Bandwidth expansion of a pressure control system for pneumatic anti-vibration apparatuses in presence of dead time

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1 Bulletin of the JSME Journal of Advanced Mechanical Deign, Sytem, and Manufacturing Vol.9, No.3, 25 Bandwidth exanion of a reure control ytem for neumatic anti-vibration aaratue in reence of dead time Yukinori NAKAMURA, Hiroki KAWAKAMI and Shinji WAKUI Intitute of Engineering, Tokyo Univerity of Agriculture and Technology Naka, Koganei, Tokyo , Jaan yukino-n@cc.tuat.ac.j Faculty of Engineering, Tokyo Univerity of Agriculture and Technology Naka, Koganei, Tokyo , Jaan Received 5 March 25 Abtract Thi aer conider the reure control of air ring ued for a neumatic anti-vibration aaratu AVA), which iolate vibration tranmitted to emiconductor lithograhy equiment. In the control of the neumatic AVA, reure feedback i often emloyed o a to kee the inner reure of air ring contant. By exanding the bandwidth of reure control ytem, fat reone can be obtained, though thi bandwidth i retricted due to anti-reonance and reonance in a neumatic ytem. To eliminate the anti-reonance and reonance, relative dilacement derivative RDD) oitive feedback control ha been reented. However, reviou RDD oitive feedback control cannot be imlemented under the condition where the dead time of air ring exit. In order to comenate effect of the dead time, a Smith redictor i utilized. Through exeriment, it i hown that in the reence of the dead time, the anti-reonance and reonance can be eliminated by mean of the RDD oitive feedback control combined with Smith rediction. The bandwidth of the reure control ytem increae with the gain of reure PI comenator. Furthermore, effect of rooed aroach on vibration tranmiibility are invetigated. Although the vibration tranmiibility i the erformance metric for the develoment of the neumatic AVA, the cae where the dead time i reent wa not ufficiently conidered. In thi aer, it i hown that the vibration tranmiibility change in accordance with the dead time. Key word : Pneumatic equiment, Vibration control, Vibration iolation, Preure control, Dead time. Introduction A neumatic anti-vibration aaratu AVA) Ryaboy, 24; Heertje et al., 23; Bulter, 2; Pu et al., 2; Shin et al., 2) lay a crucial role in the manufacturing roce of integrated circuit. By uing thi aaratu, it i oible to reduce effect of the vibration tranmitted to the XY tage in lithograhy equiment. Air ring of the neumatic AVA uort the iolation table where the XY tage i mounted. The driving force of the air ring i regulated by ervo valve ued for the exhaut and intake of comreed air. However, ince the value of arameter of each air ring are different, generated driving force are alo different. For thi reaon, the tructure in the lithograhy equiment i deformed. Moreover, ince the XY tage move on the iolation table during lithograhy roce, the oition of the neumatic AVA center of gravity change. Conequently, the iolation table i inclined. The tructure deformation of lithograhy equiment and the inclination of iolation table caue the large oitioning error and long ettling time of the XY tage Wakui and Fuhima, 26). One aroach for the above two roblem i to ue reure feedback control Shin and Kim, 29; Wakui, 23; Canon Inc., 2a). Thi feedback control i imlemented in the commercial neumatic AVA o a to avoid the tructure deformation of lithograhy equiment. In addition to the reure feedback control, when the reure feedforward Paer No.5-73 [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer

2 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) control Shin and Kim, 29; Canon Inc., 2b) i alo ued, the inclination of the iolation table i raidly corrected. Although the fat reone of a reure control ytem can be achieved by exanding it bandwidth, it i neceary to conider anti-reonance and reonance in a neumatic ytem, which correond to the air ring. When the gain of reure comenator increae o a to exand the bandwidth, the reonance eak become large. Thi caue the detabilization of the control ytem. Thu, it i difficult to exand the control bandwidth of the reure control ytem under the condition that the anti-reonance and reonance exit. Wakui and Fuhima 26) rooed the relative dilacement derivative RDD) oitive feedback control. Thi control cheme ue the information on the derivative of the relative dilacement between the iolation table and the floor. Due to the feedback of the above information, the anti-reonance and reonance can be eliminated. However, in Wakui and Fuhima 26), the dead time of the air ring i.e., the time delay in the reone of the neumatic ytem) wa not conidered. In the indutry of emiconductor manufacturing, a large neumatic AVA i needed o a to uort heavy lithograhy equiment. For thi reaon, it take time to inect the ervo valve attached to each air ring. To avoid thi difficulty, all ervo valve are gathered and et aart from the air ring Nakamura et al., 23). Conequently, the ie connecting the ervo valve and the air ring become long, reulting in the increae of the dead time. A mentioned in ubection 3., when the dead time exit, it i neceary to know arioriinformation on the dilacement derivative of the iolation table o a to ue the RDD oitive feedback control. In other word, the future outut of a dilacement enor ha to be obtained. For thi reaon, the ytem with the RDD oitive feedback control