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A Simple Learning Control to Eliminate RF-MEMS Switch Bounce Jill C. Blecke, Member, IEEE, Student Member, ASME, David S. Epp, Hartono Sumali, and Gordon G. Parker, Member, IEEE
Abstract—A learning control algorithm is presented that reduces the closing time of a radio-frequency microelectromechanical systems switch by minimizing bounce while maintaining robustness to fabrication variability. The switch consists of a plate supported by folded-beam springs. Electrostatic actuation of the plate causes pull-in with high impact velocities, which are difficult to control due to parameter variations from part to part. A single degree-of-freedom model was utilized to design a simple learning control algorithm that shapes the actuation voltage based on the open/closed state of the switch. Experiments on three different test switches show that after 5–10 iterations, the learning algorithm lands the switch plate with an impact velocity not exceeding 0.20 m/s, eliminating bounce. Simulations show that robustness to parameter variation is directly related to the number of required iterations for the device to learn the input for a bounce-free closure. [2008-0198] Index Terms—Electrostatic devices, learning control systems, microelectromechanical devices.
I. I NTRODUCTION
R
ADIO-FREQUENCY microelectromechanical systems (RF-MEMS) have numerous benefits and applications that are extensively reviewed in [1] and [2]. In this paper, a simple learning algorithm is employed to eliminate bounce during operation of an RF-MEMS switch, building off work presented in [3] and [4]. The device investigated is a galvanic, electrostatically actuated, and parallel-plate RF-MEMS switch [5]. Electrostatically actuated RF-MEMS devices without feedback control are limited in their stable operating range. In order to increase the functionality of such devices, much effort has been devoted to modeling dynamic behavior and designing control schemes that allow electrostatic actuation in the unstable region. Mechanical design considerations to improve performance past pull-in include the use of nonlinear stiffness elements [6], additional electrode coatings [7], and alternative locations and shapes of the electrodes [8]–[10]. While these methods increase the stable operating range, the result is larger devices due to additional components. Manuscript received July 31, 2008. First published February 27, 2009; current version published April 1, 2009. This work was conducted at Sandia National Laboratories. Sandia is a multiprogram laboratory operated under Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94-AL85000. Subject Editor C. Hierold. J. C. Blecke and G. G. Parker are with Michigan Technological University, Houghton, MI 49931-1295 USA (e-mail:
[email protected]; ggparker@ mtu.edu). D. S. Epp and H. Sumali are with Sandia National Laboratories, Albuquerque, NM 87123 USA (e-mail:
[email protected]; hsumali@ sandia.gov). Digital Object Identifier 10.1109/JMEMS.2008.2007243
Adding capacitance in series with the switch has also been shown to increase the stable operating range of electrostatic MEMS. There are some side effects to be dealt with in using this technique, including increased voltage requirements as well as parasitic capacitances that limit actuator displacement. The effects of both of these have been reduced by employing charge control [11] or by placing an inductor in series with the capacitor and switch [12]. Effects from parasitic capacitances have also been reduced using a “folded-capacitor” design [13]. The large actuation voltages associated with the series capacitor design can be reduced by implementing special current control strategies [14]. Theoretically, the stable operating range is larger when employing charge control rather than voltage control [15]; however, successful extension of stability has been shown with the use of voltage feedback. A voltage-controlled closedloop strategy, employing a position feedback control law, was shown to stabilize the actuator beyond the pull-in position [16]. Employing a dynamic voltage drive to actuate a microstructure was also shown to increase the operating range without relying on position feedback [17]. Open-loop configurations have been used to control devices through the unstable operating range as well. A carefully designed input waveform allows for gentle pull-in resulting in bounce reduction [3], [4]. A significant drawback of this particular strategy is that the waveform is only applicable to a single device and does not account for parameter variation from part to part. In order to better control this device despite parameter variation, a simple learning control algorithm is presented. Fundamental development of learning control is described in [18]. Learning control is heavily used and referenced in robotic control systems but has yet to be widely employed for MEMS applications. The following paper describes a simple binary learning control algorithm that is applied to an electrostatically actuated RF-MEMS switch. The initial input is based on the soft landing in [3] and [4] and allows real-time input shaping based on the response of the switch. The control is experimentally demonstrated on three devices. II. S WITCH D YNAMICS The device considered here is an electrostatically actuated parallel-plate RF-MEMS switch, as shown in Fig. 1. The movable electrode is suspended above the fixed electrodes with four folded-beam springs. The springs are attached to the substrate by two anchors and connect to each corner of the switch plate. The electrostatic force to close the switch is generated by a
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Fig. 3. Single-degree-of-freedom RF-MEMS switch model during impact.
