Tacking of High-spee, Non-smooth an Micoscale-amplitue Wave Tajectoies Jiaech Kongthon Depatment of Mechatonics Engineeing, Assumption Univesity, Suvanabhumi Campus, Samuthpakan, Thailan Keywos: Abstact: High-spee Tacking, Invesion-base Contol, Micoscale Positioning, Reuce-oe Invese, Tacking. In this aticle, an invesion-base contol appoach is popose an pesente fo tacking esie tajectoies with high-spee (Hz), non-smooth (tiangle an sawtooth wave, an micoscale-amplitue ( micon) wave foms. The inteesting challenge is that the tacking involves the tajectoies that possess a high fequency, a micoscale amplitue, shap tunaouns at the cones. Two iffeent types of wave tajectoies, which ae tiangle an sawtooth waves, ae investigate. The moel, o the tansfe function of a piezoactuato is obtaine expeimentally fom the fequency esponse by using a ynamic signal analyze. Une the invesion-base contol scheme an the moel obtaine, the tacking is simulate in MATLAB. The main contibutions of this wok ae to show that () the moel an the contolle achieve a goo tacking pefomance measue by the oot mean squae eo (RMSE) an the maximum eo (Emax), () the maximum eo occus at the shap cone of the tajectoies, () tacking the sawtooth wave yiels lage RMSE an Emax values,compae to tacking the tiangle wave, an () in tems of obustness to moeling eo o unmoele ynamics, Emax is still less than % of the peak to peak amplitue of micon if the inceases in the natual fequency an the amping atio ae less than 5% fo the tiangle tajectoy an Emax is still less than % of the peak to peak amplitue of micon if the inceases in the natual fequency an the amping atio ae less than. % fo the sawtooth tajectoy. INTRODUCTION A piezo stage is wiely use in positioning an actuating motions in nano/micoscale isplacements o amplitues. Seveal woks have use a piezoactuato to achieve the goals. Fo example, the woks one by Kongthon et al., (, an ) employe a piezo-base positioning system to ive the biomimetic cilia-base evice so that the mixing pefomance in a mico evice was impove. Moallem et al., () use piezoelectic evices fo the flexue contol of a positioning system. The tacking of a tajectoy is vey common in contol poblems such as the woks by Beschi et al., () an Matin et al., (99). Tacking can be challenging in high-fequency applications with vey small isplacements. The challenge in this wok is that the tajectoies ae of high-spee (Hz), nonsmooth (tiangle an sawtooth wave, an micoscale-amplitue ( micon) wave foms. The goal is to popose a contolle that can tack pescibe tajectoies popely with a goo tacking pefomance. The tacking pefomance can be measue by the oot mean squae eo (RMSE) an the maximum eo (E max ). The est of this aticle is stuctue as follows. Section intouces the two tajectoies. The piezoactuato moel is obtaine in section. The contol scheme is popose in section. Section 5 shows the esults. In section, the obustness is investigate. Section 7 conclues the aticle. TRAJECTORIES. The Tajectoies to Be Tacke In this wok, thee ae two types of wave fom tajectoies use to investigate the tacking pefomance of the piezoactuato moel: tiangle wave, shown in Fig. an sawtooth wave, shown in Fig.. 99 Kongthon, J. Tacking of High-spee, Non-smooth an Micoscale-amplitue Wave Tajectoies. DOI:.5/597979957 In Poceeings of the th Intenational Confeence on Infomatics in Contol, Automation an Robotics (ICINCO ) - Volume, pages 99-57 ISBN: 978-989-758-98- Copyight c by SCITEPRESS Science an Technology Publications, La. All ights eseve
