Linear-quadratic and norm-bounded differential game combined guidance strategy against active defense aircraft in three-player engagement

Tao CHAO, Xintao WANG, Songyan WANG, Ming YANG

Control and Simulation Center, Harbin Institute of Technology, Harbin 150080, China

KEYWORDS Active defense aircraft;Acceleration;Energy optimization;Game theory;Three-player engagement

Abstract A new type of guidance strategy, combining linear quadratic and norm-bounded game theory, is proposed for the scenario of an attacker against active defense aircraft in three-player engagement.The problem involves three players, an attacker, a defender and a target.The differential game theory and the solution of Hamiltonian equation are utilized to obtain the combined guidance strategy for each player with arbitrary-order dynamics.The game process is divided into 4 phases, C1-C4, according to the switching time.The linear quadratic differential game guidance scheme is employed to reduce the fuel cost in the game parts of C1 and C3.The norm-bounded game guidance strategy is adopted to satisfy the constraint of control input in the game stages C2 and C4.Furthermore, zero-effort miss distance is introduced to meet the constraints of game space and defender’s killing radius in the guidance strategy, which guarantees that the attacker is able to avoid the interception of the defender and hit the target with lower fuel cost and maximum acceleration.And it is proved that the proposed guidance strategy satisfies the Nash equilibrium condition.Finally, the feasibility and superiority of combined guidance strategy are respectively illustrated by nonlinear numerical simulation and verified by comparing with linear quadratic and norm-bounded differential game guidance strategies.

1.Introduction

In recent decades, as countries attach more importance to capabilities of aircraft penetration, the issue of three-player conflict has become the focus of many scholars’research.1The three-player conflict problem is that a defense vehicle is launched to intercept the enemy’s vehicle when the target aircraft realizes the threat of the attacker, which is also referred to as the Target-Attacker-Defender (TAD) problem.Unlike the traditional one-to-one game of pursuit-evasion,there are three roles, which are the target aircraft (referred to as the target), the attacking vehicle (referred to as the attacker), and a defense vehicle (referred to as the defender),to fight against each other.The TAD problem was first proposed by Boyell of the American Radio Corporation in 1976,2which analyzed the minimum launch distance of successful interception according to the kinematic relationship of target, attacker and defender.Afterwards, the method of Line-of-Sight (LOS) guidance was presented by Yamasaki and Balakrishnan3of National Defense Academy of Japan to study the TAD issue.Based on the previous research,Ratoon and Shima4of Israel Institute of Technology also considered the TAD problem from the perspective of LOS guidance scheme.However, the guidance strategy of the defender was only designed by virtue of the classical approach of LOS guidance, which ignored the cooperation of the target and the defender.

In recent years,TAD problem has become the hot point of many scholars’research.Rubinsky and Gutman5employed the vector guidance approach to study the TAD issue, which presented the method adopting full three-dimensional vector kinematics suitable for an exoatmospheric engagement.One and two-way cooperative guidance strategies were proposed by Kumar and Dwaipayan6,7using the method of LOS guidance and the theory of Sliding-Mode Control (SMC) to solve the TAD problem, which ensured that the defender remained on the LOS between the attacker and the target to intercept the attacker.On this basis, the method of LOS guidance was utilized by Luo et al.8to present the cooperative LOS guidance scheme for the TAD confrontation.The TAD issue with the constraint of the impact time was considered in Ref.9to design the cooperative guidance law for the target-defender team to safeguard the target from an attacker.Yan and Lyu10designed the two-side cooperative interception guidance strategy for the target and the defender with relative time-to-go deviation by virtue of LOS guidance method.

For the sake of improving the cooperation of the target and the defender, Differential Game (DG) theory is proposed to cope with the TAD problem.DG theory was founded by American scholar Bather,11which was utilized to handle the dynamic game issue of two-player or multi-player.Compared with the LOS guidance method, the guidance strategies of the target and the defender were designed by means of the DG theory,which didn’t require the assumption of maneuvering strategy for the attacker.Therefore, cooperative characteristic between the target and the defender was enhanced through the DG theories.On the basis of these researches, Rusnak12Ratnoo and Shima,13Rubinsky and Gutman14employed the DG theory to study the TAD issue and concluded that the main approaches are two DG theories: Linear Quadratic Differential Game (LQDG)15and Norm-bounded Differential Game (NDG).16

