February 8, 2017 Vehicle System Dynamics ForceEstimationcorected [623344]

February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
To appear in Vehicle System Dynamics
Vol. 00, No. 00, Month 20XX, 1–12
Vehicle Longitudinal Force Estimation Using Adaptive Neural
Network Nonlinear Observer
Mourad Boufadenea∗, Mohamed Belkheiria,b, Abdelhamid Rabhi and Ahmed El Hajjajic
aLaboratoire de T´ el´ ecommunications, Signaux et Syst´ eme s. Universit´ e Amar Telidji de
Laghouat BP G37, Route de Ghardaia Laghouat 03000, Alg´ erie ;bUniversit´ e Amar Telidji de
Laghouat BP G37,Route de Ghardaia Laghouat 03000, Alg´ erie ;cLaboratoire de Modelisation,
Information et Systemes, 33 Rue de Saint Leu, Amiens 80000, F rance
(v4.0 released October 2014 )
This paper presents an adaptive neural network nonlinear ob server for the estimation of
the longitudinal tire forces as well as the lateral speed whi ch are difficult to be measured
using sensors . The proposed adaptive NN observer uses the lo ngitudinal speed, yaw rate and
the steering angle dynamics of the vehicle as measured states . Hence the adaptive nonlinear
observerforthestatesandthelongitudinaltireforceswhi charesupposedunknowndynamics,
is used to estimate them with high performance. Simulations and the obtained results show
the effectiveness of the proposed neural network nonlinear o bserver.
Keywords: Vehicle force estimation; neural network estimators; vehi cle state estimation;
adaptive nonlinear observer
1. Introduction
The vehicle dynamics is affected by the longitudinal and late ral tire forces. Hence the
estimationoftheseforcesplayscentralroleinimprovingt hevehicleperformanceinsafety
and comfort terms. Therefore. engineering have developed m any control systems rely on
the knowledge of the longitudinal Tire forces such as Anti Lo ck Braking system (ABS),
active front steering and electronic stability program and recently collision avoidance
systems [1]
To improve the safety of the vehicle, therefore the use of sen sors is very important,
currently available inexpensive sensors for longitudinal velocity, yaw rate and the acceler-
ation. However it is not the case for longitudinal tire/road forces which are more difficult
to be measured for both economics and technical reasons. The refore, these forces became
an important task to be estimated or observed.
Recently many studies on the estimation of the longitudinal and lateral forces have
been introduced in literature [2, 3, 5]. An Extended Kalman F ilter is used In [8] to
estimate longitudinal and lateral tire forces using a ten de gree of freedom model; where
the tire forces supposed to be bounded. In [16] an unscented k alman filter is used for
a planar vehicle mathematical model to estimate the vehicle states where the steering
angle , the four wheel velocities, the yaw rate and the latera l acceleration are estimated
∗Corresponding author. Email: [anonimizat]
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February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
using a standard vehicle dynamics control (VDC) sensors; ho wever the longitudinal and
lateral tyre/road forces are described using a simplified em pirical Magics Formula model
[17, 18], which contains many unknown variables that should be estimated accurately
such as the tire grip condition variable. A fuzzy logic techn ique is combined with a
linear kalman filter is used In [19, 20], for a one mass vehicle model for vehicle speed
and acceleration estimations , where the obtained results h ave shown good estimation
performance; however the authors does not use a more complic ated vehicle model. In
[21] a dual extended kalman filter is used to estimate state an d the parameters of a
more complicated vehicle dynamical models, thus the obtain ed results have shown a
good estimation performance, however the tire forces are es timated using TMeasy [22]
model that calculates the resultant forces. In [2] the later al and longitudinal tire forces
are estimated using an extended kalman filter applied to a fou r wheel dynamics model;
the longitudinal forces are lumped into a single force and us ed as a dynamics (bounded
dynamics ) whereas the lateral force dynamics are written in terms of the relaxation
length and a quasi static model. .A nonlinear observer is des igned in [25] to estimate
the longitudinal and lateral velocities of a 3 degree of free dom vehicle models; thus the
obtained results shows a good performance of the proposed ob server; The authors used
a Dugoffs tire model to represent the longitudinal forces; ho wever this model subjected
to many varying parameters such as cornering stiffness and ti re road coefficient friction.
