Research on threat assessment problems of island air defence system based on the leader-follower model

A model without leader is used to investigate the multi-agent system with all agents in equal status.

The elements in the set N = {1, 2, ..., n} denotes n agents, and x_i ∈ R represents the location of an individual i in the one-dimensional space, that is, the opinion of each agent. In a continuous time situation (t ∈ R+), the opinion change process of an agent i can be expressed as below: dxidt=α∑j=1,j≠iN(xj−xi), i,j∈N {{d{x_i}} \over {dt}} = \alpha \sum\limits_{j = 1,j \ne i}^N ({x_j} - {x_i}),\quad i,j \in N

The initial condition is set as: xi(0)=xi0∈ℝ, i∈N, {x_i}(0) = {x_{i0}} \in \mathbb{R},\quad i \in N,

Here, α is a positive constant, and its value can be adjusted according to actual need.

The model with leader is used to investigate the multi-agent system where the relationship of leading and being led exists. When the entire system has a leader, the opinion variation process of an agent i is as: {dxNdt=0,dxidt=α∑j=1n−1(xj(t)−xi(t))+γ(xN(t)−xi(t)) \left\{ {\matrix{ {{{d{x_N}} \over {dt}} = 0,} \hfill \cr {{{d{x_i}} \over {dt}} = \alpha \sum\limits_{j = 1}^{n - 1} ({x_j}(t) - {x_i}(t)) + \gamma ({x_N}(t) - {x_i}(t))} \hfill \cr } } \right. where, i = 1, 2, ..., n − 1. γ > 0 represents the influencing factor of the leader n to the other individuals. The more the γ is, the greater the leader's influence will be.

An island air defence system without leader refers to that each unit is equal in status during target threat assessment, and there is no system that any authority leader could affect the assessment results of each unit. See detailed mathematical model as below [19,20,21,22].

All combat units of the whole system are represented by m agents in the set M = {1, 2, ..., m}, x_i ∈ R denotes the position that the agent i is located in a one-dimensional space, that is, the assessed threat value of each unit. Then, in a continuous time (t ∈ R+), the mathematical model of the island air defence threat assessment system is as follows: dxidt=α∑j=1,j≠im(xj−xi), i,j∈M {{d{x_i}} \over {dt}} = \alpha \sum\limits_{j = 1,j \ne i}^m ({x_j} - {x_i}),\quad i,j \in M

The initial condition is set as: xi(0)=xi0∈ℝ, i=1, 2, …, m, {x_i}(0) = {x_{i0}} \in \mathbb{R},\quad i = 1,{\kern 1pt} 2,{\kern 1pt} \ldots ,{\kern 1pt} m, where, α is a positive constant, which can be adjusted automatically according to the need.

In the normal condition, there is a command post (CP) unit in the island anti-missile system, which collects the combat information returned by each combat unit, makes comprehensive analysis, and formulate battle plan [23,24,25]. This command procedure conforms to the idea of the opinion dynamics model with a single leader, so its mathematical model is built as follows: {dxmdt=0,dxidt=α∑j=1,j≠im−1(xj(t)−xi(t))+γ(xm(t)−xi(t)), \left\{ {\matrix{ {{{d{x_m}} \over {dt}} = 0,} \hfill \cr {{{d{x_i}} \over {dt}} = \alpha \sum\limits_{j = 1,j \ne i}^{m - 1} ({x_j}(t) - {x_i}(t)) + \gamma ({x_m}(t) - {x_i}(t)),} \hfill \cr } } \right. where i = 1, 2, ..., m − 1. γ > 0 represents the impact factor of the leader m to all other combat units.

The larger the value of γ is, the greater the leader's impact will be.

Considering the basic composition of one island air defence combat system, a combat system model is built, consisting of a CP, an air early warning (AEW), three anti-missile vehicles (AMV), two electronic countermeasures vehicles (ECV), and an air-defence radar (ADR). Each unit is connected by the communication link to form a network. Limited by the natural conditions of the island, construction funds and equipment compatibility, a model of centralised island air defence combat command system is set up, as shown in Figure 1.

