Prescribed-Time Cooperative Guidance Law for Multi-UAV with Intermittent Communication

Abstract

Networked cooperation of multi-unmanned-aerial-vehicle (UAV) is significant, but their cooperative guidance presents challenges due to weak cross-domain communication. Given the difficulties in information transmission security, this paper proposes a prescribed-time cooperative guidance law (PTCGL) for networked multiple UAVs with intermittent communication. Supported by the directed internal communication, the proposed PTCGL can ensure the simultaneous arrival in the case of the pinning external communication is time-triggered intermittent. In the first stage, PTCGL is designed to combine with a time-scaling function and second-order intermittent pinning consensus. With the desired convergence performance, the convergence time can be pre-specified independent of initial conditions and parameter tuning in theory. In the second stage, begin with the suitable initial conditions at the prescribed convergence time, the multiple UAVs are organized by proportional navigation guidance, so as to ensure the cooperation regardless of weak communication. Finally, simulations are conducted to verify the effectiveness and robustness of the proposed cooperative guidance law.

1. Introduction

Relying on information sharing and resource integration, networked cooperation organizes multi-unmanned-aerial-vehicle (UAV) leading to prominent effectiveness, which has become increasingly prevalent and plays an important role in aviation and aerospace [1,2,3]. However, due to the unideal communication environment, multi-UAV is always subject to security threats, especially in the process of cross-domain cooperation [4]. Therefore, given the mission requirements and communication constraints, there is a need to design a reliable and robust multi-UAV cooperative guidance law that can effectively support the cooperation.

The concept of cooperative guidance was first proposed by Jeon, I.S. et al. [5,6], and since then, consensus-based networked cooperative guidance has been widely studied over the last decade [7,8,9,10]. Ref. [7] introduced a two-stage consensus-based cooperative guidance law that consists of undirected consensus protocol and switching scheme. Ref. [8] developed a group-consensus-based cooperative guidance law, which can undertake complex multi-task. Ref. [9] organized the distributed cooperative guidance with a two-direction scheme, wherein the impact time asymptotically converges in LOS direction. For cooperative guidance problems, simultaneous arrival within the short guidance window is the priority. Indeed, insufficient convergence performance of asymptotic consensus may lead to efficiency degradation, or even a failure to reach cooperative saturation. In this regard, fast convergence property and robustness of guidance law is crucial to multi-UAV cooperation.

Obviously, the regular asymptotic stability is difficult to ensure the convergence ahead of arrival time. By combining the finite-time theory, the convergence rate of the cooperative guidance law is improved [11,12,13,14,15,16,17,18,19,20]. Based on an integrated leader-follower framework, Ref. [11] developed a finite-time cooperative guidance law. Ref. [14] proposed a finite-time super-twisting strategy for leader-follower cooperative salvo guidance. Considering the singularity of consensus-based design, Ref. [18] desgined a finite-time decentralized two-stage cooperative guidance law in 2-dimensional and 3-dimensional engagement, respectively. In Ref [20], Liu. S et al. investigated a finite-time cooperative active defence guidance law. Whereas, the bound estimation of finite convergence time always depends on the initial states and tuning parameters. Once the initial conditions are unavailable or the guidance parameters are unsuitable, the finite-time convergence is hard to ensure the sufficient convergence before arrival.

Fortunately, superior to the finite-time theory, fixed-time consensus can predetermined the boundedness of convergence time regardless of initial condition [21,22,23,24,25,26,27]. Li et al. [21,22] proposed fixed-time cooperative guidance law along the LOS direction, so as to reach consensus within a fixed time before interception. By incorporating fixed-time consensus protocol, Ref. [23] proposed an extended biased proportional navigation cooperative guidance law. Based on the cooperative guidance law designed in Ref. [27], the consensus of multi-UAV is guaranteed by the virtue of the fixed-time convergent error dynamic. Although the fixed-time convergence allows the per-assigned time priori to the arrival, demands on tuning parameters may degrade the guidance performance and lead to capability challenges. Therefore, it is necessary to enhance performance advantages in parameter independence when studying the fixed-time convergent guidance methods.

Accordingly, focusing on the convergence performance of consensus-based cooperative guidance, relative researches are continuously facilitating for prescribed-time convergence performance. Ref. [28] proposed a two-stage prescribed-time optimal cooperative guidance law based on Pontryagin’s principle. Incorporated with a motion-planning design concept, the time-varying appropriate time sequence was designed to assign the convergence time. In Ref. [29], Sinha et al. proposed a predefined-time consensus-based cooperative guidance law for simultaneously caption. Ref. [30] organized a cooperative guidance with the second-order prescribed-time pinning consensus, which can provide a suitable condition for the two-stage scheme. To overcome the implicit time constraints, Ref. [31] investigated a novel prescribed-time integral sliding mode guidance law for salvo. Theoretically, the continuously developed convergence performance shows its superiority in independence of initial conditions and tuning parameters. Benefit from the parameter independence, impact-time synchronization within a pre-defined time can be guaranteed.

To be specific, the aforementioned researches are engaged on a safe communication environment, wherein the reliable, accurate and real-time network provides desired information supporting for the cooperative guidance. In the view of difficulties in practical engineering, however, cooperation faces the challenges of unideal communication environment, considering limited communication capability, external disturbance and active cyber-attack. Weak communication, which refers to the intermittent or time delay in the process of networked cooperation, conducts a heavy burden on the network connection, and the cooperative guidance law would become fragile. Previous studies have contributed to the robustness against typical network connection problems. In Ref. [32], the multi-UAV cooperation was ensured by asymptotic convergence performance, wherein the intermittent weak communication were dealt as a system disturbance without analysing the communication mechanism. Refs. [33,34] considered time-delay problem. Considering the weak communication connectivity, Ref. [35] promoted a finite-time cooperative guidance with topology switching. Although the above metioned researches can guarantee the robustness in weak communication, but the insufficient convergence performance may lead to the performance degrades or failure occurs. It is worth mentioning that how to keep a desired convergence performance with weak communication is a challenge topic.

Motivated by the limitations of convergence performance with intermittent communication, an intermittent pinning prescribed-time cooperative guidance law (PTCGL) is proposed. Firstly, a networked cooperative guidance law is proposed for multi-UAV cooperation. Theoretically, compared with existing finite/fixed cooperative guidance laws, the convergence time can be pre-assigned regardless of initial conditions and parameter tuning. Then, the three-dimensional two-stage PTCGL is further developed based on characteristic of the proposed intermittent pinning prescribed-time consensus-based cooperative guidance law. Communicated with intermittent pinning-directed network, the sufficient conditions of prescribed-time consensus are simplified.

The rest of this paper is organized as follows. Section 1 introduces the cooperative guidance problem with the unideal intermittent communication conditions; Section 2 designs the cooperative guidance law based on the prescribed-time pinning group consensus protocol; Section 3 further analyses the application scheme for two-stage cooperative guidance law; Section 4 performs numerical simulations to validate the effectiveness, robustness and superiority of the proposed PTCGL; Section 5 concludes this paper.

2. Problem Formulation

2.1. Cooperative Guidance Engagement

Consider the three-dimensional (3-D) engagements with n UAVs flying to the target simultaneously, as shown in the Figure 1.

Neglecting the lag introduced by autopilot dynamics, the mass-point motion model between $M_i$ and its target can be expressed as follows [36]:

$$\left\{\begin{array}{cc}{\dot{R}}_i\hfill & =-Vcos{\theta }_{i}cos{\varphi }_i\hfill \\ {\dot{\theta }}_{Li}\hfill & =-{\displaystyle \frac{V}{R_i}}sin{\theta }_i\hfill \\ {\dot{\varphi }}_{Li}\hfill & =-{\displaystyle \frac{V}{R_{i}cos{\theta }_{Li}}}cos{\theta }_{i}sin{\varphi }_i\hfill \\ {\dot{\theta }}_i\hfill & ={\displaystyle \frac{a_{zi}}{V}}-{\dot{\varphi }}_{Li}sin{\theta }_{Li}sin{\varphi }_i-{\dot{\theta }}_{Li}cos{\varphi }_i\hfill \\ {\dot{\varphi }}_i\hfill & ={\displaystyle \frac{a_{yi}}{Vcos{\theta }_i}}+{\dot{\varphi }}_{Li}sin{\theta }_{Li}tan{\theta }_{i}cos{\varphi }_i-{\dot{\theta }}_{Li}tan{\theta }_{i}sin{\varphi }_i-{\dot{\varphi }}_{Li}cos{\theta }_{Li}\hfill \\ {\sigma }_i\hfill & =arccos\left(cos{\theta }_{i}cos{\varphi }_i\right),{\sigma }_i\in [0,\pi )\hfill \end{array}\right.$$

(1)

where $R_i$ is the relative distance between the UAV $M_i$ and its target; the speed V are assumed to be invariant; $a_{yi}$ and $a_{zi}$ represnets adjustable perpendicular acceleration components in the pitch and yaw channels, respectively; ${\theta }_{Mi}$, ${\varphi }_{Mi}$ are the Euler angles from the line-of-sight (LOS) coordinate system to the body coordinate system; ${\theta }_{Li}$ and ${\varphi }_{Li}$ are LOS angles in azimuth and elevation directions, respectively; heading error ${\sigma }_i$ represents the angle between the velocity vector and the LOS.

2.2. Communication Network Topology

The communication network topology among the multi-UAV can be described by a pinning-group graph $\overline{\mathcal{G}}=({\mathcal{G}}_{\mathit{d}},\mathcal{G})$, where ${\mathcal{G}}_{\mathit{d}}$ and $\mathcal{G}$ represent the external pinning communication and internal group communication, respectively.

$\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$ is constructed to described the communication relationship among n-UAV, which contains the UAV set $\mathcal{V}=\{m_1,m_2,\cdots ,m_n\}$, edge set $\mathcal{E}=\left\{\right(i,j)\in \mathcal{V}\times \mathcal{V}\}$ and weighted adjacency matrix $\mathcal{A}=\left\{a_{ij}\right\}\in {\mathbb{R}}^{n\times n}$. If there exist a directed communication link from $M_i$ to $M_j$, $a_{ij}=1$; otherwise, $a_{ij}=0.$ The graph Laplacian matrix $\mathcal{L}=\left[l_{ij}\right]\in {\mathbb{R}}^{n\times n}$ is denoted as $l_{ii}={\sum }_{j=1,j\ne i}^n{a}_{ij},l_{ij}=-a_{ij}$, for $i\ne j$. The internal communication network is satisfied that (i) $\mathcal{G}$ is strongly connected, wherein for any pair of UAVs, there exist a directed path; (ii) n-UAV are divided into g-group $(g\ge 2)$ and the sub-graph is denoted as ${\mathcal{G}}_{\mathit{k}}=({\mathcal{V}}_{\mathit{k}},{\mathcal{E}}_{\mathit{k}},{\mathcal{A}}_{\mathit{k}})(k=1,2,\cdots ,g)$, where for $M_i\in {\mathcal{G}}_k\subseteq \mathcal{G}$, its group number is $g_i=k^{*}$ and flying to the group target $T_k^{*}$ [37].

${\mathcal{G}}_{\mathit{d}}=({\mathcal{V}}_{\mathit{d}},{\mathcal{E}}_{\mathit{d}},\mathcal{D})$ represents the external pinning communication, wherein ${\mathcal{V}}_{\mathit{d}}$ is pinned-UAV set, ${\mathcal{E}}_{\mathit{d}}$ is the pinning edge set and pinning matrix $\mathcal{D}=diag\left[d_i\right]\in {\mathbb{R}}^{n\times n}$. If $m_i$ is selected to be pinned, $d_i=1$; otherwise, $d_i=0$.