ICINCO - th Intenational Confeence on Infomatics in Contol, Automation an Robotics contolle then nees to tack the esie tajectoy of each type. PIEZO ACTUATOR MODEL Figue : Oiginal tajectoy fo tiangle wave of μ m amplitue an Hz fequency A piezo-base positioning system, o piezo stage, can be use in applications that equie vey small isplacements an lage fequency anges. A piezoactuato can geneate an extemely small isplacement own to the subnanomete ange. The numbe of vibation moes fo the piezo stage is infinite since the beam mechanism insie the piezo stage has an infinite imension. In geneal, an infinite imensional plant can be appoximate by a finite imensional moel, an in pactice, it is possible to take the fist few moes of vibation to epesent the total ynamics of the plant.. Fequency Response Expeiment To obtain the moel of the piezoactuato shown in Fig., the piezoactuato an the ynamic signal analyze shown in Fig., togethe with an inuctive senso an a powe amplifie ae connecte as shown in Fig. 5 to get the fequency esponse, an the moel is then obtaine. Figue : Oiginal tajectoy fo sawtooth wave of μm amplitue an Hz fequency. Filtee an Desie Tajectoies It can be seen in Figs. an that the oiginal tajectoies contain vey shap tunaouns at the cones. In pactice, an actuato cannot tack a tajectoy with a vey shap cone popely as it has a limite banwith. In oe to achieve a goo tacking pefomance, the oiginal tajectoies theefoe nee to be smoothen by a secon-oe filte with the filteing tansfe function of the fom. G f ω f ( s + ω f ω f s + ω f whee ω f is the beak fequency of the filte, an In this wok, ω f of Hz, o (π ) a/s is selecte to get the tajectoies filtee. The filtee tajectoy is heeafte efee to as the esie tajectoy. The Figue : Piezoactuato use to pouce mico-scale amplitues of oscillations with high fequencies. Figue : Dynamic signal analyze use to get the fequency esponse to obtain the moel of the actuato. 5
Tacking of High-spee, Non-smooth an Micoscale-amplitue Wave Tajectoies Figue 5: Block iagam use fo obtaining the fequency esponse of the piezoactuato.. Tansfe Function an Time Scaling In this wok, the poles, the zeos, an the gain of the piezoactuato ae foun expeimentally an the expeimental esult fom the fequency esponse shows that the moel is compose of poles an zeos in the fequency ange of to Hz. The poles ae locate in the complex s-plane at p, p p, p p5, p -.8 ± 95.7i -9. ± 9.i - 5.9 ± 5.i The zeos ae locate in the complex s-plane at z, z z, z The constant gain is -59. ± 7.i - 9.7 ± 597.i () () K.879x 7 () The poles an the zeos specify an efine the popeties of the tansfe function, thus escibing the input-output system ynamics. The poles, the zeos, an the gain K all togethe completely povie a full esciption of the system an chaacteize the system ynamics an the esponse. The tansfe function can now be constucte by using the poles, the zeos, as well as the gain, an the esulting tansfe function G ( is foun to be of the fom. G (.88( ) s s 7 +.( ) s +.97( ) s 5 9 7 + 5.( ) s 7 + 5.95( +.58(.97( ) s + 5.( ) 7.85( ) s +.9( ) s +.95( ) s ) + ) s () The DC gain of the system in B is equal to log (5.x /.95x ) 9. B. The inspection of Eq.() inicates that the system esponse is vey fast with the settling time in millisecons. To avoi numeical poblems with simulations in MATLAB, the time unit nees to be change fom secon to millisecon. To o this, each vaiable, s, in the tansfe function in Eq.( ) is eplace by s,an the new tansfe function G ms ( in tems of millisecon is obtaine as.88s +.97s + 59.5s + G ms ( 5 s +.s + 5.s + 5.8s 9.7s + 5 85.s + 9.s + 95 (5) The s vaiable in the new tansfe function in Eq.(5) has the unit in aian/millisecon. In MATLAB, the time axis must theefoe be escale to millisecon. The Boe iagam that epesents the fequency esponse of the piezoactuato is plotte by using G ms ( an illustate in Fig.. Figue : Boe iagam of the piezoactuato. In this wok, the sixth-oe moel of the actuato is ecompose into thee moes of seconoe systems by using the paallel state space ealization metho, shown in Fig.7 so that obustness can be investigate by poviing each moe with vaiations in the natual fequency an the amping atio. Figue 7: Diagam fo paallel state space ealization, an appoach to ecoupling the moes of oscillations. 5
