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Joint Optimization of Radar and Jammer Space-time Cooperative Beamforming for a Multitasking Dynamic Scene
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摘要: 现代雷达对抗形势复杂多变，体系与体系的作战已成为基本特点，而体系整体性能关乎着战场的主动权乃至最终的胜负。通过优化体系中雷达与干扰波束资源可以提升整体性能，获得在空间、时间域优效的低截获探测性能，然而空时域协同波束联合优化是一个复杂多参数耦合的非凸问题。该文针对空时域多任务动态场景，建立了以雷达探测性能为优化目标，以干扰性能以及能量限制为约束条件的优化模型。为求解该模型，该文提出了基于迭代优化的空时协同波束联合设计方法，即以雷达发射、接收、多干扰机发射波束交替迭代优化。其中，针对多干扰机协同优化的不定矩阵二次约束二次规划(QCQP)问题，该文基于可行点追踪-连续凸逼近(FPP-SCA)算法，在SCA算法的基础上，通过引入松弛变量与惩罚项，保证算法在合理松弛度下的可行性，解决了矩阵不定情况下难以获得可行解的问题。仿真表明，在一定的干扰机能量约束下，该文所提方法在保证雷达高性能探测目标且不受干扰情况下，同时实现了多干扰机在空时域干扰对方每个平台以掩护我方雷达探测的效果；相比传统算法，在动态场景中基于FPP-SCA算法的协同干扰具有更优效果。
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关键词:
- 雷达与干扰资源 /
- 空时波束联合优化 /
- 多参数耦合非凸问题 /
- 迭代优化 /
- 可行点追踪-连续凸逼近(FPP-SCA)
Abstract: The modern radar confrontation situation is complex and changeable, and inter-system combat has become a basic feature. The overall system performance affects the initiative on the battlefield and even the final victory or defeat. By optimizing the beam resources of radar and jammers in a system, the overall performance can be improved, and the effective low-intercept detection effect can be obtained in the spatial and temporal domains. However, joint optimization of cooperative beamforming in the spatial and temporal domains is a nonconvex problem with complex multiparameter coupling. In this paper, an optimization model is established for a multitasking dynamic scene in the spatial and temporal domains. Radar detection performance is the optimization goal, while the interference performance and energy limitation of jammers are the constraints. To solve the model, a joint design method of space-time cooperative beamforming based on iterative optimization was proposed; that is, iterative optimization of radar transmitting, receiving, and multiple jammers transmitting beamforming vectors was alternately optimized. To solve the Quadratically Constrained Quadratic Programs (QCQP) problem with indefinite matrices for multijammer collaborative optimization, this paper is based on the Feasible Point Pursuit Successive Convex Approximation (FPP-SCA) algorithm. In other words, on the basis of the SCA algorithm, algorithm feasibility is ensured through reasonable relaxation by introducing relaxation variables and a penalty term, which solves the difficulty of obtaining a feasible solution when a problem contains indefinite matrices. Simulation results show that under the constraint of certain jammer energy, the proposed method achieves the effect of multiple jammers interfering with each enemy platform in the spatial and temporal domains to cover our radar detection. This effect is achieved while ensuring high-performance radar detection of the target without interference. Compared with traditional algorithms, the collaborative interference based on the FPP-SCA algorithm exhibits a better performance in the dynamic scene. -
图 2 基于迭代的波束资源联合优化算法总体框架
Figure 2. The overall framework of the joint optimization algorithm of beamforming resources based on iteration
图 3 各平台相对地面的位置及运动路线
Figure 3. The position and movement route of each platform relative to the ground
图 4 各平台相对雷达的位置及运动路线
Figure 4. The position and movement route of each platform relative to the radar
图 5 各时刻的目标函数优化过程
Figure 5. The optimization process of the objective function at each moment
图 7 各时刻干扰机发射波束权矢量(功率归一化)
Figure 7. Transmit beamforming vectors of jammers at each moment (power normalization)
图 8 侦察平台各角度上的接收功率
Figure 8. The received power of the reconnaissance platforms at all angles
图 10 对方平台在干扰机方向的接收功率
Figure 10. The received power of the opposing platforms in the direction of the jammers
图 11 雷达探测目标的信干噪比
