Active Disturbance Rejection Control for an Electro-statically Actuated MEMS Device

INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS VOL.6NO.Setember/December 6-69 Active Diturbance Rejection Control for an Electro-tatically Actuated MEMS Device Lili DONG and Jaon EDWARDS Abtract An active diturbance rejection control (ADRC trategy i develoed for extending the travel range of a MEMS electrotatic actuator to a deired level in the reence of external diturbance and noie. Two controller deign are contructed on the actuator. One i a ingle-loo claic ADRC deign which i able to achieve the travel range of 97% of the full ga between the two late of the actuator. The other i multi-loo controller deign with an ADRC in an inner loo and a PI controller in an outer loo for charge and dilacement control reectively. The multi-loo controller deign can drive and tabilize the dilacement outut of the actuator to % of it full ga. Both controller deign are uccefully imulated onto a arallel-late electrotatic actuator model. The imulation reult verify the effectivene of the controller through extending the travel range of the actuator to the deired value in the reence of noie and diturbance. The controller erformance of the claic ADRC i comared with that of the multi-loo control. Fruency-domain analye roved the tability and robutne of the two ADRC controller deign. Index Term Electrotatic actuator MEMS Active Diturbance Rejection Control Stability Robutne. M I. INTRODUCTION EMS electrotatic actuator alo termed a microactuator are the ey device allowing MEMS to erform hyical movement []. They have the advantage of mall ize low cot and low ower conumtion. An imreive range of alication demontrate the electrotatic actuator utility. Some examle of the alication are: otical grate and hutter micro-mirror variable caacitor and micro-accelerometer []. A one-degree-of-freedom arallel-late micro-actuator conit of a movable late and a fixed late in an electric field. When the movable late i dilaced from it original oition the caacitance formed between the two late will be varying. The dilacement of the movable late i uually changed through a voltage (or charge control of the ga of the caacitor. However a the ga between the two late i increaed to one third of original ga a ull-in henomenon caue the intability of the ytem and drag the movable late immediately to the fixed late [ ] cauing the failure of oeration of the actuator. Therefore extending the traveling range of the movable late beyond the ull-in limit ha become a central toic in the control of electrotatic actuator. A method or mechanim to extend that travel range referably to the extent of the entire initial ga i highly deirable eecially for otical alication []. L. Dong i with the Deartment of Electrical & Comuter Engineering Cleveland State Univerity Cleveland OH 5 USA. J. Edward i with the NASA Glenn Reearch Center Cleveland OH 5 USA. Correonding author. Email: L.Dong@cuohio.edu. Electrotatic actuator oerate in two energy domain electrical and mechanical []. It i thu oible to extend the uable range of the actuator both mechanically and electrically. In the mechanical domain one traightforward aroach i to deign the ga o large that the actuator i table over the oerating range []. The drawbac of thi aroach i that the maximum ga i comletely deendent on micro-fabrication technology and can t be changed by the deigner. The other aroach i to ulement the elatic retoring force of the uort o a to avoid the ull-in henomena. Thi aroach ha been imlemented by uing leverage and nonlinear tiffening ring [-5]. However trengthening the elaticity of the uort will ruire increaed voltage to enhance the travel ditance thu raiing the ower aumtion and the cot. In the electrical domain both oen and cloed-loo control trategie have been alied to the actuator. The reult in [] demontrated an increaed travel range u to 8% of the initial ga uing an oen-loo charge controller. However ince the oen-loo control i enitive to external diturbance the cloed-loo controller baed on outut feedbac for which additional electrode or enor are needed to meaure the oition of the movable late of the actuator have been reorted in [6-]. A an alternative to charge control voltage control i more oular and effective [6]. A ecial voltage ource with a erie caacitor [6-8] i utilized to change the travel ditance of the actuator. The technique how table oeration of the actuator at % 6% and 9% of the initial ga. The diadvantage of the aroach are that it i deendent on an accurate mathematical model of the actuator