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

Y. Wang; Y. Cheng; X. Shan; J. J. Mou; L. Zhou; Mohamed Salama

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

Dettagli della rivista

Formato: Rivista
eISSN: 2444-8656
Prima pubblicazione: 01 Jan 2016
Frequenza di pubblicazione: 2 volte all'anno
Lingue: Inglese

As the performance of modern air-raid weapons and equipment and their attacking modes advance continuously, the requirements for threat assessment capability of air defence system are getting higher and higher. To solve this problem, many scholars have made researches from various angles. Literature [1] makes an overall consideration of the relative value of strategic places, and the attack ability, attack intention and arrival time of targets, and builds the threat assessment model of air defence operations in naval strategic places. According to the characteristics of target threat assessment problems, such as high dimension and time-variation, Sun et al. [2] proposes a multi-moment threat assessment method based on the dynamic grey principal component analysis. Literature [3] aims at the target threat assessment problems in air defence operations, and raises a literature assessment approach which can fuse the multi-moment decision information and make adjustment according to a decision-maker's risk preference.

The abovementioned studies have improved and innovated the threat assessment method, but few focus on the investigations of threat assessment process of a system. The target threat assessment of air defence system, as one main content of the eight tactics-level command activities, is greatly affected by the command system. A reasonable command system is the precondition that the threat value of an attacking target can be assessed accurately and efficiently. This paper employs the opinion dynamics model in the multi-agent theory to build a command model of island air defence system under centralised and distributed command systems, and integrates with some examples to simulate. This provides a new way to study the command system of air defence system.

Two opinion dynamics models

The consistency issue is always a hotspot in the self-organising system research field, which has been applied in physics [4,5,6], chemistry [7], biology [8,9,10,11,12], computer science [13,14], humanistic sociology, managing science [15], etc. Therein, the opinion consensus issue has attracted more and more scholars due to its wide application prospect. At present, its main research application in military science involves: in extreme conditions, how could multiple unmanned aerial vehicles quickly and efficiently collaborate to complete combat formation and execute a specific search and rescue task; how could multiple agents reach to an assigned underwater position within limited time and form a specific array mode; etc. [16,17,18].

In a multi-agent system, how could the agents impact each other and reach a consensus in the end? For a system with leader, its influencing mechanism can be described by the leader-follower opinion dynamics model.

2.1

Opinion model without leader

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: $\frac{d x_{i}}{dt} = α \sum_{j = 1, j \neq i}^{N} (x_{j} - x_{i}), i, j \in N$ {{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: $x_{i} (0) = x_{i 0} \in ℝ, i \in N,$ {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.

2.2

Opinion model with leader

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: ${\begin{array}{l} \frac{d x_{N}}{dt} = 0, \\ \frac{d x_{i}}{dt} = α \sum_{j = 1}^{n - 1} (x_{j} (t) - x_{i} (t)) + γ (x_{N} (t) - x_{i} (t)) \end{array}$ \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.

Threat assessment model of island air defence system based on opinion dynamics

Model hypothesis: (1)

Each unit of the island air defence system is mainly composed of command and control staffs of each level and equipment, and the properties of each node are similar to those of agents, so we consider to deem m nodes of the entire island anti-missile system as m agents. That's to say, the whole air defence system is a multi-agent system.

(2)

In the air defence operations, each combat unit gives their own threat assessment values of the attacking targets respectively according to their own intelligence mastered, their tasks undertaken, natural environment, etc.

(3)

The less the target information that each combat unit masters is, the lower the confidence level of target threat estimation that they give will be, and the lower the target threat estimation will be.

(4)

Once the target threat estimation given by each combat unit is collected, discussed and agreed on, finally, an accurate target threat estimation will be obtained.

Referring to the opinion dynamics models with and without a leader in Section 1, two kinds of threat assessment models of island air defence system are built, respectively.

3.1

Threat assessment model of island air defence system without leader

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: $\frac{d x_{i}}{dt} = α \sum_{j = 1, j \neq i}^{m} (x_{j} - x_{i}), i, j \in M$ {{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: $x_{i} (0) = x_{i 0} \in ℝ, i = 1, 2, \dots, m,$ {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.

3.2

Threat assessment model of island air defence system with a leader

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: ${\begin{array}{l} \frac{d x_{m}}{dt} = 0, \\ \frac{d x_{i}}{dt} = α \sum_{j = 1, j \neq i}^{m - 1} (x_{j} (t) - x_{i} (t)) + γ (x_{m} (t) - x_{i} (t)), \end{array}$ \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.

Example simulation

4.1

Simulation on threat assessment of an island air defence system under centralised command and control system

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.

Network model of centralised island air defence combat system. AMV, anti-missile vehicles; ADR, air-defence defence radar; AEW, air early warning; CP, command post; ECV, electronic countermeasures vehicles.

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: $x_{i} = \frac{1}{l^{norm}} \cdot a,$ {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.

Table 1

The number of connected edges of each node in the command and control network.

