Investigation of sliding surface design

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL 57, NO 2, MARCH 2008 747 Investigation of Sliding-Surface Design on the Performance of Sliding Mode Controller in Antilock Braking Systems Taehyun Shim, Sehyun Chang, and Seok Lee Abstract—Sliding mode control (SMC) has widely been employed in the development of a wheel-slip controller because of its effectiveness in applications for nonlinear systems as well as its performance robustness on parametric and modeling uncertainties The design of a sliding surface strongly influences the overall behavior of the SMC system due to the discontinuous switching of control force in the vicinity of a sliding surface that produces chattering This paper investigates the effects of sliding-surface design on the performance of an SMC-based antilock braking system (ABS), including a brake-torque limitation, an actuator time delay, and a tire-force buildup Different sliding-surface designs commonly used in ABS were compared, and an alternative sliding-surface design that improves convergence speed and oscillation damping around the target slip has been proposed An 8-degree-of-freedom (dof) nonlinear vehicle model was developed for this paper, and the effects of brake-system parameter variations, such as a brake actuator time constant, target slip ratios, an abrupt road friction change, and road friction noises, were also assessed Index Terms—Antilock braking system (ABS), sliding mode control (SMC), sliding-surface design N OMENCLATURE Vehicle mtot Vehicle total mass (in kilograms) Vehicle sprung mass (in kilograms) ms Jroll Roll inertia (in kilograms meter square) Jyaw Yaw inertia (in kilograms meter square) L Length of wheel base (in meters) Distance of center of gravity (c.g.) of sprung mass from La front axle (in meters) Distance of c.g of sprung mass from rear axle Lb (in meters) Track width at front axle (in meters) tf Track width at rear axle (in meters) tr c.g height of sprung mass (in meters) hcg Average roll center distance below sprung mass c.g ho (in meters) Manuscript received January 25, 2006; revised November 29, 2006, April 15, 2007, and April 19, 2007 This work was supported by the Institute for Advanced Vehicle Systems (IAVS), University of Michigan—Dearborn The review of this paper was coordinated by Dr M Abul Masrur T Shim and S Lee are with the Department of Mechanical Engineering, University of Michigan—Dearborn, MI 48128 USA (e-mail: tshim@umich.edu; eoklee@umd.umich.edu) S Chang is with the Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: sehyun@umich.edu) Digital Object Identifier 10.1109/TVT.2007.905391 hl hf hr U/V Distance of sprung mass c.g to ho (in meters) Front roll center height at front axle (in meters) Rear roll center height at rear axle (in meters) Longitudinal/lateral velocities of c.g in body-fixed coordinate (in meters per second) Initial longitudinal velocity of c.g (in meters per uo second) ϕ Roll angle (in radians) ϕ˙ Roll angular velocity of c.g in body-fixed coordinate (in radians per second) Ω Yaw angular velocity of c.g in body-fixed coordinate (in radians per second) Ω˙ Yaw acceleration of c.g in body-fixed coordinate (in radians per second square) ax Longitudinal acceleration at c.g (in meters per second square) Lateral acceleration at c.g (in meters per second ay square) Suspension/tire muf Vehicle unsprung mass at front axle (in kilograms) mur Vehicle unsprung mass at rear axle (in kilograms) Unsprung mass c.g height at front axle (in meters) huf Unsprung mass c.g height at rear axle (in meters) hur Suspension roll stiffness at front axle (in newton meters κrf per degree) Suspension roll stiffness at rear axle (in newton meters κrr per degree) κroll Roll stiffness (in newtons per meter) Broll Roll damping coefficient (in newton seconds per meter) Rotational inertia of each wheel (in