In recent years, the rapid development of fixed-wing and rotary-wing small tactical unmanned aerial vehicles (UAVs) has posed significant challenges to air defense systems. The emergence of swarm UAV attacks has become a major threat to existing anti-aircraft defenses, demanding more efficient and intelligent target assignment strategies. Traditional anti-aircraft guns often struggle with low combat effectiveness when engaging swarm UAVs, and evaluating target assignment schemes remains complex. This article addresses these issues by proposing a target assignment model for anti-aircraft guns intercepting swarm UAVs, based on a hybrid simulated annealing-discrete particle swarm optimization (SA-DPSO) algorithm. The model establishes a comprehensive evaluation index system for anti-UAV target assignment, optimizing damage probability while considering fire resource consumption. Through instance analysis, the rationality and effectiveness of the model are validated, providing a decision-making basis for efficient anti-UAV operations.
The battlefield situation in anti-UAV warfare requires commanders to perceive both friendly and enemy equipment statuses, understand the environment, and implement dynamic solving strategies. Thus, battlefield situation awareness, estimation, and prediction are crucial for gaining informational advantage. In practical scenarios, commanders must integrate battlefield态势 to formulate打击 strategies. Based on damage database predictive capabilities, effectively acquiring battlefield态势 and understanding equipment status are prerequisites for implementing打击 strategies. Under this premise, optimization methods are used for evaluation to derive optimal打击 strategies, guiding battlefield operations as the ultimate goal of the damage assessment system.
Different打击 strategies yield varying damage efficiencies. For a specific target, selecting the appropriate打击 involves using various operational characteristic data as independent variables in optimization methods. By optimizing within a certain range, the optimal value is obtained, leading to the best打击 strategy. Extending this to multiple targets, optimization iterations can be performed among them, resulting in optimal打击 decisions based on the damage assessment system and optimization strategy. This approach enhances the overall anti-UAV capability, particularly in countering swarm无人机 threats.
Air defense target assignment is a critical component of command and control, directly influencing combat effectiveness. The goal is to rationally and scientifically assign incoming UAVs to various air defense fire units, maximizing the system’s operational效能. This process involves information processing, threat assessment, data fusion, and comprehensive battlefield situational awareness to achieve target assignment and battlefield指挥. In the context of anti-UAV operations, this becomes even more complex due to the high mobility and swarm nature of UAVs.
To address this, we first consider the threat assessment of UAV targets. Threat analysis in air defense is a multi-factor decision-making problem, involving factors such as the location and importance of defended targets, the type, number, altitude, speed, and approach direction of aerial targets, and the distance and course捷径 from aerial targets to defended targets. Let $A = [a_1, a_2, \ldots, a_j]$ be the set of UAV target schemes, and $U = [u_1, u_2, \ldots, u_m]$ be the set of evaluation indicators. The threat decision matrix for UAV target $a_j$ based on indicator $u_m$ can be derived, leading to the threat matrix $[w_1, w_2, \ldots, w_j]$ for the UAV targets. This matrix is essential for prioritizing targets in anti-UAV engagements.
Next, we examine the firing shift cycle. Shift firing refers to the method where an anti-aircraft gun, after completing fire on one UAV entering the defense zone, redirects its barrel to engage other UAVs based on instructions. The shift firing cycle $t_{zh}$ is the time interval from when the fire control computer receives the firing command for one target to when it receives the command for the next target after shifting. It depends on the current firing time $t_{sj}$ and the shift preparation time $t_{zb}$. The firing time $t_{sj}$ is fixed and includes the time from receiving the firing command to shell impact and damage assessment. It is given by:
$$ t_{sj} = t_{fs} + (n – 1) t_{ls} + t_{pg}, $$
where $t_{fs}$ is the time from receiving the firing command to shell exit, $t_{ls}$ is the total time interval for firing multiple shells in a burst, and $t_{pg}$ is the shell flight time and damage assessment time. After a single firing, the shift preparation time $t_{zb}$ includes the time for issuing shift commands and barrel redirection, typically around 3 seconds. Thus, the final shift cycle for the anti-aircraft gun is:
$$ t_{zh} = t_{fs} + (n – 1) t_{ls} + t_{pg} + t_{zb}. $$
This cycle is critical in anti-UAV operations, as it affects the timeliness of target assignment and overall system responsiveness.
