Sliding Mode Tracking Control of an Electrostatic Parallel-Plate MEMS

Marc Gronle, Guchuan Zhu, and Lahcen Saydy

Abstract— This work aims at developing tracking control al-gorithms based on sliding mode control (SMC) for the manipu-lation of electrostatically actuated parallel-plate MEMS. By thisapproach, stable full range operation with high performanceis achieved, even in the presence of parametric uncertainties.Moreover, SMC allows employing simple actuation mechanismsby pure switching control signals, which is particularly suitedfor the implementation by high-speed flip-flop circuits. It isshown in the present work that a power supply with two discreteoutput levels is sufficient to realize stable operation beyond thepull-in position for both state and output feedback control.System performance using developed SMC schemes is assessedby numerical simulations.

I. INTRODUCTION

Electrostatic actuators are some of the most popularMEMS (micro-electromechanical systems) devices. Due totheir versatility, reduced size, high speed, and low powerconsumption, electrostatic micro-actuators are rapidly findingtheir way into a variety of scientific, commercial, and defenseapplications [3], [24], [18], such as adaptive optics [30],optical network switching [5], [6], [10], projection systems[11], [15], resonators [1], [14], and inter-satellite laser com-munications [26] among others.

Electrostatic actuation has the advantage of simple struc-tural geometry, flexible operation, and easy fabrication fromstandard and well-understood materials [13], [28]. However,this actuation scheme results in highly nonlinear dynamics,giving rise to a saddle-node bifurcation, called “pull-in,”which limits the stable open-loop operation to a small portionof the whole physically available range [25], [8], [29],[32]. Large operation range, accurate positioning, and fastresponses are essential to many high-performance applica-tions. For example, for all-optical switches, these propertiesimply the capability of supporting large port-count cross-connection with short optical path, low insertion loss, andshort switching time [5], [6]. Closed-loop control is con-sidered to be a viable solution for extending the stableoperational range and further enhancing the performance ofelectrostatic MEMS, which motivated the majority of thework on this topic in the field of MEMS.

The work reported in the literature has addressed theapplication of a variety of techniques, in particular, non-linear control techniques, to control parallel-plate electro-static micro-actuators, such as static and dynamic feedback[19], passivity-based design [20], flatness-based control [36],[37], control Lyapunov function (CLF) synthesis [35], and

Department of Electrical Engineering, Ecole Polytechnique de Montreal,P.O. Box 6079, Station Centre-Ville, Montreal, QC, Canada H3C 3A7.This work was supported in part by the Natural Science and EngineeringResearch Council of Canada (NSERC).

Lyapunov-based design [21]. Robust and adaptive control ofMEMS in the presence of parasitics and parametric uncer-tainties has also been considered [38], [27]. These workshave demonstrated potential benefits of applying advancedcontrol techniques to enhance the performance of MEMS.Other issues related to implementation, such as modularintegration of MEMS and control systems into on-chipsolutions, have also been addressed [17].

The present work is motivated by a practical consid-eration related to the realization of electrostatic MEMS.More specifically, actuation of electrostatic MEMS requiresa quite high voltage, typically several hundreds of volts.Moreover, for guaranteeing desired performance, the voltagesource must be linear in the whole range with sufficientlyhigh gain-bandwidth product. Implementing such devicesrequires also big footprint. It is known that with the availablesemiconductor technology, generating discontinuous pulsesignals requires much simpler circuitry. The frequency ofswitching can also be extremely high. Therefore, it is ofpractical interest to consider techniques well adapted forthe control with switching signals. Obviously, sliding modecontrol possesses the required nature. In fact, SMC hasalready been used in MEMS control [34], [9], [33]. However,the model used in these works did not contain the dynamicsof driving circuits and hence, it does not capture the mostimportant behavior of the nonlinearity. In addition, theremight be performance issues when the bandwidth of controlloop is comparable with that of driving circuits.

