... Braking System. Andon V. Topalov, Erdal Kayacan, Yesim Oniz, and Okyay Kaynak ..... [9] Inteco, âUser's manual: The laboratory antilock braking system con-.
Proceedings of the 7th Asian Control Conference, Hong Kong, China, August 27-29, 2009
FrA3.4
Adaptive Neuro-Fuzzy Control with Sliding Mode Learning Algorithm: Application to Antilock Braking System Andon V. Topalov, Erdal Kayacan, Yesim Oniz, and Okyay Kaynak be satisfactory. Moreover, sensor signals are usually highly uncertain and noisy [2].
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Because of the highly nonlinear and uncertain structure of ABS, many difficulties arise in the design of a wheel slip regulating controller. Sliding mode control is a preferable option, as it guarantees the robustness of the system against changing working conditions. Kachroo and Tomizuka proposed a sliding mode controller in [3] that can maintain the wheel slip at any desired value. Unsal, et. al.[4] proposed a sliding mode observer to track the reference wheel slip, and a PI-like controller is used to reduce the chattering problem. A self learning fuzzy controller is combined with a sliding mode controller in [5]. The stability of the system is guaranteed, as the tuning algorithms for the controller are derived in the Lyapunov sense. Besides sliding mode control, there are several intelligent control schemes including fuzzy logic control, adaptive control, and neural network approach[6]-[7]. In [8], an observerbased direct adaptive fuzzy-neural controller for an ABS is developed under the assumption that only the wheel slip is measurable. To track the varying desired slip value, an observer-based output feedback control law and update law for online tuning of the weighting factors of the direct adaptive fuzzy-neural controller are derived.
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Abstract— A neuro-fuzzy adaptive control approach for nonlinear dynamical systems, which are coupled with unknown dynamics, modeling errors, and various sorts of disturbances, is proposed and used to design a wheel slip regulating controller. The implemented control structure consists of a conventional controller and a neuro-fuzzy network-based feedback controller. The former is provided both to guarantee global asymptotic stability in compact space and as an inverse reference model of the response of the controlled system. Its output is used as an error signal by an on-line learning algorithm to update the parameters of the neuro-fuzzy controller. In this way the latter is able to gradually replace the conventional controller from the control of the system. The proposed learning algorithm makes direct use of the variable structure systems theory and establishes a sliding motion in terms of the neuro-fuzzy controller parameters, leading the learning error toward zero. In the simulations, it has been tested on the control of antilock breaking system model and the analytical claims have been justified under the existence of uncertainty and large nonzero initial errors.
I. INTRODUCTION
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During an emergency stop, tractive forces are generated between the road and the tire surface which may possibly cause a rapid increase in slip resulting in locking up of the wheels, i.e. the vehicle can not be steered by the driver. Antilock Braking Systems (ABS) are electronically controlled systems which sense the tendency towards a wheel lock up, and reduce the braking force to avoid skidding while braking, especially in slippery conditions. Although many attempts have been made over the decades, an accurate mathematical model of ABS has not been obtained yet. One of the shortcomings is that the controller must operate at an unstable equilibrium point for optimal performance. A small perturbation of the controller input may induce a drastic change in the output. Furthermore at present, there are no affordable sensors which can accurately identify the road surface, and make these data available to ABS controller. Additionally, an accurate measurement of vehicle’s absolute velocity is required to calculate the wheel slip. To solve this problem, number of estimators are proposed throughout the literature. For instance, in [1], recursive least squares method is used to estimate the real time vehicle velocity. Regarding the fact that the system parameters highly depend on the road conditions and vary over a wide range, the performance of ABS may not always
A. V. Topalov, E. Kayacan, Y. Oniz and O. Kaynak are with the Department of Electrical and Electronics Engineering, Bogazici University, 34342, Bebek, Istanbul, Turkey {topalov, erdal.kayacan,
yesim.oniz, o.kaynak}@ieee.org
A. V. Topalov is on leave from Technical University of Sofia, campus Plovdiv, Bulgaria
978-89-956056-9-1/09/©2009 ACA
The present paper addresses the design of an adaptive neuro-fuzzy controller that can be used to track the varying desired slip value in ABS. The proposed controller uses a new variable structure systems-based (VSS-based) on-line learning algorithm for parameters adaptation. It controls the error dynamics. The latter is defined as the control signal produced by a conventional controller connected in parallel and it is described using a differential equation. Differently from the gradient-based learning methods which aim to minimize an error function, here the learning parameters are tuned by the proposed algorithm in a way to enforce the error to satisfy this stable equation. The paper is organized as follows. Section II starts with an introduction to the quarter vehicle model describing longitudinal motion of the vehicle and angular motion of the wheel under braking, and continues with the proposed adaptive neuro-fuzzy control approach used to design an intelligent controller for tracking of the varying slip value. Then, the developed new variable structure systems-based method for parametric adaptation of fuzzy rule-based neural networks with a scalar output is presented. Section III is devoted to the obtained results from simulations. The concluding remarks are given in section IV.