LQDG theory was developed by many scholars to achieve a series of researches for the issue of TAD.17–26The loss of fuel is considered to propose the guidance strategies since there is the energy consumption term∫t0 u2（ε)dε in LQDG, which is suitable for the long-term flight of aircraft.A sort of cooperative differential game guidance strategy was studied by LQDG method,and the solutions of game for continuous and discrete systems under arbitrary-order dynamics were given by Perelman et al.17LQDG guidance strategy was proposed by Shaferman and Shima18for the TAD problem with the terminal angle constraint,which guaranteed that the attacker was intercepted by the defender at the desired terminal angle.On the basis of this research, Shaferman and Shima19,20investigated the cooperative guidance scheme to settle the game issue that multiple missiles attacked the single target with the terminal angle constraint.The guidance strategy was designed in Ref.21by virtue of the dead zone function to solve the issue that the attacker evaded the static obstacle and hit the target in the three-player conflict scenario.On this basis,Ref.22studied a guidance strategy for the TAD scene with the constraints of multiple static obstacles and the terminal angle.Novel evasion guidance strategy was presented in Ref.23by the method of LQDG to study the TAD game for air-breathing hypersonic vehicles.Liu et al.24put forward the LQDG strategies for the scenario of two-pursuit versus single-evader and proved the Nash equilibrium condition of the guidance schemes.A scenario of multiple attackers trying to intercept the active defense aircraft was considered to derive the cooperative differential game guidance strategies by means of LQDG theory in Ref.25Nonlinear active defense guidance strategy26was proposed by LQDG and SMC theories to weaken the influence of time-to-go estimation error and improve the successful rate of interception.Although the energy consumption was considered by virtue of LQDG to design the guidance strategy, the acceleration constraint was ignored in the abovementioned literatures.The maximum maneuver of the attacker is not used to avoid the defender or pursue the target at the terminal time,which will lead to failure of the attack mission.

NDG theory was proposed to make up for the shortcomings of LQDG theory27–32.Since NDG contains the term umax, the maximum acceleration is utilized to maneuver at the terminal time, which improves the success rate of attacking.NDG-method-based cooperative pursuit-evasion strategies were derived for the target and the defender in Ref.27.A class of NDG guidance strategies were proposed by Sun et al.28–30for three-player conflict scenario, which analyzed the game space with regard to the initial Zero-Effort-Miss(ZEM) between the attacker and the defender, but the game space between the defender and the target wasn’t discussed.Differential game guidance strategies31were designed to overcome the difficulty of choosing parameters caused by traditional guidance strategies based on NDG theory, and enable that the attacker avoided the defender with different killing radii and hit the target.However, many types of guidance strategies were obtained in Ref.31according to the sign of initial ZEM distances, which were too cumbersome to apply in practical combat.Sun et al.32put forward the NDG guidance scheme, which guaranteed that the target lured the attacker close to the defender based on the attacker’s reaction to the target.It was worthy to point out that maneuverability constraint was satisfied to design the guidance strategy in the above literatures, but energy consumption wasn’t taken into account.NDG guidance strategy is not suitable for the entire game stage, because the application of maximum acceleration results in excessive energy consumption.Moreover, chattering in control signals will occur on account of the sign function in NDG.If this method is adopted in the whole game stage, the chattering phenomenon will affect the entire process of game.

A Combined Differential Game (CDG) theory was proposed in view of the merits of LQDG and NDG.This theory divides the game into different stages according to the switching time.Before the switching time,the energy consumption is considered and LQDG is employed to deal with the issue of differential game.NDG guidance strategy is adopted to cope with the problem of the game in the condition of bounded input after the switching time.CDG guidance strategy was designed to meet the constraints of energy consumption and input bounded for one-to-one game in Refs.33–35which identified the game space from the switching time to the terminal time.Yan et al.36proposed the CDG guidance strategy and employed the method of machine learning to estimate the unknown system dynamics under the framework of one-toone pursuit-evasion game.

A novel type of CDG guidance strategy is proposed for the TAD problem in this paper.The game is divided into 4 stages,C1-C4, by switching time, and the design process of the guidance strategy is stated as follows:firstly,the relative kinematic equations of the engagement scenario are established and linearized around the collision triangle based on assumptions.Secondly, the corresponding cost functions are given and the issue of TAD is divided into linear quadratic and normbounded differential game problems according to the switching time.Then,CDG guidance scheme based on the analytical solution of the TAD problem is developed to evade the defender and attack the target.The proposed guidance strategy enables that the attacker accomplishes the penetration mission with lower fuel cost and maximum acceleration by virtue of combining LQDG and NDG methods.Moreover,it is proved that the guidance strategy satisfies the Nash equilibrium condition.Compared with the aforementioned literatures, the main contributions of this paper are stated as follows:

(1) Compared with the LQDG guidance strategy,the CDG guidance strategy proposed in this paper considers the constraint of input bounded to satisfy the limitation of overload in the C2 and C4 stages, and adopts the maximum acceleration to overcome the difficulty of the attacker’s lack of maneuverability.

