A. EL HAJJAJI, A. CIOCAN and D. HAMADCentre de Recherche de Robotique, d’Electrotechnique et d’Automatique, 7, Rue du Moulin Neuf,80000 Amiens, France; e-mail: [email protected]
Abstract. This study introduces a fuzzy four-wheel steering control design method for automotivevehicles. After the analysis of some stability aspects of the vehicle lateral motion, including frontsteering angle variations, the representation of vehicle nonlinear model by Takagi–Sugeno (T–S)fuzzy model is presented. Next, based on the fuzzy model, a fuzzy controller is developed to improvethe stability of the vehicle. Sufficient conditions for stability and stabilization of the T–S fuzzy modelusing fuzzy feedback controllers is given. To demonstrate the effectiveness of the proposed fuzzycontroller, simulation results are given showing the performance improvements of the vehicle interms of the stability and the maneuverability in critical situations.
These last years, automobile security has been of considerable attention, and manysafety systems have been developed and installed in the vehicle in order to improveits performance in terms of road safety (ABS, ESP, etc.). However, the numberof road accidents remains relatively significant. One of the principal causes ofthese accidents is related to the change of vehicle dynamic behavior in criticalsituations produced by a change of the road state (low friction road) or the skid incornering. To improve the performance of the vehicle, active control systems mustbe developed and installed. These systems must particularly improve the security interms of handling in an emergency. In this sense, many works have been developedin the literature [6, 9, 11, 12, 18]. Thus, the problems of modeling and controlof automated vehicles were studied in [18] and [11]. The robust stabilization ofside dynamics was developed in [12]. In [6], the sliding mode was applied tocontrol the rate of slip of the tires whereas in [9], the problem of the anti-skidwas treated by fuzzy logic. In this paper, we present a fuzzy control design methodbased on the Takagi–Sugeno (T–S) fuzzy model of four-wheel drive vehicles. Thiscontrol type has been successfully applied to the stabilization of nonlinear systems[4, 5, 7, 8, 10, 14, 15, 17]. In most of these applications, the fuzzy systems werethought of as universal approximators for nonlinear systems. The T–S fuzzy modelhas been proved to be a very good representation for certain classes of nonlineardynamic systems. In our studies, a nonlinear plant was represented by a set oflinear models interpolated by membership functions and then a model-based fuzzy
142 A. EL HAJJAJI ET AL.
controller was developed to stabilize the T–S fuzzy model. In Section 2, we exposethe mathematical model of the vehicle used as well as an analysis of this model inthe phase plan. In Section 3, the strategy of control employed will be presented.We give a structure of the fuzzy controller. We then present a technique allowingone to determine the fuzzy controller parameters by analyzing the stability of theclosed loop system. A new approach, which makes it possible to obtain less con-servative stability sufficient conditions than Lyapunov’s approach combined withLMI approach [3], will be presented. Lastly, to illustrate our approach, we showsimulation results highlighting the improvements made to the vehicle in terms ofhandling in critical situations.
2. Vehicle Model Analysis
Assuming that a vehicle has a constant velocity, a two-dimensional model withnonlinear tire characteristics of the four wheels vehicle behavior can be describedby differential equations (cf. Figure 1) [13]:
(β
r
)=
Ff + Fr
mU− r
afFf − arFr
Iz
cos(β)
, (1)
where β denotes the side slip angle, r is the yaw velocity, Ff is the cornering forceof the two front tires, Fr is the corning force of the two rear tires. U is the vehiclevelocity, Iz is the yaw moment of inertia, m is the vehicle mass.
Using the Magic formula [1], the cornering forces Ff and Fr are given as func-tions of tire slip angles by the following expressions:
Ff = Df sin[Cf tan−1{Bf(1 − Ef)αf + Ef tan−1(Bfαf)
where δf is the front steer angle, δr is the rear steer angle, αf is the slip angle of thefront tires and αr is the slip angle of the rear tires. The parameters of the modelsare given in Table I.
Figure 2 shows the cornering force characteristics as functions of the front tireslip in a dry road. To justify the necessity to improve the vehicle stability, we willanalyze its behavior in the phase plan. Thus, Figures 3, 4 and 5 respectively showthe vehicle state trajectories traveling to 20 m/s on a low-friction road for 2 s frominitial states, for three different front steering angles, the rear steering angle beingzero. We remark that the vehicle has one stable equilibrium point at the origin (0, 0)
and two unstable equilibrium points when the front steer angle is equal to zero asillustrated in Figure 3. However, we also remark that when the front steering angleincreases, the stable equilibrium point moves to one of the two unstable equilibriumpoints (see Figure 4) and when it becomes very important (>0.08 rad) the stable
144 A. EL HAJJAJI ET AL.
Figure 3. State trajectories with δf = 0 rad, U = 20 m/s.
Figure 4. State trajectories with δf = 0.06 rad, U = 20 m/s.
equilibrium point disappears as shown in Figure 5. From arbitrary initial states, thevehicle falls in spin at δf �= 0.08 rad. These figures shows that some trajectoriesthat are stable for zero steering angle become unstable when a constant nonzerofront steering angle is considered.
