## 2 Multidisciplinary Dynamic System Design Optimization and Uncertainty  
In the design of a complex system, the coupling of static and dynamic disciplines is almost unavoidable. MDSDO theory achieves the unification of two types of discipline design optimization through decomposition and coordination. An MDSDO problem can be expressed as the following mathematical expression:  
\[ \begin{array}{ll}\underset{\mathbf{d},\mathbf{x}(\mathbf{t}),\mathbf{u}( \mathbf{t})}{\text{min}}&\phi(\mathbf{d},\mathbf{x}(\mathbf{t}),\mathbf{u}( \mathbf{t}),\mathbf{t})\\ subject\ to&\dot{x}-f(\mathbf{d},\mathbf{x}(\mathbf{t}),\mathbf{u}(\mathbf{t}), \mathbf{t})=\mathbf{0}\\ &h(\mathbf{d},\mathbf{x}(\mathbf{t}),\mathbf{u}(\mathbf{t}),\mathbf{t})= \mathbf{0}\\ &g(\mathbf{d},\mathbf{x}(\mathbf{t}),\mathbf{u}(\mathbf{t}),\mathbf{t})\geq \mathbf{0}\end{array}  \tag{1}\]  
where \(\phi(\cdot)\) is the optimization objective, \(\dot{x}(\cdot)\) is the state equation, \(h(\cdot)\) is the equality constraint, \(g(\cdot)\) is the inequality constraint, \(\mathbf{d}\) represents the design variable, \(\mathbf{x}(\mathbf{t})\) represents the state variable, and \(\mathbf{u}(\mathbf{t})\) represents the control variable. An MDSDO problem is commonly solved using the co-design framework in MDF, which is a two-loop structure flow in which the outer loop is design variable optimization and the inner loop is control optimization [31]. The flow chart of the co-design framework is shown in Figure 1.  
The outer loop of static discipline design optimization can be expressed as follows:  
Figure 1: The algorithm flow chart of MDSDO. \[ \begin{array}{ll}\underset{\mathbf{d}}{min}&\phi^{*}(\mathbf{d},\mathbf{x}( \mathbf{t}),\mathbf{u}(\mathbf{t}),\mathbf{t})\\ \text{subject to}&g(\mathbf{d},\mathbf{x}(\mathbf{t}),\mathbf{u}(\mathbf{t}), \mathbf{t})\geq\mathbf{0}\end{array}  \tag{2}\]  
where the state variable \(\mathbf{x}(\mathbf{t})\) and the control variable \(\mathbf{u}(\mathbf{t})\) are derived from the inner loop. The optimization problem of the inner loop is expressed in the following, where the design variable \(\mathbf{d}\) is provided by the outer loop. \[ \begin{array}{ll}\underset{\mathbf{x}(\mathbf{t}),\mathbf{u}(\mathbf{t})} {min}&\phi(\mathbf{d},\mathbf{x}(\mathbf{t}),\mathbf{u}(\mathbf{t}),\mathbf{t })\\ \text{subject to}&\dot{x}-f(\mathbf{d},\mathbf{x}(\mathbf{t}),\mathbf{u}( \mathbf{t}),\mathbf{t})=\mathbf{0}\\ &h(\mathbf{d},\mathbf{x}(\mathbf{t}),\mathbf{u}(\mathbf{t}),\mathbf{t})= \mathbf{0}\\ &g(\mathbf{d},\mathbf{x}(\mathbf{t}),\mathbf{u}(\mathbf{t}),\mathbf{t})\geq \mathbf{0}\end{array}  \tag{3}\]  
As mentioned in the introduction, the internal optimization problem expressed above is also known as the OCP. The OCP is commonly solved by DT, which transforms the infinite-dimensional original dynamic problem into a finite-dimensional NLP using discrete variables on the time interval. Sequential quadratic programming (SQP) is employed to solve the NLP as expressed below. \[ \begin{array}{ll}\underset{\mathbf{M}_{x},\mathbf{M}_{u}}{min}&\phi( \mathbf{d},\mathbf{M}_{x},\mathbf{M}_{u})\\ \text{subject to}&\xi(\mathbf{d},\mathbf{M}_{x},\mathbf{M}_{u})=0\\ &h(\mathbf{d},\mathbf{M}_{x},\mathbf{M}_{u})=0\\ &g(\mathbf{d},\mathbf{M}_{x},\mathbf{M}_{u})\geq 0\end{array}  \tag{4}\]  
where \(\mathbf{M}_{x}\) represents the discrete state variable matrix, \(\mathbf{M}_{u}\) represents the discrete control variable matrix, and \(\xi\) represents the discrete equation of state vectors. Using the co-design framework, the problem of the traditional split sequential approach not being able to obtain a system-level optimal solution was solved by researchers. However, as a result of uncertainty in manufacturing, certain physical design parameters cannot reach an ideal value. Under such conditions, the performance of static disciplines is affected, and the state trajectory of dynamic disciplines coupled with design variables changes drastically. The continued application of the deterministic MDSDO solution may lead to unexpected failures.