Qilong SUN,Naiming QI,Zheyao XU,Yanfang LIU,Yong ZHANG

Department of Aerospace Engineering,Harbin Institute of Technology,Harbin 150001,China

An optimal one-way cooperative strategy for two defenders against an attacking missile

Qilong SUN,Naiming QI,Zheyao XU,Yanfang LIU*,Yong ZHANG

Department of Aerospace Engineering,Harbin Institute of Technology,Harbin 150001,China

This paper investigates a cooperative strategy for protecting an aerial target.The problem is solved as a game among four players(a target,two defenders,and a missile).In this scenario,the target launches two defenders(defender-1 and defender-2)simultaneously,to establish a one way cooperation system(OCS)against an attacking missile.A new optimal evasion strategy for the target is also derived.During the engagement,the target takes into account the reaction of the attacking missile,and guides defender-1 to the interception point by receiving information from defender-1.Depending on the control effort of the target,defender-2 can choose appropriate launch conditions and use very limited maneuvering capability to intercept the missile.For adversaries with first-order dynamics,simulation results show that the OCS allows two defenders to intercept the missile.During the engagement,even if one defender or communication channel is broken,the OCS still allows an interception to be made,thus increasing the target’s survivability.

?2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

With the rapid development of interception technology,threats to aircraft are growing.Thus,aircraft need to establish a defense system to protect themselves.The problem of an aircraft defense system involves multiple agents,including an aircraft,an attacking missile,and a defending missile.In this scenario,the aircraft,the attacking missile,and the defending missile are referred to as the target,the attacker,and the defender,respectively.The interception problem conventionally involves a single missile pursuing a target.Over the years,a variety of guidance laws have been proposed to describe such one-on-one engagements while assuming perfect information.1,2

In recent years,various pursuit-evasion scenarios involving multiple agents have been developed.A pursuit-evasion scenario in which several pursuers try to capture a single evader has been described.3–6A game including two pursuers and one evader was investigated.7–9In this game,the two pursuers attempted to minimize their distances from the evader,while the evader attempted to maximize its distances from the two pursuers.The game was investigated using differential theories.Rusnak10proposed a differential game for three persons,namely,a lady,a bandit,and a bodyguard.In order to solve the differential game,an approach based on multiple-objective optimization theory was presented,and the required conditions for the solution were discussed.Ratnoo and Shima11proposed an approach for protecting the target in the case where the defender used the line-of-sight guidance law to pursue the attacker at a speed at least as fast as that of the attacker.A different approach was used by Yamasaki et al.12,13to address a scenario in which the target protected itself by launching a defender that used a new line-of-sight guidance law.Ratnoo and Yamasaki proposed a similar method,wherein the purpose was to command the defender to be in the line of sight between the attacker and the target at all times.This approach performs better than the one where the defender pursues the attacker using a linear guidance law.Rubinsky and Gutman14–16investigated an approach wherein the attacker could evade the defender and continue pursuing the target in a scenario in which the target and the defender were independent and provided no help to each other.Further,Gutman and Rubinsky17,18,in order to reduce the time error,used a new method to calculate the interception time and proposed a three-dimensional nonlinear vector guidance strategy.Ratnoo and Shima19analyzed a three-player problem in which the attacker and the defender used different guidance laws.

Recently,different types of models of cooperation between the target and the defender against the attacker have been presented.Perelman et al.20reported a cooperative targetdefender guidance law with unbounded controls based on a two-team linear quadratic differential game against a homing missile.In this scenario,the target assists the defender in intercepting the missile.Meanwhile,the target also performs evasive maneuvers.Shaferman and Shima21analyzed a multiplemodel adaptive control scheme to propose a possible homing-missile guidance law.Further,Shima proposed twoway linear quadratic strategies for optimal cooperation between the target and the defender against a homing missile.The missile used a known linear guidance law,but the defender had limited maneuverability to intercept the attacker.22However,it has been shown that two-way cooperation between the defender and the target requires installation of additional equipment on the defender,which may often be unavailable.23In the same study,a one-way cooperation strategy was presented wherein only one defender protected the target.In this strategy,the defender sends information independently to the target,which lures the missile in a manner to assist the defender to intercept it.The target sends a message to the defender that allows it to turn toward the predicted intercept point.However,the navigation gains and launch conditions of the defender were not analyzed.A differential cooperative game wherein the target and the defender cooperate against the attacker was reported by Garcia et al.24,25These studies provided optimal strategies for each one of the players and performed additional analyses of the target escape regions for a given target/attacker speed ratio.

