vehicle parameter estimation and stability enhancement...

230 Int. J. Vehicle Design, Vol. 48, Nos. 3/4, 2008

Vehicle parameter estimation and stability

enhancement using sliding modes techniques

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 20E-mail: [email protected]∗Corresponding author

Leonid Fridman

Department of Control, Division of Electrical Engineering,National Autonomous University of Mexico (UNAM),Ciudad Universitaria, 04510, DF, MexicoE-mail: [email protected]

Monica Romero

Facultad de Cs. Exacts, Ingenieria y Agrimensura,Departemento de Electronica,Universidad Nacional de Rosario – ArgentinaE-mail: [email protected]

Abstract: In this paper, tyres longitudinal forces, vehicle side slip angleand velocity are identified and estimated using sliding modes observers.Longitudinal forces are identified using higher order sliding mode observers.In the estimation of the vehicle side slip angle and vehicle velocity, anobserver based on the broken super – twisting algorithm is proposed.Validationswith the simulatorVE-DYNApointedout the goodperformanceand the robustness of the proposed observers. After validating theseobservers, controller design for the braking is accomplished using a reducedstate space model representing the movement of the vehicle centre ofgravity in the (X, Y ) plane. Driver’s reactions are taken into account.The performance of the closed loop system is carried out by means ofsimulation tests.

Keywords: automotive; estimation; sliding mode observer; vehicleparameters.

Reference to this paper should be made as follows: Shraim, H.,Ouladsine, M., Fridman, L. and Romero, M. (2008) ‘Vehicle parameterestimation and stability enhancement using sliding modes techniques’,Int. J. Vehicle Design, Vol. 48, Nos. 3/4, pp.230–254.

Copyright © 2008 Inderscience Enterprises Ltd.

Vehicle parameter estimation and stability enhancement 231

Biographical notes: Hassan Shraim received his Diploma in the MechanicalEngineering from the Lebanese university 2003. In 2004 he received themaster degree from the University Paul Cézanne, polytechnic Marseille inthe control and simulation of complex systems, PhD in the diagnosis andcontrol for non-linear systems and their applications in the vehicle domain.He is now an industrial scientific consultant.

Mustapha Ouladsine received his PhD in Nancy 1993 in the estimationand identification of non-linear systems. In 2001, he joined the laboratoryof sciences of information and systems in Marseille France. His researchinterests include non-linear estimation and identification, neural networks,diagnosis and control and their applications in the vehicle and aeronauticdomains, he published more than 50 technical papers.

Leonid Fridman received his MS in Mathematics from Kuibyshev (Samara)State University, Russia, PhD in Applied Mathematics from Institute ofControl Science (Moscow), and DSc in Control Science from MoscowState University of Mathematics and Electronics in 1976, 1988 and 1998correspondingly. In 2002, he joined theDepartment ofControl andRobotics,Division of Electrical Engineering of Engineering Faculty at National ofAutonomous University of Mexico, Mexico. His research interests includevariable structure systems, nonlinear observation, singular perturbations,and systems with delay. He is an editor of three books and five special issueson sliding modes. He published over 200 technical papers.

Monica Romero is Electronic Engineer Graduated at National University ofRosario (UNR), Argentina, and PhD Graduated at National University ofLaPlata,Argentina.Now, she isAdjunt Professor at theControlDepartmentofFacultaddeCs. Ex., Ingenieria yAgrimensura,UNR.Her research interestincludes motion control, power electronics, and fault detection and isolationsystems with applications in mechatronics.

1 Introduction

1.1 Preliminaries and motivations

In the recent years, important research has been undertaken to investigate the safedriving conditions in both normal and in critical situations. Safe driving requiresthe driver to react extremely quickly in dangerous situations, which is generally verydifficult unless for experts, that result the instability of the system. Consequently, theimprovement of the vehicle dynamics by active chassis control is necessary for suchcatastrophic situations. Increasingly, commercial vehicles are being fitted with microprocessor based systems to enhance the safety and to improve driving comfort, increasetraffic circulation, and reduce environmental pollution associated with vehicles.

Considerable attention has been given to the development of the control systemsover the past few years, authors have investigated and developed different methodsand different strategies for enhancing the stability and the handling of the vehiclesuch as, the design of the active automatic steering (You and Chai, 1999) and the

232 H. Shraim et al.

wheel ABS control (Petersen, 2003; Unsa and Kachroo, 1999), or the concept ofa four-wheel steering system (4WS) which has been introduced to enhance vehiclehandling. Some researchers have shown disadvantages on (4WS) vehicles (Nalecz andBindemann, 1989).

