Integrated Design of Cooperative Detection and Guidance Considering Equal Numbers of Aircraft on Both Sides

Abstract

In the scenario where the number of interceptors is equal to the number of target aircraft, and recognizing that the geometric configuration of interceptors during their maneuver towards targets affects detection effectiveness and guidance accuracy, we propose a Cooperative Detection and Guidance (CDG) method rooted in optimal control theory. This method optimizes detection by adjusting the line-of-sight (LOS) angle to minimize errors, and leverages the Fast Multiple Model Adaptive Estimation (Fast MMAE) algorithm to enhance interceptors’ ability to estimate the motion states and maneuver switching times of target aircraft, thereby boosting guidance accuracy. Results from 500 Monte Carlo simulations reveal that, compared to the Augmented Proportional Navigation (APN) guidance law, our integrated detection and guidance approach exhibits superior target recognition capabilities and achieves higher interception accuracy.

1. Introduction

The Cooperative Detection and Guidance (CDG) method has been extensively studied. Since its proposal, this method has been mostly applied to scenarios where two interceptors engage one target aircraft or the number of interceptors exceeds that of the target aircraft [1,2]. Compared with the traditional one-to-one interception mode, although this method can improve guidance precision and interception success rate, it cannot achieve the maximum interception effectiveness. Therefore, based on the CDG method, studying the cooperative detection and guidance problem when the number of interceptors is equal to that of the target aircraft is of great significance.

Currently, cooperative detection is mostly applied in research on spacecraft orbital docking and aerial vehicle guidance [3,4,5,6,7]. Both types of research aim for precise target positioning, and their theoretical applications are interoperable. The three-line-of-sight (3-LOS) cooperative positioning method has been widely used. Its principle is to form a detection triangle by acquiring the line-of-sight (LOS) angles of two of our aerial vehicles to the target aircraft and the positions of the two aerial vehicles, thereby estimating the target aircraft’s position, velocity, and acceleration information. However, this type of method is rarely used in scenarios such as autonomous rendezvous of non-coplanar spacecraft [8,9] and precise positioning, tracking, and identification of target aircraft by interceptors.

To address the scenario where a single aircraft launches multiple cooperative interceptor missiles simultaneously to counter incoming targets, Reference [10] proposed a cooperative reduced-order estimation method based on information sharing. By enabling each defensive missile to share LOS angle data with other team members, this method achieves the estimation of relative states and unknown parameters of individual members within the group, thereby improving interception efficiency. To solve the problem of cooperative strikes against multiple maneuvering targets and enhance multi-missile detection efficiency, Reference [11] proposed a group cooperative midcourse guidance law for heterogeneous missile formations. It introduced a super-twisting disturbance observer to estimate target acceleration, designed a cooperative guidance law using a group consensus protocol in the LOS angle direction, and developed a time-varying LOS angle formation tracking midcourse guidance law in the vertical direction using observer-estimated information. Its stability was proven via Lyapunov theory. Various noise errors are unavoidable during the detection and identification process. The multi-model adaptive estimation (MMAE) algorithm can ensure filter stability, improve filter tracking performance and estimation precision, and overcome the theoretical limitations of the extended Kalman filter (EKF) [12,13].

The original concept of cooperative guidance relies on the information interaction mechanism and coordinated control strategy among multiple aerial vehicle platforms to achieve cooperative strike operations under terminal spatiotemporal constraints. Divided by the constraint dimensions of technical implementation, the aerial vehicle swarm cooperative guidance technology under terminal spatiotemporal constraints can be further categorized into three types [14,15,16]:

(1)

Cooperative guidance technology with only strike time synchronization as the core constraint. Its core goal is to precisely control the flight speed and trajectory planning of each vehicle to ensure all strike platforms complete target strikes synchronously within a preset time window. It is mostly used in operational scenarios requiring “instantaneous saturation strikes” [17,18].

(2)

Cooperative guidance technology with only strike angle precision as the core constraint. It focuses on precisely controlling the projectile-target intersection angle by optimizing parameters such as the pitch angle and yaw angle of the vehicle’s terminal trajectory to improve the damage effect on specific parts of the target, for example, top strikes against armored targets to meet high-precision damage requirements [19,20,21].

(3)

Dual-constraint cooperative guidance technology that simultaneously considers strike time synchronization and strike angle precision. This technology needs to comprehensively balance the priorities of temporal and angular constraints, coordinating the vehicle’s speed, attitude, and trajectory through a multivariable coupled control algorithm. It is suitable for complex operational missions with strict requirements on both time synchronization and damage precision, such as multi-platform joint precision strikes against high-value fixed targets [22,23].

Cooperative guidance technology with strike time synchronization as the core constraint lies in constructing physically compatible, consistent coordination parameters. Based on this, a dynamic model of remaining flight time error is established, followed by the design of a feedback control module, ultimately achieving synchronized strikes against targets by the aerial vehicle swarm.

In terms of technical performance, some studies have proposed an adaptive cooperative guidance scheme in the LOS direction based on fast fixed-time consensus theory, flight time parameters, and undirected topology. By designing a novel nonsingular terminal sliding mode method to construct an adaptive fixed-time guidance law, this study does not require the target’s maneuvering information. The stability is verified via Lyapunov theory, and simulation results show that the scheme can achieve time-synchronized attacks and meet the desired impact angle [24].

To address the cooperative guidance challenge of multiple hypersonic glide vehicles, relevant research has proposed a method based on parametric design and analytical solutions of flight time: converting reentry trajectory optimization into parameter optimization to determine the angle of attack profile and reentry time; deriving an analytical formula for remaining flight time to satisfy cooperative constraints and calculate yaw angle control commands; optimizing parameters using the bee colony algorithm. Simulation results verify the time constraint satisfaction and cooperative precision of the method, and its robustness is confirmed through the Monte Carlo method [25]. For maneuvering targets in three-dimensional space that require multi-missile cooperative interception with specified attack time and desired LOS angle, a study has proposed a three-dimensional leader-follower guidance law. Optimized performance indicators are designed for different directions, and theoretical proofs and multiple sets of numerical simulations have verified the effectiveness, superiority, and robustness of the guidance law [26].

In addition, some studies focus on multi-vehicle spatiotemporal cooperative guidance technology that does not rely on remaining flight time, proposing a two-stage strategy combining cooperative guidance and proportional navigation to meet two-dimensional constraints. In three-dimensional scenarios, by adding a plane tracking guidance link, the scheme can still satisfy spatiotemporal constraints even when speed changes, and numerical simulation results have confirmed its effectiveness and application advantages [27]. To solve the high-precision interception problem of aerial maneuvering targets under limited attack time, a study has proposed a guidance law based on a nonlinear virtual relative model for targets near the origin. Relevant coefficients are determined through polynomial functions to meet constraint requirements. Simulation results show that the errors of the guidance law in different scenarios are within an acceptable range, and its performance is superior to existing similar guidance laws [28].

To solve the optimal initial and terminal guidance state problem for multi-missile interception of multi-targets, relevant research has proposed a midcourse guidance law considering the combined effect of flight time: constructing a three-dimensional model, designing an adaptive disturbance observer integrating finite-time theory and radial basis function, and setting intra-group and inter-group consensus protocols. Numerical simulation results have verified the effectiveness and superiority of the guidance law [29].

Cooperative guidance technology with strike angle precision as the core constraint adopts a core approach: optimizing key parameters such as the pitch angle and yaw angle of the vehicle’s terminal trajectory to precisely regulate the intersection angle between the missile and the target, thereby improving the damage effectiveness to specific parts of the target.

In the relevant research field, some scholars have integrated the “bias term” design concept with a multi-phase composite approach, proposing a novel two-stage guidance method. This method stipulates that when the continuous time integral of the bias term reaches a preset threshold, the guidance mode is switched to pure proportional guidance, ultimately achieving precise control of the impact angle constraint [30]. Another study proposed a multi-missile cooperative three-dimensional guidance law; although it can meet the requirement of impact angle constraint, this guidance law is only applicable to intercepting stationary targets [31]. Additionally, researchers have designed a three-dimensional guidance law incorporating impact angle constraint based on optimal control theory, but this guidance law is mainly targeted at slow-moving targets such as warships [32]. From a theoretical perspective, the optimal control method has the prominent advantage of “optimality.” On the basis of satisfying various constraints, it can achieve the optimization of specific performances, and naturally offers convenience in handling various constraints, such as minimum energy consumption and specific impact angle. For scenarios where relative velocity measurement data cannot be obtained, a study specifically explored the corresponding cooperative strike guidance scheme, successfully solving the technical problems caused by the lack of measurement information [22]. Another study proposed a novel three-dimensional preset-time cooperative guidance law (3-D PTCGL) based on the dynamic event-triggered (DET) mechanism. This guidance law can be used for multi-missile salvo attacks to intercept maneuvering targets with impact angle constraints [33]. Meanwhile, a study designed a two-phase impact angle control guidance scheme command controller suitable for air-to-air combat scenarios. This controller integrates two correction strategies: one based on the solution characteristics of the first phase, and the other, a prediction-based correction mechanism in the second phase. Incorporating these two correction strategies into the two-phase guidance scheme enables the interception of high-speed aerial targets at a specified impact angle [34].

For the scenario where the number of interceptors equals that of the target aircraft, this paper designs a CDG method based on optimal control theory. Through two interceptors adjusting the LOS separation angles, the detection errors of the two target aircraft are reduced. Meanwhile, the Fast MMAE algorithm is introduced to estimate the flight states of target aircraft, obtaining accurate information such as their position, velocity, and acceleration, as well as identifying their maneuver switch times. This achieves the tracking, interception, and simultaneous hit of target aircraft by the two interceptors.

The innovations of this paper are mainly reflected in two aspects: (1) For the scenario where the number of interceptors equals that of target aircraft, this paper designs a CDG method based on optimal control theory. This method innovatively achieves the comprehensive integration of cooperative detection and cooperative guidance, enabling interceptors to adjust the LOS angle and flight trajectory during flight, thereby improving their detection precision of target aircraft. (2) This paper introduces the Fast MMAE algorithm to estimate the motion states and maneuver switch times of target aircraft, thereby enhancing the guidance precision of interceptors.

