Modelling and control of an electrostatically actuated torsional micromirror

Abstract

This work aims at developing control algorithms for an electrostatically actuated torsional micromirror, extending the operational range of the device to a full 90 • tilt angle. The analytical model of the micromirror equipped with an additional vertical electrode is established. Since the geometrical extent of the device is comparable to the air gap, the effect of the fringing field is also incorporated into the model. It is shown that the considered system is differentially flat and, based on this property, a closed-loop control scheme is constructed for both scanning control and set-point control. In addition, the desired performance can be specified through reference trajectories, allowing the control system tuning to be performed in a systematic way. The simulation results demonstrate the advantage of the developed control scheme over the constant voltage control.

Full text

INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING
J. Micromech. Microeng. 16 (2006) 2044–2052 doi:10.1088/0960-1317/16/10/017
Modelling and control of an
electrostatically actuated torsional
micromirror
Guchuan Zhu1, Muthukumaran Packirisamy2, Mehran Hosseini3
and Yves-Alain Peter3
1Department of Electrical Engineering, ´
Ecole Polytechnique de Montr´
eal, C.P. 6079,
Succursale centre-ville, Montreal, QC H3C 3A7, Canada
2Department of Mechanical and Industrial Engineering, Concordia University,
1455 de Maisonneuve Blvd. West, Montreal, QC H3G 1M8, Canada
3Engineering Physics Department, ´
Ecole Polytechnique de Montr´
eal, C.P. 6079,
Succursale centre-ville, Montreal, QC H3C 3A7, Canada
E-mail: guchuan.zhu@polymtl.ca
Received 7 June 2006
Published 25 August 2006
Online at stacks.iop.org/JMM/16/2044
Abstract
This work aims at developing control algorithms for an electrostatically
actuated torsional micromirror, extending the operational range of the device
to a full 90◦tilt angle. The analytical model of the micromirror equipped
with an additional vertical electrode is established. Since the geometrical
extent of the device is comparable to the air gap, the effect of the fringing
field is also incorporated into the model. It is shown that the considered
system is differentially flat and, based on this property, a closed-loop control
scheme is constructed for both scanning control and set-point control. In
addition, the desired performance can be specified through reference
trajectories, allowing the control system tuning to be performed in a
systematic way. The simulation results demonstrate the advantage of the
developed control scheme over the constant voltage control.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
With its increased reliability and decreased manufacturing
cost, the technology of microelectromechanical systems
(MEMS) has been leveraged to benefit many industries,
such as medicine, telecommunications, automotive and
aerospace. Micromirror is one of the most widely used
microoptoelectromechanical systems (MOEMS) devices.
Among its many interesting applications, examples include
laser printing, bar code reading, laser surgery and fine pointing
mechanisms for inter- and intra-satellite laser communication
systems, to name only a few.
Among the numerous approaches for actuating micro
devices, for example, piezoelectric, thermopneumatic,
electromagnetic and electrostatic forces, electrostatic
actuation remains the most frequently applied principle in
view of versatility and simple implementation [1]. However,
when a static control scheme is used, the stable operational
range of parallel-plate electrostatic actuators is limited by the
pull-in phenomenon, which occurs when the rate of increase
of the electrostatic torque becomes larger than that of the
mechanical restoring torque at the equilibrium point. Beyond
the pull-in position, the moveable electrode will suddenly and
catastrophically snap down to the fixed one. There is an
abundant literature relating to this topic (see, e.g., [2–9]).
In some applications, e.g., torsional micromirrors, the
continuous actuation range of the device may also be limited by
the maximum capacitance of the device. When the capacitance
of the actuator reaches its maximum value, the micromirror
cannot move further as the force direction will reverse and this
will limit the maximum angle of tilt of the micromirror.
Besides the pull-in instability issue, many applications of
MEMS also impose stringent requirements on the actuator
transient behaviour, such as, settling time, overshoot and
oscillations. It is easy to illustrate that an over-damped system
0960-1317/06/102044+09$30.00 © 2006 IOP Publishing Ltd Printed in the UK 2044