In terms of vehicle safety, and in order to develop a control law for the vehiclechassis, accurate andprecise tools suchas sensors shouldbe implementedon thevehicle,to give a correct image of its comportment. Difficulties in measuring all vehicle statesand forces, due to high costs of some sensors, or the non existence of others, makethe design and the construction of observers necessary. In the field of automotiveengineering, the estimation of vehicle side slip angle and wheel interaction forces withthe ground are very important, because of their influence on the stability of the vehicle.Many researchers have studied and estimated vehicle side slip angle, using a bicyclemodel as in Stephant (2004), or by using an observer with adaptation of a qualityfunction as in Von Vietinghoff et al. (2005) which requires a certain linearised form ofthe model. Moreover, an extended Kalman filter is used for the estimation of wheelforces (Samadi et al., 2001).

In the previous work (Shraim et al., 2005a), we have proposed a non-linear controlstrategy based on the principles of predictive control for vehicle trajectory tracking, theproposed controller was not robust to outside disturbance in a certain given horizon,and in addition to that drivers’ reactions were not considered.

In this paper we propose a robust sliding mode controller together with estimationof the states, forces and parameters, in order to ensure safety in the critical situationsand considering driver’s reactions (steering angle and torques applied at the wheels).

1.2 Methodology

The problem of observation has been actively developed within Variable StructureTheory using sliding mode approach. Sliding mode observers (see, for example, thecorresponding chapter in the textbooks by Edwards and Spurgeon (1998) and Utkinet al. (1999) and the recent tutorials by Slotine et al. (1987), Barbot and Floquet (2004),Edwards et al. (2002) and Poznyak (2003) are widely used due to their attractivefeatures:

• insensitivity (more than robustness) with respect to unknown inputs

• possibilities to use the values of the equivalent output injection for the unknowninputs identification (Utkin et al., 1999)

• finite time convergence to the reduced order manifold.

In Levant (1998, 2003) robust exact differentiators were designed ensuring finite timeconvergence to the real values of derivatives, as an application of super-twistingalgorithm (Levant, 1993). A new generation of observers based on the high ordersliding mode differentiators are recently developed (see Davila et al., 2005; Floquetand Barbot, 2006; Bejarano et al., 2007; Fridman et al., 2007, 2008). Those observers:

• provide a finite time convergence to the exact values of states variables

• allow the finite time identification the unknown inputs without filtration

• guarantee the best possible accuracy of the state estimation w.r.t. to thesampling steps and deterministic noises.


In this paper the sliding mode controller is used to ensure safety in the criticalsituations and considering driver’s reactions (steering angle and torques applied at thewheels).

In the paper the results are validated by the simulator VE-DYNAdeveloped by thegroup of companies TESIS, which is an independent expert team for the simulationof virtual vehicles in real time, their products are employed by almost all Germanautomakers and sought after worldwide. Their software comprises high precisionmodels for simulation of vehicle dynamics VE-DYNA, engine dynamics EN-DYNAand brake hydraulics (RTBrakeHydraulics). The simulator VE-DYNA that we use inour study is a software especially designed for the fast simulation of vehicle dynamicsboth in real-time applications (Hardware-in-the-Loop, Software-in-the-Loop) andconcept studies on a standard desktop PC, its computational performance enables itsusage for the optimisation e.g., for the parameter identification. VE-DYNA vehiclemodel is fully parametric and has a modular architecture with the following programmodules: Vehicle multi-body system (chassis), axles (axle kinematics, compliance),tyre model TM-Easy, transmission and drive line, engine, aerodynamics and brakingsystem. All vehicle Degrees of Freedom (DoF) are nonlinearly modelled. The numberof overall DoF depends on the number of additional bodies and the suspension type.There are at least 15 DoF up to 65 DoF in the base vehicle model; the global vehiclemodel and each sub model are validated by real experiments with different operationconditions.

1.3 Paper contribution

The main contributions of this work reside in two important points:

• The estimations of wheels contact forces with the ground, side slip angle and thevelocity of the vehicle which avoid the use of expensive sensors. Theseestimations preview also some critical situations that may occur while rolling,such as excessive rotation around Z axis and also excessive side slipping,inappropriate lateral acceleration, . . . The proposed observers are characterisedby the rapid convergence to real values, robustness and they do not requireextensive computation load.