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to Newton’s second law, the equation of the motion of the system can be written as: J1 ω˙ 1 = Ft r1 − (d1 ω1 + M10 + TB )
(1)
J2 ω˙ 2 = −(Ft r2 + d2 ω2 + M20 )
(2)
Ft in (1) and (2) stands for the road friction force which is given by Coulomb Law: Ft = μ(λ)Fn
(3)
Fn is calculated by the following equation: Fn =
d1 ω1 + M10 + TB + Mg L(sin φ − μ(λ) cos φ)
(4)
where L is the distance between the contact point of the wheels and the rotational axis of the balance lever of the upper wheel and φ is the angle between the normal in the contact point and the line L. Under normal operating conditions, the rotational velocity of the wheel would match the forward velocity of the car. When the brakes are applied, braking forces are generated at the interface between the wheel and road surface, which causes the wheel speed to decrease. As the braking force at the wheel increases, slippage will occur between the tire and the road surface. The wheel speed will tend to be lower than vehicle speed. The parameter used to specify the difference between the above two velocities is called wheel slip (λ), and it is defined as: r2 ω2 − r1 ω1 (5) λ= r2 ω2 While a wheel slip of 0 indicates that the wheel velocity and the vehicle velocity are the same, a ratio of 1 indicates that the tire is not rotating and the wheels are skidding on the road surface, i.e., the vehicle is no longer steerable. The road adhesion coefficient is a nonlinear function of some physical variables including wheel slip and it can be approximated using the following formula developed by [9]:
Schematic view of quarter vehicle model [9]
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Fig. 1.
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II. THE ADAPTIVE NEURO-FUZZY CONTROL APPROACH A. The System to be Controlled The free body diagram of the quarter vehicle model describing longitudinal motion of the vehicle and angular motion of the wheel under braking is presented in Fig. 1 [9]. Although the model is quite simple, it preserves the fundamental characteristics of an actual system. In deriving the dynamic equations of the system, several assumptions are made. First, only longitudinal dynamics of the vehicle is considered. The lateral and vertical motions are neglected. Furthermore, it is assumed that there is no interaction between the four wheels of the vehicle [10].
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Regarding the model, there are three torques acting on the upper wheel: Braking torque, friction torque in the upper bearing and the friction torque among the wheels. Similarly, two torques are acting on the lower wheel: The friction torque in the lower bearing and the friction torque between these wheels. System parameters are presented in Table I.
