Dynamic Slip-Ratio Estimation and Control of Antilock ... - IEEE Xplore

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 5, MAY 2009

Dynamic Slip-Ratio Estimation and Control of Antilock Braking Systems Using an Observer-Based Direct Adaptive Fuzzy–Neural Controller Wei-Yen Wang, Senior Member, IEEE, I-Hsum Li, Ming-Chang Chen, Shun-Feng Su, Senior Member, IEEE, and Shi-Boun Hsu

Abstract—This paper proposes an antilock braking system (ABS), in which unknown road characteristics are resolved by a road estimator. This estimator is based on the LuGre friction model with a road condition parameter and can transmit a reference slip ratio to a slip-ratio controller through a mapping function. The slip-ratio controller is used to maintain the slip ratio of the wheel at the reference values for various road surfaces. In the controller design, an observer-based direct adaptive fuzzy–neural controller (DAFC) for an ABS is developed to online-tune the weighting factors of the controller under the assumption that only the wheel slip ratio is available. Finally, this paper gives simulation results of an ABS with the road estimator and the DAFC, which are shown to provide good effectiveness under varying road conditions. Index Terms—Antilock braking systems (ABSs), observer-based direct adaptive fuzzy–neural controller (DAFC), road estimators.

I. I NTRODUCTION

B

RAKING under critical conditions, such as those encountered with wet or slippery road surfaces, panicked driver reactions, or mistakes committed by other drivers and pedestrians, can lead to wheel lock. This phenomenon is strongly undesirable, since the friction force on a locked wheel is usually considerably less than an unlocked wheel. Furthermore, while the wheels are locked, steering becomes impossible, leading to loss of control of the vehicle. Hence, preventing locking-up during the braking process is important for driver and passen-

Manuscript received January 12, 2006; revised March 20, 2007 and July 17, 2008. First published November 18, 2008; current version published April 29, 2009. This work was supported by the National Science Council, Taiwan, under Grant NSC 94-2213-E-030-011. W.-Y. Wang is with the Department of Applied Electronics Technology, National Taiwan Normal University, Taipei 106, Taiwan (e-mail: wywang@ntnu. edu.tw). I-H. Li is with the Department of Computer Science and Information Engineering, Lee-Ming Institute of Technology, Taipei 243, Taiwan (e-mail: [email protected]). M.-C. Chen is with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan (e-mail: [email protected]). S.-F. Su is with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan. S.-B. Hsu is with the Mstar Semiconductor Company, Hsinchu 302, Taiwan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2008.2009439

ger safety. For these reasons, antilock braking systems (ABSs) [1]–[5] have become one of the most common automotive technologies. The wheel slip ratio, which is the difference between the vehicle and the wheel speeds, is regarded as one of the most important process parameters affecting the quality of vehicle control. The main goal of most ABS systems is to ensure that the wheel slip ratio remains within the range of about 0.1–0.3, which is suitable for most road conditions. Most traditional ABS control strategies [6] aim to maintain the wheel slip ratio at 0.2, although this value is a compromise. In other studies [6], [7], known optimal slip ratios, with respect to various road characteristics, are utilized as reference signals, and good performance is shown. However, one main problem of these methods is determining how to find the optimal slip ratio if the road characteristics are unknown. In general, optimal slip ratios are highly dependent on particular road characteristics, such as whether the road is dry or wet. A tire/road friction estimator using only angular wheel velocity has been proposed to monitor the road characteristics [8]. In [8], a LuGre model [9], which effectively models tire/road friction based on the relative contact velocity, is used to form a tire/road friction estimator. In this paper, the LuGre tire/road estimator is applied to estimate the road characteristics and then supply the reference slip ratio to an ABS controller through a mapping function. A fuzzy–neural network (FNN) [10]–[12] is used to identify the mapping function from road characteristics to reference slip ratios. Hence, this ABS, based on the LuGre tire/road friction estimator, called LuGre-based ABS, can recognize the road characteristics and obtain a reference slip ratio. Regarding controllers, researchers have greatly improved the performance of ABS [13], [14] by utilizing many algorithms, such as sliding mode control [15], fuzzy logic control [16], adaptive control [17], genetic neural control [18], etc. All these methods assume that the system states are available for measurement. In traditional ABSs, certain states, such as the wheel acceleration and the brake-line pressure, are difficult to obtain or are prone to noise and other measurement errors. To resolve this problem, a state observer is required. Thus, the observer-based direct adaptive fuzzy–neural controller (DAFC) [19]–[22] is applied to ABS control, and its stability can be guaranteed by the universal approximation theorem. The DAFC also can overcome system uncertainties and disturbances. In this paper, we propose an ABS based on a road estimator, using the DAFC to force the wheel slip ratio to follow a reference slip ratio obtained by the road estimator.

0278-0046/$25.00 © 2009 IEEE

WANG et al.: DYNAMIC SLIP-RATIO ESTIMATION AND CONTROL OF ANTILOCK BRAKING SYSTEMS

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can be described as a function of the slip ratio λ, which is defined as λ=

ν − Rω . ν

(3)

A. Friction Model

Fig. 1.

