Application of parsimonious learning feedforward control to mechatronic systems

Transcription

1 Application of parimoniou learning feedforward control to mechatronic tem T.J.A.de Vrie, W.J.R.Velthui and L.J.Idema Abtract: For motion control, learning feedforward controller (LFF) hould be applied when accurate proce modelling i difficult. When controlling uch procee with LFF in the form of multidimenional B-pline network, large network ize and a poor generaliing abilit ma reult, known a the cure of dimenionalit. Therefore, a parimoniou (reduced dimenionalit) LFF i required. Empirical modelling method are not uited to obtain parimoniou network for highl nonlinear procee becaue large data et are needed. Alternativel, (qualitative) proce knowledge can be ued to contruct parimoniou LFF controller. In the reearch reported, a parimoniou LFF ha been applied to a linear motor motion tem. Experiment howed fat learning, good network parimon, and mall tracking error for a range of motion. 1 Introduction Modelling and=or identification difficultie arie with procee that uffer from uncertaint in proce tructure or in man parameter, trong nonlinearitie, or timevariance in proce tructure or in man parameter. With thee procee, conventional controller and even nonlinear or adaptive controller can how poor performance becaue their deign relie heavil on the availabilit of an accurate proce model. Even when a reaonabl accurate model i available a priori, feedback controller uffer from a trade-off between high performance and robut tabilit. In thi reearch, we focu on motion application. When accurate proce knowledge i available, application of a control tem that conit of a feedforward and a feedback controller hould be conidered. Thi tpe of control tem doe not uffer from the mentioned trade-off. The feedforward controller obtain a high tracking performance while the feedback controller compenate diturbance. When no accurate proce model i available a priori, the conventional feedforward controller can be replaced b a learning feedforward controller (LFF) (Fig. 1) [1, 2]. The LFF ha a function approximator whoe input output mapping can be adapted during control and i intended to become equal to 1. In thi reearch, the function approximator i implemented a a B-pline network (B) becaue it combine fat and accurate learning with mall computational cot [3]. The learning mechanim i baed on the output of the feedback controller. Thi ignal attempt to decreae the tracking error and i therefore conidered to be a good indication of how the feedforward ignal hould be adapted uch that a maller tracking error reult. everal example of ucceful application of LFF have been realied, e.g.: path tracking for an autonomou mobile robot [1]; accurate poitioning with a linear motor motion tem [4]; control of a flight imulator tick [5]. When deigning an LFF, the election of network input and the network reolution (i.e. the number of bai function) are crucial choice. The traightforward approach, uing a multidimenional B, confront u with the cure of dimenionalit, reulting in large network ize and a poor generaliing abilit. Thi i due to the fact that a multidimenional B doe not adhere to the principle of parimon [3, 6], which a that the bet model are obtained uing the implet poible, acceptable tructure that contain the mallet number of parameter. In general, parimoniou model can be obtained automaticall b emploing empirical modelling technique [7], or b hand uing qualitative proce knowledge. In thi paper, a parimoniou LFF for a linear motor i deigned and teted uing both approache. # IEE, 21 IEE roceeding online no DOI: 1.149/ip-cta:21556 aper firt received 23rd October 2 and in revied form 24th Ma 21 T.J.A. de Vrie and W.J.R. Velthui are with the Drebbel Intitute, EL-RT, Univerit of Twente,.O. Box 217, L-75 AE Enchede, The etherland [email protected] L.J. Idema i with Océ, Venlo, The etherland u F r e B u Fig. 1 LFF configuration 318 IEE roc.-ontrol Theor Appl., Vol. 148, o. 4, Jul 21

