Non-linear Control and Observer Design for the Self-energizing Electro-Hydraulic Brake

Transcription

1 9. Non-inear Contro and Oberver Deign for the Sef-energizing Eectro-Hydrauic Brake Michae Kühnein, Juian Ewad, Hubertu Murrenhoff RWTH achen Univerity, Germany, Intitute for uid Power Drive and Contro (IS) Matthia Liermann merican Univerity of Beirut, Lebanon BSTRCT The paper preent the Sef-energizing Eectro- Hydrauic Brake (SEHB) with a non-inear controer, with focu on the deign of the oberver. The objective for deigning a non-inear contro agorithm for the innovative Sef-energizing Eectro-Hydrauic Brake (SEHB) i to obtain a controer that i inenitive concerning poibe parameter change of the contro path and epeciay the friction coefficient. The deigned autarkic brake contro unit on which the non-inear coed-oop contro wi be impemented, i ocated coe-by the brake actuator and i to be operated by impe open oop command from a uperior contro unit. The paper horty expain the brake principe of the SEHB and a impified mode of the brake ytem. The deign of the inearizing feedback and the oberver i preented. The controer how good diturbance rejection in imuative tudie uing a ophiticated, previouy verified imuation mode of the brake ytem. INTRODUCTION Hydrauic ervo ytem generay have high demand on the contro agorithm. Linear contro method are conventionay vaid for a imited operating range ). In order to faciitate a impe contro trategy and to fufi the requirement of ervo ytem, non-inear controer may be utiized. Many non-inear contro agorithm are baed on the tate pace repreentation. or the cae of the SEHB, the input-output inearization method ) ha been ued to inearize the non-inear contro path ). With thi approach, a impe fu tate feedback contro can be ued. In a previou pubication ), a tate feedback controer wa preented for the SEHB. or the SEHB a fu tate feedback i ued with an oberver. irt of a, a functiona overview of the SEHB i given in the next chapter. PRINCIPLE O THE SEHB There are numerou way to reaize a hydrauic efenergizing brake. The brake ytem ayout of the ytem under tudy i hown in ig.. The brake ytem i dedicated for raiway appication and mainy conit of the brake pad, a brake caiper, and a brake actuator. The brake actuator camp the brake hydrauicay. In contrat to conventiona brake, the SEHB feature an additiona upporting cyinder. The functionaity ha be decribed at the exampe of a reguar braking proce: To tart the braking proce, the eft contro vave i opened from neutra poition poitivey. Preurized oi fow toward the piton face ide chamber of the brake actuator. The troke of the brake actuator caue the brake pad to be preed onto the brake dic. Thereby, a norma force i appied and a friction force deveop. The brake caiper i upended movabe and due to the friction force it advance in tangentia direction. By the movement the brake caiper exert a force on the upporting cyinder. In doing o, preure buid up in the chamber of the upporting cyinder. the contro vave i ti open in poitive direction, the preurized oi i continued to be fed to the piton face ide chamber of the brake actuator, increaing the camping force. In thi way the oop of ef-energization i coed. diadvantage of fu tate feedback contro in practice i that a tate variabe have to be meaured. cot effective oution to reduce intrumentation i the ue of an oberver, which etimate the tate 4), 5). Sometime thi i the ony viabe approach a many dynamic variabe are hard to meaure ).