The tire force includes the normal, longitudinal, and lateral forces. The normal force Fz is a function of the weight of the vehicle or the component of the weight acting in a vertical plane relative to the road surface. The longitudinal force Fx is effective at road-surface level; it allows the driver to apply throttle and then brakes to accelerate and slow down the vehicle as the vehicle is being steered. The friction coefficient is defined as μ = Fx /Fz . In previous research [9], [23], a road condition θ was introduced into the LuGre friction model. The friction force is shown as follows:

Dynamics of wheel rotational motion.

z˙ = νr − θ

δ0 |νr | z g(νr )

(4)

Fx = (δ0 z + δ1 z˙ + δ2 νr )Fz

(5)

g(νr ) = Fc + (Fs − Fc )e−| νs |

(6)

where νr 1/2

Fig. 2. for θ.

Static view of the distributed LuGre model under different values

The remainder of this paper is organized as follows. In Section II, the braking and vehicle systems are modeled. Section III introduces the concept of the road estimator. The observer-based DFAC is presented in Section IV. Section V gives the details of the ABS with DAFC and simulation results. Finally, Section VI states the conclusions. II. P ROBLEM F ORMULATION Considering the free-body diagram of a rolling tire with applied driving torque Tt shown in Fig. 1, the equation required to simulate the driving process and compute the wheel speed can be written as ω˙ =

Kb P i − Fx R Tb − Tt = I I

(1)

and the differential equation of vehicle longitudinal dynamics is ν˙ =

Fx mq

(2)

where ω is the angular velocity of the wheel; ν is the vehicle velocity; Fx is the longitudinal reactive force; Tb is the braking torque; mq is the mass of the quarter of the vehicle supported by the wheel; R is the tire rolling radius; I is the moment of inertia; and Kb is the gain between the pressure of the ABS, Pi , and the braking torque Tb . The control objective of the ABS is to regulate wheel slip in order to maximize the coefficient of friction between the wheel and the road for any road condition. In general, in Fig. 2, the coefficient of friction during braking

δ0 is the rubber longitudinal lumped stiffness; δ1 is the rubber longitudinal lumped damping; δ2 is the viscous relative damping; Fc is the normalized Coulomb friction; Fs is the normalized static friction; νs is the Stribeck relative velocity; z is the internal friction state; and νr is a relative velocity defined as Rω − ν, which is equal to −λν. The parameter θ is introduced to capture the changes in the road conditions. Fig. 2 shows that there are various μ versus λ curves under different values for θ, where μ is the friction coefficient and λ is the slip ratio. Therefore, as shown in Fig. 2, by defining the parameter θ, we can find the various μ versus λ curves. Furthermore, the various road conditions have the different μ versus λ curves. Hence, the road conditions can be obtained by observing the relationship between θ and the μ versus λ curves. We use (9)–(12) to calculate the friction coefficients μ under various road conditions θ for a fixed velocity νr . Curves for four values of θ are shown in Fig. 2. The curve λd is a best fit of the maximum values of μ for each curve θ. B. Second-Order Dynamic Slip-Ratio System In traditional ABSs, certain states, such as the wheel acceleration and the brake-line pressure, are difficult to obtain or are prone to noise and other measurement errors. To resolve this problem, we will use an observer-based DAFC introduced in Section IV to control the ABS following the reference slip ratio. By first differentiating (3) along with (1), we find the differential of λ to be 1 R (Fx R − Kb Pi ) + (1 − λ)ν˙ . (7) λ˙ = ν I

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˙ and z = [z z˙ z¨], aν means acceleraDefining λ = [λ λ] tion and a˙ ν means the derivative of aν , and then, the time derivative function of (7) can be expressed as 2 ˙ ˙ ν F (1 − λ) a ˙ − λa R X ν ¨= + λ Iν ν RKb P˙i + d(λ, ν, ν, ˙ Pi , z) Iν ˙ ν, z) + b(ν)u + d(λ, ν, ν, ≡ f˜(λ, λ, ˙ Pi , z) −

Defining the state vector x and output variable yr , respectively, as ⎡ ⎤ η x = ⎣χ⎦ z

⎡

Assumption 1: The road condition θ for a given section of road is assumed to be unchanging (i.e., θ is constant), and changes in road conditions between two different road sections are assumed to be instantaneous. Dynamic friction models can be used to suitably describe the tire/road contact friction. One of these models is the LuGre friction model. The potential advantages of this model are its ability to closely describe some of the physical phenomena found in tire/road friction and to possess a parameter θ directly related to road conditions. By substituting (4) and (5) into (1) and (2), respectively, and adding viscous rotational friction δω , with Assumption 1, we can derive a one-wheel vehicle model with the LuGre friction model used in previous studies [24], [25], called the LuGre-based ABS (9) (10) (11) (12)

where ur is an input variable for the road estimator and we have neglected the term δ2 in (10). In (9)–(12), we assume that only ω is measurable. To set our systems in the same framework as the classical dynamic system, the following change of coordinates is introduced: η = Rmq ν + Iω (13)