kilograms meter Jw square) Tire radius (in meters) Rw λ Tire longitudinal slip α Tire lateral slip (in radians) ω Angular velocity of wheel rotation (in radians per second) δ Road wheel steer angle (in radians) External torque applied at wheel (in newton meters) Tb I I NTRODUCTION I N RECENT years, the use of electronic control systems has increasingly become popular in passenger vehicles, resulting in a significant improvement in driver satisfaction, comfort, and safety These systems combine the existing hardware with 0018-9545/$25.00 © 2008 IEEE 748 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL 57, NO 2, MARCH 2008 new electronic components and maximize their performance An antilock braking system (ABS) was the first system used, but now, the systems also include traction control, vehicle dynamic control, active steering control, direct yaw moment controls, etc These systems are activated when the vehicle is at the physical limit of adhesion between the tires and the road, allowing the driver to keep control of the vehicle For the ABS, various control methodologies, such as feedback linearization [1], fuzzy logic [2], neural network [3], sliding mode control (SMC) [4]–[7], and hybrid control, which uses two different controls in a combined way [8]–[12], have been developed since its introduction in the 1950s All of these methods try to accurately control the wheel slip to maximize the effectiveness of the tire force in the longitudinal direction, reducing the stopping distance by preventing wheel lockup as well as providing a steering capability in the lateral direction An ABS has been proven to be extremely effective during braking on a slippery road condition Among the different control methodologies used in ABS, the SMC has widely been investigated [4]–[7] because of its robust characteristics for model uncertainties as well as its effectiveness for applying control to nonlinear systems Reference [4] showed the SMC for a wheel-slip control using a one-wheel model with an inclusion of tire relaxation length Reference [5] developed the wheel-slip controller for a full-vehicle model using the SMC in which an extended Kalman filter (EKF) [20] was used to estimate the brake pad friction and the tire braking force Reference [6] demonstrated an eddy-currentbraking system for a hybrid electric vehicle using a nonlinear 8-degree-of-freedom (dof) vehicle model and an SMC-based wheel-slip controller Other approaches to obtain a controlled braking torque using the SMC can be found in [7]–[10] In [7], it is assumed that the tire force is available using a smart tire concept The hybridization of SMC and a fuzzy method to improve the braking performance were presented in [8]–[10] The SMC with a pulsewidth modulation (PWM), which is useful to control the conventional hydraulic brake system, was demonstrated by vehicle tests on an in-door test bench in [27] Reference [28] presented an extremum seeking control via sliding mode approach to maximize the braking force In the development of SMC, the design of the sliding surface dictates the behavior of the overall control system Due to the discontinuous switching of control force in the vicinity of the sliding surface, it produces a chattering which can cause system instability and damage to both actuators and plants During the application to actual brake systems, the performance of theoretically determined switching control is degraded due to the brake-torque limitation, brake actuator time delay, and tire-force buildup, resulting in an oscillation near the sliding surface Thus, a reduction of a chattering and a rapid convergence to the sliding surface in the actual brake systems are of importance In order to reduce the chattering, a higher order SMC [13] and a boundary-layer method [14] with moderate tuning of a saturation function have extensively been used In addition, the