Fig. 1.
RF-MEMS switch.
the initial distance between the plate and substrate (with zero input) is given by dg . The travel range the plate is required to move to close the switch is represented by dt . The force exerted on the plate is an electrostatic force Fe and is defined by Fe =
εAVa2 2(dg − x)2
(1)
where ε is the permittivity of air, A is the cross-sectional area of the switch plate, and Va is the actuation voltage. The dynamic equation for the model shown in Fig. 2 is given by the following equation, where m is the mass of the plate: 2 x ¨ + 2ζf ωnf x˙ + ωnf x=
Fig. 2.
Single-degree-of-freedom RF-MEMS switch model.
Fe . m
(2)
An electrostatic constant Ke is defined by voltage potential applied between the plate and the electrodes on the substrate directly below it. Upon actuation, the electrostatic force overcomes the spring restoring force and the plate moves toward the contacts. Under static conditions, stability is limited to one-third of the gap between the plate and the contacts. The instability is an inherent result of the nonlinear electrostatic actuation force. When the plate reaches the unstable position, it snaps down to the contacts and closes the switch. This phenomenon is referred to as pull-in [19]–[21]. Pull-in dynamics result in excessive impact velocities of the plate, causing wear on the electrodes and reducing the lifetime of the switch (see, e.g., [4] and [22]). Another side effect of a large impact velocity is the plate bouncing and interrupting the transmission after initial closure. Since switch closure time is defined as the time after all bounce has decayed, this results in long closure times. The goal of this paper is to maintain minimum closing time of the switch by reducing the bounce at impact, eliminating interruptions subsequent to switch closing. III. M ODELING AND S YSTEM I DENTIFICATION A single-degree-of-freedom model of the switch, as shown in Fig. 2, was created for simulation-based development of the control strategy. While the nonlinearities associated with damping in MEMS devices have been investigated [23], to maintain the simplicity of the model, constant damping is assumed. The natural frequency and the damping ratio are denoted by ωnf and ζf , respectively. The position of the mass is denoted by x, while
Ke =
εA . 2m
(3)
Substituting (1) and (3) into (2), the dynamic equation can be rewritten as 2 x ¨ + 2ζf ωnf x˙ + ωnf x=
Ke Va2 . (dg − x)2
(4)
A secondary single-degree-of-freedom system, adding stiffness and damping, was used to model an impact (Fig. 3), which is similar to the work presented in [22]. The natural frequency and damping ratio of the system during contact are represented by ωnc and ζc , respectively. This secondary system was implemented in the simulation only when the dynamics of the system indicated contact between the electrodes. The linear parameters, ωnf and ζf , in the model were determined experimentally using base motion to excite the switch [24]. Fig. 4 shows a frequency response function between the plate mass and the substrate. The first significant mode ωnf of the system was found to be at 21 kHz with a damping ratio ζf of 0.02. The gap dg was obtained from the work in [3] and [4], performed on the same type of switch. A gradient-based optimization algorithm was used to tune the remaining parameters such as dt , Ke , ωnc , and ζc to a measured data set. The final list of model parameters is displayed in Table I. A time-domain simulation was created using the parameters identified in Table I. When the switch is in an open position, the
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Fig. 4. Base excitation frequency response function of the mass plate in reference to the device substrate. TABLE I NOMINAL PARAMETERS FOR MODEL SIMULATION
simulation runs using the system described by 2 x ¨ + 2ζf ωnf x˙ + ωnf x=
Ke Va2 . (dg − x)2
(5)
An idealized voltage across the contacts is simulated in the model as well. The voltage is high when the switch is open and low when the position of the plate x is less than a threshold. The threshold to signal contact was set at 0.1 μm in the simulation. While the simulated voltage is low, the simulation runs the impact model described by 2 x= x ¨ + 2ζc ωnc x˙ + ωnc
Ke Va2 . (dg − x)2
(6)
The system reverts to the system model in (5) when the electrodes are no longer in contact (simulated voltage high). Fig. 5 compares the simulation to measured data from a test part, including bounce. Single-point velocity measurements were used to collect the data using a laser Doppler vibrometer and did not capture tilt or bending of the plate that can cause one location of the plate to be at a different displacement than the exact point of the contact. When the location of the laser is not placed directly over the contact, the displacement can surpass the travel range of the switch, dipping below a position of 0 μm. IV. L EARNING C ONTROL D ESIGN Learning control utilizes the system response to a known input to adjust the control law after each cycle [18], [25], [26]. Different implementations of learning control are em-
Fig. 5.