In this work, we consider one degree-of-freedom parallel-plate MEMS. The development of control algorithms com-bines the technique of flat systems and sliding mode trackingcontrol. We start by designing state feedback control, andthen move onto output feedback using state observers. Theperformance of the developed algorithms is evaluated bynumerical simulations.

The paper is organized as follows. Section II presents thedynamic model of the device under consideration. Section IIIis dedicated to synthesis of state feedback control. Section IVpresents state observer design and analysis of output feed-back control. The results of simulation study are reportedin Section V. Finally, Section VI presents some concludingremarks.

II. DYNAMIC MODEL OF ELECTROSTATIC

PARALLEL-PLATE MEMS

The schematic representation of 1 DOF electrostaticparallel-plate actuators is shown in Fig. 1 where m is themass of the moveable plate, k is the stiffness coefficient, b

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Fig. 1. Schematic representation of 1DOF parallel-plate electrostaticactuator. The top structure is fixed for sustaining the moveable plate.

is the damping coefficient, G is the air gap, G0 is the zero-voltage gap, x = 1−G/G0 is the normalized displacement,R is the loop resistance, Va is the voltage across the actuator,Vs and Is are the source voltage and the source current,respectively. If the moveable plate is supposed to be a rigidbody without deformation and only the main electric fieldwithin the gap is considered, then the actuator capacitanceis given by Ca = εA/G [25], where A is the effective areaof the electrodes and ε is the permittivity in the gap.

By scaling the system variables with respect to a criticalbias voltage Vpi (the so-called pull-in voltage) [23], [36]:

q =2Qa

3C0Vpi

, u =Vs

Vpi

, i =Is

Vpiω0C0, r = ω0C0R,

where q is the normalized charge, C0 = εA/G0 is thecapacitance at the zero-voltage position, and ω0 is theundamped natural frequency, the normalized bias voltage canbe expressed by ua = 3q(1 − x)/2 and the dynamics ofnormalized charge become q = 2i/3. The state-space modelof the system in a normalized time scale with respect to ω0

is then given by [36]:

x = v (1a)

v = −2ζv − x +1

3q2 (1b)

q = −1

rq(1 − x) +

2

3ru (1c)

where v is the displacement speed and ζ is the dampingratio. System (1) is defined on a restricted state spaceX =

{(x, v, q) ∈ R

3 | x < 1}

. Note that x = 1 signifiesthat the moveable and fixed plates come into contact. Atthis point, the system exhibits switching behavior [19]. Forsimplicity, this work will not take into account the contactdynamics.

III. SLIDING MODEL CONTROL SYNTHESIS

To tackle the closed-loop tracking control of the consid-ered device, we take the deflection as output, y = x. Weobtain then by direct computation:

y = v, (2a)

y = −2ζy − y +1

3q2 (2b)

y(3) = −2ζy − y +2

3qq (2c)

which leads to

q = ±√

3 (y + 2ζy + y), (3a)

u =9r

4q

(y(3) + 2ζy + y

)+

3

2q(1 − y) (3b)

Since the states as well as the input can be expressed byoutput and its derivatives, we can conclude that System (1)is differentially flat, except for q = 0, and y = x is a flatoutput [16], [36]. The system is therefore exactly linearizableby state feedback and a diffeomorphism [16] in the region

X = {(x, v, q) | (x, v, q) ∈ (−∞, 1) × R × R\{0}} .

In fact, by choosing a new input, u, of the form

u =4q

9r

(u −

3

2q(1 − y)

)− 2ζy − y, (4)

the system (1) is equivalent to

y(3) = u. (5)

The tracking control can then be carried out easily in theframework of flat systems by dealing with the linearizedsystem (5) which is under the Brunovsky normal form.

