# Precise Mission Process Control Based on a Novel Dual-Code Group Network Plan Diagram
## 5 Mission Process Control Strategy
### Control Strategy Solution
#### 5.2.4 Solving of Control Strategy  
According to the maximum anti-disturbance capability \(e_{\max}(t)\) obtained in Section 5.2.3, the system controllability condition is obtained as  
\[e(t)\leq e_{\max}\left(t\right) \tag{57}\]  
According to the system controllability condition in Equation (57), the mission is not fully controllable if the actual system output \(y(t)\) provides an output greater or equal to the current maximum acceptable output \(y_{\max}(t)\). If \(y_{\max}(t)-y_{c}>0\), it means that the maximum error that can be eliminated by the system is \(y_{\max}(t)\); if \(y_{\max}(t)-y_{c}\leq 0\), it means that the system could not eliminate any error. If the controllable moment \(t_{c}\) satisfies the system controllability condition, it means that the error can be eliminated by applying the control quantity \(\mathbf{u}(t_{c})\), so the control strategy is to solve the appropriate \(\mathbf{u}(t_{c})\). Furthermore, the solving of control quantity can be divided into two steps: the first step is to apply control quantity to which controllable missions,and the second step is to apply how much control quantity. Based on the conclusion of Theorem 5, we can obtain the valid controllable missions and invalid controllable missions. According to (50), the output depends on the critical path length, and all valid controllable missions belong to the critical path, so as long as the maximum anti-disturbance capability can be achieved without any change in the critical path, the error can be eliminated just by applying the control quantity to the current valid controllable missions. Next, we give the critical path conservation theorem to prove that changing the critical path does not improve the maximum anti-disturbance capability of the current critical path. Assume that for the current controllable moment \(t_{c}\), the critical path is \(\hat{\mathcal{K}}^{*}(t_{c})\), and the maximum anti-disturbance can be achieved without changing the critical path after applying the control quantity \(\mathbf{u}(t_{c})\). That is, \(\hat{\mathcal{K}}^{*}(t_{c}+\Delta t)=\hat{\mathcal{K}}^{*}(t_{c})\), which is referred to as the \"critical path conservation theorem\". We give a scenario. As shown in Figure 12(a), the red path is the critical path, and the mission \(x_{(8,9)}\) is the constant critical mission, thus \(y(t)=T_{PS}^{(8,9)}(t)\). The missions \(x_{(2,3)}\), \(x_{(4,5)}\), and \(x_{(6,7)}\) are parallel to each other. Since \(x_{(2,3)}>x_{(6,7)}>x_{(4,5)}\), it makes \(m_{(2,3)}\) a critical mission and \(m_{(4,5)}\) and \(m_{(6,7)}\) are non-critical missions. If \(m_{(2,3)}\), \(m_{(4,5)}\) and \(m_{(6,7)}\) are controllable missions, \(y(t)\) can be reduced by shortening their mission durations. Now, \(m_{(2,3)}\), \(m_{(4,5)}\), and \(m_{(6,7)}\) are all compressed to the shortest mission duration. As shown in Figure 12(b), \(y(t)\) is minimized when \(x_{\min}^{(2,3)}=\max(x_{\min}^{(6,7)},x_{\min}^{(4,5)})\). As shown in Figure 12(c), when \(x_{\min}^{(2,3)}<\max(x_{\min}^{(6,7)},x_{\min}^{(4,5)})\), the critical paths change, and \(m_{(2,3)}\) no longer belongs to the core paths, and \(m_{(6,7)}\) becomes the new critical path mission. However, \(y(t)\) does not decrease compared to Figure 12(b). This indicates that \(y(t)\) can be minimized without changing the critical path. When the core path reaches the switching threshold, \(y(t)\) reaches the minimum exactly. This completes the Proof of Theorem 5. Based on the conclusion of Theorem 5, we can then realize disturbance elimination by calculating the control volume to change the critical path length under the premise that the critical path remains unchanged.