Multi-UAV counter-game model based on uncertain information

Applied Mathematics and Computation 366 (2020) 124684

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Multi-UAV counter-game model based on uncertain information Jiwei Xu a,1, Zhenghong Deng a,∗, Qun Song a,1, Qian Chi b,∗, Tao Wu a,c, Yijie Huang a, Dan Liu d,e, Mingyu Gao f a

School of Automation, Northwestern Polytechnical University, Xi’an 710072, China College of Mechanical and Electronic Engineering, Northwest A&F University, Yangling 712100, China c Equipment Management and UAV Engineering College, Air Force Engineering University, Xi’an 710051, China d School of Economics and Management, Chang’an University, Xi’an 710064, China e Department of Civil, Environmental, and Geomatics Engineering, Florida Atlantic University, Boca Raton, FL 33431, USA f College of Electronic Information, Hangzhou Dianzi University, Hangzhou 310018, China b

a r t i c l e

i n f o

Article history: Received 23 June 2019 Revised 7 August 2019 Accepted 18 August 2019

Keywords: Multi-UAV counter-game Interval decision Dominant equilibrium Situation matrix Uncertain information

a b s t r a c t When a multi-UAV is performing a mission, the information obtained by the commander could be highly uncertain. How to choose strategies based on uncertain information will directly affect the success or failure of the UAV mission. In this paper, we present a multiUAV confrontation model based on uncertain information, which can provide a powerful reference for strategy selection. We established a multi-UAV game model, and proposed a game payment function, and introduce a situation matrix to imitate the uncertainty of war information. In order to complete the calculation of uncertain information, the paper introduces a complex calculation method and establishes an interval decision model. The simulated experiment results show that the situation matrix proposed in this paper could represents the uncertainty in the battle field, and the interval decision method could process data effectively. Furthermore, the proposed algorithm could provide optimal strategy for decision support in battle field with uncertain information, and provide new research direction for game with uncertain information. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Since the emergence of game theory, its application in military operations has become a very important research hotspot, providing a new method for the research on the increasingly complex battlefield [1–12]. Meanwhile, in order to cope with the increasingly diverse battlefield situation, the form of UAV operations has become more and more complicated. In order to make the UAV have stronger capability for attacking and higher probability for mission completion, multi-UAV cooperative combat is also becoming a hotspot in the research community [13], including attack strategy selection, task assignment and control, and so on. In order to cope with these problems, the game theory is introduced into the research of UAVs, especially in the area of UAV communication. Based on the game theory, a fast and efficient deployment method for UAV networks is given in [14]. ∗

1

Corresponding authors. E-mail addresses: [email protected] (Z. Deng), [email protected] (Q. Chi). These authors contributed equally to this work

https://doi.org/10.1016/j.amc.2019.124684 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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J. Xu, Z. Deng and Q. Song et al. / Applied Mathematics and Computation 366 (2020) 124684

