TELK OMNIKA Indonesian Journal of Electrical Engineering V ol. 12, No . 6, J une 2014, pp . 4190 4199 DOI: http://dx.doi.org/10.11591/telk omnika.v12.i6.4380 4190 Optimal Vibration Contr oller f or V ehic le Active Suspension System under Road Roughness Disturbances Shi-Y uan Han *1 , Gong-Y ou T ang 2 , Y ue-Hui Chen 1 , Guang-P eng Li 1 , and Xi-Xin Y ang 3 1 Shandong Pro vincial K e y Labor ator y of Netw or k Based Intelligent Computing , Univ ersity of Jinan, 336 W est of Nanxinzhuang Road, Jinan, 250022, China 2 College of Inf or mation Science and Engineer ing, Ocean Univ ersity of China, 238 So ngling Road, Qingdao 266100, China 3 Softw are T echnical College , Qingdao Univ ersity , 308 Ningxia Road, Qingdao , 266100 , China * Corresponding author , e-mail: ise hansy@ujn.edu.cn Abstract The prob lem of optimal vibr ation control f or v ehicle activ e suspension systems under road rough- ness disturbance is considered. First, the models f or tw o-deg ree-fre edom quar ter-car suspension system under road roughness disturbance are presented, and road disturbances are considered as the output of an e xosystem. Then, the f eedf orw ard and f eedbac k optimal vibr ation control (FFO VC) la w f or v ehicle activ e sus- pension systems is obtained and the e xistence and uniqueness of the FFO VC is pro v ed. A state obser v er is designed to solv e the prob lem of the ph ysically realizab le f or the f eedf orw ard compensator . Numer ical sim ulations illustr ate the eff ectiv eness of the FFO VC la w . K e yw or ds: v ehicle suspension system, roughness road disturbance , f eedf orw ard and f eedbac k control, vibr ation control, optimal control Cop yright c 2014 Institute of Ad v anced Engineering and Science . All rights reser v ed. 1. Intr oduction With the de v elopment of machines technology and inf or mation techn ology , the v ehicle suspension system underw ent three stages: passiv e [1, 2], semi-activ e [3], and activ e suspension systems [4, 5, 6], in the last f e w decades . As is w ell kno wn, v ehicle activ e suspension systems , compared with passiv e and semi-activ e suspension system, could diminish the vibr ation of the v ehicle body b y using po w er sources more eff ectiv ely . Compared to passiv e suspension system, activ e suspension system could meet the requirement more closely about dr iving saf ety , v ehicle handling, and r ide comf or t. By using po w er sources (e .g., compressors and h ydr aulic pumps) activ e suspension systems ha v e f e w er limitations on the optimization procedures , where t he sus- pension char acter istics can be adjusted while dr iving to accommodate the profile of the road[7]. By using lo w consumption elements and minimizing the required energy le v el, activ e suspension system can compensate f or the lak e of higher production consumption and tur n into more pr ac- tical ones . Recently , a consider ab le amount of theoretical and e xper imental research eff or t has been aimed at impro ving v ehicle suspension systems , such as fuzzy control [8], adaptiv e control [9], sliding mode theor y [10], and H 1 control [6, 11]. The essential elements in an y activ e v ehicle suspension design and cont rol alw a ys in- clude r ide comf or t, tire deflection, and suspension deflection. Ho w e v er , the vibr ation of v ehicle is mainly caused b y the road disturbances , which ma y result in deter ior ation of r ide comf or t, v ehi- cle