i not caual in the reence of the dead time, and the control bandwidth of reure control ytem cannot be exanded. Shin and Kim 29) reented the feedback and feedforward control cheme which ue the inner reure of air ring. Kawahima et al. 27) develoed reure differentiator in order to decreae the natural frequency of the neumatic AVA. However, thee work did not addre the roblem of the bandwidth exanion of reure control ytem with the dead time. Thu, the dead time of the air ring i a ractical iue that hould be olved. In thi aer, we conider how to exand the bandwidth of the reure control ytem with the dead time of the air ring. The firt contribution of thi aer i to reent the method for the elimination of the anti-reonance and reonance. To reduce effect of the dead time, a Smith redictor Smith, 959) i emloyed. By uing the RDD oitive feedback combined with the Smith rediction, the anti-reonance and reonance can be eliminated even if the dead time exit. Furthermore, the bandwidth of the reure control ytem can be exanded by tuning the gain of a reure PI comenator. Exeriment demontrate that a well-known Smith redictor aroach ued for dead time comenation i effective in term of the bandwidth exanion of reure control ytem. The econd contribution of thi aer i to invetigate effect of our rooed method on vibration tranmiibility. In the develoment of the neumatic AVA and lithograhy equiment, the vibration tranmiibility i necearily emloyed a a erformance metric e.g, ASML Inc., 24; Nikon Cor., 2; Ryaboy, 24). Although our reviou work Nakamura et al., 23) conidered the iolation of the vibration caued by the dead time, the vibration tranmiibility wa not dicued. We how that the vibration tranmiibility change in accordance with the dead time. The ret of the aer i organized a follow. Section 2 exlain the neumatic AVA, the anti-reonance and reonance, and the RDD oitive feedback control. Section 3 decribe the drawback of the reviou RDD oitive feedback control and reent the rooed method for the bandwidth exanion of reure control ytem with the dead time. Section 4 how effect of the dead time on the vibration tranmiibility. Finally, concluion are drawn in Section Preure control of neumatic anti-vibration aaratu 2.. Pneumatic anti-vibration aaratu and it control ytem In thi ubection, we firt exlain the neumatic AVA and it control ytem. Figure how the hotograh of the neumatic AVA that i conidered in thi aer, and Fig. 2 illutrate the exerimental etu. The AVA ha the air ring which utain the iolation table. The air ring and comreor are connected with a ie. Through thi ie, the air generated from the comreor i ulied to the air ring. Since thi AVA i an active tye iolator, the amount of the air i dynamically adjuted by uing the feedback control. Due to the adjutment, the vibration of the iolation table can be attenuated. A reure enor Setra, 24) i emloyed for the control of the reure in the air ring. Moreover, a dilacement enor SDL, NS23/A) i attached to the AVA in order to meaure the relative dilacement between the iolation table and the floor. An acceleration enor JAE, JA-5V) i alo attached o a to meaure the acceleration of the iolation table. The outut ignal of thee enor are fed back to a digital ignal roceor DSP), where control law i imlemented. In accordance with the control law, the inut voltage to the ervo valve i decided. [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 2

3 A IN N D AD72- ADC A OUT N D AD72- A 2 DAC N AD72- D 3 DIO AND AD-795 ANDORO MEDA Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) Acceleration enor Iolation table Preure enor Fig. Air lider Air ring Servo valve Dilacement enor Pneumatic anti-vibration aaratu. The actuator of thi aaratu i a neumatic ring, whoe inner reure i ket contant by adjuting the amount of the air. Air comreor Oilfree Air Regulator Iolation table Servo valve Pie Air ring AVA Valve driver Preure ignal Poition ignal Acceleration ignal DSP Fig. 2 Exerimental etu for neumatic anti-vibration aaratu. The outut of reure, acceleration, and dilacement enor are ued a feedback ignal in the control ytem for the neumatic AVA. The inut voltage to the ervo valve i determined in accordance with the control law. Pneumatic AVA r k k ar +T a T a r Preure control ytem Preure PI comenator +T u k r T G q c V A f air D M K x ~ k g k re Preure FB A k a k acc k o Acceleration FB Dilacement FB Fig. 3 Block diagram of neumatic anti-vibration aaratu and it control ytem. Block inide the dahed box how the neumatic AVA, and block inide the blue olid box how the reure control ytem. The PI comenator i emloyed o a to manage the inner reure of air ring. Figure 3 how the block diagram of the neumatic AVA and it control ytem. The arameter are ummarized in Table and 2. In Wakui and Fuhima 26), the neumatic AVA wa modeled a a linearized ytem around equilibrium oition of the iolation table. Under the aumtion that the dilacement x of the floor i zero, the tranfer function P xu ) from the inut voltage u to the dilacement x of iolation table i exreed a P xu ) = x) u) = M 2 + D + K + A 2 β V + c A G q β V + c. ) A G q /β V +c) in ) correond to the neumatic ytem of the AVA, and M 2 +D+K correond to the mechanical ytem of the iolation table. The control ytem for the neumatic AVA conit of dilacement, acceleration, and reure feedback loo. Firt, the dilacement feedback i emloyed for the levitation of the iolation table. Then, the acceleration feedback control i ued to increae the daming of the control ytem. Finally, to kee the inner reure of air ring contant, the reure feedback control i imlemented in the control ytem Anti-reonance and reonance caued by air ring Then, we how that in the neumatic ytem of the AVA, there exit anti-reonance and reonance, which revent the bandwidth exanion of the reure control ytem. From Fig. 3, the tranfer function P u from the inut voltage u to [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 23