In [23] High gain sling mode observer is used to estimate the l ongitudinal forces using
LuGre friction model; however this model contain many varyi ng parameters that has an
effect on the longitudinal forces. A longitudinal force esti mation function is derived from
an energy function In [11] using a simple angular velocity dy namics at each wheel. In
[3] a nominal observer is used to estimate the longitudinal f orce at each wheel and the
braking torque or engine of a vehicle model. In [4] Sliding mo de observer with unknown
input Is used to estimate the longitudinal force by consider ing the dynamical equations of
the wheels, where the brake engine and the cylinder pressure of the wheels are supposed
to be measured or estimated.
In this paper adaptive neural network nonlinear observer is designed to estimate the
unknown longitudinal forces of four freedom vehicle model u sing online update law radial
basis neural network function to approximate appropriatel y the unknown longitudinal
forces; the main advantages of the neural network especiall y in identification of nonlinear
dynamics that they do not require any mathematical descript ion of the forces. The stabil-
ity of the proposed observer is proven using a Lyaponouv base d function. The following
assumptions are used:
•The lateral speed, yaw rate and the steering angle dynamics a re used as a measured
signals
•The longitudinal forces are supposed to be unknown nonlinea r dynamics and estimated
using an adaptive neural network function approximation
•The proposed observer could be used to estimate systems subj ected to an unknown
disturbance as well as for parameter uncertainty
•The convergence of the unknown function (unknown dynamics) are guarantied by on
online update law weights of the neural network function
The rest of this paper is organized as follows, section (2)de scribes the problem formula-
tion and the adaptive neural network observer, section (3)d escribes the stability of the
considered nonlinear observer, section (4)describes the d ynamical model of the vehicle,
section (5)is devoted to the application of the proposed obs erver for the estimation of
the longitudinal forces, section (6) shows simulation and r esults, at section (7) we end
up with conclusion.
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February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
2. Problem Formulation
Let’s consider the following nonlinear systems with the fol lowing structure:
˙x=Ax+B1u+Φ(x,y,u)+B2f(x,u,y) (1)
y=Cx
WhereA∈ ℜn×nis the system matrix, B1∈ ℜn×m1andB2∈ ℜn×m2are a known
vector ,u∈ ℜmare the inputs to the system, whereas C∈ ℜp×nis the output matrix,
f(x,u,y) :ℜn×ℜm2→ ℜm2is partially unknown function, Φ( t,u,y) :ℜn×ℜm1→ ℜm2
which is a known nonlinear function. To complete the descrip tion of the system, the
following assumptions are hold:
Assumption 1. The pairs (A,C) is observable
Assumption 2. The function f(y,u,x)could be represented in a parametric form as
φ(y,u)θ(t,x)whereφ:ℜn×ℜl:→ ℜn×l, which is a known function
Assumption 3. The function θ(t,x)it could be a bounded parameter or unbounded
function which will be estimated
Assumption 4. The signals y and u are measured signals
Under these assumptions, the system of equation (1) can take the following form:
/bracketleftbigg
˙x1
˙x2/bracketrightbigg
=/bracketleftbigg
a11a12
a21a22/bracketrightbigg/bracketleftbigg
x1
x2/bracketrightbigg
+/bracketleftbigg
b11
b12/bracketrightbigg
u+/bracketleftbigg
b21
b22/bracketrightbigg
φθ (2)
y=C+/bracketleftbigg
x1
x2/bracketrightbigg
Where the parameters a11,a12,a21anda22are known linear parameters that construct
the system matrix A, b11,b12,b21andb22are known parameters, C∈ ℜp×nis the corre-
sponding output matrix.
2.1.Adaptive Neural Network Nonlinear Observer
An adaptive neural network observer will be designed to esti mate the state vectors based
on the over mentioned assumptions, combined with bounded or unbounded function θ
estimation algorithm. Thus the adaptive nonlinear observe r takes the following form:
˙ˆx=Aˆx+B1u+Φ(ˆx,y,u)+B2φ(u,y)ˆθ+L(y−Cˆx)
ˆθ=ˆWTσ(ˆx) (3)
˙ˆW=γσ(ˆx)φ(y,u)TFe
Wwhere ˆxis the estimated state vector, the unknown function or param eterθ(t,x) is
estimatedboundedorunboundedfunction, γistheadjustablegainfortheneuralnetwork
weight. thus in order to complete the description of the prop osed observer the following
assumptions are used:
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February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
Assumption 5. There exist a matrix L∈ ℜn×psuch that the matrix Ac:=A−LCis
Hurwitz.
