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When threatened by a pursuer, an evading aircraft launches two defenders to accomplish a cooperative evasion, constituting a four-aircraft interception engagement. Under the assumption that the pursuing aircraft adopts the augmented proportional guidance law and first-order dynamics, a cooperative intercept mathematical model with an intercept angle constraint is established, allowing for the cooperative maneuvering of the evader. Based on the differential linear matrix inequality (DLMI), a controller design method of input and output finite time stability (IO-FTS) is proposed and applied to the aircraft’s cooperative intercept scenario. A cooperation performance analysis is carried out for two cases: (1) two defenders intercept the pursuer with various intercept angle constraints, and (2) the evader acts in a lure role to cooperate with two defenders. The simulation results indicate that the proposed method of controller design has the ability to guarantee that the two defenders intercept the pursuer at the preassigned intercept angles. The cooperative intercept scenario with a lure role is shown to be a very effective method for reducing the maximum required acceleration for defenders, which confirms the availability and advantage of cooperation. The strong adaptability and robustness of the cooperative guidance law with respect to various initial launch conditions is also verified.

A lone wolf cannot defeat a lion, but a group of wolves can easily contend against a lion, the reason for which is that the explicit division of labor, coordinated in cooperation, makes wolves much more powerful. There is an analogous case in an interception confrontation between multiple aircraft in which cooperation is also particularly important. The multiple-aircraft cooperative intercept scenario has become a subject of great interest in recent years. This paper addresses the tactics of cooperative defense in an engagement between four aircraft: an evader, two defenders, and a pursuer. For simplicity, the participants in this scenario are described by the terms Evader, Defender 1, Defender 2, and Pursuer, as shown in Figure

Schematic diagram of a four-aircraft engagement.

According to the different strategies and roles in multiple-aircraft cooperative engagement, the existing cooperative scenario generally can be classified into two modes: one is the cooperative scenario of different collaborative tasks and roles and the other is the cooperative scenario of the same collaborative tasks and roles. In the first category, the current research focuses on the evader-attacker-defender scenario (an evader, a pursuer, and a defender), which means that the evading aircraft carries a defensive missile (defender) by itself, and the evader acts as a lure and the defender acts as an interception to cope with the incoming interceptor together. This scenario was proposed by Boyell [

The second cooperative scenario is developed on the basis of the traditional one-to-one interception. The assignment and role of each aircraft in the interception process are basically the same. The core idea of this scenario is the consensus of flight status or impact time under different flight conditions (e.g., communication delay, different topologies, and salvo attack with intercept angle constraint). Recently, the research focus lies in the many-to-one cooperative intercept scenario based on two-to-one (namely, attacker-attacker-target) scenario, which is very useful and critical for the cluster operations. For the attacker-attacker-target scenario, Shaferman and Shima [

A multiple-aircraft engagement is a typical finite time process: the intercept process must be completed within a finite time interval rather than over an infinite time period. The guidance command is also meaningful in a finite time interval [

Inspired by the first scenario, we carried out the exploratory research based on the different roles in cooperation. As mentioned above, the aforementioned research focuses on the design of the optimal cooperative guidance law for a two-aircraft cooperative scenario based on optimal control theory, under the assumption that either the evader or defender possesses certain perfect information. Research into cooperative strategies between an evader and multiple defenders under intercept angles constraints is relatively lacking. An intercept scenario different than scenarios addressed in [

The remainder of this paper is organized as follows. In Section

A planar engagement between the four aircraft is considered. A schematic view of this engagement is shown in Figure

Schematic planar geometry of the four-aircraft engagement.

The four aircraft have first-order dynamics.

Pursuer adopts augmented proportional navigation.

Evader, Defender 1, and Defender 2 can share information with each other, and any delay is ignored.

According to the relative motion of the particles and the coordinate transformation, the relative motion equations between Evader and Pursuer can be derived as

Similarly, the LOS rotation motions of Defender 1-Pursuer and Defender 2-Pursuer satisfy

During the engagement, the velocities of the four agents are assumed to be constant, and intercept time can be approximated by

Then the corresponding times-to-go

Obviously, the critical condition to ensure cooperative evasion is

The dynamic equation can be expressed by (

According to Assumption

By (

The relative movement of the four aircraft constitutes a system in which we describe the cooperative guidance law of Evader, Defender 1, and Defender 2. Therefore, the design of the acceleration commands

In this case, Evader is considered to have a special mission that precludes its ability to freely maneuver in cooperation with Defender 1 and Defender 2 against Pursuer. The acceleration command of Evader is known by its defenders and is regarded as the external input vector