ICINCO - th Intenational Confeence on Infomatics in Contol, Automation an Robotics To fin the state space epesentation by the paallel state space ealization metho, the tansfe function can be ewitten in the fom of patial factions. y( + u( s p s p 5 + + + s p s p 5 + s p k s + s p + () whee,, ae the esiues, p, p, p ae the poles of the system, an ks is the iect tem. The iect tem is equal to zeo fo a stictly pope tansfe function. The poles locate at s p, p,, p ae shown in Eq.(), an it follows that G y( Gms, ( + Gms,( Gms, ( (7) u( ms ( + Whee G, ( ms, G, ( ms, an G, ( ms ae obtaine as follows..5s +. G ms,( fo moe s +.98s +.9.97s +.898 G ms,( fo moe s +.88s + 7..s.5 G ms,( fo moe s +.5s + 8.77 Now, the system is ecouple to thee moes, an each moe is epesente by a secon-oe tansfe function. The system in Eq.(7) epesents the oiginal system escibe by Eq.(5) an peseves the oiginal system esponse chaacteistics. A secon-oe system possesses a pai of complex conjugate poles an the pole location etemines the natual fequency an the amping atio. Fo a secon-oe system, the location of the poles s, s is elate to the natual fequency ω n an the amping atio by s, s ωn ± iωn (8) Fom the pole locations an Eq.(8) above, the natual fequency ω n an the amping atio fo each moe of vibation can be foun an shown in Table. It is note that the system is stable since all the poles have a negative eal pat, an the moe numbe is etemine by ealizing that the highe moe numbe will have a geate natual fequency. Table : Pole location, natual fequency an amping atio fo each moe. Pole Location ω n (Hz) Moe p -.8 95.7i 5.5.75 + p -.8 95.7i 5.5.75 p -9. + 9.i.9. p -9. 9.i.9. p 5-5.9 + 5.i 85.9. p -5.9 5.i 85.9. CONTROL SCHEME The notions an the evelopments of invesionbase contol have attacte eseaches in the fiel an have been aoun fo moe than fou ecaes. The ealy an emakable woks on invesion-base appoach wee pesente by Silveman (99) an Hischon (979). Late on, many evelopments an contibutions wee mae by means of invesionbase contol, o feefowa contol methos such as the woks by Peng et al., (99), Meckl et al., (99), Piazzi et al., (), Devasia (), Dunne et al., (), Yang et al., () an Boekfah et al., (). The stana invesion contol theoy is base on a known o pe-escibe tajectoy. In this wok, the tajectoies ae pescibe o known a pioi an the system is a minimum phase type an is stable. The invesion-base contol appoach is hence suite an popose fo tacking the esie tajectoies.. State Space Repesentation It is well known that fo a linea time-invaiant system (LTI system), the plant ynamics can be epesente by the state equation of the fom. x ( Ax( + Bu( (9) an the output equation of the fom. y ( Cx( + Du( () Fo a stictly pope system such as the case hee, D is equal to zeo. Now G, ( ms, G, ( ms, an G, ( ms in Eq.(7) can be cast into the state space fom of Eq. (9) an Eq. (), an matices A, B, C, an D ae as follows. 5
