## 4 Main Result
### Numerical Study
#### 4.4.4 Case of Erasure Communication Channel  
Thirdly, let us study how the erasure of data in the communication channel affects the data transmission accuracy and the regulation time for the relative position of the satellites.  
For evaluating the data transmission accuracy, the transients of the transmission procedure were excluded by consideration only the last \(0.3t_{\mathrm{fin}}\) interval of the simulation time. For this interval, the standard deviations \(\sigma_{\varepsilon_{x_{1}}}\) and \(\sigma_{\varepsilon_{x_{1}}}\) of the data transmission errors \(e_{x_{1}}[k]=x_{1}(t_{k})-\hat{x}_{1}(t_{k})\), and \(e_{\hat{x}_{1}}[k]=\hat{x}_{1}(t_{k})-\hat{x}_{1}(t_{k})\) (respectively) were calculated. Logarithmically scaled functions \(\sigma_{\varepsilon_{x_{1}}}\), \(\sigma_{\varepsilon_{x_{1}}}\) versus transmission rate \(R\) for various values of erasure probability \(p\) are plotted in Figures 8 and 9. The plots show that the data transmission errors decrease monotonically, looking like inversely proportional functions on communication rate \(R\) and are practically negligibly small as \(R>1\) bit/s for all considered values of \(p\).  
A summary graph of the dependence of regulation time \(T^{*}\) on the data transmission rate \(R\) (for each one channel) at various erasure probabilities \(p\in\{0,0.1,0.2,0.3\}\) is shown in Figure 10. The curves in Figure 10 make an impression about the required load of the communication channel and the quality of the stabilization process with its use. It is seen from the plot that for significantly high data transmission rate (exceeding 2 bit/s in our example) erasure of data in the channel with probability up to 0.3 does not have an effect on the regulation time. This time is defined by the system dynamics regardless of the communication channel capacity.  
Figure 10: Regulation time \(T^{*}\) vs transmission rate \(R\) for various erasure probability \(p\) values.  
Illustrations of the adaptive coding procedure are given in Figures 13 and 14. Adaptively tuning quantizer range \(M[k]\) in accordance with algorithm (5) is depicted in Figure 13. The plot shows how the range is automatically increased at the initial stage of the process and decays then, which leads to reducing the data transmission error. Figure 14 illustrates an influence of data erasure on the codewords, transmitted over the channel. Signal \(s[k]\) from the coder is received in the form of \(s_{\zeta}[k]\) on the decoder side. The difference between these signals causes the additional data transmission errors.  
Figure 14: Time histories of \(s[k]\) on the coder side (marked by the red circles) and \(s_{\zeta}[k]\) (marked by the blue dots) on the decoder side. Case of quantization and erasure, \(R=1.5\) bit/s, \(p=0.2\). \(x_{12}(0)=200\) m, \(\dot{x}_{12}(0)=0.025\) m/s, \(z_{12}(0)=-50\) m, \(\dot{z}_{12}(0)=-0.025\) m/s; \(T^{*}=7.07\) h.  
Figure 12: Satellites relative trajectories on the (XZ) plane for \(x_{12}(0)=200\) m, \(\dot{x}_{12}(0)=0.025\) m/s, \(z_{12}(0)=-50\) m, \(z_{12}(0)=-0.025\) m/s. Case of quantization and erasure, \(R=1.5\) bit/s, \(p=0.2\). \(T^{*}=7.07\) h.