## 3 Selected Research Results
### Analysis of Obtained Results  
The usefulness of the optimal diagnostic symptoms may be determined by the values of the correlation coefficient [9] between the individual cylinders and the sensitivity index (8) calculated for all cylinders separately, as well as the optimization performed considering the selected criteria. The sensitivity coefficient \(w_{w}\) was determined from the following relationship:  
\[w_{w}=\frac{y_{imax}-y_{imin}}{\overline{y}} \tag{21}\]  
where \(y_{imax}\) is the maximum value of the diagnostic parameter in the set of observations, \(y_{imin}\) is the minimum value of the diagnostic parameter in the collection of observations, and \(\overline{y}\) is the average value of a diagnostic parameter in a group of observations.  
The usefulness of the most diagnostically oriented symptoms was evaluated based on the determined correlation coefficient values and the sensitivity index between individual cylinders and for all cylinders separately (Figure 15).  
The highest values of these coefficients were obtained for decimation, windowed time-domain, and Haar wavelet analyses.  
Diagnostic tests can be optimized to select the most advantageous parameters for a given object and to determine the order of checks using the criteria [3]. Optimization was carried out in this work considering selected criteria such as the measurement sensor's location, signal analysis domain, correlation, and sensitivity of diagnostic symptom values with technical state features to minimize energy losses due to friction and, thus, reduce the fuel consumption of the diagnosed tribological pairs [9; 29].  
In terms of the output of vibration acceleration signals, the diagnostic parameter can be determined using the following formula:  
\[a_{p}(\theta,\tau)=O[a_{o}(\theta,\tau)]=O[g_{o}(F,E_{s},Z,Y),\tau] \tag{22}\]  
where \(a_{p}(\theta,\tau)\) is the parameter of vibration acceleration processes; \(\tau\) is the dynamic time; \(\theta\) is the time of operating element; \(F\) is the fuel feed vector, refrigerant feed, etc.; \(X\) is the vector of state features; \(E_{s}\) is the vector of control (fuel setting, engine load, and injection time); \(Z\) is the vector of disturbances; \(Y\) is the vector of output quantities; and \(O\) is the operator of the object performance.  
The vector of vibration acceleration symptoms \(S_{a}(r,\theta)\) is obtained during the monitoring of operation, whether the diagnostic parameters \(a_{p}(\theta,\tau)\) are correlated (or uncorrelated) and sensitive to the characteristics of the technical states; whether they exceed the limit values is determined according to the following relationship:  
\[S_{all}\leq S_{a}(r,\theta)=\Psi[r,X(\theta)]+Z(r,\theta)=A(w_{w})[X(\theta) ]+[Z(r,\theta)]\leq S_{all} \tag{23}\]  
Figure 15: Assessment of the usefulness of diagnostic symptoms based on the correlation coefficient and sensitivity index value: \(k\)—correlation coefficient, \(w_{w}\)—sensitivity index.  
where \(A(w_{w})\) is the sensitivity matrix of the linear expansion of the operator \(\Psi[\cdot]\); \(\Psi[\cdot]\) is the matrix operator of assignment; \(r\) is the distance resulting from the distribution of the propagated damage and the measurement sensor; \(S_{all}\) is the lower limit value of the diagnostic symptom; and \(S_{all}\) is the upper limit value of the diagnostic symptom.  
The model (24) allows for improving the current decisions on the state of the injector based on the experience gained, which is a combination of the results of signal analysis and diagnosed objects.  
The experimental results show that the presented methods can accurately monitor fatigue crack growth. Industry 4.0 requires testing methods with advanced diagnostic and predictive algorithms to improve the accuracy of models. In addition, classification models were proposed to evaluate damage growth to enhance classification accuracy based on a machine learning matrix.