Sliding mode observers to replace vehicles expensive sensors and to preview driving critical situations

Int. J. Vehicle Autonomous Systems, Vol. 5, Nos. 3/4, 2007 345

Copyright © 2007 Inderscience Enterprises Ltd.

Sliding mode observers to replace vehicles expensive sensors and to preview driving critical situations

Hassan Shraim* and Mustapha Ouladsine Laboratory of Sciences of Informations and of Systems, LSIS UMR 6168 University of Paul Cézanne, Aix-Marseille III Av., escadrille de Normandie, Niemen 13397 Marseille Cedex 20, France E-mail: [email protected] E-mail: [email protected] *Corresponding author

Leonid Fridman Engineering Faculty, Department of Control, Division of Electrical Engineering, National Autonomous University of Mexico (UNAM), Ciudad Universitaria, 04510, D.F, Mexico E-mail: [email protected]

Abstract: In this paper, tyre longitudinal forces, vehicle side-slip angle and velocity are identified and estimated using a sliding mode observer. For this purpose and in order to insure the observability of the system, the model is decoupled in two parts. In the first part, longitudinal forces are identified using a second order sliding mode observer based on a modified super-twisting algorithm, while in the second part, vehicle velocity and side-slip angle are estimated using a classical sliding mode observer. Validations with the simulator VE-DYNA pointed out the good performance and the robustness of the proposed observers.

Keywords: estimation; sliding mode observer; automotive; vehicle parameters.

Reference to this paper should be made as follows: Shraim, H., Ouladsine, M. and Fridman, L. (2007) ‘Sliding mode observers to replace vehicles expensive sensors and to preview driving critical situations’, Int. J. Vehicle Autonomous Systems, Vol. 5, Nos. 3/4, pp.345–361.

Biographical notes: Hassan Shraim received his Diploma in Mechanical Engineering from the Lebanese University, 2003. In 2004 and 2007 respectively, he received his Master and PhD Degrees from the University Paul Cézanne Aix Marseille III in the modelling, control and estimation of complex systems. His research interests include estimation, diagnosis and control for non-linear and complex systems and their applications on real systems.

Mustapha Ouladsine received his PhD in Nancy, 1993 in the estimation and identification of non-linear systems. In 2001, he joined the laboratory of sciences of information and of systems in Marseille France. His research interests include non-linear estimation and identification, neural networks, diagnosis and control and their applications in the vehicle and aeronautic domains. He has published many technical papers in this field.

346 H. Shraim, M. Ouladsine and L. Fridman

Leonid Fridman received his PhD from Institute of Control Science (Moscow), and Doctor of Science degrees from Moscow State University of Mathematics and Electronics in 1988 and 1998, respectively. Now, he is with the Department of Postgraduate Study and Investigations at the Chihuahua Institute of Technology, Chihuahua, Mexico. In 2002, he joined the Department of Control at National of Autonomous University of Mexico. He is an editor of two books and four special issues on sliding mode control and an author of over 170 technical papers.

1 Introduction

In order to implement an advanced vehicle control system to obtain a desired vehicle motion, and to provide safety, global vehicle traction control should be realised. For this purpose a mathematical model representing, with a good precision, the real behaviour of the states describing the real system should be written; in addition, accurate and precise tools such as sensors should be implemented on the vehicle, in order to give a correct image of its comportment. Due to the fact that it is not possible to measure all the states and all the forces, due to the high costs of some sensors, or the non-existence of others, observers should be designed and constructed.

Recently, a great deal of research has been performed on the study of traction control by Petersen (2003) and Unsa and Kachroo (1999). But, for the complicated analytical models representing the vehicle (Shraim et al., 2005), it is seen that the study of the observation and the control for the global vehicle is not evident, due to the complicated form of the contact forces. These complications make the identification and the estimation of such forces of substantial interest, especially since the sensors for these forces are expensive, and their estimation is still an open problem in the literature. Another important thing that should be noticed is that, in the field of automotive engineering, the estimation of the side slip angle at the centre of gravity and the wheels’ interaction forces with the ground are very important, because of their influence on the stability of the vehicle. Many researchers have studied and estimated the side slip angle (Stephant, 2004), in which a bicycle model is used for the vehicle; this model is limited to the cases where the lateral acceleration is small, and in von Vietinghoff et al. (2005), an observer with adaptation of a quality function is used for the estimation of the side slip angle, but this observer requires the linearisation of the model. Moreover, an extended Kalman filter is used for the estimation of wheel forces (Samadi et al., 2001).

