Nonlinear Control and Design Methodologies for
Electrostatic MEMS Devices
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
Phillip M. Ozmun
Certificate of Approval:
Ramesh Ramadoss
Assistant Professor
Electrical and Computer Engineering
John Y. Hung, Chair
Professor
Electrical and Computer Engineering
Robert N. Dean
Assistant Professor
Electrical and Computer Engineering
Joe F. Pittman
Interim Dean
Graduate School
Nonlinear Control and Design Methodologies for
Electrostatic MEMS Devices
Phillip M. Ozmun
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
August 04, 2007
Nonlinear Control and Design Methodologies for
Electrostatic MEMS Devices
Phillip M. Ozmun
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Phillip was born on December 4, 1979 in South-Eastern Idaho to Michael and Joan
Ozmun. After graduating from Idaho Falls High School, he continued his education at the
University of Idaho. While earning his B.S. degree in Mechanical Engineering he enjoyed
socializing with friends and playing on the university?s club baseball team. After graduation
he worked for Havlovick Engineering Services in Mobile, Alabama. After the projects he
worked on in Mobile were completed, he pursued graduate work at Auburn University.
iv
Thesis Abstract
Nonlinear Control and Design Methodologies for
Electrostatic MEMS Devices
Phillip M. Ozmun
Master of Science, August 04, 2007
(B.S.M.E, University of Idaho, 2003)
62 Typed Pages
Directed by John Y. Hung
A method to extend the travel range of electrostatically actuated MEMS is presented.
A gap closing actuator (GCA) is used to demonstrate the method. An output (position)
feedback controller is presented, along with two variable structure controllers. The forced-
damping variable structure controller uses two stable structures, and the sliding mode (hy-
brid) controller uses an unstable structure and a stable structure. An adaptive controller
is also presented for devices that have adequate natural damping. A design methodol-
ogy for nonlinear mechanical springs is presented. The mechanical nonlinearity offsets the
electrostatic nonlinearity to extend the device travel range without feedback.
v
Acknowledgments
First and foremost, I must thank my parents for all their encouragement. Without
their support, it is unlikely I would have ever received my bachelor?s degree in Mechanical
Engineering at the University of Idaho.
Next, I?d like to thank Bradley Havlovick and Jim Richardson at Havlovick Engineering
Services. Their expertise, guidance, and patience helped me develop as an engineer, and
the experience I gained working with them will stay with me the rest of my career.
I must also thank Dr. John Y. Hung, my adviser, without whose direction and assis-
tance, I could have never completed my degree. His control courses were some of the most
informative and stimulating classes I have taken.
Working with Dr. Ramesh Ramadoss was also an integral part of my Auburn experi-
ence. His guidance and knowledge in the area of MEMS was instrumental in the formulation
of the method developed in this thesis.
vi
Style manual or journal used Bibliography conforms to those in the IEEE Transactions.
Computer software used The document preparation package TEX (specifically LATEX)
together with the departmental style-file aums.sty.
vii
Table of Contents
List of Figures x
1 Introduction 1
1.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Chapter 2: Variable Structure Technique . . . . . . . . . . . . . . . 2
1.1.3 Chapter 3: Adaptive Controller . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 Chapter 4: Device Function Design . . . . . . . . . . . . . . . . . . . 3
1.1.5 Chapter 5: Extension to Other Systems . . . . . . . . . . . . . . . . 3
1.2 Nonlinear Electrostatic MEMS Actuators . . . . . . . . . . . . . . . . . . . 3
1.3 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Previous Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Input and Device Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Static Analysis: Equilibrium and Stability . . . . . . . . . . . . . . . . . . . 9
1.7.1 Open Loop Considerations . . . . . . . . . . . . . . . . . . . . . . . 10
1.7.2 Closed Loop Considerations . . . . . . . . . . . . . . . . . . . . . . . 11
1.8 Dynamic Analysis: Equilibrium and Stability . . . . . . . . . . . . . . . . . 12
1.8.1 Static Force Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8.2 Force Potential Barriers . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 Design of a Linear Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.9.1 Satisfying Device-Input Static Requirements . . . . . . . . . . . . . 15
1.9.2 Positive Region Operation . . . . . . . . . . . . . . . . . . . . . . . . 16
1.9.3 Negative Region Operation . . . . . . . . . . . . . . . . . . . . . . . 17
1.9.4 ??s Effect on Localized Stiffness . . . . . . . . . . . . . . . . . . . . . 19
2 Variable Structure Technique 21
2.1 Available Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 GCA Variable Structure Controllers . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Stable-Stable (Forced Damping) Switching . . . . . . . . . . . . . . 24
2.2.2 Stable-Unstable (Sliding Mode) Switching . . . . . . . . . . . . . . . 26
2.2.3 Switching Surface Considerations . . . . . . . . . . . . . . . . . . . . 28
3 Adaptive Controller 29
3.1 Robustness Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
viii
4 Device Function Design 34
4.1 Nonlinear Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Extension to Other Systems 40
5.1 Series Capacitor Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Torsional Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6 Conclusion and Discussion 46
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2 Output vs State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.3 Multi-Dimensional Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.4 Final Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Bibliography 50
ix
List of Figures
1.1 Gap Closing Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Device Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Device and Input Functions in Open Loop . . . . . . . . . . . . . . . . . . . 11
1.4 Device and Input Functions with Closed Loop . . . . . . . . . . . . . . . . . 12
1.5 Net Force Function for Fig 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Energy Barriers for Dynamic Stability . . . . . . . . . . . . . . . . . . . . . 14
1.7 Output Position for Linear Controller, Positive Region . . . . . . . . . . . . 16
1.8 Input Voltage for Linear Controller, Positive Region . . . . . . . . . . . . . 17
1.9 Output Position for Linear Controller, Negative Region . . . . . . . . . . . 18
1.10 Input Voltage for Linear Controller, Negative Region . . . . . . . . . . . . . 18
1.11 Device-Input Function for Various Chi Values . . . . . . . . . . . . . . . . . 19
1.12 Net Force Functions for Various Chi Values . . . . . . . . . . . . . . . . . . 20
2.1 Low Stiffness Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 High Stiffness Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Mixed System Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Constant Voltage, Zero Damping Phase Portrait . . . . . . . . . . . . . . . 23
2.5 Stable-Stable Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Stable-Stable: Position vs Time . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Stable-Stable: Control Voltage vs Time . . . . . . . . . . . . . . . . . . . . 25
2.8 Stable-Unstable Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . . . 26
x
2.9 Stable-Unstable: Postion vs Time . . . . . . . . . . . . . . . . . . . . . . . . 27
2.10 Stable-Unstable: Control Voltage vs Time . . . . . . . . . . . . . . . . . . . 27
3.1 Adaptive Controller Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Adaptive Controller: Position vs Time . . . . . . . . . . . . . . . . . . . . . 31
3.3 Adaptive Controller: Control Voltage vs Time . . . . . . . . . . . . . . . . . 31
3.4 Robustness Study: Position vs Time . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Robustness Study: Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 System Response with Nonlinear Spring . . . . . . . . . . . . . . . . . . . . 36
4.2 Beam Element for Nonlinear Spring Design . . . . . . . . . . . . . . . . . . 37
4.3 Interference and Deflection Profiles . . . . . . . . . . . . . . . . . . . . . . . 38
4.4 Discrete Nonlinear Spring Device Function . . . . . . . . . . . . . . . . . . . 39
5.1 Series Capacitor Device-Input Plot . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Series Capacitor Net Force Plot . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Torsional Device-Input Plot: Open Loop . . . . . . . . . . . . . . . . . . . . 44
5.4 Net Torque Plot: Open Loop . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.5 Torsional Device-Input Plot: Closed Loop . . . . . . . . . . . . . . . . . . . 45
5.6 Net Torque Plot: Closed Loop . . . . . . . . . . . . . . . . . . . . . . . . . 45
xi
Chapter 1
Introduction
Micro-Electro Mechanical Systems or MEMS devices combine sensors, actuators, me-
chanical structures, electronics, and optics on a single substrate and combine technological
advances from fields that were previously unrelated, such as biology and microelectronics
[1]. MEMS have the advantage of being smaller, having a higher Q, low insertion loss,
and lower power consumption compared to traditional construction, among several other
advantages [2].
