Luận án tiến sĩ: Design and Control of Cost-Effective Electronic Suspension Systems

ofOlSLA} SE]rtAdATA AA BA ALAELS] 412] 9t alo} Design and Control ofCost-Effective Electrostatic Suspension Systems SAL sa aS 7\ AAS A ##t†Truyen The Le... Optical levitation1.4 Squeeze

ofOlSLA} SE]rtAd

ATA AA BA ALAELS] 412] 9t alo}

Design and Control ofCost-Effective Electrostatic Suspension Systems

SAL sa aS

7\ AAS A ##t†Truyen The Le

Design and Control ofCost-Effective Electrostatic Suspension Systems

Aen AS

o] EES FHV} Bo] HOE AS

2011 xỉ 02 4

SAL] Sa a}7]2]2'§3} BSS

Truyen The Le

Truyen The Le 2]

^Atlld4+ Bata (21)AAA A ASF (21)

Design and Control ofCost-Effective Electrostatic Suspension Systems

By

Truyen The Le

Supervisor: Prof Jong Up Jeon

Department of Mechanical and Automotive Engineering

Graduate SchoolUniversity of Ulsan

February 2011

Design and Control ofCost-Effective Electrostatic Suspension Systems

By

Truyen The Le

Supervisor: Prof Jong Up Jeon

Submitted tothe Graduate School of the University of Ulsan

In partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

AtDepartment of Mechanical and Automotive Engineering,

University of Ulsan, Ulsan, Korea

February 2011

Design and Control ofCost-Effective Electrostatic Suspension Systems

This certifies that the dissertationof Truyen The Le is approved by

Committee Chairman Prof KYU YEOL PARK

Committee Member Prof JONG UP JEON

Committee Member Prof OCK TAECK LIM

Committee Member Prof YOUNG SOO SUH

Committee Member Dr KEE BONG CHOI

I would like to express heartfelt gratitude to my supervisor, Prof Jong Up Jeon for hisguidance, advice and support during my study in University of Ulsan I would like tothank Professors in the committee, Prof Kyu Yeol Park, Prof Ock Taeck Lim, Prof.Young Soo Suh and Dr Kee Bong Choi for their suggestion and comments throughoutthe research

I would like to express special gratitude to my wife and my children, Linh Giang & AnhQuan, whose patience and support have been invaluable during my study

I would like to express gratitude to my parents, sisters and brothers for theirencouragement and support

I would like to thank all members in MEMS Lab., University of Ulsan for theirfriendship

February, 2011

TRUYEN THE LE

CONTENTS1 INTRODUCTION

1.1 Aerodynamic levitation1.2 Acoustic levitation1.3 Optical levitation1.4 Squeeze film air bearing1.5 Magnetic levitation1.6 Electrostatic levitation1.7 Scope of the Work and Outline of the Thesis

1l2

4

55610

DYNAMIC MODEL OF ONE DEGREE OF FREEDOM ELECTROSTATICSUSPENSION SYSTEM

2.1 Electrostatic force2.2 Damping force2.3 Dynamic equation

THE COVENTIONAL ELECTROSTATIC SUSPENSION SYSTEM

3.1 Proportional-Integral-Derivative (PID) controller3.1.1 Linearization

3.1.2 PID controller3.2 Design of feedback control by using technique back-stepping3.3 Sub-conclusion |

12121417181919202330

DEVELOPMENT OF COST-EFFECTIVE ELECTROSTATIC SUSPENSION SYSTEMS

4.1 Principle of operation4.2 Simple on-off control4.3 Bang-bang time optimal control

4.3.1 State space representation4.3.2 Recoverable set

4.3.3 Switching criteria4.3.4 Stability analysis4.3.5 Simulation results4.3.6 Experiments

333436363841434647

4.4.1 Introduction 584.4.2 Design of delay controller and stability analysis 584.4.3 Simulation results 674.4.4 Experiments 724.4.5 Sub-conclusion 2 764.5 Variable structure controller 784.5.1 Design of variable structure controller 794.5.2 Simulation results 874.5.3 Discussion 904.5.4 Experiments 934.6 Sub-conclusion 3 97

