Fixed-Time Distributed Event-Triggered Cooperative Guidance Methods for Multiple Vehicles with Limited Communications to Achieve Simultaneous Arrival

Abstract

Aiming at the salvo-attack problem of multiple missiles, a distributed cooperative guidance law based on the event-triggered mechanism is proposed, which enables missiles with large differences in spatial location and velocity to achieve simultaneous attacks with only a few dozen information exchanges. It effectively reduces the generation of control commands and communication frequency, thereby reducing channel load and improving communication efficiency and reliability. Compared to traditional periodic sampling communication, the number of communications has been reduced by over $90\%$. The guidance process is divided into two stages. The first stage is the cooperative guidance stage, where missiles achieve consensus of the time-to-go estimates through information exchange. In this stage, each missile is designed with an event-triggered function based on its own state error, and the missile only updates and transmits its information in the communication network when the error meets the set threshold, effectively reducing the occupancy rate of missile-borne resources during the cooperation process. The second stage is the independent guidance stage, where missiles can hit the target simultaneously while keeping the communication network silent. This is achieved by ensuring that the time-to-go estimates of missiles can represent the real time-to-go after achieving consensus. By the design of the two-stage guidance law and the replacement of the event-triggered function, the cooperative guidance system can be ensured to remain stable in scenarios where the leader missile is present and destroyed, and this excludes Zeno behavior. The stability of the cooperative guidance law is rigorously proved by algebraic graph theory, matrix theory, and the Lyapunov method. Finally, the numerical simulation results demonstrate the validity of the algorithm and the correctness of the stability analysis.

1. Introduction

With the rapid development of air defense and missile defense technology, important military targets usually possess multi-layered air defense and missile defense capabilities at long range, medium range, and short range. There are generally two strategies for attacking these targets. One is to increase the missile penetration speed and terminal maneuverability to make it difficult for enemy air defense systems to track and intercept. The other feasible means is to rely on existing missile platforms to build an inter-missile communication network, so that multiple missiles can penetrate the enemy’s multi-layered and tight air defense network through inter-missile information interaction, thus causing effective destruction to the target [1]. Each attack strategy has its advantages, and combining them can more effectively respond to the increasingly powerful anti-missile threats. In addition, multiple missiles cooperative operation is also an crucial means to achieve the concept of “dispersion of forces and concentration of firepower” in distributed operations. Missiles launched from various locations are connected through a communication network, enabling them to support and cooperate with each other. This interconnectivity significantly enhances the missiles’ search and recognition capability, battlefield situational awareness capability, and electronic countermeasures capability while reducing the risks faced by one’s own military forces. Multiple missiles cooperative operation is one of the technologies that must be developed for future warfare.

Early missile swarm cooperative operation was mainly achieved through offline trajectory planning, with no communication between missiles, which is essentially a problem of a single missile attacking a target with time constraints. Although this mode of operation can be a good solution to the problem of the simultaneous arrival of multiple missiles, it cannot adapt to the increasingly complex battlefield environment. The ideal missile swarm cooperative operation should network the missiles to enable them to cooperate in reconnaissance, guidance, threat avoidance, and autonomous decision making after launch. The simultaneous attack is an important manifestation of cooperative operation for multiple missiles, where all members of the missile swarm arrive simultaneously to carry out saturation attacks on the target or arrive according to a preset timing sequence to break through the target’s cover in layers. Due to the uncertainty of the target position in the mid-guidance segment before the missile acquires the target, it is challenging to achieve simultaneous attacks on the target. Therefore, simultaneous attacks are mainly achieved in the terminal guidance segment, and the design of cooperative terminal guidance law directly affects the missile’s penetration success rate and the damaging effect on the target. The development of simultaneous attacks terminal guidance technology has always kept pace with the change in the concept of multiple missiles cooperative operation and has developed the impact-time-control guidance approach (ITCG) and the cooperative guidance approach. ITCG was first proposed by Jeon et al. [2] in 2006 to solve the problem of simultaneous attacks of anti-ship missiles. Subsequently, scholars considered different practical situations [3] and adopted different control methods [4] based on Jeon’s research results, resulting in different impact-time-control guidance laws from guidance time [5], minimum control force [6], and angle of attack constraints [7]. ITCG was developed in the early stage of research on multiple missiles cooperative operation technology and is essentially an independent guidance approach. It requires all missiles to be assigned a uniform time-to-go before the terminal guidance stage, and each missile aligns its real time-to-go with the command by adjusting its flight trajectory. To ensure mission success, the largest time-to-go estimates in the missile swarm should be selected as the time-to-go command. However, time-to-go estimates of the missiles at the initial moment of the terminal guidance stage are inaccurate, which may cause some members of the missile swarm to be unable to complete the cooperative operation mission. Even if with accurate time-to-go estimates, all missiles except the one with the longest time-to-go estimates will increase their trajectory curvature to extend their arrival time, which will waste a lot of control energy and increase the risk of interception.

The cooperative guidance approach is developed in conjunction with the rise of the consensus theory of multi-agent systems, and its core role is to network the missiles. The missiles share their time-to-go estimates via the communication network and generate new guidance commands based on the information acquired in real time to achieve simultaneous attacks. Adopting the cooperative guidance approach in the terminal guidance phase not only solves the problems associated with the ITCG but also enables the missile swarm to respond flexibly to sudden changes in the battlefield situation. The application of multi-agent consensus theory in the field of terminal guidance enables the missiles to attack as a networked whole, constructing the prototype of missiles’ intelligent cooperative operation. Consensus theory was first proposed in the field of computer science and has a long research history, but it was not until Fiedler [8] introduced algebraic graph theory in 1973 that consensus theory had a suitable research tool and began to attract the attention of a large number of scholars. It has since been widely studied in fields such as unmanned combat [9], deep space exploration [10], and smart grids [11]. The premise of the cooperative guidance approach based on multi-agent consensus theory to achieve simultaneous attacks of multiple missiles is that the time-to-go estimates must reach consensus before any missile hits the target, which proposes strict requirements on the convergence rate of the consensus variables. Currently, research on the convergence rate of consensus variables in the cooperative guidance approach is mostly based on the finite-time consensus of multi-agent systems. Although the finite-time consensus algorithm has better dynamic characteristics, higher accuracy, and a faster convergence rate than the asymptotic-time consensus, the achievement of finite-time consensus is highly dependent on the initial values of the system state. For a high-speed missile swarm, the initial value of the time-to-go estimates is large in the terminal guidance phase and cannot be accurately obtained in advance, so the finite-time consensus algorithm is not applicable. To overcome this limitation, some scholars have developed fixed-time consensus algorithms based on the finite-time consensus algorithm. The fixed-time consensus algorithm allows a multi-agent system to reach stability independently of its initial state. At present, some research results have proved the effectiveness of the fixed-time consensus theory in multi-agent systems [12,13,14,15], but few scholars have applied the fixed-time consensus algorithm to the design of cooperative guidance laws to meet the strict requirements of high-speed missile swarm for the convergence time of consensus variables.

In addition, to the best of the authors’ knowledge, current research on the guidance law for cooperative simultaneous attacks relies on the assumption of “continuous communication” [16,17,18,19]. This assumption requires that the missile swarm be supported by powerful computing resources and the ideal communication environment. In the actual battlefield environment, missiles generally rely on their internal power supply, and the computing capabilities of processors and the communication bandwidth are limited. Frequent communication introduces latency and packet loss and consumes a large number of computing resources, which will lead to the instability of the control system and shorten the available time for executing missions. The main focus of this paper is on how to reduce the frequency of communication among missiles while ensuring the completion of simultaneous attacks. Introducing the event-triggered mechanism can provide a solution to the above problems, and its core idea is to decide whether a missile communicates based on whether the defined event is satisfied. The event-triggered mechanism has already been studied in the multi-agent field [20,21,22,23,24]. Ref. [20] studied the fixed-time consensus of a first-order multi-agent system based on the event-triggered mechanism under an undirected communication topology and provided two event-triggered strategies: centralized and distributed. The conclusion is that distributed event-triggered strategy should be given priority in the case of a large number of swarm members. Ref. [21] solved the global stability problem of a general linear system by limited control of the event-triggered mechanism. Since actual systems are nonlinear, it is necessary to study the event-triggered mechanism of nonlinear multi-agent systems, and Refs. [22,23], respectively, studied the event-triggered strategies of nonlinear systems with dynamic disturbances. Ref. [24] studied the adaptive event-triggered mechanism of multi-agent systems with randomly switching communication topologies.

