Neural network control of a MEMS torsion micro mirrorKhaled Al-Aribe and George K. Knopf, Member, IEEE

Abstract - High-performance optical communicationnetworks require stable switches to rapidly redirect a lightbeam from one port, or fiber, to another without converting the optical signal into an electrical signal. Micro-electromechanical(MEMS) optical switches are miniature devices that use tinymirrors to alter the path of light. These switches enable all-optical capability, are small and compact, and cheap tofabricate. However, many torsion mirror optical switches areactivated by electrostatic actuators that exhibit a pull-inphenomena which greatly effects system stability. The pull-in phenomena occurs when the gap between the electrodes in theactuator is reduced to less than two thirds of the original valuethereby causing an uncontrolled contraction between the twosides the capacitor. This paper describes how a backpropagation neural network can be used to control anelectrostatically actuated optical switch without using stiff suspension systems or mechanical stops. The two-layernetwork is applied to both single input and dual input MEMStorsion mirror optical switches. Simulation studies are presented to demonstrate how the proposed scheme will allowthe switching mechanism to operate in a stable range and avoid the effect of pull-in phenomena.

I. INTRODUCTIONRecent advances in micro-electromechanical system

(MEMS) technology and micro fabrication have enabled thedevelopment of high-performance optical communicationnetworks. The MEMS devices often exploit the principlesof electrostatic actuation because of the low power considerations. A key application of these miniature deviceshas been the design of active micro mirrors for redirectingthe path of light beams from one fibre or port to another.

The various beam steering micro mirrors described in theliterature can be categorized according to the type of motiongenerated [1]. Deformable micro mirrors are flexiblestructures that are able to alter their shape, whereas piston micro mirrors translate perpendicular to the mirror plane. Incontrast, torsion micro mirrors are able to rotate about one-axis. This last type of micro mirror also exhibits good dynamic response and, therefore, has been used for a varietyof applications that require high operating speed such as optical switching [2].

Manuscript received February 20, 2005. This work was supported in part by Natural Sciences and Engineering Research Council of Canada(NSERC).

K. Al-Aribe is with the Department of Mechanical & Materials Engineering, The University of Western Ontario, London N6A 5B9 Canada (e-mail: kalaribe@uwo.ca).

G.K. Knopf is with the Department of Mechanical & MaterialsEngineering, The University of Western Ontario, London N6A 5B9 Canada (e-mail: gknopf@engga.uwo.ca; tel: 519-661-2130; fax: 519-661-3757).

Although the mechanical and dynamic characteristics of electrostatically actuated torsion micro mirrors have been extensively studied, these devices continue to exhibitundesirable characteristics such as the pull-in phenomena.The pull-in effect greatly influences the performance of the device if used to switch between multiple ports in an opticalcommunication network.

The pull-in phenomena occurs when the gap between theelectrodes in the parallel-plate electrostatic actuator comewithin two third’s of the original separation distance. Oncethe gap reaches this point, an uncontrolled contraction of the electrodes occur and direct contact is quickly achieved. When the two electrodes come into contact, the systemmight become a short-circuited and localized heating mayoccur. The heat rapidly causes the contact surfaces to meltand device to fail.

Several techniques have been developed to restrict theangular rotation of the moveable mirror so that it does notextend beyond the stable zone. The most direct approach isto construct a complex stiff suspension system with amechanical stop to prevent the electrode rotation angle, or stoke, from extending beyond the known region of stability [3],[4]. The use of stiff suspension works well in limitingthe magnitude of the electrode stroke but requires a higherdriving voltage to overcome the structure stiffness, and therefore occurs at a significant energy cost. As well, thismechanical solution works best for applications that requireonly two discrete positions (ON-OFF).

Alternatively, other researchers and engineers have struggled with developing analytical models to accuratelypredict the pull-in angle for broad range of torsion micromirror designs based on key parameters such as electrodegeometry and micro beam dimensions [1],[5]. However,predicting the micro mirror’s angular response to a specific driving voltage is complex because of the nonlinear dynamics. In this context, a robust controller that adaptivelylearns the input-output mapping of the MEMS micro mirrorsystem and then regulates the angle of the parallel plate electrodes . Stability of the micro system can be ensuredgiven a lower driving voltage and, thereby, providingsmoother switching action.

In the following Section, a mathematical description ofthe relationship between driving voltage and rotation angleis presented. A feedforward neural network is brieflydiscussed in Section III and the results of the simulation are presented in Section IV. The backpropagation neural network adaptively develops a mapping between the

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

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required mirror angle and driving voltage for theelectrostatic torsion micro mirror and, thereby, can be used as an inverse controller for regulating an optical switch.Finally, general observations and conclusions are presentedin Section V.

