MODELING AND CONTROL OF THE ACTIVE SUSPENSION SYSTEM USING PROPORTIONAL INTEGRAL SLIDING MODE APPROACH

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1 Asian Journal of Conrol Vol. 7 No. pp June 5 91 MODELING AND CONROL OF HE ACIVE SUSPENSION SYSEM USING PROPORIONAL INEGRAL SLIDING MODE APPROACH Yahaya Md. Sa and Johari Hali Shah Bin Osan ABSRAC he purposes of his paper are o presen a new ehod in odeling an acive suspension syse for half-car odel in sae space for and o develop a robus sraegy in conrolling he acive suspension syse. Proporional inegral sliding ode conrol sraegy is proposed for he syse. A siulaion sudy is perfored o prove he effeciveness and robusness of he conrol approach and perforance of he conroller is copared o he linear quadraic regulaor and he exising passive suspension syse. KeyWords: Acive suspension syses sliding ode conrol isached condiion. I. INRODUCION An ideal suspension should isolae he car body fro road disurbances and inerial disurbances associaed wih cornering and aking or acceleraion [1]. Furherore he suspension us be able o iniize he verical force ransied o he passengers for passengers cofor. hese objecives can be achieved by iniizing he verical car body acceleraion. An excessive wheel ravel will resul in non-opiu aiude of yre relaive o he road ha will cause poor handling and adhesion. Furherore o ainain good handling characerisic he opiu yre-o-road conac us be ainained on four wheels. An early design for auoobile suspension syses focused on unconsrained opiizaions for passive suspension syse which indicae he desirabiliy of low suspension siffness reduced unsprung ass and an opiu daping raio for he bes conrollabiliy []. hus he passive suspension syse which approach opial characerisics had offered an aracive choice for a vehicle suspension syse and had been widely used for passengers. However he suspension spring and daper do no provide energy o he suspension syse and conrol only he oion of he car body and wheel by liiing he suspension velociy according o he rae deerined by he designer. o overcoe he above proble acive suspension syses Manuscrip received April 1 3; revised July 9 3; acceped May 4 4. he auhors are wih he Faculy of Elecrical Engineering Universiy echnology of Malaysia 8131 UM Skudai Johor Malaysia (e-ail: yahaya@fke.u.y). have been proposed by various researchers [3-5]. Acive suspension syses dynaically respond o changes in he road profile because of heir abiliy o supply energy ha can be used o produce relaive oion beween he body and wheel. ypically acive suspension syses include sensors o easure suspension variables such as body velociy suspension displaceen wheel velociy and wheel or body acceleraion. An acive suspension is one in which he passive coponens are augened by acuaors ha supply addiional forces. hese addiional forces are deerined by a feedback conrol law using daa fro sensors aached o he vehicle. Various conrol sraegies such as opial sae-feedback [35] backseeping ehod [4] fuzzy conrol [6] and sliding ode conrol [7] have been proposed in he pas years o conrol he acive suspension syse. In his paper we will consider a new odeling echnique o represen a half-car oion equaion in he sae space for. Usually only one highes degree of differenial variable exiss in he oion equaion. In he previous odeling i is difficul o represen he oion equaion ino he sae space for if ore han one highes degree of differenial variable occurred in he oion equaion. his new echnique is capable o overcoe such siuaion. hen we discuss a conrol schee ha will iprove furher he ride cofor and road handling of he acive suspension syse. he proposed conrol schee differs fro he previous sliding ode echniques in he sense ha he sliding surface is based on he proporional-inegral sliding ode conrol () sraegy. he addiional inegral in he proposed sliding surface provides one ore degree of