Let now e1 = y − yr be the position tracking error whereyr(t) is the desired trajectory, the dynamics of tracking erroris then by

e1 = e2, (6a)

e2 = e3, (6b)

e3 = u −...y r. (6c)

where e2 = y − yr and e3 = y − yr. Considering e3 as avirtual input to e1-e2 subsystem, a stabilizing law will be ofthe form

e3 = −k1e1 − k2e2, k1 > 0, k2 > 0. (7)

The sliding surface can then be chosen as [12]:

s = e3 + k2e2 + k1e1 = 0. (8)

The equivalent control that keeps the trajectory of trackingerrors on the sliding manifold must satisfy

s = e3 + k2e2 + k1e1 = k1e2 + k2e3 + ueqv −...y r = 0,

from which one can derive the equivalent control in theoriginal coordinates:

ueqv =9r

4q

(−

2

3rq2e1 + e2(1 − k1) + e3(2ζ − k2)

+2

3rq2(1 − yr) + yr + 2ζyr +

...y r

)(9)

which satisfies −M < ueqv < M with M being a positiveconstant. Clearly, ueqv exists only in a domain depending onM and the reference trajectory D = X ∩ {|q| > q(M,yr)}.

For sliding mode control, we firstly consider a schemecomposed of the equivalent control and a switching compo-nent:

u1 = ueqv − Msgn(s)sgn(q). (10)

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To verify the attractivity of the sliding manyfold, we considera Lyapunov function candidate V (s, t) = 1

2s2. By directcomputation, we obtain

V = ss = −4

9rsqMsgn(s)sgn(q) = −

4

9rM |q||s|

≤ −4

9rMq(M,yr)|s| < 0,∀s �= 0.

Therefore, the time to reach the sliding manifold is given by[12]:

treach ≤9r

√V (s0, 0)

2Mq(M,yr)

which is finite for trajectories bounded away from the originand can be rendered small for a sufficiently big M .

Note that due to the term ueqv in (10), the implementationof such a control requires a high voltage linear amplifierwhich is power consuming and costly. An alternative is toproceed with designing a pure switching control law of thefollowing form

u2 = −Msgn(s)sgn(q). (11)

We take again V (s, t) = 12s2 as a Lyapunov function

candidate which leads to

V = ss = −4

9rqs(ueqv + Msgn(s)sgn(q))

= −4

9r|q||s|(ueqvsgn(s)sgn(q) + M).

For trajectories sufficiently bounded away from the origin,one can always bound the equivalent control in such a waythat M ± ueqv ≥ δ > 0. Therefore

V ≤ −4δ

9r|q||s| ≤ −

4δq(M,yr)

9r|s| < 0,∀s �= 0

and the corresponding reaching time to the sliding manyfoldis then given by

treach ≤9r

√V (s0, 0)

2δq(M,yr).

As the implementation of the control given in (11) requiresonly a two-level switching amplifier, the complexity and thecost of driving circuitry can be considerably reduced. Thisrepresents an advantage of SMC in comparison to otherapproaches.

IV. OUTPUT FEEDBACK CONTROL

The implementation of state feedback control algorithmsrequires the measurement of all the state variables, includingthe speed v. Usually, the gap between the electrodes andthe charge on the device can be deduced from the inputcurrent, the voltage across the device, and the capacitance(see, e.g., [2]). However, direct sensing of velocity duringnormal operations for micro-devices is extremely difficult.When the speed measurement is not available, we can use acertainty-equivalent implementation of state feedback designby replacing v by its estimate obtained from an observerbased on the measurement of deflection and charge. The

observer design and the output feedback control synthesisare presented in the following.

