Analysis of Active Suspension Systems with Hydraulic Actuators

Transcription

1 Proceeding o the 3 IAVSD conerence, Atugi, Jaan, Augut 3. Analyi o Active Suenion Sytem with Hydraulic Actuator XIAOMING SHEN and HUEI PENG SUMMARY Mot o the exiting active uenion tudie aume that an ideal orce actuator exit and will carry out the commanded orce accurately. In reality, due to the interaction between the hydraulic actuator and uenion ytem, the lightly damed mode o the lant will generate lightly damed zero (LDZ) or the ervo-loo ytem, which conine the cloed-loo erormance to a low requency range. Converting the active uenion roblem ormulation into a dilacement control roblem, a uggeted by everal recent aer, will not olve thi roblem. Four candidate remedie to reduce the eect o the LDZ are tudied and their otential and limitation are dicued. INTRODUCTION Active Suenion (AS) have been widely tudied over the lat year, with hundred o aer ublihed (ee [] or an extenive review). Mot o the ublihed reult ocu on the main-loo deign, i.e., on the comutation o the deired control orce, a a unction o vehicle tate and the road diturbance. It i commonly aumed that the commanded orce will be roduced accurately. Simulation o thee main-loo deign were requently done without conidering actuator dynamic, or with highly imliied ub-loo dynamic. In reality, actuator dynamic can be quite comlicated, and interaction between the actuator and the vehicle uenion cannot be ignored. Thi i eecially true or hydraulic actuator, which remain to be one o the mot viable choice due to their high ower-to-weight ratio, low cot and the act orce can be generated over a rolonged eriod o time without overheating. However, hydraulic actuator alo have everal advere attribute: they are nonlinear and their orce generation caabilitie are highly couled to the vehicle body motion []. Few exerimental veriication on AS algorithm have been reorted and all are limited to bandwidth o -4 Hz [3][4][5]. In other alication uch a durability tet, hydraulic actuator have achieved bandwidth o 5Hz or higher ucceully. However, in thee alication, the cloedloo ytem are uually deigned to track a dilacement ignal, rather than a orce ignal. Some reearcher thu eculate that or hydraulic actuator, orce tracking i harder than dilacement tracking. Several aer have been ublihed to recat the orce control roblem in AS deign to a dilacement control roblem [6][7]. But do we really have a undamental limit in uing hydraulic actuator or orce control in general? Or i it an alication-eciic henomenon? For examle, AS ytem have to ettle or a maller actuator, while or durability tet rig more owerul actuator and higher uly reure can be ued. AS actuator are commonly laced in arallel with exiting ring/damer, while the durability actuator uually are connected in erie with Graduate tudent, School o Mechanical and Power Energy Engineering, Shanghai Jiao Tong Univerity, China Correonding author, Aociate Proeor, Deartment o Mechanical Engineering, Univerity o Michigan, Ann Arbor, MI 489-5, , heng@umich.edu

