ADAPTIVE ROBUST TRACKING CONTROL OF PRESSURE TRAJECTORY BASED ON KALMAN FILTER *

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1 附件 1 英文版论文写作参考格式 1 CAO Jian ZHU Xiaocong AO Guoliang State Key Laoratory of Fluid Power ranmiion and Control, Zhejiang Univerity, Hangzhou 31007, China YAO Bin State Key Laoratory of Fluid Power ranmiion and Control, Zhejiang Univerity, Hangzhou 31007, China School of Mechanical Engineering Purdue Univerity Lafayette IN 47907, USA ADAPIVE ROBUS RACKING CONROL OF PRESSURE RAJECORY BASED ON KALMAN FILER * Atract: For the adative rout control (ARC) trategy aed on ackteing deign, tracking control of oth force trajectory and oition trajectory need the high-reciion reure control in neumatic ytem. In thi aer, an adative rout reure controller i deigned to enhance the tracking accuracy of reure trajectory in the chamer when the neumatic cylinder i moving. Off-line fitting of the effective throttling area and on-line arameter etimation of flow coefficient are ued to imrove the model comenation in the adative term, and rout feedack and Kalman filter are ued to achieve trong routne. Exerimental reult how that the rooed adative rout reure controller not only achieve good tracking of reure trajectory even when the neumatic cylinder i moving, ut alo otain very mooth control inut which indicate the effectivene of model comenation. Key word: Pneumatic ervo control Adative rout control Kalman filtering Effective throttling area 0 INRODUCION he comreiility of air and the inherent nonlinearity of neumatic ytem continue to make neumatic ervo ytem a reearch focu [1]. he high-reciion oture control of arallel maniulator driven y neumatic mucle i realized through alying adative rout control trategy into it []. herefore, it i an attemt that alying thi trategy to the neumatic cylinder with large friction force for achieving the high-reciion trajectory tracking of force and oition. For the adative rout control trategy aed on ackteing deign, tracking control of oth force trajectory and oition trajectory need the high-reciion reure control in neumatic ytem [3]. For the contant reure control or te reone of reure control in neumatic ytem, only PID algorithm controlling a roortional valve i needed to achieve etter teady-tate error. However, for the tracking control of inuoidal reure trajectory and even aritrary reure trajectory, the PID algorithm would ring aout large tracking error and evere control inut chattering, o thi algorithm could not realize the high-reciion trajectory tracking of force and oition. herefore, it i neceary to utilize the adative model comenation for imroving the tracking accuracy of reure trajectory and ue rout feedack for guaranteeing the trong routne of neumatic ytem. he key to model comenation lie in realizing the nonlinear comenation of ma flow rate and reure dynamic. Richer and Hurmuzlu rooed the accurate model of reure dynamic in neumatic cylinder, time delay and attenuation in neumatic line, valve dynamic, flow nonlinearitie through the valve orifice, iton friction force, and identified unknown arameter of the aove model through uing ecially deigned exeriment [4]. Moreover, a nonlinear liding mode controller i deigned on the ai of the aove model to achieve the high-reciion tracking control of hi roject i uorted y National Natural Science Foundation of China(No ); Received May 30, 007; received in revied form Augut 8, 008; acceted Augut 15, 008 inuoidal force trajectory. However, it i very difficult to otain o much a numer of geometric and functional characteritic or arameter of neumatic cylinder and roortional directional valve. herey, many reearcher ue off-line identification for controlling the force or oition on the ai of otaining common geometric and functional characteritic or arameter [1,5]. In thi aer, the off-line curve fitting and on-line identification for the model are comined to achieve etter adative rout tracking control of reure trajectory. Recently, the tendency of alying neumatic control to ionic root i more and more oviou [6,7]. For the future, neumatic ervo control would not e limited to tracking the determinitic trajectory, and would extend to tracking the aritrary trajectory. In thi way, the tatitical roerty of noie may e unknown and time-varying. herefore, Kalman filter would e ued to attenuate the influence of roce noie and meaurement noie and imrove the filtering accuracy. Moreover, the firt-order difference of reure could e otained y Kalman filter, which would e convenient for deigning an adative rout motion controller in the future. 