Moreover, in Figures 6 and 7, the effect of the speed on the behavior of thevehicle is examined. Figure 6 shows the phase trajectories of the vehicle when thespeed is equal to 25 m/s and the steering angle is equal to 0.05 rad. We can notethat all the trajectories are instable. Figure 7 shows the trajectories of phase when
FOUR WHEEL STEERING CONTROL BY FUZZY APPROACH 145
Figure 5. State trajectories with δf = 0.08 rad, U = 20 m/s.
Figure 6. State trajectories with δf = 0.05 rad, U = 25 m/s.
we considered a steering angle equal to 0.09 rad and a speed equal to 15 m/s.We remark that some trajectories become stable even if the front steering angle ishigher than 0.08 rad. From these figures, we can also notice that the behavior ofvehicle is degraded when its speed increases.
In conclusion, one can say that the stability domain of vehicle is reduced whenthe speed and/or the steering angle increase.
146 A. EL HAJJAJI ET AL.
Figure 7. State trajectories with δf = 0.08 rad, U = 15 m/s.
Figure 8. Control system structure.
This analysis highlights the importance of integrating a control and stabilizationsystem to making it possible to avoid the skidding of the vehicle. For this, wepropose a control system operating on the rear wheel-axle of the vehicle to improveits stability in this kind of situation. The block diagram of the control system isshown in Figure 8.
3. Control strategy
In this work, the control strategy is developed from a T–S fuzzy model of thevehicle obtained from model equations given in Section 2. In this approach, anonlinear model given in (1) is approximated by a set linear local models (T–Sfuzzy model) interpolated by membership functions and then a fuzzy controllerwill be developed to stabilize a T–S fuzzy model.
FOUR WHEEL STEERING CONTROL BY FUZZY APPROACH 147
3.1. FUZZY MODEL OF VEHICLE
As is well known, the tires are the dominant source of nonlinearities in the vehiclemodel. So in specifying our T–S fuzzy model, we will consider cornering forcecharacteristics. In general, the cornering tires are supposed to be proportional to theslip angles [12]. However, as can be seen in Figure 2, this approximation is onlyvalid for low slip. In order to take into account the nonlinearity of the corneringtires, we have defined two slip regions: high slip region (slip angle > 0.09 rad) andlow slip region (slip angle < 0.09 rad). We have then supposed in each region thatthe tire’s cornering stiffness parameters ∂F(f,r)/∂α(f,r) are constant (see Figure 2).Moreover, using the following approximations:
αf∼= β + afr
U− δf and αr
∼= β − arr
U+ δr.
We obtain two linear models. We refer to one model as the big slip model and theother as the small slip model. Both models are derived from (1):
If |αf| is small then X = A1X + Bf1δf + Br1δr,
If |αf| is big then X = A2X + Bf2δf + Br2δr,
where
A1 =
−Cf1 + Cr1
mU−1 − afCf1 − arCr1
mU 2
−afCf1 − arCr1
Iz
−a2f Cf1 + a2
r Cr1
UIz
,
A2 =
−Cf2 + Cr2
mU−1 − afCf2 − arCr2
mU 2
−afCf2 − arCr2
Iz
−a2f Cf2 + a2
r Cr2
UIz
,
Bf1 =
Cf1
mUafCf1
Iz
, Br1 =
Cr1
mU
−arCr1
Iz
,
Bf2 =
Cf2
mUafCf2
Iz
, Br2 =
Cr2
mUarCr2
Iz
,
where Cf1 = dFf/dαf, Cr1 = dFr/dαr in low slip area and Cf2 = dFf/dαf,Cr2 = dFr/dαr in high slip area. Then, given a pair (X(t), U(t)), the resulting
148 A. EL HAJJAJI ET AL.
fuzzy system model is inferred as the weighted average of the linear models andhas the form:
A sufficient condition that guarantees the stability of the fuzzy system describedby Equation (5), is obtained using Lyapunov’s direct method. The above equationis asymptotically stable if there exists a common positive definite matrix P suchas [16]:
We have here to determine the matrix P and the fuzzy controller parameters K1
and K2. To solve this kind of the problem, one can use the well-known LMI ap-proach [3]. The idea of this approach consists in supposing that the feedback state
FOUR WHEEL STEERING CONTROL BY FUZZY APPROACH 149
parameters are determined beforehand by linear feedback techniques. We then usethe LMI approach to check the existence of the matrix P satisfying inequalities (6),(7) and (8). In [2], another design technique has been proposed. Contrary to the pre-vious approach, we have supposed that the definite positive matrix P exists. Then,we have determined the admissible areas of the fuzzy controller parameters check-ing inequalities (6), (7) and (8). However, these two design techniques generateconservative stability conditions (pessimists conditions) because of the existenceof the crossover models in inequality (8). To have less conservative conditions ofstability, we propose a new technique. Thus, we will take as a starting point thefirst approach and we will give a sufficient condition of the stability for the closedloop system (5), even if the sufficient condition (8) is not satisfied.