MEMS actuators can be driven a multitude of ways. Thermal, piezoelectric, and elec-
trostatic are the most common. In electrostatic actuation there are two classes, linear
actuators and nonlinear actuators. In linear actuators the electrostatic force is only depen-
dent on the drive voltage. In nonlinear devices, the electrostatic force varies nonlinearly with
respect to position. These nonlinear devices are the focus of this thesis. For a typical device
the actuation range in open loop operation is limited to a fraction of the potential range
due to a phenomenon called ?pull-in?, where the nonlinear electrostatic force overwhelms
the mechanical spring force. Although research has been performed to extend this travel
range by eliminating pull-in there are still physical limitations [3]. Nonlinear electrostatic
MEMS devices can be used in the communication industry to replace traditional switches,
varactors, phase shifters, filters, and resonators. They also have potential in many optical
applications and are currently being used in high definition televisions.
1
1.1 Chapter Overview
1.1.1 Chapter 1: Introduction
The first sections in chapter 1 cover the governing equations of nonlinear electrostatic
MEMS devices, introduce the device-input formulation for such systems, and review some
of the previous research done to improve the performance of such devices.
The last sections in chapter 1 apply the device-input formulation to a gap closing ac-
tuator (GCA). Static stability (force balance) is used to determine the slope-intersection
requirements of the device-input functions to ensure stability. Input function design con-
siderations are covered and a simple, linear controller is proposed. Lastly, an analysis is
presented that illustrates how modifying the linear controller gains can modify the local
stiffness of the system.
1.1.2 Chapter 2: Variable Structure Technique
Chapter 2 covers two variable structure techniques applied to the GCA using the lin-
ear controller from chapter 1. The first technique is based on switching between two stable
structures with varying stiffness. The second technique is based on stable-unstable switch-
ing, and forms a sliding mode under certain circumstances. The two variable structure
controllers are shown to improve the dynamic performance of the GCA system, even in the
absence of natural damping.
1.1.3 Chapter 3: Adaptive Controller
Chapter 3 introduces an adaptive technique based on the GCA system and linear
controller from chapter 1. A proportional-integral (PI) method is used to adjust the slope
2
gain of the linear controller. This chapter concludes with a robustness analysis of the
controllers presented in chapters 2 and 3.
1.1.4 Chapter 4: Device Function Design
The first 3 chapters were concerned with modifying the input function to improve
device performance. Chapter 4 looks at how the device function can be modified to im-
prove system performance. The first section explores design considerations for closed loop
operation. Next, a nonlinear mechanical force equation is determined that allows for full
range operation without feedback. The nonlinearity is introduced using an interference
profile that limits beam deflection. A beam analysis method is proposed for determining
the interference profile required.
1.1.5 Chapter 5: Extension to Other Systems
Chapter 5 shows the utility of the method by showing how it can be applied to other
nonlinear MEMS. The method is used to demonstrate how the series capacitor method can
be used to extend the travel range of the GCA at the cost of higher actuation voltage. Next
the method is applied to a torsional device, with similar conclusions to that of the GCA
example.
1.2 Nonlinear Electrostatic MEMS Actuators
Electrostatic MEMS actuators can be divided into two types, linear and nonlinear.
For both cases the electrostatic force is determined by the energy-displacement relationship
associated with the device capacitance:
3
Fe(x) = 12V 2 ddxC(x) (1.1)
For linear devices like comb-drive actuators the capacitance varies linearly with dis-
placement so that the electrostatic force is independent of displacement. For nonlinear
devices the capacitance is inversely proportional to displacement so that the electrostatic
force contains inverse-squared terms.
1.3 Governing Equations
A generalized form of the governing equation for all electrostatic MEMS devices is
given.
m?x+b(x, ?x)?x+k(x, ?x)x = f0(x)V(x, ?x)2 (1.2)
Where x is the dependent variable (position or rotation), m is a constant inertia term,
b is the damping term, k is the spring term, f0 is an input term, and V is the input voltage.
In systems with multiple degrees of freedom, equations in the form of (1.2) typically apply
to each DOF.
In the bulk of this thesis, the idea of force balance or static equilibrium is employed
regularly. Equation (1.2) can be expressed under steady state equilibrium as the intersection
of two curves.
fe(x, ?x) def= V(x, ?x)2 (1.3)
4
fd(x, ?x) def= k(x, ?x)xf
0(x)
(1.4)
fe(x, ?x) = fd(x, ?x) (1.5)
Where fe is defined as the feedback or input function of the system. In open loop it
is the constant value of the input voltage squared, V 2. The second function, fd, can be
defined as the device function, and is determined by device geometry and material. By
manipulating these two functions, devices and controllers can be designed to perform as
required.
1.4 Previous Research
As MEMS technology has developed many control and design methodologies have de-
veloped to improve performance. Feedback control was first introduced in MEMS sensors
in order to enhance measurement accuracy. In some cases designers modified the open-loop
control signals using dynamic models. These pre-shaped control methods improved the
dynamic behavior without the need of feedback but are highly tailored to each device, have
limited application range, and require accurate models [4].
