Micro, Nanosystems and Systems on Chips E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 397

Veröffentlichungsjahr: 2013

Beschreibung

Micro- and nanosystems represent an area of major scientific and technological opportunity and challenge, with actual and potential applications in almost all fields of human activity. The aim of this book is to present the central concepts of dynamic control systems (modeling, estimation, observation, identification, feedback control) and to show how they can be adapted and applied to the development of novel very small-scale systems and their associated human interfaces. The application fields presented here come from micro- and nano-robotics, biochips, near-field microscopy (AFM and STM) and nano-systems networks. Alina Voda has assembled contributions from leading experts at top research universities to produce the first overview of the major role that control systems science will play in the development of micro and nano-science and technologies.

Leseprobe

Contents

Introduction

PART I. MINIAND MICROSYSTEMS

Chapter 1. Modeling and Control of Stick-slip Micropositioning Devices

1.1. Introduction

1.2. General description of stick-slip micropositioning devices

1.3. Model of the sub-step mode

1.4. PI control of the sub-step mode

1.5.Modeling the coarsemode

1.6. Voltage/frequency (U/f) proportional control of the coarse mode

1.7. Conclusion

1.8.Bibliography

Chapter 2. Microbeam Dynamic Shaping by Closed-loop Electrostatic Actuation using Modal Control

2.1. Introduction

2.2. Systemdescription

2.3. Modal analysis

2.4. Mode-based control

2.5. Conclusion

2.6.Bibliography

PART II. NANOSYSTEMSAND NANOWORLD

Chapter 3. Observer-based Estimation of Weak Forces in a Nanosystem Measurement Device

3.1. Introduction

3.2. Observer approach in an AFM measurement set-up

3.3. Extension to back action evasion

3.4. Conclusion

3.5. Acknowledgements

3.6.Bibliography

Chapter 4. Tunnel Current for a Robust, High-bandwidth and Ultraprecise Nanopositioning

4.1. Introduction

4.2. Systemdescription

4.3. Systemmodeling

4.4. Problemstatement

4.5. Tools to deal with noise

4.6. Closed-loop requirements

4.7.Control strategy

4.8.Results

4.9. Conclusion

4.10.Bibliography

Chapter 5. Controller Design and Analysis for High-performance STM

5.1. Introduction

5.2. General description of STM

5.3. Control design model

5.4. H∞ controller design

5.5. Analysis with system parametric uncertainties

5.6. Simulation results

5.7. Conclusions

5.8.Bibliography

Chapter 6. Modeling, Identification and Control of a Micro-cantilever Array

6.1. Introduction

6.2. Modeling and identification of a cantilever array

6.3. Semi-decentralized approximation of optimal control applied to a cantilever array

6.4. Simulation of large-scale periodic circuits by a homogenization method

6.5.Bibliography

6.6. Appendix

Chapter 7. Fractional Order Modeling and Identification for Electrochemical Nano-biochip

7.1. Introduction

7.2. Mathematical background

7.3. Prediction error algorithm for fractional order system identification

7.4. Fractional order modeling of electrochemical processes

7.5. Identification of a real electrochemical biochip

7.6. Conclusion

7.7.Bibliography

PART III. FROM NANOWORLDTO MACROAND HUMAN INTERFACES

Chapter 8. Human-in-the-loop Telemicromanipulation System Assisted by Multisensory Feedback

8.1. Introduction

8.2. Haptic-based multimodal telemicromanipulation system

8.3. 3D visual perception using virtual reality

8.4. Haptic rendering for intuitive and efficient interaction with the micro-environment

8.5. Evaluating manipulation tasks through multimodal feedback and assistance metaphors

8.6. Conclusion

8.7.Bibliography

Chapter 9. Six-dof Teleoperation Platform: Application to Flexible Molecular Docking

9.1. Introduction

9.2. Proposed approach

9.3. Force-position control scheme

9.4. Control scheme for high dynamical and delayed systems

9.5. From energy description of a force field to force feeling

9.6. Conclusion

9.7.Bibliography

List of Authors

Index

To Anaïs and Raphaël

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

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© ISTE Ltd 2010

The rights of Alina Voda to be identified as the author of this work have been asserted by her in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Micro, nanosystems, and systems on chips : modeling, control, and estimation / edited by Alina Voda.

  p.; cm.

 Includes bibliographical references and index.

