GA-based Sliding Mode Controller for Yaw Stability Improvement Norhazimi Hamzah
Yahaya Md Sam, Hazlina Selamat
Faculty of Electrical Engineering Universiti Teknologi MARA Pulau Pinang, Malaysia
[email protected]
Faculty of Electrical Engineering Universiti Teknologi Malaysia Johor, Malaysia
[email protected],
[email protected]
M Khairi Aripin Control, Instrumentation & Automation Department. Faculty of Electrical Engineering, UTeM Melaka, Malaysia
[email protected]
Abstract— The vehicle handling and stability can be enhanced with direct yaw moment control (DYC). In this paper, a sliding mode controller (SMC) with genetic algorithm optimization is proposed for the yaw moment control. A single track vehicle model with nonlinear tire forces is utilized for the controller design. The effectiveness of the proposed controller is compared to conventional sliding mode controller by co-simulations in CarSim and Matlab/Simulink. The simulation results of a step steer maneuvers shows that the proposed controller give better performance in tracking the desired yaw rate and control other necessary response for the vehicle handling and stability. In addition, the chattering phenomenon is also reduced, giving a smooth tracking trajectory. Keywords— sliding mode control, genetic algorithm, direct yaw moment control, yaw stability control, chattering
I. INTRODUCTION In automotive engineering, handling and stability of road vehicle is very important especially during critical or cornering maneuver. One of the common and efficient approach to improve road vehicle handling and stability is through direct yaw moment control (DYC). The DYC works by generating the required corrective yaw moment by differential braking where the braking forces on all four wheels are controlled individually such that the required corrective yaw moment is obtained. The differential longitudinal forces generated at each tire will results in yaw moment around the centre of gravity (CG), thus compensating the saturated tire force to stabilize the vehicle [1]. This in turn controls the lateral motion of the vehicle to stabilize its behavior. Various control strategies have been proposed for controlling the vehicle stability by DYC. [2] applied fuzzy
logic control for the yaw moment control to improve the vehicle handling and stability. A simple logic is used to distribute the obtained yaw moment to the front left or right tire. In [3], generalized predictive control (GPC) law is derived based on the prediction of the yaw rate error growth. By applying optimal predictive control to non-linear dynamic model, the control law ensures minimal control input that satisfies the actuator constraint were generated [4]. This will prevent undesirable effect such as excessive braking which will slow down the vehicle. In addition, optimal controller based on linear quadratic regulation (LQR) is proposed by [5]. On the other hand, [6] and [7] applied sliding mode control (SMC) for the DYC to enhance the control system robustness. This is due to the fact that the vehicle system is a highly nonlinear and uncertain system due to uncertainty of the roadtire friction coefficient, the vehicle loading, the nonlinear tire forces, various uncertain operating conditions, external disturbances as well as other unmodeled dynamics. The application of SMC shows very good result on handling and stability performances. However, SMC optimal performance and inherent robustness to matched uncertainties caused by unmodeled dynamics and external disturbance depends on appropriate tuning of the controller parameters. As the vehicle system operates under a variety of operating condition, the controller gain should be optimized to be able to operate under all condition. The tuning of the SMC controller parameter is extremely tedious and time consuming [8] because most of the times the controller parameters are tuned by trial and error approach. Therefore, it is no guarantee that the parameter selected is the optimal or sub-optimal solution [9]. The control energy sometimes is overly defined to ensure accurate tracking
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of the desired value which then creates higher chattering in the system. In place of this conventional approach, the genetic algorithm (GA) is used to optimize the parameters of the sliding mode controller which is the switching gain. GA is a global optimization approach based on the concept of evolutional process and natural genetics [10]. GA does not require the derivative of the objective function to find the optimum and therefore can minimize naturally the objective function without complex mathematical operations [11]. With GA, a trade off between accurate tracking and control energy can be established to automatically improve the performance of the system as well as minimizing chattering. This paper is organized as follow. Section II describes the vehicle dynamics model while Section III gives the design strategy of the sliding mode control and the genetic algorithm optimization. The simulations and analysis part in Section IV provides the comparison of proposed strategy and the conventional strategy and conclusion is presented in Section V.
⎛
α f = δ f − arctan ⎜⎜ β + ⎝
⎛
l f ψ ⎞ ⎟ v x ⎟⎠
(3)
⎞ ⎟⎟ ⎝ ⎠ where δ f is the front wheel steering angle.