However,in these studies,the defense system only had one defender and the navigation gains of the players were not analyzed.Thus,if the defender or communication channel was to break,the interception would fail.Meanwhile,because the navigation gains of the players were not analyzed,the control effort of the defender would be easily saturated,as it would not be able to select a reasonable navigation gain.In this paper,we set an aircraft’s protection as the background when the aircraft faces an interceptor’s threat.In order to increase the level of the aircraft’s protection and survivability,we design a system with two defenders to protect the aircraft.The defenders and the aircraft exhibit one-way cooperation,and communication between the aircraft and the two defenders is complementary.Thus,even if one of the defenders or communication channels is broken,the system can still intercept the missile.The navigation gain of the defenders is analyzed,and one can select a reasonable navigation gain for the defenders while avoiding the need for control efforts larger than the defenders’maximum control ability.

2.Problem formulation

2.1.Problem description

The endgame scenario and problem flow chart are displayed in Fig.1.The problem consists of four entities:an attacking missile(M),an aircraft described as a target(T),and two defenders(D1andD2).It is assumed that defender-1 uses a linear guidance strategy to intercept the attacking missile while sending information to the target,and that defender-2 receives the control effort information from the target regarding the predicted intercept point.The target launches the defenders to form an OCS to protect itself.It is assumed that the missile attacks the target using a linear guidance strategy that is known to the target.The ranges between the target and the missile,the missile and defender-1,and the missile and defender-2 are denoted byRMT,RMD1,andRMD2,respectively.Further,VM,VT,VD1,andVD2are the velocities of the missile,the target,defender-1,and defender-2,respectively.Their lateral accelerations are denoted asaM,aT,aD1,andaD2,respectively.The lines of sight between the target and the missile,the missile and defender-1,and the missile and defender-2 are denoted by LOSMT,LOSMD1,and LOSMD2,respectively.Further, λMT,λMD1,and λMD2are the angles between LOSMT,LOSMD1,and LOSMD2and theX-axis,respectively.The flight path angles of the missile,the target,defender-1,and defender-2 are denoted as γM,γT,γD1,and γD2,respectively.The subscript 0 represents the initial state of the corresponding parameter.uM,uT,uD1,anduD2are the controllers of the agents normal to the corresponding LOS.

Fig.1 Engagement geometry and problem flow chart.

2.2.Nonlinear kinematics

The range rates are

The LOS rate relations satisfy the following expressions:

The path angle relations satisfy the following expression:

During the engagement,it is assumed that the dynamics of each player correspond to those of a linear time-invariant system,which can be represented by the following equations:

where ηiis the vector of the internal state of each agent with dim(ηi)=ni(i=M,T,D1,D2)andrepresents its controller.Ai,bi,Ci,diis the player’s dynamics state-space model matrices.

2.3.Linearized kinematics

The relative displacement between the missile and the target normal to LOSMT0is denoted byyMT.The relative displacement between the missile and defender-1 normal to LOSMD10is denoted byyMD1,and the relative displacement between the missile and defender-2 normal to LOSMD20is denoted byyMD2.The accelerations of the missile and the target normal to LOSMTare denoted asuMLanduTL,respectively.The accelerations of defendersD1andD2normal to LOSMD1and LOSMD2are denoted asuDL1anduDL2,respectively.uML,uTL,uDL1,anduDL2can be linearized at the initial lines of sight LOSMT0,LOSMD10,and LOSMD20by the following expressions:

uM,uT,uD1,anduD2are defined in the following relations:

For the entire duration of the game,we assume that the velocity vectors of the four players are not perpendicular to the corresponding LOS.The state vector of the linearized engagement is then expressed by

where the dimension of x is 6+nM+nT+nD1+nD2.

The equations of motion corresponding to Eq.(12)are as follows:

wherea=cos(γT0+ λMT0),b=cos(γD10+λMD10),c=cos(γD20+λMD20),d=cos(γM0- λMT0),e=cos(γM0- λMD10), andf=cos(γM0- λMD20).