II. MEMS TORSION MICRO MIRRORSA MEMS torsion micro mirror has a reflective surface

affixed to one side of a rotating electrode. The miniaturemirror is fabricated in silicon and coated with metal such as gold or aluminum to increase the surface’s reflectiveproperties, minimize surface roughness, and eliminate light transmission through the surface [6]. Surface roughness plays an important role because minute defects can cause the impinging light beams to scatter.

Electrostatic actuated torsion micro mirrors can have either single electrode or dual electrode configurations. The mirror in a single electrode optical switch is suspended by atorsion microbeam that acts as a spring that creates an opposing force to the electrostatic force. The directionalswitching of the incident light beam is accomplished bymaintaining an appropriate balance between the forcesgenerated by the torsion spring and electrostatic actuation.

In contrast, the dual electrode configuration suspends themirror on a low stiffness torsion microbeam and adjusts the rotation angle by controlling the driving voltage supplied totwo spatially separated electrodes located below the mirror,Fig. 1. The torsion microbeam in this case provides stabilityand small amount of damping to the rotating system. Duringoperation, only one actuator electrode is activated. If V1 is ON and V2 is OFF, then the mirror rotates counterclockwise.Alternatively, if V1 is OFF and V2 is ON, then the mirrorrotates clockwise.

The performance characteristics of the MEMS device area direct function of the various component dimensions. The length of the moveable micro mirror is given by theparameter a, and the width is L. Note that the mirror widthL is essentially the effective length of the electrode. The total length of the torsion microbeams is denoted as l, and the gap between the micro mirror and the electrode, as measured from torsion beam centre line, is g. For descriptive purposes, it is often desirable to define the spread betweenthe base electrodes using parameters a1 and a2, where

a1 = a (1a)a2 = a (1b)

The parameters above are normalized by the mirror length aso that and are independent of the actual micro mirrorsize.

When a voltage potential V is applied across the micromirror and one of the base electrodes, the electrostatic attraction will cause the mirror to rotate through a specificangle . In the context of a parallel-plate capacitor model[1], the micro mirror and base electrode can be viewed as the integration of infinitesimally small capacitors with

individual widths of dx. The electrostatic torque, e, used to create the angular displacement of the two torsion beamsgenerates an elastic recovery torque r. The micro mirrorreaches a static equilibrium condition when these torques are balanced; that is e r.

Fig. 1. Diagram of a dual input torsion micro mirror. Onlyone driving voltage input, V1 or V2, is activated at any instant in time.

At equilibrium the driving voltage at either side, V = (V1

or V2), can be mathematically modeled as V= K0 F( , , ) (2)Equation (2) is the normalized form that represents the static relationships between the rotation angle, the driving voltageand all the structural parameters of the torsion micro mirror.In its expanded form [1], the above equation can rewritten as

21

30

11ln

11

11

KV (3)

where the normalized rotation angle is

max(4)

and the maximum possible rotation angle at electrode

contact is given byag2sin 1

max .

Micro mirror

TorsionMicrobeam

Electrode

Base Substrate

a

l/2

L

Anchor

Torsion Microbeam

BaseSubstrate

a2

gV1

ActuatorElectrode

dF

dx a1

V r

738

The combination of all the structural parameters of the torsion micro mirror, with the exception of the normalizedelectrode parameters [1], is given by

21

3max0

02

LS

K (5)

where is the permittivity of air, and S0 is the variable thatdenotes the stiffness of the torsion beams. The parameter K0

acts as a sensitivity coefficient for the microbeam system.The stiffness of the torsion beam [1] is given by

lGI

S P20

(6)

The parameter G is the shear modulus, l is the length of each of the two sections of the torsion beam, and Ip is the polar moment of inertia of the square beam cross-section given by

4

43

12121.0

31

b

hbh

bhI P if h b (7a)

4

43

12121.0

31

h

bhb

hbI P if h > b (7b)

The parameters b and h in Eqn. (7) are the thickness andwidth of the torsion microbeams, respectively.

The typical relationship [5] between the driving voltageand rotation angle of the torsion micro mirror is illustrated inFig. 2. The value 0 is the normalized angle when thedriving voltage is at maximum Vmax. Based on Eq. (3), it isevident that the discontinuous mapping relationship is, inlarge part, determined by the normalized electrode parameters ( ). Consequently, the electrode geometrydirectly influences the behavior of the torsion micro mirror.