Model comparison to measured data including bounce.
ployed depending on the knowledge of the system [26]. If the desired path is known, the error function is generally a function of the difference between desired and actual system states at each point in the trajectory. For the RF-MEMS switch, good variables to monitor would be displacement and velocity. However, obtaining velocity measurements on MEMS parts is currently impractical outside of a laboratory setting, and displacement measurements would likely require the inclusion of microelectronics at the RF-MEMS device. To eliminate the need for any changes to the switch and to make control of the switch practical, the control algorithm developed in this paper is based only on the continuity across the switch contacts (if the switch is closed or open). The voltage was sampled at 4 MHz—a rate faster than the switch dynamics. Measuring the continuity across the switch gives no significant information about the path of the switch—it only indicates contacting and noncontacting positions. In this particular case, there are two critical components of the trajectory: 1) the timing of the initial closure relative to the application of the switch close command and 2) whether an interruption exists in the continuity after the initial closure (switch reopening due to bounce). The desired end-point conditions of the switch are that of minimal close time with a soft landing—ideally, the plate reaches closure with zero velocity, makes contact, and stays in contact. To achieve these goals, an iterative method was developed for generating the switch input voltage time history that results in a bounce-free closure and is robust to parameter variation. The shaped time history input was first introduced in [3] and [4]. A sample input is shown in Fig. 6 and described in functional form by ⎧ 0 ≤ t < t1 , ⎪ ⎪ ⎪ t 1 ≤ t < t2 , ⎪ ⎪ ⎨ t 2 ≤ t < t3 , Va (t) = ⎪ t 3 ≤ t < t4 , ⎪ ⎪ ⎪ ⎪ t ⎩ 4 ≤ t < t5 , t 5 ≤ t < t6 ,
1 2 Vp [1 − cos ω1 t] Vp 1 2 Vp [1 − cos ω1 (t
− t2 + t1 )]
0 1 2 Vh Vh .
(7)
[1 − cos ω2 (t − t4 )]
Each time designation in Fig. 6 and (7) is defined in Table II. In general, the input is characterized by an initial
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TABLE III SIMULATION INPUT PARAMETERS
Fig. 6.
Sample shaped input waveform. TABLE II INPUT WAVEFORM BREAKPOINTS
iterations. Upper and lower limits are set at the beginning of the control algorithm for tp and tc . Both variables begin at their lower limits, 5 and 1 μs, respectively. Once the magnitude of the input is fixed, if the switch does not land within a region of when Vh is applied, tp is incrementally increased using (8) until the switch lands within the region or lands early (before Vh is applied). The current value of tp is denoted by tp,n . The fraction by which tp,n is adjusted for the next iteration is denoted by Δtp , and the new value of tp,n is denoted by tp,n+1 tp,n+1 = (1 + Δtp ) · tp,n .