Both observers are validated by the simulator VE-DYNA. Several validationswere made to cover most of the driving cases, such as a double lane trajectory,straight line motion with strong variation in acceleration and deceleration,strong change in the steering angle.

• The control of the yaw rate, side slip angle and velocity of the vehicle bycontrolling wheels braking systems. In this part, a reference value is generatedfor each of the controlled parameters at each time step, and then a robust slidingmode control is applied. This controller functions only in the critical situationsand can generate only braking torques on the four wheels.

The paper is organised as follows: In Section 2, problem statement is proposed andthe model used for the vehicle is shown. In Section 3, observer design is made, in thissection, two slidingmode observers are proposed: a third order slidingmode is used forthe estimation of wheels velocities and the identification of the longitudinal forces, anda sliding mode observer based on broken super-twisting algorithm (Davila et al., 2005)


for the estimation of the side slip angle and the velocity of the vehicle. In Section 4,the controller design is made and the equations of reference trajectories are shown.In Section 5 simulations are shown and finally a conclusion for the work is presented.

All the solutionsof thedifferential equationsmentioned in thepaper areunderstoodin the sense of Filippov (1988).

2 Problem statement

The presented problem is the problem of assistance for the driver in critical situations,i.e., when the vehicle goes out from its safety regions and enters in the dangeroussituations. So accurate tools to represent vehicle states and parameters are required.These accurate representations need many precise and expensive sensors. The use ofmany sensors requires an important diagnosis system to avoid false data. To overcomethese problems, robust virtual sensors are proposedwhich estimate vehicle parameters,forces and states. These virtual sensors are embedded in a closed sliding mode controlloop. The model used for estimation and control is a non-linear one obtained byapplying the fundamental principles of dynamics at the centre of gravity on Figure 1(Von Vietinghoff et al., 2005):

vCOG =1M

(cos(β)

∑FL − sin(β)

∑FS

)(1)

β =1

MvCOG

(cos(β)

∑FS − sin(β)

∑FL

)− ψ (2)

with ∑FL = Fxwind + cos(δf )(Fx1 + Fx2) + cos(δr)(Fx3 + Fx4)

− sin(δf )(Fy1 + Fy2) − sin(δr)(Fy3 + Fy4)

and ∑FS = sin(δf )(Fx1 + Fx2) + sin(δr)(Fx3 + Fx4)

+ cos(δf )(Fy1 + Fy2) + cos(δr)(Fy3 + Fy4)

IZ ψ = tfcos(δf )(Fx2 − Fx1) + sin(δf )(Fy1 − Fy2)+ L1sin(δf )(Fx2 + Fx1) + cos(δf )(Fy1 + Fy2)+ L2sin(δr)(Fx3 + Fx4) − cos(δr)(Fy3 + Fy4)+ trcos(δr)(Fx4 − Fx3) + sin(δr)(Fy3 − Fy4). (3)

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

IriΩ = −r1iFxi + torquei i = 1 : 4. (4)

In this paper, the task is to design a virtual sensor (observer) for the vehicle to estimatethe states, parameters and forces which need expensive sensors for their measurement.


Figure 1 2D vehicle representation

Figure 2 Wheel and its contact with the ground

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 high order sliding mode observers are proposed foreach equation (4) to observe the angular velocity and to identify the longitudinalforce of each wheel. After having the longitudinal forces, we substitute their values inequations (1)–(3). From these equations, it is seen that if we substitute the longitudinalforces, we will still have as complex terms the lateral forces. In order to model thelateral force, we use a brush model for the contact with the ground (Shraim et al.,2005b). The brush model divides the surface of contact into two parts: a sliding partand an adherent part. Then the lateral force generated at the surface of contact willbe the sum of forces generated at each part of the surface. These lateral forces are


represented as in Shraim et al. (2005b):

Fyi =tan(αi)µiFziXslidingi√(

tan(αi)2 +(

Ωi

V xi

((K1i +

K2i

pi

)× Fzi

1000

)− 1

)2)× Xti

+Ai

3× X3

adherencei (5)

with

Ai =1B

2Ki tan(αi) − Fyslidingi × ϕ

XadherenceiXslidingi(6)

and

B = C + D (7)

C =1β1

(1 + exp(−β1Xadherencei)(sin(β1Xadherencei) − cos(β1Xadherencei))) (8)

