Particle Filters and Auto-Encoders Combination for Damage Diagnosis on. . . 5 where wτ r represents the hysteretic displacement process noise. Note that this equation is specialized to MDOF systems with degrading hysteretic behavior on the first degree of freedom only. All the variables which, in a healthy structure, are assumed to be invariant over time with respect to variations of external conditions other than temperature are modeled as an augmented state with constant value in time, to which some process noise is added to represent the effects of unmodeled dynamics. For instance, the equation governing the evolution of the spring stiffness value over time is: kτ+1 i =kτ i +wτ spring; ∀i =1, . . . ,n+1 (7) where wτ spring is the process noise related to stiffness. Similar equations determine the evolution of the other invariant parameters included in the state-space, i.e., β, βBW, γ, andn, with the only difference of replacing the process noise related to stiffness to the corresponding one, i.e., wτ damping, wτ βBW , wτ γ, and wτ n. Moreover, stiffness is the only parameter which is assumed to vary with temperature. Hence, rescaling the relationship reported in [25] in order to comply with the stiffness values considered in this work, the discretized equation governing the stiffness evolution in time due to eventual change in temperature is represented by: ki(T) =(0.1(T) 2 −18T +10000) · ki(0◦C) T ; ∀i =1, . . . ,n (8) where T identifies the external temperature measured in◦C. The equations shown above represent the evolution of the hidden Markov states of the MDOF system considered. Those hidden states are estimated within the PF framework processing the information from some measurements z, which are related to the system states through a model which is assumed to be known, which is determined by the following general function: zτ =h xτ, vτ (9) whereh(·) is a generic, possibly non-linear function andvτ is the Gaussian measurement noise vector at time stepτ,which is assumed to be distributed according to a multivariate distribution N(0, v). Assuming that only acceleration measurements are available, the explicit form of Eq. (9) for the i-th degree of freedom at time step τ is: zτ i = Fτ i mi − yτ i mi (cτ i +c τ i+1) + yτ i−1 mi cτ i + yτ i+1 mi cτ i+1 − xτ i mi (kτ i +k τ i+1) + xτ i−1 mi kτ i + xτ i+1 mi kτ i+1+ − rτ i mi kτ i δi1 +v τ i ; ∀i =1, . . . ,n (10) wherevτ i represents the acceleration measurement noise at time stepτ, so that v =diag(σz1 . . .σzn). When the acceleration equation for the extreme degrees of freedom, i.e., i =1,n, is considered, Eq. (10) has to be modified deleting those terms including xτ i−1 and yτ i−1 for i =1, those withxτ i+1 , yτ i+1 , kτ i+1 and cτ i+1 for i =n. The augmented state-space representation of the MDOF system presented above is considered within the PF posterior estimation framework for performing state estimation. Particle filtering represents a widely used model-based parameters identification and state estimation Bayesian method which can deal with complex scenarios, including non-Gaussian noise and non-linear dynamics. In particular, the selected PF algorithm is the sample importance resampling (SIR) algorithm, the main operative steps of which have already been described by the authors [42]. For brevity’s sake, in this work no detailed mathematical treatment of the topic is given. The interested reader is referred to the works in [28, 43, 44] for further details about the functioning of the PF SIR algorithm. However, as already mentioned above, exploiting PF posterior estimation only may not reveal successful in performing damage detection in case varying external and operational conditions arise, leading to inaccurate results. Thus, it is employed the original, unique framework already presented in the work in [42], to which the reader is referred to for the detailed description of the procedure adopted in this work. That framework consists of the integration of the PF posterior estimation with neural network auto-encoders, which allow to offset the effects of confounding factors from the damage detection and localization procedure. Auto-encoders are deep neural networks characterized by a symmetric structure of layers, consisting of one dimensionality reduction side, which reduces the input dimensionality, followed by a reconstructing side, which aims at producing in the output layer the same values fed in input. In particular, one auto-encoder per spring in the MDOF system is considered in this work, all sharing the same structure: one input layer, three hidden layers, i.e., mapping, bottleneck, and demapping layers, and one output layer (Fig. 2). The relationship between layers is set up during the so-called training phase,
RkJQdWJsaXNoZXIy MTMzNzEzMQ==