Substituting (13) into (9)–(12), the following are obtained: IFz δ2 Fz δ 2 η˙ = − η+ + R 2 Fz δ 2 − δ ω ω + u r mq mq δ0 δ0 χ˙ = − χ + I · − δω ω + u r δ1 δ1 δ0 |νr | z z˙ = νr − θ g(νr ) θ˙ = 0. (14)

⎤

⎡ ⎤ 0 ⎥ 0 0 ⎦ x + ⎣ 0 ⎦ θφ(yr , ur , x) 1 −1 − Rmq 0 0 ⎤ ⎡ 2 ⎡ ⎤ R Fz δ2 + I Fmz δq2 − δω 1 ⎥ ⎢ I δδ10 − δω +⎣ ⎦ yr + ⎣ 1 ⎦ u r 0 R+ I

(8)

⎢ x˙ = ⎣

− Fmz δq2

0

0

− δδ01

Rmq

III. E STIMATOR FOR T IRE /R OAD C ONTACT F RICTION

χ = Iω + RFz δ1 z.

(15)

we can then rewrite (14) as follows:

˙ ν )/ν); ˙ ν, z) = (R2 F˙X /Iν) + ((1 − λ)a˙ ν − λa where f˜(λ, λ, ˙ Pi , z) represents b(ν) = −(RKb /Iν); and u = P˙ i . d(λ, ν, ν, the nonlinearities obtained by differentiating λ˙ with respect to ν on the premise that other parameters are treated as constants.

˙ + F z δ 2 νr mq ν˙ = Fz (δ0 z + δ1 z) I ω˙ = − RFz (δ0 z + δ1 z) ˙ − δω ω + u r δ0 |νr | z z˙ = νr − θ g(νr ) θ˙ = 0

yr = ω

yr = [ 0

−RFz δ1 I

1 I

]x

(16)

where φ(yr , ur , x) = ν=

δ0 |Ryr − ν| z g(Ryr − ν)

(17)

η − I · yr . Rmq

(18)

Under the hypotheses from previous methods [8], an estimator structure is proposed for the LuGre-based ABS shown in (14), and it can be expressed as ⎡

⎤

⎡ ⎤ 0 ⎥ ˆ r , ur , x) 0 − δδ01 0 ⎦ x ˆ + ⎣ 0 ⎦ θφ(y 1 −1 − Rmq 0 0 ⎤ ⎡ 2 ⎡ ⎤ ⎡ ⎤ R Fz δ2 +I Fmz δq2 −δω 1 k1 ⎥ ⎢ I δδ10 −δω +⎣ ⎦yr + ⎣ 1 ⎦ur + ⎣ k2 ⎦(yr − yˆr ) I k3 0 R+ Rm q ⎡ ⎤ 0 + ⎣ 0 ⎦ 2θmax (fmax +f (ˆ x) sgn(yr − yˆr ) −1

⎢ ˆ˙ = ⎣ x

− Fmz δq2

0

0

νr | δ0 |ˆ ˙ zˆ(ω− yˆr ) θˆ= γ g(ˆ νr ) yˆr = [ 0

1 I

−RFz δ1 I

ˆ ]x

(19)

where δ0 f (ˆ x) = Fc

I R+ Rmq

|ˆ η| ysup + Rmq

|ˆ z |.

(20)

ηˆ, χ, ˆ zˆ, yˆr , and νˆr are the estimated values of η, χ, z, yr , and νr , respectively; θmax , fmax , and ysup are the maximum values of θ, f (ˆ x); and yr , respectively, and γ is the learning rate. The estimator gain vector of the road estimator is K = [k1 k2 k3 ].


Theorem 1 [8]: The road estimator can ensure that ˆ = x under the following limt→∞ θˆ = θ and limt→∞ x conditions, with ⎤ ⎡ Fz δ2 0 0 − mq ⎥ ⎢ − δδ01 0 ⎦ Ar = ⎣ 0 1 − Rm 0 0 q ⎡ ⎤ 0 Br = ⎣ 0 ⎦ −1 ⎤ ⎡ 2 R Fz δ2 + I Fmz δq2 − δω ⎥ ⎢ I δδ10 − δω Er = ⎣ ⎦ I R + Rm q ⎡ ⎤ 0 Rr = ⎣ 0 ⎦ −1 Cr = [ 0

1 I

−RFz δ1 I

].

and using (1)–(3), we have ˜ T Qr x ˆ )|− y˜r ν1 ˜ +2 |˜ yr θφ(yr , ur , x)−φ(yr , ur , x V˙ r ≤− x ˜ +2 |˜ ˜ T Qr x ˜ |− y˜r ν1 yr θ ρ(yr , ur ) x ≤− x ˜ +2|˜ ˜ T Qr x yr ||θ| (f (x)+f (ˆ ≤− x x))− y˜r ν1 ˜ +|˜ ˜ T Qr x yr | [−2θmax (fmax +f (ˆ ≤− x x))+sgn(˜ yr )ν1 ] (27) which suggests that ν1 should be defined to have a high-gain component ν1 = 2θmax (fmax + f (ˆ x)) sgn(˜ yr ).