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TABLE I N OMENCLATURE
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Name ω1 ω2 TB r1 r2 J1 J2 d1 d2 Fn μ λ λd Ft M10 M20 Mg
c4 λp + c3 λ3 + c2 λ2 + c1 λ (6) a + λp The corresponding μ-λ curve for different road conditions can be seen in Fig. 2 μ(λ) =
Description Angular velocity of the upper wheel Angular velocity of the lower wheel Braking torque Radius of the upper wheel Radius of the lower wheel Moment of inertia of the upper wheel Moment of inertia of the lower wheel Viscous friction coefficient of the upper wheel Viscous friction coefficient of the lower wheel Total normal load Road adhesion coefficient Wheel slip Desired slip Road friction force Static friction of the upper wheel Static friction of the lower wheel Moment of gravity acting on balance lever
B. The Control Scheme and the Neuro-Fuzzy Network Structure The proposed control scheme is depicted on Fig. 3. The conventional proportional plus derivative (PD) controller is provided both as an ordinary feedback controller to guarantee global asymptotic stability in compact space and as an inverse reference model of the response of the system under control. The PD control law is described as follows: τc = kD e˙ + kP e
During deceleration, a braking torque is applied to the upper wheel, which causes wheel speed to decrease. According
(7)
where e = λd − λ is the feedback error, λd is the desired slip value, kP and kD are the controller gains. Consider a neuro-fuzzy network with two inputs and one output implemented as a feedback controller (the NFFC
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Rij : If x1 is Ai and x2 is Bj , then fij =ai x1 +bj x2 +dij
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where i = 1, ...I and j = 1, ...J It is further assumed that the output of each fuzzy if-then rule consists of the constant dij only, which is a widely used simplification. Each Gaussian membership function is defined by two parameters: its center c and the distribution σ which are among the tunable parameters of the fuzzy-neural structure. The strength of the rule Rij is obtained as a T-norm of the membership functions in the premise part (by using a multiplication operator):
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0.3 0.25 0.2 0.15 0.1 Dry Asphalt Packed Snow Rough Ice
0.05 0
0
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0.6
0.8
Wij = μAi (x1 )μBj (x2 )
1
Wheel Slip
Fig. 2.
(8)
The output signal of the fuzzy-neural network τn (t) is calculated as a weighted average of the output of each rule:
μ-λ curve for different road conditions
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=
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τn (t) =
J i=1 j=1 Wij fij I J i=1 j=1 Wij
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Road Adhesion Coefficient
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I J
¯ ij fij W
(9)
i=1 j=1
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¯ ij is the normalized value of the output signal of where W the neuron ij from the second hidden layer of the network:
i=1
Wij J j=1
(10)
Wij
The following vectors have been specified: T W (t) = W 11 (t) W 12 (t) ... W 21 (t) ... W IJ (t) : vector of the normalized output signals of the neurons from the second hidden layer, T σA = [σA1 σA2 ... σAi ... σAI ] : vector of the parameters defining the distribution of the Gaussian membership functions relevant to the first input of the network, T σB = σB1 σB2 ... σBj ... σBJ : vector of the parameters defining the distribution of the Gaussian membership functions relevant to the second input of the neuro-fuzzy network, T cA = [cA1 cA2 ... cAi ... cAI ] : vector of the parameters defining the centers of the Gaussian membership functions relevant to the first network input, T cB = cB1 cB2 ... cBj ... cBJ : vector of the parameters defining the centers of the Gaussian membership functions relevant to the second network input, The following assumptions have been used in this investigation: Both, the input signals x1 (t) and x2 (t), and their time derivatives x˙ 1 (t) and x˙ 2 (t) will be considered bounded:
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Fig. 3. Block diagram of the proposed adaptive neuro-fuzzy control scheme for ABS slip tracking
¯ ij = W I
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block on Fig. 3). Its structure is presented on Fig. 4. The ˙ are fuzzified incoming signals, x1 (t)=e(t) and x2 (t)=e(t), by using Gaussian membership functions, and are associated accordingly with I and J numbers of fuzzy labels which are determined by their corresponding membership functions μ.
Fig. 4.
The fuzzy-neural network
A fuzzy if-then rule base of Takagi-Sugeno type is used where the fuzzy sets are included in the premise part only. The corresponding rule Rij can be expressed as:
|x1 (t)| ≤ Bx , |x2 (t)| ≤ Bx
∀t
(11)
|x˙ 1 (t)| ≤ Bx˙ , |x˙ 2 (t)| ≤ Bx˙
∀t
(12)
where Bx and Bx˙ are known positive constants. The vectors defining the distributions and the centers of the Gaussian membership functions are assumed to be bounded as follows:
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(20)
σ˙ Ai = −
sA i αsgn (τc ) sTA sA
(21)
σ˙ Bj = −
sBj α sgn (τc ) sTB sB
(22)
¯ ij W f˙ij = − ¯ T ¯ αsgn (τc ) W W
(23)
c˙Bj = − σA ≤ Bσ , σB ≤ Bσ , cA ≤ Bc , cB ≤ Bc
(13)
where Bσ and Bc are some known positive constants. It will be assumed that, due to physical constraints, the time variable weight coefficients of the connections between the neurons in the second hidden layer and the output neuron are also bounded, i.e., ∀t
(14)
for some positive constant Bf . ¯ < 1. In addition From (8) to (13) it follows that 0 < W I ijJ ¯ it can be easily seen from (10) that i=1 j=1 Wij = 1. The control input of the system τ is determined as follows:
T
α>
τ = τc − τn
∀t
(16)
where Bτ and Bτ˙ are some known positive constants.
