Nomenclature
-
${a_{i\;}}$
-
acceleration of the
$i$
th interceptor in the inertial coordinate frame (
${{m}}/{{{s}}^2}$
) -
${a_T}$
-
acceleration of the target in the inertial coordinate frame (
${{m}}/{{{s}}^2}$
) -
$\boldsymbol{A}$
-
adjacency matrix of graph
$\boldsymbol{G}$
-
$\boldsymbol{G}$
-
communication network topology between interceptors
-
$\boldsymbol{L}$
-
Laplace matrix of the graph
$\boldsymbol{G}$
-
$N$
-
number of interceptors
-
${P_i}$
-
position of the
$i$
th interceptor in the inertial coordinate frame (
${{m}}$
) -
${\tilde P_{xi}}$
-
the virtual interception point (
${{m}}$
) -
${\bar P_{xi}}$
-
position of the interceptor after the flight the time-to-go
${t_{goi}}$
according to the current position -
and speed (
${{m}}$
) -
${P_T}$
-
position of the target in the inertial coordinate frame (
${{m}}$
) -
${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over P} _x}$
-
average position of virtual interception points of all interceptors (
${{m}}$
) -
${\boldsymbol{P}_{old}}$
-
position of the old target (
${{m}}$
) -
${\boldsymbol{P}_{new}}$
-
position of the new target (
${{m}}$
) -
${\boldsymbol{P}_{virtual}}$
-
virtual target position (
${{m}}$
) -
${t_{go}}$
-
time-to-go of the interceptor (
${{s}}$
) -
${t_{goi}}$
-
time-to-go of the
$i$
th interceptor (
${{s}}$
) -
${V_i}$
-
velocity of the
$i$
th interceptor in the inertial coordinate frame (
${{m}}/{{s}}$
) -
${V_T}$
-
velocity of the target in the inertial coordinate frame (
${{m}}/{{s}}$
) -
${\bar V_{xi}}$
-
error between the interceptor velocity and the target velocity at the end of midcourse guidance (
${{m}}/{{s}}$
) -
${\bar V_{xi}}$
-
real-time velocity error between interceptor and target (
${{m}}/{{s}}$
) -
${\boldsymbol{V}_{approach}}$
-
approach speed of the virtual target approaching the new target (
${{m}}/{{s}}$
) -
${\boldsymbol{V}_e}$
-
relative velocity between the old target and the new target (
${{m}}/{{s}}$
) -
${\boldsymbol{V}_{virtul}}$
-
virtual target velocity (
${{m}}/{{s}}$
)
Greek Symbol
-
${\alpha _{xi}}$
-
the coordination coeffcient
-
$\lambda $
-
parameter of the target handover law
-
$\dot \gamma $
-
heading angular velocity (
${{deg}}/{{s}}$
) -
${{\Delta }}{\gamma _{{{max}}}}$
-
maximum allowable heading error (
${{deg}}$
) -
$\eta $
-
the coefficients of the handover law
-
${q_\varepsilon }$
-
elevation angle of LOS (
${{deg}}$
) -
${q_\beta }$
-
azimuth angle of LOS (
${{deg}}$
)
1.0 Introduction
With the rapid development of artificial intelligence, unmanned systems, materials and other technologies in recent years, the performance of target manoeuvering, decision-making and stealth has been greatly improved. This brings significant challenges to the interception and strike missions of a single interceptor. To improve the interceptor hit probability, most scholars apply cluster technology to interceptors to perform coordinated interception and strike missions. Through information sharing and functional complementation, multiple interceptors achieve coordinated strikes on targets and perform missions that are difficult to complete with a single interceptor.
Over the past few decades, scholars have conducted extensive research on cooperative guidance laws. Depending on the characteristics of the target, cooperative guidance laws can be categorised into those designed for low-mobility (stationary, uniform-speed, low-speed ground/sea targets) and high-mobility (high-speed, high-overload aerial targets) targets.
In the case of cooperative guidance laws for low-mobility targets, for the salvo attack of anti-ship missiles, a new guidance problem with impact time constraint was investigated in Jeon et al. [Reference Jeon, Lee and Tank1] Xu et al. [Reference Xu and Lv2] investigated a time cooperative guidance law for multiple UAVs with the same strike time. Zhang et al. [Reference Zhang, Tang and Guo3] addressed the optimal decentralised three-dimensional cooperative guidance problem for multiple interceptors against a stationary target considering dynamically changing and directed communication topologies. To solve the problem of the three-dimensional (3D) cooperative guidance of multiple interceptors attacking a stationary target, two nonsingular distributed cooperative guidance strategies were proposed in Yang et al. [Reference Yang and Song4] Zhai et al. [Reference Zhai and Yang5] proposed an event-triggered distributed cooperative guidance law for multiple missiles to simultaneously attack a stationary target with consideration of autopilot lag and unknown disturbances. An impact angle, speed and acceleration control guidance (IASAG) law against the stationary target was designed in Chen et al. [Reference Chen, Liu, Shan and Wang6], which was critical for the effectiveness of the air-to-surface guided weapons.
Regarding cooperative guidance laws for high-mobility targets, for the three-dimension terminal guidance problem of multiple cooperatively intercepting a manoeuvering target, Song et al. [Reference Song, Song and Xu7] investigated a novel finite-time cooperative guidance law with impact angle constraints. Kumar et al. [Reference Kumar and Mukherjee8] proposed a new approach to control the interception time for intercepting manoeuvering targets, using deviated pursuit. Aiming at the problem of multi-missile cooperative interception of manoeuvering targets, based on the time-to-go, the two different cooperative guidance laws were designed in Ma et al. [Reference Ma and Song9] To address the issue of intercepting maneuvering targets at a specific time, a polynomial guidance method for impact-time control was proposed in Li et al. [Reference Li, Liu, Li and Liang10] In order to improve the multiple-missile cooperative attack capability and penetration capability, You et al. [Reference You, Chang, Zhao, Wang and Zhang11] investigated two three-dimensional impact-angle-constrained cooperative guidance strategies against manoeuvering target for controllable thrust missiles. Tao et al. [Reference Tao, Lin, Song and Li12] proposed an optimal 3-D spatial-temporal cooperative guidance (STCG) law for intercepting a manoeuvering target with impact angle and time constraints. Zhang et al. [Reference Zhang, Dong, Yv, Li and Ren13] designed finite-time distributed state estimation algorithm and cooperative guidance law for high manoeuvering target salvo attack.
From the above, it can be seen that currently, whether the target is low-mobility or high-mobility, the design of guidance laws focuses on the terminal guidance stage. Moreover, the related theories are relatively mature and can be extended to the midcourse guidance stage for designing cooperative midcourse guidance laws.
During cooperative strike missions, factors such as position changes, sensor performance limitations, and signal interference can lead to unstable communication links between interceptors, resulting in changes to the communication topology and affecting mission efficiency. Therefore, scholars have conducted extensive research on the cooperative strike problem of interceptor formations under topology switching. The consensus problem in directed networks with arbitrary finite time-varying communication delays under both fixed topology and switching topologies was investigated in Sun et al. [Reference Sun, Hou, Zhou and Wu14] By using non-smooth analysis and fixed time stability techniques, Ning et al. [Reference Ning, Jin and Zheng15] proposed the distributed protocols to achieve the fixed time consensus over fixed and switching topology. For leader-following missiles to intercept a manoeuvering target with switching communication topologies, Zhao et al. [Reference Zhao, Dong, Song and Ren16] investigated the cooperative time-varying formation guidance (CTRG) problem. Based on an extended state observer (ESO), a cooperative guidance method for multiple flight vehicles (MFV) cooperative interception with switching topologies was proposed in Zhao et al. [Reference Zhao, Yang, Chao and Wang17] To address the practical fixed-time average consensus problems of first-order nonlinear multi-agent systems with external disturbances under switching topologies, Sun et al. [Reference Sun, Ren, Liu and Sun18] proposed a dynamic event-triggered fixed-time consensus protocol. Currently, research on cooperative guidance laws concerning topology switching mainly focuses on the terminal guidance stage, and the related theories are quite mature. However, there is still a gap in the design research of cooperative midcourse guidance laws for topology switching. Therefore, it is necessary to extend these theories to the midcourse guidance stage to address the design of cooperative midcourse guidance laws when multiple interceptors dynamically change network communication topology based on operational requirements.
With the rapid development of military equipment, medium- to long-range interceptors have been widely deployed for strategic strike missions. For medium- to long-range interceptors, with attack ranges extending over hundreds of kilometers, relying solely on terminal guidance during flight makes hitting targets challenging. To enhance accuracy, guidance in the midcourse phase is crucial. The transition between midcourse and terminal guidance techniques is particularly important. Many scholars have conducted extensive research on this technology. Concurrently, cooperative guidance techniques studied above have also undergone extensive simulation and validation on medium- to long-range interceptors.
Currently, there are relatively few research achievements regarding cooperative midcourse guidance. By using earliest intercept geometry (EIG) concept and the optimal control theory for area air defense, Shin et al. [Reference Shin, Tsourdos, Ménec, Markham and White19] proposed a cooperative midcourse missile guidance law. To impose a desired impact angle against stationary targets, based on sliding-mode control, nonlinear robust guidance strategies were proposed in Kumar et al. [Reference Kumar, Tsalik and Shima20] For multiple missiles to intercept a target under the condition of large detection errors, Wang et al. [Reference Wang, Yao, He and Liu21] designed a novel cooperative midcourse guidance scheme. For providing suitable initial conditions of successful terminal salvo attack, a novel finite-time cooperative midcourse guidance law with terminal handover constraints was investigated in Zhang et al. [Reference Zhang, Tang and Guo22] By combining the traditional trajectory shaping guidance law and the cooperative parameter, Wu et al. [Reference Wu, Fang, Fu, Wang and Ma23] proposed a novel cooperative midcourse guidance law. By combining the cooperative term with a distributed consensus protocol including communication delay under the directed communication topology, a novel distributed cooperative midcourse guidance (DCMG) law with communication delay was designed in Wu et al. [Reference Wu, Ren, Luo, Fang and Fu24] The above considerations, in designing cooperative midcourse guidance laws, include factors such as detection errors, terminal switching constraints and communication delays. During the midcourse phase, interceptors are typically far from the target, allowing ample time to adjust target attacks based on current situations, thereby achieving the objective of targeting high-value targets. Therefore, researching cooperative midcourse guidance related to target switching in this phase is crucial in this paper.