Combat unit	Connected edge number (l)	Normalised edge number (l^norm)

CP	7	1
AEW	1	0.14
ADR	6	0.86
AMV 1	4	0.57
AMV 2	4	0.57
AMV 3	4	0.57
ECV 1	4	0.57
ECV 2	4	0.57

ADR, air-defence defence radar; AEW, air early warning; AMV, anti-missile vehicles; CP, command post; ECV, electronic countermeasures vehicles.

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.

Simulation of threat assessment process of island air defence system without leader. ADR, air-defence defence radar; AEW, air early warning; AMV, anti-missile vehicles; CP, command post; ECV, electronic countermeasures vehicles.

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.

Simulation of threat assessment process of island air defence system with a leader. ADR, air-defence defence radar; AEW, air early warning; AMV, anti-missile vehicles; CP, command post; ECV, electronic countermeasures vehicles.

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.

4.2

Simulation on threat assessment of island air defence system under distributed command and control system

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.

Network model of distributed air defence operation system for islands. ADR, air-defence defence radar; AEW, air early warning; AMV, anti-missile vehicles; CP, command post; ECV, electronic countermeasures vehicles.

Table 2

The number of connected edges of each node in the command and control network.

Combat unit	Number of connected edges (l)	Normalized edge number (l^norm)

CP	7	1
AEW	6	0.86
ADR	6	0.86
AMV 1	6	0.86
AMV 2	6	0.86
AMV 3	5	0.71
ECV 1	6	0.86
ECV 2	6	0.86

ADR, air-defence defence radar; AEW, air early warning; AMV, anti-missile vehicles; CP, command post; ECV, electronic countermeasures vehicles.

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.

Simulation of threat assessment process of leader free distributed island air defence and anti-missile system. ADR, air-defence defence radar; AEW, air early warning; AMV, anti-missile vehicles; CP, command post; ECV, electronic countermeasures vehicles.

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.

Simulation of threat assessment process of leader distributed island air defence and anti-missile system. ADR, air-defence defence radar; AEW, air early warning; AMV, anti-missile vehicles; CP, command post; ECV, electronic countermeasures vehicles.

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.

Conclusions

This paper uses the opinion dynamics models with and without a leader in the multi-agent theory as a reference, builds an island air defence threat estimation model, and simulates the target threat assessment process. Through the simulation results, we can know that under a command and control system with unsound communication network, the CP acts as an absolute leader and leads the whole air defence and anti-missile system to perform threat assessment. Though it takes a longer time, it can get an accurate estimated threat value of a target; while under the distributed command and control system with a wholesome communication network, each combat unit can get an overall feature information of the target, so the command and control rights are passed down. This reduces the impact of the CP on the target threat assessment process, helps to accurately and quickly obtain an estimated threat value of the target, and greatly improves the operational effectiveness of the entire air defence system.

From the abovementioned results, the proposed target threat assessment model can simulate an island air defence threat assessment process, and the results obtained basically conform to the real situation. It furnishes a new method to study the strategy generation mechanism of island air defence command system, and has certain reference significance to the construction of island air defence command and control system. Next, we will consider building a new model to simulate the impact of communication delay on the threat assessment process of air defence system.

Figure e tabelle

The number of connected edges of each node in the command and control network.

Combat unit	Number of connected edges (l)	Normalized edge number (l^norm)

CP	7	1
AEW	6	0.86
ADR	6	0.86
AMV 1	6	0.86
AMV 2	6	0.86
AMV 3	5	0.71
ECV 1	6	0.86
ECV 2	6	0.86

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Applied Mathematics and Nonlinear Sciences

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

Y. Wang,
Y. Wang,
Y. Cheng,
Y. Cheng,
X. Shan,
X. Shan,
J. J. Mou,
J. J. Mou,
L. Zhou e
L. Zhou e
Mohamed Salama
Mohamed Salama

Pubblicato online: 05 Sep 2022

Volume & Edizione: Volume 7 (2022) - Edizione 2 (July 2022)

Pagine: 921 - 930

Ricevuto: 13 Dec 2021

Accettato: 28 Apr 2022

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

Anteprima PDF

Articolo

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Figure e tabelle

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

The number of connected edges of each node in the command and control network.

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Applied Mathematics and Nonlinear Sciences

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

Y. Wang,Y. Wang,Y. Cheng,Y. Cheng,X. Shan,X. Shan,J. J. Mou,J. J. Mou,L. Zhou eL. Zhou eMohamed SalamaMohamed Salama

Pubblicato online: 05 Sep 2022

Volume & Edizione: Volume 7 (2022) - Edizione 2 (July 2022)

Pagine: 921 - 930

Ricevuto: 13 Dec 2021

Accettato: 28 Apr 2022

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

Anteprima PDF

Articolo

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Figure e tabelle

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

The number of connected edges of each node in the command and control network.

Riferimenti

Articoli recenti

Pianifica la tua conferenza remota con Sciendo

Y. Wang,
Y. Wang,
Y. Cheng,
Y. Cheng,
X. Shan,
X. Shan,
J. J. Mou,
J. J. Mou,
L. Zhou e
L. Zhou e
Mohamed Salama
Mohamed Salama