1 SYSEM DYNAMICS 1.1 Schematic diagram of neumatic ytem he chematic diagram of neumatic ytem i hown in Fig. 1. wo roortional directional valve(mpye-5-1/8 HF-010B y FESO) earately control two chamer of rodle neumatic cylinder(dgpil k-kf-au y FESO), three reure tranducer(sde--d10-g14-i-m1 y FESO) are ued to meaure the reure of chamer A, chamer B and uly reure reectively, and a oition tranducer (RPS0500MD601V y MS) i ued to meaure the oition and velocity of rodle neumatic cylinder. Since two valve earately control the two chamer of cylinder, there exit two control degree of freedom in the ytem, thu two different trajectorie can e controlled. For examle, motion trajectory and reure trajectory, motion trajectory and energy trajectory, or motion trajectory and tiffne trajectory. Moreover, the ytem would e controlled comletely ince there i no zero

2 dynamic in it. In thi aer, chamer A i ued to track a motion trajectory y oen-loo control and chamer B track a reure trajectory y cloed-loo control Preure dynamic in neumatic cylinder heoretical exreion for reure dynamic in chamer can e otained y euation of tate for ideal ga, continuity euation for teady one-dimenional flow, and firt law of thermodynamic. However, the aove theoretical exreion need the temerature in chamer for the dicharging roce. In fact, it have een found y reearcher that the exerimental value of the temerature were cloe to the adiaatic curve only for the charging roce and they were cloe to the iothermal aumtion for the dicharging roce [4]. herefore, Richer and Hurmuzlu reented an integrated exreion of reure dynamic(chamer B) hown in E.(1). herein, α in = 1.4, α = 1.0, α = 1. out Fig.1 Schematic diagram of neumatic ytem 1.air uly.air treatment unit 3.reure tranducer 4.temerature tranducer 5.dSPACE controller 6.oition tranducer 7.rodle neumatic cylinder 8.roortional directional valve R A = + x + d t (1) ( α α ) α () in in out out where R i the erfect ga contant, i the utream temerature of roortional directional valve, V i the volume of air in chamer B, i the reure of air in chamer B, d () t i the lumed diturance of whole ytem. 1.3 Ma flow rate of roortional directional valve When the roortional directional valve i in the dead zone of neutral oition, the leakage ma flow rate i iggih. Moreover, the valve would work in the dead zone when the reure control i teady at certain contant value. herefore, it i neceary to model accurately the ma flow rate in the dead zone in order to reduce control inut chattering and enhance the accuracy of reure control. When the valve i at neutral oition, the leakage ma flow rate i decried y annular flow model that i roortional to differential reure. When the valve i oen, the ma flow rate through orifice i regarded a the nozzle flow model. o facilitate the deign of nonlinear controller, the leakage ma flow rate i alo decried y the nozzle flow model and then thi imlie the curve fitting in the ene of leakage throttling area. hrough meauring the ma flow rate under different control voltage, the relation etween control voltage and effective throttling area could e otained a hown in Fig.. he critical reure ratio P in theoretical formula i not cr eual to the critical reure ratio in flow formula of Fig. Relation etween control voltage and effective throttling area ISO6358. For guaranteeing the effective model comenation in the controller, the exerimental value of critical reure ratio would e regarded a the tranition oint etween onic flow and uonic flow. At the ame time, for keeing the egment function continuou, the following euation of ma flow rate will e ued (charging roce) [9]. = AC e C () m where A i the effective