2 Proceeding o the 3 IAVSD conerence, Atugi, Jaan, Augut 3. vehicle uenion. Were AS reearcher attacking the wrong roblem by olving a orce tracking roblem? I there i a undamental iue related to orce control, can we avoid thi undamental limit by ormulating a dilacement control roblem? Are there other remedie? The main uroe o thi aer i to rovide anwer to thee quetion. Firt, we comare the AS ervo-loo control roblem with that o durability tet rig. For both AS and durability tet rig, linearized equation are ued o that traner unction and requency reone can be ued or the analyi. Since the lightly-damed zero (LDZ) o the cloed-loo ytem are the main ource o erormance limit [7], we will olve the cloed-loo traner unction o the two tracking roblem (both orce and dilacement) analytically. The ole and zero o the cloed-loo traner unction will then be analyzed. By doing o, the undamental limit imoed by the LDZ will be clearly undertood. It i hown that the dilacement control roblem alo ha it own air o LDZ. Furthermore, while the natural requency o the dilacement control LDZ i a little higher, their daming ratio i lower. Thereore, witching to a dilacement control roblem i not a comlete anwer. Subequently, remedie to reduce the advere eect o the LDZ are tudied. Four candidate aroache are analyzed new actuator; uenion arameter otimization; add-on mode uch a vibration aborber; and advanced control algorithm. Analyi and imulation reult are reented to how the eect o the rooed remedie. MODEL DEVELOPMENT. Quarter-Car Active Suenion Model with a Hydraulic Actuator In thi ection, we will reent the model o a quarter-car AS ytem equied with a hydraulic actuator. The governing equation o the quarter-car uenion ytem (Fig. ) are reented. The two degree o reedom (DOF) are the vertical motion o the rung ma ( m ) and unrung ma ( m u ), reectively. Thi DOF ytem ha one external inut, z, which i the rate o change o road urace elevation. A orce ( ) i alied by the active uenion actuator on m and m u. The tate ace model i: zu z ku c cu k c zu z c u z u mu mu mu m d u z u m u = mu dt z z u z z u z c k c z m m m m The traner unction rom actuator orce to uenion troke i then z z ( m m ) c k G = = z () u u u u 4 3 mmu ( cmu mc mcu ) ( muk ccu km mku ) ( kuc kcu ) kku By uing tandard ervo-valve dynamic equation, the hydraulic ytem i decribed by: () A kxd A β β = x ignqrt( P ign( x) / A) ( z u z ) V V x = ( x kviv ) τ (3)

3 Proceeding o the 3 IAVSD conerence, Atugi, Jaan, Augut 3. where A i the iton area, β i the luid bulk modulu, k xd i the oriice low coeicient, x i the ervo valve dilacement, ignqrt( y) ign( y) y, P i the uly reure, V i the cylinder chamber volume, and k v i the valve gain. Eq.(3) i then linearized, which enable u to do a more in-deth analyi: V Cx x v A( z u z ) = β A (4) τ x x = K i v v v v z m k c z u m u k u c u z o Fig.. Quarter-car active uenion model In Fig., the requency reone o the linear model i lotted againt inuoidal imulation reult o the nonlinear ytem at inut magnitude o 7mA (~4N orce at Hz). It can be een that Eq.(4) i a good aroximation o Eq.(3) under reaonable inut magnitude. 8 6 Bode Diagram Linearized V. Nonlinearized (Poition Control) linearized nonlinearized 6 5 Bode Diagram Linearized V. Nonlinearized (Force Control) linearized nonlinearized Magnitude (db) - Magnitude (db) Frequency (Hz) Fig.. Validation o the linearization reult (uenion troke and orce). 7 - Frequency (Hz) Uing the linear aroximation o Eq.(4), the dynamic o the overall lant with ervo valve current i v a the control inut, ha the ollowing orm ku c cu k c zu z mu mu mu mu mu zu z z c u z u u d z z mu u c k c z z u = dt z m z m m m iv z βa βa c xβa x k v v V V V x v τ τ (5) 3

4 Proceeding o the 3 IAVSD conerence, Atugi, Jaan, Augut 3.. Active Suenion Force Control Problem Given Eq.(5), the traner unction rom i v to can be obtained: 4 3 mm u ( cm u cm mc u ) ( km u km cc u mk u ) ( kc u ck u ) kk u Gi = Cx βak v (6) iv D where the 6 th -order denominator olynomial i 4 3 D= ( τ )[ mvm u ( cvm u cvm mvc u ) (muβa kvm cvc u mkv u βa m kvm u ) (7) ( cuβa cukv kuvc ) kuβa kukv ] Similarly, the traner unction rom i v to uenion dilacement, z zu, i z x v[( u) u u] z C β A K m m c k u Gzi = = (8) iv D Zhang and Alleyne [7] ointed out that the numerator o the traner unction hown in Eq.(6) i the ame a the denominator olynomial o the quarter-car uenion lant hown in Eq.(). Thi i not a coincidence, and can be exlained by ormulating a block diagram o the augmented ytem. Fig. 3 how a decomoed orm o Eq.(5). The eedback block H () doe not rereent a eedback control action. Rather, it i an inherent couling between uenion motion and hydraulic orce generation. From Fig. 3 and Eq.(4)(5), it can be een that G() i the uenion dynamic hown in Eq.(). The orce generation dynamic, Eq.(4), can be written a = Gi a v GHy a, where β ACK x v A ( τ ) Ga = and H =. V τ CK i v _ actuator G a () Force x v lant G () y: outut ( z z u ) H () eedback Fig. 3. Block diagram o the uenion dilacement ub-loo. When an engineer i deigning the ervo-loo control algorithm to enure roer orce generation rom the hydraulic actuator, he/he i acing a dynamic ytem hown in Fig. 4. Again, the two block hown in the eedback loo were not rom a eedback control algorithm, but rather i a hyical eedback the control deigner ha to deal with. i v _ actuator G a () Force H() G () eedback Suenion Fig. 4. Block diagram o the actuator orce ub-loo. Given Fig. 4, the lant to be controlled i: 4

