Nonlinear feedback control of Vehicle Speed

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Nonlinear feedback control of Vehicle Speed Y. ABBASSI ∗ Y. AIT-AMIRAT ∗∗ R. OUTBIB ∗∗∗ Universit´e de Technologie de Belfort-Montb´eliard (UTBM), 90000 Belfort (e-mail: [email protected]). ∗∗ Universit´e de Franche-Comt´e, Departement AS2M of FEMTO-ST (e-mail: [email protected]) ∗∗∗ Laboratoire LSIS - Universit´e d’ Aix-Marseille, France, (e-mail: [email protected]) ∗

Abstract: This paper deals with the development of a nonlinear static state feedback control applied to the vehicle speed. The control is obtained by suitable change of variable and after some preliminary feedback. The main objective in this study is the stabilization of vehicle velocities, i.e longitudinal, lateral and yaw rate using Lyapunouv stability theory and LaSalle invariance principle. Keywords: Nonlinear control, stabilization, vehicle dynamics, simulation. NOMENCLATURE M g CG a, b h kax , kay , f fr Fl Cf Cr δ ψ Vψ = ψ˙ V x , Vy Iz

linear methods, such pole placement. Several authors were interested in the problem of control of vehicle dynamics. D.C. Liaw (2008) has proposed a feedack linearization for the control of the lateral dynamic. However, deficiencies associated with the linear approaches appears when one moves away from the set point of linearization.

Vehicle mass. Gravitational acceleration. Centre of gravity of the vehicle. Distances from CG to front and rear wheel axes Height of CG Aerodynamic coefficients in x and y directions. Rolling friction coefficient. Force distribution coefficient. Longitudinal (Acceleration or braking) forces. Cornering stiffness coefficients of front tires. Cornering stiffness coefficients of rear tires. Steering front wheels angle. Yaw angle. Yaw rate. Longitudinal and lateral velocity in x and y directions. Yaw moment inertia.

In this work, we are interested by the problem of the control of the vehicle speed by feedback. The originality here consist to consider all the inherent non linearities of the system without any kind of linearization. Note that, the problem of the control in nonlinear systems has been a subject of research for several authors. Thus, a great number of results were proposed. These results are in the form of necessary, as in R. W. Brockett et al. (1983) and J. Tsinias (1989), or sufficient conditions. Sufficient ones, are closed to some classes of systems, like homogeneous systems or partially linear systems, see Z. Arstein (1983), or dissipative systems in V. Jurjevic et al. (1978) and R. Outbib et al. (1999) etc.

1. INTRODUCTION Models considered to describe the vehicle dynamics are highly nonlinear and do not raise any kind of the traditional classes of the nonlinear systems studied in the literature. Indeed, the vehicle dynamics has various properties which make that their control is a difficult and interesting problem. There exist several types of models for the longitudinal, lateral and yaw motions, see Abbassi et al . (2007) and B. D’andrea Novel et al. (2005). The majority of the movements are characterized by an absolutely nonlinear behavior which does not allow to use the traditional reliable 978-3-902661-93-7/11/$20.00 © 2011 IFAC

The paper is organized as follows. We present a nonlinear vehicle model suitable for the present study. Next, some variables transformations are proposed to simplify the structure of the model. Then, a feedback nonlinear control for the stabilization of the velocities is proposed. Simulation results showing various speeds of the controlled vehicle are presented. 2. VEHICLE MODEL There exist several different types of models for the lateral and yaw motions of a vehicle, but, a model suitable for the present study is derived from Abbassi et al . (2007) and A. Alloum et al. (1995). The vehicle dynamics are developed in two coordinate directions x, y and one rotation around the z axis. Besides, the vehicle is considered as a front wheels driving and steering. The equations of motion are given by :

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

   ˙ x = 1 Fl + c1 + M Vy Vψ + Cf δ Vy  V   M Vx     Vψ  2  + aCf δ − kax Vx      Vx   Vy 1   Fl δ − M Vx Vψ + Cf δ + c2 V˙ y =    M Vx  Vψ 2 + c3 − kay Vy    Vx    1 Vy   V˙ ψ = aFl δ + c4 Vx Vψ + aCf δ + c5    Iz  Vx    Vψ   +c6    Vx 