The effectiveness of the method is proved by experimental numerical analysis. In [15], they considered a differential game theoretic approach to compute optimal strategies by a team of UAVs to evade the attack of an aerial jammer on the communication channel. In [16], the game theory method is used to study the dynamic channel problem in the construction of wireless networks by UAVs. In [17], the Bayesian game model is used to study the trade-off between intrusion detection and overhead in the UAV network, and finally address the dilemma between the intrusion detection rate and the false positive rate. The algorithm implements an accurate detection of attacks with low overhead. In [18], the spectrum access problem of multi-UAV networks is studied from the perspective of game theory. In [19], the Bayesian-Stackelberg method is used to study the anti-interference transmission problem of UAV communication network. The performance of the algorithm is verified by simulation, and the influence of incomplete information and observation error on user utility is given. In [20], an evolutionary game-based UAV-assisted vehicle network mode selection method is proposed, and an evolutionary stabilization strategy is obtained. With this algorithm, the proportion of vehicles in different communication modes can quickly converge to the evolutionary equilibrium state. The algorithm has fast convergence speed, controllable convergence speed, high transmission reliability and low resource utilization cost. In [21], the distributed learning problem of high dynamic heterogeneous intelligent UAV network relay decision-making is studied by using the alliance game. Furthermore, the research community studied UAV countermeasure based on game theory. In [22], based on the unreliable characteristic of battle field, a method to obtain reliable information in battle field with help of multi-UAV consensus is proposed. In this study, different information in battle field is compared, and the similarity among the information is used to determine the reliability of the information, which lay the foundation for designing game strategy. In [23], the game model of dynamic weapon allocation for multi-UAV is proposed, which is based on the high precision and real time characteristic of multi-UAV fighting. In [24], the multi-UAV fighting is modeled as differential game problem, and the Nash differential game problem is converted to the corresponding differential vibrational inequality problem. However, in real-life battle field, due to the limitation in reconnaissance means, and the changing situation in battle field, there are always uncertainties existing in the information got by the commander. The uncertain information is not considered in the researches discussed above. In this paper, a countermeasure game model for multi-UAV is proposed based on uncertain information. On one hand, it could provide reference for decision making in battle filed with uncertain information. On the other hand, it could provide reference to game problem with uncertain information. Firstly, we designed the game model for multi-UAV, and proposed our payoff function. Secondly, we introduced situation matrix, so as to simulate the main resource of uncertain information under real-life scenario. Thirdly, interval data processing is applied, which transforms the uncertain information into certain information. The simulated experiment result shows that the uncertainty in the battle field could be represented by the proposed algorithm, and the optimal strategy of both the attacker and the defender could be estimated by this algorithm, which makes the proposed mode providing strong support to decision making in battle field. 2. Multi-UAV game model 2.1. Model analysis In the UAV game model, it is necessary to satisfy the basic three elements of game theory. The game model can be expressed as:

G={N, S, U }

(1)

Here, N represents the players of the game, N = {f, d}, with f for the attacker, and d for the defender. S represents the player’s strategies set, S = {sf ,sd }, with sf for the attacker’s strategies, and sd for the defender’s strategies. Generally, there are two strategies for sf and sd , one for attack and one for evasion. Hence, we have sf = {attack, evasion}, sd = {attack, evasion}. U is the payoff function interval between two parties, U={uf (si ),ud (sj )}. Here, uf (si ) represents the attacker’s payoff value under strategy si , and ud (sj ) represents the defender’s payoff value under strategy sj . 2.2. Payoff function In the multi-UAV game, it is assumed that the attacking party has m combat units and the defender has n units. Therefore, assuming that attacker j is attacking defender i, the attacker’s payoff function could be defined as:

uifj = pifj ∗ wdi − c f

(2)

f

Here, ui j is the attacker’s payoff function for attacker j attacking defender i. f pi j represents the probability of the attacker j destroying wdj represents the value of defender i. cf represents the value consumed by the attacker, mainly

defender i.

including air-to-air missiles and air-to-ground missiles. Similarly, the defender’s payoff function can be obtained as:

ubji = pbji ∗ w jf − cd

(3)

J. Xu, Z. Deng and Q. Song et al. / Applied Mathematics and Computation 366 (2020) 124684

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Suppose that the number of attacker’s strategy is Pand the number of defender’s strategy is Q, so the payoff function can be expressed as:

gi j =

P 

xi j ∗ uifj −

Q 

i=1

yi j ∗ udji

(4)

j=1



0, the ith attacker does not attack the jth defender

xi j =

1, the ith attacker attack the jth defender

(5)

 yi j =

1, the jth defender attack the ith attacker 0, the jth defender does not attack the ith attacker

(6)

Therefore, the attacker’s payoff matrix can be obtained as

s1

s1d s2d

G= . .. sQd

⎡f

...

s2f

g11 ⎢ g21 ⎢ . ⎣ .. gQ1

sPf ... ... .. . ...

g12 g22 .. . gQ2

⎤

g1P g2P ⎥ .. ⎥ ⎦ . gQP

(7)

3. Uncertain information processing 3.1. Source of uncertain information The information uncertainty of the game is mainly caused by the situation of both parties. Situation is an assessment of enemy’s threat to us based on the destruction ability, maneuver ability and behavior intention of enemy aircraft, which is the basis of air combat decision. There are many factors influencing the situation, such as angle, distance and speed, which make it impossible for both sides to accurately calculate the real-time situation. Therefore, it is necessary to estimate the situation, the situation matrix can be expressed as:

⎡

s11 ⎢ s21 S=⎢ . ⎣ .. sm1

s12 s22 .. . sm2

⎤

... ... .. . ...

s1n s2n ⎥ .. ⎥ ⎦ . smn

(8)

The element sij in the matrix can be seen as the advantage of the UAV i to the UAV j, which is an estimated value, defined ij

ij

ij

ij

as the interval form: si j = [smin , smax ]and satisfies 0 < smin < smax < 1. As can be seen from the above, the value of the payoff function is an interval and the data distribution inside this interval is not clear. In real-life scenarios, the value of the payoff function is not evenly distributed, and the situational function is not evenly distributed. One of the reasons is that the situational function changes as the position of the UAV changes. During the quantitative process of situation matrix, the interval representation is used, which could be looked as the sum of certain information and uncertain information. 3.2. Uncertain information In real-life applications, there exists uncertainty in the measurement result due to the limitation of measuring tools, the changing measurement environment, and so on [25,26]. The uncertainty could be represented by interval data. However, the interval data could be used in game computing without further processing to the interval data. Therefore, in this paper, we introduce a method to process the interval data, which provide a new way for interval decision making. Suppose an interval data x = [x1 ,x2 ]. x could be represented by plural form:

u = a + bi

(10)

Here, a = x1 ,b = x2 − x1 , and i ∈ [0, 1]. Therefore, it can be seen that uand x are equivalent. Converting a plural form to a triangle form, we get:

u = r (cos θ + i sin θ )



(11) x −x

where r = (x1 ) + (x2 − x1 ) , θ = arctan 2x 1 . 1 For two intervals u1 = r1 (cos θ 1 + isin θ 1 ) and u2 = r2 (cos θ 2 + isin θ 2 ), their production could be defined as 2

2

u1 u2 = r1 r2 (cos (θ1 + θ2 ) + i sin (θ1 + θ2 ) )

(12)

4

J. Xu, Z. Deng and Q. Song et al. / Applied Mathematics and Computation 366 (2020) 124684 Table 1 The description of the attacker and defender. Unit

Weapon

Attack ability

Information type

f1 , f2 , f3 f4 , f5 d1 , d2 , d3 d4 , d5 T

Air-to-air missile Air-to-ground missile Air-to-air missile Ground-to-air missile No

Air attack capability Ground attack ability Air attack capability Air attack capability No

Uncertain Uncertain Uncertain Uncertain Uncertain

Table 2 Unit value of attackers and defenders. UAV value of the attacker

Value interval

UAV value of the defender Air defense position value of the defender Target value of the defender Air-to-air missile value of the defender Ground-to-air missile value of the defender UAV value of the attacker Air-to-air and air-to-surface missile value of the attacker

[2000,2100] [2050,2150] [3000,3200] [95,100] [100,120] [300,310] [5000,5300]

Therefore, the general value is defined as:

M ( Sk ) =

N 

r1k r2k . . .rNk (cos (θ1k + θ2k + · · · θNk ) + i sin (θ1k + θ2k + · · · θNk ) )

(13)

i=1

In particular, whencos (θ 1k + θ 2k + θ Nk ) + i sin (θ 1k + θ 2k + θ Nk ) = 1, we get:

M ( Sk ) =

N 

r1k r2k . . .rNk

(14)

i=1

The model defined in formula (14) is named as integrated principal value model.

4. Numerical results 4.1. Data Here, we study the military scenario in which the attacker has 5 UAVs, with three of the UAVs carrying 1 air-to-air missile, and two UAVs carrying 1 air-to-ground missile. The defender has three UAVs, with each UAV carrying 1 air-to-air missile. The defender has two air defense bases, with each base being able to launch 1 ground-to-air missile. In addition, defender has a high value target which named T. Here, we write f = {f1 ,f2 ,f3 ,f4 ,f5 } for the attacker, and d = {d1 ,d2 ,d3 ,d4 ,d5 ,T} for the defender. The description of the attacker and defender are shown in Table 1, and the unit values of both the attacker and the defender are shown in Table 2. Currently, the hit rate of air-to-air missile without interference is about 90%, it is 50−60% when there exists interference. In this paper, we set the hit rate to about 70%. According to figures publicly available, the first hit rate of ground-to-air missiles is about 80−90%, and in this paper, we also take this value for the corresponding interval. Generally, for air-toground missiles, the hit rate is high, so we set the value to about 90%. When defining the hit rate of both the attacker and the defender, the hit rate value of the target that cannot be attacked is defined as 0. Therefore, in our simulation, we set the hit rate matrix for the attacker and the defender as illustrated in formula (15) and formula (16), respectively.