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Using the sliding mode control theory principles [11] the zero value of the learning error coordinate τc (t) can be defined as time-varying sliding surface, i.e.,
1 2 τ (t) 2 c
Vc =
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C. The Sliding Mode Learning Algorithm
(24)
then given an arbitrary initial condition τc (0), the learning error τc (t) converges to zero during a finite time th , and a sliding motion sustained on τc (t) = 0 for all t > th . Proof: Consider the following Lyapunov function candidate:
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|τ (t)| ≤ Bτ , |τ˙ (t)| ≤ Bτ˙
4Br Bf + Bτ˙ 1 − 8Bq Bf
m
(15)
It will be also adopted that τ and τ˙ are also bounded signals, i.e.
T
where , sA = [sA1 sA2 ... sAI ] , sB = [sB1 sB2 ... sBJ ] sAi = x1 − cAi , sBj = x2 − cBj and α is a sufficiently large positive design constant satisfying the inequality:
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|fij | ≤ Bf
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The time derivative of Vc is given by: V˙ c
(17)
= τc τ˙c = τc (τ˙n + τ˙ ) = ⎞ ⎛ J I ¯˙ ij + τ˙ ⎠ (26) ¯ ij + fij W f˙ij W = τc ⎝
rd
Sc (τn , τ ) = τc (t) = τn (t) + τ (t) = 0
.e
which is the condition that the neuro-fuzzy network is trained to become a nonlinear regulator to obtain the desired response during the tracking-error convergence movement by compensation for the nonlinearity of the controlled plant. The sliding surface for the nonlinear system under control ˙ is defined as: Sp (e, e)
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It can be easily shown that
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Sp (e, e) ˙ = e˙ + χe
c˙Ai = −
σ Ai αsgn (τc ) Tσ σA A
¯ ij ¯˙ ij = −W ¯ ij K˙ ij + W W
(18)
with χ being a constant determining the slope of the sliding surface. Definition: A sliding motion will have place on a sliding manifold Sc (τn , τ ) = τc (t) = 0 after a time th , if the condition Sc (t)S˙ c (t) = τc (t) τ˙c (t) < 0 is satisfied for all t in some nontrivial semi-open subinterval of time of the form [t, th ) ⊂ (−∞, th ). It is desired to devise a dynamical feedback adaptation mechanism, or online learning algorithm for the neuro-fuzzy network parameters such that the sliding mode condition of the above definition is enforced. Theorem 1: If the adaptation law for the parameters of the considered neuro-fuzzy network is chosen respectively as:
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J I i
¯ ij K˙ ij W
˙ A = x1 −cAi and B = where K˙ ij =2(AA˙ + B B), σ Ai Then V˙ c can be further expressed as follows:
(19)
V˙ = τc
.
¯ ij + fij (−W ¯ ij K˙ ij + f˙ij W
¯ ij +W
j
I J i
¯ ij K˙ ij W
) + τ˙ =
j
I J i
¯ ij − 2 f˙ij W
I J
j
i
j
¯ ij (AA˙ + B B)f ˙ ij + W
I J J I ¯ ¯ ˙ ˙ Wij fij Wij (AA + B B) + τ˙ = +2 i
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I J i
= τc
(27)
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The tracking performance of the slip control system in ABS can be analyzed by introducing the following Lyapunov function candidate:
I J A ¯ Wij fij = τc −αsgn(τc ) − 2 ˙ 1 σ Ai + 2 (x σ Ai i j
B + 2αsgn(τc )) + 2 x˙ 2 σBj + 2α sgn (τc ) + σBj J J I I A ¯ ¯ Wij fij Wij +2 ˙ 1 σAi + 2 (x σ Ai i j i j
B + τ˙ = + 2α sgn(τc )) + 2 (x˙ 2 σBj + 2α sgn(τc )) σB j J I ¯ ij fij rij − W = τc −α sgn(τc ) − 2
i
+ 4τn α sgn(τc )
¯ ij fij qij + 2τn W
j J I i
¯ ij qij + τ˙ W
J I i
¯ ij rij + W
j
=
Remark: The obtained result means that, assuming the sliding mode control task is achievable, using τc as a learning error for the NFFC together with the adaptation laws (1923) enforces the desired reaching mode followed by a sliding regime for the slip control system in ABS.