According to the second equation (4), we have   Fl γΘ = + 1 Cf δΘ Cf

(1)

By using the first equation of (4), the equation (5) becomes   Fl + 1 (σ − Fl ) (6) γΘ = Cf or   Fl2 σ + Fl 1 − + γΘ − σ = 0 Cf Cf

where kax = 12 ρCa Sf and kay = 21 ρCa Sl are the aerodynamic drag. ρ is the mass density of air, Ca is the aerodynamic drag coefficient. Sf and Sl are respectively the frontal and lateral area of the vehicle. The constants ci are defined as: c1 = −M f g, c2 = −(Cf + Cr ) c3 = c5 = −(aCf − bCr ), c4 = −M f h c6 = −(a2 Cf + b2 Cr )

By a similar computation, we obtain for δ Cf Θδ 2 − δ(Cf + σ) + γ = 0

∆1 =

3. VARIABLES TRANSFORMATION For technical reasons of computation as in R. Outbib et al (2000), we carry out the following transformation   a Iz (2) V˜ψ = Vψ − M Vy ⇔ Vψ = V˜ψ + M Vy a Iz model (1) becomes  Ma ˜ aM 2 2 1 ˙ c1 + Vy Vψ + V − kax Vx2 + Vx = M Iz y " Iz #! Vy a2 V˜ψ a2 Vy Fl + Cf δ + +M Vx Iz Vx Iz Vx  2 M a M a Vy 1 − Vx V˜ψ − Vx Vy + c2 V˙ y = M Iz Iz Vx M a Vy a V˜ψ 2 + c3 − kay Vy + +c3  Iz Vx Iz Vx    Fl δ + Cf δ)      a  c4 aM  c4  ˙   V˜ ψ = + M Vx V˜ψ + + M Vx .   Iz a Iz a   c V  V˜ c   a y ψ 6 5   − c2 + − c3 Vy +   a V I a V  x z x V   aM  c6 y   + − c3 + kay Vy2 Iz a Vx

The preliminary feedback is defined by:  σ = Fl + Cf δΘ γ = Fl δ + Cf δ

where Θ indicates the following quantity a2 V˜ψ a2 Vy Vy + +M Θ= Vx Iz Vx Iz Vx

(7)

(8)

Finally, the control variables Fl and δ can be calculated according to σ and γ to leave equations (7) and (8). However, it will be necessary to consider only the operating ranges where equations (7) and (8) have real and pertinent solution to the physical field. Precisely, the equation (7) has a real solution if its discriminant (equation considered as a polynomial of degree two in Fl ) is positive, then :

with Fl and δ denote the control inputs of the system.

The                          

(5)



1−

σ Cf

2

−

4 (γΘ − σ) ≥ 0 Cf

(C1)

In other words, it is necessary for all the following study to ensure that the condition (C1) is satisfied. Then, we have to choose among the two possible solutions according to physical criteria. Fl =

Cf p σ − Cf ∆1 ± 2 2

(S1)

The equation (8), considered as a polynomial of second degree in δ, has a real solution if its discriminant is positive 2

∆2 = (Cf + σ) − 4γCf Θ ≥ 0

(C2)

The real solution will be given by √  (C + σ) − Sgn (Cf + σ) ∆2   f if Θ 6= 0 2Cf Θ δ= γ   elsewhere Cf + σ (S2)

(3)

where Sgn indicates the sign function. It is introduced to keep only the acceptable solution in steering angle δ (when Θ 6= 0). Notice that δ is defined as continuous function. Indeed, to prove it, we can use the following approximation (4) √

a−x≈

√

a−

1 x √ 2 a

for x in a neighborhood of zero and a one positive constant.