2 9 Asian Journal of Conrol Vol. 7 No. June 5 freedo and also reduce he seady sae error. In he convenional sliding ode he sliding ode gain is deerined solely by he desired closed loop poles. herefore he sliding surface is fully dependen on he sliding ode gain. In he he sliding surface gain is deerined by he desired closed loop gain and he design paraeer ha can be adjused o fulfill he sliding surface requireen. o deonsrae he effeciveness and robusness of he proposed conrol schee copuer siulaion was perfored and presened in his paper. x x wr b I b wr c L r x b k θ o L f k bf f r k bf c bf ff x x bf x bf II. DYNAMIC MODEL OF HE ACIVE SUSPENSION SYSEM Consider he odel of a passenger s car subjec o irregular exciaion fro a road surface as shown in Fig. 1. his odel has been used by [6] in he design of acive suspension syse. he oion equaions of he half-car odel is represened in a vecor arix for as follows: MX + SX + X = Df + Ew (1) where he sae acive conrol and exciaion vecors are respecively given by bf wr f r X= ( x x x x ) f = ( f f ) w = ( w w w w ) f f r r and he arices M S D and E respecively are given as follows: L r b / L Lfb / L Ib / L Ib / L M = wr cbf cbf c c Lc f bf Lc f bf Lc r Lc r S = cbf cbf c c kbf kbf k k Lk f bf Lk f bf Lk r Lk r = kbf kbf + k k k + kwr 1 1 Lf Lr D = and E =. k kwr w r x k wr k Fig. 1. A half car acive suspension odel. x w f I b is he ass oen of ineria for he vehicle body b is he ass for he vehicle body and wr are he ass of he fron and rear wheels respecively x c is he verical displaceen of he vehicle body a he cener of graviy x bf and x are he verical displaceens of he vehicle body a he fron and rear suspension locaions respecively x and x wr are he verical displaceens of he vehicle body a he fron and rear wheels respecively θ is he roary angle of he vehicle body a he cener of graviy f f and f r are he acive conrols a he fron and rear suspensions respecively w f and w r are he irregular exciaions fro he road surface L f and L r are he disances of he fron and rear suspension locaions respecively wih reference o he cener of graviy of he vehicle body and L f + L r = L. Define he N sae variables for he syse as x i = X and = Ẋ where i = 1 3 N/ he half-car x N i + odel in Eq. (1) can be rewrien in sae space for as follows: x () = Ax () + Bu() + B z() () i i p N N I N N where A = A p1 = M A p = M S Ap1 Ap N N B = B p = u() and z() are he conrol M D M E inpu and he disurbance inpu respecively. Appendix 1 deails he sae equaion of he half-car odel in he sae space for wih he sae variables defined as x 1 = x bf x = x x 3 = x x 4 = x x5 = x bf x6 = x x7 = x and x 8 = x wr. he sae equaion shows ha he disurbance inpu is no in phase wih he

3 Y.M. Sa and J.H.S. Osan: Modeling and Conrol of he Acive Suspension Syse Using Proporional Inegral 93 acuaor inpu i.e. rank[b] rank[b B p ] herefore he syse does no saisfying he aching condiion. Le sar he analysis wih rewriing Eq. () ino he following for x() = Ax() + Bu () + f() (3) where x() R n is he sae vecor u() R is he conrol inpu and he coninuous funcion f() represens he uncerainies wih he isached condiion i.e. rank[b f()] rank [B]. he following assupions are aken as sandard: Assupion i: here exiss a known posiive consan β such ha f() β where denoes he sandard Euclidean nor. Assupion ii: he pair (A B) is conrollable and he inpu arix B has full rank. III. SWICHING SURFACE AND CONROLLER