128 A. Gannon et al. A better metric to compare baselines is mean squared error. We follow a method baseline selection using mean squared error introduced in [7]. First the test data, y[n], is normalized to produce y0[n] y0 Œn D y Œn rXN 1 nD0 y2 Œn (12.1) and the baselines xl[n] for baselines l D1,2, : : : ,L x0l Œn DA xl Œn (12.2) where ADXN 1 nD0 xl Œn y0 Œn X N 1 nD0 x2 l Œn : (12.3) The optimal baseline is that which minimizes the mean squared error between the test data and that baseline lopt Dargmin l N 1 XnD0 y0 Œn x0l Œn 2 : (12.4) Once the optimal baseline set has been selected, we must determine whether or not damage has actually occurred. If the average magnitude of the time history difference exceeds a detection threshold, the system deduces that damage has indeed occurred. This detection threshold is determined in the following manner: each baseline set in the database is compared to each other baseline set. The threshold is equal to the average magnitude of the differences between each pair of baseline time-histories, plus two standard deviations of those differences. Figure 12.4 shows an example of a test waveform (top), the corresponding baseline waveform (middle), and the resultant waveform from the baseline subtraction (bottom). Note that the resultant waveform is plotted on a smaller y-scale to more clearly exhibit its shape. 12.3.4.3 Rayleigh Maximum Likelihood Estimate As mentioned in Sect. 12.3.4.2, the time-sample at which the difference time-history first becomes anything more than noise should represent the time of flight corresponding to the shortest path from actuator to damage to sensor. This makes sense intuitively, because the sensor shouldn’t record anything different until the wave has had time to scatter from the damage location. The task is now to determine which time sample corresponds to the first scattered arrival. This will be accomplished using a Rayleigh Maximum Likelihood Estimate filter, which assigns to each time sample in the feature vector a relative likelihood that the first scattered arrival occurs at that time-sample. Our first step is to obtain an analytical signal from the difference waveform by applying a complex matched filter with the actuation waveform. In the figure below, the red waveform is the difference between the test data and the baseline data for a given actuator-sensor pair, while the blue curve is the envelope of that data difference produced by the matched filter. The envelope for each actuator-sensor pair becomes the feature vector that will be analyzed in order to provide a good estimate of the time-of-flight for the first scattered arrival at the sensor. Assuming one single damage location, the difference time-history for each actuator-sensor pair can be divided into two parts: (i) the time before any waves scattered from the damage reach the sensor, and (ii) the arrival at the sensor of the first directly scattered wave and all the subsequent echoes of the scattering. For the waveform below, the dividing point can be identified by visual inspection to be near time-sample 2,500. This location coincides with the peak of the green curve at the bottom of the figure. This green curve is obtained by applying a Rayleigh Maximum Likelihood Estimate (RMLE) filter to the signal envelope in blue (Fig. 12.5). The Rayleigh Maximum Likelihood Estimate works as follows: for each time-sample m on the feature vector, two Rayleigh parameters (¢1 and ¢2) are estimated based on the distribution of signal magnitudes ¤m according to
RkJQdWJsaXNoZXIy MTMzNzEzMQ==