Sliding mode observer designs have been proposed by various authors, and they have received much attention and have shown good effectiveness when applied to non-linear systems by Shtessel et al. (2003), Edwards and Spurgeon (1998), Edwards et al. (2002) and Poznyak (2003). These types of observers are widely used due to the finite time convergence, robustness with respect to uncertainties and the possibility of uncertainty estimation. The new generation of differentiators and observers based on the second order sliding mode algorithms have been recently developed and used as in Shtessel et al. (2003). In Levant (1998), a robust exact differentiator was designed ensuring finite time convergence, as an application of the super-twisting algorithm (Levant, 1993). These differentiators are, for example, successfully used in Sira-Ramirez (2002) and Bartolini et al. (2003).

Sliding mode observers to replace vehicles expensive sensors 347

Another type of observer that also uses the sliding mode has been reported. In the early work of Slotine et al. (1987), the observer was constructed for a second order non-linear dynamic system involving only single measurement. The extension of such observers to nth order and multi output systems have also been addressed in the literature. Further development in this field of sliding observers was made by Misawa (1988) who applied such observers in robot manipulators.

In this paper, two classes of the sliding mode observers are used:

• In the first part, the observation of angular velocities of the four wheels of the vehicle and the identification of the longitudinal forces, which are supposed to be unknown elements, are made using a second order sliding mode observer based on the modification of the super-twisting algorithm. Only partial knowledge of the system model is required.

• In the second part, the identified longitudinal forces are used as inputs. The side slip angle and the velocity of the centre of gravity of the vehicle are estimated by a classical sliding mode observer. All the other forces and parameters are then deduced by using direct equations.

The first order sliding mode observer is not used in the first part because of the uncertainties and parameter identification two successive filtrations are necessary leading to a bigger corruption of results.

So, realising the standard first order sliding modes observer one filtration is needed to reconstruct the velocity and two successive filtrations are necessary to identify unknown inputs, while for the second order sliding mode observer the filtration is needed once when we would like to identify the unknown inputs.

Simulation results are compared by the results of the simulator VE-DYNA, which is developed by the group of companies TESIS. This simulator specially designed for fast simulation of vehicle dynamics and it is validated by real measures. The main contributions of this work reside in the estimations of wheels contact forces with the ground, the side slip angle and the velocities of the vehicle’s centre of gravity. This estimation previews some critical situations that may occur while rolling, such as excessive rotation around the Z axis and also excessive side slipping, inappropriate lateral acceleration, etc. For all these estimations, sliding mode observers are used. This observer is characterised by its rapid convergence to real values, its robustness for bounded modelling errors can be guaranteed and extensive computation load is not required.

The paper is organised as follows: In Section 2, the problem statement is proposed. In this section, the model of the vehicle is presented to show the complexity that it has and the problem of the observability. In Section 3 steps of the work are proposed by a graph showing at each step the proposed solution. In Section 4, the observer is designed and two classes of sliding mode observer are proposed, and at each step validation with the simulator VE-DYNA and comments are made. Finally in Section 5 a conclusion of the work will be shown.


