# Reliability and Maintenance Analysis of Unmanned Aerial Vehicles +
## 6 The UAV as a Complex Maintenance System
### Uncertainty of Degradation in Corrective Maintenance  
If \((n-1)\cdot\tau\ \leq C_{i}<n\tau\) we have two occurrences for the maintenance decision at time \(n\tau\): _preventive_ maintenance (PM) (see Figure 4a) and _corrective_ maintenance (CM) (see Figure 4b) according to:  
\[\left\{\begin{array}{l}\text{{Preventive}}\ if:\ C_{i}\ \ \leq X_{i}(n\tau)<H_{i}\\ \text{{Corrective}}\ if:\ X_{i}(n\tau)\geq H_{i}\end{array}\right. \tag{9}\]  
The probability that a preventive maintenance happens at the specific time \(n\tau\) after the degradation level of the \(i\)th component has reached the control limit \((n-1)\cdot\tau\ \leq C_{i}<n\tau\) is [63]:  
\[P_{\tau}\{PM\ \ at\ \ n\tau\}\ =\ P_{\tau}\{T_{H_{i}}>n\tau,\ (n-1)\tau\leq T_{C_{i} }<n\tau\} \tag{10}\]  
Now we consider the monotonic expression in the preventive maintenance: consider, for example, what is happening around the monotone function immediately after the \(3t\) maintenance interval (see Figure 5a).  
Figure 4: (**a**) Maintenance limit of preventive maintenance; (**b**) maintenance limit corrective maintenance.  
Figure 5: Uncertainty evaluation of corrective maintenance: the inspection point at 3t in detailed and expanded as a confidence interval.  
After the maintenance interval, since one of the returns from the field of intervention is the degradation state of the systems and subsystems, we know exactly what the degradation status of the systems and subsystems is. In other words, we can quantify as \(X_{i}(3t)\) as the status of the probability at the moment we are studying.  
Immediately after the \"3t moment\", we have a view of the _drift in time_ of the value: a band of uncertainty affects the probabilistic function (supposedly monotonous). The uncertainty is due to the capability of controlling the state of degradation of systems (and subsystems) limited by our confidence interval (see Figure 5a-c) in terms of knowledge of the complete system.  
Now we consider \(X_{1},\,\ldots\,\,,\,X_{n}\) as samples of the subsystems degradation status of normal density after the revision \(X_{i}(3t)\) with unknown mean \(m\) and variance \(\sigma^{2}\) (known) and sample average \(\overline{X}\). We have:  
\[\int\limits_{Xi(t)-Xi(t)\,\,\alpha}^{Xi(t)+Xi(t)\,\,\alpha}\frac{1}{\sqrt{2\pi }}e^{\frac{X(t)\,\,2}{2}}\ dX\ =\ Xi(t)\ (1-\alpha) \tag{11}\]  
Therefore, we have:  
\[Xi(t)(1-\alpha)=\ P_{r}\bigg{\{}\bigg{|}\frac{\sqrt{n}}{\sigma}\cdot( \overline{X}-m)\bigg{|}\ \leq\ Xi(t)\ \alpha\bigg{\}} \tag{12}\]  
Explaining the second member (see Figure 6):  
\[Xi(t)(1-\alpha)=\ P_{r}\bigg{\{}\overline{X}-\ \frac{\sigma}{\sqrt{n}}\cdot Xi(t )\alpha\leq m\ \leq\overline{X}+\ \frac{\sigma}{\sqrt{n}}\cdot Xi(t)\ \alpha\bigg{\}} \tag{13}\]  
We have for the confidence interval of level \(Xi(t)(1-\alpha)=\) for \(m\):  
\[\bigg{[}\overline{X}-\ \frac{\sigma}{\sqrt{n}}\cdot Xi(t)\alpha,\overline{X}+ \ \frac{\sigma}{\sqrt{n}}\cdot Xi(t)\ \alpha\ \bigg{]} \tag{14}\]  
It is necessary to treat the limitation of this method: the term \(Xi(t)-Xi(t)\ \alpha\) can never be lower than \(X_{i}(3t)\). This is because the system, despite monotonic evolution in a more optimistic than linearized way (Figure 5b--green line), cannot, for logical reasons of entropy, improve over time, or have a _negative_ degradation. This condition is only theoretically possible, and is due to the uncertainty of the state of knowledge of the system. Therefore, to restore the physical consistency of the uncertainty evaluation, it is necessary to add the condition:  
\[\left\{\begin{array}{c}\Big{[}\overline{X}-\ \frac{\sigma}{\sqrt{n}}\cdot Xi (t)\alpha,\overline{X}+\ \frac{\sigma}{\sqrt{n}}\cdot Xi(t)\ \alpha\ \Big{]}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  
Our prediction of the state of health of our system cannot disregard the knowledge of the state of the subsystems: their number and state influence the possible uncertainty of the value \(X_{i}(t)\).