Closed-loop Voltage Control of a Parallel-plate MEMS Electrostatic Actuator

Transcription

1 Amerian Control Conferene Marriott Waterfront, Baltimore, MD, USA June -July, ThB8. Cloed-loo Voltage Control of a Parallel-late MEMS Eletrotati Atuator Lili Dong, *, Member, IEEE, Jaon Edward Abtrat Thi aer addree the ontrol roblem of extending the travel range of a MEMS eletrotati atuator through a loed-loo voltage ontrol heme. Sine the eletrotati atuator i inherently untable due to it ull-in limit, one of the major ontrol goal i to tabilize the atuator ytem beyond the limit. In addition, the ontroller ha to be robut againt external diturbane and noie. After omaring and analyzing the advaned ontroller being reorted, an ative diturbane rejetion ontroller (ADRC) i originally emloyed to the miro atuator to olve the ontrol roblem. The ADRC mainly onit of an extended tate oberver (ESO) and a PD ontroller. The ESO i ued to etimate ytem tate and the external diturbane, whih an be taen a an augmented tate of the ESO. The PD ontroller baed on the oberved tate drive the dilaement outut of the atuator to a deired level. The ADRC i uefully imulated onto a arallel-late eletrotati atuator. The imulation reult verified the effetivene of the ontroller through extending the travel range of the atuator to 99% of the initial ga between two late in the reene of noie and diturbane. Index Term Eletrotati atuator, MEMS, ADRC, ESO, tability, robutne. M I. INTRODUCTION EMS eletrotati atuator, alo termed a miroatuator, are the ey devie allowing MEMS to erform hyial movement []. They have the advantage of mall ize, low ot, and low ower onumtion. An imreive range of aliation demontrate the eletrotati atuator utility. Some examle of the aliation are: otial grate and hutter, miro-mirror, variable aaitor, and miroaelerometer []. A one-degree-of-freedom, arallel-late miro-atuator onit of a movable late and a fixed late in an eletri field. When the movable late i dilaed from it original oition, the aaitane formed between the two late will be varying. The dilaement of the movable late i uually hanged through a voltage (or harge) ontrol of the ga of the aaitor. However, a the ga between the two L. Dong i with the Deartment of Eletrial & Comuter Engineering, Cleveland State Univerity, Cleveland, OH 5, USA. J. Edward i with the NASA Glenn Reearh Center, Cleveland, OH 5, USA. *Correonding author. L.Dong@uohio.edu. late i inreaed to one third of original ga, a ull-in henomenon will aue the intability of the ytem and drag the movable late immediately to the fixed lated [, ], auing the failure of oeration of the atuator. Therefore, extending the traveling range of the movable late beyond the ull-in limit ha beome a entral toi in the ontrol of eletrotati atuator. A method or mehanim to extend that travel range, referably to the extent of the entire initial ga, i highly deirable, eeially for otial aliation []. Eletrotati atuator oerate in two energy domain, eletrial and mehanial []. It i thu oible to extend the uable range of the atuator both mehanially and eletrially. In the mehanial domain, one traightforward aroah i to deign the ga o large that the atuator i table over the deired oerating range [].The drawba of thi aroah i that the maximum ga i omletely deendent on miro-fabriation tehnology and an not be hanged by the deigner. The other aroah i to ulement the elati retoring fore of the uort o a to avoid the ull-in henomena. Thi aroah ha been imlemented by uing leverage and nonlinear, tiffening ring [-5]. However, trengthening the elatiity of the uort will ruire inreaed voltage to enhane the travel ditane, thu raiing the ower onumtion and the ot. In the eletrial domain, both oen and loed-loo ontrol trategie have been alied to the atuator. The wor of [] demontrated an inreaed travel range u to 8% of the initial ga uing an oen-loo harge ontroller. However, ine the oen-loo ontrol i enitive to the external diturbane, the loed-loo ontroller baed on outut feedba, for whih additional eletrode or enor for meauring the oition of the movable late of the atuator are needed, have been reorted in [6-]. A an alternative to harge ontrol, voltage ontrol i oular and effetive [6]. A eial voltage oure with a erie aaitor [6-8] i utilized to hange the travel ditane of the atuator. The tehnique howed table oeration of the atuator at %, 6%, and 9% of the initial ga. The diadvantage of the aroah are that it i deendent on an aurate mathematial model of the atuator and it ruire large atuation voltage. A linear time-varying roortional gain ontroller i develoed in [9], where a quantitative feedba theory (QFT) [] i utilized to deign the //$6. AACC 9

