## 1 Introduction
### Modeling Rules and Diagnostic Bond Graph  
Figure 6 illustrates the traditional diagnostic bond graph of the system. According to the literature [37], the following guidelines were used to determine the causal relationship:  
* There is a sole causal relationship between the potential source and the flow source, with the potential source's causality stroke on the side closest to the tip and the flow source's causality stroke on the side furthest from the tip.
* Both inertial and capacitive elements have precedence for defining integral causality, but the causality of the inertial element is on the side closest to the tip, whereas the causality of the capacitive element is on the side farthest from the tip.
* The resistive component prioritizes admittance-type causality; the causality stroke is on the tip side, but coordination with other bond graph parts must be taken into account throughout the designation process, and the causality is extended in the direction of flow at the node. Each bond and node are numbered according to the following principles:
* Starting from number 1, each bond is numbered in turn, counterclockwise, centered on the node, and the signal bonds are not numbered.
* Starting from number (1), each node is numbered according to the direction of energy flow, and the converter \(TF\) node is not numbered.  
In Figure 6, the pre-defined fault nodes and the nodes where multiple bonds converge in the system are chosen as the virtual sensor placement locations. To detect the pressure signal, the pressure sensor \(SSp\) is placed at the common potential node (0-node), and to detect the flow signal, the flow sensor \(SSQ\) is placed at the common flow node (1-node). Table 2 displays the correspondence between the sensor and analytical redundancy relations (ARRs).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Meaning** \\ \hline \(p_{Rolless}Q_{Rolless}\) & Pressure and flow of rodless cavity \\ \(p_{Rod}Q_{Rod}\) & Pressure and flow of rod cavity \\ \(A_{Ralless}\)’\(Ralless\) & Acting area and force of piston on one side of rodless cavity \\ \(A_{Rad}F_{Rod}\) & Acting area and force of piston on one side with rod cavity \\ \(v\) & Piston velocity \\ \(R_{Ex-l}\) & Hydraulic resistance effect of external leakage of actuator \\ \(C_{Ex-Ralless}C_{Ex-Rad}\) & Liquid capacity effect of rodless cavity and rodless cavity \\ \(R_{In-l}\) & Hydraulic resistance effect of internal leakage of actuator \\ \(R_{Eq-f}\) & Equivalent friction liquid resistance effect of reciprocating motion outside the actuator \\ \(F_{ex}(x)\) & Other external forces acting on external loads \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters definition of actuators bond graph.  
Figure 5: Actuator bond graph.  
Due to the simplified construction model in this paper, there are only four residuals generated, of which two sensors generate one ARR. When the model is complex and delicate, dozens or even hundreds of residuals will be generated, and each residual will show a very different detection effect. There are not many residuals with good detection effects. Based on convex optimization, [PERSON] and others [38] investigated selecting residuals with better detection performance from a large number of residuals.