 ISBN 978-1-84821-190-2

1. Microelectromechanical systems. 2. Systems on a chip. I. Voda, Alina.

 TK7875.M532487 2010

 621.381--dc22

2009041386

British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-190-2

Introduction

Micro and nanosystems represent a major scientific and technological challenge, with actual and potential applications in almost all fields of human activity. From the first physics and philosophical concepts of atoms, developed by classical Greek and Roman thinkers such as Democritus, Epicurus and Lucretins some centuries BC at the dawn of the scientific era, to the famous Nobel Prize Feynman conference 50 years ago (“There is plenty of room at the bottom”), phenomena at atomic scale have incessantly attracted the human spirit. However, to produce, touch, manipulate and create such atomistic-based systems has only been possible during the last 50 years as the appropriate technologies became available.

Books on micro-and nanosystems have already been written and continue to appear. They focus on the physics, chemical, technological and biological concepts, problems and applications. The dynamical modeling, estimation and feedback control are not classically addressed in the literature on miniaturization. However, these are innovative and efficient approaches to explore and improve; new small-scale systems could even be created.

The instruments for measuring and manipulating individual systems at molecular and atomic scale cannot be imagined without incorporating very precise estimation and feedback control concepts. On the other hand, to make such a dream feasible, control system methods have to adapt to unusual systems governed by different physics than the macroscopic systems. Phenomena which are usually neglected, such as thermal noise, become an important source of disturbances for nanosystems. Dust particles can represent obstacles when dealing with molecular positioning. The influence of the measuring process on the measured variable, referred to as back action, cannot be ignored if the measured signal is of the same order of magnitude as the measuring device noise.

This book is addressed to researchers, engineers and students interested in the domain of miniaturized systems and dynamical systems and information treatment at this scale. The aim of this book is to present how concepts from dynamical control systems (modeling, estimation, observation, identification and feedback control) can be adapted and applied to the development of original very small-scale systems and to their human interfaces.

All the contributions have a model-based approach in common. The model is a set of dynamical system equations which, depending on its intended purpose, is either based on physics principles or is a black-box identified model or an energy (or potential field) based model. The model is then used for the design of the feedback control law, for estimation purposes (parameter identification or observer design) or for human interface design.

The applications presented in this book range from micro-and nanorobotics and biochips to near-field microscopy (Atomic Force and Scanning Tunneling Microscopes), nanosystems arrays, biochip cells and also human interfaces.

The book has three parts. The first part is dedicated to mini-and microsystems, with two applications of feedback control in micropositioning devices and microbeam dynamic shaping.

The second part is dedicated to nanoscale systems or phenomena. The fundamental instrument which we are concerned with is the microscope, which is either used to analyze or explore surfaces or to measure forces at an atomic scale. The core of the microscope is a cantilever with a sharp tip, in close proximity to the sample under analysis. Several chapters of the book treat different aspects related to the microscopy: force measurement at nanoscale is recast as an observer design, fast and precise nano-positioning is reached by feedback control design and cantilever arrays can be modeled and controlled using a non-standard approach. Another domain of interest is the field of biochips. A chapter is dedicated to the identification of a non-integer order model applied to such an electrochemical transduction/detection cell.

The third part of the book treats aspects of the interactions between the human and nanoworlds through haptic interfaces, telemanipulation and virtual reality.

Alina Voda

Grenoble

January 2010

PART I

Mini and Microsystems

Chapter 1

Modeling and Control of Stick-slip Micropositioning Devices1

The principle of stick-slip motion is highly appreciated in the design of micropositioning devices. Indeed, this principle offers both a very high resolution and a high range of displacement for the devices. In fact, stick-slip motion is a step-bystep motion and two modes can therefore be used: the stepping mode (for coarse positioning) and the sub-step mode (for fine positioning). In this chapter, we present the modeling and control of micropositioning devices based on stick-slip motion principle. For each mode (sub-step and stepping), we describe the model and propose a control law in order to improve the performance of the devices. Experimental results validate and confirm the results in the theoretical section.

1.1. Introduction

In microassembly and micromanipulation tasks, i.e. assembly or manipulation of objects with submillimetric sizes, the manipulators should achieve a micrometric or submicrometric accuracy. To reach such a performance, the design of microrobots and micromanipulators is radically different from the design of classical robots. Instead of using hinges that may introduce imprecision, active materials are preferred. Piezoelectric materials are highly prized because of the high resolution and the short response time they can offer.