Note that, as the measurement of charge is available, theobserver design can be carried out by considering the reducedorder system:(

xv

)=

(A11 A12

A21 A22

)(xv

)+

(B1

B2

)q2 (12a)

y =(1 0

) (xv

)(12b)

with A11 = 0, A12 = 1, A21 = −1, A22 = −2ζ, B1 = 0,and B2 = 1

3 . As the only state to estimate is the speed v,we consider a reduced observer of the form

v = zc + Ly, (13a)

zc = Azc + Bq2 + Ly. (13b)

By choosing

A = A22 − LA12 = −2ζ − L

B = B2 − LB1 =1

3L = A21 + LA − LA11 = −1 + L(−2ζ − L)

the dynamics of estimation error ev = v − v satisfy

ev = (A22 − LA12)ev = (−2ζ − L)ev (14)

which is exponentially stable for all L > −2ζ.Note that the dynamics of micro-actuators are significantly

affected by the pressure due to the surrounding air whichcannot escape immediately as the moveable plate movesagainst the fixed one, creating the so-called squeezed filmdamping force. This force is a highly nonlinear function ofstate variables [4]. Therefore, we consider a sliding modeobserver which is robust vis-a-vis the variation of dampingratio. More specifically, the observer candidate for System(12) is of the following form:(

yv

)=

(A11 A12

A21 A22

)(yv

)+

(B1

B2

)q2 +

(L1

L2

)sgn(y − y)

(15)where (L1, L2) is the observer gain to be determined. Theerror dynamics are then given by(

ey

ev

)=

(A11 A12

A21 A22

)(ey

ev

)−

(L1

L2

)sgn(ey) (16)

with ey = y − y and ev = v − v. According to [31], theobserver sliding surface can be chosen as

σ = y − y = ey = 0.

Considering a Lyapunov candidate V = 12σ2, we have

V = σσ = σ(ev − L1sgn(σ)) = −|σ|L1 + σev.

If L1 is chosen in such a way that L1 > supt>0 |ev|, thenσσ < 0,∀t > 0, and the sliding manifold can be reached

1001

in finite-time. On the sliding manifold, σ = ey = 0 and,consequently,

L1sgn(ey) = A12ev = ev,

ev =

(A22 −

L2

L1A12

)ev.

Therefore, if

L2 >A22L1

A12= −2ζ0L1,

where ζ0 is the nominal damping ratio, then ev will droptowards zero as t → ∞. Furthermore, it is straightforward toverify that the uncertainty related to ζ in System (12) satisfiesthe so-called observer matching condition [22]. Thereforeany deviation from the nominal value of ζ will be completelyrejected.

Finally, when the state estimation is used to implementcontrol algorithms, the control sliding surface becomes:

s = e3 + k2e2 + k1e1 = 0

where e1, e2, and e3 are computed in the same way as e1, e2,and e3 in state feedback by using estimated state variables.It can be proved by means of standard techniques (see, e.g.,[7]) that the closed-loop error dynamics of the sliding modecontrol with sliding mode observer are asymptotically stable.

V. SIMULATION

Numerical simulations have been carried out to validatethe control algorithms developed in this work. Simulationstudy is also helpful for determining parameters, such asswitching level M , and for assessing system performancewhich can not be easily performed by theoretical analysis.

Set-point control is considered in simulation. The gener-ation of reference trajectories connecting set-points is basedon the algorithm described in [16], which results in apolynomial of the following form:

yr(t) = y(ti) + (y(tf ) − y(ti))τ5(t)

4∑i=0

aiτi(t), (17)

where y(ti) is the deflection at time ti, y(tf ) is the desiredone at time tf , and τ(t) � (t−ti)/(tf −ti). The coefficientsin (17) can be determined by imposing the initial and finalconditions

y(ti) = y(tf ) = y(ti) = y(tf ) = y(3)(ti) = y(3)(tf ) = 0,

which yields a0 = 126, a1 = −420, a2 = 540, a3 = −315,and a4 = 70.

For all simulations, the system parameters are set toζ = 0.5 and r = 0.8. Furthermore, the actuator is supposedto be driven by a bipolar voltage source, whose amplitude islimited by a saturation of ±2Vpi.