pulse, followed by a hold voltage separated from the pulse by a short zero voltage. The initial pulse of the waveform is used to move the switch toward the contacts. Before it reaches pull-in, the voltage is set to zero, and the momentum in the switch plate allows it to coast toward the contacts. At the end of the coast period, the hold voltage is applied. The waveform utilizes a haversine function to transition between the different voltage levels. A piecewise continuous function (7) compiles the waveform based on the values of Vh , Vp , tp , and tc . In the learning process, any of the four variables can be changed: the magnitude of the initial pulse Vp , the hold voltage Vh , the coast time tc , and the end time of the initial pulse tp . The input waveform must start out conservative so as to not break the device. The device will fail when enough voltage is applied such that the displacement of the mass equals or exceeds the total gap, welding the switch in the closed position. The correct voltages may not be known prior to operation. This type of learning control can determine those voltages, as long as the initial attempts are conservative enough to avoid breaking the device. Nominal values of 98 and 72 V for Vp and Vh were chosen, respectively. These values are 65% of the magnitudes suggested in [3] and [4] for a similar device (Vp = 150 V and Vh = 110 V). If the switch does not make contact during the first iteration, the magnitudes are increased to 70% of the suggested waveform. The waveform is increased by 5% at each iteration until contact is made. Once contact is made for the first time, the magnitudes in the waveform are fixed, and only tp and tc are adjusted on future
(8)
After the first iteration that records an early initial closure, the current and previous values of tp are used as the new limits, and the next value of tp is assigned the central value in that interval. Each adjustment after this iteration employs the same bisectional-type method of adjustment (9), where the upper and lower limits are determined by the values used in the previous two input waveforms. The upper and lower bounds are denoted by tp,u and tp,l , respectively, tp,n+1 =
1 · (t+ tp,u ). 2
(9)
Once contact is recorded within the region of when Vh is applied, tc is increased incrementally (10) until bounce is eliminated. The current value of tc is denoted by tc,n . The fraction by which tc,n is adjusted is denoted by Δtc , and the new value of tc,n is denoted by tc,n+1 tc,n+1 = (1 + Δtc ) · tc,n .
(10)
At any point in the learning process, if closure no longer occurs when Vh is applied, the control reverts back to adjusting tp . V. S IMULATION R ESULTS By using the set of nominal model parameters listed in Table I and the input parameters in Table III, the simulation required ten iterations to eliminate bounce. In the final iteration of the simulation, Vh was equal to 82.5 V and Vp was equal to 112 V. Fig. 7 shows the adjustments made to tp and tc throughout the simulation. Fig. 8 shows the position time history of the initial response, an intermediate response, and the response on the final iteration. As mentioned in Section IV, devices can fail if too much voltage is applied. Even among parameter variations, this type of learning control can determine an initial waveform that is conservative enough to not break the switch. However, increased
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TABLE IV AVERAGE NUMBER OF ITERATIONS FOR EACH SIMULATION SET
Fig. 7. Value of (a) tc and (b) tp , in microseconds, for each iteration.
Fig. 9. Simulation position time history of the response of the switch to show the elimination of bounce.
the strategy would require 240 ms (24 iterations at 100 μs per iteration) to learn the correct input for zero bounce closure. VI. E XPERIMENTAL R ESULTS Fig. 8. Simulation position time history of the response of the switch to show the elimination of bounce.
robustness of the system comes at a cost of iterations—more time will be required to find a solution that has eliminated bounce. An analysis looked briefly at the relationship between the number of iterations, the number of failed devices, and parameter uncertainty. By using a random uniform distribution, each system parameter (ζf , ωnf , Ke ) was varied up to ±5%, ±25%, and ±50% of the nominal value using a uniform random distribution. Fifteen simulations were completed for each case, once using the suggested waveform as an initial input and second using 50% of the magnitude of the suggested waveform as the initial input. Table IV displays the average number of iterations for each simulation set. The simulations show that the more conservative the initial waveform is, the larger the number of required iterations, but the number of failed devices is also reduced. Zero devices failed using the reduced initial waveform, as shown in the second row of Table IV. With the control strategy described in Section V, 225 simulations were completed with the parameters varied up to ±15% using a uniform random distribution. In all cases, the control found a solution that eliminated bounce with an average of 14.8 iterations. The simulations ranged from 1 to 24 iterations. Fig. 9 shows the iteration distribution over the range. At most,
The control scheme described in Section IV was implemented experimentally on three test switches. Since it was known that a value of Vh equal to 110 V and Vp equal to 150 V were appropriate for this type of switch based on the work in [3] and [4], the control implemented in the tests began with these magnitudes as the nominal values in order to reduce the number of iterations and, therefore, wear on the actual test pieces. The test bed consisted of a four-probe station with laser Doppler vibrometer for measuring switch displacement. Two probes were used to provide switch actuation voltage, while two more probes were used to measure continuity across the switch by measuring the voltage across a resistor and constant voltage source placed in series with the switch. The controller logic, programmed in LabView, relied solely on the binary continuity measurement. Results were recorded for three different switches referred to as devices 1, 2, and 3. The input parameters, listed in Table V, when applied to device 1, required seven iterations to eliminate bounce. Fig. 10 shows the actuation waveform for the first and final iterations. Fig. 11 shows the associated time history of the continuity measured across the switch as well as the position measured from the laser vibrometer for the response of the switch under the initial waveform and the final waveform in the learning process. The interesting points to note are the lack of interruption in the continuity in the final iteration in Fig. 11(a) and the reduction of closing time between the two iterations shown in Fig. 11(b).