However, the solution based on event-triggered mechanism is usually no longer effective for cooperative guidance systems that take the time-to-go estimates as a consensus variable, because the time-to-go estimates will still change after reaching consensus, which leads to the occurrence of Zeno behavior (Zeno behavior refers to the event-triggered function being triggered infinitely within a finite time interval). This paper cleverly solves this problem by proposing a two-stage cooperative guidance strategy, successfully applying the event-triggered mechanism to the design of cooperative guidance laws for missile swarm with the time-to-go estimates as the cooperative variable. The main contributions of this paper are as follows.

(1) The cooperative guidance law in this paper achieves consensus of real time-to-go of the missiles. As the missile’s flight conditions are usually unknown, only an expression for the time-to-go estimates can be established. In the proposed guidance law, once the time-to-go estimates of the missiles reach consensus, the navigation ratio will become a special constant, meaning that the time-to-go estimates of the missile will represent the real time-to-go, which is a prerequisite for the effects of the two-stage guidance strategy.

(2) In the two-stage guidance strategy, the first stage is the cooperative guidance stage, where the missiles achieve consensus of time-to-go estimates through information exchange, and the event-triggered mechanism is designed in this stage to reduce the communication frequency of the missile swarm. The second stage is the independent guidance stage, where each missile disconnects from communication and guides independently towards the target once the consensus error of time-to-go estimates converges to zero. Under the framework of the two-stage cooperative guidance law, the problem that the event-triggered mechanism does not apply to the missile swarm with the time-varying leader is solved by ensuring that no Zeno behavior occurs before consensus is reached and the missiles can hit the target simultaneously without communication after consensus is reached. In addition, the two-stage guidance strategy can also improve the missile’s ability to respond to communication interference and facilitates silent attacks on the target.

(3) The design of the event-triggered mechanism is completely distributed. A threshold is set for the time-varying error of the time-to-go estimates of each missile, and the calculation of the threshold only requires obtaining information about the adjacent missile’s event-triggered moment. Communication and guidance command updates will only occur when the time-varying error of the time-to-go estimates of the missile reaches the threshold, effectively solving the problems caused by the limited missile-borne computing resources and communication bandwidth.

(4) The time-to-go estimates of the missiles can quickly converge to the consensus within a fixed time. The fixed-time consensus algorithm can ensure that the missile swarm with large initial position differences and high flight speeds achieves consensus of the real time-to-go in a short time, and meets the strict time requirements of the high-speed missile swarm.

(5) Considering practical operation scenarios, the cooperative guidance law is first designed for the missile swarm with the leader and then extended to cases where the leader is destroyed. In the above two cases, the stability of the guidance system is ensured by the replacement of the event-triggered function.

In this paper, distributed cooperation, two-stage guidance, and event-triggered mechanism are designed to reduce the computation and communication burden during the cooperation process, which at the same time brings the challenge to the stability analysis of the guidance system.

The remainder of this paper is organized as follows: Section 2 presents frequently used symbols, knowledge related to algebraic graph theory, and some lemmas required for proof. Section 3 first describes the operational scenario studied in this paper, then establishes the equations of engagement kinematics and introduces the fixed-time convergence criterion and the event-triggered mechanism. In Section 4, a fixed-time distributed cooperative guidance law based on the event-triggered mechanism for multiple missiles to achieve simultaneous attacks is designed. Section 5 extends to the case where the leader is destroyed. Section 6 analyzes the performance of the cooperative guidance law through numerical simulation, and the conclusion of this paper is given in Section 7.

2. Preliminaries

2.1. Notations

This section explains some common symbols used in the paper. ${\mathbf{R}}_{+}$, ${\mathbf{R}}^N$, and ${\mathbf{R}}^{N\times N}$ denote the space of positive real numbers, the space of N dimensional real vectors, and the space of $N\times N$ dimensional real matrices, respectively. ${\mathbf{1}}_N\in {\mathbf{R}}^N$ denotes the column vector with all elements equal to one. $\left|\right|·\left|\right|$ denotes the Euclidean norm of a vector or the 2-norm of a matrix. The relation $\mathit{M}>0$ for any square matrix $\mathit{M}$ means that the matrix $\mathit{M}$ is positive definite in this paper.

2.2. Algebraic Graph Theory

The interconnection between missiles can be modeled using a graph, where the vertices of the graph represent individual missiles, and the edges represent the topological relationships (such as communication and sensing). We denote the graph $\mathit{G}=(\mathit{V},\mathit{E},\mathit{A})$, where $\mathit{V}=(v_1,v_2,\cdots v_N)$ represents the set of nodes consisting of N missiles, and $v_i$ represents the node of the i-th missile. $\mathit{E}\subset \mathit{V}\times \mathit{V}=\{(v_i,v_j):v_i,v_j\in \mathit{V}\}$ represents a set of edges, and $(v_i,v_j)\in \mathit{E}$ means that the j-th missile can receive the information from the i-th missile. The neighborhood includes the indegree neighborhood and the outdegree neighborhood. The indegree neighborhood of the i-th missile is the set of missiles that send messages directly to it, which is defined as $N_i=\left\{v_j\right|(v_j,v_i)\in E\}$. The outdegree neighborhood of the i-th missile is the set of missiles that can directly receive information from it, which is defined as $N_j=\left\{v_i\right|(v_i,v_j)\in E\}$.

Algebraic graph theory is the theory that studies the relationship between the structure of a graph and its matrix representation, and the two most important concepts used in multi-agent systems are the adjacency matrix and the Laplacian matrix. The adjacency matrix stores the relationships between all vertices of the graph in the form of a matrix, and the adjacency matrix of the graph $\mathit{G}$ can be written as $\mathit{A}=\left[a_{ij}\right]\in {\mathbf{R}}^{N\times N}$, where its diagonal elements $a_{ii}=0$, $a_{ij}>0$ when $(v_j,v_i)\in \mathit{E}$, and $a_{ij}=0$ when $(v_j,v_i)\notin \mathit{E}$. $a_{ij}$ represents the weight of the edge $(v_i,v_j)\in \mathit{E}$, which is used to define the importance of communication links. The Laplacian matrix is a matrix representation of the graph, and its i-th row represents the accumulation of gains generated when the i-th node perturbs other nodes. The Laplacian matrix of the graph $\mathit{G}$ can be represented as $\mathit{L}=\left[l_{ij}\right]\in {\mathbf{R}}^{N\times N}$, where $l_{ii}={\sum }_{j\in N_i}{a}_{ij}$ and $l_{ij}=-a_{ij},\forall i\ne j$.

We assume that the communication topology of the missile swarm consists of one leader and N followers. The variables associated with the leader missile are denoted by the subscript r, and the variables associated with the followers are denoted by the subscript $1,2,\dots ,N$. The adjacency matrix of the leader missile is defined as $\mathit{B}=\mathrm{diag}(b_i)\in {\mathbf{R}}^{N\times N}$, where $b_i>0$, if the i-th follower can receive information from the leader missile, and $b_i=0$ otherwise. We define $\mathit{M}=\left[m_{ij}\right]\in {\mathbf{R}}^{N\times N}=\mathit{L}+\mathit{B}$. If all follower missiles can directly access information from the leader missile, or indirect access to information about the leader missile from other follower missiles, then the leader missile is said to be globally reachable.

2.3. Some Lemmas

Lemma 1

([12]). For a system $\dot{\mathit{x}}=f(\mathit{x},t)$ with an initial state of $\mathit{x}\left(0\right)=x_0$, if there exists a continuous positive definite and radially unbounded function $\mathit{V}\left(\mathit{x}\right):{\mathbf{R}}^N\to {\mathbf{R}}_{+}\bigcup 0$ that satisfies the following conditions:

(1) 

$\mathit{V}\left(\mathit{x}\right)=0\iff \mathit{x}=0$.

(2) 

$D^{*}\mathit{V}\left(\mathit{x}\right)\le -a{V}^p\left(\mathit{x}\right)-b{V}^q\left(\mathit{x}\right)$.

where a, b, p, and q are positive constants, and $p\in (0,1)$, $q\in (1,+\infty )$, then the system can achieve global fixed-time stability, and the stability time T satisfies

$$T\le T_{max}={\displaystyle \frac{1}{a\left(1-p\right)}}+{\displaystyle \frac{1}{b\left(q-1\right)}}$$

(1)

Lemma 2

([20]). The following inequalities hold for any non-negative real numbers $x_1,x_2,\cdots ,x_n$.

$$\begin{array}{c}\hfill n^{1-\tilde{p}}{\left(\sum _{i=1}^n{x}_i\right)}^{\tilde{p}}\ge \sum _{i=1}^n{x}_i^{\tilde{p}}\ge {\left(\sum _{i=1}^n{x}_i\right)}^{\tilde{p}}\end{array}$$

(2)

where $\tilde{p}\in (0,1]$.

$$\begin{array}{c}\hfill n^{1-\tilde{q}}{\left(\sum _{i=1}^n{x}_i\right)}^{\tilde{q}}\le \sum _{i=1}^n{x}_i^{\tilde{q}}\le {\left(\sum _{i=1}^n{x}_i\right)}^{\tilde{q}}\end{array}$$

(3)

where $\tilde{q}\in [1,+\infty )$.

