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Nonlinear Dynamics

, Volume 95, Issue 1, pp 541–555 | Cite as

Consensus of fractional multi-agent systems by distributed event-triggered strategy

  • Guojian Ren
  • Yongguang YuEmail author
  • Conghui Xu
  • Xudong Hai
Original Paper
  • 209 Downloads

Abstract

In this manuscript, the problem of event-triggered consensus for fractional general linear multi-agent systems is investigated, which include integer-order general linear multi-agent systems as the special case. A distributed event-triggered strategy is proposed to utilize in fractional multi-agent systems, under which the network can achieve consensus. Also, Zeno behavior can be precluded to ensure the feasibility of the devised event-triggered strategy. Furthermore, in order to avoid keeping track of the measurement errors continuously, a self-triggered strategy is designed, in which the next update time instant of each agent can be computed by using its local history state information. Finally, some numerical simulations are presented to indicate the validity of the devised control strategies.

Keywords

Fractional systems Distributed consensus Event-triggered control Multi-agent systems Self-triggered algorithm 

1 Introduction

Multi-agent systems have a lot of practical applications. For example, wireless sensor networks as a well-known class of multi-agent systems have been adopted widely in modern industry, such as process control and machine health monitoring. Other applications include consensus, formation stabilization [1, 2], attitude alignment, flocking [3]. Among them, the consensus problem focuses on developing appropriate controller to achieve general agreement among all agents in the networks. So far, there have been a large number of literature discussing the consensus problem in the first-order, second-order, and high-order networked systems [4, 5, 6, 7, 8, 9, 10, 11, 12]. For instance, in Ref. [11], several sufficient conditions depending on the generalized algebraic connectivity were given to ensure that consensus could be reached in directed networks. And in Ref. [12], the topic of leader-following consensus was investigated with the pinning control strategy in second-order multi-agent systems.

However, the most of the exiting references related to the consensus problem have been confined to the integer-order systems, in which the dynamics for all agent are modeled by the classical differential equations. It should be noted that a variety of phenomena with memory properties cannot be described by the integer-order dynamics [13, 14, 15], including the flocking of agents in porous media [16] and microorganisms moving in macromolecule fluids or mud layers. Indeed, the distributed coordination problem of the fractional networked systems has been already investigated in [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. In Ref. [17], several sufficient criteria on the fractional order were obtained to achieve coordination in fractional networked systems. By using the Laplace transform and frequency domain theory, the consensus problem was discussed in fractional delayed multi-agent systems [18]. The LMI methods are adopted in Ref. [19] to obtain sufficient conditions for ensuring robust consensus in fractional multi-agent systems with positive real uncertainty. With the help of the Mittag-Leffler stability theory, it is shown in Ref. [20] that all agents could convergence to a small region when the fractional multi-agent systems is subjected to external disturbances. A fractional observer is designed in Ref. [21] to reach leader-following consensus in multi-agent systems with second-order dynamics. In Ref. [22], an adaptive controller was developed to implement consensus in Lipschitz nonlinear fractional multi-agent systems. And Yu et al. [23] studied the leader-following consensus issue by using adaptive pinning control techniques in fractional linear and nonlinear multi-agent systems. Also, the problem of consensus is investigated in Ref. [28] for fractional multi-agent with any bounded input time delay. Two novel sufficient conditions are obtained in Ref. [30] to realize coordination for fractional nonlinear multi-agent systems with distributed impulsive control strategy. And Wang et al. [33] adopted a heterogeneous impulsive method to study leader-following exponential consensus in fractional nonlinear multi-agents system.

Nevertheless, the protocols devised in above-mentioned references must continuously update. In practice, the frequent communication tends to generate high and unnecessary energy consumption. It is desperately required to develop the energy-efficient controllers, especially in the systems with resource-limited equipment. Moreover, the fixed allotment of bandwidth limits the number of communications between different nodes in networks. To overcome these problems, some energy-efficient controllers, such as impulsive controllers [30, 33, 34, 35, 36, 37], sampled-data control protocols [38] and intermittent controllers [39], have been proposed. Particularly, some energy-saving impulsive control strategies have been designed to realize consensus in integer-order and fractional-order multi-agent systems [30, 33, 34, 35, 36, 37]. Also, a sampled-data control protocol is developed in Ref. [40] to ensure consensus for both leaderless case and leader-following case, where the system only updates the controller at some sampled time instants. The control approach used in Ref. [40] is time-scheduled control. In such control strategy, there might be a large increase in the number of controller updates in order to avoid some undesired results including oscillation and instability behavior in the systems. Other disadvantages of time-scheduled strategy are discussed in Ref. [41]. For purpose of settling these limitations of time-scheduled control, Astr\(\ddot{o}\)m et al. developed the event-triggering strategy in Ref. [42]. Note that the topic of event-triggered consensus for integer-order networked systems has been discussed in [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62]. However, in fractional systems, the incorporation of weakly singular kernels by fractional-order derivative makes it difficult to implement event-triggering strategy. As far as we know, only a few authors study the consensus problem of fractional multi-agent systems with event-triggering strategy [63, 64], and the topic of event-triggering consensus for fractional multi-agent systems still remains an open problem.

According to the above discussion, the event-triggered consensus problem is considered for fractional multi-agent systems in this manuscript. A distributed event-triggered protocol is designed grounded on the combinational measurement, Also, a rigorous proof is given to preclude Zeno behavior. With the devised mechanism, consensus can be achieved in the undirected networks. Several sufficient conditions are established by utilizing the tools including Lyapunov functional method, the fractional inequality technique and the Laplace transform, to implement consensus. Moreover, in order to avoid continuous monitoring of measurement errors, a self-triggered strategy is first proposed in fractional multi-agent systems. In such self-triggered algorithm, no state information or error measurement is needed in between two consecutive triggering time instants. Finally, several numerical simulations are exploited to indicate the accuracy of the obtained theory. It should be pointed out that the theoretical results obtained in this paper is not a simple extension from integer-order systems to fractional systems. We use some properties of the Caputo fractional derivative and the integral inequality to overcome the adverse effects from the incorporation of weakly singular kernels in fractional derivative.

We arrange the rest of this manuscript as below. Some basic definitions and lemmas are introduced in Sect. 2. Several sufficient conditions to ensure event-triggered consensus and exclude Zeno behavior in the devised strategy are presented in Sect. 3. In Sect. 4, a self-triggered controller is further designed to avoid continuously checking the triggering condition. Then, we give some numerical examples to indicate validity of the derived results in Sect. 5. Finally, a short conclusion and some future works are given in Sect. 6.

Some mathematical symbols will be introduced as below, which will be adopted throughout the manuscript. The notation \(A \otimes B\) represents the Kronecker product of matrices A and B. \(I_N\) stands for the \(N \times N\) identity matrix. \({\mathbf {0}}\) refers to the null matrix with compatible dimension. \({{\mathbf {1}}_N}\) is the N-dimensional column vector with all entries being 1.

2 Preliminaries and problem formulation

2.1 Graph theory

We will use a graph \({\mathscr {G}} = ({\mathscr {V}},{\mathscr {W}})\) with the node set \({\mathscr {V}} = \left\{ {{v_1},{v_2},\ldots ,{v_N}} \right\} \) and the edge set \({\mathscr {W}} \subseteq {{\mathscr {V}}^2}\) to model the communication topology for all agents. If agent j can communicate with agent i, there will be an edge denoted by \(({v_i},{v_j})\) in the graph \({\mathscr {G}}\). And agent i is called a neighbor of agent j. All neighbors of agent i are denoted by \(N_i\). The following two types of matrices will be used in the following sections: (1) the adjacency matrix \(A = \left[ {{a_{ij}}} \right] \in {{\mathbb {R}}^{N \times N}}\) with \({a_{ij}} = 1\) if \(({v_i},{v_j}) \in {\mathscr {W}}\) and \({a_{ij}} = 0\) otherwise, (2) the symmetric Laplacian matrix \(L = \left[ {{l_{ij}}} \right] \in {{\mathbb {R}}^{N \times N}}\) with \({l_{ii}} = \sum \nolimits _{j \in {N_i}} {{a_{ij}}} \) and \({l_{ij}} = - {a_{ij}}\), \(i \ne j\).

Lemma 1

[65] If \({\mathscr {G}}\) is an undirected graph, then L is a symmetric matrix with \({{\mathbf {1}}_N}\) as an eigenvector corresponding to the eigenvalue zero, and has N real eigenvalues, in an ascending order:
$$\begin{aligned} 0 = {\lambda _1}\left( L \right) \le {\lambda _2}\left( L \right) \le \cdots \le {\lambda _N}\left( L \right) , \end{aligned}$$
and
$$\begin{aligned} \mathop {min}\limits _{x \ne 0,{1^T}x = 0} \frac{{{x^T}Lx}}{{{{\left\| x \right\| }^2}}} = {\lambda _2}\left( L \right) , \end{aligned}$$
where \(\min \left( f \right) \) denotes the minima of the function f and \({\lambda _2}\left( L \right) \) is called the algebraic connectivity of \({\mathscr {G}}\). If \({\mathscr {G}}\) is connected, then \({\lambda _2}\left( L \right) >0\).

2.2 Caputo fractional operator

Definition 1

[14] The fractional integral of order \(\alpha \) for a function f is defined as
$$\begin{aligned} {I^\alpha }f\left( t \right) = \frac{1}{{\varGamma \left( \alpha \right) }}\int _{{0}}^t {{{\left( {t - \tau } \right) }^{\alpha - 1}}f\left( \tau \right) \mathrm{d}\tau } , \end{aligned}$$
where \(\varGamma \left( \cdot \right) \) is Gamma function with \(\varGamma \left( \alpha \right) = \int _0^{ + \infty } {{t^{\alpha - 1}}{e^{ - t}}\mathrm{d}t} \) and \(t \ge 0\) and \(\alpha > 0\).