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TABLE V EXPERIMENTAL INPUT PARAMETERS
Fig. 12. Value of (a) tc and (b) tp , in microseconds, at each iteration for device 1.
Fig. 10. Device 1 open-loop voltage command for the first and final iterations.
Fig. 13. Device 1 velocity response.
Fig. 11. Device 1 (a) continuity measurement and (b) position time history to show elimination of bounce.
During the experiment, tp changed from 5 to 9.38 μs and tc remained at 1 μs, as shown in Fig. 12. On the first iteration, the switch plate initially landed at approximately 20 μs (Fig. 11), after Vh was applied at 13 μs (Fig. 10). Since the landing occurred too late, tp was incrementally increased. This continued through iteration 4. Iteration 5 recorded an initial closure too early (before Vh was applied). By using the value of tp on iteration 4 as the lower bound and the value of tp on iteration 5 as the upper bound, the value of tp was adjusted using a bisectional method on iteration 7 described by (9), after resetting its value in iteration 6. On the final iteration, the initial
closure of the switch coincided with the application of the hold voltage Vh at approximately 16 μs. Fig. 13 shows the measured velocity response of device 1 for the first and final iterations. Two markers on the plot indicate initial landing of the switch plate. The learning algorithm reduced the initial contact velocity from 0.29 m/s on the first iteration to 0.11 m/s on the final iteration. This reduction in velocity between the two iterations eliminated the bounce of the switch plate. Without bounce, the total closure time of the switch was reduced from 34 to 16 μs. The higher frequency content shown in Fig. 13 is attributed to higher modes of the system that could appear if the measurement location is not on the exact contact location. The same initial parameters were applied to device 2 with the exception that Δtp was set to 40%. Table VI compares the values of tp and tc for devices 1 and 2. Despite similar initial inputs to the system, the tests resulted in different waveforms. This indicates that there is variation within one or more parameters that affect the dynamics of the switch; however, the strategy was still able to find a waveform for each of that eliminated bounce.
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TABLE VI WAVEFORM PARAMETER RESULTS FOR THREE DEVICES
TABLE VII IMPACT VELOCITY FOR THREE DEVICES
Since the control algorithm relies only on the continuity measurement across the contacts, it is imperative that this measurement be accurate. Contamination between the two electrodes could cause the continuity to degrade, even at impact, resulting in a larger number of iterations to find a solution, or no solution at all if the contamination is large enough and is not removed with the motion of the device. However, the fact that only a continuity measurement is required makes the control practical. For design and monetary cost, the continuity measurement could be placed in the circuitry on the chip such that the device could be calibrated at the demand of the user while in service, not necessarily in a laboratory setting. R EFERENCES
The velocity at impact for devices 1 and 2 is compared in Table VII. In all cases, the final waveform reduced the impact velocity, thus reducing the bounce event. Two different input waveforms were applied to device 3, referred to as set A and set B, respectively, to explore the effect that changing the initial waveform has on the control algorithm output. Set A uses the parameters described in Table V. Set B uses those parameters in Table V, as well, but begins with a tp value of 10 μs. The values of tp and tc in the two tests on device 3 are compared in Table VI. Upon elimination of bounce, the total closure time was reduced to the same value as set A, 17 μs. The velocity at impact for device 3 is also compared in Table VII. The results of the experiments demonstrated that the learning algorithm could be implemented to reduce the velocity of the initial impact and thus remove any significant bounce of the switch plate. The elimination of bounce greatly reduces the time required for closure of the switch.
VII. C ONCLUSION A simple strategy was developed to generate an open-loop input to an RF-MEMS switch that is subject to parameter variation from part to part. The algorithm uses the continuity measured across the switch to adjust the parameters. Simulations show that the strategy can account for parameter uncertainty as well as a direct relationship between the number of iterations and the magnitude of variation. Future work could involve further investigation into this relationship to find an optimal compromise between iterations and uncertainty. The algorithm was also tested experimentally on three different switches and shown to eliminate bounce. The elimination of bounce due to the reduction of impact velocity to less than 0.20 m/s allowed the total closure time to be no larger than 20 μs. For the experimental data sets, 5–10 iterations were required to find a bounce-free solution. It is also shown that changing the initial parameters of the waveform still results in a similar solution when applied to the same device.