Various geometrical methods have also been researched. These fall mostly into two
categories; multiple electrode arrangements [5] and leveraged bending methods [6]. While
these methods have their benefits, complications can arise due to complex control and
switching requirements for a large number of electrodes, and high actuation voltage using
leveraged bending. Closely related to leveraged bending is the use of strain-stiffening springs
5
to increase travel range. This effect introduces a mechanical nonlinearity to compensate
for the electrical nonlinearity that causes pull-in. Current methods rely on residual axial
stresses in the mechanical springs and are highly problematic.
A wide variety of feedback techniques have been proposed. Some of the more popular
methods include capacitor feedback, current/charge control [7] (also an open loop method),
and voltage controllers [4]. Force balance methods have also been used in designing feedback
algorithms in micro-mirrors [8]. Traditional nonlinear techniques have also been explored
[9]. Force balance techniques were also the basis for the method discussed here. Original
developments concerned bi-directional operation of a GCA fabricated at Auburn University
[10].
In order to familiarize the reader with the control methodology being used, the tech-
nique will be demonstrated on a gap closing actuator (GCA) comprised of interlocked
comb-like structures. Figure 1.1 shows the device layout along with sign conventions.
As a voltage is applied across the moving and stationary combs an electrostatic force
moves the actuator in the positive x direction. A balancing mechanical spring force acts in
the opposite direction if the actuator is in static equilibrium.
1.5 Governing Equation
The dynamic model of the GCA system is given by:
m?x =?B?x?Kx+
epsilon1oepsilon1rAV 2
2
parenleftbigg 1
(xo ?x)2 ?
1
(yo +x)2
parenrightbigg
(1.6)
6
Figure 1.1: Gap Closing Actuator
Where m is the proof mass, B is the damping constant, epsilon1o is the free-space permittivity,
epsilon1r is the relative permittivity of air, A is the actuator area, V is the applied voltage, xo is
the nominal positive gap distance, yo is the nominal negative gap distance, and K is the
system spring constant. Values of model parameters are shown in Table 1.1.
1.6 Input and Device Functions
By rewriting the force equation under static equilibrium, the following is observed:
V 2(x) = 2Kxepsilon1
oepsilon1rA
parenleftbigg 1
(xo ?x)2 ?
1
(yo +x)2
parenrightbigg?1
(1.7)
fe(x) = fd(x)
7
Table 1.1: Nomenclature and parameter values
Variable Parameter Value Units
m mass 3.6519?10?7 kg
B damping 2.45?10?4 N-sec/m
epsilon1o free space permittivity 8.854?10?12 F/m
epsilon1r relative permittivity 1 NA
A area 5.85?106 ?m2
xo nominal small gap 10 ?m
yo nominal large gap 25 ?m
K spring constant 103.125 N/m
The left hand side of (1.7) is the input function, and is referred to as fe(x). The right
hand side of (1.7) is the device function, and is referred to as fd(x). The intersection(s)
of these two functions are the locations of the equilibrium points occurring in the travel
region. Figure 1.2 shows the device function for the given device. The input function is
arbitrary and is not shown; however for a constant voltage, it would be a horizontal line.
?2.5 ?2 ?1.5 ?1 ?0.5 0 0.5 10
100
200
300
400
500
600
Device Function
x/x0
V2
fd
Figure 1.2: Device Function
8
1.7 Static Analysis: Equilibrium and Stability
Stability of equilibrium points can be determined by examining properties of fe(x)
and fd(x). By looking at the net force vs. displacement graph, the equilibrium points are
located where the net force is equal to zero (Fnet = 0). The stable equilibrium points are
equilibrium points that have negative slopes ( ddxFnet < 0). This implies that a perturbation
in one direction yields a net force in the opposite direction to push the actuator back toward
the equilibrium point. Requirements for the static stability of the system can be expressed
in terms of fe(x) and fd(x).
Fnet = epsilon1oepsilon1rA2
parenleftbigg 1
(xo ?x)2 ?
1
(yo +x)2
parenrightbigg
V 2 ?Kx (1.8)
Fnet = epsilon1oepsilon1rA2
parenleftbigg 1
(xo ?x)2 ?
1
(yo +x)2
parenrightbigg
(fe ?fd) (1.9)
Isolating (fe?fd) to one side of the equation and taking the derivative with respect to
x:
d
dx(fe ?fd) =
dFnet
dx
2
epsilon1oepsilon1rA
parenleftbigg 1
(xo ?x)2 ?
1
(yo +x)2
parenrightbigg?1
+
Fnet 2epsilon1
oepsilon1rA
d
dx
parenleftbigg 1
(xo ?x)2 ?
1
(yo +x)2
parenrightbigg?1
(1.10)
9
Since the net force is zero at an equilibrium point and ddxFnet < 0, equation (1.10) can
be simplified to the following.
d
dx(fe ?fd)
parenleftbigg 1
(xo ?x)2 ?
1
(yo +x)2
parenrightbigg
< 0 (1.11)
The sign of the augmenting function depends on the value of x, but can easily be
determined, yielding the following result:
For
x < x
2o ?y2o
2(yo +xo) ?
d
dx(fe ?fd) > 0
For
x > x
2o ?y2o
2(yo +xo) ?
d
dx(fe ?fd) < 0
So for equilibrium points in the positive region of motion to be stable, dfddx > dfedx , for
equilibrium points in the negative region to be stable, dfedx > dfddx .
1.7.1 Open Loop Considerations
During open loop operation the input function is constant with respect to displacement.
Because dfedx = 0, the only stable equilibrium points exist in the positive operation range
where the slope of the device function is greater than zero. Figure 1.3 shows the positions
of the equilibrium points and how they move with increasing voltage input.
When a voltage is applied, two equilibrium points are formed in the positive range of
motion. As the voltage increases, these stable and unstable equilibrium points gravitate
10
?2.5 ?2 ?1.5 ?1 ?0.5 0 0.5 10
100
200
300
400
500
600
Open Loop Device and Input Functions
x/x0
V2
fd
fe10
fe20
fe25
Stable Equilibria
Unstable Equilibria
Figure 1.3: Device and Input Functions in Open Loop
toward each other and combine into an unstable equilibrium point when dfddx = 0, resulting
in pull-in.
1.7.2 Closed Loop Considerations
One method of extending the travel range of the device is to employ feedback to the
system by forcing the voltage to be a function of x. Figure 1.4 shows the stable and unstable
equilibrium points for the device with an arbitrary periodic input function.
