2.1 Hypersonic gliding vehicle model

By ignoring the rotation and curvature of the Earth, we describe the three-degree-of-freedom point-mass equations of re-entry flight phase for GHGVs as follows:

(1) \begin{align} \left\{\begin{array}{lll}{\dot{x}}_{i}={V}_{i}\mathrm{cos}{\theta }_{i}\mathrm{cos}{\sigma }_{i} {} {} \\ {\dot{y}}_{i}={V}_{i}\mathrm{sin}{\theta }_{i} {} {} \\ {\dot{z}}_{i}={V}_{i}\mathrm{cos}{\theta }_{i}\mathrm{sin}{\sigma }_{i} {} {} \\ {\dot{V}}_{i}=-\frac{{C}_{{D}_{i}}{\rho }_{i}{V}_{i}^{2}{S}_{i}}{2{m}_{i}}-g\mathrm{sin}{\theta }_{i} {} , {} \\ {\dot{\theta }}_{i}=\frac{{C}_{{L}_{i}}{\rho }_{i}{V}_{i}{S}_{i}\mathrm{cos}{\upsilon }_{i}}{2{m}_{i}}-\frac{g}{{V}_{i}}\mathrm{cos}{\theta }_{i} {} {} \\ {\dot{\sigma }}_{i}=\frac{{C}_{{L}_{i}}{\rho }_{i}{V}_{i}{S}_{i}\mathrm{sin}{\upsilon }_{i}}{2{m}_{i}\mathrm{cos}{\theta }_{i}} {} {} \end{array}\right.\end{align}

where $ i=0$ represents the HGV as a leader, and $ i=\mathrm{1, 2},\cdots ,K$ represent the HGVs as followers; $ {V}_{i}$ is the velocity; $ {\theta }_{i}$ is the flight path angle; $ {\sigma }_{i}$ represents the heading angle; $ {x}_{i},{y}_{i},{z}_{i}$ stand for the position in the ground coordinate system; $ {\upsilon }_{i}$ is the bank angle; $ {m}_{i}$ means the mass, $ {S}_{i}$ means the reference area; g represents the gravitational acceleration; $ {\rho }_{i}$ represents dimensional atmospheric density, denoted by $ {\rho }_{i}={\rho }_{0}{e}^{-\frac{{h}_{i}}{{H}_{MCP}}}$ , where $ {h}_{i}={y}_{i}$ , $ {\rho }_{0}=1.225$ , $ {H}_{MCP}=7100$ . The terms $ {C}_{{D}_{i}}$ and $ {C}_{{L}_{i}}$ are designated to denote the coefficients of lift and drag described as $ {C}_{{L}_{i}}={C}_{{L}_{i}}^{\alpha }{\alpha }_{i}$ , $ {C}_{{D}_{i}}={C}_{{D}_{i}}^{{\alpha }_{i}^{2}}{\alpha }_{i}^{2}+{C}_{{D}_{i}}^{{\alpha }_{i}}{\alpha }_{i}+{C}_{{D}_{i}}^{0}+{C}_{{D}_{i}}^{{\delta }_{e}}{\delta }_{e,i}$ , where $ {\delta }_{e,i}$ is the deflection angle of speed brake; $ {\alpha }_{i}$ is the angle-of-attack; $ {\upsilon }_{i}$ is the bank angle; $ {C}_{{L}_{i}}^{\alpha }$ , $ {C}_{{D}_{i}}^{{\alpha }_{i}^{2}}$ , $ {C}_{{D}_{i}}^{{\alpha }_{i}}$ , $ {C}_{{D}_{i}}^{0}$ , and $ {C}_{{D}_{i}}^{{\delta }_{e}}$ are known coefficients.

To ensure the validity of the established motion model,the following fundamental assumptions are introduced:

Define state variables as $ {\boldsymbol{\Psi }}_{i,1}={\left[{x}_{i},{y}_{i},{z}_{i}\right]}^{T}$ , $ {\boldsymbol{\Psi }}_{i,2}={\left[{V}_{i},{\theta }_{i},{\sigma }_{i}\right]}^{T}$ , control input as $ {\mathbf{u}}_{i}=\left[-{\delta }_{e,i},\right.{\left.{C}_{L,i}\mathrm{cos}{\upsilon }_{i},{C}_{L,i}\mathrm{sin}{\upsilon }_{i}\right]}^{T}$ , then the actual control signal can be inversely solved by (2)

(2) \begin{align} {\upsilon }_{i}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{{\mathbf{u}}_{{i}_{\left(3\right)}}}{{\mathbf{u}}_{{i}_{\left(2\right)}}}\right),{\alpha }_{i}=\frac{{\mathbf{u}}_{{i}_{\left(2\right)}}}{\mathrm{cos}{\upsilon }_{i}},{\delta }_{e,i}=\frac{-{\mathbf{u}}_{{i}_{\left(1\right)}}-\left({C}_{{D}_{i}}^{{\alpha }_{i}^{2}}{\alpha }_{i}^{2}+{C}_{{D}_{i}}^{{\alpha }_{i}}{\alpha }_{i}+{C}_{{D}_{i}}^{0}\right)}{{C}_{{D}_{i}}^{{\delta }_{e}}}\end{align}

Considering the effects of uncertainties and disturbances to which the system is subjected in engineering practice, we obtain the time-based non-affine nonlinear system:

(3) \begin{align} \left\{\begin{array}{lll}{\dot{\boldsymbol{\Psi }}}_{i,1}={\boldsymbol{\Pi }}_{i,1}\left({\stackrel{-}{\boldsymbol{\Psi }}}_{i,2}\right)+{\Delta }{\boldsymbol{\Omega }}_{i,1}\left({\boldsymbol{\Psi }}_{i,1}\right) {} {} \\ {\dot{\boldsymbol{\Psi }}}_{i,2}={\boldsymbol{\Xi }}_{i}\left({\stackrel{-}{\boldsymbol{\Psi }}}_{i,2}\right)+{\boldsymbol{\Pi }}_{i,2}\left({\stackrel{-}{\boldsymbol{\Psi }}}_{i,2},{\mathbf{u}}_{i}\right)+{\Delta }{\boldsymbol{\Omega }}_{i,2}\left({\boldsymbol{\Psi }}_{i,1}\right) {} , {} \\ {\omega }_{i}={\boldsymbol{\Psi }}_{i,1} {} {} \end{array}\right.\end{align}

where $ {\stackrel{-}{\boldsymbol{\Psi }}}_{i,j}={\boldsymbol{\Psi }}_{i,1},\cdots ,{\boldsymbol{\Psi }}_{i,j}$ ; $ {\boldsymbol{\Xi }}_{i}$ is known nonlinear function; $ {\boldsymbol{\Pi }}_{i,1}$ and $ {\boldsymbol{\Pi }}_{i,2}$ are known non-affine items; $ {\Delta }{\boldsymbol{\Omega }}_{i,1}$ and $ {\Delta }{\boldsymbol{\Omega }}_{i,2}$ stand for unknown internal uncertainties; $ {\omega }_{i}$ is the system output of the $ i-th$ follower.

The following assumptions and lemmas are put forward, which will be utilised soon afterwards.

Assumption 5. For functions in non-affine form, there is positive lower bound $ \underset{-}{\Pi }$ and upper bound $ \stackrel{-}{\Pi }$ satisfying the following form

(4) \begin{align} 0 \lt \underset{\mathrm{}-\mathrm{}}{\Pi }{ \le }||\frac{\partial {\boldsymbol{\Pi }}_{i,j}\left({\stackrel{-}{\boldsymbol{\Psi }}}_{i,j+1}\right)}{\partial {\boldsymbol{\Psi }}_{i}}||{ \le }\stackrel{-}{\Pi } \lt \infty \end{align}

Lemma 1. [Reference Cui, Xu, Chen and Yu27] For any positive constants $ {p}_{1}$ , $ {p}_{2}$ and $ {\chi }_{a}$ , it produces

(5) \begin{align} {\left|{x}_{1}\right|}^{{p}_{1}}{\left|{x}_{2}\right|}^{{p}_{2}}{ \le }\frac{{p}_{1}{\chi }_{a}}{{p}_{1}+{p}_{2}}{\left|{x}_{1}\right|}^{{p}_{1}+{p}_{2}}+\frac{{p}_{2}{\chi }_{a}^{\frac{-{p}_{1}}{{p}_{2}}}}{{p}_{1}+{p}_{2}}{\left|{x}_{2}\right|}^{{p}_{1}+{p}_{2}},\end{align}

Lemma 2. [Reference Cui, Xu, Chen and Yu27] For any $ {x}_{2}\le {x}_{1}$ , $ {x}_{3}\ge 0$ , $ {x}_{4} \gt 0$ and $ p \gt 0$ , it yields

(6) \begin{align} \left\{\begin{array}{l}{x}_{2}^{1+p}-{x}_{1}^{1+p}{ \le }({x}_{1}-{x}_{2}{)}^{1+p}\\ {x}_{3}^{p}({x}_{4}-{x}_{3}){ \le }\frac{{x}_{4}^{1+p}-{x}_{3}^{1+p}}{1+p}\end{array}\right.\end{align}

Lemma 3. [Reference Polycarpou and Ioannou28] For any $ \zeta \gt 0$ and $ x\in \mathbb{R}$ , we have

(7) \begin{align} 0{ \le }\left|x\right|-x\,\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{x}{\zeta }\right){ \le }\iota \zeta \end{align}

where $ \iota $ is a constant satisfying $ \iota ={e}^{-(\iota +1)}$ , $ \iota =0.2785$ .

Lemma 4. [Reference Tong and Li29] For any $ x\in \mathbb{R},\mu \in {\mathbb{R}}^{+}$ , we have

(8) \begin{align} \left|x\right|-\frac{{x}^{2}}{\sqrt{{x}^{2}+{\mu }^{2}}}{ \le }\mu \end{align}

Lemma 5. For any matrix $ \mathit{X}\in {\mathbb{R}}^{m\times 1}$ and positive parameter $ \tau $ meeting $ 0 \lt \tau \lt 1$ , there exists

(9) \begin{align} -\tau \mathbf{X}{ \le }\tau {\left[\left|{X}_{i,1}\right|,\left|{X}_{i,2}\right|,\cdots ,\left|{X}_{m,1}\right|\right]}^{T}\end{align}

2.2 Graph theory

In the MAS, a communication network between agents is described by a graph $ \mathcal{R}=\left(\mathcal{V},\epsilon ,\mathcal{A}\right)$ , where $ \mathcal{V}=\left\{\mathrm{1, 2},\dots ,n\right\}$ is set of nonempty nodes, $ \epsilon =\left\{\left({n}_{i},{n}_{j}\right)\right\}\in \mathcal{V}\times \mathcal{V}$ represents the set of edges, $ \mathcal{A}=\left[{a}_{ij}\right]\in {\mathbb{R}}^{n\times n}$ is adjacency matrix. If $ \left({n}_{i},{n}_{j}\right)\in \epsilon $ , $ {a}_{ij} \gt 0$ , then the agent $ i$ can pass information to agent $ j$ . If $ \left({n}_{i},{n}_{j}\right)\notin \epsilon $ , then $ {a}_{ij}=0$ . Assume that agents do not have access to information about themselves, which means $ {a}_{ii}=0$ . For undirected graphs, the adjacency matrix exhibits symmetry implying $ {a}_{ij}={a}_{ji} \gt 0$ . For directed graph, the adjacency matrix is asymmetric.

The degree matrix $ \mathcal{P}=diag\left({p}_{i}\right)\in {R}^{n\times n}$ of graph $ \mathcal{R}$ represents the sum of the outdegree and indegree of a node. Note that $ \mathcal{P}$ is a diagonal array and the diagonal elements are $ {p}_{i}=\sum _{j\in {N}_{i}}\,{a}_{ij},i=\mathrm{1, 2},\dots ,n$ . Define the Laplacian matrix as $ \mathcal{L}=\mathcal{P}-\mathcal{A}$ , we have

(10) \begin{align} {\mathcal{L}}_{ij}=\left\{\begin{array}{l@{\quad}l}\sum\limits _{j\in {n}_{i}}\,{a}_{ij},\,i=j {} \\[4pt] -{a}_{ij},i\ne j {} . \end{array}\right.\end{align}

Consider the information sharing between leaders and followers to be unidirectional, i.e. the leaders can pass information to the followers, but the followers’ information cannot influence the leaders. Define $ \mathcal{D}=diag\left({d}_{1},{d}_{2},\dots ,{d}_{n}\right)$ , note that $ {d}_{i}=1$ when followers can get information from the leader, otherwise $ {d}_{i}=0$ . Denote $ \mathcal{H}=\mathcal{L}+\mathcal{D}$ as the communication topology matrix, which represents the information transmission relation between leaders and followers. When at least one follower can get information from the leader and the undirected graph is connected, $ \mathcal{H} \gt \gt 0$ .

