## 4 Results and Discussion
### Creation of the Model of the Dynamic Model Section  
The model creation was based on experimental measurements of pressures in AMs and actuator rotation at various loads and at large step changes in the input pressure. Results of measurements are listed in Table 1 and graphical responses are in Figure 7. Results of measurements at various loads and at small step changes in the input pressure are listed in Table 2 and graphical responses are in Figure 8. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Measurement** & \(p_{1}\) & \(p_{2}\) & \(p_{svd}\) & **Load** & \(\beta\) & \(\beta\) \\
**No.** & **[kPa]** & **[kPa]** & **[kPa]** & **[kg]** & **[Increment]** & **[deg]** \\ \hline
1 & 487 & 0 & 487 & 0 & 253 & 36.43 \\
2 & 0 & 489 & \(-489\) & 0 & \(-254.4\) & \(-36.63\) \\
3 & 490 & 0 & 490 & 1.2 & 255.5 & 36.80 \\
4 & 0 & 464 & \(-464\) & 1.2 & 252 & \(-36.29\) \\
5 & 454 & 0 & 454 & 2.14 & 250 & 36.00 \\
6 & 0 & 455 & \(-455\) & 2.14 & \(-245\) & \(-35.28\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of pressures in MAS 20–250 and actuator rotation at large step changes of the input pressure. The comparison of actuator responses at low, moderate, and high input pressure values clearly indicates that in the case of first two types of input, the responses are monotonous, whereas at low inputs and the concurrent impact of a failure, these responses contain oscillations prior to the stabilisation. They are probably caused by the effects of the failure gravitation with a constant effect acting in the same direction causing, together with the air compressibility in the AM, the formation of a short-term damped marginal cycle \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Measurement** & \(p_{1}\) & \(p_{2}\) & \(p_{sd}\) & Load & \(\beta\) & \(\beta\) \\
**No.** & **[kPa]** & **[kPa]** & **[kPa]** & **[kg]** & **[Increment]** & **[deg]** \\ \hline
7 & 216 & 0 & 216 & 2.14 & 183.3 & 26.40 \\
8 & 0 & 226 & \(-\)226 & 2.14 & \(-\)192 & \(-\)27.65 \\
9 & 220 & 0 & 220 & 1.2 & 183.3 & 26.39 \\
10 & 0 & 229 & \(-\)229 & 1.2 & \(-\)190 & \(-\)27.36 \\
11 & 227 & 0 & 227 & 0 & 183.3 & 26.39 \\
12 & 0 & 232 & \(-\)232 & 0 & \(-\)197 & \(-\)28.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(Cont.\)  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Measurement** & \(p_{1}\) & \(p_{2}\) & \(p_{sd}\) & Load & \(\beta\) & \(\beta\) \\
**No. ** & **[kPa]** & **[kPa]** & **[kPa]** & **[kg]** & **[Increment]** & **[deg]** \\ \hline
13 & 247 & 160 & 87 & 2.14 & 121.43 & 17.48 \\
14 & 155 & 198 & \(-\)43 & 2.14 & \(-\)48.21 & \(-\)6.94 \\
15 &  
prior to the stabilisation of the moment equilibrium on the actuator axis. Therefore, the determination of the mathematical model of the actuator dynamics was carried out while considering mainly the responses at great input changes, without the effects of failures; however, a general expression of the model must contain also the properties manifested at small inputs and failures. The resulting transfer function would have the following form:  
\[G_{1}(s)=\frac{K_{p}}{Ts+1}\text{ or }G_{2}(s)=\frac{K_{p}}{(T_{1}s+1)(T_{2}s+1)} \tag{24}\]  
The model might be identified while applying the method of least squares (in a discrete form), provided that the differential equation is as follows:  
\[y(t)=-a_{1}y(t-1)-\cdots-a_{n}y(t-n)+b_{1}u(t-1)+\cdots+b_{m}u(t-m) \tag{25}\]  
where \(y(t)\) is the output from the system, \(a_{1},\ldots,a_{n}\) are the output coefficients, \(u(t)\) is the input to the system, and \(b_{1},\ldots,b_{m}\) are input coefficients. The vector form is then as follows  
\[ \begin{array}{c}\theta=\left[\begin{array}{cccccc}a_{1}&\cdots&a_{n}&b_{1 }&\cdots&b_{m}\\ \phi(t)=\left[\begin{array}{cccccc}-y(t-1)&\cdots&-y(t-n)&u(t-1)&\cdots&u(t-m )\\ \end{array}\right]^{T}\end{array}\right]^{T}\end{array}  \tag{26}\]  
where \(\theta\) represents a _parametric vector_ and \(\phi(t)\) represents a _regression vector_.