throttling area of valve, e C i flow coefficient, i the uly reure of valve, and flow arameter C i exreed a m γ + 1 γ γ 1 0 < R γ + 1 (3) γ Cm = Ct 1 < λ R( γ 1) 1 1 γ λ Ct 1 λ < 1 λ 1 R ( γ 1) 1 in which λ i the minimal reure ratio to have a laminar flow, C t = i a correction factor etween theoretical value and exerimental value of critical reure ratio. Flow coefficient C i ued to include extra loe due to local friction and lo of kinetic energy, it i related to the ratio etween downtream reure and utream reure, and aumed to e olynomial function of the reure ratio according to Perry law [9]. Being enlightened y hi method, the flow coefficient i exreed a the olynomial function of reure ratio to reflect that the flow coefficient varie according to the reure. Moreover, coefficient of olynomial function will e identified on-line for reducing model error in the adative rout controller. C = a0 + a1 + a (4) where a0, a1, a are all unknown arameter and will e identified on-line. DESIGN OF ADAPIVE ROBUS PRESSURE CONROLLER here till exit model error in the ytem from reure dynamic in neumatic cylinder, ma flow rate of valve and temerature in chamer. hu, model error will e lumed into

3 the diturance d () t in E.(1) and d() t = d0 + d () t. herein, d i the low time-varying art of lumed 0 diturance in model (alo nominal value of lumed diturance), and d () t i the fat time-varying art of lumed diturance. hen, E.(1) could e rewritten a E.(5). A the length of aer i limited, the adative rout reure controller for charging roce i deigned in chamer B while the controller for dicharging roce i imilar to thi rocedure. R A = ( αinin αoutout ) + α x + d0 + d () t (5).1 Adative rout reure controller A can hown in E. () and E. (5), unknown arameter C, d in ytem model will e udated on-line through 0 adative arameter etimation to imrove control accuracy. Meanwhile, a dicontinuou rojection maing i utilized to guarantee that the arameter etimate and their derivative remain in the known ounded region all the time [10]. hen the adative rout controller aed on ackteing deign i yntheized a follow. (1) Ste 1 Denote reure error a e = d in which d i deired reure trajectory. From E. (5), the dynamic of e i e = d (6) R A = ( αinin αoutout ) + α x d + d0 + d () t Define virtual ma flow rate a m = αinin αout, then out E.(6) can e rewritten a R A e = m + α x d + d0 + d () t (7) he deired virtual ma flow rate i md md = mda + (8) md A ˆ mda = α x d0 + (9) d R R R where i the model comenation term in ARC and mda md i the rout feedack term in ARC, which conit of the following linear art and nonlinear art. md = md1+ (10) md md1 = ke (11) R e md = k (1) 1 η R where k, k 1, η are all oitive arameter. Let d ˆ 0 = d0 d and define the oitive emi-definite 0 Lyaunov function a 1 V = e, one otain R V = ke + e md d 0 + d () t (13) i yntheized to dominate the uncomenated model md uncertaintie coming from oth arametric uncertaintie and uncertain nonlinearitie, which i choen to atify the following condition. R e md d 0 + d () t ε (14) R e md 0 where ε i a oitive deign arameter that may e aritrarily mall. Sutituting E.(14) into E.(13), one otain the following euation for guaranteeing Lyaunov taility. R V = ke + e md d 0 + d () t ke + ε (15) d i udated y an adatation law a follow. ˆ0 dˆ 0 = Proj ˆ ( Γ1σ 1) (16) d 0 with the adatation function given y σ1 = ϕ1e (17) Define augmented Lyaunov function a 1 Va = V + d, 0 the time derivative of V i a R V a = V + d 0d 0 = ke + e md + d 0( d 0 e (18) ) Due to d ˆ 0 = d, the regreor i 0 ϕ 1 = 1. () Ste C i alo udated y on-line arameter etimation according to the reure dynamic of ytem E. (5) for further reducing the influence of model error. o ya the rolem that numerical differentiation of reure feedack ignal would reult in evere noie if it were adoted y arameter etimation, a tale filter with a relative degree larger than or eual to 1 would e emloyed in reure dynamic and d () t = 0 would e aumed [11], and then a tandard linear regreion model for arameter etimation could e otained. y = f = ϕ f β (19) where i the filtered outut of f, ϕ i the f correonding regreor of β, and β = [ a0, a1, a]. he leat uare tye etimation algorithm i ued to etimate unknown arameter β [1], and a dicontinuou rojection maing i ued to kee arameter etimate ounded. β i udated y an adatation law a follow. ˆ β = Proj ˆ β ( Γσ ) (0) with the adatation function given y 1 σ = ϕ f Γ (1) 1+ vtr { ϕ f Γϕ f} and the adatation rate matrix given y 1 Γ = αγ Γϕ fϕ f Γ 1+ vtr { ϕ f ϕ f} () where α i the forgetting factor, v 0 (3) Ste 3 After 0 and md C i calculated, the control inut of Fig.3 Princile of reure tracking control

