A Sliding Mode (SM) Block Control is proposed to control an Antilock Brake System (ABS). The control problem is to achieve reference tracking for the slip rate, such that, the friction between tyre and road surface is good enough to control the car. The closed-loop system is robust in presence of matched and unmatched perturbations. To show the performance of the proposed control strategy, a simulation study is carried on, where results show good behavior of the ABS under variations in the road friction.CINVESTAVUniversidad de Guadalajar

Galicia, Marcos

Loukianov, Alexander

Rivera, Jorge

Sánchez-Torres, Juan D.

Repositorio Institucional del ITESO

Instituto Tecnológico y de Estudios Superiores de Occidente 2010-09-08 Sliding mode control for Antilock BrakeSystem Galicia, Marcos; Loukianov, Alexander; Rivera, Jorge; Sánchez-Torres,Juan D. M. I. Galicia, J. D. Sánchez, A. G. Loukianov and J. Rivera, "Sliding mode control for Antilock BrakeSystem," Electrical Engineering Computing Science and Automatic Control (CCE), 2010 7thInternational Conference on, 2010, pp. 87-92.  Enlace directo al documento: http://hdl.handle.net/11117/3328 Este documento obtenido del Repositorio Institucional del Instituto Tecnológico y de Estudios Superiores deOccidente se pone a disposición general bajo los términos y condiciones de la siguiente licencia:http://quijote.biblio.iteso.mx/licencias/CC-BY-NC-2.5-MX.pdf (El documento empieza en la siguiente página)Repositorio Institucional del ITESO rei.iteso.mxDepartamento de Matemáticas y Física DMAF - Artículos y ponencias con arbitrajeSliding Mode Control for Antilock Brake SystemMarcos I. Galicia∗, Juan Diego Sa´nchez∗, Alexander G. Loukianov∗ and Jorge Rivera†∗Departmento de Control Automa´tico, CINVESTAV-IPN Unidad Guadalajara, Zapopan, Jalisco, 45015 Me´xicoEmail: [mgalicia, dsanchez, louk]@gdl.cinvestav.mx†Centro Universitario de Ciencias Exactas e Ingenierı´as de la Universidad de Guadalajara, Guadalajara, Jalisco, 44430 Me´xicoEmail: jorge.rivera@cucei.udg.mxAbstract—A Sliding Mode (SM) Block Control is proposed tocontrol an Antilock Brake System (ABS). The control problemis to achieve reference tracking for the slip rate, such that, thefriction between tyre and road surface is good enough to controlthe car. The closed-loop system is robust in presence of matchedand unmatched perturbations. To show the performance of theproposed control strategy, a simulation study is carried on,where results show good behavior of the ABS under variationsin the road friction.Keywords—Brake Control, Antilock Braking Systems (ABS),Sliding Mode Control, Automotive Control.I. INTRODUCTIONThe ABS control problem consists in imposing a desiredvehicle motion and as a consequence, provides adequatevehicle stability. The main difficulty arising in the ABSdesign is due to its high nonlinearities and uncertaintiespresented in the mathematical model. Therefore, the ABShas become an attractive research area in nonlinear systemscontrol framework. There are several works reported in theliterature using the sliding mode technique [1], [2], [3], [4],[5]. In this work we design a new controller on the basisof sliding mode (SM) [6]. In order to achieve robustnessto matched, and unmatched perturbations, and ensure outputtracking. Theorically, this SM control can guarantee therobustness of the system through the entire response startingfrom the initial time instance. In spite of the mentioned aboveworks we consider a real situation: the control input can takeonly two values ”0” or ”1” that corresponds to the controlvalve position.The work is organized as follows. The mathematical modelfor the longitudinal movement of a vehicle, including thebrake system is presented in Section 2. In Section 3 aSM controller for ABS is designed. The simulation resultsare presented in Section 4 to verify the robustness andperformance of the proposed control strategy. Finally, someconclusions are presented in Section 5.II. MATHEMATICAL MODELIn this section, the dynamic model of a vehicle is showed.Here we use a quarter of vehicle model, this model considersthe pneumatic brake system, the wheel motion and the vehiclemotion. We study the task of controlling the wheels rotation,such that, the