18 A Bayesian State-Space Approach for Damage Detection and Classification 173 Table 18.1 Test cases and damage scenarios for structural models (a) Column structure (b) 3 story 2 bay structure Test case Damage scenario Test case Damage scenario 1 Intact column 1 Intact column 2 Minor damage, lower joint 2 Minor damage at 17 3 Major damage, lower joint 3 Major damage at 17 4 Minor damage, upper joint 4 Minor damage at 1 5 Major damage, upper joint 5 Major damage at 1 6 Major damage at 1 and 17 structure is instrumented with four accelerometers, one at each connection, including the connection with the foundation, and at the top of the structure. In order to excite the cantilever beam, it is displaced by approximately 5 cm and then released and allowed to freely vibrate for 10 s, during which data was collected. There are ten test sequences for each damage scenario, and they are summarized in Table 18.1a. The second structure is a 3 story 2 bay configuration with a footprint of 120 60cm as shown in Fig. 18.1c. The structure consists of steel columns and beam frames of similar dimensions for each story that are bolted together to form each story. Damage is similarly introduced on the bolted connections with the minor and major damage cases by removing two bolts or loosening all four at connections 1 and 17, which are on opposite corners of the structure, with 1 being on the first story, and 17 being on the second. This structure is instrumented with 18 triaxial accelerometers at each of the connections between elements. For this structure the excitation is a small shaker with a weight of 0.91 kg and a piston weight of 0.17 kg that was attached to the top corner of the structure at connection 18, which provided a random white Gaussian noise excitation in the frequency range of 5–350 Hz in the flexible direction. Test measurements lasted for 30 s, during which the shaker is always exciting the structure, thus there is no ramp up or unforced section of the data. The damage scenarios are summarized in Table 18.1b. For each damage scenario, ten sequences were acquired. 18.3 Theory We describe the state-space switching interaction model (SSIM) of [6] in Sect. 18.3.1 and its modification for the application to classification of time-series in Sect. 18.3.2. The relevant background for this paper are probabilistic graphical models (Bayesian networks and dynamic Bayesian network in particular) and principles of Bayesian inference. Graphical models are a language that uses graphs to compactly represent families of joint probability distributions among multiple variables that respect certain constraints dictated by a graph. In particular, a Bayesian network (BN) consists of a directed acyclic graphGD.V;E/, whose nodes X1;X2; : : : ;XN represent random variables, and a set of conditional distributions p.Xi j pa.Xi //, i D 1; : : : ;N, where pa.Xi / is a set of variables that correspond to the parent nodes (parents) of node Xi . Dynamic Bayesian networks (DBNs) are Bayesian networks that model sequential data, such as time-series. Each signal in a model is represented with a sequence of random variables that correspond to its value at different indices, or discrete time points. Edges are allowed only from a variable with a lower index to a variable with a higher index (i.e., they must “point” forward in time). An introduction to the Bayesian approach and Bayesian networks can be found in [7]. An introduction to dynamic Bayesian networks can be found in [8]. 18.3.1 State-Space Switching Interaction Model (SSIM) We assume that N multivariate signals evolve according to a Markov process over discrete time points t D0;1; : : : ;T. The value of signal i at time point t > 0 depends on the value of a subset of signals pa.i; t/ at time point t 1. We refer to pa.i; t/ as a parent set of signal i at time point t. While the preceding implies a first-order Markov process, the approach extends to higher-ordered Markov processes. A collection of directed edges Et Df.v; i/I i D1; : : : ;N; v 2pa.i; t/g forms a dependence structure (or so-called interaction graph) at time point t, Gt D.V;Et /, where V Df1; : : : ;Ng is the set of all signals. That is, there is an edge fromj toi inGt if and only if signal i at time point t depends on signal j at time point t 1. Let Xi t denote a (multivariate) random variable that describes the latent state associated to signal i at time point t. Then, signal i depends on its parents at time t according to a probabilistic model p.Xi t jX pa.i;t/ t 1 ; i t / parametrized by i t , where
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