Eq.(13)can be written as

where

By linearizing the kinematics around the collision triangles,the closing speeds between the attacking missile and the evading target,the attacking missile and defender-1,and the attacking missile and defender-2 can be regarded as being constant.The closing speeds are defined asVMT,VMD1,andVMD2,respectively,which are given by

whereRMT0,RMD10,andRMD20are the initial ranges between the attacking missile and the target,the attacking missile and defender-1,and the attacking missile and defender-2,respectively.

It is required that the defenders intercept the attacking missile before it hits the target.It is assumed that the defenders disappear after timeis defined by wheretrepresents the time,is the time before the attacking missile hits the evading target,is th-e time before defender-1 intercepts the missile,andis the time before defender-2 intercepts the missile.

2.4.Missile guidance law

Over the years,a number of guidance laws have been developed.When deriving a guidance law,perfect information,linear kinematics,and unbounded controls are usually assumed.Under these assumptions,the derived guidance laws have a common form.1For the entire duration of the game,we assume that the guidance of the attacking missile is given by the following linear form:

K1,K2,KT,KMare the matrix elements to the corresponding guidance law.The most well-known among these linear guidance laws are the proportional navigation(PN)guidance law,the augmented proportional navigation(APN)guidance law,and the optimal guidance law(OGL),which have the following form:

whereNjis the navigation gain andZjis the zero-effort-miss(ZEM)distance corresponding to the guidance law.

3.Optimal target evasion strategy

In this section,a new optimal target evasion strategy is proposed.It is assumed that a success of the defending missiles is not guaranteed.Thus,the game will terminate atIt is also assumed that the two defenders are launched at the same time and that once defender-1 intercepts the attacking missile,the target evades the missile,whose guidance law is known to the target.Then,a new optimal target evasion strategy is designed.

The equations for the evasive motion are given by

Eq.(22)can be written as

In Section 2,it has been stated that the guidance law of the attacking missile is in a linear form.By substituting the attacking missile’s guidance law from Eq.(18)into Eq.(23),a new equation for the evasive motion can be written as follows:

where

whereZMTis the ZEM distance between the missile and the target,t1refers to the time available for the target to evade the attacking missile, α and ρ are nonnegative weights,andb1is a constant parameter related to the maneuverability of the target and is designed to reduce the variability of the control commands.26

To solve this optimal control problem,the well-known ZEM distance transformation is used as follows:

In order to solve the transition matrix,the running timetis substituted withand

Consequently,we get where φ1,1,φ1,2,φ1,3,φ1,4are the matrix elements in the first line of

It is assumed that the missile and the target exhibit firstorder acceleration dynamics,and that their time constants are 0.05 s and 0.1 s,respectively.It is also assumed that the missile uses the PN guidance law,and that the parameters are satisfied by whereNPNis the navigation gain of the missile.

ZMT(t)is a function of the transition matrix elements multiplied by the current state vector.The time evolutions of the elements corresponding to Eq.(29)for differentNPNare shown in Fig.2.Note that,for the same time value,the absolute values of the elements increase with a decrease inNPN.IfNPNreaches a certain value,the elements corresponding toapproaching zero will have both positive and negative values.NPNrepresents the attacking missile’s control gain for pursuing the target.The control effort of the missile to pursue the target increases,and the evasion distance from the missile for the target decreases with an increase inNPN.

On differentiating Eq.(26)with respect to time and by combining the derivative of the transition matrix from Eq.(28),we obtain

The Hamiltonian function of the problem corresponding to Eq.(25)is

where

Fig.2 Time evolutions of the evasion-related transition matrix elements for different NPN.

The solution of Eq.(33)is given by

The optimal strategy,uT,minimizes the Hamiltonian function and satisfies

Then,the open-loop optimal,uT,is obtained as

On substituting Eq.(36)into Eq.(31)and integrating fromt1toin the following form:

Finally,on substituting Eq.(38)into Eq.(36),uTcan be determined as

By introducing Eq.(38)and Eq.(39)into Eq.(25),the cost function can be written as a quadratic equation ofb1.The extreme point of the quadratic equation is given by

On substituting Eq.(40)into Eq.(39),we get the open-loop guidance law as

By introducing Eq. (41) intoZMT(t)=ZMT(t1)+and combining Eq.(41),the closed-loop optimal evasion guidance law can be written as