handling, dr iving saf ety , and e v en str uctur al damage . Theref ore , the influence from the road disturbances m ust be considered in an y activ e suspension design. It should be noted that the road disturbances are mainly caused b y road roughness and v ar iab le v elocity . Then, the vibr a- tion control f or v ehicle activ e suspension system could be vie w ed as an optimal vibr ation prob lem where one w ould attempt to k eep the r ide comf or t, tire deflection, and suspension deflection at an acceptab le le v el [11, 12]. In order to analyz e the dynamic beha vior of a v ehicle under road Receiv ed September 12, 2013; Re vised December 21, 2013; Accepted J an uar y 13, 2014 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA ISSN: 2302-4046 4191 disturbances , road disturbances are typically considered as a r andom process with a g round dis- placement po w er spectr al densit y (PSD)[13]. Based on the char acters of road roughness and v ar iab le v elocity , the road disturbances are f or m ulated as an e xosystem in this paper . This paper in v estigates the optimal vibr ation control f or an activ e v ehicle suspension sys- tem under road disturbances . The model of activ e v ehicle suspension syste m is b uilt and the road disturbance is vie w ed as an e xosystem. Then, the or iginal vibr ation control is f or m ulated as the optimal vibr ation control f or v ehicle activ e suspension system under road disturbances . By using the optimal control theor y , the FFO VC la w of the v ehicle activ e suspension system under road disturbances is obtained, and the e xistence and uniqueness of the FFO VC is pro v ed. In order to solv e the ph ysically realizab le prob lem of the f eedf orw ard compen s a tor , a reduced-order obser v er is constr ucte d. A n umer ical e xample of the FFO VC la w f or an activ e v ehicle suspension with under road disturbances is presented to demonstr ate the eff ectiv eness of the FFO VC la w . The rest of paper is organiz ed as f ollo ws . Section 2 presents the descr iptions of the v ehicle activ e suspension system, e xosystem of the ro ad disturbance , and q uadr atic perf or mance inde x. In Section 3, the main results of this paper are presented, in which the FFO VC la w is obtained based on the optimal control theor y and the e xistence and uniqueness of the FFO VC is pro v ed. A reduced-order obser v er is constr ucted in Section 4 to solv e the ph ysically realizab le prob lem. Numer ical e xamples are giv en in Section 5 to demonstr ate the eff ectiv eness of the FFO VC la w . Finally , w e conclude our findings in Section 6. 2. Pr ob lem f orm ulation 2.1. System f orm ulation The tw o-deg ree-freedom quar ter-car activ e suspension system is sho wn in Fig.1 [11, 12]. Figure 1. The tw o-deg ree-freedom quar ter-car activ e suspension system The dynamic equation f or v ehicle activ e suspension system is descr ibed as: m s z s ( t ) + b s [ _ z s ( t ) _ z u ( t )] + k s [ z s ( t ) z u ( t )] = u ( t ) ; m u z u ( t ) + b s [ _ z u ( t ) _ z s ( t )] + k s [ z u ( t ) z s ( t )] + k t [ z u ( t ) z r ( t )] = u ( t ) ; (1) where m s is the spr ung mass; m u is the unspr ung mass; k s and b s are the stiffness and damp- ing of the passiv e v ehicle suspension system, respectiv ely; k t stands f or compressibility of the pneumatic tire; z s ( t ) and z u ( t ) are the displacements of the v ehicle spr ung mass and unspr ung masses , respectiv ely; z r ( t ) is the road displacement input; u ( t ) represents the activ e control f orce of the v ehicle suspension system, which is produced b y h ydr aulic or other shoc k absorber . Defining the f ollo wing state v ar iab les: x 1 ( t ) = z s ( t ) z u ( t ) ; x 2 ( t ) = z u ( t ) z r ( t ) ; x 3 ( t ) = _ z s ( t ) ; x 4 ( t ) = _ z u ( t ) ; (2) Vibr ation Control f or V ehicle Suspension System under Disturbances (Shi-Y uan Han) Evaluation Warning : The document was created with Spire.PDF for Python.
