Robust Dwell Time Allocation for Multiple Ballistic Reentry Target Tracking in Phased Array Radar

Phased array radar (PAR) is shown to provide an enhanced performance for target tracking due to its beam agility and ability for time resource allocation. Existing algorithms for PAR resource allocation often consider standard and simplified dynamic models for target motion, which are not suitable for complex scenarios such as for ballistic target tracking. To expand the applications of resource allocation algorithms in this context, this paper proposes a robust dwell time allocation algorithm for multiple ballistic reentry target (BRT) tracking in PAR. We aim to achieve an online dwell time allocation by establishing and solving optimization problems at each tracking interval. The motion of BRT is characterized by using the flat earth model, and the predicted Cramér-Rao lower bound (PCRLB) for BRT is derived to quantify the performance of tracking. By minimizing the weighted sum of time consumption of the PAR and cost of attaining the desired performance, a robust dwell time allocation approach is established. Finally, the superiority of the proposed algorithm is demonstrated through simulation.

SECTION I. Introduction

Ballistic reentry targets (BRTs) have garnered significant attention due to their importance to both military operations and safety of critical facilities [1]–​[4]. These targets typically include ballistic missiles, decoys, and space debris. To prevent these objects from causing economic losses and casualties upon reentry to earth, their accurate tracking has become a critical issue. To this end, existing researchers have attempted to improve tracking accuracy by developing more precise kinematic models, as shown in [3], [5]–​[9]. Additionally, studies in [1], [10], [11] have explored different tracking algorithms for BRTs, enhancing tracking accuracy by selecting the most effective algorithms. We note that the methods above rely on conventional sensor systems with fixed working modes or parameters. These works aim to reduce tracking errors by only refining the filtering process, freedom to manage the specific sensor types has not been utilized.

Meanwhile, the capability and flexibility of sensor systems have gained significant improvements due to the development of integrated circuit and signal processing techniques. Phased array radar (PAR), in particular, is widely used in multi-target tracking (MTT) due to its superior properties of transmitted beams and ability to manage time resources [12], [13]. Unlike traditional sensors and mechanical scanning radars, PAR implements MTT by synthesizing multiple agile tracking beams in different directions. PAR can freely control the dwell time at each beam position to manage the signal-to-noise ratio (SNR) of the echoes [14], [15]. The high degree of freedom for beam control has led to the development of PAR resource allocation methods, which aim to obtain the optimal resource allocation scheme to enhance the overall MTT performance. In this context, Wang et al. presented a joint revisit and dwell time management strategy for target tracking based on the PCRLB in PAR [16]. Yan et al. proposed a resource allocation algorithm to facilitate PAR's ability to manage track and search (TAS) tasks simultaneously [17]. Existing research has conducted in-depth and extensive discussions on the time management issue of phased array radar. However, they often employ simple target motion models, e.g., the constant velocity model [18], and have not yet considered the resource allocation for tracking complex moving targets such as multiple BRTs.

Given the importance of BRT tracking and the lack of suitable resource allocation researches on this topic, this paper proposes a robust dwell time allocation approach for PAR in the context of multiple BRT tracking. We employ the flat earth-based model to characterize the motion of BRT and derive the PCRLB for the model to quantify the performance. By minimizing the time consumption of the PAR while achieving the desired tracking performance, the robust dwell time allocation result is obtained.

SECTION II. System Description

A. System Configurations and Constraints

We focus on a PAR working in TAS mode that tracks multiple BRTs with fast beam agility. The radar achieves MTT by employing agile tracking beams at different time instants. Consequently, the primary challenge lies in the allocation of dwell time for each tracking beam to enhance the overall performance given the limited time resources. To simplify the problem, we assume that the number of BRTs is known in advance and does not exceed the maximum capacity of the radar. Track initiation has been implemented. Moreover, each BRT moves independently in the radar's surveillance region which means the multi-BRT tracking tasks can be divided into multiple independent single BRT tracking tasks.

Suppose that at the k-th frame, there exist Q point-like BRTs within the surveillance area of the PAR. The dwell time allocated to each BRT at the k-th frame is denoted as

$$\begin{equation*}{{\mathbf{D}}^k} = \left[ {d_1^k, \ldots ,d_Q^k} \right],\tag{1}\end{equation*}$$ where the dwell time allocated to each BRT satisfies $$\begin{equation*}{d_{{\text{min}}}} \leq d_q^k \leq {d_{{\text{max}}}}.\tag{2}\end{equation*}$$

In (2), the minimum d_min ensures the effectiveness of the pulse accumulation and the maximum d_max is constrained by physical constraints such as thermal dissipation and power consumption. [14], [19], [20].