2 2 Linear motor motion tem 2.1 tem decription Linear motor [8, 9] are deigned to perform linear motion with ubmillimetre accurac. The linear motor ued in thi reearch i the o-called linear motor motion tem (LIMM), deigned b hilip. The motor configuration conit of a baeplate covered with permanent magnet and a liding part, the tranlator, which hold the electric coil and their iron core (Fig. 2). The deign team ha choen to ue relative low-cot magnet that have coniderable magnetic tolerance. Furthermore the magnet are not placed with the highet poible accurac o the magnetic propertie of the motor are not full predictable. onequentl, the tranlator experience a ine-haped diturbance force during operation, known a the torque ripple, which i caued b two phenomena cogging: a magnetic interaction between the permanent magnet and the iron core trie to align magnet and pole to table poition of the tranlator (Fig. 3). ogging depend on the relative poition of the tranlator with repect to the magnet; it i independent of the motor current and i oberved in a zero-current ituation a well. error in commutation: moving a coil through a varing electromagnetic field, generate a back EMF, depending on the velocit of the coil in a linear wa. When the current in the coil i not proportional to the back EMF, a torque ripple will reult which depend on the poition and the velocit of the tranlator. Other diturbance are friction phenomena in the ball bearing that are preent between the tranlator and the guiding rail. We expect that the friction characteritic can be decribed b the tribeck curve, i.e. the friction conit of oulomb friction, tiction and vicou friction. 2.2 reviou reearch In previou reearch, the LIMM wa controlled b a welltuned ID controller. Experiment howed tracking error ranging from 5 to 75 mm, depending on the velocit and acceleration of the motion. Becaue the dominant diturbance have a reproducible nature, LFF control wa applied to enhance the tracking performance [4]. A robutl tuned D feedback controller ha been realized on the bai of a crude fourth-order dnamic model. It wa aumed that the diturbance could depend on the tranlator poition, velocit and acceleration or an combination of thee. Therefore a multidimenional B ha been applied having the reference poition, velocit and acceleration a input. Due to the highl nonlinear behaviour of the LIMM the following network reolution ha been choen: 42 firt-order B-pline on the poition domain, 3 on the cogging force cogging force velocit domain and five on the acceleration domain (ection 3.2). The network ha 63, network weight. The performance obtained for training trajectorie wa good ( 1mm). However, the large amount of weight caued a poor generaliing abilit. onequentl, extenive training i needed to obtain an acceptable performance for a wide range of motion. Thi can be overcome b a parimoniou LFF. 3 arimoniou LFF net force = a net force b Fig. 3 Example of a table and an untable poition of the tranlator. Detent tranlator poition are found ever 16 mm a table b Untable 3.1 Obtaining parimon A parimoniou LFF can be obtained in an automated wa uing empirical modelling technique, like EIT [1] oramod[11], or b hand uing qualitative proce knowledge. In motion application, the proce can often be modelled b a linear tate pace model and additive nonlinear function, hown in Fig. 4. The tate vector can be choen uch that it conit of poition and their correponding velocitie, i.e. x ¼ x 1 I where _x 1 ¼ x 2 uch that A ¼ ; x 2 A 21 A 22 B ¼ and hðþ ¼ ð1þ B 2 hðxþ nonlinear (time-variant) realit φ A φ B permanent magnet tranlator φ bae plate 1 mm Fig. 2 Working principle of LIMM howing flux-line of permanent magnet f A, f B, and f are phae of three-phae motor current φ A v u B x A h( ) Fig. 4 Decompoition of a nonlinear proce into linear proce decribed b {A, B, } and nonlinearitie h() x IEE roc.-ontrol Theor Appl., Vol. 148, o. 4, Jul

