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1 INTRODUCTION
A hypersonic gliding vehicle (HGV) usually refers to an aircraft with a speed above 5 Ma and flying airspace between 20 and 100 km [1, 2]. Due to the rapid flight speed and strong penetration ability, it poses severe challenges to the existing early warning detection system. As an important sensor in the battlefield perception system, radar plays a significant role in the searching and tracking of HGV. According to the guidance of early warning information, radar can quickly and effectively detect these targets, providing a longer time window for the defence system to improve the success rate of interception. However, owing to the limited resources (such as time and energy) of radar, it is necessary to reasonably allocate time and energy resources between tracking tasks and search tasks to achieve better detection efficiency. Therefore, reasonably allocating radar search resources based on early warning information is an important issue that needs to be addressed.
To the best of our limited knowledge, many scholars have researched the detection of HGV and the optimisation of radar resources, but these studies mainly focus on the HGV trajectory tracking and prediction [3, 4], radar task scheduling [5] and radar target allocation [6, 7]. There are few studies on the optimisation of radar search parameters. In fact, target search is an important task of a phased array radar. Therefore, this study mainly focusses on the search strategy of HGV with guidance information.
The search optimisation problem under guidance information mainly includes two parts: the radar search optimisation model and the optimisation algorithm. Jang et al. [8] proposes a target optimisation model that minimises the search load while maintaining the expected cumulative detection probability, and uses a semi-analytical method to solve the problem of the search parameters setting (such as dwell time and beam width). Yan et al. [9] proposes an integrated resource allocation method for radar when performing search and tracking tasks simultaneously, and obtains a multi-group Pareto optimal solution using the Pareto theory. Briheche [10] establishes a search optimisation model with the goal of minimising the search time, and expresses the objective function as an integer programme for solution. Tao et al. [11] establishes a search resource allocation model to minimise the average discovery time, and obtains the optimal search data rate by the Lagrange multiplier method. Huang et al. [12] proposes a radar search optimisation method under the early warning aircraft guidance and obtains the best search data rate by combining the Lagrange method with the obstacle method. Jiao et al. [13] optimises the search parameters of the space-based radar and proposes a parameter optimisation algorithm with the least search time, which considers the dwell time, search area and other factors. Fan et al. [14] establishes a radar search model with the objective of minimising the search airspace, and solves it with particle swarm optimisation (PSO) to achieve the planning of the search airspace. It can be found that the existing research studies mainly aim to establish a radar search model with the goal of simply increasing the cumulative detection probability or simply reducing the average discovery time. It lacks a comprehensive consideration of the cumulative detection probability, the average discovery time and the target priority level. When searching for targets, the radar should prioritise targets with high priority. Therefore, we propose a HGV search method based on early warning information guidance. The main contributions of this paper are as follows.
-
The priority judgement model of HGV is established. Combined with the early warning guidance information, three indicators, namely height, velocity and distance, are selected as the basis for priority judgement. The quantitative formulae corresponding to the three indicators are constructed, which can quickly judge the priority level of HGV.
-
A search optimisation model is proposed, which comprehensively considers the cumulative detection probability, average discovery time and target priority level. For HGV targets, it is equally vital to shorten the discovery time and increase the cumulative detection probability. Meanwhile, targets with high priority levels should be preferentially detected.
-
A hybrid optimisation method is proposed based on PSO and differential evolution (DE). The adaptive inertia weight and learning factors of PSO are designed to improve the search efficiency of particles. In addition, the mutation, crossover and selection operations of DE are used to perturb the particle swarm to enhance the population diversity of particles, thereby improving the global optimisation ability of the algorithm.
The rest of this paper is organised as follows. Section 2 describes the determination of the radar search airspace. Section 3 introduces the HGV priority quantification model and the radar search model. Section 4 details the hybrid optimisation algorithm based on PSO and DE. Section 5 verifies the effectiveness of the proposed method, and Section 6 concludes the paper.
