A novel cooperative mid-course guidance scheme for multiple intercepting missiles

Long WANG,Yu YAO,*,Fenghu HE,Ki LIU

aControl and Simulation Centre,Harbin Institute of Technology,Harbin 150090,China

bSchool of Aeronautics and Astronautics,Dalian University of Technology,Dalian 116024,China

A novel cooperative mid-course guidance scheme for multiple intercepting missiles

Long WANGa,Yu YAOa,*,Fenghua HEa,Kai LIUb

aControl and Simulation Centre,Harbin Institute of Technology,Harbin 150090,China

bSchool of Aeronautics and Astronautics,Dalian University of Technology,Dalian 116024,China

Available online 14 February 2017

*Corresponding author.

E-mail address:yaoyu@hit.edu.cn(Y.YAO).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2017.01.015

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/).

In the interception engagement,if the target movement information is not accurate enough for the mid-course guidance of intercepting missiles,the interception mission may fail as a result of large handover errors.This paper proposes a novel cooperative mid-course guidance scheme for multiple missiles to intercept a target under the condition of large detection errors.Under this scheme,the launch and interception moments are staggered for different missiles.The earlier launched missiles can obtain a relatively accurate detection to the target during their terminal guidance,based on which the latter missiles are permitted to eliminate the handover error in the mid-course guidance.A significant merit of this scheme is that the available resources are fully exploited and less missiles are needed to achieve the interception mission.To this end,first,the design of cooperative handover parameters is formulated as an optimization problem.Then,an algorithm based on Monte Carlo sampling and stochastic approximation is proposed to solve this optimization problem,and the convergence of the algorithm is proved as well.Finally,simulation experiments are carried out to validate the effectiveness of the proposed cooperative scheme and algorithm.

©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 guidance;

Handover design;

Monte Carlo sampling;

Shoot-shoot-look;

Stochastic approximation;

Stochastic optimization

1.Introduction

With the progress of science and technology,modern defense system has been developed dramatically in recent years.For most of the invading targets such as exo-atmospheric ballistic missiles,an interception by direct hit has been realized as an efficient approach.1Since an early warning system can track these targets accurately,they are easily captured by the seeker of an intercepting missile at the handover point.Moreover,the handover error is consequently within the missile’s maneuverability,which guarantees that the miss distance of a single missile is small enough to kill the target.However,some specific targets developed in recent years become more difficult to track by a ground-based radar or even a space-based target tracking system,such as supersonic cruise missile.Thus,in the mid-course guidance,an enough accurate prediction of the target’s trajectory is not available to a missile any more.This leads to increasing handover errors,which accordingly result in a large terminal miss distance with the same maneuverability.

To achieve a successful interception,cooperation among multiple missiles becomes a promising alternative.On the one hand,the performance of target tracking is improved in the terminal guidance stage via information sharing among the intercepting missiles.2,3On the other hand,the intercept zone of the target can also be expanded,4which can alleviate the dilemma between improving the accuracy of an early warning system and the missile’s maneuverability.According to specific combat missions,there are different cooperative schemes,such as the salvo attack and shoot-look-shoot.The salvo attack scheme is a mechanism in which multiple missiles cooperatively intercept and hit a target simultaneously.5This scheme is usually adopted in the scenario of attacking a marine target that moves in a relatively low speed.6,7In recent years,the salvo attack scheme has been applied to the interception of high-speed targets.In Ref.8,two missiles were f i red to intercept one invading target under a detection suffering from exorbitant errors and a cooperative guidance law was proposed to improve the interception probability.In Ref.9,a guidance law for multiple missiles in near-space was designed to guarantee that all missiles hit the target simultaneously.The salvo attack scheme was applied to the interception with large handover errors in Ref.10,wherein the optimal locations of missiles at the handover moment were designed to maximize the interception performance.

In the shoot-look-shoot scheme,11the intercepting missiles are launched and encounter with the target at different moments.The first group of missiles is launched once the invading targets are detected.Then after a damage assessment,another group of missiles is launched to intercept the targets that leak through the previous round of interception.If there is not sufficient time to perform the damage assessment,the successive group of missiles is launched before the previous round of interception is completed.This kind of scheme is referred to as shoot-shoot-look,12which is applied to protecting a battleship against a salvo of invading missiles in Ref.13.

