For the case of noise you can inject or propagate the type of noise you want, in Matlab there are several types of already built noise realizations with the simplest being the gaussian. As for the disturbance, what do you precisely mean? Is it mechanical aggression, thermal heating etc. If thats what you mean you can create a similar noise configuration which will take the hand of these disturbances. Besides, Fault diagnosis is very big research domain it starts with the platform you are considering you fault in. You mean machines, power system, or wiring cables. Because each has specific detection techniques. thus you shall specify your domain of interest. I hope I am clear.

Hello John,

Assume a linear state space form described by the matrices (A,B,C,D).

In model-based fault detection, the objective is to derive a certain model which is assumed to be given in the above linear state space (A,B,C,D). 

Furthermore, you would like to estimate some parameters. 

There's quite plenty of algorithms that estimate the parameters of interest, given the models' correctness.

The complexity of the model is a major issue here ( please see S. Qin, “Data-driven fault detection and diagnosis for complex industrial processes").

The algorithm may be really good but not good enough to estimate your parameters in a really complex model, thus leading to parameter uncertainty.

As for noise, well, all practical systems have noise within most devices (examples ADCs and amplifiers). Stochastically speaking, the noise (in most cases) is assumed to be a random process independent from the parameters of interest. 

Thus while expressing the input output relation of the system, we write:

y(k) = Cx(k) + Du(k) + noise

The disturbance is also a noise process, due to the model, algorithm and anything that perturbs the following equation:

x(k+1) = Ax(k) + Bu(k) + disturbance.

Hope this helps :-)