In this section, we will prove that there is no Zeno behavior for the considered FOMAS with the EIFC. Then, some impulsive quasiconsensus criteria are established for FOMAS (1).

### Theorem 1

*Consider the FOMAS* (1) *with the checked period* \(T>0\), *impulsive instants* \(t_{k}\) *for* \(k=1,2,\ldots \) *determined by the Algorithm* 1. *If Assumptions* 1*–*3*hold*, *and there are positive matrices* *P*, \(\Psi _{1i}\), \(\Psi _{2i}\), \(\Xi _{1i}\), \(\Xi _{2i}\) *and constants* \(a_{i}\), *positive constants* \(\xi _{i}\), \(i=1,2,\ldots,N\), *such that*

$$\begin{aligned}& \Psi _{1i}\leq \Psi _{2i}, \end{aligned}$$

(7)

$$\begin{aligned}& \Xi _{1i}-\Xi _{2i}\leq \xi _{i}I_{n}. \end{aligned}$$

(8)

*Then*, *there is no Zeno behavior for the concerned FOMAS*, *that is*, *there is a constant* \(\tau >0\) *such that* \(\inf \{t_{k}-t_{k-1}\}\geq \tau >0\), *where*

\begin{array}{c}{\mathrm{\Psi}}_{1i}=\left(\begin{array}{ccc}P{A}_{i}+{A}_{i}^{T}P+{Q}^{T}{\mathrm{\Lambda}}_{g}Q& P{B}_{i}& P\\ \ast & -{\mathrm{\Lambda}}_{g}& 0\\ \ast & \ast & -{\mathrm{\Xi}}_{1i}\end{array}\right),\hfill \\ {\mathrm{\Psi}}_{2i}=\left(\begin{array}{ccc}{a}_{i}P& 0& 0\\ \ast & 0& 0\\ \ast & \ast & -{\mathrm{\Xi}}_{2i}\end{array}\right),\hfill \end{array}

*and* \(a=\max_{1\leq i\leq N}\{a_{i}\}\), *Q* *and* \(\Lambda _{g}\) *are defined in Remark *1.

### Proof

Choose a Lyapunov function as \(V(t)=\sum_{i=1}^{N}e_{i}^{T}(t)Pe_{i}(t)\), according to Lemma 1 and Remark 1 for any \(t\in (t_{k-1},t_{k}]\), \(k=1,2,\ldots \) , one has

$$ \begin{aligned} {}_{t_{k}}\mathcal{D}^{\alpha}V(t)|_{\text{(5)}} \leq{}& 2\sum_{i=1}^{N}e_{i}^{T}(t)P \bigl[A_{i}e_{i}(t)+B_{i}f \bigl(e_{i}(t) \bigr)+ \varphi _{i} \bigl(x_{0}(t) \bigr) \bigr] \\ &{}\times \sum_{i=1}^{N} \bigl[\eta _{i}(t)^{T}\Psi _{1i}\eta _{i}(t)+ \varphi _{i}^{T} \bigl(x_{0}(t) \bigr) \Xi _{1i}\varphi _{i} \bigl(x_{0}(t) \bigr) \bigr], \end{aligned} $$

(9)

where \(\eta _{i}(t)=[e_{i}^{T}(t), f^{T}(e_{i}(t)), \varphi _{i}^{T}(x_{0}(t))]^{T}\), according to (7) and (8), one has

$$ {}_{t_{k}}\mathcal{D}^{\alpha}V(t)\leq aV(t)+\sum _{i=1}^{N}\xi _{i} \varpi _{i}^{2}. $$

(10)

Noting that, \({}_{t_{k}}\mathcal{D}^{\alpha}C=0\) for any constant *C*, then, we have

$$ {}_{t_{k}}\mathcal{D}^{\alpha} \biggl(V(t)+ \frac {\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a} \biggr) \leq a \biggl(V(t)+ \frac {\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a} \biggr).$$

According to Lemma 2, one has

$$ V(t)\leq -\frac {\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a}+ \biggl(V \bigl(t_{k-1}^{+} \bigr)+ \frac {\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a} \biggr)E_{\alpha} \bigl(a(t-t_{k-1})^{ \alpha} \bigr), \quad t\in (t_{k-1},t_{k}]. $$

(11)