The fire allocation zone for anti-aircraft guns is another key aspect. During anti-swarm UAV operations, accurately timing target assignment is vital. Assigning too early may increase computational load due to target maneuvers, while too late may reduce damage effectiveness. The fire allocation zone defines the spatial area where assigned targets and shells meet within the kill zone. The far boundary of the fire allocation zone is:
$$ \rho_{yj} = \left[ \rho_{\text{max}}^2 + v^2 (t_{zh} – t_{pg})^2 + 2v(t_{zh} – t_{pg}) \sqrt{\rho_{\text{max}}^2 – h^2 – p^2} \right]^{\frac{1}{2}}, $$
and the near boundary is:
$$ \rho_{jj} = \left[ \rho_{\text{min}}^2 + v^2 (t_{zh} – t_{pg})^2 + 2v(t_{zh} – t_{pg}) \sqrt{\rho_{\text{min}}^2 – h^2 – p^2} \right]^{\frac{1}{2}}, $$
where $\rho_{\text{max}}$ and $\rho_{\text{min}}$ are the far and near boundaries of the gun’s damage zone, respectively, $h$ is the incoming UAV’s altitude, $p$ is the course捷径, and $v$ is the UAV’s speed. These formulas help optimize target assignment by ensuring engagements occur within effective ranges.
The damage probability calculation for UAVs is fundamental to evaluating anti-UAV effectiveness. For an anti-aircraft gun system, the damage probability $P$ is the product of the hit probability $P_{mz}$ and the conditional destruction probability $P_{jh}$ given a hit:
$$ P = P_{mz} P_{jh}. $$
In practice, these probabilities are often assumed independent for simplicity. Assuming each gun’s hit events are independent, with destruction probability $p_i$ for $i = 1, 2, \ldots, m$, the overall damage probability for multiple guns engaging a target is:
$$ G(k) = 1 – \prod_{i=1}^{m} (1 – p_i). $$
This formula is used to assess the cumulative effect of multiple anti-aircraft guns on a single UAV target, which is common in anti-swarm scenarios.
Building on these concepts, we establish the target assignment model. In air defense operations, optimized fire allocation principles prioritize fire units with high damage probability and threat index, while considering fire resource consumption. Assume $m$ anti-aircraft guns and $n$ incoming UAV targets in a batch. Let $w_j$ be the threat degree of the $j$-th UAV target, $p_{ij}$ be the damage probability of the $i$-th gun against target $j$, and $x_{ij}$ be the decision variable (1 if assigned, 0 otherwise). The damage probability for UAV target $j$ is:
$$ P_j = 1 – \prod_{i=1}^{m} (1 – p_{ij})^{x_{ij}}. $$
The allocation效能 $B(x)$ is maximized as:
$$ B(x) = \max \sum_{j=1}^{n} w_j \left[ 1 – \prod_{i=1}^{m} (1 – p_{ij})^{x_{ij}} \right]. $$
However, in reality, besides minimizing threat, we must balance weapon usage代价 to avoid excessive consumption. Therefore, we refine the objective function by introducing a satisfaction function $\mu(P_i)$ that peaks at an ideal damage probability. When damage probability is below the ideal, satisfaction increases slowly; above, it decreases rapidly. The enhanced objective function is:
$$ B_1(x) = \max \sum_{j=1}^{n} w_j \left[ 1 – \prod_{i=1}^{m} (1 – p_{ij})^{x_{ij}} \right] \cdot \mu(P_j). $$
Constraints include shift target数量限制:
$$ \sum_{j=1}^{n} x_{ij} \leq M_i, \quad i = 1, 2, \ldots, m, $$
and single UAV占用 gun数量约束:
$$ 0 < \sum_{i=1}^{m} x_{ij} \leq N \leq m, \quad j = 1, 2, \ldots, n. $$
The shift time condition is mathematically described as:
$$ \sum_{j_1=1}^{n} \sum_{j_2=1}^{n} x_{ij_1} x_{ij_2} t^{j_1 j_2}_i = \frac{\sum_{j=1}^{n} x_{ij} \left( \sum_{j=1}^{n} x_{ij} – 1 \right)}{2}, $$
where $j_1$ and $j_2$ are any two incoming UAV targets, and $t^{j_1 j_2}_i$ indicates whether the time difference $\Delta t^{j_1 j_2}_i$ between targets $j_1$ and $j_2$ entering the fire range of gun $i$ is greater than the shift time $t_{zh}$. If $|\Delta t^{j_1 j_2}_i| \geq t_{zh}$, then $t^{j_1 j_2}_i = 1$, meaning the gun can sequentially engage both UAVs; otherwise, $t^{j_1 j_2}_i = 0$. This ensures practical feasibility in anti-UAV engagements.
