PSI - Issue 78

A. Romanazzi et al. / Procedia Structural Integrity 78 (2026) 777–784 779 In equation (2), is the vector of the unknown system parameters of the size ( + ) and ( ) is the vector which includes the known input and output samples. The solution of the general form can be estimated by using the Least Square regression method, as: ̂ = [∑ ( ) ( ) = 1 ] −1 ∑ ( ) ( ) = 1 (5) In off-line applications, an approach to address time-variant system over non-stationary data is the segmentation of the input-output signals, for which a nonlinear time-variant system can be assumed as a sequence of time invariant system piecewise linear as in Lennart (1999), Bassevill and Nikiforov (1993), and Gustafsson (2001). Therefore, considering a general case in which the system is time varying, the AR-order, , X-order, , and delay order, , can be time-dependent, and equation (2) results as: ( ) = ( ) ( ) (6) With the special condition that the system parameters are piecewise constant, and change only at certain time instant , for which: ( ) = , < ≤ +1 (7) Leading to a set of consecutive linear models varying over the time signal, as: ( ) = ( ) (8) Where is a -dimensional parameter vector, is the regressor vector, and the measurement vector ( ) is assumed to have dimension . The objective is determining the instants , referred to as the change time, in which changes in occurs. However, determining the exact instant at which the changes occur can be cumbersome and time consuming due to the number of data points to be analysed, in particular when high sampling frequency is adopted, as in the case of seismic signals. In this framework, a newly approach for dynamic identification of time variant systems by means of ARX segmentation of non-stationary signals is proposed. The approach is based on recursive finding of ARX models to describe a non-linear system as a sequence of piecewise linear systems, which are iteratively fitted over large combination of model order. Following, the proposed algorithm is validated on experimental dataset of a masonry prototype. In particular, data recorded during shake table tests at different input intensity are considered, and the outcomes of the analysis performed by using the proposed algorithm are compared with the dynamic identification conducted in controlled environments. 2. Algorithm for dynamic identification of non-linear systems from non-stationary signals 2.1. Algorithm overview A general overview of the algorithm for the segmentation of SISO data by means of ARX methods is illustrated in Figure 1. At first, single input and single output (SISO) records of a system are gathered, and preprocessed by means of offset and trend removing, filtering and down-sampling of the time series. In order to identify the pre-seismic dynamic properties of the structure, the initial section of the data series is isolated, while the remaining data are considered for the segmentation. With this regard, a threshold of characteristic values of the time series can be set; in particular, three methods are proposed, considering, i ) normalized Arias Intensity ( ̅̅̅ ) of the input signal, for which lower and upper boundary of input ̅̅̅ values are required to determine the beginning and the end of the pre seismic sections, as [ ̅̅̅ ̅̅̅ ] ; ii ) acceleration limit of the output record, ̈ for which the upper boundary of the pre-seismic segment is defined as [0 ̈ ] , or iii ) mixed ̅̅̅ -based on the input data with control on the maximum output accelerations in the segment of the pre-seismic segments, for which lower and upper boundary of input ̅̅̅ values are set and additionally control output acceleration limit, ̈ , is required to check the