In modern warfare, the proliferation of unmanned aerial vehicles (UAVs), particularly small drones, has introduced significant challenges for defense systems due to their low-altitude, slow-speed, and small-size characteristics. These UAVs can be easily exploited by adversaries for illicit reconnaissance and harassment, often bypassing traditional defense barriers. As a key countermeasure, laser weapon systems offer rapid response, precision strikes, reusability, strong anti-jamming capabilities, and minimal collateral damage, making them a cost-effective solution for anti-UAV operations. However, single laser weapons have limited detection and interception ranges, struggling to counter multi-directional low-altitude threats. Thus, networking multiple laser weapons to form an integrated combat capability is essential. Fire allocation, a core element of anti-UAV decision-making, involves optimally assigning targets to available weapons to maximize system effectiveness under dynamic battlefield conditions. This article explores the application of particle swarm optimization (PSO) algorithms to address fire allocation problems in anti-UAV scenarios, focusing on laser weapon systems. I will analyze the combat characteristics of laser weapons, establish a fire allocation model, and propose an improved PSO algorithm with efficient encoding and diversity control strategies to enhance real-time performance and solution quality.
The operational process of laser weapons against UAVs involves several constraints and factors that influence interception success. From a first-person perspective, I consider the following aspects in analyzing laser weapon combat characteristics for anti-UAV missions. First, resource constraints include weapon status, communication connectivity, and available lasing time. Only laser weapons in ready state, with functional communication links, unengaged, and sufficient remaining lasing time can be allocated. Second, spatial constraints depend on the relative position and motion between the UAV and laser weapon, limited by fire control radar, electro-optical device slewing rates, and engagement range. Assuming a UAV flies at a constant altitude $h$ and speed $v_T$, traversing the lethal zone of a laser weapon with radius $r_W$, the geometric relationship is defined in a path coordinate system $Oqs$, where the $s$-axis aligns with $v_T$. The lateral range $q_T$ must satisfy $q_T \leq r_W$, and the altitude and speed must be within bounds: $h_{\min} \leq h \leq h_{\max}$ and $v_T \leq v_{\max}$. Third, guidance probability $p_C$ reflects the likelihood of successfully directing the fire control radar to acquire the target, expressed as $p_C = p_A p_D$, where $p_A$ is the probability of target position falling within radar coverage and $p_D$ is the detection probability. If the radar autonomously detects the target, $p_D = 1$ and guidance time $t_D = 0$. Fourth, time constraints ensure the target remains within the lethal zone during weapon engagement. The extrapolated distance $s_{TE}$ after allocation command is $s_{TE} = s_{TC} + v_T t_E$, where $t_E = t_D + t_R$ (with $t_R$ as average laser aiming and firing time). The remaining allocation time $t_I$ and dwell time $t_S$ in the lethal zone are calculated as:
$$ t_I = \begin{cases} 0 & \text{if } s_{\min} \leq s_{TE} \leq s_{\max} \\ (s_{\min} – s_{TE}) / v_T & \text{if } s_{\min} > s_{TE} \\ +\infty & \text{if } s_{TE} > s_{\max} \end{cases} $$
$$ t_S = \begin{cases} (s_{\max} – s_{TE}) / v_T & \text{if } s_{\min} \leq s_{TE} \leq s_{\max} \\ (s_{\max} – s_{\min}) / v_T & \text{if } s_{\min} > s_{TE} \\ 0 & \text{if } s_{TE} > s_{\max} \end{cases} $$
Fifth, kill probability $p_K$ depends on target type, weather, laser power, and dwell time. For simplification, I model it as $p_K = p_{K0} (1 – e^{-t_S/\tau})$, where $p_{K0}$ is a reference kill probability and $\tau$ is a time constant. This indicates that longer dwell times increase kill probability, crucial for anti-UAV effectiveness. These characteristics form the basis for quantifying interception feasibility and building the fire allocation model.
To formalize the fire allocation problem for anti-UAV operations, I define sets: $S_W = \{W_1, W_2, \dots, W_{n_W}\}$ for $n_W$ laser weapons and $S_T = \{T_1, T_2, \dots, T_{n_T}\}$ for $n_T$ UAV targets. Let $p_{Kij}$ be the kill probability of weapon $i$ against target $j$, and $p_{Cij}$ be the guidance probability. A binary decision variable $z_{ij}$ is introduced, where $z_{ij} = 1$ if weapon $i$ is assigned to target $j$, else $z_{ij} = 0$. The expected damage efficiency for target $j$ is:
$$ p_{K\sum,j} = 1 – \prod_{i=1}^{n_W} (1 – p_{Cij} p_{Kij})^{z_{ij}} $$
Considering threat weights $w_{T,j}$ for each target, with $\sum_{j=1}^{n_T} w_{T,j} = 1$, I aim to minimize a combined objective function balancing kill efficiency and early interception. The optimization model is:
$$ \min F = \sum_{j=1}^{n_T} w_{T,j} \left\{ \lambda (1 – p_{K\sum,j}) + (1 – \lambda) \sum_{i=1}^{n_W} \frac{t_{Iij}}{t_C + t_{I\max}} \right\} $$
Subject to constraints:
$$ \sum_{j=1}^{n_T} z_{ij} \leq 1 \quad \text{for } i = 1, 2, \dots, n_W $$
$$ \sum_{i=1}^{n_W} z_{ij} \leq 2 \quad \text{for } j = 1, 2, \dots, n_T $$
$$ p_{Kij} > p_{KC} $$
Here, $t_{I\max}$ is the maximum remaining allocation time among assignments, $t_C$ is a time constant, and $\lambda \in [0,1]$ weights the trade-off between maximizing damage and early interception. The constraints ensure each weapon engages at most one target, each target is assigned to at most two weapons (for balanced resource use), and kill probability exceeds a threshold $p_{KC}$. This model represents a nonlinear integer programming problem with multiple constraints, common in anti-UAV scenarios, requiring efficient optimization algorithms.