(2) Energy consumption term ∫t0 u2（ε)dε is taken into account to reduce the fuel cost in contrast to NDG in the game phases C1 and C3.There is no chattering phenomenon since the sign function is not used at the stages.

(3) The CDG guidance strategy takes the killing radius of the defender as a design parameter to guarantee that the attacker avoids the interception of the defender with different killing radii.Compared with the guidance strategies proposed in Ref.31, the guidance strategy is more conveniently utilized in practical combat because it doesn’t include the sign of initial ZEM between the attacker and the defender.

(4) Limitation of the game space is taken into account in the CDG guidance strategy in order that the attacker successfully hits the target according to ZEM between the attacker and the target.

The structure of the remainder of this paper is shown as follows: Section 2 introduces the nonlinear and linear models of the TAD problem.The framework of CDG, including the game decomposition and the order reduction of system, is elaborated in Section 3.By virtue of the analytical solution of TAD game problem, the proposed guidance strategy is designed in Section 4.The satisfaction of the Nash equilibrium condition of the guidance scheme is proven in Section 5.Section 6 verifies the effectiveness of the guidance strategy through nonlinear simulation, and compares it with the LQDG and NDG guidance strategies to illustrate the superiority.

2.Preliminaries

2.1.Nonlinear kinematics

Fig.1 Kinematic geometry.

2.2.Linear kinematics

3.Game framework

Remark 1.The switching time ts1?（0,tfAD), ts2?（tfAD,tfAT)are prescribed according to the terminal time of attacker flight.The fuel cost of the attacker decreases with the increase of the switching time ts1?（0,tfAD), ts2?（tfAD,tfAT) in the C1 and C3 phases.On the contrary,the energy consumption of the attacker grows with the reduction of ts1?（0,tfAD), ts2?（tfAD,tfAT) due to employing the maximum acceleration in C2 and C4.

3.1.Game decomposition

Fig.2 Flowchart of guidance strategy.

Assume that the killing radius of the defender R is known,and δ is the positive value for the sake of satisfying yAD（tfAD)>R.

The cost function between the attacker and the target is proposed to reduce the distance yATand increase the fuel cost of the target.

3.2.Order reduction

In order to lower the complexity of the problem,ZEM is introduced to reduce the system order in this article, which means the miss distance if none of the adversaries in engagement apply any control from the current time onward.The ZEMs of attacker-target and attacker-defender are represented as ZAT, ZAD, respectively.

According to terminal projection transformation37, ZATand ZADare obtained as follows:

4.Solutions of game

According to the cost functions Eqs.(31)–(35),the solutions of the game are given using the differential game theory11in the phases of C1-C4.In this section, the corresponding Hamiltonian functions of the cost functions Eqs.(31)–(35)are acquired in order to handle the game issue in three-player conflict.And the CDG guidance strategy is obtained by means of the analytical solutions of Hamiltonian functions.

(1) The corresponding Hamiltonian function is obtained in C1.

Substituting Eqs.(28)and(29)into Eq.(36),we can obtain

Remark 4.For ｜ZAD（t)｜≥（R+δ), the following formula is yielded by substituting Eqs.(57) and (58) into Eq.(29), and integrating from t to tfAD.

(3) When the defender disappears,and the combat becomes one-to-one game between the attacker and the target.The optimal guidance strategy is given in order to analyze the boundary of the game space in C4.The Hamiltonian function is

Fig.3 Game space.

Remark 6.The following conditions are satisfied in order to ensure that Eq.90 holds.

5.Nash equilibrium condition

Combining inequalities Eqs.(128) and (130) acquires

It can be obtained from inequality Eq.(131) that the optimal guidance schemes u*A（t) given by Eq.(57) and u*D（t) given by Eq.(58) satisfy the Nash equilibrium condition in C2.

Similarly, u*A（t) in Eq.(57) and u*D（t) in Eq.(58) meet the Nash equilibrium condition for ZAD（t)≤-（R+δ) according to Case 2.

And these completed the proof.■.

Corollary 2.The optimal guidance strategies of u*A（t) in Eq.(57) and u*T（t) in Eq.(66) satisfy the Nash equilibrium condition in C2.