4192 ISSN: 2302-4046 x 1 ( t ) is th e suspension deflection, x 2 ( t ) denotes the tire deflection, x 3 ( t ) denotes the speed of spr ung mass , and x 4 ( t ) is the speed of unspr ung mass . Then, the state v ar iab le x ( t ) of the v ehicle suspension system is introduced as: x ( t ) = [ x 1 ( t ) x 2 ( t ) x 3 ( t ) x 4 ( t )] T : (3) Riding comf or t is the k e y perf or mance cr iter ia in an y v ehicle suspension system design. Ride comf or t is usually e v aluated b y the spr ung mass acceler ation z s ( t ) in th e v er tical direction, the dynamic tr a v el of suspension system usually b y the amount of suspension def ection z s ( t ) z u ( t ) and the road holding ability usually b y the tire def ection z u ( t ) z r ( t ) . In order to satisfy the requirements of perf or mance cr iter ia, the controlled output y c ( t ) is defined as: y c ( t ) = 2 4 y c 1 ( t ) y c 2 ( t ) y c 3 ( t ) 3 5 = 2 4 z s ( t ) z s ( t ) z u ( t ) z u ( t ) z r ( t ) 3 5 = 2 4 z s ( t ) x 1 ( t ) x 2 ( t ) 3 5 : (4) It is unnecessar y and uneconomical to output all of the v ar iab les , so the measured output y m ( t ) can be e xpressed b y: y m ( t ) = z s ( t ) z u ( t ) _ z s ( t ) T (5) Then, the activ e v ehicle suspension equation of a contin uous-time syst em in the state space f or m can be e xpressed b y: _ x ( t ) = Ax ( t ) + B u ( t ) + D v ( t ) y c ( t ) = C x ( t ) + E u ( t ) y m ( t ) = C x ( t ) (6) where A = 2 6 6 4 0 0 1 1 0 0 0 1 k s m s 0 b s m s b s m s k s m u k t m u b s m u b s m u 3 7 7 5 ; B = 2 6 6 4 0 0 1 m s 1 m u 3 7 7 5 ; D = 2 6 6 4 0 1 0 0 3 7 7 5 ; C = 2 4 k s m s 0 b s m s b s m s 1 0 0 0 0 1 0 0 3 5 ; E = 2 4 1 m s 0 0 3 5 ; C = 1 0 0 0 0 0 1 0 ; (7) v ( t ) = _ z r ( t ) is the road disturbance . By using g round height sensor to predict the road surf ace shape [14], the road disturbance v ( t ) could be measur ab le . 2.2. Disturbance anal ysis In order to impro v e the r ide comf or t and v ehicle oper ation, the eff ect of road disturbance m ust be considered in activ e suspension design. In gener al, vibr ations in v ehicle suspension sys- tem are caused b y the e xistence of the road disturbances . The road disturbances are typically specified as a stochastic process with a g round displacement po w er spectr al density (PSD): G d () = 8 < : G d ( 0 ) 0 n 1 ; 0 G d ( 0 ) 0 n 2 ; > 0 (8) where is a spatial frequency and it is the reciprocal of the w a v elength, which donates the w a v e TELK OMNIKA V ol. 12, No . 6, J une 2014 : 4190 4199 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA ISSN: 2302-4046 4193 n umbers per meter , it’ s dimension is m 1 . 