- NOVEL ELECTROSTATIC SUSPENSION SYSTEM USING MOVABLE ELECTRODE

5.1 Introduction 995.2 Principle of operation 1015.3 Dynamic equation 1025.3.1 The dynamic equation of suspended object 1025.3.2 The dynamic modeling of the suspension system 1065.4 Linearization 1075.5 Control strategy 1095.5.1 Design of Genetic-PID controller 1105.5.2 Continuous proximate time optimal control 1145.6 Experimental works 1265.6.1 Design and manufacturing of mechanical amplifier 1265.6.2 Design of mechanical amplifier-piezo actuator-movable electrode 1305.6.3 Novel suspension system 1335.6.4 Design of voltage amplifier 1345.6.5 Determining of voltage supplied to electrodes 1375.6.6 Experimental results 14]5.6.6.1 Experimental results obtained by PID control 1425.6.6.2 Experimental results obtained by proximate time optimal control 143

FINAL CONCLUSIONREFERENCES

APENDIX 1APENDIX 2

145149152154

Symbols and abbreviations

Symbols

F electrostatic forceFa damping forceF Ủy damping force in case suspended object is silicon waferFa damping force in case suspended object is aluminum diskF linearized value of F”

ự electrode voltageV linearized value of V"

A area of electrodem mass of suspended object6 permittivity of air

n viscosity of airLa linearization constantky linearization constantZ0 suspended gapky damping coefficientk2 electric coefficientK time delay

Ker critical time delayŠ static displacement of PZTđãa strain coefficient

Vcc operating voltage of PZTVon charge supplied voltage

RS, _ recoverable setH gain factor of mechanical amplifier

ro radius of suspended objectP(r) _ pressure in a thin filmKp _ proportional gainKì integral gainKp _ derivative gain

Abbreviations

TOC Time optimal controlPZT Piezo actuatorGA Genetic AlgorithmHDD Hard disk driverPID _ Proportional-Integral-DerivativeDOF Degree of freedom

Chapter 1

INTRODUTION

In manufacturing processes for highly integrated semiconductor devices and liquid crystaldisplays (LCD), contamination of product surfaces by dust particles is a major factorrestricting the product yield Mechanical grips cause damage and scatter particulates that aredetrimental to the fabrication processes In addition, the direct contact and friction betweenthe object and other materials not only produce dust particles during fabrication processesbut also give the transported object a charge, which attracts dust particles from theworking environment Therefore, it is necessary to handle the sample without any physicalcontact with other materials

Especially, in recent years fabrication processes in vacuum environment becomeimportant More importantly these dry space processes remove significant sources ofcontaminants thus eliminating many manufacturing steps For instance, approximately25% of the process steps in semiconductor fabrication require vacuum process, manyMEMS devices such as inertial sensor, accelerometers and mechanical resonators need ahigh vacuum environment to improve their performance, and a packaging process in alow pressure environment improves the mechanical quality factor because it reduces airdamping significantly However, the handling of substrates having extreme small sizewhich are moved from process to process during fabrication cannot be accomplished withthe vacuum suction method which is used in ambient air Therefore, demands of handlingobject without mechanical contact in vacuum environment are also increasing

Levitation is the process by which an object is suspended against gravity, in a stableposition, without physical contact For levitation on Earth, first, a force is requireddirected vertically upwards and equal to the force of gravity Second, for any smalldisplacement of suspended object, a returning force should appear to stabilize 1t Manykind of levitation have been studied and applied to industrial such as aerodynamic,acoustic, optical, squeeze film air bearing, magnetic, and electrostatic levitation

1.1 AERODYNAMIC LEVITATION [1]

In aerodynamic levitation a spherical specimen is lifted by a fluid jet Stability in thevertical direction results from the divergence of the jet, which leads to a decreasing dragwith increasing height In the transverse direction this levitation is stable because the jetis deflected toward an off-axis specimen This asymmetry, or the increased Bernoulliforce at the side where the flow is faster, produces a centering force that leads to stablelevitation of a ball even by a tilted jet of water or air.

