The purpose of this paper is to present a new approach in controlling an active suspension system. This approach utilized the proportional integral sliding mode control scheme. Using this type of sliding surface, the asymptotic stability of the system during sliding mode is assured compared to the conventional sliding surface. The proposed control scheme is applied in designing an automotive active suspension system for a quarter-car model and its performance is compared with the existing passive suspension system. A simulation study is performed to prove the effectiveness of this control design

Abd. Ghani, Mohd. Ruddin

Md. Sam, Yahaya

Osman, Johari Halim Shah

Jurnal Teknologi

English

the purpose of this paper to present new approach in controlling active suspension system, This approach utilized the proportional integral sliding mode control scheme. using this type of sliding surface, the asymptotic stability of the system during sliding mode is assured compared to the conventional sliding surfac

MD Sam, Yahaya

Halim Shah Osman, Johari

Ruddin Abd Ghani, Moh.

Binus University Repository

SLIDING MODE CONTROL OF ACTIVE SUSPENSION SYSTEM

Universiti Teknologi Malaysia Institutional Repository

SLIDING MODE CONTROL OF ACTIVE SUSPENSION SYSTEM 1Jurnal Teknologi, 37(D) Dis. 2002: 1–10© Universiti Teknologi MalaysiaSLIDING MODE CONTROL OF ACTIVE SUSPENSION SYSTEMYAHAYA MD. SAM1 , JOHARI HALIM SHAH OSMAN2 &MOHD. RUDDIN ABD. GHANI3Abstract. The purpose of this paper is to present a new approach in controlling an active suspensionsystem. This approach utilized the proportional integral sliding mode control scheme. Using this typeof sliding surface, the asymptotic stability of the system during sliding mode is assured compared to theconventional sliding surface. The proposed control scheme is applied in designing an automotiveactive suspension system for a quarter-car model and its performance is compared with the existingpassive suspension system. A simulation study is performed to prove the effectiveness of this controldesign.Key words: Automotive control, sliding mode control, robust control, mismatched uncertainties.Abstrak. Kertas kerja ini mempersembahkan suatu pendekatan baru dalam mengawal sistemgantungan aktif. Pendekatan yang dicadangkan ialah menggunakan skim kawalan mod gelangsarperkadaran-pengkerbedaan. Menggunakan permukaan gelangsar jenis ini, kestabilan asimptotik sistemadalah lebih pasti berbanding dengan jenis permukaan gelangsar konvensional. Skim kawalan yangdicadangkan diaplikasikan penggunaannya dalam mereka bentuk sistem gantungan aktif automotifmodel kereta suku. Keupayaan sistem kawalan yang direka bentuk akan dibandingkan dengan sistemgantungan pasif yang sedia ada. Kajian dalam bentuk penyelakuan dijalankan untuk membuktikankeberkesanan rekabentuk sistem kawalan berkenaan.Kata kunci: Kawalan automotif, kawalan mod gelangsar, kawalan tegap, ketakpastian tak terpadan.1.0 INTRODUCTIONIn the past millennium, there had been widespread interest in using advanced controltechniques to improve the performance of vehicle suspension system. Performance ofthe suspension system has been greatly increased due to increasing vehicle capabilities.Several performance characteristics have to be considered in order to achieve a goodsuspension system [1]. These characteristics deal with regulation of body movement,regulation of suspension movement and force distribution. Ideally the suspensionshould isolate the body from road disturbances and inertial disturbances associatedwith cornering and braking or acceleration. Furthermore, the suspension must beable to minimize the vertical force transmitted to the passengers for passengers comfort.These objectives can be achieved by minimizing the vertical car body acceleration.1&2 Faculty of Electical