# Precise Mission Process Control Based on a Novel Dual-Code Group Network Plan Diagram
## 5 Mission Process Control Strategy
### Control Strategy Solution
#### 5.2.3 Maximum Anti-Disturbance Capability  
According to the definition of the output \(y(t)\) in (26), the output is the planned start time of \(m^{*}\). Since the output at the controllable moment \(t_{c}+\Delta t\) can be changed by applying the control quantity \(\mathbf{u}(t_{c})\) at the moment \(t_{c}\), the minimum value of the plan start time \(\min(T_{PS}^{m^{*}}(t+\Delta t))\) at the moment \(t+\Delta t\) determines the maximum error \(e_{\max}(t)\) that the system can eliminate at moment \(t\). \[e_{\max}(t)=y(t)-\min\Bigl{(}T_{PS}^{m^{*}}(t+\Delta t)\Bigr{)} \tag{53}\]  
If \(e(t)>e_{\max}(t)\), it means that the error cannot be eliminated no matter what control strategy is applied, and the system is not controllable at this time. If \(e(t)\leq e_{\max}(t)\), it means that the error can be eliminated by applying the control quantity, and the system is controllable at this time. We define \(e_{\max}(t)\) as the maximum anti-disturbance capability of the system at the moment \(t\), which provides an important decision support for the mission process manager. Based on \(e_{\max}(t)\), we can obtain the maximum output \(y_{\max}(t)\) that the system can withstand at moment \(t\). \[y_{\max}\left(t\right)=e_{\max}\left(t\right)+y_{\epsilon} \tag{54}\]  
Equation (10) gives the calculation of the earliest start time \(T_{ES}^{m_{i}}\) and earliest end time \(T_{EF}^{m_{i}}\) of each mission based on the initial value of each mission duration \(x_{m_{i}}\). Similarly, based on the shortest duration \(\min(x_{m_{i}}(t))\) of each mission, we can obtain the minimum value of the plan start time \(\min(T_{PS}^{m_{i}}(t+\Delta t))\) of each mission at the moment \(t+\Delta t\) and the minimum value of the plan end time \(\min(T_{PF}^{m_{i}}(t+\Delta t))\). The calculation of \(\min(x_{m_{i}}(t))\) is shown below:  
\[\min(x_{m_{i}}(t))=\left\{ \begin{array}{l}x_{m_{i}}(t),c_{m_{i}}(t)>0\\ x_{m_{i}}^{m_{i}},c_{m_{i}}(t)=0\end{array} \right. \tag{55}\]  
Figure 12: Diagram of adjusting the output by changing the duration of the critical path mission. (**a**) shows the dual coding system. (**b**) shows that if the plan end time of the non-critical path mission \(x_{(9,10)}\) is reduced, the length of the critical path remains unchanged. (**c**) shows that by reducing the plan end time of the critical path mission \(x_{(5,6)}\), the length of the critical path can be effectively compressed. Furthermore, \(\min(T_{PS}^{m_{i}}(t+\Delta t))\) can be expressed as  
\[\left\{ \begin{array}{l}\min\bigl{(}T_{PS}^{m_{i}}(t+\Delta t)\bigr{)}=\max_{m_{ i}\in\mathcal{E}M_{m_{i}}}\Bigl{\{}\min\Bigl{(}T_{PS}^{m_{i}}(t+\Delta t)\Bigr{)}+x_{ \min}^{m_{i}}\Bigr{\}}\\ \min\bigl{(}T_{PS}^{m_{i}}(t+\Delta t)\bigr{)}=\min\bigl{(}T_{PS}^{m_{i}}(t+ \Delta t)\bigr{)}+x_{\min}^{m_{i}}\end{array} \right. \tag{56}\]  
The maximum anti-disturbance capability of the system, \(e_{\max}(t)\), can be obtained by bringing the earliest plan start time, \(\min(T_{PS}^{m_{i}}(t+\Delta t))\), of \(m^{*}(t+\Delta t))\), into (53). The controllability condition has no relation to the cost function but only with the group network plan diagram itself and, more specifically, with the controllability of each sub-mission, their duration, and the cooperative relationships between them. According to Algorithm 4, if a group network plan diagram contains more controllable sub-missions, then the plan is more resistant to perturbation.