NON-MINIMUM PHASE MODEL OF VERTICAL POSITION ELECTRO-HYDRAULIC CYLINDER FOR TRAJECTORY ZPETC

Transcription

1 NON-MINIMUM PHASE MODEL OF VERTICAL POSITION ELECTRO-HYDRAULIC CYLINDER FOR TRAJECTORY ZPETC Norlela Isha, Maidah Tajjudi,Hashimah Ismail,Mihael Patri,Yahaya Md Sam,Ramli Ada Faulty of Eletrial Egieerig, Uiversiti Teologi MARA (UiTM, 445 Shah Alam, Selagor, Malaysia Faulty of Egieerig, UNISEL, Selagor, Malaysia Faulty of Eletrial Egieerig, UTM Malaysia, Johor, Malaysia s: Submitted: Apr. 6, Aepted: May, Published: Jue, Abstrat Hydrauli atuator has bee widely used i idustrial appliatios due to its fast respose ad apability of movig heavy load. The oliear properties of hydrauli ylider had halleged researhers to desig a suitable otroller for positio otrol, motio otrol ad traig otrol. Based o these problems, we had doe a real-time digital traig otrol studies o eletrohydrauli ylider usig trajetory ero phase error traig otrol (ZPETC without fatoriatio of eros polyomial algorithm. With the proposed strategy, the otroller parameters are determied usig omparig oeffiiets methods. The eletro-hydrauli system mathematial model is approximated usig system idetifiatio tehique with o-miimum phase system beig osidered. The real-time experimetal result will be ompared with simulatio result usig a model from a real plat. Keywords Feedforward Cotrol, ZPETC, System idetifiatio, pole plaemet, o-miimum phase 54

2 Norlela Isha, Maidah Tajjudi,Hashimah Ismail,Mihael Patri,Yahaya Md Sam, ad Ramli Ada No-miimum Phase Model of Vertial Positio Eletro-hydrauli Cylider for Trajetory ZPETC I. INTRODUCTION Hydrauli atuator has bee widely used i idustrial equipmets ad proesses due to its liear movemets, fast respose ad aurate positioig of heavy load. This is priipally due to its high-power desity ad system solutio that it a provided [,]. The atural oliear property of hydrauli ylider has halleged researhers i desigig suitable otroller for positioig otrol [], motio otrol ad traig otrol [4]. With itetio to improve the motio or traig performae effetively, may researhers have used advaed otrol strategies to otrol hydrauli ylider [5,6]. Classial feed-forward otroller based o pole-ero aellatio for miimum phase system, maes the overall trasfer futio be uity thus perfet traig otrol (PTC is ahieved [7]. Ufortuately, this otroller aot be implemeted to o-miimum phase system as this would result a ustable traig otrol. The ero-phase error- traig otrol (ZPETC was the proposed by Tomiua [8] ad has attrated attetio may researhers as a effetive ad simple remedy to the problem due to ustable eros. By elimiatig phase error aused by oaelled eros, ZPETC displays good traig performae. The gai error, whih aot be elimiated by ZPETC beomes larger for fast traig otrol ad auses udesirable effet o the traig performae. I resolvig these problems, there has bee may researh wors i this area [9-]. Based o these searios, this paper disusses the implemetatio of real-time digital traig otrol of eletro-hydrauli ylider usig trajetory ero phase error traig otrol without fatoriatio of eros polyomial where the otroller parameters are determied usig omparig oeffiiets method. Simulatio ad real-time experimetal results will be ompared ad disusses o their traig performaes. This paper was orgaied i the followig maer: Setio II desribes ZPETC without fatoriatio of eros; Setio III desribes plat ad model idetifiatio; Setio IV desribes otroller desig; Setio V desribes result ad disussio; ad Setio VI is the olusio. II. ZPETC WITHOUT FACTORIZATION OF ZEROS The traig otrol system with two-degrees-of-freedom that is osistig of feedba ad feedforward otrollers is give i figure. I traig otrol system without feedforward 55