A further application of aerodynamic levitation is the contact-free positioning of nonconducting samples under the condition of microgravity 1n space

Nozzle

Fig 1.1 Aerodynamic Levitation [1]

1.2 ACOUSTIC LEVITATION [1, 5]Acoustic levitation uses sound traveling through a fluid, usually a gas, to balance theforce of gravity On Earth, this can cause objects and materials to hover unsupported inthe air A basic acoustic levitator has two main parts, a transducer which is a vibratingsurface that makes sound, and a reflector The transducer and reflector have concavesurfaces to help focus the sound A sound wave travels away from the transducer andbounces off the reflector

First, the wave, like all sound, is a longitudinal pressure wave In a longitudinal wave,movement of the in the waves is parallel to the direction the waves travels Second, thewave can bounce off of surfaces It follows the law of reflection, which states that theangle of incidence, the angle at which something strikes a surface, equals the angle ofreflection, the angle at which it leaves the surface In other words, a sound wave bouncesoff a surface at the same angle at which it hits the surface A sound wave that hits a

Finally, when a sound wave reflects off of a surface, the interaction between itscompressions and rarefaction causes interference Compressions that meet othercompressions amplify one another, and compressions that meet rarefactions balance oneanother out Sometimes, the reflection and interference can combines to create a standingwave Standing sound waves have defined nodes, or areas of minimum pressure, andantinodes, or areas of maximum pressure Imagine a river with rocks and rapids Thewater is calm in some part of the river, and it is turbulent in others Floating debris andfoam collect in calm portion of the river In order for a floating object to stay still in afast-moving part of the river, it would need to be anchored or propelled against the flowof the water This is essentially what an acoustic levitator does, using sound movingthrough a gas in place of water.

By placing a reflector the right distance away from transducer, the acoustic levitatorcreates a standing wave When the orientation of the wave is parallel to the pull ofgravity, portions of the standing wave have a constant downward pressure and other havea constant upward pressure The nodes have very little pressure

In space, where there is little gravity, floating particles collect in the standing wave’snodes, which are calm and still On Earth, objects collect just below the nodes, where theacoustic radiation pressure, or the amount of pressure that a sound wave, can exert on asurface, balance the pull of gravity

SampleObject

PRESSURE DISTRIBUTION

Transducer

Acoustic Reflector

e ye Levitatec

sample|

“fm Resonant

plate

ma Ba

Xx >2 transducer

Fig 1.3 Acoustic levitation [1]

In typical experiments, the high-intensity ultrasonic field is that of a standing wave of20 to 40 kHz generated by a piezoelectric transducer

1.3 OPTICAL LEVITATION [1]Optical levitation was developed in the early seventies by Athhur Ashkin and is theprecursor of the optical tweezers The governing physical processes are the same,springing from photon momentum transfer In optical levitation, the gravitational force isbalanced by photon pressure produced by a vertically directed focused laserbeam Closeto the focus the intensity 1s strong enough to slow down and hold tiny particles in a stabletrap Particles can then be manipulated and moved by moving the laserbeam or thesurrounding

Levitated sampleLens

Laserbearr

1.4 SQUEEZE FILM AIR BEARING [1]A squeeze film bearing 1s a special kind of air cushion which seems well suited tomicro machine application It simply operates by a rapid oscillation of one of twosurfaces with a gas between them.