4 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) Table Parameter of neumatic AVA. Table 2 Parameter of control ytem for neumatic AVA. Symbol Decrition Nominal value and unit M Ma of iolation table 2.33 kg D Vicou daming coefficient 479 N /m K Sring contant N/m G q Flow gain.6 5 m 3 //V β Comreibility /Pa c Flow conductance 4.6 m 3 //Pa V Volume of air ring.5 3 m 3 A Effective area m 2 u Inut voltage to ervo valve -V Inner reure of air ring -Pa f air Driving force of air ring -N x Dilacement of iolation table -m x Dilacement of floor -m Symbol Decrition Value and unit k r Gain of 2 reure PI comenator T Time contant of 8.93 reure PI comenator k g Preure feedback gain 3 k ar Gain of acceleration PI comenator T a Time contant of.3 acceleration PI comenator k a Acceleration feedback gain 37.5 k Gain of.5 dilacement P comenator k re Preure enor enitivity.43 5 V/Pa k acc Acceleration enor enitivity.5 V/m/ 2 k o Dilacement enor enitivity 667 V/m r Reference of -V dilacement control ytem r Reference of -V reure control ytem Outut of k g -V reure feedback gain the inner reure i given by P u ) = u) = G q M 2 + D + K) 2) M 2 + D + K)β V + c) + A 2. Since the flow conductance c i ufficiently mall Wakui and Fuhima, 26), the tranfer function 2) i aroximated a follow: G q M 2 + D + K P u ), 3) β V + c M 2 + D + K + K air where K air = A 2 /β V ) denote the air ring tiffne [N/m] Lee and Kim, 27). Note that effect of M 2 + D + K in the numerator of 3) give rie to the anti-reonance. On the other hand, effect of M 2 + D + K + K air in the denominator of 3) caue the reonance Effect of RDD oitive feedback Finally, we decribe effect of the RDD oitive feedback control Wakui and Fuhima, 26) on the frequency characteritic of the neumatic AVA. Thi control cheme i ued to eliminate the anti-reonance and reonance. It lead to the bandwidth exanion of the reure control ytem. In the following, we briefly exlain thi control cheme. Figure 4 illutrate the block diagram of the RDD oitive feedback control. k tun denote the tunable arameter of the RDD oitive feedback control, and ũ denote the outut of the reure PI comenator. From Fig. 3 and 4, the tranfer function from the outut ũ of reure PI comenator to the inner reure of air ring i exreed a ũ) = G q M 2 + D + K) M 2 + D + K)β V + c) + A A G q k tun k o ). 4) The tunable arameter k tun i determined o that the anti-reonance and reonance do not exit. By chooing k tun = A G q k o, 5) 4) become ũ) = G q β V + c. 6) Since 6) i a firt order lag ytem, the anti-reonance and reonance of the AVA can be eliminated by mean of the RDD oitive feedback. Now, by uing the air ring tiffne K air, 4) can be aroximated a follow: ũ) G q β V + c M 2 + D + K M 2 + D + K + K air G ). 7) qk tun k o A [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 24

5 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) By ubtituting 5) into 7), K air { G q k tun k o )/A } in 7) become zero. Thu, the hyical meaning of the RDD oitive feedback control i that effect of the air ring tiffne are cancelled. Figure 5 how Bode lot of the tranfer function from the outut ũ of reure PI comenator to the outut k re of reure enor. In thi exeriment, a eudo-differentiator i emloyed intead of a differentiator hown in Fig. 4 o a to obtain the dilacement derivative ẋ. Moreover, a inuoidal ignal i ued a the reference r of the dilacement control ytem. When the tunable arameter i et a k tun = A /G q k o ), the anti-reonance and reonance are ureed. However, due to the large arameter error of the flow gain G q, the anti-reonance and reonance cannot be ureed in exeriment even if nominal value of G q i ued for the etting of the tunable arameter k tun. For thi reaon, we continuouly change the value of the tunable arameter k tun and meaure frequency reone. In accordance with the tunable arameter k tun, the anti-reonance and reonance eak are changed. Furthermore, when the anti-reonance and reonance are ureed, the hae of the tranfer function /ũ become about -9 deg in the frequency region higher than break-oint frequency. During the continuou change of the tunable arameter, we determine the value of k tun o that i) the anti-reonance and reonance eak are reduced and ii) the hae become about -9 deg. Due to thi method, the tunable arameter i et to k tun = 23. The value of arameter ued in thi exeriment are hown in Table 2. From Fig. 5, we can find that the anti-reonance and reonance are eliminated by uing the RDD oitive feedback control. From the reult of Wakui and Fuhima 26), the cloed-loo tranfer function of the reure control ytem in Fig. 4 i written a r ) =. 