DESIGN In his sudy we uilized he PI sliding surface define as follows: σ () = Cx() ( CA + CBK ) x( τ) dτ (4) where C R n and K R n are consan arices. he arix K saisfies λ(a + BK) < and C is chosen so ha CB is nonsingular. I is well known ha if he syse is able o ener he sliding ode hence σ() =. herefore he equivalen conrol u eq () can hus be obained by leing σ () = [9] i.e. σ () = Cx() { CA + CBK} x() = (5) If he arix C is chosen such ha CB is nonsingular his yields ueq = Kx() ( CB) C f() (6) Subsiuing Eq. (6) ino Eq. (3) gives he equivalen dynaic equaion of he syse in sliding ode as: x() = (A + BK) x() + {I B(CB) C} f() (7) n heore 1. If F() = In B(CB) C β β1 he uncerain syse in Eq. (7) is boundedly sable on he sliding surface σ() =. Proof. For sipliciy we le A = ( A+ BK) (8) F() = {I B(CB) C} f( ) (9) n and rewrie (7) as x() = Ax() + F() (1) Le he Lyapunov funcion candidae for he syse is chosen as V() = x () Px() (11) aking he derivaive of V() and subsiuing Eq. (7) gives V() = x () [A P + PA]x() + F () Px() + x () PF() = x () Qx() + F () Px() + x () PF () (1) where P is he soluion of A P+ PA = Q for a given posiive definie syeric arix Q. I can be shown ha Eq. (1) can be reduced o: V() λ (Q) x() P x() (13) in + β1 Since λ in (Q) > consequenly V() < for all and x B c (η) where B c (η) is he copleen of he closed ball β1 P B(η) cenered a x = wih radius η= λ in ( Q ). Hence he syse is boundedly sable. Reark. For syses wih uncerainies saisfying he aching condiion i.e. rank[b f()] = rank[b] Eq. (7) can be reduced o x() = (A + BK)x() [1]. hus asypoic sabiliy of he syse during sliding ode is always assured. Now he design of he conrol schee ha drives he sae rajecories of he syse in Eq. (3) ono he sliding surface σ() = and he syse reains in i hereafer is presened. For he uncerain syse in Eq. (3) saisfying assupions (i) and (ii) he following conrol law is proposed: σ() u ( ) = ( CB) [ CAx() +φσ( )] k( CB ) σ () +δ (14) where φ R is a posiive syeric design arix k and δ are posiive consans. heore. he hiing condiion of he sliding surface (4) is saisfied if A + BK x() f() (15) Proof. In he hiing phase σ ()σ() >. Using he 1 Lyapunov funcion candidae V() = σ () σ() i can be shown ha:

4 94 Asian Journal of Conrol Vol. 7 No. June 5 V() =σ () σ () kσ() =σ () ( CA + CBK) x() φσ() + C f() σ ( ) +δ k φ + σ() σ ( ) +δ + { C A+ BK x() C f() } σ() (16) Heigh () Road profile: 5c and 11 c bups I follows ha V() < if condiion (15) is saisfied. hus he hiing condiion is saisfied. IV. SIMULAION AND DISCUSSION Fig.. ypical road disurbance. he aheaical odel of he syse as defined in Eq. (3) and he proposed proporional inegral sliding ode conroller () in Eq. (14) were siulaed on copuer. For coparison purposes he perforance of he is copared o he linear quadraic regulaor () conrol approach. We assue a quadraic perforance index is in he for of: ( () () () ()) 1 J = x Qx + u Ru d (17) where Q is a syeric posiive sei-definie arix and R is a syeric posiive definie arix. hen he opial linear feedback conrol law is obained as u = Kx () (18) where K is he designed arix gain. he nuerical values for he odel paraeers are aken fro [6] and are as follows: b = 43kg I b = 6kg = 3kg wr = 5kg k bf = 1kN/ k = kN/ k = k wr = 15kN/ c bf = 5N/(/s) c = 4N/(/s) L f =.871 L r = v = /s. In he sudy he following ypical road disurbance is used: where a denoes he bup apliude (see Fig. ). his ype a(1 cos(8 π)) / if.5.75 and w () = oherwise of road disurbance has been used by [411] in heir sudies. Furherore he axiu ravel disance of he suspension ravel used is ±8c as suggesed by [4]. In he design of he conroller he weighing arices Q and R are