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

Now we try to solve the initial problem of the stabilization by a feedback control on the system (3) where the control variables are given by (σ, γ) and after that, we use the equations (S1) and (S2) to deduce the values of the origin variables, Fl and δ. However, during the synthesis of the feedback control (σ, γ), it is necessary to check the both conditions (C1) and (C2). By using the expressions of Fl and δ involving σ and γ, the system (3) becomes    Ma ˜ aM 2 2 1  2 ˙  c1 + Vy Vψ + V − kax Vx + σ Vx =   M Iz Iz y    2  aM Vy Ma ˜ 1   Vx Vψ − Vx Vy + c2 − V˙ y =   M I I V  z z x !    ˜ψ  a V M a V y  2  + c3 + c3 − kay Vy + γ Iz Vx Iz Vx (9)     a  c4 aM  c4 ˙  ˜ ˜  Vψ = + M V x Vψ + + M Vx Vy   Iz a Iz a         a c6 Vy V˜ψ c5   + − c2 − c3 +   a Vx Iz a Vx    Vy aM  c6   − c3 + kay Vy2 + Iz a Vx

Let

σ1 = and

1 M



c1 +

Ma ˜ aM 2 2 Vy Vψ + V − kax Vx2 + σ Iz Iz y

Ma ˜ M 2a Vy − Vx V ψ − Vx Vy + c2 + Iz Iz !Vx M a Vy a V˜ψ + c3 − kay Vy2 + γ c3 Iz Vx Iz Vx

1 γ1 = M



the system (9) becomes  V˙ x = σ1     V˙ y = γ1          V˜˙ = a c4 + M V V˜ + aM c4 + M .  ψ x ψ Iz a Iz a c c V  V˜  a 5 6 y ψ   V V + − c + − c x y 2 3   a V I a V  z x  V x  aM  c6  y  + − c3 + kay Vy2 Iz a Vx



(10)

gation by transforming the third equation of the system (11). We introduce the function F defined by   1 a  c6 a  c4 + M Vx + − c3 (12) F (Vx ) = Iz a Iz a Vx

A simple reasoning shows that the system (11) can be written in the following form  ˙   Vx = σ1 ˙ Vy = γ1  (13)    V˜˙ ψ = G Vx , Vy , V˜ψ

with    i h G Vx , Vy , V˜ψ = F (Vx ) − F (Vx0 ) V˜ψ + M Vy + h i c V 5 y F (Vx0 ) V˜ψ + M Vy + − c2 + kay Vy2 a Vx The expression of the derivative for Vx = Vx0 is:   h i V˜˙ ψ = G Vx0 , Vy , V˜ψ = λ1 Vy2 + λ2 Vy + λ3 V˜ψ

with  λ1 = kay    c  1   1 aM  c4 5   + M Vx0 + − c2 λ2 =   λ1 Iz a a Vx0     aM c6 1 + − c3  0  I a V  z  x           λ3 = 1 a c4 + M Vx0 + a c6 − c3 1 λ1 Iz a Iz a Vx0

Let ∆Vx0 = λ22 − 4λ3 V˜ψ , then ! (i+1) p 2   Y λ2 + (−1) ∆Vx0 0 ˜ G Vx , Vy , Vψ = λ1 Vy + 2 i=1

Given the following Lyapunov function W defined by: p   λ2 − Sgn(λ2 ) ∆Vx0 1 Vy W Vy , V˜ψ = Vy2 + 2 2 2 q + λ2 − Sgn(λ2 ) ∆Vx0

(11)

The method consists now to stabilize this system under the conditions (C1) and (C2). The procedure is displayed in two steps. The first one consists to check that for σ1 = γ1 = 0, both conditions (C1) and (C2) are satisfied. In the second step, one synthesizes a control law with |σ1 | ≤ ǫ and |γ1 | ≤ ǫ where ǫ is a real positive number. So, the conditions (C1) and (C2) remain satisfied for all t > 0.

A simple computation shows that ! p   0 λ + Sgn(λ ) ∆ ∂W 2 2 V x G Vx0 , Vy , V˜ψ = λ1 Vy + 2 ∂Vy Next, the system (13) becomes  ˙   Vx = σ1   ˙  Vy = γ1     V˜˙ ψ = G Vx0 , Vy , V˜ψ + G Vx , Vy , V˜ψ        −G V 0 , Vy , V˜ψ x

First, we will prove that: G(Vx , Vy , V˜ψ ) − G(Vx0 , Vy , V˜ψ
) = 0 ˜ G1 (Vx , Vy , Vψ ) Vx − Vx

(14)

where G1 is a regular function.