In more recent research,12–15the scheme of shoot-lookshoot or shoot-shoot-look is often used for intercepting multiple targets.The successive group of missiles is launched to intercept the targets that are missed in the previous round of interception.The transmission of target movement information between two successive rounds of interception is not taken into consideration.This information is definitely beneficial to improve the performance of the successive interceptions in the case where the early warning system cannot provide accurate movement information of a target.Inspired by this,we apply the shoot-shoot-look scheme for the mid-course guidance of multiple missiles to intercepting a target with large detection errors.The target movement information obtained by the early launched missiles will be transmitted to the successive group of missiles.In the light of this mechanism,the latter missiles can maneuver in advance to eliminate the guidance error.In order to make full use of the latter missiles’maneuverability,the time interval Δtbetween two rounds of interception should be as long as possible.However,since the target is moving,the error of the target movement information grows with the time interval Δt,which implies a larger Δtbrings in bigger handover errors for the latter missiles.Thus,in this kind of cooperative scheme,the time interval between two rounds of interception needs to be designed properly.Based on the preliminary work in Ref.16,a more general scenario is considered in this paper,and an algorithm based on Monte Carlo sampling and stochastic approximation17–20is proposed to obtain the optimal cooperative handover parameters.

The remainder of this paper is organized as follows:Section 2 presents the engagement process and the problem formulation of the cooperative handover parameter design.The proposed optimization algorithm and its convergence analysis will be introduced in Section 3 based on Monte Carlo sampling and stochastic approximation.Section 4 presents a series of simulation experiments to demonstrate the effectiveness of the proposed cooperative scheme and the optimization algorithm.Conclusions and potential applications of the proposed cooperative scheme are given in Section 5.

2.Problem formulation

In this section,a description of the cooperative engagement process will be presented first,and then the design of cooperative handover parameters is formulated into an optimization problem.

2.1.Engagement process

In the cooperative interception scheme,the intercepting missiles are launched at different moments.The launches at the same moment are called a round of interception.For each round of interception,one or several missiles are launched.We suppose that there areKrounds of interception,and the total number of missiles launched in thejth round of interception is denoted byNj（j=1，2，...，K）.Letdenote theith missile in thejth round of interception.The time schedule of the cooperative interception is shown in Fig.1,whereare the launch,handover and end moments of missiles in thejth round of interception,respectively.Letdenote the time interval between the terminal guidance of two consecutive interception rounds.

In this paper,we assume that the information is transmitted within two consecutive rounds of interception,i.e.,when the missiles in thejth round of interception are in the terminal guidance,the target movement information will be transmitted to the missiles in the （j+1）th round,which are still in the midcourse guidance.Ignoring the boost stage,the process of the first round of interception can be divided into two stages according to the source of target movement information,i.e.,the mid-course guidance and terminal guidance.However,the interception process of the missiles in the consecutive round of interception can be divided into three stages.For the first stagetis guided by the early warning system,and the target movement information is contaminated with large errors.For the second stagetis guided by the missiles in the previous round of interception and the target movement information is much more accurate than that in the first stage.For the last stagethomes in on the target based on the information provided by the seeker system equipped on itself.

A graphic representation of two consecutive rounds of interception is shown in Fig.2,wherexOyis a Cartesian inertial reference frame.The origin of the reference frame is set to be the launch site,thex-axis is in the horizontal plane pointing to the target,and they-axis is in the vertical plane perpendicular tox-axis.Tdenotes the target andqjis the line of sight(LOS)angle betweenTanddicted interception point ofduring its mid-course guidance.and target respectively.Due to the errors of the target movement information,the final interception point of the target is uncertain.All the possible final interception points form an area,which is called the predicted interception area.

Before the problem formulation,we have the following assumptions for simplicity.

Assumption 1.At the handover moment,the LOS angle of each missile with respect to the target in one round is approximated to be equal to each other.Besides,the LOS angle of a missile near the end of mid-course guidance and at beginning of the terminal guidance is assumed to be constant.

Assumption 2.The velocity of an intercepting missile,Vji,at the handover moment is fixed for different handover conditions.

Assumption 3.The effect of handover error on the total flight time of a missile in the terminal guidance can be neglected.

Assumption 4.The missiles in one round are assumed to f l y with a fixed formation and hit the target at the same moment,and then the total time of missiles in the terminal guidance can be assumed to be the same.