D = X(1 + exp(−β1Xadherencei) cos(β1Xadherencei)) (9)

with

β1 = 4

√Ki

E × Iri(10)

and

ϕ = 1 + F + G + H (11)

F = − exp(−β1Xslidingi) cos(β1Xslidingi) (12)

G = − exp(−β1Xadherencei) cos(β1Xadherencei) (13)

H = exp(−β1Xti) cos(β1Xti) (14)

Fyslidingi =tan(αi)µiFziXslidingi√(

tan(αi)2 +(

Ωi

V xi

((K1 +

K2

pi

)× Fzi

1000

)− 1

)2)× Xti

. (15)

From this representation of the lateral force, we need to define some of the variables,all the equations for these variables are detailed in Shraim and Ouladsine (2006) andUwe and Nielsen (2005), and they are given by:

• the coefficient of adherence µ:

µi =Fxi

Fzi(16)


• the side slip angle of each wheel:

α1 = tan−1(

vCOG sin(β) + L1ψ

vCOG cos(β) − tf ψ

)− δf (17)

α2 = tan−1(

vCOG sin(β) + L1ψ

vCOG cos(β) + tf ψ

)− δf (18)

α3 = tan−1(

vCOG sin(β) − L2ψ

vCOG cos(β) − trψ

)− δr (19)

α4 = tan−1(

vCOG sin(β) − L2ψ

vCOG cos(β) + trψ

)− δr. (20)

The velocity of each wheel,

V x1 = (vCOG cos(β) − tf ψ) cos(δf ) + (vCOG sin(β) + L1ψ) sin(δf ) (21)

V x2 = (vCOG cos(β) + tf ψ) cos(δf ) + (vCOG sin(β) + L1ψ) sin(δf ) (22)

V x3 = (vCOG cos(β) − trψ) cos(δr) + (vCOG sin(β) − L2ψ) sin(δr) (23)

V x4 = (vCOG cos(β) + trψ) cos(δr) + (vCOG sin(β) − L2ψ) sin(δr). (24)

For the vertical forces, cheap sensors can be found for their measure (or they can beestimated as shown inUwe andNielsen (2005)), the determination of the contact patchand its repartition into a sliding part and adhesion part can be found as shown inShraim and Ouladsine (2006).

The system described by the equations (1)–(3) is observable if we consider thatwe measure only the yaw rate (and by supposing the longitudinal forces as inputs).A sliding mode observer based on broken super-twisting algorithm is used to estimatevehicle side slip angle and velocity. By these estimations, the longitudinal and lateralvelocities of the centre of gravity, the lateral forces of the wheels are then directlydeduced.

By these estimations, the driver (or the controller) knows if the states andparameters 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).

In this study it is supposed that we can measure:

• angular positions of the wheels

• front wheel angle

• yaw rate

• torque applied at each wheel

and it is required to estimate:

• angular velocity of the wheels

• contact forces

• vehicle velocity

• vehicle side slip angle.


3 Observer design

3.1 Estimation of wheels angular velocities and longitudinal forces

In this part, slidingmode observers are proposed to observe the angular velocitywi andto identify the longitudinal force of each wheel Fxi. Dynamical equations of wheels(4) are written in the following form:

x1 = x2 (25)x2 = f(x1, x2, u)

wherex1 andx2 are respectively θi (which ismeasured) andwi (tobeobserved) (appearsimplicitly in Fxi), and u is torquei. In fact this torque may be measured as shown inRajamani et al. (2006), and it can also be estimated by estimating the motor and thebraking torque, the motor torque may be estimated as in Khiar et al. (2006), while thebraking torque is estimated by measuring the hydraulic pressure applied at each wheel(existing on the most of the vehicles).

3.2 Broken super-twisting observer structure

In the first part of this section, the so-called broken super-twisting algorithm observerproposed in Davila et al. (2005) will be employed (see M’Sirdi et al., 2006). This isboth inherently suited to nonlinear plant representations and does not require to haverelative degree one with repsect to unknown inputs.

˙x1 = x2 + z1 (26)˙x2 = f1(x1, x2, u) + z2

where x1 and x2 are the state estimations of the angular positions and the angularvelocities of the four wheels respectively, f1 is a nonlinear function containing onlythe known terms (which is only the torque in our case), z1 and z2 are the correctionfactors based on the broken super-twisting algorithm having the following forms:

z1 = λ|x1 − x1|1/2 sign(x1 − x1) (27)z2 = λ0 sign(x1 − x1).

