Local Hölder exponent analysis of aeroengine dynamics

2.1. EGT data acquisition and signal processing

Data available from commercial 737 aircrafts are used for this paper. The 737 aircraft data includes operational flight data from on-board flight data recorders. These recorders are part of the Aircraft Condition Monitoring System (ACMS) such as Smart ACMS Recorder (SAR) and Quick Access Recorder (QAR).

SAR data records specific flight parameters on preset data recording channels. But the QAR data is a more comprehensive that data includes an extensive list of flight parameters recorded at specific sampling intervals which are set by the manufacturer. Moreover, the set of data includes engine reports which are generated when selected parameters meet certain predefined conditions during a particular flight phase. Furthermore, the data also includes baseline values for a subset of flight parameters and the respective threshold operating values for these parameters. In addition the series information can come in handy to decide at what stage during a flight, the parameters corresponding to a system or sub-system reach abnormal operating conditions. Therefore, the QAR data is applied in this paper.

By applying flight condition data and situational parameters, diagnostic and prognostic schemes can be developed which contribute to determining engine behavior during a specific flight and predict engine performance by estimating the flight parameter conditions of future flights. Various flight parameters such as Exhaust Gas Temperature (EGT), Fuel Flow (FF), Engine Fan Speeds (N1 and N2) are usually used to estimate the engine health and performance [16].

2.2. Analytical method

The aeroengine dynamics have to be considered as nonstationary. Therefore, most of the widely used signal processing techniques based on the assumption of stationarity and globally characterise signals are not fully suitable for detecting short-duration dynamic phenomena [17]. Thus, we address the problem of estimation of the local scaling exponent.

For convenience, we describe briefly the Hölder exponent firstly. Given a function $f\left(t\right)$ with a singularity at time $t_0$, the Hölder exponent $h\left(t_0\right)$ defined as the supremum exponent such that there exists polynomial $P_n(t-t_0)$ of order $n<h\left(t_0\right)$ satisfying [8-11]:

where $t$ in the neighborhood of $t_0$, and $C$ is a constant. $P_n\left(t\right)$ is often associated with the Taylor expansion of $f\left(t\right)$ around $t_0$, but Eq. (1) is valid even if such expansion does not exist [18]. The Hölder exponent is therefore a function defined for each point of $f$, and it describes the local regularity of the function $f$. By selecting properly a wavelet kernel $\psi \left(t\right)$ that is orthogonal to polynomials up to degree $n$, the continuous wavelet transform (WT) of $f\left(t\right)$ at $t=t_0$ is expressed as:

Further,

The local Hölder exponent $h\left(t_0\right)$ of strength of singularity can be evaluated as the scaling power of the wavelet coefficient at $t=t_0$ for $a\to 0$ [19].

3.1. Example 1: Aircraft A

In order to test the robustness of the proposed fractal comparative diagnostics method, cruise mode time series of aircraft A are used in this section.

Fig. 2The Hölder exponents of (a) left engine and (b) right engine

Here, we apply the Hölder exponent analysis method to signal of left and right engine for aircraft A, see Fig. 2, where one of the engine is fracture failure. We observe that there are different scaling phenomena in the plot. And then, we can study the difference between the Hölder exponents of left and right engine by using error function. The error function is defined as:

where $h_{left}\left(t\right)$ are these Hölder exponents of left engine, $h_{right}\left(t\right)$ are Hölder exponents of right engine respectively.

Fig. 3 shows the nontrivial difference ${\mathrm{\Delta }}_f\left(t\right)=\left|h_{left}\right(t)-h_{right}(t\left)\right|$ between the Hölder exponents of left and right engine where one of the two engines is fracture failure. And then, we calculate the Hölder exponents for are normal engine as shown in Fig. 4.

Fig. 5 indicates the error function for Hölder exponents in normal engine ${\mathrm{\Delta }}_n\left(t\right)=\left|h_{left}^{normal}\right(t)-h_{right}^{normal}(t\left)\right|$, where $h_{left}^{normal}\left(t\right)$, and $h_{right}^{normal}\left(t\right)$ are the Hölder exponent of normal left and right engine. Approximate to 0, compare to the nontrivial difference in Fig. 3.

Fig. 3The error function Δf(t) for one of the two engines is fracture failure in aircraft A

Fig. 4The Hölder exponents of normal engine (a) left and (b) right for aircraft A

Fig. 5The error function Δn(t) of normal left and right engine for aircraft A. Δf(t)>>Δn(t) indicate the Hölder exponent method to diagnose engine fault is available

3.2. Example 2: Aircraft B

To examine the availability of fractal comparative diagnostics method using data in aircraft B, we apply the Hölder exponent analysis method again.

Here, the Hölder exponent are calculated for aircraft B, as show in Fig. 6, where one of the engine is fracture failure. Obviously, there are different scaling phenomena between the Hölder exponents of left and right engine. For analysing the dissimilarity, the error function defined by Eq. (4) is employed once again.

Fig. 6The Hölder exponents of (a) left engine and (b) right engine for aircraft B where one of the engine is fracture failure

Fig. 7The error function Δf(t) for one of the two engines is fracture failure in aircraft B

Fig. 8The Hölder exponents of (a) left engine and (b) right engine for aircraft B for normal engine

We represent the error function ${\mathrm{\Delta }}_f\left(t\right)=\left|h_{left}\right(t)-h_{right}(t\left)\right|$ between the Hölder exponents of left and right engine in Fig. 7. For comparison, we calculate the Hölder exponents for are normal engine as shown in Fig. 8.

In Fig. 7, we observe that the nontrivial values of error function ${\mathrm{\Delta }}_f\left(t\right)$, where one of the engines is fracture failure. For comparison, the error function:

for normal engine are shown in Fig. 9. As expected the error function ${\mathrm{\Delta }}_n\left(t\right)$ is less than the error function ${\mathrm{\Delta }}_f\left(t\right)$ distinctly. We show the negligible difference between the Hölder exponents of left and right engine in Fig. 9, where both the left and right engine are normal.

Fig. 9The error function Δn(t) of normal left and right engine for aircraft B. The error function Δf(t)>>Δn(t) indicate the Hölder exponent method is available for diagnosing engine fault