Tacking of High-spee, Non-smooth an Micoscale-amplitue Wave Tajectoies.9.98 A 7..88 8.77.5 B C D [..5.898.97.5.]. Invesion-base Contol Appoach The elative egee,, of the system is efine by the iffeence between the numbe of poles an the numbe of zeos. Fo the moel govene by Eq.(5), the elative egee is the numbe of poles minus the numbe of zeos, o -. The full oe invese can lea to a computational ift ue to numeical eos in the simulation softwae. To avoi the computational numeical poblem, the euce oe invese appoach is to be use to fin the invese input, as in the wok by Boekfah et al., (). To etemine the invese input in the invesionbase metho, it is necessay to take the th time eivative so that the input appeas, i.e., y CA x( + CA t A x( B u( ( y + y Bu( () The invese input u inv equie to tack a sufficiently smooth tajectoy y is etemine fom Eq.(),i.e., y( u inv ( By A x( y () t In the euce oe invese, some components of the state ae known when the esie output, y (.), an its time eivatives ae efine. In paticula, the following cooinate tansfomation can be mae. y ( C () y ( CA ( T z( ( ) x( x( y ( CA ( ) t ( T η η ( Tη ( η Tx( () whee (.) is the known potion of the state, an η is the unknown potion of the state, an the bottom potion T η of the cooinate tansfomation matix T is chosen such that the matix T is invetible, leaing to the invese tansfomation, i.e., [ T ] T + η x ( T l T l T () η η By taking the time eivative of η in Eq.() an using the state equation in Eq.(9),the invese input in Eq.() can be ewitten as the output of η in the following invese system. η( t ) T x ( T ( Ax( + Bu( ) (5) η η Now the state x( in Eq.() can be use in Eq.(5) to obtain the invese system. η ( A η( B Y ( () u inv inv + inv ( C η ( D Y ( (7) inv inv + whee η is teme as the intenal state, an Ainv Tη [ A ( BBy Ay )] T B C inv inv [ Tη [ A ( BBy Ay )] Tl Tη BBy inv By AyT inv y l By D [ B A T Y ( ( y ( ( ) Fo this paticula wok of the elative egee, thee ae theefoe fou moe states to be chosen, an x x η( x x whee can be chosen. 5 ] ] 5
ICINCO - th Intenational Confeence on Infomatics in Contol, Automation an Robotics Ay CA CAA By CA B CAB T T Tη y C ( T y CA z( x( x( ( T ( η T η η η an the matix T is.. T.5..898.8.97.885.5.79..5 The tacking can now be simulate in MATLAB computing softwae. mean squae eo (RMSE) can also be use as an inex of the tacking pefomance, an the oot mean squae eo is efine as RMSE N ( y N i a ( y ( ) whee N is the numbe of the ata points. () Figue 8: Tacking esults fo the tiangle wave tajectoy. 5 RESULTS AND DISCUSSIONS With the initial conitions of being zeos fo all the states at time t, the tacking esults ae illustate in Fig.8 an Fig.9. 5. Quantifying Thetacking Eos To measue the pefomance of the tacking, the eo E( of tacking, o tacking eo can be efine as E( y ( y ( (8) a whee y a ( is the actual tajectoy output an y ( is the esie tajectoy output. Eq.(8) efines an eo fo each point of time ( an the eo fo each point of time is plotte along with the tajectoy outputs in Fig. 8 an Fig.9. Anothe quantification of tacking pefomance that can be use to evaluate the tacking is the maximum tacking eo E max an the maximum tacking eo is given by E max E( (9) max To evaluate the oveall tacking pefomance fo the entie tacking time of millisecons, the oot Figue 9: Tacking esults fo the sawtooth wave tajectoy. 5. Tacking Pefomance Fom Fig.8 an Fig.9, it can be seen that the tacking eo tens to each a maximum value at the tunaouns of the waves. Table shows vey small values of the E max an the RMSE values of tacking the tiangle wave an the saw tooth wave an inicates vey goo tacking pefomances. In paticula, E max values ae vey small, compae to the wave amplitue of μm i.e.,.8 μm fo the tiangle wave an.87 μm fo the sawtooth wave. This epots that the tacking 5