Subsequently, since the total dwell time allocated to the tracking task per frame of the radar is limited, we define the maximum total dwell as D_max, which satisfies

$$\begin{equation*}\sum\nolimits_{q = 1}^Q {d_q^k \leq {D_{{\text{max}}}}} .\tag{3}\end{equation*}$$

B. Target Motion Models

During the reentry phase, the motion of BRTs can be simplified as being influenced solely by gravity and drag based on the flat earth model [1]. Consequently, we consider a classical kinematic model with known ballistic coefficient of BRT in flat earth approximation [3]. Defining the state vector of the q-th BRT at the k-th frame as ${\mathbf{s}}_q^k = {\left[ {x_q^k,\dot x_q^k,y_q^k,\dot y_q^k,z_q^k,\dot z_q^k} \right]^ \top }$, where "⊤" denotes matrix transpose, the kinematic model from the k-th frame to the k + 1-th frame can be expressed as

$$\begin{equation*}{\mathbf{s}}_q^{k + 1} = {\mathbf{Fs}}_q^k + {\mathbf{Ga}}\left( {{\mathbf{s}}_q^k} \right) + {\mathbf{v}}_q^k.\tag{4}\end{equation*}$$

In (4), F and G denote the state transition matrix and the control matrix, respectively. The term ${\mathbf{v}}_q^k\sim {\mathcal{N}}({\mathbf{0}},{\mathbf{Q}})$ is the process noise that follows a Gaussian distribution. The expressions of F, G and Q can be found in [1]. Coming back to (4), ${\mathbf{a}}\left( {{\mathbf{s}}_q^k} \right)$ is the control input vector given by

$$\begin{equation*}{\mathbf{a}}\left( {{\mathbf{s}}_q^k} \right) = \frac{{ - g\rho _q^k\sqrt {{{\left( {\dot x_q^k} \right)}^2} + {{\left( {\dot y_q^k} \right)}^2} + {{\left( {\dot z_q^k} \right)}^2}} }}{{2\beta }}\left[ {\begin{array}{l} {\dot x_q^k} \\ {\dot y_q^k} \\ {\dot z_q^k} \end{array}} \right] - \left[ {\begin{array}{l} 0 \\ 0 \\ g \end{array}} \right],\tag{5}\end{equation*}$$ where g is gravitational acceleration, β denotes ballistic coefficient. The term $\rho _q^k = {c_1}\exp \left[ { - {c_2}z_q^k} \right]$ signifies air density with c₁ and c₂ varying with altitude [1], [10], [21].

C. Radar Measurement Model

The extraction of target measurement is accomplished through threshold detection [22], [23], in which the measurement of the q-th BRT at the k-th frame can be expressed as:

$$\begin{equation*}{\mathbf{z}}_q^k = {\mathbf{h}}\left( {{\mathbf{s}}_q^k} \right) + {\mathbf{n}}_q^k = {\left[ {\begin{array}{llll} {R_q^k}&{\theta _q^k}&{\phi _q^k}&{f_q^k} \end{array}} \right]^ \top } + {\mathbf{n}}_q^k.\tag{6}\end{equation*}$$

In (6), $R_q^k,\theta _q^k,\phi _q^k$ and $f_q^k$ represent radial distance, azimuth, elevation and Doppler shift, respectively. Their expressions are given in [19], [24], [25]. As indicated in [16], [17], [26], the measurement noise ${\mathbf{n}}_q^k\sim {\mathcal{N}}\left( {{\mathbf{0}},{\mathbf{\Sigma }}_q^k} \right)$ and the covariance ${\mathbf{\Sigma }}_q^k$ are dependent on the corresponding dwell time $d_q^k$:

$$\begin{equation*}{\mathbf{\Sigma }}_q^k \propto {\text{diag}}\left\{ {\frac{1}{{d_q^k{\beta ^2}}},\frac{{{W_\theta }}}{{d_q^k}},\frac{{{W_\varphi }}}{{d_q^k}},\frac{1}{{d_q^kT_e^2}}} \right\},\tag{7}\end{equation*}$$ where β is signal effective bandwidth and T_e is the effective time duration. The terms W_θ and W_φ are the width of the receiving beam in azimuth and elevation, respectively.

Let's come back to Eq. (7), increasing the dwell time $d_q^k$ reduces the diagonal element of ${\mathbf{\Sigma }}_q^k$ and enhances the accuracy of the measurement ${\mathbf{z}}_q^k$, thereby improving the tracking performance of BRT q. However, the dwell time is constrained by the radar system, as indicated in (2) and (3). Consequently, this paper focuses on optimally allocating the beam dwell time to each BRT within these system constraints to enhance the overall multiple BRT tracking performance.