V˙ ≤ −q˜ x2 .

(ii)

∀x1 , x2

|φ(yr , ur , x)| ≤ f (x) ≤ fmax ∀|x| < ∞, ∀|yr | < ∞, |ur | < ∞.

˜ → 0 when t → ∞. Then, x ˜ and θ˜ are Thus, we can find x ˜ ˜ → 0; θ converges to zero when t → ∞. bounded, and x

t→∞

(22)

Condition 4: The map T (s): ψ → y¯r of the system ˜˙ = Ar − Kr CT ˜ + Br ψ x r x ˜ y˜r = CT rx

(23)

˜ =x−x ˆ and y ˜r = with [Ar − Kr CT r ] as Hurwitz, where x ˆ r such that yr − y T T Pr Ar − Kr CT Pr = − Qr r − Ar − Kr Cr Pr Br = Cr .

(24)

Proof: Defining the Lyapunov function ˜ T Pr x ˜+ Vr = x

1 ˜2 θ γ

(25)

where

λ˙ = Aλ + B (f (λ) + bu + d) y = CT λ where

0 A= 0

1 0

(30) 0 B= 1

θ˜ = θ − θˆ (26)

1 C= 0

˙ T = [λ1 λ2 ]T ∈ R2 is a vector of state. Deand λ = [λ λ] fine the output tracking error e = ym − y = λd − λ, the reference vector ym = [ym y˙ m ]T = [λd λ˙ d ]T , and the tracking error vector e = [e e] ˙ T = [e1 e2 ]T . Based on the certainty equivalence approach, an optimal control law is u∗ =

1 ˆ −f (λ) + y¨m + KT ce b

(31)

ˆ denotes the estimate of e and Kc = [k2c k1c ]T is the where e feedback gain vector chosen such that the characteristic polynomial of A − BKT c is Hurwitz because (A, B) is controllable. Since only the system output (the slip ratio) y = λ is assumed to be measurable and f (λ) is assumed to be unknown, the optimal control law (31) cannot be implemented. Thus, we suppose that a control input u is u = uf + usν

˜ =x − x ˆ x ˜ y˜r = yr − yˆr = CT rx

IV. O BSERVER -B ASED A DAPTIVE F UZZY –N EURAL C ONTROL

(21)

Condition 3: The trajectories φ(yr (t), ur (t), x(t)) of the system satisfy lim φ (yr (t), ur (t), x(t)) = 0.

(29)

In this paper, the control objective is to design a DAFC such that the slip ratio y = λ follows a reference slip ratio ym = λd . First, we convert the tracking problem to a regulation problem; hence, (8) is rewritten as

|φ(yr , ur , x1 ) − φ(yr , ur , x2 )| ≤ ρ(yr , ur )|x1 − x2 |

(28)

With this choice of ν1 , we have

Condition 1: (Ar , CT r ) is an observable pair. Condition 2: There exists a known function ∞ > ρ0 ≥ ρ(yr , ur ) ≥ 0 and a known upper bound fmax such that either (i)

1749

(32)

where uf is designed to approximate the optimal control law (31) and the control term usν is employed to compensate for the external disturbances and modeling error.

1750


vector, and ϕ = [ϕ1 ϕ2 · · · ϕh ]T is a fuzzy basis vector, where ϕi is defined as 2 j=1

i

ϕ (e) =

h i=1

μAij (ej )

2 j=1

.

(36)

μAij (ej )

In this paper, we set the preconditions of the membership functions on the universe of discourse in [emin , emax ], where emin and emax , respectively, represent the minimum and maximum of input variables. Moreover, the consequent singleton membership functions can be randomized in [θmin , θmax ], where θmin and θmax , respectively, represent the minimum and maximum of the output variable. The type-1 learning phase is adopted in this paper. By learning from incoming training data, the consequent singleton membership functions will approach the optimal value.

Fig. 3. Configuration of a fuzzy–neural approximator.

From (30)–(32), we have ˆ + B [bu∗ − buf − busν − d] e˙ = Ae − BKT ce e1 = CT e.

(33)

Thus, the tracking problem has been converted into a regulation problem of designing an observer for estimating the vector e in (33) in order to regulate e1 to zero. A. FNNs The configuration of the FNN is shown in Fig. 3. At layer I, input nodes stand for the input linguistic variables e1 and e2 . At layer II, nodes represent the values of the membership functions. At layer III, nodes are the values of the fuzzy basis vector ϕ. Each node of layer III performs a fuzzy rule. The links between layers III and IV are fully connected by the weighting factors θ c = [p1 p2 · · · ph ]T , i.e., the adjusted parameters. At layer IV, the output stands for the value of uf in (32). The fuzzy inference engine uses the fuzzy IF–THEN rules to perform a mapping from an input linguistic vector e = [e1 e2 ] ∈ R2 to an output linguistic variable uf ∈ R. The ith fuzzy IF–THEN rule is written as Ri : If e1 is Ai1 and e2 is Ai2 then uf is B i