Regardless of the input function, the equilibrium voltage is always on the device func-
tion curve. The equilibrium points are where the input function intersects the device func-
tion, and the slopes of the two functions at those points determine whether the points are
stable or unstable. In theory, the device range is the entire gap distance in the positive di-
rection. In the negative direction no equilibrium points can exist where the device function
11
?2.5 ?2 ?1.5 ?1 ?0.5 0 0.5 10
100
200
300
400
500
600
Arbitrary Closed Loop Device and Input Functions
x/x0
V2
fd
fe
Stable Equilibria
Unstable Equilibria
Figure 1.4: Device and Input Functions with Closed Loop
is negative (0 > x > x2o?y2o2(yo+xo)). Otherwise travel is possible throughout the negative region
with proper feedback.
1.8 Dynamic Analysis: Equilibrium and Stability
In general, by knowing the device function one can determine the required input func-
tion to establish a stable equilibrium point in the regions described (anywhere in the positive
direction, in a limited range in the negative direction). However, it is helpful to understand
the dynamics of the system in order to design an adequate input function.
1.8.1 Static Force Plots
Static equilibrium has been the heart of the analysis, so examining the net force plots
is insightful. Figure 1.5 shows the net force plot for the feedback function in Figure 1.4.
Only the positive region is shown for clarity.
12
0 0.2 0.4 0.6 0.8 1?3
?2
?1
0
1
2
3x 10?3 Net Force Plot for Fig 2.4
x/x0
Force (N)
Fnet
Stable Equilibria
Unstable Equilibria
Positive Force Moves Actuator
Negative Force Moves Actuator
Figure 1.5: Net Force Function for Fig 2.4
Figure 1.5 gives a rough picture of the dynamic stability of the system. Near the
stable equilibrium points exist regions of positive and negative force that drive the system
back toward the equilibrium point. In these regions the system can oscillate about the
equilibrium point as long as there is insufficient energy to reach an adjacent (unstable)
equilibrium point. Dynamic stability is limited by these unstable equilibria.
1.8.2 Force Potential Barriers
The area bounded under the force curve is a measure of the electro-mechanical potential
energy in the system, similar to the spring potential energy in a mechanical system. Ne-
glecting any damping, all this potential energy is converted to kinetic energy of the moving
mass. Figure 1.6 is given for an example.
Initially the actuator is at rest at a stable equilibrium point with potential barriers
Er and El to the right and left of the equilibrium point respectively. If the actuator is
13
0 0.2 0.4 0.6 0.8 1?3
?2
?1
0
1
2
3x 10?3Energy Barriers for Dynamic Stability
x/x0
Force (N)
Fnet
Stable EquilibriaUnstable Equilibria
Initial PositionE
r
El
Figure 1.6: Energy Barriers for Dynamic Stability
perturbed with an energy less than either barrier, it will oscillate about the equilibrium
point as the energy is converted back and forth between the kinetic energy of the mass
and the potential energy of the field. However, if the actuator is perturbed with an energy
greater than either barrier, it will leave the region of the current equilibrium point.
In designing an input function the position of the unstable equilibrium points and the
potential barriers need to be taken into account to avoid problems associated with the
dynamic stability of the system.
1.9 Design of a Linear Controller
A linear controller can be designed to meet the desired criteria. The general form of
the feedback function is:
14
V(x) = ?(x??) (1.12)
Where ? is the slope and ? is the x-intercept.
1.9.1 Satisfying Device-Input Static Requirements
In order to establish the desired equilibrium point, the device and input function are
equated at the desired point.
(?(xe ??))2 = 2Kxeepsilon1
oepsilon1rA
parenleftbigg 1
(xo ?xe)2 ?
1
(yo +xe)2
parenrightbigg?1
(1.13)
Solving for ?:
? =?
radicalBigg
2Kxe
epsilon1oepsilon1rA(xe ??)2
parenleftbigg 1
(xo ?xe)2 ?
1
(yo +xe)2
parenrightbigg
(1.14)
Where the sign determines whether negative or positive voltage is used. In order to
satisfy the slope condition for positive region operation xe < ? ? xo. For the negative
region ? needs to be less than xe by very little. The slope of the device function is relatively
large in the negative region, so selecting the intercept must be done carefully.
15
1.9.2 Positive Region Operation
Figure 1.7 and Figure 1.8 show the position output and voltage input for positive
operation with ? = xo. The input voltage was also limited to be in the range 0?V?26V
to ensure the system has only one (stable) equilibrium point. Large overshoots due to poor
damping require either small steps or a ramp input in order to prevent the device from
colliding with the stationary combs when x = xo. Interestingly, the linear controller is
mathematically equivalent to charge controllers like that proposed in [11] when ? = xo and
yo ??.
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1x 10?5 Position vs Time
time (s)
Position (m)
Figure 1.7: Output Position for Linear Controller, Positive Region
The choice of the x-intercept, ?, is somewhat arbitrary. However, in choosing ?, the
following facts need to be considered; 1) an intercept closer to the equilibrium point yields
larger potential barriers, 2) an intercept closer to xo yields lower local stiffness.
16
0 0.05 0.1 0.15 0.20
5
10
15
20
25
30
time (s)
Input (V)
Input Voltage vs Time
Figure 1.8: Input Voltage for Linear Controller, Positive Region
1.9.3 Negative Region Operation
Figure 1.9 and Figure 1.10 show the position output and voltage input for negative
operation with ? =?yo + 23(yo +xe). Again, large overshoots due to poor damping require
either small steps or a ramp input to prevent the device from colliding with the stationary
combs x =?yo.
Similar considerations to the positive region operation are required when choosing ?
for negative region operation. The choice of ? also needs to satisfy the slope requirements
for stable operation. One should also note that operation in the negative region will be
somewhat more difficult than operation in the positive region due to the non-existence of
equilibrium points in the region of (0 > x? x2o?y2o2(yo+xo)). Either an external force or dynamic
excitation would have to be used to initialize the device for negative region operation. In
Figure 1.9 the device was initialized at -10?m for example.
17
0 0.05 0.1 0.15 0.2
?2.5
?2
?1.5
?1
x 10?5
time (s)
Position (m)
Position vs Time
Figure 1.9: Output Position for Linear Controller, Negative Region
0 0.05 0.1 0.15 0.20
20
40
60
80
100
120
140
160
180
time (s)
Input (V)
Voltage vs Time
Figure 1.10: Input Voltage for Linear Controller, Negative Region
18
1.9.4 ??s Effect on Localized Stiffness
At this point it is beneficial to look at ??s effect on the localized stiffness of the system.
Only positive region operation will be considered, however a similar argument can be made
for negative region operation.
Figure 1.11 and Figure 1.12 show the device-plant functions and net force functions
for 3 different intercept values respectively . The input functions are half parabolas with a
saturation characteristic (fe ? 262 and fe = 0 for x ? ?). With ? = xo the input function
is the widest parabola possible to establish one stable equilibrium point. As ? ? xe, the
parabolas narrow and in the limiting case the feedback looks like a relay.