SECTION III. Performance Metric and State Estimation for BRT Tracking

A. Dwell-Time-Aware Metric for BRT Tracking Performance

In previous studies, PCRLB has been widely utilized as a metric for evaluating tracking accuracy. Theoretically, the mean square error (MSE) of the target state estimate should not be statistically less than its PCRLB [27], [28]:

$$\begin{equation*}{E_{{\mathbf{s}}_q^k}}\left[ {\left( {{\mathbf{\hat s}}_q^k - {\mathbf{s}}_q^k} \right){{\left( {{\mathbf{\hat s}}_q^k - {\mathbf{s}}_q^k} \right)}^ \top }} \right] \geq {{\mathbf{J}}^{ - 1}}\left( {{\mathbf{s}}_q^k|d_q^k} \right),\tag{8}\end{equation*}$$ where ${\mathbf{\hat s}}_q^k$ represents an unbiased estimate of the target state ${\mathbf{s}}_q^k$. The matrix operation A ≽ B signifies that A – B is a positive semidefinite matrix. The term ${\mathbf{J}}\left( {{\mathbf{s}}_q^k|d_q^k} \right)$ is the dwell-time-aware Fisher information matrix (FIM), whose inverse offers a lower bound on the error variance of unbiased estimates [19], [29]. According to [27], the expression of FIM is defined as: $$\begin{equation*}{\mathbf{J}}\left( {{\mathbf{s}}_q^k|d_q^k} \right) = {{\mathbf{J}}^P}\left( {{\mathbf{s}}_q^k} \right) + {{\mathbf{J}}^Z}\left( {{\mathbf{s}}_q^k|d_q^k} \right),\tag{9}\end{equation*}$$ where ${{\mathbf{J}}^P}\left( {{\mathbf{s}}_q^k} \right)$ and ${{\mathbf{J}}^Z}\left( {{\mathbf{s}}_q^k|d_q^k} \right)$ are the prior information matrix and data information matrix, respectively [1]: $$\begin{align*} & {{\mathbf{J}}^P}\left( {{\mathbf{s}}_q^k} \right) \approx {\left[ {{\mathbf{Q}} + {\mathbf{\hat M}}_q^k{\mathbf{J}}{{\left( {{\mathbf{s}}_q^{k - 1}|d_q^{k - 1}} \right)}^{ - 1}}{{\left( {{\mathbf{\hat M}}_q^k} \right)}^ \top }} \right]^{ - 1}},\tag{10} \\ & {{\mathbf{J}}^Z}\left( {{\mathbf{s}}_q^k|d_q^k} \right) = {\left( {{\mathbf{\tilde H}}_q^k} \right)^ \top }{\left( {{\mathbf{\Sigma }}_q^k} \right)^{ - 1}}{\mathbf{\tilde H}}_q^k,\tag{11}\end{align*}$$ with $$\begin{equation*}{\mathbf{\hat M}}_q^k = {\mathbf{F}} + {\left. {{\mathbf{G}}\frac{{d{\mathbf{a}}\left( {\mathbf{s}} \right)}}{{d{\mathbf{s}}}}} \right|_{{\mathbf{s}} = {\mathbf{\hat s}}_q^k}},{\mathbf{\tilde H}}_q^k = {\left. {\frac{{d{\mathbf{h}}\left( {\mathbf{s}} \right)}}{{d{\mathbf{s}}}}} \right|_{{\mathbf{s}} = {\mathbf{\tilde s}}_q^k}}.\tag{12}\end{equation*}$$

The term ${\mathbf{\tilde s}}_q^k$ is the state prediction of ${\mathbf{s}}_q^k$. Subsequently, we choose the square root of PCRLB's trace as the metric [19], [30]:

$$\begin{equation*}\xi _q^k\left( {d_q^k} \right) = \sqrt {{\text{Tr}}\left[ {{\mathbf{\Lambda }}{{\mathbf{J}}^{ - 1}}\left( {{\mathbf{s}}_q^k|d_q^k} \right){{\mathbf{\Lambda }}^ \top }} \right]} ,\tag{13}\end{equation*}$$ where "Tr" represents the trace operation, and the term Λ is the normalization matrix [30], [31].

B. State Estimation for BRT Tracking

Here, extended Kalman filter (EKF) is utilized to implement the state estimation for BRT tracking [18]. The filtering performance is governed by the echo SNR as indicated in (7), which is positively correlated with dwell time. Referring back to Eq. (13), the metric $\xi _q^k\left( {d_q^k} \right)$ is similarly controlled by $d_q^k$. Therefore, the allocation of dwell time has a precise impact on the target tracking accuracy.