In the process of cooperative guidance, weak communication is inevitable. Different from the internal directed communication in a relative close range, the external pinning information is easier to be targeted and interrupted, due to the long-range transmission, limited network capability, engagement interference. On the other hand, active intermittent pinning is also an effective scheme to save resources and avoid cyber-attack. Therefore, the intermittent pinning communication will result in effectiveness degradation or even failed of cooperation. The cooperative guidance with intermittent communication can be described as periodic and aperiodic, as shown in the Figure 2. The flight are divided into $n_p+1$ disjoint intervals by a time-triggered pinning sequence $\mathit{\xi }\left(t\right)=\{t_1,t_2,\cdots ,t_p,\cdots ,t_{n_p}\},p\in {\mathbb{N}}^{+}$, the p-th intermittent interval is defined as ${\tau }_p=t_p-t_{p-1}$, where $t_0$ is the initial time.

Assumption 1.

For the multi-UAV system communicated with a pinning-group graph $\overline{\mathcal{G}}$, the internal group network $\mathcal{G}$ is strongly connected and there is at least one UAV in a subgroup is pinned if the intermittent external pinning network ${\mathcal{G}}_d$ is triggered by time sequence $\mathit{\xi }\left(t\right)$.

Lemma 1

([38]). For a strongly connected network $\mathcal{G}$, there exist a diagonal matrix $\mathbf{\Lambda }=diag({\lambda }_1,$ ${\lambda }_2,\cdots ,{\lambda }_n)$, such that ${\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)$ and ${\mathsf{{\rm Y}}}_{max}\left(\mathcal{L}\right)$ be real numbers,

$$\left\{\begin{array}{c}{\mathsf{{\rm Y}}}_{max}\left(\mathcal{L}\right)=\underset{{\mathit{p}}^T\mathit{\lambda }=0,\mathit{p}\ne 0}{max}{\displaystyle \frac{{\mathit{p}}^T\mathbf{\Gamma }\mathit{p}}{{\mathit{p}}^T\mathbf{\Lambda }\mathit{p}}}\hfill \\ {\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)=\underset{{\mathit{p}}^T\mathit{\lambda }=0,\mathit{p}\ne 0}{min}{\displaystyle \frac{{\mathit{p}}^T\mathbf{\Gamma }\mathit{p}}{{\mathit{p}}^T\mathbf{\Lambda }\mathit{p}}}\hfill \end{array}\right.$$

(2)

where $\mathbf{\Gamma }={\displaystyle \frac{\mathbf{\Lambda }\mathcal{L}+{\mathcal{L}}^T\mathbf{\Lambda }}{2}}$, $\mathit{\lambda }={[{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n]}^T>0$, ${\mathit{\lambda }}^T\mathcal{L}=0$ and ${\sum }_{i=1}^n{\lambda }_i=1$.

Note 1.

For a strongly connected graph, each UAV can be viewed as a root of a directed spanning tree. Reffering to [37], if there is at least one UAV in a subgroup, the root is pinned, thus, $\mathcal{L}+\gamma \mathcal{D}(\gamma >0)$ is a M-matrix.

2.3. Guidance Objective

The design objective of cooperative guidance is to achieve the finite-time convergence, so as to ensure the simultaneously arrival, which satisfies

$$\left\{\begin{array}{c}\underset{t\to t_f^i}{lim}{R}_i\to 0\hfill \\ \underset{t\to t_f^i}{lim}{\sigma }_i\to {\sigma }_d\hfill \\ \underset{t\to t_f^i}{lim}{\dot{\sigma }}_i\to 0\hfill \\ t_f^i\to t_f^j,i,j=1,2,\cdots ,n\hfill \end{array}\right.$$

(3)

where $t_f^i$ is the arrival time of $m_i$ and ${\sigma }_d$ is the desired heading angle.

Then, for the pinning-group cooperation supported by $\overline{\mathcal{G}}$, the guidance objective can be rewritten as

$$\left\{\begin{array}{c}\underset{t\to t_f^i}{lim}{R}_i\to 0,m_i\in \mathcal{G}\hfill \\ \underset{t\to t_f^i}{lim}{\sigma }_i\to {\sigma }_d^{g_i},m_i\in {\mathcal{G}}_{\mathit{k}}\hfill \\ \underset{t\to t_f^i}{lim}{\dot{\sigma }}_i\to 0,m_i\in \mathcal{G}\hfill \\ t_f^i\to t_f^j,m_i,m_j\in {\mathcal{G}}_{\mathit{k}}\hfill \end{array}\right.$$

(4)

Thus, select the second-order state variables as $x_1^i=R_i/V$ and $x_2^i=-cos{\sigma }_i$, $i=1,2,\cdots ,n$, so that the cooperative guidance model can be transformed into

$$\left\{\begin{array}{cc}{\dot{x}}_1^i\hfill & =-{\dot{R}}_i/V=-cos{\sigma }_i=x_2^i\hfill \\ {\dot{x}}_2^i\hfill & ={\dot{\sigma }}_{i}sin{\sigma }_i={\displaystyle \frac{a_{zi}}{V}}sin{\theta }_{i}cos{\varphi }_i+{\displaystyle \frac{a_{yi}}{V}}sin{\varphi }_i+{\displaystyle \frac{V}{R_i}}{cos}^2{\theta }_i{sin}^2{\varphi }_i+{\displaystyle \frac{V}{R_i}}{sin}^2{\theta }_i\hfill \end{array}\right.$$

(5)

Hence, the cooperative guidance can be transformed into a second-order multi-agent consensus problem. Notably, the convergence should be ensured within the finite-time guidance interval, that is

Definition 1.

For multi-UAV communicated on pinning group network $\overline{\mathcal{G}}$, wherein the external pinning communication is triggered by intermittent sequence $\mathit{\xi }\left(t\right)$, the prescribed-time cooperative guidance is achieved if the solution of Equation (5) with any initial conditions

$$\left\{\begin{array}{c}\underset{t\to t_0+T}{lim}\parallel x_i\left(t\right)-x_j\left(t\right)\parallel =0,m_i,m_j\in {\mathcal{G}}_{\mathit{k}}\hfill \\ \underset{t\to t_0+T}{lim}\parallel x_i\left(t\right)-x_{g_i}\left(t\right)\parallel =0,m_i\in \mathcal{G}\hfill \end{array}\right.$$

(6)

and for $\forall t\ge t_0+T$

$$\left\{\begin{array}{c}\parallel x_i\left(t\right)-x_j\left(t\right)\parallel =0,m_i,m_j\in {\mathcal{G}}_{\mathit{k}}\hfill \\ \parallel x_i\left(t\right)-x_{g_i}\left(t\right)\parallel =0,m_i\in \mathcal{G}\hfill \end{array}\right.$$

(7)

where $x_i=[x_1^i,x_2^i]$, $x_{g_i}=[x_1^{g_i},x_2^{g_i}]$, and $T\le min(t_f^i)$ is the mission-specified finite-time convergence time.

2.4. Useful Lemmas

Lemma 2

([38]). For a nonsingular matrix $\mathcal{M}=\left\{m_{ij}\right\}\in {\mathbb{R}}^{n\times n}$, the following statements are equivalent: (i) $\mathcal{M}$ is a M-matrix; (ii) all eigenvalues of $\mathcal{M}$ have positive real part, which means $Re\left({\lambda }_i\left(\mathcal{M}\right)\right)>0,i=$$1,2,\cdots ,n$; (iii) there existing a positive diagonal matrix $\mathbf{\Lambda }=diag\left\{1/{\lambda }_1,\right.$$\left.1/{\lambda }_2,\cdots ,1/{\lambda }_n\right\}>0$, in which $\mathit{\lambda }={\left[{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_N\right]}^T={\mathcal{M}}^{-1}{\mathbf{1}}_n$, such that $\mathbf{\Lambda }\mathcal{M}+{\mathcal{M}}^T\mathbf{\Lambda }$ is positive definite.

Lemma 3

([39]). For arbitrary vector $p$ and $q$, there exist a positive constant ℓ, such that the following inequality holds,

$$\parallel p\parallel \parallel q\parallel \le {\mathit{\ell }\parallel p\parallel }^2+{\displaystyle \frac{1}{4\mathit{\ell }}}{\parallel q\parallel }^2$$

(8)

Lemma 4

([38]). For positive definite real symmetric matrix ${\mathcal{Q}}_1,{\mathcal{Q}}_2\in {\mathbb{R}}^{N\times N}$, the following inequality holds

$${\mu }_{min}\left({\mathcal{Q}}_{\mathbf{1}}^{-1}{\mathcal{Q}}_{\mathbf{2}}\right){\mathit{p}}^T{\mathcal{Q}}_{\mathbf{1}}\mathit{p}\le {\mathit{p}}^T{\mathcal{Q}}_{\mathbf{2}}\mathit{p}\le {\mu }_{max}\left({\mathcal{Q}}_{\mathbf{1}}^{-1}{\mathcal{Q}}_{\mathbf{2}}\right){\mathit{p}}^T{\mathcal{Q}}_{\mathbf{1}}\mathit{p}$$

(9)

Lemma 5

([40]). For a state vector $p$ with arbitrary initial condition $p\left(0\right)=p_0$, the dynamic system $\dot{p}=F(p,t),F(\mathbf{0},t)=0,t\in {\mathbb{R}}^{+}$, wherein $F(p,t)$ is locally bounded uniformly in time, is said to be prescribed-time stable, if there exist a Lyapunov function $V(t,p(t\left)\right)$ with $V(t,\mathbf{0})=0$ satisfying

$$\dot{V}(t,p\left(t\right))\le -bV(t,p\left(t\right))-\eta \varrho \left(t\right)V(t,p\left(t\right))$$

(10)

where $b\ge 0$, $\eta \in {\mathbb{R}}^{+}$, $\varrho \left(t\right)=\dot{\rho }\left(t\right)/\rho \left(t\right)$, and $V(t,p(t\left)\right)$ yields

$$\left\{\begin{array}{c}V(t,p\left(t\right))\le \rho {\left(t\right)}^{-\eta }{e}^{-b(t-t_0)}V\left(t_0\right),t\in [t_0,T^{*})\hfill \\ V(t,p\left(t\right))\equiv 0,t\in \left[T^{*},\infty \right)\hfill \end{array}\right.$$

(11)

where $T^{*}=t_0+T$ and the time-scaling function is

$$\rho \left(t\right)=\left\{\begin{array}{c}{\left({\displaystyle \frac{T}{T^{*}-t}}\right)}^h,t\in \left[t_0,T^{*}\right)\hfill \\ 1,t\in \left[T^{*},+\infty \right)\hfill \end{array}\right.$$

(12)

in which $h>1$, and $T>0$ is a mission-specified constant.

Note 2.

The origin of system is globally prescribed-time stable if the settling time T can be prescribed such that $p\left(t\right)\equiv \mathbf{0}$ for $t\ge T$.