2 roortional gain. The travel ditane that the roortional gain ontroller an drive i 6% of initial ga. Unfortunately the effet of noie are not onidered by the ontroller. A Lyaunov-baed nonlinear ontrol aroah ha been reorted in [-] to extend the travel range to over 9% of initial ga. Partiularly in [], the imulation reult how a maximum travel range of full ga without overhoot. In general, the aroahe in [-] are effetive, but their utility i omewhat limited by it mathematial omlexity, maing it very diffiult to imlement in the real world. The goal of thi aer i develoing a robut and eay-toimlement ontroller to extend the travel range of the eletrotati atuator to over 9% of it initial ga in the reene of external diturbane and noie. A loed-loo voltage ontrol will be utilized. In order to ahieve the ontrol goal, an ative diturbane rejetion ontroller (ADRC) i alied to the atuator. The ADRC ha been uefully emloyed in maro ytem and MEMS gyrooe [5-7]. It i the firt time that we modified and develoed it on the MEMS eletrotati atuator. One of the advantage of the ontroller i that it doe not rely on aurate mathematial model of a lant, whih for the eletrotati atuator varie greatly over it oerating range. Another advantage of the ADRC i that it only ha three tuning arameter, maing it imle to imlement in ratie. The ret of thi aer i organized a follow. The dynami modeling of the eletrotati atuator i given in etion II. The deign of the ADRC i reented in etion III. The imulation reult are hown in etion IV. Setion V give onluding remar and ugget future reearh. II. DYNAMIC MODELING OF ELECTROSTATIC ACTUATOR A imlified one-degree-of-freedom eletrotati atuator model wa eleted baed on [8] for ontroller deign. The eletro-mehanial model of the atuator i hown in Fig.. Fig.: Eletro-mehanial model of an eletrotati atuator From Fig., we an ee that an eletrotati atuator onit of a arallel-late aaitor with one fixed eletrode and one varying eletrode. The inut voltage oure V i modeled with a erie oure reitor, R. The variable I i the inut urrent. The initial ga with zero alied voltage i denoted by g. The ga g i oitive in the diretion of inreaing ga, while i the dilaement of the moving late and i oitive in the diretion of dereaing ga. The relationhi between g and i given by g g. () A. Firt Prinile Modeling A the harge Q on the two late build, the fore of attration grow, bringing the late loer together. In order to ee the late from naing down, there need to be an ual and ooite fore reiting thi motion. Thi fore i modeled by the retoring fore of a mehanial ring with ring ontant. A daming term, b, rereent the queezed-film daming oeffiient. It hould now be lear that thi devie i oerating in two energy domain, eletrial and mehanial. In mehanial domain, aording to Newton nd law, we have m& Fe Fb F () where F b b& i the linear queeze-film daming fore, F x i the linear mehanial ring fore and F e Q /εa i the nonlinear eletrotati fore where A i late area, and ε i dieletri ontant. Equation () an be rewritten a Q. () m&& b& εa Note that the ma of the uer late of the aaitor i o mall that the gravitational fore ating on the atuator an be negleted. Next, we will onider eletrial domain. Alying Kirhhoff' iruit law to the atuator give I V. () ( ) S V at R where V i the voltage aro the aaitor late. The at voltage aro the atuator i Qg V at. (5) ε A The urrent I an be olved by ubtituting (5) into (). Uing the fat that I S Q&, we have Qg Q&. (6) V R εa Subtituting () into (6) yield Q( g ) Q& VS. (7) R εa Equation () and (7) ontitute the nonlinear model of the eletrotati atuator. B. Equation Normalization For the imliity of later erformane analyi and ontroller deign for the eletrotati atuator, normalized uation of () and (7) are derived. The oition of the uer late relative to the lower late (), time (t), the harge built u on the late (Q), and the oure voltage (V ) are normalized a follow. x τ ω t Q V q v (8) g q i v i In (8), the dilaement i normalized by the ga with

