and v 1v the input voltage of the piezoelectric actuator. As v 1v is the output voltage of the voltage amplifier, the output displacement (z v ) of the piezoelectric actuator can be expressed as:

(5.2.9)

Now, the vertical distance (d) between tip and sample surface has been obtained as (see Fig. 5.1):

where z s is the unknown surface variations and z c the positioning error because of the coupling dynamics. Both will act as disturbances for the vertical system. From Eq.

(5.2.7) and Eq. (5.2.9), this vertical distance (d) can be expressed as:

The linearization of tunneling current i t with respect to the distance (d) between tip and sample surface is discussed in Chapter 3 (Eq. (3.3.6)), so the linearized tunneling current is expressed as:

where k is a constant term and i eq the tunneling current at operating point.

The output voltage v 2v of current sensor (CV C) is taken as:

where G cv is the gain of the current sensor. This output voltage (v 2v ) can be expressed as:

The measured output voltage of the vertical system is expressed as:

where n is the measurement noise. From Eq. (5.2.13), the overall vertical output voltage

where the vertical distance (d) is given by Eq. (5.2.11).

So, the overall dynamics of the vertical system is:

(5.2.15)

Global MIMO plant model

The complete MIMO plant model having two input voltages (u h and u v ) and two output voltages (v h and v v ) is:

(5.2.16) with state feedback gain matrix, K ∈ R m×n , can be applied to obtain desirable closed loop dynamics: ẋ = (A -BK)x

(5.4.3)

For LQR control, the following cost function is defined: 5.4.4) By substituting Eq. (5.4.2) into Eq. (5.4.4):

The objective of LQR control is to find a state feedback gain matrix (K), such that the cost function Eq. (5.4.5) is minimized. In Eq. (5.4.5), the matrix Q ∈ R n×n is a weighting matrix for the states and matrix R ∈ R m×m is a weighting matrix for the input signals. Q should be selected to be positive semi-definite and R to be positive definite. The optimal solution is classically given by: (5.4.6) with P obtained by solving:

This result is the Algebraic Riccati Equation (ARE).

Summarizing, the procedure to find the LQR state feedback gain matrix K is:

• Select the weighting matrices Q and R;

• Solve Eq. (5.4.7) to find P ;

• Compute K using Eq. (5.4.6).

With this, the optimal feedback u = -R -1 B T P x is obtained.

Appendix 1: Some experimental problems with tunneling current Here are some possible practical remedies to this:

• Environmental conditions play vital role while working at nanometer scale. External disturbances must be avoided as much as possible and the experiment should not be performed while there are movement of heavy vehicles in the surroundings of the experimental lab.

• The lateral position of the tip over the sample surface can be changed with the help of precision micrometer screws.

• The sample surface can be cleaned.

• If the problem as shown in Fig. 6.3 occurs, the tip and/or sample surface can be changed.