2.3 Fixed-time theory

Lemma 6. [Reference Cheng, Xin, Wang, Gan, He and Ding30] Regarding a dynamic system $ \dot{\Phi }\left(t\right)=\chi (\Phi ,t),\Phi \left(0\right)={\Phi }_{0}$ , if the Lyapunov function $ L\left(x\right)$ is positive definite meeting

(11) \begin{align} \dot{L}\left(x\right){ \le }-{c}_{1}L\left(x\right)-{c}_{2}{L}^{p}\left(x\right)-{c}_{3}{L}^{q}\left(x\right)+{\delta }_{a}\end{align}

where $ {c}_{1}$ , $ {c}_{2}$ , $ {c}_{3}$ are positive constants, $ 0 \lt {\delta }_{a} \lt +\infty $ , $ p$ and $ q$ are odd numbers meeting $ 0 \lt p \lt 1$ , $ q \gt 1$ . Then the system exhibits semi-global practical fixed-time stability, with the residual set of the solution fulfills

(12) \begin{align} x\in \left\{x\left|L\right(x){ \le }\mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{{\delta }_{a}}{(1-{{\Delta }}_{\epsilon }){c}_{1}}{\left(\frac{{\delta }_{a}}{(1-{{\Delta }}_{\epsilon }){c}_{2}}\right)}^{\frac{1}{p}}{\left(\frac{{\delta }_{a}}{(1-{{\Delta }}_{\epsilon }){c}_{3}}\right)}^{\frac{1}{q}}\right\}\right\}\end{align}

where $ {{\Delta }}_{\epsilon }$ is a constant satisfying $ 0 \lt {{\Delta }}_{\epsilon } \lt 1$ . Then the calculation of the settling time is performed as below

(13) \begin{align} {T}_{s}{ \le }\mathrm{m}\mathrm{a}\mathrm{x}\left\{\frac{\mathrm{l}\mathrm{n}\left(1+\frac{{c}_{1}}{{{\Delta }}_{\epsilon }{c}_{2}}\right)}{{c}_{1}(1-p)},\frac{\mathrm{l}\mathrm{n}\left(1+\frac{{{\Delta }}_{\epsilon }{c}_{1}}{{c}_{2}}\right)}{{{\Delta }}_{\epsilon }{c}_{1}(1-p)}\right\}+\mathrm{m}\mathrm{a}\mathrm{x}\left\{\frac{\mathrm{l}\mathrm{n}\left(1+\frac{{c}_{1}}{{{\Delta }}_{\epsilon }{c}_{3}}\right)}{{c}_{1}(q-1)},\frac{\mathrm{l}\mathrm{n}\left(1+\frac{{{\Delta }}_{\epsilon }{c}_{1}}{{c}_{3}}\right)}{{{\Delta }}_{\epsilon }{c}_{1}(q-1)}\right\}\end{align}

By utilising inequality $ \mathrm{l}\mathrm{n}(1+x) \lt x,x \gt 0$ , we obtain

(14) \begin{align} \left\{\begin{array}{lll}\mathrm{m}\mathrm{a}\mathrm{x}\left\{\frac{\mathrm{l}\mathrm{n}\left(1+\frac{{}_{1}}{{{\Delta }}_{\epsilon }{c}_{2}}\right)}{{c}_{1}(1-p)},\frac{\mathrm{l}\mathrm{n}\left(1+\frac{{{\Delta }}_{\epsilon }{c}_{1}}{{c}_{2}}\right)}{{{\Delta }}_{\epsilon }{c}_{1}(1-p)}\right\} \lt \frac{1}{{c}_{2}{{\Delta }}_{}(1-p)} {} {} \\ \mathrm{m}\mathrm{a}\mathrm{x}\left\{\frac{\mathrm{l}\mathrm{n}\left(1+\frac{{c}_{1}}{{{\Delta }}_{\epsilon }{c}_{3}}\right)}{{c}_{1}(q-1)},\frac{\mathrm{l}\mathrm{n}\left(1+\frac{{{\Delta }}_{\epsilon }{c}_{1}}{{c}_{3}}\right)}{{{\Delta }}_{\epsilon }{c}_{1}(q-1)}\right\} \lt \frac{1}{{c}_{1}{{\Delta }}_{\epsilon }(q-1)} {} {} \end{array}\right.\end{align}

According to the conclusion in Cheng [Reference Cheng, Xin, Wang, Gan, He and Ding30], the upper bound of settling time $ {\mathrm{T}}_{\mathrm{s}}$ is tighter than that in Cui [Reference Cui, Xu, Chen and Yu27].

2.4 Radial basis function neural network (RBFNN)

In this section, the RBFNN can approximate an arbitrary smooth continuous function with arbitrary accuracy is introduced. Define input variable as $ x\in \mathbb{R}$ , weight vector as $ \boldsymbol{\Theta }\in {\mathbb{R}}^{p\times m}$ , Gaussian basis function vector as $ \mathbf{F}\left(x\right)={\left[{F}_{1}\left(x\right),\cdots ,{F}_{p}\left(x\right)\right]}^{T}$ , $ \boldsymbol{\Phi }_{j}\left(x\right)={e}^{-\frac{(x-{\mu }_{j}{)}^{2}}{2{{\sigma }_{j}}^{2}}},j=1,\dots ,p$ , where $ {\mu }_{j}$ represents the centre of the basis function, $ {\sigma }_{j}$ is the width of the Gaussian basis function. The expression of the RBFNN is denoted as $ {\Omega }={\boldsymbol{\Theta }}^{\mathrm{*}T}\mathbf{F}\left(x\right)+\epsilon \left(x\right)$ , where $ {\boldsymbol{\Theta }}^{\mathrm{*}}$ is the ideal weight vector, the basic functions satisfy $ ||\mathbf{F}\left(x\right)||\le {F}_{M}$ , $ \epsilon $ is the approximation error satisfying $ \left|\epsilon \right|\le {\epsilon }_{M}$ , $ \left|\dot{\epsilon }\right|\le {\epsilon }_{m}$ .

3.1 Expanded nonlinear affine system

Evidently, the system (3) exhibits significant nonlinearity and non-affine property due to the coupling of states $ {\boldsymbol{\Psi }}_{i,1}$ , $ {\boldsymbol{\Psi }}_{i,2}$ and control input $ {\mathbf{u}}_{i}$ , which leads to high-frequency jitters as noted in Shaffer [Reference Shaffer, Ross, Oppenheimer, Doman and Bollino32]. Thus, we add an additional auxiliary differential equation $ {\dot{\mathbf{u}}}_{i}={\mathbf{u}}_{fi}$ to the system. Define $ {\boldsymbol{\Psi }}_{i,3}={\mathbf{u}}_{i}$ , we rewrite the system (3) as follows

(15) \begin{align} \left\{\begin{array}{lll}{\dot{\boldsymbol{\Psi }}}_{i,1}={\boldsymbol{\Pi }}_{i,1}\left({\stackrel{-}{\boldsymbol{\Psi }}}_{i,2}\right)+{\Delta }{\boldsymbol{\Omega }}_{i,1}\left({\boldsymbol{\Psi }}_{i,1}\right) {} {} \\ {\dot{\boldsymbol{\Psi }}}_{i,2}={\boldsymbol{\Xi }}_{i}\left({\stackrel{-}{\boldsymbol{\Psi }}}_{i,2}\right)+{\boldsymbol{\Pi }}_{i,2}\left({\stackrel{-}{\boldsymbol{\Psi }}}_{i,3}\right)+{\Delta }{\boldsymbol{\Omega }}_{i,2}\left({\boldsymbol{\Psi }}_{i,1}\right) {} {} \\ {\dot{\boldsymbol{\Psi }}}_{i,3}={\mathbf{u}}_{fi} . \\ {\omega }_{i}={\boldsymbol{\Psi }}_{i,1} {} {} \end{array}\right.\end{align}

3.2 Adaptive RBFNN fixed-time dynamic event-triggered control protocol design

(16) \begin{align} \underset{t\to {T}_{s}}{\mathrm{l}\mathrm{i}\mathrm{m}}||{\omega }_{i}\left(t\right)-{\omega }_{d}\left(t\right)-{\omega }_{i}^{e}\left(t\right)||\le {{\Delta }}_{d}\end{align}

it is believed that the follower $ \mathrm{i}$ can track the trajectory of the leader within a fixed time, and the control objective is achieved. In (16), $ {\mathrm{T}}_{\mathrm{s}}$ is the fixed convergence time, $ {\boldsymbol{\omega }}_{\mathrm{d}}$ is the observed position of the leader, $ {\boldsymbol{\omega }}_{\mathrm{i}}^{\mathrm{e}}$ is the desired relative distance between the leader and the follower $ \mathrm{i}$ , $ {{\Delta }}_{\mathrm{d}}$ is a small positive constant.

To implement the controller, the following assumptions are proposed:

Step 1. Define the position tracking error for the follower $ i$ as

(17) \begin{align} {\mathbf{z}}_{i,1}=\sum _{j=1}^{K}\,{a}_{i,j}\left({\omega }_{i}-{\omega }_{j}\right)+{d}_{i}\left({\omega }_{i}-{\omega }_{d}-{\omega }_{i}^{e}\right)\end{align}

where $ {\omega }_{i}$ and $ {\omega }_{j}$ represent the outputs of follower $ i$ and follower $ j$ , respectively. $ {d}_{i}$ represents the connection between follower $ i$ and leader 0; $ {a}_{i,j}$ is the connection between follower $ i$ and $ j$ .

Define $ {l}_{i}=\sum _{j=1}^{K}\,{a}_{i,j}$ , combining (15-17), we have

(18) \begin{align} \begin{array}{ll}{\dot{\textbf{z}}}_{i,1} {} &=\left({l}_{i}+{d}_{i}\right){\dot{\omega }}_{i}-{l}_{i}{\dot{\omega }}_{j}-{d}_{i}{\dot{\omega }}_{d}-{d}_{i}{\dot{\omega }}_{i}^{e}\\ {} &=\left({l}_{i}+{d}_{i}\right)\left({\boldsymbol{\Pi }}_{i,1}+{\Delta }{\boldsymbol{\Omega }}_{i,1}\right)-{l}_{i}\left({\boldsymbol{\Pi }}_{j,1}+{\Delta }{\boldsymbol{\Omega }}_{j,1}\right)-{d}_{i}{\dot{\omega }}_{d} \end{array}\end{align}

Define $ {\Delta }{{\Omega }}_{1}=||\left({l}_{i}+{d}_{i}\right){\Delta }{\boldsymbol{\Omega }}_{i,1}-{l}_{i}{\Delta }{\boldsymbol{\Omega }}_{j,1}||$ , RBFNN is exploited to compensate the deviation induced by unknown nonlinear function $ {\Delta }{{\Omega }}_{1}$

(19) \begin{align} {\Delta }{\stackrel{-}{{\Omega }}}_{1}={\boldsymbol{\Theta }}_{i,1}^{T}{\mathbf{F}}_{i,1}+{\epsilon }_{i,1}\end{align}

in which $ {\mathbf{F}}_{i,1}$ is the basis function vector, $ {\boldsymbol{\Theta }}_{i,1}^{T}$ is the weight vector, $ \left|{\epsilon }_{i,1}\right|\le {\zeta }_{i,1}$ . Define $ {\vartheta }_{i,1}={||{\boldsymbol{\Theta }}_{i,1}||}^{2}$ , then the estimation error $ {\stackrel{\sim }{\vartheta }}_{i,1}$ can be written as $ {\stackrel{\sim }{\vartheta }}_{i,1}={\vartheta }_{i,1}-{\widehat{\vartheta }}_{i,1}$ , where $ {\widehat{\vartheta }}_{i,1}$ is the estimation of $ {\vartheta }_{i,1}$ .

In conventional backstepping control, the control signal $ {\boldsymbol{\Psi }}_{i}^{c}$ is subject to repeated differentiation, where increasing system order necessitates proportional differentiation operations, thereby inducing the EI phenomenon. To resolve this inherent limitation, we implement a fixed-time command filter (FCF) that

(20) \begin{align} {{\Delta }}_{\tau }{\dot{\boldsymbol{\Psi }}}_{i}^{d}=\left({\boldsymbol{\Psi }}_{i}^{d}-{\boldsymbol{\Psi }}_{i}^{c}\right){||{\boldsymbol{\Psi }}_{i}^{d}-{\boldsymbol{\Psi }}_{i}^{c}||}^{p-1}+\left({\boldsymbol{\Psi }}_{i}^{d}-{\boldsymbol{\Psi }}_{i}^{c}\right){||{\boldsymbol{\Psi }}_{i}^{d}-{\boldsymbol{\Psi }}_{i}^{c}||}^{q-1},{\boldsymbol{\Psi }}_{i}^{d}\left(0\right)={\boldsymbol{\Psi }}_{i}^{c}\left(0\right)\end{align}

where $ {\boldsymbol{\Psi }}_{i}^{d}$ is the output of the FCF and $ {\boldsymbol{\Psi }}_{i}^{c}$ is the input; $ {{\Delta }}_{\tau }$ stands for a little parameter.