[1] J. J. Yao, “RF MEMS from a device perspective,” J. Micromech. Microeng., vol. 10, no. 4, pp. R9–R38, Dec. 2000. [2] J. Bryzek et al., “Marvelous MEMS,” IEEE Circuits Devices Mag., vol. 22, no. 2, pp. 8–28, Mar./Apr. 2006. [3] D. A. Czaplewski et al., “A soft-landing waveform for actuation of a single-pole single-throw ohmic RF MEMS switch,” J. Microelectromech. Syst., vol. 15, no. 6, pp. 1586–1594, Dec. 2006. [4] H. Sumali, J. E. Massad, D. A. Czaplewski, and C. W. Dyck, “Waveform design for pulse-and-hold electrostatic actuation in MEMS,” Sens. Actuators A, Phys., vol. 134, no. 1, pp. 213–220, Feb. 2007. [5] C. W. Dyck et al., “Fabrication and characterization of ohmic contacting RF MEMS switches,” in Proc. SPIE Conf. Micromachining Microfabrication, 2004, vol. 5344, pp. 79–88. [6] D. M. Burns and V. M. Bright, “Nonlinear flexures for stable deflection of an electrostatically actuated micromirror,” in Proc. SPIE: Microelectron. Struct. MEMS Opt. Process. III, Sep. 1997, vol. 3226, pp. 125–136. [7] X. Wu, R. A. Brown, S. Mathews, and K. R. Farmer, “Extending the travel range of electrostatic micro-mirrors using insulator coated electrodes,” in Proc. Int. Conf. Opt. MEMS, 2000, pp. 151–152. [8] M. A. Rosa, D. De Bruyker, A. R. Völkel, E. Peeters, and J. Dunec, “A novel external electrode configuration for the electrostatic actuation of MEMS based devices,” J. Micromech. Microeng., vol. 14, no. 4, pp. 446– 451, Apr. 2004. [9] R. Legtenberg, J. Gilbert, S. D. Senturia, and M. Elwenspoek, “Electrostatic curved electrode actuators,” J. Microelectromech. Syst., vol. 6, no. 3, pp. 257–265, Sep. 1997. [10] E. S. Hung and S. D. Senturia, “Extending the travel range of analog-tuned electrostatic actuators,” J. Microelectromech. Syst., vol. 8, no. 4, pp. 497– 505, Dec. 1999. [11] J. I. Seeger and B. E. Boser, “Charge control of parallel-plate electrostatic actuators and the tip-in instability,” J. Microelectromech. Syst., vol. 12, no. 5, pp. 656–671, Oct. 2003. [12] J. M. Kyyarainen, A. S. Oja, and H. Seppa, “Increasing the dynamic range of a micromechanical moving-plate capacitor,” Analog Integr. Circuits Signal Process., vol. 29, no. 1/2, pp. 61–70, Oct. 2001. [13] E. K. Chan and R. W. Dutton, “Electrostatic micromechanical actuator with extended range of travel,” J. Microelectromech. Syst., vol. 9, no. 3, pp. 321–328, Sep. 2000. [14] R. Nadal-Guardia, A. Dehe, R. Aigner, and L. M. Castaner, “Current drive methods to extend the range of travel of electrostatic microactuators beyond the voltage pull-in point,” J. Microelectromech. Syst., vol. 11, no. 3, pp. 255–263, Jun. 2002. [15] O. Bochobza-Degani, D. Elata, and Y. Nemirovsky, “A general relation between the ranges of stability of electrostatic actuators under charge or voltage control,” Appl. Phys. Lett., vol. 82, no. 2, pp. 302–304, Jan. 2003. [16] P. B. Chu and K. S. J. Pister, “Analysis of closed-loop control of parallelplate electrostatic microgrippers,” in Proc. IEEE Int. Conf. Robot. Autom., May 1994, pp. 820–825. [17] L. A. Rocha, E. Cretu, and R. F. Wolffenbuttel, “Using dynamic voltage drive in a parallel-plate electrostatic actuator for full-gap travel range and positioning,” J. Microelectromech. Syst., vol. 15, no. 1, pp. 69–83, Feb. 2006. [18] S. Arimoto, S. Kawamura, F. Miyazaki, and S. Tamaki, “Learning control theory for dynamical systems,” in Proc. IEEE Conf. Decision Control, Dec. 1985, pp. 1375–1380. [19] A. Fargas-Marques and A. M. Shkel, “On electrostatic actuation beyond snapping condition,” in Proc. IEEE Sensors, Nov. 2005, pp. 600–603.