To solve this target assignment problem with nonlinear constraints, we employ a hybrid intelligent optimization algorithm. Penalty functions are used to convert constrained problems into unconstrained ones, and simulated annealing (SA) is integrated into discrete particle swarm optimization (DPSO) to enhance global search efficiency and avoid local optima. The discrete particle swarm optimization (DPSO) algorithm, introduced by Kennedy and Eberhart in 1997, maps discrete problem spaces to continuous particle motion spaces, retaining the iterative rules of classic PSO. In DPSO, particle values are limited to 0 or 1, and velocity represents the likelihood of a position取值为 1. However, DPSO can suffer from local optima, so we incorporate SA to temporarily accept suboptimal solutions with probability control, enabling detailed local searches and improving convergence.
The SA-DPSO hybrid algorithm features a双层 parallel structure. It uses DPSO results as the initial population for SA, and SA solutions obtained via Metropolis sampling as the next generation’s initial population. This combination ensures optimization quality while enhancing efficiency. The SA-DPSO algorithm流程 is as follows:
- Initialize parameters: inertia weight $\omega$, learning factors $c_1$ and $c_2$, population size $R$, maximum iterations $k_{\text{max}}$, and randomly generate $R$ particles.
- Calculate each particle’s fitness $G_r$, compare with individual best $p_{\text{best}}$, and update if better.
- Compare individual best $p_{\text{best}}$ with global best $g_{\text{best}}$, and update if better.
- If termination conditions are met, end; otherwise, proceed to step 5.
- Execute SA algorithm:
- Initialize SA parameters: initial temperature $T$ and iterations $L$ per $T$ value.
- For $k = 1, 2, \ldots, L$, execute steps 3-5.
- Generate new solution $x’_r$, compute $\Delta f = G_r – G’_r$, where $G’_r$ is the new solution’s fitness.
- If $\Delta f < 0$, accept $x’_r$, setting $x_r(k+1) = x’_r$; otherwise, accept with probability $\exp(-\Delta f / T)$.
- If termination conditions are met, output current solution as optimal and end.
- Lower temperature by convergence rate $\alpha$: $T(k+1) = \lambda T(k)$. If $T \geq 0$, return to step 2; otherwise, end.
This hybrid approach effectively balances exploration and exploitation in anti-UAV target assignment optimization.
To validate the model and algorithm, we conduct a simulation实例. Consider 9 S-70 UAVs入侵 the defense zone sequentially, engaged by 6 anti-aircraft guns at different positions. Each gun performs a single burst against匀速直线运动 targets. The threat indicators for the 9 UAVs and the damage probabilities for each gun are shown in Table 1.