Definition 2

[14] Caputo’s fractional derivative of order \(\alpha \) for a function \(f \in {C^n}\left( {\left[ {0, + \infty } \right] ,{\mathbb {R}}} \right) \) is defined by
$$\begin{aligned} {D^\alpha }f\left( t \right) = \frac{1}{{\varGamma \left( {n - \alpha } \right) }}\int _{{0}}^t {\frac{{{f^{\left( n \right) }}\left( \tau \right) }}{{{{\left( {t - \tau } \right) }^{\alpha - n + 1}}}}\mathrm{d}\tau }, \end{aligned}$$
where \(t \ge 0\) and n is a positive integer such that \(n - 1< \alpha < n\). When \(\alpha =n \),
$$\begin{aligned} {D^n}f\left( t \right) = {f^{\left( n \right) }}\left( t \right) . \end{aligned}$$
Particularly, when \(0< \alpha < 1\),
$$\begin{aligned} {D^\alpha }f\left( t \right) = \frac{1}{{\varGamma \left( {1 - \alpha } \right) }}\int _{{0}}^t {\frac{{f'\left( \tau \right) }}{{{{\left( {t - \tau } \right) }^\alpha }}}\mathrm{d}\tau } . \end{aligned}$$
The Laplace transform of the Caputo fractional derivative is
$$\begin{aligned} {\mathscr {L}}\left\{ {D^\alpha f\left( t \right) ;s} \right\} = {s^\alpha }F\left( s \right) -&\sum \limits _{k = 0}^{n - 1} {{s^{\alpha - k - 1}}{f^{\left( k \right) }}\left( {{0}} \right) } , \\&n - 1< \alpha < n, \end{aligned}$$
where s is the variable in Laplace domain.
For \(\alpha ,\beta \in {\mathbb {R}}\), the Mittag-Leffler function and its Laplace transform are given as follows [14]:
$$\begin{aligned} {E_{\alpha ,\beta }}\left( z \right) = \sum \limits _{k = 0}^\infty {\frac{{{z^k}}}{{\varGamma \left( {k\alpha + \beta } \right) }}} . \end{aligned}$$
(1)
$$\begin{aligned} {\mathscr {L}}\left\{ {{t^{\beta - 1}}{E_{\alpha ,\beta }}\left( { \pm a{t^\alpha }} \right) } \right\} = \frac{{{s^{\alpha - \beta }}}}{{{s^\alpha } \mp a}}, \\ {\mathscr {R}}\left( s \right) > 0, a \in {\mathbb {C}}, \left| {a{s^{ - \alpha }}} \right| < 1,\\ \end{aligned}$$
where \({\mathscr {R}}\left( s \right) \) is the real parts of s and \({\mathbb {C}}\) is the set of all complex numbers. When \(\beta = 1\), and \(\alpha > 0\), (1) will be written as
$$\begin{aligned}{E_\alpha }\left( z \right) = \sum \limits _{k = 0}^\infty {\frac{{{z^k}}}{{\varGamma \left( {k\alpha + 1} \right) }}} . \end{aligned}$$
Exponential function can be seen as a special case of it with \(\alpha \mathrm{{ = 1}}\).

Next, we will give some useful results and lemmas.

Lemma 2

[14] Let \(\Omega = \left[ {0,b} \right] \) be an interval on the real axis \({\mathbb {R}}\), let \(n = \left[ \alpha \right] + 1\) with \([\alpha ] = \max \left\{ {m \in {\mathbb {Z}} | {m \le \alpha } } \right\} \) for \(\alpha \notin {\mathbb {N}}\) or \(n = \alpha \) for \(\alpha \in {\mathbb {N}}\). If \(y\left( t \right) \in {C^n}\left[ {0,b} \right] \), then
$$\begin{aligned} {I^\alpha } {D^\alpha } y\left( t \right) = y\left( t \right) - \sum \limits _{k = 0}^{n - 1} {\frac{{{y^{\left( k \right) }}\left( 0 \right) }}{{k!}}{{\left( {t} \right) }^k}} . \end{aligned}$$
In particular, if \(0 < \alpha \le 1\) and \(y\left( t \right) \in {C^1}\left[ {0,b} \right] \), then
$$\begin{aligned} {I^\alpha } {D^\alpha } y\left( t \right) = y\left( t \right) - y\left( 0 \right) . \end{aligned}$$

Definition 3

[66] The constant \({\bar{x}}\) is an equilibrium point of the following Caputo fractional dynamic system
$$\begin{aligned} \left\{ \begin{array}{l} {D^\alpha }x(t)=f(t,x(t))\\ x(0)=x_{0} \end{array} \right. \end{aligned}$$
(2)
where \(\alpha \in (0, 1]\), \(x\left( t \right) = {\left( {{x_1}\left( t \right) ,{x_2}\left( t \right) ,\ldots ,{x_n}\left( t \right) } \right) ^T}\) and f(tx) is piecewise continuous on t and satisfies locally Lipschitz condition on x, if and only if \(f\left( {t,{\bar{x}}} \right) = 0\).

Definition 4

[66] If \({\bar{x}} = 0\) is an equilibrium point of system (2), the solution of (2) is said to be Mittag-Leffler stable if
$$\begin{aligned} \left\| {x\left( t \right) } \right\| \le {\left[ {m\left( {{x_{{t_0}}}} \right) {E_\alpha }\left( { - \lambda {{ {t } }^\alpha }} \right) } \right] ^b}, \end{aligned}$$
where \(\lambda> 0, b > 0, m(0) = 0\), \(\left\| \cdot \right\| \) denotes an arbitrary norm and \(m\left( x \right) \ge 0\) satisfies locally Lipschitz condition on x.

Remark 1

Mittag-Leffler stability for system (2) implies asymptotic stability for any initial value, i.e., \(\left\| x \right\| \rightarrow 0\) with \(t \rightarrow + \infty \).

Lemma 3

[66] The equilibrium point \({\bar{x}} = 0\) of fractional-order system (2) is Mittag-Leffler stable if there exist positive constants \({\alpha _1},{\alpha _2},{\alpha _3},a,b\) and a continuously differentiable function V(tx(t)) satisfying
$$\begin{aligned}&{\alpha _1}{\left\| x \right\| ^a} \le V\left( {t,x\left( t \right) } \right) \le {\alpha _2}{\left\| x \right\| ^{ab}}, \end{aligned}$$
(3)
$$\begin{aligned}&{D^\beta }V\left( {t,x\left( t \right) } \right) \le - {\alpha _3}{\left\| x \right\| ^{ab}}, \end{aligned}$$
(4)
where \(t \ge 0,\beta \in \left( {0,1} \right] ,V\left( {t,x\left( t \right) } \right) :\left[ {{t_0},\infty } \right) \times D \rightarrow R\) satisfies locally Lipschitz condition on x; \(D \in {R^n}\) is a domain containing the origin. If the assumptions hold globally on \({R^n}\), \({\bar{x}} = 0\) is globally Mittag-Leffler stable.

Lemma 4

[67] Let \(x\left( t \right) \in {\mathbb {R}}^n\) be a real continuous and differentiable vector function. Then, for any time instant \(t \ge 0\), one has
$$\begin{aligned} {D^\beta } {x^T}\left( t \right) Px\left( t \right) \le 2{x^T}\left( t \right) P{D^\beta } x\left( t \right) ,\quad \forall \beta \in \left( {0,1} \right] , \end{aligned}$$
where \(P \in {\mathbb {R}}^{n \times n}\) is a positive definite matrix.

2.3 The problem description

In this manuscript, we focus on the consensus problem for a fractional multi-agent system consisting of N agents. For any agent i, the dynamics can be described as
$$\begin{aligned} {D^\alpha } {x_i} =A{x_i} + B{u_i},\quad i \in \left\{ {1,2,\ldots ,N} \right\} , \end{aligned}$$
(5)
where \(\alpha \in \left( {0,1} \right] \), \({x_i} \in {{\mathbb {R}}^n}\) is the state vector for the ith agent, \({u_i} \in {{\mathbb {R}}^m}\) is its controller, and \(A \in {R^{n \times n}}\) and \(B\in {R^{n \times m}}\) are constant matrices.
Let the combined measurement be \({q_i}\left( t \right) = \sum \nolimits _{j = 1}^N {a_{ij}}( {x_j}\left( t \right) - {x_i}\left( t \right) ) \) and the measurement error be
$$\begin{aligned} {e_i}\left( t \right) = {q_i}\left( {t_k^i} \right) - {q_i}\left( t \right) ,\quad t \in \left( {t_k^i,t_{k + 1}^i} \right] . \end{aligned}$$
(6)
Then, the control input for agent i can be designed as:
$$\begin{aligned} {u_i}\left( t \right) = K{q_i}\left( {t_k^i} \right) ,\quad t \in \left( {t_k^i,t_{k + 1}^i} \right] , \end{aligned}$$
(7)
where K is the feedback gain matrix which is given in Theorem 1. Our purpose is to devise the following triggering condition as
$$\begin{aligned} h\left( {{e_i}\left( t \right) ,{q_i}\left( t \right) } \right) = 0, \end{aligned}$$
(8)
to generate the triggering time sequence \(\left\{ {t_0^i,t_1^i,\ldots } \right\} \) with \({t_0^i \ge 0}\) for agent i, \(i \in \left\{ {1,2,\ldots ,N} \right\} \). It follows from the expressions of \({e_i}\left( t \right) \) and \({q_i}\left( t \right) \) that (8) is distributed.

