1.0 INTRODUCTION

Modern control systems are becoming more and more complex and control

algorithms more and more sophisticated. Consequently, the issues of availability,

reliability, and operating safety are of major importance. For safety-critical

systems, the consequences of faults can be extremely serious in terms of human

mortality and economic loss. Therefore, there is a growing need for the

supervision and fault diagnosis analysis to increase the reliability of such safety-

critical systems due to early indications concerning which faults are developing

can help avoid system breakdown, mission aborting and catastrophes.

In this report we analyse the sensors fault detection problem and propose to

combine the advantages of the which has the ability to decouple the disturbances

with the advantages of dedicated observers to achieve an adequate isolation,

applied to a bus suspension, since it is a very important system in the automotive

sector.

Figure 1.0: Bus’ suspension system

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2.0MATHEMATICAL MODEL

The simulation presented in this work is based on a dynamic system of a ¼ bus suspension model. The model diagram is shown in figure 2, and is described by the following equations [4].

e(t) = x(t) − xˆ(t) [1]

[2]

Where :

M1 and M2 - body mass and suspension mass respectively, k1 - the spring constant of suspension system,k2 -the spring constant of Wheel and tire, b1 - the damping constant of the suspension system,b2 - the damping constant of the wheel and tire, x1 and x2 - distances and W is reference for any road disturbance.

Figure 2.0: Bus suspension model

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Assume that all of the initial condition are zeroes, so these equations represent the situation when the bus wheel goes up a bump. The dynamic equations above can be expressed in a form of transfer functions by taking Laplace Transform of the above equations. The derivation from above equations of the Transfer Functions G1(s) and G2(s) of output,X1-X2, and two inputs,U and W, is as follows.

[3]

[4]

[5]

[6]

[7]

[8]

Find the inverse of matrix A and then multiple with inputs U(s)and W(s) on the right hand side as the following:

[9]

[10]

When we want to consider the control input U(s) only, we set W(s) = 0. Thus we get the transfer function G1(s) as the following:

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[11]

When we want to consider the disturbance input W(s) only, we set U(s) = 0. Thus we get the transfer function G2(s) as the following:

[12]

3.0SYSTEM DESCRIPTION

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The fault diagnosis process basically consists of three levels; the first level

is fault detection, which indicates when a fault occurs in the system. The second

level corresponds to fault isolation, where the location of the fault is determined.

Third and final level is fault identification that estimates the size and type or nature of

the fault. By this reason this process is also known as Fault Detection and Isolation

(FDI).

The model based FDI approach requires a system’s model, which is an

idealized assumption. In practice this assumption is not fully met. Usually the

parameters of the system are uncertain or varying in time .However, Unknown Input

Observer (UIO) design, aside from modelling system uncertainties, solves the FDI

problem in an effective way. This technique surpasses the classical approach of

hardware redundancy for a software redundancy, with an obvious cost efficiency

benefit. UIO design requires apart from the system model, detectable output stages

as well. The residual is obtained by comparison of the actual system output and the

estimated observers output. The purpose of UIO’s is to produce an estimated output

which approaches asymptotically to actual state meanwhile rejects noise effects and

modelling errors of the system. In this approach the perturbations should be

decoupled from the generated residuals. This is achieved by assuming the unknown

inputs matrix (disturbances) as known. Based on this information, the disturbances

can be decoupled.

4.0SAMPLE CALCULATION FOR UNKNOWN INPUT OBSERVER (UIO)

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The proposed UIO is based on mathematical model of the bus’ suspension

system. Table 1 lists the known parameter values of the system.

Parameter Description Value

M1 Body mass 2500 Kg

M2 Suspension mass 320 Kg

k1 Spring suspension constant 80 000 N/m

k2 Spring wheel and tire

constant

500 000 N/m

b1 Damping suspension

constant

350 Ns/m

b2 Damping wheel and tire

constant

15 020 Ns/m

Table 1.0: List of parameters

4.1Transfer function for control input U(s) :

G1(s) = 2820 s2+15020 s+500000∆

4.2 Transfer function for disturbance input W(s):

G2(s) = −37550000 s3−1.25∗e9 s2

∆

∆=[ (m1∗m 2 ) (m1∗(b1+b2 ) )+(m2∗b1 ) (m 1∗(k 1+k 2 ) )+ (m2∗k 1 )+ (b1∗b2 ) (b1∗k 2 )+(b2∗k 1 ) k1∗k 2 ]

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∆=0.0001 s4+0.0039 s 3+0.1481 s2+0.1377 s+4.0000

4.3 MIT rule block diagram

θ ymodel -

+

Uc yplant

Figure 3.0: Block diagram of Bus’s suspension system based on MIT rule

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Controller

ᴨ

∑ᴨᴨ

Plant

−γs

5.0RESULT AND DISCUSSION

5.1 Block diagram for bus’ suspension system, γ=1.5

5.2 Block diagram for bus’ suspension system, γ=2.0

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By using Matlab, both of the graph of γ=1.5 and γ=2.0 shows it is not exactly stable for the system. This occurs because the graph between two transfer functions for bus’s suspension system does not match at the end. We observe that the transfer function must be apply with another method in order to get a good result.

6.0CONCLUSION

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As a conclusion, by using the MIT rule towards the bus’ suspension system, the graph is not unlikely good because the the graph between two transfer functions for bus’s suspension system does not match at the end. Therefore, a different transform function should be applied to get more matching between transform function.

7.0REFERENCE

Juan Anzurez Marin, Luis A. Torres Salomao and Isidro I. Lázaro, Fault Detection and Isolation for a bus suspension model using an Unknown Input Observer Design.

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