Engineering, UTM, 81310 UTM Skudai, Johor, Malaysia.3  Kolej Universiti Teknikal Kebangsaan Malaysia, 75450 Melaka, Malaysia.JT37D[1].pmd 02/16/2007, 19:541YAHAYA MD. SAM , JOHARI HALIM SHAH OSMAN & MOHD. RUDDIN ABD. GHANI2An excessive wheel travel will result in non-optimum attitude of tyre relative to theroad that will cause poor handling and adhesion. Furthermore, to maintain goodhandling characteristic, the optimum tyre-to-road contact must be maintained on fourwheels.An early design for automobile suspension systems focused on unconstrainedoptimizations for passive suspension system which indicate the desirability of lowsuspension stiffness, reduced unsprung mass, and an optimum damping ratio for thebest controllability [2]. Thus the passive suspension system, which approach optimalcharacteristics had offered an attractive choice for a vehicle suspension system andhad been widely used for passengers. However, the suspension spring and damperdo not provide energy to the suspension system and control only the motion of the carbody and wheel by limiting the suspension velocity according to the rate determinedby the designer. To overcome the above problem, active suspension systems havebeen proposed by various researchers [3–7]. Active suspension systems dynamicallyrespond to changes in the road profile because of their ability to supply energy thatcan be used to produce relative motion between the body and wheel. Typically, activesuspension systems include sensors to measure suspension variables such as bodyvelocity, suspension displacement, wheel velocity and wheel or body acceleration.An active suspension is one in which the passive components are augmented byactuators that supply additional forces. These additional forces are determined by afeedback control law using data from sensors attached to the vehicle. Various controlstrategies such as optimal state-feedback [3], backsteeping method [4], optimal state-feedback [5], fuzzy control [6] and sliding mode control [7] have been proposed in thepast years to control the active suspension system. The sliding mode control has relativelysimpler structure and it guarantees the system stability.Variable Structure Control with Sliding Mode Control (SMC) was first proposedand elaborated in early 1950’s in the Soviet Union by Emelyanov and severalcoresearchers such as Utkins and Itkis [8]. During the last two decades, significantinterest on VSC and SMC have been generated in the control research community.VSC has been applied into general design method being examined for wide spectrumof system types including nonlinear system, multi-input multi-output (MIMO) systems,discrete-time models, large-scale and infinite-dimension systems, and stochasticsystems. The most eminent feature of VSC is it is completely insensitive to parametricuncertainty and external disturbances during sliding mode [9].Essentially, VSC utilizes a high-speed switching control law to drive the nonlinearplant’s state trajectory onto a specified and user-chosen surface in the state space whichis called the sliding or switching surface, and to maintain the plant’s state trajectory onthis surface for all subsequent time. This surface is called the switching surface becauseif the state trajectory of the plant is “above” the surface, a control path has one gain anda different gain if the trajectory drops “below” the surface. During the process, thecontrol system’s structure varies from one to another, thus earning the name variableJT37D[1].pmd 02/16/2007, 19:542SLIDING MODE CONTROL OF ACTIVE SUSPENSION SYSTEM 3structure control (VSC). To emphasize the important role of the sliding mode, thecontrol is also called sliding mode control (SMC).There are two steps in the SMC design. The first step is designing a sliding surfaceso that the plant restricted