Recently, with the rapid development of network and communication technology, the distributed coordination for networked systems has been studied deeply [1][5]. Cooperative control of multiagent systems has become a hot topic in the fields of automation, mathematics, computer science, etc [6][10]. It has been applied in both military and civilian sectors, such as the formation control of mobile robots, the cooperative control of unmanned spacecraft, the attitude adjustment and position of satellite, and the scheduling of smart power grid systems, etc. As a kind of distributed cooperative control problems of multiagent systems with multiple leaders, containment control regulates followers eventually converge to a target area (convex hull formed by the leaders) by designing a control protocol, which has been paid much more attention in recent years [11][13].
In the complex practical environments, many distributed systems cannot be illustrated with the integerorder dynamics and can only be characterized with the fractionalorder dynamics [14][16]. For example, flocking movement and food searching by means of the individual secretions, exploring of submarines and underwater robots in the seabed with a massive number of microorganisms and viscous substances, working of unmanned aerial vehicles in the complex space environment [17], [18]. Cao et al. have studied the coordination of multiagent systems with fractional order [19], [20], and obtained the relationship between the number of individuals and the order in the stable fractional system. Yang et al. have studied the distributed coordination of fractional order multiagent systems with communication delays [21], [22]. Motivated by the broad application of coordination algorithms in fractionalorder multiagent systems (FOMAS), the containment control of distributed fractionalorder systems will be studied in this paper.
For containment control problems, the current research works are mainly focused on integerorder systems [11][13], [23][26]. In [11], containment control problem for firstorder multiagent systems with the undirected connected topology is investigated, and the effectiveness of control strategy is proven by using partial differential equation method. In [12], secondorder multiagent systems with multiple leaders are investigated, the containment control of multiagent systems with multiple stationary leaders can be achieved in arbitrary dimensions. In [13], two asymptotic containment controls of continuoustime systems and discretetime systems are proposed for the multiagent systems with dynamic leaders, and the constraint condition for control gain and sampling period are given. Considering factors such as external disturbance and parameter uncertainty in [23], the attitude containment control problem of nonlinear systems are studied in a directed network. The impulsive containment control for secondorder multiagent systems with multiple leaders is studied in [25], [26], where all followers are regulated to access the dynamic convex hull formed by the dynamic leaders.
When agents transfer information by means of sensors or other communication devices in coordinated network, communication delays have a great impact on the behaviors of the agents. Now, the influences of communication delays on multiagent systems have also been paid more attentions [2], [7][10] where these research activities on the coordination problem are mainly concentrated on integerorder multiagent systems. In [24], containment control problem of multiagent systems with time delays is studied in fixed topology, and two cases of multiple dynamic leaders and multiple stationary leaders are discussed, respectively. As far as we know, few researches have been done on the containment consensus of fractional order multiagent systems with time delays.
In this paper, the containment control algorithms for multiagent systems with fractional dynamics are presented, and the containment consensus of distributed FOMAS with communication delays is studied under directed connected topologies. The main innovation of this paper is that the distributed containment control of fractional order multiagent systems with multiple leaders and communication delays is studied for the first time. The research presented in this paper is different from [21], where consensus of FOMAS without leader [21] is much easier than containment control of FOMAS with multiple leaders in this paper. The rest of the paper is organized as follows. In Section Ⅱ, we recall some basic definitions about fractional calculus. In Section Ⅲ, some preliminaries about graph theory are shown, and fractional order coordination model of multiagent systems is presented. Containment control of fractional coordination algorithm for multiagent systems with communication delay is studied in Section Ⅳ. In Section Ⅴ, numerical simulations are used to verify the theoretical analysis. Conclusions are finally drawn in Section Ⅵ.