5 Proceeding o the 3 IAVSD conerence, Atugi, Jaan, Augut 3. Ga G = i i = v GaGH (9) Obviouly, the ole o the active uenion lant ( G ) become zero o G i. From Eq.(6), we can ee that thi i indeed the cae. Why i thi a concern? Becaue the active uenion lant (hown in Eq.()) include two under-damed mode (at around Hz and Hz). The mall daming o the tire mode (~.-.3) and the uenion mode (~.-.3?)imlie that the lant o the orce ervo loo ( G i hown in Eq.(9)) will have a air o under-damed zero at around Hz. For eedback-only control algorithm, the under-damed zero will remain to be zero o the cloed-loo ytem. Thereore, the cloed-loo bandwidth will be limited by thi air o lightly damed zero. What i we ue other control coniguration? Ater all, nonlinear algorithm baed on the cancellation o uenion-actuator interaction have been rooed and a omewhat higher cloed-loo tracking erormance ha been reorted [8][9]. Let rerain rom analyzing eciic control algorithm, and intead examine thi roblem rom a more undamental viewoint. Auming that we know little about the ervo-loo control algorithm, excet that it i doing a good job. Baed on the erect control analyi [][], the well-behaved control algorithm will generate an invere o the lant. In other word, a air o lightly damed zero (LDZ) demand much larger control action at their reonant requency or accetable control erormance. Thi i theoretically oible but uually diicult to realize in ractice. From the above analyi, and the act that hydraulic actuator have been ued in dilacement-tracking alication uch a durability tet rig u to 5-8 Hz, it i temting to claim that reormulating the AS ub-loo roblem (rom orce control) into a dilacement tracking roblem will be the anwer. A mentioned in Section, everal rior ublication baed on thi belie were ublihed [6][7][], and two way to recat the AS ervo-loo into dilacement tracking roblem were rooed..3 Active Suenion Dilacement Control Problem In thi ection, by ollowing the ame rocedure hown in Section., we examine whether the erormance limitation imoed by the LDZ can be removed in the dilacement control roblem. The ytem coniguration i till the one hown in Fig., and the control roblem i to maniulate the ervo-valve current o that a deired uenion dilacement i ollowed. The lant o thi control roblem i hown in Fig. 3. It traner unction i: z z GG u a Gzi = = () iv GaGH Obviouly, the zero o the uenion lant ( G ) are alo zero o G zi. Thi act i conirmed by examining Eq. () and (8). From Eq.(8), it i obviou that the mall tire daming create a air o LDZ or G zi. The reonant requency o thee zero i at around 3Hz (v. Hz or the orce loo). In other word, the limitation or tracking erormance (a analyzed by the erect control analyi) i omewhat relieved, but not by much. In act, while the reonant requency o the LDZ o G zi i higher, their daming ratio i 5

