# Non-Mypopic Beam Scheduling for Multiple Smart-Target Tracking in Phased Array Radar Networks
## 4 Simulation and Analysis
### Performance Results with Multi-Dimensional States  
This subsection attempts to validate the application of the US-MP index policy to the multi-dimensional states case. \(N=8\) targets are tracked by the radar network with \(K=1,2,3\) radars. The radars share the same parameters \(P_{Tx}=1\) kw, \(\zeta=0.3^{\circ}\), and \(\eta=1\) MHz. Without loss of generality, \(K\) radars are approximately in the same location \((x_{Tx},y_{Tx})=(0,0)\). For each target, \(\zeta_{n,t}=1\) during the radar tracking time horizon  
Figure 5: The discounted cost per target in the constant ratio \(1/4\). \(T=40\) s. The MP index policy with the truncated time \(\tau=6\), the TEC index policy, and the myopic policy assess the tracking performance. In the US method when calculating the Whittle indices, \(\gamma=1\). The state transition matrices corresponding to the CV and CT model [41] are given by  
\[\mathbf{F}_{n}^{1}=\mathbf{I}_{2}\otimes \begin{bmatrix}1&T_{s}\\ 0&1\end{bmatrix} , \tag{53}\]  
\[\mathbf{F}_{n}^{2}= \begin{bmatrix}1&\dfrac{\sin(\omega T_{s})}{\omega}&0& \dfrac{\cos(\omega T_{s})-1}{\omega}\\ 0&\dfrac{\cos(\omega T_{s})}{0}&0&\sin(\omega T_{s})\\ 0&\dfrac{1-\cos(\omega T_{s})}{\omega}&1&\dfrac{\sin(\omega T_{s})}{\omega}\\ 0&-\sin(\omega T_{s})&0&\cos(\omega T_{s})\end{bmatrix} , \tag{54}\]  
where the tracking time interval \(T_{s}\) is 1 s, and the turn rate is \(3^{\circ}/\)s. \(\otimes\) represents the Kronecker product. \(\mathbf{I}_{2}\) denotes the \(2\times 2\) identity matrix. The process noise covariance matrix is  
\[\mathbf{Q}_{n}^{m}=q_{n}^{m}\mathbf{I}_{2}\otimes \begin{bmatrix}T_{s}^{3}/3&T _{s}^{2}/2\\ T_{s}^{2}/2&T_{s}\end{bmatrix} , \tag{55}\]  
where \(q_{n}^{m}\) denotes the amplitude of the process noise. For \(n=1,2,\ldots,N\), \(q_{n}^{1}=10\), \(q_{n}^{2}=20\), \(\mathbf{R}_{n,t}^{2}=\mathbf{R}_{n,t}^{1}\circ\text{diag}(4,2)\), and \(\mathbf{R}_{n,t}^{1}\) is calculated in (5) and (6). \(\circ\) means the Hadamard product. The measurement cost \(h_{n}=0\). Due to the fact that the smart target's flight trajectory changes depending on the decision made at each time step, it is impossible to predefine the target trajectory. To address this, we use Monte Carlo simulation to thoroughly test a variety of flight trajectories and scenarios, yielding average performance results. The initial dynamic state estimation \(\hat{\pi}_{n,0}\) and TEC state \(\mathbf{P}_{n,0}\) are generated by \([50\cdot 10^{4},10^{2},50\cdot 10^{4},10^{2}]^{\prime}\cdot\delta_{x}\) and \(\text{diag}(10^{4},10^{2},10^{4},10^{2})\cdot\delta_{p}\), respectively, where \(\delta_{x}\sim\text{U}(0,1)\) and \(\delta_{p}\sim\text{U}(0.5,1.5)\). Three scenarios are defined for the performance analysis: one with reckless targets, one with cautious targets, and a third scenario that includes both types of targets, with four of each type. The weight is assumed as \(d_{n}=5\), \(n=1,2,\ldots,K\), and \(d_{n}=1\), \(n=K+1,K+2,\ldots,N\). The results with \(N_{\text{mc}}=1000\) Monte Carlo simulations are shown in Table 2. As the value of \(K\) varies from 1 to 3, the number of targets \(N\) remains constant at 8, and the number of targets tracked will increase. Consequently, the tracking costs and AWRMSEs obtained by all policies naturally decrease, and the difference between the performances of the US-MP index policy and other myopic policies decreases. The WRMSEs of the three scenarios are plotted in Figures 6-8. The US-MP index policy based on the trace of TEC matrices always obtains lower tracking errors and outperforms the TEC index policy and myopic policy. Figure 6: WRMSE of policies for reckless targets. (**a**) \(K=1\) (**b**) \(K=2\) (**c**) \(K=3\). Here, the performances of policies are also analyzed while varying the number of targets and radars with a fixed ratio \(K/N=1/4\). A scenario is considered to consist of 50% reckless targets with \(d_{n}=5\) and 50% cautious targets with \(d_{n}=1\), respectively. The discounted cost per target and AWRMSE results under \(N_{\text{mc}}=1000\) Monte Carlo