In addition to the high accuracy, a large range of motion is also important in microassembly/micromanipulation tasks. Indeed, the pick-and-place of small objects may require the transportation of the latter over a long distance. To execute tasks with high accuracy and over a high range of displacement, micropositioning devices and microrobots use embedded (micro)actuators. According to the type of microactuators used, there are different motion principles that can be used e.g. the stick-slip motion principle, the impact drive motion principle and the inch-worm motion principle. Each of these principles provides a step-by-step motion. The micropositioning device analyzed and experimented upon in this chapter is based on the stick-slip motion principle and uses piezoelectric microactuators.

Stick-slip micropositioning devices can work with two modes of motion: the coarse mode which is for long-distance positioning and the sub-step mode which is for fine positioning. This chapter presents the modeling and the control of the micropositioning device for both fine and coarse modes.

First we describe the micropositioning device. The modeling and control in fine mode are then analyzed. We then present the modeling in coarse mode, and end the chapter by describing control of the device in coarse mode.

1.2. General description of stick-slip micropositioning devices

1.2.1. Principle

Figure 1.1a explains the functioning of the stick-slip motion principle. In the figure, two microactuators are embedded onto a body to be moved. The two microactuators are made of a smart material. Here, we consider piezoelectric microactuators.

If we apply a ramp voltage to the microactuators, they slowly bend. As the bending acceleration is low, there is an adherence between the tips of the microactuators and the base (Figure 1.1b). If we reset the voltage, the bending of the legs is also abruptly halted. Because of the high acceleration, sliding occurs between their tips and the base. A displacement Δx of the body is therefore obtained (Figure 1.1c). Repeating the sequence using a sawtooth voltage signal makes the body perform a step-by-step motion. The corresponding motion principle is called stick-slip. The amplitude of a step is defined by the sawtooth voltage amplitude and the speed of the body is defined by both the amplitude and the frequency. The step value indicates the positioning resolution.

While the step-by-step motion corresponds to the coarse mode, it is also possible to work in sub-step mode. In this case, the rate of the applied voltage is limited so that the legs never slide (Figure 1.1d). In many cases, this mode is used when the error between the reference position and the present position of the device is less than one step. This mode is called fine mode.

Figure 1.1.Stick-slip principle: (a–c) stepping mode and (d) scanning mode

1.2.2. Experimental device

The positioning device experimented upon in this paper, referred to as triangular RING (TRING) module, is depicted in Figure 1.2. It can perform a linear and an angular motion on the base (a glass tube) independently. Without loss of generality, our experiments are carried out only in linear motion. To move the TRING-module, six piezoelectric microactuators are embedded. Details of the design and development of the TRING-module are given in [RAK 06, RAK 09] while the piezoelectric microactuators are described in [BER 03].

(1.1)

Figure 1.2.A photograph of the TRING-module

As introduced above, two modes of displacement are possible: the fine and the coarse modes. In the next sections, the fine mode of the TRING-module is first modeled and controlled. After that, we will detail the modeling and the control in coarse mode, all with linear motion.

1.3. Model of the sub-step mode

The sub-step modeling of a stick-slip micropositioning device is highly dependent upon the structure ofmicroactuators. This in turn depends upon the required number of degrees of freedom and their kinematics, the structure of the device where they will be integrated and the structure of the base. For example, [FAT 95] and [BER 04] use two kinds of stick-slip microactuators to move the MICRON micropositioning device (5-dof) and the MINIMAN micropositioning device (3-dof). Despite this dependence of the model on the microactuator’s structure, as long as the piezoelectric microactuator is operating linearly, the sub-step model is still linear [RAK 09].

During the modeling of the sub-step mode, it is of interest to include the state of the friction between the microactuators and the base. For example, it is possible to control it to be lower than a certain value to ensure the stick mode. There are several models of friction according to the application [ARM 94], but the elastoplastic model [DUP 02] is best adapted to the sub-step modeling. The model of the sub-step mode is therefore linear and has an order at least equal to the order of the microactuator model.

1.3.1. Assumptions

During the modeling, the adhesion forces between the foot of the microactuators and the base are assumed to be insignificant relative to the preload charge. The

preload charge is the vertical force that maintains the device on the base. The base is considered to be rigid and we assume that no vibration affects it because we work in the stick mode. Indeed, during this mode, the tip of the microactuator and the base are fixed and shocks do not cause vibration.