2 RADAR SEARCH AIRSPACE
Early warning guidance information provides an important data basis for radar to search and detect HGV. It mainly includes HGV position information, velocity information and error information. The early warning system usually predicts the HGV trajectory according to the historical tracking data, and then provides the predicted trajectory information to the radar. As shown in Figure 1, the radar determines the search airspace based on the guidance information. Due to the high manoeuvrability of HGV, the prediction error will keep increasing and the search airspace will gradually become larger with the extension of prediction time. Assuming that the HGV position and velocity information estimated by the early warning system is , the radar needs to convert it to azimuth angle and elevation angle information.
Radar search based on early warning information guidance
The change curve of the priority level with height
The change curve of the priority level with velocity
The change curve of the priority level with distance
The relationship between single detection probability and SNR
The relationship between the number of pulses and SNR
Relationship between the cumulative detection probability and the number of revisits
Assuming the real azimuth angle and elevation angle of HGV are
, and the estimated azimuth angle and elevation angle of HGV are
. Then we can get the formula as follows.
(1)
where
is the estimated error corresponding to the azimuth angle and elevation angle.
Assuming the probability density of the HGV appearing in the radar search airspace
is
. Then the probability of HGV appearing in this search airspace is Ref. [15, 16].
(2)
The range of the radar search airspace is mainly affected by the error of guidance information. The larger the search airspace is, the more conducive it is to detect HGV. However, too large search airspace will also cause a waste of radar resources and reduce radar performance. Therefore, we usually determine the range of search airspace based on the Pau Ta criterion (3σ principle) [17].
3 RADAR SEARCH MODEL
The fast beam scanning capability and fast waveform agility capability of the phased array radar enable it to be flexibly adjusted between multiple working modes [18]. The main tasks of the radar include searching and tracking. The searching task is the premise for the radar to achieve other functions. The main problem discussed in this section is establishing a radar search model so that the radar can reasonably allocate time and energy resources in the search task.
3.1 Hypersonic gliding vehicle priority judgement model
The priority of HGV is different due to different manoeuvre states. Radar will improve the detection probability of high-priority HGVs when performing the search task. Therefore, how to judge the HGV priority is the primary problem in the radar searching process. We propose three indicators to judge the priority of HGV, namely height, velocity, and distance.
- (1)
Height
The lower the height of HGV, the shorter the remaining flight time, the closer to the HGV attack phase and the more difficult it is to defend against interception. Therefore, the lower the HGV height, the greater the priority. The HGV usually flies in the airspace range of 20 to100 km, and its height priority is modelled as a function in the interval [0,1].
(3)
(4)
where Hmax indicates the upper bound of the HGV flight altitude, and its value is 100 km. Hmin indicates the lower bound of the HGV flight altitude, and its value is 20 km. The change curve of the priority level with height is shown in Figure 2.
- (2)
Velocity
The faster the HGV velocity, the stronger its manoeuvrability and attack capability. The high speed also makes it difficult for the radar to detect and track HGV. Therefore, the faster the HGV speed, the greater the priority level. The manoeuvring speed of HGV is usually 5 to 20 Ma, and the priority level of HGV speed is modelled as,
(5)
where V indicates the upper bound of the HGV flight speed, and its value is 20 Ma. is the priority coefficient of speed. The range of its values is generally [0.025, 0.035]. The value of can be determined according to the needs of the system, and its value chosen in this paper is 0.03. The change curve of the priority level with velocity is shown in Figure 3.
- (3)
Distance
The distance is an important indicator for the priority level judgement. The smaller the distance, the greater the priority. Assuming that the maximum detection range of the radar is 1500 km, the distance priority level can be modelled as,
(6)
where indicates the minimum priority level and indicates the priority coefficient of distance. The distance priority level curve can be adjusted by and as shown in Figure 4. is usually set as a constant in the interval [0.1, 0.3]. is usually set as a constant in the interval [0.001, 0.003]. r indicates the distance between the HGV and radar.By computing the priority level of height, velocity and distance, we can get the comprehensive priority level of HGV as follows.
