Optimal PID Design for Control of Active Car Suspension System

Transcription

1 I.J. Informaton Technology and Computer Scence, 2018, 1, Publshed Onlne January 2018 n MECS ( DOI: /jtcs Optmal PID Desgn for Control of Actve Car Suspenson System O. Tolga Altnoz Ankara Unversty, Department of Electrcal and Electroncs Engneerng, Ankara, 06830, Turkey E-mal: taltnoz@ankara.edu.tr A. Egemen Ylmaz Ankara Unversty, Department of Electrcal and Electroncs Engneerng, Ankara, 06830, Turkey E-mal: aeylmaz@eng.ankara.edu.tr Receved: 13 October 2017; Accepted: 01 December 2017; Publshed: 08 January 2018 Abstract Ths research s based on the determnaton of the parameters of the PID and fractonal-order PID controllers desgned for quarter-car suspenson system. Intally, wthout consderng the actve suspenson structure, the performance of the passve suspenson system under dfferent wheel load ndex s presented by usng the transfer functon of the system. Then, by addng a wheel-load, the classcal PID controller s desgned and appled to the current controlled hydraulc actuator as a part of actve suspenson system. The parameters of ths controller are determned by three heurstc optmzaton algorthms; Partcle Swarm Optmzaton (PSO), Dfferental Evoluton (DE) and Gravtatonal Search Algorthm (GSA). As the second part of ths study after evaluatng the performance of classcal PID controller, fractonal-order PID controller s desgned and appled to the problem to mprove the performance of the classcal PID controller. Smlarly, the parameters of ths controller are also obtaned by usng the same optmzaton algorthms. In the paper, for modelng the road, nstead of snusodal (road wth hll) or random changes, a saw tooth sgnal s preferred as a relatvely harder condton. Implementaton results are showed that the performance of the fractonal-order PID controller s much better that PID controller and also nstead of relatvely complex and expensve controller, t s possble to use fractonal-order PID controller for the problem. Index Terms PID Control, Fractonal-order PID control, Partcle Swarm Optmzaton, Dfferental Evoluton, Gravtatonal Search Algorthm, suspenson system, quarter-car.. To absorb the vbraton due to the mperfect condtons of the way,. To comfort of the passengers from way condtons,. To transfers brakng force to the wheel and protect the ntegrty between wheel and car body. Smlar to ar platforms, wheeled vehcles are under three man oscllaton force as graphcally presented n Fg. 1. Saltaton: movement of the car from up to down caused from lght rough road at hgh speed Swng: movement of the front and end of the car caused from heavy rough road at relatvely slow speed (fast speed sn t suggested) Rollover: movement of the left and rght of the car, caused from turns. Each of these movements corresponds to the force on the suspenson system from ground to car body. In other words, the change at the poston of the wheel causes these oscllatons, and t s expected from the suspenson system to handle these condtons and guarantee (f t s possble) a safe and comfort travel. Swng Saltaton Rollover I. INTRODUCTION Suspenson systems are nstalled between car body and wheel to absorb the undesred vbraton whch occurs due to the road condton. Road handlng capablty of any transportaton vehcle (wheeled vehcles) s the key factor for safety and comfort of the passengers, and t has drect relaton wth suspenson system. The necessty of suspenson systems can be summarzed as: Fg.1. Three oscllaton behavor on the wheeled (road) vehcle. In a general manner, the models of the suspenson can be dvded nto three forms wth respect to the control perspectve: passve, sem-actve [1] and actve [2] suspenson systems. Passve suspenson systems are predesgned and plug-n devces such that almost all parameters are determned at the producton phase. Hence,