Remark.In the proposed scheme,the information transmitted to the latter missiles consists of both the position and speed of the target.If the intercepting missile is equipped with a radar seeker or a radar/infra-red compound seeker,the relative distance,approaching speed,LOS angle and LOS angle rate between the missile and target can be measured by the seeker directly.Then together with the movement information of the missile,the target’s position and speed can be obtained.When the missile is equipped with an infra-red seeker,only the LOS angle and LOS angle rate are directly measurable.However,the relative distance and approaching speed between the missile and target can be estimated via target tracking algorithm if the LOS angle rate is not zero.As a matter of fact,if a missile is unable to intercept the target due to the large handover error,the LOS angle rate cannot be zero,which satisfies the conditions of observability for the relative distance and approaching speed.21,22In summary,the information in the proposed scheme can be guaranteed.

2.2.Handover error of an intercepting missile

A schematic view of the planar interception geometry between a missile and a target at the handover moment in the vertical plane is shown in Fig.3.Fis the missile with the velocity ofVF.［xT，yT］Tand ［xF，yF］Tare the position vectors of the target and missile,respectively.velocity vectors of the target and missile,respectively.is the virtual target used for designing the handover parameters,

which is provided by the early warning system.is the vector of movement states ofat the handover moment.qis the LOS angle between the virtual target and the missile.γFand γTare the flight path angles of the missile and the virtual target,respectively.Sis the predicted interception point.R0is the distance between the missile and target at the handover moment.

VTand γTcan be provided by the early warning system andR0is determined by the detection range of the seeker system.WhenVFis fixed,tfand the interception geometry between the missile and the target are determined byqat the handover moment.

The position of the predicted interception pointScan be expressed asWith consideration of the error of target movement information,the position of the final interception point is

Since the vector of detection errorsunknown and obeys certain distributions,the position of the final interception is also unknown.Then the predicted interception area can be described as

Now,the expression of handover error of the missile with respect to a point in the predicted interception area will be presented.The zero effort miss distance(denoted byZ)of a missile with respect to the target at the handover moment can be expressed as

Substituting the movement information of the target and missile into Eq.(6)and taking Eq.(7)into consideration,we can obtain

together with Eq.(3),we can get

where［X，Y］∈ Ω andis the position of the predicted interception point.

Based on the definition of handover error,the description of interception probability for the target byNjmissiles in each round will be introduced in the next subsection.

2.3.Interception probability in each round

Considering that there is no extra source of information except the early warning system for the mid-course guidance in the first round of interception,the interception probability by the first group of missiles will be presented to begin with.

In the mid-course guidance of missiles launched in the firstception point of the target in the predicted interception area with the probability density function of?to Eq.(9),the handover error ofwith respect to the pointis

whereq1is the LOS angle between the target and the missiles launched in the first round.When the pointis reachable by

whereamaxis the maximum acceleration of a missile,which is assumed to be the same for different missiles.is the total flight time of a missile in the terminal guidance of the first round.

be expressed as

Based on Eq.(12),the total interception probability for the target byN1missiles in the first round of interception can be described as23,24

Next,we will describe interception probability in thejth（j＞ 1）round.In the mid-course guidance of thejthround of interception,letbe the predicted interception area in thekth（k=1，2）guidance stage,andbe the final interception point inwith the probability density function ofis provided by the early warning system andis provided by the missiles in the previous round of interception.is predicted based on the target movement information obtained by the missiles in the previous round of interception,the prediction errors enlarge with the time inter-denotes the predicted interception point ofin thekth guidance stage.Then the handover error ofin thekth guidance stage can be written as

whereqjis the LOS angle between the target and the missiles in thejth round of interception.A functionis defined to indicate whether a pointor not during the time interval

whered（Δtj-1）is the maximal distance thatduring the time intervaltion of Δtj-1and can be calculated as whereare the total flight time of a missile in the terminal guidance of the （j-1）th andjth round,respectively.

Then the probability that a pointis reachable to at least one of theNjmissiles within the time intervalcan be described as

Based on Eq.(18),the probability that the final interception point is within the maneuverability ofin the terminal guidance is

Furthermore,considering thatwith each other,letrewritten as

2.4.Optimization problem formulation

In order to achieve the maximum interception probability in each round of interception,the handover parameters of the missiles need to be optimized.It can be seen from Eqs.(10),(12)and(13)that the interception probability in the first round is determined by the LOS angle and the predicted interception point of each missile.Additionally,the interception probability in the successive round is also dependent on the time interval between the two rounds of interception.In practice,the predicted interception point is chosen to be the center of the predicted interception area for a one-to-one interception.