4. SPEED STABILIZATION We are interested here to the stabilization of the system describing the speeds behavior of the vehicle (11) around the set point (Vx0 , 0, 0) where Vx0 is the reference value for longitudinal speed. To do that, we continue our investi-

Indeed, a simple reasoning shows that ZVx ∂G 0 ˜ ˜ (τ, Vy , V˜ψ )dτ G(Vx , Vy , Vψ ) − G(Vx , Vy , Vψ ) = ∂Vx

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

    1
2 Vx − Vx0 + ln W (Vy , V˜ψ ) + 1 W1 Vx , Vy , V˜ψ = 2

Let consider the change of variable defined by τ = tVx + (1 − t) Vx0 We have:
dτ = Vx − Vx0 dt Then, it comes that     G Vx , Vy , V˜ψ − G V 0 , Vy , V˜ψ

A simple computation shows that W1 is a positive function
on Vx0 , 0, 0 .

x

= Vx −

Vx0




Z1 0

 ∂G  tVx + (1 − t) Vx0 , Vy , V˜ψ dt ∂Vx

So, we obtain the expression (14). Now, we have to compute explicitly    Z1 ∂G  ˜ tVx + (1 − t) Vx0 , Vy , V˜ψ dt G 1 V x , Vy , V ψ = ∂Vx 0

By definition, G is given by the expression   G Vx , Vy , V˜ψ     1  a  c6 a c4 V˜ψ + M Vx + − c3 = Iz a Vx  Iz a  c  1 aM c4 5 + + M Vx + − c2 Iz a a Vx  1  aM  c6 − c3 Vy + kay Vy2 + Iz a Vx A simple computations give Vy V˜ψ ∂G = η1 2 + η2 2 + η3 V˜ψ + η4 Vy ∂Vx Vx Vx

Consider the nonlinear state feedback given by    Vx − Vx0  ˜ψ .  σ1 = −η − G V , V , V 1 x y  2   1 + (Vx − Vx0 )    1 ∂W      ∂ V˜ψ W + 1   ! p  λ2 + Sgn(λ2 ) ∆Vx0 ∂W   γ1 = −λ1 Vy +   2  ∂ V˜ψ     ∂W 1   −η 2    ∂Vy  ∂W 1 + ∂V y with : ∂W = λ3 ∂ V˜ψ

and

Sgn(λ2 )Vy + 4Sgn(λ2 )λ2 + 4 p ∆Vx0

p

∆Vx0

(17)

!

p λ2 − Sgn(λ2 ) ∆Vx0 ∂W = Vy + ∂Vy 2

Notice that η is a positive parameter introduced for performance tuning and may be different in the two expressions. (15)

The derivative of W1 by considering the system (16) with the state feedback (17) is given by

where the constants ηi , i = 1, ...4 are given by   ˙ 1 Vx , Vy , V˜ψ = ∂W1 V˙ x + ∂W1 V˙ y + ∂W1 V˜˙ ψ W ∂Vx ∂Vy ∂ V˜ψ
0 2 V x − Vx = −η 2 (Vx − Vx0 )  1 + 2 1 ∂W −η  2 ∂Vy ∂W 1 + ∂V y ≤0

  a  c6  − c η = −  3 1   Iz a    aM  c6 c    η2 = − 5 + c2 − − c3 a a Iz a  c4  η = + M  3   Iz a    aM c4    η4 = +M Iz a

Finally, by using the expression (15), we obtain

G1 =

Z1 0

Then, the system is stable. To prove that the system is attractive, we use the invariance principle of LaSalle J P. LaSalle et al (1961).