In the midcourse and terminal guidance handover, to achieve the trajectory handover of composite guidance, Hou et al. [Reference Hou, Liu and Zhang25] proposed the concept of introducing the trajectory handover section and conducting the handover section guidance. Wang et al. [Reference Wang26] considered the problem of compound guidance about mid-long range air-to-air missile. To resolve the problem of handover between midcourse guidance and terminal guidance in the combined guidance, Li et al. [Reference Li, Song and Chen27] designed a programme. Aiming at the trajectory shift of medium- and long-range missile in the course of midcourse and terminal guidance handover, based on the idea of shifted-guidance, Zhang et al. [Reference Zhang, Fang, Gao and Diao28] designed a new shifted-law. For the trajectory shift of intercept missile in the course of midcourse and terminal guidance handover, a new shifted law was designed in Liu et al. [Reference Liu, Lv, Zhang, Duan and Yang29] to realise two-order smooth transition. For the problem of fixed target missile guidance with virtual handover point, based on the optimisation theory in Hibert space, a global energy optimal guidance law with/without terminal angle constraint was proposed in Li et al. [Reference Li, Wang, Li, He and Zhang30] For the problem of midcourse and terminal guidance transition, based on the analysis of the constraints of trajectory smooth transition, Liu et al. [Reference Liu, Chen, Chen, Duan and Liu31] proposed a simple but effective method to realise smooth transition by using a linear operator. The above content focuses on the design of midcourse-to-terminal guidance handover laws. Since this paper primarily studies the midcourse phase of interceptors, where target switching can be conducted based on the current situation, it is important to avoid abrupt changes in guidance laws due to target switching. To address this, a handover law for target information can be designed based on this concept, allowing a smooth transition from old target information to new target information.
According to the research and analysis, for a cluster of medium- and long-range interceptors performing interceptor strike missions, the cooperative guidance phase is divided into three phases, namely formation organisation, cooperative midcourse guidance and cooperative terminal guidance. In the cooperative midcourse guidance phase, according to the mission requirements, some interceptors give up the cooperative strike mission and perform other missions. The remaining interceptors adjust the network topology according to the current situation. When the target is found to have been destroyed, target changing is performed to attack another target with higher value.
Based on the above research, a cooperative midcourse guidance law with target changing and topology switching is investigated in this paper to solve the problem of target loss and communication topology switching in the case of multiple interceptors intercepting targets. Compared with the existing literatures, the main merits of this paper are concluded as follows:
For the problem of communication topology switching, an average position consistency protocol for virtual interception points is proposed, and the convergence is proved by Lyapunov stability theory.
The target handover law and handover phase guidance law in the cooperative midcourse phase are designed for the interceptor changing target from the old to the new one.
The whole cooperative midcourse guidance law including the handover law method between phases are proposed in the case of target changing and communication topology switching.
The remainder of this paper is organised as follows: Section 2 gives the problem formulation and introduces some preliminaries. In Section 3, the cooperative midcourse guidance law, target handover law and whole cooperative midcourse guidance law are given. In Section 4, simulation results illustrate the effectiveness of the proposed guidance law. Finally, Section 5 gives the conclusion.
2.0 Problem formulation and preliminaries
2.1 Guidance model
In the inertial coordinate frame, the kinematic equation of the
$i$
th interceptor can be expressed as
\begin{equation}\left\{ \begin{aligned}& {{{\dot{\boldsymbol{P}}}}_{i}}\!\left( t \right)={{\boldsymbol{V}}_{i}}\!\left( t \right) \\[5pt]& {{{\dot{\boldsymbol{V}}}}_{i}}\!\left( t \right)={{\boldsymbol{a}}_{i}}\!\left( t \right) \\\end{aligned} \right.i=1,2,...,N\end{equation}
where,
${\boldsymbol{P}_i}\!\left( t \right) = {\left[ {{x_i}\!\left( t \right),{y_i}\!\left( t \right),{z_i}\!\left( t \right)} \right]^T}$
,
${\boldsymbol{V}_i}\!\left( t \right) = {\left[ {{V_{xi}}\!\left( t \right),{V_{yi}}\!\left( t \right),{V_{zi}}\!\left( t \right)} \right]^T}$
,
${\boldsymbol{a}_i}\!\left( t \right) = {\left[ {{a_{xi}}\!\left( t \right),{a_{yi}}\!\left( t \right),{a_{zi}}\!\left( t \right)} \right]^T}$
represent the position, the velocity, the acceleration of the
$i$
th interceptor in the inertial coordinate frame, respectively.
The kinematic equation of the target can be given as
\begin{equation}\left\{ \begin{aligned}& {{{\dot{\boldsymbol{P}}}}_{T}}\!\left( t \right)={{\boldsymbol{V}}_{T}}\!\left( t \right) \\[5pt]& {{{\dot{\boldsymbol{V}}}}_{T}}\!\left( t \right)={{\boldsymbol{a}}_{T}}\!\left( t \right) \\\end{aligned} \right.\end{equation}
where,
${\boldsymbol{P}_T}\!\left( t \right) = {\left[ {{x_T}\!\left( t \right),{y_T}\!\left( t \right),{z_T}\!\left( t \right)} \right]^T}$
,
${\boldsymbol{V}_T}\!\left( t \right) = {\left[ {{V_{Tx}}\!\left( t \right),{V_{Ty}}\!\left( t \right),{V_{Tz}}\!\left( t \right)} \right]^T}$
,
${\boldsymbol{a}_T}\!\left( t \right) = {\left[ {{a_{Tx}}\!\left( t \right),{a_{Ty}}\!\left( t \right),{a_{Tz}}\!\left( t \right)} \right]^T}$
represent the position, the velocity, the acceleration of the target in the inertial coordinate frame, respectively.
2.2 Cooperative trajectory shaping guidance law
Based on the zero-effect miss distance and the optimal theory, the trajectory shaping guidance law is given in the line-of-sight coordinate frame, which can be expressed as [Reference Lukacs and Yakimenko32]
\begin{equation}\begin{aligned}{{\boldsymbol{a}}_{c}}\!\left( t \right)=\frac{4\left( {{\boldsymbol{x}}_{T}}\!\left( t \right)-\boldsymbol{x}\!\left( t \right)+\left( {{\boldsymbol{V}}_{T}}\!\left( t \right)-\boldsymbol{V}\!\left( t \right) \right){{t}_{go}} \right)}{t_{go}^{2}}+\frac{2\left( {{\boldsymbol{x}}_{T}}\!\left( t \right)-\boldsymbol{x}\!\left( t \right)+{{{\bar{\boldsymbol{V}}}}_{f}}{{t}_{go}} \right)}{t_{go}^{2}}+{{\boldsymbol{a}}_{T}}\!\left( t \right)\end{aligned}\end{equation}
where,
${\boldsymbol{x}_T}\!\left( t \right) - \boldsymbol{x}\!\left( t \right)$
indicates the zero-effect miss distance;
${\boldsymbol{V}_T}\!\left( t \right)$
and
$\boldsymbol{V}\!\left( t \right)$
represent the speed of target and interceptor, respectively;
$\bar{\boldsymbol{V}}_f = {\boldsymbol{V}_T}\!\left( {{t_f}} \right) - \boldsymbol{V}\!\left( {{t_f}} \right)$
;
${t_{go}}$
denotes the time-to-go of the interceptor;
${\boldsymbol{a}_T}\!\left( t \right)$
presents the acceleration of the target.
According to the Ref. [Reference Wu, Fang, Fu, Wang and Ma23], for the problem of cooperative interception of maneuvering or stealth targets by medium- and long-range interceptors, the cooperative term is introduced to optimise and improve the traditional trajectory shaping guidance law Equation (3). A new cooperative midcourse guidance law is proposed to achieve the purpose of cooperative interception. The acceleration of the
$i$
th interceptor on the
$x$
-axis in the inertial coordinate frame can be given as
\begin{equation}\begin{aligned}{{a}_{xi}}\!\left( t \right)=&\frac{6\left( {{{\tilde{P}}}_{xi}}\!\left( t \right)-{{{\bar{P}}}_{xi}}\!\left( t \right) \right)}{t_{goi}^{2}}+\frac{2\left( {{{\bar{V}}}_{xi}}\!\left( {{t}_{mf}} \right)-{{{\bar{V}}}_{xi}}\!\left( t \right) \right)}{{{t}_{goi}}}\\&+{{a}_{Tx}}(t)-{{\alpha }_{xi}}\!\left( {{{\tilde{P}}}_{xi}}\!\left( t \right)-{{{\overset{\scriptscriptstyle\frown}{P}}}_{x}}\!\left( t \right) \right)\ \ \ \ i=1,2,...,N\end{aligned}\end{equation}
where,
${t_{goi}}$
is the time-to-go of the
$i$
th interceptor;
${\tilde P_{xi}}\!\left( t \right)$
indicates the virtual interception point;
${\bar P_{xi}}\!\left( t \right)$
represents the position of the interceptor after the flight time-to-go
${t_{goi}}$
, based on the current position and speed;
${\bar V_{xi}}\!\left( {{t_{mf}}} \right)$
represents the error between the interceptor velocity and the target velocity at the end of midcourse guidance;
${\bar V_{xi}}\!\left( t \right)$
presents the real-time velocity error between interceptor and target;
${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over P} _x}\!\left( t \right)$
denotes the average position of virtual interception points of all interceptors;
${\alpha _{xi}}$
is the coordination coefficient;
$N$
is the number of interceptors. The expressions are given as follows
\begin{align}\begin{array}{*{20}{l}}{{{\tilde{P}}_{xi}}\!\left( t \right) = {x_T}\!\left( t \right) + {V_{Tx}}\!\left( t \right){t_{goi}}}\end{array}\end{align}
\begin{align}{\bar P_{xi}}\!\left( t \right) = {x_i}\!\left( t \right) + {V_{xi}}\!\left( t \right){t_{goi}}\end{align}
\begin{align}\begin{array}{*{20}{l}}{{{\bar V}_{xi}}\!\left( t \right) = {V_{Tx}}\!\left( t \right) - {V_{xi}}\!\left( t \right)}\end{array}\end{align}
\begin{align}\begin{array}{*{20}{l}}{{{\bar V}_{xi}}\!\left( {{t_{mf}}} \right) = {V_{Tx}}\!\left( {{t_{mf}}} \right) - {V_{xi}}\!\left( {{t_{mf}}} \right)}\end{array}\end{align}
\begin{align}\begin{array}{*{20}{l}}{{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over P} }_x}\!\left( t \right) = \left( {\mathop \sum \nolimits_{i = 1}^N {{{{\tilde P}}}_{xi}}\!\left( t \right)} \right)/N = {x_T}\!\left( t \right) + {V_{Tx}}\!\left( t \right){{\bar t}_{go}}}\end{array}\end{align}
\begin{align} {{{\bar t}_{go}} = \frac{1}{N}\mathop \sum \limits_{i = 1}^N {t_{goi}}} \end{align}
The acceleration of the interceptor on the
$x$
-axis is given above. The acceleration on the
$y$
-axis and the
$z$
-axis is the same as the
$x$
-axis principle. Do not elaborate here.