seleced as follows: Q = diag[q 1 q q 3 q 4 q 5 q 6 q 7 q 8 ] where q 1 = q = q 3 = q 4 = q 5 = q 6 = q 7 = q 8 = 1 and R = diag(r 1 r ) where r 1 = r = 1 1. hus he designed gains for he conroller for he fron and rear suspensions can be calculaed as k 1f = 49.8 k 1r = 3.8 k f =.4 k r = 1.4 k 3f =.7 k 3r = 74.6 k 4f = 4. k 4r = k 5f = k 5r = 8 k 6f = 9.3 k 6r = 6.1 k 7f = 69.1 k 7r = 6.1 k 8f = 3. and k 8r = he values of he arix K used for he is siilar o he values calculaed for he conroller i.e K = and hus λ ( A+ BK ) = { ± j.87.5 ±.69}. In his siulaion he following values are seleced for he : C = φ = diag [1 1] k 1 = k = 1 and δ 1 = δ = 1. Figures 3(a) and 3(b) show ha he sae rajecories slide ono he sliding anifold and reains in i hereafer. herefore he hiing condiion as in heore is saisfied. he conroller inpu signals ha drives he sae rajecories ono he sliding anifold are shown in Figs. 4(b) and 4(c). he resuls iply ha he proposed conrol sraegy is robus o he isach proble inheren in he syse. In order o fulfill he objecive of designing an acive suspension syse i.e. o increase he ride cofor and road handling here are wo paraeers o be observed in he siulaions. he wo paraeers are he car body acceleraion and he wheel deflecion. Figures 5(a) and 5(b) show he ravel disance for he fron and rear suspensions for an acive suspension syse using boh conrollers and a passive suspension syse for coparison purposes. he resuls show ha he suspension ravel wihin he ravel

5 Y.M. Sa and J.H.S. Osan: Modeling and Conrol of he Acive Suspension Syse Using Proporional Inegral 95.4 Sliding surface (fron).3 Sliding surface (rear)...1 Sliding surface Sliding surface Fig. 3(a). Sliding surface for fron Fig. 3(b). Sliding surface for rear. Inpu signal (fron) 15 Inpu signal (rear) Force (N) Force (N) Fig. 4(a). Inpu signal for fron Fig. 4(b). Inpu signal for rear.. Suspension ravel (fron). Suspension ravel (rear) ravel () ravel () Fig. 5(a). ravel disance for fron suspension Fig. 5(b). ravel disance for rear suspension.

6 96 Asian Journal of Conrol Vol. 7 No. June 5.3 Wheel deflecion (fron).3 Wheel deflecion (rear) Deflecion () -.1 Deflecion () Fig. 6(a). Deflecion of fron wheel Fig. 6(a). Deflecion of rear wheel..8 Body acceleraion (fron) 1.5 Body acceleraion (rear) Acceleraion (/s ) Acceleraion (/s ) Fig. 7(a). Acceleraion of fron car body Fig. 7(b). Acceleraion for rear car body. lii i.e. ±8c and he resuls also show ha he acive suspension uilizing he echnique perfor beer as copared o he ohers. Figures 6(a) and 6(b) illusrae clearly how he can effecively increase he yre o road conac in coparisons o he ehod and he passive syse. Increase he on road conac will direcly iprove he vehicle handling which can avoid he car fro skidding. Moreover he body acceleraion for he half-car odel using he proposed echnique as in Figs. 7(a) and 7(b) is slighly reduced his guaranees beer ride cofor. herefore i is concluded ha he acive suspension syse wih he sraegy will iprove he ride cofor while reaining he road handling characerisics as copared o he ehod and he passive suspension syse. V. CONCLUSION he paper presens a new ehod o ransfor he oion equaion ino he sae space for. Furherore he paper also presens a robus sraegy in designing a conroller for an acive suspension syse which is based on he variable srucure conrol heory. he proposed conroller is capable of saisfying all he pre-assigned design requireens wihin he acuaors liiaion. A deailed sudy of he proporional inegral sliding ode conrol algorih is presened. he perforance characerisics and robusness of he acive suspension syse under he proposed ehod are evaluaed and copared wih an conrol sraegy and a passive suspension syse. he resul shows ha he use of he proposed proporional ine-