In the above equations the function | · | 12 and sign(.) should be thought of as

componentwise extensions of their traditional scalar counterparts.Taking x1 = x1 − x1 and x2 = x2 − x2 we obtain the equations for the error

˙x1 = x2 − λ|x1|1/2 sign(x1) (28)˙x2 = F (t, x1, x2, x2) − λ0 sign(x1)

where F (t, x1, x2, x2) = f(x1, x2, u) − f1(x1, x2, u) is the unknown function to beidentified.

Suppose that the system states can be assumed bounded, then the existence isensured of a constant f+, such that the inequality:

|F (t, x1, x2, x2)| < f+ (29)


holds for any possible t, x1, x2 and |x2| ≤ 2 sup |x2|. As described in Davila et al.(2005), it is sufficient to choose λ0 = 1.1f+ and λ = 1.5

√f+. A sliding motion occurs

in the error system (27) which makes x1 ≡ 0 and x2 ≡ 0 in finite time. Furthermorewhilst sliding x2 = 0 and so from equation (27) we get:

α sign(x1 − x1) = F (t, x1, x2, x2) (30)

where the left hand of the above equation represents the average value of thediscontinuous term which must be taken in order to maintain a sliding motion.The quantity sign(x1 − x1) can be obtained by appropriate low-pass filtering of thediscontinuous injection signal sign(x1 − x1) and so is available in real time.

So in the broken super-twisting observer considered here the estimate of thelongitudinal forces must be obtained by filtering the discontinuous injection signal(which may cause a certain delay). In the next, a third order sliding mode observer willbe considered to obviate this necessity to filter.

3.3 A third order sliding mode observer

The proposed third order sliding mode observer has the form:

˙θi = Ωi + λ0|θi − θi|2/3 sign(θi − θ)︸︷︷︸

vo (31)˙Ωi =torquei

Iri+ z1

where θi and Ωi are the state estimations of the angular positions and the angularvelocities of the four wheels respectively, z1 is the correction factor based on thesuper-twisting algorithm having the following forms:

z1 = λ1|Ωi − vo|1/2 sign(Ωi − vo) + Z1︸︷︷︸v1 (32)

Z1 = λ2 sign(Z1 − v1).

with λ2 = 3 3√

λ0, λ1 = 1.5√

λ0 et λ0 = 2f+. The sliding occurs in (32) in finite time(see Levant, 2003). In particular, this means that the longitudinal forces can beestimated in a finite timewithout the need for lowpass filter of a discontinuous switchedsignal.

Remark 1: In order to apply a sliding mode observer, only the Euler integrationmay be used.

Remark 2: The broken super-twisting algorithm observer structure (26) and (27) isreferred to in the literature as broken super-twisting structure since in the estimationanalysis associated with (28) the disturbance term with a amplitude ε appears in thechannel where the discontinuous terms acts, whereas typically in the super-twistingcontroller formulation the disturbance term occurs in the channel associated with |ε| 1

2

(Levant, 1998).


3.3.1 Simulations and results

Simulations are made and results are compared by those provided by simulatorVE-DYNA, the operation condition corresponds to a strong variation in Fxi Figure 7and wi Figure 6 (acceleration, constant velocity, deceleration, constant velocity,acceleration, constant velocity, deceleration, constant velocity) with a zero steeringangle. The sameobserver is applied on the fourwheels, but for the similarity, we presentonly one observer corresponding to the front left wheel (wheel 1). The simulator usesa car with two rear wheel drives. Figure 3 shows the input torque for the two rearwheels and Figure 4 the torque for the two front wheels. In Figures 5 and 6 we seeθ1 and w1 (given by the simulator VE-DYNA) and those computed by the proposedobserver. In these figures we see the rapid convergence of the observer in spite ofthe initial values are: θ10 = 0 radians, θ10 = 50 radians, w10 = 0 rad/s and w10 =100 rad/s.

Figure 3 Motor and braking torque (N.m) applied at the two rear wheels (see online versionfor colours)

Figure 4 Motor and braking torque (N.m) applied at the two front wheels (see online versionfor colours)

The unknown functions computed from the equivalent output injection is supposedequal to −r1iFxi

Iri, we suppose that the radius and the moment of inertia of the wheel

are constants, the we can find the longitudinal force. Figure 7 shows a comparisonbetween the longitudinal force (computed from the observer) and that given byVE-DYNA.


