Cooperative interception with fast multiple model adaptive estimation

Sho-bo Wng , Yng Guo , Shi-cheng Wng , Zhi-guo Liu , Shui Zhng

a Xi’an Research Institution of Hi-Technology, China

b Xi’an Research Institution of Hi-Technology; Northwestern Polytechnical University, China

Keywords:Cooperative guidance Optimal control Fast multiple model adaptive estimation(fast MMAE)Bang-bang maneuver Switch time Detection configuration Estimation error

ABSTRACT For the case that two pursuers intercept an evasive target,the cooperative strategies and state estimation methods taken by pursuers can seriously affect the guidance accuracy for the target, which performs a bang For the case that two pursuers intercept an evasive target, the cooperative strategies and state estimation methods taken by pursuers can seriously affect the guidance accuracy for the target, which performs a bang-bang evasive maneuver with a random switching time. Combined Fast multiple model adaptive estimation(Fast MMAE) algorithm,the cooperative guidance law takes detection configuration affecting the accuracy of interception into consideration.Introduced the detection error model related to the line-of-sight(LOS)separation angle of two interceptors,an optimal cooperative guidance law solving the optimization problem is designed to modulate the LOS separation angle to reduce the estimation error and improve the interception performance. Due to the uncertainty of the target bang-bang maneuver switching time and the effective fitting of its multi-modal motion, Fast MMAE is introduced to identify its maneuver switching time and estimate the acceleration of the target to track and intercept the target accurately. The designed cooperative optimal guidance law with Fast MMAE has better estimation ability and interception performance than the traditional guidance law and estimation method via Monte Carlo simulation.

1. Introduction

Using the multi-interceptor cooperative intercepting maneuvering target has been extensively studied. These researches include two aspects:one is to design cooperative guidance law,and another is to estimate the state of maneuvering target.The optimal control theory [1,2], differential game theory [3], and finite time control theory have been employed for designing the linear cooperative guidance law. The sliding mode control theory [4] is often used to design the nonlinear cooperative guidance law. The intercept time in the linear cooperative guidance problem is often determined in advance, while it is usually achieved through coordination between aircraft in nonlinear cooperative guidance problem. However, earlier studies about designing the guidance law based on the assumption that interceptors have perfect information about the target, that is, ideal measurements and known inputs to the dynamic system. This assumption that satisfies certainty equivalence principle(CEP)and separation theorem(ST)can make optimal guidance control system easy to analyze [5]. The validity of the CEP has been proved for linear optimal control problems with unbounded control, quadratic cost function, and Gaussian noise (termed linear quadratic Gaussian). Under the condition of satisfying the CEP and ST,the designs of the estimator and the controller can be optimized separately for the linear Gaussian system.However, the CEP and ST cannot be proved to be feasible for nonlinear non-Gaussian systems. The generalized separation theorem (GST) [6] stating the probability density function(PDF) of the target in designing the guidance law needs to be considered is proposed to solve the problem. For single-mode target maneuvering, such as, Kalman filter, Extended Kalman filter, Unscented Kalman filter et al. are used to estimate the target state of this type of maneuver. With the complication of target maneuvering modes,the state estimation method for single-mode target maneuvers no longer adapts to multi-mode maneuvering target.Besides,during the multi-aircraft’s engagement process,the detection link can provide effective measurement information for the guidance process, and the guidance link can change the interception trajectory of the aircraft which in turn affects the cooperative detection configuration and accuracy.Because the sine theorem and the LOS angle information between the two pursuers are used to accomplish the cooperative distance measurement[7],the relative distance between the pursuers and the target cannot be calculated when the LOS separation angle is equal to zero,and the estimation error is very large when it is particularly small. Therefore,the problem of detection configuration [7-11] should also be considered in the process of multi-interceptor guidance design.