To model the TRING micropositioning device, a physical approach has been applied [RAK 09]. While physical models of stick-slip devices strongly depend upon their structure and characteristics and on their microactuators, the structure of these models does not vary significantly. Assuming the piezoelectric microactuators work in the linear domain, the final model is linear. The order of the model is equal to the microactuator’s model order added to the model order of the friction state. The substep modeling can be separated into two stages: the modeling of the microactuator (electromechanical part) and the inclusion of the friction model (mechanical part).

1.3.2. Microactuator equation

The different microactuators and the positioning device can be lumped into one microactuator supporting a body (Figure 1.4).

Figure 1.4.Schematic of the microactuator

If the microactuator works in a linear domain, a second-order lumped model is:

(1.2)

where δ is the deflection of the microactuator, ai are the parameters of the dynamic parts, dp is the piezoelectric coefficient, sp is the elastic coefficient and Fpiezo is the external force applied to the microactuator. It may be derived from external disturbance (manipulation force, etc.) or internal stresses between the base and the microactuator.

1.3.3. The elastoplastic friction model

The elastoplastic friction model was proposed by Dupont et al. [DUP 02] and is well adapted for stick-slip micropositioning devices. Consider a block that moves along a base (Figure 1.5a). If the force F applied to the block is lower than a certain value, the block does not move. This corresponds to a stick phase. If we increase the force, the block starts sliding and the slip phase is obtained.

Figure 1.5.(a) A block that moves along a base and (b) the contact between the block and the base can be approximated by a medium asperity

w. While ≠ 0, the deflection xasp continues to vary. This phase is elastic because of xasp but also plastic because of w.

If F is increased further, xasp tends to a saturation called (steady state) and the speed of the block is equal to ≠ 0. This phase is called plastic because removing the force will not reset the block to its initial position.

The equations describing the elastoplastic model are:

(1.3)

where N designates the normal force applied to the block, ρ0 and ρ2 are the Coulomb and the viscous parameters of the friction, respectively, ρ1 provides damping for tangential compliance and α (xasp) is a function which determines the phase (stick or slip). Figure 1.6 provides an example of allure of α.

Figure 1.6.An example of allure of α

(1.4)

1.3.4. The state equation

To compute the model of the stick-slip micropositioning device in a stick mode, the deformation of the microactuator (equation (1.2)) and the friction model (equation (1.4)) are used. Figure 1.7 represents the same image as Figure 1.4 with the contact between the tip of the microactuator and the base enlarged. According to the figure, the displacement xsub can be determined by combining the microactuator equation d and the friction state xasp using dynamic laws [RAK 09].

Figure 1.7.An example of allure of α

The state equation of the TRING-module is therefore:

(1.5)

where the state vector is composed of:

– the states of the electromechanical part: the deflection δ of the piezoelectric microactuator and the corresponding derivative ; and

– the states of the friction part: the deflection of a medium asperity xasp and the corresponding derivative .

The following values have been identified and validated for the considered system [RAK 09]:

(1.6)

and

(1.7)

1.3.5. The output equation

The output equation is defined as

(1.8)

where T is the friction and xsub is the displacement of the mass m during the stick mode. xsub corresponds to the fine position of the TRING device. The different parameters are defined:

(1.9)

1.3.6. Experimental and simulation curves

In the considered application, we are interested in the control of the position. We therefore only consider the output xsub. From the previous state and output equations, we derive the transfer function relating the applied voltage and xsub:

(1.10)

where s is the Laplace variable.

To compare the computed model GxsubU and the real system, a harmonic analysis is performed by applying a sine input voltage to the TRING-module. The chosen amplitude of the sine voltage is 75V instead of 150V. Indeed, with a high amplitude the minimum frequency from which the drift (and then the sliding mode) starts is low. In the example of Figure 1.8, a frequency of 2250 Hz leads to a drift when the amplitude is 150V while a frequency of 5000 Hz does not when amplitude is 75 V. The higher the amplitude, the higher the acceleration is and the higher the risk of sliding (drift). When the TRING-module slides, the sub-step model is no longer valuable.

Figure 1.8.Harmonic experiment: (a) outbreak of a drift of the TRING positioning system (sliding mode) and (b) stick mode

Figure 1.9 depicts the magnitude of the simulation (equation (1.10)) and the experimental result. It shows that the structure of the model and the identified parameters correspond well.