2 Optmal PID Desgn for Control of Actve Car Suspenson System 17 based on the road condtons, t sn't possble to obtan "robust" response from ths devce confguraton. The advantage of these systems are ther relatvely low cost, snce they generally contan less number of devces (also do not have nterconnected devces such as actuator, pneumatcs and hydraulcs related to suspenson system) when compared to other model. Sem actve suspenson systems have varable dampng force and dampng coeffcents [1], whch can be changed by usng control algorthms. These systems reman between actve and passve systems. An actve suspenson system, whch s the problem envronment of ths paper, s the enrched verson of the passve systems wth controllable actuators. These systems can able to supply energy when t s desred [3]. These actuators are controlled by usn current or voltage (based on the structure) change. Suspenson systems are dvded nto two categores as Depended Systems and Independent Systems. Depended systems ndcate a physcally connected systems. These systems do not prefer as front suspenson systems (at modern automobles) for years due to ther weght and correspondng oscllaton. But they are generally preferred as rear suspenson systems. The second group s named as ndependent suspenson system where suspensons at each wheel are ndependent from each other; only ant-roll bar s connected. The well-known ndependent suspenson system s "McPherson" developed n 1947 [4]. As the suspenson systems are dvded two parts, the correspondng models are smlarly dvded nto three groups: Quarter-car [5], half-car [6] and full-car models [7]. Quarter car model s consdered for only one wheel of the vehcle. It s assumed that the desgned suspenson s for ndependent suspenson system whch s the theme of ths paper. Half-car model s used for dependent suspenson systems, where two suspensons are consdered as one system. Full-car model s for overall car model, both front and rear suspenson systems are consdered. Ths paper organzed as two man sectons followng the ntroducton and lterature revew secton. Secton 3 s gven for explanng "whats" and "hows" of the paper. The problem (actve and passve suspenson system), soluton (PID and fractonal-order PID controllers) and used mathematcal tools (optmzaton algorthms PSO, DE and GSA) are explaned n that secton. Secton 4 s the "results" secton. In that part the performance of the controllers and mplementaton detals are presented. II. LITERATURE REVIEW The control proposals for suspenson systems are vary from classcal control approaches, robust control, optmal control, to ntellgent methods. However, among them one of the oldest and problem orented control methods s called "Prevew Control". The prevew control s based on the detecton of the road defect, and then based on these data, the control acton s formed [3]. Ths controller can be consdered as contnuous tme optmal control algorthm. Even the performance of ths controller s better than classcal controller, the necessty for extra sensors at front and rear bumper [6] makes the controller expensve for the proposed performance. Optmal control s appled to both quarter-car [8] and half car models [9]. The survey for optmal control and ts desgn s presented by Sharp and Peng [8]. Krtolca and Hrovat [9], n ther paper, they determne optmal closed-loop system egenvalues for a desred dampng raton (0.707). Model predctve control (as an adaptve control scheme) s one of the nterestng control algorthm for suspenson systems. Full-car [7] and sem-actve suspenson models [1] are evaluated as a part of ths controller. Snce the model predctve control s a relatvely tme consumng problem especally for a slow suspenson and relatvely fast car system. Canale et.al. [1] are proposed a fast model predctve control technque. They also compared ther proposed control algorthm wth the most basc controllers whch are called sky-hook (smlar to on-off controller, only two state) and clpped (state-based controller lke more stated verson of on-off controller) controllers. Snce the controller desgn dea for suspenson systems s based on the robustness of the car body, robust control s frequently appled to the problem. In general, lnear matrx nequalty (LMI) optmzaton s appled to H control algorthm for actve suspensons [5, 10, 11]. Du and Zhang [12], appled robust state feedback controller to a tme delayed suspenson system. Lke optmal control, the dea s to fnd desred closed loop poles