The rest of this paper is organized as follows: Section 2 presents the establishment of the aircraft kinematic model and the miss distance model. Section 3 focuses on the construction of the CDG method based on optimal control theory. Section 4 describes the application method and steps of the Fast MMAE algorithm in this scenario. Section 5 conducts a simulation-based comparative analysis between the IDG method and the APN guidance law under the application background of Fast MMAE. Section 6 summarizes the main conclusions of this study.

2. Problem Formulation

Two of our intercepting vehicles lock onto two target vehicles and maneuver toward them, while the target vehicles adopt the bang-bang optimal evasive maneuver strategy for evasion. While maneuvering, our two intercepting vehicles detect the positions of the target vehicles and conduct information exchange. The specific interception scenario is illustrated in Figure 1. The kinematic models of intercepting vehicles P1, P2, and target vehicles E1, E2 are established in the inertial coordinate system X-O-Y. $a_i$, $V_i$, $q_i$ ($i\in \left\{P_1,P_2,E_1,E_2\right\}$) denote the LOS normal acceleration, velocity, and flight-path angle of the vehicles, respectively. $q_{P1E1}$, $q_{P2E1}$, $q_{P1E2}$, $q_{P2E2}$ denote the LOS angles between P1 and E1, P2 and E1, P1 and E2, and P2 and E2, respectively. $r_{P1E1}$, $r_{P2E1}$, $r_{P1E2}$, $r_{P2E2}$ denote the relative distances between P1 and E1, P2 and E1, P1 and E2, and P2 and E2, respectively.

2.1. Kinematics Model

The intercepting vehicle and the target vehicle can be represented by a polar coordinate system composed of the relative distance r and LOS angle q between them:

$$\left\{\begin{array}{l}{\dot{r}}_{PiEj}=V_{PiEj}\\ {\dot{q}}_{PiEj}=V_{PiEj}^{\perp }/r_{PiEj}\end{array}\right. \left(i=1,2 j=1,2\right)$$

(1)

${\dot{r}}_{PiEj}$, ${\dot{q}}_{PiEj}$ denote the closing velocity and LOS angle rate of the intercepting vehicle relative to the target vehicle, respectively. $V_{PiEj}$, $V_{PiEj}^{\perp }$ denote the velocity of the intercepting vehicle relative to the target vehicle along the LOS direction and the relative velocity perpendicular to the LOS direction, respectively, where

$$\left\{\begin{array}{l}{V}_{PiEj}=-V_{Ej}\cos \left(q_{Ej}+q_{PiEj}\right)-V_{Pi}\cos \left(q_{Pi}-q_{PiEj}\right) \left(i=1,2 j=1,2\right)\\ V_{PiEj}^{\perp }=-V_{Ej}\sin \left(q_{Ej}+q_{PiEj}\right)-V_{Pi}\sin \left(q_{Pi}-q_{PiEj}\right)\end{array}\right.$$

(2)

During the approach process between the intercepting vehicle and the target vehicle, since the acceleration $V_i$ of each vehicle is perpendicular to its velocity $a_i$ in direction, the acceleration can only change the direction of the vehicle’s velocity but not alter its magnitude. Therefore, the entire interception process can be regarded as a constant-velocity approach process, and the interception termination time between the intercepting vehicle and the target vehicle is given by:

$$t_{fPiEj}={\displaystyle \frac{-r_{PiEj}}{{\dot{r}}_{PiEj}}}={\displaystyle \frac{r_{PiEj}}{V_{Ej}\cos \left(q_{Ej}-q_{PiEj}\right)+V_{Pi}\cos \left(q_{Pi}+q_{PiEj}\right)}}$$

(3)

Throughout the entire process, the LOS angle rate of the intercepting vehicle is given by:

$${\dot{q}}_i={\displaystyle \frac{a_i}{V_i}} i\in \left\{P_1,P_2,E_1,E_2\right\}$$

(4)

Throughout the entire interception process, to ensure more accurate detection errors of the intercepting vehicles regarding the LOS angles and relative distances to the target vehicles, two detection triangles are specifically configured: namely, the detection triangle formed by intercepting vehicles P1 and P2 for target vehicle E1, and that for target vehicle E2. Thus, the entire interception process can be linearized, and accordingly, we set:

$$x\left(t\right)={\left[\begin{array}{cccccccccccc}{x}_1\left(t\right)& x_2\left(t\right)& x_3\left(t\right)& x_4\left(t\right)& x_5\left(t\right)& x_6\left(t\right)& x_7\left(t\right)& x_8\left(t\right)& x_9\left(t\right)& x_{10}\left(t\right)& x_{11}\left(t\right)& x_{12}\left(t\right)\end{array}\right]}^T$$

(5)

where

$$\left\{\begin{array}{l}{x}_1\left(t\right)=y_{11}\\ x_2\left(t\right)=y_{12}\\ x_3\left(t\right)=y_{21}\\ x_4\left(t\right)=y_{22}\end{array}\right. \left\{\begin{array}{l}{x}_5\left(t\right)={\displaystyle \frac{\mathrm{d}{y}_{11}}{\mathrm{d}t}}\\ x_6\left(t\right)={\displaystyle \frac{\mathrm{d}{y}_{12}}{\mathrm{d}t}}\\ x_7\left(t\right)={\displaystyle \frac{\mathrm{d}{y}_{21}}{\mathrm{d}t}}\\ x_8\left(t\right)={\displaystyle \frac{\mathrm{d}{y}_{22}}{\mathrm{d}t}}\end{array}\right. \left\{\begin{array}{l}{x}_9\left(t\right)=a_{E_1}\\ x_{10}\left(t\right)=a_{E_2}\\ x_{11}\left(t\right)=a_{P_1}\\ x_{12}\left(t\right)=a_{P_2}\end{array}\right.$$

(6)

$y_{ij}$ denotes the lateral displacement between intercepting vehicle i and target vehicle j, and $\mathrm{d}{y}_{ij}/\mathrm{d}t$ denotes the lateral velocity between them. $y_{ij}$ can be expressed as:

$$\left\{\begin{array}{l}{y}_{ij}\approx \left(q_{P_i{E}_j}-q_{P_i{E}_{j0}}\right)r_{P_i{E}_j}\\ r_{P_i{E}_j}\approx V_{P_i{E}_j}{t}_i^{go}\end{array}\right. \left(i=1,2 j=1,2\right)$$

(7)

$$q_{P_i{E}_j}\approx q_{P_i{E}_{j0}}+{\displaystyle \frac{y_i}{V_{P_i{E}_j}{t}_i^{go}}}$$

(8)

$r_{P_i{E}_j}$ denotes the relative distance between each pair of vehicles, and $t_i^{\mathrm{go}}$ denotes the remaining time-to-go of the intercepting vehicle, where

$$t_i^{go}=\left\{\begin{array}{l}{t}_{f{P}_i{E}_j}-t,t\le t_{f{P}_i{E}_j}\\ 0,t>t_{f{P}_i{E}_j}\end{array}\right. \left(i=1,2 j=1,2\right)$$

(9)

Assuming the dynamic characteristics of each vehicle can be equivalent to a first-order inertial link, then:

$$\left\{\begin{array}{l}{\dot{x}}_1\left(t\right)={\displaystyle \frac{\mathrm{d}{y}_{11}}{\mathrm{d}t}}\\ {\dot{x}}_2\left(t\right)={\displaystyle \frac{\mathrm{d}{y}_{12}}{\mathrm{d}t}}\\ {\dot{x}}_3\left(t\right)={\displaystyle \frac{\mathrm{d}{y}_{21}}{\mathrm{d}t}}\\ {\dot{x}}_4\left(t\right)={\displaystyle \frac{\mathrm{d}{y}_{22}}{\mathrm{d}t}}\end{array}\right. \left\{\begin{array}{l}{\dot{x}}_5\left(t\right)=a_{E_1}-a_{P_1}\\ {\dot{x}}_6\left(t\right)=a_{E_1}-a_{P_2}\\ {\dot{x}}_7\left(t\right)=a_{E_2}-a_{P_1}\\ {\dot{x}}_8\left(t\right)=a_{E_2}-a_{P_2}\end{array}\right. \left\{\begin{array}{l}{\dot{x}}_9\left(t\right)={\displaystyle \frac{a_{E_{1}C}-a_{E_1}}{{\tau }_{E_1}}}\\ {\dot{x}}_{10}\left(t\right)={\displaystyle \frac{a_{E_{2}C}-a_{E_2}}{{\tau }_{E_2}}}\\ {\dot{x}}_{11}\left(t\right)={\displaystyle \frac{a_{P_{1}C}-a_{P_1}}{{\tau }_{P_1}}}\\ {\dot{x}}_{12}\left(t\right)={\displaystyle \frac{a_{P_{2}C}-a_{P_2}}{{\tau }_{P_2}}}\end{array}\right.$$

(10)

In the formula, ${\tau }_{E_i}$, ${\tau }_{P_i}$ (i = 1,2) denote the first-order dynamic time constants of each vehicle, and $a_{E_{i}C}$, $a_{P_{i}C}$ denote the command accelerations of each vehicle, respectively.