(34)

where Ai1 , Ai2 , and B i are fuzzy sets. Fuzzy systems with fuzzy rule base, product inference engine, singleton fuzzifier, and center average defuzzifier are of the following nonlinear mapping: h 2 i p μAij (ej ) uf =

i=1

h i=1

j=1

2

j=1

= θT c ϕ(e)

(35)

μAij (ej )

where μAij (ej ) is the membership function value of the fuzzy variable, h is the total number of IF–THEN rules, and the total number of fuzzy rules is 49. pi is the point at which μB i (pi ) = 1; θ c = [p1 p2 · · · ph ]T is an adjustable parameter

B. DAFC Design In this section, our primary tasks are to design an observer that estimates the state vector e in (33), to use the FNN to approximate the optimal control law u∗ in (31), and to develop the direct adaptive update law to adjust the parameters of the FNN in order to achieve the control objective. e) First, we replace uf in (32) by the output of the FNN θ Tc ϕ(ˆ in (35), i.e., T ˆ|θ c = θ T uf e e) (37) c ϕ(ˆ ˆ denotes the estimate of e. where e Next, consider the following observer that estimates the state vector e in (33): ˆ˙ = Aˆ ˆ + B(b¯ e e − BKT ν − busν ) + Ko (e1 − eˆ1 ) ce ˆ eˆ1 = CT e

(38)

where Ko = [ko1 ko2 ]T is the observer gain vector, chosen such that the characteristic polynomial of A − Ko CT is strictly Hurwitz because (C, A) is observable. The control term ν¯ is employed to compensate for the external disturbance d and the ˜ =e−e ˆ modeling error. We define the observation errors as e and e˜1 = e1 − eˆ1 . Subtracting (38) from (33), we have ˜˙ = (A − Ko CT )˜ e e + B [bu∗ − buf (ˆ e|θ c ) − b¯ ν − d] ˜. e˜1 = CT e

(39)

Moreover, the output error dynamics of (39) can be given as e˜1 = H(s) [bu∗ − buf (ˆ e|θ c ) − b¯ ν − d]

(40)

where s is the Laplace variable and H(s) = CT (sI − (A − Ko CT ))−1 B is the transfer function of (40). ˆ belong to compact sets Assumption 2 [19]: Let e and e e ∈ 2 : ˆ e ≤ Ue = {e ∈ 2 : e ≤ me < ∞} and Ueˆ = {ˆ meˆ < ∞}, respectively. It is known a priori that the optimal


parameter vector θ ∗c = arg minθc ∈Mθc [supe∈Ue ,ˆe∈Ueˆ |u∗ (e) − uf (ˆ e|θ c )|] lies in some convex region Mθc = {θ c ∈ h : θ c ≤ mθc }, where the radius mθc is constant. Based on Assumption 2, (39) can be rewritten as ˜˙ = A−Ko CT e ˜ +B [buf (ˆ e e|θ ∗c )−buf (ˆ e|θ c )−b¯ ν +τm −d] ˜ e˜1 = CT e

(41)

where τm = bu∗ − buf (ˆ e|θ ∗c ) is an approximation error. Based on (37), (41) can be rewritten as T ˜ ϕ(ˆ ˜ + B bθ ˜˙ = A − Ko CT e e e) − b¯ ν + τm − d c ˜ e˜1 = CT e

(42)

∗

˜ = θ ∗ − θ c . Since only the output e˜1 in (42) is aswhere θ c c sumed to be measurable, we use the strictly positive real (SPR)– Lyapunov design approach to analyze the stability of (42). Equation (42) can be rewritten as T ˜ ϕ(ˆ e˜1 = H(s) bθ e) − b¯ ν + τm − d c

(43)

where H(s) = CT (sI − (A − Ko CT ))−1 B is a known stable transfer function. In order to employ the SPR–Lyapunov design approach, (43) can be written as T ˜ ϕ(ˆ e˜1 = H(s)L(s) θ e ) − νf + τ f c

(44)

˜ T ϕ(ˆ ˜ T ϕ(ˆ where τf = L−1 (s)τT , τT = τm −d−L(s)θ e)+bθ e), c c ν ]. L(s) is chosen so that L−1 (s) is a proper and νf = L−1 (s)[b¯ stable transfer function and H(s)L(s) is a proper SPR transfer function. Suppose that L(s) = s + b1 , such that H(s)L(s) is a proper SPR transfer function. Then, the state-space realization of (44) can be written as ˜+ ˜˙ = Ac e e

˜ T ϕ(ˆ Bc θ e) c

− νf + τ f

with the projection operator Pr (κ˜ e1 ϕ(ˆ e)) = κ˜ e1 ϕ(ˆ e) − κ

2 where Ac = (A − Ko CT ) ∈ 2×2 , BT c = [0 b1 ] ∈ , and T 2 Cc = [1 0] ∈ . For the purpose of the stability analysis of the DAFC controller, the following assumptions and lemma are required. Assumption 3: The uncertain nonlinear function b is bounded by