Remark 2

In this manuscript, for the purpose of reducing the number of triggering events, the control input \({u_i}\left( t \right) = K \sum \nolimits _{j \in {N_i}} {\left( {{x_j}\left( {t_k^i} \right) - {x_i}\left( {t_k^i} \right) } \right) } \) [44, 45, 47, 52] rather than \({u_i}(t) = K \sum \nolimits _{j \in { {N}_i}}{({{x_j}({t_{k'(t)}^j}) - {x_i}({t_k^i})})} \), where \(k'(t) = \arg {\min _{l \in {Z^ + }:t \ge t_l^j}} \{ {t - t_l^j} \}\), as in [63, 64], is implemented in the network.

The research topic in this manuscript is to choose appropriate K in order to achieve consensus. Before deriving the main results, we introduce the following assumptions.

Assumption 1

(AB) is stabilizable.

Under Assumption 1, for any \(\theta > 0\), there exists a matrix \(P>0\) such that Riccati inequality holds [64]:
$$\begin{aligned} {A^T}P + PA - 2\theta PB{B^T}P + \theta I < 0. \end{aligned}$$

Assumption 2

The undirected communication graph \({\mathscr {G}}\) is connected.

3 Distributed event-triggered control design

In this subsection, condition (8) will be designed and a rigorous proof will be provided to exclude Zeno behavior.

Theorem 1

With Assumptions 1 and 2, there exists a matrix \(P>0\) such that the Riccati inequality holds:
$$\begin{aligned} {A^T}P + PA - 2\theta PB{B^T}P + \theta I < 0, \end{aligned}$$
(9)
where \(\theta = \mu \left( {{\lambda _2}\left( L \right) - {{\delta {\lambda _N}\left( {{I_N} - J} \right) }/2}} \right) \) with \(\mu > 0, ~0< \delta < {{2{\lambda _2}\left( L \right) } / {{\lambda _N}\left( {{I_N} - J} \right) }}\), \(J = \frac{1}{N}{\mathbf{1 }_N}\mathbf{1 }_N^T\), \({\lambda _2}\left( L \right) \) being the second smallest eigenvalue of the Laplacian matrix L, and \({\lambda _N}\left( {I_N-J} \right) \) being the largest eigenvalue of the matrix \(I_N-J\). Then, letting \(K=\mu B^TP\), under control input (7) with the event-triggered condition designed as:
$$\begin{aligned} h\left( {{e_i}\left( t \right) ,{q_i}\left( t \right) } \right) = \left\| {{e_i}\left( t \right) } \right\| - {\eta _i}\left\| {{q_i}\left( t \right) } \right\| = 0, \end{aligned}$$
(10)
where \({\eta _i} = \sqrt{{{\left( {1 - {\varepsilon _i}} \right) \theta \delta } / {2\mu {\lambda _N}\left( {{I_N} - J} \right) {\lambda _N}\left( {{L^2}} \right) {\lambda _N}\left( {{\hat{B}}} \right) }}} < 1 \) with \(0< {\varepsilon _i} < 1\) for \(i \in \left\{ {1,2,\ldots ,N} \right\} \), \({\hat{B}} = PB{B^T}P\), \({\lambda _N}\left( {{L^2}} \right) \) and \({\lambda _N}\left( {{{\hat{B}}}} \right) \) being the largest eigenvalues of the matrices \(L^2\) and \({\hat{B}}\), respectively, fractional multi-agent system (5) will achieve consensus.

Proof

With \(K=\mu B^TP\) and (6), system (5) under controller (7) can be written as
$$\begin{aligned} {D^\alpha }{x_i} = Ax_i \!+\!\mu BB^TP\left( {{e_i} \!+\! {q_i}} \right) , i \in \left\{ {1,2,\ldots ,N} \right\} .\nonumber \\ \end{aligned}$$
(11)
Let \(x\left( t \right) = {\left( {{x_1}\left( t \right) ,\ldots ,{x_N}\left( t \right) } \right) ^T},e\left( t \right) = {\left( {{e_1}\left( t \right) ,\ldots ,{e_N}\left( t \right) } \right) ^T}\) and \(q\left( t \right) = {\left( {{q_1}\left( t \right) ,\ldots ,{q_N}\left( t \right) } \right) ^T}.\) Then, we have \(q\left( t \right) = - \left( {L \otimes {I_n}} \right) x\left( t \right) \), thus (11) can be rewritten in a matrix form as:
$$\begin{aligned} {D^\alpha }x\left( t \right)= & {} \left( {{I_N} \otimes A - L \otimes \mu B{B^T}P} \right) x\left( t \right) \nonumber \\&+ \left( {{I_N} \otimes \mu B{B^T}P} \right) e\left( t \right) . \end{aligned}$$
(12)
Let \({\bar{x}}\left( t \right) = \frac{1}{N}\sum \nolimits _{i = 1}^N {{x_i}\left( t \right) } \), \({{{\hat{x}}}_i}\left( t \right) = {x_i}\left( t \right) - {\bar{x}}\left( t \right) \), and \(J = \frac{1}{N}{\mathbf{1 }_N\mathbf{1 }_N^T}\). Then, one can easily obtain \({\hat{x}}\left( t \right) = (\left( {{I_N} - J} \right) \otimes I_n)x\left( t \right) \), where \({\hat{x}}\left( t \right) = {\left( {{{{\hat{x}}}_1}\left( t \right) ,\ldots ,{{{\hat{x}}}_N}\left( t \right) } \right) ^T}\). This together with (12) gives
$$\begin{aligned} {D^\alpha }{\hat{x}}\left( t \right)= & {} \left( {{I_N} \otimes A - L \otimes \mu B{B^T}P} \right) {\hat{x}}\left( t \right) \nonumber \\&+ \left[ {\left( {{I_N} - J} \right) \otimes \mu B{B^T}P} \right] e\left( t \right) . \end{aligned}$$
(13)
We construct the following Lyapunov function candidate for system (13):
$$\begin{aligned} V\left( t \right) = \frac{1}{2}{{{\hat{x}}}^T}\left( t \right) \left( {{I_N} \otimes P} \right) {\hat{x}}\left( t \right) . \end{aligned}$$
(14)
Obviously, the above Lyapunov function candidate satisfies inequality (3).
Denote \({\hat{A}} = {{\left( {{A^T}P + PA} \right) } / 2}\) and \({\hat{B}} = PB{B^T}P\). With the help of Lemma 1 and Lemma 4, for \(t \in \left( {t_k^i,t_{k + 1}^i} \right] , k=0,1,2,\ldots ,\) one has
$$\begin{aligned} {D^\alpha }V\left( t \right)\le & {} {{{\hat{x}}}^T}\left( t \right) \left( {{I_N} \otimes {\hat{A}} - L \otimes \mu {\hat{B}}} \right) {\hat{x}}\left( t \right) \nonumber \\+ & {} {{{\hat{x}}}^T}\left( t \right) \left[ {\left( {{I_N} - J} \right) \otimes \mu {\hat{B}}} \right] e\left( t \right) \nonumber \\\le & {} {{{\hat{x}}}^T}\left( t \right) \left( {{I_N} \otimes {\hat{A}} - {I_N} \otimes \mu {\lambda _2}\left( L \right) {\hat{B}}} \right) {\hat{x}}\left( t \right) \nonumber \\+ & {} {{{\hat{x}}}^T}\left( t \right) \left[ {\left( {{I_N} - J} \right) \otimes \mu {\hat{B}}} \right] e\left( t \right) . \end{aligned}$$
(15)
Then, noting inequalities (10) and \(\left\| \xi \right\| \cdot \left\| \zeta \right\| \le {\delta / 2}{\left\| \xi \right\| ^2} + {1 / {\left( {2\delta } \right) }}{\left\| \zeta \right\| ^2}\) for any \(\delta > 0\) and any \(\xi ,\zeta \in {{\mathbb {R}}^n}\), we obtain
$$\begin{aligned}&{D^\alpha }V\left( t \right) \le {{{\hat{x}}}^T}\left( t \right) \left( {{I_N} \otimes {\hat{A}} - {I_N} \otimes \mu {\lambda _2}\left( L \right) {\hat{B}}} \right) {\hat{x}}\left( t \right) \nonumber \\&\qquad + \frac{\delta }{2}{{{\hat{x}}}^T}\left( t \right) \left[ {\left( {{I_N} - J} \right) \otimes \mu {\hat{B}}} \right] {\hat{x}}\left( t \right) \nonumber \\&\qquad + \frac{1}{{2\delta }}{e^T}\left( t \right) \left[ {\left( {{I_N} - J} \right) \otimes \mu {\hat{B}}} \right] e\left( t \right) \nonumber \\&\quad \le {{{\hat{x}}}^T}\left( t \right) \left( {{I_N} \otimes {\hat{A}} - {I_N} \otimes \mu {\lambda _2}\left( L \right) {\hat{B}}} \right) {\hat{x}}\left( t \right) \nonumber \\&\qquad + \frac{{\delta {\lambda _N}\left( {{I_N} - J} \right) }}{2}{{{\hat{x}}}^T}\left( t \right) \left[ {{I_N} \otimes \mu {\hat{B}}} \right] {\hat{x}}\left( t \right) \nonumber \\&\qquad + \frac{{{\lambda _N}\left( {{I_N} - J} \right) }}{{2\delta }}{e^T}\left( t \right) \left( {{I_N} \otimes \mu {\hat{B}}} \right) e\left( t \right) \nonumber \\&\quad = {{{\hat{x}}}^T}\left( t \right) \left( {{I_N} \otimes {\hat{A}} - {I_N} \otimes \theta {\hat{B}}} \right) {\hat{x}}\left( t \right) \nonumber \\&\qquad + \frac{{{\lambda _N}\left( {{I_N} - J} \right) }}{{2\delta }}{e^T}\left( t \right) \left( {{I_N} \otimes \mu {\hat{B}}} \right) e\left( t \right) \nonumber \\&\quad \le - \frac{\theta }{2}{{{\hat{x}}}^T}\left( t \right) {\hat{x}}\left( t \right) + \frac{{{\lambda _N}\left( {{I_N} - J} \right) }}{{2\delta }}{e^T}\left( t \right) \nonumber \\&\qquad \left( {{I_N} \otimes \mu {\hat{B}}} \right) e\left( t \right) . \end{aligned}$$
(16)
Furthermore, from condition (10) and the inequality \({\left( {a + b} \right) ^2} \le 2{a^2} + 2{b^2}\) for any \(a,b \in {\mathbb {R}}\), one obtains
$$\begin{aligned}&\frac{{{\lambda _N}\left( {{I_N} - J} \right) }}{{2\delta }}{e^T}\left( t \right) \left( {{I_N} \otimes \mu {\hat{B}}} \right) e\left( t \right) \nonumber \\&\quad \le \frac{{\mu {\lambda _N}\left( {{I_N} - J} \right) {\lambda _N}\left( {{\hat{B}}} \right) }}{{2\delta }}{\left\| {e\left( t \right) } \right\| ^2} \nonumber \\&\quad = \frac{{\mu {\lambda _N}\left( {{I_N} - J} \right) {\lambda _N}\left( {{\hat{B}}} \right) }}{{2\delta }}\sum \limits _{i = 1}^N {{{\left\| {{e_i}\left( t \right) } \right\| }^2}} \nonumber \\&\quad \le \frac{{\left( {1 - \varepsilon } \right) \theta }}{{2{\lambda _N}\left( {{L^2}} \right) }}\sum \limits _{i = 1}^N {{{\left\| {{q_i}\left( t \right) } \right\| }^2}} \nonumber \\&\quad = \frac{{\left( {1 - \varepsilon } \right) \theta }}{{2{\lambda _N}\left( {{L^2}} \right) }}{x^T}\left( t \right) \left( {{I_N} \otimes {L^2}} \right) x\left( t \right) \nonumber \\&\quad = \frac{{\left( {1 - \varepsilon } \right) \theta }}{{2{\lambda _N}\left( {{L^2}} \right) }}{{{\hat{x}}}^T}\left( t \right) \left( {{I_N} \otimes {L^2}} \right) {\hat{x}}\left( t \right) \nonumber \\&\quad \le \frac{{\left( {1 - \varepsilon } \right) \theta }}{2}{{{\hat{x}}}^T}\left( t \right) {\hat{x}}\left( t \right) , \end{aligned}$$
(17)
where \(\varepsilon = \mathop {\min }\limits _{1 \le i \le N} \left\{ {{\varepsilon _i}} \right\} \).
Then, together with (17), (16) further becomes
$$\begin{aligned} {D^\alpha }V\left( t \right) \le \frac{{ - \theta \varepsilon }}{2}{{{\hat{x}}}^T}\left( t \right) {\hat{x}}\left( t \right) . \end{aligned}$$
(18)
Notice that \(V\left( t \right) \le \frac{\mathrm{{1}}}{\mathrm{{2}}}{\lambda _N}\left( P \right) {{\hat{x}}^T}\left( t \right) {\hat{x}}\left( t \right) \), we have
$$\begin{aligned} {D^\alpha }V\left( t \right) \le \frac{{ - \theta \varepsilon }}{{{\lambda _N}\left( P \right) }}V\left( t \right) . \end{aligned}$$
(19)
It follows from (19) that inequality (4) is satisfied. Then, from Lemma 3, system (13) is proved to be Mittag-Leffler stable, which implies that system (13) is asymptotically stable. From the definition of \({\hat{x}}\left( t \right) \), it is evident that consensus can be achieved in system (5) asymptotically under control law (7).