and it ruire large actuation voltage. A linear time-varying roortional gain controller i develoed in [9] where a quantitative feedbac theory (QFT [] i utilized to deign the roortional gain. The travel ditance that the roortional gain controller can drive i 6% of initial ga. Unfortunately the effect of noie are not conidered by the controller. A Lyaunov-baed nonlinear control aroach ha been reorted in [-] to extend the travel range to over 9% of initial ga. Particularly in [] the imulation reult how a maximum travel range of full ga without overhoot. In general the aroache in [-] are effective but their utility i omewhat limited by it mathematical comlexity maing it very difficult to imlement in the real world. The goal of thi aer i develoing a robut and eay-toimlement control aroach to extend the travel range of the electrotatic actuator to over 9% of it initial ga in the reence of external diturbance and noie. A cloed-loo voltage controller i utilized. In order to achieve the control goal an active diturbance rejection controller (ADRC i alied to the actuator. The ADRC conit of an extended tate oberver (ESO and a PD controller. The ESO can not

Dong & Edward: Active Diturbance Rejection Control for an Electro-tatically Actuated MEMS Device 6 only etimate the internal ytem tate variable but external diturbance. The PD controller i contructed baed on the oberved tate of the ESO. It drive and tabilize the dilacement outut of the actuator to a deired value. Both claic and an alternative ADRC are develoed for the actuator. For the former ADRC all of the ytem arameter are aumed unnown while the later ADRC ue artially nown modeling information of the actuator and it i contructed on the artially nown model of the actuator. The claic ADRC ha been uccefully emloyed in macro ytem and MEMS gyrocoe [5-]. It i the firt time that we aly it to the MEMS electrotatic actuator. The alternative ADRC ha never been reorted in current literature yet. It i originally built u for the micro actuator ytem due to it excellent noie attenuation caability and outtanding tracing erformance. One of the advantage of the ADRC i that it doe not rely on an accurate mathematical model of a lant which for the electrotatic actuator 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 ractice. The ret of thi aer i organized a follow. The dynamic modeling of the electrotatic actuator i given in Section II. The deign of the ADRC i reented in Section III. The tability and robut analye are included in Section IV. The imulation reult are hown in Section V. Section VI rovide concluding remar and ugget future reearch. II. DYNAMIC MODELING OF ELECTROSTATIC ACTUATORS A imlified one-degree-of-freedom electrotatic actuator model i elected baed on [] for controller deign. The electro-mechanical model of the actuator i hown in Fig.. Fig.: Electro-mechanical model of an electrotatic actuator From Fig. we can ee that the electrotatic actuator conit of a arallel-late caacitor with one fixed electrode and one varying electrode. The inut voltage ource V i modeled with a erie ource reitor R. The variable I i the inut current. The initial ga with zero alied voltage i denoted by g. The ga g i oitive in the direction of increaing ga while X i the dilacement of the moving late and X i oitive in the direction of decreaing ga. The relationhi between g and X i given by g g X. (.. Modeling Uing Firt Princile The actuator in Fig. i oerating in two energy domain electrical and mechanical. A the charge Q on the two late build u the force of attraction grow bringing the late cloer together. In order to ee the late from naing down there need to be an ual and ooite force reiting thi motion. Thi force i modeled by the retoring force of a mechanical ring with ring contant. A daming term b rereent the queezed-film daming coefficient. In the mechanical domain according to Newton nd law we have mx& Fe Fb F ( where F b bx& i the linear queeze-film daming force F X i the linear mechanical ring force and F e Q /εa i the nonlinear electrotatic force. Equation ( can be rewritten a Q. ( mx&& bx& X εa Note that the ma of the uer late of the micro actuator i o mall that the gravitational force acting on the actuator can be neglected. Next we conider the electrical domain. Alying Kirchhoff' circuit law to the actuator give I V. ( ( S V act R where V i the voltage acro the caacitor late and V act i the ource voltage. The voltage acro the actuator i Qg V act. (5 ε A The current I can be olved by ubtituting (5 into (. Uing the fact that I S Q& we have Qg Q&. (6 V R εa