(7)
where , and are the priority level weights corresponding to the height, velocity and distance, respectively.
By normalising all the HGV priority levels, the value of the priority level corresponding to each HGV can be obtained.
3.2 Optimisation objective
When a radar is searching for a target, it usually wants to detect the target with maximum probability in the shortest time. Therefore, we take the shortest average discovery time and the largest cumulative detection probability as the optimisation criteria and optimise the search parameters of the radar.
Assuming that according to the early warning guidance information, the number of airspaces that the radar needs to search is N. The search frame period of the ith airspace is
, and the time of the target appears in the ith airspace obeys a uniform distribution. Then the average time for the target to be detected in the first search frame period can be expressed as,
(8)
Assuming that the target is discovered after waiting for k search frame periods, the time and probability of the target being detected can be expressed as,
(9)
(10)
where
is the target detection probability within a single dwell time. According to the characteristic of HGV, it is usually modelled as a Swerling III target [19]. The formula of its detection probability is taken from Ref. [20].
(11)
where
is the detection threshold,
is the incomplete gamma function, and
is the accumulated pulse number for a single dwell time. SNR is the signal-to-noise ratio.
It can be concluded that the average discovery time of the target in the ith airspace is,
(12)
The cumulative detection probability of the target in the ith airspace is,
(13)
where n is the number of radar revisits. Generally, we hope that the average discovery time is as short as possible, and the cumulative detection probability is as large as possible. Therefore, we design the optimisation criterion as,
(14)
3.3 Constraint condition
Due to the limitations of time and energy resources, the radar should also meet the following constraints when searching for targets:
- (1)
Constraint of detection probability
The cumulative detection probability is related to the single detection probability and the number of radar revisits n, and their relationship can be expressed as,
(15)
It can be seen that the cumulative detection probability can be increased by increasing the single detection probability or the number of revisits. The single detection probability is related to SNR. When the false alarm probability is constant, the relationship between the single detection probability and SNR is shown in Figure 5. Assuming that pulses are coherently accumulated, the SNR is related to the number of pulse accumulations, and their relationship is shown in Figure 6. When the single detection probability is constant, the relationship between the cumulative detection probability and the number of radar revisits is shown in Figure 7.
At the same time, the number of radar revisits n should satisfy the following equation.
(16)
where Tz represents the total time resources. - (2)
Constraint of dwell time
The equation of the radar detection range [21, 22] is as follows.
(17)
where and are the radar transmit power and the transmit antenna gain, respectively. is the receiver antenna area. is the target scattering cross-sectional area. L represents the total system loss. k = 1.38 × 10–23 J/K is the Boltzmann parameter. = 288 K is the radar system temperature. B is the radar bandwidth.The radar dwell time can be expressed as,
(18)
where is the pulse width.Combining (17) and (18), the equation can be obtained as follows.
(19)
In order to avoid distance ambiguity, the pulse recurrence period needs to meet.
(20)
where C is the light velocity, and its value is 3 × 108 m/s. - (3)
Constraint of time resource
The total time for the radar to perform the search task is,
(21)
where is the number of radar beams in the ith airspace.The phased array radar can perform searching and tracking tasks at the same time by using time segmentation technology as shown in Figure 8.
Assuming that the resource occupancy rate of the search task is , the total time for the radar searching task needs to meet the following equation:
(22)
Considering the target priority and constraints conditions, we design the optimisation equation of radar search parameters as follows.
(23)
(24)
4 HYBRID OPTIMISATION ALGORITHM
It can be seen that the above optimisation problem is a multi-objective and multi-constraint optimisation problem. In order to improve the solution efficiency, we propose a hybrid optimisation algorithm that combines DE and PSO.
4.1 Adaptive particle swarm optimisation
- (1)
Particle swarm optimisation
PSO is a simplified model of swarm intelligence [23-26], which regards the solution process of the optimisation problem as the process of birds foraging and regards the solution space as the flight space of birds. It searches for the optimal solution through the movement of particles in the solution space. PSO has the advantages of simple parameters and easy implementation. It has been widely used in the fields of function optimisation, pattern classification and control engineering.