7 Y.M. Sa and J.H.S. Osan: Modeling and Conrol of he Acive Suspension Syse Using Proporional Inegral 97 gral sliding ode conrol echnique proved o be effecive in conrolling he vehicle and is ore robus as copared o he linear quadraic regulaor ehod and he passive suspension syse. REFERENCES 1. Appleyard M. and P.E. Wellsead Acive Suspension: Soe Background Proc. Conr. heory App. Vol. 14 pp (1995).. hopson A.G. Design of Acive Suspension Proc. Ins. Mech. Engrs Vol. 185 pp (1971). 3. Alleyne A. and J.K. Hedrick Nonlinear Adapive Conrol of Acive Suspensions IEEE rans. Conr. Sys. echnol. Vol. 3 pp (1997). 4. Lin J.S. and I. Kanellakopoulos Nonlinear Design of Acive Suspension IEEE Conr. Sys. Mag. Vol. 17 pp (1997). 5. Esailzadeh E. and H.D. aghirad Acive Vehicle Suspensions wih Opial Sae-Feedback Conrol J. Mech. Sci. pp (1996). 6. Yoshiura. K. Nakainai M. Kurioo and J. Hino Acive Suspension of Passengers Cars using Linear and Fuzzy-Logic Conrols Conr. Eng. Prac. Vol. 7 pp (1991). 7. Yoshiura. A. Kue M. Kurioo and J. Hino Consrucion of an Acive Suspension Syse of a Quarer Car Model using he Concep of Sliding Mode Conrol J. Sound Vib. Vol. 39 pp (1). 8. Osan J.H.S. Decenralized and Hierarchical Conrol of Robo Manipulaors Ph.D. Disseraion Ciy Universiy London U.K. (1991). 9. Ikis U. Conrol Syse of Variable Srucure Wiley New York U.S.A. (1976). 1. Edwards C. and S.K. Spurgeon Sliding Mode Conrol: heory and Applicaions aylor and Francis London U.K. (1998). 11. D Aao F.J. and D.E. Viasallo Fuzzy Conrol for Acive Suspensions Mecharonics Vol. 1 pp (). Yahaya Md. Sa received he B.E. degree in Elecrical Engineering fro Universiy echnology of Malaysia in 1986 M.Sc. degree in Conrol Syses Engineering fro Sheffield Universiy Unied Kingdo in 1988 and he Ph.D. degree in Conrol Engineering fro Universiy echnology of Malaysia in 4. He is currenly an Associae Professor wih he Faculy of Elecrical Engineering Universiy echnology of Malaysia. His research ineress include an opial conrol robus conrol and sliding ode conrol and applicaion of hese ideas o he auooive syses. Johari Hali Shah Bin Osan obained his Bachelor of Science degree in Physics and Maser of Science Degree in Elecrical Engineering fro Souhern Illinois Universiy Carbondale Illinois USA in 1983 and 1985 respecively. He received his Ph.D. degree in Conrol Engineering in 1991 fro Ciy Universiy London UK. Currenly he is aached o he Fakuli Kejurueraan Elekrik Universii eknologi Malaysia as an acadeic saff since 1986 where he eaches Conrol Engineering heory Digial Conrol heory and Roboics. His research ineress include Roboics Robus Conrol of Uncerain Syses Adapive Conrol Variable Srucure Conrol heory and Conrol of Large Scale Syses.

8 98 Asian Journal of Conrol Vol. 7 No. June 5 APPENDIX A L LL f L LLL f r w f + O I b( Lr + Lf ) Ib( Lr + Lf ) b( Lf + Lr) Ib( Lf + Lr) f f w x = x+ + f Ap1 Ap fr k wr wr L LLL f r L Lr L + b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) k wr wr wr where O = 1 1 I = 1 1 A p1 Lkbf Lf Lkbf Lkbf Lf Lk Lk Lf LLr k Lk Lf LLr k + + b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) kbf ( kbf + k ) = Lkbf Lf Lr Lkbf Lkbf Lf Lr Lkbf Lk Lr Lk Lk Lr Lk + + b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) k ( k + kwr ) wr wr and A p Lcbf LLc f bf Lcbf LLc f Lc LLLc f r Lc LLLc f r + + b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) cbf c bf = Lcbf Lf Lr Lcbf Lcbf Lf Lr Lcbf Lc Lr Lc Lc Lr Lc + + b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) b( Lf + Lr) Ib( Lf + Lr) c c wr wr