brake-by-wire concept was investigated in recent years to overcome the chattering by precisely applying the optimum brake pressure to each wheel using the electrohydraulic Fig Schematic of the 8-dof vehicle model and the forces at a wheel during the braking and the electromechanical brakes [15] However, the effects of sliding-surface design on the overall controller performance (minimizing the chattering, improving the tracking speed, etc.) have not closely been investigated This paper studies the effects of sliding-surface design on the overall performance of the ABS Different sliding-surface designs introduced in [4]–[7], [27], and [28] for the wheelslip control on the ABS have been compared through simulation An alternative sliding-surface design that improves the convergence speed to the sliding surface has been proposed and simulated using an degree of freedom (8-dof) nonlinear vehicle model During the simulation, the effects of brakesystem parameter variations, such as a brake actuator time constant, target slip ratios, an abrupt road friction change, and road friction noises, were also assessed In the following section, a vehicle dynamic model is described and validated by comparing the simulation results with those of the CarSim vehicle model [16] The sliding mode controller used for the wheel-slip control with different sliding surfaces is presented in Section IV The simulations and discussion are given in Section V II V EHICLE -M ODEL D EVELOPMENT Fig shows the schematic of an 8-dof vehicle model used for this paper This model has dof for the chassis velocities and dof at each of the four wheels representing the wheel spin dynamics The chassis velocities at the c.g include the longitudinal velocity U , the lateral velocity V , the roll angular velocity ϕ, ˙ and the yaw angular velocity Ω The model neglects the pitch and heave motions The application of Newton’s second law to the lumped vehicle mass longitudinally, laterally, and about the longitudinal and vertical axes through the center of mass produces the following equations of motion Longitudinal motion Fx =mtot (U˙ −V Ω)= (Fxi cos δ−Fyi sin δ)+ i=1 Fxi i=3 (1) SHIM et al.: SLIDING-SURFACE DESIGN ON THE PERFORMANCE OF SLIDING MODE CONTROLLER IN ABS 749 Lateral motion Fy =mtot (V˙ +U Ω)= (Fxi sin δ+Fyi cos δ)+ i=1 Fyi (2) i=3 Yaw motion ˙ Mz =Jyaw Ω=L a (Fxi sin δ+Fyi cos δ)−Lb i=1 Fyi i=3 + tf (−1)i−1 (Fxi cos δ−Fyi sin δ)+ t2r i=1 (−1)i−1 Fxi (3) i=3 Roll motion Fig Block diagram of a vehicle model Jroll ă + Broll + roll ϕ = ms gh1 sin ϕ + ms (ay )h1 cos ϕ (4) where i in (1)–(4) is 1, 2, 3, and 4, and it represents the left front, right front, left rear, and right rear wheels, respectively The resulting accelerations affect the distribution of vertical tire forces at each of the wheels Although this model does not have a suspension between the body and the wheels, the vertical load transfer between front/rear and inside/outside due to the vehicle longitudinal and lateral accelerations has been included The vertical forces Fz can be derived from their moments about the center of mass in equilibrium with the corresponding moments due to the static weight and the longitudinal and lateral accelerations (ax and ay , respectively) F zrf m L h a h ms =−(ay cos ϕ+g sin ϕ) tfκ(κrfrfh+κ − s tbL f y −muf ay tuf rr ) f f − 12 (muf huf +ms hcg +mur hur ) aLx + 12 mtot g F zrf Lb L ms − ms La hr ay −m a hur =−(ay cos ϕ+g sin ϕ) trκ(κrrrfh+κ ur y tr tr L rr ) (7) ms + ms La hr ay +m a hur = (ay cos ϕ+g sin ϕ) trκ(κrrrfh+κ ur y tr tr L rr ) + 12 (muf huf +ms hcg +mur hur ) aLx + 12 mtot g LLa (8) In the aforementioned vertical-force equations, the first term represents the load transfer due to the sprung-mass roll moment, and the second term is the contribution from the sprung-mass lateral acceleration The load-transfer effect due to the lateral acceleration is