## 4 Fault Diagnosis Based on Traditional and Interval Analytical Redundancy Relations
### Interval Analytical Redundancy Relations Method  
Based on the traditional analytical redundancy relation method, the interval analytical redundancy relation method in Section 3 is introduced. According to Figure 13, the residual derivation under the interval analytical redundancy relations is performed. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & \(r_{1}\) & \(r_{2}\) & \(r_{3}\) & \(r_{4}\) & \(D_{b}\) & \(I_{b}\) \\ \hline \(Sp\) & 1 & 1 & 0 & 0 & 1 & 0 \\ \(R_{Ex-l}\) & 0 & 0 & 0 & 1 & 0 \\ \(R_{In-l}\) & 0 & 0 & 1 & 0 & 1 & 1 \\ \(R_{Pa}\) & 1 & 0 & 0 & 0 & 1 & 0 \\ \(R_{PB}\) & 0 & 1 & 0 & 0 & 1 & 0 \\ \(R_{AT}\) & 1 & 0 & 0 & 0 & 1 & 0 \\ \(R_{BT}\) & 0 & 1 & 0 & 0 & 1 & 0 \\ \(p_{T}\) & 1 & 0 & 0 & 1 & 0 \\ \(C_{Rad}\) & 1 & 0 & 0 & 0 & 1 & 0 \\ \(C_{Rondless}\) & 0 & 1 & 0 & 0 & 1 & 0 \\ \(A_{Rodless}\) & 0 & 1 & 0 & 1 & 1 & 1 \\ \(R_{E_{26}-f}\) & 0 & 0 & 0 & 1 & 0 \\ \(F_{ex}(x)\) & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fault signature matrix of landing gear R/E hydraulic system. Taking residual \(r_{1}\) as an example, the new residual \({r_{1}}^{\prime}\) after considering the negative effects of various uncertainties of the system is  
\[{r_{1}}^{\prime}=\frac{(1+\delta_{Sp})Sp-(SSp_{1}+\zeta_{SSp_{1}})}{(1+\delta_{ Sp_{A}})R_{PA}}+(SSQ_{1}+\zeta_{SSQ_{1}})+\frac{SSQ_{2}+\zeta_{SSQ_{2}}}{(1+ \delta_{A_{Rad}})A_{Rad}} \tag{30}\] \[-(1+\delta_{C_{Rad}})C_{Rad}\frac{d(SSp_{3}+\zeta_{SSp_{3}})}{dt}- \frac{SSp_{3}+\zeta_{SSp_{3}}}{(1+\delta_{R_{Ex-1}})R_{Ex-1}}-\frac{(SSp_{1}+ \zeta_{SSp_{1}})-(1+\delta_{PT})p_{T}}{(1+\delta_{RA})R_{AT}}\]  
wherein \(\delta\) is the corresponding of the parameter, and \(\zeta\) is the uncertainty of the measured value of the sensor. After considering the interval analytic redundancy method, \({r_{1}}^{\prime}\) becomes \([{r_{1}}^{\prime}]\). \[\begin{array}{ll}[{r_{1}}^{\prime}]=&\left[{\underline{r_{1}}^{\prime}},{ \overline{r_{1}}}^{\prime}\right]=\frac{(1+\left[\delta_{Sp}\right]Sp-(SSp_{1 }+\left[\zeta_{SSp_{1}}\right])}{(1+\left[\delta_{R_{Pal}}\right])R_{PA}}+(SSQ_ {1}+\left[\zeta_{SSQ_{1}}\right])+\frac{SSQ_{2}+\left[\zeta_{SSQ_{2}}\right]}{( 1+\left[\delta_{R_{Rad}}\right])A_{Rad}}\\ &-(1+\left[\delta_{C_{Rad}}\right])C_{Rad}\frac{d(SSp_{3}+\left[\zeta_{SSp_{ 1}}\right])}{dt}-\frac{SSp_{3}+\left[\zeta_{SSp_{2}}\right]}{(1+\left[\delta _{Ex-1}\right])R_{Ex-1}}-\frac{(SSp_{1}+\left[\zeta_{SSp_{1}}\right])-(1+ \left[\delta_{PT}\right])p_{T}}{(1+\left[\delta_{RT}\right])R_{AT}}\\ &\\ [{r_{2}}^{\prime}],\,[{r_{3}}^{\prime}],\,[{r_{4}}^{\prime}]\) can be obtained according to the same method:  
\[\left\{ \begin{array}{ll}[{r_{2}}^{\prime}]=\left[{\underline{r_{2}}^{\prime}},{ \overline{r_{2}}}^{\prime}\right]=\frac{(1+\left[\delta_{Sp}\right])Sp-(SSp_{ 2}+\left[\zeta_{SSp_{2}}\right])}{(1+\left[\delta_{R_{PB}}\right])R_{PB}}- \frac{(SSp_{2}+\left[\zeta_{SSp_{2}}\right])-(1+\left[\delta_{PT}\right])p_{T }}{(1+\left[\delta_{R_{BT}}\right])R_{BT}}\\ -(SSQ_{1}+\left[\zeta_{SSQ_{1}}\right])-(1+\left[\delta_{A_{Radless}}\right])A _{Radless}(SSQ_{2}+\left[\zeta_{SSQ_{2}}\right])-(1+\left[\delta_{C_{Radless }}\right])C_{Radless}\frac{d(SSp_{4}+\left[\zeta_{SSp_{4}}\right])}{dt}\\ [{r_{3}}^{\prime}]=\left[{\underline{r_{3}}^{\prime}},{\overline{r_{3}}}^{ \prime}\right]=(SSp_{4}+\left[\zeta_{SSp_{4}}\right])-(SSp_{3}+\left[\zeta_{SSp _{3}}\right])-(1+\delta_{R_{Im-I}})R_{Im-I}(SSQ_{1}+\left[\zeta_{SSQ_{1}} \right])\\ [{r_{4}}^{\prime}]=\left[{\underline{r_{4}}^{\prime}},{\overline{r_{4}}}^{ \prime}\right]=(1+\left[\delta_{A_{Radless}}\right])A_{Radless}(SSp_{4}+ \left[\zeta_{SSp_{4}}\right])-\frac{SSp_{3}+\left[\zeta_{SSp_{3}}\right]}{(1+ \left[\delta_{A_{Rad}}\right])A_{Rad}}\\ -(1+\left[\delta_{R_{E_{6-f}}}\right])R_{Eq-f}(SSQ_{2}+\left[\zeta_{SSQ_{2}} \right])-(1+\left[\delta_{F_{Ex}(x)}\right])F_{Ex}(x)\end{array} \right. \tag{32}\]  
See Table 5 for interval values in Equations (37) and (38).