6 Proceeding o the 3 IAVSD conerence, Atugi, Jaan, Augut 3. much lower. Thi undamental erormance limit arie rom low tire daming, and thu deign change uch a uing a larger and more owerul hydraulic actuator or a ater ervo valve will not hel. 3. DURABILITY TEST RIG Beore we dicu oible remedie to addre the negative imact imoed by the LDZ, let irt examine why thi ha not been a roblem or other hydraulic alication uch a durability tet rig. In Fig. 5, a imliied one degree-o-reedom durability tet rig i hown. The ma-ring-damer ytem hown on the to rereent the dominate mode o the teted ytem. The weight o the ground late lu the tire() i denoted a m u. It i traightorward to ind the ollowing two traner unction z m k c z u m u Fig. 5. Durability Tet Rig model. zu m c k G = = () [ mmu ( cmu mc ) mk muk ] z Cxβ AKv( m c k) u Gzi = = () ' iv D ' 4 3 where D = ( τ )[ mvm u ( cvm mvc u ) ( kvm mukv βa m ) cβa kβa ]. The zero o the lant hown in Eq.() are again zero o the dilacement ervo-loo, hown in Eq.(). Since the daming ratio o thee zero i higher than thoe o Eq.(8) (~.3 v. ~.), the imoed limitation on tracking erormance i much eaier to deal with. 4. POSSIBLE REMEDIES In thi ection, we will dicu oible remedie (deign change) to addre the advere eect o LDZ on AS ytem. Thee change aim to eliminate the LDZ, or to increae their daming ratio. Both orce and dilacement roblem are dicued. Here we are not trying to ditinguih whether the target alication i or narrow band (< 3Hz) or wide band (>Hz). Rather, we will ocu on the inluence o the working rincile o a orceroducing actuator on the ervo-loo erormance. It hould be ointed out that while LDZ roblem can be addreed by ole/zero cancellation, thi method i not conidered becaue o it oor robutne in real imlementation. 4. Alternative actuator ytem The reult reented in thi aer o ar aume a hydraulic actuator i ued. A ointed out earlier, the LDZ arie rom the uenion lant (hown in Eq.()). Thereore, 6

7 Proceeding o the 3 IAVSD conerence, Atugi, Jaan, Augut 3. modiication o the hydraulic actuator ytem (higher uly reure, larger cylinder, ater ervo valve) will not be eective. When an electromagnetic, a neumatic or a iezoelectric actuator i ued, a hyical eedback (the H() block in Fig. 3) till exit, although in dierent orm. For examle, in the voltage control mode, a DC motor will have back electromotive orce, and or neumatic actuator, volume/reure couling eect exit. The act uch hyical eedback exit imlie that LDZ are reent and oe a roblem or the control deign. When an actuator without hyical eedback i ued, the ervo-loo i no longer in the orm hown in Fig. 3, and the analyi reult dicued in the reviou two ection need to be modiied. An examle o orce actuator that do not exhibit a hyical eedback ath i electromagnetic motor running under current drive mode. Since the motor current i ulied regardle o the motor eed, the ervo-loo control ytem i indeendent o the uenion lant, which i deicted in Fig. 6. Obviouly, ince G () i outide o the ervo-loo, the LDZ no longer exit or the orce ub-loo. For the dilacement tracking roblem, however, LDZ till exit becaue G () i inide the ervo-loo. d _ C() i actuator Ga ( ) Force lant G ( ) y yd _ C() i actuator Ga ( ) Force lant G ( ) y Fig. 6. Block diagram o the orce ub-loo and dilacement ub-loo or a current-mode drive electric actuator. 4. Eect o modiied uenion arameter In thi ub-ection, we will examine the eect o modiied uenion arameter on the characteritic o the LDZ. In articular, we will ocu on their eect on the daming ratio o the LDZ. We aume that the rung ma i given and cannot be aigned. The nominal value o the uenion arameter are hown in Table. Table. Nominal value o the uenion arameter. Parameter Meaning Value Parameter Meaning Value m Srung ma 53kg m Unrung u ma c Suenion daming 4.. Dilacement control roblem Only three arameter will inluence the LDZ: c Tire daming N / m / ec u 6kg k Suenion tine N / m / ec u N / m k Tire tine 9 N / m m u, u c and k u (ee Eq.()). Aarently, c will increae the the mot imortant arameter i the tire daming c u. Increaing u daming ratio o the LDZ. However, ince thi will alo increae rolling reitance, it i quite obviou little room i let or the deigner. 4.. Force control roblem Five arameter will inluence the LDZ (numerator o Eq.(6)): c, k, m u, c u and k u. Note that there are two air o LDZ. We varied all ive arameter within a range o 7