2 Problem statement

The increasing demands for safety require accurate tools to represent states and parameters of the vehicle. These accurate representations need a lot of precise and expensive sensors, which means that an important diagnosis system should be implemented to avoid false data. To avoid these problems, which are the problems of expensive sensors and the complicated diagnosis system required, robust virtual sensors are proposed. These virtual sensors are based on a non-linear model which can be found by applying the fundamental principles of dynamics at the centre of gravity (Shraim et al., 2005) on Figure 1:

Figure 1 2D schema representation

( )COG1 cos( ) sin( )L SF Fβ β= −∑ ∑vM

(1)

( )COG

1 cos( ) sin( )S LF FM

β β β= − −∑ ∑vψ. (2)

with

wind 1 2 3 4

1 2 3 4

cos( )( ) cos( )( )sin( )( ) sin( )( )

LF F F F F FF F F F

δ δδ δ

= + + + +

− + − +∑ x f r

f r

x x x x

y y y y

and

1 2 3 4

1 2 3 4

sin( )( ) sin( )( )cos( )( ) cos( )( )

SF F F F FF F F F

δ δδ δ

= + + +

+ + + +∑ f r

f r

x x x x

y y y y


2 1 1 2

1 2 1 1 2

2 3 4 3 4

4 3 3 4

{cos( )( ) sin( )( )}{sin( )( ) cos( )( )}{sin( )( ) cos( )( )}

{cos( )( ) sin( )( )}.

I F F F FF F F FF F F FF F F F

δ δδ δδ δδ δ

= − + −+ + + ++ + − ++ − + −

Z f f f

f f

r r

r r r

t x x y y

L x x y y

L x x y y

t x x y y

ψ

(3)

The model representing the dynamics of each wheel i is found by applying Newton’s law to the wheel and vehicle dynamics Figure 2:

Figure 2 Wheel and its contact with the ground

1 torque 1 : 4I Fω = − + =ri i i ir x i (4)

In this paper, the task is to design virtual sensors (observers) for the vehicle to estimate the states, parameters and forces which need expensive sensors for their measurement. But due to the fact that it is not easy to apply an observer for the global model, equation (4) are taken at first, a second order sliding mode observer based on the modification of super-twisting algorithm is proposed for each equation (4) to observe the angular velocity of the wheel and to identify the longitudinal force. After having the longitudinal forces, we insert their values in equations (1)–(3). From these equations, it is seen that if we insert the longitudinal forces, we will still have as complex terms the lateral forces. For that reason, the wheel side forces are approximated to be proportional to the tyre side-slip angles αij (Uwe and Nielsen, 2005):

11

COGFL FL FL

LF C Cα δ β ×

= × = × − − fy

v

ψ (5)

12

COGFR FR FR

LF C Cα δ β ×

= × = × − − fy

v

ψ (6)

23

COGRL RL RL

LF C Cα β ×

= × = × − +

yv

ψ (7)

24

COG

.RR RR RRLF C Cα β

×= × = × − +

y

v

ψ (8)


For lateral accelerations above 4 m/sec2 and large tyre side-slip angles, the linear relations for the lateral forces are not sufficiently accurate any more. In this case, the tyre side-slip constant becomes time variant and the parameters CFL, CFR, CRL and CRR are written as: CFL(t), CFR(t), CRL(t) and CRR(t). As described in Uwe and Nielsen (2005), the tyre side-slip constant is calculated at every time step:

0 1 2 3( ( ( ) / )) ( ) arctan( ( ))( )

( )F F α

α−

= Zij k Zij k ij kij k

ij k

k k t k t k tC t

t (9)

k0 contains the actual friction coefficient; k1, k2 and k3 depend on the tyres characteristics, the tyre side-slip constants are adapted and inserted in the lateral forces equations. The equations of the lateral forces are validated by real measures (Uwe and Nielsen, 2005).

