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Robust control of a MEMS optical switch using fuzzy tuning sliding mode

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International Conference on Control, Automation and Systems 2010 Oct. 27-30, 2010 in KINTEX, Gyeonggi-do, Korea
1. INTRODUCTION
During the past decade, the trend of miniaturization has led to the new discipline that is well known as micro-electro-mechanical systems (MEMS). The use of MEMS for optical switching has turned to be the most attractive since this application takes fiber optic telecommunications to the next level. The optical switch is a device that switches an optical signal from one optical fiber to another, without having to first convert the optical signal into an electrical signal. Micro mirrors are fabricated to perform the switching function in an all optical medium [1]. The process of controlling the position of this micro mirrors and optimizing the operation of optical switches with guaranteed performance, stability and reliability becomes a challenging task. Controlling the device requires the establishment of its full models, including the optical, mechanical and electrical models [2]. According to the high complex and nonlinear model of the MEMS optical switch, the simplified model is proposed for control objectives [3]. In this way, the simplified nonlinear dynamic model of MEMS optical switch is derived through system identification. Moreover, they assume that the model can be considered to be linear and the linear controller is designed for the dynamics. In [4], feedback linearization controller with estimated states from a nonlinear observer is introduced to improve the dynamic closed loop performance of the MEMS optical switch. Recently, the conventional sliding mode controller has been implemented in order to control the nonlinear MEMS optical switch [5]. In this paper, a new approach of sliding mode control is presented by applying fuzzy method for the control of a MEMS optical switch. By this method, sliding mode controller gain is tuned according to real-time error which leads to the optimization of the energy consumption and accurate control process.
2. MATHEMATICAL MODEL
A general optical switch structure consisting of an electrostatic comb drive, the body of the device, and a blade or shuttle is shown in Fig. 1 [1]. A voltage applied to the comb drive actuator generates a force that moves the shuttle and attached micro-mirror that cuts a light beam exiting a transmitting fiber and being collected in a receiving and modulating its density. In order to derive a mathematical model of system dynamics it is needed to determine parameters of the relevant differential equation that describes forces acting on the shuttle. It is assumed that the shuttle has one degree of freedom and moves only in one direction. It is important to mention that there might be other degrees of freedom, like rotation around the main body axes, translational along them, as well as different vibrational modes. The influence of additional degrees of freedom should be addressed through modal analysis, as they can influence the dynamical behavior of the switch in the main degree of freedom. Subsequent control system design procedure should filter out harmonics that corresponds to those modes. They can be determined either analytically or with the aid of finite element modal analysis. However, only the main degree of freedom will be considered in this paper and related materials for modeling purpose are referred to [3]. Fig. 1. SEM image of a MEMS optical switch [1]
Robust Control of a MEMS Optical Switch Using Fuzzy Tuning Sliding mode
Amin Hassani
1
and Arash Khatamianfar
2
1
Department of Electrical Engineering, Ferdowsi University, Mashhad, Iran (E-mail: amin_hassani@ieee.org)
2
Department of Electrical Engineering, Sadjad Institute of Higher Education, Mashhad, Iran (E-mail: arash_khatamianfar@sadjad.ac.ir)
Abstract
: In this paper, a new approach of sliding mode control is presented for the control of a MEMS optical switch in which electrical, mechanical, and optical models are considered. In this method, fuzzy gain scheduling is employed to fight against uncertainties. By applying fuzzy method, sliding mode gains are tuned according to real-time error which leads to the optimization of the energy consumption and accurate control process. Robustness of the proposed controller is evaluated in the presence of uncertainties created by measurement errors, un-modeled dynamics, parametric errors, and external disturbances. This work improves upon the works previously done in this field of research by reducing the effects of modeling errors and uncertainty limits in calculating control law. Behavior of the control system is also kept robust and precise throughout. Simulation results show that the proposed approach outperforms the conventional sliding mode controller in terms of efficiency and accuracy.