The proof of Theorem 1 is thus completed. \(\square \)

Remark 3

We suppose the graph \({\mathscr {G}}\) is undirected. However, it is obvious that Theorem 1 can be extended to the case with balanced directed graphs.

Next, we will give a rigorous proof of excluding Zeno behavior to ensure the feasibility of this proposed distributed event-triggered strategy.

Theorem 2

The fractional general linear multi-agent systems (5) under control input (7) with event-triggered condition (10) will not exhibit Zeno behavior.

Proof

Suppose the triggering time for agent i, \(i \in \left\{ {1,2,\ldots ,N} \right\} \) is \(t_k^i\). We will prove that the inter-event times \(t_{k+1}^i-t_k^i\) is strictly positive.

From (6) and (7), we have
$$\begin{aligned}&{D^\alpha }{q_i}\left( t \right) \nonumber \\&\quad = \sum \limits _{j = 1}^N {{a_{ij}}\left( {{D^\alpha }{x_j}\left( t \right) - {D^\alpha }{x_i}\left( t \right) } \right) } \nonumber \\&\quad = \sum \limits _{j = 1}^N {a_{ij}}\left( A{x_j}\left( t \right) + \mu B{B^T}P{q_j}\left( {t_{k'\left( t \right) }^j} \right) \right. \nonumber \\&\qquad - \left. A{x_i}\left( t \right) - \mu B{B^T}P{q_i}\left( {t_k^i} \right) \right) \nonumber \\&\quad = A{q_i}\left( t \right) + \mu B{B^T}P\sum \limits _{j = 1}^N {a_{ij}}\left( {q_j}\left( {t_{k'\left( t \right) }^j} \right) \right. \nonumber \\&\qquad -\left. {q_i}\left( {t_k^i} \right) \right) ,t \in \left( {t_k^i,t_{k + 1}^i} \right] , \end{aligned}$$
(20)
where \(k'(t) = argmi{n_{l \in N:t \ge t_l^j}}\left\{ {t - t_l^j} \right\} \).
According to Eq. (19), we have
$$\begin{aligned} {D^\alpha }V\left( t \right) \le \frac{{ - \theta \varepsilon }}{{{\lambda _N}\left( P \right) }}V\left( t \right) \le 0. \end{aligned}$$
Then, it follows from Definition 1 and Lemma 2 that
$$\begin{aligned} {I^\alpha }{D^\alpha }V\left( t \right)= & {} V\left( t \right) - V\left( 0 \right) \nonumber \\= & {} \int _0^t {\frac{{{{\left( {t - s} \right) }^{\alpha - 1}}}}{{\varGamma \left( \alpha \right) }}{D^\alpha }V\left( s \right) \mathrm{d}s \le 0.} \end{aligned}$$
(21)
Therefore, it is easy to obtain \(V\left( t \right) \le V\left( 0 \right) \). Then, for any bounded initial condition V(0), \({\hat{x}}\left( t \right) \) is bounded. It follows from \(q\left( t \right) = - \left( {L \otimes {I_n}} \right) {\hat{x}}\left( t \right) \) and Eq. (20) that \({D^\alpha }{q_i}\left( t \right) \) is bounded. Thus, there exists a constant \(\gamma _i^k > 0\) such that \(\gamma _i^k \ge \mathop {\max }\nolimits _{t \in \left[ {t_k^i,t_{k + 1}^i} \right] } \left\| {{D^\alpha }{q_i}\left( t \right) } \right\| \). Also, it is evident that \(\sigma _i^k = \mathop {\max }\nolimits _{1 \le l \le k} \left\{ {\gamma _i^l} \right\} > 0\) exists. This together with Lemma 2 gives
$$\begin{aligned}&\left\| {{e_i}\left( t \right) } \right\| \nonumber \\&\quad = \left\| {{q_i}\left( t \right) - {q_i}\left( {t_k^i} \right) } \right\| \nonumber \\&\quad =\frac{\mathrm{{1}}}{{\varGamma \left( \alpha \right) }}\left\| {\int _{\mathrm{{0}}}^t {{{\left( {t - s} \right) }^{\alpha - 1}}{D^\alpha }{q_i}\left( s \right) \mathrm{d}s} } \right. \nonumber \\&\qquad \left. { - \int _{\mathrm{{0}}}^{t_k^i} {{{\left( {t_k^i - s} \right) }^{\alpha - 1}}{D^\alpha }{q_i}\left( s \right) \mathrm{d}s} } \right\| \nonumber \\&\quad = \frac{1}{{\varGamma \left( \alpha \right) }}\left\| \int _0^{t_k^i} \left( {{{\left( {t - s} \right) }^{\alpha - 1}} - {{\left( {t_k^i - s} \right) }^{\alpha - 1}}} \right) \right. \nonumber \\&\quad \quad \left. {D^\alpha }{q_i}\left( s \right) \mathrm{d}s \right\| \nonumber \\&\quad + \frac{1}{{\varGamma \left( \alpha \right) }}\left\| {\int _{t_k^i}^t {{{\left( {t - s} \right) }^{\alpha - 1}}{D^\alpha }{q_i}\left( s \right) \mathrm{d}s} } \right\| \nonumber \\&\quad \le \frac{1}{{\varGamma \left( \alpha \right) }}\int _0^{t_k^i} \left( {{{\left( {t - s} \right) }^{\alpha - 1}} - {{\left( {t_k^i - s} \right) }^{\alpha - 1}}} \right) \nonumber \\&\quad \left\| {{D^\alpha }{q_i}\left( s \right) } \right\| \mathrm{d}s \nonumber \\&\quad + \frac{1}{{\varGamma \left( \alpha \right) }}\int _{t_k^i}^t {{{\left( {t - s} \right) }^{\alpha - 1}}\left\| {{D^\alpha }{q_i}\left( s \right) } \right\| ds} \nonumber \\&\quad \le \frac{{\sigma _i^{k - 1}}}{{\varGamma \left( \alpha \right) }}\int _0^{t_k^i} {\left( {{{\left( {t - s} \right) }^{\alpha - 1}} - {{\left( {t_k^i - s} \right) }^{\alpha - 1}}} \right) \mathrm{d}s} \nonumber \\&\quad + \frac{{\gamma _i^k}}{{\varGamma \left( \alpha \right) }}\int _{t_k^i}^t {{{\left( {t - s} \right) }^{\alpha - 1}}\mathrm{d}s} \nonumber \\&\quad = \frac{{\sigma _i^{k - 1}}}{{\varGamma \left( {\alpha + 1} \right) }}\left[ {{{\left( {t - t_k^i} \right) }^\alpha } + {{\left( {t_k^i} \right) }^\alpha } - {{\left( t \right) }^\alpha }} \right] \nonumber \\&\quad + \frac{{\gamma _i^k}}{{\varGamma \left( \alpha +1 \right) }}{\left( {t - t_k^i} \right) ^\alpha } \nonumber \\&\quad \le \frac{{2\sigma _i^k}}{{\varGamma \left( \alpha +1 \right) }}{\left( {t - t_k^i} \right) ^\alpha } - \Delta \left( {t_k^i} \right) , t \in \left( {t_k^i,t_{k + 1}^i} \right] ,\nonumber \\ \end{aligned}$$
(22)
where \(\Delta \left( {t_k^i} \right) = \frac{{\sigma _i^{k - 1}}}{{\varGamma \left( {\alpha + 1} \right) }}\left( {{t^\alpha } - {{\left( {t_k^i} \right) }^\alpha }} \right) > 0\).
On the other hand, from (6), we have
$$\begin{aligned}&\frac{{\eta _i^2}}{{\mathrm{{2 + 2}}\eta _i^2}}{\left\| {{q_i}\left( {t_k^i} \right) } \right\| ^2}\nonumber \\&\quad = \frac{{\eta _i^2}}{{\mathrm{{2 + 2}}\eta _i^2}}{\left\| {{e_i}\left( t \right) + {q_i}\left( t \right) } \right\| ^2}\nonumber \\&\quad \le \frac{{\eta _i^2}}{{\mathrm{{1 + }}\eta _i^2}}\left( {{{\left\| {{e_i}\left( t \right) } \right\| }^2} + {{\left\| {{q_i}\left( t \right) } \right\| }^2}} \right) . \end{aligned}$$
(23)
The inequality \({\left\| {{e_i}\left( t \right) } \right\| ^2} \le \frac{{\eta _i^2}}{{\mathrm{{2 + 2}}\eta _i^2}}{\left\| {{q_i}\left( {t_k^i} \right) } \right\| ^2}\) together with (23) yields