Because we are concerned with the normal acceleration

According to (

Owing to the noncooperation between Evader and Defenders, we can only control the Defenders to intercept Pursuer, meaning the input vector

The corresponding vectors

Setting

That is,

The system described by (

In this case, the Evader is able to freely maneuver with Defender 1 and Defender 2 against Pursuer for improved intercept performance. It is then necessary to design the acceleration commands of Evader, Defender 1, and Defender 2 for cooperative evasion. At this point, according to (

The corresponding vector

The external disturbance of the system can be regarded as the external input vector

When it is desired that Defender 1 and Defender 2 intercept Pursuer at designated intercept angles, an intercept model with angle constraints is considered. The state equations with an intercept angle constraint for the Defenders can be derived based on state equations (

For system equations (

According to the state equations in Sections

As mentioned above, the four-aircraft cooperative guidance process is a typical finite time process, and the cooperative intercept model established in Section

Assume that the linear time-varying (LTV) system equations (

If the LTV system equations (

IO-FTS and Lyapunov bounded input-bounded output (BIBO) stability are two different concepts. The IO-FTS theory focuses on signals defined over a finite time interval. The input output signals of IO-FTS are all norm bounded by preassigned quantitative bounds. However, the Lyapunov BIBO stability focuses on the existence of the bounds of input and output signals during an infinite time period.

IO-FTS is usually defined in the zero initial condition, which is due to the fact that the initial conditions and input signals will affect both the system states and output simultaneously. However, the effects of system states and output can be superimposed for linear systems, such as the system described in (

The multiple-aircraft cooperative interception problem given above can also be described as follows: Given a positive time scalar

If there exists a symmetric positive definite matrix-valued function

In order to design the cooperative guidance law for the four-aircraft engagement, we need to seek an appropriate state feedback controller

If there exists a symmetric positive definite matrix-valued function

Let

(In the following proofs, one omits variable

By (

Assume

Then (

According to (

Integrating (

By (

By using Schur complements [

Let

By system equations (

Let

To determine the state feedback controller

One discretizes DLMI into several standard LMIs, and the issue of controller design is transformed into the problem of solving LMIs. Based on the Matlab control toolbox and the in-point method, the numerical solution of LMIs can be calculated after several iteration loops. Therefore, the feasibility of the solution can be guaranteed by the existing method and software. The details for this procedure can be referred to in [

In this section, a simulation analysis is conducted to evaluate the behavior of the multiple-aircraft cooperative intercept scenario. The specific process can be divided into two steps: (1) Based on the system state equations (

Owing to the fact that the intercept angles of the Defenders and cooperative maneuvering of Evader can affect flight performance simultaneously, in Sections

In this case, Evader launches Defender 1 and Defender 2 for cooperative intercept, and Evader does not cooperate with Defender 1 and Defender 2. We assume that the two Defenders hit the Pursuer from the top for a better attack effect; therefore, the intercept angles of Defender 1 and Defender 2 are set to −15° and −30°, respectively. Based on the system equations derived in Section

Simulation parameters for cooperative intercept.

Parameter | Symbol | Initial value |
---|---|---|

Initial range | 15,000 m | |

Initial range | 15,000 m | |

Velocity of Evader | 250 m/s | |

Velocity of Pursuer | 500 m/s | |

Velocity of Defenders | 500 m/s | |

Response time of Evader | 0.1 s | |

Response time of Defenders | 0.05 s | |

Response time of Pursuer | 0.05 s | |

Navigation parameter | 4 | |

Navigation parameter | 3 | |

Initial system state vector | ||

Measurement matrix function | ||

Measurement matrix |

Figure

Four aircraft flight trajectories (one-side intercept without cooperation of Evader).

Figure

Intercept angle variation of the aircraft.

Figure

The variation of LOS angle rotation rate.

The above simulation results support the conclusion that the Defenders accomplish the cooperative interception in the finite time interval, showing that the designed controller can guarantee the ability of the Defenders to cooperatively intercept at preassigned angles. Thus, the theoretical methods underpinning the controller design are verified.

In order to illustrate the effectiveness and advantages of the guidance law proposed in this paper, we conduct a comparative simulation experiment. Two defenders use the proposed guidance law and the augmented proportional navigation (APN,

The simulation results are shown in Figures

Four aircraft flight trajectories.

The acceleration variation of Defender 1.

The acceleration variation of Defender 2.

In this case, based on the system state equations derived in Section

Figure

Four aircraft flight trajectories (one-side intercept with cooperation of Evader).