4.One-way cooperation system strategies

In this section,we discuss the optimal guidance law for the OCS.In the previous section,we have assumed that the guidance of the attacking missile has a linear form that is known to the target and that defender-1 uses a linear guidance strategy to intercept the attacking missile.During the game,defender-1 sends information to the target.The target can use this information to predict the positions of the attacking missile and defender-1.The maneuvering of the missile depends on the target.Thus,the target can take into account the reaction of the attacking missile when guiding defender-1 to the interception point.Meanwhile,information related to the maneuvering of the target is sent to defender-2.Thus,defender-2 can predict the positions of the target and the missile.The initial conditions of the missile,the target,and defender-1 are known,and in order to obtain a more reasonable guidance law,the navigation gain of defender-1 is analyzed.The optimal guidance law for defender-2 is also derived,and its reasonable launch conditions are analyzed.

4.1.OCS dynamics

It is assumed that the guidance law of the attacking missile has a linear form.The guidance strategy that defender-1 uses to intercept the attacking missile is a linear one and has the following form:

whereNPND1is the navigation gain of defender-1.

By substituting Eq.(30)and Eq.(43)into Eq.(14),the dynamics of the OCS can be written as

During the game,the aim is to determine the optimal solution to the one-way cooperative problem.The traditional cost function is

In order to reduce the variability of the control commands,uT,obtained from the traditional cost function,we select the following cost function26:

In order to obtain the defender-2 optimal guidance law,we select the following cost function:

Thus,we need to obtain the optimal guidance law and minimize the cost functions.

4.2.Derivation of optimal guidance law

To solve the optimal control problem,the well-known ZEM distance transformation is used as follows:

where

On substitutingtwithwe get

Eq.(50)can be written in the following form:

where φ5,i(i=1,2,...,10)are the matrix elements in fifth line ofand φ8,i(i=1,2,...,10)are the matrix elements in eighth line of

In order to determine the ZEM dynamics,the derivatives ofZMD1andZMD2with respect to time can be written as

It can be seen that the dynamics ofZMD1depend only on the controller of the target,uT,because the guidance laws of the missile and defender-1 are known.The dynamics ofZMD2are dependent only on the controller of defender-2,uD2,becauseuThas been determined based on the cost functions and the dynamics ofZMD1andZMT.

To solve the optimal problem related toyMD1Eq.(46)and Eq.(47)can be rewritten as follows:

By using a method similar to that used in Section 3,the closed-loop optimal cooperative-luring strategy of the target corresponding to Eq.(55)can be written as

The optimal cooperative-pursuit strategy of defender-2 is determined in the following section.The Hamiltonian for this problem is

The adjoint equation and transversality condition are

which yield the following:

The open-loop optimal strategy of defender-2 minimizes the Hamiltonian is defined bythat satisfies

The closed-loop optimal strategy is

On substituting Eq.(62)into Eq.(54),we obtain

It is noted that the signs of(t)are opposite;therefore,|ZMD2(t)|is monotonically decreasing,and the guidance law converges.By replacingtwith,Eq.(63)can be expressed as follows:

By integrating Eq.(64)from any given initial conditionZMD2we can obtain the corresponding optimal trajectory.The optimal border trajectoriescan be written as

It is assumed that the acceleration command of defender-2 is limited to 3g(gis acceleration of gravity),that=3 s,and that the other parameters are the ones shown in Table 1 with γD10=-20°.Given these assumptions,the optimal ZEM trajectories of the missile and defender-2 for different initial defender-2 conditions are shown in Fig.3,in which the horizontal axis representsIt can be seen that the optimal trajectories are parallel to each other,and that the optimal border trajectories,are represented by the thick lines.If the initialZMD2(t=0)lies in the region between the optimal border trajectories,one can guarantee a zero miss distance.

5.Numerical simulations

In this section,the results of numerical simulations of the OCS guidance laws derived in the previous sections are presented.It is assumed that the missile,the target,and the two defenders exhibit first-order acceleration dynamics,with time constants of 0.05 s,0.1 s,1/30 s,and 1/30 s,respectively.The initial parameters are listed in Table 1.

Table 1 Initial parameters.

Fig.3 Optimal ZEM trajectories of the missile and defender-2.

Fig.4 Time evolution of the ZEM distance between the missile and defender-1 for different initial courses.