α r = arctan⎜⎜ − β +
lrψ vx
(4)
The nonlinear relationship of the tire lateral force to tire slip angle, normal force and surface friction coefficient can be observed in Figure 2. The tire lateral force dependence on the slip angle can be described by the Pacejka’s Magic Formula [12] as in equation 5. (5) F y f , r = D y sin(C y arctan( B y Φ y )) + S vy with Φy = (1 − E y )(α + S hy ) +
Ey By
arctan(B y (α + S hy ))
(2)
where the state variables are the vehicle side-slip angle β and the yaw rate ψ at the center of gravity (CG), m is the vehicle mass, v x is the vehicle forward velocity, l f and lr are the distance of the front and rear axle from the center of gravity (CG), F yf and F yr are the lateral force of the front and rear
curvature factor respectively and S vy , S hy denotes the vertical and horizontal shift. These coefficients depend on the type of the tire and the road condition.
4000 mu=0.9 mu=0.7 3000
mu=0.5 mu=0.3 mu=0.1
2000
Fy(N) at Fz=4000(N)
A. Vehicle Model for the Controller Design For the controller design, a two degrees of freedom vehicle model with nonlinear tire considering only the lateral and yaw dynamics is used as shown in Figure 1. The used of nonlinear vehicle model allow the analysis of the influence of lateral acceleration on the lateral vehicle dynamics to be carried out. Assuming small steering angle, the governing equations can be expressed as [4]: mvx β = −mvxψ + 2 Fyf + 2 Fyr (1)
tires and M z is the external stabilizing yaw moment.
front and rear tire slip angle ( α f , α r ).
where B y , C y , D y and E y are the stiffness, shape, peak and
II. SYSTEM MODELING
I zzψ = 2l f Fyf − 2l r Fyr + M z
The front and rear tire lateral force ( Fyf , Fyr ) depends on the
1000
0
-1000
-2000
-3000
-4000 -20
-15
-10
-5
0 slip angle(deg)
5
10
15
20
Fig. 2. The tire lateral force at pure braking/driving
ψ
ψ
Fig. 1. The two degree of freedom vehicle model
B. Vehicle Model for the Evaluation of the Controller For the evaluation of the proposed controller, the multibody vehicle dynamics simulation package CarSim is used as the vehicle simulation model. It is a full vehicle model with complex and accurate tire property as shown in Figure 3 [13]. The response characteristics produced by CarSim have shown to give comparable result with experimental data from real vehicle. Using CarSim, the evaluation of the robustness
property of the proposed controller against unmodeled dynamics can be carried out.
⎧⎪ψ , ψ t arg et = ⎨ d ⎪⎩ψ lim sgn(ψ d ),
ψ d < ψ lim ψ d ≥ ψ lim
(8)
And to ensure handling stability, the side slip angle must also be bounded as [14] β lim = tan −1 (0.02μg ) (9)
Fig. 3. CarSim vehicle model
III. CONTROLLER DESIGN The goal of the control strategy is to find the required yaw moment to minimize the yaw rate tracking error to address handling responsiveness In addition, the vehicle side slip angle also must be bounded to an acceptable region to ensure handling stability. This yaw moment will be generated by differential braking where the braking pressure is applied individually at the four wheels. Due to this, the output yaw moment must not exceed the maximum difference of the longitudinal forces that the differential braking can generate. The controller is designed based on sliding mode control to ensure robustness of the system against parameter uncertainties and unmodeled dynamics. Genetic algorithm is used to tune the gain of the sliding mode controller to improve the performance of the system. A. Desired Model Based on single track model steady state cornering, the desired yaw rate is given as [4] vx (6) ψ d = δf l (1 + k us v x 2 ) where l = l f + lr and k u s is the stability factor and define as follows [14]; ku s =
m( l r C r − l f C f ) lC f C r
The lateral acceleration of the vehicle cannot exceed the tire cornering capability, thus the desired yaw rate must be limited by the following value [15] μ.g a y (7) ψ d ≤ ψ lim = = vx vx Consequently, the control target for the yaw rate tracking is given by
B. Sliding Mode Controller Design From (2), the yaw dynamics can alternatively be written as x = f ( x) + b( x)u (10) where 1 f ( x) = 2l f Fyf − 2l r Fyr I zz 1 b( x ) = I zz
(
)
u = Mz The magnitude of the uncertainties is constrained by the following function: f ( x) − fˆ ( x) ≤ F ( x) (11)