Whereas,for a many-to-one interception,the missiles are often flying with a fixed formation,and the distance between two neighboring missiles is chosen to be twice the maximum distance that a missile can maneuver.In this case,the reachable areas of the missiles are connected with each other to cover the predicted interception area,which is illustrated in Fig.4,where Δdis the distance between two neighboring missiles.For the many-to-one interception in one round,the predicted interception point of one of the missiles is chosen to be the center of the predicted interception area,and then the predicted interception points of other missiles can be determined based on the formation.Thus,it is reasonable to assume that the predicted interception points of the missiles are predetermined.Now,the problem to be solved here is to design the LOS angles in each round and the time interval between two rounds of interceptions to achieve the maximum interception probability.

In practical engagement,trajectories of the missiles might be limited,and thus the LOS angle should be under some constraints.We assume that the lower bound and upper bound ofqjarerespectively.Then the problem of designing the cooperative handover parameters can be described as the following stochastic optimization with constraints:

The challenge here is that the stochastic variable depends on some of the decision variables.Specif i cally,the probability density function ofin Eq.(20)is a function of Δtj-1.Therefore,it is difficult to obtain the gradient of the objective function explicitly with respect to Δtj-1.To solve Eq.(22),Monte Carlo sampling and the stochastic approximation approach based on finite difference are used in this paper,which will be presented in the next section.

3.Recursive algorithm based on stochastic approximation

In this section,an algorithm to solve the optimization problem formulated in tion 2 will be presented and then the convergence property of the proposed algorithm is analyzed.

3.1.Recursive algorithm for optimization problem

The optimization problem described by Eq.(22)can be rewritten in the canonical form as

where ξ is a vector of stochastic variables with the probability density function off（ξ）,θ ∈ Rρa vector of parameters to be optimized,g（θ，ξ）a function of θ andfunction of θ,andnthe total number of constraint functions.

To begin with,a method to estimate the objective functionJ（θ）and its gradient with respect to the parameters is provided.A series of samplesξ with the sample sizeMis generated for the optimization problem described by Eq.(23),and then the objective functionJ（θ）can be estimated by

Let ∈=J（θ）-（θ）be the estimation error,and according to the Central Limit Theory,∈～ N（0，）,where σ is the standard deviation ofg（θ，ξ）,i.e.,

The gradient ofJ（θ）with respect to θi（i=1，2，...，ρ）can be estimated via finite difference,i.e.,

where θiis theith component of θ, Δia unit vector in theithdirection of θ,andcthe magnitude of difference.andare the estimation errors ofJ（θ +cΔi）andJ（θ-cΔi）,respectively.

Next,we will analyze the bias ofbe the first,second and third derivatives ofJ（θ）with respect to θi.We suppose thatous for alliand θ ∈ Rρ.Then based on the Taylor-expansion theorem,we can obtain

whereb= ［b1，b2，...，bρ］T.For the special case whereJ（θ）is a quadratic function and（θ）=0 for alliand θ,Eq.(28)indicates thatis an unbiased estimation of the gradient,i.e.,O（c2）=0.

Now,the algorithm to solve θ*will be presented,which maximizes the objective functionJ（θ）under the constraint functionslℓ（θ）≤ 0 （ℓ=1，2，···，n）.We assume thatJ（θ）has a finite optimal solution.Then,based on the Karush-Kuhn-Tucker(KKT)condition,θ*is the optimal solution of Eq.(22)if and only if there exist Lagrangian multipliers

Furthermore,a point satisfying Eq.(30)is also a saddle point of the following Lagrangian function

The recursive algorithm based on Monte Carlo sampling and stochastic approximation is given as follows:

The algorithm can be summarized as follows:

Algorithm 1(recursivealgorithmfortheoptimization problem).

(1)Initialization:set k=1,and choose an initial value of θ（k）and λ（k）;

(2)Do:{

(3)Generate a series of samplesof the random vector ξ based on the probability density distribution of ξ;

(4)Calculate the value of J（θ（k））and its gradient with respect to θ（k）;

(5)Obtain θ（k+1）and λ（k+1）according to Eq.(32);

(6)Set k=k+1}

(7)Whilethe maximum iteration is not reached.

3.2.Convergence analysis of algorithm

Before analyzing the convergence of the algorithm,we define continuous-time dynamics as the following ordinary difference equations:

The following Lemma shows that the trajectory of Eq.(36)converges to a point（θ*，λ*）satisfying Eq.(30).