Then, the model of the vehicle equations (1)–(3) is rewritten Uwe and Nielsen (2005) as:

1COG 1 2

COG

23 4 COG

2

COG

1 ( )cos( ) ( ( ) ( ))

sin( ) cos( )2

( ( ) ( )) sin( )

FL FR

L

RL RR

LF F C CM

F F C A

LC C

δ β δ β

ρδ β β

β β

= + − − + − −

− + + −

+ + − +

f f

f x x a

v x x t tv

er v

t tv

ψ

ψ

(10)

11 2

COG COG

23 4 COG

2

COG

1 ( )sin( ) ( ( ) ( ))

cos( ) sin( )2

( ( ) ( )) cos( )

FL FR

L

RL RR

LF F C CM

F F C A

LC C

β δ β δ β

ρδ β β

β β

= + − + + − −

− − + −

+ + − + −

x x f f

f x x a

t tv v

er v

t tv

ψ

ψψ

(11)

{ 11 1 2

COG

1 2 1

1

COG

22

COG

1 ( cos( ))( )sin( )

cos( )( ( ) ( ))( ) cos( ) ( )cos( )

( ( ) ( )) sin( )

( )( ( ) ( ))

Z

FL FR

FR FL

RL RR

LL F FI

C C L F F

LC C

LL C C

δ δ δ β

δ δ δ

δ β δ

β

= − + + − −

+ − + −

− − − −

− + + − + +

fl f x x f f

f lf f f x x f

f f f

lr

nv

t t n t

t t tv

n t tv

ψψ

ψ

ψ4 3( ) .F F

−

r x xt

(12)

The system described by the equations (1)–(3) is observable if we consider that we measure only the yaw rate (and by supposing the longitudinal forces as inputs). A classical sliding mode observer is used to the estimate the side slip angle and the velocity of the centre of gravity. By these estimations, the longitudinal and lateral velocities of the centre of gravity, the lateral forces of the wheels are then directly


deduced. By these estimations, the driver (or the controller) knows if the states and parameters are in the safe region or not. These regions depend on the velocity, coefficient of friction and the steering angle (Uwe and Nielsen, 2005; Gillespie, 1992).

3 The proposed solution

The steps for estimating the states, parameters and forces are described by Figure 3.

Figure 3 The proposed solution

4 Observer design

4.1 Estimation of wheels angular velocities and longitudinal forces

In this part, a second order SM observer based on modification super-twisting algorithm is proposed to observe the angular velocity wi and to identify the longitudinal force of each wheel Fxi. Dynamical equations of wheels (4) are written in the following form:

1 2

2 1 2( , , )==

x x

x f x x u (13)

where x1 and x2 are respectively θi (which is measured) and wi (to be observed) (appears implicitly in Fxi), and u is torquei. In fact, this torque may be measured as shown in Rajamani et al. (2006), and it is equal to the difference between the motor and the braking torque, the motor torque may be estimated as in Khiar et al. (2006), while the braking torque is estimated by measuring the hydraulic pressure applied at each wheel.


As described in Davila et al. (2005), the proposed super-twisting observer has the form:

1 2 1

2 1 1 2 2

ˆ ˆ

ˆ ˆ( , , )

= +

= +

x x z

x f x x u z (14)

where 1 2ˆ ˆandx x are the state estimations of the angular positions and the angular velocities of the four wheels, respectively, f1 is a non-linear function containing only the known terms (which is only the torque in our case), z1 and z2 are the correction factors based on the super-twisting algorithm having the following forms:

1/ 21 1 1 1 1

2 1 1

ˆ ˆsign( )ˆsign( ).α

= − −= −

z x x x x

z x x

λ (15)

Taking 1 1 1 2 2 2ˆ ˆand= − = −x x x x x x we obtain the equations for the error 1/ 2

1 2 1 1

2 1 2 2 1

sign( )

ˆ( , , , ) sign( )F α

= −

= −

x x x x

x t x x x x

λ (16)

where 1 2 2ˆ( , , , )F t x x x is the difference between the unknown and the known function. Suppose that the system states can be assumed bounded, then the existence is ensured

of a constant f+, such that the inequality:

1 2 2ˆ( , , , )F +<t x x x f (17)

holds for any possible t, x1, x2 and |x2| ≤ 2 sup |x2|. Which means it is sufficient to give for f + the sum of the maximum values of the

longitudinal force and the torque that may be applied. Let α and λ satisfy the inequalities:

2 ( )(1 ), ,(1 )

ααα

++

+

+ +> >−−f p

fpf

λ (18)

where p is some chosen constant, 0 < p < 1. To prove that the observer equations (3) and (4) for the system (2) ensure the finite

time convergence of estimated states to the real states, i.e., 1 2 1 2ˆ ˆ( , ) ( , ).→x x x x the same steps are considered as in Davila et al. (2005).