d1 ⎡ f1 [0.70, 0.73] f2 ⎢[0.71, 0.73] Pf = f3 ⎢[0.72, 0.75] ⎣ f4 [0, 0] f5 [0, 0]

d2 [0.69, 0.71] [0.70, 0.73] [0.71, 0.74] [0, 0] [0, 0]

d3 [0.70, 0.72] [0.71, 0.73] [0.69, 0.71] [0, 0] [0, 0]

d4

[0, 0] [0, 0] [0, 0] [0.87, 0.91] [0.87, 0.92]

d5

[0, 0] [0, 0] [0, 0] [0.87, 0.93] [0.87, 0.91]

T

⎤

[0, 0] [0, 0] ⎥ [0, 0] ⎥ ⎦ [0.93, 0.95] [0.93, 0.95]

(15)

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Fig. 1. Attacker’s strategies.

⎡ f1 d1 [0.62, 0.65] d2 ⎢[0.72, 0.73] d ⎢[0.77, 0.79] Pd = 3 ⎢ d4 ⎢[0.81, 0.84] ⎣ d5 [0.88, 0.92] T [0, 0]

f2 [0.68, 0.71] [0.70, 0.73] [0.75, 0.76] [0.79, 0.83] [0.86, 0.87] [0, 0]

f3 [0.70, 0.73] [0.71, 0.73] [0.67, 0.75] [0.83, 0.85] [0.85, 0.89] [0, 0]

f4 [0.72, 0.75] [0.69, 0.71] [0.62, 0.68] [0.81, 0.84] [0.83, 0.87] [0, 0]

f5 ⎤ [0.75, 0.79] [0.74, 0.79] ⎥ [0.72, 0.75]⎥ ⎥ [0.85, 0.87]⎥ ⎦ [0.89, 0.93] [0, 0]

(16)

Generally speaking, when the UAV is carrying out a task, the attacker will try its best to keep concealed, so as to reduce the probability of being discovered, and to obtain the situational advantage as much as possible. As far as the two sides play, the situational advantage of one side is the situational disadvantage of the other side. For convenience, the value of the situation matrix is defined as follows.



max sifj + min sidj = 1, min sifj + max sidj = 1, effectivesituation

(17)

max sifj = min sifj = max sidj = min sidj = 0, invalidsituation Therefore, the situational advantage matrix of the attacker is defined as:

d ⎡ 1 f1 [0.57, 0.63] f2 ⎢[0.59, 0.65] S f = f3 ⎢[0.78, 0.79] ⎣ f4 [0, 0] f5 [0, 0]

d2 [0.54, 0.61] [0.68, 0.73] [0.55, 0.60] [0, 0] [0, 0]

d3 [0.57, 0.60] [0.53, 0.58] [0.52, 0.56] [0, 0] [0, 0]

d4 [0, 0] [0, 0] [0, 0] [0.85, 0.89] [0.83, 0.85]

d5 [0, 0] [0, 0] [0, 0] [0.79, 0.87] [0.83, 0.85]

T ⎤ [0, 0] [0, 0] ⎥ [0, 0] ⎥ ⎦ [0.85, 0.91] [0.87, 0.90]

(18)

4.2. Attacker’s strategies and defender’s strategies From the perspective of the attacker, it has three UAVs carrying air-to-air missiles, which would attack the defender’s three UAVs at the same time. Because one UAV attacked one target with a hit rate of about 70%, the two UAVs attacked one target at the same time could have a hit rate of about 91%, and the three UAVs attacked one target at the same time could have a hit rate of about 97%. Therefore, it can be seen that 2VS1 and 3VS1 are strictly dominated strategies. When calculating the payoff matrix, we will eliminate strictly dominated strategies. Therefore, there are six strategies for f1 , f2 and f3 . The other two UAVs of the attacker have 3 strategies for each, and hence totally there are 6 strategies. Therefore, the attacker has a total of 54 strategies. The attacker’s strategies are shown in Fig. 1.