To tackle the impact of filtering error, a compensative strategy is formulated as

(21) \begin{align} {\boldsymbol{\Upsilon }}_{i,n}={\mathbf{z}}_{i,n}-{\mathbf{w}}_{i,n},n=\mathrm{1, 2},3\end{align}

The first order error compensative signal $ {\mathbf{w}}_{i,1}$ is defined as

(22) \begin{align} {\dot{\mathbf{w}}}_{i,1}=-{\tau }_{i,1}{\mathbf{w}}_{i,1}-{\mu }_{i,1}{\mathbf{w}}_{i,1}{||{\mathbf{w}}_{i,1}||}^{p-1}-{\lambda }_{i,1}{\mathbf{w}}_{i,1}{||{\mathbf{w}}_{i,1}||}^{q-1}+\left({l}_{i}+{d}_{i}\right)\left({\boldsymbol{\Psi }}_{i,1}^{d}-{\boldsymbol{\Psi }}_{i,1}^{c}\right)\end{align}

in which $ {\tau }_{i,1}$ , $ {\mu }_{i,1}$ , $ {\lambda }_{i,1}$ are positive design parameters; $ p$ and $ q$ represent odd numbers to be designed, satisfying $ 0 \lt p \lt 1$ , $ q \gt 1$ .

The virtual control signal $ {\boldsymbol{\Psi }}_{i,1}^{c}$ of the follower $ i$ in the first order subsystem is devised as follows:

(23) \begin{align} {\boldsymbol{\Psi }}_{i,1}^{c}=\frac{1}{{l}_{i}+{d}_{i}}\left[\begin{array}{l}-{\tau }_{i,1}{\boldsymbol{\Upsilon }}_{i,1}-{\chi }_{i,1}{\boldsymbol{\Upsilon }}_{i,1}{||{\boldsymbol{\Upsilon }}_{i,1}||}^{p-1}-{\lambda }_{i,1}{\boldsymbol{\Upsilon }}_{i,1}{||{\boldsymbol{\Upsilon }}_{i,1}||}^{q-1}\\[4pt] -\dfrac{{\boldsymbol{\Upsilon }}_{i,1}}{2}-\dfrac{{\boldsymbol{\Upsilon }}_{i,1}}{2{\ell}_{i,1}^{2}}{\widehat{\vartheta }}_{i,1}{\mathbf{F}}_{i,1}^{T}{\mathbf{F}}_{i,1}+{l}_{i}{\boldsymbol{\Pi }}_{j,1}+{d}_{i}{\dot{\omega }}_{d}\end{array}\right]\end{align}

where $ {\chi }_{i,1}$ and $ {\lambda }_{i,1}$ are positive design parameters.

Define the second order subsystem error as $ {\mathbf{z}}_{i,2}={\boldsymbol{\Pi }}_{i,1}-{\boldsymbol{\Psi }}_{i,1}^{d}$ , Construct the Lyapunov function $ {L}_{i,1}=\frac{{\boldsymbol{\Upsilon }}_{i,1}^{T}{\boldsymbol{\Upsilon }}_{i,1}}{2}+\frac{{\mathbf{w}}_{i,1}^{T}{\mathbf{w}}_{i,1}}{2}+\frac{{\stackrel{\sim }{\vartheta }}_{i,1}^{2}}{2{\kappa }_{i,1}}$ , in which $ {\kappa }_{i,1}$ represents a positive parameter. According to the derivative of tracking error (18) and compensative strategy (21), we get

(24) \begin{align} {\dot{L}}_{i,1} {} &={\boldsymbol{\Upsilon }}_{i,1}^{T}{\dot{\boldsymbol{\Upsilon }}}_{i,1}+{\mathbf{w}}_{i,1}^{T}{\dot{\mathbf{w}}}_{i,1}+\frac{{\dot{\stackrel{\sim }{\vartheta }}}_{i,1}{\stackrel{\sim }{\vartheta }}_{i,1}}{{\kappa }_{i,1}}\nonumber\\ {} &=\left({l}_{i}+{d}_{i}\right){\boldsymbol{\Upsilon }}_{i,1}^{T}\left({\boldsymbol{\Pi }}_{i,1}+{\Delta }{\boldsymbol{\Omega }}_{i,1}\right)-{l}_{i}{\boldsymbol{\Upsilon }}_{i,1}^{T}\left({\boldsymbol{\Pi }}_{j,1}+{\Delta }{\boldsymbol{\Omega }}_{j,1}\right)\nonumber\\ {} &\quad-{d}_{i}{\boldsymbol{\Upsilon }}_{i,1}^{T}{\dot{\omega }}_{d}-{\boldsymbol{\Upsilon }}_{i,1}^{T}{\dot{\mathbf{w}}}_{i,1}+{\mathbf{w}}_{i,1}^{T}{\dot{\mathbf{w}}}_{i,1}-\frac{{\dot{\widehat{\vartheta }}}_{i,1}{\stackrel{\sim }{\vartheta }}_{i,1}}{{\kappa }_{i,1}}\end{align}

The RBFNN weight updating law is designed as

(25) \begin{align} {\dot{\widehat{\vartheta }}}_{i,1}=\frac{{\kappa }_{i,1}{\boldsymbol{\Upsilon }}_{i,1}^{T}{\boldsymbol{\Upsilon }}_{i,1}}{2{\ell}_{i,1}^{2}}{\mathbf{F}}_{i,1}^{T}{\mathbf{F}}_{i,1}-{\xi }_{i,1}{\widehat{\vartheta }}_{i,1}-{\varsigma }_{i,1}{\widehat{\vartheta }}_{i,1}^{p}-{\varpi }_{i,1}{\widehat{\vartheta }}_{i,1}^{q}\end{align}

with $ {\xi }_{i,1}$ , $ {\varsigma }_{i,1}$ and $ {\varpi }_{i,1}$ being parameters to be designed.

According to Young’s inequality and Lemma 2, we get

(26) \begin{align} \big(({l}_{i}+{d}_{i}){\boldsymbol{\Upsilon }}_{i,1}^{T}{\Delta }{\boldsymbol{\Omega }}_{i,1}-{l}_{i}{\boldsymbol{\Upsilon }}_{i,1}^{T}{\Delta }{\boldsymbol{\Omega }}_{j,1}\big) {} &\le \frac{{\vartheta }_{i,1}{\boldsymbol{\Upsilon }}_{i,1}^{T}{\boldsymbol{\Upsilon }}_{i,1}{\mathbf{F}}_{i,1}^{T}{\mathbf{F}}_{i,1}}{2{\ell}_{i,1}^{2}}+\frac{{\ell}_{i,1}^{2}}{2}+\frac{{||{\boldsymbol{\Upsilon }}_{i,1}||}^{2}}{2}+\frac{{\zeta }_{i,1}^{2}}{2}\nonumber\\ {\stackrel{\sim }{\vartheta }}_{i,1}{\widehat{\vartheta }}_{i,1} {} &\le \frac{{\vartheta }_{i,1}^{2}}{2}-\frac{{\stackrel{\sim }{\vartheta }}_{i,1}^{2}}{2}\nonumber\\ {\stackrel{\sim }{\vartheta }}_{i,1}{\widehat{\vartheta }}_{i,1}^{p} {} &\le \frac{2{\vartheta }_{i,1}^{p+1}}{p+1}-\frac{{\stackrel{\sim }{\vartheta }}_{i,1}^{p+1}}{p+1}\nonumber\\ {\stackrel{\sim }{\vartheta }}_{i,1}{\widehat{\vartheta }}_{i,1}^{q} {} &\le \frac{2{\vartheta }_{i,1}^{q+1}}{q+1}-\frac{{\stackrel{\sim }{\vartheta }}_{i,1}^{q+1}}{q+1} {} \end{align}

Through the synthesis of (22-26), it can be inferred that

(27) \begin{align} {\dot{L}}_{i,1} {} &\le -{\tau }_{i,1}{||{\boldsymbol{\Upsilon }}_{i,1}||}^{2}-{\chi }_{i,1}{||{\boldsymbol{\Upsilon }}_{i,1}||}^{p+1}-{\lambda }_{i,1}{||{\boldsymbol{\Upsilon }}_{i,1}||}^{q+1}+{\tau }_{i,1}{\boldsymbol{\Upsilon }}_{i,1}^{T}{\mathbf{w}}_{i,1}+{\mu }_{i,1}{\boldsymbol{\Upsilon }}_{i,1}^{T}{\mathbf{w}}_{i,1}{||{\mathbf{w}}_{i,1}||}^{p-1}\nonumber\\ {} &+{\lambda }_{i,1}{\boldsymbol{\Upsilon }}_{i,1}^{T}{\mathbf{w}}_{i,1}{||{\mathbf{w}}_{i,1}||}^{q-1}-{\tau }_{i,1}{||{\mathbf{w}}_{i,1}||}^{2}-{\mu }_{i,1}{||{\mathbf{w}}_{i,1}||}^{p+1}-{\lambda }_{i,1}{||{\mathbf{w}}_{i,1}||}^{q+1}\nonumber\\ {} &+\frac{{{\ell}}_{i,1}^{2}}{2}+\frac{{\zeta }_{i,1}^{2}}{2}+\left({l}_{i}+{d}_{i}\right){\boldsymbol{\Upsilon }}_{i,1}^{T}{\mathbf{z}}_{i,2}+\left({l}_{i}+{d}_{i}\right){\mathbf{w}}_{i,1}^{T}\left({\boldsymbol{\Psi }}_{i,1}^{d}-{\boldsymbol{\Psi }}_{i,1}^{c}\right)\nonumber\\ {} &+\frac{{\xi }_{i,1}}{{\kappa }_{i,1}}\left(\frac{{\vartheta }_{i,1}^{2}}{2}-\frac{{\stackrel{\sim }{\vartheta }}_{i,1}^{2}}{2}\right)+\frac{{\varsigma }_{i,1}}{{\kappa }_{i,1}}\left(\frac{2{\vartheta }_{i,1}^{p+1}}{p+1}-\frac{{\stackrel{\sim }{\vartheta }}_{i,1}^{p+1}}{p+1}\right)+\frac{{\varpi }_{i,1}}{{\kappa }_{i,1}}\left(\frac{2{\vartheta }_{i,1}^{q+1}}{q+1}-\frac{{\stackrel{\sim }{\vartheta }}_{i,1}^{q+1}}{q+1}\right) {} \end{align}

Step 2. According to Step 1, the error in second order subsystem for the follower $ i$ can be written as

(28) \begin{align} {\mathbf{z}}_{i,2}={\boldsymbol{\Pi }}_{i,1}-{\boldsymbol{\Psi }}_{i,1}^{d}\end{align}

The compensative signal $ {\mathbf{w}}_{i,2}$ is defined as

(29) \begin{align} {\dot{\mathbf{w}}}_{i,2}=-{\tau }_{i,2}{\mathbf{w}}_{i,2}-{\mu }_{i,2}{\mathbf{w}}_{i,2}{||{\mathbf{w}}_{i,2}||}^{p-1}-{\lambda }_{i,2}{\mathbf{w}}_{i,2}{||{\mathbf{w}}_{i,2}||}^{q-1}+\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Psi }}_{i,2}^{d}-\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Psi }}_{i,2}^{c}\end{align}

The non-affine item $ {\boldsymbol{\Pi }}_{i,1}$ takes partial derivatives of $ {\boldsymbol{\Psi }}_{i,1}$ and $ {\boldsymbol{\Psi }}_{i,2}$ respectively, we have

(30) \begin{align} \frac{{\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,1}}={0}^{3\times 3}\end{align}

(31) \begin{align} \frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}=\left[\begin{array}{c@{\quad}c@{\quad}c}\mathrm{cos}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}}\mathrm{cos}{\boldsymbol{\Psi }}_{i,{2}_{\left(3\right)}} & {} {-}{\boldsymbol{\Psi }}_{i,{2}_{\left(1\right)}}\mathrm{sin}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}}\mathrm{cos}{\boldsymbol{\Psi }}_{i,{2}_{\left(3\right)}} & -{\boldsymbol{\Psi }}_{i,{2}_{\left(1\right)}}\mathrm{cos}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}}\mathrm{sin}{\boldsymbol{\Psi }}_{i,{2}_{\left(3\right)}} {} \\ \mathrm{sin}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}} {}& {\boldsymbol{\Psi }}_{i,{2}_{\left(1\right)}}\mathrm{cos}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}} & 0 {} \\ \mathrm{cos}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}}\mathrm{sin}{\boldsymbol{\Psi }}_{i,{2}_{\left(3\right)}} & -{\boldsymbol{\Psi }}_{i,{2}_{\left(1\right)}}\mathrm{sin}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}}\mathrm{sin}{\boldsymbol{\Psi }}_{i,{2}_{\left(3\right)}} & {\boldsymbol{\Psi }}_{i,{2}_{\left(1\right)}}\mathrm{cos}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}}\mathrm{cos}{\boldsymbol{\Psi }}_{i,{2}_{\left(3\right)}} {} \end{array}\right]\end{align}