3 The nonlinearitie h(x) can motl be written a a function of the poition and their firt and econd derivative hðxþ ¼hðx 1 ;_x 1 ; x 1 Þ¼h 1;1 ðx 1 Þþh 1;2 ð_x 1 Þþh 1;3 ð x 1 Þþ þ h 2;1 ðx 1 ;_x 1 Þþh 2;2 ðx 1 ; x 1 Þþh 2;3 ð_x 1 ; x 1 Þþ þ h 3;1 ðx 1 ;_x 1 ; x 1 Þ ð2þ The following tem dnamic reult: x 1 A 22 _x 1 A 21 x 1 hðx 1 ; _x 1 ; x 1 Þ¼B 2 u ð3þ The LFF i to learn the teering ignal u d F which make the actual poition of the proce x 1 track the deired one 1 u d F ¼ B 1 2 ð 1 A 22 _ 1 A 21 1 h 1;1 ð 1Þ... h 2;1 ð 1; _ 1 1 Þ... h 3;1 ð 1; _ 1; 1ÞÞ Tpicall, h 3;1 will be zero, a will be one or more term of h 2; j. In that cae a tructure of additive network will compl to the principle of parimon and one multidimenional network will not. Often, the tructure and the hape of the proce nonlinearitie are known. The deigner can make an educated gue about the required parimoniou network tructure and the necear bai function. We illutrate thi for the LIMM. The linear part of the dnamic conit of a ma m (the tranlator) and a damper d (the vicou friction in the ball-bearing), reulting in B 2 ¼ m 1 ; A 22 ¼ m 1 d; A 21 ¼ ð5þ The main nonlinear phenomenon i the cogging force, which i expected to depend on the poition x 1 of the tranlator. However, low-velocit experiment howed a torque ripple of which the pitch and the amplitude alo depend on the motion direction. In term of eqn. 1 the cogging force i given b F cogg ¼ h 1ðx 1 Þ for _x 1 < ð6þ h 2 ðx 1 Þ for _x 1 Further nonlinearitie are tiction and oulomb friction, depending on the velocit of the tranlator. tiction force ma differ when the tranlator tart to move or top, giving F nonl:fric ¼ h 3ð_x 1 Þ for x 1 < ð7þ h 4 ð_x 1 Þ for x 1 The final nonlinear term concern error in the commutation, depending on the magnetic field through the coil, i.e. the poition of the tranlator, and in a linear wa on the velocit of the tranlator F comm ¼ h 5 ðx 1 Þ_x 1 ð8þ ubtituting thee equation in eqn. 4 give u d F ¼ m 1 þ d_ 1 F cogg F nonl:fric F comm ¼ m 1 þ d_ 1 h 1 ð 1Þ h 2 ð 1Þ h 3 ð_ 1Þ h 4 ð_ 1Þ h 5 ð 1Þ_ 1 ð9þ An additive tructure coniting of a univariate network with input ẍ d 1 and a bivariate network with input xd 1, ẋd 1 offer ufficient modelling flexibilit (Fig. 5a). For proper training (explained later), we alo conider an alternative configuration with the ame modelling flexibilit, et more network. We create a parimoniou network tructure b ð4þ Fig e e x&v-net a-net a x&v-net x-net v-net a-net b arimoniou network tructure for the LIMM a Minimal network b onminimal network u incorporating a B for each collection of term in eqn. 9 that depend on the ame tate. ogging force h 1 ( 1 )andh 2( 1 ) are tored in two B that have 1 a input; one for each motion direction. The friction term dẋ d 1, h 3(ẋ d 1 )andh 4(ẋ d 1 ) require two B that have ẋ d 1 a the input; one for negative ẍd 1 and another for poitive one. For mẍ d 1 a B i created that ha ẍd 1 a the input. Finall h 5 ( 1 )ẋd 1 i compenated b a two-dimenional B that ha 1, ẋd 1 a input. The reulting network tructure i hown in Fig. 5b. An important conequence of the latter approach i that different kind of diturbance have been iolated in different network. 3.2 B-pline ditribution The B-pline ditribution determine the hape of the ignal the B i able to learn and the convergence propertie. In thi reearch, econd-order B-pline are ued. For a one-dimenional B uing econd-order B- pline, the output conit of firt-order polnomial between the B-pline knot [6]. Therefore the maller the ditance between the B-pline knot, the more accurate a ignal can be approximated. However, a too mall ditance ma caue divergence of the learning proce [2, 12]. We have choen the following B-pline ditribution: The magnet are placed at an interval of approximatel 16 mm. We expect h 1 ( 1 ) and h 2( 1 ) to be ine-haped with a pitch of approximatel 16 mm. To accuratel compenate cogging, each period hould be approximated b at leat ten piecewie firt-order polnomial. For a poition range of.7 m, thi give 1 (7=16) 5 B-pline. We aumed that the friction phenomena can be decribed b the tribeck curve, which i highl nonlinear at low velocitie and relativel mooth at large velocitie. 32 IEE roc.-ontrol Theor Appl., Vol. 148, o. 4, Jul 21

