# Precise Mission Process Control Based on a Novel Dual-Code Group Network Plan Diagram
## 4 Mission Process Control System Modeling
### System Equations
#### 4.3.3 Dynamics Equation  
We define the change of system state in the absence of an external control quantity as the system dynamics equation, i.e., the process of adaptive change of the system state in response to an external disturbance. Assuming that \(m_{i}\) has a post-mission \(m_{j}\in\mathcal{F}\mathcal{M}_{m_{i}}\), there are \(T_{PF}^{m_{i}}(t)=T_{PS}^{m_{j}}(t)=T\). When there is no disturbance, at the moment \(t=T\), the mission \(m_{i}\) ends exactly and the mission \(m_{j}\) starts exactly, when there are \(s_{m_{i}}(T)=0\) and \(s_{m_{i}}(T)=0s_{m_{j}}(T)=1\). However, when \(t=T\), there is a delayed disturbance in the mission \(m_{i}\), and according to Equation (28), the disturbance is obtained as 1 time step \(\Delta t\), i.e., \(\omega_{m_{i}}(t)=\Delta t\). Without changing the plan, we have  
\[T_{PS}^{m_{j}}(t+\Delta t) =T \tag{33}\] \[T_{AS}^{m_{j}}(t+\Delta t) =T+\Delta t\]  
When the delay moment of mission \(m_{i}\) exceeds the planned start time of the post-mission \(m_{j}\):  
\[T_{AF}^{m_{i}}(t+\Delta t)>T+\Delta t>T_{PS}^{m_{j}}(t+\Delta t) \tag{34}\]  
According to the start condition of the mission shown in (20), there must be  
\[T_{PS}^{m_{i}}(t+\Delta t)\geq T_{AF}^{m_{j}}(t+\Delta t) \tag{35}\]  
Therefore (33) and (34) contradict each other. In order to avoid this situation, the plan must be tuned at moment \(t\). At this point the delay disturbance \(\omega_{m_{i}}(t)\) should be applied to the current state \(x_{m_{i}}(t)\), delaying the plan start time of the post-mission \(m_{j}\), so that \(T_{PS}^{m_{j}}(t+\Delta t)=t+\ Deltat\). This auto-delay process is the behavior of the dynamics within the system due to disturbance excitation and can be considered a zero-input response generated by the system dynamics. Therefore, in the case of \(\mathbf{u}(t)=\mathbf{0}\), the dynamics equation of the system is  
\[\mathbf{x}(t+\Delta t)=\mathbf{x}(t)+\mathbf{\omega}(t) \tag{36}\]  
where \(\mathbf{\omega}(t)=[\omega_{m_{1}}(t),\omega_{m_{2}}(t),\dots,\omega_{m_{n}}(t)]\). \[\omega_{m_{i}}(t)=\left\{ \begin{array}{l}1,t\geq T_{PF}^{m_{i}}(t)\wedge m _{i}\in\mathcal{M}_{1}(t)\\ 0,\text{ other}\end{array} \right. \tag{37}\]  
Next, we can obtain two theorems about time parameters based on the dynamic equation. Theorem 4.3: _According to the time parameter operator \(\mathcal{P}(t)=\rho(\mathbf{x}(t))\), the obtained mission state parameters have the following time constancy:_  
\[T_{PS}^{m_{i}}(t)=T_{PS}^{m_{i}}(t-1)\mid t>T_{AS}^{m_{i}}(t) \tag{38}\]\[T_{PF}^{m_{i}}(t)=T_{PF}^{m_{i}}(t-1)\mid t>T_{AF}^{m_{i}}(t) \tag{39}\]  
Proof of Theorem 3.: First, prove (38). When \(t>T_{AS}^{m_{i}}(t)\), there is \(s_{m_{i}}(t-\Delta t)<2\). According to (20), \(\forall m_{j}\in\mathcal{EM}_{m_{i}}\), and there must be \(x_{m_{j}}(t)=x_{m_{j}}(t-\Delta t)=C_{m_{j}}\), where \(C_{m_{j}}\) is a constant. According to the equation for the time-parameter operator \(\rho(\cdot)\), \(\mathcal{EM}_{m_{i}}\) does not change over time, and therefore, \(T_{PS}^{m_{i}}(t)\) is only related to \(x_{m_{j}}(t)\). If \(x_{m_{j}}(t)=x_{m_{j}}(t-\Delta t)\), we can obtain \(T_{PS}^{m_{i}}(t)=T_{PS}^{m_{i}}(t-1)\)), and (38) is proven. Second, prove (39). If \(t>T_{AF}^{m_{i}}(t)\), we have \(x_{m_{i}}(t)=x_{m_{i}}(t-\Delta t)\). Since \(T_{AF}^{m_{i}}(t)=T_{AS}^{m_{i}}(t)+x_{m_{i}}(t)\), according to the conclusion of (39), we can obtain \(T_{PF}^{m_{i}}(t)=T_{PF}^{m_{i}}(t-\Delta t)\), (39) is proved. This completes the Proof of Theorem 3. Theorem 4.: _The time parameters of the mission satisfy the following equations:_  
\[T_{AS}^{m_{i}}(t)=T_{PS}^{m_{i}}(t)\mid m_{i}\in\mathcal{M}_{0}(t)\cup \mathcal{M}_{1}(t) \tag{40}\]  
\[T_{AE}^{m_{i}}(t)=T_{PE}^{m_{i}}(t)\mid m_{i}\in\mathcal{M}_{0}(t) \tag{41}\]  
Proof of Theorem 4.: First, prove (40). Based on the start condition of the mission \(m_{i}\) shown in (20), it can be obtained that the state of the mission \(m_{i}\) at the moment \(t\) satisfies \(s_{m_{i}}(t-\Delta t)=2\), \(s_{m_{i}}(t)=1\).