8) k g k re T c + G q k g k re k r 8) denote the tranfer function from the reference r of reure control ytem to the inner reure of air ring. A the gain k r of the reure PI comenator increae, the break-oint frequency of 8) become high. Thu, one can exand the bandwidth of the reure control ytem. r ~ Preure PI comenator +T u ~ k r T k g k re u k tun Preure FB Acceleration FB Pneumatic AVA RDD oitive FB control Dilacement FB k o x Gain [db] Phae [deg] - w/ RDD oitive FB Reonance w/o RDD oitive FB -5 Anti-reonance - 8 w/ RDD oitive FB 9 w/o RDD oitive FB Fig. 4 Block diagram of RDD oitive feedback control cheme. The blue olid box contain the RDD oitive feedback controller, whoe tunable arameter i denoted by k tun. Thi control cheme i ued for the ureion of the anti-reonance and reonance. Fig. 5 Elimination of anti-reonance and reonance with RDD oitive feedback control no dead time). By uing thi control method, the anti-reonance and reonance in the frequency region from to 3 Hz are ureed. 3. Bandwidth exanion in reence of dead time 3.. Drawback of reviou RDD oitive feedback We firt how that when the neumatic AVA ha the dead time, the reviou RDD oitive feedback cannot be imlemented. Figure 6 illutrate the block diagram of the control ytem with the dead time. T d denote the dead time of the air ring. The tranfer function from the outut ũ of reure PI comenator to the inner reure of air ring i given by ũ) = G q M 2 + D + K)e Td M 2 + D + K)β V + c) + A A G q k tun k o e T d 9) ) G q M 2 + D + K)e T d β V + c M 2 + D + K + K air G qk tun k o e T d ). ) A [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 25

6 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) Note that the denominator in 9) and ) include the dead time term e T d. In order to cancel effect of the air ring tiffne, the tunable arameter k tun of the RDD oitive feedback control hould be elected a follow: k tun = A G q k o e T d. ) From Fig. 6, the inut voltage u to the ervo valve i exreed a u) = ũ) + k tun k o x) 2) Subtituting ) into 2), we obtain u) = ũ) + A G q e T d x). 3) By invere Lalace tranform, we have ut) = ũt) + A G q ẋt + T d ). 4) To determine the inut voltage u at time t, it i required to know the future value of the dilacement derivative, i.e., ẋt + T d ), a riori. However, thi a riori information i unavailable due to the cauality of the control ytem. For thi reaon, the reviou RDD oitive feedback control cannot be emloyed in the reence of the dead time. Figure 7 how Bode lot of the tranfer function from the reference r of reure control ytem to the outut of reure feedback gain. Thi i the reult in the cae where the dead time exit. The dead time i T d =.. A mentioned in the above, ince the reviou RDD oitive feedback control cannot be imlemented in the reence of the dead time, thi feedback control cheme i not emloyed in the exeriment of Fig. 7. From thi figure, we can ee that even if the gain k r of reure PI comenator become large, the anti-reonance and reonance till remain. Moreover, the reonance eak increae with the gain k r of reure PI comenator. Thi caue the detabilization of the control ytem. Thu, even in the cae where the dead time exit, the anti-reonance and reonance have to be eliminated o a to exand the bandwidth. Outut of reure PI comenator u ~ u Dead time e - T d Pneumatic AVA RDD oitive FB control x Gain [db] Phae [deg] k r= k r= k r=5-9 k r=2-8 - Fig. 6 Block diagram of neumatic AVA and it control ytem in reence of dead time. In thi figure, T d denote the dead time of the neumatic AVA. Due to effect of the dead time, the reviou RDD oitive feedback cannot be imlemented. Fig. 7 Bandwidth of reure control ytem in reence of dead time dead time T d =. ). The red and blue line how the reult in the cae of k r = 2and k r = 5, reectively. The anti-reonance and reonance remain regardle of the gain k r of reure PI comenator. Furthermore, when the gain k r of reure PI comenator increae, the reonance eak alo increae RDD oitive feedback control combined with Smith rediction Then, we conider the bandwidth exanion of the reure feedback control ytem. Then, we conider the bandwidth exanion of the reure feedback control ytem. A mentioned in ubection 3., the reviou RDD oitive feedback control cannot be imlemented under the condition where the dead time i reent. To comenate effect of the dead time, variou control method have been reorted. For intance, a frequency-domain aroache, internal model [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 26

7 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) r Preure PI comenator + u ~ T k r T ~ k g k re k tun Smith redictor P xu T - d ) k o e - e - T d Pneumatic AVA RDD oitive FB control x Fig. 8 Block diagram of RDD control cheme combined with Smith redictor aroach. In thi figure, P xu denote the model of the neumatic AVA, and T d denote the time delay ued for Smith rediction. The Smith redictor i emloyed in the inner loo of the reure control ytem o a to comenate effect of the dead time. control Morari and Zafiriou, 989), artial model matching method Kitamori, 979; Kitamori, 2), Smith redictor Smith, 959), etc., are widely known. Meanwhile, a time-domain aroache, tate-rediction control Manitiu and Olbrot, 979; Furutani and Araki, 998), Lyaunov-Kraovkii aroach Fridman and Shaked, 22), Lyaunov- Razumikhin aroach Jankovic, 2), etc., were reented. However, the ractical alication of thee control method to the dead-time comenation of neumatic AVA wa not ufficiently dicued. On the other hand, we conidered the imlementation method of the Smith redictor aroriate for the control of neumatic AVA Nakamura et al., 23). In thi literature, the model of the