Introduce the error in third order subsystem as $ {\mathbf{z}}_{i,3}={\boldsymbol{\Pi }}_{i,2}-{\boldsymbol{\Psi }}_{i,2}^{d}$ . Thus, combining dynamics model (15) and compensative strategy (21), we take the derivative of $ {\mathbf{z}}_{i,2}$ as

(32) \begin{align} {\dot{\mathbf{z}}}_{i,2} {} &={\dot{\boldsymbol{\Pi }}}_{i,1}-{\dot{\boldsymbol{\Psi }}}_{i,1}^{d}\nonumber\\ {}&=\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,1}}{\dot{\boldsymbol{\Psi }}}_{i,1}+\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\dot{\boldsymbol{\Psi }}}_{i,2}-{\dot{\boldsymbol{\Psi }}}_{i,1}^{d}\nonumber\\ {}&=\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,1}}\left({\boldsymbol{\Pi }}_{i,1}+{\Delta }{\boldsymbol{\Omega }}_{i,1}\right)+\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}\left({\boldsymbol{\Xi }}_{i}+{\boldsymbol{\Upsilon }}_{i,3}+{\mathbf{w}}_{i,3}+{\boldsymbol{\Psi }}_{i,2}^{d}+{\Delta }{\boldsymbol{\Omega }}_{i,2}\right)-{\dot{\boldsymbol{\Psi }}}_{i,1}^{d} {} \end{align}

Similarly to step 1, define $ {\Delta }{{\Omega }}_{2}=||\sum _{k=1}^{2}\,\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,k}}{\Delta }{\boldsymbol{\Omega }}_{i,k}||$ , the estimation based on RBFNN is written as $ {\Delta }{\stackrel{-}{{\Omega }}}_{2}={\boldsymbol{\Theta }}_{i,2}^{T}{\mathbf{F}}_{i,2}+{\epsilon }_{i,2}$ , where $ \left|{\epsilon }_{i,2}\right|\le {\zeta }_{i,2}$ . Define $ {\vartheta }_{i,2}={||{\Theta }_{i,2}||}^{2}$ , the estimation of $ {\vartheta }_{i,2}$ as $ {\widehat{\vartheta }}_{i,2}$ , then the estimation error $ {\stackrel{\sim }{\vartheta }}_{i,2}$ can be written as $ {\stackrel{\sim }{\vartheta }}_{i,2}={\vartheta }_{i,2}-{\widehat{\vartheta }}_{i,2}$ .

We devise the virtual control signal $ {\boldsymbol{\Psi }}_{i,2}^{c}$ of the follower $ i$ in the second order subsystem as

(33) \begin{align} \begin{array}{l}{\boldsymbol{\Psi }}_{i,2}^{c}={\left(\dfrac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}\right)}^{-1}\left(\begin{array}{l}-{\tau }_{,2}{\boldsymbol{\Upsilon }}_{i,2}-{\chi }_{i,2}{\boldsymbol{\Upsilon }}_{i,2}{||{\boldsymbol{\Upsilon }}_{i,2}||}^{p-1}-{\lambda }_{i,2}{\boldsymbol{\Upsilon }}_{i,2}{||{\boldsymbol{\Upsilon }}_{i,2}||}^{q-1}-\frac{{\boldsymbol{\Upsilon }}_{i,2}}{2}\\[6pt] -\dfrac{{}_{i,2}}{2{\ell}_{i,2}^{2}}{\widehat{\vartheta }}_{i,2}{\mathbf{F}}_{i,2}^{T}{\mathbf{F}}_{i,2}-\dfrac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,1}}{\boldsymbol{\Pi }}_{i,1}-\dfrac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Xi }}_{i}+{\dot{\boldsymbol{\Psi }}}_{i,1}^{d}\end{array}\right)\end{array}\end{align}

Let us consider the Lyapunov function as $ {L}_{i,2}=\frac{{\boldsymbol{\Upsilon }}_{i,2}^{T}{\boldsymbol{\Upsilon }}_{i,2}}{2}+\frac{{\mathbf{w}}_{i,2}^{T}{\mathbf{w}}_{i,2}}{2}+\frac{{\stackrel{\sim }{\vartheta }}_{i,2}^{2}}{2{\kappa }_{i,2}}$ . By merging (32), we get

(34) \begin{align} {\dot{L}}_{i,2} {} &={\boldsymbol{\Upsilon }}_{i,2}^{T}{\dot{\boldsymbol{\Upsilon }}}_{i,2}+{\mathbf{w}}_{i,2}^{T}{\dot{\mathbf{w}}}_{i,2}+\frac{{\dot{\stackrel{\sim }{\vartheta }}}_{i,2}{\stackrel{\sim }{\vartheta }}_{i,2}}{{\kappa }_{i,2}}\nonumber\\ {} &={\boldsymbol{\Upsilon }}_{i,2}^{T}\left({\dot{\mathbf{z}}}_{i,2}-{\dot{\mathbf{w}}}_{i,2}\right)+{\mathbf{w}}_{i,2}^{T}{\dot{\mathbf{w}}}_{i,2}+\frac{\left({\dot{\vartheta }}_{i,2}-{\dot{\widehat{\vartheta }}}_{i,2}\right){\stackrel{\sim }{\vartheta }}_{i,2}}{{\kappa }_{i,2}}\nonumber\\ {} &={\boldsymbol{\Upsilon }}_{i,2}^{T}\left(\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,1}}{\Delta }{\boldsymbol{\Omega }}_{i,1}+\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\Delta }{\boldsymbol{\Omega }}_{i,2}\right)-{\boldsymbol{\Upsilon }}_{i,2}^{T}{\dot{\boldsymbol{\Psi }}}_{i,1}^{d}-{\boldsymbol{\Upsilon }}_{i,2}^{T}{\dot{\mathbf{w}}}_{i,2}+{\mathbf{w}}_{i,2}^{T}{\dot{\mathbf{w}}}_{i,2}-\frac{{\dot{\widehat{\vartheta }}}_{i,2}{\stackrel{\sim }{\vartheta }}_{i,2}}{{\kappa }_{i,2}}\nonumber\\[4pt] {} &\quad+{\boldsymbol{\Upsilon }}_{i,2}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,1}}{\boldsymbol{\Pi }}_{i,1}+{\boldsymbol{\Upsilon }}_{i,2}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Xi }}_{i}+{\boldsymbol{\Upsilon }}_{i,2}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Upsilon }}_{i,3}+{\boldsymbol{\Upsilon }}_{i,2}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}\left({\mathbf{w}}_{i,3}+{\boldsymbol{\Psi }}_{i,2}^{d}\right)\end{align}

Similarly, the RBFNN weight updating law is designed as

(35) \begin{align} {\dot{\widehat{\vartheta }}}_{i,2}=\frac{{\kappa }_{i,2}{\boldsymbol{\Upsilon }}_{i,2}^{T}{\boldsymbol{\Upsilon }}_{i,2}}{2{{\ell}}_{i,2}^{2}}{\mathbf{F}}_{i,2}^{T}{\mathbf{F}}_{i,2}-{\xi }_{i,2}{\widehat{\vartheta }}_{i,2}-{\varsigma }_{i,2}{\widehat{\vartheta }}_{i,2}^{p}-{\varpi }_{i,2}{\widehat{\vartheta }}_{i,2}^{q}\end{align}

By applying Lemma 1 and Lemma 2, the inequality (36) is obtained:

(36) \begin{align} {\boldsymbol{\Upsilon }}_{i,2}\bigg(\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,1}}{\Delta }{\boldsymbol{\Omega }}_{i,1}+\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\Delta }{\boldsymbol{\Omega }}_{i,2}\!\bigg) {} &\le \frac{{\vartheta }_{i,2}{\boldsymbol{\Upsilon }}_{i,2}^{T}{\boldsymbol{\Upsilon }}_{i,2}{\mathbf{F}}_{i,2}^{T}{\mathbf{F}}_{i,2}}{2{\ell}_{i,2}^{2}}+\frac{{\ell}_{i,2}^{2}}{2}+\frac{{||{\boldsymbol{\Upsilon }}_{i,2}||}^{2}}{2}+\frac{{\zeta }_{i,2}^{2}}{2}\nonumber\\ {\stackrel{\sim }{\vartheta }}_{,2}{\widehat{\vartheta }}_{i,2} {} &\le \frac{{\vartheta }_{i,2}^{2}}{2}-\frac{{\stackrel{\sim }{\vartheta }}_{i,2}^{2}}{2}\nonumber\\ {\stackrel{\sim }{\vartheta }}_{i,2}{\widehat{\vartheta }}_{i,2}^{p} {} &\le \frac{2{\vartheta }_{i,2}^{p+1}}{p+1}-\frac{{\stackrel{\sim }{\vartheta }}_{i,2}^{p+1}}{p+1}\nonumber\\ {\stackrel{\sim }{\vartheta }}_{i,2}{\widehat{\vartheta }}_{i,2}^{q} {} &\le \frac{2{\vartheta }_{i,2}^{q+1}}{q+1}-\frac{{\stackrel{\sim }{\vartheta }}_{i,2}^{q+1}}{q+1} {} \end{align}

By substituting compensative signal (29), virtual control signal (33), RBFNN weight updating law (35) and the inequality (36) into (34), we get

(37) \begin{align} {\dot{L}}_{i,2} {} &\le -{\tau }_{i,2}{||{\boldsymbol{\Upsilon }}_{i,2}||}^{2}-{\chi }_{i,2}{||{\boldsymbol{\Upsilon }}_{i,2}||}^{p+1}-{\lambda }_{i,2}{||{\boldsymbol{\Upsilon }}_{i,2}||}^{q+1}+{\tau }_{i,2}{\boldsymbol{\Upsilon }}_{i,2}^{T}{\mathbf{w}}_{i,2}+{\mu }_{i,2}{\boldsymbol{\Upsilon }}_{i,2}^{T}{\mathbf{w}}_{i,2}{||{\mathbf{w}}_{i,2}||}^{p-1}\nonumber\\ {} &\quad+{\lambda }_{i,2}{\boldsymbol{\Upsilon }}_{i,2}^{T}{\mathbf{w}}_{i,2}{||{\mathbf{w}}_{i,2}||}^{q-1}-{\tau }_{i,2}{||{\mathbf{w}}_{i,2}||}^{2}-{\mu }_{i,2}{||{\mathbf{w}}_{i,2}||}^{p+1}-{\lambda }_{i,2}{||{\mathbf{w}}_{i,2}||}^{q+1}+\frac{{\ell}_{i,2}^{2}}{2}+\frac{{\zeta }_{i,2}^{2}}{2}\nonumber\\ {} &\quad+{\boldsymbol{\Upsilon }}_{i,2}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Upsilon }}_{i,3}+{\mathbf{w}}_{i,2}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}\left({\boldsymbol{\Psi }}_{i,2}^{d}-{\boldsymbol{\Psi }}_{i,2}^{c}\right)\nonumber\\ {} &\quad+\frac{{\xi }_{i,2}}{{\kappa }_{i,2}}\left(\frac{{\vartheta }_{i,2}^{2}}{2}-\frac{{\stackrel{\sim }{\vartheta }}_{i,2}^{2}}{2}\right)+\frac{{\varsigma }_{i,2}}{{\kappa }_{i,2}}\left(\frac{2{\vartheta }_{i,2}^{p+1}}{p+1}-\frac{{\stackrel{\sim }{\vartheta }}_{i,2}^{p+1}}{p+1}\right)+\frac{{\varpi }_{i,2}}{{\kappa }_{i,2}}\left(\frac{2{\vartheta }_{i,2}^{q+1}}{q+1}-\frac{{\stackrel{\sim }{\vartheta }}_{i,2}^{q+1}}{q+1}\right)\end{align}

Step 3. According to Step 2, we express the third-order error for the follower $ i$ in terms of

(38) \begin{align} {\mathbf{z}}_{i,3}={\boldsymbol{\Pi }}_{i,2}-{\boldsymbol{\Psi }}_{i,2}^{d}\end{align}

The compensative signal $ {\mathbf{w}}_{i,3}$ is developed as

(39) \begin{align} {\dot{\mathbf{w}}}_{i,3}=-{\tau }_{i,3}{\mathbf{w}}_{i,3}-{\mu }_{i,3}{\mathbf{w}}_{i,3}{||{\mathbf{w}}_{i,3}||}^{p-1}-{\lambda }_{i,3}{\mathbf{w}}_{i,3}{||{\mathbf{w}}_{i,3}||}^{q-1}\end{align}

Thus, the derivative of $ {\mathbf{z}}_{i,3}$ becomes

(40) \begin{align} {\dot{\mathbf{z}}}_{i,3} {} &={\dot{\boldsymbol{\Pi }}}_{i,2}-{\dot{\boldsymbol{\Psi }}}_{i,2}^{d}\nonumber\\ {} &=\sum _{k=1}^{3}\,\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,k}}{\dot{\boldsymbol{\Psi }}}_{i,k}-{\dot{\boldsymbol{\Psi }}}_{i,2}^{d}\nonumber\\ {} &=\sum _{k=1}^{2}\,\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,k}}\left({\boldsymbol{\Pi }}_{i,k}+{\Delta }{\boldsymbol{\Omega }}_{i,k}\right)+\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Xi }}_{i}+\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\mathbf{u}}_{fi}-{\dot{\boldsymbol{\Psi }}}_{i,2}^{d}\end{align}

The non-affine item $ {\boldsymbol{\Pi }}_{i,2}$ takes partial derivatives of $ {\boldsymbol{\Psi }}_{i,1}$ , $ {\boldsymbol{\Psi }}_{i,2}$ and $ {\boldsymbol{\Psi }}_{i,3}$ respectively, we have

(41) \begin{align} \frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,1}}=-\frac{{\boldsymbol{\Pi }}_{i,2}\otimes \left[\begin{array}{lll}1 {} 0 {} 0\end{array}\right]}{{h}_{s}}\end{align}

where $ \otimes $ means the Kronecker product.