Levitated

Gas film object

Z= Zc sinwt

Shaker plate

P Weight External force

Fig 1.5 Squeeze film gas levitation [1]

In the Fig 1.5, the shaker plate exhibits sine wave movement

1.5 MAGNETIC LEVITATION [1,3]Truly stable levitation without consumption of energy is possible only in magneticfields Several types of magnetic levitation have been successfully tested

I)2)

3)

4)5)

Levitation by strongly repulsive magnets must be combined with rollers forhorizontal guidance or with controlled electromagnets

Levitation with superconductors may be based on the pole diamagnetism 1n theMeissner state that allows stable levitation superconductor in magnetic fieldsgenerated

The levitation may rely on the repulsion between a superconducting magnet anda conducting plate or guide way which it moves The repulsion originates fromthe eddy currents in the plate that cause both lift and drag

Levitation may use the eddy currents generated in a conducting plate by an accoil

Suspension with controlled dc electromagnets has becomes feasible with theappearance of high-power electronics

Non-contact transportation systems using electromagnetic forces have already beenstudied However, these systems cannot be used to directly handle non-ferromagneticmaterial since the electromagnetic forces cannot be used to manipulate them.

As a result, non-ferromagnetic objects to be transported such as silicon wafers, glassplates and aluminum disk must be loaded on a levitated carriage, which leads to directcontact between the object and the carriage Furthermore, air and acoustic levitationtechniques have also been developed It is apparent that these techniques are notappropriate in a vacuum environment

1.6 ELECTROSTATIC LEVITATIONCompared to electromagnetic levitation, electrostatic suspension has _ severaladvantages The main advantage is that the electrostatic suspension can suspend variousmaterials such as conductive, semiconductor and dielectric materials, in contrast toelectromagnetic levitation which can levitate only ferromagnetic materials

In 1964, W Knoebel from University of Illinois reported his study about electronicvacuum gyro [2] The electric vacuum gyro is a two-axis free-rotor gyroscope havinglong term high precision for inertial navigation A spherical metal rotor shielded fromstray magnetic field is supported or levitated without physical contact in ultra-highvacuum by servo controlled high-voltage electric fields In this experiment, the rotor isaccurately centered between electrode pairs by servo controlled high voltage A 25 gramrotor is given an acceleration of 4 g’s by 3800 volts rms at an electrode gap of 0.010 in.The field intensity is 150,000 volts per cm at a field emission current of less than 0.5microamp

The electrostatic levitation for containerless processing of metals and alloys (2-4 mmdiameter spheres) 1n vacuum under high temperature has been reported by Rhim, 1993 Aschematic diagram of the present high-temperature / high-vacuum electrostatic levitator isshown in Fig 1 6 The electrode assembly is housed in a cylindrical vacuum chamberand all the necessary equipment for levitation, heating The 30 | stainless steel chamber

can be evaluated to an ultimate vacuum of 5x10 Torr The electrode assembly shown

schematically in Fig 1.7 is located as the center of the chamber, where | is the sample, 2

which allows access to the sample storage system Damping voltages applied on theseside electrodes prevent sample oscillation in the lateral direction Numerous sample have

been successfully levitated in this research such as In (157°C), Sn (232°C), Bi (271.44°C),

Pb (327°C), Al (670°C), Cu (1083°C), Ni (1455°C).

High voltageamplifiers

Fig 1.7 Schematic diagram of an electrode assembly [1]

Many electrostatic suspension systems where the ratio between surface of suspendedobject and levitated gap is small have been reported in recent year A 3.5 inch diameteraluminum hard disk media has been suspended successfully by actively controlling

and of dielectric materials have been reported by Jeon et al [6-9] Especially, theprinciple of electrostatic force generation for dielectrics is different from that forconductors and semiconductors In Jeon’s study for electrostatic suspension of dielectricsmaterials, glass plate have been suspended electrostatically at a gap length of about 0.3mm The special feature of this pattern electrode is that it possesses many structuralboundaries over which potential differences exits Since the electric field in the vicinitiesof the boundaries is strong, a fast charge induction or fast polarization process is obtainedin these regions As a result, the suspension system characteristics, such as the suspensioninitiation time and the dynamic stability, are improved This study also show that the airhumidity has a strong influence on the resistivity of the glass plate and, hence, on thesuspension system characteristic.