The CEP has never been proved for realistic missile guidance problems,characterized by bounded control,non-Gaussian random target maneuvers,and saturated state variables.In most cases,the guidance laws designed are all based on a simple and single maneuver assumption, such as well-defined, mostly constant. But in realistic engagement, the maneuvering form of the target will be more complex and intelligent.The factors that affect the guidance accuracy are: inaccurate mathematical model, noisy measurements, target maneuvers form, and state estimation accuracy. In order to facilitate our analysis of guidance control problems, it is often based on a simplified mathematical model with a linear loworder dynamic characteristics.The simplified mathematical model will bring inevitable system error,so the accuracy of the estimated state variables will be particularly important. For linear systems characterized by zero mean and Gaussian white noise,Kalman filter[12]is the optimal minimum-variance estimator.The measurement noise used in interception simulation does satisfy this assumption,but it is only an approximate method to model the random target maneuver by shaping filter (SF) [13] driven by zero mean and Gaussian white noise. The target acceleration is also taken as the state variable in SF.In this way,the interference input is the target random acceleration instructions which may be discontinuous and represent the random jump process. Although these instructions are bounded,they certainly do not satisfy the Gaussian distribution and are not white noise processes. It has been pointed out that there is no global optimal filter for all guidance laws/interception cases [14], and there is no unique optimal filter/guidance law combination suitable for all feasible target maneuvers [15].

There are many papers on the research of tracking maneuvering target by using the idea of multiple models.The performance of the multiple model filtering method is better than that of the traditional single-model filtering method. The idea of multiple model algorithm was first proposed by Magill [16] in 1965. This kind of algorithm has become the main method to study the state estimation of the hybrid system. The multiple model adaptive estimator (MMAE) can be used to track the random maneuvering target.At this time,the evasion strategy of the maneuvering target is equivalent to the finite set of models in MMAE in which each Kalman filter corresponds to different evasion strategies and runs in parallel in filtering estimation.The estimated state variables are obtained by fusion of all Kalman filters according to minimum mean square error (MMSE) [15] or maximum a posteriori (MAP)[17]algorithm.Although the estimation accuracy of this method is high,it depends heavily on the model set used.Especially when the model set is large, the real-time performance of the multi-model filtering method is reduced and the engineering implementation is difficult limiting the application of the method in practice. Efficient algorithms have been proposed in the past in the context of maneuvering target tracking for alleviating the computational load problem. Many effective algorithms were proposed to reduce the computational burden in the process of target tracking [18-20].The algorithm proposed in Ref. [18] can make the filter gain and variance only need to be calculated once. Hexner [19] pointed out that there is a lower bound of the maneuvering instructions detection time that is independent of the estimator type.According to this lower bound,the total number of filters in MMAE can be cut and the operation speed can be improved.Based on reference[19],Shima [20] proposed a Fast MMAE algorithm which can highly reduce computational load. Using the idea of aggregation and cut,the efficient algorithm greatly reduced the total number of filters needed. This order reduction scheme can not lead to performance degradation of MMAE. Similar to the algorithm of [18], the proposed algorithm featured one-time covariance and gain computations for all filters in the bank.