Based on the kinematic models established above, the system state equation of the intercepting vehicles and target vehicles can be established as follows:

$$\dot{x}\left(t\right)=A\left(t\right)x\left(t\right)+B\left(t\right)u\left(t\right)+C\left(t\right)a_{EC}$$

(11)

$$A\left(t\right)=\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& -1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0& -1\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& -1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& -1\\ 0& 0& 0& 0& 0& 0& 0& 0& -{\displaystyle \frac{1}{{\tau }_{E_1}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& -{\displaystyle \frac{1}{{\tau }_{E_2}}}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -{\displaystyle \frac{1}{{\tau }_{P_1}}}& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -{\displaystyle \frac{1}{{\tau }_{P_2}}}\end{array}\right]$$

(12)

$$B\left(t\right)={\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{{\tau }_{P_1}}}& {\displaystyle \frac{1}{{\tau }_{P_2}}}\end{array}\right]}^T$$

(13)

$$C\left(t\right)={\left[\begin{array}{cccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{{\tau }_{E_1}}}& {\displaystyle \frac{1}{{\tau }_{E_2}}}& 0& 0\end{array}\right]}^T$$

(14)

In the formula, $u\left(t\right)$ denotes the command acceleration of the intercepting vehicle.

2.2. Measurement Model

Based on current detection technology, intercepting vehicles P1 and P2 can each detect the LOS angle information with target vehicles E1 and E2, but cannot directly measure the relative distances between the four vehicles. According to the interception scenario illustrated in Figure 1, after forming two detection triangles, assuming the two interceptors can perform a real-time exchange of the LOS angle $q_{\mathbf{P}i\mathbf{E}j}$ (i = 1, 2 j = 1, 2) detected from the target vehicles and their mutual relative position information, the relative distances between each pair of vehicles can be calculated using the law of sines:

$$\left\{\begin{array}{l}{r}_{P_1{E}_1}=r_{P_1{P}_2}{\displaystyle \frac{\sin \left(q_{P_1{P}_2}-q_{P_2{E}_1}\right)}{\sin \left(q_{P_2{E}_1}-q_{P_1{E}_1}\right)}}\\ r_{P_2{E}_1}=r_{P_1{P}_2}{\displaystyle \frac{\sin \left(q_{P_1{P}_2}-q_{P_1{E}_1}\right)}{\sin \left(q_{P_1{E}_1}-q_{P_2{E}_1}\right)}}\end{array}\right.$$

(15)

$$\left\{\begin{array}{l}{r}_{P_1{E}_2}=r_{P_1{P}_2}{\displaystyle \frac{\sin \left(q_{P_1{P}_2}-q_{P_2{E}_2}\right)}{\sin \left(q_{P_2{E}_2}-q_{P_1{E}_2}\right)}}\\ r_{P_2{E}_2}=r_{P_1{P}_2}{\displaystyle \frac{\sin \left(q_{P_1{P}_2}-q_{P_1{E}_2}\right)}{\sin \left(q_{P_1{E}_2}-q_{P_2{E}_2}\right)}}\end{array}\right.$$

(16)

where

$$r_{P_1{P}_2}=\sqrt{{\left(x_{P_1}-x_{P_2}\right)}^2+{\left(y_{P_1}-y_{P_2}\right)}^2}$$

(17)

$$q_{P_1{P}_2}=\arctan \left(y_{P_1}-y_{P_2},x_{P_1}-x_{P_2}\right)$$

(18)

Considering practical scenarios, the detected LOS angle information contains noise, where ${\widehat{q}}_{P_i{E}_j}=q_{P_i{E}_j}+{\sigma }_{P_i{E}_j}$ (i = 1, 2 j = 1, 2) denotes the noise-corrupted LOS angle. Thus, errors also exist in the aforementioned relative distance calculation. Assuming the detection error ${\sigma }_{P_i{E}_j}$ is Gaussian white noise, and all noise components are mutually independent, i.e., $E_j\left({\sigma }_{P_1{E}_j},{\sigma }_{P_2{E}_j}\right)=0$, ${\sigma }_{P_i{E}_j}~N\left(0,{\eta }_{P_i{E}_j}^2\right)$, the relative distance with error is given by:

$${\widehat{r}}_{P_i{E}_j}=H{x}_i+v_{P_i{E}_j,y}=y_{ij}+v_{P_i{E}_j,y}$$

(19)

$$H=\left[\begin{array}{cccccccccccc}1& 1& 1& 1& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

(20)

where $v_{P_i{E}_j,y}$ represents the measurement error of the relative distance, which satisfies $v_{P_i{E}_j,y}~N\left(0, {\sigma }_{P_i{E}_j,y}^2\right)$, as

$$\left\{\begin{array}{l}{\sigma }_{P_1{E}_1,y}={\displaystyle \frac{r_{P_1{P}_2}{q}_{P_1{E}_1}}{{\sin }^2\left(q_{P_1{E}_1}-q_{P_2{E}_1}\right)}}\sqrt{{\sin }^2\left(q_{P_1{P}_2}-q_{P_1{E}_1}\right){\sigma }_{P_2{E}_1}^2+{\sin }^2\left(q_{P_1{P}_2}-q_{P_2{E}_1}\right){\cos }^2\left(q_{P_1{E}_1}-q_{P_2{E}_1}\right){\sigma }_{P_1{E}_1}^2}\\ {\sigma }_{P_2{E}_1,y}={\displaystyle \frac{r_{P_1{P}_2}{q}_{P_2{E}_1}}{{\sin }^2\left(q_{P_1{E}_1}-q_{P_2{E}_1}\right)}}\sqrt{{\sin }^2\left(q_{P_1{P}_2}-q_{P_2{E}_1}\right){\sigma }_{P_1{E}_1}^2+{\sin }^2\left(q_{P_1{P}_2}-q_{P_1{E}_1}\right){\cos }^2\left(q_{P_1{E}_1}-q_{P_2{E}_1}\right){\sigma }_{P_2{E}_1}^2}\end{array}\right.$$

(21)

$$\left\{\begin{array}{l}{\sigma }_{P_1{E}_2,r}={\displaystyle \frac{r_{P_1{P}_2}{q}_{P_1{E}_2}}{{\sin }^2\left(q_{P_1{E}_2}-q_{P_2{E}_2}\right)}}\sqrt{{\sin }^2\left(q_{P_1{P}_2}-q_{P_1{E}_2}\right){\sigma }_{P_2{E}_2}^2+{\sin }^2\left(q_{P_1{P}_2}-q_{P_2{E}_2}\right){\cos }^2\left(q_{P_1{E}_2}-q_{P_2{E}_2}\right){\sigma }_{P_1{E}_2}^2}\\ {\sigma }_{P_2{E}_2,r}={\displaystyle \frac{r_{P_1{P}_2}{q}_{P_2{E}_2}}{{\sin }^2\left(q_{P_1{E}_2}-q_{P_2{E}_2}\right)}}\sqrt{{\sin }^2\left(q_{P_1{P}_2}-q_{P_2{E}_2}\right){\sigma }_{P_1{E}_2}^2+{\sin }^2\left(q_{P_1{P}_2}-q_{P_1{E}_2}\right){\cos }^2\left(q_{P_1{E}_2}-q_{P_2{E}_2}\right){\sigma }_{P_2{E}_2}^2}\end{array}\right.$$

(22)

It can be inferred from the above equation that the detection error $\left|q_{P_1{E}_j}-q_{P_2{E}_j}\right|$ increases as the relevant angle parameter decreases. When $\left|q_{P_1{E}_j}-q_{P_2{E}_j}\right|=0$, when the two intercepting vehicles are collinear with one of the target vehicles, the detection error will tend to infinity, making it impossible to effectively obtain the relative distance information of this target vehicle. Therefore, such a situation should be avoided in the process of cooperative guidance design.

2.3. Interception Indices

In addition to direct hits, an interception can also be regarded as successful if the minimum distance M between the intercepting vehicle and the target vehicle is less than the lethal range $R_k$ of the intercepting vehicle’s warhead, i.e.,

$$P_d\left(M,R_k\right)=\left\{\begin{array}{l}1 M\le R_k\\ 0 M>R_k\end{array}\right.$$

(23)

The miss distance becomes a random variable affected by target random maneuvers and detection noise. Thus, to evaluate its impact on guidance accuracy, the cumulative distribution function (CDF) is generally used for estimation. Therefore, we can judge the success of interception by the predetermined probability of successful interception based on the known lethal range of the warhead, i.e.,

$$\mathrm{SSKP}\left(R_k\right)=E\left\{P_d\left(M,R_k\right)\right\}$$

(24)

The mathematical expectation E of the miss distance can be calculated by the cumulative integral function, i.e.,

$$\mathrm{SSKP}\left(R_k\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{P}_d\left(M,R_k\right)f_M\left(m\right)}\mathrm{d}m={\displaystyle \underset{0}{\overset{R_k}{\int }}{f}_M\left(m\right)\mathrm{d}m=pr\left(M\le R_k\right)}=F_M\left(R_k\right)$$

(25)

$f_M$ denotes the probability density function (PDF), and $F_M$ denotes the CDF. According to Reference [1], the interception probability is usually set to 95%, thus:

$$J=\underset{R_k}{\arg }\left\{\mathrm{SSKP}\left(R_k\right)=95\%\right\}$$

(26)

3. Collaborative Optimal Guidance Method Design

In the interception scenario constructed in Section 2, since the target vehicles adopt bang-bang maneuvers for evasion, the intercepting vehicles will also adjust their flight trajectories in real time according to the target vehicles’ maneuver patterns to achieve successful interception. This section mainly proposes an integrated detection and guidance scheme for this interception scenario, thereby realizing the goal of two intercepting vehicles successfully intercepting two target vehicles at the same time. However, different LOS separation angles will affect detection accuracy, which in turn influences the guidance effect. To better achieve successful interception, considering the impact of different detection configurations on guidance accuracy, the design of the LOS separation angle in the guidance gain should aim to minimize detection errors, thus achieving high interception accuracy. Figure 2 shows the principle of the guidance and control method.