3 otroller, the referee sigal otiuously varyig ad mixed with the losed-loop system dyamis, whih mae futio of feedba otroller is regulatio agaist disturbae iputs. The feedforward otroller is required suh that the referee sigal a be pre-shaped by the feedforward otroller, so that more emphasis to the frequey ompoets that were ot suffiietly hadled by the feedba system a be provided [6]. r Feedforward Cotroller u + _ Feedba Cotroller Plat y Figure. Two-degree-of-freedom otroller Referrig to figure, the losed-loop trasfer futio of the system (without feedforward otrol a be represeted by the followig disrete time model: where d B ( G ( ( l A ( A a a... B ( bo b b... d = time delay a a a b a b The futio B ( - a be fatoried ito miimum phase ad o-miimum phase fators: B ( ( B ( B ( where B ( deotes the miimum phase fator ad B ( deotes the o-miimum phase fator. The ovetioal ZPETC reported i the literature [4] a be divided ito three blos as show i figure. The blo diagram of feedforward ZPETC osists of the gai ompesatio filter, phase ompesatio filter ad stable iverse. Figure shows the struture of the ZPETC feedforward otroller without eros fatoriatio. The blo diagram of feedforward ZPETC without fatoriatio of eros osists of the gai ompesatio filter, phase ompesatio filter ad losed-loop trasfer futio deomiator. 56

4 Norlela Isha, Maidah Tajjudi,Hashimah Ismail,Mihael Patri,Yahaya Md Sam, ad Ramli Ada No-miimum Phase Model of Vertial Positio Eletro-hydrauli Cylider for Trajetory ZPETC r B ( F p d B ( A ( B ( u gai ompesatio filter phase ompesatio filter stable iverse Figure. Covetioal ZPETC struture blo diagram r d F g (, F B ( p A ( u gai ompesatio filter phase ompesatio filter Closed-loop Trasfer futio deomiator Figure. ZPETC without fatoriatio of eros struture blo diagram Similar to all others ZPETC, the desig maily foused o the seletio of appropriate gais ompesatio filter to esure the overall gai is uity withi the frequey spetrum of referee trajetory. To esure the gai ompesatio filter, F g does ot itrodues ay phase error, the same approah doe by [5-6] will be followed. The FIR symmetri filter was used. The filter is represeted by equatio Fg (, ( ( where α is the order of the filter. A suitable ost futio to represet the error betwee the desired ad atual frequey respose is give by Eq. (4. J ( B ( B ( ( l (4 The desig objetive here is to fid a set of α suh that the ost futio give by Eq. (4 is miimied. For fiite α, Eq.(4 aot be made ero for all frequeies. By miimiig the ost futio of Eq. (4, B ( B ( ( (5 The optimal set of α a be obtaied by expadig Eq. (5 to polyomial of positive ad egative power of, ad the ompare the oeffiiets of the same power. 57

5 III. PLANT AND MODEL IDENTIFICATION A. Plat The experimetal equipmet that used i these real-time studies is a eletro-hydrauli system that is show i figure 4. The hydrauli ylider was held i vertial positio. This is a very hallegig problem as effet of gravity is trivial. The eletro-hydrauli system osists of sigle-eded ylider type of atuator. The bidiretioal ylider has 5 mm stroe legth; 4 mm bore sie ad 5 mm rod sie. The wire displaemet sesor is mouted at the top of ylider rod. The pressuried fluid flow is otrol by eletroi otrol valve. The iterfaig betwee the omputer ad plat was doe usig Matlab Real-Time Worshop via Advateh PCI-76 iterfae ard. Positio Trasduser PCI76 Data Aquisitio Board Bidiretioal Cylider Proportioal Valve Figure 4. Experimetal setup for eletro-hydrauli system B. Model Idetifiatio The plat model that used i these studies was obtaied through ope-loop experimet o the Eletro-hydrauli system of figure 4. The ope-loop trasfer futio of the plat was approximated usig Matlab System Idetifiatio Toolbox. The sigal give i figure 5 was used as a iput sigal to the plat for model idetifiatio. The sigal was geerated usig three differet frequeies based o Eq. (6 ad represeted by Eq. (7. u( p a os i i t s i (6 58