Other information except for the line-of-sight(LOS)angle,such as lateral relative distance, time-to-go, close velocity, and acceleration, needs to be measured and estimated so that accurate target states and excellent interception performance can be achieved.Except for the LOS angle and its derivative to time,the acquisition of almost of all the above information depends on the relative distance information between pursuers and evader. However, under the existing technical conditions,the relative distance information cannot be obtained through the pursuer’s own sensors. Therefore,using the way multi-aircraft cooperative distance measurement has become a practical method. In a previous study [21], the interceptor’s sensor is set to only measure the LOS angle information against the target.The method of cooperative detection can be used to estimate the target’s state more precisely based on interactive multiple model (IMM) filter and multiple model particle filter(MMPF) [22,23]. Although the results illustrated that the way of cooperative detection could significantly improve the estimation accuracy and interception performance, powerful real-time data processing capabilities were required.Because the two interceptors use the method of cooperative measuring distance which uses the sine theorem to obtain the relative distance with the target, the detection error will be suffered from the guidance configuration of the two interceptors. The interceptor using the method of double LOS cooperative distance measurement in Ref. [7], the relative distance information of the pursuers-to-evader was calculated by information-sharing way which will be a very effective way to increase the success rate of an attack. The detection error model affecting the accuracy of lateral relative distance detection was given and is related to the LOS separation angle.Decreasing the LOS separation angle between the two interceptors against the target will cause excessive detection errors. Based on the principle, the cooperative guidance law minimizing detection errors for evader was designed.Based on[7],Fonod and Shima[9,10]introduced the detection error model presented by Liu and extended the problem of cooperative detection and guidance to the case where the target launched two defenders to cooperative anti-intercept missiles.From the side of interceptors, Reference [9] showed that the terminal interception angle formed between pursuers can affect the accuracy of estimation and the performance of guidance for the target. When the relative angles ranging from 30°to 65°were applied at the terminal, great estimation as well as guidance performance can be generated for the target. Designing the blindingevasion guidance law for the target to protect itself from the interception of pursuers,reference[10]gave the blinding guidance law based on the detection error model presented by Liu to force the LOS separation angle between two interceptors to reach consistent such that achieving the goal of interceptor’s detection ability worse firstly, then the “evader” guidance law taking the bang-bang maneuver at the end to improve the survivability.

The rest of the paper is organized as follows: In Section 2, the cooperative interception engagement problem statement is analyzed, and the measurement error model and the performance index evaluating miss distance is introduced. In Section 3, the cooperative guidance law which considers the detection error model is designed for the bang-bang maneuvering target based on the optimal control. In Section 4, the proposed guidance law and Fast MMAE method are implemented in simulations, and the performance is verified by results. The conclusion of this paper is proposed in Section 5.

2. Problem statement

For the case that an evader is engaged by our interceptors,it will adopt the most effective evasion strategy to deal with interceptors,that is, bang-bang optimal evasion maneuver. Our interceptor needs to take the effective estimation method to detect the random change of target acceleration and accurately estimate the state variables of the target. In addition, the relative geometric configuration depended on the LOS angle between two pursuers can affect the detection accuracy for the evader during the cooperative interception engagement process.

The kinematic model of interception engagement is established in the inertial coordinate system XI- OI- YI. Fig. 1 shows the planar engagement motion of evader, and the two pursuers. In Fig.1,the evader and pursuers are denoted by the variables E and Pirespectively.The normal acceleration,speed,and flight-path angle of aircrafts are denoted by ai,Vi,γii?｛P1,P2,E｝.The ranges between interceptors and evader are denoted by rPiE,i?｛1,2｝, and the LOS angle between them is denoted by λPiE,i?｛1,2｝.

2.1. Dynamics model

The relative motion between pursuers and evader can be expressed as a polar coordinates （r,λ） formed by relative distance rPiEand LOS angle λPiE:

During the entire guidance process, the normal acceleration of the aircraft can only change the direction of velocity, not its magnitude.The change rate of flight-path angle can be obtained by the normal acceleration and speed as follows:

Fig.1. Planar engagement motion of multi-aircraft.

Remark 1. When a nominal collision triangle between the pursuer and the evader is formed, the engagement process between them can be linearized, and the deviation between the LOS angle and the flight-path angle will become very small. In simulation section linearization (every time cycle) around the nominal collision triangle is continuously performed.

After linearization under the condition of a nominal collision triangle, let

where, τPiand τEis the first-order dynamic time constant of the pursuers Pi and evader E respectively.

The equation set (6) can be expressed as a form of state-space description:

Remark 2. Obviously, when the condition t =min（tfP1E,tfP2E） is satisfied, the guidance progress will end.