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J I ¯ ij qij (fij − τn ) |τc | − W = −α − 4α i
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III. SIMULATION RESULTS
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j
¯ ij Br (Bf + Bf ) + Bτ˙ |τc | = W
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J I
¯ ij Bq (Bf + Bf )+ W
j
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= |τc | −α(1 − 8Bq Bf ) + 4Br Bf + Bτ˙ < 0
(28)
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where rij and qij are defined as follows: rij =
A B x˙ 1 + x˙ 2 , σA i σBj
qij =
In this section, a number of computer simulated dynamic responses are obtained to investigate the performance of the proposed control algorithm. The sampling time is 0.5 ms and I = J = 3 for all the simulations. All figures below show simulation results for the quarter vehicle model with initial longitudinal velocity of V = 20m/s maneuvering on a straight line. The reference wheel slip, which is the slip value that corresponds to the peak value of tire road friction coefficient, is calculated for different road conditions, i.e. rough ice, compact snow and dry asphalt. During the simulation studies, the vehicle is considered going out from rough ice road conditions to compact snow and then from compact snow to dry asphalt. The numerical values used for the parameters of the quarter vehicle model in this study can be seen in [9] Fig. 5 shows the response of the system to the PD controller alone and the NFFC coupled with a PD controller. It can be seen the NFFC learns the system dynamics in a finite time duration, and makes the system performance better compared to the conventional PD controller alone. As explained in the previous sections, to obtain a mathematical model of ABS is very difficult and sometimes impossible because of the nonlinearities and noises both coming from inside and outside of the system. In such cases, the conventional controller cannot remain always well-tuned and the adaptive neuro-fuzzy controller structure proposed will help to maintain the desired performance. The control signals for both controller structures (the PD controller alone and the adaptive NFFC coupled with a PD controller) can be seen in Fig. 6. It can be observed that the neuro-fuzzy feedback controller has been able to
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J I
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≤ −α |τc | + 4α |τc | + 2 |τc |
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J I ¯ ij rij (fij − τn ) + τ˙ τc ≤ W − 2 i
(34)
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− 4α sgn(τc )
1 V˙ p = S˙ p Sp = 2 S˙ c Sc kD
|τc | ≤ 2 −α(1 − 8Bq Bf ) + 4Br Bf + Bτ˙ kD < 0, ∀Sc , Sp = 0
j
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i J I
1 2 S (33) 2 p Theorem 2: If the adaptation strategy for the adjustable parameters of the NFFC is chosen as in (19-21), then the negative definiteness of the time derivative of the Lyapunov function in (33) is ensured. Proof : Evaluating the time derivative of the Lyapunov function in (33) yields: Vp =
A B 2 + σ2 σA Bj i
|rij | ≤ Br , |qij | ≤ Bq
(29) (30)
and the positive constants Br and Bq are bounded by the following inequalities: Bx + B c Bx + Bc , Bq ≤ 2 (31) 2 Bσ Bσ3 The inequality (28) shows that the controlled trajectories of the learning error τc (t) converge to zero in a stable manner. The relation between the sliding line Sp and the zero P , is adaptive learning error level Sc , if χ is taken as χ = kkD determined by the following equation: Br ≤ 2Bx˙
kp e = kD S p Sc = τc = kD e˙ + kP e = kD e˙ + kD
(32)
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4
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2
Dry Asphalt
Time Derivative of Error
Wheel Slip
0.25
Compact Snow
0.2
0.15 Rough Ice
0.1
Reference NFFC+PD PD
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0
0
0.1
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0.3 0.4 Time (s)
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0
−2
−4
−6
−8 0.7
−10 −0.04
−0.03
−0.02
−0.01
0
0.01
Error
Fig. 5.