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[20] Y. Nemirovsky and O. Bochobza-Degani, “A methodology and model for the pull-in parameters of electrostatic actuators,” J. Microelectromech. Syst., vol. 10, no. 4, pp. 601–615, Dec. 2001. [21] A. H. Nayfeh, M. I. Younis, and E. M. Abdel-Rahman, “Dynamic pullin phenomenon in MEMS resonators,” Nonlinear Dyn., vol. 48, no. 1/2, pp. 153–163, Apr. 2007. [22] B. McCarthy, G. G. Adams, N. E. McGruer, and D. Potter, “A dynamic model, including contact bounce, of an electrostatically actuated microswitch,” J. Microelectromech. Syst., vol. 11, no. 3, pp. 276–283, Jun. 2002. [23] T. Veijola, “Compact models for squeezed-film dampers with inertial and rarefied gas effects,” J. Micromech. Microeng., vol. 14, no. 7, pp. 1109– 1118, Jul. 2004. [24] J. Blecke, J. Berg, D. S. Epp, and H. Sumali, “RF-MEMS switch system identification for control,” in Proc. Int. Modal Anal. Conf., Jan. 2007, Paper 260. [25] Y. Tsypkin, “Self-learning—What is it?” IEEE Trans. Autom. Control, vol. AC-13, no. 6, pp. 608–612, Dec. 1968. [26] K.-S. Fu, “Learning control systems—Review and outlook,” IEEE Trans. Autom. Control, vol. AC-15, no. 2, pp. 210–221, Apr. 1970.
Jill C. Blecke (S’06–M’07) received the B.S. degree from the Mechanical Engineering Department, Michigan Technological University, Houghton, in 2005, where she is currently working toward the Ph.D. degree. Her research interests include modeling and control of MEMS, as well as dynamic testing. Ms. Blecke is a Student Member of the American Society of Mechanical Engineers.
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Hartono Sumali received the M.S. and Ph.D. degrees in mechanical engineering from Virginia Polytechnic Institute and State University, Blacksburg, in 1992 and 1997, respectively. From 1997 to 2002, he was an Assistant Professor with Purdue University, West Lafayette, IN. He was also with McDermott, Inc., Houston, TX; Caterpillar, Inc., Peoria, IL; and the U.S. Naval Research Laboratory, Washington, DC. He is currently a Technical Staff Member with Sandia National Laboratories, Albuquerque, NM. His research activities are in the mechanical dynamics of MEMS, and in structural dynamics.
Gordon G. Parker (M’95) received the B.S. degree in electrical and systems engineering from Oakland University, Rochester, MI, in 1987, the M.S. degree in aerospace engineering from the University of Michigan, Ann Arbor, in 1988, and the Ph.D. degree in mechanical engineering from the State University of New York, Buffalo, in 1994. From 1989 to 1996, he was with General Dynamics Space Systems and Sandia National Laboratories, Albuquerque, NM, working in the areas of structural dynamics and control. Since 1996, he has been with the Mechanical Engineering–Engineering Mechanics Department, Michigan Technological University, Houghton, where he is currently the John and Cathi Drake Professor of Mechanical Engineering. His research interests focus on nonlinear control system analysis and design. His application areas span a variety of different disciplines, including active structures, chemical kinetics, systems biology, MEMS devices, and spacecraft.
David S. Epp received the B.S. degree in mechanical engineering and the M.S. and Ph.D. degrees in the area of control systems from the University of Oklahoma, Norman, in 1997, 2001, and 2002, respectively. During the latter part of his doctoral studies, he was a Consultant with Scrub Oak Technologies, doing large-scale structural control work. He is currently a Technical Staff Member with Sandia National Laboratories, Albuquerque, NM, working in the areas of structural dynamics and mechanical testing.
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