Figure 5 Angular position (rad) by the simulator VE-DYNA, and that estimatedby the observer (see online version for colours)

Figure 6 Angular velocity (rad/s) by the simulator VE-DYNA, and that estimatedby the observer (see online version for colours)

Figure 7 The unknown input after filtration (N) and the longitudinal force from the simulatorVE-DYNA (see online version for colours)

3.4 Estimation of the side slip angle, velocity of the vehicle andreconstruction of the yaw rate

In this part, a sliding mode observer based on the super-twisting algorithm is used toestimate the velocity and the side slip angle at the centre of gravity. The model of thevehicle is a non-linear model and it can be written as follows:

x = f(x, u) = A(x) + B(x)u (33)


where

A(x) =

1M

(cos(β)(− sin(δf )(Fy1 + Fy2) − sin(δr)(Fy3 + Fy4))

− sin(β)(cos(δf )(Fy1 + Fy2) + cos(δr)(Fy3 + Fy4))1

MvCOG(cos(β)(cos(δf )(Fy1 + Fy2) + cos(δr)(Fy3 + Fy4))

− sin(β)(− sin(δf )(Fy1 + Fy2) − sin(δr)(Fy3 + Fy4))) − ψ

1IZ

(tf sin(δf )(Fy1 − Fy2) + L1cos(δf )(Fy1 + Fy2)+L2cos(δr)(Fy3 + Fy4) + trsin(δr)(Fy3 − Fy4))

(34)

and

B(x) =

b11 b12 b13 b14b21 b22 b23 b24b31 b32 b33 b34

(35)

with:

b11 =cos(β)

Mcos(δf ) − sin(β)

Msin(δf ),

b12 =cos(β)

Mcos(δf ) − sin(β)

Msin(δf )

b13 =cos(β)

Mcos(δr) − sin(β)

Msin(δr),

b14 =cos(β)

Mcos(δr) − sin(β)

Msin(δr)

b21 =cos(β)

MvCOGsin(δf ) − sin(β)

Mcos(δf ),

b22 =cos(β)

MvCOGsin(δf ) − sin(β)

Mcos(δf )

b23 =cos(β)

MvCOGsin(δr) − sin(β)

Mcos(δr),

b24 =cos(β)

MvCOGsin(δr) − sin(β)

Mcos(δr)

b31 =1IZ

L1 sin(δf ) − tf cos(δf ),

b32 =1IZ

L1 sin(δf ) + tf cos(δf )

b33 =1IZ

L2 sin(δr) − tr cos(δr),

b34 =1IZ

tr cos(δr) + L2 sin(δr)x = [vCOG β ψ] (36)


the input:

u = [Fx1 Fx2 Fx3 Fx4] (37)

and the measurement vector

y = [ψ]. (38)

Before the design of the sliding mode observer for the model of equation (33),the observability of the model must be investigated and tested. The observabilitydefinition is local and uses the Lie derivative (Nijmeijer and Van der Schaft, 1990)(see also Fridman et al., 2007, 2008). It is a function of the state trajectory and theinputs to themodel. For theFor the systemdescribed by equation (33) the observabilityfunction is:

observability(x, u) =

c(x)Lfc(x, u)

L2fc(x, u)

where

Lfc(x, u) =dcj(x)

dxf(x, u).

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

Jobservability =ddx

observability(x, u).

By applying these notions to the system described by equation (33), we see that its rankis 3 and hence observable.

A complete study for the observability for the system presented by equation (33)(including the cases thatwhenwehavean inputmaking thematrix singular) is presentedin Fridman et al. (2007, 2008).

So, the proposed sliding mode observer based on the hierarchical super twistingalgorithm is:

x = f(vCOG, β, ψ, u) + ∆1|(ψ − ˙

ψ)|1/2 sign(ψ − ˙ψ) + Z1

y = Cx

Z1 = ∆ sign(ψ − ˙ψ)

(39)

where ∆ and ∆1 are the gains of the sliding mode observer. The convergence of thisobserver is proved in Levant (1998).


3.4.1 Simulations results

Once again, the estimated variables, vCOG and β are compared to that provided byVE-DYNA. The operation conditions are given by a variation in δf Figure 8 andtorquei Figure 9, which constitute a significant driving situation. Fxi are estimatedfrom the third order sliding mode observer. In Figures 10–12, we see the observed

vCOG, β and ˆψ and those provided by VE-DYNA. The rapid convergence point out

the good performance of the proposed observer. The gains of the observer used are:∆ = [10, 10, 10]T ; ∆1 = [10, 10, 15]T .