2.3 Graph theories
In the multiple interceptors cooperative midcourse guidance, the communication network topology between interceptors can be described by a graph
$\boldsymbol{G} = \left( {v,\varepsilon, \boldsymbol{A}} \right)$
, where
$v = \left\{ {1,2, \cdots, N} \right\}$
represents the set of nodes;
$\varepsilon \subset v \times v = \left\{ {\left( {i,j} \right):\;i,j \in v} \right\}$
indicates edges between nodes.
$\boldsymbol{A} = \left[ {{a_{ij}}} \right] \in {\boldsymbol{R}^{N \times N}}\!\left( {i,j = 1,2, \cdots, N} \right)$
denotes the adjacency matrix of graph
$\boldsymbol{G}$
. Assume that the communication topology is undirected, we have
${a_{ij}} = {a_{ji}} = 1$
if there is communication between the interceptor nodes
$i$
and
$j$
, otherwise
${a_{ij}} = {a_{ji}} = 0$
. The Laplacian matrix of graph
$\boldsymbol{G}$
is defined as
$\boldsymbol{L} = \left[ {{l_{ij}}} \right] \in {\boldsymbol{R}^{N \times N}}$
, which can be expressed as
\begin{equation}{{l}_{ij}}=\left\{ \begin{aligned}& \sum\limits_{j=1}^{N}{{{a}_{ij}}},\ \ \ i=j \\& -{{a}_{ij}},\ \ \ \ \ i\ne j \\\end{aligned} \right.\end{equation}
In this paper, the following assumptions are made:
Assumption 1. The communication network of multiple interceptors is undirected.
Assumption 2.
The interceptor autopilot has no time delay, and the handover time
$T$
is mainly related to the handover heading error, and the handover heading error is less than the maximum allowable heading error.
2.4 Correlation definitions and lemmas
Definition 1. [Reference Hong, Xu and Jie33] Consider the following nonlinear system
\begin{align}\dot{\boldsymbol{x}}=\boldsymbol{f}\!\left( \boldsymbol{x},t \right),\boldsymbol{f}\!\left( \mathbf{0},t \right)=\mathbf{0},\boldsymbol{x}\in {{\boldsymbol{R}}^{n}}\end{align}
where,
$\boldsymbol{x} \in U \subseteq {\boldsymbol{R}^n}$
;
$f\;:\;U \to {\boldsymbol{R}^n}$
is a continuous function from the domain
$U$
containing the origin
$\boldsymbol{x} = \mathbf{0}$
to space
${\boldsymbol{R}^n}$
. For any initial condition
$\boldsymbol{x}\!\left( \mathbf{0} \right) = {\boldsymbol{x}_0} \in {U_0} \subseteq U$
, there exists
$T \geq 0$
such that the following equation holds
\begin{equation}\left\{ \begin{aligned}& \underset{t\to T\left( {{\boldsymbol{x}}_{0}} \right)}{\mathop{\lim }}\,\boldsymbol{x}\!\left( t,\mathbf{0},{{\boldsymbol{x}}_{0}} \right)=0 \\[5pt]& If\ t \gt T\left( {{\boldsymbol{x}}_{0}} \right),\ then\ \boldsymbol{x}\!\left( t,\mathbf{0},{{\boldsymbol{x}}_{0}} \right)=\mathbf{0} \\\end{aligned} \right.\end{equation}
according to Equation (13), it is known that the state of the system converges to the origin
$\boldsymbol{x} = \mathbf{0}$
in finite time. In the neighbourhood
$U$
, if the system origin
$\boldsymbol{x} = \mathbf{0}$
is Lyapunov stable and converges to a neighbourhood of the system origin in finite time, then the system origin
$\boldsymbol{x} = \mathbf{0}$
is called finite-time stable. When
$U = {\boldsymbol{R}^n}$
, the origin is called globally finite-time stable.
Lemma 1. [Reference Bhat and Bernstein34] For system (13), in the domain
$U$
, if there is a continuously differentiable and radially unbounded positive definite function
$V\left( x \right)$
that satisfies
\begin{align}\dot V\left( \boldsymbol{x} \right) \le - \beta {V^p}\!\left( \boldsymbol{x} \right),\boldsymbol{x} \in U\end{align}
where
$\beta $
is positive constant, and
$p$
satisfies
$p \in \left( {0,1} \right)$
. According to Definition 1, system (13) is finite-time stable. The system state can converge to the equilibrium point in finite time and the convergence time satisfies
\begin{align}T \le \frac{1}{{\beta \left( {1 - p} \right)}}{V^{1 - p}}\!\left( {{\boldsymbol{x}_0}} \right)\end{align}
Lemma 2. [Reference Long and Feng35] If the graph
$G$
is undirected and connected, then for any
$\boldsymbol\xi = {\left[ {{\xi _1},{\xi _2}, \cdots {\xi _n}} \right]^T} \in {\boldsymbol{R}^n}$
, we have
${\boldsymbol\xi ^T}\boldsymbol{L}\boldsymbol\xi = \left( {1/2\;\;} \right)\mathop \sum \limits_{i = 1}^n \mathop \sum \limits_{j = 1}^n {a_{ij}}{\left( {{\xi _i} - {\xi _j}} \right)^2}$
.
Lemma 3. [Reference Long and Feng35] If
${z_1},{z_2}, \cdots, {z_n} \geq 0$
and
$0 \lt \varepsilon \lt 1$
, then
\begin{align}{\left( {\mathop \sum \limits_{i = 1}^n {z_i}} \right)^\varepsilon } \le \mathop \sum \limits_{i = 1}^n z_i^\varepsilon \end{align}
Lemma 4. [Reference Zhang36] If the function
$y = f\left( x \right)$
has
$n + 1$
different nodes, which are
${x_0},{x_1}, \cdots, {x_n}$
, the function value is
${y_i} = f\left( {{x_i}} \right)$
, and the derivative value is
$y{'_i} = f'\!\left( {{x_i}} \right)$
. The polynomial
$H\!\left( x \right)$
of at most
$2n + 1$
degrees is required to satisfy
\begin{align}H\!\left( {{x_i}} \right) = {y_i},H'\!\left( {{x_i}} \right) = y{'_i}\end{align}
According to Hermite interpolation theory, then we have
\begin{align}H\!\left( x \right) = \mathop \sum \limits_{i = 0}^n \left[ {{y_i}{h_i}\!\left( x \right) + y{'_i}{H_i}\!\left( x \right)} \right]\end{align}
When
$n = 1$
, the following expressions hold
\begin{align}\!\left\{ {\begin{array}{l}{{h_0}\!\left( x \right) = \left( {1 + 2\dfrac{{x - {x_0}}}{{{x_1} - {x_0}}}} \right){{\left( {\dfrac{{x - {x_1}}}{{{x_0} - {x_1}}}} \right)}^2}}\\[12pt] {{h_1}\!\left( x \right) = \left( {1 + 2\dfrac{{x - {x_1}}}{{{x_0} - {x_1}}}} \right){{\left( {\dfrac{{x - {x_0}}}{{{x_1} - {x_0}}}} \right)}^2}}\\[12pt] {{H_0}\!\left( x \right) = \left( {x - {x_0}} \right){{\left( {\dfrac{{x - {x_0}}}{{{x_1} - {x_0}}}} \right)}^2}}\\[12pt] {{H_1}\!\left( x \right) = \left( {x - {x_1}} \right){{\left( {\dfrac{{x - {x_0}}}{{{x_1} - {x_0}}}} \right)}^2}}\end{array}} \right.\end{align}
Then, according to Equation (20), the polynomial
$H\!\left( x \right)$
of two points can be expressed as
\begin{equation}\begin{aligned}H\!\left( x \right)=&\left( 1+2\frac{x-{{x}_{0}}}{{{x}_{1}}-{{x}_{0}}} \right){{\left( \frac{x-{{x}_{1}}}{{{x}_{0}}-{{x}_{1}}} \right)}^{2}}{{y}_{0}}+\left( 1+2\frac{x-{{x}_{1}}}{{{x}_{0}}-{{x}_{1}}} \right){{\left( \frac{x-{{x}_{0}}}{{{x}_{1}}-{{x}_{0}}} \right)}^{2}}{{y}_{1}}\\[12pt]&+\left( x-{{x}_{0}} \right){{\left( \frac{x-{{x}_{0}}}{{{x}_{1}}-{{x}_{0}}} \right)}^{2}}{{{{y}'}}_{0}}+\left( x-{{x}_{1}} \right){{\left( \frac{x-{{x}_{0}}}{{{x}_{1}}-{{x}_{0}}} \right)}^{2}}{{{{y}'}}_{1}} \\\end{aligned}\end{equation}
3.0 Main results
3.1 Design of cooperative midcourse guidance law
3.1.1 Design of guidance law
For the case that the communication topology of multiple interceptors is undirected and connected, based on the finite-time theory, a cooperative guidance law with communication topology switching is designed as
\begin{equation}\left\{ \begin{aligned}& {{a}_{xi}}\!\left( t \right)=\frac{6\left( {{{\tilde{P}}}_{xi}}\!\left( t \right)-{{{\bar{P}}}_{xi}}\!\left( t \right) \right)}{t_{goi}^{2}}+\frac{2\left( {{{\bar{V}}}_{xi}}\!\left( {{t}_{mf}} \right)-{{{\bar{V}}}_{xi}}\!\left( t \right) \right)}{{{t}_{goi}}}+{{a}_{Tx}}(t)-{{\alpha }_{xi}}\!\left( {{{\tilde{P}}}_{xi}}\!\left( t \right)-{{{\hat{P}}}_{xi}}\!\left( t \right) \right) \\[5pt]& {{{\dot{\hat{P}}}}_{xi}}\!\left( t \right)\text{=}\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)sign\left[ {{{\hat{P}}}_{xj}}\!\left( t \right)-{{{\hat{P}}}_{xi}}\!\left( t \right) \right]{{\left| {{{\hat{P}}}_{xj}}\!\left( t \right)-{{{\hat{P}}}_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}}}} \\[5pt]& {{a}_{yi}}\!\left( t \right)=\frac{6\left( {{{\tilde{P}}}_{yi}}\!\left( t \right)-{{{\bar{P}}}_{yi}}\!\left( t \right) \right)}{t_{goi}^{2}}+\frac{2\left( {{{\bar{V}}}_{yi}}\!\left( {{t}_{mf}} \right)-{{{\bar{V}}}_{yi}}\!\left( t \right) \right)}{{{t}_{goi}}}+{{a}_{Ty}}(t)-{{\alpha }_{yi}}\!\left( {{{\tilde{P}}}_{yi}}\!\left( t \right)-{{{\hat{P}}}_{yi}}\!\left( t \right) \right)\\[5pt]& {{{\dot{\hat{P}}}}_{yi}}\!\left( t \right)\text{=}\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)sign\left[ {{{\hat{P}}}_{yj}}\!\left( t \right)-{{{\hat{P}}}_{yi}}\!\left( t \right) \right]{{\left| {{{\hat{P}}}_{yj}}\!\left( t \right)-{{{\hat{P}}}_{yi}}\!\left( t \right) \right|}^{{{\alpha }_{yij}}}}} \\[5pt]& {{a}_{zi}}\!\left( t \right)=\frac{6\left( {{{\tilde{P}}}_{zi}}\!\left( t \right)-{{{\bar{P}}}_{zi}}\!\left( t \right) \right)}{t_{goi}^{2}}+\frac{2\left( {{{\bar{V}}}_{zi}}\!\left( {{t}_{mf}} \right)-{{{\bar{V}}}_{zi}}\!\left( t \right) \right)}{{{t}_{goi}}}+{{a}_{Tz}}(t)-{{\alpha }_{zi}}\!\left( {{{\tilde{P}}}_{zi}}\!\left( t \right)-{{{\hat{P}}}_{zi}}\!\left( t \right) \right) \\[5pt]& {{{\dot{\hat{P}}}}_{zi}}\!\left( t \right)\text{=}\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)sign\left[ {{{\hat{P}}}_{zj}}\!\left( t \right)-{{{\hat{P}}}_{zi}}\!\left( t \right) \right]{{\left| {{{\hat{P}}}_{zj}}\!\left( t \right)-{{{\hat{P}}}_{zi}}\!\left( t \right) \right|}^{{{\alpha }_{zij}}}}} \\\end{aligned} \right.\end{equation}
where,
$0 \lt {\alpha _{xij}} \lt 1$
;
$0 \lt {\alpha _{yij}} \lt 1$
;
$0 \lt {\alpha _{zij}} \lt 1$
;
${\alpha _{xij}} = {\alpha _{xji}}$
;
${\alpha _{yij}} = {\alpha _{yji}}$
;
${\alpha _{zij}} = {\alpha _{zji}}$
;
${N_i}$
denotes the number of the
$i$
th interceptor and its neighbour interceptor.