2.2. Measurement model

Using an IR sensor can only measures LOS angle λPiEin actual combat situations. In addition, the measurement for LOS angle is disturbed by a zero-mean white Gaussian measurement noises vPiwhich are assumed mutually independent in the guidance process.The pursuer’s LOS angle measurement noise is assumed to obey the distribution:

It can be seen from Fig. 1 that during the engagement a measuring baseline relative to the evader can be formed between the two pursuers.It is assumed that the measurement information can be shared between the two pursuers.Consider the assumption that the pursuer’s states are known accurately and can be shared with the other pursuer, and the relative position information （r,λPiPj）, i,j=1,2,i≠j between them can be calculated.

Therefore, the relative distance information λPiEcan be calculated through the sine theorem and the known information that the LOS angle λPiPj,i,j=1,2,i≠j and the measurement baseline r between the pursuers as follows:

According to the(9),the displacement information between the interceptors and the evader can be calculated under the condition of only measuring LOS angle information, so the measurement equation of the pursuer against evader can be expressed as

2.3. Performance index

Successful interception of a maneuvering target requires a minimal miss distance or even a direct hit. However, due to the influence of various factors, the interceptor cannot hit the maneuvering target with high accuracy. Especially, the state estimation method for maneuvering target seriously restricts the guidance accuracy. A realistic lethality model influenced by many factors is difficult to get, so we propose a simplified lethality function to evaluate the probability of destorying the target:

where,Rkis the lethality radius(LR)of the warhead,M is the miss distance between interceptor and target.

The index of successful intercepting target is miss distance,which is influenced by random target maneuver and detection noise, and become a random variable. We usually use the cumulative distribution function(CDF)as empirical estimate to evaluate the impact of miss distance on guidance accuracy, and use it to compare the performance of different guidance laws. So we can judge whether the interception is successful or not by using the determined kill probability in advance under the given LR condition. This kill probability is defined by

where E is the mathematical expectation with respect to the miss distance random variable, and the can be calculated by the CDF, it follows that

This performance index is to be minimized by the interceptor.

3. Design of cooperative guidance law

In this section, the cooperative guidance law for the multiaircraft engagement problem described by Section 2 is proposed,where two interceptors can cooperatively intercept a target with a bang-bang evasive maneuver. Because the guidance law changes the relative position of the pursuers while changing the flight trajectory, and the relative geometry configuration can affect the accuracy of cooperative detection and thus the guidance performance, the interception configuration has both effects on cooperative detection and guidance. In the process of cooperative guidance design, the real-time modulation of the LOS separation angle can effectively constrain the interception geometry configuration, and thus maintain a better detection ability and achieve a better guidance performance.The LOS separation angle need to be controlled to reduce the detection error so that the guidance performance can be improved. If the pursuer can form a perfect LOS separation angle which can minimize detection error, then the estimation accuracy and interception performance will be enhanced.

Therefore, the optimal control method controlling the LOS separation angle will be taken to achieve the goal of interception by taking the minimized miss distance and the minimized energy consumption [24] into consideration.

3.1. Cost function

3.2. Order reduction

Before solving the optimization problem, first its order need to be reduced. Such an order reduction is often denoted as terminal projection and its base on the homogeneous solution of differential equation. Using this method, the new state variables Zi（t） is,defined as follows:

Using new state variable Zi（t）,the objection function of(25)can be expressed as

3.3. Optimal controller

The Hamiltonian function of the objection function (28) is

The time derivative of the zero-effort variables is state independent, simplifying the adjoint equations considerably

Substituting (32) and (34) into (28) respectively, we have

Integrating(35) and (35) from t to tf, we have

Substituting (38) and (39) into (32) and (34) respectively, and the optimal controller is obtained as