Figure 1.9.Comparison of the simulation of the developed model and the experimental results

1.4. PI control of the sub-step mode

The aim of the sub-step control is to improve the performance of the TRING-module during a highly accurate task and to eliminate disturbances (e.g. manipulation force, adhesion forces and environmental disturbances such as temperature). Indeed, when positioning a microcomponent such as fixing a microlens at the tip of an optical fiber [GAR 00], the manipulation force can disturb the positioning task and modify its accuracy. In addition, the numerical values of the model parameters may contain uncertainty. We therefore present here the closed-loop control of the fine mode to introduce high stability margins.

The sub-step functioning requires that the derivative dU/dt of the voltage should be inferior to a maximum slope . To ensure this, we introduce a rate limiter in the controller scheme as depicted in Figure 1.10.

Figure 1.10.Structure of the closed-loop system To

To ensure a null static error, we choose a proportional-integral (PI) controller. The parameters of the controller are computed to ensure a phase margin of 60°, required for stability in residual phase uncertainty.

First, we trace the Black-Nichols diagram of the open-loop system GxsubU, as depicted in Figure 1.11.

Figure 1.11.Black-Nichols diagram ofGxsubU

Let

(1.11)

Figure 1.12.Results of the PI control of the TRING-module in sub-step functioning

Figure 1.12b shows the Black-Nichols diagram of the closed-loop system and indicates the margin phase. According to the figure, the margin gain is 50 dB. These robustness margins are sufficient to ensure the stability of the closed-loop system regarding the uncertainty of the parameters and of the structure of the developed model. Finally, the closed-loop control ensures these performances when external disturbances occur during the micromanipulation/microassembly tasks. A disturbance may be of an environmental type (e.g. temperature variation) or a manipulation type (e.g. manipulation force).

1.5. Modeling the coarse mode

When scanning over a large distance (e.g. pick-and-place tasks in microassembly), the micropositioning device should work in coarse mode. The applied voltage is no longer limited in slope as for the fine mode, but has a sawtooth form. The resulting displacement is a succession of steps. This section, which follows that of [BOU 06], discusses the modeling and control of the coarse mode. The presented results are applicable to stepping systems.

1.5.1. The model

First, let us study one step. For that, we first apply a ramp input voltage up to U. If the slope of the ramp is weak, there is no sliding between the tip of the microactuators and the base. Using the model in the stick mode, the displacement of the device is defined:

(1.12)

To obtain a step, the voltage is quickly reduced to zero. The resulting step xstep is smaller than the amplitude xsub that corresponds to the last value of U (Figure 1.13a). We denote this amplitude . We then have:

(1.13)

Figure 1.13.(a) Motion of a stick-slip system and (b) speed approximation

If we assume that backlash Δback is dynamically linear relative to the amplitude U, the step can be written as:

(1.14)

(1.15)

where α > 0 is the static gain of Gstep.

From Figure 1.13b and equation (1.15), we easily deduce the speed:

(1.16)

The speed is therefore bilinear in relation to the amplitude U and the frequency f of the sawtooth input voltage:

(1.17)

However, the experiments show that there is a deadzone in the amplitude inside which the speed is null. Indeed, if the amplitude U is below a certain value U0, the micropositioning system does not move in the stepping mode but only moves back and forth in the stick mode. To take into account this threshold, equation (1.15) is slightly modified and the final model becomes:

(1.18)

1.5.2. Experimental results

1.5.3. Remarks

(1.19)

Figure 1.14.Speed performances of the micropositioning system (experimental results in solid lines and simulation of equation (1.18) in dashed lines): (a) speed versus the amplitude U and (b) speed versus the frequency f

Comparing equation (1.19) and the second equation of equation (1.18), we demonstrate the pseudo-linearity of the backlash in relation to U:

(1.20)

1.6. Voltage/frequency (U/f) proportional control of the coarse mode

The micropositioning device working in coarse mode is a two-inputs-one-output system. The input variables are the frequency and the amplitude of the sawtooth voltage while the output is the displacement.

A stick-slip device is a type of stepping motor, and so stepping motor control techniques may be used. The easiest control of stepping motors is the open-loop counter technique. This consists of applying the number of steps necessary to reach a final position. In this, no sensor is necessary but the step value should be exactly known. In stick-slip micropositioning devices, such a technique is not very convenient. In fact, the friction varies along a displacement and the step is not very predictible. Closed-loop controllers are therefore preferred.