6 Norlela Isha, Maidah Tajjudi,Hashimah Ismail,Mihael Patri,Yahaya Md Sam, ad Ramli Ada No-miimum Phase Model of Vertial Positio Eletro-hydrauli Cylider for Trajetory ZPETC V i ( = Cos. t s + Cos 4 t s + Cos 6 t s (7 where a i is the amplitude, i is the frequey (rad/se, t s is the samplig time (se ad is iteger. From Eq. (7, whe usig three differet frequeies for iput sigal, the model that a be obtaied is limited to seod ad third order oly. Higher-orders model may produe ustable output. I these studies, the third-order ARX model with iput-output sigals sampled at 5ms was seleted to represet the earest model of true plat. Figure 5. Iput sigals for model idetifiatio The output sigal of the plat obtaied usig the iput sigal of figure 5 ad sampled at 5 ms, is give i figure 6. The iput ad output sigals of Figure 5 ad Figure 6 were divided ito two parts, i.e. (5- samples ad (-5 samples. The first part of the iput output sigals were used to obtai the plat model ad the seod part of the iput-output sigals were used to validate the obtaied model. 59

7 Figure 6. Output sigal of the plat usig 5 ms samplig time Usig Matlab System Idetifiatio Toolbox, the first part of the iput-output sigal produes a plat model, ARX i the form of disrete-time ope-loop trasfer futio as follows: B ( o A ( o (8 From Eq. (8, its a be simplified as B ( o A ( o.87 ( (9 From Eq. (9, the eros polyomial is give by B (.4.48 ( B (.4.48 Whe Eq. ( is fatoried, the loatios of ero are at =.878 ad = -.4. This meas that the model obtaied is a o-miimum phase model with a ero situated outside the uity irle. A o-miimum phase model a be obtaied usig small samplig time whereas miimum phase model a be obtaied usig larger values of samplig time [7]. The poleero plot of Eq. ( is give i figure 7. 5

8 Displaemet (i Norlela Isha, Maidah Tajjudi,Hashimah Ismail,Mihael Patri,Yahaya Md Sam, ad Ramli Ada No-miimum Phase Model of Vertial Positio Eletro-hydrauli Cylider for Trajetory ZPETC Poles (x ad Zeros (o - 5ms Figure 7. Pole-ero plot of Eq. ( The seod part of iput-output sigals were used to validate the obtaied model of Eq. (8. The seod part of the iput sigal is used as a iput to the model ad the output from the model was ompared with the seod part of the output sigal. The result a be see from figure 8. Usig model seletio riterios, the followig iformatio were obtaied: Best Fit : 89 % Loss Futio :.9 e -5 Aaie s Fial Preditio Error, FPE:.7 e -5 Based o the smallest values riteria of FPE ad Best Fit of 89 %, this model a be aepted Model Plat Time Steps Figure 8. Compariso betwee the model ad plat output sigal 5

9 IV. CONTROLLER DESIGN A. Trajetory ZPETC Sheme This setio presets the proposed trajetory ero phase error traig otrol (ZPETC without fatoriatio of eros polyomial as give i figure 9 where the otroller parameters are determied usig omparig oeffiiets method. Referee trajetory ( d B ( A ( d B ( A ( output Figure 9. Trajetory ZPETC struture The desig objetive here is to fid a optimal set of α by miimiig the ost futio of Eq. (5. The optimal set of α a be obtaied by expadig Eq. (5 to polyomial of positive ad egative power of, ad the ompare the oeffiiets of the same power. From Eq. (9 ad (, the optimal set of α for 5 th order gai ompesatio filter is obtaied as follows: α. α ad.89.48(. 97( (α α ( 4 4 α ( 4 α α ( 5 5 ( 5 α ( ( By expadig Eq. ( to polyomial of positive ad egative power of ad the ompare the oeffiiets of the same power, the followig equatio is obtaied: 5

10 Norlela Isha, Maidah Tajjudi,Hashimah Ismail,Mihael Patri,Yahaya Md Sam, ad Ramli Ada No-miimum Phase Model of Vertial Positio Eletro-hydrauli Cylider for Trajetory ZPETC ( Solvig Eq.(, the optimal set of α is obtaied. I these studies, we osider the order of F g ( -, is α =, ad. The optimal α obtaied by miimiig the ost futio is give i Table I. Whe the filter order is irease to th ad th order, usig the same tehique, the obtaied optimal set of α is give i Table II ad Table III. The results show that the values obtaied are almost overgig to ero, as the filter order ireasig, as show i figure. Table I: Optimal α for th order gai ompesatio filter of Eq. ( α Table II: Optimal α for th order gai ompesatio filter of Eq. ( α α Table III: Optimal α for th order gai ompesatio filter of Eq. ( α α α

