Precision Tracking Control of a Piezoelectric-Actuated System

Transcription

1 Proceengs of the 5th Meterranean Conference on Control & Automaton, July 7-9, 7, Athens - Greece T3- Precson Trackng Control of a Pezoelectrc-Actuate System Jng-Chung Shen, Wen-Yuh Jywe, *Huan-Keng Chang An *Yu-Lng Shu Department of Automaton Engneerng, Natonal Formosa Unversty Huwe, Yunln, Tawan Emal: cshen@nfu.eu.tw *Department of Electrcal Engneerng, Natonal Yunln Unversty of Scence an Technology, Doulou, Yunln, Tawan. Abstract--In ths paper, precson trackng control of pezoelectrc-actuate systems s scusse. In orer to obtan precson trackng control, Prantl-Ishlnsk (PI) moel s use to moel the hysteress nonlnearty. Then, the nverse PI moel s use to reuce the hysteress nonlnearty an a slng-moe controller s use to compensate the remanng nonlnear uncertanty an sturbances. The pezoelectrc-actuate system s moele as a lnear moel couple wth a hysteress. The lnear moel s entfe then t s use to esgn the slng-moe controller. Fnally, expermental results are presente to verfy the usefulness of ths metho. I. INTRODUCTION Pezoelectrc actuators are becomng ncreasngly mportant n toay s postonng technology ue to the rurements of nanometer resoluton n splacement. It s well known that the pezoelectrc actuator has many avantages [] such as: ) there are no movng parts; ) the actuators can prouce large forces; 3) they have almost unlmte resoluton; ) the effcency s hgh; an 5) response s fast. However, t also has some ba characterstcs such as: ) hysteresys behavor; ) rft n tme; 3) temperature epenence. Hysteress characterstcs are generally nonfferentable nonlneartes an usually unknown, ths often lmts system performance va, e.g., unesrable oscllatons or nstablty. Therefore, t s ffcult to obtan an accurate traectory trackng control. Recently, several methos have been reporte for the traectory trackng control of a pezoelectrc-actuate system. Ge an Jouaneh [,3] use a combnaton of a proportonal ntegral ervatve (PID) feeback controller wth a feeforwar controller that nclue the Presach moel of hysteress. Ther expermental results showe that the trackng performance was mprove greatly. However, the result n [] s only val for a snusoal traectory an the metho n [3] nees to tran the moel by usng reference nput before the control starte. Ku et al. [7] combne a PID feeback controller wth an aaptve neural network feeforwar controller to control a nanopostoner that was actuate by a pezoelectrc actuator. Cruz- Hernanez an Haywar [] propose a varable phase metho. They utlze an operator to shft the peroc nput sgnal by a phase angle that epene on the ampltue of the nput sgnal, then use ths operator to reuce the hysteress nonlnearty. Huang an Ln [] propose a new hysteress moel base on two frst-orer transfer functons n parallel wth two parameters etermne from experment. Aaptve control s also an approach to the nverse control of plants wth hysteress behavor. Tao an Kokotovc [9] evelope an aaptve hysteress nverse an cascae t wth the system so that the effects of hysteress nonlnearty coul be reuce. Xu [] utlze an aaptve neural network nverse controller to compensate the hysteretc behavor of a pezoelectrc actuator an a PI controller n the outer loop to overcome the remanng nonlnear uncertanty. Furthermore, Hwang et al. [] utlze an offlne learne neural network moel to reuce the effect of hysteress then esgne a screte-tme varable structure controller to overcome the remanng uncertanty. They also renforce ths metho [5] by usng a recurrent neural network to mprove the control performance. However, the computaton buren of the controller that was esgne by ther metho s heavy. In the lterature [], Shen et al. utlze nteger slng-moe controller to compensate the nonlnearty an sturbances n pezoelectrc actuate systems. In ths paper, Prantl-Ishlnsk (PI) [3,] moel s use to moel the hysteress nonlnearty. Then, the nverse PI moel s use to reuce the hysteress nonlnearty an a slng-moe controller s use to compensate the remanng nonlnear uncertanty an sturbances. The man avantages of PI moel over the Presach moel are that t s less complex an ts nverse can be compute analytcally. In ths stuy, the slng-moe uncertanty (sturbance) estmaton an compensaton scheme [,,] s use to esgn the feeback controller. Fnally, ths esgn metho was apple to the moton control of a nano-stage. The expermental results are presente that verfy the usefulness of ths metho. II. EXPERIMENTAL SETUP The expermental apparatus of ths stuy s a one-axs nano-stage. Actuaton of ths nano-stage s one wth a pezoelectrc actuator (PSt5/7/ Vs, Pezomechank