2 ig. : SEHB ytem ayout ig. : Punger cyinder mode a reut, the brake draw it actuation energy out of the kinetic energy of the vehice and ony neceitate ow eectric power for the contro vave. By meauring the preure in the upporting cyinder, the brake torque i controed. Thi i a major benefit of the SEHB when compared to conventiona brake. In order to deign a mode baed controer for the SEHB, the next chapter preent the ytem mode. or the ake of impicity the brake force buid-up i decribed ony. MTHEMTICL SYSTEM DERIPTION MTHEMTICL MODEL During the brake force buid-up the ring ide chamber of the brake actuator i connected to the ow preure ine by the fuy opened contro vave on the right hand ide in ig.. ony one chamber i preurized the brake actuator ha be modeed a a punger cyinder. The upporting cyinder i ao a punger cyinder. or the contro of the piton ide chamber of the brake actuator ony one contro edge i neceary. Therefore, a /-way proportiona contro vave i ued for the mathematica mode. urthermore, if the vave i opened, a i the cae for brake force buid-up, there i a inge hydrauic capacity in the ytem. Thee impification ead to the ytem iutrated in ig.. Thi impified mode conit of a punger brake actuator, a punger upporting cyinder, a mechanica connection between the cyinder, and a /-way proportiona vave to contro the oi fow. The hydrauic capacity i concentrated in the chamber of the brake actuator. The preure buid-up equation of the brake actuator can be formuated by H p B p p x x () C If friction, damping, and pring force are negected, x can be expreed by x p () cca By taking the mechanica connection of brake actuator and upporting cyinder into account, the preure of the upporting cyinder can be expreed depending on the brake actuator preure (). p i p Inerting (), () into () and oving for p yied (4). B i p x p (4) C H cca The dynamic of the contro ervo vave i approximated a a econd order ag eement 6). Then the acceeration and the veocity of the vave poo are expreed by (5) and (6). () v K u D v x (5) x v (6)

3 The equation (4), (5), and (6) contitute the threedimeniona tate pace repreentation of the SEHB, which form the bai for the non-inear controer ynthei. fixed friction coefficient i aumed in the mathematica mode. POSSIBLE IMPROEMENTS TO THE MTHEMTICL MODEL In order to mode the SEHB ytem more preciey, the underying mathematica mode can be enhanced. With the more ophiticated modeing of the SEHB decribed in thi chapter the controer i deigned. Stiffne - The greatet inaccuracy of the impified mode decribed above i the combined tiffne of the brake pad and the brake caiper c ca. i known from meaurement, thi tiffne i not contant ike aumed above. Intead, the tiffne c ca depend on the poition of the brake pad with repect to the brake dic. The poition of the brake pad i directy reated to the poition of the brake actuator x. The reation between the tiffne c ca and the poition of the brake actuator ha been determined by meaurement and i a poynomia equation, in which x i unknown. With the baance of force of the brake actuator c ca i expreed dependent oey on the brake actuator preure. riction and pring force - nother major refinement can be appied to the mathematica mode by conidering the inner friction of the two cyinder and the pring force of the brake actuator. Then the baance of force for the brake actuator i given by (7). p c x (7) r, Sp, The baance of force for the upporting cyinder i then decribed by (8). p (8) r, Soving equation (7) for x and differentiating it yied (9). ca Determination of the frictiona force - When examining equation (), three new parameter are identified. The pring force of the brake actuator i known a a product of the pring contant and the poition of the brake actuator. The other two parameter are the inner friction force of the cyinder. In order to determine them they were both meaured. or the brake actuator a dependency of the direction of trave wa identified, () and (). r, r, 8 N x N, x () N N x, x () mm The inner frictiona force of the upporting cyinder i ower becaue of maer diameter and decribed by equation (). r, ign N 75 () mm x N x Cyinder veocitie in the equation (), (), and () are obtained by equation (9) and the mechanica connection. The tate pace decription contituted by equation (), (5), and (6) i the bai for contro deveopment in the next chapter. LINERIZING CONTROL To obtain a inear ytem for the SEHB an input-output inearizing controer i deigned. The controer deign ha not be dicued further in thi paper ince it i given in a previou artice ). ig. give an overview of the input-output inearizing controer. ig. : Structure of inearizing feedback x p r, Sp, (9) c ca Negecting the derivative with repect to time of the friction and pring force, and inerting () and () into the preure buid-up equation of the brake actuator (4), ead to equation (). p B x i p C H r, c ca Sp, r, Equation () i the ubtitution of equation (4). p () The contro path of the SEHB i non-inear due to the orifice formua of the contro vave. Therefore, the reation of the ytem output, the upporting cyinder preure p, and the ytem input, the contro vave votage u, i ao non-inear. With the feedback deigned by the input-output inearization a inear reation of the new ytem input v and the output p i