Modelling and control of an electrostatically actuated torsional micromirror
Figure 1. Scheme of a torsional micromirror.
will suffer from a long settling time, while an under-damped
one will suffer from oscillations and overshoot. Note that when
the moveable and fixed electrodes are close to each other, the
overshoot will drive the moveable electrode to hit the fixed
one. The unexpected contact could damage the surface of the
electrodes and consequently reduce the lifetime of the device.
This paper addresses the modelling and control of
an electrostatically actuated torsional mirror including an
additional vertical electrode, allowing the device to attain a full
90◦tilt angle. Such devices are demonstrated in, e.g., [10,11].
The present work concentrates on the modelling and control
of this type of actuator. More precisely, we first extend the
modelling presented in [11], which provides a more accurate
description of the capacitance of the device with respect to
the deflection angle. Since the geometric extent of the device
is comparable to the air gap, the effect of the fringing field
is also considered in the modelling. Then we develop a
nonlinear control scheme, allowing us to extend the stable
actuation range to the full 90◦range. The proposed control
scheme is based on an essential geometric property of the
system, namely differential flatness [12,13], and it combines
techniques of trajectory planning and feedback linearization
control. It is therefore suited for both open-loop and closed-
loop control, depending on application cases. Note that due
to space limitation, we will not address the fabrication process
of the device. Interested readers are referred to [10,11]and
references therein.
The rest of this paper is organized as follows. Section 2
establishes the model of a one degree of freedom (1DOF)
torsional micromirror. Section 3gives the dynamical model
of the device and performs pull-in and stress analysis.
Section 4is devoted to control system design. The simulation
results are reported in section 5. Finally, section 6contains
some concluding remarks.
2. Capacitance of 1DOF torsional micromirrors
The considered device in this work is a typical 90◦scanning
micromirror whose schematic representation is given in
figure 1. The micromirror is sustained by torsional beams
clamped at the ends as shown. The substrate acts as the
fixed electrode and the micromirror is the moving electrode.
The geometrical parameters of the device that were used for
Table 1. Parameters of the electrostatic scanning micromirror.
Parameter Value
Mirror width W600 (µm)
Mirror length L300 (µm)
Height of vertical electrodes h60 (µm)
Air gap d305 (µm)
Torsion beam width w16 (µm)
Torsion beam length l320 (µm)
Mirror and torsion beam thickness δ0.4 (µm)
Permittivity ε8.85 ×10−12 (F m−1)
Modulus of rigidity G73 (GPa)
the present analysis are given in table 1. One should note that
the dimensions of the mirror are in the range of a few hundred
micrometres.
The device is actuated by a voltage source, where Is,V
s
and Vaare the source current, the applied voltage and the
actuation voltage, respectively. The force created due to the
electrostatic field between the mirror and the substrate will
tilt the micromirror towards the substrate. The tilting will
also give rise to a counter mechanical resisting torque from
the torsional beams supporting the micromirror. In its tilted
state, the mirror can effectively reflect the light in the desired
direction.
Note that a vertical electrode, served also as a stopper, is
added on the substrate, in order to increase the force due to its
electrostatic field helping the micromirror to attend a 90◦tilt
angle [10,11]. By this way, it is possible to tilt the mirrors to
any desired angle using an appropriate control signal.
One of the most often used formulae for capacitance due
to the horizontal electrode of the torsion microstructure is of
the following form (see, e.g., [5]):
Chm =εW
θln d
d−Lθ ,(1)
where εis the permittivity in the air gap. An equivalent
model expressed as the electrical torque is given by [14]. This
model takes into account only the main electrical field and is
subjected to small tilt angles. In [11], this model is modified
to take account of the capacitance due to the counter electric
field. The modelling of a similar device can also be found in
[10]. The formulation used in the above-mentioned work is
still based on the assumption of small angular deflections and
may not be accurate for large tilt angles.
From figure 1it is easy to see that the capacitance due to
the horizontal electrode can be extended to the full deflection
range by replacing the term d−Lθ with the actual vertical
deflection of the moving plate, d−Lsin θ, which yields
Chm =εW
θln d
d−Lsin θ.(2)
In addition, for structures of which the gap separating the
two plates is comparable to the geometric extent, the effect
of the fringing field is no longer negligible (see, e.g., [15]).
Figure 2(a) shows the capacitance calculated by (1) and the
results obtained by numerical simulation using MEMS CAD
tool ConventorWare. It can be seen that for the considered
device, the actual capacitance is much higher than the one that
considered only the main field.
2045