| Incoming Target | Threat Degree | Damage Probability (Gun 1 to 6) | |||||
|---|---|---|---|---|---|---|---|
| 1 | 0.91 | 0.41 | 0.29 | 0.81 | 0.73 | 0.27 | 0.38 |
| 2 | 0.58 | 0.75 | 0.38 | 0.86 | 0.86 | 0.34 | 0.49 |
| 3 | 0.49 | 0.56 | 0.22 | 0.76 | 0.45 | 0.25 | 0.30 |
| 4 | 0.22 | 0.53 | 0.33 | 0.74 | 0.87 | 0.31 | 0.37 |
| 5 | 0.51 | 0.80 | 0.84 | 0.44 | 0.73 | 0.22 | 0.57 |
| 6 | 0.88 | 0.47 | 0.53 | 0.31 | 0.53 | 0.64 | 0.45 |
| 7 | 0.66 | 0.38 | 0.70 | 0.57 | 0.32 | 0.58 | 0.33 |
| 8 | 0.41 | 0.42 | 0.59 | 0.67 | 0.36 | 0.80 | 0.74 |
| 9 | 0.70 | 0.82 | 0.51 | 0.20 | 0.68 | 0.89 | 0.91 |
The arrival times (in seconds) for each UAV are represented in a matrix:
$$ t_{fl} = \begin{bmatrix}
61 & 42 & 47 & 70 & 76 & 66 & 57 & 49 & 55 \\
72 & 51 & 56 & 81 & 87 & 77 & 68 & 58 & 66 \\
81 & 60 & 65 & 90 & 96 & 86 & 75 & 67 & 73 \\
88 & 67 & 72 & 97 & 103 & 93 & 82 & 74 & 80 \\
99 & 78 & 83 & 108 & 114 & 104 & 93 & 85 & 91 \\
105 & 84 & 89 & 114 & 120 & 110 & 99 & 91 & 97
\end{bmatrix}. $$
Applying the anti-UAV target assignment model with an average shift time of 4 seconds, and using the SA-DPSO algorithm for optimization—with inertia weight linearly decreasing from 1.1 to 0.4, learning factors of 1.5, initial population of 100 particles, initial temperature of 10,000, and cooling coefficient of 0.9—we obtain the assignment matrix $x_{ij}$ when damage probability for each UAV is at least 90%. The satisfaction function is defined as: if $P_i \leq 0.9$, then $P_i = \log_{1.9}(P_i + 1)$; if $0.9 < P_i < 1$, then $P_i = e^{0.9 – P_i}$. The resulting assignment matrix is:
$$ x_{ij} = \begin{bmatrix}
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1
\end{bmatrix}. $$
This assignment ensures that each target is engaged within shift time constraints, with joint damage probabilities as follows:
$$ P = [84.87\%, 95.80\%, 94.20\%, 93.99\%, 96.80\%, 96.80\%, 92.05\%, 92.52\%, 94.80\%, 91.00\%]. $$
To evaluate the SA-DPSO algorithm’s performance, we compare it with classic DPSO and genetic algorithm (GA) over 20 simulations. For GA, parameters include initial population of 50, binary chromosome length of 20, maximum generations of 100, crossover probability of 0.8, and mutation probability of 0.1. Each particle represents a target assignment scheme, optimized until damage probability for each UAV exceeds 90%. The results are summarized in Table 2 and Figure 1.
| Objective Function | SA-DPSO | DPSO | GA | Optimal Solution |
|---|---|---|---|---|
| Allocation效能 $B(x)$ | 5.031 | 4.958 | 4.833 | 5.031 |
| Average Targets Engaged per Gun | 3.3 | 3.6 | 4.1 | 3.3 |
Table 2 shows that SA-DPSO achieves the highest allocation效能, indicating superior optimization capability. Compared to GA and DPSO, SA-DPSO reduces the average number of targets engaged per gun by 9.1% and 24%, respectively, meaning less operational pressure on the anti-aircraft gun system while maintaining damage probability above 90%. Figure 1 illustrates the iteration curves, where SA-DPSO converges fastest with the best convergence. These comparisons confirm that SA-DPSO is both rapid and reliable for anti-UAV target assignment.
In conclusion, anti-UAV target assignment remains a forefront research area in military technology, crucial for battlefield decision-making and equipment development. This article proposes a multi-indicator target assignment model for anti-aircraft guns engaging swarm UAVs, incorporating threat indicators, shift timing, and practical fire allocation zones. By adding shift time constraints and refining the objective function, we establish a comprehensive evaluation index system for anti-UAV operations. The integration of simulated annealing into discrete particle swarm optimization enhances global search ability, as demonstrated through实例 analysis. The results show that the model is scientifically sound and effective, providing valuable insights for engineering applications in anti-UAV defense. Future work could explore real-time adaptation and integration with other anti-UAV technologies to further improve swarm engagement capabilities.