0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
Norm x
Norm f
d
Device?Input Functions for Various Intercepts
fe6
fe8
fe10
fd
Figure 1.11: Device-Input Function for Various Chi Values
Figure 1.12 shows how the localized stiffness varies with the choice of ?. Linearizing the
system about the equilibrium point, dFdx
vextendsinglevextendsingle
vextendsinglex
e
= k. For a linear system k would be the spring
constant or stiffness. The minimal system stiffness occurs when ? = xo, as ? decreases the
stiffness increases and in the limiting case the stiffness approaches infinity.
19
0.3 0.35 0.4 0.45 0.5?0.5
0
0.5
Norm x
Norm F
Net Force Plot with Various Intercepts
chi=6e?6chi=8e?6
chi=10e?6k
6
k10
k8
Figure 1.12: Net Force Functions for Various Chi Values
Because the localized stiffness of the system can be modified by choice of controller
gains, a variable structure technique can be used to improve system performance.
20
Chapter 2
Variable Structure Technique
The control methodology up to this point does a good job of controlling the system
under ideal conditions. Even under these circumstances, stabilizing the system can be
problematic for systems with low mechanical damping. For zero damping, the system is
only marginally stable. Because many MEMS devices have low damping and any real system
has uncertainties, delays, and noise, a more robust control method may be required.
One solution is to use variable structure control. Variable structure control works
by switching between 2 or more different control structures to improve performance. For
example Figure 2.1 and Figure 2.2 show the phase response for the same system, the first
having a feedback with low stiffness (low frequency), the second having a feedback with
high stiffness (high frequency).
?5 0 5?3
?2
?1
0
1
2
3
Position
Velocity
Low Stiffness Phase Portrait
Figure 2.1: Low Stiffness Phase Portrait
21
?5 0 5?3
?2
?1
0
1
2
3
Position
Velocity
High Stiffness Phase Portrait
Figure 2.2: High Stiffness Phase Portrait
Both systems are marginally stable. However, by switching between the two structures
the system becomes asymptotically stable as shown in Figure 2.3. The switching criteria is
to use the high stiffness system if the position-velocity product is greater than 0, otherwise
the low stiffness system is used.
?2 ?1 0 1 2 3?1.5
?1
?0.5
0
0.5
1
1.5
2
2.5
3
Position
Velocity
Mixed System Phase Portrait
Figure 2.3: Mixed System Phase Portrait
22
2.1 Available Structures
There are 6 different types of structures available for second order system equilibrium
points. By linearizing the system an the equilibrium point and determining the eigenvalues
of the linearized system the type can be determined. Because the damping coefficient is
so low for the GCA system, it can be ignored without affecting the analysis. With zero
damping, there are only 2 possible structures to switch between, an unstable saddle point
and the marginally stable center point. This fact can be attributed to the physics of the
system, forces either pushing the actuator away from or toward the equilibrium point.
Figure 2.4 shows the phase portrait for the system with a constant input voltage with the
two types of equilibrium points.
Figure 2.4: Constant Voltage, Zero Damping Phase Portrait
2.2 GCA Variable Structure Controllers
Both the stable and unstable equilibrium behavior can be used to implement variable
structure control for the GCA system. Switching between two marginally stable systems
works similarly to the controller shown in Figure 2.3, where the system stiffness can be
23
varied as discussed previously. This forced damping method can be implemented using 2
or 4 different structures. Using 2 structures improves damping when the system dynamics
are well known. Using 4 structures also improves damping and also makes the system more
robust in cases where there are many unknowns present.
Switching between a marginally stable structure and an unstable structure results in a
sliding mode controller, where the phase trajectory is forced toward the switching surface.
Again, 2 or 4 structures can be used, with similar reasons as stated above. Under some
circumstances, this type of variable structure controller outperforms the forced damping
type.
2.2.1 Stable-Stable (Forced Damping) Switching
To demonstrate how the forced damping controller improves system response absent
of system uncertainties the linear controller previously described is used. The high stiffness
structure uses ? = xe + xo?xe10 , the low stiffness structure uses ? = xo. The switch sample
time was set at 1 ?s for simulations.
0 0.2 0.4 0.6 0.8 1x 10?5?0.01
?0.005
0
0.005
0.01
0.015
0.02 Phase Portrait: Stable?Stable
Position (m)
Velocity (m/s)
Figure 2.5: Stable-Stable Phase Portrait
24
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1x 10?5
time (s)
Position (m)
Position vs Time:Stable?Stable
Figure 2.6: Stable-Stable: Position vs Time
0 0.05 0.1 0.15 0.20
5
10
15
20
25
30
time (s)
Voltage (V)
Voltage vs Time:Stable?Stable
Figure 2.7: Stable-Stable: Control Voltage vs Time
25
2.2.2 Stable-Unstable (Sliding Mode) Switching
To demonstrate how the sliding mode controller improves system response absent of
system uncertainties the linear controller previously described is used. The stable structure
uses ? = xe + xo?xe4 , the unstable structure uses ? = 5xo. The switch sample time was set
at 1 ?s for simulations.
0 0.2 0.4 0.6 0.8 1
x 10?5
?0.015
?0.01
?0.005
0
0.005
0.01
0.015
0.02
Position (m)
Velocity (m/s)
Phase Portrait:Stable?Unstable
Figure 2.8: Stable-Unstable Phase Portrait
Figure 2.8 shows that the system only reaches the sliding mode on the last few equi-
librium points. The structures chosen form a hybrid system, partially forced-damping,
partially sliding-mode, depending on where in the travel range the device is located. The
switching surface also needed to be modified to ensure proper controller performance.
26
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1x 10?5
time (s)
Position (m)
Position vs Time:Stable?Unstable
Figure 2.9: Stable-Unstable: Postion vs Time
0 0.05 0.1 0.15 0.25
10
15
20
25
30
time (s)
Voltage (V)
Voltage vs Time:Stable?Unstable
Figure 2.10: Stable-Unstable: Control Voltage vs Time
27
2.2.3 Switching Surface Considerations
For the forced-damping controller, only the position-velocity product needed to be
known to determine if the high-stiffness or low-stiffness structure was engaged. In order
to make the sliding-mode controller work properly the switching criteria had to be slightly
modified. For the stable structure to be engaged:
(mx1 +x2)x1 > 0 (2.1)
Where m ? 0 is the slope of the switching surface. Otherwise the unstable structure
was engaged. For the stable-unstable method described in the last section m = 15000
was chosen using trial and error. A more rigorous method of determining the correct
slope to ensure a sliding mode would be forcing ??? < 0 where the switching surface is
? = mx1 +x2. However, understanding the shape of the phase portraits of both structures
is less cumbersome than doing the math.