 ∂G  tVx + (1 − t) Vx0 , Vy , V˜ψ dt ∂Vx

Consider a set Ω defined by ˙ 1 (Vx , Vy , V˜ψ ) = 0} Ω = {W

η1 V˜ψ + η2 Vy + η3 V˜ψ + η4 Vy = Vx Vx0

Then, the system (13) can be rewritten as:  ˙ Vx = σ 1    V˙ = γ y 1 ˙ 0 ˜ ˜  
 V ψ = G(Vx , Vy , Vψ ) +G1 (Vx , Vy , V˜ψ ) Vx − Vx0

or Ω=

(16)

n (Vx , Vy , V˜ψ ) : Vx = Vx0 and ) p λ2 − Sgn(λ2 ) ∆Vx0 Vy + =0 2

On the set Ω, we have

Thereafter and to prove the asymptotic stability of the feedback system, we introduce the following Lyapunov function: 6282

! p λ2 − Sgn(λ2 ) ∆Vx0 Vy + 2 ! p λ2 − Sgn(λ2 ) ∆Vx0 ∂W = −λ1 Vy + 2 ∂ V˜ψ d dt

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

p λ2 − Sgn(λ2 ) ∆Vx0 =0 Vy + 2   p (18) 2Sgn(λ2 )λ3 Vy + 2(λ2 − Sgn(λ2 ) ∆Vx0 ) ∗ q R =0 λ22 − 4λ3 V˜ψ R = ∗

! p λ2 + Sgn(λ2 ) ∆Vx0 Vy + 2

Finally, and according to (18), we have p  λ − Sgn(λ2 ) ∆Vx0   Vy + 2 =0 2 p   V + λ2 + Sgn(λ2 ) ∆Vx0 = 0 y 2 or  p   V + λ2 − Sgn(λ2 ) ∆Vx0 = 0 y 2 q   Vy + 2(λ2 − Sgn(λ2 ) ∆V 0 ) = 0 x

18 16 14 12 10 η=0.5 η=1

8 6 0

50

100 Time (s)150

200

250

η=0.5 η=1

1000

l

where

20

Fig. 1. Longitudinal velocity Vx from 7 m/s to 20 m/s Acceleration force F (N)

   

 p λ2 − Sgn(λ2 ) ∆Vx0    Vy + =0  2 ! p λ2 + Sgn(λ2 ) ∆Vx0  ∂W  =0 Vy +   ∂ V˜ 2 ψ

(19)

800 600 400 0

50

100

150 Temps (s)

200

250

Fig. 2. Acceleration force Fl (20)

0 Steering angle δ (deg)

or     

Longitudinal velocity Vx (m/s)

The greatest invariant set checks the following equations

The system (19) does not have a solution in physical field. The above computation is done under the assumption such that ∆Vx0 > 0, but it imposes a condition on V˜ψ . The system (16) has Vy = V˜ψ = 0 as a unique solution. Thus, the greatest invariant set contained in Ω is limited in a neighborhood of (Vx0 , 0, 0). Finally, the system is asymptotically stable. This finish the proof of the following theorem, which is the main contribution of this paper. Theorem 1. The system (16) controlled by the
state feedback (17) is asymptotically stable on Vx0 , 0, 0 .

Fig.4 show the lateral velocity in the case of acceleration, i.e., evolution of the longitudinal velocity from 7 m/s to 20 m/s.

−0.1 −0.15 −0.2 −0.25 0

50

100

150 Temps (s)

200

250

Fig. 3. Steering angle δ 1.5 Lateral velocity Vy (m/s)

5. SIMULATION RESULTS

To evaluate the performance of the proposed nonlinear control systems a various simulations are proposed by using the vehicle data given in Appendix A. The main objective of the feedback control here is the stabilization of the different velocities of the vehicle. Fig. 1, Fig. 2 and Fig. 3 show respectively the evolution of the longitudinal velocity Vx , the acceleration force Fl and the steering angle δ in the case of acceleration. We can see when η increases, the vehicle reaches the stabilization velocity more quickly. Obviously the acceleration force also increases as shown in Fig. 2. However, the parameter η was introduced in the nonlinear control (??) and its determination is done in regard of one supplement criteria as a given rise time or avoiding overshoot.