GZhuet al
020 40 60 80
0
0.05
0.1
0.15
0.2
(
b
)
Deflection (degree)
Capacitance (pF)
020 40 60 80
0
0.005
0.01
0.015
0.02
0.025
0.03
(
a
)
Deflection (degree)
Capacitance (pF)
main field capacitance
1st order approximation
2nd order approximation Cv
Ch
Ctotal
FEM based simulation
Ctotal–sim
Figure 2. Capacitances for different modelling: (a) capacitance due to the horizontal electrode, (b) capacitance of the device.
To obtain a more accurate model that takes account of
fringing capacitance, we start with the structure of micro-
strapline of width much smaller than length, WL.The
capacitance of such devices, including the effect of the fringing
field, is given by [16]
Cp=εA
x1+ 2x
πW +2x
πW ln πW
x,(3)
where xis the air gap between the electrodes and Ais the
area of the electrode. This formulation can be extended
to devices with an arbitrary shape by replacing Wwith the
effective width [17]. For rectangular devices, the effective
length is Leff =2L. Therefore, the effective width is given
by Weff =A/Leff =W/2. A modified expression of the
capacitance can then be given by
Ch=Chm 1+ 2d
πWeff
+2d
πWeff
ln πWeff
d,(4)
where Chm is the capacitance due to the main electrical field
given by (1). The simulation result shows that the fringing
capacitance is still underestimated (see figure 2(a)). It is
reported in the literature that the first-order approximations
of the form (3) often result in underestimated capacitances
(see, e.g., [15]).
To further reduce the modelling error, we adapted the
formulation of a second-order approximation proposed in [15]
to torsion devices and yield
Ch=Chmγ(θ) (5)
with
γ(θ) =1+ 2d
πWeff 1+lnπWeff
d
+d−Lsin θ
πWeff
ln d−Lsin θ
4πWeff 2
.(6)
The capacitance given by the last expression is depicted
in figure 2(a) and it can be seen that the modelling error is
considerably reduced compared to the model deduced from
the main field. Even though further improvement of modelling
accuracy is possible, this will increase the complexity of the
model and, consequently, make the implementation of control
algorithms difficult.
Usually, the height of the stopper, h, is much smaller than
the air gap, d, the capacitance due to the vertical electrode is
then given by
Cv=εW L
d−h
dx
xcos θ=εW ln L
d−h1
cos θ.(7)
Hence, the total capacitance of the device is
C=Ch+Cv=C0Cθ,(8)
where
C0=εWL
d(9)
is the main field capacitance at the zero voltage position.
Figure 2(b) shows the total capacitance calculated from
(8)(Ctotal)and that obtained by numerical simulation with
ConventorWare (Ctotal-sim ). We can see that the analytical
model has a very good agreement with the simulation result.
3. Dynamical model and pull-in analysis
The equation of motion of the device is given by
J¨
θ+b˙
θ+kθ =Te,(10)
where Jis the mass moment of inertia of the moving
electrode and bis the linear viscous damping coefficient. The
restoring mechanical torque of torsion beams is supposed to
be proportional to the tilt angle:
Tm=kθ, (11)
therefore the mechanical stiffness coefficient, k, is given as
(see, e.g., [18])
k=2Gwδ3
3l1−192
π5
δ
wtanh πw
2δ,(12)
where Gis the modulus of rigidity and the geometric
dimension of torsion beams (w, l and δ) is defined in
table 1.
The electrostatic torque can be obtained from the
derivative of the capacitance with respect to the angular
deflection which reads
Te=1
2V2
aC0
∂Cθ
∂θ =1
2V2
aC0C
θ,(13)
where C
θ=∂C/∂θ is a function of θ. Figure 3illustrates the
effect of adding a vertical electrode for obtaining tilt angles
2046