The PSO regards each particle as a potential best solution. It determines the best position according to the fitness value of each particle, and updates the current movement speed and position of each particle through its own best position and the global best position. Its update formula is as follows.
(25)
(26)
where is the current position of the ith particle in the jth dimension. is the particle best position. is the global best position. is the current particle velocity. and are the learning factors. and are the random number distributed in the interval [0, 1]. w is the inertia weight. - (2)
Design of adaptive parameters
The first term on the right side of (25) mainly represents the inertial motion of the particle, which is controlled by the inertia weight w. During the iterative process, w can be dynamically changed. In the early iteration stage, a larger value of w is beneficial to enhance the particle's inertia weight, thereby enhancing the search ability of the global best solution. In the later stage of the iteration, a smaller value of w is beneficial to the rapid convergence of the algorithm. We adopt a dynamic variable weighting strategy [27] and set w to the cosine change form. The equation is as follows.
(27)
where and are the maximum inertia weight and the minimum inertia weight, respectively. is the maximum number of iterations. Figure 9 shows the relationship between the inertia weight and the number of iterations.
The second term and third term on the right-hand side of (25) represent the particle cognitive and the social factors, respectively. c1 is mainly used to control the influence of the particle's best position on the movement of the particle. The larger the value of c1, the more the particles tend to be closer to their own best position. c2 is mainly used to control the influence of the global best position on the movement of the particle. The larger the value of c2, the more the particles tend to be closer to the global best position. In order to maximise the search range in the early stage of iteration and converge to the optimal position quickly in the later stage of iteration, we design the learning factors c1 and c2 as a trigonometric function.
(28)
(29)
where
,
,
and
are the control parameters of learning factors. Figure 10a,b shows the relationship between the learning factors and the number of iterations.
A new particle iteration formula can be obtained by introducing (27) – (29) into (25).
4.2 The proposed method
In order to overcome the shortcomings of PSO, which are the lack of population diversity and easy to fall into local optimality, we propose a hybrid optimisation algorithm based on PSO and DE.
DE is a heuristic search algorithm [28-30] that finds the best solution through population mutation, crossover and selection. It is put forward on the basis of the genetic algorithm [31]. It has a strong global search ability, and its equation is as follows.
(30)
where r1, r2 and r3 are random serial numbers, and they are different from each other. F is a scaling factor designed as an adaptive regulation parameter.
(31)
(32)
where F0 is a mutation operator.
DE is used to disturb the iteration of PSO. The formula is as follows.
(33)
where
is a random number in the interval [0,1]. When
is less than or equal to the threshold S, the particles first perform mutation operation, and then perform population crossover and selection. When
is greater than the threshold S, the particle is updated according to (25). The threshold S is designed as an adaptive function.
The relationship between the threshold S and the number of iterations is shown in Figure 11. It can be seen that at the beginning of iteration, the value of S is larger so that particles have a greater probability of performing DE to increase population diversity. At the later stage of iteration, the value of S is smaller so that the particles can converge as soon as possible and find the best value.
Time segmentation of the phased array radar
The variation curve of inertia weight
The variation curve of learning factors. (a) The variation curve of c1; (b) The variation curve of c2
The variation curve of the threshold S
5 SIMULATION
In order to verify the effectiveness of the proposed method, this section provides the results of numerical operation. The radar parameters are designed as shown in Table 1. Assuming that the cross-sectional area of HGV is 0.01 m2, and the false alarm probability is 10−6, the following two benchmark models are selected for comparison. The first model is a radar task scheduling algorithm based on the particle swarm-annealing algorithm proposed by Meng and Tian [32], which is marked as Method 1. The second model is a sensor network deployment optimisation algorithm based on modified differential evolution proposed by Cao et al. [33], which is marked as Method 2.