3.1. Objective Function

In the design of the performance index, taking into account the impact of $\left|q_{P_1{E}_j}-q_{P_2{E}_j}\right|$ on detection errors, the objective function is given by:

$$\begin{array}{ll}J=& {\displaystyle \frac{a_1}{2}}{y}_{1\left(P_1{E}_2\right)}^2+{\displaystyle \frac{a_2}{2}}{y}_{2\left(P_2{E}_2\right)}^2+{\displaystyle \frac{a_3}{2}}{y}_{3\left(P_1{E}_1\right)}^2+{\displaystyle \frac{a_4}{2}}{y}_{4\left(P_2{E}_1\right)}^2+{\displaystyle \frac{b_1}{2}}{\left[{\displaystyle \frac{y_1}{V_{P_1{E}_2}\left(t_{12}^{go}+\Delta t\right)}}-{\displaystyle \frac{y_2}{V_{P_2{E}_2}\left(t_{22}^{go}+\Delta t\right)}}-{\Delta }_{c1}\right]}^2\\ & +{\displaystyle \frac{b_2}{2}}{\left[{\displaystyle \frac{y_3}{V_{P_1{E}_1}\left(t_{11}^{go}+\Delta t\right)}}-{\displaystyle \frac{y_4}{V_{P_2{E}_1}\left(t_{21}^{go}+\Delta t\right)}}-{\Delta }_{c2}\right]}^2+{\displaystyle \frac{c_1}{2}}{\displaystyle {\int }_0^{t_{f1}}{u}_1^2\mathrm{d}t}+{\displaystyle \frac{c_2}{2}}{\displaystyle {\int }_0^{t_{f2}}{u}_2^2\mathrm{d}t}\end{array}$$

(27)

where $\Delta t$ is an infinitesimal quantity greater than 0 and infinitely close to 0; ${\displaystyle \frac{a}{2}}{y}_{\left(P_i{E}_j\right)}^2$ ensures that the interceptors have a small miss distance; ${\displaystyle \frac{b_i}{2}}{\left[{\displaystyle \frac{y_i}{V_{P_i{E}_1}\left(t_{i1}^{go}+\Delta t\right)}}-{\displaystyle \frac{y_j}{V_{P_j{E}_1}\left(t_{i1}^{go}+\Delta t\right)}}-{\Delta }_{ci}\right]}^2$ regulates the LOS separation angle by adding LOS angle constraints; and ${\displaystyle \frac{c}{2}}{\displaystyle {\int }_0^{t_f}{u}^2\mathrm{d}t}$ ensures low maneuvering energy consumption.

3.2. Model Reduction

To simplify the derivation process and obtain the analytical solution, it is necessary to perform model reduction on the performance index. A new state variable $Z^{\left(i\right)}\left(t\right)$ is introduced to represent the zero-effort miss (ZEM), then:

$$\begin{array}{c}{Z}^{\left(i\right)}\left(t\right)=D{\mathsf{\Phi }}^{\left(i\right)}\left(t_{fi},t\right)x^{\left(i\right)}\left(t\right)+D{\displaystyle {\int }_t^{t_{fi}}{\mathsf{\Phi }}^{\left(i\right)}\left(t_{fi},\tau \right)G{a}_{EC}\mathrm{d}\tau }\\ Z^{\left(i\right)}\left(t\right)=D{\mathsf{\Phi }}^{\left(i\right)}\left(t_{fi},t\right)x^{\left(i\right)}\left(t\right)+D{\displaystyle {\int }_t^{t_{fi}}{\mathsf{\Phi }}^{\left(i\right)}\left(t_{fi},\tau \right)G{a}_{EC}\mathrm{d}\tau }\end{array}$$

(28)

where ${\mathsf{\Phi }}^{\left(i\right)}\left(t_{fi},t\right)$ is the state transition matrix of the state equation; $D\in R^{1\times 12}$ is a constant vector used to separate each element in $x\left(t\right)$. From the properties of the state transition matrix, it can be derived that:

$${\dot{\mathsf{\Phi }}}^{\left(i\right)}\left(t_{fi},t\right)=-{\mathsf{\Phi }}^{\left(i\right)}\left(t_{fi},t\right)A^{\left(i\right)}\left(t\right)$$

(29)

Then, we can obtain:

$${\dot{Z}}^{\left(i\right)}\left(t\right)=D\left({\dot{\mathsf{\Phi }}}^{\left(i\right)}\left(t_{fi},t\right)x+{\mathsf{\Phi }}^{\left(i\right)}\left(t_{fi},t\right)\dot{x}\right)-D{\mathsf{\Phi }}^{\left(i\right)}\left(t_{fi},t\right)G{a}_{EC}=D{\mathsf{\Phi }}^{\left(i\right)}\left(t_{fi},t\right)B^{\left(i\right)}{u}_i$$

(30)

It can thus be seen that each element in ${\dot{Z}}^{\left(i\right)}\left(t\right)$ is mutually independent and only related to the control input $u_i$. Let $d_i=D{\mathsf{\Phi }}^{\left(i\right)}\left(t_{fi},t\right)B^{\left(i\right)}$, then ${\dot{Z}}^{\left(i\right)}\left(t\right)=d_i{u}_i$.

When $D=D_1=\left[\begin{array}{cccccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

$$Z_1\left(t\right)=D_1\mathsf{\Phi }\left(t_{f11},t\right)x\left(t\right)+D_1{\displaystyle {\int }_t^{t_{f11}}\mathsf{\Phi }\left(t_{f11},\tau \right)C{a}_{EC}\mathrm{d}\tau }$$

(31)

At this point, the first variable $y_{11}$ in $\mathit{x}\left(t\right)$ can be separated out;

Similarly,

$$\left\{\begin{array}{l}D=D_2=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\end{array}\right]\\ D=D_3=\left[\begin{array}{cccccc}0& 0& 1& 0& \cdots & 0\end{array}\right]\\ D=D_4=\left[\begin{array}{ccccccc}0& 0& 0& 1& 0& \cdots & 0\end{array}\right]\end{array}\right.$$

(32)

The variables $y_{12}$, $y_{21}$, $y_{22}$ in $x\left(t\right)$ can be separated out, respectively; thus, the objective function can be expressed as:

$$\begin{array}{ll}J=& {\displaystyle \frac{a_1}{2}}{Z}_{1\left(P_1{E}_2\right)}^2+{\displaystyle \frac{a_2}{2}}{Z}_{2\left(P_2{E}_2\right)}^2+{\displaystyle \frac{a_3}{2}}{Z}_{3\left(P_1{E}_1\right)}^2+{\displaystyle \frac{a_4}{2}}{Z}_{4\left(P_2{E}_1\right)}^2+{\displaystyle \frac{b_1}{2}}{\left[{\displaystyle \frac{Z_1}{V_{P_1{E}_2}\left(t_{12}^{go}+\Delta t\right)}}-{\displaystyle \frac{Z_2}{V_{P_2{E}_2}\left(t_{22}^{go}+\Delta t\right)}}-{\Delta }_{c1}\right]}^2\\ & +{\displaystyle \frac{b_2}{2}}{\left[{\displaystyle \frac{Z_3}{V_{P_1{E}_1}\left(t_{11}^{go}+\Delta t\right)}}-{\displaystyle \frac{Z_4}{V_{P_2{E}_1}\left(t_{21}^{go}+\Delta t\right)}}-{\Delta }_{c2}\right]}^2+{\displaystyle \frac{c_1}{2}}{\displaystyle {\int }_0^{t_{f12}}{u}_{12}^2\mathrm{d}t}+{\displaystyle \frac{c_2}{2}}{\displaystyle {\int }_0^{t_{f22}}{u}_{22}^2\mathrm{d}t}\\ & +{\displaystyle \frac{c_3}{2}}{\displaystyle {\int }_0^{t_{f11}}{u}_{11}^2\mathrm{d}t}+{\displaystyle \frac{c_4}{2}}{\displaystyle {\int }_0^{t_{f21}}{u}_{21}^2\mathrm{d}t}\end{array}$$

(33)

In the above equation, $t_{ij}^{go}$ is the remaining flight time of interceptor i against target j, and $u_{ij}$ is the acceleration of interceptor i against target j. Since in the simulation process, interceptor 1 only intercepts target 1 and interceptor 2 only intercepts target 2, $t_{12}^{go}$, $t_{21}^{go}$, $u_{12}$, $u_{21}$ are only used for auxiliary calculation and has no practical significance.

3.3. Optimal Guidance Law

The Hamiltonian function of the above-mentioned objective function is given by:

$$H={\displaystyle \frac{c_1}{2}}{u}_{12}^2+{\displaystyle \frac{c_2}{2}}{u}_{22}^2+{\displaystyle \frac{c_3}{2}}{u}_{11}^2+{\displaystyle \frac{c_4}{2}}{u}_{21}^2+{\lambda }_{Z_1}{\dot{Z}}_1\left(t_f\right)+{\lambda }_{Z_2}{\dot{Z}}_2\left(t_f\right)+{\lambda }_{Z_3}{\dot{Z}}_3\left(t_f\right)+{\lambda }_{Z_4}{\dot{Z}}_4\left(t_f\right)$$

(34)

Then, ${\lambda }_{Z_i}$ can be determined by

$$\left\{\begin{array}{l}{\lambda }_{Z_1}=a_1{Z}_1\left(t_f\right)-{\displaystyle \frac{b_1{\Delta }_{c1}}{V_{P_1{E}_2}{t}_{12}^{go}}}+{\displaystyle \frac{b_1}{V_{P_1{E}_2}{t}_{12}^{go}}}\left[{\displaystyle \frac{Z_1\left(t_f\right)}{V_{P_1{E}_2}{t}_{12}^{go}}}-{\displaystyle \frac{Z_2\left(t_f\right)}{V_{P_2{E}_2}{t}_{22}^{go}}}\right]\\ {\lambda }_{Z_2}=a_2{Z}_2\left(t_f\right)+{\displaystyle \frac{b_1{\Delta }_{c1}}{V_{P_2{E}_2}{t}_{22}^{go}}}-{\displaystyle \frac{b_1}{V_{P_2{E}_2}{t}_{22}^{go}}}\left[{\displaystyle \frac{Z_1\left(t_f\right)}{V_{P_1{E}_2}{t}_{12}^{go}}}-{\displaystyle \frac{Z_2\left(t_f\right)}{V_{P_2{E}_2}{t}_{22}^{go}}}\right]\\ {\lambda }_{Z_3}=a_3{Z}_3\left(t_f\right)-{\displaystyle \frac{b_2{\Delta }_{c2}}{V_{P_1{E}_1}{t}_{11}^{go}}}+{\displaystyle \frac{b_2}{V_{P_1{E}_1}{t}_{11}^{go}}}\left[{\displaystyle \frac{Z_3\left(t_f\right)}{V_{P_1{E}_1}{t}_{11}^{go}}}-{\displaystyle \frac{Z_4\left(t_f\right)}{V_{P_2{E}_1}{t}_{21}^{go}}}\right]\\ {\lambda }_{Z_4}=a_4{Z}_4\left(t_f\right)+{\displaystyle \frac{b_2{\Delta }_{c2}}{V_{P_2{E}_1}{t}_{21}^{go}}}-{\displaystyle \frac{b_2}{V_{P_2{E}_1}{t}_{21}^{go}}}\left[{\displaystyle \frac{Z_3\left(t_f\right)}{V_{P_1{E}_1}{t}_{11}^{go}}}-{\displaystyle \frac{Z_4\left(t_f\right)}{V_{P_2{E}_1}{t}_{21}^{go}}}\right]\end{array}\right.$$