Lemma 3

([21]). If the communication topology of missiles is undirected and connected, then its Laplacian matrix $\mathit{L}$ is symmetric, and the inequality ${\mathit{x}}^{\mathrm{T}}\mathit{Lx}\ge {\lambda }_2{\mathit{x}}^{\mathrm{T}}\mathit{x}$ holds if ${\mathbf{1}}^{\mathrm{T}}\mathit{x}=0$ is satisfied for any $\mathit{x}\in {\mathbf{R}}^N$, where ${\lambda }_2$ is the smallest nonzero eigenvalue of the matrix $\mathit{L}$.

3. Problem Description

3.1. Operational Scenario Description

The operational scenario studied in this paper involves using multiple missiles to attack a high-value military target with tight air defense systems, and the operational process is shown in Figure 1. Multiple missiles launched from various platforms converge in the predetermined airspace, forming a stable communication link between them and approaching the target in a high–low-trajectory formation to penetrate the target’s long-range and mid-range air defense network. After the missile-borne sensors detect the target’s location, the cooperative guidance law is employed to achieve the simultaneous attacks of multiple missiles against the target. It places tremendous pressure on the target’s short-range air defense system in a short period, making it overloaded and unable to intercept all of the missiles. The focus of this paper is on solving the cooperative guidance problem in the abovementioned missile swarm operation process.

When penetrating the target’s long-range and mid-range air defense networks, one missile is typically selected as the leader to fly at a high altitude and perceive the battlefield situation. The rest of the missiles follow at a low altitude, silently receiving information transmitted by the leader to reduce the possibility of the entire missile swarm being detected by the enemy. When the missile-borne sensors detect the target, the missile also faces the threat of the target’s short-range air defense system. Cooperative guidance law is required at this point to penetrate the target’s short-range air defense network. This paper considers the presence of the leader missile in the high–low-trajectory formation strategy and designs a simultaneous attack cooperative guidance law for the missile swarm. Additionally, compared to the follower missiles flying silently at a low altitude, the leader flying at a high altitude is more susceptible to be detected and destroyed. Ensuring that the missile swarm can still simultaneously attack the target in the absence of the leader is one of the key issues considered in the cooperative guidance law design in this paper.

3.2. Engagement Kinematics

We consider the scenario where one leader missile and N follower missiles simultaneously attack a stationary target. Assuming that the missiles are in favorable spatial positions for attacking the target under the control of the formation control system and that the missiles are already pointed at the target, the three-dimensional space attack time cooperation problem can be simplified into a vertical plane attack time cooperation problem. In the vertical plane, the engagement scenario between the missiles and the target is shown in Figure 2.

In Figure 2, T represents the target to be attacked, $M_i$ represents the i-th missile in the swarm; $q_i$, ${\eta }_i$, and ${\sigma }_i$, respectively, represent the angle between the LOS (line of sight) and the horizontal reference line, heading error, and fight-path angle of the i-th missile; $r_i$ represents the distance between the i-th missile and the target; $a_{n,i}$ and $a_{t,i}$, respectively, represent the normal acceleration and tangential acceleration of the i-th missile.

From Figure 2, we can obtain the missile–target engagement kinematics for missiles to attack a stationary target, which is expressed as

$$\begin{array}{c}\left\{\begin{array}{c}{\dot{r}}_i=-V_{i}cos{\eta }_i\hfill \\ {\dot{q}}_i=-{\displaystyle \frac{V_{i}sin{\eta }_i}{r_i}}\hfill \\ {\dot{V}}_i=a_{t,i}\hfill \\ {\dot{\sigma }}_i={\displaystyle \frac{a_{n,i}}{V_i}}\hfill \\ {\eta }_i={\sigma }_i-q_i, i=\{1,\cdots ,N\}\cup \left\{r\right\}\hfill \end{array}\right.\hfill \end{array}$$

(4)

The control inputs during the missile guidance process are the normal acceleration $a_{n,i}$ and tangential acceleration $a_{t,i}$ in the above equations, and the tangential acceleration is introduced to improve the maneuverability of the missiles.

3.3. Fixed Time Convergence Criterion of the Time-to-Go Estimates

One of the main purposes of this paper is to ensure that all missiles in the swarm hit the target simultaneously. By choosing the time-to-go estimates of the missile as the consensus variable, the premise of achieving simultaneous attacks is that the time-to-go estimates of the missile swarm must reach consensus before any missile hits the target. For the missiles with large differences in spatial location and velocity, the convergence rate of the consensus variable is crucial, and the fixed-time consensus algorithm can overcome the influence of the initial state of the guidance system on the convergence rate of the consensus variable. The following is the judgment criterion for the convergence of time-to-go estimates within a fixed time.

Define the time-to-go estimates of the i-th missile in the swarm as ${\widehat{t}}_{go,i},i=\{1,\cdots ,N\}\bigcup \left\{r\right\}$, then the consensus error of time-to-go estimates between the i-th follower missile and its adjacent missile can be expressed as

$$\begin{array}{c}\hfill {\xi }_i=\sum _{j\in N_i}{a}_{ij}({\widehat{t}}_{go,j}-{\widehat{t}}_{go,i})-b_i({\widehat{t}}_{go,i}-{\widehat{t}}_{go,r})\end{array}$$

(5)

Let $\mathit{\xi }={\left[\begin{array}{cccc}{\xi }_1& {\xi }_2& \cdots & {\xi }_N\end{array}\right]}^{\mathrm{T}}$. If the designed cooperative guidance law can make the vector $\mathit{\xi }$ satisfy conditions

$$\begin{array}{c}\hfill \underset{t\to T}{lim}\left|\right|\mathit{\xi }\left|\right|=0\end{array}$$

(6)

and

$$\begin{array}{c}\hfill \left|\right|\mathit{\xi }\left|\right|=0,\forall t\ge T\end{array}$$

(7)

with any initial conditions, then the missile swarm is said to achieve consensus of the time-to-go estimates within a fixed time.

The actual time required for the time-to-go estimates of the swarm to converge to consensus is T, and $T<T_{max}$. $T_{max}$ represents the latest convergence time of the system, which is also called the fixed convergence time because its magnitude is independent of the initial state of the system. Under the fixed-time cooperative guidance law, as long as the actual actuators of missiles can meet the requirements, the time-to-go estimates of the missiles will always converge to consensus within $T<T_{max}$ regardless of the initial state of the missile swarm. The fixed-time consensus algorithm can meet the strict requirements of the high-speed missile swarm for the convergence time of the consensus variable.

3.4. Event-Triggered Mechanism

In the process of cooperative guidance, missiles need to transmit parameters through inter-missile communication networks to synchronize their arrival times. Conventional cooperative guidance laws are designed based on periodic sampled information, requiring continuous communication and calculation of control parameters for the missiles to achieve the desired motion state, which consumes a lot of computing resources and communication bandwidth. In this paper, the event-triggered mechanism is designed to set a threshold for the time-varying error of the time-to-go estimates for each missile, and communication is triggered only when the error meets the set threshold. The event-triggered mechanism can greatly reduce the communication frequency of the missile swarm and effectively save limited missile-borne computing resources, but at the same time, the error introduced by the event-triggered mechanism also brings challenges to the stability analysis of the distributed guidance system.

Define t as the duration of the guidance process. Define $t_i^k,k=1,2,\cdots ,n$ as a monotonically increasing time sequence determined by the event-triggered function of the i-th missile; ${\widehat{t}}_{go,i}\left(t_i^k\right)$ represents the time-to-go estimates of the i-th missile at the event-triggered moment $t_i^k$; ${\xi }_i^k$ is the consensus error of time-to-go estimates between the i-th missile and its adjacent missiles at the event-triggered moment $t_i^k$; and ${\xi }_i^{k-1}$ is the consensus error of time-to-go estimates between the i-th missile and its adjacent missiles at the previous event-triggered moment $t_i^{k-1}$. The principle of the event-triggered mechanism is shown in Figure 3.

The design of the event-triggered function should avoid the occurrence of Zeno behavior, which refers to the event-triggered function being triggered infinitely within a finite time interval. If the event-triggered time sequence of the i-th missiles satisfies

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{t}_i^k=\sum _{k=0}^{\infty }\left(t_i^{k+1}-t_i^k\right)=t_i^{\infty }\end{array}$$

(8)

then missile i is said to exhibit Zeno behavior. The occurrence of Zeno behavior indicates that the event-triggered function violates its intended design.

