ADAPTIVE CONTROL OF MEMS DEVICES

Transcription

1 ADAPIVE CONROL OF MEMS DEVICES Afshin Izadian, Lawrence Horna and Parviz Faouri, Lane Deartent of Couter Science and Electrical Engineering, Electro Mechanical Systes LAB West Virginia University ABSRAC he anufacturing of Micro-Electro-Mechanical Systes (MEMS) involves several icro-achining rocesses, each include variations in the araeters of the device fro the desired values. hese variations change the behavior of the device and affect its erforance, therefore there is a need of eans to coensate these changes and control the device s behavior. In this aer, soe anufacturing stes that result in uncertainties in odeling of the device are investigated. An effective ethod to control their behavior is introduced and is also exerientally tested. In this regard, alication of the continuous-tie Model Reference Adative Controllers (MRAC) for tracing control of icro cob resonators is investigated. he rocedure of the controller design and odeling of icro cob resonator considering the uncertainties are investigated. he controller is ileented in a real-tie control board and the exeriental results deonstrate the caabilities of MRAC in a ulti-uncertainty MEMS otion control tas. KEY WORDS Micro cob resonator, odeling, araeter uncertainty, adative control, tracing. Introduction he increasing colexity and integration level of MEMS devices deand high level and colicated controllers to obtain the required erforance. Manufacturing rocesses of MEMS also doinate the need of the sohisticated controllers by including the uncertainties in the araeters of the device. hese rocesses also require the control theories to develo the recision and enhance their quality []. o date, the alication of control schees to MEMS have lagged in develoent coared to MEMS fabrication and structure design. here are enorous challenges to atheatically odel MEMS devices accurately and deterine their level of uncertainties in different geoetries and alications. Lateral cob resonators used in any alications such as gyroscoes and the drivers of icro-systes are of articular interest to this aer. heir anufacturing requires different stes which are recognized as bul and icro achining rocesses. Each of these fabrication stes involves ultile rocesses such as deosition, etching, and growing of aterial and layers. Different technologies aly different stes, so that the nuber of achining rocesses can be different []. Each of these achining rocesses ay include an ierfection of anufacturing in the construction of the device. herefore, the anufactured device araeters are different fro that of ideally designed. In addition to the anufacturing ierfections, oerating conditions such as teerature, huidity, and air ressure have unnown effects on the device araeters. Also, the storage condition, durability and life-tie of the device affect its erforance and reliability. Conventional controllers require the nowledge of araeters of the device to erfor a suitable control tas. he oen loo controller designed by Ernest J. Garcia et al [3] have a sile square wave driven ethod to control a icro engine to rovide enough torque for icro syste structures. Closed loo controllers such as conventional PID s are designed according to the nown araeters of syste [4]-[7]. Advanced controllers ae use of the off-line and online syste identification techniques [], [4]. Robust controllers with a nown uncertainty level are also been develoed [8]. In this case, extra anufactured caacitor sensors were rovided to easure the dislaceent [9]. his aer resents a feasibility study of alication of the continuous-tie odel reference adative control to icro cob resonators. In this aer, the reference is defined as the outut of the odel in which the controller drives the lant to follow odels outut trajectory, and to obtain zero tracing error. he echanical odel of MEMS including uncertainties of the odel and henoena such as fringing are introduced. he controller design rocedure for non-strictly Positive Systes (non-spr) is investigated. Siulations of the tracing controller show the alication of MRAC on MEMS. he controller is ileented onto the real-tie control board and exeriental results are introduced. An integrated through wafer otical data onitoring syste is used to easure the systes dislaceent [4] - [6], [8], and [0].. Modeling of MEMS Cob Resonator A tyical icro cob resonator shown in Figure, consists of the oving stage also nown as shuttle, the stationary cobs, and the susension srings. he cob resonator can be odeled as a ass-sring-daer in a second order syste. Often MEMS devices oerate according to the electrostatic force generation which is generated by alying two identical oosite sign DC voltages onto the stationary cobs at both stationary sides