(35)

The optimal cooperative guidance law can be derived from the coupled equations:

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial H}{\partial u_{11}}}=0\\ {\displaystyle \frac{\partial H}{\partial u_{22}}}=0\end{array}\right.$$

(36)

We can obtain

$$\left\{\begin{array}{l}{c}_3{u}_{11}+{\lambda }_{Z_3}{d}_3=0\\ c_2{u}_{22}+{\lambda }_{Z_2}{d}_2=0\end{array}\right.$$

(37)

The result can be obtained by solving the simultaneous equations:

$$\left\{\begin{array}{l}{u}_{12}=-{\displaystyle \frac{d_1{a}_1}{c_1}}{Z}_1\left(t_f\right)+{\displaystyle \frac{b_1{d}_1{\Delta }_{c1}}{V_{P_1{E}_2}{t}_{12}^{go}{c}_1}}-{\displaystyle \frac{b_1{d}_1}{V_{P_1{E}_2}{t}_{12}^{go}{c}_1}}\left[{\displaystyle \frac{Z_1\left(t_f\right)}{V_{P_1{E}_2}{t}_{12}^{go}}}-{\displaystyle \frac{Z_2\left(t_f\right)}{V_{P_2{E}_2}{t}_{22}^{go}}}\right]\\ u_{22}=-{\displaystyle \frac{d_2{a}_2}{c_2}}{Z}_2\left(t_f\right)-{\displaystyle \frac{b_1{d}_2{\Delta }_{c1}}{V_{P_2{E}_2}{t}_{22}^{go}{c}_2}}+{\displaystyle \frac{b_1{d}_2}{V_{P_2{E}_2}{t}_{22}^{go}{c}_2}}\left[{\displaystyle \frac{Z_1\left(t_f\right)}{V_{P_1{E}_2}{t}_{12}^{go}}}-{\displaystyle \frac{Z_2\left(t_f\right)}{V_{P_2{E}_2}{t}_{22}^{go}}}\right]\end{array}\right.$$

(38)

$$\left\{\begin{array}{l}{u}_{11}=-{\displaystyle \frac{d_3{a}_3}{c_3}}{Z}_3\left(t_f\right)+{\displaystyle \frac{b_2{d}_3{\Delta }_{c2}}{V_{P_1{E}_1}{t}_{11}^{go}{c}_3}}-{\displaystyle \frac{b_2{d}_3}{V_{P_1{E}_1}{t}_{11}^{go}{c}_3}}\left[{\displaystyle \frac{Z_3\left(t_f\right)}{V_{P_1{E}_1}{t}_{11}^{go}}}-{\displaystyle \frac{Z_4\left(t_f\right)}{V_{P_2{E}_1}{t}_{21}^{go}}}\right]\\ u_{21}=-{\displaystyle \frac{d_4{a}_4}{c_4}}{Z}_4\left(t_f\right)-{\displaystyle \frac{b_2{d}_4{\Delta }_{c2}}{V_{P_2{E}_1}{t}_{21}^{go}{c}_4}}+{\displaystyle \frac{b_2{d}_4}{V_{P_2{E}_1}{t}_{21}^{go}{c}_4}}\left[{\displaystyle \frac{Z_3\left(t_f\right)}{V_{P_1{E}_1}{t}_{11}^{go}}}-{\displaystyle \frac{Z_4\left(t_f\right)}{V_{P_2{E}_1}{t}_{21}^{go}}}\right]\end{array}\right.$$

(39)

Substituting Equations (38) and (39) into ${\dot{Z}}^{\left(i\right)}\left(t\right)$, we obtain

$$\left\{\begin{array}{l}{\dot{Z}}_1\left(t\right)=-{\displaystyle \frac{d_1^2}{c_1}}\left[a_1{Z}_1\left(t_f\right)+{\displaystyle \frac{b_1}{{\left(V_{P_1{E}_2}{t}_{12}^{go}\right)}^2}}{Z}_1\left(t_f\right)-{\displaystyle \frac{b_1}{V_{P_1{E}_2}{V}_{P_2{E}_2}{\left(t_{22}^{go}\right)}^2}}{Z}_2\left(t_f\right)-{\displaystyle \frac{b_1{\Delta }_{c1}}{V_{P_1{E}_2}{t}_{12}^{go}}}\right]\\ {\dot{Z}}_2\left(t\right)=-{\displaystyle \frac{d_2^2}{c_2}}\left[a_2{Z}_2\left(t_f\right)-{\displaystyle \frac{b_1}{V_{P_1{E}_2}{V}_{P_2{E}_2}{\left(t_{12}^{go}\right)}^2}}{Z}_1\left(t_f\right)+{\displaystyle \frac{b_1}{{\left(V_{P_2{E}_2}{t}_{22}^{go}\right)}^2}}{Z}_2\left(t_f\right)+{\displaystyle \frac{b_1{\Delta }_{c1}}{V_{P_2{E}_2}{t}_{22}^{go}}}\right]\end{array}\right.$$

(40)

$$\left\{\begin{array}{l}{\dot{Z}}_3\left(t\right)=-{\displaystyle \frac{d_3^2}{c_3}}\left[a_3{Z}_3\left(t_f\right)+{\displaystyle \frac{b_2}{{\left(V_{P_1{E}_1}{t}_{11}^{go}\right)}^2}}{Z}_3\left(t_f\right)-{\displaystyle \frac{b_2}{V_{P_1{E}_1}{V}_{P_2{E}_1}{\left(t_{21}^{go}\right)}^2}}{Z}_4\left(t_f\right)-{\displaystyle \frac{b_2{\Delta }_{c2}}{V_{P_1{E}_1}{t}_{11}^{go}}}\right]\\ {\dot{Z}}_4\left(t\right)=-{\displaystyle \frac{d_4^2}{c_4}}\left[a_4{Z}_4\left(t_f\right)-{\displaystyle \frac{b_2}{V_{P_2{E}_1}{V}_{P_1{E}_1}{\left(t_{11}^{go}\right)}^2}}{Z}_3\left(t_f\right)+{\displaystyle \frac{b_2}{{\left(V_{P_2{E}_1}{t}_{21}^{go}\right)}^2}}{Z}_4\left(t_f\right)+{\displaystyle \frac{b_2{\Delta }_{c2}}{V_{P_2{E}_1}{t}_{21}^{go}}}\right]\end{array}\right.$$

(41)

Integrating Equations (40) and (41) from $t$ to $t_f$, we have

$$\left\{\begin{array}{ll}{Z}_1\left(t_f\right)-Z_1\left(t\right)=& \left(-{\displaystyle \frac{a_1}{c_1}}{\displaystyle {\int }_t^{t_f}{d}_1^2\mathrm{d}t}-{\displaystyle \frac{b_1}{c_1{V}_{P_1{E}_2}^2}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_1}{t_{12}^{go}}\right)}^2\mathrm{d}t}\right)Z_1\left(t_f\right)+{\displaystyle \frac{b_1}{V_{P_1{E}_1}{V}_{P_2{E}_2}{c}_1}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_1}{t_{22}^{go}}\right)}^2\mathrm{d}t}{Z}_2\left(t_f\right)\hfill \\ & +{\displaystyle \frac{b_1{\Delta }_{c1}}{V_{P_1{E}_2}{c}_1}}{\displaystyle {\int }_t^{t_f}\frac{d_1^2}{t_{12}^{go}}dt}\hfill \\ Z_2\left(t_f\right)-Z_2\left(t\right)=& {\displaystyle \frac{b_1}{V_{P_1{E}_2}{V}_{P_2{E}_2}{c}_2}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_2}{t_{12}^{go}}\right)}^2\mathrm{d}t{Z}_1\left(t_f\right)}+\left(-{\displaystyle \frac{a_2}{c_2}}{\displaystyle {\int }_t^{t_f}{d}_2^2\mathrm{d}t-}{\displaystyle \frac{b_1}{V_{P_2{E}_2}^2{c}_2}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_2}{t_{22}^{go}}\right)}^2\mathrm{d}t}\right)Z_2\left(t_f\right)\\ & -{\displaystyle \frac{b_1{\Delta }_{c1}}{V_{P_2{E}_2}{c}_2}}{\displaystyle {\int }_t^{t_f}\frac{d_2^2}{t_{22}^{go}}}\mathrm{d}t\end{array}\right.$$