| Parameter | Value |
|---|---|
| Peak power | 582 kW |
| Transmitting antenna gain | 38.4 dB |
| Working frequency | 450 MHz |
| Radar system loss | 8 dB |
| Noise factor | 2.9 dB |
| Noise temperature | 290 K |
| Boltzmann constant | 1.38 × 10−23 J/K |
| Beam width | 1° |
| Scanning range of azimuth angle | (−60°, 60°) |
| Scanning range of elevation angle | (5°, 85°) |
The simulation experiments are carried out in two simulation scenarios. Simulation scenario I is the optimisation of radar search resources under three HGV targets, which is a typical scenario setting. Simulation scenario II is the optimisation of radar search resources under five HGV targets, which is mainly to verify the performance of the proposed method in complex search scenarios.
5.1 Simulation scenario I
Assuming that there are three HGV targets from different directions. The height, velocity, and distance of HGV are estimated by the early warning system. The search airspace parameters of three HGV targets are determined based on the early warning information as shown in Table 2. According to Section 3.1, the corresponding priority levels can be calculated as 0.2670, 0.3147 and 0.4183, respectively. The search resources of the three airspaces are allocated through Method 1, Method 2 and the proposed method. The relationship between the total cumulative detection probability and the search resource occupancy rate is shown in Figure 12. The relationship between the average discovery time and the search resource occupancy rate is shown in Figure 13. As the search resource occupancy rate increases, the total cumulative detection probability gradually increases and the average discovery time gradually decreases. When the search resource occupancy rate is greater than 0.7, the increasing of the total cumulative detection probability and the decreasing of the average discovery time gradually slow down. Figures 12 and 13 show that the proposed method is better than Method 1 and Method 2. Especially when the resource occupancy rate of the search task is low, the advantage of the proposed method is more prominent.
| Target | Search airspace | Target priority | |||
|---|---|---|---|---|---|
| Elevation | Azimuth | Distance | Velocity | Height | |
| Target 1 | 12°–24° | 16°–31° | 480 km | 8 Ma | 74 km |
| Target 2 | 30°–45° | −30°–−20° | 390 km | 6.5 Ma | 65 km |
| Target 3 | 47°–57° | 0°–13° | 360 km | 7 Ma | 52 km |
Total cumulative detection probability in scenario I
Average discovery time in scenario I
Figure 14a–c shows the detection probability of Method 1, Method 2 and the proposed method for the three targets under different search task resource occupancy rates. It can be found that the detection probability of target 3 is greater than that of target 1 and target 2 under the same search task resource occupancy rate. This is because the target with higher priority obtains relatively more search resources. When the search task resource occupancy rate is 0.1, both the PSO and the proposed method choose to give up the search of target 1 to maximise the detection probability of target 2 and target 3.
Detection probabilities of HGV in scenario I. (a) Detection probability of Method 1; (b) Detection probability of Method 2; (c) Detection probability of the proposed method
In order to analyse the performance of the proposed method in detail, the time resource allocation schemes corresponding to the three algorithms are given when the resource occupancy rate of the search task is 0.1 as shown in Figure 15. It can be seen that the proposed method allocates 63% of time resources to target 3%, 30% of time resources to target 2%, and 7% of time resources to target 1. Such an allocation of time resources can achieve more efficient target detection.
The time resource allocation schemes in scenario I. (a) Time resource allocation schemes of Method 1; (b) Time resource allocation schemes of Method 2; (c) Time resource allocation schemes of the proposed method
The convergence of Method 1, Method 2 and the proposed method is shown in Figure 16, when the search task occupancy rate is 0.1, 0.5 and 1, respectively. It can be found that the convergence speed of the proposed method is always better than those of Method 1 and Method 2. When the search tasks occupancy rate is lower, the convergence speed is slower.