neumatic AVA conidering ie length wa reented. The identification method of the dead time wa alo rooed. Since, from reult of Nakamura et al. 23), the model and dead time of the neumatic AVA are available, it i exected that the Smith redictor can be imlemented and emloyed for the bandwidth exanion of reure control ytem. Moreover, thi aer focue on howing that when the dead-time comenation can be achieved, effect of air ring tiffne K air are eaily cancelled by uing reviou RDD oitive feedback control without other advanced technique. For thee reaon, the Smith rediction aroach i adoted. The block diagram of our method i hown in Fig. 8. P xu denote the model of the neumatic AVA, and T d denote the time delay [] ued for Smith rediction. In thi method, the RDD oitive feedback control i combined with Smith rediction. The tranfer function from the outut ũ of reure PI comenator to the inner reure of air ring i exreed a ũ) = = G q M 2 + D + K)e Td { k tun k o P xu ) e Td )} M 2 + D + K)β V + c) + A [A G q k tun k o e T d { k tun k o P xu ) e T d )} ] P xu )M 2 + D + K)e T d A [ k tun k o P xu ) + k tun k o { P xu )e T d P xu )e T d }]. 5) By chooing P xu = P xu and T d = T d, 5) i rearranged into ũ) = P xu)m 2 + D + K)e Td. 6) A { k tun k o P xu )} From ) and 6), we obtain ũ) = G q M 2 + D + K)e Td M 2 + D + K)β V + c) + A A G q k tun k o ). 7) Note that the denominator of 7) i the ame a that of 4). A decribed in ubection 2.3, effect of the air ring tiffne are cancelled by etting the tunable arameter in 7) a k tun = A /G q k o ). For thi reaon, even if the dead time i reent, it i not neceary to emloy a riori information on the dilacement derivative ẋt + T d ). Hence, one can imlement the RDD oitive feedback for the reure control ytem with the dead time. When the Smith redictor i utilized, the cloed-loo tranfer function of the reure control ytem i exreed a r ) = G q k r M 2 + D + K)T + )e Td M 2 + D + K)[T β V + c){ k tun k o P xu ) e T d )} + G q k r k g k re T + )e T d ] +A T 2 [A { k tun k o P xu ) e T d )} G q k tun k o e T d ]. 8) [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 27

8 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) The model and time delay ued for Smith rediction are choen a P xu = P xu and T d = T d, reectively. Then, uing 5), we can rewrite 8) a r ) = G q k r T + )e Td T β V + c) + G q k r k g k re T + )e T d. 9) By etting the time contant of reure PI comenator a that of air ring i.e., T = T air ), we obtain r ) = e Td. 2) k g k re T c + e T d G q k g k re k r When the gain k r of the reure PI comenator i ufficiently large, 2) i aroximated a r ) k g k re = cont. 2) Hence, the bandwidth exanion of the reure control ytem can be achieved by mean of our aroach Exerimental validation Finally, we conduct exeriment on the frequency and time reone of the control ytem for the neumatic AVA in the reence of the dead time. In the exeriment, the value of model arameter ued in the Smith redictor are the ame a nominal value, which are hown in Table. The value of controller arameter and enor enitivitie are ummarized in Table 2. A mentioned in Section, the ervo valve i et aart from the neumatic AVA ued for the lithograhy equiment. In emiconductor manufacturing indutrie, the ie with length of everal ten meter i emloyed. Due to thi, it i neceary to reare a long ie in exeriment. However, when the ie i changed for exeriment, it lead to eriou damage to the connection art of the ie and air ring. Furthermore, in the room where the neumatic AVA i et, the floor loe lightly. Under thee exerimental condition, effect of ground vibration are larger than thoe of the dead time. A a reult, there i a difficulty in accurately evaluating effect of dead-time comenation. For thee reaon, the dead time i generated in the DSP. The value of the dead time i T d =.. The other condition of exeriment in thi ection are the ame a thoe in ubection 2.3. Under the etting decribed above, the erformance of our aroach i evaluated Sureion of anti-reonance and reonance Figure 9 how Bode lot of the tranfer function from the outut ũ of reure PI comenator to the outut k re of reure enor. From thi figure, we can ee that when the RDD oitive feedback control i ued, the anti-reonance and reonance are not oberved. In the cae with the reviou RDD oitive feedback control, a riori information on the dilacement derivative ha to be known ee ) and 4)). On the other hand, in the cae with the rooed aroach, one can elect the tunable arameter k tun uch that effect of air ring tiffne K air are cancelled even if a riori information i not available ee 7)). For thi reaon, the elimination of the anti-reonance and reonance can be achieved in the reence of the dead time. Moreover, by comaring Fig. 9 with Fig. 5, we can alo ee that when the dead time exit, the hae of Bode lot decreae in the frequency region at about Hz. Thi hae lag i due to effect of the dead time term e T d in 7) Bandwidth exanion of reure control ytem Figure how Bode lot of the tranfer function from the reference r of reure control ytem to the outut of reure feedback gain. Thi i the cae with the rooed aroach i.e., RDD oitive feedback control combined with Smith rediction). From Fig., we can find that the bandwidth of the reure control ytem increae with the gain k r of the reure PI comenator. Thi i becaue when the gain k r of the reure PI comenator i large, the gain of the cloed-loo tranfer function /r can be aroximated a a contant ee 2)). Hence, in the reence of the dead time, the bandwidth exanion of the reure control ytem can be achieved by uing our aroach. Moreover, to evaluate the erformance of our aroach in time domain, we meaure the inner reure of the air ring. Figure a) how the te reone of the control ytem in the cae without the rooed method. On the other [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 28