2 Proceengs of the 5th Meterranean Conference on Control & Automaton, July 7-9, 7, Athens - Greece T3- GmbH) wth a nomnal expanson of μ m (at 5V). Capactance-type gap sensors (D-5, Pezomechank GmbH) are use for poston measurement. The measure range of the gap sensor s extene by a factor of three. Therefore, total ranges of the gap sensors are 5 μ m wth a senstvty of.v / μm. Control of ths nano-stage s one wth a controller boar wth a Power PC central processng unt (DS3, SPACE GmbH). The samplng rate of the control algorthm was KHz. The resoluton of A/D converters s -bt. n H[ x]( k) = w H [ x, y ]( k () = r ) where w are the weghtngs, r are the threshol values wth = r < r <... < rn an y are the ntal states. The control nput threshol values r are usually chosen to be ual ntervals. To fn the parameters of hysteress moel, the responses of the actuator must be measure expermentally. Then the values of r are selecte an the weght parameters w are foun by ong a least square ft of uaton () to the response ata of actuator. Fg. 3 shows the hysteress response of the actuator an ts PI moel (n=). Fg.. Photograph of the nano-stage. III. SYSTEM MODELING AND CONTROLLER DESIGN Ths secton escrbes the way to moel the pezoelectrc actuate system an how to esgn the feeback controller. Frstly, PI hysteress moel s use to moel the hysteress nonlnearty of actuator an the nverse of ths moel s use to cancel out the hysteress nonlnearty. Then the lnearze moel of the pezoelectrc actuate system s estmate an an ntegral slng moe controller s esgne to compensate the remanng uncertantes an sturbances. 3. Prantl-Ishlns Hysteress Moel The elementary operator n the PI hysteress moel s a backlash operator (see Fg. ). A backlash operator s efne by y( k + ) = H r[ x, y]( k + ) () = max{ x( k + ) r,mn{ x( k + ) + r, y( k)}} where x s the control nput, y s the output of actuator, r s the magntue of backlash. The ntal conton of () s gven by y ( ) = max{ x() r,mn{ x() + r, y}} Where ntal state y R, an s usually but not necessarly ntalze to. Hysteretc nonlneartes can be moele by a lnearly weghte superposton of many backlash operators wth fferent threshol an weght values, Dsplacement (μm) Fg.. Backlash operator measure moel Input Voltage (V) Fg. 3. Measure hysteress nonlnearty an ts moel. 3. Inverse Moel of Hysteretc Nonlneartes The nverse of the moel () s also a PI type an s gven by H n [ y]( = w' H [ y, y' ](. (3) = r ' The nverse moel parameters can be foun by

3 Proceengs of the 5th Meterranean Conference on Control & Automaton, July 7-9, 7, Athens - Greece T3- w ' = ; w w w', = ( w + w )( w + w ) = = y' = w y + w y, = n = = + r ' = w ( r r ) ; = Kn The nverse moel (3) can be use to cancel out the hysteress nonlnearty of pezoelectrc actuator as shown n Fg.. Fg. 5 shows the nverse compensate result. It can be seen that most hysteress nonlnearty s compensate. Output splacement (μm) Inverse Hysteress moel Pezoelectrc actuator hysteress Actuator Lnearzaton Fg.. Pezoelectrc actuator lnearzaton. Lnear Plant Dynamcs - Desre splacement (μm) Fg. 5. Result of nverse compensaton. 3.3 Open Loop Characterstcs an Moel In orer to esgn the feeback controller properly, the open-loop characterstcs of the pezoelectrc actuate systems are nvestgate. Frstly, the fruency-response experments are conucte. When measurng the fruency response, a bas voltage was ae to push the stage platform to the center of the movng range. Then, a ranom exctaton sgnal was sent to the pezoelectrc actuator an the splacement s measure by the gap sensor. The test results are epcte n Fg.. As seen n Fg., the banwth s about Hz. A lnear ynamc moel, represente as a thr orer transfer functon was curve-ftte to the measure fruency response. The response of the moel also shows on Fg.. The poles of the moel are -3 an -593 ± 35 respectvely. The omnant pole s -3 an the others poles are more than tmes faster than ths pole. Therefore, t s possble to moel ths system by a frst orer transfer functon. Base on the results of nvestgaton, the lnear plant ynamcs of ths pezoelectrc-actuate system can be moele by a frst orer uncertan lnear system. Thus, the moel of ths system can be moele as the block agram shows n Fg. 7. Where u s the nput of the nverse moel, represents the sturbance an the frst orer fferental uaton T ( + Δ( ) y& + y = v, () escrbes the ynamc behavor of the system. Where T s the nomnal tme constant an Δ ( represents the uncertanty. Parameter T an the boun of Δ ( can be etermne by ong step response tests at varous workng ponts. From Fg. 7, v can be represente as v = u + N( + (, (5) where N ( represents the remanng nonlnear uncertan part of the hysteress. From () an (5), the followng ynamc uaton can be obtane: y u y& = + + φ( () T T where Δ( y u) + N + φ ( = T ( + Δ) represents the sturbance an uncertantes. Ampltue Phase (eg) u Fruency response Fruency (ra/s) moel measure Fg.. Fruency response of the system H x H v T ( + Δ( ) y& + y= v Fg. 7. Moel of the pezoelectrc-actuate system. y