Subtituting ( into (6 yield Q( g X Q& VS. (7 R εa Equation ( and (7 contitute the nonlinear model of the electrotatic actuator... Equation Normalization For the imlicity of later erformance analyi and controller deign for the electrotatic actuator normalized uation of ( and (7 have to be derived. The oition of the uer late relative to the lower late (X time (t the charge built u on the late (Q and the ource voltage (V are normalized a follow. X x g τ ω t q Q q i V v (8 v i In (8 the dilacement x i normalized by the ga without alied voltage (g time τ i normalized by the natural fruency (ω of the actuator charge q i normalized by the accumulation of charge at ull-in (q i and the ource voltage v i normalized by the ull-in voltage (v i. From [] the uation that govern the ull-in voltage the amount of charge at ull-in and the caacitance at initial ga are given by q i Cv 8g εa (9 i v i C 7C The detail of normalization can be found in []. The reult of the normalization are rereented by g

6 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMSVOL.6NO.Setember/December where b ς mω x x x q q& ( x q v ( r r r ω RC and. ( && ς & (.. Model Linearization We chooe the tate variable of the normalized model of the actuator a x(t q(t and (t where (t i the velocity ( x& ( t of the movable late of the actuator. For mall-ignal linearization the uilibrium value of the tate variable which are rereented by X 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 the normalized actuator model are ω m 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 X (/Q Q v ( Q. ( From ( we can ee that a we chooe different uilibrium dilacement X we will have different Q. The uilibrium value of X and Q correonding to different ercentage of the dilacement with reect to full ga are calculated and given in Table I... Tranfer Function Exreion and Pull-in Dilacement For the convenience of later fruency-domain analye a tranfer function rereentation of the linearized electrotatic actuator model i develoed. Conducting Lalace tranform on (5 and imlifying the tranformed uation yield x V ( Q (6 ( 9( r ( X ςr ( ς ( X r ( X where x( rereent the Lalace tranform of the dilacement outut of the actuator. From (6 we can ee that when X / the tranfer function will have a ole at the origin. Any oerating oint with a dilacement greater than / will roduce a ole in the right half lane. Thi how exlicitly how the actuator ytem become untable at the ull-in dilacement of /. Since the tranfer function rereentation of the actuator model (6 i a third-order lant it can be rewritten a P % ( b (7 ( a ( a ( a where -a -a and -a are ole and the ubcrit % in P % ( rereent the ercentage of the dilacement with reect to full ga. The value of arameter b a a and a can be obtained by comaring (7 and (6 with the uilibrium value of X and Q lited in Table I. The obtained arameter b a a and a are given in Table II. It how that the ole of a change greatly during the traveling range of the actuator while the other two ole (-a and a are almot unchanged. TABLE I: EQUILIBRIUM POINTS X.5..... Q.87.577.776.987..95 V.559.79.995.996..9859 X.5.6.7.8.9.95 Q.7.6.9.59.6.688 V.985.85.65.68.65.66 Define X X and X Q. Then the linearized model i δx& δx δx& ς X δx δv δx& ( δx X X { r r r B δx δy [ ] δx C δx A (5 According to [] we ue ζ and r.95 for the linearized model of the electrotatic actuator in (5. TABLE II: PARAMETER VALUES Plant b a a a P 5.89.7.5.9 P.56.7.8.895 P.68.689..9 P.8.669..76 P.678.66.. P.59.65.8 -.556 P 5.5798.6.5 -. P 6.6767.6. -.56 P 7.67796.597. -. P 8.777.58.9 -.999 P 9.7687.566.8 -.88 P 95.7898.559.7 -.59 III. CONTROLLER DESIGN Thi ection will introduce two ind of linear ADRC deign: claic and alternative ADRC. The tate ace rereentation of the two ADRC deign will be develoed

firt. Then the tranfer function rereentation of the ADRC will be derived for later fruency-domain analye... Claic ADRC Deign For claic ADRC all of the ytem arameter in (7 are aumed to be unnown. A fourth-order extended tate oberver (ESO will be develoed to etimate both the internal ytem tate (including dilacement velocity and charge and external diturbance. Baed on the accurate etimation of ESO the claic ADRC i derived to drive the actuator outut to a deired a dilacement. A. Introduction to Claic ADRC From (6 and (7 the electrotatic actuator can be modeled by a third-order differential uation a follow. & y && f ( y y& & y d t bu (8 In (8 the function f ( y y& & y d t which will be denoted a f in the following dicuion rereent all of the other force on the actuator lant excluding control effort y(t