Fig. 2. Typical actuation curve for the MEMS torsion micromirror. The curve represents the relationship betweendriving voltage and observed rotation angle [5].

The angle 0 is commonly referred to as the pull-in angle.Figure 2 indicates that a voltage smaller than Vmax is requiredto maintain the torque equilibrium for rotation angles lessthan 0. As voltage increases, the micro mirror rotatesbeyond 0 and rapidly “snaps-down” until the mirror’s edge

touches the surface of the substrate creating the observed pull-in phenomena [1],[5]. Consequently, the movementfrom 0 to max is almost instantaneous.

III. NEURAL NETWORK CONTROLLERArtificial neural networks can approximate any complex

nonlinear function based on observations of plant input-output behaviour [7]-[10]. The strength of neural computingapproach lies in its ability to learn and adapt to changes inthe environmental conditions.

The main signal processing element is the computationalneuron that performs a nonlinear weighted summation of theapplied inputs xi (i=1, 2, , p). All weighted inputs to a particular neuron are summed along with a bias weight wj0

to produce a single scalar result uj. Mathematically, theoutput of neuron j can be expressed as [10]

yj = f(W(n)TX(n) ) (8)

Where yj is the output of neuron j, W(n) is theinterconnection weight vector, T is the transpose, X(n) is theinput signal vector for iteration n, and f(.) is the activationfunction. The nonlinear sigmoid activation can be given by

)(1

1)(e

f (9)

The mapping of inputs to outputs is achieved byconnecting numerous simple neurons in a multi layeredfeedforward architecture. To model a complex process such as the torsion micro mirror, the neural network mustundergo both training and operation phases. During thetraining phase, the neurons in the network’s first layerreceive randomly selected inputs xi(n) with known outputvalues. Each neuron j in layer l computes an output ,based on the interconnection weight , and transmitsthe response to the next layer’s neurons as an input, and so on, until this signal is moved forward through the network.

)(ny lj

)(nwlji

At the final layer, the desired output of the system iscompared with the responses of the output neurons. Sincethe network outputs will initially not be the same as thedesired values, an error signal ei(n) is produced which ispropagated back through the two layers updating theinterconnection weights by a step change )(nwl

ji. The

iterative update equation is

= + (10))1(nwlji )(nwl

ji )(nwlji

where is the learning rate that establishes the adaptationstep size. This weight adaptation process continues until a set of connection weights and biases exist to enable the network to approximate any output given an input within therange of the training data.

NormalizedRotationAngle

max

0

Vmax

Contact Failure

Driving Voltage V1 or V2

739

In this study, a feedforward controller is created using twoparallel SISO (single-input single-output) neural networksthat approximate the inverse behavior of the torsion micromirror. The networks, Fig. 3, ensure that the systemfunctions within the stable operating region. Thebackpropagation algorithm, Fig. 4, is used to determine theweights for the rotation angle to driving voltage map.

Fig. 3. The feedforward controller constructed from twoparallel neural networks that generate the appropriatecontrol signals (V1 or V2) for the two opposite electrodes of the MEMS micro mirror.

Fig. 4. The backpropagation training structure used to adaptthe weights of each neural network. For a given drivingvoltage V, the network error is e = V – y.

IV. RESULTS AND DISCUSSIONThe role of the neural network is to model the inverse

response behaviour of the torsion mirror over the stableoperating region. So that the system can be controlled to be operated only in the stable domain. Several numerical

studies were performed to determine whether a simple two-layer neural network could mimic this nonlinear behaviour.Since only one electrode on the base substrate is activated at a particular time, it is possible to decouple the system intotwo separate SISO subsystems. Each subsystem is modeledusing a (1x6x1) network architecture. Sigmoid transferfunctions are used by the neurons in the first layer whilstlinear transfer functions are applied to the neuron in theoutput layer.

Equation (3) is used to generate the data for thesimulation and analysis. The training data set contains 670 input-output sample pairs and the test data set includes 130 samples. The system parameters used to describe theelectro-mechanical system are given in Table I. The training and testing algorithms for the neural networks wereimplemented using the MATLAB © Neural Networks ToolBox. The learning rate was set to =0.01. The weights of thenetwork were adapted for 100 cycles (Epochs) through thetraining data. The overall error, Figs. 5a and 5b, aftertraining was completed was determined to be 2.7x10-6 for electrode V1 and 9.4x10-5 for electrode V2.

TABLE IPARAMETERS OF THE TORSION MICRO MIRROR.

