# A Diagnosability-Integrated Design Approach Based on Graph Theory
## 3 Diagnosability Integrated Design Method Details
### Measurement Point Determination
#### 3.1.2 Maximal Clique Extraction  
According to Corollary 1, for a system with three states \(\left(f_{0},f_{1},f_{2}\right)\), all forms of the diagnosis plot of measurement point on this system are shown in Figure 3.  
Among the four cases shown in Figure 3, (a) indicates that the measurement point is insensitive to both faults and cannot detect the occurrence of faults; (b) indicates that the measurement point can only detect one of the faults but not the other; (c) indicates that the measurement point can detect both faults but cannot distinguish between them; (d) indicates that the measurement point can detect and isolate both faults. In the diagnosability design of this test point, the information reflected by the test point in case (a) was too little, so it was not considered in this paper. The remaining three cases were discussed in this paper, and the distribution of samples in the feature space corresponding to these three cases is shown in Figure 4.  
As can be seen in Figure 4, the different forms of diagnosis plot of measurement points corresponded to different situations that distinguish three system states. For the third case, the three states of the system were completely distinguishable, while for both the first and second cases, the presence of the fault \(f_{2}\) caused some fault states to be indistinguishable, with the difference that for the second case, the fault state \(f_{1}\) and \(f_{2}\) were mixed and it was difficult to distinguish between the two fault states. In other words, the fault can be detected, but the fault cannot be isolated and identified, while for the first case, the fault state \(f_{2}\) and \(f_{0}\) were mixed. The reason for this situation may be that the fault \(f_{2}\) was difficult to detect, or it may be that due to the existence of the system structure, the relevant information was not transmitted to the measurement point when the fault occurred, making  
Figure 3: Forms of the diagnosis plot of measurement point of three-state system. The dots in the figure represent the states of the system, and the connecting lines between the dots represent that the two states are distinguishable. The subfigures (**a–d**) show the four cases of whether the states of the three-state system are distinguishable or not, respectively.  
the signal of the measurement point in the fault state not different from that in the normal state, causing the fault \(f_{2}\) to be undetectable.  
According to the above analysis, when the diagnosis plot of measurement point presented the third case, as shown in Figure 4, the states of the system were best discriminated at this time, and the states in the space did not appear to be mixed. However, in practice, it is not always possible to achieve this; therefore, the diagnosis plot of measurement point \(\mathcal{G}_{t_{i}}=(V_{t_{i}},E_{t_{i}})\) should contain as many states as possible, and these states should satisfy the structure shown in Figure 3d as much as possible, i.e., the diagnosis plot of measurement point should be as close as possible to the structure of the maximal clique.  
In order to extract the maximal clique from the diagnosis plot of measurement point \(\mathcal{G}_{t_{i}}=(V_{t_{i}},E_{t_{i}})\), we can refer to the Bron-Kerbosch algorithm in the literature, the details of which are not described in this paper due to the length of the paper.  
It should be noted that for each diagnosis plot of measurement point, the maximal cliques contained in it may not be unique (e.g., in (c) of Figure 3, both \(\{f_{0},f_{1}\}\) and \(\{f_{0},f_{2}\}\) are maximal cliques), and the p-th maximal clique of the diagnosis plot of measurement point \(\mathcal{G}_{t_{i}}\) is noted as \(\mathcal{C}_{t_{i}}^{p}\).