Particle Filters and Auto-Encoders Combination for Damage Diagnosis on. . . 7 2.2 Definition of the Diagnostic Indicators The squared error value related to each spring in the system has to be further processed in order to eventually trigger a diagnostic alarm. Those alarms may be given comparing the SEτ i value with a properly defined threshold, as suggested in [38]. However, the threshold value definition is arbitrary and may lead to inaccurate results. Moreover, in case of a not properly defined threshold, false alarms may be given since noise affects the inputs signals to the auto-encoder, due to a combination of acceleration measurement noise, process noises simulating unmodeled dynamics and Monte Carlo errors. Hence, in order to try to solve these issues, the authors proposed an original method consisting of combining an automatic threshold definition module with the construction of both a deterministic and a probabilistic indicator [42]. In particular, each spring i =1, . . . ,n+1 is given one threshold value according to the following relationship: ATSE 99 i =p99(SE 1:τ0 i ) (14) wherep99(SE 1:τ0 i ) is the 99th percentile of the probability density function of the squared error defined in Eq. (11), computed over a fixed time window of widthτ0 time steps taken at the beginning of the operational life of the structure, when the system is assumed to be in healthy conditions. An on/off deterministic fault indicator is introduced for each spring i, which is built up, at each time step i, comparing the ATSEi 99 threshold value with the moving average over a sliding window of widthτ0 time steps of the squared errors for each spring i, i.e., μ SE τ−τ0+1,τ i . Note that the time window within which the automatic thresholds are defined coincides with the one over which the moving average is computed. Thanks to the properties of the moving average operation, the wider the window width, the more robust the algorithm to false alarms from error fluctuations, with the drawback of increasing the anomaly detection time. The deterministic fault indicator is defined according to the following relationship: Id τ−τ0+1,τ SE,i = ⎧ ⎨ ⎩ 1 if μ SE τ−τ0+1,τ i >ATSE 99 i 0 otherwise (15) Hence, the fault indicator is activated, i.e., is set to 1, whenever the signal μ SE τ−τ0+1,τ i exits the healthy region defined by the respective threshold. Therefore, only significant variations of the signal μ SE τ−τ0+1,τ i may trigger alarms, filtering out the confounding effects coming from noise and disturbances. Moreover, an additional, probabilistic fault indicator is introduced in order to further damp out possible false alarms triggered by its deterministic counterpart: Ip τ−τ0+1,τ SE,i = τ j=τ−τ0+1 Id j SE,i τ0 (16) where Idτ SE,i represents the instantaneous counterpart at time step τ of Id τ−τ0+1,τ SE,i , which is defined as: Idτ SE,i = 1 if SEτ i >ATSE 99 i 0 otherwise (17) 3 Case Study The case study presented in this work is developed considering a three degrees of freedom MDOF system on the basis of the one shown in Fig. 1. The system parameters are taken from the work in [28], according to which mi =1kg ∀i =1, . . . , 3, kj =9 N m andcj =0.25 N· s m (i.e., β =27.8·10−3) ∀j =1, . . . , 3). The first degree of freedom is assumed to be affected by non-linear hysteretic degradation with parameters βBW =2, γ =1 and n =2. The structural state variables are affected by normally distributed process noise characterized by zero mean and variances σ 2 w,position =10−16 m2, σ 2 w,speed =10−16m2 s2 , σ 2 w,spring =1.62·10−5N2 m2 , andσ 2 w,damping =8·10−13 for position, velocity, stiffness, and damping proportionality constant,
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