3.1.2 Stability analysis
Because the mathematical model of the interceptor is similar in the three directions of
$x$
,
$y$
and
$z$
, only the acceleration stability analysis in the
$x$
direction is given, and the principle of the other two directions is the same, which is not repeated here.
In the whole network topology, the average value of the virtual interception points for all distributed structures can be expressed as
\begin{align}\begin{array}{*{20}{l}}{P_{xi}^{\rm{*}}\!\left( t \right) = \dfrac{1}{n}\mathop \sum \limits_{i = 1}^n {{{{\hat P}}}_{xi}}\!\left( t \right)}\end{array}\end{align}
Then, according to Equation (21), the consensus protocol sum of all interceptors can be given as
\begin{equation}\begin{aligned}\sum\limits_{i=1}^{n}{{{{\dot{\hat{P}}}}_{xi}}\!\left( t \right)}=\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)sign\left[ {{{\hat{P}}}_{xj}}\!\left( t \right)-{{{\hat{P}}}_{xi}}\!\left( t \right) \right]{{\left| {{{\hat{P}}}_{xj}}\!\left( t \right)-{{{\hat{P}}}_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}}}}}=0\end{aligned}\end{equation}
where,
${a_{ij}}\!\left( t \right)$
represents the communication topology relationship between interceptors, and the communication topology structure is undirected and time-varying;
$n$
is the total number of interceptors, with
$n \geq {N_i}$
.
Obviously, according to Equation (23),
$\mathop \sum \limits_{i = 1}^n {\hat P_{xi}}\!\left( t \right)$
is a constant value, so we can obtain
\begin{equation}\begin{aligned}\sum\limits_{i=1}^{n}{{{{\hat{P}}}_{xi}}\!\left( t \right)}\text{=}\sum\limits_{i=1}^{n}{{{{\hat{P}}}_{xi}}\!\left( 0 \right)}\end{aligned}\end{equation}
Furthermore, according to Equation (22), we can get
\begin{equation}\begin{aligned}{{{\hat{P}}}_{xi}}\!\left( t \right)=P_{xi}^{*}\!\left( t \right)+{{\Delta }_{xi}}\!\left( t \right)=\frac{1}{n}\sum\limits_{i=1}^{n}{{{{\hat{P}}}_{xi}}\!\left( t \right)}+{{\Delta }_{xi}}\!\left( t \right) \\\end{aligned}\end{equation}
where,
${{\rm{\Delta }}_{xi}}\!\left( t \right)$
is the difference value between the average value of the virtual interception points of the
$i$
th distributed structure composed of the
$i$
th interceptor and the average value of the virtual interception points of all distributed structures. In order to enable the interceptors to reach the handover area almost simultaneously, it needs to meet
${{\rm{\Delta }}_{xi}}\!\left( t \right) \to 0$
.
Then, according to Equation (25), we have
\begin{equation}\begin{aligned}\sum\limits_{i=1}^{n}{{{\Delta }_{xi}}\!\left( t \right)}=\sum\limits_{i=1}^{n}{{{{\hat{P}}}_{xi}}\!\left( t \right)}-\sum\limits_{i=1}^{n}{\left[ \frac{1}{n}\sum\limits_{i=1}^{n}{{{{\hat{P}}}_{xi}}\!\left( t \right)} \right]}=0\end{aligned}\end{equation}
Meanwhile, combining Equations (21) and (23), according to Equation (25), we take the derivate of
${\hat P_{xi}}\!\left( t \right)$
with respect to time
$t$
as
\begin{equation}\begin{aligned}{{{\dot{\hat{P}}}}_{xi}}\!\left( t \right)&=\frac{1}{n}\sum\limits_{i=1}^{n}{{{{\dot{\hat{P}}}}_{xi}}\!\left( t \right)}+{{{\dot{\Delta }}}_{xi}}\!\left( t \right) \\[5pt]&={{{\dot{\Delta }}}_{xi}}\!\left( t \right) \\[5pt]&=\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)sign\left[ {{{\hat{P}}}_{xj}}\!\left( t \right)-{{{\hat{P}}}_{xi}}\!\left( t \right) \right]{{\left| {{{\hat{P}}}_{xj}}\!\left( t \right)-{{{\hat{P}}}_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}}}} \\\end{aligned}\end{equation}
According to Equation (27), the Lyapunov function is denoted as
\begin{equation}\begin{aligned}{{V}_{xi}}=\frac{1}{2}\sum\limits_{i=1}^{n}{{{\Delta }_{xi}}\!\left( t \right){{\Delta }_{xi}}\!\left( t \right)}\end{aligned}\end{equation}
We take the derivate of
${V_{xi}}$
with respect to time
$t$
as
\begin{align}{{\dot{V}}_{xi}}=\sum\limits_{i=1}^{n}{{{\Delta }_{xi}}\!\left( t \right){{{\dot{\Delta }}}_{xi}}\!\left( t \right)}\end{align}
Then, substituting Equation (27) into Equation (29), we can obtain
\begin{equation}\begin{aligned}{{{\dot{V}}}_{xi}}&=\sum\limits_{i=1}^{n}{{{\Delta }_{xi}}\!\left( t \right){{{\dot{\Delta }}}_{xi}}\!\left( t \right)} \\[5pt]&=\sum\limits_{i=1}^{n}{{{\Delta }_{xi}}\!\left( t \right)}\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)sign\left[ {{{\hat{P}}}_{xj}}\!\left( t \right)-{{{\hat{P}}}_{xi}}\!\left( t \right) \right]{{\left| {{{\hat{P}}}_{xj}}\!\left( t \right)-{{{\hat{P}}}_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}}}} \\\end{aligned}\end{equation}
Furthermore, substituting Equation (25) into Equation (30), we have
\begin{equation}\begin{aligned}{{{\dot{V}}}_{xi}}&=\sum\limits_{i=1}^{n}{{{\Delta }_{xi}}\!\left( t \right)}\sum\limits_{j=1}^{{{N}_{i}}}{\left(\begin{aligned}&{{a}_{ij}}\!\left( t \right)sign\left[ \frac{1}{n}\sum\limits_{j=1}^{n}{{{{\hat{P}}}_{xj}}\!\left( t \right)}+{{\Delta }_{xj}}\!\left( t \right)-\frac{1}{n}\sum\limits_{i=1}^{n}{{{{\hat{P}}}_{xi}}\!\left( t \right)}-{{\Delta }_{xi}}\!\left( t \right) \right]\\[5pt]&\cdot{{\left| \frac{1}{n}\sum\limits_{j=1}^{n}{{{{\hat{P}}}_{xj}}\!\left( t \right)}+{{\Delta }_{xj}}\!\left( t \right)-\frac{1}{n}\sum\limits_{i=1}^{n}{{{{\hat{P}}}_{xi}}\!\left( t \right)}-{{\Delta }_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}}}\end{aligned}\right)} \\[5pt]&=\sum\limits_{i=1}^{n}{{{\Delta }_{xi}}\!\left( t \right)}\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)sign\left[ {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right]{{\left| {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}}}} \\\end{aligned}\end{equation}
Then, by using Equation (26), we can rewrite Equation (31) as
\begin{equation}\begin{aligned}{{{\dot{V}}}_{xi}}&=\sum\limits_{i=1}^{n}{{{\Delta }_{xi}}\!\left( t \right)}\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)sign\left[ {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right]{{\left| {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}}}} \\[5pt]& =\sum\limits_{i=1}^{n}{\frac{1}{2}\!\left( {{\Delta }_{xi}}\!\left( t \right)-{{\Delta }_{xj}}\!\left( t \right) \right)}\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)sign\left[ {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right]{{\left| {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}}}} \\[5pt]&=-\frac{1}{2}\sum\limits_{i=1}^{n}{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)sign\left[ {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right]{{\left| {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}}}} \\[5pt]&=-\frac{1}{2}\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right)\left| {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right|{{\left| {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}}}}} \\[5pt]&=-\frac{1}{2}\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right){{\left| {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}+1}}}} \lt 0 \\\end{aligned}\end{equation}
According to Equations (28) and (32), the system is globally asymptotically stable. Meanwhile, according to Lemma 3, Equation (32) is also rewritten as
\begin{equation}\begin{aligned}{{{\dot{V}}}_{xi}}&=-\frac{1}{2}\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}\!\left( t \right){{\left| {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right|}^{{{\alpha }_{xij}}+1}}}} \\[5pt]&=-\frac{1}{2}\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{\left\{ {{a}_{ij}}^{\frac{2}{1+{{\alpha }_{xij}}}}\!\left( t \right){{\left[ {{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}^{2}} \right]}^{\frac{{{\alpha }_{xij}}+1}{1+{{\alpha }_{xij}}}}} \right\}}^{\frac{1+{{\alpha }_{xij}}}{2}}}}} \\[5pt]&\le -\frac{1}{2}{{\left\{ \sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right){{\left[ {{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}^{2}} \right]}^{\frac{{{\alpha }_{xij}}+1}{1+{{\alpha }_{0}}}}}}} \right\}}^{\frac{1+{{\alpha }_{0}}}{2}}} \\[5pt]&=-\frac{1}{2}{{\left\{ \frac{{{\Xi }_{1}}}{{{\Xi }_{2}}}\frac{{{\Xi }_{3}}}{{{V}_{xi}}}{{V}_{xi}} \right\}}^{\frac{1+{{\alpha }_{0}}}{2}}} \\\end{aligned}\end{equation}
Where
