The principle of the observer is very concrete. Let us consider the sequence of two places. The first place describes the journey of a car from the town A to B which lasts between 2 and 3 hours. The second represent the following journey from B to C with a temporization between 5 and 6 hours. If the time u1 of departure of the car is known, the arrival at the intermediate town B can obviously be estimated: [u1+2,u1+3]. Symmetrically, If the time y1 of arrival to the town C is known, the arrival at the intermediate town B can also be estimated: [y1-6,y1-5]. Consequently, the estimate of the date associated with B can be calculated by a forward-backward approach: [maximum(u1+2,y1-6), minimum(u1+3,y1-5)]. If the date is out this interval, we can conclude that there is a break-down or an unpredictable event.

But, the model can equivalently be described by the form x<=f(x) which gives : x<=minimum(u1+3,y1-5) , u1<=x-2 and y<=x+6. The first inequality allows the estimation of the greatest x. This value can be introduced in the two other inequalities: a fault is detected when they are not satisfied. This situation arrives when the model has changed: for instance, a breakdown of the car between the two towns B and C entails that the temporization associated to the second place equals 9 which does not belong to [5,6] . If the real data are u=10 and y=21, the greatest estimate x is 13 and u1<=x-2 is satisfied (10<=13-2=11) contrary to y<=x+6 (21<=13+6=19) which shows an incoherence between the used model and the evolution of the current trajectory.

Therefore, this simple example shows an application of fixed point theory in fault detection. Based on the well-known principle of redundancy, the fault detection needs the calculation of only one bound. Therefore, coherence between data (estimate, input and outputs) is checked.