The application of electrostatic levitation technology has not had the success of itselectromagnetic counterpart since the maximum attainable electrostatic force itself isweak in ambient air The reason why the electrostatic force is not strong is the magnitudeof supplied voltages to the stator electrode is limited by the Paschen’s law for breakdownvoltage of parallel plates (Fig 1.8) It is known that in the atmospheric environment, thegenerally accepted value of the breakdown field strength is 3 kV/mm For this field

strength, the available maximum electrostatic force intensity is approximately 4 mN/cm?

in the atmospheric environment However, the electrostatic forces are greatly improved invacuum environment because the discharge intensity is must higher in high vacuum

range For example, in an environment of pressure 10° torr, it has been reported that a

field intensity as high as 15 kV/mm can be achieved in a field emission current less than0.5 pA Therefore, in high vacuum environments, greater electrostatic force can beobtained It means that heavy objects can be levitated by the electrostatic suspensionsystems which are operated in vacuum environment

Since the electrostatic force per unit area is small, voltages with several hundred voltsmust be applied to the electrodes in suspending suspended objects The conventionalelectrostatic suspension systems utilize high-voltage amplifiers in a PID (proportional-integral-derivative) based feedback control scheme to generate these high voltages [6-12]However, a major disadvantage of these systems is that the high-voltage amplifiers are

potential industrial application In addition, the required number of DC high-voltageamplifiers is proportional to the number of electrodes to be controlled This would resultin extreme system costs in the case of distributed electrodes pattern used for thesuspension of large flexible objects.

In order to develop a low cost electrostatic suspension systems where suspendedobject whit a large surface area/ thickness ratio are used, several bang bang control typesare studied and reported [13, 14] This method has a property that 1t does not deploy anyhigh-voltage amplifiers Instead only single high-voltage power supplies are required thatcan deliver a constant voltage of positive or negative polarity for an arbitrary number ofindividual stator electrodes that have to be controlled Therefore, the cost of the system isreduced significantly

104 > =

|-© _

od)

&s 3

> 10

N:;

327107 l | l | |

0 1 10 10 100 1000

pd (cm Torr)

Fig 1.8 The Paschen curve for dry air, nitrogen, and hydrogen

However, electrostatic suspension systems based on relay feedback control asdescribed in the literature [13] require the presence of large air damping forces tostabilize the object’s motions since object control 1s just based on deviations from thereference positions It means that this system cannot be used 1n a vacuum environmentbecause damping forces do not exist in vacuums An electrostatic suspension systemwhich is based on a time optimal control scheme has also been proposed for use in

of the system were not considered Moreover, suspension experiments in a vacuumenvironment were not carried out in that study.

1.7 SCOPE OF THE WORK AND OUTLINE OF THE THESISThe scope of this thesis is two-fold and will focus on (1) the development of cost-effective suspension systems (which are have been studied in [13] and [14]) where low-priced high voltage supply instead of the very high costly high-voltage amplifiers areemployed, and (i1) a design a newly highly-precise suspension system with low cost

The dynamic model of electrostatic suspension system is introduced in Chapter 2 Thisis used to study different control strategies for suspension system The modeling of anelectrostatic suspension system is based on the nonlinear damping force and electrostaticforce generated between electrode and suspended object

The Chapter 3 reviews the conventional electrostatic suspension systems which havebeen studied in previous research

The Chapter 4 presents a development in this research of a cost-effective electrostaticsuspension system The main focus in this chapter is on studying the effect of time delayas introduce in Section 4.4, and the effect of damping force as introduced in Section 4.5on the performance of the suspension system

In Chapter 5 of this thesis, the new highly precise electrostatic suspension system ispresented A novel aspect of this system is that a movable electrode is used instead of astationary electrode in the previous suspension systems By using a rapid deformation ofthe piezoelectric (PZT) actuator, the movement of the movable electrodes supplied byconstant voltage is controlled As a consequence, the electrostatic forces are controlled byvarying the capacitance formed by the movable electrodes and the suspended object.The following contributions made by the thesis can be highlighted:

1) For cost-effective electrostatic suspension system- Recoverable set for suspension system

- Critical time delay which is defined maximum time delay that the system can beacceptable to ensure stability

- Sufficient conditions for system stability

- An analysis of the effect of time delay on system stability and performance ofsuspension system

- Design and implementation of delay controller for the cost effective electrostaticsuspension system

- Design of variable structure controller adapted to nonlinear characteristic dynamic ofsuspension system to improve the performance of suspension system operated inenvironment with high damping force including an analysis of effect of dampingforce

- Provide knowledge to decide whose suspension system for application in vacuum orin atmosphere suspension to have a good performance

i1) For the novel suspension system- Propose a new method to contacless suspension by using electrostatic force.- Design a novel suspension system using piezo actuator together with:

- Design a mechanical amplifier- Design a movable electrode- Design a piezo actuator-mechanical amplifier-movable electrode unit- Modeling of the novel suspension system

- Different feedback controls for the novel suspension system

Chapter 2DYNAMIC MODEL OF ONE DEGREE OF FREEDOOM ELECTROSTATIC

Suspendedobject

Fig 2.1 One degree of freedom of electrostatic suspension system

The external forces acting on the suspended object are attractive electrostatic force F šdamping force Fy, and gravity force In figure 2.1, V’ is electrode voltage.

2.1 ELECTROSTATIC FORCELet us assume that the electrical resistance of the suspended object is very small so

stator electrode and the suspended object is very small and therefore can be neglected Itis also assumed that a uniform electric field is formed between the stator electrode andthe suspended object This assumption is appropriate because the ratio of the air gaplength between the stator electrode and the object to the overlapping area between them isvery small in our suspension system Under the assumptions above, the electrostatic force

LOẠI ° +

V 1s not a function of z ), we have

oF] _ led# 7 3

27aOZ |

— “0

ý >0

where Zo and Vo are displacement at equilibrium point and bias voltage, respectively.This implies that the motion of the silicon wafer will exhibit an unstable behavior without

any active control of the force Fš

2.2 DAMPING FORCEThe electrostatic suspension system consists of two parallel plates, stator electrodeand suspended object, as shown in Fig 2.3 The resistive force to the suspended objectnormally against the stator electrode is caused by the damping pressure between the twoplates Since the gap length is significantly smaller than the area of the object, the air inthe gap can be modeled as a squeeze film when the object moves vertically The behaviorof squeeze film is in general governed by both viscous and the inertial effect within thefluid However, for the very small geometries, inertial effect is often negligible In thatcase, the behavior of the fluid is governed by the well-known Reynolds equation Thedamping pressure consists of two main components: the component to cause the viscousflow of air when the air is squeezed out of (or sucked into) the plate region and that tocause the compression of air film [25] Assuming that the compressibility of air can beneglected and considering a one-dimensional case, the Reynolds equation for squeezefilm air damping in disk plate case can be written in a polar coordinate system as

where Øø{7) is pressure distribution in a thin film between electrode and object, 7 = 18 x

10° N.s/mỸ is the viscosity of air.

The right term of above equation is not depended ofr, therefore, it can be written as

T2 =k (2.3)

ror\ or

inwhich &=- 7“

Upon integrating on Eq (2.3), we obtained as

Air <—~>

WEEE EEEEEEEEY ——

v MovementSuspended

object Damping force F.

VY VYv V |

V alr

Fig 2.4 Pressure distribution for a circular shaped object

If the suspended object is aluminum disk has a circular shaped with hole as shown inFig 2.5, the boundary conditions are as

Then, the pressure damping in that case is

Using F ” and Fy (where Fy presents for both Fagy and Fat ), the dynamic equation

of motion of the suspended object is described as follows:- In case suspended object is Silicon wafer

Chapter 3

THE CONVENTINAL ELECTROSTATIC SUSPENSION SYSTEM

This section presents a brief review on control strategy of conventional electrostaticSuspension systems which are used high voltage amplifier to generate the control voltage.This method is applied in many previous studies such as [6-12].The schematic of thesystem is presented in Fig 3.1 In that case the suspended object is assumed as 4 inchsilicon wafer