4 achieved. coordinate tranformation from x to z coordinate i crucia to deign the inearizing feedback. OBSERER DESIGN STTE SPCE CONTROLLER The inearized contro path with the new input v and the output p i iutrated in ig. 4. To contro the ytem a tate pace controer i appied. The tate pace controer create the output v, which i the new input of the contro path. Input of the tate pace controer are the reference input w (p Set ) and the vector z, coniting of the three variabe brake actuator preure p, contro vave poo veocity v, and contro vave poo poition x. The tate feedback can then be formuated by v p S k p k z k z (4) Set ig. 4: State pace controer oop ytem. Thee poe are deignated by the matrix K (8). K K b k k T k k k k k (8) By etting up the characteritic poynomia of the coed oop in the factored form with Ci root of the poynomia (9) C C C and comparing it coefficient in the expanded form with det I K, the equation for the eement of the controer vector k are expreed (). k C C C k k C C C C C C () k C C C or the cae of the SEHB the ytem matrice B and C are coumn matrice b and c (SISO ytem). Moreover, the controer ha a row matrix k. or the input of the reference igna, a tranfer factor S i required. The differentia equation for the tranformed SEHB ytem in tate pace repreentation i given by (5). (5) z z b v The ytem output y i expreed by (6). y c T z (6) The tate pace controer i deigned with the poe aignment method. The poe of the open oop SEHB contro path are etabihed by the matrix ). They are determined by oving the characteritic equation (7). det I det (7) It can be ceary een that a three poe of the contro path,,, and, are zero. im of the poe aignment i to pace a poe of the coed oop eft of thoe of the contro path ). With the knowedge of the contro path poe it i poibe to aocate the poe of the coed Equation (), ao referred to a ckermann ormua ), aow the cacuation of the controer coefficient depending on the poe of the coed oop SEHB ytem which can be choen by the deigner. three rea vaue of the controer poe are et at - /, o that the controer vector k cacuate to 8 k k k k 6 The tranfer factor S i cacuated o that the ytem output equa the reference input w under teady tate condition 7), for which () hod. z () K z b S w Thi ead to equation () for the determination of S 7). S With T c T c b K () ()

5 and k k k k k, (4) K Therefore S i cacuated to S k 8. (5) The poe of the tate pace controer are paced eft of the open oop poe ). probem of tate pace controer often i that not a ytem tate are acceibe. or the raiway appication of the SEHB it i intended to oitary meaure the preure of the upporting cyinder p. By the mechanica connection, ee equation (), the brake actuator preure p i reated to p. The tate variabe of the contro vave, v and x, are hardy acceibe through meaurement and are thu unknown. Taking the appication of the controer into account, ony two out of the three tate variabe of the tate vector z are known in practice. LUENBERGER OBSERER To overcome thi probem, an oberver i deigned. Intead of meauring the unknown two tate variabe, they are etimated. The oberver ue a mode of the contro path in parae and etimate the tate vector z, which i then termed ẑ, ig. 5. parae contro path mode, the Luenberger oberver utiize a feedback of the difference between the ytem output p and the etimated oberver output ˆp. In doing o, the obervationa error of the SEHB ytem i minimized. The obervationa error i defined a (6). e z zˆ (6) It differentia equation i given by (7). e c z zˆ c e (7) ccording to Luenberger 5), it i neceary that the eigenvaue of the oberver are negative, which caue the tate of the oberver to converge to the tate of the origina ytem. If a eigenvaue of the matrix c have negative rea part, equation (8) hod for the obervationa error ). im e t t (8) Lunze ) propoe to et the eigenvaue of the oberver ignificanty eft of the eigenvaue of the contro path repectivey the coed oop (ig. 6). ig. 6: Oberver poe are eft of coed oop poe ig. 5: SEHB tate pace controer with oberver He furthermore recommend that the aboute vaue of the rea part houd be two to ix time a arge a the aboute vaue of the dominating eigenvaue. The eigenvaue of the oberver are determined by characteritic equation (9). det c (9) det I c det T If the cacuation are carried out, () ead to (). () or the contro probem of the SEHB, a Luenberger oberver approximate the tate vector. It ue the input of the contro path v and the avaiabe output, which i the brake actuator preure p. In addition to the I c T det ()