$$\begin{aligned} {\left\| {{e_i}\left( t \right) } \right\| ^2} \le \frac{{\eta _i^2}}{{\mathrm{{1 + }}\eta _i^2}}\left( {{{\left\| {{e_i}\left( t \right) } \right\| }^2} + {{\left\| {{q_i}\left( t \right) } \right\| }^2}} \right) , \end{aligned}$$
which is equivalent to
$$\begin{aligned} {\left\| {{e_i}\left( t \right) } \right\| ^2} \le \eta _i^2{\left\| {{q_i}\left( t \right) } \right\| ^2}. \end{aligned}$$
Therefore,
$$\begin{aligned} {\left\| {{e_i}\left( t \right) } \right\| } \le \frac{{{\eta _i}}}{{\sqrt{2 + 2\eta _i^2} }}\left\| {{q_i}\left( {t_k^i} \right) } \right\| \end{aligned}$$
(24)
can ensure condition (10) sufficiently.
When the event is triggered, one has \(t = t_{k + 1}^i\). According to event-triggered condition (10), we also have \(\left\| {{e_i}\left( {t_{k + 1}^i} \right) } \right\| = {\eta _i}\left\| {{q_i}\left( {t_{k + 1}^i} \right) } \right\| \). Because \({\left\| {{e_i}\left( t \right) } \right\| } \le \frac{{{\eta _i}}}{{\sqrt{2 + 2\eta _i^2} }}\left\| {{q_i}\left( {t_k^i} \right) } \right\| \) is a sufficient condition for \(\left\| {{e_i}\left( t \right) } \right\| \le {\eta _i}\left\| {{q_i}\left( t \right) } \right\| \), it is easy to obtain \(\left\| {{e_i}\left( {t_{k + 1}^i} \right) } \right\| \ge \frac{{{\eta _i}}}{{\sqrt{2 + 2\eta _i^2} }}\left\| {{q_i}\left( {t_k^i} \right) } \right\| \). Then, together with (22), we have
$$\begin{aligned} \frac{{2\sigma _i^k}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t_{k + 1}^i - t_k^i} \right) ^\alpha }\ge & {} \left\| {{e_i}\left( {t_{k + 1}^i} \right) } \right\| + \Delta \left( {t_k^i} \right) \nonumber \\\ge & {} {\frac{{{\eta _i}}}{{\sqrt{2 + 2\eta _i^2} }}\left\| {{q_i}\left( {t_k^i} \right) } \right\| } \nonumber \\&+\, \Delta \left( {t_k^i} \right) . \end{aligned}$$
(25)
From (25), one obtains
$$\begin{aligned}&t_{k + 1}^i - t_k^i\\&\quad \ge {\left[ {\frac{{\varGamma \left( {\alpha + 1} \right) }}{{2\sigma _i^k}}\left( {\frac{{{\eta _i}}}{{\sqrt{2 + 2\eta _i^2} }}\left\| {{q_i}\left( {t_k^i} \right) } \right\| + \Delta \left( {t_k^i} \right) } \right) } \right] ^{\frac{1}{\alpha }}}\\&\quad > 0.\\ \end{aligned}$$
The proof is thus completed. \(\square \)

Remark 4

The proof of excluding Zeno behavior is not a simple extension from integer-order systems to fractional systems. Fractional derivatives are defined on an interval rather than a point, which make it very difficult to obtain inequalities in the procedure of derivation. Here, we use the integral inequality to overcome the adverse effects the incorporation of weakly singular kernels in fractional derivative and provide a rigorous proof of excluding Zeno behavior.

4 Self-triggered algorithm

In the event-triggered strategy proposed in former sections, it is required to continuously check event-triggered condition (10). For the purpose of avoiding this demand, an improved self-triggered controller will be designed in this section. Unlike the event-triggered protocol proposed in Theorem 1, there is no state or error measurements required between two consecutive triggering time instants.

Before developing such self-triggered algorithm, we can design the following more conservative triggering condition based on the discussion in the proof of Theorem 2,
$$\begin{aligned} h'\left( {{e_i}\left( t \right) ,{q_i}\left( {t_k^i} \right) } \right) = \left\| {{e_i}\left( t \right) } \right\| - s_k^i = 0, \end{aligned}$$
(26)
where \(s_k^i=\frac{{{\eta _i}}}{{\sqrt{\mathrm{{2 + 2}}\eta _i^2} }}\left\| {{q_i}\left( {t_k^i} \right) } \right\| \).
According to (20), we have
$$\begin{aligned}&{D^\alpha }{q_i}\left( t \right) \nonumber \\&\quad = - A\left( {{q_i}\left( {t_k^i} \right) - {q_i}\left( t \right) } \right) + A{q_i}\left( {t_k^i} \right) \nonumber \\&\quad \quad +\, \mu B{B^T}P\sum \limits _{j = 1}^N {{a_{ij}}\left( {{q_j}\left( {t_{k'\left( t \right) }^j} \right) - {q_i}\left( {t_k^i} \right) } \right) } \nonumber \\&\quad = -\, A{e_i}\left( t \right) + A{q_i}\left( {t_k^i} \right) \nonumber \\&\quad \quad +\, \mu B{B^T}P\sum \limits _{j = 1}^N {a_{ij}}\left( {q_j}\left( {t_{k'\left( t \right) }^j} \right) \right. \nonumber \\&\quad \quad \left. -\, {q_i}\left( {t_k^i} \right) \right) ,t \in \left( {t_k^i,t_{k + 1}^i} \right] . \end{aligned}$$
(27)
Together with \({D^\alpha }{e_i}\left( t \right) = - {D^\alpha }{q_i}\left( t \right) \), one has
$$\begin{aligned}&\left\| {{D^\alpha }{e_i}\left( t \right) } \right\| \le \left\| A \right\| \left\| {{e_i}\left( t \right) } \right\| \nonumber \\&\quad + \left\| {A{q_i}\left( {t_k^i} \right) + \mu B{B^T}P\sum \limits _{j = 1}^N {{a_{ij}}\left( {{q_j}\left( {t_{k'\left( t \right) }^j} \right) - {q_i}\left( {t_k^i} \right) } \right) } } \right\| \nonumber \\&\quad \le \frac{{{\eta _i}\left\| A \right\| }}{{\sqrt{\mathrm{{2 + 2}}\eta _i^2} }}\left( {\left\| {{q_i}\left( {t_k^i} \right) } \right\| } \right) \nonumber \\&\quad + \left\| {A{q_i}\left( {t_k^i} \right) + \mu B{B^T}P\sum \limits _{j = 1}^N {{a_{ij}}\left( {{q_j}\left( {t_{k'\left( t \right) }^j} \right) - {q_i}\left( {t_k^i} \right) } \right) } } \right\| .\nonumber \\ \end{aligned}$$
(28)
Let \({\omega _i}\left( t \right) = \left\| \mu B{B^T}P\sum \limits _{j = 1}^N {a_{ij}}\left( {{q_j}\left( {t_{k'\left( t \right) }^j} \right) - {q_i}\left( {t_k^i} \right) } \right) \right. + \left. {A{q_i}\left( {t_k^i} \right) } \right\| + \frac{{{\eta _i}\left\| A \right\| }}{{\sqrt{\mathrm{{2 + 2}}\eta _i^2} }}\left\| {{q_i}\left( {t_k^i} \right) } \right\| \) and \(\omega _k^i = {\omega _i}\left( {t_k^i} \right) \). It follows from the expression of \({\omega _i}\left( t \right) \) that it will keep constant as \(\omega _k^i\) until the combined state \({{q_j}\left( {t_{k'\left( t \right) }^j} \right) }\) is changed.
The developed self-triggered strategy is given by Algorithm 1. According to it, the main result is derived below.