whch gve the desred performance. Also Bououden et. al. [13] proposed a robust predctve control method, where quarter-car model s selected and suspenson system modeled wth Takag-Sugeno (T-S) fuzzy approach. Fuzzy controller s another controller approach for the problem. L et. al. [14] desgned state and output feedback controllers by convertng the problem nto optmzaton problem and solvng wth an optmzaton algorthm. In that paper, the system also s onstructed wth T-S fuzzy system and uncertantes are also consdered. Uncertantes at actuator, sprung and unsprung masses s a module that takng nto account for control desgn. L et. al. [15] desgned adaptve sldngmode control (nonlnear control algorthm) and solved the problem by usng T-S fuzzy approach. Adaptvely and fuzzy s merged on [16] and [17]. Chou et. al. [16] suggested a Fuzzy PID controller (the outputs of Fuzzy and PID controllers are summed at the controller module) and ther parameters are optmzed by usng PSO. Smlarly, at D'Amato and Vassolo paper [17], Fuzzy controller parameters optmzed by Genetc Algorthm (GA). GA also appled for actve suspenson system where parameters of the system are estmated [18, 19]. Optmzaton s an mportant tool for determnaton of the controller parameters as explaned prevously. It s also possble to optmze the parameters of the components formed the suspenson system. In [20], Sun et. al., n [21] Prabakar et. al. and n [22] Yao et. al. used optmzaton algorthm to fnd the MR damper parameters. As a dfferent perspectve, Prabakar et. al. preferred a multobjectve (non-domnated sortng genetc algorthm NSGA-II) for MR damper parameters. Multobjectve

3 18 Optmal PID Desgn for Control of Actve Car Suspenson System optmzaton s appled to fnd the controller parameters for sem-actve [23] and actve suspenson systems by estmatng the controller gan matrx of the state feedback controller [24]. PID controller as a classcal control algorthm s also appled to solve actve suspenson problem and optmzed wth heurstc algorthms. In [25], Dangor et. al. proposed a cascaded controller where the parameters of the PID controller are optmzed wth PSO [26, 27], GA [28] and DE [29] (even there are many paper for PID parameter optmzaton, ths paper compares three well-known optmzaton algorthms). The results of the paper showed that PSO and DE presents almost same performance and GA optmzed parameter controller presented the worst performance. Hence evaluaton of the results from ths paper, n ths research PID controller parameters are optmzed PSO and DE. Also to fnd a change to mprove the PID performance, Gravtatonal Search Algorthm (GSA) [30] s selected as thrd optmzaton algorthm. In ths paper frst the classcal PID controller parameters are optmzed by usng these three optmzaton algorthm. Then, to mprove the performance of the controller, fractonalorder PID controller s desgned and optmzed as the second part of the study. III. MATHEMATICAL DESCRIPTION OF THE PROBLEM AND TOOLSET In ths secton, the problem envronment, mathematcal descrpton of the problem, control algorthms and the optmzaton algorthms used for solvng the control problem are dscussed. A. Problem Defnton In ths paper, the controllers are desgned to regulate hydraulc actuator for mnmzng the dfference between reference load pattern and correspondng car body trajectory for safety and passenger comfort. Fg. 2 llustrates the actve suspenson system. Sprng and damper (wth coeffcents k 1 and b ) are placed between wheel and car body wth mass M 1. Mathematcal model of the system s presented n Eq.s 1 and 2. 1 M x b y x k y x Q 1 ( ) 1( ) (1) A 1 M y b y x k y x k u x Q 2 ( ) 1( ) 2( ) (2) A The necessary power s dstrbuted from hydraulc actuator by alterng the valve openng whch s proportonally equal to flow (Q). The flow s adjusted wth the current and modeled as frst-order system as gven Eq. 3. Q c Q k (3) The am of ths paper s to desgn controller for adjustng the dfference between road change and body poston. Therefore, ntally the effect of the change at the level of the wheel to the car poston s explaned from passve suspenson model. Then, controller desgned for actve suspenson system. B. Passve Model Evaluaton k 1 M 1 M 2 k 2 Fg.3. Approxmated Descrpton of the Passve Suspenson System. f b f y x u M1 y Q k 1 b M2 x k 2 u Fg.2. Approxmated Descrpton of the Actve Suspenson System Model. The parameters at the fgure 2 from bottom to top are; u poston change at the bottom of the wheel; the model of the road s appled from ths pont. The wheel s modeled as an unsprung mass M 2 wth a sprng k 2. Fg.4. Step Response of the Passve Suspenson System wth Respect to Dfferent Wheel Load Ratng. Fgure 3 llustrates quarter-car passve suspenson model, and Eq.s 4 and 5 gve the mathematcal