Particle swarm optimization is a metaheuristic inspired by bird flocking behavior, suitable for solving complex optimization problems like fire allocation in anti-UAV systems. In standard PSO, a population of $N$ particles explores an $n$-dimensional search space, with each particle $i$ having position $x_i$, velocity $v_i$, and personal best $p_i$. The global best is $p_g$. For generation $k$, the update rules for dimension $d$ are:
$$ v_{i,d}^{k+1} = w v_{i,d}^k + \phi_{1,d} (p_{i,d}^k – x_{i,d}^k) + \phi_{2,d} (p_{g,d}^k – x_{i,d}^k) $$
$$ x_{i,d}^{k+1} = \text{int}(x_{i,d}^k + v_{i,d}^{k+1}) $$
where $w$ is inertia weight, $\phi_{1,d}$ and $\phi_{2,d}$ are random numbers, and $\text{int}(\cdot)$ ensures integer values for discrete allocation. However, basic PSO faces challenges like high-dimensional search spaces and premature convergence. To address this for anti-UAV fire allocation, I propose improvements in particle encoding, negative reinforcement learning, and diversity control.
First, particle encoding is optimized by leveraging weapon interception feasible regions. Instead of mapping each weapon to all targets, I precompute feasible target sets $S_{T}^{W_i}$ for each weapon $i$ based on the constraint $p_{Kij} > p_{KC}$. This reduces the solution space from $n_T^{n_W}$ to a smaller domain, enhancing efficiency. A particle is encoded as $x = (x_1, x_2, \dots, x_{n_W})$, where $x_i \in [1, |S_{T}^{W_i}|]$ indicates the assigned target from the feasible set. For example, if weapon $i$ can intercept targets $\{T_2, T_3, T_4\}$, then $x_i = 1$ might correspond to $T_2$. This encoding inherently satisfies the kill probability constraint and reduces dimensionality. Table 1 summarizes a sample feasible region for six laser weapons against five UAV targets in an anti-UAV scenario.
| Weapon ID | Feasible Target IDs | Number of Feasible Targets |
|---|---|---|
| W1 | T2, T3, T4, T5 | 4 |
| W2 | T3, T4 | 2 |
| W3 | T1, T4, T5 | 3 |
| W4 | T4 | 1 |
| W5 | T1, T2, T5 | 3 |
| W6 | T2, T3 | 2 |
Second, I incorporate negative reinforcement learning by adding a term to the velocity update that drives particles away from historically poor positions. The modified update rule is:
$$ v_{i,d}^{k+1} = w v_{i,d}^k + \phi_{1,d} (p_{i,d}^k – x_{i,d}^k) + \phi_{2,d} (p_{g,d}^k – x_{i,d}^k) – \phi_{3,d} (\bar{p}_{i,d}^k – x_{i,d}^k) $$
where $\bar{p}_{i,d}^k$ is the worst position recorded for particle $i$. This negative feedback helps avoid local optima, balancing exploration and exploitation in anti-UAV allocation. Third, to prevent premature convergence, I implement a diversity control strategy. Define the average position $\bar{x}_d^k = \sum_{i=1}^N x_{i,d}^k / N$ and diversity measure $\rho^k = \frac{1}{N} \sum_{i=1}^N \sum_{d=1}^D |x_{i,d}^k – \bar{x}_d^k|$. If $\rho^k < \rho_{\min}$, indicating low diversity, the velocity update is perturbed:
$$ \hat{p}_{g,d}^k = \mathcal{N}(p_{g,d}^k, \rho_d^k) $$
$$ v_{i,d}^{k+1} = w v_{i,d}^k + \phi_{2,d} (\hat{p}_{g,d}^k – x_{i,d}^k) $$
where $\mathcal{N}(\cdot)$ generates a normally distributed value around $p_{g,d}^k$ with variance $\rho_d^k$, encouraging exploration. These enhancements improve PSO’s performance for real-time anti-UAV fire allocation.