0 = 1/2 is a ref erence frequency . n 1 and n 2 are road roughness constants , in gener al, n 1 = 2 and n 2 = 1 : 5 . In this paper , the road disturbances f or v ehicle activ e suspension systems are mainly caused b y the road roughness . Assume that the speed of the car is v 0 and the road displacement input z r ( t ) is an appro ximately per iodic function. Since the wheel and suspension systems ha v e the char acter istic of lo w pass filter ing (LPF), the road disturbances with lo w frequency are consid- ered . Theref ore , the road displacement input from the road irregular ities can be appro ximately sim ulated b y the f ollo wing finite sum of F our ier ser ies: z r ( t ) = p X j =1 j ( t ) = p X j =1 j sin( j ! 0 t + j ) ; j = 1 ; 2 ; : : : ; p; (9) where p is used to restr ict the r ange of frequency , ! 0 = 2 v 0 =l , l is the length of the road segment. j = p 2 G d ( j ) ,  = 2 =l , and the initial phase j 2 [0 ; 2 ) is a r andom v ar iab le f ollo wing a unif or m disturbance . In order to f acilitate d esign of the optimal control la w , the definition of road disturbance state v ector is w ( t ) = [ w 1 ( t ) ; ; w 2 p ( t )] T = [ 1 ( t ) ; ; p ( t ) ; _ 1 ( t ) ; ; _ p ( t )] T (10) The road disturbance v ector v ( t ) can be giv en b y: _ w ( t ) = Gw ( t ) ; v ( t ) = F w ( t ) ; (11) where G = 0 I ~ G 0 ; F = [ 0 ; ; 0 ; | {z } p 1 ; ; 1 | {z } p ] ~ G = diag ! 2 0 ; ; ( p! 0 ) 2 (12) One can see that the der iv ativ es of the system (9) and (11) are equiv alent. Noting that, the r ank of h F T ( F G ) T ( F G 2 p 1 ) T i T = 2 p , the pair ( F ; G ) is obser v ab le . 2.3. Optimal perf ormance inde x f orm ulation In vie w of the limited po w er of the actuator , a smaller activ e f orce u ( t ) f or the activ e sus- pension system should be chosen to red uce energy consumption in pr actical applications . Due to the persistent eff ect from road disturbance , the state v ector and the control v ector of the activ e suspension system will not con v erge to z ero synchronously . Theref ore , the tr aditional quadr atic optimal control perf or mance inde x will not be a v ailab le . In this case , the f ollo wing quadr atic a v er- age perf or mance inde x is chosen as: J ( u ( )) = lim T !1 1 T Z T 0 [ y T c ( t ) Q 0 y c ( t ) + u 2 ( t )] dt; (13) where Q 0 = di ag ( q 1 ; q 2 ; q 3 ) and it is positiv e definite matr ix. q i can be deter mined b y diff erent v ehicle’ s e xplicit requirements f or perf or mance inde x es of r ide comf or t, road holding ability , sus- pension deflection, and energy-sa ving in the activ e suspension system design. Substituting (6) Vibr ation Control f or V ehicle Suspension System under Disturbances (Shi-Y uan Han) Evaluation Warning : The document was created with Spire.PDF for Python.
4194 ISSN: 2302-4046 and (7) into (13), the quadr atic a v er age perf or mance inde x (13) is ref or m ulated as: J ( u ( )) = lim T !1 1 T Z T 0 [ x T ( t ) Qx ( t ) + 2 x T ( t ) N u ( t ) + R u 2 ( t )] dt; (14) where Q = C T Q 0 C = 2 6 6 6 6 4 k 2 s q 1 m 2 s + q 2 0 k s b s q 1 m 2 s k s b s q 1 m 2 s q 3 0 0 b 2 s q 1 m 2 s b 2 s q 1 m 2 s b 2 s q 1 m 2 s 3 7 7 7 7 5 ; N = C T Q 0 E = h k s q 1 m 2 s 0 b s q 1 m 2 s b s q 1 m 2 s i T ; R = q 1 m 2 s + 1 ; (15) where denoted as the symmetr y elements . Then, the prob lem of optimal vibr ation control f or v ehicle activ e suspension system is ref or m ulated to find a control la w u ( t ) f or the system (6) with respect to the perf or mance inde x (14) that mak es the perf or mance inde x (14) obtain the minim um v alue . 