SETIT 2005
3rd International Conference: Sciences of Electronic, Technologies of Information and Telecommunications March 27-31, 2005 – TUNISIA
Over or Under Pressure Detection of tire A. Rabhi *, N. Zbiri**, N. K. M’Sirdi * and M. Ouladsine* *
LSIS - UMR CNRS 6168 Domaine Universitaire de Saint-Jérôme Avenue Escadrille Normandie-Niemen 13397 MARSEILLE CEDEX 20 ** Laboratoire des Systèmes Complexes 40, Rue du Pelvoux - 91020 Evry Cedex
[email protected] Abstract : This paper presents a methodology which makes it possible to detect a change of the tire pressure. A vertical dynamical model of the tire is used. The concept of the method consists in estimating, on line, the variation of the tire pressure. The system detection is built on a combination of an observer with an estimator. The decision step is developed using the sequential test of Wald. Keywords: under over inflation of tire, vertical model of the tire, failure, identification, test of Wald.
1 Introduction The early cost associated to road accidents (hospitalization and destruction of goods) is valued to 3% of the world. On the other hand, some studies prove that more than 90% of the set of the automotive accidents could be avoided if the drivers of the vehicles were warned in time of an imminent risk. [1,2]: "road security "]. For that reason the systems of integrated security became a priority for all big automotive groups that always invest more in order to make their vehicles safety. The setting up detection systems, and thereafter the localization of shortcoming (mechanical organ wear, as the tire, for example…) allow for accentuating the security and to participate in saving the human lives knowing that the state of the tires determines, indeed, the behavior of the vehicle at the time of driving. This work consists in detecting over or under pressure of the wheels while being based on the construction of a software sensor to estimate the variations of the pressure of the tires. A part of this work has already been achieved [3,4,5] but the given results were not experimentally. The goal of this application is to develop this work and to set in motion a based detection system on the hypotheses testing. To elaborate this system, ones of the first ideas consist in installing of pressure sensors on every tire. But, in spite of its obvious simplicity, the setting up of such a system becomes quickly exorbitant. Indeed, without counting the cost of such sensors, the tires must be
adapted to their installation. It will be necessary to obligate the industrial therefore to sell adequate tires. Before this problem, one turned toward a less direct method. Can estimate correctly, the pressure of the tires from data that one already possesses?
All pressure changes will be able to be detected thus. An alarm, elaborated from a sequential test of hypotheses, will permit to inform the driver on the state of the wheels pressure. The interest of this system it is that it requires only the measure of the of the vehicle speed provided by the system ABS.
Figure1: Problem of under pressure
SETIT2005 This paper is composed by five parts: Section 1 gives an introduction. Section 2 presents the vertical model of vehicle and its state representation. Section 3 introduces the model-based fault detection method by using the variation parameter estimation and the parity relations [6] to generate the residual. The test of Wald [7] used for the diagnosis decision is also developed. Section 4 presents the application of the approach to vehicle pressure and us illustrated simulations results are illustrated. Conclusion and prospects of this work are given in section 5.
We can write
xa xb
=
y = (I
A 11
A 12
xa
A 21
A 22
xb
0)
+
B1 B2
U+d
(3)
xa xb X=
With
xa xb
x a is the set of the measurable states x b is the set of the non measurable states
2 Vertical model of tire Among existence of an exhaustive number of vertical models of tires, the choice seems to be difficult. We don' t yet possess the comparison of limits validations and efficiencies of each one of these models. In order to estimate the variation of the stiffness K we are interested in the vertical vibrations. For this reason, our model choice is given by figure 2.
In our case we take:
x a = ω1
ω2 θs
xb =
θ s # ω1 - ω 2
(4)
In the case where the system in the matrices vary slightly we can write:
Aij devient → Aij + ∆Aij Bi devient → Bi + ∆Bi
i, j = 1,2
(5)
The equations of state (3) write themselves then under the form:
xa A A A = 11 12 + 11 xb A21 A22 A21
Figure2. Vertical model of the wheel
A12 A22
xa B B + 1 + 1 U+d (6) xb B2 B2
While concentrating within a same matrix the various variations of parameters, the expression (4) rewrites itself as follows:
ω1
ω2 θs
xa A11 = xb A 21
!
