# Non-Mypopic Beam Scheduling for Multiple Smart-Target Tracking in Phased Array Radar Networks
## 2 Model Description and Problem Formulation
### Target Dynamic and Measurement Models  
Consider \(N\) smart targets labeled by \(n=1,2,\ldots,N\) and a radar network consisting of \(K<N\) phased array radars, as illustrated in Figure 1. Target 1 and Target \(N\) will likely change their dynamic models in response to radar tracking, while Target 2 will be likely to maintain its dynamic model. All radars are synchronized to operate over time slots \(t=0,1,\ldots\). The radar network centrally steers beams to track at most \(K\) targets at each time, as one radar beam can be only assigned to one target in a time slot. Without loss of generality, let the dynamic state of target \(n\) at time \(t\) be \(\mathbf{x}_{n,t}\triangleq[x_{n,t},\hat{x}_{n,t},y_{n,t},\hat{y}_{n,t}]^{\prime} \in\mathbb{R}^{4}\), where \([x_{n,t},y_{n,t}]^{\prime}\) is the position, and \([\hat{x}_{n,t},\hat{y}_{n,t}]^{\prime}\) is the velocity. \([\cdot]^{\prime}\) is the transpose operator. Targets follow independent and linear dynamics. Meanwhile, they are smart, that is, they react to radar radiation by switching between \(M\) dynamic models, labeled by \(m=1,2,\ldots,M\), e.g., CV, CA, CT, etc. Target \(n\)'s dynamics under model \(m\) is  
\[\mathbf{x}_{n,t}=\mathbf{F}_{n}^{m}\mathbf{x}_{n,t-1}+\mathbf{w}_{n,t-1}^{m}, \tag{1}\]  
where \(\mathbf{F}_{n}^{m}\) is the state transition matrix, and \(\mathbf{w}_{n,t-1}^{m}\) is process noise with mean being zero and covariance matrix being \(\mathbf{Q}_{n}^{m}\). Define \(\mathbf{Q}_{n}^{m}=q_{n}^{m}\mathbf{Q}_{n}\) for a covariance matrix \(\mathbf{Q}_{n}\), where \(0<q_{n}^{1}<q_{n}^{2}<\cdots<q_{n}^{M}\), and \(m=1\) corresponds to the CV model, so the uncertainty of the dynamic model increases with \(m\). At each time \(t\), a binary variable \(a_{n,t}\in\{0,1\}\) denotes the tracking action imposed on target \(n\). Let \(a_{n,t}=0\) if target \(n\) is not tracked; \(a_{n,t}=1\) otherwise. Let the detection probability of the radars be \(p_{\text{d}}\in[0,1]\)[33]. If target \(n\) is detected at time \(t\), a measurement \(\mathbf{z}_{n,t}\) is derived with the following measurement equation,  
\[\mathbf{z}_{n,t}=\text{H}_{n}(\mathbf{x}_{n,t})+\mathbf{v}_{n,t}, \tag{2}\]  
where \(\text{H}_{n}(\mathbf{x}_{n,t})=(r_{n,t}(\mathbf{x}_{n,t}),\theta_{n,t}(\mathbf{x}_{n,t}))\) with the range \(r_{n,t}(\mathbf{x}_{n,t})\) and the azimuth \(\theta_{n,t}(\mathbf{x}_{n,t})\) being defined as  
\[r_{n,t}(\mathbf{x}_{n,t})=\sqrt{(x_{Tx}-x_{n,t})^{2}+(y_{Tx}-y_{n,t})^{2}}, \tag{3}\]  
\[\theta_{n,t}(\mathbf{x}_{n,t})=\arctan\biggl{(}\frac{y_{n,t}-y_{Tx}}{x_{n,t}-x_{ Tx}}\biggr{)}, \tag{4}\]  
where \((x_{Tx},y_{Tx})\) denotes the location of the radars, \(\mathbf{v}_{n,t}\) is a zero-mean white Gaussian noise with covariance matrix \(\mathbf{R}_{n,t}=\text{diag}(\sigma_{r,n}^{2}(t),\sigma_{\theta,n}^{2}(t))\), where \(\sigma_{r,n}^{2}(t)\) and \(\sigma_{\theta,n}^{2}(t)\) are the variances in the measurement noise on the range and azimuth, respectively, and written as [34],  
\[\sigma_{r,n}^{2}(t)\propto(P_{Tx}|\xi_{n,t}|^{2}\eta^{2}/(r_{n,t}(\mathbf{x}_{n,t }))^{4})^{-1}, \tag{5}\]  
\[\sigma_{\theta,n}^{2}(t)\propto(P_{Tx}|\xi_{n,t}|^{2}\varsigma^{-2}/(r_{n,t}( \mathbf{x}_{n,t}))^{4})^{-1}, \tag{6}\]  
Figure 1: System model of a phased array radar network. where \(\propto\) is a proportional operator, \(P_{Tx}\) denotes the transmitted power, \(\varsigma\) and \(\eta\) denote the 3-dB receive beamwidth and the waveform bandwidth of the phased array radars, \(\zeta_{n,t}\) incorporates the effects of phase offsets, and the RCS impact of target \(n\) on both phase and amplitude. Note that when a smart target maneuvers, such as sudden changes in velocity or direction, errors arise from the unpredictability of the target's motion, causing increased measurement errors [35]. In general, the estimation accuracy of the target state can be improved by the adaptive filtering technology at the moment of abrupt maneuvering. It dynamically adjusts the process and measurement noise to accommodate these sudden changes in motion [36; 37].