November 27, 2016 Vehicle System Dynamics ForceEstimation [623343]

November 27, 2016 Vehicle System Dynamics ForceEstimation
To appear in Vehicle System Dynamics
Vol. 00, No. 00, Month 20XX, 1–10
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;bUniversit´ e Amar Telidji de Laghouat
BP G37,Route de Ghardaia Laghouat 03000;cLaboratoire de Modelisation, Information et
Systemes, 33 Rue de Saint Leu, Amiens 80000, France
(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 confort 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, sensor use is very impor tant, currently available
inexpensive sensors such as longitudinal velocity, yaw rat e and the acceleration. However
it is not the case for longitudinal tire/road forces which ar e more difficult to measured
for both economics and technical reasons. Therefore, these forces became an important
task to be estimated or observed,. Thus, in this work we focus on the estimation of the
longitudinal tire/road forces.
Recently many studies on the estimation of the longitudinal and lateral forces have
been introduced in literature [2, 3, 5].
AnExtandedKalmanFilterisusedIn[8]toestimatelongitud inalandlateraltireforces
using a ten degree of freedom model; where the tire forces sup posed to be bounded.
In [2] the lateral and longitudinal tire forces are estimate d using an extanded kalman
∗Corresponding author. Email: [anonimizat]
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November 27, 2016 Vehicle System Dynamics ForceEstimation
filter applied to a four wheel dynamics model; the longitudin al forces are lumped into
a single force and used as a dynamics (bounded dynamics ) wher eas the lateral force
dynamics are written in terms of the relaxation length and a q uasi static model. A
longitudinal forc estimation function is derived from an en ergy function In [11] using
a simple angular velocity dynamics at each wheel. In [3] a nom inal observer is used
to estimate the longitudinal force at each wheel and the brak ing torque or engine of a
vehicle model. In [4] Sliding mode observer with unknown inp ut Is used to estimate the
longitudinal force by considering the dynamical equations of the wheels, where the brake
engine and the cylinder pressure of the wheels are supposed t o be measured or estimated.
In this paper an adaptive neural network nonlinear observer is designed to estimate the
unknown longitudinal forces of a four freedom vehicle model using an online update law
for the radial basis neural network function to approximate appropriately the unknown
longitudinal forces. The stability of the proposed observe r is proven using a lyaponouv
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.
2. Problem Formulation
Consider the following nonlinear systems with the followin g structure:
˙x=Ax+Φ(x,y,u)+Bf(x,u,y) (1)
y=Cx
WhereA∈ ℜn×nrepresent the system matrix, B∈ ℜnis a known vector , u∈ ℜmare
the inputs to the system, whereas C∈ ℜp×nis the output matrix, f(x,u,y) is partially
unknown function, Φ( t,u,y) :ℜn×ℜm1→ ℜm2which is a known nonlinear function. To
complete the description of the system, the following assum ptions are hold:
Assumption 1The pairs (A,C) is observable; there exist a matrix L∈ ℜn×psuch that
the matrixAc:=A−LCis Hurwitz.
Assumption 2There 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
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November 27, 2016 Vehicle System Dynamics ForceEstimation
form:
AT
cP+PAc=−Q
BTP=CTFT
Where Q is a given positive definite matrix
Assumption 3The 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 4The unmodelled dynamics θ(x,y,u)is unbounded and will be estimated
using radial basis neural network function such that ˆθ(x,y,u) =ˆMTδ(x)
Assumption 5The weight ˆWof the radial basis neural network function is bounded
˙ˆW= 0
Assumption 6Signals y and u are measured signals
2.1.Adaptive Neural Network Nonlinear Observer
An adaptive neural network observer will be designed to esti mate the state vector based
on the above mentioned assumptions, combined with an update law for the radial basis
weights to estimate the unknown function ( unknown dynamics ) or parameters, based on
a radial basis neural network approximation. Thus the adapt ive neural network nonlinear
observer takes the following form:
˙ˆx=Aˆx+Φ(ˆx,y,u)+Bφ(u,y)ˆθ+L(y−Cˆx)
ˆθ=ˆWTσ(ˆx) (2)
˙ˆW=γσ(ˆx)φ(y,u)TFe
WhereFis a positive definite matrix, where ˆ xrepresents the estimated states, the
unknown function or parameter θ(t,x) is estimated using a radial basis function neural
network ˆθ(t,x) =ˆWTσ(ˆx)+ǫ,ˆWis the estimated Weights of the RBF function, where
σ(x) = exp( −/bardblˆx−c/bardbl
2b2) is the activation function, whith c is called the center vec tor and
b is a positive scalar called the width, γis the adjustable gain of the update weights.
Stability analysis of the proposed observer will be given in the next section
3. Stability Analysis
The stability of the proposed observer is proven using the fo llowing Lyapunov candidate
function:
V=eTPe
2+˜WTγ−1˜W
2(3)
Wheree=x−ˆxand˜W=W−ˆWbe the error equations of the states and radial
basis neural network weights respectively. Hence using equ ations (1) and (2); the error
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November 27, 2016 Vehicle System Dynamics ForceEstimation
dynamics is defined as:
˙e=Ace+B˜Wσ(x) (4)
Hence, the derivative of V yields:
˙V=˙eTPe
2+eTP˙e
2
+˙˜WTγ−1˜W
2+˜WTγ−1˙˜W
2(5)
Substituting the error equation and adopting the update law of the neural network
weights found in equations (2) and (4) into equation (5) give s:
˙V=(Ace+B˜Wσ(x))TPe
2
+eTP(Ace+B˜Wσ(x))
2(6)
−(γσ(ˆx)φ(y,u)TFe)Tγ−1˜W
2
−˜WTγ−1(γσ(ˆx)φ(y,u)TFe)
2
