# Precise Mission Process Control Based on a Novel Dual-Code Group Network Plan Diagram
## 4 Mission Process Control System Modeling
### System Modeling
Based on the systematic observation \(\boldsymbol{s}(t)\), three sets of missions can be obtained at time \(t\), which are the completed missions set \(\mathcal{M}_{2}(t)\), the ongoing missions set \(\mathcal{M}_{1}(t)\), and the unstarted missions set \(\mathcal{M}_{0}(t)\), i.e.,  
\[\mathcal{M}_{0}(t)=\{m_{i}\mid s_{m_{i}}(t)=0\} \tag{23}\] \[\mathcal{M}_{1}(t)=\{m_{i}\mid s_{m_{i}}(t)=1\}\] \[\mathcal{M}_{2}(t)=\{m_{i}\mid s_{m_{i}}(t)=2\}\]  
Define the set of missions that happen to end at moment \(t\) as \(\mathcal{M}_{3}(t)\), which can be denoted by  
\[\mathcal{M}_{3}(t)=\big{\{}m_{i}\mid s_{m_{i}}(t)=2,T_{AF}^{m_{i}}(t)=t\big{\}} \tag{24}\]  
The system state can be further obtained from the observations. Define the system state as \(\boldsymbol{x}(t)=[x_{m_{1}}(t),x_{m_{2}}(t),\dots,x_{m_{i}}(t)]\), where \(x_{m_{i}}(t),m_{i}\in\mathcal{M}_{s}\) is the duration of mission \(m_{i}\) at moment \(t\). \[x_{m_{i}}(t)=\left\{ \begin{array}{ll}T_{PF}^{m_{i}}(t)-T_{PS}^{m_{i}}(t),&s _{m_{i}}(t)=0\\ T_{PF}^{m_{i}}(t)-T_{AS}^{m_{i}}(t),&s_{m_{i}}(t)=1\\ T_{AF}^{m_{i}}(t)-T_{AS}^{m_{i}}(t),&s_{m_{i}}(t)=2\end{array} \right. \tag{25}\]  
The mission order in \(\boldsymbol{x}(t)\) is consistent with that in the sorted mission set \(\mathcal{M}_{s}\). The mission duration \(x_{m_{i}}(t)\) is a time \(t\)-dependent variable due to the possible bias between the mission progress and the plan. According to Hypothesis 2, the missions order in \(\mathcal{M}_{s}\) does not change since the logical relationships between missions are consistent, so the missions order in \(\mathcal{M}_{s}\) does not change. From the equations for calculating the network plan parameters shown in (10)-(15), it can be obtained that the time parameter in the network plan diagram is only related to the duration of each mission \(x_{m_{i}}(t)\), so \(\boldsymbol{x}(t)\) can reflect the real-time state of the network plan diagram. Furthermore, the parameters of the plan diagrams for each mission defined in Section 3.3 are all time \(t\)-dependent variables. Define the system output as \(y(t)\), referring to the planned start time \(T_{PS}^{m^{*}}(t)\) of some constant critical mission \(m^{*}\) at time \(t\), with  
\[y(t)=T_{PS}^{m^{*}}(t) \tag{26}\]  
A constant critical mission \(m^{*}\) is defined by Theorem 2. It lies on a certain critical path \(\mathcal{K}\) and has no parallel mission, which usually best represents the global progress of the network plan diagram. According to the conclusion of Theorem 2, \(m^{*}\) always belongs to the critical path, i.e., \(m^{*}\in\mathcal{K}\). Since there is no total time difference for the missions on the critical path, the planned start time \(T_{PS}^{m^{*}}(t)\) of \(m^{*}\) can be expressed as the sum of the mission durations of all missions on the critical path that are before \(m^{*}\), namely,  
\[y(t) =T_{PS}^{m^{*}}(t)\] \[=\sum_{m_{i}\in K(t),T_{PS}^{m^{*}_{i}}(t)<T_{PS}^{m^{*}}(t)}x_{m_{ i}}(t) \tag{27}\]  
Different constant critical missions represent different mission progress control goals. If the mission that flows into the global end node \(F\) is selected as the critical mission, \(y(t)\) represents the end time of the entire plan. The goal of mission progress control is to ensure that the entire plan ends as planned. If an intermediate mission on the critical path is selected as the constant critical mission, the goal of mission progress control is to ensure that this critical mission starts as planned. Based on the definition of a constant critical mission, the desired output of the system \(y_{e}\) is the predetermined plan start time \(T_{PS}^{m^{*}}\) for the selected constant critical mission \(m^{*}\). \(T_{PS}^{m^{*}}\) is a constant, i.e., \(y_{e}\equiv T_{PS}^{m^{*}}\). The system error \(e(t)\) is the bias between the actual output \(T_{PS}^{m^{*}}(t)\) and the desired output \(T_{PS}^{m^{*}}\).