4 Proceengs of the 5th Meterranean Conference on Control & Automaton, July 7-9, 7, Athens - Greece T3-3. Integral Slng-Moe Controller Desgn Ths subsecton escrbes how to esgn the ntegral slng-moe controller. In ths stuy, the slng moe sturbance (uncertanty) estmaton an compensaton scheme s apple to esgn the close-loop controller for the pezoelectrc-actuate system. Let y be the esre splacement, whch may be tme varyng. Defne e = y y (7) as the trackng error. From () an (7), the error ynamcs can be obtane as y u e& = y& y& = y& + φ(. () T T Let the control law be u = y + T ( λ e + y& ) + u (9) where λ s the feeback gan to be esgne so that the error ynamc wll have the esre response whle the system s free of sturbance an uncertanty, an u s the uncertanty an the sturbance compensaton component yet to be etermne by the slng moe estmator. Defnng the swtchng functon as S = z e () wth z& = λe u + ψ, z() = e() () T where z s the state varable of ths auxlary process, ψ s the swtchng acton assgne as f S > ψ = ηsgn( S), sgn( S) = f S < () f S = an the postve constant η satsfes η > φ( (3) Ensurng a slng regme S = rures conseraton of the Lyapunov canate V =.5S. Dfferentatng V wth respect to tme an substtutng (-) to obtan V& = S( z& e& ) = S[ ( η sgn( S) + φ( )] () From (3) an (), t s seen that V & < f S (5) Thus the slng conton s satsfe. Note that z ( ) = e(), therefore S = for = t () From (5) an (), t can be conclue that the slng moe exsts at all tmes,.e., t (7) Denote the uvalent value of ψ as S & = for all ψ. Snce S & =, ψ can be etermne from (), () an (): ψ = φ () Ths means that the uvalent value of ψ uals the uncertantes an sturbances. By selectng u = Tψ, the uncertantes an sturbances can be compensate. It was shown n [] that the uvalent ψ s ual to the average value measure by a frst-orer lnear flter wth the swtche acton as ts nput. Therefore, u can be wrtten as u = Tψ av = Tψ (9) wth τ ψ& av + ψ av = ψ. () The tme constant τ shoul be mae small enough that the plant an sturbance ynamcs are allowe to pass through the flter wthout sgnfcant phase elay. Substtutng (9) an (9) nto () yels e& + λe = ψ φ, () whch s uvalent to e& + λe =. Ths uaton represents the esre error ynamcs. IV. EXPERIMENTAL RESULTS In orer to entfy the nomnal tme constant T an estmate the boun of Δ, step response tests at varous workng ponts are execute. From the test results, t was foun that the tme constant of ths stage les between. an 3.5 ms. When esgnng the controller, λ s chosen as large as possble to obtan wer banwth. The fnal controller parameters are chosen as follows: λ = 55, η = an τ =.. In orer to check the performance of ths control metho, snusoal waves wth 9 μm ampltue an fferent fruences range from -Hz were use to test the trackng performance. Fg. epcts the trackng results of Hz sgnal. Table summarzes the maxmum trackng errors of trackng test at fferent fruences. Comparson was mae wth the ntegral slng-moe controller (wthout nverse moel compensaton). It can be foun from Table that the nverse moel compensaton can mprove the trackng accuracy sgnfcantly. Fg. 9 shows the trackng results of a mult-fruency, nonstatonary ynamc moton profle (moulate an Hz snusos wth varyng ampltues). The maxmum trackng error s.39μm. It can be seen that the output stll can track the reference very well. For etermnng the precson of the trackng control, a 5Hz sne wave wth nm ampltue was use to comman the stage. Fg. shows the traectory followng results of ths stage to nm ampltue nput. For very small ampltue of reference poston of nm whch s almost the same as the magntue of nose, the measure poston stll tracke the reference farly well. It can be conclue that the trackng resoluton s about nm. In orer to etermne the close-loop banwth of ths stage, a 9 μm magntue snusoal sgnal was comman to ths nano-stage. The fruency of the comman moton was slowly ncrease untl an attenuaton of.77 n the