shown in the third term The fourth and fifth terms indicate the load transfer due to the longitudinal acceleration and the static loading condition, respectively The detailed derivation can be found in [6] It should be noted that the same spring properties are used at the left and right of the front and rear axles, respectively For each of the wheels, a separate equation of motion must be derived relating the angular acceleration ω˙ to the respective wheel torque Tb and longitudinal tire force Fx Jw ω˙ i = Tb,i − Fx,i Rw , Fy = µx · Fzp · R(sR ) · sx , s2x + η(sR )2 tan2 α µy · Fzp · R(sR ) · η tan α s2x + η(sR )2 tan2 α (i = 1−4) (9) (10) where R(sR ) =D sin(C arctan (B(1−E)sR +E arctan(BsR ))) (11) sR = (6) + 12 (muf huf +ms hcg +mur hur ) aLx + 12 mtot g LLa F zlr Fx = (5) m L h a h ms = (ay cos ϕ+g sin ϕ) tfκ(κrfrfh+κ + s tbL f y +muf ay tuf rr ) f f − 12 (muf huf +ms hcg +mur hur ) aLx + 12 mtot g F zrr Lb L The longitudinal and lateral forces on each tire are calculated using the Milliken tire model [18] by taking into account the slip angle, the longitudinal slip, and the vertical force This tire model is slightly modified from the Magic Formula tire model [19] by normalizing the variables according to the vertical load Equation (10) shows the longitudinal Fx and the lateral Fy tire forces Cx sx µx Fzp + Cy tan α µy Fzp   0.5(1 + η0 ) η(sR ) = −0.5(1 − η0 ) cos(0.5sR ),  1, η0 = (12) for sR ≤ 2π for sR > 2π Cy µx Cx µy (13) µx and µy are the tire longitudinal and lateral friction coefficients, and Cx and Cy represent the longitudinal and lateral stiffness coefficients The peak vertical force Fzp is approximated by the following function according to the vertical load Fz Fzp (Fz ) = Fz + (1.5Fz /mtot · g)3 (14) These equations are combined in the Matlab/Simulink environment, as shown in Fig In order to reflect realistic wheel and brake systems, first-order dynamics of the tire- and brakeforce buildups as well as the brake-torque limit have been implemented in the wheel dynamics III V EHICLE -M ODEL V ALIDATION The responses of the vehicle model have been compared with those of the CarSim vehicle model for a fishhook maneuver A 750 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL 57, NO 2, MARCH 2008 TABLE I VEHICLE PARAMETERS Fig Comparison of the vehicle responses between the 8-dof vehicle model and the CarSim for a fishhook maneuver during a vehicle speed of u0 = 50 km/h (a) Hand wheel steering input-gear ratio (16) (b) Trajectory (c) Yaw rate Φ (d) Lateral acceleration ay steering wheel input is applied at s while a vehicle is moving at a speed of 50 km/h The steering wheel input was increased from 0◦ to 200◦ for the first 0.4083 s, which is maintained for 0.1617 s, and then applied in the opposite direction at the same magnitude within 0.817 s The same tire model [18] has been used for both vehicle models during the simulation Fig compares the responses of lateral acceleration, yaw rate, and vehicle trajectory between the 8-dof and the CarSim models Vehicle parameters of a midsize sedan, as shown in Table I, were used for this simulation As shown in Fig 3, the 8-dof vehicle-model responses closely matched those of the CarSim vehicle model The slight discrepancies shown in Fig come from the differences in complexities of the vehicle models The CarSim vehicle model has more dof compared with the 8-dof model used in this paper IV D EVELOPMENT OF A W HEEL -S LIP C ONTROLLER U SING SMC A wheel-slip controller is typically designed to achieve a target (desired) slip ratio for a given driving condition A wheel slip ratio for a braking can be defined as follows: λ = (Rw ω − uxw )/uxw (15) where uxw represents the longitudinal velocity of the wheel center along the longitudinal tire-force direction For brake application, the target slip ratio of ABS can be set to the peak slip ratio of λ = λpeak , which