To validate the approach, I conduct a numerical simulation for an anti-UAV scenario with six laser weapons and five UAV targets. The threat weights are $W_T = (0.180, 0.214, 0.286, 0.176, 0.144)$, with target T3 being a fixed-wing UAV and others rotary-wing. Parameters include $p_{K0} = 0.8$ for rotary-wing and $0.6$ for fixed-wing, $p_A = 0.98$, $p_{KC} = 0.3$, $\lambda = 0.5$, and time constants as per operational data. After feasibility checks, the weapon interception probabilities are computed, as shown in Table 2, which details kill probabilities, remaining times, and dwell times for feasible weapon-target pairs.
| Weapon ID | Target ID | Remaining Time (s) | Dwell Time (s) | Kill Probability |
|---|---|---|---|---|
| W1 | T2 | 274.2 | 220.6 | 0.797 |
| W1 | T3 | 167.1 | 107.6 | 0.559 |
| W1 | T4 | 545.2 | 231.9 | 0.797 |
| W1 | T5 | 551.1 | 159.5 | 0.785 |
| W2 | T3 | 114.3 | 47.8 | 0.418 |
| W2 | T4 | 372.2 | 162.1 | 0.786 |
| W3 | T1 | 331.0 | 214.3 | 0.796 |
| W3 | T4 | 687.1 | 229.9 | 0.797 |
| W3 | T5 | 641.2 | 229.2 | 0.797 |
| W4 | T4 | 295.9 | 221.4 | 0.796 |
| W5 | T1 | 667.1 | 183.4 | 0.792 |
| W5 | T2 | 540.9 | 155.9 | 0.783 |
| W5 | T5 | 351.6 | 188.2 | 0.793 |
| W6 | T2 | 197.4 | 156.0 | 0.783 |
| W6 | T3 | 93.2 | 94.0 | 0.543 |
Using the improved PSO with parameters $N=10$, $D=6$, $k_{\max}=30$, $c_1=0.6$, $c_2=0.6$, $c_3=0.3$, and linear decreasing $w$, the algorithm converges to an optimal allocation. The global best particle is $P_g = (2, 1, 1, 1, 3, 1)$, decoded as assignment $a = (T3, T3, T1, T4, T5, T2)$. This means weapons W1 and W2 jointly engage target T3 (high threat), W3 engages T1, W4 engages T4, W5 engages T5, and W6 engages T2. The allocation aligns with anti-UAV principles, prioritizing high-threat targets and balancing resources. Table 3 summarizes the final allocation scheme and corresponding kill probabilities.
| Weapon ID | Assigned Target ID | Individual Kill Probability | Combined Kill Probability for Target |
|---|---|---|---|
| W1 | T3 | 0.559 | 0.7433 (for T3) |
| W2 | T3 | 0.418 | |
| W3 | T1 | 0.796 | 0.796 (for T1) |
| W4 | T4 | 0.796 | 0.796 (for T4) |
| W5 | T5 | 0.793 | 0.793 (for T5) |
| W6 | T2 | 0.783 | 0.783 (for T2) |
I compare the improved PSO with basic PSO in terms of objective function value and diversity over iterations. The improved algorithm shows faster convergence and maintains higher diversity, avoiding premature stagnation. The objective function $F$ decreases rapidly, reaching a minimum of approximately 0.215, while basic PSO plateaus earlier at around 0.250. Diversity $\rho^k$ in improved PSO fluctuates above $\rho_{\min}=1$, whereas basic PSO drops below, indicating loss of exploration. These results demonstrate that the enhanced PSO efficiently handles multi-target allocation in anti-UAV systems, offering real-time solutions with robust stability.
In conclusion, laser weapons are a pivotal technology for anti-UAV defense, and effective fire allocation is critical for maximizing their networked potential. By modeling laser weapon combat characteristics and formulating a fire allocation problem with damage efficiency and early interception goals, I have developed an improved particle swarm optimization algorithm tailored for anti-UAV scenarios. The key contributions include a feasible region-based particle encoding method that reduces problem dimensionality, negative reinforcement learning to accelerate convergence, and diversity control to prevent premature convergence. Simulation results confirm that the algorithm achieves near-optimal allocations quickly, adhering to operational constraints and principles. For future work, integrating real-time target tracking data and adaptive parameter tuning could further enhance performance in dynamic anti-UAV environments. This approach provides a practical tool for command decision-support in laser-based anti-UAV systems, contributing to safer airspace and effective counter-drone operations.
Throughout this article, I have emphasized the importance of optimization in anti-UAV contexts, using mathematical models and algorithmic innovations to address complex allocation challenges. The use of particle swarm optimization, with its inherent flexibility and efficiency, offers a promising path for advancing anti-UAV capabilities, ensuring that limited resources are deployed optimally against evolving drone threats. As UAV technology continues to advance, such computational methods will become increasingly vital for maintaining defensive superiority in modern warfare and security operations.