3. Design of Cooperative Guidance Command

In the weak communication environment, the multi-UAV cooperation is supported by the intermittent pinning-group communication $\overline{\mathcal{G}}$. Combing the characteristic of prescribed-time convergence, the prescribed-time cooperative guidance command is proposed as follows,

$$\begin{array}{c}{a}_{yi}=\left\{\begin{array}{c}-{\displaystyle \frac{V^2}{R_i}}sin{\varphi }_i+{\displaystyle \frac{u_i(\mathit{\xi }\left(t\right),t)V}{2sin{\varphi }_i}},t\in \left[t_0,T^{*}\right)\hfill \\ KV{\dot{\theta }}_{Li}sin{\theta }_{i}sin{\varphi }_i+KV{\dot{\varphi }}_{Li}cos{\theta }_{Li}cos{\theta }_i,t\in \left[T^{*},+\infty \right)\hfill \end{array}\right.\hfill \\ a_{zi}=\left\{\begin{array}{c}-{\displaystyle \frac{V^2}{R_i}}sin{\theta }_{i}cos{\varphi }_i+{\displaystyle \frac{u_i(\mathit{\xi }\left(t\right),t)V}{2sin{\theta }_{i}cos{\varphi }_i}},t\in \left[t_0,T^{*}\right)\hfill \\ KV{\dot{\theta }}_{Li}cos{\varphi }_i,t\in \left[T^{*},+\infty \right)\hfill \end{array}\right.\hfill \end{array}$$

(13)

based on the intermittent pinning-group consensus protocol

$$\begin{array}{c}{u}_i(\mathit{\xi }\left(t\right),t)=p_i(\mathit{\xi }\left(t\right),t)+\alpha {\varrho }^2\left(t\right)\sum _{j=1}^n{l}_{ij}{x}_1^{g_j}+\beta \varrho \left(t\right)\sum _{j=1}^n{l}_{ij}{x}_2^{g_j}\hfill \\ +\alpha {\varrho }^2\left(t\right)\sum _{j=1,j\ne i}^n{a}_{ij}\left(x_1^j-x_1^i\right)+\beta \varrho \left(t\right)\sum _{j=1,j\ne i}^n{a}_{ij}\left(x_2^j-x_2^i\right)\hfill \end{array}$$

(14)

with intermittent pinning communication

$$p_i(\mathit{\xi }\left(t\right),t)=\left\{\begin{array}{c}-\gamma \varrho \left(t\right)\left[\alpha \varrho \left(t\right)\left(x_1^i\left(t\right)-x_1^{g_i}\right)+\beta \left(x_2^i\left(t\right)-x_2^{g_i}\right)\right],t\in {\mathit{\xi }}_c\hfill \\ 0,t\in {\mathit{\xi }}_b\hfill \end{array}\right.$$

(15)

where ${\mathit{\xi }}_c$ and ${\mathit{\xi }}_b$ respectively represent the time-interval set with and without pinning communication, which satisfy ${\mathit{\xi }}_b\bigcup {\mathit{\xi }}_c=[t_0,t_0+T)$ and $n_c+n_b=n_p+1$. Additionally, denote the last time-interval $[t_{n_p},T^{*})$ as $[t_{n_p},t_{n_p+1})$, with $T^{*}=t_0+T$.

3.1. First Stage: Prescribed-Time Stability

Theorem 1.

For multi-UAV system Equation (5) communicated with $\overline{\mathcal{G}}$, where the communication topology is directed strongly connected and when the pinning communication is triggered by intermittent sequence $\mathit{\xi }\left(t\right)$, there exist guidance parameters $h>1$, $\alpha ,\beta ,\gamma >0$ satisfying Equation (16), such that the pinning group matrix $\mathcal{L}+\gamma \mathcal{D}$ is a M-matrix, so that the proposed cooperative guidance law can ensure the prescribed-time convergence within $t_0+T$ in the first cooperative stage.

$$\left\{\begin{array}{c}{\left({\mathit{\ell }}_1^p\right)}^2-{\mathit{\ell }}_2^p(\alpha {\mathit{\ell }}_1^p+\beta {\mathit{\ell }}_2^p){\mathit{\Theta }}_0<0\hfill \\ -2\left[\alpha {\mathit{\ell }}_1^{p}h-{\mathit{\Theta }}_1(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p)\right]{\mathit{\Theta }}_0+{\mathit{\Theta }}_2<0\hfill \\ -4\kappa \beta {\mathit{\ell }}_2^{p}h{\mathit{\Theta }}_0+{\mathit{\Theta }}_3<0\hfill \\ ln{\overline{\mathcal{Y}}}^p-b{\tau }_{min}<0\hfill \\ (p+1)\eta h-2>0\hfill \end{array}\right.$$

(16)

where

$$\left\{\begin{array}{c}{\mathit{\Theta }}_0=min\left[{\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right),{\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}\right]\hfill \\ {\mathit{\Theta }}_1=\left(2+\eta h+b{T}^{*}\right)\hfill \\ {\mathit{\Theta }}_2=2{\mathit{\ell }}_1^p\kappa \left(1+\eta h+b{T}^{*}\right)\hfill \\ {\mathit{\Theta }}_3={\mathit{\ell }}_1^p(4\kappa h+1)+(2\kappa {\mathit{\ell }}_2^p+{\mathit{\ell }}_1^p)(\eta h+b{T}^{*})\hfill \end{array}\right.$$

(17)

with parameters ${\mathit{\ell }}_1^p\ge 0$, ${\mathit{\ell }}_2^p>0$, $b>0$, $\eta >0$, $\kappa >0$, ${\overline{\mathcal{Y}}}^p={max}_{i=0}^p\left[{\displaystyle \frac{{\mu }_{max}\left({\Omega }_{i+1}\right)}{{\mu }_{min}\left({\Omega }_i\right)}}{\mu }_{max}\left({\mathcal{Y}}_i^T{\mathcal{Y}}_i\right)\right]$, $p\in [1,n_p]$.

Proof. 

First, the system transformation is provided as preparation. Let $e_1^i=x_1^i-x_1^{g_i}$, $e_2^i=x_2^i-x_2^{g_i}$, the consensus protocol can be transformed into

$$u_i(\mathit{\xi }\left(t\right),t)=\left\{\begin{array}{c}-\alpha {\varrho }^2\left(t\right)\sum _{j=1}^n{l}_{ij}{e}_1^j-\beta \varrho \left(t\right)\sum _{j=1}^n{l}_{ij}{e}_2^j,t\in {\mathit{\xi }}_b\hfill \\ -\alpha {\varrho }^2\left(t\right)(\sum _{j=1}^n{l}_{ij}{e}_1^j+\gamma d_i{e}_1^i)-\beta \varrho \left(t\right)(\sum _{j=1}^n{l}_{ij}{e}_2^j+\gamma d_i{e}_2^i),t\in {\mathit{\xi }}_c\hfill \end{array}\right.$$

(18)

Let $\mathit{U}=\left[u_i(\mathit{\xi }\left(t\right),t)\right]\in {\mathbb{R}}^{n\times 1}$, $\mathcal{E}=[{\mathcal{E}}_1^T,{\mathcal{E}}_2^T]$, where ${\mathcal{E}}_1=\varrho \left(t\right){\mathit{E}}_1$, ${\mathcal{E}}_2={\mathit{E}}_2$, ${\mathit{E}}_1={[e_1^1,e_1^2,\cdots ,e_1^n]}^T$, ${\mathit{E}}_2={[e_2^1,e_2^2,\cdots ,e_2^n]}^T$. Due to ${\displaystyle \frac{\dot{\varrho }\left(t\right)}{\varrho \left(t\right)}}={\displaystyle \frac{\varrho \left(t\right)}{h}}$, substituting Equation (14) and Equation (18) into Equation (5) yields

$$\dot{\mathcal{E}}=\varrho \left(t\right)\left[\begin{array}{cc}{\displaystyle \frac{1}{h}}{\mathit{I}}_N& {\mathit{I}}_N\\ 0& 0\end{array}\right]\mathcal{E}+\left[\begin{array}{c}0\\ {\mathit{I}}_N\end{array}\right]\mathit{U}=\left\{\begin{array}{c}\varrho \left(t\right)\left[\begin{array}{cc}{\displaystyle \frac{1}{h}}{\mathit{I}}_N& {\mathit{I}}_N\\ -\alpha \varrho \left(t\right)\mathcal{L}& -\beta \varrho \left(t\right)\mathcal{L}\end{array}\right]\mathcal{E},t\in {\mathit{\xi }}_b\hfill \\ \varrho \left(t\right)\left[\begin{array}{cc}{\displaystyle \frac{1}{h}}{\mathit{I}}_N& {\mathit{I}}_N\\ -\alpha \varrho \left(t\right)(\mathcal{L}+\gamma \mathcal{D})& -\beta \varrho \left(t\right)(\mathcal{L}+\gamma \mathcal{D})\end{array}\right]\mathcal{E},t\in {\mathit{\xi }}_c\hfill \end{array}\right.$$

(19)

(1) $t\in [t_p,t_{p+1})\subset {\mathit{\xi }}_b$, without pinning communication

The Lyapunov candidate function is selected as

$$V\left(t\right)={\mathcal{E}}^T{\Omega }_p\mathcal{E}$$

(20)

where ${\mathbf{\Omega }}_p=\left[\begin{array}{cc}\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right)\mathbf{\Gamma }& {\mathit{\ell }}_1^p\mathbf{\Lambda }\\ {\mathit{\ell }}_1^p\mathbf{\Lambda }& {\mathit{\ell }}_2^p\mathbf{\Lambda }\end{array}\right]$, $\mathbf{\Gamma }={\displaystyle \frac{\mathbf{\Lambda }\mathcal{L}+{\mathcal{L}}^T\mathbf{\Lambda }}{2}}$, $\mathbf{\Lambda }$ is a positive definite diagonal matrix defined as Lemma 1.

According to the Schur’s Complement Lemma, when Equation (16) is satisfied, ${\mathit{\ell }}_2^p\mathbf{\Lambda }>0$ and $\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right)\mathbf{\Gamma }-{\mathit{\ell }}_1^p\mathbf{\Lambda }{({\mathit{\ell }}_2^p\mathbf{\Lambda })}^{-1}{\mathit{\ell }}_1^p\mathbf{\Lambda }>0$. Thus, the validity of V is ensured by Equation (16).

For $t\in [t_{p-1},t_p)\subset {\mathit{\xi }}_b$, the derivative of $V\left(t\right)$ is

$$\dot{V}\left(t\right)=\varrho \left(t\right)\left\{-2\left[\alpha {\mathit{\ell }}_1^p-{\displaystyle \frac{\beta {\mathit{\ell }}_1+\alpha {\mathit{\ell }}_2^p}{h}}\right]{\mathcal{E}}_1^T\mathbf{\Gamma }{\mathcal{E}}_1-\beta {\mathit{\ell }}_2^p{\mathcal{E}}_2^T\mathbf{\Gamma }{\mathcal{E}}_2+2{\mathit{\ell }}_1^p{\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_2+{\displaystyle \frac{{\mathit{\ell }}_1^p}{h}}({\mathcal{E}}_1^T\mathbf{\Lambda }{\mathcal{E}}_2+{\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_1)\right\}$$

(21)

Based on Lemma 1,

$$\begin{array}{c}{\mathcal{E}}_1^T\mathbf{\Gamma }{\mathcal{E}}_1\ge {\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right){\mathcal{E}}_1^T\mathbf{\Lambda }{\mathcal{E}}_1\hfill \end{array}$$

(22)

Similarly,

$${\mathcal{E}}_2^T\mathbf{\Gamma }{\mathcal{E}}_2\ge {\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right){\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_2$$

(23)

For $\mathit{\lambda }>0$, one has

$${\mathcal{E}}_1^T\mathbf{\Lambda }{\mathcal{E}}_2={\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_1\le \kappa {\mathcal{E}}_1^T\mathbf{\Lambda }{\mathcal{E}}_1+{\displaystyle \frac{1}{4\kappa }}{\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_2$$

(24)

Based on the conditions in Equation (16), $\alpha {\mathit{\ell }}_{1}h-(\beta {\mathit{\ell }}_1+\alpha {\mathit{\ell }}_2)>0$, it yields

$$\begin{array}{c}\dot{V}\left(t\right)\le -2\varrho \left(t\right)\left[\beta {\mathit{\ell }}_2^p{\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)-{\mathit{\ell }}_1^p-{\displaystyle \frac{{\mathit{\ell }}_1^p}{4h\kappa }}\right]{\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_2\hfill \\ -2\varrho \left(t\right)\left\{\left[\alpha {\mathit{\ell }}_1^p-{\displaystyle \frac{(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p)}{h}}\right]{\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)+{\displaystyle \frac{{\mathit{\ell }}_1^p\kappa }{h}}\right\}{\mathcal{E}}_1^T\mathbf{\Lambda }{\mathcal{E}}_1\hfill \end{array}$$

(25)

If Equation (16) is satisfied, $4h{\kappa }_1\beta {\mathit{\ell }}_2{\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)-(1+4h\kappa ){\mathit{\ell }}_1>0$ and $[\alpha {\mathit{\ell }}_{1}h-(\beta {\mathit{\ell }}_1+\alpha {\mathit{\ell }}_2)]{\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)+{\mathit{\ell }}_1\kappa >0$, hence, $\dot{V}<0$ and the system Equation (5) is stable.