to the sliding surface has a desired system response. Thismeans the state variables of the plant dynamics are constrained to satisfy another set ofequations which define the so-called switching surface. The second step is constructinga switched feedback gains necessary to drive the plant’s state trajectory to the slidingsurface. These constructions are built on the generalized Lyapunov stability theory.In this paper we will consider a control scheme that can improve further the ridecomfort and road handling of the active suspension system. The control of the activesuspension system is complicated further due to the inherent mismatched conditionpresent in the system. The mismatched condition exist in the system due to the roadprofile which acts as the disturbance input. The proposed control scheme differs fromthe previous sliding mode techniques in the sense that the sliding surface is based onthe proportional-integral (PI) sliding mode control. The additional integral in theproposed sliding surface provides one more degree of freedom and also reduce thesteady state error. It will be shown that the proposed controller would be able toovercome the mismatched condition provided a certain condition is satisfied. Acomputer simulation will be performed to demonstrate the effectiveness and robustnessof the proposed control scheme.2.0 DYNAMIC MODEL OF THE VEHICLE SUSPENSIONMost of the past active suspension designs were developed based on the quarter-carmodel as in Figure 1. The following state-space model (equation (1)) can be easilyobtained from Figure 1.−                  − − −         = + +        −          − − −         ?? ???a s a s a s saa us a us a us a us sx xx K / M C / M K / M x / Mu rx xx K / M C / M K / M C / M x / M1 12 23 34 40 1 0 1 0 00 1 00 0 0 1 0 11 0(1)where Ms and Mus are the masses of the car body and wheel respectively, xs and xw arethe displacements of car body and wheel respectively, Ka and Kt are the springcoefficients, Ca is the damper coefficient, ?r  is the road disturbance and ua is thecontrol force from the hydraulic actuator and assumed as the control input. The statevariables are defined as forms: x1 = xs – xw for suspension travel, = ?sx x2 for car bodyvelocity, x3 = xw – r for wheel deflection and = ?wx x4  for wheel velocity. Equation (1)can be written in vector-matrix form asJT37D[1].pmd 02/16/2007, 19:543YAHAYA MD. SAM , JOHARI HALIM SHAH OSMAN & MOHD. RUDDIN ABD. GHANI4( ) ( ) ( ) ( )= + +?x t Ax t Bu t f t (2)where ( )∈ℜnx t  is the state vector, ( )∈ℜmu t  is the control input, and the continuousfunction ( )f t represents the uncertainties with the mismatched condition. The followingassumptions are taken as standard:Assumption i: There exists a known posititve constant such that ( ) β≤? ?f t , where•? ?denotes the standard Euclidean norm.Assumption ii: The pair  is controllable and the input matrix has full rank.3.0 SWITCHING SURFACE AND CONTROLLER DESIGNIn this study, PI sliding surface define as follows is utilized:( ) ( ) ( ) ( )σ τ τ= − +∫tt Cx t CA CBK x d0(3)where ∈ℜmxnC  and ∈ ℜmxnK  are constant matrices. The matrix K satisfies( )λ + <A BK 0  and C is chosen so that CB is nonsingular. It is well known that if theFigure 1 A quarter-car modelCar BodyMSCar WheelMUSKaKtCaCauaxsxsxwrJT37D[1].pmd 02/16/2007, 19:554SLIDING MODE CONTROL OF ACTIVE SUSPENSION SYSTEM 5system is able to enter the sliding mode, hence ( )σ =t 0 . Therefore the equivalentcontrol, ( )equ t  can thus be obtained by letting ( )σ =? t 0 [8] i.e,( ) ( ) { } ( )σ = − + =? ?t Cx t CA CBK x t 0 (4)If the matrix C is chosen such that CB is nonsingular, this yields( ) ( ) ( ) ( )−= −equ t Kx t CB Cf t1 (5)Substituting equation (5) into system (2) gives the equivalent dynamic equation of thesystem in sliding mode as:( ) ( ) ( ) ( ){ } ( )−= + + −? nx