11 Alpha.9.8 = = = Figure. Optimal α of order filter B. Simulatio Studies The implemetatio of simulatio studies that based o figure 9 a be simplified to the otrol struture of figure due to the effet of poles aellatio to the trajetory ZPETC struture. From this figure, we a see that the implemetatio of traig otrol by simulatio does ot required the whole plat model trasfer futio. What is eeded oly the ero polyomial equatio of the plat model. Referee trajetory ( B ( B ( output Figure. Traig otrol struture for simulatio studies. C. Real-Time Studies For real-time studies, we proposed the otrol struture give i figure. This otrol struture osists of two parts, whih are feedforward otrol ad feedba otrol. For feedforward otrol, we used the trajetory-zpetc struture. For feedba otrol, we used the pole-plaemet method [8]. This method eable all poles of the losed loop to be plaed at desired loatio ad providig satisfatory ad stable output performae. 54

12 Norlela Isha, Maidah Tajjudi,Hashimah Ismail,Mihael Patri,Yahaya Md Sam, ad Ramli Ada No-miimum Phase Model of Vertial Positio Eletro-hydrauli Cylider for Trajetory ZPETC Figure. Traig struture for real-time studies D. Feedba Cotrol System The feedba otrol system for the proposed trajetory ZPETC system is give i figure. The otroller was desiged usig pole-plaemet method. + u K f F( _ Bo ( A ( o y G( Figure. Feedba otroller usig pole-plaemet method. This method eables all poles of the losed-loop to be plaed at desired loatio ad providig satisfatory ad stable output performae. All otroller parameters were obtaied by solvig the followig Diophatie equatio to solve for F ( - ad G ( -. The losed-loop trasfer futio of the system is give by: Y( U( A o ( K ( f B o F( B o ( G( ( 55

13 where A o( a a a a m B o( b b b b m F ( f f f f m m G ( g g g g g Usig Diophatie equatio to solve for F( ad G (, A o( F( Bo ( G( T( (4 with T ( t ad t is the loatio of a pole i a uit irle. The rage of t is t. For slow respose, t is set large ad for fast respose, t is set small. The forward Sum( T gai K. f Sum( B o I this paper, we used t. 8 sie the traig required slow respose. Attempts have bee made to use small values of t, but the resposes are very fast ad produig large traig errors. Usig developed Visual C++ osole programmig ad Matlab/Simuli, the followig parameters were omputed: T. 8 K f 5 F ( G (

14 Gai Norlela Isha, Maidah Tajjudi,Hashimah Ismail,Mihael Patri,Yahaya Md Sam, ad Ramli Ada No-miimum Phase Model of Vertial Positio Eletro-hydrauli Cylider for Trajetory ZPETC V. RESULTS AND DISCUSSION I this setio, the simulatio ad real-time results were aalyed to show the effetiveess of the desiged otroller. The simulatio ad real-time result of usig otroller parameters are give i Table I, II ad III ad applied to plat model whih resultig RMSE that give i Table IV. Table IV: RMSE performae (mm Simulatio Real-Time α RMSE The traig performaes i terms of root mea squared error (RMSE for all the simulated ad real-time results are summaried i Table IV. The results show the th, th ad th order filter traig performaes. It a be observed that the traig error has bee greatly redued whe the filter order was ireased. As we a observe from the simulated results of Table IV, by itroduig larger filter order to the system, the performae betwee th ad th order filter has improved by 8%. For the real-time result, by itroduig larger order filter, the performae betwee th ad th has improved by 78%. However, the result betwee the simulated ad real-time does ot provide similar performae due to plat-model mismath. Figure 4, shows the frequey respose for the overall system. The frequey respose a be improved if the order of F g ( -, is ireased. Based o figure 4, it a be observed that by usig th order filter, it will ot able to produe a gai that ear to uity. The gai is almost flat at uity whe a higher order filter is used Frequey (rad/s Figure 4. Frequey respose of th, th ad th order ZPETC 57