In closed-loop techniques, a natural control principle is the following basic algorithm:

(1.21)

where xc and x are the reference and the present positions of the stick-slip devices, respectively, and step is the value of one step. The resolution of the closed-loop system is equal to 1 step. If the accuracy of the sensor is lower than 1 step, a slight modification can be made:

(1.22)

It is clear that for very precise positioning, the basic algorithm must be combined with a sub-step controller (such as the PI controller presented in the previous section). In that case, equation (1.21) is first activated during the coarse mode. When the error position xc − x is lower than the value of a step, the controller is switched into the sub-step mode.

In order to avoid the use of two triggered controllers for coarse mode and fine mode, Breguet and Clavel [BRE 98] propose a numerical controller where the frequency f of the sawtooth voltage is proportional to the error. In this, the position error is converted into a clock signal with frequency equal to that of the error. When the error becomes lower than a step, the frequency tends towards zero and the applied voltage is equivalent to that applied in the fine mode. Since the amplitude U is constant, the step is also constant and the positioning resolution is constant all along the displacement.

A technique based on the theory of dynamic hybrid systems has been used in [SED 03]. The mixture of the fine mode and the coarse mode actually constitutes a dynamic hybrid system. In the proposed technique, the hybrid system is first approximated by a continuous model by inserting a cascade with a hybrid controller. The approximation is called dehybridization. A PI-controller is then applied to the obtained continuous system.

In the following section, we propose a new controller scheme. In contrast to the dehybridization-based controller, the proposed scheme is very easy to implement because it does not require a hybrid controller. The proposed scheme always ensures the stability. The resolution that it provides is better than that of the basic algorithm. It will be shown that the controller is a globalization of three existing controllers: the bang-bang controller, the proportional controller and the frequency-proportional controller cited above.

1.6.1. Principle scheme of the proposed controller

The principle scheme of the controller is depicted in Figure 1.15. Basically, the principle is that the input signals (the amplitude and the frequency) are proportional to the error. This is why the proposed scheme is referred to as voltage/frequency (or U/f) proportional control. In Figure 1.15, the amplitude saturation limits any over-voltages that may destroy the piezoelectric microactuators. The frequency saturation limits the micropositioning system work inside the linear frequential zone. The controller parameters are the proportional gains KU > 0 and Kf > 0.

Figure 1.15.Principle scheme of the U/f proportional control

1.6.2. Analysis

Because of the presence of saturation in the controller scheme (Figure 1.15), different situations can occur [RAK 08] dependent upon the frequency and/or the amplitude being in the saturation zones. In this section, we analyze these situations.

Let Us and fs be the saturations used for the voltage and the frequency, respectively.

1.6.2.1. Case a

In the first case, we assume that both the amplitude and the frequency are saturated, i.e.

(1.23)

This can be intepreted in two ways: the present position of the device is different from the reference position or the chosen proportional gains are very high. The equation of the closed-loop system in this case is obtained using the principle scheme in Figure 1.15 and equation (1.18). We have:

(1.24)

In such a case, the amplitude U is switched between Us and −Us according to the sign of the error (Figure 1.16a). This case is therefore equivalent to a sign or bangbang controller. With a sign control, there are oscillations. The frequency and the amplitude of these oscillations depend on the response time Tr of the process, on the refreshing time Ts of the controller and on the frequency saturation fs (Figure 1.16b). To minimize the oscillations, the use of realtime feedback systems is recommended.

Figure 1.16.Sign controller equivalence of the U/f controller

1.6.2.2. Case b

If the amplitude U is lower than the threshold U0 regardless of frequency, i.e. if

(1.25)

1.6.2.3. Case c

In this case, the frequency is saturated while the amplitude is not. The condition corresponding to this case is:

(1.26)

In such a case, the system is controlled by a classical proportional controller with gain KU (Figure 1.17).

Figure 1.17.Voltage proportional control

The equation of the closed loop is easily obtained:

(1.27)

(1.28)

According to equation (1.28), the closed-loop process is a first-order dynamic system with a static gain equal to unity and a disturbance U0. The static error due to the disturbance U0 is minimized when increasing the gain KU. Because the order is equal to that of the closed-loop system, this case is always stable.

1.6.2.4. Case d

Here we consider that the amplitude is saturated while the frequency is not, i.e.

(1.29)

In such a case, the frequency of the sawtooth voltage is proportional to the error. The controller is therefore a frequency proportional controller (Figure 1.18). The

Figure 1.18.Frequency proportional control

difference between this case and the controller proposed in [BRE 98] is that, in the latter, the controller is digital and based on an 8-bit counter.