\begin{equation}\begin{aligned}& {{\alpha }_{0}}={{\max }_{i,j}}{{\alpha }_{ij}} \\[5pt] & {{\Xi }_{1}}=\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right){{\left[ {{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}^{2}} \right]}^{\frac{{{\alpha }_{xij}}+1}{1+{{\alpha }_{0}}}}}}} \\[5pt] & {{\Xi }_{2}}=\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right)\left[ {{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}^{2}} \right]}} \\[5pt] & {{\Xi }_{3}}=\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right)\left[ {{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}^{2}} \right]}} \\\end{aligned}\end{equation}
Suppose
$\left( {{i_0},{j_0}} \right) = {\rm{argmax}}{\left( {{{\rm{\Delta }}_{xj}}\!\left( t \right) - {{\rm{\Delta }}_{xi}}\!\left( t \right)} \right)^2}$
, when
${\left( {{{\rm{\Delta }}_{xj}}\!\left( t \right) - {{\rm{\Delta }}_{xi}}\!\left( t \right)} \right)^2}$
takes the maximum value, that is, there is
$\left( {i,j} \right)$
taking
$\left( {{i_0},{j_0}} \right)$
. The term of
${{\rm{\Xi }}_1}$
in Equation (34) can be rewritten as
\begin{equation}\begin{aligned}{{\Xi }_{1}}&=\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right){{\left[ {{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}^{2}} \right]}^{\frac{{{\alpha }_{xij}}+1}{1+{{\alpha }_{0}}}}}}} \\[5pt] &\ge {{a}_{{{i}_{0}}{{j}_{0}}}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right){{\left[ {{\left( {{\Delta }_{x{{j}_{0}}}}\!\left( t \right)-{{\Delta }_{x{{i}_{0}}}}\!\left( t \right) \right)}^{2}} \right]}^{\frac{{{\alpha }_{x{{i}_{0}}{{j}_{0}}}}+1}{1+{{\alpha }_{0}}}}} \\\end{aligned}\end{equation}
The term of
${{\rm{\Xi }}_2}$
in Equation (34) can be rewritten as
\begin{equation}\begin{aligned}{{\Xi }_{2}}&=\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right)\left[ {{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}^{2}} \right]}} \\[5pt] &\le \sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right)}}\!\left[ {{\left( {{\Delta }_{x{{j}_{0}}}}\!\left( t \right)-{{\Delta }_{x{{i}_{0}}}}\!\left( t \right) \right)}^{2}} \right] \\\end{aligned}\end{equation}
Therefore, combining Equation (35) with Equation (36), we have
\begin{equation}\begin{aligned}\frac{{{\Xi }_{1}}}{{{\Xi }_{2}}}&=\frac{\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right){{\left[ {{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}^{2}} \right]}^{\frac{{{\alpha }_{xij}}+1}{1+{{\alpha }_{0}}}}}}}}{\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right)\left[ {{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}^{2}} \right]}}} \\[5pt] &\le \frac{{{a}_{{{i}_{0}}{{j}_{0}}}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right){{\left[ {{\left( {{\Delta }_{x{{j}_{0}}}}\!\left( t \right)-{{\Delta }_{x{{i}_{0}}}}\!\left( t \right) \right)}^{2}} \right]}^{\frac{{{\alpha }_{x{{i}_{0}}{{j}_{0}}}}+1}{1+{{\alpha }_{0}}}}}}{\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right)}}\!\left[ {{\left( {{\Delta }_{x{{j}_{0}}}}\!\left( t \right)-{{\Delta }_{x{{i}_{0}}}}\!\left( t \right) \right)}^{2}} \right]} \\[5pt] &={{K}_{1}} \\\end{aligned}\end{equation}
Then, according to Lemma 2, we can rewrite the term of
${{\rm{\Xi }}_3}$
in Equation (34) as
\begin{equation}\begin{aligned}{{\Xi }_{3}}&=\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{N}_{i}}}{{{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right)\left[ {{\left( {{\Delta }_{xj}}\!\left( t \right)-{{\Delta }_{xi}}\!\left( t \right) \right)}^{2}} \right]}} \\[5pt] &=2{{\boldsymbol \Delta }_{x}}^{T}\boldsymbol{L}\!\left( \boldsymbol{G} \right){{\boldsymbol \Delta }_{x}} \\\end{aligned}\end{equation}
where
\begin{equation}\begin{aligned}&\boldsymbol{G}=\left[ {{a}_{ij}}^{\frac{2}{1+{{\alpha }_{0}}}}\!\left( t \right) \right]\in {{\boldsymbol{R}}^{n\times n}} \\[5pt] &{{\boldsymbol \Delta }_{x}}={{\left[ {{\Delta }_{x1}},{{\Delta }_{x2}},\cdots, {{\Delta }_{xn}} \right]}^{T}} \\\end{aligned}\end{equation}
Furthermore, according to Equation (38), we can obtain
\begin{equation}\begin{aligned}\frac{{{\Xi }_{3}}}{{{V}_{xi}}}&=\frac{2{{\boldsymbol \Delta }_{x}}^{T}\boldsymbol{L}\!\left( \boldsymbol{G} \right){{\boldsymbol \Delta }_{x}}}{\frac{1}{2}{{\boldsymbol \Delta }_{x}}^{T}{{\boldsymbol \Delta }_{x}}} \\[5pt] &\ge 4{{\lambda }_{2}}\!\left[ \boldsymbol{L}\!\left( \boldsymbol{G} \right) \right] \\[5pt] & \gt 0 \\\end{aligned}\end{equation}
where
${\lambda _2}\!\left[ {\boldsymbol{L}\!\left( \boldsymbol{G} \right)} \right]$
is the second smallest eigenvalue of the Laplacian matrix of
$\boldsymbol{G}$
.
Then, combining Equation (37) with Equation (40), we can rewrite Equation (33) as
\begin{equation}\begin{aligned}{{{\dot{V}}}_{xi}}&=-\frac{1}{2}{{\left\{ \frac{{{\Xi }_{1}}}{{{\Xi }_{2}}}\frac{{{\Xi }_{3}}}{{{V}_{xi}}}{{V}_{xi}} \right\}}^{\frac{1+{{\alpha }_{0}}}{2}}} \\[5pt] &\le -\frac{1}{2}{{\left\{ {{K}_{1}}4{{\lambda }_{2}}\!\left[ \boldsymbol{L}\!\left( \boldsymbol{G} \right) \right] \right\}}^{\frac{1+{{\alpha }_{0}}}{2}}}V_{xi}^{\frac{1+{{\alpha }_{0}}}{2}} \\[5pt] &=-{{2}^{{{\alpha }_{0}}}}{{\left\{ {{K}_{1}}{{\lambda }_{2}}\!\left[ \boldsymbol{L}\!\left( \boldsymbol{G} \right) \right] \right\}}^{\frac{1+{{\alpha }_{0}}}{2}}}V_{xi}^{\frac{1+{{\alpha }_{0}}}{2}} \\\end{aligned}\end{equation}
Define
${{K}_{2}}=\min \left\{ {{K}_{1}}{{\lambda }_{2}}\!\left[\boldsymbol{L}\!\left(\boldsymbol{G} \right) \right] \right\}$
, Equation (41) can be rewritten as
\begin{equation}\begin{aligned}{{\dot{V}}_{xi}}\le -{{2}^{{{\alpha }_{0}}}}K_{2}^{\frac{1+{{\alpha }_{0}}}{2}}V_{xi}^{\frac{1+{{\alpha }_{0}}}{2}}\end{aligned}\end{equation}
According to Equations (28) and (42) and Lemma 1, we have
\begin{align}\underset{t\to {{T}_{1}}}{\mathop{\lim }}\,{{\Delta }_{xi}}\!\left( t \right)=0\end{align}
where
\begin{align}{{T}_{1}}=\frac{{{2}^{1-{{\alpha }_{0}}}}{{V}_{xi}}{{\left( 0 \right)}^{\frac{1-{{\alpha }_{0}}}{2}}}}{K_{2}^{\frac{1+{{\alpha }_{0}}}{2}}\!\left( 1-{{\alpha }_{0}} \right)} \gt 0\end{align}
Hence, combining Equation (25) with Equation (43), it is easy to get that
\begin{align}\underset{t\to {{T}_{1}}}{\mathop{\lim }}\,\left( {{{\hat{P}}}_{xi}}-{{{\hat{P}}}_{xj}} \right)=0\end{align}
In summary, when the communication topology between interceptors is undirected and the connected time satisfies Equation (44), Equation (45) can be held to achieve the purpose of cooperative interception.
3.2 Design of target handover law
For the target handover law, this paper uses the virtual target method to study. In actual flight, the interceptor changes target according to mission requirements. Based on the old target, the position error between the old target and the new target is gradually increased to form a virtual target. The virtual target gradually replaces the new target to achieve the smooth handover of the target.
3.2.1 Handover model
Assume that multiple interceptors have reselected new targets based on mission requirements. By comparing the positions of the old target and the new target, the relative position can be expressed as
\begin{align}{{\boldsymbol{P}}_{e}}\!\left( t \right)={{\boldsymbol{P}}_{new}}\!\left( t \right)-{{\boldsymbol{P}}_{old}}\!\left( t \right)\end{align}
where,
${\boldsymbol{P}_{old}}$
denotes the position of the old target;
${\boldsymbol{P}_{new}}$
represents the position of the new target.
We take the derivate of
${\boldsymbol{P}_e}\!\left( t \right)$
with respect to time
$t$
as
\begin{align}{{\dot{\boldsymbol{P}}}_{e}}\!\left( t \right)={{\boldsymbol{V}}_{e}}\!\left( t \right)\end{align}
where
${\boldsymbol{V}_e}$
is the relative velocity between the old target and the new target.
3.2.2 Design of handover method
(a) Handover time
In flight, when the interceptor performs target changing and guidance law handover, the handover time
$T$
is unknown. Therefore, it is necessary to design a suitable handover time
$T$
to achieve the smooth handover of target and guidance law.
The heading angular velocity of the interceptor can be given as
\begin{align}\dot{\gamma }=\frac{a\!\left( t \right)}{V}\end{align}
where,
$a\!\left( t \right)$
is the acceleration of the interceptor;
$V$
represents the velocity of the interceptor.
