We provide an example scenario that demonstrates our model-based methodology. The figure below shows estimation results for the case of friction damage. There was little error present in the tracking of the outputs, so we show only the estimation results of the unknown wear parameter wr, as it is the uncertainty in this value that contributes most to the uncertainty in prediction, if future inputs are known. In this case, the estimate converges within 10 cycles. The maximum and minimum values stay around 30% of the mean.
The probability mass functions of the EOL predictions (rounded to the nearest cycle) for each prediction point (every 10 cycles) are shown in the figure below. The distributions all cover the true EOL, and as time progresses, the predictions become significantly more accurate and precise.
This result is also captured by the α-λ accuracy metric, as given in the figure below. Here, α in [0,1] defines bounds as a function of the true RUL, i.e., the interval [(1-α)RUL*, (1+α)RUL*], and λ in [0,1] denotes the fraction of the time from the first prediction to the true EOL. A third parameter, β in [0,1], defines a bound on the fraction of the probability mass of a prediction that falls within the α-bounds. The α-λ metric requires that for given prediction times, at least β of the predicted RUL distribution falls within the cone of accuracy defined by α. The metric, therefore, visually summarizes both accuracy and precision. The figure below shows the RUL predictions as box plots at each prediction point. The percent of the probability mass which falls within the α-bounds is shown above each box plot, along with the outcome of the metric at that point in time. Here, we choose α=0.1 and β=0.5. In this case, the metric fails at the third prediction point, as less than half of the probability mass is contained within the α-bounds, although the mean does fall within the bounds.
Comprehensive results may be found in the related publications.