3 zero alied voltage (g ), time i normalized by the natural frueny (ω ) of the atuator, harge i normalized by the aumulation of harge at ull-in (q i ), and the oure voltage i normalized by the ull-in voltage (v i ). From [8] the uation that govern the ull-in voltage, the amount of harge at ull-in and the aaitane at initial ga are given in (9). q i Cv, 8g, εa (9) i v i C 7C g The detail of normalization an be found in []. The reult of the normalization are rereented by where ς b mω & x + ςx& + x q q& + ( x ) q v r r, r ω RC, and ω m () (). () C. Model Linearization We hooe the tate variable of the normalized model of the atuator a x(t), q(t), and (t), where (t) i the veloity ( x& (t) ) of the movable late of the atuator. For mall-ignal linearization, the uilibrium value of the tate variable, whih are rereented by, Q, and S, have to be determined. Then the nonlinear uation will be exanded in term of erturbation from thee uilibrium value. The tate uation of normalized atuator model are x& x f x& x ςx + x f x& r r ( x ) x + v f The uilibrium oint are determined by olving f, f, and f. The olution are given by S, (/)Q, Q + v () Q. () A we hooe different uilibrium dilaement, we will have different Q. The uilibrium value of and Q orreonding to different erentage of the dilaement with reet to full ga are alulated and given in Table II and III in Aendix. Define, and Q. Then the linearized model i δx& δx + δx& ς δx δv δx& ( ) δx { r r r B δx δy [ ] δx C δx A (5) Aording to [], we ue ζ and γ.95 for the linearized model of the eletrotati atuator in (5). D. Tranfer Funtion Exreion and Pull-in Dilaement For the onveniene of later frueny-domain analye, a tranfer funtion rereentation of the linearized eletrotati atuator model i develoed. Conduting Lalae tranform on (5) and imlifying the tranformed uation will yield V ( ) Q. (6) () 9( r + ( + ςr ) + ( ς ( ) + r) + ( ) From (6), we an ee that when /, the tranfer funtion will have a ole at the origin. Any oerating oint with a dilaement greater than / will rodue a ole in the right half lane. Thi how exliitly how the eletrotati atuator beome untable at the ull-in dilaement of /. Sine the tranfer funtion rereentation of the atuator model (6) i a third-order lant, it an be rewritten a P % () b, (7) ( + a )( + a )( + a ) where a, a, and a are alar, and the ubrit % in P % () rereent the erentage of the dilaement with reet to full ga. The arameter value of b, a, a, and a an be obtained by omaring (7) and (6) with the uilibrium value of and Q lited in Aendix. III. CONTROLLER DESIGN Thi etion will briefly introdue ADRC deign in it tate ae rereentation at firt. Then a tranfer funtion rereentation of the ADRC will be derived. A linear ADRC for ontinuou-time ytem will be utilized in the aer. A. Introdution to ADRC From () and (), the eletrotati atuator an be modeled by a third-order differential uation a follow. & y && f ( y, y&, & y, d, t) + bu (8) In (8), the funtion f ( y, y&, & y, d, t), whih will be denoted a f in the following diuion, rereent all of the other fore on the atuator lant exluding ontrol effort, y i ual to normalized dilaement outut x(t), d denote an external diturbane fore, b i ontroller gain, and u i ual to the V in (). A we deign the ADRC, the funtion f i aumed to be unnown and i referred to a a generalized diturbane. The ESO will etimate the generalized diturbane. We hooe tate variable a x y, x y&, x & y and x f, whih i an augmented tate. Auming h f& and h i bounded within interet, we an rewrite (8) a x Ax + Bu + Eh y Cx & (9)