Modelling and control of an electrostatically actuated torsional micromirror
0 20 40 60 80
0
0.5
1
1.5
2
2.5 x 10
−9
(
b
)
Deflection (degree)
Torque (N–m)
0 20 40 60 80
−3
−2
−1
0
1
2
3
4x 10
−10
(
a
)
Deflection (degree)
Torque (N– m)
Th
Tm
Th
Ttotal
Tv
Figure 3. Electrostatic and mechanical torques: (a) mechanical torque of torsion beams and electrostatic torque due to the horizontal
electrode for Va=150 (V), (b) electrostatic torques for Va=150 (V).
Figure 4. CoventorWare simulation of stress distribution for 88.4◦
deflection.
from 0◦to 90◦. The electrostatic torques are obtained for
a voltage Va=150 (V) and are expressed as the functions
of deflection. It can be seen from figure 3(a) that in the
absence of the vertical electrode, there are three equilibria in
the operation range. Due to the fact that when the capacitance
of the actuator reaches its maximum value, the force direction
will be reversed, the micromirror is unable to attend a full
90◦tilt. Adding a vertical electrode of sufficient height will
render the capacitance and the electrostatic torque monotonic
(see figures 2(b)and3(b)), allowing the device to achieve a
full 90◦tilt angle actuation.
Even though the dynamical model of the device depends
only on its geometry, it is necessary to verify the maximum
stress to assure a safe operation. CoventorWare has been
used to simulate the stress distribution of the device under
investigation for a deflection near 90◦and the result is shown
in figure 4. It can be seen that the maximum von Mises stress
is about 660 MPa, which is much lower than the typical yield
strength of silicon (7 GPa) [19]. Therefore, the device can
safely operate in the whole range of deflection.
It has been shown that for actuators with one degree of
freedom, the pull-in angle is only related to the geometry of the
Table 2. Pull-in angles and pull-in voltages corresponding to
different stopper heights.
Stopper height (µm) Pull-in angle (◦) Pull-in voltage (V)
20 28.09 213.29
30 27.87 208.23
40 27.65 203.34
50 27.43 198.61
60 27.21 194.03
device and does not depend on spring structures. Furthermore,
the pull-in angle can be obtained by solving the following
equation (see, e.g., [7]):
∂C
∂θ θ=θpi −θpi
∂2C
∂θ2θ=θpi =0,(14)
and the pull-in voltage is given by
Vpi =



2k
∂2C
∂θ2θ=θpi
=2k
C
θpi =2kθpi
C
θpi
,(15)
where C
θ=∂2C/∂θ2and θpi indicate the pull-in angle. By
numerically solving equation (14), we got the pull-in angle and
the corresponding pull-in voltage for different stopper heights,
asshownintable2.
It now remains to model the dynamics of the driven
circuit as shown in figure 1. Assuming that the system started
operating from an initially uncharged state at t=0, then the
charge in the electrodes at the time tis
Q(t) =t
0
Is(τ ) dτ, (16)
or equivalently
˙
Q(t) =Is(t). (17)
Since the voltage across the plates Va=Q(t)/ Ct(t ),the
current through the resistor Rcan be obtained by a simple
application of Kirchhoff’s voltage law as [20]
˙
Q(t) =1
RVs(t) −Q(t)
Ct(t ) .(18)
Note that
V2
aC0C
θ=1
C0
C
θQ2
C2
θ
.
2047

