## 4 Case Study
### Physical Model  
The physical model displays the estimated state and also a proportional number of how close the predicted value output is to the measured data. This number is the Normalized Root Mean Square Error (_NRMSE_) given in Equation (4) [28], where \(p\) is the vector from predict value, \(m\) is the vector from measured data, and both have \(n\) elements. \[NRMSE=\frac{\sqrt{\sum_{i=1}^{n}(p_{i}-m_{i})^{2}}}{\sqrt{\sum_{i=1}^{n}(p_{i} -\bar{\rho})^{2}}} \tag{4}\]  
The outlined abnormal situations considered in this case study were devised to meet complex scenarios typically found in landing-gears operation. Abnormal situations addressed in this case study have, at least, one of the following characteristics: high dimensionality, high effort to distinguish boundaries, indirect measurements, and time-variant boundaries. Table 5 has the expected system states. Those states were physically simulated in an FDI Workbench, through different release heights, changing springs, and changing weight on the first platform. The physical model can represent any linear or linearized system. Nevertheless, the focus is going to be over a generic three degree of freedom mass-spring-damper model, which is the most suitable for many landing-gear types. There is an assumption that the model parameters are known. \begin{table}
\begin{tabular}{c} \hline
**Sentence** \\ \hline \(d_{1,1}\) is an eliminated system-state \\ \(d_{1,2}\) is a candidate system-state \\ \(d_{1,3}\) is a candidate system-state \\ \(d_{1,4}\) is an acceptable indecision grade \\ \(d_{2,1}\) is an eliminated system-state \\ \(d_{2,2}\) is a candidate system-state \\ \(d_{2,3}\) is a candidate system-state \\ \(d_{2,4}\) is an acceptable indecision grade \\ \(d_{3,1}\) is a eliminated system-state \\ \(d_{3,2}\) is a candidate system-state \\ \(d_{3,3}\) is a winner system-state \\ \(d_{3,4}\) is a negligible indecision grade \\ \hline \end{tabular}
\end{table} 
Table 4: Semantic perspective. Figure 5 presents the analytical perspective of the experiment. There are three specific physical conditions occurring in different time-space, resulting in a Linear-Parameter-Varying (LPV) approach for this method. Condition 1 occurs from release moment until, just precisely, before the first impact of \(m_{3}\) over the floor. Free-fall equations rule this behavior. Unlike conditions 2 and 3, the system stabilizes very fast in \(L_{1}\) and \(L_{2}\), that is, oscillations after release are negligible. Condition 2 remains while \(m_{3}\) keeps contact to the floor. Otherwise, the equations in condition 3 describe the physics. It is relevant to notice that condition 3 can replace condition 2 if oscillations have to be considered, but this situation increases the model complexity. Conditions 2 and 3 have two great differences. One of them is that \(k_{3}\) and \(d_{3}\) are floor stiffness and floor viscous damping coefficients respectively, it is not considered for condition 3, and the second one is that condition 2 has its equilibrium position aligned with specific positions, defined in parameters \(p_{n}\), while condition 3 has its equilibrium through space-intervals defined in \(L_{1}\) and \(L_{2}\) disregarding any specific state-position. Table 6 details all useful parameters and variables of the method until this moment. Abnormal situations A1 and A2 are related to wear problems that correspond to equipment degradation. A4 and A5 indicate abnormal situations corresponding to the system operation exceeding the safe operating usage limit. It is important to highlight that the labels from A1 to A5 correspond to possible states of the target system. From this perspective, abnormal or normal situations are not relevant to the algorithm since the assigned task is just to classify the actual system state in this study case. Figure 5: System dynamics.