(42)

$$\left\{\begin{array}{ll}{Z}_3\left(t_f\right)-Z_3\left(t\right)=& \left(-{\displaystyle \frac{a_3}{c_3}}{\displaystyle {\int }_t^{t_f}{d}_3^2\mathrm{d}t}-{\displaystyle \frac{b_2}{c_3{V}_{P_1{E}_1}^2}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_3}{t_{11}^{go}}\right)}^2\mathrm{d}t}\right)Z_3\left(t_f\right)+{\displaystyle \frac{b_2}{V_{P_1{E}_1}{V}_{P_2{E}_1}{c}_3}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_3}{t_{21}^{go}}\right)}^2\mathrm{d}t}{Z}_4\left(t_f\right)\\ & +{\displaystyle \frac{b_2{\Delta }_{c2}}{V_{P_1{E}_1}{c}_3}}{\displaystyle {\int }_t^{t_f}\frac{d_3^2}{t_{11}^{go}}\mathrm{d}t}\\ Z_4\left(t_f\right)-Z_4\left(t\right)=& {\displaystyle \frac{b_2}{V_{P_2{E}_1}{V}_{P_1{E}_1}{c}_4}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_4}{t_{11}^{go}}\right)}^2\mathrm{d}t{Z}_3\left(t_f\right)}+\left(-{\displaystyle \frac{a_4}{c_4}}{\displaystyle {\int }_t^{t_f}{d}_4^2\mathrm{d}t-}{\displaystyle \frac{b_2}{V_{P_2{E}_1}^2{c}_4}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_4}{t_{21}^{go}}\right)}^2\mathrm{d}t}\right)Z_4\left(t_f\right)\\ & -{\displaystyle \frac{b_2{\Delta }_{c2}}{V_{P_2{E}_1}{c}_4}}{\displaystyle {\int }_t^{t_f}\frac{d_4^2}{t_{21}^{go}}}\mathrm{d}t\end{array}\right.$$

(43)

It can be easily derived as

$$\left\{\begin{array}{ll}{Z}_1\left(t_f\right)=& {\displaystyle \frac{\left(1+\frac{a_2}{c_2}{\displaystyle {\int }_t^{t_f}{d}_2^2\mathrm{d}t}+\frac{b_1}{V_{P_2{E}_2}^2{c}_2}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_2}{t_{22}^{go}}\right)}^2\mathrm{d}t}\right)\left(Z_1\left(t\right)+\frac{b_1{\Delta }_{c1}}{V_{P_1{E}_2}{c}_1}{\displaystyle {\int }_t^{t_f}\frac{d_1^2}{t_{12}^{go}}\mathrm{d}t}\right)}{N_1}}\\ & +{\displaystyle \frac{\left(\frac{b_1}{V_{P_1{E}_2}{V}_{P_2{E}_2}{c}_1}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_1}{t_{22}^{go}}\right)}^2\mathrm{d}t}\right)\left(Z_2\left(t\right)-\frac{b_1{\Delta }_{c1}}{V_{P_2{E}_2}{c}_2}{\displaystyle {\int }_t^{t_f}\frac{d_2^2}{t_{22}^{go}}}\mathrm{d}t\right)}{N_1}}\\ Z_2\left(t_f\right)=& {\displaystyle \frac{\left(1+\frac{a_1}{c_1}{\displaystyle {\int }_t^{t_f}{d}_1^2\mathrm{d}t}+\frac{b_1}{c_1{V}_{P_1{E}_2}^2}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_1}{t_{12}^{go}}\right)}^2\mathrm{d}t}\right)\left(Z_2\left(t\right)-\frac{b_1{\Delta }_{c1}}{V_{P_2{E}_2}{c}_2}{\displaystyle {\int }_t^{t_f}\frac{d_2^2}{t_{22}^{go}}}\mathrm{d}t\right)}{N_1}}\\ & +{\displaystyle \frac{\left(\frac{b_1}{V_{P_1{E}_2}{V}_{P_2{E}_2}{c}_2}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_2}{t_{12}^{go}}\right)}^2\mathrm{d}t}\right)\left(Z_1\left(t\right)+\frac{b_1{\Delta }_{c1}}{V_{P_1{E}_2}{c}_1}{\displaystyle {\int }_t^{t_f}\frac{d_1^2}{t_{12}^{go}}\mathrm{d}t}\right)}{N_1}}\\ N_1=& \left(1+{\displaystyle \frac{a_1}{c_1}}{\displaystyle {\int }_t^{t_f}{d}_1^2\mathrm{d}t}+{\displaystyle \frac{b_1}{c_1{V}_{P_1{E}_2}^2}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_1}{t_{12}^{go}}\right)}^2\mathrm{d}t}\right)\left(1+{\displaystyle \frac{a_2}{c_2}}{\displaystyle {\int }_t^{t_f}{d}_2^2\mathrm{d}t}+{\displaystyle \frac{b_1}{V_{P_2{E}_2}^2{c}_2}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_2}{t_{22}^{go}}\right)}^2\mathrm{d}t}\right)\\ & -\left({\displaystyle \frac{b_1}{V_{P_1{E}_2}{V}_{P_2{E}_2}{c}_1}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_1}{t_{22}^{go}}\right)}^2\mathrm{d}t}\right)\left({\displaystyle \frac{b_1}{V_{P_1{E}_2}{V}_{P_2{E}_2}{c}_2}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_2}{t_{12}^{go}}\right)}^2\mathrm{d}t}\right)\end{array}\right.$$

(44)

$$\left\{\begin{array}{ll}{Z}_3\left(t_f\right)=& {\displaystyle \frac{\left(1+\frac{a_4}{c_4}{\displaystyle {\int }_t^{t_f}{d}_4^2\mathrm{d}t}+\frac{b_2}{V_{P_2{E}_1}^2{c}_4}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_4}{t_{21}^{go}}\right)}^2\mathrm{d}t}\right)\left(Z_3\left(t\right)+\frac{b_2{\Delta }_{c2}}{V_{P_1{E}_1}{c}_3}{\displaystyle {\int }_t^{t_f}\frac{d_3^2}{t_{11}^{go}}\mathrm{d}t}\right)}{N_2}}\\ & +{\displaystyle \frac{\left(\frac{b_2}{V_{P_1{E}_1}{V}_{P_2{E}_1}{c}_3}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_3}{t_{21}^{go}}\right)}^2\mathrm{d}t}\right)\left(Z_4\left(t\right)-\frac{b_2{\Delta }_{c2}}{V_{P_2{E}_1}{c}_4}{\displaystyle {\int }_t^{t_f}\frac{d_4^2}{t_{21}^{go}}}\mathrm{d}t\right)}{N_2}}\\ Z_4\left(t_f\right)=& {\displaystyle \frac{\left(1+\frac{a_3}{c_3}{\displaystyle {\int }_t^{t_f}{d}_3^2\mathrm{d}t}+\frac{b_2}{c_3{V}_{P_1{E}_1}^2}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_3}{t_{11}^{go}}\right)}^2\mathrm{d}t}\right)\left(Z_4\left(t\right)-\frac{b_2{\Delta }_{c2}}{V_{P_2{E}_1}{c}_4}{\displaystyle {\int }_t^{t_f}\frac{d_4^2}{t_{21}^{go}}}\mathrm{d}t\right)}{N_2}}\\ & +{\displaystyle \frac{\left(\frac{b_2}{V_{P_2{E}_1}{V}_{P_1{E}_1}{c}_4}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_4}{t_{11}^{go}}\right)}^2\mathrm{d}t}\right)\left(Z_3\left(t\right)+\frac{b_2{\Delta }_{c2}}{V_{P_1{E}_1}{c}_3}{\displaystyle {\int }_t^{t_f}\frac{d_3^2}{t_{11}^{go}}\mathrm{d}t}\right)}{N_2}}\\ N_2=& \left(1+{\displaystyle \frac{a_3}{c_3}}{\displaystyle {\int }_t^{t_f}{d}_3^2\mathrm{d}t}+{\displaystyle \frac{b_2}{c_3{V}_{P_1{E}_1}^2}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_3}{t_{11}^{go}}\right)}^2\mathrm{d}t}\right)\left(1+{\displaystyle \frac{a_4}{c_4}}{\displaystyle {\int }_t^{t_f}{d}_4^2\mathrm{d}t}+{\displaystyle \frac{b_2}{V_{P_2{E}_1}^2{c}_4}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_4}{t_{21}^{go}}\right)}^2\mathrm{d}t}\right)\\ & -\left({\displaystyle \frac{b_2}{V_{P_1{E}_1}{V}_{P_2{E}_1}{c}_3}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_3}{t_{21}^{go}}\right)}^2\mathrm{d}t}\right)\left({\displaystyle \frac{b_2}{V_{P_2{E}_1}{V}_{P_1{E}_1}{c}_4}}{\displaystyle {\int }_t^{t_f}{\left(\frac{d_4}{t_{11}^{go}}\right)}^2\mathrm{d}t}\right)\end{array}\right.$$

(45)

4. Fast Multiple Model Adaptive Estimation

The Multiple Model Adaptive Estimation (MMAE) is a Bayesian technique that employs Kalman filters. It adapts to uncertainties in system states through multiple models, and dynamically adjusts the weights of these models based on current observation data to provide the most accurate state estimation. As shown in Figure 3, MMAE consists of multiple unit filters, each representing a possible hypothetical scenario. If the true unknown mode is included in the adopted model set, MMAE can serve as an optimal estimator.

In MMAE, all unit filters process the same set of measurement data and are mutually independent and operate in parallel. For the i-th unit filter, the basic state estimation output at time step k is denoted as ${\widehat{x}}_{\left.k\right|k-1}^i$, and the corresponding measurement residual is given by:

$$r_k^i=z_k-{\widehat{z}}_{\left.k\right|k-1}^i$$

(46)

Based on the conditional Bayesian formula and the law of total conditional probability, the recursive formula for the posterior probability of ${\alpha }_i$ corresponding to the i-th unit filter is constructed, i.e.,

$$p\left(\left.{\alpha }_i\right|Z^k\right)={\displaystyle \frac{p\left(\left.z_k\right|{\alpha }_i,Z^{k-1}\right)p\left(\left.{\alpha }_i\right|Z^{k-1}\right)}{{\displaystyle \sum _{j=1}^{L}p\left(\left.z_k\right|{\alpha }_j,Z^{k-1}\right)p\left(\left.{\alpha }_j\right|Z^{k-1}\right)}}}$$

(47)

where $p\left(\left.z_k\right|{\alpha }_i,Z^{k-1}\right)$ is the likelihood function of ${\alpha }_i$ at time step k, and this likelihood function follows a Gaussian distribution under the linear and Gaussian assumptions, i.e.,

$$p\left(\left.z_k\right|{\alpha }_i,Z^{k-1}\right)=p\left(r_k^i\right)=N\left(r_k^i;0,S_k^i\right)$$

(48)

where $S_k^i$ is the measurement residual covariance matrix of ${\alpha }_i$.