Keywords:
MEMS optical switch, sliding mode control, fuzzy control, gain scheduling.
978-89-93215-02-1 98560/10/$15 ©ICROS 2281
The mathematical model of switch consists of three parts; electrical, mechanical and optical. Altogether, the system can be described with a second order nonlinear differential equation as
, ,,
where
m
is the effective moving mass of the shuttle,
d
is a function describing losses such as damping and friction,
k
is the stiffness of the suspension,
f
is the electrostatic force acting on the model,
P
is intensity of light, and
x
is the shuttle position. The system exact parameters
m
,
d
,
k
, and
f
are not easy to obtain and we will go step by step to determine all of these parameters. First, the electrical model is built and followed by mechanical model.
2.1 Electrical Model
The electrical part of the model considers generation of the electrostatic force by applying voltage to the terminals of the comb drive electrodes. The capacitance of the comb drive as a function of position should be determined first. Capacitance of the comb drive can be calculated as a sum of all capacitance among pairs of its movable interdigitated fingers. Each two fingers form one parallel plate capacitor. Capacitance is given as a function of position as
2
/
Comb Drive
where
8.85 10
Fm
is the dielectric constant of vacuum,
n
is the number of the movable comb fingers (
n
=150),
T
is thickness of the structural layer (
T
=
35
),
is the gap between fingers (
2.6
) and
is the overlapped length of fingers when no voltage is applied (
15
). At rest position, the capacitance of the comb drive is
0 ,
15 0.27
which increases as force is applied and the fingers move closer. Generally, the electrostatic force of the capacitor is given as a product of squared voltage and change of capacitance with respect to position as
, 0.5
/
where
V
is voltage applied over the electrodes. By combining Eq. (2) and Eq. (3), electrostatic force can be calculated as
,
⁄
where
is defined as the input gain to the system with the value of
17.8/
. It is interesting to note that capacitance in Eq. (2) depends linearly on the position over a wide range of deflections. It is one of the most important characteristics of the comb drive. Generally, for other configurations, this is not the case and capacitance is a higher nonlinear function of position x. Consequently, electrostatic force in Eq. (4) depends only on voltage across the capacitor not on position. It should be noted that the linear relationship does not hold for extreme deflections and may cause considerable undesired results that necessitate using a robust control scheme to meet such those unmodeled disturbances.
2.2 Mechanical Model
In order to get the mechanical model of the system three parameters, have to be determined. The first one is the so-called effective moving mass, the second one is damping coefficient, and the third one is the stiffness of the employed suspension. Available models have been developed for typical types of suspensions based on measurements and FEA. Effective mass for the switch can be expressed as
0.5
2.74
. When calculated, the effective mass of the system is
2.39 10
. Stiffness is generally a nonlinear function of position
f=k(x).
For most metals and for silicon spring-like structures [6], it can be described as
. For the suspension given in this paper stiffness is assumed to be linear and its coefficient is given as
0.46 /
. Damping, or energy dissipation, is the parameter that is the most difficult to determine analytically, even through FEA. The reason lies in the number of different mechanisms that cause it, including friction, viscous forces, drag, etc [7]. We will consider viscous forces as a primary reason that causes damping. Four different mechanisms could contribute to damping, Couette flow, Poiseuille flow, Stokes flow, and Squeeze film damping [8]. Finally, they can be written as
. When actual parameters are substituted, damping is expressed as
, 0.0363 15 10
.