t A BK x t I B CB C f t1 (6)Theorem 1: If( ) ( )β β−≤ = −? nF t I B CB C11 (7)the uncertain system in equation (6) is boundedly stable on the sliding surface ( )σ =t 0 .Proof: For simplicity, we let( )= +?A A BK (8)( ) ( ){ } ( )−= −? nF t I B CB C f t1 (9)and rewrite (6) as( ) ( ) ( )= +? ??x t Ax t F t (10)Let the Lyapunov function candidate for the system is chosen as( ) ( ) ( )= TV t x t Px t (11)Taking the derivative of ( )V t  and substituting equation (6) , gives( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) = + + + = − + +? ? ? ? ?? ?T T T TT T T TV t x t A P PA x t F t Px t x t PF tx t Qx t F t Px t x t PF t (12)where P is the solution of + = −? ?TA P PA Q  for a given positive definite symmetricmatrix Q. It can be shown that equation (12) can be reduced to:( ) ( ) ( ) ( )λ β= − +? minV t Q x t P x t2 12 (13)Since ( )λ >min Q 0 , consequently ( ) <?V t 0  for all t and ( )η∈ cx B , where ( )ηcBJT37D[1].pmd 02/16/2007, 19:555YAHAYA MD. SAM , JOHARI HALIM SHAH OSMAN & MOHD. RUDDIN ABD. GHANI6is the complement of the closed ball ( )ηB , centered at =x 0  with radius ( )βηλ=minPQ12 .Hence, the system is boundedly stable. Theorem 1 shows that if the condition (7) is satisfied, the system is boundedlystable during sliding mode even in the present of the mismatched condition.Remark: For the system with uncertainties satisfy the matching condition, i.e, ,then equation (6) can be reduced to [10]. Thus asymptotic stability of the systemduring sliding mode is assured.The next step is to design the control scheme that drives the state trajectories of thesystem in equation (2) onto the sliding surface and the system remains in it thereafter.For the uncertain system in equation (2) satisfying assumptions (i) and (ii), thefollowing control law is proposed:( ) ( ) ( ) ( ) ( ) ( )( )σφσσ δ− −= − + −   +tu t CB CAx t t k CBt1 1 (14)where φ ∈ℜmxn  is a positive symmetric design matrix, and are the positive constants.Theorem 2: The hitting condition of the sliding surface (3) is satisfied if( ) ( )+ ≥A BK x t f t (15)Proof: In the hitting phase ( ) ( )σ σ >T t t 0 ; using the Lyapunov function candidate( ) ( ) ( )σ σ= TV t t t12, we obtain( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ){ } ( )σ σσσ φσσ δφ σ σσ δ= = − + − − + +      ≤ − + + + − +    ? ?TTV t t tk tt CA CBK x t t Cf ttkt C A BK x t C f t tt2(16)JT37D[1].pmd 02/16/2007, 19:556SLIDING MODE CONTROL OF ACTIVE SUSPENSION SYSTEM 7It follows that ( ) <?V t 0  if condition (15) is satisfied. Thus, the hitting condition issatisfied. 4.0 SIMULATION AND DISCUSSIONThe mathematical model of the system as defined in equation (2) and the proposed PIsliding mode controller in equation (14) were simulated on computer by using theMATLAB and SIMULINK software package. Numerical values for the modelparameters are taken from [3], and are as follows:Ms = 290kg, Mus = 59 kg, Ka = 16812 N/m, Kt = 190000 N/m, Ca = 1000 N/(m/sec)Let the set of typical road disturbance be of the form( ) ( )( )π − ≤ ≤ ≤ ≤= a t / if . t t .r tcos 8 2 0 50 0.75 and 3.00 3 250 otherwisewhere a denotes the bump amplitude (see Figure 2). This type of road disturbancehas been used by [4,11] in their studies.Furthermore, the maximum travel distance of suspension travel as suggested by [4]is ±8  has been used. The controller parameters has been chosen as follows for thesimulation: K = [16754 939.14 −1.818e5 172.67], C = [200 50 20 10], φ = 1000 , k = 1 andδ = .0 001.Figure 2 Typical road disturbance – 11 cm and 5cm bumps0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 500.020.040.060.080.10.12Road disturbance - bu mpsTime (sec)Bumps height (m)Road disturbance – bumpsTimes (sec)Bumps height (m)JT37D[1].pmd 02/16/2007, 19:557YAHAYA MD. SAM , JOHARI HALIM SHAH OSMAN & MOHD. RUDDIN ABD. GHANI8In order to fulfill the objective of