9 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) Gain [db] Phae [deg] - w/ RDD oitive FB w/o RDD oitive FB w/o RDD oitive FB w/ RDD oitive FB -8 - Fig. 9 Elimination of anti-reonance and reonance with RDD oitive feedback control dead time T d =. ). The red and blue line how the frequency characteritic in the cae without and with the RDD oitive feedback control. Since effect of the dead time are comenated due to the ue of the Smith redictor, the anti-reonance and reonance can be ureed by uing the RDD oitive feedback control. hand, Fig. b) how the te reone in the cae with the rooed method. In thee figure, the inner reure at the equilibrium oition of the iolation table i et to zero. The te inut ued a a reference ignal i alied to the control ytem at t =.5. From Fig. a), we can ee that large overhoot and long ettling time are oberved in the cae without the rooed method. A hown in Fig. 7, when the rooed method i not emloyed, the bandwidth of the reure control ytem cannot be ufficiently exanded due to the anti-reonance and reonance. For thi reaon, the overhoot and ettling time become large. On the other hand, from Fig. b), we can find that the rie time of the te reonei imroveddueto the bandwidthexanion ee reonefor.5 t ). For the ame reaon, the ettling time i alo imroved. The reult hown in Fig. b) indicate that the fat reone of the reure control ytem i obtained. Therefore, in the reence of the dead time, our rooed aroach i effective for the reure control of the neumatic AVA. Gain [db] Phae [deg] k r =5 k r = k r=2 w/o RDD FB) -3 k r=2 w/ RDD FB) k r =5 k r = k r=2 w/o RDD FB) k r=2 w/ RDD FB) - Fig. Bandwidth of reure control ytem dead time T d =. ). The dahed line and the olid line how the reult in the cae without and with the RDD oitive feedback control, reectively. The gain k r of reure PI comenator i et to 2, 5, and 22. When the gain k r become large, the bandwidth of the reure control ytem exand. Preur [kpa] 7. k r= k r =5 k r=2 2 3 Time t [] a) Inner reure of air ring w/o rooed method). Preur [kpa] 7. k r= k r =5 k r=2 2 3 Time t [] b) Inner reure of air ring w/ rooed method). Fig. Ste reone of control ytem dead time T d =. ). The red, blue, and green line how the time reone in the cae of k r = 2, 5, and 22, reectively. The uer and lower figure are the cae without and with the RDD oitive feedback control combined with Smith rediction. Since the bandwidth of the reure control ytem i exanded by mean of the rooed method, the rie time and the ettling time are imroved. 4. Vibration tranmiibility of neumatic AVA in reence of dead time In reviou ection, the dilacement x of the floor i aumed to be zero due to the imlicity of dicuion. In thi ection, we invetigate effect of floor vibration on the neumatic AVA in the reence of the dead time. A a erformance [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 29

10 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) metric for the iolation of the floor vibration, the vibration tranmiibility i widely emloyed ASML Inc., 24; Nikon Cor., 22; Ryaboy, 24). Thi metric i defined a the ratio of the dilacement amlitude of iolation table to the dilacement amlitude of floor, x/x. When the magnitude x/x i maller than one i.e., db), the vibration from the floor i attenuated. For thi reaon, it i deirable to reduce the magnitude of the vibration tranmiibility in the develoment of the neumatic AVA. Before reenting reult, we exlain the vibration tranmiibility in the cae where the dead time i not reent. In order to dicu thi cae, Wakui and Fuhima 26) adoted root locu, which i one of geometric aroache. On the other hand, we analytically how how the vibration tranmiibility i changed by the RDD oitive feedback control. In the control ytem of Fig. 3, the time contant T a of the acceleration PI comenator i taken a T a = T c G q k r k g k re. 22) Then, from Fig. 3, the vibration tranmiibility i given by T a {D + K + K air G ) } qk tun k o T T a + K di x) x ) = A T + T a + T a {M 2 + D + D e ) + K + K air G ) }, 23) qk tun k o T T a + K di A T + T a + where K di = A k ar k k o k g k re, D e = A k ar k a k acc T a k g k re, and K di denote ervo tiffne [N/m] due to the dilacement feedback control and D e denote electrical daming coefficient [N /m]. From 5) and 23), we have x) x ) = T a D + K) + K di. 24) T a {M 2 + D + D e ) + K} + K di From 23) and 24), it i found that when effect of the air ring tiffne are cancelled by uing the RDD oitive feedback, the vibration tranmiibility x/x change. Figure 2 how the imulation reult on the vibration tranmiibility of the control ytem in the cae where the dead time i not reent. The condition of thi imulation are the ame a thoe of exeriment reented in ubection 3.3. The value of arameter of the neumatic AVA ued for the imulation are lited in Table. From Fig. 2a), we can ee that the frequency region where the magnitude x/x of the vibration tranmiibility i maller than db i exanded by uing the RDD oitive feedback control. The reaon i a follow. The breakoint frequencie of the firt order lag ytem /T +) and /T a +) in 23) are /2πT ). and /2πT a ).53 Hz, reectively. On the other hand, from Fig. 2b), we can find that the reonance frequency of the vibration tranmiibility x/x i about.78 Hz, which i larger than the above breakoint frequencie. Thu, in the frequency region including the reonance frequency, the firt order lag ytem can be aroximated a /T + ) /T )and/t a + ) /T a ). Moreover, the ervo tiffne i K di = A k ar k k o )/k g k re ) 666 N/m, which i much le than the um of ring contant of the air ring and mechanical ytem i.e., K di K + K air = 267 N/m). Under thee aroximation, 23) can be exreed a x) x ) D + K + K air G ) qk tun k o A M 2 + D + D e ) + K + K air G ). 