High voltage | PID ° Reference

Fig 3.1 Schematic of conventional suspension system using high voltage amplifier

The high voltage amplifier is used as a means of amplifying the low output voltagesignals of the feedback controller to the suspension voltage of positive and negative

3.1 PROPORTIONAL-INTEGRAL-DERIVATIVE CONTROLLER (PID)3.1.1 Linearization

From here we use #¿ to present the damping force This symbol is used for common the

damping force in case suspended object is 4 inch silicon wafer and also for 3.5aluminium disk

Linearization technique using Taylor series is applied to equations (2.1) and (2.7) toobtain a linearized equation of motion of suspended object at equilibrium position Let Zo,Vo, Fa and Fo be the values at the equilibrium state and z, V, Fy, F be the linearizedvalues Then, the variables z, V, and #' can be written:

The linearized equations are obtained as:

F° =F, -k,z+kV (3.2)Fi=F,,+F, (3.3)

where k,, ky are linearization constants and Fo is the equilibrium force which are givenby:

From equations (3.1), (3.2), (3.3) we obtained the linearized equation of motion asfollows:

V(s) ms’ +k,s—k, G10)

It easily to conclude the system possesses an unstable open-loop dynamiccharacteristic The proportional integral derivative (PID) feedback control law isproposed to stabilize the system as follows

Fig 3.2 The block diagram of closed loop control system

The block diagram of closed loop control system using PID controller is shown in Fig.3.2 The coefficient Kp, Kp, and K; are the proportional, derivative, and integral gains,respectively By replacing (3.11) into (3.10), the closed-loop system can be obtained as

2(s) _ #„ s(K,p+Kps+K,/s) (3.12)r(s) om Ề E + K pk, J (Keke ~ k, } a

m m m

The characteristic equation of closed-loop system is given by

The stability of the system can be investigated by applying the Routh’s stabilitycriterion to the characteristic equation The Routh’s array can be obtained as

1 KpR, _ k,

Mmkh, + Kpk, Kk,

Based on equations (3.16) and (3.17) the gains Kp, Kp, and K; of the controller can beobtained

Figure 3.3 presents the simulation result of suspension of 4 inch silicon wafer using

PID controller The mass of 4 inch silicon wafer is about 14.7x10° kg The proportional,

derivative, and integral gains are 6.5x10°, 2x10°, and 2x10°, respectively The silicon

wafer is suspended at gap of 300 um from initial gap of 350 um

Fig 3.3 Simulation result of suspension of 4-inch silicon wafer by using PID control

3.2 DESIGN OF FEEDBACK CONTROL BY USING TECHNIQUE STEPPING

BACK-The nonlinear state space of electrostatic suspension system of one degree of freedomcan be written as follows

2

- z Vv" (3.18)

Z,=2k, xH|

-In equation (3.18), &¡ and Az are damping and electric coefficient where they arecalculated by (for 4 inch silicon wafer)

4

_ = N2 -= (3.19)m m

transformation to translate the equilibrium point (z,,0)' to the origin.

Let’s define a new variable as

Z,=2Z1-ZoZ,=z2 (3.21)

U=V”—W}

In the new coordinates the model takes the form

2 =Z,

m (3.22): : (Z, + zo) (Z, +2) (Z, + zy)

The new model 1s 1n the form

Ự =#Œ)+g(Z)øÐ=#,ŒZ.p)+g„(2.ø)U

Adding and subtracting ø(Z)®(Z)to the system (3.27) we obtain the equivalent system

A change of variable gives

Defining the augmented Lyapunov function for new system such as

We arrive at the control law

OVv =-—9(Z)-kx,k>0 3.362 g(Z) (3.36)

With &> 0 that gives

|

V(Z)= s2 (3.40)

We have

V =Z,Z, (3.41)

Thus, the control law as