β1 ≤ b ≤ β2

e) e˜1 θ T c ϕ(ˆ θc 2 θ c

(46)

(49)

where κ is a positive number called the learning rate, which determines the rate of learning, and mθ is a boundary of the adjusted parameter θ c . The supervised control law is ⎧ ρ, ⎪ ⎨ −ρ, ν¯ = ⎪ ⎩ ρ˜ e1 /α,

if e˜1 ≥ 0 and |˜ e1 | > α e1 | > α if e˜1 < 0 and |˜ where α is a positive constant if |˜ e1 | ≤ α

(50)

where ρ means supervised control. If ρ is chosen to be arbitrarily large, the control law can achieve stabilization. Then, with the supervised control law ν¯, e˜1 (t) converges as t → ∞. Proof: Consider the Lyapunov function candidate V =

1 ˜T ˜ 1 T ˜ P˜ e e+ θ θc 2 2κ c

(51)

where P = PT > 0. Differentiating (51) with respect to time and inserting (45) in the aforementioned equation yields T T 1 T T ˜c. ˜ ϕ−νf +τf + 1 θ ˜˙ θ ˜ Ac P+PAc e ˜ +˜ V˙ = e eT PBc θ c 2 κ c (52) Considering that H(s)L(s) is SPR, there exists P = PT > 0 such that AT c P + PAc = − Q PBc = Cc

(45)

(53)

where Q = QT > 0. By using (53), (52) becomes T T 1 T ˜c. ˜ ϕ − νf + τ f + 1 θ ˜˙ θ ˜ Q˜ V˙ = − e e + e˜1 θ c 2 κ c Let ν¯ be given as ⎧ ρ, ⎪ ⎨ −ρ, ν¯ = ⎪ ⎩ ρ˜ e1 /α,

if e˜1 ≥ 0 and |˜ e1 | > α e1 | > α if e˜1 < 0 and |˜ where α is a positive constant if |˜ e1 | ≤ α

(54)

where ρ ≥ (ε/β1 ). By using Assumptions 3–4, (50), and the e2 ≥ λmin (Q)|˜ e1 |2 , where λmin (Q) > 0, fact that λmin (Q)˜ we have

where gl = β1 and g u = β2 are given positive constants. Assumption 4: τT is assumed to satisfy

where ε is a positive constant.

Theorem 2: From Assumptions 2, 3, and 4 and the following adaptive law: ⎧ e1 ϕ(ˆ e), if θ ⎨ κ˜ c < mθc or θ c < mθc and e˜1 θ T e) ≥ 0 θ˙ c = c ϕ(ˆ ⎩ e1 ϕ(ˆ e)) , if θ c < mθc and e˜1 θ T e) < 0 Pr (κ˜ c ϕ(ˆ (48)

˜ e˜1 = CT ce

|τT | ≤ ε

1751

(47)

T 1 ˜c . ˜Tϕ + 1 θ ˜˙ θ V˙ ≤ − λmin (Q)|˜ e1 |2 + e˜1 θ c 2 κ c

(55)

1752


Fig. 4. Overall system of LuGre-based ABS.

Inserting (48) in (55) and after some manipulation yields 1 V˙ ≤ − λmin (Q)|˜ e1 |2 . 2

(56)

Equations (51) and (56) only guarantee that e˜1 (t) ∈ L∞ and e(t) ∈ L∞ ; however, these equations do not guarantee the convergence. Considering that all variables in the right-hand side of (45) are bound, e˜˙ 1 (t) is bound, i.e., e˜˙ 1 (t) ∈ L∞ . Integrating both sides of (56) and some manipulation yields ∞ 0

V (0) − V (∞) |˜ e1 |2 dt ≤ 1 . 2 λmin (Q)

(57)

Since the right side of (57) is bound, e˜1 (t) ∈ L2 . Using e1 (t)| = 0. This comBarbalat’s lemma [19], we have limt→∞ |˜ pletes the proof. Theorem 3: Consider the nonlinear system (30) that satisfies Assumptions 2–4. Suppose that the control law is e|θ c ) + usν u = uf (ˆ

(58)

with the state observer (38) and the adaptive law (48) for the nonlinear ABS dynamic system (8). Let usν = ν¯. Then, all signals in the ABS dynamic system are bounded, and e1 (t) converges to zero as t → ∞. e1 (t)| = 0 and Proof: From Theorem 2, we have limt→∞ |˜ ˜(t) ∈ L∞ . Using (38) and the fact that usν = ν¯, we obtain e ˆ˙ = A − BKT ˜ ˆ + Ko CT e e c e ˆ. e˜1 = CT e

(59)

Similarly, considering that A − BKT C is a Hurwitz matrix and ˜(t) is bounded, e ˆ(t) is bounded. From e ˜ =e−e ˆ, it follows e ˆ, e, and that e1 , e ∈ L∞ , and e1 (t) → 0 as t → ∞. From e ym ∈ L∞ , it follows that λ ∈ L∞ . The boundedness of y(t) follows from that of e1 (t) and ym (t). This completes the proof.