Let $\mathcal{H}\left(t\right)=\dot{V}+(b+\eta \varrho \left(t\right))V$, one has

$$\begin{array}{c}\mathcal{H}\left(t\right)=\varrho \left(t\right)\left\{-\left[2\alpha {\mathit{\ell }}_1^p-\left({\displaystyle \frac{2}{h}}+\eta \right)(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p)\right]{\mathcal{E}}_1^T\mathbf{\Gamma }{\mathcal{E}}_1-2\beta {\mathit{\ell }}_2^p{\mathcal{E}}_2^T\mathbf{\Gamma }{\mathcal{E}}_2\right.\hfill \\ \left.+(2{\mathit{\ell }}_1^p+\eta {\mathit{\ell }}_2^p){\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_2+\left({\displaystyle \frac{{\mathit{\ell }}_1^p}{h}}+\eta {\mathit{\ell }}_1^p\right)({\mathcal{E}}_1^T\mathbf{\Lambda }{\mathcal{E}}_2+{\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_1)\right\}\hfill \\ +b\left\{\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathcal{E}}_1^T\mathbf{\Gamma }{\mathcal{E}}_1+{\mathit{\ell }}_2^p{\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_2+{\mathit{\ell }}_1^p({\mathcal{E}}_1^T\mathbf{\Lambda }{\mathcal{E}}_2+{\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_1)\right\}\hfill \end{array}$$

(26)

Due to $\varrho \left(t\right)\ge {\displaystyle \frac{h}{T^{*}}}$, combing with Equations (22) and (24), it can be derived as

$$\begin{array}{c}\mathcal{H}\left(t\right)\le \left\{-{\displaystyle \frac{h}{T^{*}}}\left[2\alpha {\mathit{\ell }}_1^p-\left({\displaystyle \frac{2}{h}}+\eta \right)(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p)\right]{\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)\right.\hfill \\ \left.+b\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)+2{\mathit{\ell }}_1^p\kappa \left[{\displaystyle \frac{1+\eta h}{T^{*}}}+b\right]\right\}{\mathcal{E}}_1^T\mathbf{\Lambda }{\mathcal{E}}_1\hfill \\ +{\displaystyle \frac{h}{T^{*}}}\left[-2\beta {\mathit{\ell }}_2^p{\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)+{\mathit{\ell }}_1^p\left(2+{\displaystyle \frac{1}{2h\kappa }}\right)+\left(\eta +{\displaystyle \frac{b{T}^{*}}{h}}\right)\left({\mathit{\ell }}_2^p+{\displaystyle \frac{{\mathit{\ell }}_1^p}{2\kappa }}\right)\right]{\mathcal{E}}_2^T\mathbf{\Lambda }{\mathcal{E}}_2\hfill \end{array}$$

(27)

Obviously, based on Equation (16), $\mathcal{H}\left(t\right)\le 0$, that is

$$\dot{V}\left(t\right)\le -bV\left(t\right)-\eta \varrho \left(t\right)V\left(t\right)$$

(28)

Hence,

$$V\left(t\right)\le \rho {\left(t\right)}^{-\eta }{e}^{-b(t-t_p)}V\left(t_p\right),t\in [t_p,t_{p+1})\subset {\mathit{\xi }}_b$$

(29)

(2) $t\in [t_p,t_{p+1})\subset {\mathit{\xi }}_c$, with pinning communication

The Lyapunov candidate function is selected as

$$V\left(t\right)={\mathcal{E}}^T{\Omega }_p\mathcal{E}$$

(30)

where ${\mathbf{\Omega }}_p=\left[\begin{array}{cc}\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathbf{\Gamma }}^{\mathcal{D}}& {\mathit{\ell }}_1^p{\mathbf{\Lambda }}^{\mathcal{D}}\\ {\mathit{\ell }}_1^p{\mathbf{\Lambda }}^{\mathcal{D}}& {\mathit{\ell }}_2^p{\mathbf{\Lambda }}^{\mathcal{D}}\end{array}\right]$, ${\mathbf{\Gamma }}^{\mathcal{D}}={\displaystyle \frac{{\mathbf{\Lambda }}^{\mathcal{D}}\mathcal{M}+{\mathcal{M}}^T{\mathbf{\Lambda }}^{\mathcal{D}}}{2}}$, $\mathcal{M}=\mathcal{L}+\gamma \mathcal{D}$ is a M-matrix with a positive definite diagonal matrix ${\mathbf{\Lambda }}^{\mathcal{D}}$ defined as Lemma 2.

Similarly, when Equation (16) is satisfied, $\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathbf{\Gamma }}^{\mathcal{D}}-{\mathit{\ell }}_1^p{\mathbf{\Lambda }}^{\mathcal{D}}{\left({\mathit{\ell }}_2^p{\mathbf{\Lambda }}^{\mathcal{D}}\right)}^{-1}{\mathit{\ell }}_1^p{\mathbf{\Lambda }}^{\mathcal{D}}>0$ and ${\mathit{\ell }}_2^p{\mathbf{\Lambda }}^{\mathcal{D}}>0$, the validity of V is achieved.

For $t\in [t_p,t_{p+1})\subset {\mathit{\xi }}_c$, the derivative of $V\left(t\right)$ is

$$\begin{array}{c}\dot{V}\left(t\right)=\varrho \left(t\right)\left\{-2\left[\alpha {\mathit{\ell }}_1^p-{\displaystyle \frac{\beta {\mathit{\ell }}_1+\alpha {\mathit{\ell }}_2^p}{h}}\right]{\mathcal{E}}_1^T{\mathbf{\Gamma }}^{\mathcal{D}}{\mathcal{E}}_1-\beta {\mathit{\ell }}_2^p{\mathcal{E}}_2^T{\mathbf{\Gamma }}^{\mathcal{D}}{\mathcal{E}}_2+2{\mathit{\ell }}_1^p{\mathcal{E}}_2^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_2\right.\hfill \\ \left.+{\displaystyle \frac{{\mathit{\ell }}_1^p}{h}}({\mathcal{E}}_1^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_2+{\mathcal{E}}_2^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_1)\right\}\hfill \end{array}$$

(31)

Based on Lemmas 2 and 3, one can obtained that

$${\mathcal{E}}_1^T\mathbf{\Gamma }{\mathcal{E}}_1\ge {\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)\sum _{k=1}^n\parallel {\mathcal{E}}_1^i{\parallel }^2\ge {\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}\sum _{k=1}^n{\displaystyle \frac{1}{{\lambda }_i}}{\parallel {\mathcal{E}}_1^i\parallel }^2={\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}{\mathcal{E}}_1^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_1$$

(32)

Similarly, one has

$$\begin{array}{c}{\mathcal{E}}_1^T{\mathbf{\Gamma }}^{\mathcal{D}}{\mathcal{E}}_1\ge {\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}{\mathcal{E}}_1^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_1\hfill \end{array}$$

(33)

Based on the conditions in Equation (16), $\alpha {\mathit{\ell }}_{1}h-(\beta {\mathit{\ell }}_1+\alpha {\mathit{\ell }}_2)>0$, it yields

$$\begin{array}{c}\dot{V}\left(t\right)\le -2\varrho \left(t\right)\left[\beta {\mathit{\ell }}_2^p{\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}-{\mathit{\ell }}_1^p-{\displaystyle \frac{{\mathit{\ell }}_1^p}{4h\kappa }}\right]{\mathcal{E}}_2^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_2\hfill \\ -2\varrho \left(t\right)\left\{\left[\alpha {\mathit{\ell }}_1^p-{\displaystyle \frac{(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p)}{h}}\right]{\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}+{\displaystyle \frac{{\mathit{\ell }}_1^p\kappa }{h}}\right\}{\mathcal{E}}_1^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_1\hfill \end{array}$$

(34)

If Equation (16) is satisfied, $4h\kappa \beta {\mathit{\ell }}_2{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)-(1+4h\kappa ){\mathit{\ell }}_1{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)>0$ and $[\alpha {\mathit{\ell }}_{1}h-(\beta {\mathit{\ell }}_1+\alpha {\mathit{\ell }}_2)]{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)+{\mathit{\ell }}_1\kappa {\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)>0$, hence, $\dot{V}<0$ and the system Equation (5) is stable.

Let $\mathcal{H}\left(t\right)=\dot{V}+(b+\eta \varrho \left(t\right))V$, one has

$$\begin{array}{c}\mathcal{H}\left(t\right)=\varrho \left(t\right)\left\{-\left[2\alpha {\mathit{\ell }}_1^p-\left({\displaystyle \frac{2}{h}}+\eta \right)(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p)\right]{\mathcal{E}}_1^T{\mathbf{\Gamma }}^{\mathcal{D}}{\mathcal{E}}_1-2\beta {\mathit{\ell }}_2^p{\mathcal{E}}_2^T{\mathbf{\Gamma }}^{\mathcal{D}}{\mathcal{E}}_2\right.\hfill \\ \left.+(2{\mathit{\ell }}_1^p+\eta {\mathit{\ell }}_2^p){\mathcal{E}}_2^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_2+\left({\displaystyle \frac{{\mathit{\ell }}_1^p}{h}}+\eta {\mathit{\ell }}_1^p\right)({\mathcal{E}}_1^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_2+{\mathcal{E}}_2^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_1)\right\}\hfill \\ +b\left\{\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathcal{E}}_1^T{\mathbf{\Gamma }}^{\mathcal{D}}{\mathcal{E}}_1+{\mathit{\ell }}_2^p{\mathcal{E}}_2^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_2+{\mathit{\ell }}_1^p({\mathcal{E}}_1^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_2+{\mathcal{E}}_2^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_1)\right\}\hfill \end{array}$$

(35)

Due to $\varrho \left(t\right)\ge {\displaystyle \frac{h}{T^{*}}}$, combing with Equations (22) and (24), it can be derived as

$$\begin{array}{c}\mathcal{H}\left(t\right)\le \left\{-{\displaystyle \frac{h}{T^{*}}}\left[2\alpha {\mathit{\ell }}_1^p-\left({\displaystyle \frac{2}{h}}+\eta \right)(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p)\right]{\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}\right.\hfill \\ \left.+b\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}+2{\mathit{\ell }}_1^p\kappa \left[{\displaystyle \frac{1+\eta h}{T^{*}}}+b\right]\right\}{\mathcal{E}}_1^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_1\hfill \\ +{\displaystyle \frac{h}{T^{*}}}\left[-2\beta {\mathit{\ell }}_2^p{\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}+{\mathit{\ell }}_1^p\left(2+{\displaystyle \frac{1}{2h\kappa }}\right)+\left(\eta +{\displaystyle \frac{b{T}^{*}}{h}}\right)\left({\mathit{\ell }}_2^p+{\displaystyle \frac{{\mathit{\ell }}_1^p}{2\kappa }}\right)\right]{\mathcal{E}}_2^T{\mathbf{\Lambda }}^{\mathcal{D}}{\mathcal{E}}_2\hfill \end{array}$$