4.1.1 Simulations and results Simulations are made and the results are compared by those provided by simulator VE-DYNA, the operation condition corresponds to a strong variation in Fxi Figure 8 and wi Figure 7 (acceleration, constant velocity, deceleration, constant velocity acceleration, constant velocity, deceleration, constant velocity) with a zero steering angle. The same observer is applied to the four wheels, but for the similarity, we present only one observer corresponding to the front left wheel (wheel 1). The simulator uses a car with two rear wheel drives. Figure 4 shows the input torque for the two rear wheels and Figure 5 the torque for the two front wheels. In Figures 6 and 7 we see θ1 and w1 (given by the simulator VE-DYNA) and those computed by the proposed observer. In these figures


we see the rapid convergence of the observer in spite of the initial values being θ10 = 0 radians, 10

ˆ 50 radians,θ = w10 = 0 rad/sec and 10ˆ 100 rad / sec.=w The unknown functions 1 2 2ˆ( , , )Fi x x x are filtered by a first order filter. We see that

the filtered functions approximate Fxi given by VE-DYNA Figure 8.

Figure 4 Motor and braking torque (N.m) applied at the two rear wheels

Figure 5 Motor and braking torque (N.m) applied at the two front wheels

Figure 6 Angular position (rad) by the simulator VE-DYNA, and that estimated by the observer


Figure 7 Angular velocity (rad/sec) by the simulator VE-DYNA, and that estimated by the observer

Figure 8 The unknown input after filtration (N) and the longitudinal force from the simulator VE-DYNA

4.2 Estimation of the side slip angle, velocity of the vehicle and reconstruction of the yaw rate

In this part, a classical first order sliding mode observer is used to estimate the velocity and the side slip angle of the centre of gravity. As described in equations (10)–(12), the model of the vehicle is non-linear and it can be written as follows:

( , )( )

==x f x u

y c x (19)

where

COG[ ]β=x v ψ (20)

the input:

1 2 3 4[ ]F F F F δ= x x x x fu (21)


and the measurement vector

[ ].=y ψ (22)

Before the design of the sliding mode observer for the model of equation (19), the observability of the model must be investigated and tested. The observability definition is local and uses the Lie derivative (Nijmeijer and van der Schaft, 1990). It is a function of the state trajectory and the inputs to the model. For the system described by equation (19) the observability function is:

2

( )observability( , ) ( , )

( , )LL

=

f

f

c x

x u c x u

c x u

where

( , ) ( , ).L = jf

cc x u f x u

x

d

d

The system is observable if its Jacobian matrix Jobservability has a full rank (which is 3 in our case).

observability observability( , ).J = x uxdd

By applying these notions to the system described by equation (19), we see that its rank is 3 and hence, observable.

So, the proposed sliding mode observer is:

ˆ ˆ ˆ( , ) sign( )ˆ ˆC

= + ∆ −

=

x f x u y y

y x (23)

where ∆ is the gain of the sliding mode observer.

4.2.1 Simulations and results

In this part, the estimated vehicle velocity and side slip angle are compared to that of the simulator VE-DYNA, it is seen that the errors are practically very small. Simulations are made covering most of driving situations, but here one significant simulation is shown where the side slip angle varies strongly, which is the case when there is change in the steering angle and torques Figures 9 and 10 respectively. A run of 22 s is made, the longitudinal forces are estimated from the first part, and then inserted as inputs to this observer. In Figures 11–13, we see the observed states with those provided by the simulator; the rapid convergence shown points out the good performance of the proposed observer. The gain of the observer is ∆ = [10, 10, 10]T:


Parameter Value

M 1296 kg r1i 0.9 kg.m2 Iri 0.28 m lo –0.03 m Cij 50000 N/rad L1 0.97 m IZ 1750 tf 0.7 m l1 0.12 m AL 2.25 m2