i ual to normalized dilacement outut x(t d denote external diturbance force b i controller gain and u i ual to V in (6. A we deign the ADRC the function f i aumed to be unnown and referred to a a generalized diturbance. The ESO etimate the generalized diturbance. We chooe tate variable a x y x y& x & y and x f which i an augmented tate. Auming h f& and h i bounded within the interet (8 can be rereented by a tate-ace model. The tate-ace rereentation of (8 i where A & (9 X AX Bu Eh y Cx B b Dong & Edward: Active Diturbance Rejection Control for an Electro-tatically Actuated MEMS Device 6 E C [ ] ( T X [ x x x x ]. From [5] the augmented tate x and the other tate can be etimated uing the ESO given a follow. z& Az Bu L( y yˆ ( yˆ Cz In ( z i the etimated tate vector and z[z z z z ] T where z z z and z are the etimated x x x and x reectively. The oberver gain vector L i choen uch that all the oberver ole are located at ω o where ω o i taen a oberver bandwidth. A the oberver gain are given by ( the characteritic uation of the ESO i (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 cloely trac y y& & y& and f. We aume that i an aroximate b. Then the control inut to the actuator i choen a o u ( u z ( where u denote a control law. Equation ( reduce (8 to o & y && u ( which i a trile integrator lant. The lant can be controlled by a conventional roortional derivative controller which i u ( r z dz d z z (5 where r denote a deired dilacement outut for the actuator. In (5 d and d are controller gain and are choen a ω ω. (6 c d ω c d The controller gain above can lace all the cloed-loo ole of the controller at -ω c which i taen a controller bandwidth. From ( and (6 we can ee that the ADRC including the ESO only ha three tuning arameter ω c ω o and. The detail about the arameter tuning can be found in [6]. The few tuning arameter enable imle imlementation of the controller in ractice. Although the ADRC ha been alied to MEMS gyrocoe ([9 ] it wa never emloyed to electrotatic actuator before. The controller ha been modified in thi aer and alied to the electrotatic actuator. The detail about the alication are given a follow. B. Tranfer Function Rereentation of the Claic ADRC Combing ( and (5 we can rewrite the control inut a u ( r [ d d ] z. (7 Define controller gain vector a K[K K K K ] [ d d ]. The Laalce tranform of (7 i (8 ( ( R( KZ ( U b ˆ Auming zero initial condition for z(t and it derivative the Lalace tranform of ( i ( ( I A LC [ BU( LY( ]. (9 Z Subtituting (9 into (8 yield [ ] ( ( R( K( I A LC ( BU ( LY ( U Define matrix M a M ( I A LC. Then ( can be reorganized and imlified a U b ˆ KMB KML ( R( Y ( KMB c ( Equation ( can be rereented by a cloed-loo bloc diagram a hown in Fig. where H r ( i a re-filter the controller in the feedbac ath the actuator model to be controlled D( an external diturbance and N( meaurement noie. The lant model or P % ( i given by (7. The re-filter H r ( and controller are derived from ( and are exreed a KML. ( H ( KMB r KMB

6 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMSVOL.6NO.Setember/December Fig.: Bloc diagram of cloed-loo ADRC-controlled ytem Subtituting the controller and oberver gain into ( yield and where c K L C ( c K L K L c c c c ( d d d ( ( d d d ( K L L L L ( ( H r c K L K L K L L c K L K L K L and d K L. d K L K L d K K L K L L It i noticed that the ADRC controller rooed in [5-7] i a one degree-of-freedom (DOF controller where the i laced in a feed-forward ath and the control law directly deal with the difference between reference inut and ytem outut. However Fig. how a -DOF ADRC controller. The roblem of -DOF controller i that there i alway a tradeoff between command following and diturbance rejection [7]. The ue of a -DOF controller olve thi roblem by allowing the reference ignal r and the outut meaurement y to be treated indeendently by the controller rather than by oerating on their difference y-r a in a -DOF controller. In addition the configuration hown in Fig. allow for the derivation of traditionally defined enitivity function (S comlementary enitivity function (T and other variou cloed loo tranfer function that are ued for controller erformance analye to be conducted in the following ection... Alternative ADRC Deign The claic ADRC minimize the amount of modeling information ruired to controller deign. It aume that the ytem arameter in (7 are totally unnown. However if there i artial modeling information available it can be incororated into the ESO. We originally ue the artially available modeling information to deign an alternative