4 roortional directional valve could e otained according to euation of ma flow rate (E.()) and relation etween control voltage and effective throttling area(fig. ).. Princile of reure tracking control Putting all the develoment together, the adative rout reure controller for the neumatic ytem aed on earate control of motion trajectory and reure trajectory i chematically rereented in Fig.3. 3 KALMAN FILER For reducing influence of roce noie and meaurement noie on the meaured ignal, the following Kalman filter would e adoted to get the reure of air uly and chamer B in order to imrove the filtering accuracy of meaured ignal. Define the tate vector of ytem a x = [, ] aume the amling eriod very mall, and then the tate euation and oervation euation of the ytem could e exreed a follow reectively [8]. xk ( ) = Axk ( 1) + Bwk ( 1) (3) yk ( ) = Cxk ( ) + vk ( ) (4) where A i the tate tranition matrix, B i the inut matrix of roce noie, C i the oervation matrix, w and v are roce noie and meaurement noie reectively with the aumtion that they are all earate zero-mean guaian white noie. he Kalman filter i deigned a xkk ˆ(, 1) = Axk ˆ( 1) xk ˆ( ) = xkk ˆ(, 1) + Kk ( )[ yk ( ) Cxkk ˆ(, 1) ] (5) Pkk (, 1) = APk ( 1) A + BQB 1 Kk ( ) = Pkk (, 1) C CPkk (, 1) C + R 1 Pk ( ) = [ I KkC ( ) ] Pkk (, 1) where Pk ( ) i the covariance matrix of oervation error, K( k ) i the Kalman gain matrix, Q i the covariance of roce noie, and R i the covariance of meaurement noie. When the filtering time i enough, lim Pk ( ) = P, [ ] k lim Kk ( ) = K= α, β, and arameter of filter α, β tend to k e tale. Fig.4 Oen-loo motion tracking control of neumatic cylinder Fig.5 racking reone of te reure trajectory 4 EXPERIMENAL RESULS he effectivene of the rooed controller ha een demontrated y a numer of exeriment under the uly reure of 0.45MPa. It mut e noted that the uly reure would decreae a little during the tracking control of reure trajectory with the dro of 8.9% when tracking a mooth te trajectory and the dro of 7.4% when tracking a inuoidal trajectory. 4.1 racking of te reure trajectory through deired trajectory generation For tracking a te reure trajectory through deired trajectory generation, reone comarion etween ARC and PID i hown in Fig.5. It mut e noted that oen-loo tracking control of motion trajectory i conducted from 4 to 7 in order that erformance index under the control of two controller could e comared when the neumatic cylinder i moving (Fig.4). A can e een from Fig.6, maximum tracking error of ARC i 4.5kPa, average tracking error i 1.1kPa and teady-ate error i 1.0kPa, however, maximum tracking error of PID i 9.3kPa, average tracking error i 4.0kPa and teady-ate error i.3kpa. herefore, it can e concluded that tracking error of ARC are much maller than thoe of PID and teady-tate error of ARC i cloe to the meaurement accuracy of reure tranducer, and eecially that ARC could achieve high-reciion reure tracking when the neumatic cylinder i moving. Fig.6 racking error of te reure trajectory 4. racking of inuoidal reure trajectory For tracking a inuoidal reure trajectory, reone comarion etween ARC and PID i hown in Fig.7. It i well known from Fig.8 that maximum tracking error of ARC i 3kPa ut that of PID reache 11kPa and that average tracking error of ARC i only 1.4kPa while that of PID i 7.8kPa. Moreover, a can e een from the control inut of two controller in Fig.9, the control inut of ARC i very mooth while there exit control chattering in PID. herefore, tracking error of ARC are much maller than thoe of PID with the merit of mooth control inut.