Fig. 1

The value of x_i represents the estimated threat value of the attacking target that each unit gives, which is determined by how many target information each combat unit masters. So the formula of x_i is defined as follows: xi=1lnorm⋅a, {x_i} = {1 \over {{l^{norm}}}} \cdot a, where l^norm is the normalisation of the number of edges that such unit is connected in the air defence and anti-missile command and control system, that is, l^norm = l_i/l_max. It refers to the degree of information that the ith combat unit masters in the command and control network, a is the real threat value of a target. In the actual threat assessment, the target threat value is often relevant to such factors as flight height, velocity, heading angle and target type. The paper mainly studies the influence of the command system on the estimated threat value generated by the air defence system, thus, the threat assessment process of each combat unit is simplified. From the targets with threat values within [1,10], a target A with a known threat value a = 5 is selected, to perform a numerical simulation on the threat assessment process of the island air defence and anti-missile system.

According to the combat model shown in Figure 1, the connected edges of each node in the command and control network is given in Table 1.

The MATLAB software is used to simulate the process that the island air defence and anti-missile system assesses the threat of the target A.

4.1.1

Simulation on threat assessment of centralised island air defence system without leader

When there is no leader in an island air defence system, that is to say, the system gives its estimated threat value without the influence of CP or any command and control organ, and then the threat value is discussed, fused and determined. Set α = 1, simulate the process that the system generates an estimated target threat value. See the results shown in Figure 2.

Fig. 2

Figure 2 shows that the island air defence and anti-missile system reaches to consensus at about one time unit. The estimated target threat value is given as value. a = 3, which is greatly different from the real.

4.1.2

Simulation on threat assessment of centralised island air defence and anti-missile system with a leader

According to the general condition that the CP is the leader of an island air defence system, the process that the system assesses the target threat is simulated. Set in Figure 3: α = 1, γ = 2, and we obtain the results.

Fig. 3

As shown in Figure 3, because the CP acts as a leader in the system, so the estimated target threat value given by it will not be adjusted according to the opinions of other units. Moreover, the CP greatly affects the estimated threat value provided by other units, thereby, the combat units of the system reach to consensus on threat assessment at about 2.5 time unit. However, since the CP masters the most comprehensive battlefield information in the whole centralised command and control system, it can provide the most accurate target threat assessment. Figures 2 and 3 show that under the centralised command system, the command pivot could always provide the most accurate target threat estimation. But other combat units of the system would take a longer time to understand and accept such estimated value, which certainly will affect the combat efficiency.

With the development of communication technologies, each unit of the air defence system basically can interconnect with each other, which provides a technological support for distributed command and control mode. Suppose each combat unit of the system can communicate in real time, then the distributed command and control model of original air defence system is shown in Figure 4. The connecting condition of each unit in the network and the degree that each unit masters the target information is shown in Table 2.

Fig. 4

The MATLAB software is used to simulate the process that the island air defence system assesses the threat of target A.

4.2.1

Simulation on threat assessment of distributed island air defence system without leader

In the distributed air defence and anti-missile system, there is no leader. We simulate the process that the system assesses a target, set α = 1, and get the results as shown in Figure 5.

Fig. 5

Figure 5 shows that the system reaches to consensus on the estimated target threat value through 0.9 time unit. The estimated target threat value is given as a = 4.3, with a small difference with real value.

4.2.2

Simulation on threat assessment of distributed island air defence and anti-missile system with a leader

In the distributed system, there is a leader. We simulate the process that the system assesses the threat of a target, set α = 1, γ = 2, and obtain the results as shown in Figure 6.

Fig. 6

Figure 6 shows that, in a distributed island air defence system, thanks to the impact of CP as a leader, each combat unit provides an agreed target threat assessment at about 2.5 time unit.

Figures 5 and 6 indicate that under a distributed command and control system, the existence of a leader could also reduce the efficiency of system to assess the target threat.