An integral nested Sliding Mode (SM) Control is proposed for an Antilock Brake
System (ABS) control problem by employing integral SM and nested SM concepts. This
controller has robustness against matched and unmatched perturbations, and the capability
to reduce the sliding functions gains. Application to an ABS is presented as a simulationConsejo Nacional de Ciencia y TecnologíaLomonosov Moscow State UniversityUniversidad Autónoma Metropolitana-AzcapotzalcoUniversidad de Guadalajar

Galicia, Marcos

Kruchinin, Pavel

Loukianov, Alexander

Sánchez-Torres, Juan D.

Vázquez, Iván

IFAC Proceedings Volumes

Repositorio Institucional del ITESO

Instituto Tecnológico y de Estudios Superiores de Occidente 2010-07 Integral Nested Sliding Mode Control forAntilock Brake System Vázquez, Iván; Galicia, Marcos; Sánchez-Torres, Juan D.; Loukianov,Alexander; Kruchinin, Pavel Vázquez, I.; Galicia, M.; Sánchez-Torres, J.D.; Loukianov, A. and Kruchinin, P. (2010). IntegralNested Sliding Mode Control for Antilock Brake System. Advances in Automotive Control.  Enlace directo al documento: http://hdl.handle.net/11117/3270 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 arbitrajeIntegral Nested Sliding Mode Control forAntilock Brake System ?Iva´n Va´zquez ∗ Marcos I. Galicia ∗∗ Juan Diego Sa´nchez ∗∗Alexander G. Loukianov ∗∗ Pavel A. Kruchinin ∗∗∗∗Universidad Auto´noma Metropolitana-Azcapotzalco, Azcapotzalco,02200, D.F., Me´xico, (e-mail: iva@correo.azc.uam.mx)∗∗Department of Automatic Control, CINVESTAV UnidadGuadalajara, Zapopan, Jalisco, 45015 Me´xico,(e-mail: [mgalicia, dsanchez, louk]@gdl.cinvestav.mx)∗∗∗Department of Applied Mechanics and Control, Lomonosov MoscowState University, Moscow, 119991, Russia,(e-mail: pkruch@mech.math.msu.su)Abstract: An integral nested Sliding Mode (SM) Control is proposed for an Antilock BrakeSystem (ABS) control problem by employing integral SM and nested SM concepts. Thiscontroller has robustness against matched and unmatched perturbations, and the capability toreduce the sliding functions gains. Application to an ABS is presented as a simulation example.Keywords: Brake Control, Antilock Braking Systems (ABS), Sliding Mode Control,Automotive Control, Integral Controller1. INTRODUCTIONThe ABS control is a very important problem and itsobjective is to obtain desired vehicle motion providingadequate vehicle stability. The main difficulty arising inthe ABS design is due to the high nonlinearity anduncertainty in the problem. This problem has been widelystudied and various controllers have been proposed usingSM technique (see Tan and Chin [1991], Drakunov et al.[1995], Unsal and Pushkin [1999], Hadri et al. [2002], Ming-Chin and Ming-Chang [2003]). In this work we designa new controller on the basis of integral Sliding Mode(SM) (Utkin et al. [2009]) in combination with nested SM(see Gonza´lez Jime´nez and Loukianov [2008] and Huertaet al. [2008]) in order to achieve robustness to matched,and unmatched perturbations, and ensure output tracking.Theorically, this integral nested SM control can guaranteethe robustness of the system throughout the entireresponse starting from the initial time instance and reducethe sliding functions gains in comparison with standardSM. In spite of the mentioned above works we consider thereal situation: the control input can take only two values”0” or ”1” depending on the corresponding valve beingopen or closed.