Ⅱ. FRACTIONAL CALCULUSFractional calculus has played an important role in modern science. There are two fractional operators used widely: Caputo and RiemannLiouville (RL) fractional operators. In physical systems, Caputo fractional order operator is more practical than RL fractional order operator because RL operator has initial value problems. Therefore, in this paper we will apply Caputo fractional order operator to describe the system dynamics and analyze the stability of proposed FOMAS algorithms. Generally, Caputo operator includes Caputo fractional integral and Caputo fractional derivative. Caputo fractional integral is defined as
$ ^{C}_{t_0}D^{p}_{t}f(t)=\frac{1}{\Gamma(p)}\int^{t}_{t_0}\frac{f(\theta)}{(t\theta)^{1p}}d \theta $ 
where the integral order
$ ^{C}_{t_0}D^{\alpha}_{t}f(t)=^{C}_{t_0}D^{p}_{t}\left[\frac{d^{[\alpha]+1}}{dt^{[\alpha]+1}}f(t)\right] $  (1) 
where
$ \mathcal{L}(f^{(\alpha)})=s^{\alpha}F(s)\sum\limits_{k=1}^{[\alpha]+1}s^{\alpha1}f^{(k1)}(0) $  (2) 
where
Assume that
Let
Lemma 1 [3]:
Lemma 2 [9]: Matrix
Definition 1: The convex hull of a finite set of points
Recently, fractional order systems have been widely applied in various science fields, such as physics, hydrodynamics, biophysics, aerodynamics, signal processing and modern control. The theories of fractional order equations are studied deeply, and the relationship between the fractional order and the number of agents to ensure coordination has been presented in [19]. Assume that Caputo fractional derivative is used to indicate the dynamics of multiagent systems in the complex environments, the fractional order dynamical equations are defined as
$ x_{i}^{(\alpha)}(t)=u_i(t), \quad i=1, \ldots, n $  (3) 
where
$ u_{i}(t)=\gamma\sum\limits_{k\in N_i}a_{ik}[x_{i}(t)x_{k}(t)], \quad i\in I $  (4) 
where
Suppose the multiagent systems consisting of
$ \begin{align} u_{i}(t)=\begin{cases} \gamma\sum\limits_{k\in N_i}a_{ik}[x_{i}(t)x_{k}(t)], & i=1, 2, \ldots, n_1 \\ 0, & i=n_1+1, \ldots, n.\end{cases} \end{align} $  (5) 
The systems (3)(5) can be rewritten as
$ X^{(\alpha)}(t)=\gamma \left( \begin{array}{cc} L_{1} \ \ L_{2}\\ 0\ \ \ 0 \end{array}\right)X(t) $  (6) 
where
Remark 1: Matrix
$ l_{ik}=\begin{cases}d_ia_{ii}, & i=k\\a_{ik}, & i\neq k. \end{cases} $ 
Matrix
$ l_{ik}=a_{ik}, \quad i=1, 2, \ldots, n_1; \ \, k=n_1+1, \ldots, n. $ 
Assume the collection formed by leaders is regarded as a virtual node, if one follower agent can connect to some leader, then the follower is connected to the virtual node.
Definition 2: The containment control is realized for the system (3) under certain control input (5), if the position states of the followers are asymptotically converged to the convex hull formed by the leaders.
Assumption 1: For any one follower, there is a directed connected path to the virtual node formed by leaders.