satisfies dFx /dλ = 0, in order to minimize the stopping distance The target slip ratio that produces a maximum longitudinal tire force is not fixed and instead varies with the road-condition changes, as shown in Fig Thus, the wheel-slip controller must track the different wheel slip ratios for ABS application as well as the arbitrary target slip ratio for an application of vehicle stability control (VSC) The following section shows a development of the wheelslip controller based on the SMC for the ABS application Three different sliding-surface designs were used to assess the tracking performance of the target slip ratio Fig Tire longitudinal force Fx versus slip ratio λ on various road conditions A Conventional SMC Design For a brake-system application, the controlled brake torque Tb in the SMC consists of an equivalent control torque Tb,eq and a switching control torque Tb,sw , i.e., Tb = Tb,eq + Tb,sw The equivalent control torque can be interpreted as a control ˙ move along the desired that makes the system states (λ, λ) sliding surface It is determined by the wheel dynamics in (9) and the slip ratio in (15) The switching control torque ensures that the trajectory of the system is reached at the desired sliding surface and its magnitude can analytically be obtained by using the Lyapunov stability condition In previous research, two types of sliding-surface design, t ˜+γ x ˜dτ namely, σ = x ˜ = λ − λd [4], [5], [7] and σ = x [6], [8], were mainly used for the brake-system applications SHIM et al.: SLIDING-SURFACE DESIGN ON THE PERFORMANCE OF SLIDING MODE CONTROLLER IN ABS In these designs, the controlled torque appears in the first derivative of the slip-ratio tracking error, which is defined as the difference between a current slip ratio and a target slip ratio, i.e., x ˜ = λ − λd 1) Equivalent Control Torque Tb,eq : The equivalent control for these sliding-surface designs is determined from the condition of σ˙ = in which a target slip ratio λd is assumed constant (λ˙ d = 0) In the first sliding-surface design σ = x ˜, the first derivative of σ can be expressed as σ˙ = x ˜˙ = λ˙ = By using the wheel dynamics in (9), the first derivative of a slip ratio becomes x ˜˙ = λ˙ = TABLE II SENSOR NOISE CHARACTERISTICS IN TERMS OF STANDARD DEVIATION σ Input vectors are composed of the steering angles and the brake torques at each wheel as follows: Rw (1 + λ) (Tb − Fx · Rw ) − u˙ xw = (16) uxw Jw uxw From (16), the equivalent control torque Tb,eq can be determined as Tb,eq = Fx · Rw + Jw (1 + λ) u˙ xw Rw (17) u = [u1 , u2 ] σ˙ = x ˜˙ + γ x ˜ = λ˙ − λ˙ d + γ(λ − λd ) (1 + λ) Rw (Tb − Fx Rw ) − u˙ xw + γ (λ − λd ) = = uxw Jw uxw (18) where γ is strictly positive constant Tb,eq = Fx Rw +(1+λ)u˙ xw Jw uxw Jw −γ(λ−λd ) Rw Rw (19) In the aforementioned equations, the value of the equivalent brake torque Tb,eq is difficult to determine from direct measurement Thus, it is replaced with an approximated equivalent control torque Tˆb,eq in which the estimation of vehicle states and tire forces is used The estimation of vehicle states and tire forces is achieved using the EKF technique presented in [20], and a brief summary of its procedures for the 8-dof nonlinear vehicle model is shown next The state and measurement equations are x˙ a = f (xa , u) y = h (xa , u) The augmented nonlinear state equation consists of the tireforce term (F˙ x,y = F x,y and Făx,y = 0) and the 8-dof vehicle dynamics given in (1)–(4) and (9) Thus, the augmented state vector with approximation symbol “∧” is composed of nine vehicle states, six estimated tire forces, and six first derivatives of estimated forces ˆ , Vˆ , Ω, ˆ ϕ, ˆ˙ ω ˆ ϕ, ˆ1, ω ˆ2, ω ˆ3, ω ˆ4 xa(1 : 9) = U T xa(10 : 15) = Fˆx1 ,Fˆx2 ,Fˆx3 ,Fˆx4 , Fˆy1 + Fˆy2 , Fˆy3 + Fˆy4 ˆ˙ ˆ˙ ,F ˆ˙ ˆ˙ ˆ˙ ˆ˙ ˆ˙ ˆ˙ xa(16 : 21) = F x1 x2 ,F x3 ,F x4 , F y1 + F y2 , F y3 + F y4 T T (20) (21) where u1 = [δ1 , δ2 , δ3 , δ4 ]T , and u2 = [Tb1 , Tb2 , Tb3 , Tb4 ]T For the output equation, the measurements of the longitudinal and lateral accelerations, the yaw rate, and the roll angle at the c.g., and each wheel angular velocity were used y = [ax , ay , Ω, ϕ, ω1 , ω2 , ω3 , ω4 ]T t For the second sliding-surface design σ = x ˜+γ x ˜dτ , the derivative of the sliding surface and the equivalent control can be written as 751 (22) The system noise Q was accordingly chosen by comparing its relative magnitude order with the corresponding noise covariance in order to obtain estimation accuracy and robustness under model uncertainty In particular, the covariance for tireforce term was set with large values to adopt a fast tire-force change during transient motion [20], [21] Noise covariance in Table II was determined by assuming a uniform noise distribution with a standard deviation given in [23]–[25] Although the estimation performance using the EKF is quite dependent on system model accuracy as well as sensor accuracy, this drawback of EKF can be addressed using vehicle-parameter identification [22] or the integration of inertial sensors with GPS by compensating sensor noise and bias [25] The longitudinal acceleration at each wheel u˙ xwi (i = 1−4) is approximated using the following kinematic equations based on the vehicle geometry: ˆ˙ i cos δi + νˆ˙i sin δi , ˆ˙ wi = u u (i = 1−4) (23) where ˆ˙ i = U˙ + lu,i · Ω2 u = (ax + V · Ω) + lu,i · Ω2 , lu,i ∈ [−La , −La , Lb , Lb ] νˆ˙i = V˙ + lν,i · Ω2 = (ay − U · Ω) + lν,i · Ω2 , lν,i ∈ tf tf t r tr ,− , ,− 2 2 For simplicity, the estimation of other terms in (19), such as longitudinal wheel speed, wheel slip ratio, Rw , and Jw , was not considered to determine the approximated equivalent torque By using the estimated tire forces and the longitudinal acceleration, the approximated equivalent control torque Tˆb,eq can be used for the following two different sliding-surface designs: Case 1) σ = x ˜ = λ − λd Jw (1 + λ) ˆ Tˆb,eq = Fˆx Rw + u˙ xw Rw (24) 752 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL 57, NO 2, MARCH 2008 Case 2) σ = x ˜+γ t x ˜dτ ˆ˙ xw Jw −γ(λ−λd ) uxw Jw Tˆb,eq = Fˆx Rw +(1+λ)u Rw Rw (25) 2) Switching Control Torque Tb,sw : The role of a switching control torque is to drive the system states to the sliding surface (σ = 0) By defining the switching control torque as Tb,sw = −Ksgn(σ), where K is a switching control gain and sgn(σ) is a sign function, a controlled brake torque becomes Tb = Tˆb,eq + Tb,sw = Tˆb,eq − Ksgn(σ) (26) The switching control gain K shown in (26) can be chosen by considering a stability condition and the limitation of the actuator The stability condition can be determined by the Lyapunov stability criteria For the given sliding-surface design, the stability condition can be expressed as σ σ˙ ≤ −η|σ|, where η is a strictly positive constant (27) For the first sliding-surface design σ = x ˜, by using (24) and (27), the following condition must be met for stability: σ Rw (1 + λ) Tˆb,eq − Ksgn(σ) − Fx Rw − u˙ xw uxw Jw uxw < −η|σ| (28) In order to obtain the switching control gain, let K = (uxw Jw /Rw )(F + η) so that it will satisfy the stability condition in (27), and substitute K into (28) After applying triangular inequality, the switching control gain can be written as uxw Jw (1+λ) ˆ u˙ xw − u˙ xw + η (29) K ≥ Rw Fˆx −Fx +Jw Rw Rw t For the second sliding-surface design σ = x ˜+γ x ˜dτ , the switching control gain that satisfies the Lyapunov stability criteria can be determined similar to (28) and (29) and expressed as σ (1 + λ) Rw Tˆb,eq − Ksgn(σ) − Fx Rw − u˙ xw uxw Jw uxw < −η|σ| (30) (1 + λ) ˆ u˙ xw − u˙ xw K ≥ Rw Fˆx − Fx + Jw Rw +γ uxw Jw uxw Jw |λ − λd | + η Rw Rw (31) are proportional to the maximum error percentage of the estimation values as follows: ˆ˙ xw A1 = C1 Fˆx , and A2 = C2 u (33) where C1 and C2 represent the maximum error