A model-based diagnostic and prognostic framework designed for handling single, persistent faults of different fault magnitudes, is applied to the WRS. Fig. 5 illustrates the architecture of our diagnostic and prognostic approach.

Each individual module in Fig. 5 is described in detail below.

At each discrete time step, k, the system takes as inputs, u(k), and outputs measurements, y(k). The nominal model observer also takes as inputs the system inputs, u(k), and the system measurements, and uses the nominal system model of the system to generate estimates of distributions of nominal measurements, ŷ(k) . Any appropriate filtering scheme, e.g., Kalman filter, extended Kalman filter, unscented Kalman filter, particle filter, among others, can be adopted as the nominal observer. In this work, however, the high fidelity simulation model of the nominal WRS system is used in place of the nominal observer to simulate the nominal system measurements.

The fault detector then takes in the observed and estimated measurements, y(k) and ŷ(k) , and detects when a fault has occurred based on the residual, r(k) = y(k) - ŷ(k) is determined to be statistically significant and non-zero. In this work, a Z-test is used to determine the statistical significance. Fault detectors need to be tuned so as to minimize false alarms and missed detection while maintaining the desired level of sensitivity.Once a fault is detected, fault isolation is initiated.

Based on the first observed statistically significant measurement deviation, we also generate a set of possible fault candidates, and then, for each fault candidate, we systematically determine a fault signature for each measurement using the same tuple of qualitative symbols described above. A fault signature of a fault for a measurement is a prediction of how the measurement will deviate from nominal due to the fault. Now, given the set of fault candidates, as measurements deviate from nominal, the observed measurement deviations (captured symbolically) are checked for consistency with predicted fault signatures and measurement orderings. Any fault candidate whose predictions are inconsistent is removed from consideration. As more and more measurement deviations are observed, the candidate set will reduce, ideally resulting in a singleton.

In some cases, the qualitative fault signatures alone are not sufficient in distinguishing all faults, or fault effects may take too long to manifest, and quantitative analysis is needed to correctly diagnose the true fault. The advantage of using qualitative fault isolation is that it reduces the fault candidates very quickly, thereby improving the scalability of the overall diagnosis task. Hence, the more diagnosable the system is, the smaller is the number of possible fault candidates remaining after fault isolation is performed, and fewer will be the faults that will have to be isolated through relatively (computationally) expensive quantitative methods.

The fault identification module, for each fault, f ∈ F(k), estimates p(x_{f}(k);θ_{f}(k)|y(0:k)), where x_{f} represents the set of state variables in the faulty system model that includes all state variables of the nominal model and the faulty system parameter corresponding to the particular f ∈ F(k) that needs to be estimated. θf represents the set of all original system parameter except those that are now included in x_{f} and includes some additional fault progression model parameters that are used to model how the faulty parameter progresses over time.

For prognosis, the end of life of the system is predicted, using, for each hypothesized fault candidate, a predictor based on a fault progression model integrated with the nominal model. The prediction module takes as input p(x_{f}(k);θ_{f}(k)|y(0:k)) to make predictions of End of Life (EOL), i.e., p(EOL_{f}(k)|y(0:k)), and Remaining Useful Life (RUL), i.e., p(RUL_{f}(k)|y(0:k)). A system is said to have reached its EOL when one or more constraints that define the acceptable behavior of the system is violated. EOL is the earliest time point at which the threshold of unacceptable system behavior is reached. Given EOL_{f}(t_{P}), RUL may then be defined with RUL_{f}(t_{P}) , EOL_{f}(t_{P}) - t_{P} . Note that we need to hypothesize future inputs of the system for prediction, since fault progression is dependent on the operational conditions of the system. The choice of expected future inputs depends on the knowledge of expected operational settings.