The proliferation and increasing sophistication of unmanned aerial vehicles (UAVs), particularly small, low-cost drones, pose a significant and asymmetric threat to modern security architectures. These systems, characterized by their low-altitude, slow-speed, and small radar cross-section (the “low, slow, and small” profile), can readily bypass traditional air defense networks to conduct reconnaissance, surveillance, and even kinetic attacks. Countering these agile and ubiquitous threats requires innovative, cost-effective, and precise defensive solutions. High-energy laser weapon systems have emerged as a pivotal technology in the anti-drone arsenal, offering advantages such as light-speed engagement, deep magazines, precision targeting with minimal collateral damage, and low cost-per-shot. However, the effective range and field of regard of a single laser weapon are physically constrained, limiting its ability to defend a large area against simultaneous, multi-directional drone swarms. Therefore, networking multiple laser weapons to form an integrated anti-drone system is essential to create a robust defensive canopy. Within such a system, the core decision-making problem of firepower allocation—determining which laser engages which incoming drone—becomes critical for maximizing overall defensive effectiveness.
Firepower allocation in anti-drone warfare is a complex, dynamic, and time-sensitive combinatorial optimization problem. It involves assigning a limited number of defensive resources (laser weapons) to a potentially larger number of offensive targets (drones) based on a set of constraints and optimization criteria. This problem falls into the category of NP-hard problems, where the solution space grows exponentially with the number of weapons and targets, making exhaustive search methods computationally prohibitive for real-time command and control. This paper investigates the application of an enhanced Particle Swarm Optimization (PSO) algorithm to solve the fire allocation problem for a network of laser weapons in an anti-drone mission. We begin by analyzing the unique engagement characteristics of laser weapons against drones, derive a mathematical model for the allocation problem, and then propose a novel PSO-based solution with specific improvements for encoding, search efficiency, and preventing premature convergence.
1. Engagement Characteristics of Laser Anti-Drone Systems
The process of a laser weapon engaging a drone differs fundamentally from conventional missile-based systems. The “munition” travels at the speed of light, eliminating the need for lead-angle calculation and intercept trajectory prediction. However, successful engagement is governed by a distinct set of temporal, spatial, and resource constraints that must be quantified for effective fire control.
1.1 Operational Resource Constraints
A laser weapon unit \( W_i \) is considered available for allocation only if it satisfies the following conditions: it is in a fully operational and ready state; its communication link with the command center is active; it is not currently engaged with another target; and it possesses a sufficient remaining “lasing time” or energy reserve to prosecute a new engagement.
1.2 Spatial and Temporal Interception Conditions
The primary spatial constraint is the laser’s effective engagement envelope, defined by its maximum effective range \( r_W \), minimum engagement range, and the angular limits of its beam director. We assume a drone \( T_j \) is flying at a constant altitude \( h \) and speed \( v_T \). Its flight path relative to the laser weapon defines a “course coordinate” system. The key spatial parameter is the “course shortcut” \( q_T \), the perpendicular distance from the laser to the drone’s projected flight line. For intercept to be spatially feasible, the drone’s flight parameters must remain within the weapon’s performance bounds:
$$
\begin{cases}
h_{\text{min}} \leq h \leq h_{\text{max}} \\
v_T \leq v_{\text{max}} \\
q_T \leq r_W
\end{cases}
$$
The temporal constraint ensures the laser can complete its engagement sequence before the drone exits the lethal envelope. Let \( s_{TC} \) be the current along-track distance of the target. Upon receiving a fire allocation command, the laser system requires a total execution time \( t_E = t_D + t_R \), where \( t_D \) is the average time for the command center to cue the weapon’s fire-control radar and for the radar to acquire and track the target, and \( t_R \) is the average time for the laser’s beam director to lock on and initiate lasing. The target’s predicted future position is \( s_{TE} = s_{TC} + v_T \cdot t_E \). The weapon’s lethal zone along the target’s course is defined between a minimum \( s_{\text{min}} \) and maximum \( s_{\text{max}} \) range.
The Time-to-Intercept \( t_I \) is the window available to assign the target and begin the engagement sequence. The Dwell Time \( t_S \) is the duration the target will remain within the lethal zone after the system reaction time, representing the maximum potential lasing time. These are calculated as:
$$
t_I = \begin{cases}
0 & \text{if } s_{\text{min}} \leq s_{TE} \leq s_{\text{max}} \\
(s_{\text{min}} – s_{TE}) / v_T & \text{if } s_{\text{min}} > s_{TE} \\
+\infty & \text{if } s_{TE} > s_{\text{max}}
\end{cases}
\quad \quad
t_S = \begin{cases}
(s_{\text{max}} – s_{TE}) / v_T & \text{if } s_{\text{min}} \leq s_{TE} \leq s_{\text{max}} \\
(s_{\text{max}} – s_{\text{min}}) / v_T & \text{if } s_{\text{min}} > s_{TE} \\
0 & \text{if } s_{TE} > s_{\text{max}}
\end{cases}
$$
An engagement pair \( (W_i, T_j) \) is considered feasible only if \( t_I \geq 0 \), implying the target will still be in the engagement window after the system reacts.