Theorem 3

With Assumptions 1 and 2, the fractional multi-agent system (5) can reach consensus under control law (7) with a self-triggered strategy, in which the triggering time sequence \(\left\{ {t_0^i,t_1^i,\ldots ,t_k^i,\ldots } \right\} \) with \({t_0^i \ge 0}\) is generated in Algorithm 1.

Proof

Suppose that the current triggering time for agent i, \(i \in \left\{ {1,2,\ldots ,N} \right\} \) is \(t_k^i\). We will prove that \(\left\| {{e_i}\left( t \right) } \right\| \le s_{k}^i\) holds for \(t \in \left( {t_{k}^i,t_{k+1}^i} \right] \) and \(k\ge 0\).

For \(k=0\), one has \(t_{0}^i\ge 0\). From Algorithm 1, when \(i = \arg \mathop {\min }\nolimits _l \left\{ t_{0}^i + {{{\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_0^l}}{2{\omega _0^l}}} \right) }^{\frac{1}{\alpha }}}} \right\} \) (that is, agent i is chosen to be triggered), we have \(t_1^i = t_{0}^i + {\left( {\frac{{ \left( {\alpha + 1} \right) s_0^i}}{2{\omega _0^i}}} \right) ^{\frac{1}{\alpha }}}.\) For \(t \in \left( {t_{0}^i,t_1^i} \right] \), one has
$$\begin{aligned}&\left\| {{e_i}\left( t \right) } \right\| \nonumber \\&\quad = \left\| {{e_i}\left( t \right) - {e_i}\left( {t_0^i} \right) + {e_i}\left( {t_0^i} \right) } \right\| \nonumber \\&\quad = \left\| {{e_i}\left( t \right) - {e_i}\left( {t_0^i} \right) } \right\| + \left\| {{e_i}\left( {t_0^i} \right) } \right\| \nonumber \\&\quad = \frac{1}{{\varGamma \left( \alpha \right) }}\left\| {\int _0^t {{{\left( {t - s} \right) }^{\alpha - 1}}{D^\alpha }{e_i}\left( s \right) \mathrm{d}s}} \right. \nonumber \\&\qquad \left. {-\, \int _0^{t_0^i} {{{\left( {t_0^i - s} \right) }^{\alpha - 1}}{D^\alpha }{e_i}\left( s \right) \mathrm{d}s}} \right\| \nonumber \\&\qquad + \frac{1}{{\varGamma \left( \alpha \right) }}\left\| {\int _0^{t_0^i} {{{\left( {t_0^i - s} \right) }^{\alpha - 1}}{D^\alpha }{e_i}\left( s \right) \mathrm{d}s} } \right\| \nonumber \\&\quad \le \frac{1}{{\varGamma \left( \alpha \right) }}\int _0^{t_0^i} \left( {{{\left( {t_0^i - s} \right) }^{\alpha - 1}} - {{\left( {t - s} \right) }^{\alpha - 1}}} \right) \nonumber \\&\left\| {{D^\alpha }{e_i}\left( s \right) } \right\| \mathrm{d}s \nonumber \\&\qquad +\, \frac{1}{{\varGamma \left( \alpha \right) }}\int _{t_0^i}^t {{{\left( {t - s} \right) }^{\alpha - 1}}\left\| {{D^\alpha }{e_i}\left( s \right) } \right\| \mathrm{d}s}\nonumber \\&\qquad +\, \frac{1}{{\varGamma \left( \alpha \right) }}\int _0^{t_0^i} {{{\left( {t_0^i - s} \right) }^{\alpha - 1}}\left\| {{D^\alpha }{e_i}\left( s \right) } \right\| \mathrm{d}s} \nonumber \\&\quad \le \frac{{\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}{{\left( {t - t_0^i} \right) }^\alpha } \nonumber \\&\quad \le \frac{2{\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t_1^i- t_0^i} \right) ^\alpha }\nonumber \\&\quad = s_0^i. \end{aligned}$$
(29)
When \(h = \arg \mathop {\min }\nolimits _l \left\{ t_0^i + {{{\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_0^l}}{2{\omega _0^l}}} \right) }^{\frac{1}{\alpha }}}} \right\} ,i \ne h\), we have the following two cases:
Case 1 When \(i \in {N_h}\), assume that \({\omega _i}(t )\) changes at \({u_{0,1}^i}\), where \(u_{0,1}^i = t_1^h = t_0^i + {( {\frac{{\varGamma ( {\alpha + 1} )s_0^h}}{2{\omega _0^h}}} )^{\frac{1}{\alpha }}}\), and keeps constant during \(\left( { t_0^i ,u_{0,1}^i} \right] \) and \(\left( {u_{0,1}^i,t_1^i} \right] \), which means that agent i is the next triggered agent. Then, for \(t \in \left( { t_0^i ,t_1^i} \right] \), there are also two cases that may happen. (1) When \(t \in \left( { t_0^i ,u_{0,1}^i} \right] \), from Algorithm 1, since \(h = \arg \mathop {\min }\limits _l \{ t_0^i + {{{\left( {\frac{{\varGamma ( {\alpha + 1} )s_0^l}}{2{\omega _0^l}}} \right) }^{\frac{1}{\alpha }}}} \}\), \({\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_0^h}}{2{\omega _0^h}}} \right) ^{\frac{1}{\alpha }}} \le {( {\frac{{\varGamma ( {\alpha + 1} )s_0^i}}{2{\omega _0^i}}} )^{\frac{1}{\alpha }}}\). Based on the discussion in (29) and Algorithm 1, one has
$$\begin{aligned} \left\| {{e_i}\left( t \right) } \right\|&\quad \le \frac{2{\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t- t_0^i } \right) ^\alpha }\nonumber \\&\quad \le \frac{2{\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {u_{0,1}^i- t_0^i } \right) ^\alpha }\nonumber \\&\quad = \frac{2{\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {{{\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_0^h}}{2{\omega _0^h}}} \right) }^{\frac{1}{\alpha }}}} \right) ^\alpha }\nonumber \\&\quad \le \frac{2{\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {{{\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_0^i}}{2{\omega _0^i}}} \right) }^{\frac{1}{\alpha }}}} \right) ^\alpha }\nonumber \\&\quad = s_0^i. \end{aligned}$$
(30)
(2) When \(t \in \left( {u_{0,1}^i,t_1^i} \right] \), agent h is the first agent to be triggered at \(t= u_{0,1}^i\) and agent i is the second. After the first for loop in Algorithm 1, since \(i \in {N_h}\) and the 15th statement in Algorithm 1, we have \({s_i} = s_0^i - \frac{{\omega _0^is_0^i}}{{\omega _0^h}}\). Also, \(lasttime = u_{0,1}^i\) in 18th statement in Algorithm 1. And, according to 9th–13th statements, we can obtain \({\omega _i} = \max \left\{ {\omega _0^i,{\omega _i}\left( {u_{0,1}^i} \right) } \right\} \). It follows from the 2nd and 5th statements that \(t_1^i =time= u_{0,1}^i + \left( {\frac{{\varGamma \left( {\alpha + 1} \right) \left( {s_0^i - \frac{{\omega _0^is_0^i}}{{\omega _0^h}}} \right) }}{2{\max \left\{ {\omega _0^i,{\omega _i}\left( {u_{0,1}^i} \right) } \right\} }}} \right) ^{\frac{1}{\alpha }}\). Also, one has
$$\begin{aligned}&\left\| {{e_i}\left( t \right) } \right\| \nonumber \\&\quad = \left\| {{e_i}\left( t \right) - {e_i}\left( {u_{0,1}^i} \right) + {e_i}\left( {u_{0,1}^i} \right) } \right\| \nonumber \\&\quad = \left\| {{e_i}\left( t \right) - {e_i}\left( {u_{0,1}^i} \right) } \right\| + \left\| {{e_i}\left( {u_{0,1}^i} \right) } \right\| \nonumber \\&\quad = \frac{1}{{\varGamma \left( \alpha \right) }}\left\| {\int _0^t {{{\left( {t - s} \right) }^{\alpha - 1}}{D^\alpha }{e_i}\left( s \right) \mathrm{d}s}} \right. \nonumber \\&\qquad \left. {- \int _0^{u_{0,1}^i} {{{\left( {u_{0,1}^i - s} \right) }^{\alpha - 1}}{D^\alpha }{e_i}\left( s \right) \mathrm{d}s}} \right\| \nonumber \\&\qquad +\, \frac{1}{{\varGamma \left( \alpha \right) }}\left\| {\int _0^{u_{0,1}^i} {{{\left( {u_{0,1}^i - s} \right) }^{\alpha - 1}}{D^\alpha }{e_i}\left( s \right) \mathrm{d}s} } \right\| \nonumber \\&\quad \le \frac{1}{{\varGamma \left( \alpha \right) }}\int _0^{u_{0,1}^i} \left( {{{\left( {u_{0,1}^i - s} \right) }^{\alpha - 1}} - {{\left( {t - s} \right) }^{\alpha - 1}}} \right) \nonumber \\&\left\| {{D^\alpha }{e_i}\left( s \right) } \right\| \mathrm{d}s \nonumber \\&\qquad +\, \frac{1}{{\varGamma \left( \alpha \right) }}\int _{u_{0,1}^i}^t {{{\left( {t - s} \right) }^{\alpha - 1}}\left\| {{D^\alpha }{e_i}\left( s \right) } \right\| \mathrm{d}s}\nonumber \\&\qquad +\,\frac{1}{{\varGamma \left( \alpha \right) }}\int _0^{u_{0,1}^i} {{{\left( {u_{0,1}^i - s} \right) }^{\alpha - 1}}\left\| {{D^\alpha }{e_i}\left( s \right) } \right\| \mathrm{d}s} \nonumber \\&\quad \le \frac{{\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}\left[ {{{\left( {t - u_{0,1}^i} \right) }^\alpha } - {{\left( t \right) }^\alpha } + {{\left( {u_{0,1}^i} \right) }^\alpha }} \right] \nonumber \\&\qquad +\, \frac{{{\omega _i}\left( {u_{0,1}^i} \right) }}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t - u_{0,1}^i} \right) ^\alpha }+ \frac{2{\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {u_{0,1}^i} \right) ^\alpha }\nonumber \\&\quad \le \frac{{2\max \left\{ {\omega _0^i,{\omega _i}\left( {u_{0,1}^i} \right) } \right\} }}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t - u_{0,1}^i} \right) ^\alpha } \nonumber \\&\qquad + \frac{{2\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {u_{0,1}^i} \right) ^\alpha }\nonumber \\&\quad \le \frac{{2\max \left\{ {\omega _0^i,{\omega _i}\left( {u_{0,1}^i} \right) } \right\} }}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t_1^i - u_{0,1}^i} \right) ^\alpha } \nonumber \\&\qquad +\, \frac{{2\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {u_{0,1}^i} \right) ^\alpha }\nonumber \\&\quad = \frac{{2\max \left\{ {\omega _0^i,{\omega _i}\left( {u_{0,1}^i} \right) } \right\} }}{{\varGamma \left( {\alpha + 1} \right) }}\left( {\frac{{\varGamma \left( {\alpha + 1} \right) \left( {s_0^i - \frac{{\omega _0^is_0^h}}{{\omega _0^h}}} \right) }}{{2\max \left\{ {\omega _{\mathrm{{0}}}^i,{\omega _i}\left( {u_{0,1}^i} \right) } \right\} }}} \right) \nonumber \\&\qquad + \frac{{2\omega _0^i}}{{\varGamma \left( {\alpha + 1} \right) }}\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_0^h}}{{2\omega _0^h}}} \right) \nonumber \\&\quad = s_0^i. \end{aligned}$$
(31)
Case 2 When \(i \notin {N_h}\), according to Algorithm 1, \({\omega _i}\left( t \right) \) will not change until agent i is triggered. Then, for this case, the discussion is the same as (29).