2 of the device and a tie varying voltage (e.g. a sine wave) alied to the center shuttle. his rovides enough force for the device to ove and follow certain trajectory to erfor a secific tas deending on the alication. he electrostatic force () ideally is a function of the alied voltages and the geoetry of the device, but in fact there are so any other henoena and factors that affect the force generation rocess that are not considered in the electrostatic force equation [] t Fe = 4nε ( Vb. Vs ) () g where F e is the electrostatic force, n is the nuber of fingers, ε is the erittivity of air, t is the length of a finger, g is the ga between fingers, V b, Vs are alied body and shuttle voltages resectively. he force is also a function of soe ierfect anufacturing stes such as the roughness of the surface of olysilicon, acid releasing and etching stes. Moreover, henoena such as huidity and caillary forces result in the stiction of olysilicon which affect the force generation of the device. herefore; the exact force generation is an uncertain rocess of soe oorly nown araeters. In a ass-sring-daer syste, a sile odel for the syste can be written as: F e = y + β y + s y + Fd () where is the ass of the center cob or shuttle, y is the dislaceent, s is the sring constant, β is the daing coefficient, and F d is the force of the load. he load can be alied directly onto the shuttle. As entioned earlier the araeters of the device are highly affected by the anufacturing rocess as well as the oerating condition environent. During the fabrication, the olysilicon layers are deosited in a teerature controlled rocess in a chaber, called Low Pressure Cheical Vaor Deosition (LPCVD). he quality of the deosited layer is directly related to the theral rocess and the tie of the deosition of LPCVD. he deosited olysilicon layers roduce certain regions naed grains which have different crystalline orientations. he grains grow during theral rocess and change the roerties of olysilicon. Roughness on the surface due to isalignent of the device in anufacturing also doinates the field fringing. he layers of icro cob resonator are searated fro each other by sacrificial layers ade by silicon oxide which are reoved during the release rocess. he total ass existing in oving arts varies in different devices even if they are anufactured in the sae rocess at the sae tie. hus; the ass involves high uncertainties and result in the change of the critical araeters of the device such as resonant frequency. he daing coefficient in equation () is a function of air viscosity and geoetry of the lates facing each other. It is also related to enetration deth of airflow above the shuttle [4], [6]. Certain anufacturing stes such as hotolithograhy require exact alignents and UV exosure techniques in addition to the etching and releasing of the bare areas which are not covered by hotoresist aterial. hese stes are highly deendent on the tie of exosure, alignents and release rocess. herefore; in anufacturing of soe recise arts such as susension srings, the anufacturing tolerances result in crucial araeter variations. he sring constant of the susension srings is very sensitive to the width, geoetry and density of the olysilicon; thus, ierfections of the anufacturing rocesses result in uncertain values of the sring constants. Under these uncertainties, equation () does not contain the exact odel and the actual syste s behavior, therefore it can be re-written as F ˆ e = y ˆ + ˆ β y + ˆ s y + Fˆ d (3) where the (^) sign shows the uncertain values of the associated araeters. Figure, MEMS Micro Cob Resonator and the oenings on the shuttle designed by WVU. he fact is that the desired syste should oerate siilar to () whereas the anufactured device behaves differently as (3). hen, the ain urose is to design a controller in which regardless of uncertain and unnown henoena of MEMS syste, be able to control the device and erfor the tracing of the desired reference. 3. Controller design As Eq. () shows, icro cob resonators are odeled as a second order syste. he transfer function of this syste shows a relative degree higher than one. In the next section the controller design rocedure for non-spr systes such as MEMS is described. Controller Structure he uncertain syste (3) can be written in state sace equation for as following x = x x = ( ˆ s x ˆ Fˆ Fˆ ˆ e βx ) + ˆ he transfer function of (4) can be written as G ( P) d (4) = (5) + a + a 08

