# Precise Mission Process Control Based on a Novel Dual-Code Group Network Plan Diagram
## 4 Mission Process Control System Modeling
### System Modeling  
Next, we model the problem of the mission progress mission of a group network plan diagram as a closed-loop control system. The real-time derivation of the network plan diagram and the real-time control of the mission process are realized by constructing the state quantities, input quantities, output quantities, disturbance quantities, and system equations. The idea of system modeling is shown in Figure 10.  
For convenience, we use a single code \(m\) to denote any mission, and each time parameter of the mission \(m\) is also represented by a single code. Assuming that the sorted mission set \(\mathcal{M}_{s}\) contains a total of \(n\) missions, the single-code representation of the \(i\)th mission in the set is \(m_{i}\). Define the observations of the system at time \(t\) as \(\boldsymbol{s}(t)=[s_{m_{1}}(t),s_{m_{2}}(t),\ldots,s_{m_{n}}(t)]\). \(s_{m_{i}}(t),m_{i}\in\mathcal{M}_{s}\) is the state of each mission \(m_{i}\), which corresponds to the execution of the mission \(m_{i}\), observed from the environment, and dynamically updated with time \(t\). \(s_{m_{i}}(t)\) is defined as follows:  
\[s_{m_{i}}(t)= \begin{cases}0,&\text{mission $m_{i}$ is not started at $t$ moment}\\ 1,&\text{mission $m_{i}$ is ongoing at $t$ moment}\\ 2,&\text{mission $m_{i}$ is ended at $t$ moment}\end{cases}  \tag{19}\]  
For any mission \(m_{i}\), the plan start time \(T_{PS}^{m_{i}}(t)\) and the plan end time \(T_{PF}^{m_{i}}(t)\) are always available from the current plan at moment \(t\). Since the program parameters vary over time, both are variables related to \(t\). Based on the plan start time \(T_{PS}^{m_{i}}(t)\) of the mission \(m_{i}\) and the states of all prior-missions of the mission \(m_{i}\), we can obtain the start condition of the mission \(m_{i}\), i.e.,  
\[s_{m_{i}}(t)=\left\{ \begin{array}{l}1,t>T_{PS}^{m_{i}}(t),\forall m_{j}\in \mathcal{EM}_{m_{i}},s_{m_{j}}(t)=2\\ 0,t\leq T_{PS}^{m_{i}}(t)\end{array} \right. \tag{20}\]  
Let \(s_{m_{i}}(t)=2\), once the end-of-mission signal of mission \(m_{i}\), be observed from the environment at moment \(t\). From (20), it can be seen that the start time of the mission \(m_{i}\) is determined without changing the plan, which makes it convenient for the mission executor to start the mission. The actual start time \(T_{AS}^{m_{i}}(t)\) and the actual end time \(T_{AF}^{m_{i}}(t)\) of the mission \(m_{i}\) are obtained from the observation. Both are null if the mission has not started; once the mission has started the actual start time can be obtained; and once the mission has ended the actual  
Figure 10: Diagram of the mission process control system. end time can be obtained. Therefore, the values of \(T_{AS}^{m_{i}}(t)\) and \(T_{AF}^{m_{i}}(t)\) are given by the following equation:  
\[T_{AS}^{m_{i}}(t)= \begin{cases}t,&\text{if }s_{m_{i}}(t-\Delta t)=0\wedge s_{m_{i}} (t)=1\\ \text{None},&\text{if }s_{m_{i}}(t)=0\\ T_{AS}^{m_{i}}(t-\Delta t),&\text{otherwise}\end{cases}  \tag{21}\]  
\[T_{AF}^{m_{i}}(t)= \begin{cases}t,&\text{if }s_{m_{i}}(t-\Delta t)=1\wedge s_{m_{i} }(t)=2\\ \text{None},&\text{if }s_{m_{i}}(t)=0\\ T_{AF}^{m_{i}}(t-\Delta t),&\text{otherwise}\end{cases}  \tag{22}\]  
Based on the time parameters of each mission, we define the set of time parameters of the whole plan at time \(t\) as \(\mathcal{P}(t)=\{p_{m_{1}}(t),p_{m_{2}}(t),\dots,p_{m_{n}}(t)\}\), where \(p_{m_{i}}(t)=\{T_{PS}^{m_{i}}(t),T_{PF}^{m_{i}}(t),T_{AS}^{m_{i}}(t),T_{AF}^{ m_{i}}(t)\},m_{i}\in\mathcal{M}_{s}\) represents the set of time parameters of the mission \(m_{i}\). We further divide the time parameters into two categories, calling \(T_{PS}^{m_{i}}(t)\) and \(T_{PF}^{m_{i}}(t)\) schedule parameters and \(T_{AS}^{m_{i}}(t)\) and \(T_{AF}^{m_{i}}(t)\) progress parameters.