Assumption 6. There exists a positive vector F∈ ℜn; positive definite matrix P∈ ℜn×n
which is a unique solution to the following Layponuv equatio n that takes the following
form:
AT
cP+PAc=−Q
BT
2P=CTFT
Where Q is a given positive definite matrix
Assumption 7. The function θ(x,y,u)is unbounded and will be estimated using radial
basis neural network function such that ˆθ(x,y,u) =ˆWTδ(x), whereδ(x) = exp(−/bardblx−c/bardbl
2b2)
is the activation function, with c is called the center vecto r and b is a positive scalar
called the width
Assumption 8. The weight ˆWof the RBF is bounded such that˙ˆW= 0
Let ˜e=x−ˆxand˜W=W−ˆW, we can define the observer error dynamics from
equations (1) and (3) as:
˙e=(A−LC)e+B2φ˜WTσ(ˆx) (4)
=Ace+B2φ˜WTσ(ˆx)
In the next section the stability analysis of the proposed ob server is given
Figure 1. The block diagram describing the proposed nonlinear observ er
3. Stability Analysis
Theorem 1. If the nonlinear system of equation (1) satisfies assumption s (1)−(8), an
adaptive observer (3) can be designed to estimate the unmeas ured states as well as the
unknown parameters θusing online neural network estimator in equation (3)
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February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
Proof.The stability of the proposed observer is proven using the fo llowing Lyapunov
candidate function:
V=eTPe
2+˜WTγ−1˜W
2(5)
Hence, the derivative of V yields:
˙V=˙eTPe
2+eTP˙e
2
+˙˜WTγ−1˜W
2+˜WTγ−1˙˜W
2(6)
Substituting the error equation and adopting the update law of the neural network
weights found in equations (3) and (4) into equation (6) give s:
˙V=(Ace+B˜Wσ(x))TPe
2
+eTP(Ace+B˜Wσ(x))
2(7)
−(γσ(ˆx)φ(y,u)TFe)Tγ−1˜W
2
−˜WTγ−1(γσ(ˆx)φ(y,u)TFe)
2
rearranging equation (7) gives:
˙V=eT(AcP+PAc)e
2
+(B˜Wσ(x))TPe
2+eTPB˜Wσ(x)
2(8)
−(γσ(ˆx)φ(y,u)TFe)Tγ−1˜W
2
−˜WTγ−1(γσ(ˆx)φ(y,u)TFe)
2
After simplification of equation (8) yields:
˙V=−eTQe
≤0 (9)
Therefore the observer is stable in the sense of Lyaponouv. T he effectiveness of the
proposed observed on state and unknown function is tested on a real word vehicle appli-
cation.
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February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
4. Vehicle Dynamical Model
A mathematical modeling is required for control and state ob servation of a vehicle. Many
works have been dealt with modeling of a vehicle which are mor e complex for control
applications.
The model used in this paper takes into consideration the lat eral and longitudinal
dynamics of the vehicle by defining: the translational motio n that defines the rear and
front forces acting on the system, and the rotational motion that describes the yaw rate
dynamics. For symmetry of the vehicle the front and rear forc es in this model, are lumped
to a single front and rear forces respectively [2, 15]; which are given by:
/braceleftbigg
Fxf=Fxfr+Fxfl
Fxr=Fxrr+Fxrl(10)
Where subscripts r and l defined the right and left forces resp ectively. The complete
dynamical model is given by:
m(˙vx−vy˙ψ) =Fxf+Fxr−Fyfδf
m(˙vy+vx˙ψ) =Fxfδf+Fyf+Fyr (11)
Iz¨ψ=a(Fxfδf+Fyf)−bFyr
τ˙δf=−δf+u
Wherevxandvyare the longitudinal and lateral vehicle speed respectivel y, ,˙ψis the
vehicle yaw rate, δfis the steering angle; which are used as the state variables o f the
model, u is the input vector. m is the vehicle mass, Izis the inertial moments of the
vertical axis. The dynamical equation of δfin equation (11) takes into consideration the
actuator dynamics; where τis the system time constant. A linear tire model is used in
this work which is the most simplified model, where the latera l force is is described as a
linear function of the slip angle [2] This function is then ex pressed as:
Fyf=Cfαf (12)
Fyr=Crαr
WhereCf,rrepresent the stiffness of the longitudinal tire forces; , an d the slip angle αf,r