Lemma 1.Suppose that the optimization problem Eq.(23)has fi nite optimal solution,lℓ（θ）is convex,and J（θ）is concave and continuouslydifferentiablewithnegativedefiniteHessian matrix.The gradient of J（θ）with respect toθis globally Lipschitz continuous,i.e.,there exists a constantφ＞0such thatThen the trajectory of Eq.(36)is bounded and converge to its equilibrium point under any finite initial value.

The following theorem presents the boundedness of θ（k）and λ（k）under the recursive algorithm of Eq.(32).

Theorem 1.Suppose thatis bounded,and then the trajectories ofθ（k）andλ（k）generated by Eq.(32)are bounded with probability one with any fi nite initial value.

The detailed proofs of Lemma 1 and Theorem 1 are presented in Appendix A.

Now,it will be shown that the sequence （θ（k），λ（k））generated by Eq.(32)will converge to the trajectory of Eq.(36)in the average sense.

Theorem 2.Let the sequence（θ（k），λ（k））be produced by Eq.(32)with any initial value（θ（0），λ（0））.Then

whereθ*is both the root of Eq.(30)and the optimal point of the optimization problem Eq.(23),and a.s.is short for‘almost sure”.

Proof.Let ω be any sample path,and the set of all sample paths is denoted by Ψ.For almost all sample paths,the convergence of θ（k）to θ*will be proved in the following steps.

First,time scales and processes are interpolated within［k，k+1］,and the equicontinuity of the interpolated processes is analyzed.

Definet0=0 s,andtk=and define the interpolations Θ0（t）（t≥ 0 s）of the stochastic approximation process Θ（k）to be

Define sequence of shifted process

According to the definition of the interpolated processes,we have

ρk（t）is the difference between

Next,the limit of convergent subsequences is analyzed.

Based on the boundedness of Θ（k）,for all ω ∈ Ψ,Θsk（ω，t）and Θ（ω，t）are bounded on each interval in ［0，∞）andH（Θ（k））in Eq.(39)is a continuous vector function.Then,by the Lebesgue dominated convergence theorem,we obtain

Then,according to Lemma 1,for all ω ∈ Ψ,Θ（ω，t）converges to the equilibrium point Θ*.

Now,it will be proved that limk→∞Θ（k）= Θ*a.s..Suppose that there exists a subsequence limk→∞Θ（sk）=Θ≠Θ*,and then there existsk1＞0 such that

Since ｛Θsk（t）｝ is bounded and equicontinuous in the extended case,based on the Arzela-Ascoli theorem,Θsk（ω，t）has a subsequence denoted by ｛Θsk（t）｝ that converges to Θ（ω，t）uniformly on each bounded interval.Then there existsk2＞0 such that

As for all ω ∈ Ψ, Θ（ω，t）satisfies Eq.(43)and converges to Θ*,and then there existst1＞0 s such that

Together with Eq.(45),we obtain

Thus,

Therefore,there existsk3≥k1such that

Hence,we have

which results in a contradiction.Therefore,limk→∞Θ（k）= Θ*a.s.,which implies that limk→∞θ（k）= θ*a.s.,and θ*is the equilibrium satisfying Eq.(30).□

4.Simulation experiments and analysis

In this section,some simulation experiments are carried out to demonstrate the effectiveness of the proposed cooperative scheme,as well as the convergence of the recursive algorithm.

4.1.Simulation setup

In order to show the advantage of the proposed scheme,we consider the following four different cooperative interception scenarios.

Case 1.Two missiles are deployed and hit the target simul

taneously under the salvo attack scheme.

Case 2.Three missiles are deployed and hit the target simul

taneously under the salvo attack scheme.

Case 3.Two missiles are deployed with two rounds of inter

ceptions,and one missile is launched in each round of

interception.

Case 4.Three missiles are deployed with two rounds of

interceptions,and one missile is launched in the first round

and two missiles in the second.

