## 3 The Proposed C-MDSDO
### Algorithm Flow of the Proposed Approach  
In C-MDSDO, the available co-design algorithm is first employed to obtain a deterministic solution, which is applied for the reliability assessment in the following stage. The reliability assessment is based on a constructed sub-optimization problem, in which the  
Figure 2: The influence of uncertainty on the system state trajectory. (**a**) Influence of the uncertainty samples with deviations of \(1\sigma\), \(2\sigma\), and \(3\sigma\). (**b**) Influence of the MCS-based uncertainty samples.  
stochastic optimization algorithm is used to obtain the shifting vector. With the assistance of the shifting vector, the constraints of the deterministic MDSDO problem are gradually modified, and the constraints are used for the next iteration. When the objective and reliability index converge, the solution to the MDSDO problem satisfying the reliability index is obtained. This decoupled sequence framework effectively avoids nesting reliability assessments in complex dynamic discipline optimization. The flowchart of C-MDSDO is shown in Figure 3.  
The details of each C-MDSDO step are as follows:  
1. The boundary constraints of the design variables, state variables, and control variables are established given the initial guesses;
2. The co-design framework is employed to obtain the solution of deterministic MDSDO of the step \(k=1\);
3. MCS population is established for calculating the failure probability based on the uncertainty information of design variables. The state trajectories of the MCS population are deduced based on the Runge-Kutta method and the control trajectories \(\mathbf{u}_{k}(\mathbf{t})\). The failure probability is calculated for the state trajectory of the MCS population, and the objective performance is calculated.  
In this study, the MCS population number was set to 1e5, and the state trajectory deduction method is described in Section 3.2. Figure 2 in the previous section shows the deduction trajectory \(\mathbf{X}_{M}(\mathbf{t})\) of a test case. In addition, the failure probability and reliability index are calculated as shown in Formula (6).  
\[\left\{\begin{array}{l}\beta=\Phi^{-1}(pf)\\ pf=\frac{num[g(\mathbf{d}_{h\mathbf{X}_{M}}(\mathbf{t}),u(\mathbf{t}),A)\geq 0 ]}{N_{M}}\end{array}\right. \tag{6}\]  
Figure 3: The algorithm flowchart of the proposed approach.  
where \(num[g(\mathbf{d}_{M},\mathbf{X}_{M}(\mathbf{t}),u(\mathbf{t}),\mathbf{t})\geq 0]\) represents the number of MCS populations that violate the constraint.  
1. If the objective performance is convergence and the reliability index is satisfied, go to \(\mathbf{g}\); otherwise, go to \(\mathbf{e}\);
2. The design variables \(\mathbf{d}_{k}\) and control trajectories \(\mathbf{u}_{k}(t)\) in step k are substituted into the formulated sub-optimization problem for reliability assessment, and the shifting vector \(\mathbf{v}_{k}\) is obtained;
3. The shifting vector is used to obtain the initial values of the design variables of the next iteration \(\mathbf{k}+1\) with \(\mathbf{d}_{k+1}=\mathbf{d}_{k}+\mathbf{v}_{k}\). Go back to \(\mathbf{b}\);
4. The algorithm is terminated, and the reliability-based MDSDO result is output.  
This section describes the main framework of the proposed approach and elucidates the relationship between the main problem and the sub-optimization problem. In the next section, the formulation of the sub-optimization problem is described. On the basis of the formulated sub-optimization problem, the problem-solving method of the shifting vector is presented, which is employed to improve MDSDO reliability.