η=0.5 η=1

−0.05

η=0.5 η=1

1 0.5 0 −0.5 −1 0

2

Time (s) 4

6

8

Fig. 4. Lateral velocity Vy with η = 0.5 Fig.5 shows the longitudinal velocity in two deceleration cases from 30 m/s to 20 m/s and from 30 m/s to 25 m/s with η = 0.5. the parameter η can be selected for soft acceleration/braking, i.e the driver does not accelerate so fast or braking so hard.

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Longitudinal velocity Vx (m/s)

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

REFERENCES

30 V0x =25m/s

28

V0=20m/s x

26 24 22 20 0

50

100

150 Temps (s)

200

250

Fig. 5. stabilization of Vx at the two desired velocities Vx 0 = 25 m/s and Vx 0 = 22 m/s Fig.6 shows the yaw rate Vψ for η = 0.5. −4

x 10

2

ψ

Yaw rate V (deg/s)

4

0

−2 0

5

Temps (s) 10

15

20

Fig. 6. Yaw rate Vψ

6. CONCLUSION We presented here a nonlinear approach method to design a nonlinear state feedback control. A stabilization of the different velocities of vehicle dynamics is chosen as an application. The stabilization is achieved following two steps. The first one consists to change variables in order to simplify the system structure. The second step consists to design a nonlinear feedback based on the combined Lyapunov-LaSalle approach applied to the simplified system structure.

Y. Abbassi, Y. A¨ıt-Amirat and R. Outbib, A global model for vehicle dynamics in Proceeding of AVCS, Argentina , pages 285-290, 2007. B. D’Andr´ea Novel and H. Chou, Total vehicle control using differential braking torques and activates suspension forces in Vehicle System Dynamics , volume 43, Number.4, pages 259-282, 2005. R. W. Brockett, R. S. Millmann and H. J Sussmann, Asymptotic stability and feedback stabilization in Differential Geometric Theory control, edition Progress in Mathematics, volume 27, pages 181-191, 1983. Der-Cherng Liaw, Wen-Ching Chung A feedback linearization design for the control of vehicle’s lateral dynamics in Nonlinear Dynamics, Springer Netherlands, volume 52, N4, pages 313-329, 2008. J. Tsinias, Sufficient Lyapunov-like conditions for stabilization in Math. Control Signals Systems , volume 2, pages 343-357, 1989. Z. Arstein, Stabilzation with relaxed controls in Nonlinear Analysis Theory Method Appli. , volume 7, pages 11631173, 1983. V. Jurjevic and J. P. Quinn, Controllability and stability in Newspaper of Differential Equations , volume 28, pages381389, 1978. R. Outbib, W. Aggoune, Feedback stabilzation of continuous systems by adding year integrator in International Newspaper of Applied Mathematics and Computer Science , pages 871-882, 1999. A. Alloum, A. Charara and Mr. Rombaut, Vehicle dynamic safety system by nonlinear control in Proceedings of the International IEEE Symposium one Intelligent Control , volume 6, pages 525-530, 1995. R. Outbib and A. Rachid, Control of Vehicle Speed: Nonlinear Approach in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia , 2000. J P. LaSalle and S. Lefschetz, Stability of Lyapunov’s direct method with applications in Academic Press, New York , 1961.

Notice that, the change on variables introduce two polynomial relations of second degree between the new and the old control variables. So, the final solution, in original control variables, must consider the operating ranges where the defined equation admits real solutions and respect the real handling of the vehicle. We showed that for constant longitudinal and lateral speeds and given an operating range, each of the two polynomial relations between the controls has a positive discriminant and consequently a real solutions. The simulation results show the performance of the suggested result. This result let us to consider in for coming work the problem of trajectory tracking. Appendix A. VEHICLE DATA M = 1480 kg, Iz = 1950 N.m/s2 Cf = 95000 N/rad, Cr = 50000 N/rad, h = 0.42 m, a = 1.421 m, b = 1.029 m, kax = 0.41 kg.m/s2 , kay = 0.54 kg.m/s2 . 6284