is given by the following formula:
αf=δf−Cf(vy+a˙ψ)
vx(13)
αr=(vy−b˙ψ)
vx
The parameters a and b represents the distance from the cente r of gravity to front and
rear axles of the vehicle. Hence the linear tire the front and rear lateral tire forces Fyf
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February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
andFyrrespectively are given by substituting equation (13) into ( 12) as:
Fyf=Cfδf−Cf(vy+a˙ψ)
vx(14)
Fyr=Cr(vy−b˙ψ)
vx
Substituting equation (14) into equation (11) Thus the comp lete dynamical model of
the vehicle will be written as:
˙vx=vy˙ψ+Fxf+Fxr
m−Cf
mδ2
f+(Cf(vy+a˙ψ)
mvx)δf
˙vy=−vx˙ψ+Fxfδf
m+Cf
mδf−Cf(vy+a˙ψ)
mvx+Cr(vy−b˙ψ)
mvx(15)
¨ψ=aFxfδf+aCf
mδf−aCf(vy+a˙ψ)
mvx−bCr(vy−b˙ψ)
mvx
Iz
˙δf=−δf+u
τ
5. Observer Design For Longitudinal Force Estimation
Since that the lateral velocity is not measured as well as the longitudinal forces which
are unknown nonlinear dynamics; so should be measured using sensors, and since sensors
are more expensive and some times are not available as in the c ase of longitudinal forces,
hence a good solution could be the use of non linear observer; therefore and adaptive
neural network nonlinear observer is designed to estimate t he longitudinal forces as well
as the unmeasured states.
In order to achieve these objectives the proposed observer i s used in cascade form;
where the dynamical model of the vehicle in (15) is divided in to tow parts one for the
front force and the second for the rear force estimation resp ectively, with the following
assumptions are hold:
(1)vx,˙ψandδfare measured signals.
(2) u is the input steering angle δ.
(3)FxfandFxrare unknown nonlinear dynamics that will be estimated using neural
network function approximations.
5.1.The estimation of the front force Fxf
The following subsystem is used to estimate the front force a nd the lateral velocity of
the vehicle:
/braceleftBigg¨ψ=a
IzFxfδf+aCf
Izδf−aCf(vy+a˙ψ)
Izvx−bCr(vy−b˙ψ)
Izvx
˙vy=−vx˙ψ+Fxfδf
m+Cf
mδf−Cf(vy+a˙ψ)
mvx+Cr(vy−b˙ψ)
mvx(16)
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February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
With the following state variables x1= [x11,x12] = [˙ψ,vy]; and the input vectors are
u1= [u11,u12] = [vx,δf], then from the observer structure of equation (3), equatio n (16)
will be written in compact form as:
˙ˆx1=A11x1+B11u+Φ1(ˆx1,y1,u1) (17)
+B12φ1(y1,u1)θ1(t)+L1(y1−C1ˆx1)
Where the system matrices A11andB11are defined as:
A11=/bracketleftbigg
0 1
0 0/bracketrightbigg
;B11=/bracketleftbigg
0
0/bracketrightbigg
;B12=/bracketleftbigga
Iz1
m/bracketrightbigg
(18)
The known function, and the unknown nonlinear front force ar e defined as:
φ1(y1,u1) =/bracketleftbigg
u12
u12/bracketrightbiggT
ˆθ1(t) =ˆFxf=ˆW1σ(ˆx)y1=C1x1=x11 (19)
Matrix Φ 1(ˆx1,y1,u1) is given by:
Φ1=/bracketleftBiggaCf
Izu12−aCf(ˆx12+ay1)
Izu11−bCr(ˆx12−by1)
Izu11−ˆx12
−u11y1+Cf
mu12−Cf(ˆx12+ay1)
mu11+Cr(ˆx12−by1)
mu11/bracketrightBigg
(20)
5.2.The estimation of the rear forces
The estimation of the rear force will be achieved using the fo llowing vehicle sub model:
/braceleftBigg
˙vx=vy˙ψ+Fxf+Fxr
m−Cf
mδ2
f+(Cf(vy+a˙ψ)
mvx)δf
˙δf=−δf+u
τ(21)
With the following state variables x2= [x21,x22] = [vx,δf]; and the input vector
u2= [u,u21,u22,u23] = [δ,˙ˆψ,ˆvy,ˆFxf], then from the observer structure of equation (3),
equation (21) will be written in compact form as:
˙ˆx2=A22x2+B21+Φ2(ˆx2,y2,u2) (22)
+B22φ2(y2,u2)θ2(t)+L2(y2−C2ˆx2)
Where the system matrices A2andB2are defined as:
A2=/bracketleftbigg0 0
0−1
τ/bracketrightbigg
;B21=/bracketleftbigg
0
0/bracketrightbigg
;B22=/bracketleftbigg1
m
0/bracketrightbigg
(23)
The known function, and the unknown nonlinear rear force are defined as:
φ2(y2,u2) =/bracketleftbigg
1
0/bracketrightbiggT
ˆθ2(t) =ˆFxr=ˆW2σ(ˆx) (24)