(42) \begin{align} \frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {}_{i,2}}=\left[\begin{array}{c@{\quad}c@{\quad}c}\dfrac{{\boldsymbol{\Psi }}_{i,{3}_{\left(1\right)}}{\rho }_{i}{\boldsymbol{\Psi }}_{i,{2}_{\left(1\right)}}{S}_{i}}{{m}_{i}} {}& 0 {}& 0 {} \\[4pt] \dfrac{{\boldsymbol{\Psi }}_{i,{3}_{\left(2\right)}}{\rho }_{i}{S}_{i}}{2{m}_{i}} {}& 0 {} &0 {} \\[4pt] \dfrac{{\boldsymbol{\Psi }}_{i,{3}_{\left(3\right)}}{\rho }_{i}{S}_{i}}{2{m}_{i}\mathrm{cos}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}}} {}& \dfrac{{\boldsymbol{\Psi }}_{i,{3}_{\left(3\right)}}{\rho }_{i}{S}_{i}{\boldsymbol{\Psi }}_{i,{2}_{\left(1\right)}}\mathrm{t}\mathrm{a}\mathrm{n}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}}}{2{m}_{i}\mathrm{cos}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}}} {}& 0 {} \end{array}\right]\end{align}

(43) \begin{align} \frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}=diag\left(\begin{array}{l}\dfrac{{\rho }_{i}{{\boldsymbol{\Psi }}_{i,2}^{2}}_{{\mathrm{}}_{\left(1\right)}}{S}_{i}}{2{m}_{i}}+{{\mathrm{s}ech}_{}}^{2}\left({{\boldsymbol{\Psi }}_{i,3}}_{{\mathrm{}}_{\left(1\right)}}\right),\dfrac{{\rho }_{i}{{\boldsymbol{\Psi }}_{i,2}}_{{\mathrm{}}_{\left(1\right)}}{S}_{i}}{2{m}_{i}}+{{\mathrm{s}ech}_{}}^{2}\left({{\boldsymbol{\Psi }}_{i,3}}_{{\mathrm{}}_{\left(2\right)}}\right),\\[12pt] \dfrac{{\rho }_{i}{{\boldsymbol{\Psi }}_{i,2}}_{{\mathrm{}}_{\left(1\right)}}{S}_{i}}{2{m}_{i}\mathrm{cos}{{\boldsymbol{\Psi }}_{i,2}}_{{\mathrm{}}_{\left(2\right)}}}+{{\mathrm{s}ech}_{}}^{2}\left({{\boldsymbol{\Psi }}_{i,3}}_{{\mathrm{}}_{\left(3\right)}}\right)\end{array}\right)\end{align}

Similarly, define $ {\Delta }{{\Omega }}_{3}=||\sum _{k=1}^{2}\,\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,k}}{\Delta }{\boldsymbol{\Omega }}_{i,k}||$ , exploit RBFNN to compensate the deviation:

(44) \begin{align} {\Delta }{\stackrel{-}{{\Omega }}}_{3}={\boldsymbol{\Theta }}_{i,3}^{T}{\mathbf{F}}_{i,3}+{\epsilon }_{i,3}\end{align}

where $ {\mathbf{F}}_{i,3}$ is the basis function vector, $ {\boldsymbol{\Theta }}_{i,3}^{T}$ is the weight vector, $ \left|{\epsilon }_{i,3}\right|\le {\zeta }_{i,3}$ . Define $ {\vartheta }_{i,3}={||{\boldsymbol{\Theta }}_{i,3}||}^{2}$ , then the estimation error $ {\stackrel{\sim }{\vartheta }}_{i,3}$ can be written as $ {\stackrel{\sim }{\vartheta }}_{i,3}={\vartheta }_{i,3}-{\widehat{\vartheta }}_{i,3}$ , where $ {\widehat{\vartheta }}_{i,3}$ is the estimation of $ {\vartheta }_{i,3}$ .

We devise the consensus control protocol $ {\boldsymbol{\Psi }}_{i,3}^{c}$ of the follower $ i$ as

(45) \begin{align} \begin{array}{l}{\boldsymbol{\Psi }}_{i,3}^{c}={\left(\dfrac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}\right)}^{-1}\left(\begin{array}{l}-{\tau }_{i,3}{\boldsymbol{\Upsilon }}_{i,3}-{\chi }_{i,3}{\boldsymbol{\Upsilon }}_{i,3}{||{\boldsymbol{\Upsilon }}_{i,3}||}^{p-1}-{\lambda }_{i,3}{\boldsymbol{\Upsilon }}_{i,3}{||{\boldsymbol{\Upsilon }}_{i,3}||}^{q-1}-\dfrac{{\boldsymbol{\Upsilon }}_{i,3}}{2}\\ -\dfrac{{\boldsymbol{\Upsilon }}_{i,3}}{2{\ell}_{i,j}^{2}}{\widehat{\vartheta }}_{i}{\mathbf{F}}_{i,3}^{T}{\mathbf{F}}_{i,3}-\sum _{k=1}^{2}\,\dfrac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,k}}{\boldsymbol{\Pi }}_{i,k}-\dfrac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Xi }}_{i}+{\dot{\boldsymbol{\Psi }}}_{i,2}^{d}\end{array}\right)\end{array}\end{align}

In terms of the designed real-time continuous control law $ {\boldsymbol{\Psi }}_{i,3}^{c}$ causing unnecessary waste of communication resources, a dynamic event-triggered controller is developed as

(46) \begin{align} {\mathbf{f}}_{i}\left(t\right)={\boldsymbol{\Psi }}_{i,3}^{c}\left(t\right)-\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\boldsymbol{\Upsilon }}_{i,3}{\widehat{\mathit{\eta }}}_{i}^{T}{\widehat{\mathit{\eta }}}_{i}/\sqrt{{\left({\widehat{\mathit{\eta }}}_{i}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\boldsymbol{\Upsilon }}_{i,3}\right)}^{2}+{\Lambda }^{2}}\end{align}

Considering the specific format of the error variables as $ {\mathbf{e}}_{i}\left(t\right)={\mathbf{f}}_{i}\left(t\right)-{\mathbf{u}}_{fi}\left(t\right)$ , define $ {\mathbf{e}}_{i}={\left[{e}_{i,1},{e}_{i,2},{e}_{i,3}\right]}^{T}$ , we accessorily design the dynamic event-triggered condition as

(47) \begin{align} \left\{\begin{array}{llllllllllllllllllll}{\mathbf{u}}_{fi}\left(t\right)={\mathbf{f}}_{i}\left({t}_{n}\right),t\in \left[{t}_{n},{t}_{n+1}\right) {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} {} \\ {t}_{n+1}=\mathrm{i}\mathrm{n}\mathrm{f}\left\{t\in ,\left|{e}_{i,k}\left(t\right)\right|\ge {\wp }_{i,k}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left|\frac{{u}_{fi,k}\left(t\right)}{{\hslash }_{i,k}}\right|+{\eta }_{i,k}\left(t\right),k=\mathrm{1, 2},3\right\},\end{array}\right.\end{align}

in which $ {\wp }_{i}={\left[{\wp }_{i,1},{\wp }_{i,2},{\wp }_{i,3}\right]}^{T}$ , $ {\hslash }_{i}={\left[{\hslash }_{i,1},{\hslash }_{i,2},{\hslash }_{i,3}\right]}^{T}$ , both are positive parameter vectors to be set; $ {\widehat{\mathit{\eta }}}_{i}={\wp }_{i}+{\mathit{\eta }}_{i}\left(t\right)$ , $ {\mathit{\eta }}_{i}\left(t\right)={\left[{\eta }_{i,1},{\eta }_{i,2},{\eta }_{i,3}\right]}^{T}$ satisfies

(48) \begin{align} {\dot{\eta }}_{i,k}\left(t\right)=-{\partial }_{i,k}{\eta }_{i,k}\left(t\right)-{B}_{i,k}{\eta }_{i,k}^{p}\left(t\right)-{\aleph }_{i,k}{\eta }_{i,k}^{q}\left(t\right)+{\wp }_{i,k}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left|\frac{{u}_{fi,k}\left(t\right)}{{\hslash }_{i,k}}\right|-\left|{e}_{i,k}\left(t\right)\right|\end{align}

In (48), $ {\mathit{\eta }}_{i}\left(t\right)$ means a dynamic internal trigger variable with $ {\mathit{\eta }}_{i,k}\left(0\right) \gt 0$ ; $ {\partial }_{i}={\left[{\partial }_{i,1},{\partial }_{i,2},{\partial }_{i,3}\right]}^{T}$ , $ {\mathbf{B}}_{i}={\left[{B}_{i,1},{B}_{i,2},{B}_{i,3}\right]}^{T}$ , $ {\aleph }_{i}={\left[{\aleph }_{i,1},{\aleph }_{i,2},{\aleph }_{i,3}\right]}^{T}$ , they denote positive design parameter vectors.

Remark 1: On the basis of switching condition (47), it can be delivered when $ t\in [0,+{\infty })$ , always satisfy $ {\wp }_{i,k}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left|\frac{{u}_{fi,k}\left(t\right)}{{\hslash }_{i,k}}\right|-\left|{e}_{i,k}\left(t\right)\right| \gt -{\eta }_{i,k}\left(t\right)$ . Then, (48) can be expressed alternatively as $ {\dot{\eta }}_{i,k}\left(t\right) \gt -\left({\partial }_{i,k}+1\right){\eta }_{i,k}\left(t\right)-{B}_{i,k}{\eta }_{i,k}^{p}\left(t\right)-{\aleph }_{i,k}{\eta }_{i,k}^{q}\left(t\right)$ . According to the fixed time theory, $ \underset{t\to {T}_{s}}{{\mathrm{l}\mathrm{i}\mathrm{m}}_{}}\mathrm{}{\eta }_{i}=0,\underset{{\eta }_{i}\to {0}^{+}}{{\mathrm{l}\mathrm{i}\mathrm{m}}_{}}\mathrm{}{\dot{\eta }}_{i}=0$ . By utilising switching condition (47), it has $ \left|{e}_{i,k}\left(t\right)\right| \lt {\wp }_{i,k}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left|\frac{{u}_{fi,k}\left(t\right)}{{\hslash }_{i,k}}\right|+{\eta }_{i,k}\left(t\right) \lt {\wp }_{i,k}+{\eta }_{i,k}\left(t\right)$ when $ t\in [{t}_{n},{t}_{n+1})$ . Define $ {\mathbf{u}}_{fi}\left(t\right)={\mathbf{f}}_{i}\left(t\right)-{\tau }_{i}\left({\wp }_{i}+{\mathit{\eta }}_{i}\left(t\right)\right)$ , where $ {\tau }_{i}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\frac{{f}_{i,k}\left(t\right)-{u}_{fi,k}\left(t\right)}{{\wp }_{i,k}+{\mathit{\eta }}_{i,k}\left(t\right)}\right\},k=\mathrm{1, 2},3$ is continuous meeting $ \left|{\tau }_{i}\right| \lt 1$ . Then, choose the Lyapunov function $ {L}_{i,3}=\frac{{\boldsymbol{\Upsilon }}_{i,3}^{T}{\boldsymbol{\Upsilon }}_{i,3}}{2}+\frac{{\mathbf{w}}_{i,3}^{T}{\mathbf{w}}_{i,3}}{2}+\frac{{\stackrel{\sim }{\vartheta }}_{i,3}^{2}}{2{\kappa }_{i,3}}$ , we take the derivative of $ {L}_{i,3}$ combining (21) and (40) as