Parameter Value Parameter Value Parameter Value

M 1296 kg L1 0.97 m L2 1.53 m

r1i 0.28 m IZ 1750 H 0.52 m

Iri 0.9 kg.m2 tf 0.7 m tr 0.75 m

lo −0.03 m l1 0.12 m ρ 1.25 kg/m3

Cij 50000 N/rad AL 2.25 m2

Figure 8 Front steering angle (rad) (see online version for colours)

Figure 9 Motor and braking torque (N.m) applied at the two rear wheels (see online versionfor colours)


Figure 10 Estimated vehicle velocity (m/s) and that of the simulator VE-DYNA(see online version for colours)

Figure 11 Estimated side slip angle (rad) using sliding modes and that of the simulatorVE-DYNA (see online version for colours)

Figure 12 Reconstructed yaw rate (rad/s) and that of the simulator VE-DYNA(see online version for colours)

Figure 13 Estimated Vy (m/s) and that of the simulator VE-DYNA (see online versionfor colours)


By estimating the slip angle and the velocity of the centre of gravity, the velocities ofthe centre of gravity in (X, Y ) can be found by Uwe and Nielsen (2005): the lateralvelocity (see Figure 13):

V y = vCOG sin(β) (40)

and the longitudinal velocity which coincides with that of the simulator (see Figure 14):

V x = vCOG cos(β) (41)

and the lateral force (rear left) (see Figure 15):

Figure 14 Estimated Vx (m/s) and that of the simulator VE-DYNA (see online versionfor colours)

Figure 15 Estimated lateral force (N) for the front left wheel and that of the simulatorVE-DYNA (see online version for colours)

4 Controller design

In this section a sliding mode control strategy is presented. For the design process,the model presented by equation (33) is used. As we have described, the state vectoris x = [vCOG, β, ψ]T and the input u = [Fx1, Fx2, Fx3, Fx4]T (the steering angle issupposed given by the driver). As the only available actuators are the brakes, thenonly braking torques can be generated by the controller (Uwe and Nielsen, 2005;


Alvarez et al., 2005).Thesebraking torques aredirectly related to thehydraulic pressureapplied at each wheel (Uwe and Nielsen, 2005):

torquecontroller = −r1ikbipBRi i = 1 : 4. (42)

To design a sliding mode controller, a sliding surface is proposed as:

s = x − xref (43)

with xref = [vCOG ref , βref , ψref ]T represents the reference states.The control objective is to derive the state vector x to the reference state vector xref .In order to ensure the stability, let us suppose that Lyapunov candidate is given by

Utkin (1992):

V =12sT s > 0.

Its derivative can be written as:

V = sT s

where

s = x − xref . (44)

Substituting equation (33) into equation (44):

s = A(x) + B(x)u − xref . (45)

Then we have:

V = sT (A(x) + B(x)u − xref) (46)

with

xref = A(xref) + B(xref)uref (47)

let us suppose the following control law:

∆u = u − uref = −K sign(B(x)T s) (48)

substituting ∆u in the above equation, we get:

V = sT (A(x − xref) − B(x − xref)K sign(B(x)T s)).

In order to satisfy the conditions of stability in the Lyapunov sense, V should benegative, that means:

sT (A(x − xref) − B(x − xref)K sign(B(x)T s)) < 0


since K is scalar, we can write:

sT A(x − xref) − K sT B(x − xref) sign(B(x)T s) < 0K sT B(x − xref) sign(B(x)T s) > sT A(x − xref)

the dimensions of

dim(sT B(x − xref) sign(B(x)T s)) = dim(sT A(x − xref)) = 1

then we can write:

k >sT A(x − xref)

sT B(x − xref) sign(B(x)T s)

and then

k >sT A(x − xref)|sT B(x − xref)|

a necessary condition for the existence of the control law is that sT B(x − xref) = 0.where sgn(s) is a sign function which equals to 1 when s > 0 and −1 if s < 0.The chattering of the function sgn(s) may be reduced by sat

(sΦ

), where Φ is a design

parameter denoting the boundary layer thickness (Stephant, 2004).

4.1 Reference values

The estimated vCOG, β and ψ should follow reference values. But due the fact that thecontroller is designed to assist the driver only in the critical situations, the referencevalues are chosen to be equal to the estimated values when the vehicle is in the safetyregion and equal to certain defined values other wise. They can be described as (Uweand Nielsen, 2005):

For the β:

βmax = 10–7 × v2COG

(40m/s2)(49)

βref = β if |β| ≤ |βmax|(50)

βref = ±βmax otherwise.