percentages of a longitudinal tire force and a wheel center acceleration, respectively In addition, in order to avoid the chattering problem due to the imperfect switching control under the physical limits of the actuator or model uncertainty, the sign function is substituted by the following saturation function with the boundary-layer thickness Φ around the sliding surface: sat σ = Φ sgn(σ), σ Φ, if |σ| ≥ Φ otherwise (34) B Alternative Sliding-Surface Design Among the two sliding-surface designs previously introduced, the first sliding-surface design σ = x ˜ corresponds to the bang-bang control [26] In this design, an error dynamics between the tracking error and its derivative, such as an exponential convergence of tracking error, is not incorporated For t the second sliding-surface design σ = x ˜+γ x ˜dτ , the exponential error convergence can be found in σ˙ = x ˜˙ + γ x ˜ = However, there is a drawback for this design since the accumulated error of the sliding surface makes the switching control active after the tracking performance, i.e., x ˜=x ˜˙ = 0, is achieved In order to improve the convergence rate, the following sliding-surface design had been adopted in [26] and [27], i.e., σ=x ˜˙ + γ x ˜ However, the SMC approaches in [26] and [27] were not designed by analytical design procedure to determine an equilibrium control and a switching control; for instance, the different control rule for positive and negative sides of sliding surface was applied in [26], and the SMC approach in [27] was limited to the wheel-slip control using the PWM In this paper, we propose an analytical design procedure for the sliding surface σ = x ˜˙ + γ x ˜ The equivalent brake torque can be determined from the sliding condition of σ = and the Lyapunov stability condition for the slip-ratio error The equivalent torque for the proposed sliding surface can be derived similar to that of the second sliding-surface design, i.e., σ˙ = x ˜˙ + γ x ˜ = 0, as shown in (18) In order to determine the magnitude of the switching control gain, the Lyapunov method is applied on the slip-ratio error domain using the candidate of the Lyapunov function as V = x ˜ (35) In (29) and (31), it is assumed that the approximation errors of Fx and u˙ xw are bounded within A1 and A2 as follows: For stability, the time derivative of V (˜ x) should be less than or equal to zero By differentiating (35), we obtain ˆ˙ xw − u˙ xw ≤ A2 Fˆx − Fx ≤ A1 , and u V˙ = x ˜x ˜˙ ≤ (32) In general, A1 and A2 are considered as the design parameters, and the smaller boundaries mean more expensive estimations for the exact values Thus, in the robust design viewpoint, it is assumed that these approximation boundaries (36) From the sliding-surface equation, (36) can be rewritten as 1 ˙ ˙ V˙ = (σ − x ˜˙ )x ˜˙ = σ x ˜ − |x ˜| ≤ γ γ γ SHIM et al.: SLIDING-SURFACE DESIGN ON THE PERFORMANCE OF SLIDING MODE CONTROLLER IN ABS Since − (1/γ)|x ˜˙ |2 ≤ 0, (36) can be reduced as σx ˜˙ < 753 t ˜+γ Case 2) σ2 = x (37) x ˜dτ ˆ˙ xw Tb = Fˆx Rw + (1 + λ)u From (37) σx ˜˙ = σ − γ(λ − λd ) Rw (1+λ) (Tb −Rw Fx )− u˙ xw uxw Jw uxw Rw (1+λ) Tˆb,eq −Ksgn(σ)−Rw Fx − =σ u˙ xw uxw Jw uxw Rw ˆxw Jw ˆ˙ xw Jw −γ (λ−λd ) u =σ Rw Fˆx +(1+λ)u uxw Jw Rw Rw − Ksgn(σ)−Rw Fx − =σ (1+λ) u˙ xw uxw Rw sgn(σ) ≤ uxw Jw (1 + λ) ˆ Rw Fˆx − Fx + u˙ xw − u˙ xw uxw Jw uxw − γ(λ − λd ) − F sgn(σ) ≤ Then σ K2 = Rw A1 + Jw (1 + λ) uxw Jw A2 + γ |λ − λd | Rw Rw uxw Jw η Rw (42) + Case 3) σ3 = x ˜˙ + γ x ˜ − K3 sat K3 =Rw A1 +Jw Let K = (uxw Jw /Rw )F , and substitute it into the aforementioned equation, yielding σx ˜˙ = σ uxw Jw σ2 − K2 sat Rw Φ ˆ˙ xw Jw −γ(λ−λd ) uxw Jw Tb = Fˆx Rw +(1+λ)u Rw Rw Rw (1+λ) ˆ Fˆx−Fx + u˙ xw − u˙ xw −γ(λ−λd ) uxw Jw uxw −K Jw Rw (1 + λ) ˆ Rw Fˆx − Fx + u˙ xw − u˙ xw uxw Jw uxw σ3 Φ (1+λ) uxw Jw A2 +γ |λ−λd | Rw Rw (43) ˆ˙ xw | ≥ |u ˆ˙ xw − where A1 = C1 |Fˆx | ≥ |Fˆx − Fx |, A2 = C2 |u u˙ xw |, and