5 Proceengs of the 5th Meterranean Conference on Control & Automaton, July 7-9, 7, Athens - Greece T3- trackng magntue. The measure banwth was approxmately Hz. V. CONCLUSIONS Ths artcle reporte the precson trackng control of a pezoelectrc actuate moton stage. Frstly, Prantl- Ishlnsk (PI) moel s use to moel the hysteress nonlnearty of the actuator. Then, the nverse PI moel was use to reuce the hysteress nonlnearty an a slng-moe controller was use to compensate the remanng nonlnear uncertanty an sturbances. The most mportant features of PI moel are that t s less complex an ts nverse can be compute analytcally. Experments on trackng the snusoal waveforms an nonstatonary ynamc moton profle were carre out. From the result t can be seen that the performance of ths controller s goo an nm trackng resoluton can be obtane. ACKNOWLEDGEMENTS The authors want to thank the Natonal Scence Councl of Tawan for fnancal support prove uner grant no. NSC95--E-5-. REFERENCES [] J. Cruz-Hernanez an V. Haywar, Phase Control Approach to Hysteress Reucton, IEEE Transacton on Control Systems Technology, Vol. 9, pp. 7-,. [] P. Ge an M. Jouaneh, Trackng Control of a Pezoceramc Actuators, IEEE Transactons on Control System Technology, Vol., pp. 9-, 99. [3] P. Ge an M. Jouaneh, Generalze presach moel for hysteress nonlnearty of pezoceramc actuators, Precson Engneerng, Vol., pp. 99-, 997. [] Y. C. Huang an D. Y. Ln, Ultra-Fne Trackng Control on Pezoelectrc Actuate Moton Stage Usng Pezoelectrc Hysteretc Moel, Asan Journal of Control, Vol., pp.-,. [5] C. L. Hwang an C. Jan, A Renforcement Dscrete Neuro-Aaptve Control for Unknown Pezoelectrc Actuator Systems Wth Domnant Hysteress, IEEE Transactons on Neural Networks, Vol., pp. -7, 3. [] C. L. Hwang, C. Jan an Y. H. Chen, Pezomechancs Usng Intellgent Varable- Structure Control, IEEE Transactons on Inustral Electroncs, Vol., pp. 7-59,. [7] S. S. Ku, U. Pnsopon, S. Cetnkunt an S. Nakama, Desgn, Fabrcaton, an Real-Tme Neural Network Control of a Three Degrees of Freeom Nanopostoner, IEEE/ASME Transactons on Mechatroncs, Vol. 5, pp. 73-,. [] J. C. Shen, H Control an Slng Moe Control of Magnetc Levtaton System, Asan Journal of Control, Vol., pp ,. [9] G. Tao an P. V. Kokotovc, Aaptve Control of Plants wth Unknown Hysteress, IEEE Transactons on Automatc Control, Vol., pp. -, 995. []V. Utkn, J. Gulner an J. Sh, Slng Moe Control n Electromechancal Systems, Taylor & Francs, Pastow, 999. []J. H. Xu, Neural Network Control of a Pezo Tool Postoner, Proceengs of Canaan Conference on Electrcal an Computer Engneerng, pp , 993. []J. C. Shen, W. Y. Jywe, C. H. Lu, Y. T. Jan an Y. F. Deng, Integral Slng-Moe Control of a Pezoelectrc-Actuate Moton Stage, Proceengs of the th IFAC Worl Congress, Prague, 5. [3]W. T. Ang, F. A. Garmon, P. K. Khosla an C. N. Rvere, Moelng Rate-epenent Hysteress n Pezoelectrc Actuators, Proceengs of the 3 IEEE/RES Intl. Conference on Intellgent Robots an Systems, Las Vegas, pp , 3. []K. Kuhnen, Moelng, Ientfcaton an Compensaton of Complex Hysteretc Nonlneartes A Mofe Prantl-Ishlnsk Approach, European Journal of Control, Vol. 9, No., pp. 7-7, 3. Error(μm) Dsplacement(μm) Table. Results of trackng test. Maxmum Trackng Error ( μ m ) Fruency (Hz) Wthout Inverse Moel Wth Inverse Moel.(.7%).(.9%) 5.9(%).3(.33%).(.5%).59(.%) 5.9(.%).(.9%).(.%).3(.5%) mea. ref Tme(secon) Fg.. Result of trackng a Hz sne wave.

6 Proceengs of the 5th Meterranean Conference on Control & Automaton, July 7-9, 7, Athens - Greece T3- Error(μm) Dsplacement(μm).. -. Nonstatonary Profle Trackng.... Ref. Mea Tme(secon) Fg. 9. Result of trackng a mult fruency, nonstatonary ynamc moton profle. Dsplacement(μm) Tme(secon) Fg.. Trackng resoluton of the nano-stage.