106 P. Villalpando et al. The results obtained with these algorithms need to be evaluated under multiple system orders. For this purpose, we have used the methodology of Fast multi-order [5], which reduces the computational complexity with mathematical reformulation reducing considerably the calculation time. A stability diagram [6] helps in determining representative modal parameters of the structure from the defined orders. Stable modal parameters of identifications (MOESP or SSI-COV) under different test conditions should give similar modal parameters within a limited range of variability. Model-based damage assessment requires solving a nonlinear inverse problem that can be solved with supervised learning algorithms [7, 8]. The most traditional methods are based on artificial neural networks, but its application stumbles on the time needed to train the model and the need to have a lot of parameters to be selected. Alternatively Gupta [9], presented a nonparametric method of supervised learning, which generalizes the linear approximation by using the principle of maximum entropy [10], by means of statistical inference. The first investigations using maximum entropy to the detection, location and quantification of damage have been applied to one-dimensional structural elements such as a cantilever beam [11], an eightDOF spring-mass system, an experimental beam and a car exhaust system [8], in addition this algorithm has been applied to a two-dimensional plate structure to assess debonding in honeycomb aluminum panels [7]. Experimental results have shown to be successful in identifying the assigned damages, moreover calculation time was similar to a neural network, but with the precision of a model updating algorithm based on global optimization, making their use attractive to damage detection in complex structures. This paper presents the identification of damage in a three-dimensional structure, by means of a supervised learning algorithm based on a linear approximation using the principle of maximum entropy in conjunction with the structure’s modal properties (frequencies and modal shapes). The performance of the proposed algorithm is validated with experimental tests in a laboratory structure. The structure has six levels, and it is 2 m tall. The structure was attached to a horizontal shaking table and it was tested with different seismic records (Valparaiso, Chile 1985, Maule, Chile 2010, Loma Prieta, USA 1989 Northridge, USA 1994) and colored noise (filtered from 0 to 15 Hz). Modifying the structure from the undamaged condition to a condition with damage is done by introducing a progressive reduction in the cross-section of a column located in the second level. Additionally a condition of disturbance in the structure was induced with the addition of 0.50 % of the total mass of the structure at the fourth level. The purpose of this, is to observe the behavior of the structure and of the damage identification algorithm to anomalous conditions which do not correspond to damage or accumulation to damage in the structure. 12.2 Linear Approximation to Maximum Entropy Principle Let the observation vector Yj D nYj 1; Y j 2; : : : ; Yj m o 2 Rm represent the j-th damage state of the structure, where mis the number of structural elements and Yj m is percentage of damage of element m for structural state j. Notice that it does not represent a single damage scenario but to an element global change. Let the feature vector Xj D nXj 1; X j 2; : : : ; Xj n o 2 Rn, represents a set of vibration characteristics of the structure associated with a state of damage Yj. The variables Xand Yhave joint probability distribution PX,Y. Let a set of k independent and identically distributed samples are drawn fromPX,Y; these samples represent the database (X1, Y1), (X2, Y2), : : : , (Xk, Yk). The main problem in supervised learning algorithms is the best estimateP Yˇ ˇ ˇ X , i.e., given a feature vector X, estimate the corresponding observationY. Let Yˆ denotes the estimated value of Y. Linear approximation takes the N nearest neighbors to a test point X and uses a linear combination of them to represent Xas: XD N X jD1 !j .X/ Xj .X/ ; N X jD1 !j .X/ D1; (12.1) where !1, !2, : : : , !N are weighting functions, and X1(X), X2(X), : : : , XN(X) are the Nnearest neighbors to a test point X, out of database set. The equations given in (12.1) can be expressed as the following system of linear system: A wDb; w 0 (12.2)
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