In Cases 1 and 2,only the LOS angle needs to be optimized,while in Cases 3 and 4,both the LOS angles and time interval between the two rounds of interceptions have to be optimized.The cooperative scheme adopted in Cases 1 and 2 is the same as that in Ref.10,which is used to verify the advantage of the scheme proposed in this paper.The performance index in Ref.10is to maximize the radius of a circle which is totally within the interception envelope of the missiles in terminal guidance,while for the latter missiles in the proposed cooperative scheme,the guidance error caused by the early warning systems is eliminated in the mid-course guidance and the guidance error caused by the missiles in previous round of interception is eliminated in the terminal guidance.Then,it is more appropriate to adopt the probability that the final interception point is within the reachable area of the missiles to verify the advantage of the proposed cooperative scheme.

Considering the engagement in the vertical plane,we assume that the maximal acceleration of the missileamaxis 30 m/s2in both mid-course guidance stage and terminal guidance stage.The speeds of the target and the intercepting missile are 2720 m/s and 2040 m/s,respectively.The flight path angle of the target γTis-180°,and the initial distance between the missile and the target at the handover moment is 50 km.We assume that the errors of the early warning system are uniformly distributed with the mean of zero,and ｜ΔxT｜≤ 10 km,the flight path angle of the missile,we assume that the LOS angle satisfies-15°≤qj≤ -5°.

For the parameters in Eq.(32),αkandckare chosen as follows:20

where η1,η2,β and μ are strictly positive andA≥ 0.Here these parameters are chosen as η1= η2=1, β =1, μ =1/6 andA=0.

For the missiles in the second round of interception,we assume that the information errors at the handover moment predicted by missiles in the first round satisfy

In order to verify the effectiveness of the cooperative scheme,an interception scenario is considered.We assume that the errors of target movement information provided by the early warning system are ΔxT=5 km, ΔyT=3 km,ΔT=100 m/s and ΔT=100 m/s.The position of the target at the handover moment is（80，30）km and the velocity vector of the target is ［0，-2720］m/s.The missile is guided under the differential-game-based guidance law,which is expressed as follows26:whereZis the guidance error defined by Eq.(6),and sign（·）a symbolic function.The initial states of the missile at the handover moment are calculated based on the interception geometry between the missile and the target with information errors.

4.2.Results and analysis

The convergence trajectories of LOS angles and interception probability in Cases 1 and 2 are presented in Fig.5,which show that the optimal LOS angles are about-15°and the maximum interception probabilities are 42%and 63%,respectively.This implies that at least four missiles are needed to achieve the interception probability of 100%under the salvo attack scheme.

The convergence trajectories of LOS angles,Δt1and interception probability in Cases 3 and 4 in the second round of interception are presented in Fig.6,respectively.In Case 3,the optimal LOS angle and time interval are-5°and 5.4 s,and the corresponding maximum interception probability is about 97%,which is much higher than that by three missiles in Case 2.In Case 4,when Δt1=1 s,the interception probability reaches 100%.Thus,it shows that the interception probability can be improved significantly under the proposed cooperative scheme.

From the simulation results,it can be seen that under the proposed recursive algorithm,the trajectories of states will converge to the optimal solution,and the objective function converges to the maximum value accordingly,which demonstrates that the proposed algorithm is effective in solving the optimization problem.Besides,by the comparison of Case 1 and Case 3,we can conclude that a higher interception probability can be achieved under the proposed scheme for a given number of missiles.Furthermore,we can also conclude that fewer missiles are needed to achieve the interception probability of 100%under the proposed scheme by the comparison of Case 2 and Case 4.

In fact,in the second round of interception,the maximum maneuver distance of missileis an increasing function of Δt1,and thus the probability described by Eq.(17)also increases with Δt1.Then there exists Δt′＞ 0 s such thatΔt1,in Eq.(19)is a non-increasing function of Δt1,i.e.,there exists Δt′＞ 0 s such that=1 if Δt1≤ Δt′.If Δt′≤ Δt′,thenP2=1 for all Δt1∈ ［Δt′，Δt′］.Otherwise,there exists an optimal Δt1that maximizesP2.This can be verified by the results presented in Fig.7,which shows the relationship between the interception probability and Δt1under a fixed LOS angle.

The trajectories of the missiles and the target for each case are shown in Fig.8.Under the differential-game-based guidance law,the terminal miss distance of each missile is presented in Table 1.

Table 1 Terminal miss distance of each missile in different cases.

From the trajectories of the missiles and terminal miss distances,it can be noticed that the salvo attack scheme will result in a large terminal miss distance due to the handover error,while in the proposed cooperative scheme,the missiles in the second round(,)can maneuver in advance to eliminate the handover errors and the terminal miss distance is small enough to kill the target by direct hit.