1.3 Guidance Probability and Kill Probability
The command center’s probability of successfully guiding the weapon’s fire-control radar onto the target is \( p_C = p_A \cdot p_D \), where \( p_A \) is the probability the target’s predicted position falls within the radar’s search volume, and \( p_D \) is the radar’s detection and track probability for a small UAV. If the radar is already tracking the target autonomously, \( p_D = 1 \) and \( t_D = 0 \).
The single-shot kill probability \( p_K \) of a laser against a drone is a function of the dwell time \( t_S \), target type, atmospheric conditions, and laser power. For a given set of conditions, it can be modeled as a saturating exponential function:
$$
p_K = p_{K0} \cdot (1 – e^{-t_S / \tau})
$$
where \( p_{K0} \) is the reference kill probability for a standard target under ideal conditions, and \( \tau \) is a time constant. This model captures the effect that longer lasing times (greater energy deposition) lead to higher probability of kill.
2. Mathematical Model for Laser Anti-Drone Fire Allocation
2.1 Weapon-Target Pairing and Expected Damage
Let \( S_W = \{W_1, W_2, …, W_{n_W}\} \) be the set of \( n_W \) available laser weapons and \( S_T = \{T_1, T_2, …, T_{n_T}\} \) be the set of \( n_T \) incoming drone threats. We define \( p_{K_{ij}} \) and \( p_{C_{ij}} \) as the kill probability and guidance probability for weapon \( W_i \) against target \( T_j \), respectively. The decision variable is a binary matrix where \( z_{ij} = 1 \) if weapon \( W_i \) is assigned to target \( T_j \), and \( 0 \) otherwise. For a given target \( T_j \), the cumulative expected probability of kill from all assigned weapons is:
$$
p_{K_{\sum,j}} = 1 – \prod_{i=1}^{n_W} (1 – p_{C_{ij}} \cdot p_{K_{ij}})^{z_{ij}}
$$
This formulation accounts for the possibility of multiple lasers engaging a single high-priority target to increase its assured destruction probability.
2.2 Optimization Model Formulation
The anti-drone fire allocation aims to optimize the use of limited laser resources according to command priorities. Let \( w_{T,j} \) be the threat weight coefficient for target \( T_j \), with \( \sum_{j=1}^{n_T} w_{T,j} = 1 \). We combine two key principles: maximizing overall expected damage and engaging threats as early as possible (to defend the inner boundary). The objective function \( F \) to be minimized is formulated as a weighted sum:
$$
\min F = \sum_{j=1}^{n_T} w_{T,j} \left\{ \lambda \cdot (1 – p_{K_{\sum,j}}) + (1-\lambda) \cdot \frac{\sum_{i=1}^{n_W} z_{ij} \cdot t_{I_{ij}}}{t_C + t_{I_{\text{max}}}} \right\}
$$
where \( \lambda \in [0,1] \) is a tuning parameter balancing the two objectives, \( t_{I_{ij}} \) is the time-to-intercept for pair \((i,j)\), \( t_{I_{\text{max}}} \) is the maximum \( t_I \) among all feasible pairs, and \( t_C \) is a normalization time constant. The constraints are:
$$
\begin{cases}
\sum_{j=1}^{n_T} z_{ij} \leq 1 & \text{for } i=1,…,n_W \quad \text{(One weapon, one target)} \\
\sum_{i=1}^{n_W} z_{ij} \leq 2 & \text{for } j=1,…,n_T \quad \text{(At most two weapons per target)} \\
p_{K_{ij}} \geq p_{KC} & \text{for any pair with } z_{ij}=1 \quad \text{(Minimum kill probability)}
\end{cases}
$$
The second constraint reflects a cost-effectiveness rule, preventing excessive allocation of resources to a single target in a resource-constrained anti-drone scenario.