First, we test state feedback control u1 given in (10). Inthe simulation, the reference trajectory starts at yi = 0 andgoes up to different final positions. The controller parametersare k1 = 3.5, k2 = 3.2, and M = 0.5. The position of themoveable plate is shown in Fig 2(a) and the control signal

(a)

(b)

Fig. 2. Simulation with control scheme u1: (a) deflections; (b) controlsignal for full gap operation.

corresponding to the full gap operation (yf = 1) is depictedin Fig 2(b). It can be seen that the SMC can ensure a stablefull gap operation, while the maximal rise speed of the plateis limited by the input saturation and the system parameters.It can also be seen that once the system reaches the slidingsurface, the input oscillates around the equivalent input ueqv.If ueqv exceeds the saturation boundary, the system passesinto the reaching phase. After a while, the gradient of thereference decreases and the system is able to reach the slidingmanifold again.

(a)

(b)

Fig. 3. Simulation with control scheme u2: (a) deflections; (b) controlsignal for full gap operation.

The simulation results for state feedback control u2 givenin (11) is shown in Fig 3. The system parameters arethe same as those used in the previous test except thatM = 2 which corresponds to the boundaries of the input

1002

saturation. Although the equivalent control is not used incontrol synthesis, it is depicted in Fig 3(b) to illustratedifferent operation phases. A better illustration of differentphases related to sliding mode control is given in Fig. 4 inwhich the evolution of trajectories of tracking errors e1, e2,and e3 with respect to sliding surface are shown. Again, thiscontrol can ensure a full gap operation.

Fig. 4. Trajectories of tracking errors and sliding manyfold.

Different observer gains have been used to illustrate thetuning of sliding mode observer. The simulation results inFig. 5 show that a careful tuning is necessary for obtainingthe desired performance.

Fig. 5. Influence of sliding mode observer tuning.

The simulation results corresponding to output feedbackcontrol are given in Fig. 6. The test is carried out using thecontrol law given in (11) and two different observers, namelythe reduced order observer and the sliding mode observer.As in this test, it is supposed that there is no parametricuncertainty, the two control algorithms deliver quite similarperformance.

The last test is to asses the robustness of the control laws inthe presence of uncertainties at the level of ζ. In simulation,we add a variation Δζ = 0.5. It can been from Fig. 7 thatsliding mode observer delivers a better performance overreduced order observer.

VI. CONCLUDING REMARKS

This work dealt with the control of a parallel-plate elec-trostatic actuator, which is a basic component of manyMEMS applications. The main objective was to developa tracking controller by using sliding mode method and

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

time (t)

norm

aliz

ed p

ositi

on (

x)

sliding mode observerreduced order observerreference

(a)

0 1 2 3 4 5 6 7 8

−2

−1

0

1

2

time (t)

cont

rol s

igna

ls

input (u)equivalent input (u

eqv)

(b)

0 1 2 3 4 5 6 7 8

−2

−1

0

1

2

time (t)

cont

rol s

igna

ls

input (u)equivalent input (u

eqv)

(c)

Fig. 6. Output feedback control: (a) references and deflections; (b) controlsignal for full gap operation with reduced order observer; (c) control signalfor full gap operation with sliding mode observer.

(a)

(b)

Fig. 7. Closed-loop position errors corresponding to reduced order observerand sliding mode observer (a) Δζ = 0; (b) Δζ = 0.5.

1003

considering both the mechanical and electrical dynamics. Asthe system is differentially flat, it can be transformed, bya suitable state feedback, into the Brunovsky normal formin a domain covering nearly the whole physically availableoperation range. Sliding mode tracking control can then becarried out easily. It is shown that the sliding mode controlcan be implemented by using a two-level switching signal,requiring significantly cheaper and less complex voltagesources. Another benefit of pure switching control is itslow power-consumption. Output feedback control is alsorealized by means of state observers. The simulation resultsshow that the developed control algorithms can reach astable full gap operation, except for some singular pointdue to a controllability problem. The control using slidingmode observer is quite robust vis-a-vis certain parametricuncertainties. Nevertheless, overcoming singularities in thisproblem has not been addressed in this work and will remaina topic for future studies.

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