GZhuet al
Therefore by denoting
αθ=C
θ
C2
θ
,(19)
the electrostatic torque Tecan be expressed as
Te=1
2C0
αθQ2.(20)
To make the system analysis and control design easier, the
system (10)–(18) is transformed into normalized coordinates
by changing the time scale, τ=ωnt, and performing a
normalization as follows:
q=βQ
Qpi
,u=βVs
Vpi
,
i=Is
VpiωnC0
,r=ωnC0R,
(21)
where Qpi =C0Vpi is the pull-in charge corresponding to
the pull-in voltage, ωn=√k/J is the undamped natural
frequency, ζ=b/2Jω
nis the damping ratio and β=
C0/C
θ(θpi).
Let ω=˙
θbe the angular velocity of the moveable
electrode, then the system (10) and (18) can be written in
the normalized coordinates as





˙
θ=ω
˙ω=−2ζω −θ+αθq2
˙
q=− q
rCθ
+1
ru,
(22)
which is defined in state space X={(θ,ω,q) ⊂R3|θ∈
[0,π/2]}.
Finally, in the following study, the tilt angle of the
micromirror is considered as the output of the system
y=θ. (23)
Since the system analysis and control design will be
performed in the normalized coordinates, we can use tto
denote the time and omit the qualifier ‘normalized’, if no
confusion will be introduced.
4. Flatness-based control of torsional micromirrors
4.1. Differential flatness and trajectory planning
In order to deduce the control law, we compute the time
derivatives of the output until that the input appears and we
obtain
˙
y=ω,
¨
y=˙ω=−2ζω −θ+αθq2,
y(3)=−2ζ˙ω−˙
θ+˙αθq2+2αθq˙
q
=−2ζ¨
y−(1−α
θq2)˙
y+2αθq−q
rCθ
+1
ru,
where α
θ=∂αθ/∂θ is clearly a function of θand hence, a
function of the output. Note that the length of integration
channel from the input to the output is exactly the dimension
of the dynamical model. Furthermore, all the state variables,
as well as the input can be explicitly expressed as functions of
the output yand its derivatives ˙
y, ¨
yand y(3), and are given as
θ=y, (24)
ω=˙
y, (25)
q=¨
y+2ζ˙
y+y
αθ(y) ,(26)
u=r
2αθ(y)q (y (3)+2ζ¨
y+˙
y) −qrα
θ(y)
αθ(y) ˙
y−1
Cθ(y) .
(27)
Therefore system (22) is said to be differentially flat with
y=θas flat output [12,13].
One of the basic properties of flat systems is that they are
linearizable by a (dynamic) state feedback. In fact, it can be
seen from (27) that
u=r
2αθ(y)q (v +2ζ¨
y+˙
y) −qrα
θ(y)
αθ(y) ˙
y−1
Cθ(y) (28)
is a linearizing feedback control law that, with a
diffeomorphism, renders system (22) into the Brunovsky
canonical form
y(3)=v(29)
in the new coordinates, which is obviously linear with vas its
input.
Another important feature of flat systems is that it is
possible to compute any trajectory of the system without
integrating the corresponding differential equations. This
makes the trajectory planning simple and straightforward, as
control signals can easily be deduced from the desired outputs,
allowing the system to track arbitrary trajectories, up to some
conditions of smoothness and feasibility. This property can
also be used to construct control laws and it will be seen that
the micromirror can operate beyond the pull-in position.
Note that reference trajectories can be expressed by
any suitable function. To steer the system from an initial
point xiin state space at time tito a desired point xfin
state space at time tf, one only needs to find a sufficiently
smooth trajectory t→ yr(t), such that the initial and final
conditions are verified. Since the trajectory t→ yr(t )
does not need to verify any differential equations, it can be
simply constructed, for example, by polynomial interpolations
[21]. To illustrate the construction of trajectories, we consider
a single-input/single-output (SISO) system. Extending the
algorithm to multiple-input/multiple-output (MIMO) systems
will be straightforward. Consider at time ti,
y(ti),...,y(r+1)(ti)
and at time tf,
y(tf),...,y(r+1)(tf)
are known and they define a total of 2(r +2)conditions for the
output yr(t). Denote T=tf−tiand τ(t) =(t −ti)/T , yr(t )
can then be expressed as a polynomial of time of order equal
to 2r+3:
yr(t) =
2r+3