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,some conclusions are presented in Section 5.? This work was supported by the Consejo Nacional de Ciencia yTecnolog´ıa de Me´xico (CONACYT) under grant 46069.2. MATHEMATICAL MODELIn this section, the model of a pneumatic brake systemis under consideration. The specific configuration of thissystem includes the next: brake disks, which hold thewheels, as a result of the increment of the air pressurein the brake cylinder (Fig. 1).Fig. 1. Pneumatic brake modelThe entrance of the air trough the pipes from the centralreservoir and the expulsion from the brake cylinder to theatmosphere is regulated by a common valve. This valveallows only one pipe to be open, when 1 is open 2 is closedand vice versa. The time response of the valve is consideredsmall, compared with the time constant of the pneumaticsystems.We study the task of control of the wheels rotation, suchthat the longitudinal force, due to the contact of the wheelwith the road, is near from the maximum value in theperiod of time valid for the model. This effect is reachedas a result of the ABS valve throttling.2.1 Wheel motion equationsTo describe the wheels motion we will use a partialmathematical model of the dynamic system (Novozhilovet al. [2000], Kruchinin et al. [2001] and Magomedov et al.[2001]), the dynamics of the angular momentum changerelative to the rotation axis are given byIydΩydt= FR− L (1)where Ωy is the wheel angular velocity, Iy is the wheelinertia moment, R is the wheel radius, F is the contactforce and L is the brake torque.Fig. 2. Model for the contact element of the tire.The expression for longitudinal component of the contactforce in the motion plane according to experimental results(Pacejka [1981]) is equalF = νNφ (s) (2)where ν is the friction coefficient between the wheel andthe road, N is the normal reaction and s is the slip rates =Vx − ΩyR− dξˆdtVx(3)Vx is the longitudinal velocity of the wheel mass centerand ξˆ is the longitudinal deformation of the tire contactarea element.In (2), the function φ(s) represents a friction/slipcharacteristic relation between the tyre and road surface.Here, we use the Pacejka model (Bakker et al. [1989]),defined as followsφ (s) =D sin (C arctan (Bs− E (Bs− arctan (Bs)))) . (4)In general, this model produces a good approximationof the tyre/road friction interface. With the followingparameters B = 10, C = 1.9, D = 1 and E = 0.97 thatfunction represents the friction relation under a dry surfacecondition. A plot of this function is shown in Fig. 3.The motion equation of the contact element with massMc is described by the tire longitudinal deformation. Theinteraction between this element and the rigid part of thewheel can be described by the following viscoelastic forcesmodelMcddt(Vx − ΩyR+ ddtξˆ)= F − cx ddtξˆ − kxξˆ (5)where cx and kx are the longitudinal constants of viscousand elastic behavior of tire model, respectively. The model(5) to be used is similar to the description of the firstwaveform model in (Jansen et al. [1998]).The equations (1)-(5) characterize the wheel motion.Fig. 3. Characteristic function φ (s)2.2 Pneumatic brake system equationsWe suppose that the brake torque L is proportional to thepressure Pm in the brake cylinder:L = kLPm (6)with kL > 0.For the brake system we use an approximated model ofpressure changes in the brake cylinder due to the openingof the valve with a first order relation (Clover and Bernard[1998]):TedPmdt+ Pm = P∗ (7)where P∗ is the valve input signal.We suppose that opening and closing of the valves aremomentary and the parameters of