3. FFO VC la w By using the f ollo wing v ar iab le tr ansf or mation, u ( t ) = u ( t ) + R 1 N T x ( t ) ; (16) System (6) and the perf or mance inde x (14) are ref or m ulated to equiv alent f or ms as f ollo w ed, re- spectiv ely _ x ( t ) = ( A B R 1 N T ) x ( t ) + B u ( t ) + D v ( t ) ; (17) J ( u ( )) = lim T !1 1 T Z T 0 [ x T ( t )( Q N R 1 N T ) x ( t ) + R u 2 ( t )] dt; (18) where Q N R 1 N T = 2 6 6 6 6 4 k 2 s q 1 q 1 + m 2 s + q 2 0 k s b s q 1 q 1 + m 2 s k s b s q 1 q 1 + m 2 s q 3 0 0 b 2 s q 1 q 1 + m 2 s b 2 s q 1 q 1 + m 2 s b 2 s q 1 q 1 + m 2 s 3 7 7 7 7 5 (19) One can see that Q N R 1 N T is positiv e semidefinite matr ix. Let Q N R 1 N T = D T D , b y means of the Cholesky matr ix decomposition function in Matlab , w e can get: D = 2 6 6 6 6 4 q k 2 s q 1 q 1 + m 2 s + q 2 0 k s b s q 1 q 1 + m 2 s r k 2 s q 1 q 1 + m 2 s + q 2 k s b s q 1 q 1 + m 2 s r k 2 s q 1 q 1 + m 2 s + q 2 0 p q 3 0 0 0 0 q b 2 s q 1 q 2 k 2 s q 1 + q 1 q 2 + q 2 m 2 s q b 2 s q 1 q 2 k 2 s q 1 + q 1 q 2 + q 2 m 2 s 3 7 7 7 7 5 : (20) It’ s easy to v er ify that ( A B R 1 N T ; B ) is controllab le and ( D ; A B R 1 N T ) is obser v ab le . TELK OMNIKA V ol. 12, No . 6, J une 2014 : 4190 4199 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA ISSN: 2302-4046 4195 Based on abo v e analysis , w e obtain the f ollo wing results . Theorem 1: Consider the optimal vibr ation control f or v ehicle activ e suspension system (1) under the pe rsistent eff ect (1) from road disturbance v ector with respect to the quadr atic per- f or mance inde x es (13). The optimal control la w uniquely e xists and can be f or m ulated as: u ( t ) = R 1 [( B T P + N T ) x ( t ) + B T P 1 w ( t )] ; (21) where P is the unique positiv e definite solution of the Riccati matr ix equation, ( A B R 1 N T ) T P + P ( A B R 1 N T ) P B R 1 B T P + Q N R 1 N T = 0 (22) and P 1 is the unique solution of the Sylv ester matr ix equation ( A B R 1 B T P B R 1 N T ) T P 1 + P 1 G + P D F = 0 (23) Proof: According to the abo v e analysis , th e optimal vibr ation control f or v ehicle activ e suspension system (1) with respect to the perf or mance inde x (13) is equiv alent to the optimal control f or system (17) with respect to the perf or mance inde x (18). According to the optimal control theor y , the optimal control f or system (17) with respect to the perf or mance inde x (18) can induce the f ollo wing tw o-point boundar y v alue (TPBV) prob lem. Applying the necessar y condition of the optimal control, the optimal control la w can be descr ibed as: u ( t ) = R 1 [ B T ( t ) + N T x ( t )] : (24) Let ( t ) = P x ( t ) + P 1 w ( t ) : (25) By calculating the der iv ativ es of (26) and substituting the second equation of (24) and (11) into (26), one gets: _ ( t ) = P _ x ( t ) + P 1 _ w ( t ) = P [ Ax ( t ) + B u ( t ) + D F w ( t )] + P 1 Gw ( t ) = ( P A P B R 1 N T P B R 1 B T P ) x ( t ) + ( P B R 1 B T P 1 + B T P 1 ) w ( t ) : (26) By compar ing (27) and the first equation of (24), one gets: ( A B R 1 N T ) T P + P ( A B R 1 N T ) P B R 1 B T P + Q N R 1 N T x ( t )+ ( A B R 1 B T P B R 1 N T ) T P 1 + P 1 G + P D F w ( t ) = 0 (27) Since (28) can be pro v ed f or all x ( t ) and w ( t ) , w e can obtain the Riccati matr ix equation (22), the Sylv ester mat r ix equation (23) and the optimal control la w (21). Since ( A B R 1 N T ; B ; D ) is controllab le and obser v ab le , according to the linear optimal regulator theor y , P is the unique positiv e definite solution of the Riccati matr ix equation (22) and ( A B R 1 B T P B R 1 N T ) is the Hurwitz matr ix equation descr ibed as: Re[ i ( A B R 1 B T P B R 1 N T )] < 0 : (28) Vibr ation Control f or V ehicle Suspension System under Disturbances (Shi-Y uan Han) Evaluation Warning : The document was created with Spire.PDF for Python.