Design Algorithm: Step 1: Select the feedback and observer gain vectors Kc and T Ko such that the matrices A − BKT c and A−Ko C are Hurwitz matrices. Step 2: Choose an appropriate value for ρ and κ. Step 3: Solve the state observer in (38), using ν¯ in (48). ˆ(t). Then, from (36), comStep 4: Construct fuzzy sets for e pute the fuzzy basis vector ϕ. Step 5: Obtain the control (58) and update (48) laws.

V. O VERALL C ONTROL S YSTEM AND S IMULATION R ESULTS Optimal slip ratios vary according to the individual road surfaces, such as a dry or wet road. However, traditional slipratio control methods preset a slip ratio, which is a compromise value for different road conditions. The ABS with the LuGre friction model (LuGre-based ABS) proposed in this paper can recognize the road condition and then supply the corresponding reference slip ratio to the vehicle system through a mapping function. To monitor the road situation in the LuGre-based ABS, it is only necessary to have the angular wheel velocity ω, which is easily measured by actual sensors. The overall control system is shown in Fig. 4. The mapping function λd = g(θ) is shown in Fig. 2. It maps the road condition parameter θ to the reference slip ratio λd . The relationship of θ and λd is given in Table I. FNNs are well-known nonlinear approximators [6], [7], particularly for function approximation. Thus, to quickly and accurately learn the mapping function g(θ), we adopt FNNs as the learning mechanism. We will discuss the details of the FNN in the simulation examples. This section presents the simulation results of the proposed LuGre-based ABS to show that the tracking error of the closedloop system can approach an arbitrarily small value. The data used for the simulation are shown in Table II. An FNN is utilized to approximate the mapping function g(θ) based on the data shown in Table I. The initial weight vector of the FNN is random in [−2, 2]. Each input of the FNN has six membership functions with centers [0 0.4 0.8 1.2 1.6 2] and a width of 0.35. After training, the weighted vector is [0.2951 0.2 0.3 0.008 -0.522 0.196], with an mse of


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TABLE I RELATION OF ROAD CONDITION PARAMETER θ AND R EFERENCE S LIP R ATIO λd

TABLE II DATA USED FOR SIMULATION

Fig. 6. Vehicle velocity (ν) and rotation rate (ω).

Fig. 7. Estimated slip ratio λd and tracking slip ratio λ.

Fig. 5.

Estimated parameter θˆ and curve of θ.

10−12 . Two examples are used to test the proposed LuGrebased ABS. The first assumes that the road condition θ is 0.4 (dry road) and the parameters relating to the road estimator are θmax = 0.5, fmax = 800, ysup = 140, and γ = 1100. The parameters relating to DAFC are κ = 50, ρ = 4000, [e1 min , e1 max ] = [−0.0333 0.2333], [e2 min , e2 max ] = [−0.866 0.4], [θmin , θmax ] = [0 1], Kc = [144 24]T , and Ko = [115 115]T .

Fig. 5 shows the curve for the dry road condition θ = 0.4 and ˆ Fig. 5 also shows that the road estimator is of the estimate θ. not effective when the relative velocity is too small. Since the LuGre model models tire/road friction based on the relative contact velocity, a smooth variation over the whole learning process and good parameter tracking is obtained, as long as the relative contact velocity is sufficiently different from zero. Fig. 6 shows the curves of the vehicle speed, as it is reduced from 33.3 m/s in 2.1 s, and the corresponding velocity of the wheel. Fig. 7 shows that after a very short period of transient response, the wheel slip λ has approached the reference slip ˆ = 0.28). Fig. 8 shows the control signal for ratio (λd = g(θ) the dry road. In the second example, the road conditions are altered as follows: wet surface for 1.5 s and dry surface for the rest of the braking time, with corresponding road conditions of 0.8 and 0.4. The parameters relating to the road estimator are θmax = 0.9, fmax = 800, ysup = 140 and γ = 1100. Fig. 9 shows the evolution of the estimate θˆ when the road surface varies. As we know, wet surfaces need more time and are more difficult to estimate than dry surfaces. Again, since the LuGre model models tire/road friction based on the relative contact velocity, Fig. 9 shows that the road estimator is not effective when the

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Fig. 11.

Estimated slip ratio λd and tracking slip ratio.

Fig. 12.

Control input u.

Fig. 8. Control input u.