Moreover, the discussion is similar when \({\omega _i}\left( t \right) \) changes finite times at \(u_{0,1}^i,u_{0,2}^i,\ldots ,u_{0,{m_0}}^i\left( {{m_0} > 1} \right) \) during time interval \(\left( { t_0^i ,t_1^i} \right] \).

In the following, we will prove \(\left\| {{e_i}\left( t \right) } \right\| \le s_{k}^i\) holds for \(t \in \left( {t_{k}^i,t_{k+1}^i} \right] \) and \(k>0\).

Let \(\kappa _{k - 1}^i = \max \Big \{ \omega _0^i,{\omega _i}\left( {u_{0,1}^i} \right) ,\ldots ,{\omega _i}\left( {u_{0,{m_0}}^i} \right) ,\ldots ,\omega _{k - 1}^i, \omega _i\left( {u_{k - 1,1}^i} \right) ,\ldots ,{\omega _i}\left( {u_{k - 1,{m_{k - 1}}}^i} \right) \Big \}\). From Algorithm 1, when \(i = \arg \mathop {\min }\limits _l \left\{ {t_{k'\left( t \right) }^l + {{\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_{k'\left( t \right) }^l}}{{2\max \left\{ {\kappa _{k'\left( t \right) - 1}^l,\omega _{k'\left( t \right) }^l} \right\} }}} \right) }^{\frac{1}{\alpha }}}} \right\} \), we have \(t_{k + 1}^i = t_k^i + {\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_k^i}}{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i} \right\} }}} \right) ^{\frac{1}{\alpha }}}\). For \(t \in \left( {t_{k}^i,t_{k+1}^i} \right] \), since \(e_i(t_{k}^i)=0\), based on the technique used in (31), one has
$$\begin{aligned}&\left\| {{e_i}\left( t \right) } \right\| \nonumber \\&\quad = \left\| {{e_i}\left( t \right) - {e_i}\left( {t_k^i} \right) } \right\| \nonumber \\&\quad = \frac{\mathrm{{1}}}{{\varGamma \left( \alpha \right) }}\left\| {\int _{\mathrm{{0}}}^t {{{\left( {t - s} \right) }^{\alpha - 1}}{D^\alpha }{e_i}\left( s \right) \mathrm{d}s} } \right. \nonumber \\&\quad \left. { - \int _{\mathrm{{0}}}^{t_k^i} {{{\left( {t_k^i - s} \right) }^{\alpha - 1}}{D^\alpha }{e_i}\left( s \right) \mathrm{d}s} } \right\| \nonumber \\&\quad \le \frac{{\kappa _{k - 1}^i}}{{\varGamma \left( {\alpha + 1} \right) }}\left[ {{{\left( {t - t_k^i} \right) }^\alpha } - {{\left( t \right) }^\alpha } + {{\left( {t_k^i} \right) }^\alpha }} \right] \nonumber \\&\quad + \frac{{\omega _k^i}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t - t_k^i} \right) ^\alpha }\nonumber \\&\quad \le \frac{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i} \right\} }}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t_{k + 1}^i - t_k^i} \right) ^\alpha }\nonumber \\&\quad = s_k^i. \end{aligned}$$
(32)
When \(h = \arg \mathop {\min }\limits _l \!\left\{ \! {t_{k'\left( t \right) }^l {+} {{\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_{k'\left( t \right) }^l}}{{2\max \left\{ {\kappa _{k'\left( t \right) - 1}^l,\omega _{k'\left( t \right) }^l} \right\} }}} \!\right) }^{\frac{1}{\alpha }}}}\! \right\} \!,i \ne h\), similar to the proof for the case \(k=0\), we have the following two cases:
Case 1 When \(i \in {N_h}\), assume that \({\omega _i}\left( t \right) \) remains unchanged during \(\left( {t_k^i,u_{k,1}^i} \right] \) and \(\left( {u_{k,1}^i,t_{k+1}^i} \right] \), respectively. Then, for \(t \in \left( {t_{k}^i,t_{k+1}^i} \right] \), based on above discussion, there are also two cases that may happen. (1) When \(t \in \left( {t_{k}^i,u_{k,1}^i} \right] \), according to Algorithm 1, we obtain \(u_{k,1}^i = t_{k'\left( t \right) }^h + {\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_{k'\left( t \right) }^h}}{{2\max \left\{ {\kappa _{k'\left( t \right) - 1}^h,\omega _{k'\left( t \right) }^h} \right\} }}} \right) ^{\frac{1}{\alpha }}} \le t_k^i + {\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_k^i}}{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i} \right\} }}} \right) ^{\frac{1}{\alpha }}}\). It follows from the discussion in (32) and Algorithm 1 that
$$\begin{aligned}&\left\| {{e_i}\left( t \right) } \right\| \nonumber \\&\quad \le \frac{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i} \right\} }}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {u_{k,1}^i - t_k^i} \right) ^\alpha }\nonumber \\&\quad = \frac{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i} \right\} }}{{\varGamma \left( {\alpha + 1} \right) }}\nonumber \\&\quad \times {\left( {t_{k'\left( t \right) }^h + {{\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_{k'\left( t \right) }^h}}{{2\max \left\{ {\kappa _{k'\left( t \right) - 1}^h,\omega _{k'\left( t \right) }^h} \right\} }}} \right) }^{\frac{1}{\alpha }}} - t_k^i} \right) ^\alpha }\nonumber \\&\quad \le \frac{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i} \right\} }}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t_k^i + {{\left( {\frac{{\varGamma \left( {\alpha + 1} \right) s_k^i}}{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i} \right\} }}} \right) }^{\frac{1}{\alpha }}} - t_k^i} \right) ^\alpha }\nonumber \\&\quad = s_k^i. \end{aligned}$$
(33)
(2) When \(t \in \left( {u_{0,1}^i,t_1^i} \right] \), according to Algorithm 1, we have \(t_{k + 1}^i ={u_{k,1}^i}+{\left( \! {\frac{{\varGamma \left( {\alpha + 1} \right) \left( {s_k^i - \frac{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i} \right\} {{\left( {u_{k,1}^i - t_k^i} \right) }^\alpha }}}{{\varGamma \left( {\alpha + 1} \right) }}} \right) }}{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i,{\omega _i}\left( {u_{k,1}^i} \right) } \right\} }}}\!\right) ^{\frac{1}{\alpha }}}\). Also, one has
$$\begin{aligned}&\left\| {{e_i}\left( t \right) } \right\| \nonumber \\&\quad \le \left\| {{e_i}\left( t \right) - {e_i}\left( {u_{k,1}^i} \right) } \right\| + \left\| {{e_i}\left( {u_{k,1}^i} \right) - {e_i}\left( {t_k^i} \right) } \right\| \nonumber \\&\quad \le \frac{1}{{\varGamma \left( \alpha \right) }}\int _0^{u_{k,1}^i} \left( {{{\left( {u_{k,1}^i - s} \right) }^{\alpha - 1}} - {{\left( {t - s} \right) }^{\alpha - 1}}} \right) \nonumber \\&\quad \left\| {{D^\alpha }{e_i}\left( s \right) } \right\| \mathrm{d}s \nonumber \\&\quad + \frac{1}{{\varGamma \left( \alpha \right) }}\int _{u_{k,1}^i}^t {{{\left( {t - s} \right) }^{\alpha - 1}}\left\| {{D^\alpha }{e_i}\left( s \right) } \right\| \mathrm{d}s} \nonumber \\&\quad + \frac{1}{{\varGamma \left( \alpha \right) }}\int _0^{t_k^i} \left( {{{\left( {t_k^i - s} \right) }^{\alpha - 1}} - {{\left( {u_{k,1}^i - s} \right) }^{\alpha - 1}}} \right) \nonumber \\&\quad \left\| {{D^\alpha }{e_i}\left( s \right) } \right\| \mathrm{d}s \nonumber \\&\quad + \frac{1}{{\varGamma \left( \alpha \right) }}\int _{t_k^i}^{u_{k,1}^i} {{{\left( {u_{k,1}^i - s} \right) }^{\alpha - 1}}\left\| {{D^\alpha }{e_i}\left( s \right) } \right\| \mathrm{d}s} \nonumber \\&\quad \le \frac{{\max \left\{ {\kappa _{k - 1}^i,\omega _k^i} \right\} }}{{\varGamma \left( {\alpha + 1} \right) }}\left[ {{{\left( {t - u_{k,1}^i} \right) }^\alpha } - {{\left( t \right) }^\alpha } + {{\left( {u_{k,1}^i} \right) }^\alpha }} \right] \nonumber \\&\quad + \frac{{\kappa _{k - 1}^i}}{{\varGamma \left( {\alpha + 1} \right) }}\left[ {{{\left( {u_{k,1}^i - t_k^i} \right) }^\alpha } - {{\left( {u_{k,1}^i} \right) }^\alpha } + {{\left( {t_k^i} \right) }^\alpha }} \right] \nonumber \\&\quad + \frac{{{\omega _i}\left( {u_{k,1}^i} \right) }}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t - u_{k,1}^i} \right) ^\alpha } + \frac{{\omega _k^i}}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {u_{k,1}^i - t_k^i} \right) ^\alpha }\nonumber \\&\quad \le \frac{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i,{\omega _i}\left( {u_{k,1}^i} \right) } \right\} }}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {t_{k+1}^i - {u_{k,1}^i}} \right) ^\alpha }\nonumber \\&\quad + \frac{{2\max \left\{ {\kappa _{k - 1}^i,\omega _k^i} \right\} }}{{\varGamma \left( {\alpha + 1} \right) }}{\left( {u_{k,1}^i - t_k^i} \right) ^\alpha }\nonumber \\&\quad =s_k^i. \end{aligned}$$
(34)
Case 2 When \(i \notin {N_h}\), using Algorithm 1, \({\omega _i}\left( t \right) \) remains to be the same as \(\omega _k^i\) until agent i is triggered. Then, for this case, the proof is the same as (32).