ADRC in thi aer. The detail of the alternative ADRC are given a follow. A. Introduction to Alternative ADRC From Table II we can ee that the lant gain b and one of the ytem ole a varie ignificantly over the electrotatic actuator oerating range. However the other two arameter a and a are almot unchanged. Therefore we can aume a and a are nown arameter while b and a the unnown arameter. Then the model (7 can be divided into nown and unnown art a hown in (5. P ( X U ( b ( ( a ( a ( a nown unown (5 Equation (5 can be alo rewritten a ( a a a && x ( a a a a a a x ( a a a x bu &&& x & (6 Let f( include all the unnown term on the right ide of (6. That i f ( a & x a ( a a x& a a a x ( b u (7 Then (6 can be rewritten a ( a a && x ( a a x& f (&& x x x ( b u u & x&& & (8 We chooe the tate variable a x x x x& x & x and x f. The tate ace rereentation of (8 i where A aa ( a a & (9 X AX Bu Eh y Cx B E C [ ] ( T X [ x x x x ]. A dicued in Section. the ESO ued to oberve the tate variable in (9 i rereented by z & ( A LC z Bu Ly ( where L i given by ( and the matrice A B and C are denoted by (. The generalized diturbance f( can be etimated by the ESO. With the accurate etimation of the ESO the control law i contructed a u [ K ( r x K x& K & x fˆ ( ] ( Subtituting ( into (8 reult in & x && ( a a K && x ( a a K x& K x K. ( r A tated in Section. the controller gain K K and K in ( are choen to roduce real reeated ole ω c for the deired tranfer function of the cloed-loo ytem. The controller gain are given by K ωc K ω c aa K ω c ( a a ( The controller gain for the alternative ADRC are maller than the one for the claic ADRC that are given in (6. B. Tranfer Function Rereentation of the Alternative ADRC Again we uoe that the controller i in the feedbac ath of the ADRC controlled cloed loo ytem a hown in Fig.. The controller and re-filter are rereented by (. Subtituting the controller gain ( and oberver gain (6 into ( yield C ( N N N N ˆ b B B B and ( ( H (5 K A A A A (6 ˆ b B B B

where N K L K L K L L N K L K L K L N K L K L a a N K L A a a A L A a a L L ( a a L ( a a L ( a a A L ( KL K L L ( a a ( K L L ( a a K L B B L B ( K A ( K KL A ( K K L K L A a a L K. Claic and Alternative ADRC Deign with Secific Controller Parameter Value For the convenience of later comarion tudie the oberver bandwidth controller bandwidth and lant gain etimate are choen to be the ame value for both deign. The controller arameter are ω c rad/ ω o rad/ and b ˆ.65. The tranfer function of the claic ADRC controller deigned in ( i rereented by 5868 (.5(..97 ( 5.(.8 8 G C (7 The tranfer function of the refilter for the claic ADRC controller deigned in ( i exreed a H. ( ( 5.(.8 8 (8 According to Table II we chooe a.559 a.7 a the nown lant arameter for (7. The tranfer function of the alternative ADRC controller i given by 79 Dong & Edward: Active Diturbance Rejection Control for an Electro-tatically Actuated MEMS Device 65 (.55(..6 ( 5.(.8 8 G C (9 From (7 and (9 it can be een that both claic and alternative ADRC have ame controller ole. However they have different zero. The tranfer function of the refilter for the alternate ADRC deign i rereented by.( 9.7( 5.9( 9.7 78.9 (5 H ( 5.(.8 8 U H ( ( R( D( N( S( L( L( T ( L( (5 (55 The tranfer function from the noie inut N( to the control ignal U( i denoted a S( and i ued to invetigate the effect of enor noie and the tranfer function from the diturbance inut D( to the dilacement Y( i denoted a S( and will be ued to dicu the diturbance rejection caability. The SS( and S( are C P ( ( ( C S( L( ( P S( L( (56 (57 In thi ection the tability and robutne of the claic and alternative ADRC are invetigated on a linearzied model of the electrotatic actuator lant with the dilacement of 95% of full ga. The linearized model i given a below. P 95.7898 (.7(.559(.59 (58. Stability Analye The Bode diagram of the loo tranmiion function L(jω ((5 for both deign are hown in Fig. where P rereent the lant model (58 L(ADRC the loo tranmiion function of the claic ADRC and L(Alt the loo tranmiion function of the alternative ADRC. The tability margin obtained from the Bode diagram for the two deign are lited in Table III. The table how that both deign rovide ufficient gain and hae margin to enure the tability of the ytem. Comared to the claic ADRC the alternative ADRC deign ha imroved gain margin but ha a reduced hae margin. In addition the alternative ADRC deign ha a much maller bandwidth than the claic ADRC deign. However the relatively mall bandwidth of the alternative ADRC i beneficial when noie ource are conidered. IV. STABILITY AND ROBUSTNESS ANALYSES In the fruency domain the loo tranmiion function i a ey tool in acceing the erformance of a control ytem. The loo tranmiion function L( in Fig. i defined by From (7 and (5 we have L ( b b ˆ o L ( (5 ( a ( a ( a ( d d d c c c c (5 From Fig. the meaurement outut Y( and the control ignal U( can be rereented by (5 and (5. The enitivity function S( and comlementary enitivity function T( are defined by (55. H ( Y ( R( D( N ( (5 Fig.: Bode Plot of Actuator Model and the Loo Tranmiion Function for Both Claic and Alternative ADRC Deign