3 where a, a are the odel s desired coefficients, is the gain. Let r( be the inut signal to the odel, y ( the resultant outut signal, and the Lalace oerator. Considering general coefficients for uncertain araeter values result in the following non-spr second order transfer function G ( ) = (6) + a + a he lant s transfer function W ( ) contains the unnown and uncertain araeter values such as a, a, with outut y ( and control inut u( signals. he controller consists of soe additional control coands such as, which are called filtering signals. Each of these signals reresents an (n-) th order dynaic and are described by [] Γ + hu (7) = where in a ( n ) state vector, Γ is a ( n ) ( n ) atrix, h is a constant vector such that ( Γ, h) is controllable. Γ is chosen to lace sae oles as zeros of odel, or det[ si Γ] = N ( s) (8) Model he sae dynaic is taen for with following [] y as inut, as = Γ + hy (9) In adatation technique the araeters of the controller naely θ, θ, θ 0, each containing ( n ) araeters, are udated recursively. is defined as the gain ratio of the odel and lant and that is = (0) he coefficients of θ dynaic cancel the zeros of the lant and θ,θ 0 lace the oles of the control syste to the location of the reference odel oles []. herefore the control coand is a linear cobination of the reference signal inut, the filtered control coand vector signal, the filtered lant s outut vector, and control coand signal as [] u ( = r + θ + θ + θ () 0 y where the () sign shows the estiation of araeters. In a erfect tracing and adjusting araeter technique, zeros of the lant are cancelled by oles of the controller in a iniu hase lant case; thus, the outut of the lant due to control coand should be identical to that of reference and arbitrary bounded inut as y ( = G ( s) u ( G ( s) r( () = where and stand for the lant and odel transfer functions resectively. he adatation technique results in the following control coand [] u = ( r + θ ( + θ ( + ( (3) ( θ 0 Figure shows controller structure of the odel reference adative for systes with relative degree higher than one. Let θ be a n vector of controller araeters and be a n vector of the corresonding signals as following [ θ θ ] θ ( ) = θ t 0 [ r( ( ( y ( t ] t ) = ) (4) ( he control coand in atrix for is written as y u = θ ( ( (5) he estiation error is defined as the difference between the actual value and its estiation which is Substituting (6) in (5) results in θ ( = θ + Φ( (6) u = θ + Φ ( (7) Considering the estiation error and transfer function (6), the lant s outut is written [] Φ ( y ( = G ( s) r + G ( s) (8) he tracing error is the error between the lant outut and that of the odel. Subtracting y ( for the odels yields the error as Φ ( ( e = G ( s) (9) In this case the adatation technique is written as [] θ = sgn( ) γ e ( t ) ( t ) (0) where γ is a ositive nuber and is called the adatation gain, and also sign of is the sae as that of. Design of an adative control syste, when just the lant s inutoutut signals are available and the relative degree of the syste is grater than one, can be accolished by using the augented error technique. he basic of this ethod is adding an augenting signal with an adative gain α ( to the tracing error e (. hen the augented error signal is as followings [] where ε ( = e( + α( () is defined as = φ G G φ () 09

4 Let consider α ( as a function of tie as following α ( = + e ( α (3) or eα ( = α( (4) Substituting (4) into () yields relacing fro (9) results in ε ( = e( + ( + e ( ) α (5) ε ( = e( + + e ( α (6) φ ε ( = G + + e ( α (7) relacing fro () and defining ϖ = G (8) yields ε ( = φ ϖ + e ( α (9) he augentation error gain adjusting technique is according to the followings [4] sgn( ) γεϖ θ = (30) + ϖ ϖ γεη α = (3) + ϖ ϖ Figure 3 shows the adatation law using the augented error technique. Figure, Model reference adative controller for systes with higher orders than one. he auxiliary error signals, filters and control coand are shown 4. Dislaceent Otical Monitoring echnique he osition feedbac is obtained by the otical onitoring technique which has been used to validate the recise dislaceent onitoring of the device. hrough wafer otical dislaceent onitoring syste, shown in Figure 4, contains a laser wave-guide of wavelength 30 n eitted fro a laser diode. wo bul lenses focus the bea on the device fro the botto side which is olished by a 0. icroeter olisher in order to reduce the bea scattering and ray loss. he laser coing out of the die is collected and directed to the detector through a ulti-ode detector fiber otic robe which houses a single ode detector fiber. As the shuttle center cob oves, its body cuts the laser bea through the oenings and roduces ulses. he detector then exeriences the intensity changes of the bea due to the shuttle dislaceent, so