(49) \begin{align} {\dot{L}}_{i,3} {} &={\boldsymbol{\Upsilon }}_{i,3}^{T}{\dot{\boldsymbol{\Upsilon }}}_{i,3}+{\mathbf{w}}_{i,3}^{T}{\dot{\mathbf{w}}}_{i,3}+\frac{{\dot{\stackrel{\sim }{\vartheta }}}_{i,3}{\stackrel{\sim }{\vartheta }}_{i,3}}{{\kappa }_{i,3}}\nonumber\\ {}& ={\boldsymbol{\Upsilon }}_{i,3}^{T}\left({\dot{\mathbf{z}}}_{i,3}-{\dot{\mathbf{w}}}_{i,3}\right)+{\mathbf{w}}_{i,3}^{T}{\dot{\mathbf{w}}}_{i,3}+\frac{\left({\dot{\vartheta }}_{i,3}-{\dot{\widehat{\vartheta }}}_{i,3}\right){\stackrel{\sim }{\vartheta }}_{i,3}}{{\kappa }_{i,3}}\nonumber\\ {}& ={\boldsymbol{\Upsilon }}_{i,3}^{T}\left(\sum _{k=1}^{2}\,\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,k}}\left({\boldsymbol{\Pi }}_{i,k}+{\Delta }{\boldsymbol{\Omega }}_{i,k}\right)+\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Xi }}_{i}\right)+{\boldsymbol{\Upsilon }}_{i,3}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\boldsymbol{\Psi }}_{i,3}^{c}\nonumber\\ {}&\quad -{\boldsymbol{\Upsilon }}_{i,3}^{T}{\dot{\boldsymbol{\Psi }}}_{i,2}^{d}-{\boldsymbol{\Upsilon}}_{i,3}^{T}{\dot{\mathbf{w}}}_{i,3}+{\mathbf{w}}_{i,3}^{T}{\dot{\mathbf{w}}}_{i,3}-\frac{{\dot{\widehat{\vartheta }}}_{i,3}{\stackrel{\sim }{\vartheta }}_{i,3}}{{\kappa }_{i,3}}\nonumber\\ &\quad +{\boldsymbol{\Upsilon }}_{i,3}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}\left(-{\tau }_{i}\left({\wp }_{i}+{\mathit{\eta }}_{i}\left(t\right)\right)-\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\boldsymbol{\Upsilon }}_{i,3}{\widehat{\mathit{\eta }}}_{i}^{T}{\widehat{\mathit{\eta }}}_{i}/\sqrt{{\left({\widehat{\mathit{\eta }}}_{i}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\boldsymbol{\Upsilon }}_{i,3}\right)}^{2}+{\Lambda }^{2}}\right)\end{align}

The RBFNN weight updating law is designed as

(50) \begin{align} {\dot{\widehat{\vartheta }}}_{i,3}=\frac{{\kappa }_{i,3}{\boldsymbol{\Upsilon }}_{i,3}^{T}{\boldsymbol{\Upsilon }}_{i,3}}{2{\ell}_{i,3}^{2}}{\mathbf{F}}_{i,3}^{T}{\mathbf{F}}_{i,3}-{\xi }_{i,3}{\widehat{\vartheta }}_{i,3}-{\varsigma }_{i,3}{\widehat{\vartheta }}_{i,3}^{p}-{\varpi }_{i,3}{\widehat{\vartheta }}_{i,3}^{q}\end{align}

With the help of Young’s inequality and Lemma 2, we get

(51) \begin{align} {\boldsymbol{\Upsilon }}_{i,3}^{T}\sum _{k=1}^{2}\,\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,k}}{\Delta }{\boldsymbol{\Omega }}_{i,k} {} &\le \frac{{\vartheta }_{i,3}{\boldsymbol{\Upsilon }}_{i,3}^{T}{\boldsymbol{\Upsilon }}_{i,3}{\mathbf{F}}_{i,3}^{T}{\mathbf{F}}_{i,3}}{2{\ell}_{i,3}^{2}}+\frac{{\ell}_{i,3}^{2}}{2}+\frac{{||{\boldsymbol{\Upsilon }}_{i,3}||}^{2}}{2}+\frac{{\zeta }_{i,3}^{2}}{2}\nonumber\\ {\stackrel{\sim }{\vartheta }}_{i,3}{\widehat{\vartheta }}_{i,3} {} &\le \frac{{\vartheta }_{i,3}^{2}}{2}-\frac{{\stackrel{\sim }{\vartheta }}_{i,3}^{2}}{2}\nonumber\\ {\stackrel{\sim }{\vartheta }}_{i,3}{\widehat{\vartheta }}_{i,3}^{p} {} &\le \frac{2{\vartheta }_{i,3}^{p+1}}{p+1}-\frac{{\stackrel{\sim }{\vartheta }}_{i,3}^{p+1}}{p+1}\nonumber\\ {\stackrel{\sim }{\vartheta }}_{i,3}{\widehat{\vartheta }}_{i,3}^{q} {} &\le \frac{2{\vartheta }_{i,3}^{q+1}}{q+1}-\frac{{\stackrel{\sim }{\vartheta }}_{i,3}^{q+1}}{q+1}\end{align}

Obviously, $ -{\tau }_{i}\left({\wp }_{i}+{\mathit{\eta }}_{i}\left(t\right)\right)=-{\tau }_{i}{\widehat{\mathit{\eta }}}_{i}$ . Based on Lemma 4 and Lemma 5, one obtains

(52) \begin{align} {\boldsymbol{\Upsilon }}_{i,3}^{T}&\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}\left(-{\tau }_{i}\left({\wp }_{i}+{\mathit{\eta }}_{i}\left(t\right)\right)-\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\boldsymbol{\Upsilon }}_{i,3}{\widehat{\mathit{\eta }}}_{i}^{T}{\widehat{\mathit{\eta }}}_{i}/\sqrt{{\left({\widehat{\mathit{\eta }}}_{i}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\boldsymbol{\Upsilon }}_{i,3}\right)}^{2}+{\Lambda }^{2}}\right)\nonumber\\ &\le {\boldsymbol{\Upsilon }}_{i,3}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\left[\left|{\widehat{\eta }}_{i,1}\right|,\left|{\widehat{\eta }}_{i,2}\right|,\left|{\widehat{\eta }}_{i,3}\right|\right]}^{T}-{\boldsymbol{\Upsilon }}_{i,3}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\boldsymbol{\Upsilon }}_{i,3}{\widehat{\mathit{\eta }}}_{i}^{T}{\widehat{\mathit{\eta }}}_{i}/\sqrt{{\left({\widehat{\mathit{\eta }}}_{i}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,2}}{\partial {\boldsymbol{\Psi }}_{i,3}}{\boldsymbol{\Upsilon }}_{i,3}\right)}^{2}+{\Lambda }^{2}}\nonumber\\ &\le {\Lambda }\end{align}

Substituting Equations (39), (45) and (50-52) into Equation (49) yields

(53) \begin{align} {\dot{L}}_{i,3} {} &\le -{\tau }_{i,3}{||{\boldsymbol{\Upsilon }}_{i,3}||}^{2}-{\chi }_{i,1}{||{\boldsymbol{\Upsilon }}_{i,3}||}^{p+1}-{\lambda }_{i,1}{||{\boldsymbol{\Upsilon }}_{i,3}||}^{q+1}+{\tau }_{i,3}{\boldsymbol{\Upsilon }}_{i,3}^{T}{\mathbf{w}}_{i,3}+{\mu }_{i,3}{\boldsymbol{\Upsilon }}_{i,3}^{T}{\mathbf{w}}_{i,3}{||{\mathbf{w}}_{i,3}||}^{p-1}\nonumber\\ {} &+{\lambda }_{i,3}{\boldsymbol{\Upsilon }}_{i,3}^{T}{\mathbf{w}}_{i,3}{||{\mathbf{w}}_{i,3}||}^{q-1}-{\tau }_{i,3}{||{\mathbf{w}}_{i,3}||}^{2}-{\mu }_{i,3}{||{\mathbf{w}}_{i,3}||}^{p+1}-{\lambda }_{i,3}{||{\mathbf{w}}_{i,3}||}^{q+1}+{\Lambda }+\frac{{\ell}_{i,3}^{2}}{2}+\frac{{\zeta }_{,3}^{2}}{2}\nonumber\\ {} &+\frac{{\xi }_{i,3}}{{\kappa }_{i,3}}\left(\frac{{\vartheta }_{i,3}^{2}}{2}-\frac{{\stackrel{\sim }{\vartheta }}_{i,3}^{2}}{2}\right)+\frac{{\varsigma }_{i,3}}{{\kappa }_{i,3}}\left(\frac{2{\vartheta }_{i,3}^{p+1}}{1+p}-\frac{{\stackrel{\sim }{\vartheta }}_{i,3}^{p+1}}{1+p}\right)+\frac{{\varpi }_{i,3}}{{\kappa }_{i,3}}\left(\frac{2{\vartheta }_{i,3}^{q+1}}{1+q}-\frac{{\stackrel{\sim }{\vartheta }}_{i,3}^{q+1}}{1+q}\right)\end{align}

3.3 Stability and feasibility analysis

Proof. Firstly, based on Equations (27), (37) and (53), we build the Lyapunov function as

(54) \begin{align} {L}_{i}=\sum _{k=1}^{3}\,{L}_{i,k}=\sum _{k=1}^{3}\,\frac{{\boldsymbol{\Upsilon }}_{i,k}^{T}{\boldsymbol{\Upsilon }}_{i,k}}{2}+\frac{{\mathbf{w}}_{i,k}^{T}{\mathbf{w}}_{i,k}}{2}+\frac{{\stackrel{\sim }{\vartheta }}_{i,k}^{2}}{2{\kappa }_{i,k}}\end{align}

Combined with the above analysis, we have

(55) \begin{align} \begin{array}{c@{\quad}c}{\dot{L}}_{i} {} \le \sum _{k=1}^{3}\,\left(\begin{array}{c}-{\tau }_{i,k}{||{\boldsymbol{\Upsilon }}_{i,k}||}^{2}-{\chi }_{i,k}{||{\boldsymbol{\Upsilon }}_{i,k}||}^{p+1}-{\lambda }_{i,k}{||{\boldsymbol{\Upsilon }}_{i,k}||}^{q+1}+{\tau }_{i,k}{\boldsymbol{\Upsilon }}_{i,k}^{T}{\mathbf{w}}_{i,k}\\[6pt] +{\mu }_{i,k}{\boldsymbol{\Upsilon }}_{i,k}^{T}{\mathbf{w}}_{i,k}{||{\mathbf{w}}_{i,k}||}^{p-1}+{}_{i,k}{\boldsymbol{\Upsilon }}_{i,k}^{T}{\mathbf{w}}_{i,k}{||{\mathbf{w}}_{i,k}||}^{q-1}+\dfrac{{\ell}_{i,k}^{2}}{2}+\dfrac{{\zeta }_{i,k}^{2}}{2}\\[6pt] -{\tau }_{i,k}{||{\mathbf{w}}_{i,k}||}^{2}-{\mu }_{i,k}{||{\mathbf{w}}_{i,k}||}^{p+1}-{\lambda }_{i,k}{||{\mathbf{w}}_{i,k}||}^{q+1}\\[6pt] +\dfrac{{\xi }_{i,k}}{{\kappa }_{i,k}}\left(\dfrac{{\vartheta }_{i,k}^{2}}{2}-\dfrac{{\stackrel{\sim }{\vartheta }}_{i,k}^{2}}{2}\right)+\dfrac{{\varsigma }_{i,k}}{{\kappa }_{i,k}}\left(\dfrac{2{\vartheta }_{i,k}^{p+1}}{p+1}-\dfrac{{\stackrel{\sim }{\vartheta }}_{i,k}^{p+1}}{p+1}\right)+\dfrac{{\varpi }_{i,k}}{{\kappa }_{i,k}}\left(\dfrac{2{\vartheta }_{i,k}^{q+1}}{q+1}-\dfrac{{\stackrel{\sim }{\vartheta }}_{i,k}^{q+1}}{q+1}\right)\end{array}\right)\\[50pt] {} +\left({l}_{i}+{d}_{i}\right){\boldsymbol{\Upsilon }}_{i,1}^{T}{\mathbf{z}}_{i,2}+\left({l}_{i}+{d}_{i}\right){\mathbf{w}}_{i,1}^{T}\left({\boldsymbol{\Psi }}_{i,1}^{d}-{\boldsymbol{\Psi }}_{i,1}^{c}\right)\\[12pt] {} +{\boldsymbol{\Upsilon }}_{i,2}^{T}\dfrac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Upsilon }}_{i,3}+{\mathbf{w}}_{i,2}^{T}\dfrac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}\left({\boldsymbol{\Psi }}_{i,2}^{d}-{\boldsymbol{\Psi }}_{i,2}^{c}\right)+{\Lambda }\end{array}\end{align}