L2 1.53 m H 0.52 m rr 0.75 m

ρ 1.25 kg/m3

Figure 9 Front steering angle (rad)

Figure 10 Motor and braking torque (N.m) applied at the two rear wheels


Figure 11 Estimated vehicle velocity (m/sec) and that of the simulator VE-DYNA

Figure 12 Estimated side slip angle (rad) using sliding modes and that of the simulator VE-DYNA

Figure 13 Reconstructed yaw rate (rad/sec) and that of the simulator VE-DYNA

4.3 Longitudinal velocity, lateral velocity, wheel sideslip angle and the lateral forces

In this section, and after estimating the slip angle and the velocity of the centre of gravity, the velocities of the centre of gravity in (X, Y) can be found by Uwe and Nielsen (2005), the lateral velocity (see Figure 14):

COG sin( )V β=y v (24)

and the longitudinal velocity which coincides with that of the simulator (see Figure 15):

COG cos( ).V β=x v (25)


Figure 14 Estimated V y (m/sec) and that of the simulator VE-DYNA

Figure 15 Estimated Vx (m/sec) and that of the simulator VE-DYNA

The lateral forces can be found by applying equations (5)–(8), and results can be shown, for example, the lateral force of the front left wheel (see Figure 16):

Figure 16 Estimated lateral force (N) for the front left wheel and that of the simulator VE-DYNA

5 Conclusions

Sliding mode observers are proposed in this work to estimate vehicle parameters and states which are not easily measured. Their short time of convergence and their robustness make their usage very encouraging in automotive applications, especially when the automobile is subjected to outside disturbance and some uncertain parameters.


After having these estimations, the other parameters and states are found using validated equations. Comparison with the simulator VE-DYNA at each step shows the efficacy of the proposed observers. The important idea that also may be taken from this work, is by these estimations, we can preview the driver or the controller when a parameter is around the maximum acceptable limits or it leaves the safety region. These regions depend on the velocity and the steering angle.

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sliding control to mechanical systems’, Int. J. Control, Vol. 76, pp.875–892. Davila, J., Fridman, L. and Levant, A. (2005) ‘Second-order sliding mode observer for mechanical

systems’, IEEE Transactions on Automatic Control, Vol. 50, No. 2, pp.1785–1789. Edwards, C. and Spurgeon, S.K. (1998) Sliding Mode Control, Taylor and Francis, London. Edwards, C., Spurgeon, S.K. and Hebden, R.G. (2002) ‘On development and applications of sliding

mode observers’, in Xu, J. and Xu, Y. (Eds.): Variable Structure Systems: Towards XXIst Century, ser. Lecture Notes in Control and Information Science, Springer Verlag, Berlin, Germany, No. 274, pp.253–282.

Gillespie, T.D. (1992) Fundamentals of Vehicle Dynamics, Published by Society of Automotive Engineers, Inc., USA.

Khiar, D., Lauber, J., Floquet, T., Guerra, T., Coline, G. and Chamillard, Y. (2006) Estimation of the Instantaneous Torque of a Gasoline Engine, CIFA, French.

Levant, A. (1993) ‘Sliding order and sliding accuracy in sliding mode control’, Int. J. Control, Vol. 58, pp.1247–1263.

Levant, A. (1998) ‘Robust exact differentiation via sliding mode technique’, Automatica, Vol. 34, No. 3, pp.379–384.

Misawa, E.A. (1988) ‘Non linear state estimation using sliding observers-a state of the art survey’, ASME J. Dyn. System Measurment Control, Vol. 109, pp.344–352.

Nijmeijer, H. and van der Schaft, A.J. (1990) Nonlinear Dynamical Control Systems, Springer-Verlag, Berlin.

Petersen, I. (2003) Wheel Slip Control in ABS Brakes using Gain Scheduled Optimal Control with Constraints, Thesis submitted for the degree of doctor engineer Department of Engineering Cybernetics, Norwegian University of Science and Technology Trondheim, Norway.