Suppose the acceleration at the beginning of the handover is
${a_1}\!\left( {{t_0}} \right)$
and the acceleration at the end is
${a_2}\!\left( {{t_0} + T} \right)$
. The acceleration error for the handover phase is denoted as
\begin{align}\Delta a={{a}_{2}}\!\left( {{t}_{0}}+T \right)-{{a}_{1}}\!\left( {{t}_{0}} \right)\end{align}
Integrating Equation (49), with integration time
$T$
, we have
\begin{align}\int_{{{t}_{0}}}^{{{t}_{0}}+T}{\Delta adt}\approx 0.5T\left( {{a}_{2}}-{{a}_{1}} \right)\end{align}
Then, according to Equations (48) and (50), the heading error of the interceptor during the handover can be expressed as
\begin{equation}\begin{aligned}\Delta {{\gamma }_{m}}&=\left| \int_{{{t}_{0}}}^{{{t}_{0}}+T}{\Delta adt} \right| \\[5pt] &\approx \left| \frac{0.5T\left( {{a}_{2}}-{{a}_{1}} \right)}{V} \right| \\\end{aligned}\end{equation}
Assume that the maximum allowable heading error is
${\rm{\Delta }}{\gamma _{{\rm{max}}}}$
. According to Equation (51), we can obtain
\begin{align}T\le \frac{2\Delta {{\gamma }_{\max }}V}{\left| {{a}_{2}}-{{a}_{1}} \right|}\end{align}
By selecting the handover time
$T$
satisfying Equation (52), the successful handover of target and guidance law can be realised.
(b) Target handover law
When changing target, the virtual target gradually approaches the new target on the basis of the old target. To achieve the overlap between the virtual target and the new target, the following expression needs to be satisfied.
\begin{align}\int_{0}^{T}{{{\boldsymbol{V}}_{approach}}\!\left( t \right)}={{\boldsymbol{P}}_{e}}\!\left( t \right)\end{align}
where
${\boldsymbol{V}_{approach}}\!\left( t \right)$
is the approach speed of the virtual target approaching the new target.
The virtual target position can be defined as
\begin{align}{{\boldsymbol{P}}_{virtual}}\!\left( t \right)={{\boldsymbol{P}}_{old}}\!\left( t \right)+\lambda \left( t \right){{\boldsymbol{P}}_{e}}\!\left( t \right)\end{align}
where
$\lambda \left( t \right)$
is the parameter of the target handover law. The target handover law parameter
$\lambda \left( t \right)$
is designed for two cases.
(a) Case 1: The relative distance between the old target and the new target is a constant value
In this case,
${\boldsymbol{P}_e}\!\left( t \right)$
is a constant value. Suppose that the virtual target moves from the old target position to the new target position at a constant velocity, we have
\begin{align}{{\boldsymbol{V}}_{approach}}\!\left( t \right)=\frac{{{\boldsymbol{P}}_{e}}\!\left( t \right)}{T}=const\end{align}
where
$T$
denotes the time required to complete the handover of target.
Then, by using Equation (55), the virtual target velocity is denoted as
\begin{equation}\begin{aligned}{{\boldsymbol{V}}_{virtual}}\!\left( t \right)&={{\boldsymbol{V}}_{old}}\!\left( t \right)+{{\boldsymbol{V}}_{approach}}\!\left( t \right) \\[5pt] &={{\boldsymbol{V}}_{old}}\!\left( t \right)+\frac{{{\boldsymbol{P}}_{e}}\!\left( t \right)}{T} \\\end{aligned}\end{equation}
Furthermore, integrating Equation (56), the virtual target position can be expressed as
\begin{align}{{\boldsymbol{P}}_{virtual}}\!\left( t \right)={{\boldsymbol{P}}_{old}}\!\left( t \right)+\frac{t}{T}{{\boldsymbol{P}}_{e}}\!\left( t \right)\end{align}
By comparing Equation (57) with Equation (54), we can get
\begin{align}\lambda \left( t \right)=\frac{t}{T},t\in \left[ 0,T \right]\end{align}
According to Equation (57), the virtual target overlaps with the new target at time
$T$
, the target changing is achieved.
(b) Case 2: The relative distance between the old target and the new target is a time-varying value
In this case,
${\boldsymbol{P}_e}\!\left( t \right)$
is a time-varying value. In order to realise smooth handover of interceptor acceleration, it is required that the velocity derivatives at the beginning time
${t_0}$
and the end time
${t_0} + T$
of the handover phase exist and are continuous, and the acceleration derivatives exist and are continuous, which is called the second-order smooth handover. Compared with the first-order smooth handover, the phenomenon of excessive acceleration or divergence at the handover time can be avoided.
Therefore, the following expressions need to be satisfied at time
${t_0}$
and time
${t_0} + T$
.
\begin{equation}\left\{ \begin{aligned}&{{\boldsymbol{P}}_{virtual}}\!\left( {{t}_{0}} \right)={{\boldsymbol{P}}_{old}}\!\left( {{t}_{0}} \right) \\[5pt] &{{{\dot{\boldsymbol{P}}}}_{virtual}}\!\left( {{t}_{0}} \right)={{{\dot{\boldsymbol{P}}}}_{old}}\!\left( {{t}_{0}} \right) \\[5pt] &{{{\ddot{\boldsymbol{P}}}}_{virtual}}\!\left( {{t}_{0}} \right)={{{\ddot{\boldsymbol{P}}}}_{old}}\!\left( {{t}_{0}} \right) \\[5pt] &{{{\dddot{\boldsymbol{P}}}}_{virtual}}\!\left( {{t}_{0}} \right)={{{\dddot{\boldsymbol{P}}}}_{old}}\!\left( {{t}_{0}} \right) \\\end{aligned} \right.\end{equation}
\begin{equation}\left\{ \begin{aligned}&{{\boldsymbol{P}}_{virtual}}\!\left( {{t}_{0}}+T \right)={{\boldsymbol{P}}_{old}}\!\left( {{t}_{0}}+T \right) \\[5pt] &{{{\dot{\boldsymbol{P}}}}_{virtual}}\!\left( {{t}_{0}}+T \right)={{{\dot{\boldsymbol{P}}}}_{old}}\!\left( {{t}_{0}}+T \right) \\[5pt] &{{{\ddot{\boldsymbol{P}}}}_{virtual}}\!\left( {{t}_{0}}+T \right)={{{\ddot{\boldsymbol{P}}}}_{old}}\!\left( {{t}_{0}}+T \right) \\[5pt] &{{{\dddot{\boldsymbol{P}}}}_{virtual}}\!\left( {{t}_{0}}+T \right)={{{\dddot{\boldsymbol{P}}}}_{old}}\!\left( {{t}_{0}}+T \right) \\\end{aligned} \right.\end{equation}
Meanwhile, according to Equation (54), we take the derivate of
${\boldsymbol{P}_{virtual}}\!\left( t \right)$
with respect to time
$t$
as
\begin{equation}\left\{ \begin{array}{l} {{\boldsymbol{P}}_{virtual}}\!\left( t \right)={{\boldsymbol{P}}_{old}}\!\left( t \right)+\lambda \left( t \right){{\boldsymbol{P}}_{e}}\!\left( t \right) \\[5pt] {{{\dot{\boldsymbol{P}}}}_{virtual}}\!\left( t \right)={{{\dot{\boldsymbol{P}}}}_{old}}\!\left( t \right)+\dot{\lambda }\!\left( t \right){{\boldsymbol{P}}_{e}}\!\left( t \right)+\lambda \left( t \right){{{\dot{\boldsymbol{P}}}}_{e}}\!\left( t \right) \\[5pt] {{{\ddot{\boldsymbol{P}}}}_{virtual}}\!\left( t \right)={{{\ddot{\boldsymbol{P}}}}_{old}}\!\left( t \right)+\ddot{\lambda }\!\left( t \right){{\boldsymbol{P}}_{e}}\!\left( t \right)+2\dot{\lambda }\!\left( t \right){{{\dot{\boldsymbol{P}}}}_{e}}\!\left( t \right)+\lambda \left( t \right){{{\ddot{\boldsymbol{P}}}}_{e}}\!\left( t \right) \\[5pt] {{{\dddot{\boldsymbol{P}}}}_{virtual}}\!\left( t \right)= \left(\begin{array}{l} {{{\dddot{\boldsymbol{P}}}}_{old}}\!\left( t \right)+\dddot{\lambda }\!\left( t \right){{\boldsymbol{P}}_{e}}\!\left( t \right)+3\ddot{\lambda }\!\left( t \right){{{\dot{\boldsymbol{P}}}}_{e}}\!\left( t \right)\\[5pt]+2\dot{\lambda }\!\left( t \right){{{\ddot{\boldsymbol{P}}}}_{e}}\!\left( t \right)+\dot{\lambda }\!\left( t \right){{{\ddot{\boldsymbol{P}}}}_{e}}\!\left( t \right)+\lambda \left( t \right){{{\dddot{\boldsymbol{P}}}}_{e}}\!\left( t \right)\end{array}\right)\\\end{array} \right.\end{equation}
At the beginning time
${t_0}$
of the handover phase, we can rewrite Equation (61) as
\begin{equation}\left\{ \begin{array}{l} {{\boldsymbol{P}}_{virtual}}\!\left( t_{0} \right)={{\boldsymbol{P}}_{old}}\!\left( t_{0} \right)+\lambda \left( t_{0} \right){{\boldsymbol{P}}_{e}}\!\left( t_{0} \right) \\[5pt] {{{\dot{\boldsymbol{P}}}}_{virtual}}\!\left( t_{0} \right)={{{\dot{\boldsymbol{P}}}}_{old}}\!\left( t_{0} \right)+\dot{\lambda }\!\left( t_{0} \right){{\boldsymbol{P}}_{e}}\!\left( t_{0} \right)+\lambda \left( t_{0} \right){{{\dot{\boldsymbol{P}}}}_{e}}\!\left( t_{0} \right) \\[5pt] {{{\ddot{\boldsymbol{P}}}}_{virtual}}\!\left( t_{0} \right)={{{\ddot{\boldsymbol{P}}}}_{old}}\!\left( t_{0} \right)+\ddot{\lambda }\!\left( t_{0} \right){{\boldsymbol{P}}_{e}}\!\left( t_{0} \right)+2\dot{\lambda }\!\left( t_{0} \right){{{\dot{\boldsymbol{P}}}}_{e}}\!\left( t_{0} \right)+\lambda \left( t_{0} \right){{{\ddot{\boldsymbol{P}}}}_{e}}\!\left( t_{0} \right) \\[5pt] {{{\dddot{\boldsymbol{P}}}}_{virtual}}\!\left( t_{0} \right)= \left(\begin{array}{l} {{{\dddot{\boldsymbol{P}}}}_{old}}\!\left( t_{0} \right)+\dddot{\lambda }\!\left( t_{0} \right){{\boldsymbol{P}}_{e}}\!\left( t_{0} \right)+3\ddot{\lambda }\!\left( t_{0} \right){{{\dot{\boldsymbol{P}}}}_{e}}\!\left( t_{0} \right)\\[5pt]+2\dot{\lambda }\!\left( t_{0} \right){{{\ddot{\boldsymbol{P}}}}_{e}}\!\left( t_{0} \right)+\dot{\lambda }\!\left( t_{0} \right){{{\ddot{\boldsymbol{P}}}}_{e}}\!\left( t_{0} \right)+\lambda \left( t_{0} \right){{{\dddot{\boldsymbol{P}}}}_{e}}\!\left( t_{0} \right)\end{array}\right)\\\end{array} \right.\end{equation}
According to Equations (59) and (62), we can obtain
\begin{equation}\left\{ \begin{aligned}& \lambda \left( {{t}_{0}} \right)=0 \\[5pt] & \dot{\lambda }\!\left( {{t}_{0}} \right)=0 \\[5pt] & \ddot{\lambda }\!\left( {{t}_{0}} \right)=0 \\[5pt] & \dddot{\lambda }\!\left( {{t}_{0}} \right)=0 \\\end{aligned} \right.\end{equation}
At the end time
${t_0} + T$
of the handover phase, combining Equation (60), and referring to Equations (61) and (62), we have
\begin{equation}\left\{ \begin{aligned}& \lambda \left( {{t}_{0}+T} \right)=1 \\[5pt] & \dot{\lambda }\!\left( {{t}_{0}+T} \right)=0 \\[5pt] & \ddot{\lambda }\!\left( {{t}_{0}+T} \right)=0 \\[5pt] & \dddot{\lambda }\!\left( {{t}_{0}+T} \right)=0 \\\end{aligned} \right.\end{equation}
Suppose
${t_0} = 0$
, the time interval of the handover phase
$\left[ {{t_0},{t_0} + T} \right]$
is rewritten as
$\left[ {0,T} \right]$
, it is easy to get that
\begin{equation}\begin{aligned}\lambda \left( t \right)=&{{\eta }_{1}}\sin \frac{\pi }{T}t+{{\eta }_{2}}\cos \frac{\pi }{T}t+{{\eta }_{3}}\sin \frac{2\pi }{T}t+{{\eta }_{4}}\cos \frac{2\pi }{T}t \\[5pt] & +{{\eta }_{5}}\sin \frac{3\pi }{T}t+{{\eta }_{6}}\cos \frac{3\pi }{T}t+{{\eta }_{7}}\sin \frac{4\pi }{T}t+{{\eta }_{8}}\cos \frac{4\pi }{T}t \\\end{aligned}\end{equation}
where
${\eta _i}\!\left( {i = 1,2, \cdots, 8} \right)$
are the coefficients of the handover law.