4. Fast MMAE

In this section,an efficient MMAE called Fast MMAE is presented to identify and track the maneuvering target. In principle, the execution of MMAE algorithm requires the use of as many elemental filters as the assumed number. The MMAE algorithm needs to use as many assumptions of target maneuver strategy as possible,which will cause the number of filters to increase linearly with the number of assumptions,and the calculation load will also increase at the same time.Using Fast MMAE algorithm can reduce the number of filters used and improve the calculation speed. The Fast MMAE includes two processes: one is elemental filter aggregation,the other is filter bank cut.The escape strategy is assumed to be a sequence of maneuver commands for the target.Therefore,the difference of each elemental filter in MMAE is only leaded to the different assumptions of target maneuvering switching time. At any time t of the interception terminal, all elemental filters KF（αi）corresponding to the assumption that the maneuver switching time will occur in the future,that is,tisw>t,can be represented by a single filter.For the assumption that the maneuver switching time has already occurred, that is, tisw?t, if MMAE has not recognized that the assumption is correct, then MMAE does not need to contain the elemental filter KF（αj） corresponding to the assumption.

4.1. Traditional MMAE

Then, using the conditional Bayes formula and the conditional total probability formula, the posterior probability of the i-th elemental filter corresponding to the hypothesis αican be recursively calculated as follows

where, Sikis the measurement residual covariance matrix corresponding to the hypothesis αi.

Therefore, in the multi-model framework, the state estimation can be calculated according to the weighted average fusion of the state estimation of each elemental filter in which the weight is taken as the posterior probability of the relevant assumption.

4.2. Elemental filter aggregation

Aggregate all the elemental filters in MMAE that are indistinguishable up to the current moment.Because the state estimation,the estimation error covariance and the corresponding posterior probability of these element filters corresponding to the assumption that maneuver switching has not yet occurred at the current time are complete all the same,the elemental filter can be replaced by an aggregation filter. In this way, at time t, the number of elemental filters represented by the aggregation filter is

j（t）is the number of models in which the maneuver switching has occurred before time t. Obviously, Lag（t） decreases monotonically with j（t）.

When time t reaches the switching time corresponding to the ith model,initialize the elemental filter KF（αi）corresponding to the model;Its initial value is the state estimation and estimation error covariance matrix of the aggregation filter at this time.Because the aggregation filter represents Lag（t）identical models,the elemental filter KF（αi）has a posteriori probability, that is

4.3. Filter bank cut

Hexner et al. [19] pointed out that there is a lower bound of detection time for maneuvering command independent of estimator type. Therefore, no matter when the maneuver command switching occurs, it is assumed that the MMAE can recognize the maneuver in the subsequent time interval Tid. If, in the Tidperiod after the occurrence of a maneuver switch, the maneuver is not detected by the corresponding elemental filter (that is, the posterior probability of the elemental filter is always less than the predetermined threshold),the maneuver will not be recognized as the correct mode in the future. Therefore, the elemental filter can be removed from MMAE. According to Ref. [18], Tidcan be computed as

This shows that the physical lower bound of the achievable MMAE bank cut is determined by the detection time Tidand the terminal guidance duration time tf.

4.4. Fast MMAE filter process

3) Calculation of measurement residual and its covariance matrix

Fig. 2. Sample run planar trajectories of interception engagement.

Fig. 3. Acceleration profiles of pursuer1 and pursuer2.

Fig. 4. The posteriori probabilities of elemental filters for pursuer1.

Fig. 5. Acceleration estimation performance of Fast MMAE and KF/SF for pursuer1.

Set k=k+1 and return to step 2.

5. Simulation study

In this section,numerical simulations including two pursers and an evader are used to analyze the proposed cooperative guidance law and Fast MMAE method. First, the simulation parameters are set and the engagement scenario between multi-aircraft is analyzed. Then, the interaction between the Fast MMAE and the designed guidance law is evaluated by using Monte Carlo (MC)simulations. The estimation accuracy and guidance performance are mainly affected by two factors: one is the detection and response of the Fast MMAE to the target’s maneuvering switching time; the other is the restriction of the geometry of two interceptors on the detection capability. These results are compared with the proposed cooperative guidance law with a SF and augmented proportional navigation(APN)guidance law with a Fast MMAE.

Fig. 6. Estimation errors of pursuer1’s position, velocity, and acceleration.

Fig. 7. Estimation errors of pursuer2’s position, velocity, and acceleration.