2.3 Optical Model
The optical model is simply a function that connects the intensity of light to the position of the blade. Setup is shown on Fig. 2. Light beam is intercepted by the blade, increasing and decreasing the throughput of light. The Rayleigh-Sommerfeld model is based on a Gaussian distribution of the intensity across the light beam. Fig. 2. Optical Model [1]. Transmitted power can be described as
0.51 erf
√
where
10.9
and
11.2
. Consequently, integrating created model for each section and applying procedures done for increasing
(1) (2) (3) (4) (5) (6)
978-89-93215-02-1 98560/10/$15 ©ICROS 2282
accuracy of the model in [3], results in the nonlinear mathematical model of the switch as
2.35 10
0.0363x 4.5 10
x 0.6x 1.9 10
V
2. FUZZY SLIDING MODE CONTROLLER
According to the previous section, The nonlinear mathematical model of the switch is presented as Eq. (6). Two types of problems encountered in the controller design process are as follows [3]: 1) The quadratic term in voltage as input of the system (
u
=
V
) 2) Limited voltage available for control (0-35V) The state space representation of the system can be written as:
x
1.9 10
u 0.6x
0.0363x
4.5 10
x
/2.35 10
Let us consider the sliding surface (
s
=
0) using states error as:
in which
and
. So we can write:
0
The states of the system should approach the sliding surface. In order to guarantee this we must have:
0
0
With substitution of this in state space equations, the equivalent control is now can be obtained:
.
0.6x
0.0363x
4.5 10
x
2.35 10
Considering the impact of modeling uncertainties and other disturbances, the control law is modified by adding the switching term to Eq. (11) which is known as sliding mode control in Eq. (12).
.
0.6x
0.0363x
4.5 10
x
2.35 10
where
is the sign function and
is the switching gain which is the function of state variables and calculated via Lyapunov stability theory (
||
). We use
⁄
instead of
in order to eliminate the effect of chattering where
is a boundary layer thickness, and its value is calculated through the allowable error in the control system (
||
, 2
)[9]. It should be noted that in calculating switching gain, the extent of system uncertainties should be identified in advance, which is difficult to estimate precisely. Therefore, one appropriate approach to calculate
would be the application of fuzzy logic. Combination of fuzzy logic and sliding mode control is used in the variety of control applications in order to improve the performance of the sliding mode controllers such as eliminating the chattering effect [10], regulating boundary layer [11] and reducing the reaching time by moving the sliding surface [12] via fuzzy logic. In this work, we use fuzzy logic to adaptively tune switching gain. The inputs of fuzzy controller are
and
, and the output is
k
fuz
(
x
). In a fuzzy logic system, outputs are computed by a mechanism of If-Then rules. The general type of If-Then rules that is utilized in this paper is in the following form.
If
S
is
MF
i
and
is
MF
j
Then
k
fuz
(
x
) is
MF
k
for i=1,2 , j=1,2 and k=1,2.
where
MF
i
,
MF
j
and
MF
k
are membership functions of the inputs (
S
,
)
and the output (
k
fuz
(
x
)) respectively. The fuzzy rule base is shown in Table 1 where P is positive, N is negative, H is high and L is low. The fuzzy rule base is shown in Table 1. The schematic block diagram of the entire system is shown in Fig.3. Inputs and output membership functions of the fuzzy subsystem is as Figs. 4~5. Moreover, in Fig.6 fuzzy mapping surface is shown. Consequently, the fuzzy-sliding mode control law is obtained as follows:
.
0.6x
0.0363x
4.5 10
x
2.35 10
Table 1 Fuzzy Rule Base
)(
xk
fuz
S
N P
S
N H L P L H
U
x
d
x
e
)(
x
fuz
k
U
V
V
Fig .3 Schematic block diagram of the entire system Fig.4. Input membership functions
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 100.20.40.60.81Input Membership FunctionsS and ds/dt
D e g r e e o f m e m b e r s h i p
N P
(7) (9) (10) (11) (13) (12) (6) (8)
978-89-93215-02-1 98560/10/$15 ©ICROS 2283
Fig.5. Output membership functions Fig.6. Fuzzy Mapping Surface
4. SIMULATION RESULTS
In order to evaluate the performance of the control system, two desired mirror positions are considered
5µm ,23µm
. Disturbance and uncertainties are also assumed as
with
10
and
. The Figs. 7~10 illustrate the response of the control system to two desired positions, the input voltage, and state trajectory respectively. The value of constant parameters in control system is selected as
1500,0.001,
5,
100
, where
and
are scaling factor of the fuzzy controller. Fig.7 Response of the control system for
23µm
Fig.8 Response of the control system for
5µm
Fig.9 Input Voltage Fig.10 State trajectory on state phase plane
5. CONCLUSION
In this paper, the performance of applying fuzzy logic in combination with sliding model control in order to improve the process of controlling the mirror position in MEMS optical switch has been studied. To be more realistic, the effects of uncertainties created by measurement errors, un-modeled dynamics, parametric errors and external disturbances are considered. Simulation results illustrate that this approach outperforms the works previously done in this field of research.