The quadrant switching (m = 0) used with the stable-stable controller could also
be modified. There might be certain cases where such switching is beneficial, although
theoretically it is not necessary.
28
Chapter 3
Adaptive Controller
The control methods described thus far have assumed perfect knowledge of system dy-
namics. Due to fabrication imperfections, modeling uncertainties, and other similar effects,
the system is never perfectly known. For example, deep reactive ion etching is one of the
best fabrication techniques to generate vertical side-walls. However, even small angular
errors can affect the capacitance, [12] notes that even the relatively small aspect angle of
? < 1? can have a dramatic effect on capacitance. The equation for a simple (not quite)
parallel plate capacitor is given:
C(?)
C(0) =
d
2T tan? ln
d
d?2T tan? (3.1)
Where d top side gap width, ? is the under-etch angle, and T is the depth of the etch
(device thickness). An attempt was made to determine exactly how (3.1) modifies the device
function, but the form of the equation makes that analysis very difficult. Despite this fact,
it should be clear that the effects of aspect ratios, fringe fields, and other effects generate
some relatively large uncertainties in the device function, so any proposed controller should
be quite robust. The linear controller works well because the input function slope and
intercept can be varied to compensate for any uncertainties.
29
V(x) = ?(x??) (3.2)
Solving for ? is not possible when the device function is unknown. To compensate
for this a simple adaptive algorithm can be implemented using a PI (proportional-integral)
method:
? = P
parenleftbigg
xe ?x+I
integraldisplay t
0
xe ?x d?
parenrightbigg
(3.3)
In order for the adaptive controller to work properly, natural damping must be present
since the controller cannot force damping like the controllers previously discussed. In cases
with low damping the adaptive gains need to be decreased so that the natural damping
has time to eliminate transients. Simulations use the damping given in Table 1.1, and an
intercept of ? = xo, and PI gains of (1?1012,1?103) respectively.
0 0.2 0.4 0.6 0.8 1x 10?5?0.03
?0.02
?0.01
0
0.01
0.02
0.03
Position (m)
Velocity (m/s)
Phase Portrait:Adaptive Controller
Figure 3.1: Adaptive Controller Phase Portrait
30
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1x 10?5
time (s)
Position (m)
Position vs Time:Adaptive Controller
Figure 3.2: Adaptive Controller: Position vs Time
0 0.05 0.1 0.15 0.20
5
10
15
20
25
30
time (s)
Voltage (V)
Voltage vs Time:Adaptive Controller
Figure 3.3: Adaptive Controller: Control Voltage vs Time
31
Table 3.1: Approximate parameter values for robustness study
Variable Parameter Value Units
m mass 3.561?10?7 kg
B damping 2.39?10?4 N-sec/m
epsilon1o free space permittivity 8.854?10?12 F/m
epsilon1r relative permittivity 1 NA
A area 5.042?10?6 m2
xo nominal small gap 9.563 ?m
yo nominal large gap 27.23 ?m
K spring constant 115.2 N/m
3.1 Robustness Comparison
In order to compare the robustness of the different controllers discussed, the controller
parameters were randomly varied by ?20%. Control signals were generated using the pa-
rameters given in Table 3.1. System dynamics were still set by the parameter values given
in Table 1.1. Figure 3.4 and Figure 3.5 show the output responses and phase portraits for
all three controllers respectively.
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1x 10?5
Controller Robustness:Position vs Time
time (s)
Position (m)
stable?stablestable?unstable
adaptive
Figure 3.4: Robustness Study: Position vs Time
32
While the adaptive controller provides zero steady state error, the dynamic response
leaves something to be desired. The two variable structure controllers respond quickly,
but have steady state errors. The reason being the variable structure controllers switch
between two structures with feedback functions that intercept the device function at two
different locations. For any desired equilibrium point, the adaptive algorithm has only
one equilibrium point it settles upon, however, the variable structure algorithms have two
equilibrium points they switch between. This switching causes a very complicated dynamic
that can cause instability if the equilibrium points are too far away from each other.
0 0.2 0.4 0.6 0.8 1
x 10?5
?0.02
?0.01
0
0.01
0.02
0.03
Position (m)
Velocity (m/s)
Controller Robustness:Phase Portrait
stable?stablestable?unstable
adaptive
Figure 3.5: Robustness Study: Phase Portrait
33
Chapter 4
Device Function Design
Considering all the aspects that would need to be addressed when designing a MEMS
device is too much to cover here. However, there are a handful of items to keep in mind
when designing a system to be controlled using feedback. Once again the GCA will be used
for example, but similar arguments can be make for any nonlinear electrostatic device. The
input-device function for the GCA can be written:
V 2 = 2Fmepsilon1
oepsilon1rA
parenleftbigg 1
(xo ?x)2 ?
1
(yo +x)2
parenrightbigg?1
(4.1)
Where Fm = Kx is the mechanical force for the typical linear spring model. When
designing the spring, minimizing the spring constant, K, has two benefits. The first benefit
is that it minimizes the actuation voltage required for both open loop and closed loop
operation. The second benefit is it minimizes the forces (mechanical and electrical) acting
on the system mass when using feedback control. If there are any time delays in the loop,
then smaller forces prevent instability due to slower system dynamics. Increasing the mass
or inertia of the device has a similar effect. However, the resultant lowering of the natural
frequency of the device can make it susceptible to external vibration [13].
34
4.1 Nonlinear Springs
Theoretically, the open loop travel range can be extended by offsetting the electrical
force nonlinearity with a mechanical force nonlinearity. This can be done by designing the
mechanical force curve Fm(x) so that:
d
dx
2Fm(x)
epsilon1oepsilon1rA
parenleftbigg 1
(xo ?x)2 ?
1
(yo +x)2
parenrightbigg?1
> 0 (4.2)
One possible solution is:
Fm(x) =
??
?
??
Kx (0 < x < xl)
?(x??)
parenleftBig
1
(xo?x)2 ?
1
(yo+x)2
parenrightBig
(xl < x < xo)
(4.3)
Where xl is in the stable range of open loop operation for the linear mechanical force.
The variables ? and ? are used to ensure continuity in the force function and set the slope
of the device function. To ensure force continuity:
? = xl ? Kxl?
parenleftbigg 1
(xo ?xl)2 ?
1
(yo +xl)2
parenrightbigg?1
(4.4)
Using the GCA device as a template, a nonlinear spring system was simulated for
xl = 3?m, ? = 7.5?10?9Nm, and ? = 2.46?m.