66 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMSVOL.6NO.Setember/December TABLE III: STABILITY MARGINS OF CLASSIC AND ALTERNATIVE ADRC DESIGNS Deign GM (db PM(degree BW(rad/ec Claic 9. 59.6. Alternative.8 5. 8.8 The alternative ADRC deign ruire the artial modeling information and ue thi information to reduce the need for high controller gain. Thi mae the alternative ADRC controller lightly le ucetible to enor noie comared to the claic ADRC deign while till maintaining the imlicity of imlementation through a ingle loo deign.. Noie Attenuation and Diturbance Rejection Fig. how the Bode lot of the tranfer function rereented by (56 which decribe the enitivity of the controller outut to enor noie for both claic and alternative ADRC deign. Fig.5: Noie Amlification at the Controller Outut of Claic ADRC and Alternative ADRC Deign Fig.: Bode Diagram of Noie Senitivity Tranfer Function for the Claic and Alternative ADRC Deign From Fig. we can ee that the alternative ADRC deign acrifice a little hae lead (of around 8. o in order to decreae the high fruency gain of the controller noie enitivity tranfer function (S( in (56. Comared to the claic ADRC deign the alternative ADRC deign rovide an extra 5 db of noie attenuation at high fruencie. Fig. 5 how the noie amlification of the two deign at the controller outut a white noie i added to the ytem. We can ee from Fig. 5 that the alternative ADRC deign rovide a better noie reduction than the claic one. The amlitude of the alternative ADRC control ignal i much maller than the one of the claic ADRC deign in the reence of noie. The reult hown in Fig. 5 confirm our concluion made from Fig.. It i hown in Fig. 6 that the Bode lot of the cloedloo inut diturbance tranfer function (S( in (57 from an inut diturbance to the meaured outut x for both deign. From Fig. 6 we can ee that both deign how excellent inut diturbance rejection caabilitie over the entire fruency range. The ea magnitude reone for the claic ADRC deign i -. db at.5 rad/. The ea magnitude reone for the alternative ADRC deign i -.8 db at.5 rad/. The claic ADRC deign eae the burden on the control ytem deigner by ruiring le modeling information than the alternative ADRC deign. However the benefit of demanding le modeling information for the claic ADRC ruire that a high DC gain be ued which reult in increaed noie enitivity when comared with the alternative ADRC. Fig.6: Bode Diagram of the Cloed-loo Tranfer Function between Inut Diturbance and Outut Dilacement for the Claic and Alternative ADRC Deign V. SIMULATION RESULTS During the imulation the enor noie in Fig. 7 i added to the control ytem hown in Fig.. The two ADRC deign are imulated on the linearized model of the electrotatic actuator whoe arameter value are given in Table II. Both deign utilize an oberver bandwidth of ω o rad/ a lant gain etimate b ˆ. 65 and a controller bandwidth of ωc rad. The reduced nominal model of the actuator for the claic ADRC i.65/. The reduced nominal model for the alternative ADRC deign i.65 (59 P n ( ( a ( a (.76(.566

Dong & Edward: Active Diturbance Rejection Control for an Electro-tatically Actuated MEMS Device 67 A we et the deired travel range a % % 5% 7% and 9% of the full ga the dilacement outut for both claic and alternate ADRC deign are hown in Fig. 8. Both deign have hown excellent tracing erformance. The te reone for thee two deign at mall dilacement are demontrated in Fig. 9 where the te reone are at % of the full ga. From Fig. 9 we can ee that the claic ADRC deign exhibit 8.% overhoot while the alternate ADRC deign ha 7.5% overhoot. The overhoot at mall dilacement i accetable. However the overhoot at large dilacement could caue the late of the actuator come into contact with each other. A the deired travel range i choen a 9% of the full ga the dilacement outut (or te reone of the claic and alternative ADRC are dilayed in Fig. where we can ee that the claic ADRC deign ha a larger overhoot ercentage (.