## 5 Simulation  
### Case 1  
In this section, two simulation examples were used to test the performance of ISTUF, one was a constant-turn (CT) target tracking system and the other was a Singer acceleration model-based target tracking system. The results were compared to improved UKF (IUKF) in Reference [21] and RSTUF in Reference [13]. The CT target tracking system was in two dimensional space [13]. The detailed model is as follows:  
\[\mathbf{x}_{k}=\left[ \begin{array}{cccc}1&\frac{\sin\Omega T}{\Omega}&0&\frac {\cos\Omega T-1}{\Omega}&0\\ 0&\cos\Omega T&0&-\sin\Omega T&0\\ 0&\frac{1-\cos\Omega T}{\Omega}&1&\frac{\sin\Omega T}{\Omega}&0\\ 0&\sin\Omega T&0&\cos\Omega T&0\\ 0&0&0&0&1\end{array} \right]\  
where \(\mathbf{x}=\left[ \begin{array}{cccccc}\zeta&\dot{\zeta}&\zeta&\dot{\zeta}&\Omega \end{array} \right]^{\mathrm{T}}\); \(\zeta\) and \(\zeta\) are the position values; \(\dot{\zeta}\) and \(\dot{\zeta}\) denote the velocity values; \(\Omega\) is the turn rate; and \(T\) is the sampling interval of measurements, and \(T=0.1\)s. The values of \(\mathbf{Q}\) and \(\mathbf{R}\) are given as:  
\[\mathbf{Q}=0.1\times\left[ \begin{array}{cccccc}\frac{T^{3}}{3}&\frac{T^{2}}{2}&0 &0&0\\ \frac{T^{2}}{2}&T&0&0&0\\ 0&0&\frac{T^{3}}{3}&\frac{T^{2}}{2}&0\\ 0&0&\frac{T^{2}}{2}&T&0\\ 0&0&0&0.018T\end{array} \right] \tag{80}\]  
\[\mathbf{R}=\left[ \begin{array}{cc}\sigma_{r}^{2}&0\\ 0&\sigma_{\theta}^{2}\end{array} \right] \tag{81}\]  
where \(\sigma_{r}=10\)m and \(\sigma_{\theta}=\sqrt{7}\)mrad. The initialization is given as:  
\[\mathbf{x}_{0}=\left[ \begin{array}{cccccc}1000\text{m}&30\text{m}/\text{s}&100 0\text{m}&0\text{m}/\text{s}&-3^{\circ}/\text{s}\end{array} \right]^{\mathrm{ T}} \tag{82}\]  
\[\mathbf{P}_{0}=\text{diag}\Big{(}\left[ \begin{array}{cccccc}200\text{m}^{2}&20 \text{m}^{2}/\text{s}^{2}&200\text{m}^{2}&20\text{m}^{2}/\text{s}^{2}&100\text {mrad}^{2}/\text{s}^{2}\end{array} \right]\Big{)}. \tag{83}\]  
The probability of \(\gamma_{k}=1\) was set as \(p_{k}=0.2\). The maneuvering target was estimated for 10 s by IUKF, RSTUF and ISTUF. The root mean square errors (RMSEs) of the estimation results were used as the testing standard of estimated accuracy, which were computed by 100 Monte-Carlo (MC) runs. The comparison between the true trajectory and estimated trajectories is shown in Figure 1. It is obvious that IUKF failed to track the target and even diverged in some points. The estimated trajectory of ISTUF was more convergent to the true trajectory than that of RSTUF. The RMSEs of estimated results are shown in Figures 2-4. From the figures, it could be seen that the RMSE of IUKF was much larger than that of RSTUF and ISTUF. Especially, the position RMSE of IUKF diverges in most points. The RMSEs of RSTUF and ISTUF were similar during the beginning 4 s, and then the RMSE of ISTUF is less than RSTUF. As the time-correlated heavy-tailed noises were against the assumption of the GF, IUKF cannot track the target well and the RMSEs diverge. As time goes on, the occurrence of the one-step random delay affects the estimating accuracy of RSTUF, so RSTUF cannot perform as well as the proposed ISTUF. The averaged RMSE (ARMSE) and time-consuming of the three filters are listed in Table 2. From Table 2, we can observe that the ARMSE of ISTUF was less than IUKF and RSTUF, and the consumption time of calculation. of the three filters was similar. In summary, ISTUF was able to provide much accurate estimation with little increase in computational cost. Figure 1: Comparison of the trajectories. **Remark 4**. : _From Figures 2-4, it can be seen that the error values of RSTUF and ISTUF were similar because the probability of time-delayed measurement was small and the effect of it was not large enough. Moreover, the RSTUF was also robust to heavy-tailed noise. As a result, the two methods had a similar performance._  
**Remark 5**.