We firstly analyze the convergence of the proposed guidance corresponding to Eq.(57).DefiningNPND1=0.8,NPN=3,α → ∞,and ρ =1 of Eq.(57),Fig.4 shows the time evolution ZEM distance between the missile and defender-1 for different initial courses of defender-1.It is noted that when using Eq.(57)’s guidance law,|ZMD1|monotonically decreases,and the guidance law converges.

Fig.5 Initial ZEM distance between the missile and defender-1 for different NPND1.

Then,we analyze the influences of the parameters on the derived guidance law for γD10=-20°because the influences are similar for other engagement scenarios.Fig.5 shows the initial ZEM distance between the missile and defender-1 for differentNPND1.It is clear that the initial ZEM distance decreases with an increase inNPND1,which represents defender-1's control gain for pursuing the attacking missile.Specifically,the control effort of defender-1 will increase and the ZEM distance will decrease asNPND1increases.This is because whenNPND1increases,defender-1 will use more control effort to pursue the missile.In order to examine the engagement between the missile and defender-1,we set α → ∞ and ρ =1 in the cost functions.Fig.6 shows the time evolutions of the control efforts of the target for differentNPND1.Fig.6(a)and(b)are obtained using the designed optimal guidance law while Fig.6(c)and(d)are obtained using the traditional cost function.Note that the control effort of the target based on the designed optimal guidance law is more stable than the outcome for the traditional guidance law.Thus,the designed guidance law has an advantage for the target’s control.

Fig.7 shows the control efforts of defender-1 under the designed optimal guidance law for differentNPND1with α→ ∞ and ρ =1.From Figs.6(a)and 7,it can be noted that the control effort of defender-1 decreases and that of the target increases asNPND1decreases.The fact that the control effort of defender-1 decreases with a decrease inNPND1means that the target needs to lure the missile to a greater degree in order to ensure the engagement.Defender-1 does not exhibit a control effort whenNPND1=0.In this case,the engagement between the missile and defender-1 only depends on the luring action of the target,whose control capability is significant.The time history of the ZEM distance under the designed optimal guidance law for differentNPND1with α → ∞ and ρ =1 is shown in Fig.8.It can be seen that,when the derived guidance law is used,ZMD1decreases over time.Further,at the interception time,the ZEM distance decreases to 0,and interception occurs.

Fig.6 Control efforts of the target for different NPND1.

Fig.7 Time evolution of defender-1's control efforts for different NPND1(α → ∞,ρ=1).

Fig.8 Time evolution of the ZEM distance between the missile and defender-1 for different NPND1(α→∞,ρ=1).

Fig.9 Time evolution of the ZEM distance and control efforts of the target and defender-1 for different α (ρ =1).

The time evolution of the ZEM distance and the control efforts of the target and defender-1 for different α calculated using Eq.(55)withNPND1=0.8 are shown in Fig.9.It can be seen that the ZEM distance decreases and the control effort of the target increases when the relative penalty increases.This is because when α increases,in order to make the cost function have the minimum value,the target needs to use more control effort to lure the missile,and thus the ZEM distance decreases.We note that when α is not very large,the final miss distanceZMD1is not equal to zero,and defender-1 cannot achieve a hit-to-kill.According to defender-1's linear guidance law,whentapproachesapproaches zero,and the control effortuD1increases to a large value which is not convergent.In reality,the control efforts of the players are bounded.If the engagement between the missile and defender-1 is to terminate,the target will switch actions to evade the missile by using the optimal evasion strategy.During the entire engagement,it is assumed that the accelerations of the target,defender-1,and defender-2 are bounded and limited to 8g,8g,and 3g,respec-tively.The ZEM distance between the target and the missile and the control effort of the target for α→∞,ρ=1,NPN=3,andNPND1=0.8 areshown in Fig.10.It can be seen that,at the final interception time,the ZEM distance is about 0.01 m.The value is small because the control effort of the missile is not limited and unbounded.However,in reality,the control ability of the missile is bounded.In this study,we focused on the cooperative guidance law for intercepting the missile so we did not limit the missile’s control ability.

Fig.10 Time evolution of the ZEM distance and the control effort of the target(α → ∞,ρ =1,NPN=3,NPND1=0.8).

Fig.11 Permitted range of the initial course of defender-2(γD20)and corresponding ZMD2(t=0).