that the outut voltage of the detector contains the oveent inforation of the MEMS. Figure 3, Model Reference Adative Control for non-spr systes, adatation technique and error augentation technique. he alignent of the outut laser bea and the detector fiber otic coing out of the oenings on the device is a critical oint. he Gaussian distribution of the laser bea allows us to concentrate on the oenings and reduce the sot size on the device [5]. Figure 4, Dislaceent feedbac syste, through waver otical onitoring, including laser eitter, bul lenses and icro cob resonator are shown. he collected otical data then is transferred to the data recovery algorith which deterines the instantaneous dislaceent of the shuttle [4], [5]. he control coand then is generated by adative controller and is alied to the device. 5. Siulation and Exeriental Results MRAC is used to coensate the uncertainties of the syste by following siulations. Different control cases are siulated and the tracing error and control effort are shown in each case. Siulation Results Siulations of the designed controller under uncertainty of ass, sring constant and daer coefficients are investigated in this section. Different tyes of uncertainty iose different reflections on the control effort of the controller. he ideal odel contains the following araeter values =.090e 0 ( g ), β = 7.09e 7, = Uncertainties on Mass of the device As entioned earlier the ass of the shuttle of icro cob resonator varies according to the variations of the hotolithograhy and LPCVD rocess. Large articles fall down onto the device ight also change the ass of the shuttle. Mass changes have an effect on the resonant frequency and result in different erforance of the device. Outut tracing rofile of icro cob resonator with a high value of ass change of ten ties, is shown in s 0

5 Figure 5. he tracing error as an indicator of the tracing erforance is shown in Figure 6. his figure shows that the error reaches zero as tie increases. value. As Figure 9 shows, the tracing error reaches zero as tie increases raidly after the adatation is coleted. Figure 0 shows higher control efforts to coensate the sring constant uncertainties. Figure 5, Mass uncertainty effect odeling and tracing erforance of MRAC for 0 ties over-estiation Figure 7, odeling the effect of ass uncertainties on control coand generated by MRAC. Figure 6, racing error rofile of MRAC in ass uncertainty case Uncertainties on daing coefficient Daing coefficient uncertainties are caused by oerating condition environent such as air viscosity, teerature and huidity changes. Under-daed systes usually show longer oveents and slower oscillation decays. In the case of a change of the daing as five ties of the actual daing of the syste the controller can still rovide zero tracing error at an accetable control effort to coensate the variations. he tracing rofile in this case is shown in Figure 8. racing error reaches zero raidly as the adatation is coleted. As it is seen the daing coefficient uncertainties are odeled as hase shift in the outut of the actual syste. racing error shows no difference and a colete hase shift coensation using MRAC controllers. he control effort is shown in Figure 8. As it is seen the controller does not require too uch effort to coensate the hase shift. Uncertainties on sring constant Sring constant variations result in a significant behavioral change on the device. Syetric dislaceent and resonant frequency are highly deendent on the delicate structure and geoetry of MEMS sring structure and sring constants and as a consequence affect the erforance of oeration. Deviations fro the desired values iose an asyetric oveent or the size change which burdens extra effort on the controller. Figure 9 shows the tracing erforance of the device with sring constant change twice as uch as the desired Figure 8, Daing coefficient uncertainty odeling, erforance of the tracing control, and tracing error rofile are shown. Figure 9, Sring constant uncertainty odeling, tracing control erforance and tracing error signal of icro cob resonator. Exeriental Results he actual device under test contained a cobination of all anufacturing uncertainties which caused drastic variations fro the desired behavior. he test device was released for the silicon oxide several onths before MRAC tracing test and had been stored in a glass container. herefore the huidity aging and wearing effects were considered unnown. he controller was ileented on the real-tie control board dspace- PPC-DS 03 and ade use of the through wafer otical onitoring dislaceent technique. Alying the control coand, tracing rofile of the outut is shown in Figure.