4. Fixed-Time Distributed Cooperative Guidance Law Design Based on the Event-Triggered Mechanism

Assume that during the cooperative guidance, the missile swarm consists of one leader and N followers. The leader is responsible for constructing the battlefield situation at a high altitude and transmitting relevant information to the low-altitude flying followers in real time. To maintain stealth, the follower missiles generally do not send information to the leader, so the movement status of the leader missile is independent and is not affected by the follower missiles. In this case, the movement status of all followers must be synchronized with the leader to achieve simultaneous attacks.

Design a fixed-time distributed cooperative guidance law based on the event-triggered mechanism for the leader and followers, respectively, as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{a}_{n,r}=K_r{V}_r{\dot{q}}_r\hfill \\ a_{t,r}=0\hfill \end{array}\right.\end{array}$$

(9)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{a}_{n,i}=K_r\left(1-\alpha |{\xi }_i^k{|}^a\mathrm{sgn}\left({\xi }_i^k\right)\right)V_i{\dot{q}}_i\hfill \\ a_{t,i}=-\beta |{\xi }_i^k{|}^b\mathrm{sgn}\left({\xi }_i^k\right), i=1,2,\cdots ,N\hfill \end{array}\right.\end{array}$$

(10)

where $K_r$ is the navigation ratio of the leader, which is usually selected as $K_r\in (2,6)$. $\alpha$ and $\beta$ are the feedback gains to be designed, and satisfy $\alpha >0$ and $\beta >0$. Parameters $a\in (0,1)$, $b\in (1,\infty )$. ${\xi }_i^k$ is the consensus error of time-to-go estimates at the event-triggered moment of the i-th follower missile, which is expressed as

$$\begin{array}{c}\hfill {\xi }_i^k=\sum _{j\in N_i}{a}_{ij}\left({\widehat{t}}_{go,j}\left(t_i^k\right)-{\widehat{t}}_{go,i}\left(t_i^k\right)\right)-b_i\left({\widehat{t}}_{go,i}\left(t_i^k\right)-{\widehat{t}}_{go,r}\right)\end{array}$$

(11)

It is worth noting that when there is a leader in the cluster, the ${\widehat{t}}_{go,r}$ term in Equation (11) indicates that missiles directly communicating with the leader need to obtain the leader’s information in real time, but this does not mean the failure of the event-triggered mechanism. Especially when the scale of the cluster is large enough, the event-triggered mechanism designed in this paper can greatly reduce the communication frequency of the follower missiles during the collaboration process.

Obviously, the control command of the leader missile is designed in the framework of proportional navigation, and the navigation ratio $K_r$ is constant. The normal acceleration of the follower is also designed in the framework of proportional navigation, and the navigation ratio $K_i=K_r(1-\alpha |{\xi }_i^k{|}^a\mathrm{sgn}\left({\xi }_i^k\right))$ is related to the consensus error of time-to-go estimates of their event-triggered moment. In the given guidance command, the normal acceleration of the missile is used to adjust the curvature of the flight trajectory, so that the missiles with longer time-to-go estimates fly along a flatter trajectory to reduce the time to reach the target, and the missiles with shorter time-to-go estimates fly on a curved trajectory to extend the time to reach the target. The tangential acceleration of the missile is used to adjust the flight speed so that the missile with longer time-to-go estimates accelerates and the missile with shorter time-to-go estimates decelerates. The introduction of tangential acceleration improves the maneuverability of the missiles.

Because the cooperative guidance law is designed under the PN structure, the time-to-go estimates of the missiles can be obtained by calculating the arc length of the flight path divided by the velocity, which can be expressed as

$$\begin{array}{c}\hfill {\widehat{t}}_{go,i}={\displaystyle \frac{r_i}{V_i}}\left(1+{\displaystyle \frac{{{\eta }_i}^2}{2\left(2K_i-1\right)}}\right), i=\{1,\cdots ,N\}\cup \left\{r\right\}\end{array}$$

(12)

In the guidance process, the time-to-go estimates ${\widehat{t}}_{go,r}$ always represent the real time-to-go for the leader missile, because the navigation ratio $K_r$ in the equation is for the leader. However, for the follower missile i, the navigation ratio $K_i$ is constant $K_r$ only when the consensus error of the time-to-go estimates with adjacent missiles is zero and the tangential acceleration is zero. After this moment, the time-to-go estimates of the follower missile represent its real time-to-go.

Within the time interval $[t_i^k,t_i^{k+1})$, the time-varying error of the time-to-go estimates of the i-th follower missile is defined as

$$\begin{array}{c}\hfill e_i={\widehat{t}}_{go,i}\left(t_i^k\right)-{\widehat{t}}_{go,i}\left(t\right)\end{array}$$

(13)

The event-triggered moment of the missiles is determined by the event-triggered function, which is designed for the follower missiles as

$$\begin{array}{c}\hfill f_i\left(t\right)=\left|\right|e_i\left|\right|-{\displaystyle \frac{c}{g}}\left|\right|{\xi }_i^k\left|\right|\end{array}$$

(14)

where

$$g=\sum _{j=1}^N|m_{ij}|$$

and c is a positive constant satisfying $c<1$. The $\left|\right|e_i\left|\right|$ term is used to measure the time-varying error of the time-to-go estimates of the i-th follower missile, and only the state information of the missile itself is needed to calculate this item. There is no need to use the communication network. The $(c/g)\left|\right|{\xi }_i^k\left|\right|$ term is a threshold, and when calculating it, only the information of the adjacent follower missiles at the event-triggered moment needs to be obtained. For those connected with the leader missile, the information transmitted from the leader missile also needs to be obtained. When the time-varying state error exceeds the set threshold, the follower missiles activate the inter-missile communication to transmit their time-to-go estimates of the event-triggered moment, as well as obtain the information of the adjacent follower missiles at the event-triggered moment, and update their control input to reset the time-varying error of the time-to-go estimates to zero. Under the constraint of this event-triggered function, the communication frequency required by the follower missiles in the cooperative guidance process can be greatly reduced, effectively saving the computing resources and communication bandwidth of the missiles.

Theorem 1.

Under the effect of guidance commands (9) and (10), all members of the missile swarm can achieve the consensus of the time-to-go estimates within a fixed time and simultaneously hit the target.

Proof of Theorem 1.

The proof of Theorem 1 consists of four steps. First, we construct a Lyapunov function that incorporates the time-to-go estimates of the leader and followers. Then, we prove that the time-to-go estimates of the leader and followers can asymptotically converge to consensus. Next, the fixed-time consensus of the cooperative guidance law is proved based on the asymptotic consensus. Finally, we prove that the designed event-triggered mechanism excludes Zeno behavior.

We define the combined error of the time-to-go estimates of the i-th follower missile as

$$\begin{array}{c}\hfill e^i=\sum _{j\in N_i}{a}_{ij}\left(e_j-e_i\right)-b_i\left(e_i\right)\end{array}$$

(15)

Let ${\mathit{e}}_i={[e_1,e_2,\cdots ,e_N]}^{\mathrm{T}}$ and ${\mathit{e}}^i={[e^1,e^2,\cdots ,e^N]}^{\mathrm{T}}$. The relationship between the combined error and the time-varying error of the time-to-go estimates can be represented by the matrix form as

$$\begin{array}{c}\hfill {\mathit{e}}^i=-\mathit{M}{\mathit{e}}_i\end{array}$$

(16)

Let ${\mathit{\xi }}^{\mathbf{k}}={\left[{\xi }_1^k,{\xi }_2^k,\cdots ,{\xi }_N^k\right]}^{\mathrm{T}}$ and $\mathit{\xi }={\left[{\xi }_1,{\xi }_2,\cdots ,{\xi }_N\right]}^{\mathrm{T}}$. The relationship between the consensus error of the time-to-go estimates at the current moment and the event-triggered moment can be represented by the matrix form as

$$\begin{array}{c}\hfill {\mathit{\xi }}^k={\mathit{e}}^i+\mathit{\xi }\end{array}$$

(17)

In general, the heading error ${\eta }_i$ of the missile is very small, so we can assume that $sin{\eta }_i={\eta }_i$ and $cos{\eta }_i=1-{\eta }_i^2/2$. Substituting control input (10) into kinematics (4), we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{r}}_i=-V_i\left(1-{\displaystyle \frac{{\eta }_i^2}{2}}\right)\hfill \\ {\dot{V}}_i=-\beta {\xi }_i^k\hfill \\ {\dot{\eta }}_i=-{\displaystyle \frac{\left(K_r-K_r\alpha {\left|{\xi }_i^k\right|}^a\mathrm{sgn}\left({\xi }_i^k\right)-1\right)V_i{\eta }_i}{r_i}}\hfill \end{array}\right.\end{array}$$