(36)

Obviously, based on Equation (16), $\mathcal{H}\left(t\right)\le 0$, that is

$$\dot{V}\left(t\right)\le -bV\left(t\right)-\eta \varrho \left(t\right)V\left(t\right)$$

(37)

Hence, for $t\in [t_p,t_{p+1})\subset {\mathit{\xi }}_c$, one has

$$V\left(t\right)\le \rho {\left(t\right)}^{-\eta }{e}^{-b(t-t_p)}V\left(t_p\right),$$

(38)

(3) switching at $t=t_p,p=1,2,\cdots ,n_p$

Due to the ${\mathbf{\Omega }}_p$ is positive definite symmetric, it can be deduced that

$${\mu }_{min}\left({\mathbf{\Omega }}_p\right){\mathcal{E}}^T\mathcal{E}\le V\left(t_p\right)\le {\mu }_{max}\left({\mathbf{\Omega }}_p\right){\mathcal{E}}^T\mathcal{E}$$

(39)

Considering the time sequence is left continuous, if the pinning communication is break at $t_p$,

$$\mathcal{E}\left({\mathit{t}}_{\mathit{p}}^{+}\right)=\mathcal{E}\left({\mathit{t}}_{\mathit{p}}^{-}\right)+\varrho \left(t\right)\left[\begin{array}{cc}0& 0\\ -\alpha \mathcal{L}& -\beta \mathcal{L}\end{array}\right]\mathcal{E}\left(t_p^{-}\right)=\left[\begin{array}{cc}{\mathit{I}}_N& 0\\ -\alpha \varrho \left(t\right)\mathcal{L}& {\mathit{I}}_N-\beta \varrho \left(t\right)\mathcal{L}\end{array}\right]\mathcal{E}\left(t_p^{-}\right),t_p\in {\mathit{\xi }}_b$$

(40)

otherwise,

$$\mathcal{E}\left({\mathit{t}}_{\mathit{p}}^{+}\right)=\mathcal{E}\left({\mathit{t}}_{\mathit{p}}^{-}\right)+\varrho \left(t\right)\left[\begin{array}{cc}0& 0\\ -\alpha \mathcal{M}& -\beta \mathcal{M}\end{array}\right]\mathcal{E}\left(t_p^{-}\right)=\left[\begin{array}{cc}{\mathit{I}}_N& 0\\ -\alpha \varrho \left(t\right)\mathcal{M}& {\mathit{I}}_N-\beta \varrho \left(t\right)\mathcal{M}\end{array}\right]\mathcal{E}\left(t_p^{-}\right),t_p\in {\mathit{\xi }}_c$$

(41)

Hence, at $t_p$, the Lyapunov function can be rewritten as

$$V\left(t_p^{+}\right)\le {\mu }_{max}\left({\mathbf{\Omega }}_p\right){\mu }_{max}\left({\mathcal{Y}}_p^T{\mathcal{Y}}_p\right){\mathcal{E}}^T\left(t_p^{-}\right)\mathcal{E}\left(t_p^{-}\right)\le {\overline{\mathcal{Y}}}_{p}V\left(t_p^{-}\right)$$

(42)

where ${\overline{\mathcal{Y}}}_p={\displaystyle \frac{{\mu }_{max}\left({\mathbf{\Omega }}_p\right)}{{\mu }_{min}\left({\mathbf{\Omega }}_{p-1}\right)}}{\mu }_{max}\left({\mathcal{Y}}_p^T{\mathcal{Y}}_p\right)$.

$${\mathcal{Y}}_p=\left\{\begin{array}{c}\left[\begin{array}{cc}{\mathit{I}}_N& 0\\ -\alpha \varrho \left(t\right)\mathcal{L}& {\mathit{I}}_N-\beta \varrho \left(t\right)\mathcal{L}\end{array}\right],t_p\in {\mathit{\xi }}_b\hfill \\ \left[\begin{array}{cc}{\mathit{I}}_N& 0\\ -\alpha \varrho \left(t\right)\mathcal{M}& {\mathit{I}}_N-\beta \varrho \left(t\right)\mathcal{M}\end{array}\right],t_p\in {\mathit{\xi }}_c\hfill \end{array}\right.$$

Substituting Equations (29) and (38) into Equation (42) yields

$$\begin{array}{c}V\left(t\right)\le \rho {\left(t\right)}^{-\eta }{e}^{-b(t-t_p)}{\overline{\mathcal{Y}}}_{p}V\left(t_p^{-}\right)\hfill \\ \le \rho {\left(t\right)}^{-2\eta }{e}^{-b(t-t_{p-1})}{\overline{\mathcal{Y}}}_{p}V\left(t_{p-1}^{+}\right)\hfill \\ \le \rho {\left(t\right)}^{-2\eta }{e}^{-b(t-t_{p-1})}{\overline{\mathcal{Y}}}_p{\overline{\mathcal{Y}}}_{p-1}V\left(t_{p-1}^{-}\right)\hfill \\ \le \rho {\left(t\right)}^{-3\eta }{e}^{-b(t-t_{p-2})}{\overline{\mathcal{Y}}}_p{\overline{\mathcal{Y}}}_{p-1}{\overline{\mathcal{Y}}}_{p-2}V\left(t_{p-2}^{-}\right)\hfill \\ \le \rho {\left(t\right)}^{-4\eta }{e}^{-b(t-t_{p-3})}{\overline{\mathcal{Y}}}_p{\overline{\mathcal{Y}}}_{p-1}{\overline{\mathcal{Y}}}_{p-2}V\left(t_{p-3}^{+}\right)\hfill \\ \le \rho {\left(t\right)}^{-4\eta }{e}^{-b(t-t_{p-3})}{\overline{\mathcal{Y}}}_p{\overline{\mathcal{Y}}}_{p-1}{\overline{\mathcal{Y}}}_{p-2}{\overline{\mathcal{Y}}}_{p-3}V\left(t_{p-3}^{-}\right)\hfill \\ \cdots \hfill \\ \le \rho {\left(t\right)}^{-(p+1)\eta }{e}^{-b(t-t_0)}\prod _{i=1}^p{\overline{\mathcal{Y}}}_{i}V\left(t_0\right)\hfill \\ \le \rho {\left(t\right)}^{-(p+1)\eta }{e}^{-b(t-t_0)}{\overline{\mathcal{Y}}}^{p}V\left(t_0\right),t\in [t_p,t_{p+1})\hfill \end{array}$$

(43)

Let ${\overline{\mathcal{Y}}}^p={max}_{i=0}^p\left({\overline{\mathcal{Y}}}_i\right),p\in [1,n_p]$, with the minimum intermittent interval ${\tau }_{min}$, it can be obtained that

$$V\left(t\right)\le \rho {\left(t\right)}^{-(p+1)\eta }{e}^{pln{\overline{\mathcal{Y}}}^p-b(t-t_0)}V\left(t_0\right)\le \rho {\left(t\right)}^{-(p+1)\eta }{e}^{p(ln{\overline{\mathcal{Y}}}^p-b{\tau }_{min})-b(t-t_p)}V\left(t_0\right)$$

(44)

For $t\in [t_{n_p},+\infty )$, one has

$$\begin{array}{c}V\left(t\right)\le \rho {\left(t\right)}^{-(n_p+1)\eta }{e}^{n_p(ln{\overline{\mathcal{Y}}}^{n_p}-b{\tau }_{min})-b(t-t_{n_p})}V\left(t_0\right)\hfill \end{array}$$

(45)

Due to $0<\rho {\left(t\right)}^{-(p+1)\eta }\le 1$ and $0<e^{p(ln{\overline{\mathcal{Y}}}^p-b{\tau }_{min})-b(t-t_p)}<1$, the prescribed-time convergence can be ensured.

(4) convergence of error

According to Equation (45), the system is stable within the last interval $t\in [t_{n_p},T^{*})$, reveals that

$${\parallel \mathcal{E}\left(t\right)\parallel }^2\le {\displaystyle \frac{{\mu }_{max}\left({\Omega }_0\right)}{{\mu }_{min}\left({\Omega }_{n_p}\right)}}\rho {\left(t\right)}^{-(n_p+1)\eta }\overline{e}{\parallel \mathcal{E}\left(t_0\right)\parallel }^2$$

(46)

where $\overline{e}=e^{p(ln{\overline{\mathcal{Y}}}^p-b{\tau }_{min})-b(t-t_p)}$.

It is easy to obtain that

$${\parallel \mathcal{E}\left(t\right)\parallel }^2=\parallel {\mathcal{E}}_1{\left(t\right)\parallel }^2+\parallel {\mathcal{E}}_2{\left(t\right)\parallel }^2\ge 2\parallel {\mathcal{E}}_1\left(t\right)\parallel \parallel {\mathcal{E}}_2\left(t\right)\parallel$$

(47)

Simultaneously,

$$\parallel {\mathcal{E}}_1{\left(t\right)\parallel }^2+\parallel {\mathcal{E}}_2{\left(t\right)\parallel }^2\ge \chi (\parallel {\mathit{E}}_1{\left(t\right)\parallel }^2+\parallel {\mathit{E}}_2\left(t\right){\parallel }^2)$$

(48)

where $\chi =min\left(1,{\displaystyle \frac{h^2}{T^2}}\right)$

Hence,

$$(\parallel {\mathit{E}}_1\left(t\right)\parallel +\parallel {\mathit{E}}_2{\left(t\right)\parallel )}^2\le 2{\displaystyle \frac{{\mu }_{max}\left({\Omega }_0\right)}{\chi {\mu }_{min}\left({\Omega }_{n_p}\right)}}\rho {\left(t\right)}^{-(n_p+1)\eta }\overline{e}{\parallel \mathcal{E}\left(t_0\right)\parallel }^2$$

(49)

When $t\to T^{*-}$, noting that $0<\overline{e}<1$ and ${lim}_{t\to T^{*-}}\rho {\left(t\right)}^{-(n_p+1)\eta }=0$, it yields $\parallel {\mathit{E}}_1\left(t\right)\parallel \to 0$ and $\parallel {\mathit{E}}_2\left(t\right)\parallel \to 0$, that is, Equation (6) in Definition 1 holds.

