# Robust Static Structural System Identification Using Rotations
## 5 Conclusions  
This article proposes a statistical analysis to illustrate the theoretical advantage of measuring rotations rather than deflections in static structural system identification, which is also different from the system identification using dynamic data [50; 51; 52]. With this aim, the analytical expressions for the target parameters are derived with structural system identification using the constrained observability method. Combining these expressions and the inverse distribution theory, a procedure for obtaining the distributions of the estimations is proposed for the very first time. The statistical analyses were carried out in two structures. Its effectiveness was verified by comparison with a Monte Carlo analysis. From the numerical examples, the distributions of the estimations using only deflection measurements are very sensitive to errors. These distributions explain the reason for the biased or even negative estimations when only deflections are measured in the case of essential sets. In the comparative study of measuring deflections or rotations, it is justified that the estimations using rotations are always less sensitive to measurement errors than those using deflections, with respect to the unbiasedness and the extent of variation in the estimations. Taking account of the theoretical and practical advantages of using rotations, four strategies for using redundant rotations to improve the accuracy of estimations were proposed and compared in different structures under static loading. In the simply supported bridge example where the compatibility conditions were obtainable, the strategy of using compatibility conditions outperformed the rest. The performance of strategy 4 (averaging with outliers ruled out) was entirely satisfactory. Furthermore, strategy 4 can be applied in frame structures where the compatibility conditions are not available. In the numerical analysis of the high-rise frame, the satisfactory accuracy and robust performance achieved by using rotations along with strategy 4 justify the effectiveness and versatility of the proposed method. The theoretical basis of this method has been established in this paper and it has been proved that with numerically generated data the method is able to provide sensible results and improve the predictions obtained with just deflections. However, empirical validation is lacking, and this has to be checked to confirm that the practical application of the method is feasible. Moreover, applications based on field measurement will also be carried out in future work. Conceptualization, J.T. ; methodology, J.L. ; software, J.L. ; validation, J.L. ; formal analysis, J.L. ; writing--original draft preparation, J.L. ; writing--review and editing, J.T., F.-L.Z. and J.A.L.-G.; supervision, D.X., J.T., F.-L.Z. ; project administration, J.T., F.-L.Z. All authors have read and agreed to the published version of the manuscript. This work was partially funded by the Scientific Research Fund of the Institute of Engineering Mechanics, China Earthquake Administration (grant no. 2019 EEEVL0401), National Natural Science Foundation of China (grant nos. 51878484), Spanish Ministry of Economy and Competitiveness and the FEDER fund through the projects BIA2013-47290-R and BIA2017-86811-C2-1-R directed by [PERSON] and BIA2017-86811-C2-2-R, and the Natural Science Foundation of Shenzhen (grant no. JCYJ20190806143618723). 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). Part of this work was carried out through a collaborative agreement between Tongii University (China) and the Technical University of Catalonia, UPC. This included an exchange of faculty financed by the Chinese government. The financial support from the Chinese High-End Foreign Experts program (GDW20143100115) is greatly appreciated. Funding for this research has been provided to [PERSON] by the Chinese Scholarship Council through its program No.