15 Displaemet (i Displaemet Error (i Displaemet (i Displaemet Error (i Displaemet (i Displaemet Error (i Figure 5(a ad (b show poor traig performae whe th order filter was used. The result is already beig expeted due to the frequey respose give i figure 4. The traig RMSE error is.46 mm. It is lear that usig th order filter will ot produe satisfatory traig performae. Figure 6(a ad (b resultig traig RMSE error of.445 mm. The traig error is muh smaller as ompared to figure 5(a,b. This a be observed from frequey respose give i figure 4, whe th order filter was used. The uity gai lie obtaied is muh better ompared to th order filter.figure 7(a ad (b resultig traig RMSE error of.665 mm, whih is muh better that the oe give i figure 6(a,b. I fat, the overlappig of referee ad output sigals aot be see. The RMSE traig error a be redued further by ireasig the order filter. Higher order filter a approximate the overall trasfer futio of the system very lose to uity for all frequeies effetively. This is explaied by the frequey respose of figure output ref rmse =.46 mm - PCNE Time Steps PCNE Time Steps Figure 5(a. Experimetal result usig th order ZPETC Figure 5(b. Traig Error usig th order ZPETC 5 4 output ref rmse =.445 mm - PCNE Time Steps PCNE Time Steps Figure 6(a. Experimetal result usig th order ZPETC Figure 6(b. Traig Error usig th order ZPETC 5 4 output ref rmse =.665 mm - PCNS Time Steps Figure 7(a. Experimetal result usig th order ZPETC PCNE Time Steps Figure 7(b. Traig Error usig th order ZPETC 58

16 Norlela Isha, Maidah Tajjudi,Hashimah Ismail,Mihael Patri,Yahaya Md Sam, ad Ramli Ada No-miimum Phase Model of Vertial Positio Eletro-hydrauli Cylider for Trajetory ZPETC VI. CONCLUSIONS A otroller desig usig trajetory ZPETC without fatoriatio of eros polyomial has bee preseted. The proposed feedforward otroller desig has bee suessfully tested by simulatio ad validated by real-time digital otrol of eletro-hydrauli ylider. Simulatio ad experimetal results show good traig performaes whe higher order filter was used i the desig. A muh smaller traig error aot be ahieved due to plat-model mismath ad eletroi valve ope-lose apability. ACKNOWLEDGMENT The authors would lie to tha ad aowledge the FRGS-RMI-UiTM (6-RMI/ST/FRGS 5//Fst(85/ for fiaial support of this researh REFERENCES [] Shao, J., Wag, Z., Li, J. ad Ha, G., Model Idetifiatio ad Cotrol of Eletro- Hydrauli Positio Servo System, Pro. It.Cof. O Itelliget Huma-Mahie Systems ad Cyberetis (IHMSC 9, Haghou, Zhejiag, Chia, 9, pp. - [] Che, H.M., Re, J.C. ad Su, J.P., Slidig Mode Cotrol with Varyig Boudary Layers for a Eletro-Hydrauli Positio Servo System, It. J. of Advaed Maufaturig Tehology, vol. 6, 5, pp. 7 [] Kaddissi, C., Kee, J-P. Ad Saad, M., Idetifiatio ad Real- Time Cotrol of a Eletrohydrauli Servo System Based o Noliear Basteppig, IEEE/ASME J. Mehatrois, vol., o., 7, pp. [4] Zhu, W.H. ad Piedboeuf, J.C., Adaptive Output Fore Traig Cotrol of Hydrauli Cylider with Appliatio to Robot Maipulators, ASME J. of Dy. Syst., Meas., ad Cotrol, vol. 7, 5, pp. 6 7 [5] Eryilma, B. ad Wilso, B.H., Improved Traig Cotrol of Hydrauli Systems, ASME J. of Dy. Syst., Meas., ad Cotrol, vol.,, pp [6] Tsao, T.C. ad Tomiua, M., Robust Adaptive ad Repetitive Digital Traig Cotrol ad Appliatio to a Hydrauli Servo for Noirular Mahiig, ASME J. of Dy. Syst., Meas., ad Cotrol, vol. 6, 994, pp. 4 59

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