5 and average tracking error i 1.1kPa, and teady-tate error of ARC i cloe to the meaurement accuracy of reure tranducer. For tracking a inuoidal reure trajectory, maximum tracking error of ARC i 3kPa and average tracking error i 1.4kPa. herefore, tracking error of ARC are much maller than thoe of PID with the merit of very mooth control inut, which verifie the effectivene of the adative rout reure controller. Reference Fig.7 racking reone of inuoidal reure trajectory Fig.8 racking error of inuoidal reure trajectory [1] N. Shu, G. M. Bone et al. Exerimental comarion of two neumatic ervo oition control algorithm. In: Proc. of the IEEE International Conference on Mechatronic & Automation, Niagara Fall, Canada, 005,7: 37~4 [] Zhu Xiaocong,ao Guoliang, Yao Bin,Cao Jian. Adative rout oture control of a arallel maniulator driven y neumatic mucle. Automatica, 008, 44(9). ( 将发表 ) [3] B. Yao,D. CHRIS. Energy-aving adative rout motion control ingle-rod hydraulic cylinder with rogrammale valve. In: Proc. of the American Control Conference, Anchorage, USA, 00,5: 4819~ 484 [4] E. Richer,Y. Hurmuzlu. A high erformance neumatic force actuator ytem: Part Ⅰ-- nonlinear mathematical model. ranaction of the ASME, Journal of dynamic ytem, meaurement, and control, 000, 1(9): 416~45. [5] B. W. McDonell, J. E. Borow. Modeling, identification, and control of a neumatically actuated, force controllale root. IEEE ranaction on Rootic and Automation, 1998, 14(5): 73~74 [6] B. Verrelt, R. V. Ham, B. Vanderorght et al. he neumatic ied Lucy actuated with leated neumatic artificial mucle. Autonomou Root, 005,18(): 01~13 [7] B. ondu, S. Iolito, J. Guiochet et al. A even- degree-of-freedom root arm driven y neumatic artificial mucle for humanoid root. International Journal of Rootic Reearch, 005,4(4): 57~74. [8] QIN Yongyuan, ZHANG Hongyue, WANG Shuhua. Kalman filtering and integrated navigation theory[m]. Xi an: Northwetern Polytechnical Univerity Pre, 1998, (in Chinee). [9] Z. MOZER,A. AJI, V. SZENE. Exerimental invetigation on neumatic comonent. In: Proc. of the 1 th International Conference on Fluid Flow echnologie, Budaet, Hungary, 003,9: 517~54. [10] B. Yao and M. omizuka. Adative rout control of MIMO nonlinear ytem in emi-trict feedack form. Automatica, 37(9): , 001. [11] B. Yao. Integrated direct/indirect adative rout control of SISO nonlinear ytem in emi-trict feedack form. In: Proc. of the American Control Conference, Denver, USA, 003,6: 300~305. [1] B. Yao and R. G. Dontha. Integrated direct/indirect adative rout reciion control of linear motor drive ytem with accurate arameter etimation. In: Proc. of the nd IFAC Conference on Mechatronic Sytem, Berkeley, USA, 00,1: 633~638 Biograhical note CAO Jian i currently a PhD candidate in State Key Laoratory of Fluid Power ranmiion and Control, Zhejiang Univerity, China. Hi reearch interet include fluid ower tranmiion and control and mechatronic control, etc.. el: ; caojianjiaowang@ina.com ZHU Xiaocong i currently a ot-doctor in State Key Laoratory of Fluid Power ranmiion and Control, Zhejiang Univerity, China. Her reearch interet include the alication of novel neumatic actuator, uch a neumatic mucle actuator and artificial control method in the field of root, etc.. el: ; zhuxiaoc@zju.edu.cn Fig.9 Control inut of tracking a inuoidal reure trajectory 5 CONCLUSIONS An adative rout reure controller i rooed to enhance the tracking accuracy of reure trajectory in the chamer when the neumatic cylinder i moving. Off-line fitting of the effective throttling area and on-line arameter etimation of flow coefficient are ued to imrove the model comenation in the adative term, and rout feedack and Kalman filter are ued to achieve trong routne. For tracking a te reure trajectory with the movement of neumatic cylinder, maximum tracking error of ARC i 4.5kPa AO Guoliang i currently a rofeor in State Key Laoratory of Fluid Power ranmiion and Control, Zhejiang Univerity, China. He received hi PhD degree from Zhejiang Univerity, China, in 000. Hi reearch interet include fluid ower tranmiion and control and mechatronic control, etc.. el: ; gltao@zju.edu.cn. YAO Bin i currently a rofeor in State Key Laoratory of Fluid Power ranmiion and Control, Zhejiang Univerity, China, and alo in the School of Mechanical Engineering, Purdue Univerity, Lafayette, USA. Hi reearch interet include nonlinear control, nonlinear oerver deign, data fuion and fault detection & diagnotic, etc.. el: ; yao@urdue.edu

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