(18)

Differentiating the time-to-go estimates ${\widehat{t}}_{go,i}$ of the follower missiles and substituting the above equation into it, we can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\widehat{t}}}_{go,i}\left(t\right)& ={\displaystyle \frac{{\dot{r}}_i{V}_i-r_i{\dot{V}}_i}{V_i^2}}\left(1+{\displaystyle \frac{{\eta }_i^2}{2\left(2K_r-1\right)}}\right)+{\displaystyle \frac{r_i{\eta }_i{\dot{\eta }}_i}{V_i\left(2K_r-1\right)}}\hfill \\ & ={\displaystyle \frac{K_r\alpha {\left|{\xi }_i^k\right|}^{a}sgn\left({\xi }_i^k\right){\eta }_i^2}{2K_r-1}}-1+{\displaystyle \frac{\beta r_i{\left|{\xi }_i^k\right|}^{b}sgn\left({\xi }_i^k\right)}{V_i^k}}\left(1+{\displaystyle \frac{{\eta }_i^2}{2\left(2K_r-1\right)}}\right)\hfill \end{array}\end{array}$$

(19)

We express the time-to-go estimates of the followers in vector form, that is ${\widehat{\mathit{t}}}_{go}={[{\widehat{t}}_{go,1}\left(t\right),{\widehat{t}}_{go,2}\left(t\right),\cdots ,{\widehat{t}}_{go,N}\left(t\right)]}^{\mathrm{T}}$. We construct the Lyapunov function as

$$\begin{array}{c}\hfill V_1\left(t\right)={\displaystyle \frac{1}{2}}{\left({\widehat{\mathit{t}}}_{go}-{\widehat{\mathit{t}}}_{go,r}\mathbf{1}\right)}^{\mathrm{T}}\mathit{M}\left({\widehat{\mathit{t}}}_{go}-{\widehat{\mathit{t}}}_{go,r}\mathbf{1}\right)\end{array}$$

(20)

Since the information of the leader is globally reachable and the communication topology among followers is undirected, $\mathit{M}$ is a positive definite matrix, that is $V_1\left(t\right)\ge 0$, $t\in [0,+\infty )$. Differentiating the Lyapunov function and substituting the Equation (17) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)& ={\left({\widehat{\mathit{t}}}_{go}-{\widehat{t}}_{go,r}\mathbf{1}\right)}^{\mathrm{T}}\mathit{M}\left({\dot{\widehat{\mathit{t}}}}_{go}-{\dot{\widehat{t}}}_{go,r}\mathbf{1}\right)\hfill \\ & =-{\mathit{\xi }}^{\mathrm{T}}\left({\dot{\widehat{\mathit{t}}}}_{go}-{\dot{\widehat{t}}}_{go,r}\mathbf{1}\right)\hfill \\ & =-{\left({\mathit{\xi }}^k\right)}^{\mathrm{T}}{\dot{\widehat{\mathit{t}}}}_{go}+{\left({\mathit{e}}^i\right)}^{\mathrm{T}}{\dot{\widehat{\mathit{t}}}}_{go}\hfill \end{array}\end{array}$$

(21)

For convenience, let $n_1$ and $n_2$ represent the two terms on the right side of the above equation, respectively. According to the event-triggered function, we know that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ∥{\mathit{e}}^i∥& =∥-\mathit{M}{\mathit{e}}_i∥=\sqrt{\sum _{i=1}^N{∥\sum _{j=1}^N{m}_{ij}{\mathit{e}}_j∥}^2}\hfill \\ & \le \sqrt{\sum _{i=1}^N\left(\sum _{i=1}^N{\left|m_{ij}\right|}^2\right)\sum _{j=1}^N{∥{\mathit{e}}_j∥}^2}\hfill \\ & =\sqrt{\sum _{i=1}^N\left(\sum _{j=1}^N{\left|m_{ji}\right|}^2\right){∥{\mathit{e}}_i∥}^2}\hfill \\ & \le \sqrt{\sum _{i=1}^N{\left(\sum _{j=1}^N\left|m_{ji}\right|\right)}^2{∥{\mathit{e}}_i∥}^2}\hfill \\ & \le \sqrt{c^2\sum _{i=1}^N{∥{\xi }_i^k∥}^2}=c∥{\mathit{\xi }}^k∥\hfill \end{array}\end{array}$$

(22)

Then, using the conclusion of the above equation, we can bound $n_2$ as

$$\begin{array}{c}\hfill n_2\le ∥{\mathit{e}}^i∥∥{\dot{\widehat{\mathit{t}}}}_{go}∥\le c∥{\mathit{\xi }}^k∥∥{\dot{\widehat{\mathit{t}}}}_{go}∥=c∥{\left({\mathit{\xi }}^k\right)}^{\mathrm{T}}{\dot{\widehat{\mathit{t}}}}_{go}∥\end{array}$$

(23)

And since $c<1$, Equation (21) can be rewritten in the following form:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)\le & -{\left({\mathit{\xi }}^k\right)}^{\mathrm{T}}{\dot{\widehat{\mathit{t}}}}_{go}+c∥{\left({\mathit{\xi }}^k\right)}^{\mathrm{T}}{\dot{\widehat{\mathit{t}}}}_{go}∥\hfill \\ \hfill =& -(1-c)\left({\displaystyle \frac{K_r\alpha }{2K_r-1}}\sum _{i=1}^N{\eta }_i^2{\left|{\xi }_i^k\right|}^{a+1}\right)\hfill \\ & -(1-c)\left(\beta \sum _{i=1}^N\left(1+{\displaystyle \frac{{\eta }_i^2}{2\left(2K_r-1\right)}}\right){\displaystyle \frac{r_i{\left|{\xi }_i^k\right|}^{b+1}}{V_i^2}}\right)\hfill \end{array}\end{array}$$

(24)

Before the consensus of the time-to-go estimates of the missile swarm is achieved, ${\eta }_i$ and $r_i$ are both nonzero, that is, there exist constants such that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{r_i}{V_i^2}}\ge p_{\mathrm{con}}\hfill \\ \left|{\eta }_i\right|\ge {\eta }_{\mathrm{con}}\hfill \end{array}\right.\end{array}$$

(25)

Define

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_1=\left(1-c\right){\displaystyle \frac{K_r\alpha }{2K_r-1}}{\eta }_{\mathrm{con}}^2\hfill \\ k_2=\left(1-c\right)\left(1+{\displaystyle \frac{{\eta }_{\mathrm{con}}^2}{2\left(2K_r-1\right)}}\right)\beta p_{\mathrm{con}}\hfill \end{array}\right.\end{array}$$

(26)

Substituting $k_1$ and $k_2$ into Equation (24) and according to Lemma 2, the differential form of the Lyapunov function can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)& \le -k_1\sum _{i=1}^N{\left|{\xi }_i^k\right|}^{1+a}-k_2\sum _{i=1}^N{\left|{\xi }_i^k\right|}^{1+b}\hfill \\ & \le -k_1{\left({∥{\xi }^k∥}^2\right)}^{(1+a)/2}-k_2{N}^{(1-b)/2}{\left({∥{\xi }^k∥}^2\right)}^{(1+b)/2}\hfill \end{array}\end{array}$$

(27)

By analyzing the above equation, we know that ${\dot{V}}_1\left(t\right)=0$ holds only when all elements of the vector ${\mathit{\xi }}^k$ are 0, and ${\dot{V}}_1\left(t\right)<0$ always holds for other cases, which indicates that the time-to-go estimates of the leader and followers can asymptotically converge to consensus. Next, we prove that they can converge to consensus within a fixed time.