The next step is to examine the engagement between the missile and defender-2.The control effort of the target determines reasonable initial launch conditions for defender-2,which are used to complete the interception with highly limited maneuvering capabilities.The permitted range of the initial course of defender-2(γD20)leading toZMD2(t=0)in the region between the optimal border trajectoriesis fitted.The result is shown in Fig.11.If γD20satisfies the condition-25.9°≤ γD20≤ -29.1°,defender-2 can intercept the missile with a very limited control ability.Thus,the permitted range of the initial course of defender-2 is narrow,if defender-2's control ability is very limited.To expand the permitted range of γD20,the primary task is to increase the maximal control ability of defender-2.The optimal guidance law of defender-2 is examined for different γD20.A zero miss distance between the missile and defender-2 is achieved,as shown in Fig.12(a).Fig.12(b)shows the time evolution of|B2|.It can be seen from the figure thatZMD2converges rapidly to zero if γD20is located between-27°and-28°.This can provide enough time for fine-tuning when the target adjusts its control effort,causingZMD2to change.Further,the figure also shows that based on Eq.(54),the convergence speed ofZMD2decreases over time because|B2|decreases.IfZMD2(t=0)has a large value such as 110 m,based on γD20=-29.2°,thenZMD2cannot be decreased to zero by using the optimal guidance law,,for defender-2.This is becauseZMD2converges primarily in the initial stage when|B2|has a relatively large value.After a while,the value of|B2|decreases,˙ZMD2becomes small,and the convergence speed ofZMD2is very small.As a result,ZMD2cannot decrease to zero.Hence,γD20must be assigned a reasonable value,given that defender-2 has an extremely limited control ability.Fig.13 shows the trajectories of the four players for γD20=-28°.

Fig.13 Trajectories of the four players for γD20=-28°.

Fig.12 Time evolutions of ZMD2and|B2|for different γD20.

6.Conclusions

In previous works,an active defense system has one defender.The defender launched from the target sometimes cannot intercept the attacker because of the limits on its own guidance and the high maneuverability of the attacker.Meanwhile,once the communication between the target and the defender breaks,the target will be intercepted.In this paper,the proposed OCS containing two defenders can increase the target’s survivability.The problem is described for two defenders launched simultaneously from the target to establish the OCS.In the system,defender-1 sends information to the target,and defender-2 receives the control effort information from the target regarding the predicted intercept point;thus,the information forms a closed loop.The performance of the derived guidance law is analyzed through numerical simulations.The following conclusions can be drawn from the results.

(1)The OCS guidance law allows two defenders to intercept the missile with a zero miss distance,and the control effort of the target based on the designed optimal guidance law is more stable than the outcome for the traditional guidance law.

(2)The effect of the navigation gain of defender-1 is also analyzed.As a result,the control efforts required can be kept below the maximum control ability.Based on the control effort of the target,which depends on the guidance law of defender-1,defender-2 can choose reasonable launch conditions to intercept the missile with a highly limited maneuvering capability.

(3)Further,during the engagement,even if one of the defenders or communication channels is broken,the OCS can still complete the interception mission and hence increase the survivability of the target.The proposed guidance law has potential for use with defense systems.

Acknowledgements

This work was co-supported by the National Natural Science Foundation of China(No.11672093)and the Shanghai Aerospace Science and Technology Innovation Foundation(No.SAST2016039).

1.Zhang Y,Ma G,Liu A.Guidance law with impact time and impact angle constraints.Chinese J Aeronaut2013;26(4):960–6.

2.Song J,Song S.Three-dimensional guidance law based on adaptive integral sliding mode control.Chinese J Aeronaut2016;29(1):202–14.

3.Huang H,Zhang W,Ding J,Stipanovic DM,Tomlin CJ.Guaranteed decentralized pursuit-evasion in the plane with multiple pursuers.Proceedings of the 50th IEEE conference on decision and control and European control conference;2011 Dec 12–15;Orlando,USA.Piscataway(NJ):IEEE Press;2011.p.4835–40.

4.Zhao S,Zhou R.Cooperative guidance for multimissiles using cooperative variables.Acta Aeronaut et Astronaut Sin2008;29(6):1605–11[Chinese].

5.Zhao S,Zhou R.Cooperative guidance for multimissile salvo attack.Chinese J Aeronaut2008;21(6):533–9.