4196 ISSN: 2302-4046 According to (12), Re[ i ( G )] = 0 , theref ore i ( A B R 1 B T P B R 1 N T ) + j ( G ) 6 = 0 ; i = 1 ; 2 ; ; n ; j = 1 ; 2 ; ; 2 p; (29) So P 1 is the unique solution of Sylv ester matr ix equation (23)[15]. 4. Ph ysicall y realizab le pr ob lem In System (11), the f eedf orw ard compensator of the optimal vibr ation control la w u ( t ) in (21) contains state v ector w ( t ) s inf or mation. As w ( t ) itself is unkno wn and unmeasured as sho wn in Fig.2, the f eedf orw ard compensator is ph ysically unrealizab le . Fur ther more , as w e ha v e already pointed ou t in section 2.1 th at w e choose y m ( t ) in System (5) to estimate the state of the measured output, it is unnecessar y and uneconomical to output the state f eedbac k of the optimal vibr ation control la w u ( t ) . Theref ore , w e tr y to propose a state obser v er to reconstr uct system state v ector x ( t ) and disturbance state v ector w ( t ) . F or the simplicity of statement, w e constr uct the state obser v er to solv e the ph ysically realizab le prob lem of the optimal control la w . The design of the state obser v er is descr ibed as: _ z 1 ( t ) = ( A L 1 C ) z 1 ( t ) + L 1 y m ( t ) + B u ( t ) + D v ( t ) ; _ z 2 ( t ) = ( G L 2 F ) z 2 ( t ) + L 2 v ( t ) ; (30) where L 1 and L 2 are deter mined b y eigen v alues of the prescr ibed f or m ulas ( A L 1 C ) and ( G L 2 F ) , respectiv ely . Hence , w e can get the ph ysically realizab le f eedf orw ard and f eed- bac k dynamic optimal vibr ation control la w: _ z 1 ( t ) = ( A L 1 C ) z 1 ( t ) + L 1 y m ( t ) + B u ( t ) + D v ( t ) _ z 2 ( t ) = ( G L 2 F ) z 2 ( t ) + L 2 v ( t ) u ( t ) = R 1 [( B T P + N T ) z 1 ( t ) + B T P 1 z 2 ( t )] : (31) 5. Sim ulation In this section, sim ulation e xper iments are sho wn to illustr ate the eff ectiv eness of the FFO VC la w f or the activ e suspension systems . The par ameters of v ehicle activ e suspension sys- tem model are listed as [16]: the spr ung mass m s = 180 kg, the unspr ung mass m u = 25 kg, the stiffness of the activ e suspension system k s = 16000 N/m, the compressibility of the pneumatic tire k t = 190000 N/m, the damping of the activ e suspension system b s = 1000 N/m, and the di- mension of the control f orce is N . Hence , the matr ix v alues of the activ e suspension system (6) are giv en b y: A = 2 6 6 4 0 0 1 1 0 0 0 1 88 : 89 0 5 : 556 5 : 556 640 7600 40 40 3 7 7 5 ; B = 2 6 6 4 0 0 0 : 00556 0 : 04 3 7 7 5 ; D = 2 6 6 4 0 1 0 0 3 7 7 5 ; C = 2 4 88 : 89 0 5 : 556 5 : 556 1 0 0 0 0 1 0 0 3 5 ; E = 2 4 0 : 00556 0 0 3 5 : (32) In (9), v 0 = 20m = s ; l = 200 m; p = 200 are selected. The a v er age v alue of PSD is G d ( 0 ) = 64 10 6 m 3 . The compar ison betw een the road disturbance v ector v ( t ) in (9) and the estimation (11) is sho wn in Fig. 2. Assume th at spr ung mass acceler ation z s ( t ) , suspension deflection z s ( t ) z u ( t ) and tire deflection z u ( t ) z r ( t ) are of equal impor tance in r ide comf or t. So w e select q 1 = q 2 = q 3 = 10 6 in perf or mance inde x (13). TELK OMNIKA V ol. 12, No . 