the equation (7) aregiven by the following rules:• If P∗ = Pc = const then Te = Tin• If P∗ = Pa = 0 then Te = Toutwhere Pc is the pressure inside the central reservoir, Pa isthe atmospheric pressure, that we consider equal to zero;Tin and Tout are the time constants of internal and externalpipelines, respectively.2.3 The vehicle motion equationThe vehicle longitudinal motion dynamics are representedbyMdVxdt= Px − Fax +MVyΩz.Assuming that there is no the lateral motion, Vy = 0, thelast equation evolves inMdVxdt= Px − Fax (8)where M is the vehicle mass; Fax is the aerodynamic dragforce, which is proportional to the vehicle velocity and isdefined asFax =12ρCdAf (Vx + Vwind)2where ρ is the air density, Cd is the aerodynamiccoefficient, Af is the frontal area of vehicle, Vwind is thewind velocity; the contact force Px is modeled of the formPx = −νNϕ (s)The dynamic equations of the whole system (1)-(8) can berewritten in a more useful formIydΩydt= −νNRφ (s)− LMcd2ξˆdt2+ cxdξˆdT+ kxξˆ =−McRIyL−(McR2Iy+ 1)νNφ (s)TedLdt− kLP∗ + L = 0(9)withs = 1− Ωy RVx− 1Vxdξˆdt. (10)3. INTEGRAL NESTED SM CONTROL FOR ABSTo design an integral nested SM control for (9) and (10),the system is represented using the state variablesx = [x1, x2, x3, x4, x5]T=[Ωy, Pm, ξˆ,dξˆdt, Vx]Twith initial conditions x0 = x(0) results the following form:x˙1 = c1 (RF − kLx2)x˙2 = −c2x2 + bux˙3 = x4x˙4 = −a41x3 − a42x4 − a43x2 − f4(x1, x4)x˙5 = −c3 (F + f5(x5))(11)with outputy = s = h(x) = 1−Rx1x5− x4x5where b = c2kLPc, c1 = 1/Iy, c2 = 1/Te, c3 = 1/M , a41 =kx/Mc, a42 = cx/Mc, a43 = (RkL)/Iy f4(x1) = kνNφ(x1)and f5(x5) = d(x25 + Vwind)2, with k =(R2/Iy + 1/Mc)and d = (ρCdAf )/(2M).To calculate the control law, we obtain the outputdynamicsy˙ = a1(x1, x5) + a2(x5)x2 + f1(x)where a1(x1, x5) , a1 = RIyx5(RF +FIyx1Mx5), a2(x5) ,a2 = (RkL)/(Iyx5) and f1(x4, x5) = −d(x4/x5)dt +∆(ν). Theterm ∆(ν) contains variations of the friction parameter ν.Throughout the development of the controller, we will usethe following assumptions:A1) All the state variables are available for measurement.A2) The term f1(x) includes the friction force variation,it is considered as unmatched bounded perturbation‖f1(x)‖ < β1 <∞ (12)A3) The sign function can be approximated by thesigmoid function as is shown by the following limit:limε→∞ sigm (εS) = sign (S) . (13)The Fig. 3 shows the approximation for various values ofthe sigmoid function slope.Fig. 4. Sigmoid function for various values of the slope ε3.1 Control formulationThe control objective is to design an Integral NestedSliding Mode controller to obtain output trajectorytracking in despite of the system perturbations. Defineyref (t) as the desired trajectory of the relative slip.Let yref (t) be a twice differentiable function, but withunknown derivatives, now define the output tracking erroras e1 = y (t)− yref (t) , α1(x, t) then its derivative ise˙1 = a1 + a2x2 + g1(x, t) (14)where g1 (x, t) is the unmatched perturbation term definedbyg1(x, t) = f1(x)− y˙ref . (15)Considering x2 as virtual control in (14), we proposex2ref = x2,0 + x2,1 (16)where x2,0 is the nominal part of the virtual controland x2,1 is the part which will be designed to reject theperturbation in (14) (see Utkin et al. [2009]). Now, wedefine a new error variable e2 ase2 = x2 − x2ref , α2(x, t) (17)and two auxiliar variables σ1 and σ2 of the formσ1 = e1 + w1 (18)σ2 = e2 + w2 (19)where x2ref is the desired value of x2 to obtain the controlaim, w1 and w2 are integral variables used to reducethe control