6.Zhang Y,Ma G,Wang X.Time-cooperative guidance for multimissiles:A leader-follower Strategy.Acta Aeronaut et Astronaut Sin2009;30(6):1109–18[Chinese].

7.Kumkov SS,Me′nec SL,Patsko VS.Solvability sets in pursuit problem with two pursuers and one evader.IFAC Proc Volumes2014;47(3):1543–9.

8.Kumkov SS,Me′nec SL,Patsko VS.Level sets of the value function in differential games with two pursuers and one evader.Interval analysis interpretation.Math Comp Sci2014;8(8):443–54.

9.Liu Y,Qi N,Tang Z.Linear quadratic differential game strategies with two-pursuit versus single-evader.Chinese J Aeronaut2012;25(6):896–905.

10.Rusnak I.The lady,the bandits and the body-guard game.Proceedings of the 44th Israel annual conference on aerospace science;2004 Feb 25–26;Haifa,Israel.2004.p.1–16.

11.Ratnoo A,Shima T.Line-of-sight interceptor guidance for defending an aircraft.J Guid Control Dyn2011;34(2):522–32.

12.Yamasaki T,Balakrishnan SN,Takano H.Modified CLOS intercept guidance for aircraft defense against a guided missile.Proceedings of AIAA guidance,navigation,and control conference;2011 Aug 8–11;Portland,USA.Reston:AIAA;2011.p.1–15.

13.Yamasaki T,Balakrishnan SN,Takano H.Modified command to line-of-sight intercept guidance for aircraft defense.J Guid Control Dyn2013;36(3):898–902.

14.Rubinsky S,Gutman S.Three-body guaranteed pursuit and evasion.Proceedings of AIAA guidance,navigation,and control conference;2012 Aug 13–16;Minneapolis,USA.Reston:AIAA;2012.p.1–24.

15.Rubinsky S,Gutman S.Three-player pursuit and evasion conflict.J Guid Control Dyn2014;37(1):98–110.

16.Rubinsky S,Gutman S.Vector guidance approach to three-player conflict in exoatmospheric interception.J Guid Control Dyn2015;38(12):2270–86.

17.Gutman S,Rubinsky S.3D-nonlinear vector guidance and exoatmospheric interception.IEEE Trans Aerosp Electron Syst2015;51(4):3014–22.

18.Gutman S,Rubinsky S.Exoatmospheric thrust vector interception via time-to-go analysis.J Guid Control Dyn2016;39(1):86–97.

19.Ratnoo A,Shima T.Guidance strategies against defended aerial targets.J Guid Control Dyn2012;35(4):1059–68.

20.Perelman A,Shima T,Rusnak I.Cooperative differential game strategies for active aircraft protection from a homing missile.J Guid Control Dyn2011;34(3):761–73.

21.Shaferman V,Shima T.Cooperative multiple-model adaptive guidance for an aircraft defending missile.J Guid Control Dyn2010;33(6):1801–13.

22.Shima T.Optimal cooperative pursuit and evasion strategies against a homing missile.J Guid Control Dyn2011;34(2):414–25.

23.Prokopov O,Shima T.Linear quadratic optimal cooperative strategies for active aircraft protection.J Guid Control Dyn2013;36(3):753–64.

24.Garcia E,Casbeer DW,Pham K,Pachter M.Cooperative aircraft defense from an attacking missile.Proceedings of the 53th IEEE conference decision and control;2014 Dec 15–17;Los Angeles,USA.Piscataway(NJ):IEEE Press;2014.p.2926–31.

25.Garcia E,Casbeer DW,Pachter M.Active target defense differential game with a fast defender.Proceedings of american control conference;2015 Jul 1–3;Chicago,USA.Piscataway(NJ):IEEE Press;2015.p.3752–7.

26.Weiss M,Shima T.Optimal linear-quadratic missile guidance laws with penalty on command variability.J Guid Control Dyn2015;38(2):226–37.

18 September 2016;revised 7 November 2016;accepted 21 January 2017

Available online 21 June 2017

*Corresponding author.

E-mail address:liuyanfang_hit@163.com(Y.LIU).

Peer review under responsibility of Editorial Committee of CJA.

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http://dx.doi.org/10.1016/j.cja.2017.06.007

1000-9361?2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Cooperative strategy;

Interception;

Increased survivability;

Navigation gain;

One-way cooperation system