4 Optmal PID Desgn for Control of Actve Car Suspenson System 19 descrpton for ths model. Then, transfer functon s obtaned from dfferental equatons. x bx ( k k x by k y k u (4) M1 1 2) 2 1 y by k y bx k x (5) M2 2 2 Intally, mpact of the car body s nvestgated from step response of the system. For ths purpose, by consderng the wheel load ndex, varous car body weghts are appled to the transfer functon and correspondng step response plots are gven n Fg. 4. To obtan Fg. 4 only the weght of car body ( M 1 ) s changed and step responses are presented. From fgure (and also t s possble to observe from transfer functon) as the sze of the body ncreases, the performance ndcators for the transent response also decreases, whch are numercally presented n Table 1. overshoot and settlng tme of the transent response. Table 2 gves the summary of these effects. KI GC ( s) KP KDs (6) s The contrbuton of the PID controller parameters sn t accepted for fractonal-order PID controller (FOPID). However, FOPID controller has 2 more degrees flexble than classcal PID controller whch ncreases the flexblty of the controller and t s possble to better adjust the dynamc propertes of the systems. Gan:2.63 Pole: Dampng: Overshoot: 99.6% Table 1. Transent Response Propertes of the Passve Suspenson System wth Respect to Dfferent Wheel Load Ratng Wheel Load Ratng: 265kg 400kg 600kg 900kg 1320kg Rse Tme: 0,1267 0,1606 0,2003 0,2579 0,3178 Settlng Tme: 2,2203 3,6691 5,2068 7, ,4728 Overshoot: 0,5794 0,6211 0,6742 0,7165 0,7513 Peak Tme: 0,3697 0,4737 0,5684 0,6968 0,8445 (a) Gan:2.63 Pole: Dampng: Overshoot: 99.6% Even PID controller are desgned for actve suspenson system, a classcal root locus desgn by usng only a proportonal gan K s made and results are graphcally demonstrated n Fg. 5. The am of ths desgn s to show and answer the queston that Why a detaled/complex controller algorthm s needed for ths problem?. Fg.5 gves the root locus desgn of the only proportonally controlled suspenson system. Fg. 5a gves the value of the gan for crtcally stable system, and gan larger than 2.63 makes the system unstable. In contrary, Fg. 5b gves the desred dampng raton value (0.707) and correspondng gan value In other words, wth only proportonal gan t sn t possble to obtan a stable system wth a desred closed loop poles as demonstrated n Fg.5c. Hence, n ths paper, PID controllers are desgned and parameters are optmzed. In the next secton, PID controller (both classcal and fractonalorder) are explaned wth the optmzaton algorthms. C. PID and Fractonal-order PID Controller The classcal PID controller s presented n Eq. 6. Three parameters are needed to be optmzed for the desred performance. In bref, proportonal parameter (K P ) decreases the rse tme and steady state error. However, for a relatvely large steady state error ntegral term (K I ) s needed to elmnate ths error. The dsadvantage of the ntegral term s the ncrease at the overshoot. But ths ncrease and decreases s always depended on the structure of the plant. Therefore, n some cases the ncrease at overshoot and settlng tme s relatvely small. In other cases, dervatve term s needed to decrease Gan:4.09 Pole: Dampng:0.708 Overshoot: 4.3% (b) (c) Fg.5. Root Locus Plots for the Passve Suspenson System a) crtcally stable condton, b) desred dampng rato lne and c) correspondng unstable system for a desred response