From Lemma 1, it can be derived $ {\boldsymbol{\Upsilon }}_{i,k}^{T}{\mathbf{w}}_{i,k}\le \frac{{||{\mathbf{w}}_{i,k}||}^{2}}{2}+\frac{{||{\boldsymbol{\Upsilon }}_{i,k}||}^{2}}{2}$ , $ {\boldsymbol{\Upsilon }}_{i,k}^{T}{\mathbf{w}}_{i,k}{||{\mathbf{w}}_{i,k}||}^{p-1}\le \frac{{||{\boldsymbol{\Upsilon }}_{i,k}||}^{p+1}}{p+1}+\frac{p{||{\mathbf{w}}_{i,k}||}^{p+1}}{p+1}$ , $ {\boldsymbol{\Upsilon }}_{i,k}^{T}{\mathbf{w}}_{i,k}{||{\mathbf{w}}_{i,k}||}^{q-1}\le \frac{{||{\boldsymbol{\Upsilon }}_{i,k}||}^{q+1}}{q+1}+\frac{q{||{\mathbf{w}}_{i,k}||}^{q+1}}{q+1}$ , $ {\boldsymbol{\Upsilon }}_{i,1}^{T}{\mathbf{z}}_{i,2}\le \left({||{\boldsymbol{\Upsilon }}_{i,1}||}^{2}+\frac{{||{\boldsymbol{\Upsilon }}_{i,2}||}^{2}}{2}+\frac{{||{\mathbf{w}}_{i,1}||}^{2}}{2}\right)$ , $ {\boldsymbol{\Upsilon }}_{i,2}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}{\boldsymbol{\Upsilon }}_{i,3}\le \stackrel{-}{\Pi }\frac{{||{\boldsymbol{\Upsilon }}_{i,2}||}^{2}+{||{\boldsymbol{\Upsilon }}_{i,3}||}^{2}}{2}$ . Define $ ||{\boldsymbol{\Psi }}_{i,k}^{d}-{\boldsymbol{\Psi }}_{i,k}^{c}||\le {\iota }_{i,k}$ , where $ k=\mathrm{1, 2}$ . Then it obtains $ {\mathbf{w}}_{i,1}^{T}\left({\boldsymbol{\Psi }}_{i,1}^{d}-{\boldsymbol{\Psi }}_{i,1}^{c}\right)\le \frac{{||{\mathbf{w}}_{i,1}||}^{2}+{\iota }_{i,1}^{2}}{2}$ , $ {\mathbf{w}}_{i,2}^{T}\frac{\partial {\boldsymbol{\Pi }}_{i,1}}{\partial {\boldsymbol{\Psi }}_{i,2}}\left({\boldsymbol{\Psi }}_{i,2}^{d}-{\boldsymbol{\Psi }}_{i,2}^{c}\right)\le \stackrel{-}{\Pi }\frac{{||{\mathbf{w}}_{i,2}||}^{2}+{\iota }_{i,2}^{2}}{2}$ .

Finally, Equation (55) is further simplified as

(56) \begin{align} {\dot{L}}_{i} {} &\le -\sum _{k=1}^{3}\,\frac{{\tau }_{i,k}}{2}{||{\boldsymbol{\Upsilon }}_{i,k}||}^{2}-\sum _{k=1}^{3}\,\left({\chi }_{i,k}-\frac{{\mu }_{i,k}}{p+1}\right){\left({||{\boldsymbol{\Upsilon }}_{i,k}||}^{2}\right)}^{\frac{p+1}{2}}-\sum _{k=1}^{3}\,\left({\lambda }_{i,k}-\frac{{\lambda }_{i,k}}{q+1}\right){\left({||{\boldsymbol{\Upsilon }}_{i,k}||}^{2}\right)}^{\frac{q+1}{2}}\nonumber\\ {} &-\sum _{k=1}^{3}\,\frac{{\tau }_{i,k}}{2}{||{\mathbf{w}}_{i,k}||}^{2}-\sum _{k=1}^{3}\,\frac{{\mu }_{i,k}}{p+1}{\left({||{\mathbf{w}}_{i,k}||}^{2}\right)}^{\frac{p+1}{2}}-\sum _{k=1}^{3}\,\frac{{\lambda }_{i,k}}{q+1}{\left({||{\mathbf{w}}_{i,k}||}^{2}\right)}^{\frac{q+1}{2}}\nonumber\\ {} &-\sum _{k=1}^{3}\,{\xi }_{i,k}\frac{{\stackrel{\sim }{\vartheta }}_{i,k}^{2}}{2{\kappa }_{i,k}}-\sum _{k=1}^{3}\,\frac{{\varsigma }_{i,k}}{p+1}\frac{{2}^{\frac{p+1}{2}}}{{\kappa }_{i,k}^{\frac{1-p}{2}}}{\left(\frac{{\stackrel{\sim }{\vartheta }}_{i,k}^{2}}{2{\kappa }_{i,k}}\right)}^{\frac{p+1}{2}}-\sum _{k=1}^{3}\,\frac{{\varpi }_{i,k}}{q+1}\frac{{2}^{\frac{q+1}{2}}}{{\kappa }_{i,k}^{\frac{1-q}{2}}}{\left(\frac{{\stackrel{\sim }{\vartheta }}_{i,k}^{2}}{2{\kappa }_{i,k}}\right)}^{\frac{q+1}{2}}\nonumber\\ {} &+\sum _{k=1}^{3}\,\left(\frac{{\ell}_{i,k}^{2}}{2}+\frac{{\zeta }_{i,k}^{2}}{2}+\frac{{\xi }_{i,k}}{{\kappa }_{i,k}}\frac{{\vartheta }_{i,k}^{2}}{2}+\frac{{\varsigma }_{i,k}}{{\kappa }_{i,k}}\frac{2{\vartheta }_{i,k}^{p+1}}{p+1}+\frac{{\varpi }_{i,k}}{{\kappa }_{i,k}}\frac{2{\vartheta }_{i,k}^{q+1}}{q+1}\right)\nonumber\\ {}&+\left({l}_{i}+{d}_{i}\right)\left({||{\boldsymbol{\Upsilon }}_{i,1}||}^{2}+{||{\mathbf{w}}_{i,1}||}^{2}+\frac{{||{\boldsymbol{\Upsilon }}_{i,2}||}^{2}+{\iota }_{i,1}^{2}}{2}\right)\nonumber\\ {}&+\stackrel{-}{\Pi }\frac{{||{\boldsymbol{\Upsilon }}_{i,2}||}^{2}+{||{\boldsymbol{\Upsilon }}_{i,3}||}^{2}+{||{\mathbf{w}}_{i,2}||}^{2}+{\iota }_{i,2}^{2}}{2}+{\Lambda }\end{align}

By defining

(57) \begin{align} {\Gamma }_{i,1} {} &=\mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{{\tau }_{i,k}}{2}\right\},{\Gamma }_{i,2}=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\chi }_{i,k}-\frac{{\mu }_{i,k}}{p+1}\right\},{\Gamma }_{i,3}=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\lambda }_{i,k}-\frac{{\lambda }_{i,k}}{q+1}\right\}\nonumber\\ {\Gamma }_{i,4} {} &=\mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{{\mu }_{i,k}}{p+1}\right\},{\Gamma }_{i,5}=\mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{{\lambda }_{i,k}}{q+1}\right\},{\Gamma }_{i,6}=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\xi }_{i,k}\right\}\nonumber\\ {\Gamma }_{i,7} {} &=\frac{{2}^{\frac{p+1}{2}}}{p+1}\mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{{\varsigma }_{i,k}}{{\kappa }_{i,k}^{\frac{1-p}{2}}}\right\},{\Gamma }_{i,8}=\frac{{2}^{\frac{q+1}{2}}}{q+1}\mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{{\varpi }_{i,k}}{{\kappa }_{i,k}^{\frac{1-q}{2}}}\right\} \end{align}

\begin{align} {\Gamma }_{i,9} {} &=\sum _{k=1}^{3}\,\left(\frac{{\ell}_{i,k}^{2}}{2}+\frac{{\zeta }_{i,k}^{2}}{2}+\frac{{\xi }_{i,k}}{{\kappa }_{i,k}}\frac{{\vartheta }_{i,k}^{2}}{2}+\frac{{\varsigma }_{i,k}}{{\kappa }_{i,k}}\frac{2{\vartheta }_{i,k}^{p+1}}{p+1}+\frac{{\varpi }_{i,k}}{{\kappa }_{i,k}}\frac{2{\vartheta }_{i,}^{q+1}}{q+1}\right)\nonumber\\ {} &\quad+\left({l}_{i}+{d}_{i}\right)\left({||{\boldsymbol{\Upsilon }}_{i,1}||}^{2}+{||{\mathbf{w}}_{i,1}||}^{2}+\frac{{||{\boldsymbol{\Upsilon }}_{i,2}||}^{2}+{\iota }_{i,1}^{2}}{2}\right)\nonumber\\ {} &\quad+\stackrel{-}{\Pi }\frac{{||{\boldsymbol{\Upsilon }}_{i,2}||}^{2}+{||{\boldsymbol{\Upsilon }}_{i,3}||}^{2}+{||{\mathbf{w}}_{i,2}||}^{2}+{\iota }_{i,2}^{2}}{2}+{\Lambda }\end{align}

we get

(58) \begin{align} {\dot{L}}_{i} {} \le &-{\Gamma }_{i,1}\sum _{k=1}^{3}\,{||{\boldsymbol{\Upsilon }}_{i,k}||}^{2}-{\Gamma }_{,2}\sum _{k=1}^{3}\,{\left({||{\boldsymbol{\Upsilon }}_{i,k}||}^{2}\right)}^{\frac{p+1}{2}}-{\Gamma }_{i,3}\sum _{k=1}^{3}\,{\left({||{\boldsymbol{\Upsilon }}_{i,k}||}^{2}\right)}^{\frac{q+1}{2}}\nonumber\\ {} &-{\Gamma }_{i,1}\sum _{k=1}^{3}\,{||{\mathbf{w}}_{i,k}||}^{2}-{\Gamma }_{i,4}\sum _{k=1}^{3}\,{\left({||{\mathbf{w}}_{i,k}||}^{2}\right)}^{\frac{p+1}{2}}-{\Gamma }_{i,5}\sum _{k=1}^{3}\,{\left({||{\mathbf{w}}_{i,k}||}^{2}\right)}^{\frac{q+1}{2}}\nonumber\\ {} &-{\Gamma }_{i,6}\sum _{k=1}^{3}\,\frac{{\stackrel{\sim }{\vartheta }}_{i,k}^{2}}{2{\kappa }_{i,k}}-{\Gamma }_{i,7}\sum _{k=1}^{3}\,{\left(\frac{{\stackrel{\sim }{\vartheta }}_{i,k}^{2}}{2{\kappa }_{i,k}}\right)}^{\frac{p+1}{2}}-{\Gamma }_{i,8}\sum _{k=1}^{3}\,{\left(\frac{{\stackrel{\sim }{\vartheta }}_{i,k}^{2}}{2{\kappa }_{i,k}}\right)}^{\frac{q+1}{2}}+{\Gamma }_{i,9}\end{align}

Herein, we introduce $ L=\sum _{i=1}^{K}\,{L}_{i}$ , it can be obtained that

(59) \begin{align} \dot{L}\le -{c}_{1}L-{c}_{2}{L}^{\frac{p+1}{2}}-{c}_{3}{L}^{\frac{q+1}{2}}+{\delta }_{a}\end{align}

with $ {c}_{1}=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\Gamma }_{i,1},{\Gamma }_{i,6}\right\}$ , $ {c}_{2}=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\Gamma }_{i,2},{\Gamma }_{i,4},{\Gamma }_{i,7}\right\}$ , $ {c}_{3}=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\Gamma }_{i,3},{\Gamma }_{i,5},{\Gamma }_{i,8}\right\}$ , $ {\delta }_{a}=\sum _{i=1}^{K}\,{\Gamma }_{i,9}$ .

