# On Site Investigation and Health Monitoring of a Historic Tower in Mantua, Italy
## 4 Continuous Dynamic Monitoring of the _Gabbia_ Tower
### Temperature Effects on Natural Frequencies: Dynamic Regression and ARX Models  
As measuring the dynamic response using few accelerometers provides information mainly on the natural frequencies, these parameters have to be necessarily used as features sensitive to abnormal structural changes and damage. However, the natural frequencies of masonry towers are also affected by factors other than damage, such as temperature [9; 10; 11; 12; 13]. It should be noticed that, as reported in [9], the humidity might also affect the modal frequencies of masonry towers, with the increase of water absorption (in the rainy season) inducing a temporary decrease of natural frequencies; on the other hand, no appreciable effects of air humidity on natural frequencies have been detected in [11; 13]. Since masonry towers are generally subjected to low levels of ambient excitation, it has to be expected that the modal frequencies are mainly affected by changing temperature. Hence, the correlation between natural frequencies and temperature needs to be investigated for early detection of structural anomalies: once the normal response of a structural system to changes in its environmental conditions has been explored and can be filtered out to normalize the response data, any further changes in the sensitive features should rely to changes of the structural condition. In order to characterize the effects of temperature on the natural frequencies, dynamic regression (DR) models were established for each identified frequency. DR models [22] assume that the value of the dependent variable (_i.e._, natural frequency) at time instant \(k\) is influenced by the values of the model input (_i.e._, temperature) at current time \(k\), as well as at (\(p\)-\(I\)) previous time instants:  
\[y_{k}=b_{1}\times x_{k}+b_{2}\times x_{k-1}+\ldots+b_{p}\times x_{k-(p-1)}+ \varepsilon_{k} \tag{1}\]  
The linear model (1) can be expressed in matrix form as:  
\[\mathbf{y}\,=\,\mathbf{X}\mathbf{b}\,+\,\mathbf{\varepsilon} \tag{2}\]  
where \(\mathbf{y}_{\wp}R^{n}\) is the vector containing \(n\) observations \(y_{k}\) of the dependent variable, \(\mathbf{b}_{\wp}R^{p}\) is the vector formed by the \(p\) parameters weighting the contribution of the input, \(\varepsilon_{\wp}R^{n}\) is the vector of the random errors and \(\mathbf{X}_{\wp}R^{n\times p}\) is the matrix appropriately gathering the values of the input. Models (1) and (2) can be further generalized so that predictions are computed which also consider previous values of the dependent variables. Among the dynamic methods described in the system identification literature (see e.g., [23]), the Auto-Regressive models with eXogenous input (ARX) are probably the most widely used. ARX models [23] consist of an auto-regressive output and an exogenous input part and can be expressed in the following form:  
\[y_{k}+a_{1}\times y_{k-1}+\ldots+a_{n_{k}}\times y_{k-n_{k}}=b_{1}\times x_{k -n_{k}}+b_{2}\times x_{k-n_{k}-1}+\ldots+b_{n_{b}}\times x_{k-n_{b}+1}+ \varepsilon_{k} \tag{3}\]  
Of course, the linear model (3) can be expressed in the matrix form (2). It should be noted that in Equation (3) only one input (e.g., temperature) and one output (e.g., natural frequency) were considered. Equation (3) can be easily generalized to the case of multiple inputs by replacing \(y_{k}\) and \(x_{k}\) with corresponding row and column vectors, respectively. ARX models are characterized by three model orders: the auto-regressive order \(n_{a}\) (corresponding to the number of the considered past measures of the dependent variable), the exogenous order \(n_{b}\) (corresponding to the number of previous model inputs taken into account), and the time delay between input and output \(n_{k}\). Orders \(n_{a}\) and \(n_{b}\) determine the number of model parameters.