where fˆ ( x) is the estimation of f (x) and 0 ≤ bmin ( x) ≤ b( x) ≤ bmax ( x) (12) The control gain bˆ( x) can be estimated by the geometric mean of the lower and upper bounds of the gain. bˆ( x) = bmin ( x)bmax ( x) (13) And the bound can be written in the following form bˆ Β −1 ≤ ≤ Β where Β = (bmax / bmin )1 / 2 (14) b For the sliding mode controller design, the first step is to choose a suitable sliding surface. Choosing the sliding variable as the error between the actual yaw rate and the desired yaw rate σ = x − x d = ψ − ψ d (15) The sliding surface is defined as σ =0 (16) Thus, if σ can be forced to zero, then the error can converged to zero so that the desired dynamic can be attained. The second step is then to design the control law to drive the system trajectories to the sliding surface so that the closed loop dynamics are completely governed by the equation that define the surface. The SMC control law consist of two components which are the nominal equivalent control law ( u eq ) and the switching control law ( u sw ). The switching control law is to drive the system states toward the selected sliding surface and the equivalent control law guarantees the system state to keep sliding on the surface and converge to zero along the sliding surface. u = u eq + u sw (17)
Taking the time derivative of σ gives σ = x − x d = ψ − ψd (18) Substitute (10) into (18) and equating to zero, the equivalent control law is given by u eq = bˆ( x) −1 x d − fˆ ( x) (19)
(
)
In order to satisfy the sliding condition regardless of the uncertainty on f (x) , the switching control law is given as
u sw = −b( x) −1 K sgn(σ )
(20)
Finally, the sliding mode control law is given as u = bˆ( x) −1 x − fˆ ( x) − K sgn(σ )
[
]
(21) The control law must be chosen such that the closed-loop response always satisfies the following condition: 1 d 2 (22) σ ≤ −η σ 2 dt where η > 0 is the design parameter that guarantees the system trajectory to reach the sliding surface in finite time [16]. The switching gain, K is normally made sufficiently large to compensate the matched model uncertainties and disturbances where K is selected as K ≥ b( x) −1 (F + η ) (23) d
C. Genetic Algorithm The performance and robustness of the SMC control system depends on appropriate tuning of the switching gain ( K ). The tuning of K by trial and error method not only tedious and time consuming, but could create higher chattering in the system if overly defined. Genetic algorithm (GA) is a parallel and global search algorithm based on the concept of evolutional process and natural genetics [10]. GA emulates biological evolution by means of genetic operation such as reproduction, crossover and mutation. The basic element of a GA is the chromosome which contains the genetic information of a given solution. Genetic algorithms work by using the information obtained from testing a current population of chromosomes at each iteration to direct its search into promising areas of the search space. Genetic algorithm begins by randomly generating an initial population of candidate solution. The algorithm then calculates the fitness value for each member of the current population and then sorted them into ascending order. Based on the ranking of the fitness value, reproduction is conducted based on elitist principle where the chromosome with lower fitness value (top 20% of the present generation) is passed to the next generation to ensure the maximum fitness to keep on increasing and not fluctuated by the large crossover rate [17]. The remaining 80% of the chromosome on the other hand is rejected and replaced by a newly formed chromosome in the next generation through crossover and mutation operations. In crossover operation, two parents are randomly selected to exchange genetic information to form two chromosome offspring which are variations of the parents. This process is repeated until there are enough new chromosomes to replace the rejected 80% of the present population with the poorest
fitness value. Next, the mutation operation is conducted where the chromosome from the new population is randomly selected and the genes values are randomly changed (between 0 and 9 in decimal coding). The process is repeated until a specified percentage of the genes have been altered [8]. The process of fitness evaluation, sorting, reproduction, crossover and mutation is repeated again on the new population until one of the termination conditions is satisfied. Finally, the best chromosome is presented as the approximately optimal solution for the problem considered. The GA optimization result depends on a few fundamental parameters such as chromosome length, population size, number of generations, crossover rate and mutation rate. Large crossover rate and small mutation rate will ensure population diversities and prevent premature convergence of maximum individual fitness [17]. In this study, the population size is 20, generation size is 40, crossover rate is 0.95 and mutation rate is 0.05. The integral square error (ISE) of the yaw rate is selected as the fitness value. I=