The wheel slip Fig. 7.
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emphasized is the computational simplicity of the proposed approach. V. ACKNOWLEDGMENTS
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The work of O. Kaynak was supported in part by the Bogazici University Research Fund Project 08A204 and in part by the TUBITAK Project 107E284.The work of A. V. Topalov was supported by the Ministry of Education and Science of Bulgaria Research Fund Project BY-TH108/2008.
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take over the control operation, thus becoming the leading controller after a short time period. This results in zero output from the conventional PD controller. The output of the PD controller becomes nonzero only during the time intervals when the vehicle goes from one road condition to the other. Fig. 7 shows the phase space behavior. It figures out that Sp =0 line is attracting invariant. Clearly the error vector is guided towards the sliding manifold and due the design presented, it is forced to remain in the vicinity of the attracting loci without explicitly knowing the analytical details of the equations of motion of the slip control system.
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Control Inputs
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0
τ
c
τ
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−2
0
0.1
0.2
Fig. 6.
0.3 0.4 Time (s)
0.5
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The control signal
IV. CONCLUSIONS A novel approach for generating and maintaining sliding regime in the behavior of a system with uncertainties in its dynamics is introduced. The system under control is under a closed-loop simultaneously with a conventional PD controller and an adaptive variable structure neuro-fuzzy controller. The presented results from a simulated control of the slip in ABS have demonstrated that the predefined sliding regime could be generated and maintained if the NFFC parameters are tuned in such a way that the reaching is enforced. Another prominent feature that should be
Phase space behaviour
R EFERENCES
[1] Q. Zhang, G. Liu, B. Liu, and X. Xie, “Sensor fusion based estimation technology of vehicle velocity in anti-lock braking system,” in Proc. of the 2007 International Conference on Information Acquisition, Jeju City, Korea, 2007, pp. 106–111. [2] M. R. Akbarzadeh, K. J. Emami, and N. Pariz, “Adaptive discrete-time fuzzy sliding mode control for anti-lock braking systems,” in Proc. of the Annual Meeting of the North American, Las Vegas, USA, 2002, pp. 554–559. [3] P. Kachroo and M. Tomizuka, “Sliding mode control with chattering reduction and error convergence for a class of discrete nonlinear systems with application to vehicle control,” in Proc. of the International Mechanical Engineering Congress and Expo, vol. 57, Chicago, USA, 1995, pp. 225–233. [4] C. Unsal and P. Kachroo, “Sliding mode measurement feedback control for antilock braking systems,” Control Systems Technology, vol. 7, pp. 271–281, 1999. [5] C. Lin and C. F. Hsu, “Self-learning fuzzy sliding-mode control for antilock braking systems,” Control Systems Technology, vol. 11, pp. 273–278, 2003. [6] A. B. Will, S. Hui, and S. H. Zak, “Sliding mode wheel slip controller for an antilock braking system,” International Journal of Vehicle Design, vol. 19, pp. 523–539, 1998. [7] Y. Lee and H. S. Zak, “Genetic neural fuzzy control of anti-lock brake systems,” in Proceedings of the 2001 American Control Conference, Arlington, Virginia, 2001, pp. 671–676. [8] G. Chen, W. Wang, T. Lee, and C. W. Tao, “Observer-based direct adaptive fuzzy-neural control for anti-lock braking systems,” International Journal of Fuzzy Systems, vol. 8, pp. 208–218, 2006. [9] Inteco, “User’s manual: The laboratory antilock braking system controlled from pc,” Inteco Ltd., Poland, Tech. Rep., 2006. [10] Y. Oniz, E. Kayacan, and O. Kaynak, “A dynamic method to forecast the wheel slip for antilock braking system and its experimental evaluation,” IEEE Transactions on Systems, Man, and Cybernetics Part B: CYBERNETICS, vol. 39, pp. 551–560, 2009. [11] V. I. Utkin, Sliding Modes in Control Optimization. Springer-Verlag, 1992.
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