gain, σ1 and σ2 are pseudo-sliding functionsproposed to attenuate the perturbation terms. All thevariables will be designed later. Using equations (16)-(19)we obtainx2 = σ2 + x2,0 + x2,1 − w2. (20)Taking the derivative of σ1 results inσ˙1 = a1 + a2x2 + w˙1 + g1(t). (21)Substituting (20) in (21) yieldsσ˙1 = a1 + a2 (σ2 + x2,0 + x2,1 − w2) + w˙1 + g1(t)orσ˙1 = a1 + a2 (e2 + x2,0 + x2,1) + w˙1 + g1(t). (22)Choosing the dynamics of the integral variable w1 asw˙1 = −a1 − a2 (e2 + x2,0) (23)with w1(0) = −e1(0), the nominal part x2,0 is designed toeliminate the old known dynamics in (14) and assign thedesired dynamics −k10e1, k10 > 0x2,0 = − 1a2(a1 + k10e1) . (24)Substituting (23) and (24) in (22) yieldsσ˙1 = a2x2,1 + g1(t). (25)To attenuate the perturbation term g1 (x, t) in (25) usingthe integral SM technique and to enforce a sliding motionon σ1 = 0 the virtual control x2,1 is chosen asx2,1 = −k11sigm (ε1σ1)with k11 > 0.Using (11) and (17), straightforward calculations reveale˙2 = −c2e2 + bu+ f2e(x) (26)wheref2e(x) = −c2x2ref + ∂α2(x, t)∂x1x˙1 +∂α2(x, t)∂x3x˙3 + . . .+∂α2(x, t)∂σ1σ˙1 +∂α2(x, t)∂x5x˙5.To induce sliding mode in (26) we choose the control signalasu = 0.5sign (−e2) + 0.5. (27)3.2 Stability analysisUsing the new variables e1, e2 and σ1 the extended closedloop system (11), (22) and (27) is presented ase˙1 = −k10e1 − a2k11sigm(ε1σ1) + g1(x, t) (28)σ˙1 = −k11sigm(ε1σ1) + g1(x, t) (29)e˙2 = −c2e2 + f2e(x) + 0.5bsign (−e2) + 0.5b (30)x˙3 = x4x˙4 = −a41x3 − a42x4 − a43x2 − f4(x1, x4)x˙5 = −c3 (F + f5(x5)) .(31)The stability of (28) - (31) can be is studied step by step:A) SM stability of the projection motion (30);B) SM stability of the projection motion (29);C) SM dynamics (28) stability in the neighborhood of SMmanifold e2 = 0 and σ1 = 0.We use the following assumptions:|f2e(x)| 6 α2 |e2|+ β2 (32)|g1(x, t)| 6 α1 |σ1|+ β1 (33)|g˙1(x, t)| 6 α0 |σ2|+ β0ς = σ˙1(34)with α0 > 0, α1 > 0, α2 > 0, β0 > 0, β1 > 0, β2 > 0,c2 > α2 and b > |f2e(x)| .A) The system (30) can be presented as follows:Case 1, e2 > 0, thene˙2 = −c2e2 + f2e(x). (35)In this case, under condition (32) the solution of (30) isultimately bounded by (Khalil [2001])|e2(t)| 6 δ0, δ0 = β2c2 − α2 . (36)Case 2, e2 < 0, thene˙2 = −c2e2 + f2e(x) + b. (37)In this case e˙2 > 0, therefore, there is a time t1 such thate2(t1) = 0.B) To analyze stability of (29) we use V1 = (1/2)σ21 . Using(29) we haveV˙1 = σ1 (−k11sigm(ε1σ1) + g1(x, t)) . (38)Now we establishe the following equality:sigm(ε1σ1) = sign (σ1)−∆ (ε1, σ1) (39)where ∆ (ε1, σ1) is the difference between the sign andsigmoid functions. It is evidently that ∆ (ε1, σ1) isbounded. That is, there is a constant 0 < γ1 < 1 suchthat |∆ (ε1, σ1)| < γ1. Then using (39) the derivative (38)becomesV˙1 6 − |σ1| (k11 (1− |∆ (ε1 , σ1)|)− |g1(x, t)|)6 − |σ1| (k11 (1− γ1)− β1)− α1 |σ1|) . (40)Under the condition k1 (1− γ1) > β1 we have V˙1 < 0, andhence σ1 (t) converges to the region given by‖σ1(t)‖ 6 δ1, δ1 = k11 (1− γ1)− β1)α1. (41)Now, consider the derivative ς = σ˙1 (29) described byς˙ = −k2ς + g˙1(x, t), k2 = k11ε1(1− tanh2(ε1σ1(1)) (42)and V2 = (1/2)ς2. Then using (42) and (34) thestraightforward calculations givesV˙2 = ς (−k2ς + g˙1(x, t)) 6 − |ς| ((k2 − α0) |ς| − β1) (43)Under the condition k2 > α0 the derivative ς = σ˙1 isultimately bounded by|ς(t)| 6 δ2, δ2 = β1k2 − α0 . (44)C) Stability of the equation (28) in the neighborhood ofthe sliding manifold e2 = 0 and σ1 = 0 is studied byV0 = (1/2)e21. Using (36) and (43) we haveV˙1 = e1[−k10e1 − a2k11sigm(ε1σ1) + g1(x, t)]6 − |e1| (k10 |e1| − δ0 − δ2) (45)If k10 > 0, then the control error e1 (t) converges to anarbitrary small neighborhood of the equilibrium point,defined by |e1 (t)| < δ, δ = (δ0 + δ2) /k10. Moreover it ispossible to show that an equilibrium point of the residualdynamics (31) is exponentially stable, therefore the controlobjectives are fulfilled, and the desired performance of theclosed-loop system is obtained.4. SIMULATION RESULTSTo show the effectiveness of the proposed control law,simulations have been carried out on one wheel modeldesign example, the system parameters used are listed inTable 1. In order to maximize the friction force, in thesimulations we suppose that slip tracks a constant signal.yref = 0.205which produces a value close to the maximum of thefunction φ(s). The parameters used in the control law arek10 = 8700, k11 = 1000 and ε1 = 100.TABLE 1Values of ParametersParameter Value Parameter ValueCx 10 Vwind -5Kx 9000 v 0.8M 2396 B 10Iy 18.9 C 1.9R 0.535 D 1KL 1000 E 0.97ρ 1.225 N 23504.76Cd 0.65 Pa 0Af 6.6 Pc 8On other hand, to show robustness property of thecontrol algorithm in presence of parametric variationswe introduce a change of the friction coeficient ν whichproduces a diferent contact force, namely Fˆ . Then, ν = 0.5for t < 1 s, ν = 0.52 for t ∈ [1, 2) s, and ν = 0.5 for t > 2 s.It is worth mentioning that just the nominal values wereconsidered in the control design.In the figures 5 and 6 are shown, respectively, the slip andthe friction function φ in the braking processFig. 5. Slip performance in the braking processFig. 6. Performance of φ in the braking processwhile Figs. 7 and 8 summarize the behavior of the errorvariables e1 and e2 respectively.Fig. 7. Tracking error, e1 = y − yrefFig. 8. Tracking error, e2 = x2 − x2refIn Fig. 9 the longitudinal speed Vx and the linear wheelspeed ΩyR are showed; it is worth noting that the slipcontroller should be turn off when the longitudinal speedVx is close to zero. In Fig. 10 the control action is shown.Fig. 9. Longitudinal speed Vx (solid) and linear wheelspeed ΩyR (dashed)Fig. 10. Control inputFinally, in Fig. 11 the nominal F , and the Fˆ contact forceare shown.Fig. 11. Nominal F (solid) and Fˆ (dashed) contact forces5. CONCLUSIONIn this work an integral nested sliding mode controlfor ABS has been proposed. The simulation resultsshow good performance and robustness of the closed-loop system in presence of both the matched andunmatched perturbations, namely, parametric variationsand neglected dynamics.REFERENCESE. Bakker, H. Pacejka, and L. Lidner. A new tire modelwith application in vehicle dynamic studies. SAE PaperNo. 890087, 01:101–113, 1989.C.L. Clover and J.E. Bernard. Longitudinal tire dynamics.Vehicle System Dynamics, 29:231–259, 1998.S.V. Drakunov, U. Ozguner, P. Dix, and B. Ashrafi. ABScontrol using optimum search via sliding modes. IEEETransactions on Control Systems Technology, 3(1):79–85, 1995.L.E Gonza´lez Jime´nez and A.G. Loukianov. Integralnested sliding mode control for robotic manipulators. InProc.of the 17th IFAC World Congress pp. 9899-9904,Seoul, Korea, July 6-11, 2008.A.El. Hadri, J.C. Cadiou, and N.K. MSirdi. Adaptivesliding mode control for vehicle traction. 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Integral Nested Sliding Mode Control for Antilock Brake System

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Integral Nested Sliding Mode Control for Antilock Brake System

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