∫
tt
0
e 2 dt
(24)
D. Differential braking The yaw moment is realized through differential braking where tw tw Mz = ( Fxfr − Fxfl ) + ( Fxr r − Fxrl ) (25) 2 2 The braking force for the front wheel is given by 2M z ΔFxf = (26) tw The braking pressure at the front wheel is related to the front braking force as the following [7] 1 Pbf = ΔF xf (27) KB where K B is the linearized gain from the wheel cylinder pressure to braking force. The braking pressure on the front and rear wheels is assumed to follow a fixed 0.4 proportioning ratio Pbr = 0.4 × Pbf (28) The braking force is applied to only one side of the wheels depending on the sign of the yaw moment in equation (21). When the yaw moment has positive sign, the braking force is applied to the left wheels and when the sign is negative, the braking is applied to the right wheels. IV. SIMULATION AND RESULT ANALYSIS The proposed control strategy is evaluated through cosimulation using Matlab/Simulink and vehicle simulation package CarSim. The simulations are carried out for the step steer driving maneuver as in Figure 4. The constant speed step steer maneuver is a simple handling procedure used to evaluate both the steady-state and the transient response of the
vehicle. The responses of the proposed controller are compared and analyzed to system with conventional SMC and without control. Figure 5-7 show the response of the system for the respective steering input. Based on Figure 5, the system using GA optimization performs better than the system using conventional SMC in tracking the desired yaw rate with minimal error. In addition, chattering phenomenon can be clearly seen in system with conventional SMC control.
pressures applied to each wheel for the system using GA optimize SMC and conventional SMC are shown in Figure 7. Due to the braking pressure applied, the longitudinal speed of the vehicle reduces as illustrated in Figure 8. 4
1
The control input
x 10
SMC+GA SMC
0.8
External yaw moment (N.m)
0.6 Step steer steering input 180 160
steering input (deg)
140 120
0.4 0.2 0 -0.2 -0.4
100
-0.6
80
-0.8
60
-1
0
1
2
3
4
5
6
7
t (sec) 40 20 0
Fig. 6. The external yaw moment 0
1
2
3
4
5
6
7
t (sec) Brake pressure (MPa)
Braking pressure at each wheel using SMC+GA
Fig. 4. Step steer steering input yaw rate 25 desired SMC+GA SMC no control
20
Brake pressure (MPa)
yaw rate (deg/s)
15 22 21 10
20 19
5
2.4
2.6
2.8
3
6 FL FR RL RR
4
2
0
0
0.5
1
1.5 2 2.5 t (sec) Braking pressure at each wheel using SMC
6 FL FR RL RR
4
2
0
0
0.5
1
1.5 t (sec)
2
0
-5
Fig. 7. Braking pressure at each wheel 0
1
2
3
4
5
6
7
t (sec)
Fig. 5. The yaw rate response
Figure 6 shows the external yaw moment produced by the controllers. Based on the figure, the external yaw moment produced by both type of controller is quite high and even worse the conventional SMC control system shows large chattering throughout the whole steering maneuver. For the GA optimized SMC, the chattering reduce after some time during the steady state part of the steering maneuver. This external yaw moment is then converted into respective braking pressure based on the concept of differential braking. The braking
3
2.5
3
The longitudinal speed 80 SMC+GA SMC
vehicle speed (km/h)
[5] 75
[6] 70
[7] 65
0
1
2
3
4
5
6
7
time (s)
Fig. 8. The vehicle longitudinal speed
[8]
V. CONCLUSION A GA optimize sliding mode controller is proposed in this study to improve the vehicle handling and stability performance. The genetic algorithm is applied to search the optimum value of the switching gain. The concept of direct yaw moment control applied is proved as an efficient method for tracking the desired yaw rate and minimizes the error. In addition, an optimize switching gain also reduce chattering problem while still maintaining the accuracy and robustness feature of the sliding mode control, thus is appropriate to use in the automotive field. However, for future study optimum distribution of the yaw moment is desirable to prevent excessive braking which can slow down the vehicle. . ACKNOWLEDGMENT The authors would like to thank UiTM, UTM and MoHE for supporting the present work. REFERENCES [1]
[2]
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