Taking any \((t_{k-1},t_{k}]\), let us consider the event at \(t=t_{k}\). Based on Algorithm 1, if the first condition “\(\exists t\in (t_{k-1},t_{k-1}+T]\) such that \(V(t)\geq \theta _{1}V(t_{k-1}^{+})\)” is not met, then \(t_{k}-t_{k-1}=T>0\), it is obvious that there is no Zeno behavior. Consequently, we should investigate the case that “\(\exists t\in (t_{k-1},t_{k-1}+T]\) such that \(V(t)\geq \theta _{1}V(t_{k-1}^{+})\)”, if this event occurs at \(t_{k}\), we have \(V(t_{k})=\theta _{1}V(t_{k-1}^{+})\), combined with \(\theta _{1}>1\) and (11), we have

$$ \begin{aligned} V \bigl(t_{k-1}^{+} \bigr)+ \frac {\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a}&< \theta _{1}V \bigl(t_{k-1}^{+} \bigr)+ \frac {\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a} \\ &\leq \biggl(V \bigl(t_{k-1}^{+} \bigr)+ \frac {\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a} \biggr)E_{\alpha} \bigl(a(t_{k}-t_{k-1})^{ \alpha} \bigr). \end{aligned} $$

Thus, we have \(E_{\alpha}(a(t_{k}-t_{k-1})^{\alpha})>1\), then, we obtain \(t_{k}-t_{k-1}>0\). That is, Zeno behavior is excluded for the system. The proof is completed. □

### Theorem 2

*Consider the FOMAS* (1) *with the checked period* \(T>0\), *impulsive instants* \(t_{k}\) *for* \(k=1,2,\ldots \) *determined by Algorithm* 1. *If Assumptions* 1*–*3, (7), (8) *hold*, *and parameters of the FOMAS are satisfied by*

$$\begin{aligned}& \sigma _{\max}^{2} \biggl(I_{N}- \frac {c\mu _{\nu}}{\Gamma (\alpha +1)}(L+D)^{T} \biggr) \leq \rho , \quad \nu =1,2, \end{aligned}$$

(12)

$$\begin{aligned}& \rho \theta _{1}\leq \theta _{2}, \end{aligned}$$

(13)

*then*, *the trajectory of the error system* (5) *can exponentially converge into a ball* \(\mathbb{M}\) *with a convergence rate* \(\frac {\ln (\theta _{2})}{2T}\), *where* \(\mathbb{M}= \{e(t) |\| e(t)\| \leq \sqrt{ \frac {(\eta -1)\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a\lambda _{\min}(P)}} \}\), *in which*,

$$\eta = \textstyle\begin{cases} E_{\alpha}(a\tau ^{\alpha})& a\leq 0, \\ E_{\alpha}(aT^{\alpha})& a>0. \end{cases} $$

### Proof

Choose a Lyapunov function as \(V(t)=\sum_{i=1}^{N}e_{i}^{T}(t)Pe_{i}(t)\). If “\(\exists t \in (t_{k-1},t_{k-1}+T]\) such that \(V(t)\geq \theta _{1}V(t_{k-1}^{+})\)”, according to (12) and definition of \(t_{k}\), we have

$$ \begin{aligned} V \bigl(t_{k}^{+} \bigr)={}&e^{T} \bigl(t_{k}^{+} \bigr) (I_{N}\otimes P)e \bigl(t_{k}^{+} \bigr) \\ ={}&e^{T}(t_{k}) \biggl( \biggl(I_{N}- \frac {c\mu _{1}}{\Gamma (\alpha +1)}(L+D) \biggr) \otimes I_{n} \biggr)^{T}(I_{N} \otimes P) \\ & {}\times \biggl( \biggl(I_{N}-\frac {c\mu _{1}}{\Gamma (\alpha +1)}(L+D) \biggr) \otimes I_{n} \biggr) e(t_{k}) \\ ={}&\rho e^{T}(t_{k}) \biggl(\biggl(\biggl(I_{N}- \frac {c\mu _{1}}{\Gamma (\alpha +1)}(L+D)^{T} \biggr) \\ & {}\times \biggl(I_{N}-\frac {c\mu _{1}}{\Gamma (\alpha +1)}(L+D) \biggr)\biggr) \otimes P \biggr)e(t_{k}) \\ \leq {}&\sigma _{\max}^{2} \biggl(I_{N}- \frac {c\mu _{1}}{\Gamma (\alpha +1)}(L+D)^{T} \biggr)e^{T}(t_{k}) (I_{N} \otimes P)e(t_{k}) \\ \leq {}&\rho V(t_{k})\leq \rho \theta _{1}V \bigl(t_{k-1}^{+} \bigr)\leq \theta _{2}V \bigl(t_{k-1}^{+} \bigr). \end{aligned} $$