4 Therefore the ditance between the B-pline knot i choen mall at low velocitie and large at high velocitie. In total 3 B-pline have been defined on the velocit domain [ 1m=, 1 m=]. The acceleration network ha to learn a linear function, mẍ d 1. Therefore three B-pline are defined of which the knot are placed at { 5m= 2,m= 2,5m= 2 }. The weight of the middle B-pline i fixed to, to enure that at ẍ d 1 ¼ the output of the B i zero. For the B that compenate error in the commutation (Fig. 5b), two B-pline ditribution mut be choen; one on the poition and one on the velocit domain. ince little i known about the wa in which the torque ripple depend on the poition, we have choen a dene B-pline ditribution: 5 B-pline on [.61 m,.11 m]. Thi enure that an tpe of ignal can be learned accuratel. The error in the commutation linearl depend on the velocit. Therefore onl two B-pline are defined on the velocit domain [ 1m=, 1 m=]. 4 Evaluation Firt, AMOD wa ued to deign an LFF that ha the reference poition, velocit and acceleration a input. Experiment howed that creating an empirical model of a three-dimenional data et that i highl nonlinear would take everal da of calculation. Therefore empirical modelling technique were rejected for practical application. ubequentl, the parimoniou network tructure of Fig. 5a wa created and the ubnetwork were trained while performing repetitive motion of the following tpe: 1. low canning motion: ẋ d 1 2 [.1 m=,.1 m=] and ẍ d 1 2 [.1 m=2,.1m= 2 ]. 2. Medium velocit motion: ẋ d 1 2 [.4 m=,.4 m=] and ẍ d 1 2 [ 2m=2,2m= 2 ]. 3. High velocit motion: ẋ d 1 2 [ 1m=, 1 m=] and ẍ d 1 2 [ 4m=2,4m= 2 ]. Table 1 give the maximum tracking error after training. A before [4], the LFF wa able to coniderabl improve the tracking performance of the ID controller. To validate whether the ubnetwork actuall compenate a intended, the weight were fixed and the LFF wa ued to track untrained motion. Thee experiment howed poor tracking performance. Thi inabilit to generalie i probabl caued b a lack of excitation. Training the LFF ma be een a a form of tem identification. Identifing the parameter of an nth-order model tpicall require that the input ignal i peritentl exciting of about order 2n [13]. For the LIMM, peritenc of excitation mean that at e.g. each poition, a whole range of velocitie and acceleration i realied, o that (dẋ d 1 7 h 1( 1 ) 7 h 2( 1 ) 7 h 3(ẋ d 1 ) 7 h 4(ẋ d 1 ) 7 h 5( 1 )ẋd 1 ) i learned b the bivariate network and incidental phenomena Table 1: Tracking accurac of LIMM for repetitive motion; LFF a in Fig. 5a cancel out. However, it i difficult to realie peritentl exiting motion, e.g. high velocit motion near both end of the LIMM. Furthermore, it reult in a long training phae. The problem can be circumvented b uing the parimoniou tructure of Fig. 5b and b equentiall training the network uing trategic training experiment. During thee experiment, reference path are generated that caue pecific not-et-learned effect that can be compenated b one particular network to be dominantl preent. With the LIMM, the following training procedure ha been carried out: 1. When performing motion with ver low velocit and acceleration over the full poition range, all phenomena except cogging and oulomb friction are about zero, and hence thee motion were ued to train the poitionindexed network onl. ince the LFF contain one poition network for each direction of motion, the network will alo learn to compenate the oulomb friction. After training, the poition network weight are fixed. 2. The velocit and acceleration network are trained imultaneoul during motion with medium velocit and acceleration. The acceleration network wa trained ver quickl. The velocit network were trained with motion that cover the full velocit range and then fixed. Becaue velocitie were realied at variou poition, the effect of error in the commutation were uppoed to cancel out. 3. Finall, the bivariate network wa trained b performing motion that realied a range of velocitie for each poition. The generaliing abilit of the obtained LFF i examined b performing the previoul mentioned untrained motion. Table 2 give the maximum tracking error; the indicate the accurac achievable for an motion. A far a we know, LFF outperform an other approach for control of nchronou permanent magnet linear motor in nonrepetitive application [14]. ince the LFF obtain a mall tracking error for a wide range of motion, it ma be concluded that it i trained correctl and ha a good generaliing abilit. Examining the input output relation of the ubnetwork can further validate thi. The input output mapping of the poition network (Fig. 6 and 7) indeed dipla the cogging force a a multiple inuoidal function. Furthermore, the oulomb friction force appear a an offet between Fig. 6 and Fig. 7. In the velocit network (Fig. 8 and 9) the vicou friction can be recognied a the lope of the curve. The peak are the reult of tick-lip friction at ver low velocitie. ince the frictional behaviour around zero velocit i dependent on the ign of the acceleration, a ingle network indeed cannot compenate thee force. The acceleration network learn the intended imple linear relation, Fig. 1. Table 2: Tracking accurac of LIMM for untrained motion; LFF a in Fig. 5b Repetitive motion Tpe Velocit [m=] Acceleration [m= 2 ] Max. tracking error D LFF Untrained motion Tpe Velocit [m=] Acceleration [m= 2 ] Max. tracking error D LFF canning Medium Fat canning Medium Fat IEE roc.