Then, according to Equations (63), (64) and (65), we can obtain
\begin{equation}\left\{ \begin{aligned}& {{\eta }_{1}}={{\eta }_{3}}={{\eta }_{5}}={{\eta }_{7}}=0 \\[5pt] & {{\eta }_{2}}=-\frac{9}{16},{{\eta }_{4}}=\frac{2}{3},{{\eta }_{6}}=\frac{1}{16},{{\eta }_{8}}=-\frac{1}{6} \\\end{aligned} \right.\end{equation}
Therefore, the target handover law can be expressed as
\begin{align}\lambda \left( t \right)=-\frac{9}{16}\cos \frac{\pi }{T}t+\frac{2}{3}\cos \frac{2\pi }{T}t+\frac{1}{16}\cos \frac{3\pi }{T}t-\frac{1}{6}\cos \frac{4\pi }{T}t\end{align}
Then, substituting Equation (67) into Equation (54), we can obtain
\begin{equation}\begin{aligned}{{\boldsymbol{P}}_{virtual}}\!\left( t \right)={{\boldsymbol{P}}_{old}}\!\left( t \right)+\left( \begin{aligned}-\frac{9}{16}\cos \frac{\pi }{T}t+\frac{2}{3}\cos \frac{2\pi }{T}t+\frac{1}{16}\cos \frac{3\pi }{T}t-\frac{1}{6}\cos \frac{4\pi }{T}t \end{aligned}\right){{\boldsymbol{P}}_{e}}\!\left( t \right)\end{aligned}\end{equation}
According to Equation (68), the virtual target overlaps with the new target at time
$T$
, the target changing is achieved.
3.3 Design of whole cooperative midcourse guidance law
3.3.1 Analysis of guidance law in each phase
During combat operations, the constantly changing battlefield situation may necessitate changes in the attack targets. When a target switch occurs, the guidance commands for the interceptors may change abruptly, which needs to be avoided. To address this, we have designed a handover protocol for target information to ensure a smooth transition from the old target information to the new target information, thus preventing sudden changes in guidance commands. During the transition phase of target information, to reduce communication costs and improve accuracy, multiple interceptors will no longer engage in cooperative operations but will independently guide themselves based on the transitional target information. Once the target information update is complete, these interceptors will adjust the network communication topology according to the current battlefield situation, transitioning from single-interceptor guidance to cooperative multi-interceptor attacks on the new target. To avoid abrupt changes in the guidance law, we have also designed a smooth handover protocol from single-interceptor guidance to cooperative multi-interceptor guidance. This design ensures a smooth transition of the guidance law.
Therefore, the design of the guidance law in this paper is mainly divided into four phases, as shown in Fig. 1.
Whole course guidance law.
Phase 1
$\left[ {0,{t_1}} \right]$
: In this phase, multiple interceptors use the old communication network structure to attack the old target and use guidance law
${a_1}\!\left( t \right) = a\!\left( t \right)$
for guidance.
Phase 2
$\left[ {{t_1},{t_2}} \right]$
: This phase is the target changing phase, and there is no coordination between interceptors. In this phase, it is necessary to design the guidance law of the target handover phase. At the beginning time
${t_1}$
of the handover phase, the guidance law
${a_1}\!\left( {{t_1}} \right)$
is satisfied, and the guidance law
${a_2}\!\left( {{t_2}} \right)$
is satisfied at the end time
${t_2}$
.
Phase 3
$\left[ {{t_2},{t_3}} \right]$
: In this phase, it is necessary to design the guidance law of the handover phase. At the beginning time
${t_2}$
of the handover phase, the guidance law
${a_2}\!\left( {{t_2}} \right)$
is satisfied, and the guidance law
${a_3}\!\left( {{t_3}} \right)$
is satisfied at the end time
${t_3}$
.
Phase 4
$\left[ {{t_3},{t_f}} \right]$
: In this phase, multiple interceptors use the new communication network structure to attack the new target and use guidance law
${a_4}\!\left( t \right) = a\!\left( t \right)$
for guidance.
3.3.2 Design of guidance law in each phase
(a) Phases 1 and 4 guidance laws
The structures of guidance laws
${a_1}\!\left( t \right)$
and
${a_4}\!\left( t \right)$
used in phases 1 and 4 are shown in Equation (21). The communication network structures used in these two phases are different, and the consistency protocol in Equation (21) changes. Because of the target changing or mission change, multiple interceptors need to update the network topology.
(b) Phases 2 and 3 guidance laws
In phases 2 and 3, the guidance laws
${a_2}\!\left( t \right)$
and
${a_3}\!\left( t \right)$
of the handover phase need to be designed, and the designed guidance laws meet the requirements of smooth acceleration handover. The guidance laws are designed by Hermite interpolation. This method requires that the interpolation function and the interpolated function have equal function values at the node
${x_i}$
, equal derivative values, and equal higher-order derivatives.
The guidance law parameters for the handover phase are shown in Table 1.
Guidance law parameters in handover
Substituting the data in Table 1 into Equation (20), and by Lemma 4, the handover guidance law of phase 2 is designed as
\begin{align}{{a}_{2}}\!\left( t \right)=\left( 1+2\frac{t-{{t}_{1}}}{{{T}_{1}}} \right){{\left( \frac{t-{{t}_{2}}}{-{{T}_{1}}} \right)}^{2}}{{a}_{1}}\!\left( {{t}_{1}} \right)+\left( 1+2\frac{t-{{t}_{2}}}{{{T}_{1}}} \right){{\left( \frac{t-{{t}_{1}}}{{{T}_{1}}} \right)}^{2}}{{a}_{2}}\!\left( {{t}_{2}} \right)\end{align}
where,
${T_1} = {t_2} - {t_1}$
denotes the target handover time of phase 2;
${a_1}\!\left( {{t_1}} \right)$
is the size of the guidance law designed in Section (a) at time
${t_1}$
;
${a_2}\!\left( {{t_2}} \right)$
can be expressed as
\begin{align}{{a}_{2}}\!\left( {{t}_{2}} \right)={{a}_{1}}\!\left( {{t}_{1}} \right)+\Delta a\end{align}
where
${\rm{\Delta }}a = {a_0}\!\left( {{t_2}} \right) - {a_0}\!\left( {{t_1}} \right)$
is the acceleration variation of the trajectory shaping guidance law in phase 2.
Referring to phase 2, the handover guidance law of phase 3 is designed similarly as
\begin{align}{{a}_{3}}\!\left( t \right)=\left( 1+2\frac{t-{{t}_{2}}}{{{T}_{2}}} \right){{\left( \frac{t-{{t}_{3}}}{-{{T}_{2}}} \right)}^{2}}{{a}_{2}}\!\left( {{t}_{2}} \right)\left( 1+2\frac{t-{{t}_{3}}}{{{T}_{2}}} \right){{\left( \frac{t-{{t}_{2}}}{{{T}_{2}}} \right)}^{2}}{{a}_{3}}\!\left( {{t}_{3}} \right)\end{align}
where,
${T_2} = {t_3} - {t_2}$
is the target handover time of phase 3;
${a_3}\!\left( {{t_3}} \right)$
indicates the size of the guidance law designed in Section (a) at time
${t_3}$
for the interceptor in the new network topology.
4.0 Simulation and analysis
In order to verify the effectiveness and performance of the cooperative midcourse guidance law with target changing and topology switching in the case of target loss and communication topology switching of multiple interceptors intercepting targets, the numerical examples are provided in this section.
In the simulation, four interceptors are selected for cooperative interception verification. The parameters of four interceptors are shown in Table 2.
Simulation initial conditions for interceptors
In phase 1, four interceptors intercept the old target cooperatively. In phase 2, according to the mission requirements, the fourth interceptor intercepts the old target, and the remaining three interceptors intercept the new target. The parameters of cooperative guidance law are selected as
${\alpha _{xi}} = {\alpha _{yi}} = {\alpha _{zi}} = 0.01$
and
${\alpha _{xij}} = {\alpha _{yij}} = {\alpha _{zij}} = 1$
. At the end of the midcourse guidance phase, the elevation angle of LOS and the azimuth angle of LOS constraints of all interceptors are selected as
${q_\varepsilon } = {q_\beta } = {20^ \circ }$
. The midcourse and terminal handover distance is set 10km. The target handover time is selected as
$t = 10{\rm{s}}$
. According to Equation (52), the maximum allowable heading error is selected as
${\rm{\Delta }}{\gamma _{{\rm{max}}}} = {15^ \circ }$
, we have
$T \le 20.94{\rm{s}}$
. Therefore, the phase 2 time is selected as
${T_1} = 20{\rm{s}}$
and the phase 3 time is selected as
${T_2} = 10{\rm{s}}$
.