k=0
akτk(t). (30)
By deriving yr(t) (r +1)-times and imposing the initial
conditions, the first r+ 2 coefficients a0,...,a
r+1 are given
by
ak=Tk
k!y(k)(ti), k =0,...,r +1.(31)
2048

Modelling and control of an electrostatically actuated torsional micromirror
While the rest r+2 coefficients can be determined by the initial
and final conditions, and are given by









11··· 1
r+2 r+3 ··· 2r+3
(r +1)(r +2)(r+2)(r +3)··· (2r+2)(2r+3)
.
.
..
.
.··· .
.
.
(r +2)!(r +3)!
2··· (2r+3)!
(2r+2)









×





ar+2
.
.
.
.
.
.
a2r+3





=









y(tf)−r+1
l=0
Tl
l!y(l) (ti)
.
.
.
Tky(k)(tf)−r+1
l=k
Tl−k
(l−k)!y(l) (ti)
.
.
.
Tr+1(y(r +1)(tf)−y(r+1)(ti))









.
(32)
The technique of trajectory planning can also be applied
to stabilize the system around setpoints. In this case, the
reference trajectory will connect two equilibrium points. It
results consequently that the derivatives of the reference
trajectory should vanish at the initial and final positions, y(ti)
and y(tf), and hence, the planned trajectory becomes
yr(t) =y(ti)+(y(tf)−y(ti))τr+2 (t )
r+1

k=0
akτk(t), (33)
where the coefficients a0,...,a
r+1 can be obtained by solving
the following linear equation system:









11··· 1
r+2 r+3 ··· 2r+3
(r +1)(r +2)(r+2)(r +3)··· (2r+2)(2r+3)
.
.
..
.
.··· .
.
.
(r +2)!(r +3)!
2··· (2r+3)!
(2r+2)









×




a0
a1
.
.
.
ar+1




=




1
0
.
.
.
0





.(34)
As all the time derivatives of y(t) vanish at the equilibria,
the desired trajectory can be represented by a polynomial of
arbitrary finite order. This allows the addition of more degrees
of freedom in tuning the trajectory and obtaining the desired
behaviour (e.g. fast rise time, low overshoot and well-damped
oscillations).
For the micromirror under investigation, we can consider
˙
yi=˙
yf=0,¨
yi=¨
yf=0,y
(3)
i=y(3)
f=0.
The desired trajectory t→ y(t) is then of the following form:
yr(t) =yi+(yf−yi)τ 5(t)
4

