## 4 Case Study
### Active Fault-Tolerant Inversion Result Analysis  
With the aid of the surrogate model, the fault-tolerant interval inversion model is solved by NSGA-II, with the following parameter settings: population size = 500, maximum number of generations = 200, crossover probability = 0.8, and mutation probability = 0.01. Testing has demonstrated that these parameter settings provide good robustness in finding the Pareto front. This section presents the results of the active fault-tolerant inversion analysis for the configuration transformation process. To improve assembly fault tolerance, the Pareto solution with the largest fault tolerance for each construction stage is selected. This solution guides the construction process for that stage, with subsequent construction parameters initially set to nominal values. As real-time data become available from the current stage, these data are incorporated into the fault-tolerant inversion model for the next stage, and the optimization model is updated accordingly. This iterative process continues throughout the construction, ensuring that the construction parameters are continuously adjusted to account for construction errors and environmental disturbances. The Pareto front of the optimization results for each construction stage is shown in Figures 9-16, with a selection of Pareto solutions presented in Table 11. In these stages, stage 1 refers to the construction using only the SLB device for the configuration transformation of the main cable, while stage 6 refers to the construction of the long hangers. These two stages do not involve the inversion of the short hangers, so only two objective functions are considered in these stages. The partial Pareto solution in Table 11 clearly demonstrates the trade-offs between design objective optimality and tolerance capability. Under the constraint of structural reliability, the value of \(W\) is directly proportional to the tolerance range of the assembly controllable parameters. Increasing \(W\) helps to improve the feasibility of configuration transformation and speeds up construction. However, as the tolerance range increases, there is a slight sacrifice in the optimality of design objectives, as indicated by the combined effects of \(f_{1}\) and \(f_{2}\). From the table, it is evident that increasing \(W\) leads to a rise in \(f_{2}\), suggesting a reduction in hanger tension uniformity. Therefore, decision-makers must  
Figure 8: Precision verification of the surrogate model. Figure 14: Pareto solution for stage 6. Figure 12: Pareto solution for stage 4. Figure 13: Pareto solution for stage 5. For each construction stage, the solution with the maximum \(W\) value is selected as the construction plan for that stage. Interval inversion for each stage is performed using the active fault-tolerant control method, resulting in the determination of the active fault-tolerant parameters for the entire configuration transformation process. These parameters are updated and refined in each stage, ensuring that the construction process remains within the necessary safety and performance margins. The final active fault-tolerant results for the entire construction process are shown in Figure 17, which provides a comprehensive overview of the adjustments made to the construction parameters at each stage. Furthermore, the maximum angular deviation of the hangers, calculated at each stage, is presented in Figure 18. For example, in stage 3, 300 sample points were randomly selected within the allowable tolerance range for the tension parameters, and corresponding structural responses were calculated using FEA. After calculating the angular deviations for hangers H25-H24 in stage 3 and H23 in stage 4, the maximum angular deviation was obtained, as shown in Figure 18c. The results show that the maximum angular deviations in all stages are less than the allowable angular tolerance of \(6^{\circ}\) for the hanger ducts, demonstrating the effectiveness of the active fault-tolerant control method in preventing hanger bending damage.