Supposing that there exists a positive definite matrix P so that
We denote λi the minimum eigenvalues of Qi and λ12 the minimum eigenvaluesof Q12.
THEOREM. Supposing that
∃P > 0, (Ai − BriKi)TP + P(Ai − BriKi) = −Qi, i = 1, 2 with Qi > 0.
The closed loop system described by Equation (4) is asymptotically stable if thematrix
� =(
λ1 λ12
λ12 λ2
)> 0.
Proof. Let V (X) = XTPX Lyapunov function system (4) is stable if V (X) � 0,
V (X) = XTPX + XTPX,
V (X) = ω1ω1XT(
(A1 − B1rK1)TP + P(A1 − B1rK1)
)X
+ ω1ω2XT((A1 − B1rK2 + A2 − B2rK1)
TP + P((A1 − B1rK2)
+ (A2 − B2rK1)))
X + ω2ω2XT((A2 − B2rK2)
TP
+ P(A2 − B2rK2))X,
V (X) = −(ω1ω1XTQ1X + ω1ω2X
TQ12X + ω2ω2XTQ2X).
Using the fact that for any symmetric matrix
λmin(M)‖X‖2 � XTMX,
150 A. EL HAJJAJI ET AL.
Figure 8. Front steering angle.
Figure 9. Side slip angle.
where λmin is the smallest eigenvalue of M, we obtain:
V � −(ω21λ1 + ω1ω2λ12 + ω2λ2)‖X‖2
= −(
( ω1 ω2 )
(λ1 λ12
λ12 λ2
)(ω1
ω2
))‖X‖2,
V (X) � −(ωT�ω)‖X‖2.
FOUR WHEEL STEERING CONTROL BY FUZZY APPROACH 151
Figure 10. Yaw rate.
Figure 11. Side slip angle.
4. Simulation Results
Considering the two feedback controller parameters K1 = [0.04,−0.88]; K2 =[0.03,−0.44] that were obtained by a simple pole assignment, the matrix P check-ing inequalities (5), (6) and (7) is
P =[
3.18 −0.06−0.06 0.12
]� 0.
A series of computer simulations was carried out to examine the performanceof the proposed nonlinear control system. We note that all the simulations were
152 A. EL HAJJAJI ET AL.
Figure 12. Yaw rate.
Figure 13. State trajectories.
conducted on the nonlinear model given in Section 2. In the first one, we considerthat the vehicle velocity is constant and equal to 20 m/s. In Figures 9 and 10,we have superposed the state variable evolutions (side slips, yaw rate) with andwithout control when the front steering angle is as given in Figure 8. We remarkthat yaw rate is noticeably reduced by our control (solid line) as compared with thatof the vehicle without control (dashed line) and the side slip response with controlis better than that without control. At the same time, in Figures 12 and 13, wegive the state variable evolutions when the longitudinal velocity U = 25 m/s. We
FOUR WHEEL STEERING CONTROL BY FUZZY APPROACH 153
Figure 14. Front steering angle.
Figure 15. Vehicle velocity.
remark that the amplitudes of state variables remain low with rear control whereasthe amplitude of these variables without control becomes very important. Whatis more, to show the improvements made to the vehicle in terms of handling incritical situations, we consider a front steering angle equal to double the angle usedin Figure 5 (δf = 0.12) and the same initial conditions. Let us bear in mind thatthe vehicle without rear control has no equilibrium point, as is shown in Figure 5.Figure 14 shows the state trajectories in the phase plan with our control. We remarkclearly the reappearance of the stable equilibrium point and all the trajectories areattracted to this point even if the front steering angle is more than 0.06 rad. It iseasy to see from this figure that all the trajectories of the controlled vehicle are
154 A. EL HAJJAJI ET AL.
Figure 16. Yaw rate.
Figure 17. Side slip angle.
now stable. In other words, the vehicle with the proposed controller gives a stableresponse in such severe maneuvering. To evaluate the robustness of the developedcontroller when the velocity varies, other simulations were carried out. Thus, wesupposed that the vehicle velocity varied as indicated in Figure 15. In Figures 16and 17, we respectively show the side slip and yaw rate evolutions with our rearcontrol when the front steering angle is as given in Figure 14. One can notice thatthe vehicle with the designed control is stable in spite of velocity variations. Thisshows the important improvements brought to the stability of the vehicle usingthe designed control. This control, which operates on the rear wheels, protects the
FOUR WHEEL STEERING CONTROL BY FUZZY APPROACH 155
vehicle from spin even if the driver gives an important steering angle to avoid adangerous situation, for example.
5. Conclusion
In this paper, we have presented a new technique allowing one to improve theperformances of the vehicle in terms of stability and maneuverability in criticalsituations. The design method uses the T–S fuzzy model of the vehicle and thestability analysis of the closed loop fuzzy system. The less conservative stabilitysufficient condition is given. The simulation results have been shown to illustratethe performances of the proposed algorithm in case of a vehicle driving on a dryroad.
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