Tacking of High-spee, Non-smooth an Micoscale-amplitue Wave Tajectoies pefomance is vey goo. Table also inicates that tacking the saw tooth wave (with shape tunaouns at the cone, compae to the tiangle wave) yiels lage values of the Emax an the RMSE values, compae to tacking the tiangle wave. In othe wos, tacking of the sawtooth wave is moe ifficult than that of the tiangle wave. Table : Maximum Eo ( E max ) an Root Mean Squae Eo (RMSE). Wave Type E max ( m) Tiangle.8 Sawtooth.87 μ RMSE ( μ m).5. 5 ROBUSTNESS OF TRACKING To elve into the obustness against unmoele ynamics, moeling eo, o istubance, fo this paticula stuy, it is assume that the natual fequency ω n an the amping atio ae incease by some pecentage. Thee ae fou cases that ae investigate fo each tajectoy fo the stuy of obustness. Case []: ω n an ae not change an thei numeical values ae shown in Table.The stuy of this case was complete in Section 5. The plots wee shown in Fig.8 fo the tiangle case an in Fig.9 fo the sawtooth case. The tacking eos wee quantifie an shown in Table. This is a efeence case fo the othe thee cases. Case []: ωn an ae incease by. %. The plots ae shown in Fig. fo the tiangle case an in Fig. fo the sawtooth case. Case []: ωn an ae incease by 5. %. The plots ae shown in Fig. fo the tiangle case an in Fig. fo the sawtooth case. Case []: ωn an ae incease by. %. The plots ae shown in Fig. fo the tiangle case an in Fig.5 fo the sawtooth case. Fo all cases, the tacking eos ae quantifie an shown in Table fo the case of the tiangle tajectoy an Table fo the case of the sawtooth tajectoy. Fom Fig. to Fig.5, Table an Table, it is obseve that. E max is still less than % of the peak to peak amplitue of micon if the inceases in the natual fequency an the amping atio ae less than 5% fo the tiangle tajectoy, an E max is still less than % of the peak to peak amplitue of micon if the inceases in the natual fequency an the amping atio ae less than. % fo the sawtooth tajectoy.. E max an RMSE incease as the pecentage of change in the natual fequency an the amping atio is incease.. The tacking is quite sensitive to the change in the natual fequency an the amping atio. Paticulaly, if the inceases by %, the tacking gets wose.. The maximum eo tens to occu at the shap cone of the tajectoies. 5. The actual tajectoy of the sawtooth oscillates obviously if thee ae changes in the natual fequency an the amping atio. Figue : Tacking esults fo the tiangle wave tajectoy in case []. Figue : Tacking esults fo the tiangle wave tajectoy in case []. 55
ICINCO - th Intenational Confeence on Infomatics in Contol, Automation an Robotics Figue : Tacking esults fo the tiangle wave tajectoy in case []. Figue 5: Tacking esults fo the sawtooth wave tajectoy in case []. Table : Maximum Eo ( E max ) an Root Mean Squae Eo (RMSE) fo tiangle tajectoy. Case E ( μ m) RMSE ( μ m) max [].8 5.5 []. 7. []. 99. []. 7. Figue : Tacking esults fo the sawtooth wave tajectoy in case []. Table : Maximum Eo ( E max ) an Root Mean Squae Eo (RMSE) fo sawtooth tajectoy. Case E ( μ m) RMSE ( μ m) max [].87. []. 97 8. []. 88. []. 8. 7 CONCLUSIONS Figue : Tacking esults fo the sawtooth wave tajectoy in case []. This aticle pesents an invesion-base contol appoach to tacking wave tajectoies. The inteesting challenge is that the tacking involves the tajectoies that possess a high fequency, a micoscale amplitue, shap tunaouns at the cones. The moel o tansfe function of a piezoactuato is obtaine expeimentally fom the fequency esponse by using a ynamic signal analyze. Une the invesion-base contol scheme an the moel obtaine,the tacking is simulate in MATLAB. The main contibutions of this wok ae to show that () the moel an the contolle achieve a goo tacking pefomance measue by the oot 5