Lemma 3: With Assumption 1, matrix
Proof: From Lemma 2, matrix
$ \begin{align*} L_1^{1} =&\ (dI_{n_1}Q)^{1}\\ =&\ d^{1}(I_{n_1}+d^{1}Q+(d^{1}Q)^2+\cdots). \end{align*} $ 
Then, we obtain
From Lemma 1, Laplacian matrix
$ L_1 X_{01}+L_2 X_{02}=0 $ 
where
$X_{01}=L_1^{1} L_2 X_{02}.$ 
Therefore,
Theorem 1: Consider a directed dynamic system of
Proof: Based on the system (6), we have
$ X_1^{(\alpha)}(t)=\gamma ( L_{1} X_1(t)+ L_{2}X_2(t) )\notag \\ X_2^{(\alpha)}(t)=0. $  (7) 
Let
$ \bar{X}_1^{(\alpha)}(t)=\gamma L_{1} \bar{X}_1(t)\notag \\ X_2^{(\alpha)}(t)=0. $  (8) 
It is known that the fractional differential system (8) is asymptotically stable iff
$\lim\limits_{t\rightarrow \infty}X_1(t)=L^{1}_1L_2X_2(t).$ 
Since matrix
Remark 2: If FOMAS of
Remark 3: If the fractional order
In this section, we assume that there are communication delays in the dynamical systems, and containment control of the fractionalorder agent systems with communication delays will be studied. Under the influence of communication delays, we can get the following algorithm
$ x_{i}^{(\alpha)}(t)=u_i(t\tau), \quad i=1, \ldots, n $  (9) 
where
$ X_1^{(\alpha)}(t)=\gamma ( L_{1} X_1(t\tau)+ L_{2}X_2(t\tau) ) \notag\\ X_2^{(\alpha)}(t)=0. $  (10) 
Let
$ \bar{X}_1^{(\alpha)}(t)=\gamma L_{1} \bar{X}_1(t\tau)\notag \\ X_2^{(\alpha)}(t)=0. $  (11) 
Theorem 2: Suppose that multiagent systems are composed of
$ \tau<\frac{(2\alpha)\pi}{2(\bar{\lambda} \gamma)^{\frac{1}{\alpha}}} $  (12) 
where
Proof: By applying Laplace transformation to system(11), we can obtain the characteristic equation of the system
$ \det(s^\alpha I_n+\gamma e^{\tau s} L_1)=0. $ 
Since the Laplacian matrix
When
$ G(j\omega)=\omega^{\alpha}e^{j(\omega\tau+\frac{\alpha\pi}{2})}\gamma L_1 $  (13) 
we have the eigenvalues of
$ \begin{align*} \lambda I_{n_1}G(j\omega)& =\left\lambda I_{n_1}(\omega^{\alpha}e^{j(\omega\tau+\frac{\alpha\pi}{2})}\gamma L_1)\right\\ & =\prod\limits_{i=1}^{n_1}\left(\lambda\gamma \lambda_i \omega^{\alpha}e^{j(\omega\tau+\frac{\alpha\pi}{2})}\right) \end{align*} $ 
where
$\tau<\min\left\{\frac{(2\alpha)\pi} {2(\lambda_i \gamma)^\frac{{1}}{\alpha}}, \ \ i=1, 2, \ldots, n_1\right\}$ 
the point
$\tau<\frac{(2\alpha)\pi}{2(\bar{\lambda} \gamma)^ {\frac{1}{\alpha}}}$ 
where
Corollary 1: Suppose multiagent systems are composed of
$ \tau<\frac{(2\alpha)\pi}{2(\lambda_{\max} \gamma)^\frac{1}{\alpha}} $  (14) 
where
Corollary 2: Suppose multiagent systems are composed of
$ 2\gamma\tau<\frac{\pi}{\lambda_{\max}} $  (15) 
where
Remark 4: If the fractional order
Remark 5: The consensus result in Corollary 2 for
Consider the dynamic topology with 5 followers and 3 leaders (illustrated as
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Fig. 1 Network topology of multiagent systems. 
From the communication topology of FOMAS, the system matrix can be obtained
$ \begin{align} L_1 = \left[ \begin{array}{rrrrr} 3 &1 & 0 & 0 &1 \\ 1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 &1 & 0 \\ 0 & 0 & 1 & 3 & 1 \\ 1 & 0 & 0 & 1 & 3 \end{array} \right]. \end{align} $  (16) 
Assume that the control parameter of system is taken
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Fig. 2 Moving track of FOMAS without communication delays. 
Next, we will verify the results of FOMAS with time delays. The maximum eigenvalue of
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Fig. 3 Moving track of FOMAS with communication delay 
Then, we will enlarge the time delays in FOMAS. Let
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Fig. 4 Moving track of FOMAS with communication delay 
This paper studies containment control of fractional multiagent systems with communication time delays. Containment consensus of multiagent systems with directed network topology is studied. By applying the stability theory of frequency domain, FOMAS with delay is analyzed, and the relationship between the control gain of multiagent systems and the upper bound of time delays is derived. Suppose the orders of the fractional dynamical systems are all 1, the extended conclusion in this paper is the same with ordinary integer order systems. The containment control of fractional order multiagent systems with dynamical topologies and linear timevarying (LTV) systems will be investigated in the future works.
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