5.1. Sample run example

5.2. Estimation performance evaluation

Under the assumption that the target maneuver switch occurs at 1.6 s,we compare the estimation ability of the Fast MMAE estimator with that of the SF estimator,and explore the influence of guidance configuration on detection efficiency.

Fig.6 and Fig.7 present the estimation errors of position,speed,and acceleration for pursuers.It can be seen from Figs.6 and 7 that estimation error of acceleration using Fast MMAE has a large fluctuation which is caused by the filter delay before 2 s, and it can quickly converge to zero at 2 s. Compared to the KF/SF, a lower estimation error for acceleration is owned by Fast MMAE. Better ability to estimate acceleration owned by Fast MMAE also makes it have a lower estimation error for position and velocity.This shows that Fast MMAE has better estimation accuracy and convergence speed.

Fig. 8. Lateral detection error of pursuer1 in the APN with a Fast MMAE, proposed guidance law with a KF/SF and a Fast MMAE.

Fig.9. LOS separation angle in the APN with Fast MMAE,proposed guidance law with KF/SF and fast MMAE.

Fig.10. CDF of pursuer1’s miss distance in the APN with Fast MMAE, proposed guidance law with KF/SF, and proposed guidance law with Fast MMAE.

Fig.11. CDF of pursuer2’s miss distance in the APN with Fast MMAE, proposed guidance law with KF/SF, and proposed guidance law with Fast MMAE.

Table 1 Required WLR of APN with Fast MMAE,Proposed Guidance Law with KF/SF,and Fast MMAE for a successful interception with SSKP = 0?95.

In Fig. 8, the displacement detection error of pursuer1 against the evader is proposed for a combination of different guidance laws and estimation methods. Fig. 9 shows the curve of the LOS separation angle formed by two pursuers when using a combination of different guidance laws and estimation methods.It can be observed from the two figures that,the APN with Fast MMAE cannot control the LOS separation angle throughout the guidance process, which results in a large detection error,especially ranging from 1.5 s to 3 s.The proposed guidance law with KF/SF or Fast MMAE using the method of controlling the LOS separation angle make its all gradually increases and reach the predetermined value,which results in declining in detection error with an increase of the LOS separation angle. The comparison indicates the proposed guidance law can reduce the detection error for state estimation in the process of cooperative interception.

Fig.12. Average miss distance of pursuer1 for different switch time in the APN with Fast MMAE, proposed guidance law with KF/SF, and proposed guidance law with Fast MMAE.

Fig. 13. Average miss distance of pursuer2 in the APN with Fast MMAE, proposed guidance law with KF/SF, and proposed guidance law with Fast MMAE.

5.3. Miss distance evaluation

In this section, the intertwined guidance-estimation performance of the APN with Fast MMAE, the proposed guidance law with KF/SF and Fast MMAE are analyzed through 100 MC simulations.

Fig. 14. Average miss distance of pursuers for different LOS angle in the proposed guidance law with Fast MMAE when the switch time is 3.8 s.

Fig. 10 and Fig. 11 show the empirical cumulative distribution function (CDF) of the pursuers homing performance for different guidance laws and estimation methods. The required warhead lethality ranges (WLR) for a successful interception with SSKP =0?95 is usually regarded as the main criterion Rk（0?95） to analyze and compare the interception performance between the different guidance law and estimation methods. The WLR for a successful interception with SSKP =0?95 is summarized in Table 1.Under the conditions that the SSKP =0?95 is required,then the pursuer1 must have the following warhead lethality ranges: 1) WLR≈1?46 m for the proposed guidance law with Fast MMAE, 2) WLR≈7?01 m for the proposed guidance law with KF/SF,3)WLR≈5?63 m for the APN with Fast MMAE.In summary,the proposed guidance law with Fast MMAE achieves a more favorable miss distance distribution than the proposed guidance law with KF/SF or the APN with Fast MMAE.Compared to the Fast MMAE,the main source of interception miss distance using KF/SF method is caused by its inaccurate acceleration estimation for the evader.Compared to the proposed guidance law, the main source of interception miss distance using APN is caused by its unsuitable guidance configuration between the two pursuers. In addition, the pursuer2 required WLR of proposed guidance law with KF/SF is smaller than the pusuer1 from Figs.10 and 11, compared to Fig. 3, which verify that the required control effort of pursuers can affect the interception performance.