6 By poe aignment of the oberver eigenvaue O, O, and O and a comparion of coefficient with equation () foow the ckermann ormua (). Tabe : Parameter Simuation Parameter aue O O O O O O O O O O O O () ave Eigenfrequency f 55 Hz riction Coefficient µ.5 Nomina ave 5 bar Q N /min If the poe of the contro path of the SEHB are known, the coefficient of the Luenberger matrix are determined by poe pacement of the oberver eigenvaue. They are et to -6 /, o that the Luenberger matrix cacuate to The differentia equation of the oberver yied (). zˆ zˆ b v zˆ zˆ zˆ v y yˆ z z z zˆ zˆ zˆ () ave Damping Coefficient D.7 Rea aue of Controer Poe Ci - / Rea aue of Oberver Poe Oi -6 / variation are competed by changing one parameter at a time, whit the other are kept at the vaue ited in Tabe. Mot importanty, the controer and oberver poe, repectivey the aigned poe, are not changed. The brake force decreae i ao controed by the noninear controer. The imuation mode of the brake wa vaidated by tet rig meaurement data of the SEHB o that imuation and prototype behavior fit very we (ig. 7). The reut were obtained with a witching controer. With the verified imuation, the non-inear contro i invetigated by imuation. ig. 7: Meaurement and imuation of SEHB ytem With the given coefficient it i 8 z zˆ zˆ zˆ zˆ 8 z zˆ v 6 z zˆ It can be een that the etimated tate vector ue ony one meaurabe input z, which i the et point of the brake actuator preure. The aboute vaue of the rea part of the oberver poe are paced three time eft of the coed oop poe. ig. 8 how the tep repone of the imuation mode controed by the decribed tate pace controer in combination with the Luenberger oberver. The figure iutrate the et point of the brake force and the imuated brake force with the non-inear controer. Beide the et point for the brake force, the oitary input of the controer i the preure of the upporting cyinder p, from which the brake actuator preure neceary for the controer feedback cacuation i obtained by oving equation () for p. SIMULTION CONTROLLER PERORMNCE The controer with the oberver i impemented in a imuation environment together with a mode of the whoe SEHB ytem. The imuation mode i buit uing the ytem imuation too DSHpu. The parameter for the imuation deignated Reference are hown in Tabe.

7 ig. 8: Simuation reut of controer with oberver In a econd tet, the oop gain of the SEHB contro path i conidered. Two parameter, µ and Q N, are repreentative for the contro path oop gain. ig. iutrate change of the friction coefficient µ between the brake pad and the brake dic. ig. : Step repone for different friction coefficient ig. 8 ao how the tep repone of a we tuned P- controer for tep of the et point igna from N to N and N to N. The P-controer i tuned for the firt tep repone. t the firt tep repone, the P-controer reache 9% of the brake force after 4 m and the non-inear controer after 479 m. t the econd tep repone the time to reach 9% of the brake force i m for the P-controer, wherea the non-inear controer need 49 m. Thi mean that the dynamic of the non-inear controer are more contant within the working range of the SEHB, which reut in imiar time contant. ig. 8 i ony an extract of the working range. or ower brake force the P-controer i ower than the non-inear controer. Thi i not intended for the contro of the SEHB a the time contant houd be imiar within the working range to meet the comfort requirement during braking of the train. The reut confirm that the SEHB can be controed by a impe tate pace controer once the ytem i inearized. Time imitation for the brake force buid-up of a raiway brake are typicay at about. decreae of the friction coefficient i equivaent to a reduced oop gain. Therefore, the tep repone i protracted. n increaed friction coefficient, on the other hand, horten the time for the brake force buid-up. To improve the performance, future work wi focu on the automatic prediction of the friction coefficient. variation of the nomina vave fow infuence the fow rate ampification and thereby the oop gain of the SEHB. variation of the vave nomina fow at 5 bar preure drop per contro edge i hown in ig.. ig. : Nomina vave fow ROBUSTNESS - During operation of the brake variou parameter can change. or intance, the wiftne of the ervo vave increae for maer opening. Hence, invetigation are carried out to verify the performance of the deigned contro agorithm with a ignificanty ower and fater ervo vave (ig. 9). ig. 9: Senitivity concerning vave frequency the nomina vave fow i we-known, a variation of ±% i carried out. Once again a reduced ampification, repreented by the nomina vave fow, proong the time neceary for brake force buid-up (9%) to 84 m. On the contrary, examining the tep repone with a contro vave of raied nomina fow, a fat repone can be oberved. or the cae of a nomina fow of /min, the brake force overhoot the et point by.78% and ha a peak time of 689 m. rapid reaction ervo vave doe not affect the controer tep repone adverey. The ow ervo vave i parameterized at a frequency of 5 Hz and the non-inear controer how contant dynamic. CONCLUSION ND OUTLOOK Thi paper how the deign of a fu tate feedback contro for the SEHB. In a firt tep a mathematica mode of third order with one preure buid-up equation i decribed. The yieded mode i input-output