Fig. 9. Estimated parameter θˆ and curve of θ.


relative velocity is too small. Although Fig. 9 seems to show that an estimation error exists, Fig. 10 shows the curves of the vehicle speed (as it is reduced from 33.3 m/s in 2.93 s) and the corresponding velocity of the wheel. Fig. 11 shows that the estimated slip ratios are close to the desired values and can be used for reference signals. In Fig. 11, for both the wet ˆ = 0.2) and the dry (λd = g(θ) ˆ = 0.28) surfaces, (λd = g(θ)

it can be observed that after a very short period of transient response, the wheel slip ratio λ has approached the reference slip ratios. Fig. 12 shows the control signal for this example. Finally, we make some experiments to compare the proposed controller with a controller without a road estimator. Table III compares our system to a system without a road estimator, in two different scenarios. In the first scenario, the road is wet (θ = 0.8) for 30 m and then changes to dry (θ = 0.4). In the second scenario, the road is snowy (θ = 1.0) for 30 m, then wet. As shown by the braking times and distances, using a road estimator improves performance. Figs. 13–16 show various aspects of system performance for our system in the second scenario (snowy/wet). Fig. 13 shows the values for the road parameter θˆ as the road condition θ changes. Fig. 14 shows the estimated slip ratio λd versus the tracking slip ratio λ. Fig. 15 shows the vehicle speed ν and angular wheel velocity ω. Finally, Fig. 16 shows the control input u. VI. C ONCLUSION In this paper, we have introduced a road estimator based on the LuGre friction model into an ABS system, called the LuGre-based ABS. This solves the traditional problem of slipratio control when the road condition is unknown. In ABSs, certain states are difficult to obtain by real physical sensors or are prone to noise and other measurement errors. To resolve this problem, the observer-based DAFC is applied to ABS control,


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TABLE III COMPARISON FOR THE PROPOSED SYSTEM TO A SYSTEM WITHOUT A ROAD ESTIMATOR

Fig. 13. Estimated parameter θˆ and curve of θ.

Fig. 16. Control input u.

and its stability is guaranteed by the universal approximation theorem. The simulation results show that the road estimator effectively and quickly captures the road conditions and that the DAFC can force the output to track the reference slip ratio obtained by the road estimator. R EFERENCES

Fig. 14. Estimated slip ratio and tracking slip ratio.


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Wei-Yen Wang (SM’04) received the M.S. and Ph.D. degrees in electrical engineering from National Taiwan University of Science and Technology, Taipei, Taiwan, in 1990 and 1994, respectively. From 1990 to 2006, he was, concurrently, a Patent Screening Member of the National Intellectual Property Office, Ministry of Economic Affairs, Taiwan, where, in 2003, he was a Patent Agent. In 1994, he was an Associate Professor with the Department of Electronic Engineering, St. John’s and St. Mary’s Institute of Technology, Taipei. From 1998 to 2000, he was with the Department of Business Mathematics, Soochow University, Taipei. From 2000 to 2004, he was with the Department of Electronic Engineering, Fu Jen Catholic University, Taipei, where he was a Full Professor in 2004. In 2006, he was a Professor and Director of the Computer Center, National Taipei University of Technology, Taipei. Currently, he is a Professor with the Department of Applied Electronics Technology, National Taiwan Normal University, Taipei. His current research interests and publications are in the areas of fuzzy logic control, robust adaptive control, neural networks, computeraided design, digital control, and charge-coupled-device-camera-based sensors. He has authored or coauthored over 100 refereed conference and journal papers in the aforementioned areas. He is an Associate Editor of the International Journal of Fuzzy Systems; a member of the Editorial Board of the International Journal of Soft Computing and of the Open Cybernetics and Systemics Journal; and an Area Editor of the International Journal of Intelligent Systems Science and Technology. Dr. Wang serves as an Associate Editor of the IEEE TRANSACTIONS ON S YSTEMS , M AN , AND C YBERNETICS —P ART B: C YBERNETICS , and an Associate Editor of the IEEE Computational Intelligence Magazine.


I-Hsum Li was born in Taipei, Taiwan, in 1975. He received the M.S. degree in electronic engineering from Fu Jen Catholic University, Taipei, in 2001, and the Ph.D. degree from National Taiwan University of Science and Technology, Taipei, in 2007. Currently, he is an Assistant Professor with the Department of Computer Science and Information Engineering, Lee-Ming Institute of Technology, Taipei. His research interests include genetic algorithms, fuzzy logic systems, adaptive control, system identification, and antilock braking systems.

Ming-Chang Chen was born in Taipei, Taiwan, in 1981. He received the M.S. degree in electrical engineering from Fu Jen Catholic University, Taipei, Taiwan, in 2005. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei. His research interests include fuzzy logic systems and adaptive and intelligent controls.

Shun-Feng Su (S’89–M’91–SM’05) received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1983, and the M.S. and Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 1989 and 1991, respectively. He is currently a Professor with the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan. He has published more than 90 refereed journal and conference papers in the areas of robotics, intelligent control, fuzzy systems, neural networks, and nonderivative optimization. His current research interests include neural networks, fuzzy modeling, machine learning, virtual-reality simulation, intelligent transportation systems, data mining, and intelligent control.

Shi-Boun Hsu was born in Kaohsiung, Taiwan, in1979. He received the M.S. degree in electrical engineering from Fu Jen Catholic University, Taipei, Taiwan, in 2004. He is currently with Mstar Semiconductor Company Hsinchu, Taiwan, as a Senior System Engineer. His research interests include mixed-mode signal and system-on-a-chip control engineering.