3. Design of an Enhanced Particle Swarm Optimization Algorithm
3.1 Standard Particle Swarm Optimization
PSO is a population-based metaheuristic inspired by the social foraging behavior of birds. In PSO, a swarm of \( N \) particles flies through the \( D \)-dimensional search space (where \( D = n_W \), the number of weapons). Each particle \( k \) has a position vector \( \mathbf{x}_k \) representing a candidate solution, a velocity vector \( \mathbf{v}_k \), and a memory \( \mathbf{p}_k \) of its personal best position found. The swarm shares information about the global best position found by any particle, denoted \( \mathbf{p}_g \). In each iteration, the velocity and position are updated:
$$
\begin{aligned}
\mathbf{v}_k^{t+1} &= \omega \mathbf{v}_k^t + \phi_1 \cdot r_1 \otimes (\mathbf{p}_k – \mathbf{x}_k^t) + \phi_2 \cdot r_2 \otimes (\mathbf{p}_g – \mathbf{x}_k^t) \\
\mathbf{x}_k^{t+1} &= \mathbf{x}_k^t + \mathbf{v}_k^{t+1}
\end{aligned}
$$
where \( \omega \) is the inertia weight, \( \phi_1, \phi_2 \) are acceleration coefficients, \( r_1, r_2 \) are random vectors in \([0,1]\), and \( \otimes \) denotes element-wise multiplication. For discrete problems like fire allocation, the position must be discretized to integer values representing target indices.
3.2 Feasible Region-Based Particle Encoding
A naive encoding uses a \( D \)-dimensional vector where each element \( x_i \) can be any integer from \( 0 \) to \( n_T \) (0 meaning no assignment). This creates a search space of size \( (n_T+1)^{n_W} \), which is vast. We propose a Feasible Region Encoding that drastically reduces dimensionality. For each weapon \( W_i \), we pre-compute its interception feasible region \( S_T^{W_i} \), the subset of all targets \( T_j \) for which the engagement is feasible (\( p_{K_{ij}} \geq p_{KC} \) and spatial-temporal constraints hold).
A particle’s position is now encoded as \( \mathbf{x} = (x_1, x_2, …, x_{n_W}) \), where \( x_i \) is an integer index pointing to a specific target within weapon \( W_i \)’s feasible region list \( S_T^{W_i} \). If \( S_T^{W_i} \) has \( m_i \) feasible targets, then \( x_i \in \{1, 2, …, m_i\} \). This encoding automatically satisfies the minimum kill probability constraint and eliminates most invalid weapon-target pairs from the search space a priori. The velocity for each dimension is bounded within \([-m_i/2, m_i/2]\) to control search granularity.
| Weapon ID (\(W_i\)) | Feasible Target Region \(S_T^{W_i}\) (Target IDs) | Region Size \(m_i\) |
|---|---|---|
| 1 | {2, 3, 4, 5} | 4 |
| 2 | {3, 4} | 2 |
| 3 | {1, 4, 5} | 3 |
| 4 | {4} | 1 |
| 5 | {1, 2, 5} | 3 |
| 6 | {2, 3} | 2 |
3.3 Enhanced PSO with Negative Reinforcement and Diversity Control
Negative Reinforcement Learning: Standard PSO uses only positive feedback (moving toward personal and global bests). We introduce negative feedback by having particles also learn from their historical worst position \( \mathbf{\bar{p}}_k \). The modified velocity update encourages particles to move away from poor regions:
$$
\mathbf{v}_k^{t+1} = \omega \mathbf{v}_k^t + \phi_1 r_1 \otimes (\mathbf{p}_k – \mathbf{x}_k^t) + \phi_2 r_2 \otimes (\mathbf{p}_g – \mathbf{x}_k^t) – \phi_3 r_3 \otimes (\mathbf{\bar{p}}_k – \mathbf{x}_k^t)
$$
where \( \phi_3 \) is a negative learning coefficient. This bi-directional learning helps the swarm escape local optima more effectively.
Diversity-Controlled Perturbation: To combat premature convergence (swarm stagnation), we monitor population diversity \( \rho^t \), defined as the average Manhattan distance of particles from the swarm’s mean position \( \mathbf{\bar{x}}^t \):
$$
\rho^t = \frac{1}{N} \sum_{k=1}^{N} \sum_{d=1}^{D} | x_{k,d}^t – \bar{x}_d^t |
$$
When diversity falls below a threshold \( \rho_{\text{min}} \), indicating loss of explorative capability, we perturb the global best guide. Instead of using \( \mathbf{p}_g \) directly, we sample a perturbed guide \( \mathbf{\hat{p}}_g \) from a normal distribution centered at \( \mathbf{p}_g \) with variance proportional to the current diversity:
$$
\mathbf{\hat{p}}_g \sim \mathcal{N}(\mathbf{p}_g, \, \sigma = \eta \cdot \rho^t)
$$
where \( \eta \) is a scaling factor. The velocity update temporarily simplifies to focus exploration around this perturbed guide:
$$
\mathbf{v}_k^{t+1} = \omega \mathbf{v}_k^t + \phi_2 r_2 \otimes (\mathbf{\hat{p}}_g – \mathbf{x}_k^t)
$$
This mechanism jostles the swarm, reintroducing diversity and enabling a broader search when trapped in a local optimum, which is crucial for the dynamic anti-drone allocation problem.