(5) boundedness of command

$$\begin{array}{c}\parallel \mathit{U}\parallel =\left\{\begin{array}{c}\parallel \alpha \varrho \left(t\right)\mathcal{L}{\mathcal{E}}_1\left(t\right)+\beta \varrho \left(t\right)\mathcal{L}{\mathcal{E}}_2\left(t\right)\parallel ,t\in {\mathit{\xi }}_b\hfill \\ \parallel \alpha \varrho \left(t\right)(\mathcal{L}+\gamma \mathcal{D}){\mathcal{E}}_1\left(t\right)+\beta \varrho \left(t\right)(\mathcal{L}+\gamma \mathcal{D}){\mathcal{E}}_2\left(t\right)\parallel ,t\in {\mathit{\xi }}_c\hfill \end{array}\right.\hfill \\ =\varrho \left(t\right)\omega ({\mathcal{E}}_1\left(t\right)\parallel +\parallel {\mathcal{E}}_2\left(t\right)\parallel )\hfill \end{array}$$

(50)

where $\omega =max(\alpha ,\beta )max(\parallel \mathcal{L}\parallel ,\parallel \mathcal{L}+\gamma \mathcal{D}\parallel )\parallel$

Due to $\varrho \left(t\right)={\displaystyle \frac{h}{T}}{\rho }^{{\displaystyle \frac{1}{h}}}\left(t\right)$

$$\begin{array}{c}\parallel \mathit{U}\parallel \le {\displaystyle \frac{h}{T}}\omega \sqrt{{\displaystyle \frac{2{\mu }_{max}\left({\Omega }_0\right)}{{\mu }_{min}\left({\Omega }_{n_p}\right)}}}\rho {\left(t\right)}^{-{\displaystyle \frac{(n_p+1)\eta }{2}}+{\displaystyle \frac{1}{h}}}\sqrt{\overline{e}}\parallel \mathcal{E}\left(t_0\right)\parallel \hfill \end{array}$$

(51)

Thus,

$$\parallel \mathit{U}\left(t\right)\parallel \in L_{\infty }$$

(52)

which means that the control command is bounded over $[t_0,T^{*})$

(6) $t\in [T^{*},+\infty )$

When $t\ge T^{*}$, following the steps in Equations (30)–(38), it yields

$$V\left(t\right)\le -bV\left(t\right)\le 0$$

(53)

For $t=T^{*}$, it follows that

$$V\left(T^{*}\right)=\underset{t\to T^{*-}}{lim}V\left(t\right)=0$$

(54)

Considering the continuity of the Lyapunov function, it can be deduced that

$$0\le V\left(t\right)\le V\left(T^{*}\right)=0,\forall t\in [T^{*},+\infty )$$

(55)

Therefore, it has $V\left(t\right)\equiv 0$ and $\mathcal{E}=0$ satisfying Equation (7). Thus, when $t\in [T^{*},+\infty )$, the prescribed-time consensus can be kept. □

3.2. Second Stage: Keep Cooperation

According to the rate of LOS angle, the second-order acceleration command can be organized as Equation (13). Substituting the second-order guidance command into the system Equation (5), one can derive that

$$\begin{array}{c}{\dot{\sigma }}_i={\displaystyle \frac{1}{sin{\sigma }_i}}\left(sin{\theta }_{i}cos{\varphi }_i{\dot{\theta }}_i+cos{\theta }_{i}sin{\varphi }_i{\dot{\varphi }}_i\right)\hfill \\ =-{\displaystyle \frac{\left(K-1\right)}{sin{\sigma }_i}}{\displaystyle \frac{V}{R_i}}\left({cos}^2{\varphi }_i-{cos}^2{\theta }_i{cos}^2{\varphi }_i+{sin}^2{\varphi }_i\right)\hfill \\ =-{\displaystyle \frac{\left(K-1\right)V}{R_i}}sin{\sigma }_i\hfill \end{array}$$

(56)

Furthermore,

$${\displaystyle \frac{\partial R_i}{\partial {\sigma }_i}}={\displaystyle \frac{R_{i}cot{\sigma }_i}{K-1}}$$

(57)

Then,

$$R_i\left(t\right)={\displaystyle \frac{R_i\left(0\right){\left|sin{\sigma }_i\left(t\right)\right|}^{1/(K-1)}}{{\left|sin{\sigma }_i\left(0\right)\right|}^{1/(K-1)}}}$$

(58)

That is, when the initial condition $R_i\left(0\right)$ and $|{\sigma }_i\left(0\right)|$ are the same, the shaping of trajectory are the same. It can be obtained that

$${\dot{\sigma }}_i=-{\displaystyle \frac{(K-1)V_m}{R_i\left(0\right)}}{\left|sin{\sigma }_i\left(0\right)\right|}^{{\displaystyle \frac{1}{K-1}}}{\left|sin{\sigma }_i\left(t\right)\right|}^{{\displaystyle \frac{K-2}{K-1}}},{\sigma }_i\in [0,\pi )$$

(59)

when $K\ge 2$, the guidance can be achieved.

4. Design of Two-Stage Cooperative Guidance Law Design

Unideal communication makes the multi-UAV cooperative guidance challengeable. Although the superior convergence performance can organize the cooperation in the limited guidance interval, convergent states are more likely be broken by the unideal intermittent network. Considering the weakness of networked cooperation, especially the intermittent communication, individual guidance after convergence shows robust without network, which can keep the convergent states while avoiding network interference and significantly reduces the cost of network integration, so as to achieve simultaneous interception.

Compared with existing scheme executing switch based on tolerable state error judgement, the desired prescribed-time convergence approve the more specific time switch flag for the design of cooperative guidance law. Further considering the characteristic of prescribed-time convergence, the prescribed-time cooperative guidance law is provided as follows,

$$\begin{array}{c}{a}_{yi}=\left\{\begin{array}{c}-{\displaystyle \frac{V^2}{R_i}}sin{\varphi }_i+{\displaystyle \frac{u_i(\mathit{\xi }\left(t\right),t)V}{2o_1}},t\in \left[t_0,T^{*}-{ϵ}_3\right)\hfill \\ KV{\dot{\theta }}_{Li}sin{\theta }_{i}sin{\varphi }_i+KV{\dot{\varphi }}_{Li}cos{\theta }_{Li}cos{\theta }_i,t\in \left[T^{*}-{ϵ}_3,+\infty \right)\hfill \end{array}\right.\hfill \\ a_{zi}=\left\{\begin{array}{c}-{\displaystyle \frac{V^2}{R_i}}sin{\theta }_{i}cos{\varphi }_i+{\displaystyle \frac{u_i(\mathit{\xi }\left(t\right),t)V}{2o_2}},t\in \left[t_0,T^{*}-{ϵ}_3\right)\hfill \\ KV{\dot{\theta }}_{Li}cos{\varphi }_i,t\in \left[T^{*}-{ϵ}_3,+\infty \right)\hfill \end{array}\right.\hfill \end{array}$$

(60)

where $o_1=max(sin{\varphi }_i,{ϵ}_{1}sgn(sin{\varphi }_i))$ and $o_2=max(sin{\theta }_{i}cos{\varphi }_i,{ϵ}_{2}sgn(sin{\theta }_{i}cos{\varphi }_i))$ with small terms ${ϵ}_1$, ${ϵ}_2$ and ${ϵ}_3$, which are designed to overcome the inherent singularity problem.

The block diagram of the proposed PTCGL is outlined in Figure 3.

The practical implementation is listed as follows.

Step. 1 Networking. The cooperation of multi-UAV is organized by a pinning group network, wherein the internal communication is strongly connected directed topology $\mathcal{L}$ and the external intermittent pinning communication $\mathcal{D}$. Pinning information can intermittent transferred and is triggered by $\mathit{\xi }\left(t\right)$, which is provided by priori detected information.

Step. 2 Networked cooperation. By tuning guidance parameters $\alpha$, $\beta$, $\gamma$ and h, the proposed cooperative guidance law Equation (60) are utilized to ensure the prescribed-time convergence of range-to-go and heading error in the first-stage.

Step. 3 Stage switching. Taking the prescribed convergence time $T^{*}$ as the switching flag, the second stage begins with suitable convergent initial conditions and all UAVs governed by proportional navigation guidance.

Remark 1.

To simplify the parameter tuning of cooperative guidance law, the parameters involved in Theorem 1 can be divided into three categories:

• network-related parameter: internal communication network $\mathcal{L}$, external pinning network $\mathcal{D}$ and time-triggered communication sequence $\mathit{\xi }\left(t\right)$, which are influenced by mission environment, detection information and communication capability.
• prescribed-time parameter: parameter $t_0$ and T are pre-specified by mission or user’s requirements.
• guidance parameter: h, K, α, β and γ are strongly affected the performance of the cooperative guidance law.

Remark 2.

In theory, the convergence time of the first-stage can be arbitrarily pre-assigned, independent of initial conditions and tuning parameters. However, limited by the physical actuators and practical engagement, the cooperation can only be achieved with initial conditions within a certain range. Thus, the assignment of convergence time is limited by multiple practical factors. Although the desired prescribed-time performance can provided a specific switching point, the time-scaling gain with tiny error may lead to a sharp increasing of command, causing the saturation of actuators. Hence, setting a ${ϵ}_3$ will help to switching with a tolerable error, which benefits to application.

Based on the Theorem 1, a corollary is further deduced to simplify the parameter tuning.

Corollary 1.

With guidance parameters $h,\alpha ,\varpi >0$ satisfying Equation (61), when pinning communication is connected, if there existing $\gamma >0$ make $\mathcal{L}+\gamma \mathcal{D}$ be a M-matrix, the first-stage of cooperative guidance law can ensure the prescribed-time convergence.

$$\left\{\begin{array}{c}h>max\left({\displaystyle \frac{1}{\eta }},1\right)\hfill \\ \alpha >{\displaystyle \frac{1}{{\mathit{\Theta }}_0}}max\left\{{\displaystyle \frac{{\mathit{\Theta }}_3}{4\varpi \kappa {\mathit{\ell }}_2^{p}h}},{\displaystyle \frac{{\left({\mathit{\ell }}_1^p\right)}^2}{{\mathit{\ell }}_2^p({\mathit{\ell }}_1^p+\varpi {\mathit{\ell }}_2^p)}},{\displaystyle \frac{{\mathit{\Theta }}_2}{2[{\mathit{\ell }}_1^{p}h-{\mathit{\Theta }}_1(\varpi {\mathit{\ell }}_1^p+{\mathit{\ell }}_2^p)]}}\right\}\hfill \\ -b{\tau }_{min}+ln{\overline{\mathcal{Y}}}^p<0\hfill \end{array}\right.$$

(61)

where $\varpi =\beta /\alpha$, $\overline{\mathcal{L}}$ and $\overline{\mathcal{M}}$ are invertible, ${\overline{\mathcal{Y}}}^p={max}_{i=0}^p\left\{{\displaystyle \frac{max({\lambda }_i{\widehat{\mathcal{J}}}_p^c){\mho }_c}{min\left\{{\lambda }_{i}max\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}(\mathcal{L}),{\mathit{\ell }}_2^p\right]\right\}}}\right.$,

$\left.{\displaystyle \frac{max({\lambda }_i{\widehat{\mathcal{J}}}_p^b){\mho }_b}{min\left({\lambda }_i{\overline{\mathcal{J}}}_p^c\right)}}\right\}$, $p\in [1,n_p]$.

Proof. 

Substituting $\beta =\varpi \alpha$ into Equation (16), it is easy to obtained that

$$\begin{array}{c}\alpha >{\displaystyle \frac{1}{{\mathit{\Theta }}_0}}max\left\{{\displaystyle \frac{{\mathit{\Theta }}_3}{4\varpi \kappa {\mathit{\ell }}_2^{p}h}},{\displaystyle \frac{{\left({\mathit{\ell }}_1^p\right)}^2}{{\mathit{\ell }}_2^p({\mathit{\ell }}_1^p+\varpi {\mathit{\ell }}_2^p)}},{\displaystyle \frac{{\mathit{\Theta }}_2}{2[{\mathit{\ell }}_1^{p}h-{\mathit{\Theta }}_1(\varpi {\mathit{\ell }}_1^p+{\mathit{\ell }}_2^p)]}}\right\}\hfill \end{array}$$

(62)

Due to $n_p\ge 1$, the condition $(p+1)\eta h-2>0$ can be transformed into

$$h>{\displaystyle \frac{1}{\eta }}\ge {\displaystyle \frac{2}{(n_p+1)\eta }}$$

(63)

Hence, $h>max\left({\displaystyle \frac{1}{\eta }},1\right)$ can simplify the parameter tuning.