SECTION IV. Robust Dwell Time Allocation

As mentioned above, dwell time can affect tracking accuracy of a single BRT. For this reason, we present a robust dwell time allocation algorithm for multiple BRTs tracking in this section. The time allocation problem is formulated as a set of optimization problems by minimizing the utility function that involves the dwell time at each tracking interval. Then, an external point penalty function-based algorithm is proposed to obtain a robust solution that is applicable to both scenarios of sufficient and insufficient dwell time budget.

A. Utility Function Based on Overall System Performance

In practical scenarios, each target often has its own unique tracking accuracy requirement [19], [32]. Specifically, different types of targets necessitate varying levels of tracking accuracy; for example, lower accuracy is acceptable for bait, whereas higher accuracy is required for warheads.

To ensure that as many targets as possible satisfy their respective accuracy requirements, efficient resource usage is desirable [33]. The principle aim is to conserve system resources while attaining the desired tracking performance of each BRT. In this context, we denote the total dwell time as the utility function to quantify the overall system performance:

$$\begin{equation*}{{{\Phi }}^k}\left( {{{\mathbf{D}}^k}} \right) = \sum\nolimits_{q = 1}^Q {d_q^k} ,\tag{14}\end{equation*}$$ and the tracking accuracy requirements are defined as $$\begin{equation*}{{\mathbf{\eta }}^k} = \left[ {\eta _1^k, \ldots ,\eta _Q^k} \right],\tag{15}\end{equation*}$$ where $\eta _q^k$ denotes the tracking accuracy requirement of the q-th BRT at the k-th frame. When $\xi _q^k\left( {d_q^k} \right) \leq \eta _q^k$, it indicates that the q-th BRT meets its tracking accuracy requirements.

B. Dwell Time Allocation Problem Formulation

Combining system constraints and the utility function, the dwell time allocation problem for multiple BRT tracking in PAR can be formulated as follows:

$$\begin{equation*}\begin{array}{ll} {\mathop {\min }\limits_{{{\mathbf{D}}^k}} }&{\left[ {{\Phi ^k}\left( {{{\mathbf{D}}^k}} \right)} \right]} \\ {{\text{s}}.{\text{t}}.}&{{d_{\min }} \leq d_q^k \leq {d_{\max }}{\text{,}}} \\ {}&{\sum\nolimits_{q = 1}^Q {d_q^k} \leq {D_{\max }},} \\ {}&{\xi _q^k\left( {d_q^k} \right) \leq \eta _q^k,\forall q = 1, \ldots ,Q.} \end{array}\tag{16}\end{equation*}$$

By solving this problem, we can get the optimized dwell time allocation strategy, so as to guide beamforming and multi-target tracking of the k-th frame.

TABLE I Initial States of BRTs

C. Robust Solution Based on External Point Penalty Function

In order to solve the problem in (16), some studies employ the interior point penalty function method [16], [20], which ensures that the tracking accuracy strictly satisfies the accuracy requirements. However, when the target accuracy requirements (the 3-rd constraint) cannot be met due to exhausted resources, the system may fail to produce feasible resource allocation results, indicating a lack of robustness in the system [29]. To address this issue, we utilize an external point penalty function to transform the utility function [34], denoted as

$$\begin{equation*}{{{\Gamma }}^k}\left( {{{\mathbf{D}}^k}} \right) = {{{\Phi }}^k}\left( {{{\mathbf{D}}^k}} \right) + \sigma \sum\nolimits_{q = 1}^Q {{\text{max}}} \left\{ {\frac{{\xi _q^k\left( {d_q^k} \right)}}{{\eta _q^k}},1} \right\},\tag{17}\end{equation*}$$ where σ is penalty factor. After this transformation, the problem in (16) can be rewritten as $$\begin{equation*}\begin{array}{ll} {\mathop {{\text{min}}}\limits_{{{\mathbf{D}}^k}} }&{\left[ {{{{\Gamma }}^k}\left( {{{\mathbf{D}}^k}} \right)} \right]} \\ {{\text{s}}.{\text{t}}.}&{{d_{{\text{min}}}} \leq d_q^k \leq {d_{{\text{max}}}},\forall q = 1, \ldots ,Q} \\ {}&{\sum\nolimits_{q = 1}^Q {d_q^k \leq {D_{{\text{max}}}}} } \end{array}\tag{18}\end{equation*}$$

Obviously, when σ is sufficiently large, the system will allocate as much dwell time as possible to the targets whose tracking accuracy does not meet the requirements, ensuring they achieve the required accuracy as quickly as possible. Conversely, for targets that already meet the accuracy requirements, the system conserves resources. This approach results in a robust dwell time allocation strategy suitable for complex real-world scenarios. Finally, as indicated in [14], [25], this problem is convex, allowing the use of optimization techniques such as gradient descent and Newton's method for its solution. In this work, we employ CVX tools to solve the problem [35].