25) qk tun k o A Let f denote the reonance frequency [Hz] of the vibration tranmiibility 25). Then, from 25), the reonance frequency i exreed a f = {K + K air G )} qk tun k o D + D e) 2. 26) 2π M A 4M 2 [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 2

11 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) Tranmiibility [db] Tranmiibility [db] 3 Decreae of reonance frequency w/o RDD oitive FB -3 w/ RDD oitive FB Fig. 2 Cloe-u a) Overview. Region where tramiibiliy x/x i maller than db w/o RDD oitive FB - 5x x b) Cloe-u view. Vibration tranmiibility x/x no dead time). The red and blue line how the cae without and with the RDD oitive feedback control cheme. The lower figure i the cloe-u view of the uer figure. When the RDD oitive feedback control i ued, effect of air ring tiffne are cancelled. Conequently, the reonance frequency i decreaed, reulting in the imrovement of the vibration tranmiibility. Tranmiibility [db] Tranmiibility [db] Tranmiibility [db] 3-3 T =.5 d w/ RDD oitive FB w/o RDD oitive FB T d =. w/o RDD oitive FB w/ RDD oitive FB T d =.5 w/o RDD oitive FB w/ RDD oitive FB Fig. 3 Vibration tranmiibility x/x dead time T d =.5,., and.5 ). The red and blue line how the reult in the cae without and with the RDD oitive feedback control combined with Smith rediction. When the rooed aroach i emloyed, the magnitude x/x in the cae of T d =.5 and. become maller than db at about 2 to 3 Hz. Meanwhile, in the cae of T d =.5, the magnitude exceed db at the above frequency region. Due to effect of the dead time, the vibration tranmiibility i changed. Note that 26) for k tun = correond to the reonance frequency of the vibration tranmiibility 25) in the cae without the RDD oitive feedback control. In thi cae, the reonance frequency i exreed a f = K + Kair 2π M D + D e) 2, if k 4M 2 tun =. 27) In the cae with the RDD oitive feedback control, from 5) and 26), the reonance frequency i written a f = K 2π M D + D e) 2, if k 4M 2 tun = A. 28) G q k o Thu, when the dead time i not reent, the reonance frequency f i decreaed by uing the RDD oitive feedback ee 27) and 28)). Conequently, the frequency region where the magnitude x/x i maller than db i exanded, reulting in the imrovement of the vibration tranmiibility ee Fig. 2a)). Figure 3 how imulation reult of the effect of the dead time T d. In thi imulation, the dead time i et to T d =.5,., and.5. The other condition of Fig. 3 are the ame a thoe of Fig. 2. For the meaurement of the vibration tranmiibility, it i neceary to generate the floor vibration whoe ectrum i rich o a to excite the iolation table. However, due to the tructure of the laboratory room, in which the neumatic AVA i et, the ower ectrum of the generated floor vibration i mall in low frequency region. For thi reaon, the vibration tranmiibility cannot be accurately evaluated by exeriment. Alternatively, we erformed the imulation. From Fig. 3, we can find that when the RDD oitive feedback control combined with Smith rediction i emloyed, the vibration tranmiibility at about 2 to 3 Hz in the cae of T d =. i maller than that in the cae of T d =.5. Meanwhile, in the cae of T d =.5, the vibration tranmiibility exceed db at the above frequency region. Thi reult indicate that effect of iolation of floor vibration change in accordance with the dead time. In the following, we [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 2

12 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) exlain the detail. From Fig. 3 and 8, the vibration tranmiibility in the reence of the dead time i exreed a where x) x ) = T a [D + K){β V + c)t + G q k r k g k re T + )H)} + A {A G q k tun k o H)}T 2 ] T a [M 2 + D + K){β V + c)t + G q k r k g k re T + )H)} + A {A G q k tun k o H)}T 2 ] +G q A k r k ar k k o T + )T a + )H) +G q A k r k ar T + )T a + )k a k acc 2 + k k o )H), 29) H) = e T d k o k tun e T d ) P xu ). The model and time delay ued for Smith rediction are taken a P xu = P xu and T d = T d. From ), 5), and 29), we obtain where x) x ) = D + K){T 2 c2 2 L) + G q k r k g k re T ce Td } + A G 2 q k2 r k ark k o k g k re T a + )e Td M 2 + D + K)T 2 c2 2 + G q k r k g k re T ce T d ) + A G 2 q k2 r k ark a k acc 2 + k k o )T a + )e T d L) = M2 + D + K)β V + c) + A 2 e T d + {M 2 + D + K)A 2 /D + K)} e T d ) M 2 + D + K)β V + c) + A 2. From 22) and 3), we have 3) Let x) x ) = T a D + K)T a L) + e Td ) + K di T a + )e Td T a M 2 + D + K)T a + e T d ) + T a D e 2 + K di )T a + )e, 3) T d T a D + K) + K di N ) =, 32) T a {M 2 + D + D e ) + K} + K di { e T d Ta D + K)L) + K di e T } d + T a T a D + K) + K di N 2 ) = [ Ta e T {M 2 + D + D e e Td ) + K} + K di e T ]. 