Using Figure 1.18 and model (1.18), we have the non-linear differential model:

(1.30)

(1.31)

According to equation (1.31), the closed-loop process is a first-order system. Because the static gain is unity, there is no error static.

1.6.2.5. Case e

In this case, we consider that both the amplitude and the frequency are not saturated:

(1.32)

Using Figure 1.15 and equation (1.18), we have:

(1.33)

The previous expression is equivalent to:

(1.34)

Hence, the closed-loop system is equivalent to a first-order pseudo-linear system. Indeed, equation (1.34) has the form:

(1.35)

1.6.3. Stability analysis

– Phase 1: concerns the amplitude and the frequency in saturation. This corresponds to the error (xc − x) being initially high (case a). The speed is then constant.

– Phase 2: the error becomes smaller and the speed is not yet constant (equivalent to the rest of the cases).

Figure 1.9.Division of the displacement into two phases

According to equation (1.18), the device works in a quasi-static manner. Hence, there is no acceleration and any one case does not influence the succeeding case. Conditions relative to initial speed are not necessary so we can analyze phase 2 independently of phase 1. In phase 2, there are two sub-phases:

– Phase 2.1: either the frequency is in saturation but not the amplitude (case c) or the amplitude is in saturation but not the frequency (case d).

– Phase 2.2: neither the frequency nor the amplitude are in saturation (case e).

Because phase 2.1 is stable and does not influence phase 2.2, we can analyze the stability using the latter. For that, equation (1.34) is used. Applying the conditions

(1.36)

(1.37)

(1.38)

Phase 2.2 (which corresponds to case e) is therefore asymptotically stable. When the error still decreases and the condition becomes (KUx − U0) < 0, case b occurs and the device stops. The static error is therefore given by KUx.

1.6.4. Experiments

According to the previous analysis, three existing controllers are merged to form the U/f proportional controller. These are the sign controller (case a), the classical proportional controller (case c) and the frequency proportional controller proposed in [BRE 98] (case d).

As for the classical proportional controller, the choice of KU is a compromise. A low value of KU leads to a high static error (case b) while a high value of KU may generate oscillations (case a).

The first experiment concerns high values of Ku and Kf . They have been chosen such that phase 2 never occurs and only case a occurs. The controller was implemented using Labview software and the Windows-XP operating system. The refreshing time is not relatively high so oscillations appear in the experimental results (Figure 1.20).

Figure 1.20.High values of KU and Kf: case a

1.7. Conclusion

In this chapter, the modeling and control of a stick-slip micropositioning device, developed at Franche-Comté Electronique Mécanique Thermique et Optique — Science

Figure 1.21.(a) Low KU and high Kf and (b) high KU and low Kf

et Technologie (FEMTO-ST) Institute in the AS2M department, has been discussed. Based on the use of piezoelectric actuators, this device can be operated either in coarse mode or in sub-step mode.

In the sub-step mode, the legs never slide and the obtained accuracy is 5 nm. This mode is suitable when the difference between the reference position and the current position is less than 1 step.

The coarse mode allows step-by-step displacements; long-range displacements can therefore be achieved. The voltage/frequency (U/F) proportional control presented in this chapter is easy to implement and demonstrates a good performance. The stability

Figure 1.22.Acceptable values of KU and Kf

of the controller has been proven. The performances of the coarse mode are given by the hardware performances. Combining the sub-step mode and the coarse mode is a solution for performing high-stroke/high-precision positioning tasks. The coarse mode will be used to drive the device close to the reference position and the sub-step mode will provide additional displacement details required to reach the reference. However, this approach requires the use of a long-range/high-accuracy position sensor, which is not easy to integrate. This will be an area of future research.

1.8. Bibliography

[ARM 94] ARMSTRONG-HÉLOUVRY B., DUPONT P., CANUDAS-DE-WIT C., “A survey of models, analysis tools and compensation methods for the control of machines with friction”, IFAC Automatica, vol. 30, num. 7, p. 1083–1138, 1994.

[BER 03] BERGANDER A., DRIESEN W., VARIDEL T., BREGUET J. M., “Monolithic piezoelectric push-pull actuators for inertial drives”, IEEE International Symposium on Micromechatronics and Human Science, p. 309–316, 2003.

[BER 04] BERGANDER A., DRIESEN W., VARIDEL T., MEIZOSO M., BREGUET J., “Mobile cm3-microrobots with tools for nanoscale imaging and micromanipulation”, Proceedings of IEEE International Symposium on Micromechatronics and Human Science (MHS), Nagoya, Japan, p. 309–316, 2004.