JAT（uA,u*T)≤JAT（u*A,u*T)≤JAT（u*A,uT)

Remark 7.Inspired by Ref.17, the energy optimization is considered, but the constraint of control input is ignored in the guidance strategies Eqs.(50)–(52).There will be no chattering phenomenon because the sign function fails to be used in LQDG,which is suitable for the long-term flight.However, since the maximum acceleration of the attacker is not utilized to maneuver at the final time, it may not complete the attack task.

Remark 8.The NDG guidance strategies Eqs.(132)–(134)given in Ref.27are used in the whole phase of game.Since the maximum acceleration is employed to maneuver,the attack mission is accomplished.Nevertheless,the energy consumption isn’t considered for NDG to cause the waste of energy in the long-term flight.Moreover, chattering in control signals will occur on account of the sign function in NDG.The optimal guidance schemes are described as follows:

6.Simulations

The feasibility of CDG guidance strategy is illustrated by numerical examples in this section.An engagement scenario,where a defensive vehicle is released by airplane to intercept the attack missile, is established.Missile evades the interception of the defensive vehicle and attacks the airplane in order to complete the penetration mission.Thereinto, airplane, missile and defensive vehicle are regarded as the maneuvering target, the attacker and the defender, respectively.It is assumed that the information required for guidance schemes of game participants is obtained by sensors onboard.The simulation is given to demonstrate the performance of the proposed strategy in contrast to LQDG and NDG.

In the simulated scenario, the attacker is able to evade the defender at the terminal time tfADand hit the target at the final time tfAT.LQDG guidance schemes Eqs.(142),(141)and(140)are adopted for each player in C1.The attacker and the defender utilize the NDG guidance strategies Eqs.(143) and (144),and the target uses the LQDG guidance strategy Eq.(145) in C2.It is one-to-one game between the attacker and the target because the defender has disappeared in C3 and C4.The attacker employs the LQDG guidance scheme Eq.(147) and switches the guidance strategy to satisfy the constraint of game space according to the value of ZAT（t).The optimal guidance strategy Eq.(146)is adopted for the target in C3.The guidance strategies Eqs.(149)and(150)are used for the attacker and the target to complete the attack mission in C4.Two simulation scenarios are established, where Cases 1 and 2 are given to illustrate the CDG guidance strategy.The objective of Case 1 is the feasibility verification of the CDG guidance strategy.Case 2 is employed to demonstrate the superiority of the guidance strategy.

The simulation parameters are listed in Table 1.

Case 1.The killing radius of the defender is 40 m, and δ is 50;the constant ζ is 3;the weight parameters are selected separately as ｛αA=9.5, αD=2.8; α1=3, α2=5｝.It can be seen from Table 1 that the terminal time are tfAD= 6.0967 s,tfAT=10.5815 s; the switching time ts1and ts2are chosen as 3 s and 8 s,respectively.Simulation results of Case 1 are presented in Figs.4-12.

The engagement trajectories of the CDG guidance strategy are shown in Figs.4 and 5 at the terminal time tfATand tfAD.The attack mission is completed to avoid the defender at the final time tfADand hit the target at the final time tfATaccording to Figs.4 and 5.The attacker avoids the interception of the defender and heads towards the target directly.

Fig.6 shows the evolution of ZATand ZADwith respect to time between the attacker and the target, and between the attacker and the defender.The value of ZADand ZATgradually decreases and increases due to the adoption of LQDG in C1.ZADgradually increases and remains at 90 m since NDG is employed in C2.The guidance strategy is switched to NDG at ts1= 3 s, and it is illustrated that the attacker evades the interception of the defender by ZAD≥（R+δ)=90 m at the final time tfADaccording to Fig.6.

The boundary value ZS, which is calculated based on the above parameters, is 199.9287 m; it can be seen that the constraint of game space is satisfied by ｜ZAT｜=12 mR at t= 5.75 s, and RAT= 1.133 m at tfATbased on Fig.7.

The time evolution of ZEMs with different R for Case 1 is shown in Figs.10 and 11.It is illustrated by Fig.10 that ZADgradually decreases at the switching time ts1=3 s and converges to （R+δ) for different R at the final time tfAD.Similarly, Fig.11 shows that the value of ZATgradually increases in C1 and C2, and converges to 0 m for different R at the terminal time tfAT.