By bounding the Lyapunov function $V_1\left(t\right)$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_1\left(t\right)& \le {\displaystyle \frac{1}{2}}{\lambda }_{max}\left(\mathit{M}\right){∥{\widehat{\mathit{t}}}_{go}-{\widehat{\mathit{t}}}_{go,r}\mathbf{1}∥}^2\hfill \\ & ={\displaystyle \frac{1}{2}}{\lambda }_{max}\left(\mathit{M}\right){\displaystyle \frac{{\parallel \mathit{\xi }\parallel }^2}{{\parallel \mathit{M}\parallel }^2}}\hfill \end{array}\end{array}$$

(28)

Under the designed control input, the consensus error of the time-to-go estimates of the missile swarm is always decreasing, that is, $\left|\right|\mathit{\xi }\left|\right|\le \left|\right|{\mathit{\xi }}^k\left|\right|$ holds within the event-triggered interval $[t_i^k,t_i^{k+1})$. Furthermore, let

$$\begin{array}{c}\hfill k_3=k_1{\left({\displaystyle \frac{2}{{\lambda }_{max}\left(\mathit{M}\right)}}\right)}^{\left(1+a\right)/2}{\parallel \mathit{M}\parallel }^{1+a}\end{array}$$

(29)

and

$$\begin{array}{c}\hfill k_4=k_2{N}^{\left(1-b\right)/2}{\left({\displaystyle \frac{2}{{\lambda }_{max}\left(\mathit{M}\right)}}\right)}^{\left(1+b\right)/2}{\parallel \mathit{M}\parallel }^{1+b}\end{array}$$

(30)

Then, Equation (27) can be rewritten as

$$\begin{array}{c}\hfill {\dot{V}}_1\left(t\right)\le -k_3{V}_1{\left(t\right)}^{\left(1+a\right)/2}-k_4{V}_1{\left(t\right)}^{\left(1+b\right)/2}\end{array}$$

(31)

According to Lemma 1, the Lyapunov function $V_1\left(t\right)$ will converge to 0 within a fixed time, which also means that the consensus of the time-to-go estimates between the followers and the leader will be achieved within a fixed time, and the convergence time satisfies

$$\begin{array}{cc}\hfill T\le T_{max}=& {\displaystyle \frac{2}{k_3\left(1-a\right)}}+{\displaystyle \frac{2}{k_4\left(b-1\right)}}\end{array}$$

(32)

The next proof is that Zeno behavior does not occur during missiles communication. After the consensus of time-to-go estimates is achieved, the time-to-go estimates of the missiles begin to represent the real time-to-go. From this moment on, communication between missiles is no longer necessary, and the missile swarm can achieve simultaneous attacks on the target while keeping the communication network silent. Therefore, it is only necessary to prove that the Zeno behavior of the swarm is excluded before the consensus of time-to-go estimates is achieved. Before the consensus of the swarm is achieved, there is

$$\begin{array}{c}\hfill \left|\right|{\mathit{\xi }}^k\left|\right|>\vartheta ,\forall \vartheta >0\end{array}$$

(33)

At the event-triggered moment $t=t_i^k$, the missile i resets its time-varying error of time-to-go estimates to 0, that is, $e_i=0$.

Define

$$\begin{array}{c}\hfill \mu =sup\left\{\right|\left|{\mu }_1\right||,|\left|{\mu }_2\right||,\cdots ,|\left|{\mu }_N\right|\left|\right\}\end{array}$$

(34)

Within the event-triggered interval $[t_i^k,t_i^{k+1})$, there is

$$\begin{array}{c}\hfill \left|\right|e_i\left|\right|\le \left|\right|{\int }_{t_i^k}^t{\dot{e}}_i\left(s\right)\mathrm{d}s\left|\right|\le {\int }_{t_i^k}^t\left|\right|{\dot{e}}_i\left(s\right)\left|\right|\mathrm{d}s\le {\int }_{t_i^k}^t\left|\right|{\mu }_i\left|\right|\mathrm{d}s\le \mu \left(t-t_i^k\right)\end{array}$$

(35)

At the event-triggered moment $t=t_i^k$, the event-triggered function $f_i\left(t\right)>0$, so we obtain

$$\begin{array}{c}\hfill \left(t-t_i^k\right)\ge {\displaystyle \frac{c}{g\mu }}\left|\right|{\xi }_i^k\left|\right|\end{array}$$

(36)

Therefore, under the event-triggered mechanism designed in this paper, Zeno behavior of the missile swarm is excluded. □

5. Extension to the Case Where the Leader Missile Is Destroyed

When the missile swarm penetrates the air defense network of the target, the leader flies at a high altitude and is responsible for building the battlefield situation and searching for targets. Therefore, the probability of being detected and shot down by the enemy is high. The leader missile needs reconnaissance capabilities; hence, there are certain differences in equipment between the leader missile and the follower missiles. To enhance the success rate of the mission, the cluster of missiles typically includes several spare leader missiles. If the leader missile is destroyed before identifying the target, a spare leader missile will take its place. Assuming that the swarm has confirmed the exact location of the target, it is unnecessary to hold another election for the leader if the leader is destroyed. The followers can synchronize their time-to-go estimates through the inter-missile communication network to achieve the requirement of simultaneous attacks on the target. This section considers the case where the leader missile is shot down and studies the problem of the missile swarm simultaneous attacks under the event-triggered mechanism. The following theorem is given.

Theorem 2.

If the leader missile is shot down, the followers can still achieve simultaneous attacks on the target by replacing the event-triggered function with the control input (10).

Proof of Theorem 2.

In the case of losing the lead missile, the adjacency matrix $\mathit{B}$ of the leader missile becomes a zero matrix. This means that Equations (11) and (15) in the previous section no longer include information about the leader.

At this point, the relationship between the combination error and the time-varying error of time-to-go estimates can be represented by a matrix form as

$$\begin{array}{c}\hfill {\mathit{e}}^i=-\mathit{L}{\mathit{e}}_i\end{array}$$

(37)

The relationship between the consensus error of time-to-go estimates at the current moment and the event-triggered moment can still be represented by a matrix form as

$$\begin{array}{c}\hfill {\mathit{\xi }}^k={\mathit{e}}^i+\mathit{\xi }\end{array}$$

(38)

To ensure the stability of the swarm system in the guidance process, a new event-triggered function is designed for the missile swarm as follows.

$$\begin{array}{c}\hfill f_i\left(t\right)=\left|\right|e_i\left|\right|-{\displaystyle \frac{c}{h}}\left|\right|{\xi }_i^k\left|\right|\end{array}$$

(39)

where

$$h=2\sum _{j=1}^N{a}_{ij}$$

and the design of the $\left|\right|e_i\left|\right|$ term and the parameter c are the same as in the previous section. However, the design of the threshold term $\left(c/h\right)\left|\right|{\xi }_i^k\left|\right|$ is different. To ensure the stability of the guidance system, the constant term g is replaced by h, and the $\left|\right|{\xi }_i^k\left|\right|$ term does not include the time-to-go estimates information of the leader, so when calculating this term, only the state information of adjacent missiles’ event-triggered time is needed.

A new Lyapunov function is constructed as

$$\begin{array}{c}\hfill V_2\left(t\right)={\displaystyle \frac{1}{2}}\sum _{i=1}^N{a}_{ij}{\left({\widehat{t}}_{go,j}-{\widehat{t}}_{go,i}\right)}^2={\displaystyle \frac{1}{2}}{\widehat{\mathit{t}}}_{go}^{\mathrm{T}}\mathit{L}{\widehat{\mathit{t}}}_{go}\end{array}$$

(40)

Since the communication topology between followers is undirected, the matrix $\mathit{L}$ is a positive definite matrix, that is, $V_2\left(t\right)\ge 0$, $t\in [0,+\infty )$. By differentiating the above equation and substituting the Equation (38) into it, we can obtain

$$\begin{array}{c}\hfill {\dot{V}}_2\left(t\right)={\widehat{\mathit{t}}}_{go}^{\mathrm{T}}\mathit{L}{\dot{\widehat{\mathit{t}}}}_{go}=-{\mathit{\xi }}^{\mathrm{T}}{\dot{\widehat{\mathit{t}}}}_{go}=-{\left({\mathit{\xi }}^k\right)}^{\mathrm{T}}{\dot{\widehat{\mathit{t}}}}_{go}+{\left({\mathit{e}}^i\right)}^{\mathrm{T}}{\dot{\widehat{\mathit{t}}}}_{go}\end{array}$$

(41)

Referring to the derivation process of the Equation (22), and from the event-triggered function (39), we can obtain

$$\begin{array}{c}\hfill ∥{\mathit{e}}^i∥=∥-\mathit{L}{\mathit{e}}_i∥\le c∥{\mathit{\xi }}^k∥\end{array}$$

(42)

Combining with Lemma 2, Equation (41) can be rewritten as

$$\begin{array}{c}\hfill {\dot{V}}_2\left(t\right)\le -k_1{\left({∥{\mathit{\xi }}^k∥}^2\right)}^{(1+a)/2}-k_2{N}^{(1-b)/2}{\left({∥{\mathit{\xi }}^k∥}^2\right)}^{(1+b)/2}\end{array}$$

(43)

Since $\mathit{L}$ is a real symmetric matrix, ${\mathbf{1}}^{\mathrm{T}}\mathit{L}\mathbf{1}={\left({\mathit{L}}^{1/2}\mathbf{1}\right)}^{\mathrm{T}}\left({\mathit{L}}^{1/2}\mathbf{1}\right)=0$, that is, ${\mathit{L}}^{1/2}\mathbf{1}=0$, and it is obvious that ${\mathbf{1}}^{\mathrm{T}}{\mathit{L}}^{1/2}{\widehat{\mathit{t}}}_{go}=0$. According to Lemma $3,{\widehat{\mathit{t}}}_{go}^{\mathrm{T}}\mathit{L}\mathit{L}{\widehat{\mathit{t}}}_{go}\ge {\lambda }_2{\widehat{\mathit{t}}}_{go}^{\mathrm{T}}{\widehat{\mathit{t}}}_{go}$ hold, and we can obtain

$$\begin{array}{c}\hfill {\left|\right|\mathit{\xi }\left|\right|}^2\ge 2{\lambda }_2{V}_2\left(t\right)\end{array}$$

(44)

As analyzed in the previous section, it is always true that $\left|\right|\mathit{\xi }\left|\right|\le \left|\right|{\mathit{\xi }}^k\left|\right|$ until the consensus of the time-to-go estimates of the swarm is achieved.