Convergence comparison of different algorithms in scenario I. (a) Comparison of the convergence speed when the search task resource occupancy rate is 0.1; (b) Comparison of the convergence speed when the search task resource occupancy rate is 0.5; (c) Comparison of the convergence speed when the search task resource occupancy rate is 1
5.2 Simulation scenario II
Suppose there are five HGV targets from different directions. The altitude, velocity and distance of HGV are estimated by the early warning system. According to the guidance information, the scope of five search airspaces is determined. The parameters of the search airspace are shown in Table 3. The corresponding priorities of the five HGV targets are 0.1570, 0.1773, 0.2072, 0.2144, and 0.2441, respectively. Method 1, Method 2 and the proposed method are used to optimise the radar search resources. The relationship between the total cumulative detection probability and the search task resource occupancy rate is shown in Figure 17, and the relationship between the average discovery time and the search task resource occupancy rate is shown in Figure 18. It can be seen that the proposed method performs better than the other two algorithms.
| Target | Search airspace | Target priority | |||
|---|---|---|---|---|---|
| Elevation | Azimuth | Distance | Velocity | Height | |
| Target 1 | 20°–30° | 10°–25° | 600 km | 10 Ma | 82 km |
| Target 2 | 30°–45° | −30°–−20° | 460 km | 9 Ma | 70 km |
| Target 3 | 45°–55° | −15°–0° | 300 km | 7 Ma | 56 km |
| Target 4 | 55°–69° | 21°–32° | 330 km | 6 Ma | 49 km |
| Target 5 | 10°–22° | 32°–45° | 250 km | 8 Ma | 50 km |
Total cumulative detection probability in scenario II
Average discovery time in scenario II
Figure 19 shows the variation of the detection probability with the search task resource occupancy rate. With the decline of the search tasks occupancy rate, the detection probability of target 1 and target 2 decreases most obviously, while the detection probability of target 4 and target 5 decreases more smoothly due to the higher priority.
Cumulative detection probabilities in scenario II. (a) Detection probability of Method 1; (b) Detection probability of Method 2; (c) Detection probability of the proposed method
Figure 20 shows the time allocation schemes of the three algorithms for different targets when the search task occupancy rate is 0.1. It can be seen that the DE has a low detection probability for the target with higher priority, and the PSO emphasises the target with higher priority and ignores target 1, target 2, and target 3. The resource allocation scheme of the proposed method is relatively more reasonable, so that the cumulative detection probability and average discovery time are better than DE and PSO.
The time resource allocation schemes in scenario II. (a) Time resource allocation schemes of Method 1; (b) Time resource allocation schemes of Method 2; (c) Time resource allocation schemes of the proposed method
The convergence of Method 1, Method 2 and the proposed method is shown in Figure 21, when the search task occupancy rate is 0.1, 0.5 and 1, respectively. It can be found that the convergence speed of the algorithms in scenario II is slightly slower than that in scenario I. However, the proposed method still maintains the fastest convergence rate, which is superior to the other two algorithms.
Convergence comparison of different algorithms in scenario II. (a) Comparison of the convergence speed when the search task resource occupancy rate is 0.1; (b) Comparison of the convergence speed when the search task resource occupancy rate is 0.5; (c) Comparison of the convergence speed when the search task resource occupancy rate is 1
6 CONCLUSION
This paper studies the problem of HGV search under the guidance of early warning information. Firstly, we choose the height, velocity and distance of HGV as the priority evaluation indexes, and design the priority quantitative formulae. Secondly, considering the cumulative detection probability, average discovery time and target priority, the multi-objective search optimisation model of HGV is established. Finally, a hybrid optimisation algorithm based on DE and PSO is proposed to solve the multi-objective and multi-constraint problems. The proposed method improves the search efficiency of PSO by designing adaptive weight and learning factors, and improves the diversity of population by introducing the DE algorithm. The performance of the proposed method is verified through experiments in two scenarios. The results show that the proposed method is better than the existing mainstream methods.
ACKNOWLEDGEMENTS
This study was supported by the Military Postgraduate Funding Project of China (No. JY2019B138).
CONFLICT OF INTEREST
The authors declare that they have no conflict of interest to disclose.
REFERENCES
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