Tacking of High-spee, Non-smooth an Micoscale-amplitue Wave Tajectoies mean squae eo (RMSE) an the maximum eo (E max ), () the maximum eo tens to occu at the shap cone of the tajectoies, () tacking the sawtooth wave yiels lage RMSE an E max values, compae to tacking the tiangle wave, an () in tems of obustness against moeling eo o unmoele ynamics, E max is still less than % of the peak to peak amplitue of micon if the inceases in the natual fequency an the amping atio ae less than 5% fo the tiangle tajectoy an E max is still less than % of the peak to peak amplitue of micon if the inceases in the natual fequency an the amping atio ae less than. % fo the sawtooth tajectoy. Thee is still oom fo eveloping the tacking an impoving the tacking pefomance, in paticula fo the obustness against unmoele ynamics o istubances by means of aing a feeback contol to the invesion-base contol. ACKNOWLEDGEMENTS The autho of the aticle woul like to sinceely thank Assumption Univesity of Thailan fo suppoting the eseach. REFERENCES Beschi, M., Domio, S., Sanchez, J., Visioli, A., an Yeba, L. J.,() Event-Base PI plus Feefowa Contol Stategies fo a Distibute Sola Collecto fiel, IEEE Tansactions on Contol Systems Technology, Vol., No., July,pp. 5. Boekfah, A., an Devasia, S.,() Output-Bounay Regulation Using Event-Base Feefowa fo Nonminimum-Phase Systems, IEEE Tansactions on Contol Systems Technology, Vol.,No., Januay, pp5-75. Devasia, S., (). Shoul Moel-Base Invese Inputs Be Use as Feefowa Une Plant Uncetainty?, IEEE Tansactions on Automatic Contol, Vol.7, No., Novembe,pp85-87. Dunne, F., Pao, L. Y., Wight, A. D., Jonkman, B., an Kelley, N.,() Aing Feefowa Blae Pitch Contol to Stana Feeback Contolles fo Loa Mitigation in Win Tubines, Mechatonics, Vol., No., June, pp. 8 9. Hischon, R. M., (979). Invetibility of Multivaiable Nonlinea Contol Systems, IEEE Tansactions on Automatic Contol, Vol.AC-, No., Decembe 979,pp855-85. Kongthon, J., Chung, J.-H., Riley, J., an Devasia, S.,() Dynamics of Cilia-Base Micofluiic Devices, ASME Jounal of Dynamic Systems, Measuement an Contol,Vol., Septembe,pp.5--5-. Kongthon, J., an Devasia, S., () Iteative Contol of Piezoactuato fo Evaluating Biomimetic, Cilia Base Micomixing, IEEE/ASME Tansactions on Mechatonics, Vol.8, No., June, pp 9-95. Kongthon, J., McKay, B.,Iamatanakul, D., Oh, K., Chung, J.-H., Riley, J., an Devasia, S.,() Ae-Mass Effect in Moeling of Cilia-Base Devices fo Micofluiic Systems, ASME Jounal of Vibation an Acoustics, Vol., No., Apil, pp.5 5 7. Matin, P., Devasia, S., an Paen, B., (99). A Diffeent Look at Output Tacking: Contol of a VTOL Aicaft, Automatica, Vol., No.,pp. - 7. Meckl, P. H.,an Kincele, R.,(99) Robust Motion Contol of Flexible Systems Using Feefowa Focing Functions, IEEE Tansactions on Contol Systems Technology, Vol., No., Septembe 99,pp. 5 5. Moallem, M., Kemani, M. R., Patel, R. V., an Ostojic, M.,() Flexue Contol of a Positioning System Using Piezoelectic Tansuces, IEEE Tansactions on Contol Systems Technology., Vol., No. 5,Septembe,pp. 757 7. Peng, H., an Tomizuka, M., (99) Peview Contol fo Vehicle Lateal Guiance in Highway Automation, ASME Jounal of Dynamic Systems, Measuement an Contol, Vol. 5, No., Decembe 99, pp.79 8. Piazzi, A., an Visioli, A.,() Optimal Invesion- Base Contol fo the Setpoint Regulation of Nonminimum-Phase Uncetain Scala Systems, IEEE Tansactions on Automatic Contol,Vol., Octobe, No., pp. 5 59. Silveman, L. M.,(99) Invesion of Multivaiable Linea Systems, IEEE Tansactions on Automatic Contol, Vol.AC-,No.,June 99,pp7-7. Yang, X., Gaatt, M., an Pota, H., () Flight Valiation of a Feefowa Gust-Attenuation Contolle fo an Autonomous Helicopte, Robotics an Autonomous Systems, Vol. 59, No., Decembe, pp. 7 79. 57