We substitute Equation (43) into Equation (43) and define

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{k}_5=k_1{\left(2{\lambda }_2\right)}^{\left(1+a\right)/2}\hfill \\ k_6=k_2{N}^{\left(1-b\right)/2}{\left(2{\lambda }_2\right)}^{\left(1+b\right)/2}\hfill \end{array}\right.\end{array}$$

(45)

Equation (43) can be rewritten in the following form:

$$\begin{array}{c}\hfill {\dot{V}}_2\left(t\right)\le -k_5{V}_2{\left(t\right)}^{\left(1+a\right)/2}-k_6{V}_2{\left(t\right)}^{\left(1+b\right)/2}\end{array}$$

(46)

According to Lemma 1, the constructed Lyapunov function $V_2\left(t\right)$ will converge to 0 within a fixed time, which also means that in the absence of the leader, the consensus of time-to-go estimates of all followers will be achieved within a fixed time, and the convergence time satisfies

$$\begin{array}{cc}\hfill T\le T_{max}=& {\displaystyle \frac{2}{k_5\left(1-a\right)}}+{\displaystyle \frac{2}{k_6\left(b-1\right)}}\end{array}$$

(47)

The proof that the swarm system excludes Zeno behavior is the same as in Theorem 1. □

6. Simulation

This section verifies the effectiveness of the proposed fixed-time event-triggered distributed cooperative guidance algorithm through numerical simulation. The simulation considers a “reconnaissance-attack integration” system consisting of three follower missiles and one leader missile. It is assumed that the initial state of the missile swarm is shown in

Table 1. Initial state of the missile swarm.

Parameters	Leader	Missile 1	Missile 2	Missile 3
$X/\mathrm{m}$	5000	6000	7000	8000
$Y/\mathrm{m}$	12,000	4000	3500	3000
$Z/\mathrm{m}$	0	1000	−1000	2000
$V\left(\mathrm{m}·{\mathrm{s}}^{-1}\right)$	210	160	225	180
$\sigma /$(°)	40	15	10	15

with the target position as the origin during the cooperative guidance handover. The simulation analysis is performed for the following two scenarios.

Scenario 1: The leader missile flies at a high altitude during the cooperative guidance handover, and the inter-missile communication topology is shown in Figure 4a.

Scenario 2: The leader missile is shot down during the cooperative guidance handover, and the communication topology between the missiles is shown in Figure 4b.

As shown in Figure 4, the Laplacian matrix of the followers is

$$\begin{array}{c}\hfill \mathit{L}=\left[\begin{array}{ccc}2& -1& -1\\ -1& 1& 0\\ -1& 0& 1\end{array}\right]\end{array}$$

(48)

And the adjacency matrix of the leader is

$$\begin{array}{c}\hfill \mathit{B}=diag\left(\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \end{array}\right)\end{array}$$

(49)

Therefore, the matrix $\mathit{M}$ is

$$\begin{array}{c}\hfill \mathit{M}=\left[\begin{array}{ccc}3& -1& -1\\ -1& 1& 0\\ -1& 0& 1\end{array}\right]\end{array}$$

(50)

6.1. Effectiveness Verification

The feedback gains $\alpha$ and $\beta$ in fixed-time event-triggered cooperative guidance law (10) are used to adjust the relationship between the convergence rate of the consensus variable and the control inputs. $\alpha$ and $\beta$ have a very flexible adjustment range. Keeping $\alpha$ unchanged, increasing $\beta$ will reduce the convergence time of the consensus variable, but it will also increase the tangential control input of the missiles. Keeping $\beta$ unchanged, increasing $\alpha$ will also reduce the convergence time of the consensus variable, and the normal control input of the missiles will be larger. The design of a and b is to make the consensus variable converge to consensus within a fixed time, where $a\in (0,1)$ and $b\in (1,+\infty )$. Reducing a will speed up the convergence of the consensus variable. However, continuing to reduce a will not significantly increase the convergence rate of the consensus variable when $a\in (0,0.5)$. Therefore, $a\in (0.5,0.9)$ is chosen in the simulation. b is generally not greater than 2, as a larger b will cause a rapid increase in the initial control input.

To ensure the stability of the guidance system, c in the event-triggered functions (14) and (39) should be less than 1, and the larger the value of c, the less frequently the missile communicates. However, at the same time, the error introduced in the guidance system will be larger and needs to be corrected at the next communication moment, which can cause control input oscillation. Therefore, in simulations, the value of c is generally around 0.5, which ensures that the communication frequency is not too frequent and meets the requirements of the actual actuators of missiles.

Based on the above analysis, in the numerical simulation of this section, the feedback gains are set as $\alpha =1$, $\beta =5$, and the exponent $c=0.8$, $d=1.2$. The simulation sampling interval is 0.01 s. Considering the execution capability of of the missile, we set the threshold for tangential acceleration to be $-20\mathrm{m}/{\mathrm{s}}^2$ to $-80\mathrm{m}/{\mathrm{s}}^2$ and the threshold for normal acceleration to be $-80\mathrm{m}/{\mathrm{s}}^2$ to $80\mathrm{m}/{\mathrm{s}}^2$. Figure 5 shows the flight parameters of the missile swarm with the leader under the effect of fixed-time event-triggered distributed cooperative guidance law, and the flight parameters of the missile swarm without the leader are shown in Figure 6.

From Figure 5a,b, it can be seen that the leader and followers with large spatial position differences achieved simultaneous attacks on the target at 62 s. In Figure 5c, the time-to-go estimates of the missiles converge to consensus when it reaches 7 s. This is due to the fixed-time convergence algorithm used in this paper, which ensures that the convergence time of the consensus variable is not affected by the initial state of the missiles. In the designed guidance law, once the time-to-go estimates of the missile reach consensus, it can represent the real time-to-go. After this moment, all missiles can independently guide to the target and achieve simultaneous attacks on the target without communication. Additionally, it can be seen from the figure that the leader missile has the longest time-to-go estimates due to performing reconnaissance missions at a high altitude and being far from the target. In the case that the leader does not obtain information from the followers, to maintain the stealthiness of the followers and to achieve simultaneous attacks, all followers have to synchronize the time-to-go estimates with the leader. Furthermore, it can be seen from the figure that at the initial guidance moment, the leader’s time-to-go estimate is 62 s, and the actual time to complete the cooperative guidance mission is also around 62 s. This indicates that the time-to-go estimates expression given in this paper is very accurate.

Figure 5d shows the variable curves of the time-to-go estimates of the missiles at the event-triggered moment. In the cooperative guidance stage, the follower missiles only updated time-to-go estimates information when their error of time-to-go estimates reached the set threshold. Each abrupt change of the time-to-go estimates in the figure represents a communication of the missiles to obtain information from adjacent missiles and generate the new time-to-go estimates as the calculation parameter for the control input until the next event-triggered time arrives. In the independent guidance stage, there is no communication between the missiles, so the event-triggered mechanism is no longer effective.

Figure 5e shows the variable curves of the navigation rate of the missiles with time. The leader’s navigation rate remained at 4, and the follower’s navigation rate converged to a constant 4 at 7 s. This means that from this moment on, the time-to-go estimates of the missile represent the real time-to-go.

Figure 5f,g show the required normal and tangential accelerations of the missiles during cooperative guidance. It can be seen from the figures that the ranges of normal and tangential accelerations are $-46\mathrm{m}·{\mathrm{s}}^{-2}\le a_{n,i}\le 7\mathrm{m}·{\mathrm{s}}^{-2}$ and $-65\mathrm{m}·{\mathrm{s}}^{-2}\le a_{n,t}\le 0\mathrm{m}·{\mathrm{s}}^{-2}$, respectively. The required overload is smaller than the available overload of the conventional missile.