4 where A, B b, E, C [ ]. () From [5], the augmented tate x and the other tate an be etimated uing the ESO given a follow. z& Az + Bu + L( y yˆ ) () yˆ Cz In (), z i the etimated tate vetor and z[z, z, z, z ] T, where z, z, z, and z are the etimated x, x, x, and x reetively. The oberver gain vetor L i hoen o that all the oberver ole are loated at ω o, where ω o i taen a oberver bandwidth. A the oberver gain are given by (), the harateriti uation of the ESO will be (+ω o ). [ L L L L ] [ ω 6ω ω ω ] T L () o o With a well tuned oberver, the etimated tate z, z, z, and z will loely tra y, y&, & y& and f. We aume that i the etimated b. Then the ontrol inut to the atuator i hoen a u ( u z ), () where u denote a ontrol law. Equation () redue (8) to & y && u, () whih i a trile integrator lant. The lant an be ontrolled by a onventional roortional derivative ontroller, whih i o o u ( r [ d d ] z). (7) Define ontroller gain vetor a K[K, K, K, K ] [, d, d, ]. The Lalae tranform of (7) i (8) () ( R() KZ () ) U b ˆ Auming zero initial ondition for z(t) and it derivative, the Lalae tranform of () i Z ( ) ( I A + LC) [ BU ( ) + LY ( ) ] (9) where Z() [Z (), Z (), Z (), Z ()] T i an etimated tate vetor. Subtituting (9) into (8) yield [ ] () () R() K( I A + LC) ( BU () LY() ) U + Define matrix M a M ( I A + LC). Then () an be reorganized and imlified a U b ˆ + KMB KML () R() Y () + KMB () Equation () an be rereented by a loed-loo blo diagram a how in Fig., where H r () i a re-filter, C() the ontroller in feedba ath, P() the atuator model to be ontrolled, D() an external diturbane, and N() meaurement noie. The lant model P(), or P % () i given by (7). The re-filter H r () and ontroller C() are derived from () and are exreed a C KML + KMB () H () r + KMB. () u ( r z ) dz d z z. (5) In (5),, d, and d are ontroller gain and are hoen a ω ω. (6), d ω, d The ontroller gain above an lae all the loed-loo ole of the ontroller at ω, whih i taen a ontroller bandwidth. From (), (), and (6), we an ee that the ADRC inluding the ESO only ha three tuning arameter, ω, ω o, and. The few tuning arameter will enable imle imlementation of the ontroller in ratie. Although the ADRC ha been alied to MEMS gyrooe ([7]), it ha never been emloyed to eletrotati atuator before. The aer modified the ontroller and initially alied it to the eletrotati atuator. The detail about the aliation are given a follow. B. Tranfer Funtion Rereentation of the ADRC Combing () and (5), we an rewrite the ontrol inut a Fig.: Blo diagram of loed-loo ADRC-ontrolled ytem Subtituting the ontroller and oberver gain into () yield and where C () ( + d + d + d ) ( ) ( + d + d d ), () K + L + L + L + L (), () H r + K L + K L + K L + L K L + K L + K L K L + K L K L, and d K + L. d K + L + K L d K + K L + K L + L

5 IV. SIMULATION RESULTS In thi etion, the ADRC with two different et of tuning arameter will be diued. The value of the two et of tuning arameter are given in Table I. TABLE I: Deign ˆb TWO SETS OF TUNING PARAMETERS ω ( rad / ) ω ( rad / ) In Table I, the ontroller bandwidth ω wa hoen baed on the deired tranient reone of the ytem. The oberver bandwidth ω o i ontrained by the amlifiation of enor noie. The aroximation of the inut gain an be ued to fine tune the frueny reone of the loo tranmiion funtion of the ytem to maximize the tability margin of the ytem. From Table I, we an ee that the ontroller bandwidth i hoen fixed (ω rad/) ine thi value give the bet omromie between ontrol erformane and noie attenuation for the atuator. The detail about the tuning roe of the ADRC an be found in [6]. During the imulation, enor noie (random noie) i added to the ontrol ytem hown in Fig.. The two ADRC deign are imulated on the normalized model of the eletrotati atuator, whoe arameter value are given in Table IV of Aendix. Fig. omare the traing erformane of the two ADRC deign. In Fig., the atuator i ommanded to tra everal deired travel range whih are et to %, %, 5%, 7% and 9% of the full ga. Both deign have hown exellent traing erformane. Fig. and Fig.5 give loe-u view of the reone of the two deign at different travel range. A we hooe lant P5, the ea error in deign i about 7% while the ea error in deign i around %. So the larger ω o in deign lead to maller traing error a hown in Fig.. Fig. 6 how the te reone of the two deign with the dilaement being 99% of full ga. One an ee that the te reone for deign how a maximum overhoot erentage of %. Thi overhoot will limit the effetive travel range of the atuator to 99% of the full ga. Fig.7 illutrate the dilaement of the two deign a an external diturbane D() i added to the ytem at t5 