longitudinal force due to the contact of thewheel with the road, is near from the maximum value in theperiod of time valid for the model. This effect is reached asa result of the ABS valve throttling.Fig. 1. Pneumatic brake systemA. Pneumatic brake system equationsThe specific configuration of this system considers brakedisks, which hold the wheels, as a result of the increment ofthe air pressure in the brake cylinder (Fig. 1). The entranceof the air trough the pipes from the central reservoir andthe expulsion from the brake cylinder to the atmosphere isregulated by a common valve. This valve allows only onepipe to be open, when 1 is open 2 is closed and vice versa.The time response of the valve is considered small, comparedwith the time constant of the pneumatic system.Lets consider Figure 1, we suppose the brake torque Tb isproportional to the pressure Pb in the brake cylinderTb = kbPb (1)with kb > 0. For the brake system we use an approximatedmodel of pressure changes in the brake cylinder due to theopening of the valve with a first order relation [7], thisrelation can be represented asτdPbdt+ Pb = Pcu (2)where τ is the time constant of the pipelines, Pc is thepressure inside the central reservoir, u is the valve inputsignal. We suppose that opening and closing of the valve ismomentary and the parameters of the equation (2) are givenby the following rules:• When pipe 1 is opened and 2 is closed then u = 1 andτ = Tin• When pipe 2 is opened and 1 is closed then u = 0 andτ = Toutwhen pipe 2 is open the pressure into the brake cylinderis the atmospheric pressure Pa which is considered equal tozero.B. Wheel motion equationsTo describe the wheels motion we will use a partialmathematical model of the dynamic system [8], [9], [10] and[11].bNm = mgf = µNmφ (s)VTb ωFig. 2. Wheel forces and torquesConsider Fig. 2, the dynamics of the angular momentumchange relative to the rotation axis are given byJdωdt= rf − Tb (3)where ω is the wheel angular velocity, J is the wheelinertia moment, r is the wheel radius and f is the contactforce of the wheel.The expression for longitudinal component of the contactforce in the motion plane isf = νNmφ (s) (4)where ν is the nominal friction coefficient between thewheel and the road, Nm is the normal reaction force in thewheelNm = mgwith m equal to the mass supported by the wheel and gis the gravity acceleration. The function φ(s) represents afriction/slip characteristic relation between the tyre and roadsurface. Here, we use the Pacejka model [12], defined asfollowsφ (s) =D sin (C arctan (Bs− E (Bs− arctan (Bs))))(5)in general, this model produces a good approximation ofthe tyre/road friction interface. With the following parametersB = 10, C = 1.9, D = 1 and E = 0.97 that functionrepresents the friction relation under a dry surface condition.A plot of this function is shown in Fig. 30 0.2 0.4 0.6 0.8 1 1.200.10.20.30.40.50.60.70.80.91sφ(s)Fig. 3. Characteristic function φ (s)The slip rate s is defined ass =V − rωV(6)where V is the longitudinal velocity of the wheel masscenter. The equations (3)-(5) characterize the wheel motion.C. The vehicle motion equationThe vehicle longitudinal dynamics without lateral motionconsidered are represented asMdVdt= −F − Fa (7)where M is the vehicle mass; Fa is the aerodynamic dragforce, which is proportional to the vehicle velocity and isdefined asFa =12ρCdAf (V + Vw)2where ρ is the air density, Cd is the aerodynamic coef-ficient, Af is the frontal area of vehicle, Vw is the windvelocity; the contact force of the vehicle F is modeled ofthe formF = µNMwhere NM is the normal reaction force of the vehicle,NM = Mg with M equal to the vehicle mass and µ =νφ (s).The dynamic equations of the whole system (2)-(7) can berewritten using the state variablesx = [x1, x2, x3]T = [ω, Pb, V ]Twith initial conditions x0 = x(0) results the followingform:x˙1 = a1f − a2x2x˙2 = −a3x2 + bux˙3 = −a4F − f3(x3)(8)with outputy = s = h(x) = 1− rx1x3where a1 = r/J , a2 = kb/J , a3 = 1/τ , a4 = −1/M , b =Pc/τ and f3(x3) = d1 (x3 + Vw)2 with d1 = 12M (ρCdAf ).III. SLIDING MODES CONTROL FOR ABSBased on system (8) the considered problem is to design anSliding Mode Block controller that obtains reference trackingin despite of the perturbations in the