6 Figure 0, Modeling the effect of sring constant uncertainty on control coand generated by MRAC. According to the ulse tye behavior of feedbac signal to the syste the controller coensated a art fro the otical data variations and resulted a close tracing of the reference. Figure shows the effort of the controller in this case. Figure, Exerient results and tracing erforance of a icro cob resonator with ultile uncertainties controlled by MRAC. Reference Signal Control Effort Figure, Exerient results of a) Reference signal, b) Control Effort; Multile uncertainties controlled by MRAC. he control effort was %57 higher than that of the reference wavefor, which according to the siulation results show the uncertainty on stiffness of the susension srings of the anufactured icro cob resonator. Most of the effects of this tye of uncertainty cae fro a cobination of all the uncertainties on the device. After releasing the MEMS fro the silicon oxide layer the caillary forces roduced ore stiction and resulted in uncertainties on the force generation rocess, therefore higher control efforts were required. 6. Conclusion his aer showed the feasibility study of alication of adative controllers to MEMS icro cob resonators. We have successfully alied a continuous-tie MRAC for tracing control of icrosystes. Modeling of the structure of icrosyste with uncertain araeters was shown. he augentation technique was used in the design of the controller. Different uncertainties were odeled and their effects on control effort were studied. he laboratory test results of the controller on real-tie control board showed satisfactory erforance of the controller. We have shown both by siulation and exerient results, that the odel reference adative controllers can control the icro cob resonators in the existence of ulti-uncertainty araeter values. Acnowledgent his wor was suorted in art by NASA and NSF. Authors would lie to than WVnano facilities. References [] J. Bryze. E. Abbot, Control issues for MEMS. 4 nd Proceedings of the IEEE Conference on Decision and Control, Maui, Hawaii, USA, Deceber 003. [] S. Par, R. Horowitz and Chin-Woo an. Adative controller design of MEMS gyroscoe, IEEE intelligent transortation Syste Conference Proceeding- Oaland (CA), USA- August 5-9, 00. [3] Garcia, et al., Surface icro achined icro engine. Sensors and Actuators, Vol. 48 (3). 03-4, 995. [4] L. Wang, J. M. Dawson, L. A. Horna, P. Faouri, R. Ghaffarian, Real ie ransitional control of a MEMS cob resonator. IEEE transaction on Aerosace and electronics. Vol. 40, No., Aril 004. [5] J. M. Dawson, J. Chen, K.S. Brown, P. Faouri, L. A. Horna, hrough wafer otical robe characterization for icro electroechanically systes ositional state onitoring and feedbac control, Otical engineering, Vol. 39, No., Deceber 000 P [6] J. Dawson, L. Wang, J. Chen, P. Faouri, and L. A. Horna, MEMS Feed bac Control Using through Wafer Otical Device Monitoring, Proceeding of SPIE, he international Society for Otical Engineering Volue Seteber 000, Santa Clara, USA. [7] L. Wang, J. M. Dawson, J. Chen, P. Faouri, and L. A. Horna, stroe-length control of a MEMS device. ISIE000, Cholula, Puebla, Mexico. [8] J. Chen, P. Faouri, L. A. Horna, nonlinear control of MEMS: icroengine sliding ode control siulation. Proceeding of the IASED international Conference, intelligent systes and control, October 8-30, 999, Santa Barbara, California, USA. [9] P. Chueung, R. Horowitz, R.. Howe, Identification, Position sensing and control of an Electrostatically Driven Polysilicon Micro actuator. Proceeding of the 34th Conference on Decision and Control, New Orleans, LA- Deceber 995. [0] M. B. Khaesse, N. Kato,. Naaaura, Perforance iroveent of a agnetically levitated icro robot using an adative control. Proceeding of the international Conference on MEMS, NANO and sart Systes (ICMENS 03) 003 IEEE. [] J. J. Sniegowsi, E. J. Garcia, Micro-fabricated actuators and their alication to otics. Proc SPIE Vol. 384, P [] J. Jacques, E. Slotine, W. Li, Alied nonlinear control (Prentice Hall international, 99). [3] W. S. Levine, he Control Handboo (CRC & IEEE Press 996). [4] R. V. Monooli, odel reference alied control with an augented error signal. IEEE rans. on Autoat. Contr. Vol. AC Oct [5] J. M. Dawson. PhD dissertation West Virginia University, Integrated through Wafer Otical Monitoring of MEMS for closed loo Control (WVU, Morgantown, fall 00). [6] X. Zhang, WC. ang, Viscous air daing in laterally driven icro resonators. IEEE, Micro Electro Mechanical Systes Conference, 994,