35
0 2 4 6 80
1x 10?5
Position (m)
Nonlinear Spring Simulation:Input/Output vs Time
0
20
40
time (s)
Voltage (V)
Figure 4.1: System Response with Nonlinear Spring
Once Fm(x) has been determined, the nonlinear spring has to be designed. The design
is based on determining an interference profile to prevent beam deflection once the displace-
ment reaches a set value. An iterative method was used along with standard beam theory.
Using the beam variables from the GCA example where the beam length L = 1200?m,
the beam width w = 15?m, and the beam depth d = 75?m. The modulus of elasticity for
silicon is E = 176GPa. Using the beam element in Figure 4.2 the displacement and slope
equations can be determined.
y(l) = FEI
bracketleftbigg1
6l
3 ? 1
4Ll
2
bracketrightbigg
+sl
bracketleftbigg
1? l2L
bracketrightbigg
+yo (4.5)
dy(l)
dl =
F
EI
bracketleftbigg1
2l
2 ? 1
2Ll
bracketrightbigg
+s
bracketleftbigg
1? lL
bracketrightbigg
(4.6)
36
Figure 4.2: Beam Element for Nonlinear Spring Design
Where y(l) is the beam deflection along the beam element, yo and s are the element
deflection and slope at the left end of the element, and EI is the sectional stiffness of
the beam. The iteration method starts with determining the deflection and slope at the
first interference point using the force from equation (4.3) at x = xl. These new boundary
conditions are then applied to a new element with a desired force from equation (4.3) making
sure the displacement at the right end of the beam matches the displacement input to the
force equation. The results are shown in Figure 4.3. The blue line is the interference profile,
the black lines are the beam profiles for different mass deflections, and the red line is an
example of a beam profile without interference (linear spring).
4.2 Implementation Issues
Modifying the device function using nonlinear springs looks promising because the
travel range is extended to the entire range of motion in open loop operation. However,
implementation would be problematic. First of all, higher stresses might cause mechanical
failure in the beams. Secondly, device function uncertainty would make designing the
interference profile difficult. A more aggressive interference profile could be designed to
compensate (say by shifting the entire profile towards the proof mass), but this would
37
0 0.2 0.4 0.6 0.8 1 1.2
x 10?3
?10
?8
?6
?4
?2
0
x 10?6
m
m
Interference Geometry:Nonlinear Spring
Figure 4.3: Interference and Deflection Profiles
likely increase the actuation voltage. Thirdly, depending on fabrication technology, not all
nonlinear electrostatic devices could use interference profiles.
Planar SOI (silicon on insulator) devices would be prime candidates for implementa-
tion (like the GCA example). However, even this technology would have implementation
difficulties. Photolithography (mask) restrictions would limit the resolution of the inter-
ference profile. Fabrication issues would further degrade the profiles (over-etching, etc.).
Stiction could also be a problem, a continuous interference profile providing a large con-
tact area. The profile could be approximated using discrete contact points, but even this
causes problems. Discrete contact points can cause an intermediate pull-in phenomenon.
Figure 4.4 shows the device function for such an arrangement. When the slope of the device
function is zero a localized pull-in occurs to the next intersection point. This would cause
a complicated hysteresis behavior that would extend the travel range in a limited manner
with erratic behavior near pull-in locations.
38
0 0.2 0.4 0.6 0.8 1
x 10?5
0
500
1000
1500
2000
2500
Position (m)
Device Function (V
2 )
Nonlinear Spring Device Funtion:Discrete Interference Profile
Figure 4.4: Discrete Nonlinear Spring Device Function
Other methods have been shown to produce nonlinear mechanical springs. For example,
instead of using contact forces from an interference profile, additional electrodes could be
implemented to create forces on the beam that changes the spring characteristic. Another
example is given in [14], where controlling motion in a cross direction effects the spring
constant in the direction of interest.
39
Chapter 5
Extension to Other Systems
In order to demonstrate the scope of the method, it will be used to demonstrate how
the series capacitor method extends the travel range of the GCA example. The method will
also be applied to a tilt actuator example.
5.1 Series Capacitor Method
The series capacitor method is an open loop approach to extend the travel range of
nonlinear electrostatic MEMS actuators. The method works by forming a voltage divider
that acts like a closed loop system. The voltage across the device, Vd, is given:
Vd = Vs1+ C
dC
s
(5.1)
Where Vs is the supply voltage across the series pair, Cd is the variable capacitance
of the device, and Cs is the series capacitor. Using a series capacitor of 1/4 the nominal
capacitance of the device:
Cs = epsilon1oepsilon1rA4
parenleftbigg 1
xo +
1
yo
parenrightbigg
Vd(x) = Vs(xo ?x)(yo +x)(x
o ?x)(yo +x)+4xoyo
(5.2)
40
Where V 2d (x) is the input function that meets the slope criteria previously discussed
for stable equilibrium points. Figure 5.1 is the device-input plot and Figure 5.2 is the net
force plot for 3 different supply voltages.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Norm x
Norm f
d
Input?Device Function Plot:1/4 C
o Series Capacitor
fe125V
fe175V
fe185V
fd
Figure 5.1: Series Capacitor Device-Input Plot
The plots show that the series capacitor shown is too large to provide full range of
motion. Decreasing the value of the series capacitor would increase the travel range, but
at the cost of higher actuation voltage. Using the series capacitance given, the actuation
voltages are relatively high, pull-in occurring around Vs = 182.5V at a displacement roughly
x = 8?m. Work has been done by Dr. Robert N. Dean and Dr. John Y. Hung at Auburn
University to improve the series capacitor feedback using analog circuitry to increase travel
range without increasing the supply voltage. Improved feedback has also been reported
using MOS capacitors [15].
41
0 0.2 0.4 0.6 0.8 1?0.1
?0.05
0
0.05
0.1
Norm x
Norm F
Net Force Plot: 1/4 C
o Series Capacitor
125 V175 V
185 V
Figure 5.2: Series Capacitor Net Force Plot
5.2 Torsional Devices
The method can also be applied to torsional devices. The only functional difference
between a torsional device with GCA is the electrostatic force nonlinearity. The capacitance
of a torsional device is given [16] as:
C(?) = epsilon1oepsilon1rh? ln
parenleftbigga+b
a
parenrightbigg
(5.3)
Where ? is the angle between the two electrodes, a is the lesser radial electrode dimen-
sion, b is the electrode length, and h is the out of plane dimension of the electrode. For
most torsional devices, rotations are assumed to occur about a fixed point located a gap
distance, g, above the stationary electrode. In order for this fixed point to remain constant:
42
a(?) = gsin? ?b (5.4)
The electrostatic torque can then be found:
Te(?) = 12V 2 dd?C(?)
d
d?C(?) =
epsilon1oepsilon1rh
?
bracketleftbigg b cos?
g?b sin? ?