system. Define s∗ as thedesired trajectory of the relative slip, which must maximizethe function φ(s).Throughout the development of the controller, we willuse the following assumption:A1) All the state variables are available for measurement.A. Control DesignLet s∗ (t) be a twice differentiable function, but withunknown derivatives, now we define the output tracking erroras e1 , s− s∗ then its derivative ise˙1 = c1(x) + c2(x)x2 + fy(x)− s˙∗ (9)wherec1(x) , c1 = −r(a1x3f + a4Fx1x23)c2(x) , c2 = ra2x3fy(x) , fy = −rf3(x3)x1x3+∆(ν)fy(x) will be considered as an unmatched and boundedperturbation term‖fy (x, t) ‖ < β <∞The term ∆(ν) contains the variations of the frictionparameter ν.Considering the variable x2 as virtual control in (9) wedeterminate the desired value x2ref asx2ref = −1c2[c1 + k0e0 + k1e1] (10)where k0 > 0, k1 > 0 and e0 is the integral of the trackingerror e1 that ise˙0 = e1 (11)The variable x2ref is used to put the desired dynamic fore1 and obtain the control aim. Now we define a new errorvariable e2 , α2(x, t) in the forme2 = x2ref − x2 (12)Using (8) and (10), straightforward calculations reveale˙2 = −a3e2 − bu+ f2e(x) (13)wheref2e(x) = a3x2ref +∂α2(x, t)∂x1x˙1 + · · ·+∂α2(x, t)∂x3x˙3To induce sliding mode on the sliding manifold e2 = 0 wechoose the control signal asu = 0.5sign (e2) + 0.5 (14)B. Stability analysisUsing the new variables e0, e1 and e2 the extended closedloop system (9), (11) and (13) is presented ase˙0 = e1 (15)e˙1 = −k0e0 − k1e1 + c2e2 + g1(x, t) (16)e˙2 = −a3e2 + f2e(x) − 0.5bsign (e2)− 0.5b (17)with g1(x, t) = fy(x) − s˙∗.The stability of (16) - (17) can be is studied step by step:A) SM stability of the projection motion (17);B) SM stability of the projection motion (15)-(16);We use the following assumptions:|g1(x, t)| 6 α1 |e1|+ β1 (18)|f2e(x)| 6 α2 |e2|+ β2 (19)with α1 > 0, α2 > 0, β1 > 0, β2 > 0, a3 > α2 andb > |f2e(x)| .A) The system (17) can be presented as follows:CASE 1, e2 < 0, thene˙2 = −a3e2 + f2e(x) (20)CASE 2, e2 > 0, thene˙2 = −a3e2 + f2e(x) − b (21)we use the Lyapunov candidate function V2 = 12e22to analyzethe stability conditions. The derivative of V2 with respect totime in Case 1 isV˙2 = e2 (−a3e2 + f2e(x)) (22)under condition (19) we haveV˙2 ≤ |e2| (−a3 |e2|+ α2 |e2|+ β2)In this case, the solution of (17) is ultimately bounded by[13]|e2(t)| 6 δ0, δ0 =β2a3 − α2(23)that is similar in case 2.B) To analyze stability of the sliding mode equations(15)-(16) with e2 = 0, that system can be regarded as alinear system with nonvanishing perturbation in the form:ξ˙ = Aξ +D(ξ) (24)whereξ =[e1 e2]T;A =[0 1−k0 −k1];D =[0g1(t)]Now we use the following Lyapunov candidate function:V1 =12ξTPξ (25)with P positive definite. With the correct selection of theelements k0 and k1 the matrix A is Hurwitz, then exists oneunique solution (P > 0) to the Lyapunov equationATP + PA = −Qwhere Q = QT , Q > 0.Lyapunov equation satisfies:λmin(P ) ‖ξ‖22≤ ξTPξ ≤ λmax(P ) ‖ξ‖22(26)∂V1∂ξAξ = −ξTQξ ≤ −λmin(Q) ‖ξ‖22and the perturbation term is bounded by ‖D(ξ)‖ ≤α1 ‖ξ‖2 + β1.Derivating (25) we obtainV˙1 = −ξTQξ − 2ξTPD(ξ) (27)substituting the bounds (26) in (27), we haveV˙1 = −ξTQξ − 2ξTPD(ξ)≤ −λmin(Q) ‖ξ‖22+ 2λmax(P ) ‖ξ‖2 (α1 ‖ξ‖2 + β1)≤ (−λmin(Q) + 2α1λmax(P )) ‖ξ‖22+ 2β1λmax(P ) ‖ξ‖2= −α (1− θ) ‖ξ‖22− αθ ‖ξ‖22+ β ‖ξ‖2where α = λmin(Q)− 2α1λmax(P ) and β = 2β1λmax(P ),thenV˙1 ≤ −α (1− θ) ‖ξ‖22(28)for ∀ ‖ξ‖2> βαθ= δ.Thus, the nominal system ξ˙ = Aξ has an exponentially stableequilibrium point ξ = 0, the solution ξ(t) of (24) is ultimatelybounded and the ultimate bound is given by‖ξ‖2≤ δ√λmax(P )√λmin(P )(29)Finally, considering the absolute value of the wind speed in(8), the remaining dynamics x3 is locally stable.IV. SIMULATION RESULTSTo show the effectiveness of the proposed control law,simulations have been carried out on one wheel model designexample, the system parameters used are listed in Table 1.TABLE 1Values of ParametersParameter Value Parameter ValueAf 6.6 Vw -6Pc 8 v 0.5M 1800 B 10J 18.9 C 1.9R 0.535 D 1m 450 E 0.97ρ 1.225 g 9.81Cd 0.65 Pa 0In order to maximize the friction force, we suppose thatslip tracks a constant signal during the simulations.yref = 0.203which produces a value close to the maximum of thefunction φ(s). The parameters used in the control law arek0 = 700 and k1 = 120.On the other hand, to show robustness property of thecontrol algorithm in presence of parametric variations weintroduce a change of the friction coeficient ν which producesdiferent contact forces, namely Fˆ and fˆ . Then, ν = 0.5 fort < 1 s, ν = 0.52 for t ∈ [1, 2.5) s, and ν = 0.5 for t ≥ 2.5s. It is worth mentioning that just the nominal values wereconsidered in the control design.In Figure 4 the slip performance trough the simulation isshowed, Figure 5 shows the friction function behavior φ(s)during the braking process0 1 2 3 400.10.20.3t [s]SlipFig. 