Since the target aircraft in this paper adopts a bang-bang maneuver, each occurrence time when the target aircraft randomly switches its maneuver direction to the opposite direction can be regarded as a maneuver mode. The acceleration identification problem of such randomly maneuvering targets belongs to the state estimation problem of hybrid systems under unknown and time-invariant modes. Under this maneuver mode, the traditional MMAE algorithm often requires a large number of unit filters related to escape strategy assumptions, and the more unit filters there are, the greater the computational load. It is therefore necessary to utilize an aggregated filter to replace the unit filters corresponding to the scenario assumptions before the moment when the aircraft’s maneuver switches, thereby simplifying the computational process and reducing the computational load. The terminal flight duration $t_{\mathrm{f}}$ of the aircraft is divided into L equal parts. Assuming that the aircraft’s maneuver has switched before time t, the number of unit filters contained in the aggregated filter at this moment is:

$$\left\{\begin{array}{l}{L}_{ag}\left(t\right)=L-j\left(t\right)\\ j\left(t\right)=\mathrm{int}\left(t-\Delta t_{sw}\right)\end{array}\right.$$

(49)

where $j\left(t\right)$ is the number of models where maneuver switches occurred before time t; $\Delta t_{sw}$ is $1/L$ of $t_{\mathrm{f}}$. It can be seen from this that $L_{ag}\left(t\right)$ is monotonically decreasing with $j\left(t\right)$.

When a maneuver switch occurs, the unit filter corresponding to the i-th model reached at time t is initialized, and the initial values are the state estimation value and estimation error covariance matrix of the aggregated filter. Since the state estimation, estimation error covariance, and corresponding model posterior probabilities of the unit filters replaced by the aggregated filter are identical, the posterior probabilities of these unit filters are:

$$p\left(\left.{\alpha }_i\right|Z^k\right)=\left[{\displaystyle \frac{1}{L_{ag}\left(t\right)}}\right]p\left(\left.{\alpha }_{ag}\right|Z^k\right)$$

(50)

where $p\left(\left.{\alpha }_{ag}\right|Z^k\right)$ is the posterior probability of the aggregated filter before initializing the unit filter, and the probability of the unit filter after initialization should be subtracted from that of the aggregated filter.

As indicated in Reference [1], there exists a lower bound of the detection time for maneuver commands. Thus, any maneuver switch occurring at any time within the subsequent time interval of MMAE can be detected by the unit filter. If the maneuver is not detected by the unit filter within the time period after a certain maneuver switch, it indicates that the model posterior probability of this unit filter is always less than the predetermined threshold, and this maneuver switch will not be successfully identified in the future. Therefore, the unit filter corresponding to this maneuver should be eliminated.

After time period $T_{id}$, the number of unit filters corresponding to maneuver switch models whose maneuver detection time has not yet reached $T_{id}$ is $\mathrm{int}\left(T_{id}/\Delta t_{sw}\right)$. Together with the aggregated filter, the total number of all valid unit filters in MMAE is:

$$L_s=\mathrm{int}\left({\displaystyle \frac{T_{id}}{\Delta t_{sw}}}\right)+1$$

(51)

The total number of unit filters required is:

$$L={\displaystyle \frac{t_f}{\Delta t_{sw}}}$$

(52)

Thus, the number of unit filters in MMAE is reduced to $L_s/L\cong T_{id}/t_f$ of the original. This result indicates that the lower bound of the detection time for maneuver commands in MMAE is determined by $T_{id}$ and $t_f$.

The implementation process of Fast MMAE is as follows:

First, initialize the model set $M={\left\{m_j\right\}}_{j=1}^{L_s}$, the base state ${\left\{x_{\left.0\right|0}^j,P_{\left.0\right|0}^j\right\}}_{j=1}^{L_s}$, and the model probabilities ${\left\{{\mu }_0^j\right\}}_{j=1}^{L_s}$;

$$\left\{\begin{array}{l}{m}_j={\alpha }_j\\ x_{\left.0\right|0}^j=x_{\left.0\right|0} j=1,\cdots ,L_s\\ P_{\left.0\right|0}^j=P_{\left.0\right|0}\end{array}\right.$$

(53)

$$\left\{\begin{array}{l}{\mu }_0^{L_s}\stackrel{\Delta }{=}p\left(\left.m^{L_s}\right|Z^0\right)=1-{\displaystyle \frac{\left(L_s+1\right)}{L}}\\ {\mu }_0^i\stackrel{\Delta }{=}p\left(\left.m_i\right|Z^0\right)={\displaystyle \frac{1}{L}}\end{array}\right. i=1,\cdots ,L_s-1$$

(54)

Then, calculate the mean and covariance:

$$\left\{\begin{array}{l}{\widehat{x}}_{\left.k\right|k-1}^j={\displaystyle \sum _{i=0}^{2n_x}{w}_i^{\left(m\right)}{\chi }_{i,\left.k\right|k-1}^j}\\ P_{\left.k\right|k-1}^j={\displaystyle \sum _{i=0}^{2n_x}{w}_i^{\left(c\right)}\left({\chi }_{i,\left.k\right|k-1}^j-{\widehat{x}}_{\left.k\right|k-1}^j\right){\left({\chi }_{i,\left.k\right|k-1}^j-{\widehat{x}}_{\left.k\right|k-1}^j\right)}^T+Q_k}\end{array}\right.$$

(55)

The measurement residual and its covariance matrix are given by:

$$\left\{\begin{array}{l}{\widehat{z}}_{\left.k\right|k-1}^j=H{\widehat{x}}_{\left.k\right|k-1}^j\\ S_k^j=H{P}_{\left.k\right|k-1}^j{H}^{\mathrm{T}}+R_k\end{array}\right.$$

(56)

Finally, calculate the gain, and update the state mean and covariance:

$$W_k^j=P_{\left.k\right|k-1}^j{H}^T{\left(S_k^j\right)}^{-1}$$

(57)

$$\left\{\begin{array}{l}{\widehat{x}}_{\left.k\right|k}^j={\widehat{x}}_{\left.k\right|k-1}^j+W_k^j\left(z_k-{\widehat{z}}_{\left.k\right|k-1}^j\right)\\ P_{\left.k\right|k}^j=P_{\left.k\right|k-1}^j-W_k^j{S}_k^j{\left(W_k^j\right)}^{\mathrm{T}}\end{array}\right.$$

(58)

To avoid numerical underflow when calculating model probabilities, let ${\mu }_{k-1}^j=e^{-a_{k-1}^j}$, ${\mu }_{k-1}^j=e^{-a_{k-1}^j}$, $j=1,\cdots ,L_{\mathrm{s}}$, we can obtain

$$\left\{\begin{array}{l}{a}_{k-1}^j=-\ln {\mu }_{k-1}^j\\ b_k^j=-\ln L_k^j={\displaystyle \frac{1}{2}}{\left({\mathrm{r}}_k^j\right)}^T{\left(S_k^j\right)}^{-1}{r}_k^j+{\displaystyle \frac{1}{2}}\ln \left|2\pi S_k^j\right|\end{array}\right.$$

(59)

The maximum model probability can be expressed as

$${\mu }_k^m={\displaystyle \frac{\exp \left[-\left(a_{k-1}^m+b_k^m\right)\right]}{{\displaystyle \sum _{j=1}^{L_s}\exp \left[-\left(a_{k-1}^j+b_k^j\right)\right]}}}={\displaystyle \frac{1}{{\displaystyle \sum _{j=1}^{L_s}\exp \left[\left(a_{k-1}^m+b_k^m\right)-\left(a_{k-1}^j+b_k^j\right)\right]}}}$$

(60)

$$m=\arg \underset{j}{\min }\left(a_{k-1}^j+b_k^j\right)$$

(61)

The updated model probabilities can be derived as

$${\mu }_k^j=\exp \left[\left(a_{k-1}^m+b_k^m\right)-\left(a_{k-1}^j+b_k^j\right)-a_k^m\right]$$

(62)

$$a_k^m=-\ln {\mu }_k^m$$

(63)

Therefore, the output overall state mean and covariance can be obtained as

$$\left\{\begin{array}{l}{\widehat{x}}_{\left.k\right|k}={\displaystyle \sum _{j=1}^{L_s}{\mu }_k^j{\widehat{x}}_{\left.k\right|k}^j}\\ P_{\left.k\right|k}={\displaystyle \sum _{j=1}^{L_s}{\mu }_k^j\left[P_{\left.k\right|k}^j+\left({\widehat{x}}_{\left.k\right|k}-{\widehat{x}}_{\left.k\right|k}^j\right){\left({\widehat{x}}_{\left.k\right|k}-{\widehat{x}}_{\left.k\right|k}^j\right)}^{\mathrm{T}}\right]}\end{array}\right.$$

(64)

If the current time reaches the switching instant $i\Delta t_{sw}$ corresponding to ${\alpha }_i$ of the i-th model, where $i=L_s,L_s+1,\cdots ,L$ holds, and simultaneously $p\left(\left.m_1\right|Z^k\right)$ is less than a predetermined threshold, then the model set is updated; otherwise, the model set is filtered again.