5 20 Optmal PID Desgn for Control of Actve Car Suspenson System G CF ( s) K' K p P K' s I ' K s K' K For a dfferent value of λ and µ, fractonal-order PID controller becomes classcal PID controller, where for λ'=1 and µ =1 correspond to the classcal PID controller, for λ =1 and µ =0 correspond to the PI controller, for λ =0 and µ =1 correspond to the PD controller, and for λ =0 and µ =0 correspond to the proportonal controller. Table 2. PID Parameters Effects on Transent Response Propertes Closed Steady Rse Settlng Loop Overshoot State Tme tme Response Error K P Decrease Increase - Decrease d D s s ' K I Decrease Increase Increase Elmnate K D - Decrease Decrease - IV. OPTIMIZATION One of the objectves of ths paper s to present a comparatve study between well-known nature nspred optmzaton algorthms, whch are Partcle Swarm Optmzaton [27], Dfferental Evoluton [29] and Gravtatonal Search Algorthm [30]. These algorthms are brefly explaned at the subsectons gven below. The detaled nformaton related to the optmzaton algorthms and ther behavor can be read from the references at each sub-secton. Even the algorthm s changed at each mplementaton, the objectve (ftness) functon s remaned the same as mean square of the error whch s defned as dfference between reference trajectory and the output of the overall system. 1 J n n y ref y out 1 where n s the number of data collected from smulaton envronment, y ref s the reference trajectory (n other words the desred behavor of the suspenson system wth respect to the normalzed reference poston) and y out s the output of the overall system. A. Fundamentals of Partcle Swarm Optmzaton Partcle Swarm Optmzaton (PSO) s proposed by Kennedy and Eberhart n 1995 [26]. The algorthm s nspred from the nterconnected behavors of the anmal swarms. The algorthm s based on the motons of the overall populaton and members of the populaton (called partcles ). Each partcle has three propertes and the overall swarm has one. Each partcle has poston, velocty, and personal best poston whch s the record of the mnmum objectve poston of each partcle. Also swarm has record of the best partcle and ts objectve value. The algorthm begns wth the randomly ntalzaton of the partcles poston and velocty. Then by usng the poston of each partcle objectve values are 2 (7) (8) calculated. By usng the objectve value vector, the partcle whch has mnmum objectve value s recorded. At the same tme, personal best postons and ther objectve value are also saved to a vector. At the last step of the algorthm, the poston and velocty of each partcle s updated by usng the functon defned n Eq. 9. In ths functon, two optmzaton parameters, postons, personal best poston and global best solutons are appled to the functon. In other words, each partcle goes to a random poston between personal best poston and global best poston. v( k 1) v( k) c1rand1 ( pbest x( k)) c2rand 2 ( gbest x( k)) B. Fundamentals of Dfferental Evoluton Dfferental Evoluton (DE) algorthm was proposed by Storn and Prce n 1995 [29]. Smlarly, to evolutonary algorthms (lke GA [28]), DE has four operators: ntalzaton, mutaton, recombnaton and selecton. The algorthm begns wth randomly selected ntal vector (x). Then ths ntal vector appled to a mutaton operator. The mutaton operator s a functon that takes the ntal (x) vector and form a new vector (v) wth the same sze. The followng equaton (Eq. 10) corresponds one of the possble mutaton operator. v ( r1 r2 r3 k (9) k 1) x ( k) F x ( k) x ( ) (10) where three vectors (r 1, r 2 and r 3 ) are selected randomly and dfference of two vectors are multply by an optmzaton parameter F. After two vector sets (x and v) have obtaned, they appled to recombnaton operaton. From these two vector sets, only one set of vector s selected (u) by usng the optmzaton parameter (CR) and a random value from recombned set, selecton from recombned vector s made by comparng the value based on "s t smaller or larger than ths optmzaton parameter (CR)". The last operator s the selecton. Among two vector set (x and u) the vectors whch has the smallest objectve value s selected as survved to the next teraton. C. Fundamentals of Gravtatonal Search Algorthm Gravtatonal Search Algorthm (GSA) s developed by Rashed et. al. n 2009 [30]. The algorthm s based on the nteracton between masses by takng the law of gravtaton as the bass of the algorthm, the GSA method was developed. The GSA algorthm s a sequental process of four steps: ) ntalzaton of the populaton (same for three optmzaton algorthms). Postons and veloctes are assgned randomly at the begnnng of the algorthm; ) objectve functons are calculated from the poston vector. The calculated objectve functon values are stored n a vector and then best and worst objectve values are fnd and recorded.; ) update and physcal law calculatons are evaluated. The gravtatonal constant G(t), velocty V and poston X