4. Slip performance in the braking process0 1 2 3 40.90.9511.051.1t [s]φ(t)Fig. 5. Performance of φ(s) in the braking processwhile Figures 6 and 7 summarize the behavior of the errorvariables e1 and e2 respectively.0 1 2 3 4−0.1−0.0500.050.1t [s]e1Fig. 6. Tracking error e1 = s− s∗0 1 2 3 4050300600900t [s]e2Fig. 7. Error variable e2 = x2ref − x20 1 2 3 40510152025t [s]V,rωFig. 8. Longitudinal speed V (solid) and linear wheel speed rω (dashed)In Figure 8 the longitudinal speed V and the linear wheelspeed rω are showed; it is worth noting that the slip controllershould be turn off when the longitudinal speed V is close tozero. Figure 9 the control action is shown.0 1 2 3 4−0.5011.5t [s]uFig. 9. Control input uFinally, in Figure 10 the nominal F , and the Fˆ contactforce are shown.0 1 2 3 488008900902091009200t [s]F,F∗Fig. 10. Nominal F (solid) and Fˆ (dashed) vehicle contact forcesV. CONCLUSIONIn this work an sliding mode block control for ABS hasbeen proposed. The simulation results show good perfor-mance and robustness of the closed-loop system in presenceof both the matched and unmatched perturbations, namely,parametric variations and neglected dynamics.REFERENCES[1] H. Tan and Y. Chin, “Vehicle traction control: variable structure controlapproach,” Journal of Dynamic Systems, measurement and Control,vol. 113, pp. 223–230, 1991.[2] S. Drakunov, U. Ozguner, P. Dix, and B. Ashrafi, “ABS control usingoptimum search via sliding modes,” IEEE Transactions on ControlSystems Technology, vol. 3, no. 1, pp. 79–85, 1995.[3] C. Unsal and P. Kachroo, “Sliding mode measurement feedback controlfor antilock braking systems,” IEEE Transactions on Control SystemsTechnology, vol. 7, no. 2, pp. 271–278, 1999.[4] A. Hadri, J. Cadiou, and N. MSirdi, “Adaptive sliding mode controlfor vehicle traction,” in IFAC World Congress, Barselonna, Spain, July22-26 2002.[5] W. Ming-Chin and S. Ming-Chang, “Simulated and experimentalstudy of hydraulic anti-lock braking system using sliding-mode PWMcontrol,” Mechatronics, vol. 13, pp. 331–351, 2003.[6] V. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electro-Mechanical Systems, Second Edition (Automation and Control Engi-neering), 2nd ed. CRC Press, 5 2009.[7] C. Clover and J. Bernard, “Longitudinal tire dynamics,” Vehicle SystemDynamics, vol. 29, pp. 231–259, 1998.[8] I. Novozhilov, P. Kruchinin, and M. Magomedov, “Contact force rela-tion between the wheel and the contact surface,” Collection of scientificand methodic papers Teoreticheskaya mekhanika, MSU, vol. 23, pp.86–95, 2000, (In Russian).[9] P. Kruchinin, M. Magomedov, and I. Novozhilov, “Mathematical modelof an automobile wheel for antilock modes of motion.” Mechanics ofSolids, vol. 36, no. 6, pp. 52–57, 2001.[10] I. Petersen, A. Johansen, J. Kalkkuhl, and J. Ludemann, “Wheel slipcontrol in ABS brakes using gain scheduled constrianed LQR,” in Proc.European Contr. Conf., Porto., 2001.[11] M. Magomedov, V. Alexandrov, and K. Pupkov, “Robust adaptive sta-bilization of moving a car under braking with ABS in control circuit,”in Automotive and Transportation Technology Congress and ExpositionProceedings - Chassis and Total Vehicle, Barcelona, SPAIN., S. S.ATTCE, Ed., vol. 6, 2001.[12] E. Bakker, H. Pacejka, and L. Lidner, “A new tire model with applica-tion in vehicle dynamic studies,” SAE Paper No. 890087, vol. 01, pp.101–113, 1989.[13] H. K. Khalil, Nonlinear Systems (3rd Edition), 3rd ed. Prentice Hall,12 2001.

Sliding mode control for Antilock Brake System

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Sliding mode control for Antilock Brake System

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