To clarify the existence of ${\tau }_{min}$ condition, the proof is divided into the following steps:

(1) $t\in [t_p,t_{p+1})\subset {\mathit{\xi }}_b$

Due to the Lemma 1, one has

$${\overline{\mathbf{\Omega }}}_p^b\le {\mathbf{\Omega }}_p\le {\widehat{\mathbf{\Omega }}}_p^b$$

(64)

where ${\overline{\mathbf{\Omega }}}_p^b=\left[\begin{array}{cc}\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)& {\mathit{\ell }}_1^p\\ {\mathit{\ell }}_1^p& {\mathit{\ell }}_2^p\end{array}\right]\otimes \mathbf{\Lambda }$, ${\widehat{\mathbf{\Omega }}}_p^b=\left[\begin{array}{cc}\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{max}\left(\mathcal{L}\right)& {\mathit{\ell }}_1^p\\ {\mathit{\ell }}_1^p& {\mathit{\ell }}_2^p\end{array}\right]\otimes \mathbf{\Lambda }$. Due to ${\left({\mathit{\ell }}_1^p\right)}^2-{\mathit{\ell }}_2^p(\alpha {\mathit{\ell }}_1^p+\beta {\mathit{\ell }}_2^p){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)<0$,${\widehat{\mathbf{\Omega }}}_p^b,{\overline{\mathbf{\Omega }}}_p^b,{\mathbf{\Omega }}_p$ are all positive definite symmetric matrices, hence ${\mu }_{min}\left({\overline{\mathbf{\Omega }}}_p^b\right)>0$

The characteristic equation is computed as

$$\begin{array}{c}det|s{\mathit{I}}_{2n}-{\overline{\Omega }}_p^b|=\left|\begin{array}{cc}s{\mathit{I}}_n-\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)\mathbf{\Lambda }& -{\mathit{\ell }}_1^p\mathbf{\Lambda }\\ -{\mathit{\ell }}_1^p\mathbf{\Lambda }& s{\mathit{I}}_n-{\mathit{\ell }}_2^p\mathbf{\Lambda }\end{array}\right|\hfill \\ =s^2{\mathit{I}}_n-s\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)+{\mathit{\ell }}_2^p\right]\mathbf{\Lambda }+\left[{\mathit{\ell }}_2^p\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)-{\left({\mathit{\ell }}_1^p\right)}^2\right]{\mathbf{\Lambda }}^2\hfill \\ =\prod _{i=1}^n\left\{s^2-s\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)+{\mathit{\ell }}_2^p\right]{\lambda }_i+\left[{\mathit{\ell }}_2^p\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)-{\left({\mathit{\ell }}_1^p\right)}^2\right]{\lambda }_i^2\right\}\hfill \end{array}$$

(65)

The following inequality holds

$$\begin{array}{c}\overline{\Psi }{\lambda }_i^2={\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)+{\mathit{\ell }}_2^p\right]}^2{\lambda }_i^2-4\left[{\mathit{\ell }}_2^p\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)-{\left({\mathit{\ell }}_1^p\right)}^2\right]{\lambda }_i^2\hfill \\ =\left\{{\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)-{\mathit{\ell }}_2^p\right]}^2+4{\left({\mathit{\ell }}_1^p\right)}^2\right\}{\lambda }_i^2\hfill \\ >0\hfill \end{array}$$

(66)

Denote ${\overline{\mathcal{J}}}_p^b=\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right)+{\mathit{\ell }}_2^p\right]$, the eigenvalue can be calculated as

$${\overline{s}}_p^i=\left\{\begin{array}{c}{\overline{s}}_p^{i-}={\displaystyle \frac{{\lambda }_i}{2}}({\overline{\mathcal{J}}}_p^b-\sqrt{\overline{\Psi }})\hfill \\ {\overline{s}}_p^{i+}={\displaystyle \frac{{\lambda }_i}{2}}({\overline{\mathcal{J}}}_p^b+\sqrt{\overline{\Psi }})\hfill \end{array}\right.$$

(67)

Obviously, ${\lambda }_{i}max\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right),{\mathit{\ell }}_2^p\right]<{\overline{s}}_p^{i-}<{\overline{s}}_p^{i+}<{\overline{\mathcal{J}}}_p^b{\lambda }_i$

Similarly, the characteristic function of ${\widehat{\Omega }}_p^b$ is

$$\begin{array}{c}det|s{\mathit{I}}_{2n}-{\widehat{\Omega }}_p^b|\hfill \\ =\prod _{i=1}^n\left\{s^2-s\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{max}\left(\mathcal{L}\right)+{\mathit{\ell }}_2^p\right]{\lambda }_i+\left[{\mathit{\ell }}_2^p\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{max}\left(\mathcal{L}\right)-{\left({\mathit{\ell }}_1^p\right)}^2\right]{\lambda }_i^2\right\}\hfill \end{array}$$

(68)

Thus,

$$\begin{array}{c}\widehat{\Psi }{\lambda }_i^2={\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{max}\left(\mathcal{L}\right)+{\mathit{\ell }}_2^p\right]}^2{\lambda }_i^2-4\left[{\mathit{\ell }}_2^p\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{max}\left(\mathcal{L}\right)-{\left({\mathit{\ell }}_1^p\right)}^2\right]{\lambda }_i^2\hfill \\ =\left\{{\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{max}\left(\mathcal{L}\right)-{\mathit{\ell }}_2^p\right]}^2+4{\left({\mathit{\ell }}_1^p\right)}^2\right\}{\lambda }_i^2\hfill \\ >0\hfill \end{array}$$

(69)

Denote ${\widehat{\mathcal{J}}}_p^b=\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{max}\left(\mathcal{L}\right)+{\mathit{\ell }}_2^p\right]$, one can imply that

$${\widehat{s}}_p^i=\left\{\begin{array}{c}{\widehat{s}}_p^{i-}={\displaystyle \frac{{\lambda }_i}{2}}({\widehat{\mathcal{J}}}_p^b-\sqrt{\overline{\Psi }})\hfill \\ {\widehat{s}}_p^{i+}={\displaystyle \frac{{\lambda }_i}{2}}({\widehat{\mathcal{J}}}_p^b+\sqrt{\overline{\Psi }})\hfill \end{array}\right.$$

(70)

Therefore, ${\lambda }_{i}max\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{max}\left(\mathcal{L}\right),{\mathit{\ell }}_2^p\right]<{\overline{s}}_p^{i-}<{\overline{s}}_p^{i+}<{\widehat{\mathcal{J}}}_p^b{\lambda }_i$.

Hence, the eigenvalue of ${\mathbf{\Omega }}_p$ satisfies that

$$min\left\{{\lambda }_{i}max\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right),{\mathit{\ell }}_2^p\right]\right\}<\mu \left({\mathbf{\Omega }}_p\right)<max\left({\lambda }_i{\widehat{\mathcal{J}}}_p^b\right)$$

(71)

(2) $t\in [t_p,t_{p+1})\subset {\mathit{\xi }}_c$

According to the Equation (32),

$${\overline{\mathbf{\Omega }}}_p^c\le {\mathbf{\Omega }}_p\le {\widehat{\mathbf{\Omega }}}_p^c$$

(72)

where ${\overline{\mathbf{\Omega }}}_p^c=\left[\begin{array}{cc}\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}& {\mathit{\ell }}_1^p\\ {\mathit{\ell }}_1^p& {\mathit{\ell }}_2^p\end{array}\right]\otimes {\mathbf{\Lambda }}^{\mathcal{D}}$, ${\widehat{\mathbf{\Omega }}}_p^c=\left[\begin{array}{cc}\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\displaystyle \frac{{\mu }_{max}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{min}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}& {\mathit{\ell }}_1^p\\ {\mathit{\ell }}_1^p& {\mathit{\ell }}_2^p\end{array}\right]\otimes {\mathbf{\Lambda }}^{\mathcal{D}}$. Due to ${\left({\mathit{\ell }}_1^p\right)}^2-{\mathit{\ell }}_2^p(\alpha {\mathit{\ell }}_1^p+\beta {\mathit{\ell }}_2^p){\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}<0$, ${\widehat{\mathbf{\Omega }}}_p^c,{\overline{\mathbf{\Omega }}}_p^c,{\mathbf{\Omega }}_p$ are all positive definite symmetric matrices, hence ${\mu }_{min}\left({\overline{\mathbf{\Omega }}}_p^c\right)>0$.

Following the steps from Equation (65) to Equation (71), the eigenvalue of ${\mathbf{\Omega }}_p$ can be derived as

$$min\left({\lambda }_i{\overline{\mathcal{J}}}_p^c\right)<\mu \left({\mathbf{\Omega }}_p\right)<max\left({\lambda }_i{\widehat{\mathcal{J}}}_p^c\right)$$

(73)

where ${\overline{\mathcal{J}}}_p^c=max\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\displaystyle \frac{{\mu }_{min}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{max}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}},{\mathit{\ell }}_2^p\right]$, ${\widehat{\mathcal{J}}}_p^c=\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\displaystyle \frac{{\mu }_{max}\left({\mathbf{\Gamma }}^{\mathcal{D}}\right)}{{\mu }_{min}\left({\mathbf{\Lambda }}^{\mathcal{D}}\right)}}+{\mathit{\ell }}_2^p\right]$

(3) $t=t_p\in {\mathit{\xi }}_b$

At the switching point $t=t_p$, one has

$${\mathcal{Y}}_p^T{\mathcal{Y}}_p=\left[\begin{array}{cc}{\mathit{I}}_n+{\alpha }^2\varrho {\left(t\right)}^2{\mathcal{L}}^T\mathcal{L}& -\alpha \varrho \left(t\right){\mathcal{L}}^T\overline{\mathcal{L}}\\ -\alpha \varrho \left(t\right){\overline{\mathcal{L}}}^T\mathcal{L}& {\overline{\mathcal{L}}}^T\overline{\mathcal{L}}\end{array}\right]$$

(74)

where $\overline{\mathcal{L}}={\mathit{I}}_N-\beta \varrho \left(t\right)\mathcal{L}$.

The matrix ${\mathcal{Y}}_p^T{\mathcal{Y}}_p$ is symmetric. Considering that ${\overline{\mathcal{L}}}^T\overline{\mathcal{L}}>0$, it can be obtained that

$$({\mathit{I}}_n+{\alpha }^2\varrho {\left(t\right)}^2{\mathcal{L}}^T\mathcal{L})-{\alpha }^2{\varrho }^2\left(t\right){\mathcal{L}}^T\overline{\mathcal{L}}{\left({\overline{\mathcal{L}}}^T\overline{\mathcal{L}}\right)}^{-1}{\overline{\mathcal{L}}}^T\mathcal{L}={\mathit{I}}_n>0$$

(75)

Referring to Schur’s Complement Lemma, ${\mathcal{Y}}_p^T{\mathcal{Y}}_p>0$. Due to ${\mathcal{Y}}_p^T{\mathcal{Y}}_p$ is positive definite symmetric matrix, all the eigenvalues are existing and bigger than zero. Accordingly, denote the max eigenvalue as ${\mho }_b=max\left[{\mu }_{max}\left({\mathcal{Y}}_p^T{\mathcal{Y}}_p\right)\right],t_p\in {\mathit{\xi }}_b$.

(4) $t=t_p\in {\mathit{\xi }}_c$

When time goes to the switching point $t=t_p$, it has

$${\mathcal{Y}}_p^T{\mathcal{Y}}_p=\left[\begin{array}{cc}{\mathit{I}}_n+{\alpha }^2\varrho {\left(t\right)}^2{\mathcal{M}}^T\mathcal{M}& -\alpha \varrho \left(t\right){\mathcal{M}}^T\overline{\mathcal{M}}\\ -\alpha \varrho \left(t\right){\overline{\mathcal{M}}}^T\mathcal{M}& {\overline{\mathcal{M}}}^T\overline{\mathcal{M}}\end{array}\right]$$

(76)

where $\overline{\mathcal{M}}={\mathit{I}}_N-\beta \varrho \left(t\right)\mathcal{M}$.