8 K /3 K Proceeding o the 3 IAVSD conerence, Atugi, Jaan, Augut 3. their nominal value, deending on our judgment o what could be achieved in ractice. It wa ound that erturbing uenion daming ( c ) or uenion tine ( k ) are mot eective. The eect o c and k on the maller o the two daming ratio i hown in Fig. 7. The eect o c i not monotonic becaue the ater mode become le damed than the lower mode at high requency. We can increae LDZ daming by either increaing c or decreaing k. For examle, i we increae c by a actor o 7, the daming ratio will become.6, high enough that the zero no longer oe a roblem or the control deign. Smaller imrovement can be achieved by reducing k. Changing thee two arameter need to be done careully by weighing againt their advere eect reduced actuator energy eiciency and increaed rattle ace, reectively daming ratio ζ.6.4. Inital Value:348.5 daming ratio ζ.5..5 Inital Value:.e C (N/m/ec) Fig. 7. Eect o erturbing uenion daming and tine on daming o LDZ K (N/m) x 4 To tudy the eect o c, an AS eiciency i deined a η = Ewated / Etotal, where the total inut energy E total and the energy aborbed by c, Ewated are t Etotal ( t) = ( z z u ) dτ, ( z z u ) > (3) t Ewated = min{ ( z z u ), c ( z z u ) } dτ, ( z z u ) > The orce ued i commanded rom a main-loo LQ algorithm, where the control gain are re-comuted or each erturbed c. The relationhi between c and η i hown in Fig. 8(a). Increaing c reult in an almot monotonic reduction in η. I we increae c by a actor o 7, η will reduce rom 6% to %. Thi may be unaccetable becaue ower conumtion and cooling requirement. The mot obviou advere eect o a oter uenion (maller k ) i increaed rattle ace (hown in Fig. 8(b)) and body leaning during cornering. In other word, in ractice there i limited room or a control engineer to imrove the daming o LDZ by reducing k..8 C-η relationhi Rattle Sace (C=5N/m/ec).7.6 Inital Value η.5.4 Magnitude (db) C (N/m.ec) (a) Fig. 8. Eect o modiied uenion daming and tine. - Frequency (Hz) (b) 8

9 Proceeding o the 3 IAVSD conerence, Atugi, Jaan, Augut Vibration aborber A vibration aborber (VA) reer to a ingle degree o reedom ma-ring ocillator with low (no) daming. The baic idea or uing a vibration aborber i that the mode aociated with the additional energy torage buer create a air o zero at the deigned requency. Here we introduce a lightly-damed vibration aborber attached to the unrung ma (ee Fig. 9). The uenion dynamic become 4 3 z zu ma ( mu m ) ( mc a u muca cm a a cm a ) ( km km m k k m c c) ( c k k c) k k G = = d d ( m a c a ka)[( mu m ) cu ku ] (4) ma << m d where, by uing the act ma << m, ca<< c and ka << ku a a a u a u a u a u a u a u a d k c m m c k m c k k c m c m k m 3 ( a a a )( )( u u u ) ( a a a )( ) 3 ( ka ca ma )[( m c k)( mu cu ku ) km cm ] = (5) Notice that both the numerator and denominator aroximately contain a vibration mode ( ka ca ma ), which reult in a near-ole-zero cancellation or G i and G zi. Thereore, the eect o VA i localized. Simulation in Fig. how that the vibration aborber i not eective or the orce control loo. For the dilacement control roblem, the vibration aborber can reduce the advere eect o LDZ by almot 3dB (at 3Hz), by roerly tuning the VA daming c a. z z u k m u m ma c ka ca za k u c u Fig. 9. Quarter-car uenion model with vibration aborber. z o ( q y, g ) Poition Control with VA Poition Control without VA Force Control with VA Force Control without VA 5 Magnitude (db) 5-5 Fig.. Frequency reone o vibration aborber. - Frequency (Hz) 4.4 Advanced control algorithm By emloying advanced control trategie, we can cancel the hyical eedback introduced by the hydraulic actuator (or other actuator). Thereore, the eect o thi 9