Poznyak, A.S. (2003) ‘Stochastic output noise effects in sliding mode estimations’, Int. J. Control, Vol. 76, pp.986–999.

Rajamani, R., Piyabongkarn, D., Lew, J.Y. and Grogg, J.A. (2006) ‘Algorithms for real time estimation of individual wheel tire road friction coefficients’, American Control Conference, USA, Minneapolis, Minnesota, pp.4682–4687.

Samadi, B., Kazemi, R., Nikravesh, K.Y. and Kabganian, M. (2001) ‘Real-time estimation of vehicle state and tire friction forces’, Proceedings of the American Control Conference (ACC), USA, Minneapolis, Minnesota, pp.3318–3323.

Shraim, H., Ouladsine, M., El Adel, M. and Noura, H. (2005) ‘Modeling and simulation of vehicles dynamics in presence of faults’, 16th IFAC World Congress, Prague, Czech Republic, pp.687–692.

Shtessel, Y.B., Shkolnikov, I.A. and Brown, M.D.J. (2003) ‘An invariance principle for discontinuous dynamic systems with application to a coulomb friction oscillator’, An Asymptotic Second Order Smooth Sliding Mode Control, pp.860–865.

Sira-Ramirez, H. (2002) ‘Dynamic second order sliding mode control of the hovercraft vessel’, IEEE Trans. on Control System Technology, Vol. 10, pp.860–865.


Slotine, J.J., Hedrik, J.K. and Misawa, E.A. (1987) ‘On sliding observers for nonlinear systems’, ASME J. Dynam. Syst. Measvol., Vol. 109, pp.245–252.

Stephant, J. (2004) Contribution à l’étude et à la Validation Expérimentale D’observateurs Appliqués à la Dynamique du Véhicule, Thesis Presented to have the Doctor Degree from the UTC, University of Technology, Compiègne.

Unsa, C. and Kachroo, P. (1999) ‘Sliding mode measurement feedback control for antilock braking systems’, IEEE Transactions on Control System Technology, Vol. 7, No. 2.

Uwe, K. and Nielsen, L. (2005) Automotive Control System, Springer-Verlag, Berlin, Germany. von Vietinghoff, A., Hiemer, M. and Uwe, K. (2005) ‘Non-linear observer design for lateral

vehicle dynamics’, 16th IFAC World Congress, Prague, Czech Republic.

Website Simulator VE-DYNA, http://www.tesis.de/index.php.

Nomenclature

Symbol Physical signification

iw Angular velocity of the wheel M Total mass of the vehicle

ir Radius of the wheel i COG Centre of gravity of the vehicle

1ir Dynamical radius of the wheel i FZij Vertical force at wheel i j (i: front, rear, j: right, left) F ix Longitudinal force applied at the wheel i F iy Lateral force applied at the wheel i

fiC Braking torque applied at wheel i

miC Motor torque applied at wheel i torquei Cmi – Cfi

ZI Moment of inertia around the Z axis ψ Yaw angle ψ Yaw velocity δ f Front steering angle δr Rear steering angle δ i Deflection in the tyre i Vx Longitudinal velocity of the centre of gravity Vy Lateral velocity of the centre of gravity

riI Moment of inertia of the wheel i vCOG Total velocity of the centre of gravity L1 Distance between COG and the front axis L2 Distance between COG and the rear axis L L1 + L2 Cij Tyre side slip constants (i: front, rear, j: right, left)


αij Slip angle of the wheel I hCOG Height of COG h1 Distance between the COG and the Pitch axis

ft Front half gauge

rt Rear half gauge l tf + tr

windFx Air resistance in the longitudinal direction

windFy Air resistance in the lateral direction AL Front vehicle area ρ Air density Caerr Coefficient of aerodynamic drag nL1 Caster effect front nL2 Caster effect rear cpresi Parameter to correct for tyre pressure distribution αij Slip angle of the wheel i β Side slip angle at the COG

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Sliding mode observers to replace vehicles expensive sensors and to preview driving critical situations