Figure 11. The signal-to-interference plus noise ratio of the target detected by our radar
图 13 对方平台在各方向上的接收功率
Figure 13. The received power of the opposing platforms at all angles
1 基于FFP-SCA算法的干扰机发射波束权矢量优化过程
1. Optimization process of jammers’ transmit beamforming vectors based on FFP-SCA algorithm
步骤1 初始化： 设迭代次数$q = 0$，初始化M维向量${{\boldsymbol{z}}_0}$； 步骤2 循环执行： (1) 求解下式凸优化问题： $ \begin{gathered} \mathop {\min }\limits_{{{\boldsymbol{w}}_{{{\mathrm{t}}}}},{\boldsymbol{s}}} {\text{ }}{\boldsymbol{w}}_{\rm{t}}^{\rm{H}}{\boldsymbol{H}}{{\boldsymbol{w}}_{\text{t}}} + \mu \sum\limits_{m = 1}^{K + L + N} {{s_m}} \\ {\text{s}}{\text{.t}}{\text{.}}\left\{ \begin{aligned} & {{\text{ }}2{{\mathrm{Re}}} \left\{ {{\boldsymbol{z}}_q^{\text{H}}( - {{\boldsymbol{P}}_k}){{\boldsymbol{w}}_{\rm{t}}}} \right\} \le - {\beta _k} + {\boldsymbol{z}}_q^{\text{H}}( - {{\boldsymbol{P}}_k}){{\boldsymbol{z}}_q} + {s_k},{\text{ }}k = 1,2, \cdots ,K} \\ & {{\text{ }}2{{\mathrm{Re}}} \left\{ {{\boldsymbol{z}}_q^{\text{H}}( - {{\boldsymbol{Q}}_l}){{\boldsymbol{w}}_{\rm{t}}}} \right\} \le - {\beta _l} + {\boldsymbol{z}}_q^{\text{H}}( - {{\boldsymbol{Q}}_l}){{\boldsymbol{z}}_q} + {s_{K + l}}{\text{, }}l = 1,2, \cdots ,L} \\ & {{\text{ }}{\boldsymbol{w}}_{\rm{t}}^{\rm{H}}{{\boldsymbol{R}}_n}{{\boldsymbol{w}}_{\rm{t}}} \le {\rho _n} + {s_{K + L + n}},{\text{ }}n = 1,2, \cdots ,N} \\ & {{\text{ }}{s_m} \ge 0,{\text{ }}m = 1,2, \cdots ,K + L + N} \end{aligned} \right.\end{gathered} $ 得到当前最优解$ {\boldsymbol{w}}_{\rm{t}}^{\text{*}} $； (2) 将第q次得到的$ {\boldsymbol{w}}_{\rm{t}}^{\text{*}} $赋值给${{\boldsymbol{z}}_{q + 1}}$； (3) 令$q = q + 1$； 直到目标函数收敛，循环结束。 
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2 基于迭代的空域波束资源联合优化算法
2. Joint optimization algorithm of beamforming resources based on iterative optimization in the spatial domain
步骤1 初始化： 设置场景数据(如平台位置、阵元个数、阵元间距等)，给定$ {{\boldsymbol{w}}_{o{\rm t}}} $, $ {{\boldsymbol{w}}_{n{\rm t}}} $初值； 步骤2 执行循环： (1) 求解式(17)，得到当前最优解$ {\boldsymbol{w}}_{o{\rm r}}^{\text{*}} $：对矩阵${{\boldsymbol{B}}^{ - 1}}{\boldsymbol{A}}$进行特征值分解，最大特征值对应的特征向量即所求$ {\boldsymbol{w}}_{o{\rm r}}^{\text{*}} $； (2) 求解如式(22)所示的凸优化问题，得到当前最优解${\boldsymbol{w}}_{o{\rm t}}^{\text{*}}$； (3) 根据算法1，得到当前最优${\boldsymbol{w}}_{n{\rm t}}^{\text{*}}$； 直到目标函数收敛和优化变量收敛，循环结束； 步骤3 输出结果： 最终波束权矢量$ {\boldsymbol{w}}_{o{\rm r}}^{\text{*}} $, $ {\boldsymbol{w}}_{o{\rm t}}^{\text{*}} $和$ {\boldsymbol{w}}_{n{\rm t}}^{\text{*}} $的值即为波束资源联合优化的最优结果。
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3 基于迭代的空时协同波束资源联合优化方法
3. Joint optimization method of space-time cooperative beamforming resources based on iterative optimization
步骤1 初始化： 设置场景数据(如平台运动轨迹、阵元个数、阵元间距等)，给定$ {{\boldsymbol{w}}_{o{\rm t}}}(\tau ) $, $ {{\boldsymbol{w}}_{n{\rm t}}}(\tau ) $初值。 步骤2 执行循环： (1) 固定当前$ {{\boldsymbol{w}}_{o{\rm t}}}(\tau ) $, $ {{\boldsymbol{w}}_{n{\rm t}}}(\tau ) $，化简并求解式(2)，得到最优$ {{\boldsymbol{w}}_{o{\rm r}}^*}(\tau ) $； (2) 固定当前$ {{\boldsymbol{w}}_{o{\rm r}}}(\tau ) $, $ {{\boldsymbol{w}}_{n{\rm t}}}(\tau ) $，化简并求解式(2)，得到最优$ {{\boldsymbol{w}}_{o{\rm t}}^*}(\tau ) $； (3) 固定当前$ {{\boldsymbol{w}}_{o{\rm r}}}(\tau ) $, $ {{\boldsymbol{w}}_{o{\rm t}}}(\tau ) $，采用FPP-SCA算法求解式(35)，得到最优$ {{\boldsymbol{w}}_{n{\rm t}}^*}(\tau ) $； 直到目标函数和优化变量收敛，循环结束； 步骤3 输出结果： 当前波束权矢量$ {\boldsymbol{w}}_{o{\rm r}}^ * (\tau ) $, $ {\boldsymbol{w}}_{o{\rm t}}^ * (\tau ) $和$ {\boldsymbol{w}}_{n{\rm t}}^ * (\tau ) $的值即为运动平台的波束资源联合优化最优结果。
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表 1 各时刻的平台坐标(km)
Table 1. The coordinates of each platform at each time (km)
平台 $\tau = 1$时刻 $\tau = 2$时刻 $\tau = 3$时刻 $\tau = 4$时刻 目标 (5,100) (10,100) (15,100) (20,100) 对方雷达 (10,75) (16,70) (22,65) (28,60) 侦察平台1 (60,90) (60,85) (60,80) (60,75) 侦察平台2 (60,60) (60,55) (60,50) (60,45) 我方雷达 (0,0) 干扰机1 (–3,20) 干扰机2 (12,10) 干扰机3 (20,10)
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