designing an active suspension system i.e. toincrease the ride comfort and road handling, there are two parameters to be observedin the simulations. The two parameters are the car body acceleration and wheeldeflection. Figure 3a shows the suspension travel of both the active and passivesuspension systems for comparison purposes. The result shows that the suspensiontravel within the travel limit. Figures 3b and 3c illustrate clearly how the active suspensioncan effectively absorb the vehicle vibration in comparison to the passive system. Thebody acceleration in the active suspension system is reduced significantly, whichguarantee better ride comfort. Moreover the wheel deflection is also smaller in theactive suspension system. Therefore it is concluded that the active suspension systemimproves the ride comfort while retaining the road handling characteristics, comparedto the passive suspension system.Figure 3(a) Suspension travel0 1 2 3 4 5 6 7 8 9 10-0.08-0.06-0.04-0.0200.020.040.06Suspens ion travelTime  (sec)Displacement  (m)PassiveActive Times (sec)Displacement (m)Suspension t lPas iActivJT37D[1].pmd 02/16/2007, 19:558SLIDING MODE CONTROL OF ACTIVE SUSPENSION SYSTEM 9Figure 3(c ) Wheel deflectionFigure 3(b) Body acceleration0 1 2 3 4 5 6 7 8 9 10-10-8-6-4-202468Body accelerat ionTime  (sec)Acceleration (m/s2 )PassiveA ctive 0 1 2 3 4 5 6 7 8 9 10-6-4-2024681012x 10-3 Wheel deflectionTime  (sec)Deflection (m)PassiveActive Times (sec)Times (sec)Deflection (m) accelerationAcceleration (m/s2 )Wh el d flectionactiectiJT37D[1].pmd 02/16/2007, 19:559YAHAYA MD. SAM , JOHARI HALIM SHAH OSMAN & MOHD. RUDDIN ABD. GHANI105.0 CONCLUSIONIn this paper, the PI sliding mode control technique is proposed for controlling anactive suspension system. It has been shown mathematically and through computersimulations that the proposed control scheme is capable of improving the ride comfortand road handling characteristics of the active suspension system.REFERENCES[1] Appleyard, M., and P. E. Wellstead. 1995. Active suspension : some background. Proc. Control TheoryApplications. (142): 123-128.[2] Thompson, A. G. 1971. Design of Active Suspension. Proc. Inst. Mech. Engrs. (185): 553-563.[3] Alleyne, A., and J. K. Hedrick. 1995. Nonlinear Adaptive Control of Active Suspensions. IEEE Trans.Control System Technology. (3): 94-101.[4] Lin, J. S., and I. Kanellakopoulos. 1997. Nonlinear Design of Active Suspension. IEEE Control SystemMegazine. (17): 45-59.[5] Esmailzadeh, E., and H. D. Taghirad. 1996. Active Vehicle Suspensions with Optimal State-FeedbackControl. Journal of Mechanical Engineering Science. 200(4): 1-18.[6] Yoshimura, T., K. Nakaminami, M. Kurimoto, and J. Hino. 1991. Active suspension of passengers carsusing linear and fuzzy-logic controls. Control Engineering Practice. (7): 41-47.[7] Yoshimura, T., A. Kume, M. Kurimoto, and J. Hino. 2001. Construction of an active Suspension system ofa quarter car model using the concept of sliding mode control. Journal of Sound and Vibration. (239): 187-199.[8] Itkis, U. 1976. Control System of Variable Structure. New York: Wiley.[9] Hung, J. Y., W. Gao, and J. C. Hung. 1993. Variable Structure Control: A Survey. IEEE Transaction onIndustrial Electronics. 40(1): 2-22.[10] Edward, C., and S. K. Spurgeon. 2000. Sliding Mode control: Theory and Applications. London: Taylor andFrancis.[11] D’Amato, F. J., and D. E. Viasallo. 2000. Fuzzy Control for Active Suspensions. Mechatronics. (10): 897-920.JT37D[1].pmd 02/16/2007, 19:5510

Sliding mode control of active suspension system

http://eprints.utm.my/1429/1/JT37D%5B1%5D.pdf

Sliding mode control of active suspension system

Abstract

Similar works

Full text

Available Versions

Binus University Repository

Binus University Repository

Universiti Teknologi Malaysia Institutional Repository