6 Optmal PID Desgn for Control of Actve Car Suspenson System 21 vector s updated n ths step (Eq.s. 11 and 12). Each member assgned to a mass value n ths secton. The mass of members are the normalzed objectve values. By usng these masses, the forces upon on each mass (member) are calculated. The force s proporton to the acceleraton wth the mass values. At the end of ths step the acceleratons are calculated. In other words, any member wth the smallest objectve value has the largest mass. Therefore, other masses are dragged to ths mass. But also there are some larger masses and correspondng movement depends on the objectve values of masses. The poston and veloctes are updated by usng the followng equatons; v) repeat steps 2 and 3 untl the termnaton condton(s) are met. v ( t 1) rand v ( t) a ( t) (11) x ( t 1) x ( t) v ( t) (12) V. IMPLEMENTATION RESULTS AND DISCUSSION In the prevous sectons, the performance of the passve suspenson system s nvestgated wth respect to the dfferent wheel load ndex. Then to ndcate the mportance of the controller and necessty for a relatvely complex control algorthm, a smple proportonal coeffcent s appled to unty feedback system and wth the ad of root locus, the performance of the system dscussed. In the followng sectons, frst classcal PID controller s desgned for actve suspenson system. Then to mprove the performance of the PID controller, a fractonal-order PID (FOPID) controller s appled to the overall system nstead of classcal PID controller. A. Classcal PID Controller In Eq. 6, the laplace transform of the PID controller s presented. The controller has three parameters that need to determne for a desred performance. In the prevous studes [25], t was showed that the conventonal PID tunng algorthms are present undesred transent response for many cases when compared wth the optmzaton algorthms. Therefore, n ths paper three optmzaton algorthms are appled to fnd controller parameters. Fg 6 shows the comparson between one of the solutons and the open-loop (no controller) performance. In Fg. 2, actve suspenson system model s presented. The change at the level of the wheel s appled as the nput of u. In ths paper, the saw tooth nput s preferred as the change at the road. In prevous papers, step change, snusodal change and random change are consdered as road model. The reference pont n the fgures ndcates omttng the heght dfference. The car body and the top of the wheel reman at the dfferent heghts of the car (for some tme nterval the dstance becomes negatve for ths reason). However, n the fgures these dfferences are cancelled to present graphcs effcently. wthout control acton PSO optmzed PID control Fg.6. The change at the road and correspondng car body vertcal poston change wth and wthout PID controller. GSA DE PSO (a) PSO-GSA DE-GSA DE-PSO (b) Fg.7. a) The trajectory of the suspenson system for dfferent optmzaton algorthms and b) error dfference between optmzaton algorthms. Fg. 7a llustrates the trajectory of the wheel system controlled va classcal PID controller optmzed wth PSO, DE and PSO algorthms. The mnmum mean square error for DE, PSO and GSA are calculated as , and respectvely. The best objectve value s obtaned from PID controller optmzed wth DE algorthm. Even the DE gves the best result, the performances are almost the same for all optmzaton algorthms. Fg. 7a gves the trajectory for all optmzaton algorthms and Fg. 7b gves the error dfferences wth respect to the poston change. The performances of PSO and DE are almost the same but GSA presents relatvely bgger error.