8 inearized and thereafter a tate pace controer i deigned. The cacuation for the controer coefficient on bai of the ckermann ormua i preented. ony one of the three variabe of the tate vector, the brake actuator preure p, i avaiabe through meaurement, an oberver i required. Therefore, a Luenberger oberver to etimate the unknown tate of the contro vave i deigned. The determination of the Luenberger matrix coefficient i hown o a to aign the oberver poe eft of the controer poe. The performance of the non-inear controer with oberver i verified by imuation. The ued imuation mode wa vaidated by tet rig meaurement. Step repone of the non-inear controer confirm a more contant dynamic behavior when compared to a proportiona controer. urthermore, the non-inear controer performance for changing parameter of the contro path i uccefu. The imuation reut of the non-inear controer are promiing and future work wi focu on the vaidation of the controer by meaurement with the SEHB prototype. Moreover, an automatic prediction of the friction coefficient i in need. CKNOWLEDGMENTS The reut pubihed within thi paper have been acquired in the context of the project Sebtvertärkende Eektro-Hydrauiche Breme funded by the German reearch foundation (DG). The author expre their thank to the DG for funding the reearch work. REERENCES. Lunze, J., Regeungtechnik, 5 th ed., Springer, Berin, 8, pp Iidori,., Noninear Contro Sytem, rd ed., o. 4, Springer, London,, pp Kühnein, M., Liermann, M., Murrenhoff, H., Simpified Modeing of a Sef-energiing Hydrauic Brake, Proceeding of the 6 th PNI PhD Sympoium, o.,, pp Ogata, K., Modern Contro Engineering, 5 th ed., Prentice Ha, Upper Sadde River, 9, pp Luenberger, D.G., n Introduction to Oberver, IEEE Tranaction on utomatic Contro, o. C- 6, No. 6, Dec 97, pp Murrenhoff, H., Servohydrauik Geregete Hydrauiche ntriebe, rd ed., Shaker, achen, 8, pp be, D., Regeungtechnik, rd ed., Mainz, achen, 9, pp N.N., DIN EN 45-: Raiway appication - Braking - Ma tranit brake ytem - Part : Performance requirement, German-European Standard,. 9. Liermann, M., Sef-energizing Eectro-Hydrauic Brake, Shaker, achen, 8, pp Luenberger, D.G., Oberver for Mutivariabe Sytem, IEEE Tranaction on utomatic Contro, o. C-, No., pr 966, pp CONTCT Dip.-Ing. Michae Kühnein, RWTH achen Univerity, Germany, Intitute for uid Power Drive and Contro (IS), Steinbachtr. 5, 574 achen, Germany; Phone +49 () , E-Mai: Internet: CRONYMS : Brake ctuator NL: Non-Linear : Supporting Cyinder SEHB: Sef-energizing Eectro Hydrauic Brake