Therefore, all signals of GHGVs system (3) are SPFTS on the basis of Lemma 6. An upper bound on the convergence time is given by the following form:

(60) \begin{align} {T}_{s}\le \mathrm{m}\mathrm{a}\mathrm{x}\left\{\frac{2}{{c}_{2}{{\Delta }}_{\epsilon }(1-p)}\right\}+\mathrm{m}\mathrm{a}\mathrm{x}\left\{\frac{2}{{c}_{1}{{\Delta }}_{\epsilon }(q-1)}\right\}\end{align}

Additionally, by using Lemma 6 we obtain the tracking error vector $ {\mathbf{z}}_{i,1}$ is bounded satisfying

(61) \begin{align} ||{\mathbf{z}}_{i,1}||\le \mathrm{m}\mathrm{i}\mathrm{n}\left\{\frac{{\delta }_{a}}{\left(1-{{\Delta }}_{\epsilon }\right){c}_{1}}{\left(\frac{{\delta }_{a}}{\left(1-{{\Delta }}_{\epsilon }\right){c}_{2}}\right)}^{\frac{2}{p+1}}{\left(\frac{{\delta }_{a}}{\left(1-{{\Delta }}_{\epsilon }\right){c}_{3}}\right)}^{\frac{2}{q+1}}\right\}\end{align}

Proof completed.

Proof. Based on the DETM (46), (47) and Assumption 9, we get the following inequality in the interval $ \left[{t}_{n},{t}_{n+1}\right)$ :

(62) \begin{align} \frac{d\left|{e}_{i,k}\left(t\right)\right|}{dt}=\left|{\dot{f}}_{i,k}\left(t\right)-{\dot{u}}_{fi,k}\left(t\right)\right|=\left|{\dot{f}}_{i,k}\left(t\right)-{\dot{f}}_{i,k}\left({t}_{n}\right)\right|=\left|{\dot{f}}_{i,k}\left(t\right)\right| \lt {{\Delta }}_{f}\end{align}

with $ {{\Delta }}_{f} \gt 0$ . It is known that $ \left|{e}_{i,k}\left({t}_{n}\right)\right|=0$ , $ \left|{e}_{i,k}\left(t\right)\right|\le {\int }_{{t}_{n}}^{t}\,{{\Delta }}_{f}dt$ , which means $ \left|{e}_{i,k}\left(t\right)\right|\le \left(t-{t}_{n}\right){{\Delta }}_{f}$ . Ulteriorly, when the event trigger condition $ \left|{e}_{i,k}\left(t\right)\right|\ge {\wp }_{i,k}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left|\frac{{u}_{fi,k}\left(t\right)}{{\hslash }_{i,k}}\right|+{\eta }_{i,k}\left(t\right)$ is destroyed, the control signal will be updated. It means that $ \underset{t\to {t}_{n+1}}{{\mathrm{l}\mathrm{i}\mathrm{m}}_{}}\mathrm{}\left|{e}_{i,k}\left(t\right)\right|={\wp }_{i,k}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left|\frac{{u}_{fi,k}\left(t\right)}{{\hslash }_{i,k}}\right|+{\eta }_{i,k}\left(t\right)$ . According to Equation (62), it gets

(63) \begin{align} {T}_{n}^{i}\ge \frac{{\wp }_{i,k}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left|{u}_{fi,k}\left(t\right)/{\hslash }_{i,k}\right|+{\eta }_{i,k}\left(t\right)}{{{\Delta }}_{f}}\end{align}

Therefore, for $ {T}_{p}^{i} \gt 0$ , there exists $ {T}_{p}=\mathrm{m}\mathrm{i}\mathrm{n}\{{T}_{p}^{1},\dots ,{T}_{p}^{K}\}$ that satisfies $ {T}_{n}^{i}\ge {T}_{p} \gt 0$ , where $ i=\mathrm{1, 2},\cdots ,K$ . Then Zeno behaviour can be avoided. Proof completed.

4.1 Effectiveness verification

To validate the efficacy of the formation control strategy developed in this study, two experimental cases were implemented under identical operational conditions, differentiated solely by their distinct initial state configurations.

Table 1.

Initial conditions of the HGVs in Example 1

The uncertainty is chosen as $ {\Delta }{\boldsymbol{\Omega }}_{i,1}={\left[3.8\times 1{0}^{-5}{\boldsymbol{\Psi }}_{i,{1}_{\left(1\right)}},1.5\times 1{0}^{-4}{\boldsymbol{\Psi }}_{i,{1}_{\left(2\right)}},4.6\times 1{0}^{-3}{\boldsymbol{\Psi }}_{i,{1}_{\left(3\right)}}\right]}^{T}$ , $ {\Delta }{\boldsymbol{\Omega }}_{i,2}={\left[2.5\times 1{0}^{-3}{\boldsymbol{\Psi }}_{i,{2}_{\left(1\right)}},0.8\times 1{0}^{-4}{\boldsymbol{\Psi }}_{i,{2}_{\left(2\right)}},7\times 1{0}^{-3}{\boldsymbol{\Psi }}_{i,{2}_{\left(3\right)}}\right]}^{T}$ .

The gains in the virtual control signals are set as $ {\tau }_{i,1}={\chi }_{i,1}={\lambda }_{i,1}=0.03,i=\mathrm{1,3},4$ ; $ {\tau }_{\mathrm{2,1}}={\chi }_{\mathrm{2,1}}={\lambda }_{\mathrm{2,1}}=0.06$ ; $ {\tau }_{i,2}={\chi }_{i,2}={\lambda }_{i,2}=0.05$ , $ {\tau }_{i,3}={\chi }_{i,3}={\lambda }_{i,3}=0.08$ , $ i=\mathrm{1, 2},\mathrm{3, 4}$ ; $ p=7/9\mathrm{}\mathrm{}$ , $ q=9/7\mathrm{}\mathrm{}$ ; Choose the design parameters in error compensative signals as $ {\mu }_{i,k}=0.03$ , $ {\lambda }_{i,k}=0.01$ , $ i=\mathrm{1, 2},\mathrm{3, 4},k=\mathrm{1, 2},3$ . The parameters in adaptive update laws are designed as $ {\kappa }_{i,k}=0.5,{\xi }_{i,k}=0.02,{\varsigma }_{i,k}=0.05,{\varpi }_{i,k}=0.03$ , $ i=\mathrm{1, 2},\mathrm{3, 4},k=\mathrm{1, 2},3$ ; parameter vectors in DETM equation are $ {\wp }_{i}=0.015{\mathbf{I}}^{3\times 1}$ , $ {\hslash }_{i}={\mathbf{I}}^{3\times 1}$ , $ {\partial }_{i}=0.03{\mathbf{I}}^{3\times 1}$ , $ {\mathbf{B}}_{i}=0.04{\mathbf{I}}^{3\times 1}$ , $ {\aleph }_{i}=0.02{\mathbf{I}}^{3\times 1}$ . The filter parameters are selected as $ {{\Delta }}_{\tau }=0.1$ .

Considering the communication frequency to be $ 100$ Hz, we set the simulation step as $ {\Delta }t=0.1$ s. The communications topology graph $ \mathcal{R}$ used in the simulation is shown in Fig. 2.

Figure 2.

Communication topology for a leader and four followers.

The GHGVs cluster tight-formation configuration is shown in Fig. 3.

Figure 3.

GHGVs cluster formation configuration view from behind.

Table 2.

Initial conditions of the HGVs in Example 2

Figures 4 and 5 show the formation of flight trajectories in two examples. Obviously, by employing the proposed dynamic event-triggered control law, the follower HGVs can track the leader HGV within a safe distance to achieve the desired formation configuration.

Figure 4.

The formation of flight trajectories in Example 1.

Figure 5.

The formation of flight trajectories in Example 2.

Figures 6 and 7 shows the norm of position tracking error for each follower varying with time in two examples. Figure 8 displays the adaptive RBFNN parameters in Example 1. From Fig. 6, we discover that during a total flight of $ 300$ s, the formation configuration has basically been formed before $ t=100$ s. By comparing the simulation results in Figs 6 and 7, it can be proved that despite distinct initial relative distances of 3500 m (example 1) and 5000 m (example 2), the convergence time of the formation is consistently constrained within 100s. This observation empirically validates the independence of the fixed-time control strategy from the initial state of the system. The invariance property demonstrates the robustness of the prescribed-time stability framework, where the upper bound of convergence duration remains decoupled from initial configuration parameters through the designed time-base generator. In more detail, the tracking errors $ ||{\omega }_{i}-{\omega }_{d}||$ are limited in $ 15$ m and no longer increase after the formation is formed. This phenomenon demonstrates that the proposed control method can handle multi-source uncertainties, ensuring the stability of the formation.

Figure 6.

The norm of position tracking error varying with time in Example 1.

Figure 7.

The norm of position tracking error varying with time in Example 2.

Figure 8.

$ {\widehat{\vartheta }}_{\mathrm{i}}$ varying with time in Example 1.

The simulation results presented in Figs 9, 10, 11, 12, 13 and 14 demonstrate the temporal evolution of critical control parameters, including elevator deflection angle $ {\delta }_{e,i}$ , angle-of-attack $ {\alpha }_{i}$ , and bank angle $ {\upsilon }_{i}$ , under the proposed control strategy. As is evidenced that the speed brake deflection angle $ {\delta }_{e,i}$ for four follower HGVs is bounded within $ [{0}^{\circ },4{5}^{\circ }]$ , the angle-of-attack $ {\alpha }_{i}$ maintains adherence to the $ [{0}^{\circ },3{0}^{\circ }]$ envelope, and the bank angle $ {\upsilon }_{i}$ operates within the $ [\!-\!8{0}^{\circ },8{0}^{\circ }]$ range. These signals exhibit smoothness without high-frequency chatter and are consistent with practical engineering requirements, which demonstrates the potential application value of the designed control approach.

Figure 9.

The deflection angle of speed brake $ {\delta }_{\mathrm{e},\mathrm{i}}$ varying with time in Example 1.

Figure 10.

The deflection angle of speed brake $ {\delta }_{\mathrm{e},\mathrm{i}}$ varying with time in Example 2.

Figure 11.

The angle-of-attack $ {\alpha }_{\mathrm{i}}$ varying with time in Example 1.

Figure 12.

The angle-of-attack $ {\alpha }_{\mathrm{i}}$ varying with time in Example 2.

Figures 15 and 16 show the event triggering time of all follower HGVs in example 1. It is clear that the event triggering time of each HGV is not at the same time, which verifies that the triggering mechanism designed in this paper is distributed, which will reduce the burden of network communication. What’s more, we find that the event triggering interval becomes smaller after the system is stabilised. This is an interesting result, mainly because the virtual signals of the system tend to 0 after the stabilisation, then the actual input $ {\upsilon }_{i}$ tends to 0 according to the inverse solution in Equation (2). If the virtual signal is not updated in real time, a small deviation will cause jumping change of $ {\upsilon }_{i}$ , which is not in line with the engineering practice.

Table 3 provides the event-triggered times and ratio for follower HGVs. Statistical analysis indicates that each vehicle maintains a total number of triggers below 2500 during the interval $ t\in \left[{0,300}\right]\mathrm{s},$ which demonstrates that the introduction of dynamic event-triggered mechanism can save at least $ 74{\%}$ of communication resources.

4.2 Superiority verification

In this section, we choose the fixed-time ETM control strategy in Yang and Niu [Reference Yang and Niu33] as a comparison to declare the superiority of the strategy proposed in this paper. For both methods, we use the same initial flight conditions and control parameters as in example 1. The consistent tacking errors of followers in the two methods are shown in Figs 17 and 18. The comparison of event-triggered times and ratio of the two methods are shown in Table 4.

This is evident from Figs 17 and 18 that the stabilisation time of the proposed method is $ {T}_{s}=100\mathrm{}$ s, and that of the control strategy presented in Yang and Niu [Reference Yang and Niu33] is $ {T}_{s}=128$ s. The consensus tracking error of the proposed method is restricted to $ 15\mathrm{}$ m, and that of the control strategy presented in Yang and Niu [Reference Yang and Niu33] is $ 50\mathrm{}$ m. The comparison reveals that the control approach presented in this paper exhibits superior tracking performance and achieves a quicker convergence rate.

Upon reviewing Table 4, it is clear that the introduced method reduces the event-triggered times for each follower by $ 25$ % approximately compared to the approach outlined in Yang and Niu [Reference Yang and Niu33]. In other words, the method in this paper saves $ 10$ % more communication resources than that in Yang and Niu [Reference Yang and Niu33]. Synthesise the above analysis, the effectiveness and the advantages of the proposed distributed fixed-time dynamic event-triggered formation control method can be verified.

Figure 13.

The bank angle $ {\upsilon }_{\mathrm{i}}$ varying with time in Example 1.

Figure 14.

The bank angle $ {\upsilon }_{\mathrm{i}}$ varying with time in Example 2.

Table 3.

Event-triggered times and ratio for follower HGVs in Example 1

Figure 15.

Triggering moment and triggering interval for follower 1 and 2 in two control methods.

Figure 16.

Triggering moment and triggering interval for follower 3 and 4 in two control methods.

Table 4.

Event-triggered times and ratio for follower HGVs in two methods

Figure 17.

Consistent tacking errors of follower 1 and 2 in two control methods.

Figure 18.

Consistent tacking errors of follower 3 and 4 in two control methods.