7 22 Optmal PID Desgn for Control of Actve Car Suspenson System The trajectory of the wheel s graphcally llustrated on Fg. 7. Even the small overshoot occurs and results s much better for open loop response (n Fg. 6), the small overshoot and undershoot decreases the comfort of the passenger. Therefore, to mprove the performance of the controller as the next step, FOPID controller parameters are optmzed by usng the same algorthms. B. Fractonal-order PID Controller Fractonal-order PID s a non-nteger-order of the conventonal PID controller. Lke PID controller, there are PID parameters are needed to be tuned whch are K P, K I and K D. Besde them, non-nteger orders of the s- parameter are needed to be tuned. Therefore, there are fve parameters (addtonal parameters are λ and µ) are optmzed. Fg. 8 gves the performance of the FOPID controller wth optmzed parameters. The mnmum errors are obtaned for PSO, DE, and GSA are 5.61x10-3, 1.15x10-3, and 1.04x10-3. From the fgure the best performance s obtaned for PSO algorthm wth a very small overshoot. The performance of the DE almost remans the same for classcal and fractonal order PID mplementatons. However, GSA presents a much better performance and error reduced from 3.69x10-3 to 1.04x10-3. From the results, the best performance s obtaned from PSO algorthm. GSA DE PSO Fg.8. The change at the road and correspondng car body vertcal poston change wth and wthout PID controller. VI. CONCLUSION GSA DE PSO In ths study, actve suspenson system problem s solved wth optmal PID controllers. As the frst step of the study, the passve suspenson system s nvestgated and showed wth the ad of classcal control desgn method that only a complete control algorthm can able to handle to problems of the comfort of passengers and the road handlng. Then, classcal PID controller s appled to the problem. The parameters of PID controller are obtaned by three optmzaton algorthms; PSO, DE and GSA. From prevous studes t s expected that PSO and DE presents almost same performance. These fndngs are also verfed n ths study. But to mprove the mpact one more optmzaton algorthm GSA s appled and also almost same performance s reach for all optmzaton algorthms. Even the PID performances s acceptable, t s possble to mprove the obtaned performance by slghtly change the controller structure. For ths purpose, fractonal-order PID algorthm s desgned. Smlarly, parameters are optmzed by usng optmzaton algorthm. The results are more promsng but dfferences between optmzaton algorthms become clearer. The best performance s obtaned from PSO algorthm. For all mplementaton GSA presents the worst performance. The best results from FOPID controller s a very small overshoot and bgger undershoot at the hll fall moment. The results suggest that nstead of usng more complex and expansve control algorthms, t s possble to use low-cost FOPID algorthm wth a proper selecton of control parameters wthout need of any addtonal hardware. REFERENCES [1] M. Canale, M. Mlanese, and C. Novara, Sem-actve suspenson control usng "fast" model-predctve technques, Ieee Transactons on Control Systems Technology, vol. 14, pp , Nov [2] J. T. Cao, P. L, and H. H. Lu, An Interval Fuzzy Controller for Vehcle Actve Suspenson Systems, Ieee Transactons on Intellgent Transportaton Systems, vol. 11, pp , Dec [3] A. Hac, Optmal Lnear Prevew Control of Actve Vehcle Suspenson, Proceedngs of the 29th Ieee Conference on Decson and Control, Vols 1-6, pp , [4] S. Frk and M. Hller, Knematcs and Dynamcs of a Mcpherson Front Wheel Suspenson wth Elastc Rear Transverse Pvot Bearng, Zetschrft Fur Angewandte Mathematk Und Mechank, vol. 69, pp. T398-T399, [5] H. M. Solman, A. Benzaoua, and H. Yousef, Saturated robust control wth regonal pole placement and applcaton to car actve suspenson, Journal of Vbraton and Control, vol. 22, pp , Jan [6] J. Marzbanrad, G. Ahmad, H. Zohoor, and Y. Hojjat, Stochastc optmal prevew control of a vehcle suspenson, Journal of Sound and Vbraton, vol. 275, pp , Aug [7] Y. L. Hu, M. Z. Q. Chen, and Z. S. Hou, Multplexed model predctve control for actve vehcle suspensons, Internatonal Journal of Control, vol. 88, pp , Feb [8] R. S. Sharp and H. E. Peng, Vehcle dynamcs applcatons of optmal control theory, Vehcle System Dynamcs, vol. 49, pp , [9] R. Krtolca and D. Hrovat, Optmal Actve Suspenson Control Based on a Half-Car Model - an Analytcal

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