5.Conclusions

(1)A novel cooperative scheme is proposed for the midcourse guidance of multiple missiles to intercept a target with large detection errors.The advantage of this scheme is that fewer missiles are needed to achieve the interception task,which can be verified by the results of simulation experiments.

(2)An algorithm based on Monte Carlo sampling and stochastic approximation is proposed to design the time interval between two rounds of interception.Simulations results show that this algorithm is effective in solving the stochastic optimization problem in which the stochastic variables are dependent on the states to be optimized.

(3)This cooperative scheme can also be applied to the scenario of intercepting targets accompanied with multiple decoys.The previous intercepting missile can identify the real target27and guide the following missiles to intercept the target.Thus,the total number of missiles can be subsequently reduced.

(4)In order to guarantee the information transmission between two rounds of interception,the estimation of target movement information based on the seeker system of a missile will be the emphasis of future work.

Acknowledgements

This study was partially supported by the National Natural Science Foundation of China(Nos.61333001 and 61473099).

Appendix A

Lemma 2(Appendix28).Assume that the dynamics have a compact,positively invariant setΓ(i.e.,the trajectories staring inΓand stay inΓ)and a function V（z，ϖ）that decreases along the trajectories inΓ.Then every trajectory inΓconverges toζ,the maximum positively invariant set withinΓwith trajectory satisfying

Proof of Lemma 1.Since L（θ，λ）is concave in θ and convex in λ,we have

Thus,forthesaddlepoint （θ*，λ*） ofL,wehave L（θ，λ*）≤ L（θ*，λ*）≤ L（θ*，λ）.Take a Lyapunov function as follows:

Thus,V1（θ，λ）is a non-increasing function,which indicates that the trajectories are bounded.Meanwhile,it can be found that（θ*，λ*）is also a Lyapunov stable equilibrium point of Eq.(36).

Define the set of indices of active constraints in Eq.(36)to be ϖ = ｛ℓ:λℓ=0，lℓ（θ）＜ 0｝ and consider another Lyapunov function

Then,during a time interval with fixed ϖ,the derivative ofV2（θ，λ）along the trajectory of Eq.(36)is

Due to the fact thatJ（θ）is concave in θ andlℓ（θ）is convex in θ,2（θ，λ）≤0.In fact,the set ϖ may change with the variation of θ and λ.The following analysis shows thatV2（θ，λ）is still non-increasing at the moment when ϖ changes.

(i)The first case is that ϖ is reduced from t-to t+,in which a constraint lℓ（θ）goes through zero,from negative to positive at time t.Then an extra term is added to Eq.(A4);however,the initial value of this term is zero,at time t and˙V2（θ，λ）≤0 for t≥t+.

(ii)The second case is that ϖ is enlarged from t-to t+if a constraint satisfies lℓ（t-）≤ 0.Then the corresponding multiplier λℓdecreases from positive to zero.In this case,V2（θ，λ）will lose a term and V2（θ，λ，t-）≥ V2（θ，λ，t+）.Thus,V2（θ，λ）is decreasing but discontinuous.

UtilizingV2in Eq.(A4),we can see that the trajectory satisfying(i)and(ii)in Lemma 2 is an equilibrium.By condition(i)and the strict concavity ofJ,Eq.(A5)implies that˙θ=0 for fixed ϖ,and then θ=θ*.Iflℓ（θ*）＞0 for some ℓ,the corresponding λℓwill diverge to infinity which contradicts the boundedness of the trajectories.Therefore,l

ℓ（θ*）≤ 0（ℓ=1，2，···，n）.Iflℓ（θ*）＜ 0,thenλℓwill decrease to zero,which contradicts the continuity ofV2.Therefore,ℓ=0.

Thus,the trajectories converge to the point （θ*，λ*）such that=0，=0,which also satisfies the KKT condition Eq.(30).□

Proof of Theorem 1.Let（θ*，λ*）be the root of Eq.(30).Take

as a Lyapunov function.Then from Eq.(32),we have

Considering Eq.(A3),we have

which yields

E［V（θ（k+1），λ（k+1））］≤V（θ（k），λ（k））

where

where φ1and φ2are positive constants.Considering thatE［δ（k）］=bk=O）and wherel= ［l1，l2，...，ln］T.

From Eqs.(A7)and(A10),we have

E［V（θ（k+1），λ（k+1））］

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17 August 2016;revised 11 September 2016;accepted 10 November 2016