Next, update the model set and the unit filter bank.

$$\left\{\begin{array}{l}l=l+1\\ m_j={\alpha }_{j+1}\end{array}\right. j=1,2,\cdots ,L_s-2$$

(65)

$$\left\{\begin{array}{l}{\widehat{x}}_{\left.k\right|k}^i={\widehat{x}}_{\left.k\right|k}^{i+1\left(old\right)},P_{\left.k\right|k}^i=P_{\left.k\right|k}^{i+1\left(old\right)}\\ {\widehat{x}}_{\left.k\right|k}^{L_s-1}={\widehat{x}}_{\left.k\right|k}^{L_s\left(old\right)},P_{\left.k\right|k}^{L_s-1}=P_{\left.k\right|k}^{L_s\left(old\right)}\\ {\widehat{x}}_{\left.k\right|k}^{L_s}={\widehat{x}}_{\left.k\right|k}^{L_s\left(old\right)},P_{\left.k\right|k}^{L_s}=P_{\left.k\right|k}^{L_s\left(old\right)}\end{array}\right.$$

(66)

$$\left\{\begin{array}{l}{\mu }_k^i={\mu }_k^{i+1\left(old\right)}\\ {\mu }_k^{L_s-1}={\displaystyle \frac{{\mu }_k^{L_s\left(old\right)}}{L_{ag}}}\\ {\mu }_k^{L_s-1}={\displaystyle \frac{{\mu }_k^{L_s\left(old\right)}\left(L_{ag}-1\right)}{L_{ag}}}\end{array}\right.$$

(67)

where ${\widehat{x}}^{j\left(old\right)}$, $P^{j\left(old\right)}$ and ${\mu }^{j\left(old\right)}$ denote, respectively, the basic state mean, covariance, and model probability before the model set update.

The normalized model probability is given by:

$${\mu }_k^i={\displaystyle \frac{{\mu }_k^i}{{\displaystyle \sum _{j=1}^{L_s}{\mu }_k^j}}} j=1,\cdots ,L_s$$

(68)

5. Simulation Analysis

In this section, the engagement scenario involving two interceptors and two target aircraft will be analyzed based on the guidance model established earlier and the Fast MMAE algorithm. The interaction relationship between the Fast MMAE algorithm and the guidance method designed in this paper will be simulated and evaluated using the Monte Carlo algorithm, and the analysis results will be compared with the simulation results of the Augmented Proportional Navigation (APN) method for the same scenario.

5.1. Parameter Settings and Trajectory Simulation

In this section, the engagement zone distance is set to 12,000 m, denoted as $r_{PiEj}$ = 12,000 m, which represents the distance between the interceptor and the target vehicles. Upon entering the engagement zone, the initial vertical distances of the two interceptor vehicles are $y_{11}=50 \mathrm{m}$ and $y_{22}=50 \mathrm{m}$. According to the flight speeds of various types of aircraft in Reference [1], the flight speeds of the two interceptor vehicles are $V_{{\mathrm{P}}_1}=2000 \mathrm{m}/\mathrm{s}$ and $V_{{\mathrm{P}}_2}=1900 \mathrm{m}/\mathrm{s}$, while the flight speeds of the two target vehicles are $V_{{\mathrm{E}}_1}=1000 \mathrm{m}/\mathrm{s}$ and $V_{{\mathrm{E}}_2}=1100 \mathrm{m}/\mathrm{s}$. The final LOS angles are ${\Delta }_{\mathrm{c}1}={50}^{°}$ and ${\Delta }_{\mathrm{c}2}={40}^{°}$. The overload response time for all interceptor and target vehicles is 0.2 s. The maximum acceleration of the interceptor vehicles is $u_{\mathrm{P}}^{\max }=40 \mathrm{g}$, and the maximum acceleration of the target vehicles is $u_{\mathrm{E}}^{\max }=6 \mathrm{g}$. The sampling interval time is $\Delta t=0.001 \mathrm{s}$. In the MMAE, the durations of the terminal flight phases for the vehicles are $t_{\mathrm{f}}=4 \mathrm{s}$ and $\Delta t_{\mathrm{s}\mathrm{w}}=0.01 \mathrm{s}$, from which $L=t_f/\Delta t_{\mathrm{sw}}=40$ can be calculated. Since in the Fast MMAE, unit filters that identify ineffective maneuvers of the vehicles are integrated into an aggregate filter, the total number of actually effective unit filters is $L_{\mathrm{s}}=7$. The number of random target maneuver switching times calculated using the Monte Carlo method is $n=500$.

Figure 4 shows the flight trajectory simulation results. It can be seen from the figure that under the condition of target aircraft evasive maneuvers, the two interceptors have accurately hit their respective target aircraft. To meet the requirements of the preset final LOS separation angles $50°$ and $40°$ as well as simultaneous target engagement, the two interceptors performed circling flight in different directions along the y-axis, avoiding the problem of infinite detection error caused by the collinearity of three aircraft. Figure 5 presents the accelerations of the two interceptors. Due to the higher speed of interceptor P1, it generates a larger overload when adjusting the LOS separation angle, with a maximum acceleration of $382 m/s^2$; the maximum acceleration of interceptor P2 is $362 m/s^2$, both of which are less than $u_P^{\max }=40 g$.

5.2. Computational Performance Evaluation

In this part, based on the Fast MMAE algorithm, simulations have been conducted on the guidance law designed in this paper and APN, and a comparative analysis has been performed on the detection performance of interceptors on target aircraft under different guidance algorithms.

Figure 6 shows the posterior probabilities of interceptors P1 and P2 when using the Fast MMAE algorithm and the guidance algorithm designed in this paper, with the maneuver switch time set to 1.6 s. It can be clearly seen from the figure that the aggregated filter maintains a high probability during the period when no maneuver switch occurs, indicating that the target aircraft has not undergone a maneuver switch during this phase. The interceptors successfully detected the maneuver switch in only about 0.4 s. Due to a significant change in the acceleration of interceptor P1 after 2 s, some unit filters in Figure 6 experienced chattering, but they fully converged within 0.2 s.

Figure 7 and Figure 8, respectively, show the performance estimation of the target aircraft’s acceleration by interceptors using the CDG method and the APN guidance law. It can be seen from the figures that when CDG is adopted, the maneuver switch is successfully identified at 2 s, and the acceleration value quickly converges to the command acceleration value. Whereas when the APN guidance law is used, both the successful identification of the maneuver switch and the rapid convergence time of the acceleration are at 2.65 s, with a relatively slower identification speed.

Figure 9 and Figure 10, respectively, show the estimation errors of position, velocity, and acceleration when interceptor P1 targets target aircraft E1 and interceptor P2 targets target aircraft E2, using the CDG method and the APN guidance law. It can be seen from the figures that the estimation errors of position, velocity, and acceleration of both guidance methods can converge to zero after 2 s. However, compared with the APN guidance law, the estimation error curves of position and velocity for target aircraft using the CDG method exhibit smaller chattering and higher precision. For the acceleration estimation error, the CDG method converges to zero as early as 1.4 s (for P1-E1) and 1.2 s (for P2-E2), indicating that this guidance law has better acceleration detection precision.

Figure 11 shows the variation curves of the LOS separation angles of interceptors using the two guidance laws. Figure 12 and Figure 13 present the displacement detection errors of interceptor P1 for target aircraft E1 and E2, and those of interceptor P2 for target aircraft E1 and E2, when the CDG method is adopted. Figure 14 depicts the displacement detection errors of interceptor P1 for target aircraft E1 and interceptor P2 for target aircraft E2 when the APN guidance law is used. It can be observed from the figures that, since the APN guidance law cannot control the LOS separation angles of interceptors in this scenario, the displacement detection errors of interceptors for target aircraft are larger than those using the CDG method. The CDG method can adjust the LOS separation angles to achieve the preset final line-of-sight separation angles.

5.3. Miss Distance Evaluation

This section compares and analyzes the interception performance of an interceptor aircraft using the CDG method and the APN guidance law against target aircraft, based on the Fast MMAE algorithm and 500 Monte Carlo simulations.

Figure 15 and

Table 1. The comparison of miss distances when the CDFs of interceptors reach 95% under the two guidance methods.

Guidance Method	P1	P2
CDG	0.52 m	0.83 m
APN	1.31 m	2.24 m

show the comparison of the CDFs of miss distances for interceptors P1 and P2 using the two guidance laws. With the kill probability of a single interceptor set to SSKP = 95%, interception is deemed successful when the CDF reaches 95%, where the horizontal axis represents the miss distance. It can be seen from the figure that the CDFs of the two interceptors reach 95% at miss distances of 0.52 m and 0.83 m, respectively—much smaller than the 1.31 m and 2.24 m when the APN guidance law is adopted—exhibiting better guidance effectiveness and precision. Due to the similar accelerations of the two interceptors using the CDG method, their kill ranges are also comparable. In contrast, when the APN guidance law is used, the two interceptors cannot effectively interact with the detection information of the two target aircraft, resulting in larger kill ranges.

6. Conclusions

Based on optimal control theory, this paper proposes an integrated design method for cooperative detection and guidance (CDG) to solve the interception problem in scenarios where the number of interceptors equals that of target aircraft. This method comprehensively designs the detection and guidance stages, fully considering the impact of different configurations on detection precision. It adjusts the flight trajectories of interceptors by designing a guidance law, thereby improving their detection precision of target aircraft. Meanwhile, the Fast MMAE algorithm is introduced to estimate the motion states and maneuver switch times of target aircraft under both the CDG method and the APN guidance law in this scenario. Finally, through comparative simulation analysis, compared with the APN guidance law, the CDG method possesses the ability to adjust the LOS angle, demonstrating better detection precision and guidance precision. Additionally, the miss distance of the CDG method is significantly smaller than that of the APN guidance law. In the future, this method can be applied to military fields such as aerospace defense, it can improve the interception effectiveness of interceptors, reduce the requirement for the number of on-duty interceptors, and achieve optimal interception performance when responding to saturation attacks.

Due to the complexity of the calculation process of this method, when the number of interceptors is large, the computational power requirements for their on-board computers are relatively high. Therefore, simplifying the calculation process and designing a straightforward interception strategy will become the focus of future research.