Mirror width a 100 mMirror length L 100 m

Torsion beam length l 130 m

Torsion beam thickness b 1.5 m

Torsion beam width h 2.0 m

Electrode parameter 0.06

Electrode parameter 0.78Gap g 2.75 m

Shear modulus for poly silicon G 66 GPa

Permittivity of air 8.85 pF m

Figure 6 shows the normalized rotation angle of thetorsion micro mirror as a function of the two appliedelectrode potentials, V1 and V2. The figure also illustratesthe basic operating sequence for the controller. When thefirst electrode is activated by V1, the rotational displacementof the micro mirror starts at = 0 and moves to the largestpossible angle while still in the stable zone, 0, where

0 < max. The torsion beam’s rotation in the reversedirection is realized by reducing the applied voltage to thefirst electrode, V1, until zero volts exists. At this point, theelectrical potential across the second electrode is graduallyincreased to introduce a negative rotation in the micro mirrororientation.

DrivingVoltageV1 = y1

Plant(MEMS TorsionMicro Mirror) Actual

Rotation Angle

DrivingVoltageV2 = y2

r = x

RequiredRotationAngle

r r

r

V1

0

V2

0

FeedforwardController

ljiw

DrivingVoeV

Vltag

y

e System Error

TorsionMirrorDynamics

+

-

740

V1V2

(a) Training error for electrode V1.

(b) Training error for electrode V2.

Fig. 5. The training error of the (1x6x1) neural network over 100 Epochs.

In general, the neural network satisfactorily mimics thebehaviour of the micro mirror response. The curves for theoriginal system, Fig. 6, and the trained neural network, Fig.7, are nearly identical. The test data used for systemverification are represented as ‘*’ on Fig. 7. Upon closeinspection, small distortions can be observed at both the startand end points of the electrode activations. Training and test data based on experimental observation would,however, result in some additional minor distortions in theresult.

The system error during operation, derror, can be expressed as the difference between neural network output and truedriving voltage required to create the “desired” angulardisplacement. Figure 8 shows that the system error for theMEMS torsion mirror starts with high values at theswitching point between the two electrodes, (angulardisplacement equals to zero) and then rapidly decreases untilit approaches zero at the largest possible value for stableangular displacement. The maximum value of system error is approximately 0.082. The maximum values of the systemerror can be related to the points of maximum differencebetween the forward and backward stroke, Fig. 6.

Fig. 6. The response of the original system in terms ofnormalized rotation angle and applied electrode potentials(V1 and V2). Note that for inverse control the input is thereference angle and output is the required electrodepotential.

Fig. 7. The response of the trained neural network. The testdata is represented on the graph as ‘*’.

Fig. 8. System error (derror) over a range of angulardisplacements for the two electrodes.

TrainingError

Epochs0 10 20 30 40 50 60 70 80 90 100

10-6

10-5

10-4

10-3

10-2

10-1

100

101

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

NormalizedElectrodeVoltageV1 , V2

Normalized Angle

0 10 20 30 40 50 60 70 80 90 10010-5

10-4

10-3

10-2

10-1

100

101

TrainingError

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1V1V2

NormalizedElectrodeVoltageV1 , V2

Epochs

Normalized Angle

V2 V1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Error inNormalizedVoltagesderror

Normalized Angle

741

Neural networks provide a viable mechanism for modeling a variety of important control strategies [8],[10]. The neural network can be embedded in a simple feedforward control structure, as above, or more complex controllers that utilizes time-varying error feedback. Alternatively, it is possible to continually adapt the weights on-line to correct for local disturbances or changes in the plant or operating environment.

V. CONCLUSIONSA two-layer artificial neural network was constructed and

trained to model the complex relationship between the angular displacement and electrode voltages of a MEMS torsion micro-mirror with dual inputs. The effective elements of the decoupled system perform together to generate an integrated optical switching system. The network was trained and tested using sample data generated by an analytical model.

The results have shown that the methodology could create a model that follows plant’s dynamics over the stable operating zone. A smoother switching action is possible over this region because mechanical suspension systems are avoided. The sharp discontinuity arising from the rapid pull-in phenomena is not part of the model because this process is almost instantaneous and will cause immediate failure upon contact. Clearly, the goal of the controller is to avoid this situation.

A key recommendation for future studies is to investigate the effect of electrode geometry and dynamics on the system performance, and investigate how the neural network impacts switching times as the beam is redirected. Furthermore, the artificial neural network can be used as the foundation for other model-based control systems that regulate the driving voltage.

ACKNOWLEDGMENT

This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors also wish to acknowledge the academic scholarship provided to K. Al-Aribe from the Higher Institute of Electrical and Electronic Technology (Libyan Cultural Section).

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