# Structural Health Monitoring of 2D Plane Structures
## 4 Conclusions  
The observability technique has been studied in the literature for the structural system identification of 1D element structures (such as beams and trusses). Nevertheless, the application of this method to 2D element structures (such as dams, tunnels, and culverts) has not been studied yet. Despite how it may appear, this application is not straightforward, as the unknown variables in the system are not only located in the numerator. In fact, the location of these variables also in the denominator makes the system of equations nonlinear and prevents the application of traditional observability techniques for this type of structure. To fill this gap, an algorithm for the observability analysis of 2D element structures was developed in this paper. The main contribution of this algorithm refers to the development of a new change in variables to linearize the system of equations and makes OM applicable for 2D element structures. The proposed methodology is characterized by its generality, as similar changes in variables can be used to linearize other nonlinear problems when the unknown variables appear in both the numerator and denominator of the system of equations. To illustrate the applicability of the proposed algorithm, a state-of-the-art example (cantilever plate) was analyzed with the plane strain theory. This analysis included a step-by-step mathematical review of the system of equations, as well as a statistical study of the structural system identification with different measurement sets. The obtained results show that the unknown parameters (such as the Young's Modulus, E, and Poisson's Ratio, v) are successfully calculated with the proposed methodology. Conceptualization, H.M., J.A.L.G. and J.T. ; methodology, B.M. and J.A.L.G. ; software, B.M. ; validation, H.M., J.A.L.G. and J.T. ; formal analysis, B.M. ; investigation, B.M. and J.A.L.G. ; resources, H.M., J.A.L.G. and J.T. ; data curation, J.A.L.G. and J.T. ; writing--original draft preparation, B.M. and J.A.L.G. ; writing--review and editing, H.M., J.A.L.G. and J.T. ; visualization, B.M. ; supervision, J.A.L.G. and J.T. ; project administration, J.A.L.G. ; funding acquisition, J.A.L.G. and J.T. All authors have read and agreed to the published version of the manuscript. This research was funded by the Spanish Ministry of Economy and Competitiveness (grant number BIA2013-47290-R, BIA2017-86811-C2-1-R, and BIA2017-86811-C2-2-R), the Universidad de Castilla La Mancha (grant number 2018-COB-9092), and the Secretaria d' Universitats i Recerca de la Generalitat de Catalunya (grant number 2017 SGR 1481). Figure 5: Required number of measurements for having the partial (**a**,**b**,**c**) and full observability (**b1**,**c1**) of the system. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** All data during the study appear in the submitted article. **Acknowledgments:** The authors are indebted to the Spanish Ministry of Economy and Competitiveness for the funding provided through the research projects BIA2013-47290-R, BIA2017-86811-C2-1-R, and BIA2017-86811-C2-2-R founded with FEDER funds. Funding for this research was provided to [PERSON] by the Spanish Ministry of Economy and Competitiveness through its program for his PhD. Part of this work was done through grant number 2018-COB-9092 from the Universidad de Castilla La Mancha (UCLM). The authors are also indebted to the Secretaria d' Universitats i Recerca de la Generalitat de Catalunya for the funding provided through Agaur (2017 SGR 1481). **Conflicts of Interest:** The authors declare no conflict of interest.