For the ψ:

ψmax =1

vCOG cos(β)(aY max − vCOG sin(βref)) (51)

with

aY max = µ.8m/s2. (52)

The derivative of the velocity is found by using a robust exact differentiator proposedby Levant (1998).


For the reference value of the ψ, two cases are considered, in the case of oversteering:

ψref = ψ if |ψ| ≤ |ψmax|(53)

ψref = ±ψmax otherwise.

In the case of under steering, the rear tyre side slip angle shall be used as a reference todetermine when the front tyre side slip angle reach a critical value (Uwe and Nielsen,2005).

ψref = ±ψmax if

∣∣∣∣αi i = 1 or 2αi i = 3 or 4

∣∣∣∣ ≥ 1.5(54)

ψref = ψ otherwise.

5 Simulation results

In this section, computer simulations are carried out to verify the effectiveness of theproposed observers and controller. Simulation is made, in which driver’s inputs aregiven by the simulator VE-DYNA. The vehicle model used is validated by VE-DYNA(Shraim and Ouladsine, 2006). The driver wants to move on a ‘chicane’ (double lane)trajectory described by driver’s steering angle Figure 8 and wheel torques Figure 9.In fact, in this simulation we see that driver’s inputs make the yaw rate exceeds itslimit value. In Figure 16 we see three curves, the reference yaw rate, the yaw rate forthe system without the controller and that with the sliding mode controller, it is seenhow the controller pushes the controlled yaw rate to its reference value. In Figure 17,three curves also are shown for the side slip angle which are: the reference side slipangle, the side slip angle without the controller and that with the sliding mode control.In Figure 18, the response of the controller is shown, it is seen that in the normal caseswhere the side slip angle and the yaw rate are in their safety regions, the controllergives zero, other wise, when they exceed their limits, the controller tries to regulate thisproblem giving different torques on the different wheels.

Figure 16 Yaw rate with and without the controller and the reference yaw rate (see onlineversion for colours)


Figure 17 Side slip angle with and without the controller and the reference side slip angle(see online version for colours)

Figure 18 The four outputs of the controller (see online version for colours)

6 Conclusions

Sliding mode observers are proposed in this work to estimate vehicle parameters andstates which are not easily measured. These observers have shown a short time ofconvergence and robustness in the automotive applications that we have proposed.The validation of the proposed observers is realised by comparing the observers outputwith the outputs of the simulator VE-DYNA, reasonable and acceptable results havebeen shown. In the second part of this work, sliding mode controller is designed.This controller shows its strong and fast reactions on the braking systems in the criticalsituations where we need the controller to work.

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Website

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

Nomenclature

Symbol Physical signification

Ωi Angular velocity of the wheel

M Total mass of the vehicle

ri Radius of the wheel i

COG Centre of gravity of the vehicle

r1i Dynamical radius of the wheel i

Fzi Vertical force at wheel i

Fxi Longitudinal force applied at the wheel i

Fyi Lateral force applied at the wheel i

Cfi Braking torque applied at wheel i

Cmi Motor torque applied at wheel i

torquei Cmi − Cfi

IZ Moment of inertia around the Z axis

ψ Yaw angle

ψ Yaw velocity

δf Front steering angle

δr Rear steering angle

δi Deflection in the tyre i

Vx Longitudinal velocity of the centre of gravity

Vy Lateral velocity of the centre of gravity

Iri 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

hCOG Height of COG

tf Front half gauge


tr Rear half gauge

l tf + tr

Fxwind Air resistance in the longitudinal direction

Fywind Air resistance in the lateral direction

AL Front vehicle area

ρ Air density

Caer Coefficient of aerodynamic drag

αi Slip angle at the wheel i

β Side slip angle at the COG

µi Friction coefficient at the wheel i

Xt Length of the contact patch for the wheel i

Xadherencei Length of the adhesion patch for the wheel i

Xslidingi Length of the sliding patch for the wheel i

Vxi Longitudinal velocity of the wheel i

pBRi Braking pressure at the wheel i

kbi Brake coefficient of the wheel i

pi Inflation pressure of the tyre i

K1i Constant depending on the deformation of the tyre

K2i Constant depending on the deformation of the tyre

vehicle parameter estimation and stability enhancement...

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