[BOU 06] BOURLÈS H., Systèmes Linéaires: de la Modélisation à la Commande, Hermès–Lavoisier, 2006.

[BRE 98] BREGUET J., CLAVEL R., “Stick and slip actuators: design, control, performances and applications”, IEEE International Symposium on Micromechatronics and Human Science, p. 89–;95, 1998.

[DRI 03] DRIESEN W., BERGANDER A., VARIDEL T., BREGUET J., “Energy consumption of piezoelectric actuators for inertial drives”, IEEE International Symposium on Micromechatronics and Human Science, p. 51–58, 2003.

[DUP 02] DUPONT P., HAYWARD V., ARMSTRONG B., ALTPETER F., “Single state elastoplastic friction models”, IEEE Transactions on Automatic Control, vol. 47, num. 5, p. 787–792, 2002.

[FAT 95] FATIKOW S., MAGNUSSEN B., REMBOLD U., “A piezoelectric mobile robot for handling of micro-objects”, Proceedings of the International Symposium on Microsystems, Intelligent Materials and Robots, p. 189–192, 1995.

[GAR 00] GARTNER C., BLUMEL V., KRAPLIN A., POSSNER T., “Micro-assembly processes for beam transformation systems of high-power laser diode bars”, MST news I, p. 23–24, 2000.

[RAK 06] RAKOTONDRABE M., Design, development and modular control of a microassembly station, PhD thesis, University of Franche-Comté, 2006.

[RAK 08] RAKOTONDRABE M., HADDAB Y., LUTZ P., “Voltage/frequency proportional control of stick-slip micropositioning systems”, IEEE Transactions on Control Systems Technology, vol. 16, num. 6, p. 1316–1322, 2008.

[RAK 09] RAKOTONDRABE M., HADDAB Y., LUTZ P., “Development, sub-step modelling and control of a micro/nano positioning 2DoF stickslip device”, IEEE/ASME Transactions on Mechatronics, 2009, DOI 10.1109/TMECH.2009.2011134.

[SED 03] SEDGHI B., Control design of hybrid systems via dehybridization, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland, 2003.

1 Chapter written by Micky RAKOTONDRABE, Yassine HADDAB and Philippe LUTZ.

Chapter 2

Microbeam Dynamic Shaping by Closed-loop Electrostatic Actuation using Modal Control1

A contribution to flexible microstructure control is developed in this chapter using large arrays of nanotransducers. The distributed transduction scheme consists of two sets of N electrodes located on each side of the microstructure for electrostatic driving and capacitive detection. Since accurate point-to-point control requires a large number of controllers, modal control is proposed to limit integration complexity. This is carried out by projecting the measured displacements on the n (<< N) modes to be controlled before calculating the stresses that must be distributed throughout the beam. Although simple PID control can be used, fabrication tolerances, parameter variations and model simplifications require robust specifications ensured by sophisticated control laws. An example of the combination of the Loop Transfer Recovery (LTR) method with the Full State Feedback (FSF) control extended standard models is presented, showing high robust stability and performances.

2.1. Introduction

Smart materials and intelligent structures are a new rapidly growing technology embracing the fields of sensor and actuator systems, information processing and control. They are capable of sensing and reacting to their environment in a predictable and desired manner and are used to carry mechanical loads, alleviate vibration, reduce acoustic noise, monitor their own condition and environment, automatically perform precision alignments or change their shape or mechanical properties on command. While active structural control may be described as seeking a distributed control actuation such that a desired spatial distribution of the structure displacement is reached, in dynamic shape control the desired shape has to be additionally prescribed as a function of time.

In astronomical sciences, adaptive optical elements such as deformable mirrors used to correct for atmospheric aberrations provide a good example of shape control structures [LIA 97, ROO 02]. As the high cost of piezoelectric actuated mirrors prevented the broader adoption of this technology, deformable mirrors based on microelectromechanical systems (MEMS) have recently emerged [KEN 07]. These micromirrors are less costly and have enabled many new applications in bio-imaging including retina imaging, optical coherence tomography and wide-field microscopy. They were first used by [DRE 89] for a membrane mirror with 13 actuators in a scanning laser ophthalmoscope. More recently, adaptive aberration correction was investigated using membrane mirrors having 37 actuators [FER 03], while retinal images were obtained by a 140-actuator micromachined mirror [DOB 02].