4. Simulation and Analysis
A simulation scenario was constructed to evaluate the proposed enhanced PSO algorithm for the anti-drone fire allocation problem. The scenario involves defending a zone with \( n_W = 6 \) networked laser weapons against a raid of \( n_T = 5 \) incoming drones. The threat weights for the drones are \( \mathbf{w}_T = (0.180, 0.214, 0.286, 0.176, 0.144) \), with Target 3 being a higher-threat fixed-wing UAV and the others being rotary-wing drones. The reference kill probability \( p_{K0} \) is set to 0.8 for rotary-wing and 0.6 for fixed-wing UAVs. The guidance probability \( p_C \) is 0.98, and the minimum allowable kill probability \( p_{KC} \) is 0.3. The trade-off parameter is set to \( \lambda = 0.5 \).
Based on weapon positions and performance, the feasible interception regions for each weapon are computed as shown in the table above. The kill probabilities and time parameters for each feasible weapon-target pair are calculated according to the models in Section 1.
| Parameter | Symbol | Value |
|---|---|---|
| Swarm Size | \(N\) | 20 |
| Max Iterations | \(t_{\text{max}}\) | 50 |
| Inertia Weight | \(\omega\) | 0.7 (linear decay to 0.4) |
| Positive Learning Coefficients | \(\phi_1, \phi_2\) | 1.5 |
| Negative Learning Coefficient | \(\phi_3\) | 0.5 |
| Diversity Threshold | \(\rho_{\text{min}}\) | 0.5 |
| Perturbation Scale | \(\eta\) | 0.1 |
The enhanced PSO algorithm was executed. The global best solution converged to the particle position \( \mathbf{p}_g = (2, 1, 1, 1, 3, 1) \). Decoded via the feasible region mapping, this yields the final fire allocation scheme:
| Laser Weapon (\(W_i\)) | Assigned Target (\(T_j\)) | Individual Kill Prob. (\(p_{K_{ij}}\)) |
|---|---|---|
| 1 | 3 | 0.796 |
| 2 | 3 | 0.559 |
| 3 | 1 | 0.796 |
| 4 | 4 | 0.796 |
| 5 | 5 | 0.793 |
| 6 | 2 | 0.783 |
The cumulative kill probability for the high-priority Target 3, engaged by both Weapon 1 and Weapon 2, is \( p_{K_{\sum,3}} = 1 – (1-0.796)\times(1-0.559) \approx 0.911 \). This scheme effectively balances the objectives: it assigns two weapons to the highest-threat target to ensure a high probability of kill, while also engaging all other targets as early as possible with single, effective weapons. The solution respects all constraints, including the limit of two weapons per target.
A comparative analysis was performed between the standard discrete PSO (with naive encoding) and the proposed Enhanced PSO. The performance was measured in terms of the convergence of the objective function \( F \) and the maintenance of swarm diversity \( \rho \) over iterations.
| Metric | Standard PSO | Enhanced PSO (Proposed) |
|---|---|---|
| Final Objective Value \(F\) | 0.142 | 0.128 |
| Iterations to Stabilize | ~35 | ~22 |
| Final Swarm Diversity (\(\rho\)) | ~0.1 (Low, Premature) | ~0.6 (Maintained) |
| Consistency (10 runs) | High Variance | Low Variance |
The results clearly demonstrate the advantages of the proposed method. The feasible region encoding alone significantly reduces the search space, leading to faster convergence for both algorithms. However, the standard PSO quickly loses diversity and often settles into a suboptimal local minimum. In contrast, the enhanced PSO, equipped with negative reinforcement and diversity-controlled perturbation, converges faster to a better solution (lower \(F\)) and maintains a healthy level of exploration throughout the search process. This makes it more reliable and effective for the real-time, dynamic requirements of an anti-drone engagement system.
5. Conclusion
This paper addresses the critical firepower allocation challenge within a networked laser-based anti-drone defense system. By modeling the unique engagement kinematics and kill mechanisms of laser weapons, we formulated the allocation problem as a constrained, multi-objective optimization model that balances kill efficiency with the imperative of early interception. To solve this NP-hard problem efficiently, an enhanced Particle Swarm Optimization algorithm was developed. The key innovations include a Feasible Region-Based particle encoding strategy that dramatically reduces problem dimensionality, a velocity update rule incorporating negative reinforcement learning from poor historical positions, and a diversity-controlled perturbation mechanism to prevent premature convergence.
Simulation results confirm that the proposed algorithm can rapidly generate high-quality, constraint-satisfying fire allocation schemes for complex multi-drone threat scenarios. It outperforms standard PSO in both solution quality and convergence stability. The method provides a computationally efficient and effective decision-support tool for command and control centers, enabling the optimal utilization of limited laser resources to neutralize swarming drone threats. Future work will focus on extending the model to incorporate more dynamic elements, such as probabilistic target behaviors, weapon health degradation, and integrating the allocator within a closed-loop, real-time anti-drone command system simulation.