Fig.12 and Fig.13 present the average miss distance of pursuers for different switch time. For the APN with Fast MMAE, the pursuers influenced by the detection error which is caused by LOS separation angle can yield miss distance when the target performs a maneuver switch between 1.5 s and 4 s. For the proposed guidance law with KF/SF,the miss distance is generated when the target performs a maneuver switch between 2.3 s and 4 s for pursuer1,between 3 s and 4 s for pursuer2. For the proposed guidance law with Fast MMAE,only the pursuer1 can yield a larger miss distance compared to the pursuer2.It can be seen from Figs.12 and 13 that the proposed guidance law with Fast MMAE has the best guidance accuracy among them. The reasons that the miss distance can be produced for the APN with Fast MMAE mainly due to the influence of detection error,and for the proposed guidance law with KF/SF is the estimation accuracy of the target acceleration. In addition, as the target switching time increases, the miss distance first increases and then decreases for the proposed guidance law. This is because the closer the target is to maneuver switch at the end of the guidance,the less response time is left for the guidance system,the filter delay makes the guidance system unable to change the command in time,which eventually leads to a large miss distance.if the target maneuvering switch occurs too late,the target does not have enough time to escape,and then the miss distance is reduced.

It can be seen from Figs.12 and 13 that the target will only yield miss distance when the switching maneuver is performed at the appropriate time. If the target performs the no-switching maneuver or the timing of the switching maneuver is incorrect, no miss distance will be generated.Fig.14 present the average miss distance of pursuers for different LOS separation angle in the proposed guidance law with Fast MMAE when the switch time is 3.8 s.It can be seen from Fig.14 that the average miss distance of pursuer1 and pursuer2 reach the maximum which are 2.48 m and 3.05 m respectively when the LOS separation angle is zero. As the LOS separation angle between the two pursuers increases,their average miss distance will decrease and eventually tend to a stable value.It should be noted that when the LOS separation angle of the two interceptors increases from 0°to 4°, the average miss distance of pursuer2 decreases particularly quickly, and reaches the lowest value when the LOS separation angle is at 4°. When the LOS separation angle increases from 0°to 15°,the average miss distance of pursuer1 gradually decreases and tends to a stable value after 15°.The results indicate that the designed guidance law can achieve good interception and estimation performance when the LOS separation angle is greater than 15°.But blindly increasing the LOS separation angle will cause the two interceptors to pay a considerable control cost, and consequently, choosing a appropriate LOS separation angle can ensure the highest interception accuracy within the overload limit.

6. Conclusion

(1) This paper proposes a cooperative guidance law which introduces the detection error model for the target that can perform a bang-bang evasive maneuver with a random switching time.

(2) The optimal control is used to design the cooperative guidance law that can modulate the LOS separation angle to reduce the detection error and improve the guidance performance.The Fast MMAE is used for identify the maneuver switch time and estimate the state of the target.

(3) Using MC simulations,this paper compares the guidance and estimation performance between the APN with Fast MMAE,proposed guidance law with KF/SF, and Fast MMAE. The results indicate that the proposed cooperative guidance law can reduce the detection error by controlling the increase of the LOS separation, and the Fast MMAE has a lower estimation error and faster convergence speed compared to the KF/SF. The combined Fast MMAE and cooperative guidance law have better estimation accuracy and guidance performance.

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.

Research background (Selective item)

This work was supported by the National Natural Science Foundation (NNSF) of China under grant no. 61673386, 62073335 and the China Postdoctoral Science Foundation (2017M613201,2019T120944).