-ontrol Theor Appl., Vol. 148, o. 4, Jul

5 x-po mapping x-neg mapping x, m Fig. 6 B content after training: x-net for poitive velocitie x, m Fig. 7 B content after training: x-net for negative velocitie.4 5 oncluion Application of multidimenional B in LFF ma be difficult due to the cure of dimenionalit. plitting up a multidimenional B into everal lower-dimenional B offer a olution. Thi o-called parimoniou network tructure can be deigned on the bai of qualitative knowledge of the proce and the diturbance. For the LIMM, the proce knowledge conited of a imple qualitative model of the motor tranlator and the diturbance, i.e. cogging, oulomb friction, tiction and commutation error. When uing a parimoniou LFF, pecial attention hould be given to training the ubnetwork. When all ubnetwork are trained imultaneoul, the LFF ma have a poor generaliing abilit. In that cae, the LFF i unable to obtain a high tracking performance for motion other than the training motion. To prevent thi, a tructure with more network than required for modelling flexibilit onl hould be created and trategic training experiment hould be performed in which deignated ubnetwork are trained equentiall. The training phae can be evaluated b comparing the input output mapping of the ubnetwork to the expect input-output mapping. For the LIMM, the trategic training procedure reulted in an LFF that obtained a high tracking performance for a wide range of motion. Furthermore, the content of the ubnetwork correponded to the expected one. v-acc mapping.2.2 Fig. 8 v-dec mapping Fig. 9 mapping v, m/ B content after training: v-net for accelerating motion Fig v, m/ B content after training: v-net for decelerating motion a, m/ 2 B content after training: a-net 6 Reference 1 TARREBURG, J.G., VA LUEE, W.T.., OELE, W., and VA AMEROGE, J.: Learning feedforward controller for a mobile robot vehicle, ontrol Eng. ract., 1996, 4, (9), pp VELTHUI, W.J.R.: Learning feedforward control theor, deign and application. hd diertation, Facult of Electrical Engineering, Univerit of Twente, Enchede, etherland, BROW, M., and HARRI,.J.: eurofuzz adaptive modelling and control (rentice Hall, Hemel-Hemptead, UK, 1994) 4 OTTE, G., DE VRIE, T.J.A., VA AMEROGE, J., RAKER, A.M., and GAAL, E.W.: Linear motor motion control uing a learning feedforward controller, IEEE=AME Tran. Mechatron., 1997, 2, (3), pp VELTHUI, W.J.R., DE VRIE, T.J.A., VRIELIK, K.H.J., WIERDA, G.J., and BORGHUI, A.: Learning control of a flight imulator tick, J. A, 1998, 39, (3), pp BOELY, K.M., FREH, M., ROGER, E., HARRI,.J., and BROW, M.: Recent advance in neurofuzz control tem theor and deign. roceeding of ViAM, EFAM, Guimarãe, ortugal, 1996, pp WAG, L.-X.: Adaptive fuzz tem and control, deign and tabilit anali (rentice-hall, Englewood liff, J, UA, 1994) 8 AAR,.A., and BOLDEA, I.: Linear electric motor: theor, deign and practical application (rentice-hall, Englewood liff, J, UA, 1987) 9 BAAK, A.: ermanent-magnet D linear motor (larendon, Oxford, UK, 1996) 1 BERGER,..: Linear pline with adaptive meh ize for modelling nonlinear dnamic tem, IEE roc., ontrol Theor Appl., 1994, 141, (5), pp KAVLI, T.: AMOD, an algorithm for adaptive pline modelling of obervation data, Int. J. ontrol, 1993, 58, (4), pp VELTHUI, W.J.R., DE VRIE, T.J.A., HAAK,., and GAAL, E.: tabilit anali of learning feedforward control, Automatica, 2, 36, (12), pp ÖDERTRÖM, T., and TOIA,.: tem identification (rentice Hall, ew York, UA, 1989) 14 HRIJVER, E., and VA DIJK, J.: Hinf deign of diturbance compenator for cogging force in a linear permanent magnet motor, J. A, 1999, 4, (4), pp IEE roc.-ontrol Theor Appl., Vol. 148, o. 4, Jul 21