Figure

Intercept angle variation of the aircraft.

LOS angle rotation rate variation.

Comparing the accelerations of Defender 1 and Defender 2 in the cooperation and noncooperation scenarios presented in Cases A and B, respectively, we can obtain the comparative accelerations illustrated in Figures

Acceleration variation of Defender 1 (one-side intercept).

Acceleration variation of Defender 2 (one-side intercept).

From the point of view of probability, if the Defenders attempt to intercept Pursuer from two sides, it will be more difficult for Pursuer to avoid the interception [

Figure

Four aircraft flight trajectories (two-side intercept without cooperation of Evader).

Four aircraft flight trajectories (two-side intercept with cooperation of Evader).

Figures

Intercept angle variation of the aircraft.

LOS angle rotation rate variation.

Figures ^{2}, and that of Defender 1 reaches 338.4 m/s^{2}, indicating that compared with Defender 1, Defender 2 is more difficult to maneuver for successful intercept. Consequently, the simulation results indicate that for the two-side cooperative intercept case, Evader tends to cooperate more with the defender who demands a greater required acceleration, illustrating that the cooperative intercept scenario can reduce the required acceleration of the defender with whom Evader cooperates.

Acceleration variation of Defender 1 (two-side intercept).

Acceleration variation of Defender 2 (two-side intercept).

The initial launch conditions of the Defenders may be affected by the various flying situations of Evader. Additionally, while engaging with the different requirements of the combat mission, the Defenders must confront different battlefield launch conditions in which the validity and stability of the guidance law is critical to ensure accurate attack. Based on the cooperative guidance law proposed in Section

The simulation scenario is set up as follows: Evader cooperates with the two Defenders for cooperative evasion, and Defender 1 and Defender 2 intercept the Pursuer from different sides. The intercept angle of Defender 1 is set to 30°, and the intercept angle of Defender 2 is set to −30°. The initial value of the LOS angle rotation rate

Initial launch parameters of the two Defenders.

Case | Initial value |
Initial value |
---|---|---|

1 | 0.01 | −0.01 |

2 | 0.03 | −0.03 |

3 | 0.06 | −0.06 |

4 | 0.09 | −0.09 |

5 | 0.1 | −0.1 |

Figures

Intercept angle variation of Defender 1.

Intercept angle variation of Defender 2.

LOS angle rotation rate variation of Defender 1.

LOS angle rotation rate variation of Defender 2.

This study focuses on a four-aircraft intercept engagement scenario. Compared with previous research, this study primarily addresses two novel aspects of the intercept engagement scenario: (1) the cooperative intercept performance analysis of multiple defenders under intercept angle constraints and (2) the cooperative intercept performance analysis of the aircraft with the cooperation of the Evader in a lure role. Based on the relative motion and kinetic equations of the four-aircraft cooperative intercept models, three different cases of engagement are established. The design method for the state feedback controller is proposed based on the input and output finite time stability theory and is applied in the controller solution for the four-aircraft engagement. The simulation results show that the proposed method can guarantee that Defender 1 and Defender 2 intercept Pursuer at the preassigned intercept angles, and that the Defenders can achieve improved interception performance with the cooperation of Evader. It was also shown that the two Defenders can intercept Pursuer from either the same side or both sides, as a function of the desired intercept angle. The one-side cooperative intercept scenario reduces the maximum required acceleration of both defenders, while the two-side cooperative intercept scenario increases the intercept probability. When cooperating with the Defenders in a two-sided interception scenario, Evader tends to cooperate more with the defender who demands a greater required acceleration. Finally, the simulations demonstrate that the proposed cooperative guidance law possesses strong adaptability and robustness in the face of different initial launch conditions.

Normal acceleration of vehicle ^{2})

Normal acceleration command (m/s^{2})

State-space representations of the dynamics

The augmented system state matrix

Defender 1 and Defender 2

Evader

The state feedback matrix

The augmented proportional navigation parameters

Pursuer

Symmetric positive definite matrix-valued function

LOS angle from the vehicle

LOS angle rotation rate from the vehicle

Range between the vehicle

Initial range between the vehicle

Measurement matrix

Measurement matrix function

The total flight time (s)

Positive time scalar

System control input vector

Velocity (m/s)

System external input vector

System state variables

The initial system state variables

The augmented system state variables

System evaluation output vector

The augmented system evaluation output vector

Dynamic response time constants of the vehicle

The augmented vector of system state,

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation (NNSF) of China under grant number 61673386 and the Aviation Foundation of China under grant number 201651U8006.