% than the alternate ADRC (almot zero. The Integral Squared Error (ISE of the claic and alternative ADRC deign at % and 9% of full ga are hown in Table IV. From the table we can ee that the tracing erformance of the claic ADRC deign i better at mall dilacement but wore at the larger dilacement comared to the alternative ADRC. Thi i attributable to the higher controller gain of the claic ADRC than the alternative one. Fig.9: Dilacement Outut for Both Claic and Alternative ADRC Deign at % of Full Ga Fig. 7: Normalized enor noie Fig.: Ste Reone for the Alternate and Claic ADRC at 9% of Full Ga TABLE IV: ISES OF CLASSIC AND ALTERNATIVE ADRC DESIGNS Dilacement ISE of Claic ADRC ISE of Alternative ADRC % of full ga..5 9% of full ga.97.9 Fig.8: Dilacement outut of two ADRC deign The reone of both deign to a reference of 99% of the full ga in the reence of a te diturbance with a magnitude of.5 at t 5 time unit are hown in Fig.. From it we can ee that the alternative ADRC deign how much maller overhoot ercentage (.5% at maximum than the claic ADRC (.5% at maximum. The large overhoot ercentage of.5% for the claic ADRC controller could caue the uer and lower late of the electrotatic actuator to crah into each other and therefore reult in failure of oeration in thi deign cenario. However the diturbance rejection ability of the claic ADRC i a bit better than the alternative ADRC.

68 INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMSVOL.6NO.Setember/December One ignificant advantage of the alternative ADRC deign over the claic one i the attenuation of enor noie. Fig. how the control ignal of both deign in the reence of enor noie. From it we can ee that although the control ignal for both cae are noiy the alternative deign i uerior to the claic one. The electrical charge acting a the control ignal to the mechanical ortion of the actuator lant for both deign are hown in Fig.. The et-oint are choen a % % 5% 7% and 9% of the full ga. The figure clearly how how the control ignal react to the commanded reone. After the filtering effect of the electrical ortion of the actuator it i een that the charge control inut to the mechanical ortion i accetable in the alternate ADRC deign while the claic one i till fairly noiy. However it i imortant to note that thi noie doe not have a dramatic effect on the dilacement outut (x for the claic ADRC. The imulation reult demontrate that when it come to tracing erformance and noie minimization the alternative ADRC i a better controller deign for the electrotatic actuator than the claic one. However the claic ADRC ruire minimum modeling information and i more robut againt external diturbance than the alternative one. Fig. : Dilacement Outut for Alternate and Claic ADRC at 99% of Full Ga with Inut Diturbance Fig. : Control Signal of Claic and Alternate ADRC in the Preence of Noie Fig.: Charge Control Signal of Alternate and Claic ADRC in the Preence of Senor Noie VI. CONCLUSIONS Thi aer ha originally deigned an alternative ADRC and alied both claic and the alternative ADRC to a normalized electrotatic actuator model. Simulation reult demontrate their effectivene through extending the travel range of the actuator to over 9% of it full ga in the reence of enor noie and diturbance. Fruency-domain analye rovide theoretical uort to the tability and robutne of the controller. Comared to the other advanced controller in the current 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 excellent control erformance while bridging the ga between the imle and ractical controller utilized in indutry and the advanced theoretical aroache reented in academia. With the attenuation of noie effect being uch an imortant iue in the electrotatic actuator it would be rudent to conduct a more thorough analyi of the noie ource in the micro-cale environment in the future. A ti-in henomenon in the actuator caued by micro-fabrication imerfection will be invetigated a well. REFERENCES [] H. Fujita Microactuator and Micromachine in Proc. of the IEEE vol. 86 no. 8 Aug. 998. 7-7. [] J. Seeger Charge Control of Parallel-late Electrotatic Actuator and the Ti-in Intability Journal of MEMS vol. no. 5. 656-67 Oct.. [] L. M. Cataner and S. D. Senturia Seed-energy Otimization of Electrotatic Actuator Baed on Pull-in Journal of MEMS vol. 8 no.. 9-98 Se. 999. [] Y. Nemirovy A Methodology and Model for the Pull-in Parameter of Electrotatic Actuator Journal of Microelectromechanical Sytem vol. no.. 6 65 Dec.. [5] E. S. Hung and S. D. Senturia Extending the Travel Range of Analog- Tuned Electrotatic Actuator Journal of MEMS vol. 8 no.. 97 55 Dec. 999.