Before switching the target, the old network communication topology is selected as
\begin{equation*}\begin{aligned}\boldsymbol{A}=\left[ \begin{matrix} 0\;\;\;\;\; & 1\;\;\;\;\; & 1\;\;\;\;\; & 0 \\[5pt] 1\;\;\;\;\; & 0\;\;\;\;\; & 0\;\;\;\;\; & 1 \\[5pt] 1\;\;\;\;\; & 0\;\;\;\;\; & 0\;\;\;\;\; & 1 \\[5pt] 0\;\;\;\;\; & 1\;\;\;\;\; & 1\;\;\;\;\; & 0 \\\end{matrix} \right]\end{aligned}\end{equation*}
After switching the target, the new network communication topology is selected as
\begin{equation*}\begin{aligned}\boldsymbol{A}=\left[ \begin{matrix} 0\;\;\;\;\; & 1\;\;\;\;\; & 1 \\[5pt] 1\;\;\;\;\; & 0\;\;\;\;\; & 0 \\[5pt] 1\;\;\;\;\; & 0\;\;\;\;\; & 0 \\\end{matrix} \right]\end{aligned}\end{equation*}
4.1 Case 1: The relative distance between the old target and the new target is a constant value
The old target parameters are shown in Table 3.
Parameters of the old target
The new target parameters are shown in Table 4.
Parameters of the new target
The relative distance between the new target and the old target is shown in Fig. 2. It can be seen from Fig. 2 that the relative distance between the old target and the new target is always a constant value.
The relative distance between the new target and the old target.
The trajectories of multiple interceptors under the cooperative guidance law Equation (21) and the guidance law Equation (3) are shown in Fig. 3. It can be seen from Fig. 3 that in the initial stage of flight, the trajectories of the four interceptors all point to the old target. When the target changes, the trajectories of interceptors 1, 2 and 3 change significantly, all pointing to the new target. The interceptor 4 still attacks the old target, and it is obvious that its trajectory direction has not changed.
The trajectories of interceptors and targets.
The relative distances between interceptors and targets and the times-to-go of interceptors are shown in Figs. 4–5. It can be seen from Figs. 4–5 that in phase 2
${T_1}$
and phase 3
${T_2}$
, the relative distance and the times-to-go of interceptors 1, 2 and 3 with target change are obviously different from those of interceptor 4 without target change. When the target change is completed, the relative distances of the interceptors 1, 2, 3 can still reach the constraint range of the midcourse and terminal handover at the same time according to the designed cooperative guidance law, and the times-to-go can reach the same. However, it is evident that under the guidance law Equation (3), the relative distances and times-to-go of all interceptors cannot converge, making cooperation unattainable.
The relative distances between interceptors and targets.
The times-to-go of interceptors.
The accelerations of interceptors 1, 2, 3, 4 are shown in Figs. 6–9, respectively. It can be seen from Figs. 6–9 that the accelerations of the interceptors 1, 2, 3 change significantly, and the smooth handover can be achieved in phase 2
${T_1}$
and phase 3
${T_2}$
without abrupt change. The acceleration of interceptor 4 only achieves a smooth handover in phase 2
${T_1}$
. Because the guidance command of interceptor 4 is the handover from the cooperative guidance command of four interceptors designed at the beginning to the guidance command of a single interceptor. Subsequent flight interceptor 4 does not need to cooperate with the other three interceptors, so interceptor 4 does not have the handover time of phase 3
${T_2}$
.
The accelerations of interceptor 1.
The accelerations of interceptor 2.
The accelerations of interceptor 3.
The accelerations of interceptor 4.
It is evident that the acceleration designed according to Equation (3) is much smaller than the acceleration designed according to Equation (21). This is mainly because if the interceptors aim to achieve cooperation, the cooperation term
${\alpha _{xi}}\!\left( {{{\tilde P}_{xi}}\!\left( t \right) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over P} }_x}\!\left( t \right)} \right)$
must play a major role, thereby making the state quantities of all interceptors converge in each direction to achieve cooperation.
The elevation angles and the azimuth angles of the LOS and the velocities of the four interceptors are shown in Figs. 10–12, respectively. It can be seen from Figs. 10–12 that the elevation angles, azimuth angles of the LOS, and the velocities of the four interceptors can reach the constraint range for midcourse and terminal handover. When the target changes, the elevation angles and azimuth angles of the LOS and the velocities of interceptors 1, 2, and 3 do not change abruptly during phase 2
${T_1}$
and phase 3
${T_2}$
. The elevation angle, azimuth angle of the LOS, and velocity of interceptor 4 do not change abruptly in phase 2
${T_1}$
.
The elevation angles of LOS.
The azimuth angles of LOS.
The velocities of interceptors.
It can be seen that under the guidance law Equation (21), the speed of the interceptors increases significantly, reducing the relative distances and the differences in the times-to-go between the interceptors, thereby achieving cooperation. Under the guidance law Equation (3), there is no significant difference in the speed of the interceptors, and thus the relative distances and differences in the times-to-go between the interceptors cannot be reduced, resulting in a failure to achieve cooperation.
Obviously, when the interceptors change target, the proposed guidance law can achieve satisfactory performance. The simulation results show that the accelerations achieve a smooth handover without abrupt change in phases 2 and 3. The designed guidance law can ensure that each interceptor completes the midcourse guidance mission and meets the midcourse and terminal handover constraints.
4.2 Case 2: The relative distance between the old target and the new target is a time-varying value
The old target parameters are shown in Table 5.
Parameters of the old target
The new target parameters are shown in Table 6.
Parameters of the new target
The relative distance between the new target and the old target is shown in Fig. 13. It can be seen from Fig. 13 that the relative distance between the old target and the new target varies linearly.
The relative distance between the new target and the old target.
The trajectories of multiple interceptors under the cooperative guidance law Equation (21) and the guidance law Equation (3) are shown in Fig. 14. It can be seen from Fig. 14 that in the initial stage of flight, the trajectories of the four interceptors all point to the old target. When the target changes, the trajectories of interceptors 1, 2 and 3 change significantly, all pointing to the new target. The interceptor 4 still attacks the old target, and it is obvious that its trajectory direction has not changed.
The trajectories of interceptors and targets.
The relative distances between interceptors and targets and the times-to-go of interceptors are shown in Figs. 15–16. It can be seen from Figs. 15–16 that in phase 2
${T_1}$
and phase 3
${T_2}$
, the relative distance and the times-to-go of interceptors 1, 2 and 3 with target change are obviously different from those of interceptor 4 without target change. When the target change is completed, the relative distances of the interceptors 1,2,3 can still reach the constraint range of the midcourse and terminal handover at the same time according to the designed cooperative guidance law, and the times-to-go can reach the same. However, it is evident that under the guidance law Equation (3), the relative distances and times-to-go of all interceptors cannot converge, making cooperation unattainable.
The relative distances between interceptors and targets.
The times-to-go of interceptors.
The accelerations of interceptors 1, 2, 3, 4 are shown in Figs. 17–20, respectively. It can be seen from Figs. 17–20 that the accelerations of the interceptors 1, 2, 3 change significantly, and the smooth handover can be achieved in phase 2
${T_1}$
and phase 3
${T_2}$
without abrupt change. The acceleration of interceptor 4 only achieves a smooth handover in phase 2
${T_1}$
. Because the guidance command of interceptor 4 is the handover from the cooperative guidance command of four interceptors designed at the beginning to the guidance command of a single interceptor. Subsequent flight interceptor 4 does not need to cooperate with the other three interceptors, so interceptor 4 does not have the handover time of phase 3
${T_2}$
. It is evident that the acceleration designed according to Equation (3) is much smaller than the acceleration designed according to Equation (21). This is mainly because if the interceptors aim to achieve cooperation, the cooperation term
${\alpha _{xi}}\!\left( {{{\tilde P}_{xi}}\!\left( t \right) - {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over P} }_x}\!\left( t \right)} \right)$
must play a major role, thereby making the state quantities of all interceptors converge in each direction to achieve cooperation.
The accelerations of interceptor 1.
The accelerations of interceptor 2.
The accelerations of interceptor 3.
The accelerations of interceptor 4.
The elevation angles and the azimuth angles of the LOS and the velocities of the four interceptors are shown in Figs. 21–23, respectively. It can be seen from Figs. 21–23 that the elevation angles, azimuth angles of the LOS, and the velocities of the four interceptors can reach the constraint range for midcourse and terminal handover. When the target changes, the elevation angles and azimuth angles of the LOS and the velocities of interceptors 1, 2, and 3 do not change abruptly during phase 2
${T_1}$
and phase 3
${T_2}$
. The elevation angle, azimuth angle of the LOS, and velocity of interceptor 4 do not change abruptly in phase 2
${T_1}$
.
The elevation angles of LOS.
The azimuth angles of LOS.
The velocities of interceptors.
It can be seen that under the guidance law Equation (21), the speed of the interceptors increases significantly, reducing the relative distances and the differences in the times-to-go between the interceptors, thereby achieving cooperation. Under the guidance law Equation (3), there is no significant difference in the speed of the interceptors, and thus the relative distances and differences in the times-to-go between the interceptors cannot be reduced, resulting in a failure to achieve cooperation.
The simulation results show that in case 2, the cooperative guidance law method can still ensure smooth acceleration handover for each interceptor, successfully complete the midcourse guidance mission, and meet the midcourse and terminal handover constraints.
The above simulation results demonstrate that, under the guidance law Equation (21) designed in this paper, multiple interceptors can achieve cooperative flight to the midcourse and terminal handover area. Furthermore, a comparison with the simulation results of interceptors under guidance law Equation (3) verifies the superiority of the guidance method designed in this paper.
5.0 Conclusion
In this paper, the cooperative midcourse guidance law method with target changing and topology switching has been proposed for distributed interceptor clusters. The average position consistency protocol of virtual interception points has been designed based on cooperative trajectory shaping guidance law. Then, the convergence of the cooperative states in the guidance system has been proved by Lyapunov stability theory, and the cooperative interception mission under the condition of communication topology switching has been completed. Furthermore, based on the acceleration smoothing theory, the target handover law and the handover guidance law have been designed to realise the smooth handover between each phase. The numerical simulation results show that when the interceptor changes the target, the proposed guidance law method can ensure the smooth handover of acceleration and meet the constraints of midcourse and terminal handover, and further show the effectiveness and superiority of the proposed method.
For future research, a more in-depth analysis will be conducted regarding the differences between coordinated (organised but non-reactive) and cooperative (organised and reactive to environmental changes) interceptor formations.
Data availability statement
The data used to support the findings of this study are included within the article.
Competing interests
The authors declared that they have no conflicts of interest to this work.
Acknowledgements
The authors would like to thank the associate editor and reviewers for their valuable comments and constructive suggestions that helped to improve the paper and presentation significantly.