1
? ln
parenleftbigg
1+ b sin?g?b sin?
parenrightbiggbracketrightbigg
(5.5)
Figure 5.3 and Figure 5.4 show the normalized device-input plots and net torque plots
respectively. The shape of the device function does differ slightly from that of the GCA,
but the same slope conditions apply for stability.
The same controller used for the GCA can be used for a torsional device:
V(?) = ?(???) (5.6)
Setting ? to be the maximum angular displacement of the device, the normalized
device-input plot and net torque plots were generated using various values of ?.
Like the GCA, the travel range of a torsional actuator can be extended to the full range
using the linear controller. No further analysis has be done, however the variable structure
and adaptive techniques described earlier could be applied to torsional systems.
43
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Norm Ang
Norm V
2
Torsional Device?Input Plot: Open Loop
V1
V2
V3
fd
Figure 5.3: Torsional Device-Input Plot: Open Loop
0 0.2 0.4 0.6 0.8 1
?1
?0.5
0
0.5
1
Norm Ang
Norm T
Net Torque Plot: Open Loop
T1
T2
T3
Figure 5.4: Net Torque Plot: Open Loop
44
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Norm Ang
Norm V
2
Torsional Device?Input Plot: Closed Loop
V1
V2
V3
fd
Figure 5.5: Torsional Device-Input Plot: Closed Loop
0 0.2 0.4 0.6 0.8 1
?1
?0.5
0
0.5
1
Norm Ang
Norm T
Net Torque Plot: Closed Loop
T1
T2
T3
Figure 5.6: Net Torque Plot: Closed Loop
45
Chapter 6
Conclusion and Discussion
6.1 Overview
Static stability is at the heart of the method discussed. Using static stability, the device-
input formulation was developed by setting the derivative terms to zero in the governing
equation and isolating the input voltage term V 2 to one side of the resultant equality. The
utility of the method rests in the fact that the input voltage can be manipulated given any
device to ensure static stability. The method also provides insight into how the device itself
can be designed to improve both open and closed loop performance.
Once the device-input formulation was established, a simple, linear controller was pro-
posed. This controller guaranteed the existence of only one stable equilibrium point for the
system as long as xe < ? ? xo (positive operation). An adaptive controller was proposed
based on the linear controller the can adapt to the correct slope, ?, for a given intercept
with no additional system information required.
The proposed linear controller was also used to develop two variable structure meth-
ods to improve system performance. Controller gains determined the structures that were
switched between. The first variable structure proposed used two stable systems to force
damping. By switching between two systems with high or low localized stiffness overall per-
formance was improved dramatically. The second variable structure proposed used stable
and unstable systems to form a hybrid sliding mode controller. Switching was done between
two input functions, the first where xe < ? ? xo, had only one stable equilibrium point.
46
The second, where ? >> xo had two equilibrium points, one stable, one unstable. The rea-
son the system is best described as hybrid is because for equilibrium points where dfddx > 0
a stable-stable switching is present, where dfddx ? 0 stable-unstable switching is present and
a sliding mode controller was observed.
Next, focus was shifted from the input function to the device function. Design consid-
erations for devices that are intended to operate under closed loop control were covered.
Then, a method that extends the positive slope range of the device function was proposed.
Extending the positive slope range of the device function extends the open loop travel range
by offsetting the electrostatic nonlinearity with a mechanical nonlinearity.
After showing how the method can be used to design controllers and devices based on
the GCA example, the method was extended to other MEMS in order to demonstrate the
utility of the method. The first system was the GCA in series with a dummy capacitor. In
open loop, this system extends the travel range of the device at the cost of higher actuation
voltage. The second system was a 1-dimensional torsional actuator. The device function
was determined and a similar analysis showed that input functions just like the GCA?s could
be used to extend the travel range of the system. Variable structure and adaptive methods
could also be applied to the torsional actuator.
Aside from the numerous benefits of the method, there are many issues that would
need to be addressed in any future work. The following sections attempt to introduce the
most predominate of these issues.
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6.2 Output vs State Feedback
The bulk of the method relies on output feedback, assuming the system output is the
displacement of the actuator. There are two reasons for this: 1) the static stability analysis
only requires this type of feedback, and 2) velocity measurements can be difficult on this
scale [17]. Parasitic capacitance, fabrication uncertainties, and measurement issues make
position measurement difficult enough. The only proposed controller that requires accurate
velocity measurement is the sliding mode controller where the switching surface has been
rotated in the phase plane. The other forced-damping technique only required the sign of
the velocity-position product, which is easier to estimate. The method is robust in terms
of workability without accurate velocity information. However, the output measured from
most of these devices is in terms of capacitance, not position. While the transformation
between capacitance and position should be fairly simple for 1-1 relationships, it might be
enlightening to formulate the device-input functions where capacitance is the independent
variable. This is one potential direction of future research.
6.3 Multi-Dimensional Actuators
It would be interesting to apply the method to multi-dimensional devices such as micro-
mirrors with two rotational axes. Cross axis coupling could make the work-energy formula-
tion of the electrostatic torques difficult, but it should be possible. An additional problem
would be choosing the number of actuation electrodes. With luck the method might be
powerful enough to allow full range of motion with only 4 electrodes. Instinctively, it makes
sense that the device function would be a surface with the two independent variables being
the rotation axes. Input function(s) would also likely be surface(s) over the two rotation
48
variables, however the input-output function relations for equilibrium and stability are not
immediately clear. Research along these lines would be interesting, however, there is no
guarantee this method would work for systems with multiple degrees of freedom.
6.4 Final Discussion
Overall, the methods described are based on static equilibrium analysis. These meth-
ods were developed in a somewhat ad-hoc manner, with the intent to tie them into a more
traditional type of control system analysis. However, as the method developed, the choice
was made to keep things simple by not complicating the analysis using Lyapunov or a lin-
earization technique. Static stability based controllers ensure that for a given displacement,
a constant voltage is converged upon. While there are cases where a state of dynamic
stability can be reached with wild input voltage fluctuations (like the switching controllers
with device function uncertainty), thorough understanding of these complex dynamics are
unnecessary for slow actuation requirements.
Most implementation aspects have not been addressed. Quantization and time delays
do cause considerable problems for the designed GCA device. Simulations show that these
problems could be reduced by increasing the mass of the device, or decreasing the stiffness.
Another aspect that has not been modeled is the RC dynamics of the system, finite charge
rates will no doubt effect overall system dynamics.
The methods described show much promise, but additional work needs to be done.
Implementation, using analog or digital circuitry, is the biggest hurdle. Implementation
issues will likely present problems not addressed in this analysis, however, these methods
should improve performance for most devices.
49
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