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February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
And the output vector y2is given by:
y2=/bracketleftbigg
y21
y22/bracketrightbigg
=C2x2=/bracketleftbigg
1 0
0 1/bracketrightbigg/bracketleftbigg
x12
x22/bracketrightbigg
(25)
Matrix Φ 2(ˆx2,y2,u2) is given by:
Φ2=/bracketleftBigg
u22u21+u23
m−Cf
my2
22+(Cf(u22+u21a)
my21)y22
−y22+u
τ/bracketrightBigg
(26)
L1andL2in both sub models are the observer gain that has to be chosen a dequately to
achieve better performance
6. Simulations And Results
Simulation have been performed under Matlab Simulink by con sidering vehicle sub-
models, in order to test and validate the proposed observer. The input steering angle
shown in figure (2) is used as an input u to the system: The param eters of the vehicle
0 5 10 15 20 25−1−0.500.51
Time (s)δ (rad)
Figure 2. The Input Steering Angle u
model used are given in Table (1)
Table 1. Vehicle Parameters
m (Kg)CrCfa(m)b(m)Iz(Kg2.m)
700 1990010001.25 1.51550
6.1.Vehicle State Estimations:
We note from Figures (3) that represents the estimation of th e lateral velocity and the
yaw rate respectively which has been estimated using equati on (16), where the observer
gainL1is chosen so that the observer poles are chosen as pol1= [−10−500] ; thus the
estimated states converges rapidly to their measured ones i n a short period of time; and
Figures (4) that shows the convergence of the unknown front a nd rear forces in a short
transient period of time by updating the weights of the neura l network.
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February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
0 5 10 15 20−0.6−0.4−0.200.20.40.60.8
Time (s)vy (m/s)

The Measured vy
The Estimated vy
(a) Lateral Velocity Estimation vy.0 5 10 15 20−0.4−0.3−0.2−0.100.10.20.3
Time (s)ψ (rad/s)

The Measured ψ
The Estimated ψ
(b) The Yaw Rate Estimation ˙ψ.
Figure 3. Lateral velocity and yaw rate estimation.
02468101214161820−500−400−300−200−1000100
Time (s)Fxf (N)

The Measured F
xf
The Estimated F
xf
(a) Estimated Front Force ˆFxf.0 5 10 15 20−1000010002000300040005000
Time (s)Fxr (N)

The Measured Fxr
The Estimated Fxr
(b) The Estimated Rear Force ˆFxr.
Figure 4. Longitudinal front and rear estimation using neural netw ork function approximation.
0 5 10 15 200510152025
Time (s)vx (m/s)

The Measured vx
The Estimated vx
(a) Estimated Longitudinal Velocity vx.0 5 10 15 20−1−0.500.51
Time (s)δf (rad)

The Measured δf
The Estimated δf
(b) Estimated Steering Angle δf.
Figure 5. Longitudinal velocity and steering angle estimation.
Figures (5) that shows the estimated longitudinal velocity and the steering angle where
the observer gain L2is chosen so that the observer poles are chosen as pol2= [−150−7];
which has been estimated using equation (21); which shows a v ery fast convergence of
the estimated states to their measured states with no over sh ot. Therefore and from
the obtained simulation results a good estimation performa nce is achieved through the
application of the proposed observer.
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February 8, 2017 Vehicle System Dynamics Force˙Estimation˙corected
7. Conclusion
This paper presents an adaptive neural network nonlinear ob server to solve the problem
of unmeasured states as well as the estimation of an unknown d ynamics due to mod-
eling errors. A radial basis function neural network has bee n used to approximate the
unknown functions or dynamics (disturbance). The stabilit y of the proposed observer is
proven using Lyaponouv function. The proposed nonlinear ob server is applied to solve
the problem of longitudinal force estimation. The converge nce of the longitudinal forces
in a short period of time shows the effectiveness of the observ er to systems subjected to
unknown dynamics or disturbances.
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