i=0
aiτi(t). (35)
The coefficients in (35) can be obtained by solving (34)and
are given as a0=126,a
1=−420,a
2=540,a
3=−315 and
a4=70.
4.2. Closed-loop control
In an open-loop control scheme, the control signals are
computed on the basis of the model of system dynamics.
Clearly, if the model is perfectly known and environmental
disturbances are negligible, the output of the system should
follow the planned one. However, in the presence of
system modelling error, parameter variations and external
perturbations, the output of the system, in general, will
not coincide with or even may diverge from the reference
trajectory. A closed-loop control driven by instantaneous
measurements of entire or partial state of the system will render
the system more robust and help tracking reference trajectories.
Being differentially flat, the tracking problem of
micromirror can be solved through the linearized system under
the form of (29). Let yr(t ) be the desired trajectory, and
e=y−yrbe the tracking error. A closed-loop control
corresponding to system (29) can be chosen as
v=y(3)
r−k2(¨
y−¨
yr)−k1(˙
y−˙
yr)−k0(y −yr). (36)
Then, the tracking error esatisfies
e(3)+k2¨
e+k1˙
e+k0e=0 (37)
and will be exponentially stable around the origin if the
characteristic equation
p(s) =s3+k2s2+k1s+k0(38)
is Hurwitz stable. Consequently, ugivenby(28) will be a
feedback control law archiving a local exponential tracking of
the reference trajectory.
Usually, the controller gains in (36), k0,k
1and k2,are
determined from the performance requirement. However,
it is often not evident how the performance specified in
the linearized coordinates will affect that in the original
coordinates. In the absence of any specific consideration,
controller parameters can be chosen in a way that the location
of closed-loop poles satisfies the Butterworth configuration.
The control is thus optimal in the sense that as control effort
becomes increasingly less expensive, the closed-loop poles
should tend to radiate out from the origin along the spokes of
a wheel in the left half-plane [22] as given by the roots of
s
2k
=(−1)k+1,(39)
where kis the number of poles in the left half-plane and is the
radius of the cycle on which the poles are placed. For example,
the third-order Butterworth polynomial B3(z), z =s/,isof
the form
B3(z) =z3+2z2+2z+1,(40)
and the corresponding pole location is shown in figure 5.
Now controller gains become
k0=3,k
1=22,k
2=2,
and is the only tuning parameter, whose value can be chosen
on the basis of desired decay rate and control effort.
2049

GZhuet al
Figure 5. Pole replacement of the third-order Butterworth
polynomial.
4.3. Speed observer design
Usually, the charge on the device and the angular deflection
can be deduced from the input current, the voltage across the
device and the capacitance. However, directly sensing the
angular velocity during the normal operation of the device is
extremely difficult, if not impossible. Therefore we need to
construct a speed observer in order to provide the estimate of
ωrequired for implementing the closed-loop control schemes
described previously. A reduced-order speed observer can be
constructed as follows. Let
z=ω−kωθ, (41)
where kωis an arbitrary positive real number. Differentiating
(41) and using (22), we get
˙
z=−((2ζ+kω)kω+1)θ −(2ζ+kω)z +α(θ)q2.(42)
Thus, if we set
ˆ
z=ˆω−kωθ,
where ˆωis the required estimate of ωand ˆ
zis the estimate of
z, then
˙
ˆ
z=−((2ζ+kω)kω+1)θ −(2ζ+kω)ˆ
z+α(θ)q2.(43)
Letting e=ˆ
z−z=ˆω−ωthe estimation error, and
d
dt(ˆ
z−z) =d
dt(ˆω−ω) =˙
e, the error dynamics can be deduced
from (42) and (43)as
˙
e=−(2ζ+kω)e, (44)
which is globally exponentially stable at the origin with a
decay rate defined by kω. This implies that
ˆω=ˆ
z+kωθ(45)
and (43) forms an exponential observer.
5. Simulation results
In the following, we use the numerical simulation to verify
the performance of the proposed control algorithm. Two cases
will be dealt in simulation studies: scanning control and set-
point control. The parameters of the considered micromirror
are [11]
ζ=0.164,ω
n=5.7681 ×103(rad s−1),
and the stopper height is chosen to be 60 µm.
050 100 150 200
0
20
40
60
80
Normalized Time
Deflection (degree)
(
a
)
050 100 150 200
−0.5
0
0.5
1
1.5
(
b
)
Normalized Time
Normalized Input
A = 80o
A = 60o
A = 40o
A = 20o
Figure 6. Responses of closed-loop scanning control for different
scanning ranges: (a) system responses, (b) control signals.
5.1. Scanning control
Due to the nonlinearity associated with the micromirror,
scanning controls deduced from the static relation between
the position and actuation voltage would exhibit important
distortions. In addition, such a control is limited by the pull-in
angle, which is about 30◦in our case (see table 2). In the
simulation, the desired scanning curves are chosen as a sine
wave form with a different scanning range Aand normalized
frequency 1/Ts:
yr(τ ) =A
21+sin2π
Ts
τ+3
2π,(46)
where τ=ω0tis the normalized time. Figure 6shows the
simulation results for scanning range Aequal to 20◦,40
◦,
60◦and 80◦, and figure 7shows the responses corresponding
to different scanning periods, Ts=100,T
s=50 and
Ts=25. Clearly, the output of the system follows the
reference trajectories quite well. We have observed that further
increasing the scanning amplitude or frequency could cause
instability. This is mainly due to the fact that those reference
trajectories are not feasible. The control signals corresponding
to different scanning curves are also shown. Obviously, the
relationship between the input and output is highly nonlinear.
Therefore, it will be extremely difficult to generate desired
scanning curves using the method of static voltage inversion.
Note that the nominal damping ratio used in the controller
design is ζ=1, while its actual value is 0.164. This shows
the robustness of the closed-loop control. We have observed
that the performance of the system is not significantly affected
by stopper height.
5.2. Set-point control
For set-point control, we tested the deflection of 10◦,45
◦and
89◦. The reason to choose 89◦as a test case is to show that the
system is indeed stable near a 90◦deflection. The travelling
time from yr(ti)to yr(tf)is set to 10 (normalized time units),
or about 3 ms in the original timescale. We can see from
figure 8that the closed-loop set-point control scheme allows a
full 90◦operation and provides an excellent performance, even
in the presence of system parameter uncertainty (mismatch of
damping ratio).
2050

