PSI - Issue 84

Alberto Barontini et al. / Procedia Structural Integrity 84 (2026) 352–359

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(Kiureghian & Ditlevsen, 2009). OSP was initially formulated within the frameworks of structural identification and control. One of the earliest methods, the so-called Effective Independence (EfI) approach, has been shown to provide a robust strategy for optimising sensor networks in support of dynamic identification and model updating by pursuing a heuristic optimisation of the Fisher Information Matrix (FIM), while progressively reducing the number of sensors (Kammer, 1991). The original version of this method relies on deterministic preliminary models, nonetheless, their limitations, due to possible errors and sources of uncertainty, were addressed at an early stage by the same author. In particular, a bound was established on the number of sensors beyond which the removal of additional sensors may compromise the identification of the true target modes in the presence of an expected level of error (Kammer, 1992). Few subsequent developments have adopted a similar strategy, considering intervals for parameter values to compensate for insufficient statistical information on the sources of uncertainty (Yang et al., 2020). More studies, by contrast, have implemented strategies based on sampling uncertain parameters from probability distributions producing a stochastic optimisation framework. In this case, the different methods have treated either the optimisation outcome (Castro-Triguero et al., 2013) or only its input (Kim et al., 2018) as an aleatory variable. The approaches analysed in the present work are briefly introduced in Section 2. To ensure a fair comparison, the original formulations have been suitably modified, where necessary, to enable implementation using objective functions compatible with the FIM determinant and a heuristic backward sequential sensor placement strategy consistent with the EfI method, both of which are widely regarded as well-established references in the field. In Section 3, the methodology is exemplified through a case study application, and the main outcomes are discussed, while Section 4 presents conclusions and future scopes. 2. Methodology This section introduces six well-established methods which are then analysed and compared. Owing to space limitations, symbol definitions are provided in the Nomenclature section, while the interested reader is referred to the cited references for further methodological details. The reference optimisation algorithm adopted in this study is the EfI method. Under this approach, the contribution of the selected sensors to the linear independence of the target mode partitions is optimised by maximising the determinant of the FIM, defined as follows: = −1 (1) The EfI method pursues this objective in a heuristic way. Starting from the entire set of candidate sensor locations, it calculates the effective independence distribution of the sensor set: = ( ) =[∑ 1, = 1 ,∑ 2, = 1 ,…,∑ , = 1 ] , = [ ]⨂[ ] −1 (2) to iteratively remove the location presenting minimum value, thus minimum contribution, until reaching a predefined number of available sensors (Kammer, 1991). As previously mentioned, some methods have explored the effects of modelling error in a non-probabilistic way, namely avoiding the identification of a suitable description of the sources of uncertainty as stochastic variables. Among them, two solutions account for the effect of model error by formulating a covariance matrix that depends on the relative positions of the sensors. In the approach hereafter referred to as A1, the following formulation is used (Papadimitriou & Lombaert, 2012): = (− / ) (3) While in the approach henceforth named as A2, the covariance matrix is (Vincenzi & Simonini, 2017): = 1 | | (| |;| |) | | (| |;| |) (− / ̅) (4)