The engagement trajectories of the CDG guidance strategy for different ts1and the time evolution of ZEMs with different ts1are shown in Figs.12 and 13, respectively.The switching time ts1is chosen as 0.5 s and 5.8 s when ts2is selected as 8 s.The stage C2 is almost extended to the whole engagement between the attacker and the defender due to ts1=0.5 s.It can be seen from Fig.12 that the attacker is able to evade the interception of the defender and pursue the target in the case of the switching time ts1=0.5 s.For the switching time ts1=5.8 s,the phase C1 is almost extended to the entire engagement between the attacker and the defender.The attacker is intercepted by the defender at the final time tfAD, according to Fig.12, when the switching time ts1is chosen as 5.8 s.

For ts1=0.5 s, it is illustrated that the attacker evades the interception of the defender by virtue of ZAD≥（R+δ)=90 m at the terminal time tfADbased on Fig.13.In Fig.13, the attacker successfully hits the target according to ZAT（tfAT)=0 m in the case of the switching time ts1=0.5 s.However, the fuel cost of the attacker is increased for the case of ts1=0.5 s since the stage C2 is extended.It is demonstrated by virtue of ZAD=30 m

The engagement trajectories of the CDG guidance strategy for different ts2and the time evolution of ZEMs with different ts2are illustrated in Figs.14 and 15,respectively.The switching time ts2is selected as 6.5 s and 10 s when ts1is given as 3 s.

Table 1 Simulation parameters in Case 1.

Fig.4 Trajectories of CDG guidance strategy in Case 1.

Fig.5 Trajectories of CDG guidance strategy in tfAD.

Fig.6 Time evolution of ZEM in Case 1.

Fig.7 Relative distance of A and T, D.

The switching time ts2is selected as 6.5 s and 10 s when ts1is given as 3 s.The stage C4 is almost extended to the whole engagement between the attacker and the target due to ts2=6.5 s.Fig.14 shows that the attacker avoids the defender and hits the target at the terminal time tfADand tfATin case of the switching time ts2=6.5 s.For the switching time ts2=10 s, the phase C3 is almost extended to the entire engagement between the attacker and the target.The purpose that the attacker pursues the target at the final time isn’t able to be achieved according to Fig.14,when the switching time ts2is chosen as 10 s.

Fig.8 Time-to-go.

Fig.9 Acceleration of attacker.

Fig.10 Time evolution of ZAD with different R.

Fig.11 Time evolution of ZAT with different R.

Fig.12 Trajectories of CDG guidance strategy for different ts1.

Fig.13 Time evolution of ZEM with different ts1.

Fig.15 demonstrates,for the switching time ts2=6.5 s and ts2=10 s,that the attacker successfully evades the interception of the defender according to ZAD≥（R+δ)=90 m at the final time tfAD.It can be seen from Fig.15 that the attacker successfully hits the target by ZAT（tfAT)=0 m in the case of the switching time ts2=6.5 s.The fuel cost of the attacker pursuing the target is increased for the case of ts2=6.5 s since the stage C4 is extended.In Fig.15, the mission that the attacker hits the target isn’t completed,according to ZAT（tfAT)=40 m,when the switching time ts2is chosen as 10 s.

Fig.14 Trajectories of CDG guidance strategy for different ts2.

Fig.15 Time evolution of ZEM with different ts2.

Case 2.In order to illustrate the superiority of the CDG guidance strategy, comparisons of simulation results for LQDG given by Eqs.(140)–(142) and NDG given by Eqs.(132)–(134)guidance schemes are presented based on Table 2.

Fig.16 Time evolution of ZAD with different umaxD.

Fig.17 Time evolution of ZAT with different umaxT.

Fig.18 Time evolution of ZAT and ZAD with different umaxA.

Fig.19 ZAT for different ZAT（tfAD).

Fig.20 Trajectories of CDG guidance strategy.

Fig.21 Time evolution of ZEM with the first-order dynamics.

Table 2 Simulation parameters in Case 2.

Figs.22-25 show the trajectories of CDG, LQDG, NDG and NNDG guidance strategies.The CDG,NDG and NNDG guidance strategies guarantee that the attacker is able to evade the interception of the defender and hit the target according to Figs.22, 24 and 25, simultaneously.

For NDG and NNDG guidance strategies, the collision point between the attacker and the target is higher compared to CDG guidance scheme due to the adoption of maximum acceleration in the whole phase of game.It is illustrated that the attacker is intercepted by the defender in Fig.23 since maximum acceleration fails to be used for the LQDG guidance scheme at the terminal time tfAD.Therefore,the attack mission that the attacker avoids the interception of the defender and hits the target isn’t completed at the terminal time tfADand tfAT.