eond. The figure demontrate the robutne of the two deign with a travel range of 97% of initial ga in the reene of the diturbane. In addition, figure to 7 alo verified the effetivene of the ontroller in the reene of the noie. o without the diturbane, the travel range an be extended to 99% of full ga. Comared to the other advaned ontroller in urrent literature with multile tuning arameter, the ADRC only ha three tuning arameter maing it imle to imlement in the real world. The two ADRC deign rooed in the aer offer exellent ontrol erformane while bridging the ga between the imle ontroller utilized in indutry and the advaned theoretial aroahe reented in aademia. With the attenuation of noie effet being uh an imortant iue in the eletrotati atuator, it would be rudent to ondut a more thorough analyi of the noie oure in the miro-ale environment in the future. A tiin henomenon in the atuator aued by miro-fabriation imerfetion will be invetigated a well. Fig.: Dilaement outut of two ADRC deign Fig.: Ste reone of the firt deign V. CONCLUSIONS Thi aer originally alied a modified ADRC to a linearized eletrotati atuator model. Simulation reult demontrate the effetivene of the ADRC through extending the travel range of the atuator to 97% of it full ga in the reene of enor noie and diturbane. If Fig.5: Ste reone of the eond deign

6 P P P P P P Fig.6: Ste reone of two deign with the dilaement of 99% of full ga Fig.7: Dilaement outut of two deign with te inut diturbane at t5 TABLE II: APPENDI EQUILIBRIUM POINTS PART I Q V TABLE III: EQUILIBRIUM POINTS PART II Q V TABLE IV: PARAMETER VALUES Plant b a a a P P P P P P REFERENCES [] H. Fujita, Miroatuator and Miromahine, in Pro. of the IEEE, vol. 86, no. 8, Aug. 998, [] J. Seeger, Charge Control of Parallel-late, Eletrotati Atuator and the Ti-in Intability, Journal of MEMS, vol., no. 5, , Ot.. [] L. M. Cataner and S. D. Senturia, Seed-energy Otimization of Eletrotati Atuator Baed on Pull-in, Journal of MEMS, vol. 8, no.,. 9-98, Se [] Y. Nemirovy, A Methodology and Model for the Pull-in Parameter of Eletrotati Atuator, Journal of Miroeletromehanial Sytem, vol., no.,. 6 65, De.. [5] E. S. Hung and S. D. Senturia, Extending the Travel Range of Analog-Tuned Eletrotati Atuator, Journal of MEMS, vol. 8, no., , De [6] J. Seeger and S. Crary, Stabilization of Eletrotatially Atuated Mehanial Devie, IEEE International Conferene on Solid-State Senor and Atuator, June 997,. -6. [7] E. Chan and R. Dutton, Eletrotati Miromehanial Atuator with Extended Range of Travel, Journal of MEMS, vol. 9, no.,. -8, Se.. [8] P. B. Chu, and K. S. J. Piter, Analyi of Cloed-loo Control of Parallel-late Eletrotati Mirogrier, in Pro. IEEE Intl. Conf. Roboti and Automation, 99, [9] M. Lu, M. Hirano, G. Fedder, Poition Control of Parallel-late Miroatuator for Probe-baed Data Storage, Journal of MEMS, vol., no. 5, , Ot.. [] I. Horowitz, Quantitative Feedba Deign Theory, QFT Publiation, Boulder Colorado, 99. [] G. Zhu, J. Penet, and L. Saydy, Robut ontrol of an eletrotatially atuated MEMS in the reene of araiti and arametri unertainty, in Pro. of the 6 Amerian Control Conferene,. -8, Jun. 6. [] G. Zhu, J. Levine, and L. Praly, Imroving the Performane of an Eletrotatially Atuated MEMS by Nonlinear Control: Some Advane and Comarion, in Pro. of the th IEEE Conferene on Deiion and Control and Euroean Control Conferene, Seville, Sain, De. 5, [] G. Zhu, L. Saydy, Robut Outut Feedba Control of an Eletrotati Miro-Atuator, in Pro. Of Amerian Control Conferene, NYC, NY, July 7, [] G. Zhu, J. Lévine, L. Praly, and Y. A. Peter, Flatne-baed Control of Eletrotatially Atuated MEMS with Aliation to Adative Oti: a Simulation Study, Journal of MEMS, vol. 5, no.5,.65-7, Ot. 6. [5] Z. Gao, From Linear to Nonlinear Control Mean: a Pratial Progreion, ISA Tration, vol., no.,. 7 5, Ar.. [6] Z. Gao, Saling and bandwidth-arameterization baed ontroller tuning, in Pro. of Amerian Control Conferene, Denver, CO, June, vol. 6, [7] L. Dong, Q. Zheng, and Z. Gao, On Control Sytem Deign for the Conventional Mode of Oeration of Vibrational Gyrooe, IEEE Senor Journal, vol. 8, no., , Nov. 8. [8] S. D. Senturia, Miroytem Deign, Kluwer Aademi Publiher, November, ISBN