REFERENCES
[1]
A. Q. Liu, X. M. Zhang, C. Lu “Optical and Mechanical Models for a Variable Optical Attenuator Using a Micromirror Drawbridge,”
J. Micromech. Microeng.,
Vol. 13, 2003. [2]
J. Li, Q. X. Zhang, A. Q. Liu, “Advanced Fiber Optical Switches Using Deep RIE (DRIE) fabrication”,
Sensors and Actuators
A 102, pp 286-295, 2003. [3]
B. Borovic, C. Hong, A. Q. Liu, L. Xie, and F. L. Lewis, “Control of a MEMS Optical Switch,”
43rd IEEE Conf. Decision and Control
, 2004. [4]
L. Owusu, Borovic, Liu, “Nonlinear Control of a MEMS Optical Switch,” presented at 45th
IEEE Conference on Decision & Control
, 2006. [5]
M. B. Behrouz Ebrahimi, “Robust sliding-mode control of a MEMS optical switch,” presented at
International MEMS Conference
, 2006. [6]
H. Baruh, ‘‘Analytical Dunamics,’’ WCB/McGraw-Hill, 1999.
[7]
M. Elswenpoek, R. Wiegerink, ‘‘Mechanical Microsensors,’’
Springer-Verlag, Berlin
, 2001.
[8]
S. D. Sentura,‘‘Microsystem Design,’’
Kluwer Academic Publishers
, 2000.
[9]
Jean-Jacques Slotine, Weiping Li,
Applied Nonlinear Control
, Prentice Hall , 1991
[10]
M.M. Abdelhameed, “Enhancement of sliding mode controller by fuzzy logic with application to robotic manipulators,”
Mechatronics
, vol. 15, pp. 439-458, 2005.
[11]
H. Lee, E. Kim, H.J. Kang and M. Park, "A new sliding-mode control with fuzzy boundary layer,"
Fuzzy Sets and Systems
vol. 120, pp. 135–143, 2001.
[12]
Q.P. Ha, D.C. Rye and H.F. Durrant-Whyte, "Fuzzy moving sliding mode control with application to robotic manipulators,"
Automatica
, vol. 35, no. 4, pp. 607-616, 1999.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.20.40.60.81Output Membership Functionsk
D e g r e e o f m e m b e r s h i p
L H
-1-0.8 -0.6-0.4 -0.20 0.20.40.6 0.81-1-0.500.510.30.350.40.450.50.550.60.650.7SdS/dt
k
00.0010.0020.0030.0040.0050.0060.0070.0080.0090.0100.511.522.5x 10
-5
Time (sec)
O u t p u t P o s i t i o n ( m )
Real PositionDesired Position00.0010.0020.0030.0040.0050.0060.0070.0080.0090.010123456x 10
-6
Time (sec)
O u t p u t P o s i t i o n ( m )
Real PositionDesired Position
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01010203040Time (sec)
I n p u t V o l t a g e ( V )
Input Voltage (23 micro meter)Input Voltage (5 micro meter)
00.511.522.5x 10
-5
-20246x 10
-3
Phase Planex1
x 2
State Trajectory (5 micro meter)State Trajectory (23 micro meter)
978-89-93215-02-1 98560/10/$15 ©ICROS 2284

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