Fig.22 Trajectories of CDG guidance strategy in Case 2.

Fig.25 Trajectories of guidance strategy NNDG.

Fig.23 Trajectories of guidance strategy LQDG.

Fig.24 Trajectories of guidance strategy NDG.

The simulation results of ZEM with respect to time are demonstrated for CDG, LQDG, NDG and NNDG guidance strategies by Figs.26-29, respectively.

In Fig.26, the value of ZAD, ZATdecrease and increase in C1 and C3, respectively.The ZEM between the attacker and the defender is 90 m≥（R+δ) at the final time tfAD; the ZEM between the attacker and the target is 0 m, which explains that the defender is avoided and the target is attacked.Fig.27 shows that the ZEM between the attacker and the defender decreases at 2 s and increases to 0 m at the terminal time tfAD.And the attacker is intercepted by the defender and ZADis 0 m at the terminal time tfADaccording to Fig.27.Based on the above analysis of results, the attacker is more likely to be intercepted by the defender since the maximum maneuver fails to be used at the final time tfAD.

Fig.26 Time evolution of ZEM for CDG in Case 2.

Fig.27 Time evolution of ZEM for LQDG.

Based on Figs.28 and 29, the values of ZEM between the attacker and the defender increase at 2 s, 0.3 s and reduce to 250 m, 110 m at the terminal time tfAD.The ZEM between the attacker and the target increases to 0 m at the final time tfATaccording to Figs.28 and 29.From Figs.28 and 29, it is seen that the ZEMs between the attacker and the defender are 250 m > （R+δ) and 110 m > （R+δ) to demonstrate that the attacker can avoid the defender and the ZEMs between the attacker and the target are 0 m to illustrate that the attacker hits the target.However, it will cause the waste of energy because the maximum acceleration is utilized in all stages of flight.

Fig.28 Time evolution of ZEM for NDG.

Fig.29 Time evolution of ZEM for NNDG.

Fig.30 Time evolution of energy consumption.

Table 3 Results comparison.

The advantage of CDG is expressed by comparing with LQDG,NDG and NNDG in Table 3.For the CDG guidance strategy, the attacker evades the interception of the defender and satisfies the acceleration constraint.The energy consumption is lower for LQDG guidance strategy utilized in the entire game stage.However, the attacker is more likely to be intercepted by the defender since the maximum acceleration fails to be adopted in the game process.The attacker which employs the NDG and NNDG guidance schemes is able to evade the interception of the defender and hit the target with maximum maneuver.But the circumstance of excessive fuel cost occurs due to the existence of termin the whole game stage.

7.Conclusions

In this paper,a combined guidance strategy for the TAD issue is proposed by means of zero-effort miss and differential game theory.This guidance strategy divides the game process into 4 stages,C1-C4,where the attacker avoids the defender and hits the target with lower energy and input bounded.With the CDG compared with LQDG and NDG, the conclusions are drawn as follows:

(1) The acceleration constraint is considered to meet the limitation of overload in the CDG guidance strategy.The maximum acceleration of the attacker is adopted to complete the penetration mission in C2 and C4 and overcome the difficulty of the attacker’s lack of maneuverability with the arbitrary-order dynamics.

(2) The energy consumption is taken into account for CDG guidance strategy to reduce the fuel cost in contrast to NDG.Therefore, the guidance strategy saves the fuel for penetration mission.The guidance strategy proposed in this paper alleviates chattering phenomenon in C1 and C3 since the sign function fails to be included.

(3) The killing radii of the defender are taken as a parameter and the constraint of game space is considered in the CDG guidance strategy.The guidance scheme guarantees that the attacker evades the interception of the defender with different killing radii and ZEM satisfies the requirement of game space in order to pursue the target.The proposed guidance strategy accomplishes the attack task with lower fuel cost and maximum acceleration.

(4) The effectiveness and superiority of the proposed guidance strategy are illustrated by the simulations.The results show that the guidance strategy is able to satisfy the constraint of control input and reduce the energy consumption,which verifies that the attacker avoids the interception of the defender and hits the target at the terminal time.

The combined guidance strategy proposed in this paper will be expanded to cope with the issue of multiple attackers against an active defense target in the future research.In addition, considering the constraints of obstacle avoidance and terminal intercept angle, novel combined guidance strategy based on the research of this paper will be designed to study the TAD problem, which will guarantee that the attacker avoids the obstacle and the defender,and hits the target at the desired terminal angle.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Nos.62273119 and 61627810).