33) d d + T a T a {M 2 + D + D e ) + K} + K di Then, the vibration tranmiibility 3) i exreed a x) x ) = N )N 2 ). 34) Note that N ) correond to 24), i.e., the vibration tranmiibility in the cae where the dead time i not reent. On the other hand, N 2 ) include the dead time term e T d. For thi reaon, in the reence of the dead time, the vibration tranmiibility i changed due to effect of N 2 ) in 34), where the dead time term i included. Therefore, when the dead time exit, there i trade-off between the imrovement of vibration tranmiibility and the bandwidth exanion of reure control ytem. 5. Concluion In thi aer, we conidered the bandwidth exanion of the reure control ytem with the dead time of the air ring. To extend the control bandwidth, the anti-reonance and reonance in the neumatic ytem are eliminated by mean of the RDD oitive feedback control combined with the Smith rediction. The advantage of our method i that the RDD oitive feedback control can be imlemented without a riori information on the dilacement derivative. From exerimental reult, we found that when the rooed method i emloyed in the reence of the dead time, the control bandwidth can be increaed by tuning the gain k r of the reure PI comenator. Moreover, we alo found that due to the dead time, the vibration tranmiibility increae in a certain frequency region. Our future work i to achieve the bandwidth exanion conidering the imrovement of vibration tranmiibility. [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 22

13 Nakamura, Kawakami and Wakui, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.9, No.3 25) Acknowledgement Thi work i uorted in art by JSPS KAKENHI, Grant-in-Aid for Scientific Reearch C) Grant Number Reference ASML Inc., Lithograhic aaratu comriing a vibration iolation uort device, United State atent dicloure US8255B2 22) Butler, H., Poition Control in lithograhic equiment; an enabler for current-day chi manufacturing, IEEE Control Sytem Magazine, Vol.3, No.5 2) Canon Inc., Anti-vibration aaratu and exoure aaratu uing the ame, United State atent dicloure US623443B 2a) Canon Inc., Active anti-vibration aaratu and exoure aaratu, Euroean atent dicloure EP2478A2 2b) Fridman, E. and Shaked, U., A decritor ytem aroach to H control of linear time-delay ytem, IEEE Tranaction on Automatic Control, Vol.47, No.2 22), Furutani, E. and Araki, M., Robut tability of tate-redictive and Smith control ytem for lant with a ure delay, International Journal of Robut and Nonlinear Control, Vol.8, No. 998), Heertje, M. F., Sahin, I. H., van de Wouw, N. and Heemel, W. P. M. H., Switching control in vibration iolation ytem, IEEE Tranaction on Control Sytem Technology, Vol.2, No.3 23) Jankovic, M., Control Lyaunov-Razumikhin function and robut tabilization of time delay ytem, IEEE Tranaction on Automatic Control, Vol.46, No.7 2), Kawahima, K., Kato T., Sawamoto, K. and Kagawa, T., Realization of virtual ub chamber on active controlled neumatic iolation table with reure differentiator, Vol.3, No.2 27) Kitamori, T., Partial model matching method conformable to hyical and engineering actualitie, Proceeding of the IFAC Symoium on Sytem Structure and Control 2), Kitamori T., A method of control ytem deign baed uon artial knowledge about controlled rocee, Tranaction of the Society of Intrument and Control Engineer, Vol.5, No.4 979), in Jaanee) Lee, J. H. and Kim, K. J., Modeling of nonlinear comlex tiffne of dual-chamber neumatic ring for reciion vibration iolation, Journal of Sound and Vibration, Vol.3, No ) Manitiu, A. and Olbrot, A. W., Finite ectrum aignment roblem for ytem with delay, IEEE Tranaction on Automatic Control, Vol.24, No.4 979), Morari, M. and Zafiriou, E., Robut Proce Control 989),.75-84, Prentice-Hall, Inc. Nakamura, Y., Goto, S. and Wakui, S., Tuning method of a Smith redictor for neumatic active anti-vibration aaratue, Journal of Advanced Mechanical Deign, Sytem, and Manufacturing, Vol.7, No.4 23) Nikon Cor., Stage aembly with lightweight fine tage and low tranmiibility, United State atent dicloure US7869B2 2) Pu, H., Luo, X. and Chen, X., Modeling and analyi of dual-chamber neumatic ring with adjutable daming for reciion vibration iolation, Journal of Sound and Vibration, Vol.33, No.5 2) Ryaboy, V. M., Static and dynamic tability of neumatic vibration iolator and ytem of iolator, Journal of Sound and Vibration, Vol.333, No. 24).3-5. Shin, Y. H. and Kim, K. J., Performance enhancement of neumatic vibration iolation table in low frequency range by time delay control, Journal of Sound and Vibration, Vol.32, No ) Shin, Y. H., Kim, K. J., Chang, P. H. and Han, D. K., Three degree of freedom active control of neumatic vibration iolation table by neumatic and time delay control technique, Tranaction of ASME, Journal of Vibration and Acoutic, Vol.32, No.5 2),.-2. Smith, O. J. M., A controller to overcome dead time, ISA Journal, Vol.6, No.2 959) Wakui, S., Incline comenation control uing an air-ring tye active iolated aaratu, Preciion Engineering Vol.27, No.2 23) Wakui, S. and Fuhima C., Wide frequency band of reure feedback uing oitive feedback for an air ring tye antivibration aaratu, Tranaction of the Jaan Society of Mechanical Engineer, Serie C, Vol.72, No.79 26) in Jaanee) [DOI:.299/jamdm.25jamdm3] 25 The Jaan Society of Mechanical Engineer 23

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