E1 dynamical
ω1 −D/ J1 D/ J1 −K / J1 ω1 0 ω2 = D/ J2 −D/ J2 K / J2 ω2 + −1/ J2 Td θs 1 −1 0 θs 0
(1)
The state representation associated to these equations is given by:
X = A.X + BU + d y = CX
xa B E + 1 U+ 1 w xb B2 E2
(7)
With
"
Mechanical equations describing movement of this model are given by:
A 12 A 22
E2
w=
∆A11
∆A12
∆A12
∆A 22
Here, E1 and E 2 represent a set of the constant values, and the w set of the variations ∆ and disruption. When the pressure varies, one also gets a variation of K and D.
( −∆ K θ s − ∆ Dθ s ) / J 1 E .w = ( ∆ Kθ s + ∆ Dθ s − Td ) / J 2 0
(2)
xa ∆B1 + U + d (8) xb ∆B 2
(9)
SETIT2005 3.2 The observer of disruption in the wheel pneumatic
While puting
−∆K θ s − ∆Dθ s
w=
=
∆K θ s + ∆ Dθ s − Td
one gets
E=
1/ J1
0
0
1/ J 2
0
0
In order to estimate reduced order observer
w1
(10)
w2
From the system (4), we can develop a reduced order observer to estimate the state variable. Lut us note that
(11)
xb and w one will design a
w D / J2 zˆ =
w≈0
The new state vector V expresses itself in the following way.
V = xb
{
A11
A12
E1
xa
x b = A12
A 22
E2
xb + B 2 U (13)
0
0
w
0
0
0
0
0
}
zˆ
(18)
(19)
The variations ∆D and ∆K are deduced then by the least squares method as follows
0
∆K
y = ( I 0 0 ) xb
∆D
w
=
ζ (k )ζ (k )
N
ζ (k ) wˆ (k )
k =1
θˆs
(20) (21)
θˆs
For the diagnosis changes we consider the Wald sequential Average Test. At every time instant k, two thresholds, Θ ! and Ψ ! are defined, and according to the value of the likelihood $ ! a decision is made:
0
A 12 E1 (14) A 12 .(A 11 + A 22 ) E1 .A 11 + E 2 .A 12
$ ! %Θ ! Ψ ! %$ ! Θ ! ($ ! (Ψ
&' & ! )
*
The thresholds Θ and Ψ control the errors linked to the decision procedure. They are given by:
the so Q is of full rank system is observable.
Θ=
A 12 E1 A 12 .(A 11 + A 22 ) E1 .A 11 + E 2 .A 12
−1 T
3.3 Decision strategy
The observer' s construction is realizable and only if the previous system is observable. The observability matrix Q, with the considered sensors can be written 0
N k =1
ζ (k ) =
3.1 System observability
Q 22 =
−1
ω1 # − D / J 1 .ω1 + (D / J 1 − K / J 1 0)zˆ
B1
xa
I
1/ J2
Where
The new extended state equation is given by:
A 11 2 A 11 + A 12 .A 21
K / J2
ω1 +
1
+ G ω1 − ω 1
(12)
w
Q=
−D / J2
0
xa
0
(17)
The observer' s equation is as follows:
The estimation of w is got while increasing system (3) and introducing the vector w like an additional state variable. However one will make the following:
w
ω2 z = θs
While putting
3. Observer
xa
(16)
w = ∆K .θ s + ∆D.θ s − Td
(15)
Pn 1 − Pf
Θ=
Pn 1 − Pf
(22)
Where and are respectively the probabilities of no detection and false alarm.
SETIT2005 Velocity of tire
For independent Gaussian observations a signal with standard variance σ and mean µ' under &' and µ under & the test becomes: &'
&
(23)
M =
+
µ1 − µ 0 2
S0 =
σ2 ln Θ ( k ) µ1 − µ 0
S1 =
σ2 ln Ψ ( k ) µ1 − µ 0 k
S (k ) =
j =1
Figure3 : Estimation of velocity of tire
rj − µ0 Velocity of the rim
r is the tested signal (the residual in our work). S(k) is initialized after each detection. For independent Gaussian observations of signal of a constant mean µ and standard variance σo under H0 and σ1 under H1, the test becomes:
ln Θ(k ) + k ln S3 with:
S3 =
σ1 σ0
H0
H1
ln Ψ (k ) + k ln
< S (k )