η, γ, Φ, C1 , and C2 are the design parameters From (41)–(43), in addition to the difference of the sliding surface for each case, the switching term of Case 2) has one more term (γ) than Case 1) and one more term (η) than Case 3) Regarding Case 3), if the Lyapunov stability condition is defined using η as V˙ = x ˜x ˜˙ ≤ −η instead of (36), the switching term of Case 3) is the same as that of Case 2) This additional term (η) of Case 3) can be interpreted as the accelerating switching force at the beginning of controller activation However, in this paper, this term was not considered because it tends to attenuate the performance of slip-ratio tracking by causing a relatively large oscillation around a desired slip ratio − γ(λ − λd ) ≤ F |σ| (38) V S IMULATION /D ISCUSSION F can be obtained by using the triangle inequality (1 + λ) ˆ Rw Fˆx − Fx + u˙ xw − u˙ xw + γ |λ − λd | ≤ F uxw Jw uxw (39) By using the error boundary given in (32), the switching gain can be written as K ≥ Rw A1 + Jw (1 + λ) uxw Jw A2 + γ |λ − λd | Rw Rw (40) In summary, the control torque input with respect to each sliding-surface design is as follows: ˜ Case 1) σ1 = x σ1 Jw (1 + λ) ˆ Tb = Fˆx · Rw + u˙ xw − K1 sat Rw Φ K1 = Rw A1 + Jw (1 + λ) uxw Jw A2 + η Rw Rw (41) In order to assess the effect of controller performance due to the different sliding-surface designs, the three designs presented in the previous section have been compared through simulation For each case, the design parameters of the wheel-slip controller are tuned offline using a commercial optimization software, iSIGHT [17], for various target slip ratios and road conditions A sequential-quadratic-programming (NLPQL) algorithm in iSIGHT was used in order to minimize the performance index that is defined as the summation of the absolute error between the target slip ratio and the actual slip ratio during the simulation The smaller performance index can be interpreted as less chatter (oscillation) around the target slip ratio that reduces the brake load as well as increases the durability of the brake hardware Among the design parameters needed for a robust wheel-slip controller design, a sliding-surface design parameter γ and a switching-gain design parameter η are optimized to minimize the slip-ratio tracking error with preselected values for the rest of the design parameters The reason for only using γ and η as 754 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL 57, NO 2, MARCH 2008 the tuning parameters is to reduce computation load for lots of simulation case studies, because even though other parameters are fixed with an optimized parameter, tuning tends to lead to the similar result of the tracking performance regardless of the change of other parameters This point will be discussed later in the case study of the effect of the approximation error boundary As an example of the preselected design parameters, the maximum error boundaries for the estimated longitudinal tire force and the longitudinal wheel acceleration are each set at 50%, and the boundary-layer thickness Φ is set as the value which can prevent the chattering problem In particular, the boundary-layer thickness Φ affects on a guaranteed tracking precision ε for Case I) (ε = Φ = x ˜boundary ) and Case III) (ε = Φ = γ x ˜boundary ) [14] so that a preselected value, i.e., x ˜boundary = 0.025, was used in order to keep the tracking precision consistent for Cases I) and III) However, the integral sliding surface in Case II) has a freedom to determine the boundary-layer thickness because the tracking error can theoretically be zero as follows even though the boundary layer may have a steady-state value within the boundary layer This steady-state boundary-layer value σss can be derived by t ˜dt and considering the dynamics x˙ = x ˜ and letting x0 = x x ˜˙ from (16) and (42) Then, inside the boundary thickness, i.e., 0