SECTION V. Simulation Experiments

In this section, we examine the effectiveness of the proposed algorithm with simulation experiments. Firstly, a PAR multiple BRT tracking scenario is established. The location of radar is set to (100,100,0) km. In its surveillance area, there are 3 BRTs moving independently. Their initial states are shown in Table I and simulation parameters are given in Table II. The total number of tracking frames is 50 . Based on the above parameters, the deployment of the BRTs and radar nodes is shown in Fig. 1.

Subsequently, we divide the surveillance area into two distinct parts (Areas 1 and 2) based on the radial distance between the radar and the BRTs. When the distance is less than 90 km (Area 1), the accuracy requirement is 50 m (Accuracy requirement 1). When the distance exceeds 90 km (Area 2), the accuracy requirement is 100 m (Accuracy requirement 2). This partitioning is represented by the hemispherical surface shown in Fig. 1 (a). Throughout the simulation process, BRT 1 remains inside Area 1, BRT 3 remains outside Area 2, and BRT 2 moves from Area 2 to Area 1 in the 21st frame. Subsequently, the root mean square error (RMSE) of target tracking is calculated by Monte Carlo experiments, whose detailed expression can be found in [19].

TABLE II Simulation parameters

Fig. 1.

The deployment of three BRTs and the radar. (a) Scenario in flat earth model; (b) Top view of the scenario.

Setting the number of Monte Carlo experiments as 1000, the multiple BRT tracking performance is illustrated in Fig. 2, where the uniform distribution algorithm is utilized for comparison. This algorithm equally allocates total dwell time resources to each target. From Fig. 2(a), we can find that the PCRLB of BRT 1 remains below its tracking accuracy requirement (50 m) from the start until the 45 th frame, with a similar behavior for BRT 3 after the 34th frame. This results in wasted resources that could have been better utilized for other targets, e.g., enabling BRT 3 to meet its accuracy requirement (100 m) throughout the entire process. In brief, the uniform distribution algorithm fails to control the tracking accuracy of each target, leading to significant discrepancies between the tracking accuracy and its corresponding requirements, thereby wasting a portion of resources.

Conversely, the proposed algorithm is always driving the tracking accuracy close to the accuracy requirements as shown in Fig. 2(b). In order to further demonstrate the superiority of the proposed algorithm, we denote normalized dwell time as $d_q^k/{d_{{\text{max}}}}$ and dwell time utilization rate as Φ^k(D^k)/D_max and illustrate the dwell time allocation results of the proposed algorithm in Fig. 3. With the combination of Fig. 2(b) and Fig. 3, we can find that before the 13th frame, the PCRLB of BRTs 1 and 2 do not meet the requirements initially. During this period, system resources are insufficient. To ensure that tracking accuracy converges as quickly as possible, the system utilizes all available resources, resulting in a dwell time utilization rate equal to 1 . A similar situation occurs between frames 21 and 30. During this time, due to changes in the accuracy requirements of BRT 2 , resources become insufficient again. Moreover, as shown in Fig. 3(a), due to the influence of distance, after the 30th frame, the dwell time allocated to BRTs 1 and 3 increases to maintain tracking accuracy, while the dwell time allocated to BRT 2 decreases. Finally, it is worth noting that at the 21 st frame, the algorithm preferentially meets the accuracy requirements of BRTs 1 and 2, while slightly sacrificing the tracking accuracy of BRT 3. This is closely related to the design of the transform utility function [19], [29], which satisfy the tracking accuracy of targets with higher requirements with priority. Overall, throughout the entire tracking process, the proposed algorithm consistently achieves robust dwell time allocation results.

Fig. 2.

Tracking PCRLBs and RMSEs of three BRTs. (a) The uniform distribution algorithm; (b) The proposed algorithm.

Fig. 3.

Dwell time allocation results. (a) Normalized dwell time; (b) Dwell time utilization rate.

SECTION VI. Conclusion and Future Work

In this paper, the resource scheduling algorithm was applied to the multiple BRT tracking scenario, and a robust dwell time allocation algorithm for PAR was proposed. This algorithm achieves robust results under both insufficient and sufficient resource conditions, as demonstrated by simulation experiments. In future work, we will consider more complex ballistic models and further explore the practical application of resource scheduling algorithms in multi-sensor network.