The matrix ${\mathcal{Y}}_p^T{\mathcal{Y}}_p$ is symmetric. According to ${\overline{\mathcal{M}}}^T\overline{\mathcal{M}}>0$ and Schur’s Complement Lemma, ${\mathcal{Y}}_p^T{\mathcal{Y}}_p>0$. For the positive definite symmetric matrix ${\mathcal{Y}}_p^T{\mathcal{Y}}_p$, all the eigenvalues are positive and the max eigenvalue can be obtained as ${\mho }_c=max\left[{\mu }_{max}\left({\mathcal{Y}}_p^T{\mathcal{Y}}_p\right)\right],t_p\in {\mathit{\xi }}_c$.

When pinning communication is connected at $t_p$, the switching parameter can be rewritten as

$${\overline{\mathcal{Y}}}_p={\displaystyle \frac{max\left({\lambda }_i{\widehat{\mathcal{J}}}_p^c\right){\mho }_c}{min\left\{{\lambda }_{i}max\left[\left(\beta {\mathit{\ell }}_1^p+\alpha {\mathit{\ell }}_2^p\right){\mathsf{{\rm Y}}}_{min}\left(\mathcal{L}\right),{\mathit{\ell }}_2^p\right]\right\}}}$$

(77)

If pinning communication is disturbed at $t_p$, the switching parameter can be calculated as

$${\overline{\mathcal{Y}}}_p={\displaystyle \frac{max\left({\lambda }_i{\widehat{\mathcal{J}}}_p^b\right){\mho }_b}{min\left({\lambda }_i{\overline{\mathcal{J}}}_p^c\right)}}$$

(78)

Overall, the existence of parameter ${\overline{\mathcal{Y}}}^p={max}_{i=0}^p\left({\overline{\mathcal{Y}}}_i\right),p\in [1,n_p]$ is proved. □

5. Simulation

To analyse the feasibility of the proposed cooperative guidance law, the initial conditions of multi-UAV are set as shown in

Table 1. Initial conditions of multi-UAV.

Group ${\mathit{g}}_{\mathit{i}}$ and $\mathit{T}(\mathit{x},\mathit{y},\mathit{z})$ [km]	${\mathit{m}}_{\mathit{i}}$	$(\mathit{x},\mathit{y},\mathit{z})$ [km]	$({\mathit{\theta }}_{\mathit{i}},{\mathit{\varphi }}_{\mathit{i}}) [\circ ]$
$g_i=1^{*}(0,0,0)$	1	(−9.5, 8.2, 8.0)	(13, 13)
	2	(−10.3, −5.2, 7.5)	(7, 6)
	3	(−7.5, −9.6, 8.4)	(7, −8)
$g_i=2^{*}(0,-5,0)$	4	(−9.3, −12.9, 7.4)	(10, −7)
	5	(−5.5, −17.6, 6.9)	(2, −9)

.

In this basis case, five UAVs are divided into two groups, respectively fly to their own target $T_1^{*}$ and $T_2^{*}$ at a speed of 300 m/s. As shown in Figure 4, the Laplacian matrix of internal group communication topology can be written as

$$\mathcal{L}=\left[\begin{array}{ccccc}1& 0& -1& 0& 0\\ -1& 1& 0& 0& 0\\ 0& -1& 2& 0& -1\\ -1& 0& 0& 1& 0\\ 0& 0& 0& -1& 1\end{array}\right]$$

(79)

wherein $m_1$ and $m_5$ are pinned with $x_1^{1^{*}}=50,x_1^{2^{*}}=52,x_2^{1^{*}}=x_2^{2^{*}}=-1$.

To verify the applicability and superiority of the PTCGL, five cases were simulated: various fixed-frequency intermittent communication, prescribed-time convergence performance, variable-frequency intermittent communication, comparison, variable pinning scheme. Simulation step is 1 ms. The command limitation is 20 g, wherein the gravitational acceleration g = 9.81 m/s². Considering the physical constrains of leading angle, ${\sigma }_i<{45}^{\circ }$, which means $x_2<-\sqrt{2}/2$. The parameters of the cooperative guidance law Equation (60) are listed as follows: $h=4$, $\alpha =2.7$, $\beta =4.5$, $\gamma =1$, $K=3$, ${ϵ}_1={ϵ}_2=0.01$, ${ϵ}_3=1$ ms. To quantitatively analyse the advantages of the proposed PTCGL, the energy consumed by multi-UAV during cooperative guidance is defined as $J={\sum }_{i=1}^n{\int }_0^t\left|a_i\right|dt$, where $J_y={\sum }_{i=1}^n{\int }_0^t\left|a_{yi}\right|dt$ and $J_z={\sum }_{i=1}^n{\int }_0^t\left|a_{zi}\right|dt$ [41]. Cooperative guidance can be achieved with miss distance less than 1 m.

5.1. Case 1: Fixed-Frequency Intermittent Communication

To evaluate detailed insights on the effect of fixed-frequency intermittent pinning communication, set influential factor ${\tau }_p$ as [0.25 s, 0.5 s, 0.75 s, 1 s, 1.25 s, 1.5 s, 2 s] with a fixed convergence time of $T^{*}=44$ s, respectively.

The simulation results in Figure 5 depict cooperative guidance in an unideal communication environment, where UAVs are divided into two subgroups starting from different locations and flying to the subgroup’s target simultaneously. In the first-stage with poor connected pinning group communication, the prescribed-time convergence performance can ensure the state variables converge at the pre-assigned time $T^{*}=44$ s and provide suitable convergent conditions for the cooperation. Switching to the second-stage, the proposed PTCGL can ensure the success of group arrival time difference less than $0.01$ s, wherein the maximum value, average value and standard deviation of miss distance are less than $0.3$ m, $0.16$ m and $0.09$ m, respectively.

As revealed from Figure 6, for a fixed-frequency intermittent pinning communication, the consumption indicators increase when the intermittent interval ${\tau }_p$ becomes narrow. Due to the more frequently communication transition, more control energy is consumed. As the parts of the commands are provided in Figure 6, the commands all satisfy the execution limitation.

5.2. Case 2: Prescribed-Time Convergence Performance Verification

Based on the Case 1, the simulations of Case 2 are performed with influential factors: pre-specified convergence time $T^{*}$ and fixed-frequency intermittent interval ${\tau }_p$. Figure 7 and Figure 8 illustrates the superiority of prescribed-time performance.

The proposed PTCGL shows a desired prescribed-time performance advantage in terms of intermittent communication. The robustness for influential factors can be demonstrated: (i) the convergence time $T^{*}$ can be arbitrarily pre-specified from 28 s to 36 s; (ii) the miss distance is less than $0.3$ m with average value and standard deviation less than $0.18$ m and $0.1$ m, respectively.

Considering the dead zone of seekers, capability constraints of actuators and unideal communication environment, converging at a specified time is helpful for group simultaneous arrival. It can be observed from Figure 8, with the increase of specified time, the consumption indicators reduce. Due to the fast-convergence demand, the earlier convergence required, the more command energy is consumed. If there exists surplus control capability, the proposed PTCGL can bear even worse intermittent communication. Additionally, a relatively frequently intermittent pinning communication challenges the actuators’ execution capability, or even hinders the accomplishment of the mission. This indicates that the robustness of the proposed PTCGL can ensure the cooperative guidance with desired prescribed-time convergence performance.

5.3. Case 3: Variable-Frequency Intermittent Communication

In this subsection, cooperative guidance with variable-frequency intermittent communication is verified.

Communicated with intermittent pinning sequence in Figure 9, the results illustrated in Figure 10 can verify the effectiveness and robustness of the proposed PTCGL. Although the intermittent pinning communication are variable-frequency, the convergence can be achieved at the pre-assigned time 44 s and the time difference of the group saturation are less than $0.1$ s. The miss distance is less than $0.3$ m with average value and standard deviation less than $0.16$ m and $0.1$ m, respectively. It can be verified that the proposed PTCGL shows robustness for the intermittent communication.

5.4. Case 4: Comparative Simulation

In this subsection, the comparative simulation are performed. In order to further validate the superiority of the proposed PTCGL, compared it with the existing consensus-based cooperative guidance law (CGL). Referred to [7] combined with [37], the cooperative guidance law Equation (60) is organized with intermittent group consensus protocol as follows,

$$\begin{array}{c}{u}_i(\mathit{\xi }\left(t\right),t)={\widehat{p}}_i(\mathit{\xi }\left(t\right),t)+\widehat{\alpha }\sum _{j=1}^n{l}_{ij}{x}_1^{g_j}+\widehat{\beta }\sum _{j=1}^n{l}_{ij}{x}_2^{g_j}\hfill \\ +\widehat{\alpha }\sum _{j=1,j\ne i}^n{a}_{ij}\left(x_1^j-x_1^i\right)+\widehat{\beta }\sum _{j=1,j\ne i}^n{a}_{ij}\left(x_2^j-x_2^i\right)\hfill \end{array}$$

(80)

where

$${\widehat{p}}_i(\mathit{\xi }\left(t\right),t)=\left\{\begin{array}{c}-\widehat{\gamma }\left[\widehat{\alpha }\left(x_1^i\left(t\right)-x_1^{g_i}\right)+\widehat{\beta }\left(x_2^i\left(t\right)-x_2^{g_i}\right)\right],t\in {\mathit{\xi }}_c\hfill \\ 0,t\in {\mathit{\xi }}_b\hfill \end{array}\right.$$

(81)

with control parameters $\widehat{\alpha }=0.04$, $\widehat{\beta }=0.36$ and $\widehat{\gamma }=1$, switching requires state variables’s error less than 0.1 and 0.01, respectively.

As depicted in Figure 11, the proposed PTCGL shows its superiority in group cooperation. By utilizing the consensus-based CGL, the error of $x_2$ cannot satisfy the switching conditions at 36 s. Although the consensus-based CGL can achieve a similar convergence performance at 42 s by tuning parameters, the fast convergence leads to the saturation lasting for nearly 5 s, which challenges the actuator execution. Through the comparisons of simulation, the proposed PTCGL demonstrates a better cooperative convergence, providing a more significant prescribed-time performance advantage.

5.5. Case 5: Pinning Scheme

According to the Corollary 1, the potential application of pinning scheme can be extended. With the pinning scheme, the active intermittent pinning provided a new way to prevent cyber attacks in an unideal denied communication environment.

The intermittent pinning scheme and corresponding simulation results are shown in Figure 12. With the intermittent communication combined with pinning scheme, the multi-UAV fly to the group’s target simultaneously. The prescribed-time convergence can be achieved at 44 s. The miss distance of the multi-UAV are 0.22 m, 0.14 m, 0.24 m, $0.15$ m and $0.28$ m, respectively.

6. Conclusions

In this paper, to address the group cooperation in weak communication environment, a pinning group prescribed-time cooperative guidance law is proposed. Based on intermittent pinning group communication, the two-stage cooperative guidance law can be ensured by switching at a prescribed instant with convergent state. Simulation shows the effectiveness, robustness and superiority of the proposed PTCGL. The desired convergence time can be pre-specified independently in simulations with different intermittent sequences. To face real-world challenges, our future work will further consider integrated guidance-and-control and focus on more complex cross-domain communication engagements, such as cooperative with random switching communication under cyber attack, random packet loss or jamming.