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Fig. 2. Defender’s strategies.

From the point of view of the defender, its UAV and air defense base can attack the attacker’s UAV. According to the analysis of the previous section, 2VS1 and 3VS1 are strictly dominated strategies. When the defender chooses a strategy, choosing the 1VS1 strategy is a rational choice. Thus, the defender has 120 strategies. The defense’s strategies are the full alignment of the attackers, which is shown in Fig. 2. 4.3. Numerical experiment Different i values were selected in our experiment, including i = 0, i = 0.5, i = 1 and cos θ + isin θ = 1. For i = 0, i = 0.5, and i = 1, the value of i keeps constant, and for cos θ + isin θ = 1, the value of i changes for each interval. The experiment results are shown in Figs. 3–6. In these figures, the abscissa and the ordinate represents the strategies of the defender, and the attacker, respectively. The pixels in the graph represent the attacker’s payoff under both strategies, and the value of the payoff refers to the legend of the figure as this is a zero-sum game, only the payoff of the attacker is shown in the figure, and the defender’s pay-off is the negative of the attacker’s pay-off. As can be seen from Figs. 3–6, there is a significant difference in the pay-off matrix of attacker in each of the graphs for different value of i. That is to say, different values of i can reflect different pay-off of attackers. Therefore, it is shown that the uncertain information processing method presented in this paper can reflect the uncertainties of information. It is also shown in Figs. 3–6 that under the same strategy, the changes among the benefit of the attacker is not obvious. That is to say, at the same pixel position of Figs. 3–6, the difference between the values of the four figures is small, and the maximum difference is about 167. The reason for this result is that the difference between the upper and lower limits of the data we used here is small. This shows that the interval data processing method given in this paper can accurately reflect the changes of data. As can be seen from Figs. 3–6, there are significant differences in pay-offs under different strategies. In order to improve their own payoff, both players can choose the appropriate strategy, which also verifies the effectiveness of the method given in the article. The reason is that when setting up the situation matrix, the attacker occupies a first mover advantage, hence takes an advantage position, which in turn improves the payoff of the attacker. It is also shown in Figs. 3–6 that the attacker has a strictly dominated strategy, which is obviously in the first row, the fifth row, the ninth row, the tenth row, etc., in each figure. There are 18 strictly dominated strategies. Among these strategies, there are two UAVs (f4 and f5 ) that attack the same target (d4 ,d5 and T). The reason is consistent with the analysis in Section 4.2, and it is also consistent with our intuition. Through the comparative analysis of the data in Figs. 3–6, it is shown that there is a dominant strategy for the attacker If the attacker adopts strategy 48, the attack order of the attacker is (3, 2, 1, 4, 6), and the attacker will obtain the maximum payoff. If the defender adopts strategy 89, the attack order of the defender is (2, 3, 1, 5, 4), and the defender will obtain the maximum payoff. When the attacker selects strategy 48 and the defender selects strategy 89, the game has a dominant

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Fig. 3. Pay-off matrix of attacker, i = 0.

Fig. 4. Pay-off matrix of attacker, i = 0.5.

equilibrium that is a Nash equilibrium of the game. In the dominant equilibrium, the rewards of the attacker returns are: 7487, 7963, 8439, and 7482, respectively, and the rewards of the defender are the opposite of the attacker. Therefore, the model can provide an optimal decision strategy for both attacker and defender.

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Fig. 5. Pay-off matrix of attacker, i = 1.

Fig. 6. Pay-off matrix of attacker, cos θ + isin θ = 1.

5. Conclusion In the process of multi-UAV battles, there is serious uncertainty in the battlefield information. How to control the uncertainty of the battlefield and adopt the correct strategy are crucial to the victory. This paper aims at modeling and analyzing the optimal strategy selection of multi-UAV under uncertain information, and gives a multi-UAV counter-game model with uncertain information. The simulation analysis shows that the algorithm proposed in this article can reflect the uncertainty

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