## 5 Steady-State Criterion Derivation from Flight Data
We then use a steady-state criterion to filter the data before applying the diagnostic algorithm. The criterion employed is that the absolute value of the slope of variation in the exhaust gas temperature over time should be less than a given threshold. The slope at any given point is considered to be the one derived from a best-fit line applied to a window of points preceding that point. The idea of how this slope is calculated is demonstrated in Figure 12. Two points are used as an example, one for which the operation is transient and one for which steady state has been achieved. Figure 11: The exhaust gas temperature deltas calculated from measured values using values from a steady-state operation model as a reference. When this criterion is applied, two parameters must be chosen by the user:  
* The magnitude of the slope that is used as a threshold, a choice that has been substantiated in our analysis in the preceding sections. * The size of the window for calculating the slope, since the data collected contain noise. An example of how the window size affects the magnitude of the slope is shown in Figure 13. It can be claimed that when the slope does not change with time, a larger number of points will produce less uncertainty in the slope value estimated. Additionally, the size of the window may interact with the form of the transient. For example, a window that is very long may indicate that the operation is still transient, even though steady state has been achieved, because the data used extend far back to incorporate part of the transient. In this respect, short windows are desirable for avoiding such overlaps. Taking these comments into account, the result of the application of the slope criteria for different magnitudes of the parameters involved is examined. The success rate of the diagnosis applied to points filtered with different slopes as thresholds and with different window sizes for obtaining the slopes is shown in Figure 14. Short time windows produce lower success rates, reflecting the fact that the slope obtained is more sensitive to noise content. The trend reverses for windows larger than a certain size (30 s for the figure), now reflecting the overlap of transient and steady-state regions. Figure 12: Steady-state and transient operation characterized through the slope of the time evolution of the exhaust gas temperature. Figure 13: Obtaining the slope from windows of different sizes from time series of data. It is interesting to see how many data points are found to obey the steady-state criterion, which thus forms the population in which diagnosis can be affected. The number of points both in absolute numbers and as a percentage is shown in Figure 15. It is observed that when a \"tight\" slope criterion is applied, the number of points decreases significantly. A trade-off between the requirements for a high success rate and a substantial number of points that can be diagnostically exploited thus has to be decided. The procedure of selecting steady-state data points can be further improved if the criterion used is modified. A variant that immediately becomes a possibility is not only examining the slope itself, but also the way this slope varies along the time series. If, for example, it is required that this slope is not allowed to change abruptly from point-to-point, the scores presented above using just the slope magnitude are modified, as shown in Figures 16 and 17. The results of these figures have been derived by adding the requirement that the slope magnitude does not vary from point-to-point more than half the maximum allowed slope value. It is observed that scores of 100% are now achievable with certain combinations of the window and slope threshold. At the same time, however, the number of available points significantly decreases. Inspecting these figures shows that, for the example taken here, an optimum combination of a window 30 s wide and a slope threshold of 0.5 to 1 K/min exists.