Figure 5h shows the event-triggered moment information of the missiles. It can be seen from the figure that the follower missiles achieved the consensus of time-to-go estimates after only a few dozen communications. Communication between the missiles was unnecessary after the consensus of time-to-go estimates was achieved, and the missiles disconnected their communication links and independently guided to the target. Since follower missile 1 is a node in the communication topology and receives information from the leader and other followers, its communication is most frequent.

From Figure 6a,b, it can be seen that even in the absence of the leader missile, the three follower missiles are still able to achieve simultaneous attacks on the target. Furthermore, because there is no need to synchronize the time-to-go estimates with the leader, the missile swarm completed the guidance process at 43 s, saving 19 s compared to the case with the leader missile. The variable curves of the time-to-go estimates are shown in Figure 6c. Under the condition that the parameters of the guidance law remained unchanged, the follower missiles achieved the consensus of the time-to-go estimates at only 4 s. From that moment on, the missiles could disconnect from the communication links and navigate to the target independently to cope with complex battlefield situations, such as possible electromagnetic interference. At the beginning of the guidance phase, the time-to-go estimates of the three follower missiles are 46 s, 35 s, and 49 s, respectively, while the time required to complete the guidance mission is 42 s, which implies that the real time-to-go of the missile swarm converged towards the median value of the initial time-to-go estimates.

Figure 6d,e, respectively, show the variable curves of the time-to-go estimates at the event-triggered moment and the navigation ratio with time, both of which converged to consensus at 4 s. From Figure 6f,g, it can be seen that the range of the controlled input for the normal and tangential accelerations is $-40\mathrm{m}·{\mathrm{s}}^{-2}\le a_{n,i}\le 27\mathrm{m}·{\mathrm{s}}^{-2}$ and $-85\mathrm{m}·{\mathrm{s}}^{-2}\le a_{t,i}\le 57\mathrm{m}·{\mathrm{s}}^{-2}$, respectively. The required maximum control input is slightly larger than that in scenario 1, which is because the parameters of the guidance law are not adjusted for a more intuitive comparison with the simulation results of scenario 1. Figure 6h shows the event-triggered moment information of the missiles. Because follower missile 1 is still a node of the missile swarm in this scenario, its communication is the most frequent. However, compared to scenario 1, the number of communications is significantly reduced due to the absence of the leader missile.

6.2. A Scheme to Reduce the Time Required for the Guidance Process

Based on the comparative analysis of the simulation results of the two scenarios mentioned above, it can be concluded that the presence of the leader will greatly increase the time required for the guidance process due to the high–low-ballistic-penetration strategy adopted in this paper, which will increase the risk of interception of the missile swarm. There are two solutions to this problem.

(1) In the guidance phase, the cooperation with the leader is abandoned, and the leader only performs reconnaissance missions or guides individually as needed to reduce the time required for the cooperative guidance process. The simulation results are the same as for scenario 2.

(2) Inspired by the results analysis of scenario 2, in the case where the inter-missile communication topology is bidirectional, the real time-to-go of the missile swarm will converge toward the middle value of the initial time-to-go estimates. Therefore, it is considered to establish a bidirectional communication connection between the leader and the follower missile 1 to reduce the time required for cooperative guidance.

Obviously, if there are fewer members in the missile swarm when executing the mission, it is not a suitable choice for the leader missile to not participate in the cooperative operation mission. If Scheme 2 is adopted, the time required for the cooperative guidance of the leader and follower missiles will be greatly reduced, but the dozens of information exchanges may also reduce the stealthiness of the follower missiles. Therefore, flexible choices need to be made according to the nature of the mission.

When the leader establishes a bidirectional communication connection with follower 1, the leader can be regarded as a member of the followers, and the guidance input of the leader does not need to be designed separately and applies to the event-triggered function (39).

As can be seen in Figure 7a,b, with the follower missile 1 establishing two-way communication with the leader, the missile swarm achieves simultaneous attacks on the target at 47 s, saving 15 s of guidance time compared to scenario 1. Figure 7c shows that the time-to-go estimates of the swarm converge to consensus at 3.6 s with no change in the cooperative guidance law parameter settings, indicating that establishing additional communication link also speeds up the convergence time of the consensus variable. Figure 7d,e, respectively, show the variable curves of the time-to-go estimates at the event-triggered moment and navigation ratio with time, which also converge to consensus at 3.6 s.

From Figure 7f,g, it can be seen that the range of the controlled input for the normal and tangential accelerations are $-45\mathrm{m}·{\mathrm{s}}^{-2}\le a_{n,i}\le 70\mathrm{m}·{\mathrm{s}}^{-2}$ and $-107\mathrm{m}·{\mathrm{s}}^{-2}\le a_{t,i}\le 150\mathrm{m}·{\mathrm{s}}^{-2}$, respectively. The convergence rate of the time-to-go estimates of the missile swarm can also be reduced by setting the parameters in the cooperative guidance law, thereby reducing the required control input. Figure 7h shows the event-triggered moment information of the swarm, which indicates that the missile swarm can still complete the cooperative guidance process through only tens of communications when the leader and follower 1 have bidirectional communication.

6.3. Event-Triggered Mechanism Performance Verification

To verify the impact of the error introduced by the event-triggered mechanism on the guidance system, the following simulation analysis is carried out for the cooperative guidance law in scenario 1 without the introduction of the event-triggered mechanism. To disable the event-triggered mechanism, it is set to zero in the event-triggered functions (14) and (39).

In Figure 8, except that no event-triggered mechanism is introduced, all initial conditions and parameters of the distributed cooperative guidance law are the same as those in scenario 1. By comparing Figure 5 and Figure 8, it can be seen that both achieve simultaneous attacks on the target at 62 s. The time-to-go estimates for both converge to consensus at 7 s, and the required normal and tangential accelerations are in the range of $-46\mathrm{m}·{\mathrm{s}}^{-2}\le a_{n,i}\le 7\mathrm{m}·{\mathrm{s}}^{-2}$ and $-65\mathrm{m}·{\mathrm{s}}^{-2}\le a_{n,t}\le 0\mathrm{m}·{\mathrm{s}}^{-2}$, which shows that the design of the event-triggered mechanism does not affect the key performance indicators of the cooperative guidance process.

The only difference is that the required control input of the missiles in Figure 5 oscillates, which is caused by the error introduced by the event-triggered mechanism. The amount of error accumulated before a certain event-triggered moment needs to be corrected by the updated control input at the subsequent event-triggered moment, and the amplitude of control input oscillation can be adjusted by the parameter c to meet the requirements of the actuators of missiles.

7. Conclusions

In this paper, we take into account the actual operation scenarios and the limited missile-borne resources to design the simultaneous attacks cooperative guidance law for the missile swarm based on the “leader–follower” penetration strategy, and extend it to the scenario where the leader is destroyed. The main conclusions are as follows.

(1) The time-to-go estimates of the missile swarm can quickly converge to consensus in the case of large initial position and velocity differences, and represent the real time-to-go at later times, ensuring the realization of the simultaneous attacks.

(2) By designing a two-stage guidance law, the event-triggered mechanism is successfully applied to the design of the simultaneous attacks cooperative guidance law for the missile swarm with the leader. Numerical simulation results show that the designed event-triggered mechanism can significantly reduce the communication frequency required during the cooperative process of the missile swarm, and does not affect the performance of the guidance system. By the replacement of the event-triggered function, the guidance system’s effectiveness and stability can be maintained even in the event of losing the leader.

8. Limitation of the Study

This paper greatly reduces the communication exchanges in the cooperative guidance process of the missile cluster through the methods of distributed cooperation, two-stage guidance, and event-triggered mechanism. However, there are some research limitations that need to be noted.

(1) The model’s accuracy is inadequate. In order to emphasize the primary focus of the research, which is achieving simultaneous attacks on a target by multiple missiles with a minimal number of communication exchanges, this paper simplified the kinematic model of the missiles to a certain extent, disregarding the impact of external disturbances on the cooperative guidance process of the missiles. Improving the modeling accuracy and considering the effects of environmental disturbances caused by external factors is one of the future tasks to be undertaken.

(2) Some control parameters are empirically determined, such as the selection of parameters c in the event-triggered function. Mathematically, to ensure stability of the guidance system, the value of parameter c must be less than 1. The larger the value of c, the fewer communication exchanges that are needed within the cluster. However, at the same time, the error introduced in the guidance system will be larger and needs to be corrected at the next communication moment, which can cause control input oscillation. Establishing a rational evaluation function to determine suitable values for parameter c is one of the tasks that need to be addressed next.