In addition, when \({\omega _i}\left( t \right) \) changes finite times at \(u_{k,1}^i,u_{k,2}^i,\ldots , u_{k,{m_k}}^i\left( {{m_k} > 1} \right) \) during time interval \(\left( {t_{k}^i,t_{k+1}^i} \right] \), the proof is similar. Thus, the discussion is omitted here.

Based on the preceding discussions, (26) is given to ensure (10) sufficiently. Thus, this theorem can be proved directly using the proof of Theorem 1. \(\square \)

Remark 5

From Algorithm 1 and the proof of Theorem 2, for agent i, we have \(t_{k + 1}^i - t_k^i \ge \Big [ \frac{{\varGamma \left( {\alpha + 1} \right) }}{{2\sigma _i^k}}\left( {s_k^i + \Delta \left( {t_k^i} \right) } \right) \Big ]^{\frac{1}{\alpha }} > 0\), where \(\sigma _i^k\) is defined in the proof of Theorem 2. It is in evidence that the time between two consecutive triggering time is strictly positive, which can prove that Zeno behavior is precluded in the self-triggered strategy.

Remark 6

As far as we know, it is the first time to design self-triggered algorithm in fractional multi-agent systems. Due to the incorporation of weakly singular kernels in fractional derivative, it is difficult to implement self-triggered strategy in fractional systems. In Theorem 3, by utilizing the fractional integral inequalities, it can be proved that the fractional multi-agent system will reach consensus in the sense of (7) with the triggering time sequence determined by Algorithm 1.

5 Numerical simulations

Here, several numerical examples are derived to indicate the validity of the given results.

Consider the consensus problem of multi-agent system (5) which is assumed to be with \(A={\mathbf {0}}\) and \(B=I_3\). Figure 1 shows an undirected communication graph \({\mathscr {G}}\) with six nodes, which will be used in this section.
Fig. 1

Communication graph \({\mathscr {G}}\) of system (5)

Thus, Assumptions 1 is satisfied. Let \(\alpha = 0.9,\mu = 5,\delta = 0.4384\), and it can be computed that \({\lambda _2}\left( L \right) = 0.4384,{\lambda _N}\left( {{I_N} - J} \right) = 1\) and \({\lambda _N}\left( {{L^2}} \right) = 20.8078\). Then, by working out LMI equation (9) with the software named YALMIP (R20171121) in MATLAB, we calculate \(P = \left( {\begin{array}{*{20}{c}} {0.7085}&{}\quad 0&{}\quad 0\\ 0&{}\quad {0.7102}&{}\quad 0\\ 0&{}\quad 0&{}\quad {0.7077} \end{array}} \right) \).

Firstly, the simulations results are shown to verify Theorem 1 in Sect. 3. We randomly chose the initial conditions of fractional multi-agent system (5) as below, \({x_1}\left( 0 \right) = \left[ {6.2944, 8.1158, -7.4602} \right] ^T\), \({x_2}\left( 0 \right) = [ 8.2675, 2.6471, -4.04 91 ]^T\), \({x_3}\left( 0 \right) = [ -4.4300, 0.9376, 9.1501]^T\), \({x_4}\left( 0 \right) = [ 9.2977, -6.8477, 9.4118]^T\), \({x_5}\left( 0 \right) = \left[ { 9.1433, -0.2924, 6.0056} \right] ^T\), and \({x_6}\left( 0 \right) = [ -7.1622, -1.5647, 8.3147]^T\). Choose \({\eta _i} = 0.038\) and \(t_0^i=0, i=1,\ldots ,6\). Figures 2, 3 show the simulation results. Figure 2 shows the state trajectories of \(x_i(t) (i = 1,\ldots , 6)\) of system (5) under control law (7) with condition (10), respectively, the first, the second and the third element of the state. Evidently, consensus can be reached in the system. Furthermore, Fig. 3 shows the evolution of the measurement errors \({\left\| {{e_i}\left( t \right) } \right\| }\left( {i = 1,\ldots ,6} \right) \), which are consistent with event-triggered condition (10). It is evident that the devised distributed controller has a good performance.
Fig. 2

The state \(x_i(t)\) in system (5) under control law (7) with condition (10)

Fig. 3

Evolution of the measurement error \({\left\| {{e_i}\left( t \right) } \right\| }\left( {i = 1,\ldots ,6} \right) \)

Then, a comparison is made to discuss how the choices of different values of fractional order \(\alpha \) affect the number of triggering times for every agent. Table 1 records the numbers of triggering times for each agent before \(t=2\). It is obvious that the smaller the value of fractional order \(\alpha \) is, the less the number of triggering times is required. In order to study the reason for that, we also plot the combined measurement \(\left\| {{q_i}\left( t \right) } \right\| \) (\(i=1,\ldots , 6\)) for all agents in system (5) in Fig. 4. It is easy to see that there exists \(T > 0\) such that the smaller the value of fractional order \(\alpha \) is, the faster the decaying speed of \(\left\| {{q_i}\left( t \right) } \right\| \) (\(i=1,\ldots , 6\)) is for \(t \in \left( {0,T} \right) \). This result is in agreement with Refs. [17, 69, 70].
Fig. 4

The combined measurement \(||q_i(t)||\) with different alpha

Table 1

Performance comparison with different \(\alpha \)

\(\alpha \)

Triggering numbers for agents

1

2

3

4

5

6

1

163

156

242

192

172

165

0.9

114

117

151

131

132

117

0.8

96

100

125

113

109

98

0.7

83

86

121

106

91

84

0.6

71

73

141

134

77

72

Secondly, the simulations results are shown to verify Theorem 3 in Sect. 4. Figure 5 shows the state trajectories of \(x_i(t) (i = 1,\ldots , 6)\) of fractional network (5) under the self-triggered control strategy proposed in Algorithm 1, respectively, the first, the second and the third element of the state. Obviously, the system can realize consensus. The obtained simulation results have verified the effectiveness of the designed self-triggered strategy obviously.
Fig. 5

The state \(x_i(t)\) in system (5) via the self-triggered control scheme

6 Conclusion

In this manuscript, the consensus problem of fractional multi-agent systems is investigated by distributed event-triggered mechanism. By utilizing the Lyapunov functional method and the fractional inequality technique, some sufficient criteria are obtained to achieve consensus. Also, a rigorous proof is given to verify the exclusion of Zeno behavior. However, in such event-triggered control scheme, the measurement error is required to continuously check. In order to relax this requirement, a self-triggered mechanism is further designed, where no state or error measurement is needed in between two consecutive triggering time instants. Such a self-triggered control is first developed for fractional multi-agent systems. Its feasibility is proved rigorously and also demonstrated by some numerical examples.

In the future, we will extend the obtained results in this paper to some more practical case including discussing in time-varying directed topology and considering stochastic disturbance in networks.

Notes

Compliance with ethical standards

Conflict of interest

We declare that we have no conflict of interest.

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceBeijing Jiaotong UniversityBeijingChina

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