10 Proceeding o the 3 IAVSD conerence, Atugi, Jaan, Augut 3. remedy i imilar to introducing a non-eedback actuator dicued in Section 4.. We can even tate that the current-mode motor control concet i a ecial cae o the advanced control baed algorithm. Inide the current-mode drive i a eedback law that cancel the back electromotive orce. Thi uing eedback control to cancel hyical eedback idea can be alied to other actuator ytem. For hydraulic actuator, we exect thi to be much harder to achieve becaue the large bulk modulu o hydraulic luid rereent a rigid couling. Fat ening o hydraulic reure and ubequent maniulation o um/valve to react to reure increae/dro i harder comared with current-mode motor drive. Thereore, thi remedy may not work well with hydraulic actuator. No matter which actuator ytem i ued, the ervo-loo control algorithm and hardware need to be deigned to work with the urge in control ignal magnitude at the reonant requency o LDZ. 5. CONCLUSION The inherited challenge and oible remedie o the ervo-loo control deign or active uenion ytem are reented. It i hown that or both orce ervo-loo and dilacement ervo-loo, we have lightly damed zero (LDZ) in the lant, which imoe erormance limit in ractical imlementation. It wa ound that recating the active uenion ervo-loo into a dilacement control roblem doe not rovide igniicant beneit. Four oible remedie were tudied dierent actuation ytem, modiied uenion arameter, vibration aborber, and advanced control algorithm. It wa ound thee remedie can either comletely eliminate the LDZ, or moderately increae their daming ratio o that the imoed erormance limit i reduced. REFERENCES. Hrovat, D., (997) Survey o Advanced Suenion Develoment and Related Otimal Control Alication, Automatica, v 33 n, Alleyne, A., Liu, R., (999) On the limitation o orce tracking control or hydraulic ervo ytem, Journal o Dynamic Sytem, Meaurement and Control, Tran. o the ASME, Vol., n., 999, Alleyne, A., Hedrick, J.K., (995) Nonlinear Adative Control o Active Suenion, IEEE Tran. on Control Sytem Technology, Vol. 3, n., March, Goran, M. B., Bachrach, B. I., Smith, R. E., (99) The Deign and Develoment o a Broad Bandwidth Active Suenion Concet Car, ImechE 99, Rajamani, R., Hedrick, J.K., (994) Perormance o Active Automotive Suenion with Hydraulic Actuator: Theory and Exeriment, Proc. o the American Control Conerence, June 994, Peng, H., Strathearn, R., Uloy, A.G., "A Novel Active Suenion Deign Technique--Simulation and Exerimental Reult," Proceeding o the 997 American Control Conerence, Albuquerque, New Mexico, June 4-6, Zhang, Y., Alleyne, A., A Practical and Eective Aroach to Active Suenion Control, Proceeding o the 6th International Symoium on Advanced Vehicle Control, Hirohima, Jaan, Setember. 8. Chantranuwathana, S. and Peng, H., Practical Adative Robut controller or Active Suenion, Proceeding o the ASME International Congre and Exoition, Orlando, Florida. 9. Chantranuwathana, S., Adative Robut Force Control For Vehicle Active Suenion, Ph.D. Diertation, Univerity o Michigan,.. Skogetad, S. and Potlethwaite, I., Multivariable Feedback Control Analyi and Deign, Wiley, Morari, M., Deign o Reilient Proceing Plant III, A General Framework or the Aement o Dynamic Reilience, Chemical Engineering Science, Vol.38, No.,.88-89, Zhang, Yiheng, Alleyne, A., A New Aroach to Hal-Car Active Suenion Control, Proceeding o the 3 American Control Conerence, Denver, Colorado.