6, J une 2014 : 4190 4199 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA ISSN: 2302-4046 4197 The compar ison be tw een the open-loop system and the FFO VC la w f or activ e suspension system is sho wn in this paper . The cur v es of spr ung mass acceler ation, suspension deflection, and tire deflection are presented in Figs . 3-5, respectiv ely . In order to sho w clear ly the compar ison results betw een the closed-loop system and the open-loop system, the root-mean-square (RMS) v alues are compared in T ab le 1 f or spr ung mass acceler ation, suspension deflection, and tire deflection. 0 1 2 3 4 5 6 7 8 9 10 −0.1 −0.05 0 0.05 0.1 v(t) (m/s) Random Road Disturbance 0 1 2 3 4 5 6 7 8 9 10 −0.1 −0.05 0 0.05 0.1 Time (sec) The Estimation of Road Disturbance Figure 2. Displacement of road disturbance 0 1 2 3 4 5 6 7 8 9 10 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Time (sec) Z" s  (m/s 2 )  Sprung Mass Acceleration     FFOVC Law Open−Loop Figure 3. The cur v e of spr ung mass acceler ation It can be seen from Figs . 3-5 and T ab le 1 that the optimal vibr ation control la w w e ha v e designed in this ar ticle f o r v ehicle activ e suspension system is ab le to e ff ectiv ely control the spr ung mass acceler ation, suspension deflection, and tire deflection in lo w er v alues . Theref ore , the de- signed controller is efficient to impro v e the perf or mance inde x of r ide comf or t. 6. Conc lusion This paper has been concer ned with the de v elopment of optima l vibr ation control f or the v ehicle activ e suspension under road disturbances . This paper has presented that the or iginal vibr ation control is f or m ulated as the optimal vibr ation control f or v ehicle activ e suspension system under road disturbances . Another significant impro v ement is on the FFO VC la w . FFO VC la w can eliminate the negativ e eff ects of the road disturbances and maintain economical oper ation in an optimal f ashion. Vibr ation Control f or V ehicle Suspension System under Disturbances (Shi-Y uan Han) Evaluation Warning : The document was created with Spire.PDF for Python.
4198 ISSN: 2302-4046 0 1 2 3 4 5 6 7 8 9 10 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 Time (sec) Z s −Z u  (m) Suspension Deflection     FFOVC Law Open−Loop Figure 4. The cur v e of suspension deflection 0 1 2 3 4 5 6 7 8 9 10 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 Time (sec) Z u −Z r  (m) Tyre Deflection     FFOVC Law Open−Loop Figure 5. The cur v e of tire deflection Ac kno wledg ement This w or k w as par tially suppor ted b y the National Natur al Science F oundation of China under Gr ant (61074092, 61070130, 61203105, 61302090), the Natur al Science F oundation of Shandong Pro vince under Gr ant (ZR2010FM019, ZR2011FZ003, ZR2012FQ016), and the Doc- tor al F oundation of Univ ersity of Jinan under Gr ant (XBS1318). Ref erences [1] R. Alkhat ib , G. Nakhaie J azar , and M. F . Golnar aghi, “Optimal design of passiv e linear sus- pension using genetic algor ithm”, Jour nal of Sound and Vibr ation , v ol. 275, pp . 665-591, A ug 2004. [2] J . A. T amboli and S . G. Joshi, “Optim um design of a passiv e suspension system of a v ehicle T ab le 1. Compar ison of RMS v alues of perf or mance cr iter ia Coefficient z s ( m 2 =s ) z s z u ( m ) z u z r ( m ) Optimal Control 0.3022 0.0682 0.0283 Open-Loop 0.6903 0.1622 0.0719 Reduced Rate (%) 56.22 57.95 60.77 TELK OMNIKA V ol. 12, No . 6, J une 2014 : 4190 4199 Evaluation Warning : The document was created with Spire.PDF for Python.
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