Dong & Edward: Active Diturbance Rejection Control for an Electro-tatically Actuated MEMS Device 69 [6] J. Seeger and S. Crary Stabilization of Electrotatically Actuated Mechanical Device IEEE International Conference on Solid-State Senor and Actuator June 997 Chicago IL. -6. [7] E. Chan and R. Dutton Electrotatic Micromechanical Actuator 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 Electrotatic Microgrier in Proc. IEEE Intl. Conf. Robotic and Automation May 99 San Diego CA.8-85. [9] M. Lu M. Hirano G. Fedder Poition Control of Parallel-late Microactuator for Probe-baed Data Storage Journal of MEMS vol. no. 5. 759-769 Oct.. [] I. Horowitz Quantitative Feedbac Deign Theory QFT Publication Boulder Colorado 99. [] G. Zhu J. Penet and L. Saydy Robut control of an electrotatically actuated MEMS in the reence of araitic and arametric uncertainty in Proc. of the 6 American Control Conference Jun. 6 Minneaoli MN. -8. [] G. Zhu J. Levine and L. Praly Imroving the Performance of an Electrotatically Actuated MEMS by Nonlinear Control: Some Advance and Comarion in Proc. of the th IEEE Conference on Deciion and Control and Euroean Control Conference Seville Sain Dec. 5. 75-759. [] G. Zhu L. Saydy Robut Outut Feedbac Control of an Electrotatic Micro-Actuator in Proc. Of American Control Conference NYC NY July 7. 9-97. [] G. Zhu J. Lévine L. Praly and Y. A. Peter Flatne-baed Control of Electrotatically Actuated MEMS with Alication to Adative Otic: a Simulation Study Journal of MEMS vol. 5 no.5.65-7 Oct. 6. [5] Z. Gao From Linear to Nonlinear Control Mean: a Practical Progreion ISA Traction vol. no.. 7 5 Ar.. [6] Z. Gao Scaling and bandwidth-arameterization baed controller tuning in Proc. of American Control Conference Denver CO June vol. 6. 989-996. [7] G. Tian and Z. Gao Fruency Reone Analyi of Active Diturbance Rejection Baed Control Sytem in Proc. of IEEE International Conference on Control Alication Oct. 7 Montreal Quebec Canada. 595 599. [8] B. Sun and Z. Gao A DSP-Baed Active Diturbance Rejection Control Deign for a KW H-Bridge DC-DC Power Converter IEEE Tran. on Indutrial Electronic Vol..5 No.5. 7-77 Oct. 5. [9] L. Dong Q. Zheng and Z. Gao On Control Sytem Deign for the Conventional Mode of Oeration of Vibrational Gyrocoe IEEE Senor Journal vol. 8 no.. 87-878 Nov. 8. [] L. Dong and D. Avaneian Drive-mode Control for Vibrational MEMS Gyrocoe IEEE Tranaction on Indutrial Electronic vol. 56 no.. 956 96 9. [] S. D. Senturia Microytem Deign Kluwer Academic Publiher November ISBN 79768. Jaon Edward received the B.S. and M.S. degree from the Deartment of Electrical and Comuter Engineering at Cleveland State Univerity Cleveland OH USA in and 9 reectively. He currently wor at NASA Glenn Reearch Center Cleveland OH a a Comuter Engineer. Hi reearch interet include control of MEMS embedded ytem deign digital ignal roceing data acquiition intrumentation and digital control ytem. Lili Dong (M received the M.S.E.E. from Changchun Intitute of Otic Fine Mechanic and Phyic Chinee Academy of Science Changchun China and the Ph. D. degree in Electrical Engineering from the Univerity of Alabama Tucalooa AL USA in and 5 reectively. She joined the Deartment of Electrical and Comuter Engineering (ECE at Cleveland State Univerity (CSU a an aitant rofeor in Aug. 5. She i currently an aociate rofeor with the Deartment of ECE at CSU. Her reearch interet include the modeling and control of MEMS device adative control of linear time-varying ytem modeling and control of ower and automobile ytem and engineering education reearch. She i currently an editor for the Proceeding of American Control Conference and the chair of IEEE Control Sytem Society Cleveland Chater. She i a reviewer of multile IEEE journal and roceeding.