If “\(\exists t\in (t_{k-1},t_{k-1}+T]\) such that \(V(t)\geq \theta _{1}V(t_{k-1}^{+})\)” is not met, but “\(\exists t \in (t_{k-1},t_{k-1}+T]\) such that \(V(t)\geq \theta _{2}V(t_{k-1}^{+})\)”, similarly, we have

$$ V \bigl(t_{k}^{+} \bigr)\leq \theta _{2}V \bigl(t_{k-1}^{+} \bigr).$$

If “\(\exists t\in (t_{k-1},t_{k-1}+T]\) such that \(V(t)\geq \theta _{1}V(t_{k-1}^{+})\)” is not met, and “\(\exists t \in (t_{k-1},t_{k-1}+T]\) such that \(V(t)\geq \theta _{2}V(t_{k-1}^{+})\)” is also not met, one can conclude that

$$ V \bigl(t_{k}^{+} \bigr)=V(t_{k})\leq \theta _{2}V \bigl(t_{k-1}^{+} \bigr).$$

According to (11), one has

$$ \textstyle\begin{cases} V(t)\leq \eta V(t_{k-1}^{+})+\zeta ,& t\in (t_{k-1},t_{k}], \\ V(t_{k}^{+})\leq \theta _{2} V(t_{k-1}^{+}), \end{cases} $$

(14)

where \(\zeta =\varepsilon (\eta -1)\), \(\varepsilon =\frac {\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a}\). By mathematical induction, we can derive that

$$ V(t)\leq \eta \theta _{2}^{k}V(0)+\zeta ,\quad t\in (t_{k-1},t_{k}].$$

Noting that \(\tau \leq t_{k}-t_{k-1}\leq T\) and for any *t*, there must be *k* such that \(t\in (t_{k-1},t_{k}]\), one has \(\frac {t}{T}\leq k\leq \frac {t}{\tau}\), which implies that

$$ V(t)\leq \eta V(0)e^{\frac{\ln \theta _{2}}{T}t}+\zeta .$$

Therefore, one can conclude that

$$ \bigl\Vert e(t) \bigr\Vert \leq \sqrt{ \frac {\eta V(0)}{\lambda _{\min}(P)}}e^{\frac{\ln (\theta _{2})}{2T}t}+ \sqrt{\frac {\zeta}{\lambda _{\min}(P)}}.$$

Then, as \(t\rightarrow +\infty \), the error \(e(t)\) converges exponentially into the ball \(\mathbb{M}= \{e(t) |\| e(t)\| \leq \sqrt{ \frac {(\eta -1)\sum_{i=1}^{N}\xi _{i}\varpi _{i}^{2}}{a\lambda _{\min}(P)}} \}\) at a convergence rate \(\frac {\ln (\theta _{2})}{2T}\). The proof is completed. □

### Remark 2

Note that conditions in Theorem 1 are independent of the order *α*; however, *α* effects the value of \(E_{\alpha}(a(t_{k}-t_{k-1})^{\alpha})\), which implies that *α* will impact the time interval of two successive triggers. In addition, \(E_{\alpha}(a\tau ^{\alpha})\) is also related with *α*, which is significant in Theorem 2. Consequently, the consensus results in this paper are closely related to the order *α*.

### Remark 3

In the above, the topology structure of the network is considered as a directed graph. When the topology is undirected, one has a symmetric Laplacian matrix *L*, then, the condition (12) can be replaced as \(\lambda _{\max}^{2}(I_{N}-\frac {c}{\Gamma (\alpha +1)}(L+D)^{T}) \leq \rho \). In addition, if the FOMAS is homogeneous, which means that all nodes are identical, then it is easy to obtain \(\varpi _{i}=0\), \(i=1,2,\ldots,N\), according to the above, one can obtain the complete exponential consensus.

### Remark 4

More detailed results about error estimation, optimization for quasiconsensus of heterogeneous dynamic networks via distributed impulsive control have been discussed in [31], in which, the pinning strategy also has been investigated. Some similar results also can be derived in this paper, therefore, we omit them here.

### Remark 5

Compared with some existing results about impulsive control or the distributed impulsive control method, this paper has considered the event-triggered mechanism. Conditions in this manuscript are unrelated to the checked period *T*, which is important, the checked period *T* just effects the converge rate. Furthermore, due to the event-triggered mechanism, some unnecessary impulsive jumping can be avoided, which would be verified in the simulation part.

### Remark 6

There are some results about impulsive control with an event-triggered mechanism. In [32–35], the event-based impulsive control method has been investigated, in which, the impulsive instants are determined by a certain event. However, the feedback controllers are also used in the systems, which is different from this paper. Distributed impulsive control for heterogeneous multiagent systems based on an event-triggered scheme has been studied in [36], compared with which, events and impulsive controllers are simpler. Furthermore, this paper has discussed a FOMAS with fractional-order dynamics. Of course, letting \(\alpha =1\), the corresponding results about consensus of integer-order multiagent system can be obtained.

### Remark 7

The consensus problem has been analyzed in this paper, results about synchronization of a coupled dynamical network or master–slave system can be derived easily. For example, if there is only one follower, then the consensus problem converts to the synchronization problem of a master–slave system directly, an example will be given in the simulation part.