rearranging equation (6) gives:
˙V=eT(AcP+PAc)e
2
+(B˜Wσ(x))TPe
2+eTPB˜Wσ(x)
2(7)
−(γσ(ˆx)φ(y,u)TFe)Tγ−1˜W
2
−˜WTγ−1(γσ(ˆx)φ(y,u)TFe)
2
After simplification of equation (7) yields:
˙V=−eTQe
≤0 (8)
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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November 27, 2016 Vehicle System Dynamics ForceEstimation
4. Vehicle Dynamical Model
A mathematical modelling is required for control and state o bservation of a vehicle.
Many works have been dealt with modeling of a vehicle which ar e more complex for
control applications. The model used in this paper takes int o consideration the lateral
andlongitudinaldynamicsofthevehiclebydefining:thetra nslationalmotionthatdefines
the rear and front forces acting on the system, and the rotati onal motion that describes
the yaw rate dynamics. The front and rear forces in this model , are lumped to a single
front and rear forces respectively [2, 15]; which are given b y:
/braceleftbigg
Fxf=Fxfr+Fxfl
Fxr=Fxrr+Fxrl(9)
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 (10)
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. FyfandFyrare the front and rear lateral tire forces respectively, whi ch are
given by:
Fyf=Cfδf−Cf(vy+a˙ψ)
vx(11)
Fyr=Cr(vy−b˙ψ)
vx
WhereCf,rrepresent the stiffness of the longitudinal tire forces; a an d b reprsents the
distance from the center of gravity to front and rear axles of the vehicle.
Substituting equation (11) into equation (10) Thus the comp lete dynamical model of
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November 27, 2016 Vehicle System Dynamics ForceEstimation
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(12)
¨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 unmeasered states.
In order to achieve these objectives the proposed observer i s used in cascade form;
where the dynamical model of the vehicle in (12) 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:
-vx,˙ψandδfare measured signals.
– u is the input steering angle δ.
-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(13)
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 (2), equatio n (13)
will be written in compact form as:
˙ˆx1=A1x1+Φ1(ˆx1,y1,u1) (14)
+B1φ1(y1,u1)θ1(t)+L1(y1−C1ˆx1)
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November 27, 2016 Vehicle System Dynamics ForceEstimation
Where the system matrices A11andB11are defined as:
A1=/bracketleftbigg
0 1
0 0/bracketrightbigg
B1=/bracketleftbigga
Iz1
m/bracketrightbigg
(15)
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 (16)
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
(17)
5.2.The Estimation Of The Rear Froces
The estimation of the rear force will be achieved using the fo llowing vehicle submodel:
/braceleftBigg
˙vx=vy˙ψ+Fxf+Fxr
m−Cf
mδ2
f+(Cf(vy+a˙ψ)
mvx)δf
˙δf=−δf+u
τ(18)
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 (2),
equation (18) will be written in compact form as:
˙ˆx2=A2x2+Φ2(ˆx2,y2,u2) (19)
+B2φ2(y2,u2)θ2(t)+L2(y2−C2ˆx2)
Where the system matrices A2andB2are defined as:
A2=/bracketleftbigg0 0
0−1
τ/bracketrightbigg
B2=/bracketleftbigg1
m
0/bracketrightbigg
(20)
The known function, and the unknown nonlinear rear froce are defined as:
φ2(y2,u2) =/bracketleftbigg
1
0/bracketrightbiggT
ˆθ2(t) =ˆFxr=ˆW2σ(ˆx) (21)
And the output vector y2is given by:
y2=/bracketleftbigg
y21
y22/bracketrightbigg
=C2x2=/bracketleftbigg
1 0
0 1/bracketrightbigg/bracketleftbigg
x12
x22/bracketrightbigg
(22)
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November 27, 2016 Vehicle System Dynamics ForceEstimation
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
(23)
L1andL2in both submodels 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 (1) is used as an input u to the system:
0 5 10 15 20 25−1−0.500.51
Time (s)δ (rad)
Figure 1. The Input Steering Angle u
6.1.Vehicle State Estimations:
We note from Figures (2) that represents the estimation of th e lateral velocity and the
yaw rate respectively which has been estimated using equati on (13); thus the estimated
states converges rapidly to their measured ones in a short pe riod of time; and Figures
(3) that shows the convergence of the unknown front and rear f orces in a short transient
period of time by updating the weights of the neural network.
Figures (4) that shows the estimated longitudinal velocity and the steering angle; which
has been estimated using equation (18); which shows a very fa st convergence of the esti-
mated states to their measured states with no over shot. Ther efore and from the obtained
simulation results a good estimation performance is achiev ed through the application of
the proposed observer.
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November 27, 2016 Vehicle System Dynamics ForceEstimation
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) The 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 2. The lateral velocity and yaw rate estimation.
0 5 10 15 20−500−400−300−200−1000100
Time (s)Fxf (N)

The Measured Fxf
The Estimated Fxf
(a) The 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 3. The Longitudinal front and rear estimation using neural network function approximation.
0 5 10 15 200510152025
Time (s)vx (m/s)

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

The Measured δf
The Estimated δf
(b) The Estimated Steering Angle δf.
Figure 4. The longitudinal velocity and steering angle estimation.
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
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November 27, 2016 Vehicle System Dynamics ForceEstimation
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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