Modelling and control of an electrostatically actuated torsional micromirror
050 100 150 200
0
20
40
60
80
Normalized Time
Deflection (degree)
(
a
)
050 100 150 200
−0.5
0
0.5
1
1.5
(
b
)
Normalized Time
Normalized Input
Figure 7. Responses of closed-loop scanning control for different
scanning periods: (a) system responses, (b) control signals.
0 5 10 15 20
0
20
40
60
80
Normalized Time
Deflection (degree)
(
a
)
0 5 10 15 20
−0.1
0
0.1
0.2
(
b
)
Normalized Time
Tracking Error (degree)
10o deflection
45o deflection
89o deflection
Figure 8. Responses of closed-loop set-point control: (a)system
responses, (b) tracking errors.
Finally, note that the shorter the expected travelling
time, the larger the effort that is required to control the
device. Therefore, in practice, an appropriate compromise
must be made in order to obtain a satisfactory performance
while not using excessive control forces. Note also that
further diminution of planned travelling time could render
the reference trajectory infeasible and make the system
unstable.
6. Concluding remarks
This paper addressed the modelling and control of a 1DOF
torsional micromirror. A capacitance-based modelling of a
micromirror is presented. This model incorporates capacitance
due to different electrostatic fields into a single analytical
expression and is indispensable for control system analysis
and design. The effect of the fringing field is also considered.
Note that the highly nonlinear capacitance given by (6)is
still an approximative expression. In practice, more accurate
expressions are very difficult to obtain and might render control
algorithms extremely complicated. A possible solution for
tackling this problem is to use robust control schemes.
It has been shown that electrostatically actuated 1DOF
torsional micromirrors are differentially flat and, based on this
property, a closed-loop control scheme has been constructed.
The proposed control scheme allows us to extend the
operational range to a full 90◦tilt angle and improve the
system performance in terms of transient response, precision
and robustness.
Acknowledgments
This work was supported in part by the ´
Ecole Polytechnique
de Montr´
eal under a program of start-up funds. The authors
gratefully acknowledge Professor Jean L´
evine of the Centre
Automatique et Syst`
emes of the ´
Ecole des Mines de Paris for
pointing out the potential applications of flat system theory in
the control of MEMS.
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