Distributed Containment Control of Networked Fractional-Order Systems with Delay-Dependent Communications

This paper is concerned with a containment problem of networked fractional-order system with multiple leaders under a ﬁxed directed interaction graph. Based on the neighbor rule, a distributed protocol is proposed in delayed communication channels. By employing the algebraic graph theory, matrix theory, Nyquist stability theorem, and frequency domain method, it is analytically proved that the whole follower agents will ﬂock to the convex hull which is formed by the leaders. Furthermore, a tight upper bound on the communication time-delay that can be tolerated in the dynamic network is obtained. As a special case, the interconnection topology under the undirected case is also discussed. Finally, some numerical examples with simulations are presented to demonstrate the e ﬀ ectiveness and correctness of the theoretical results.


Introduction
In recent years, coordination of multiagent systems has attracted considerable interest in the control community due to their wide application areas in formation control 1-3 , flocking/swarming 4, 5 , consensus 6-9 , sensor networks 10, 11 , synchronization of complex networks 12, 13 , and distributed computation 14 . A common character of these applications is that each individual agent lacks a global knowledge of the whole system and can only send and/or obtain state information from its neighbors through local communications. Significant progress has been made in the coordination problem see, e.g., 11,15,16 and the references therein .

Journal of Applied Mathematics
As a special case of coordination control, containment control means to drive the followers to be in the convex hull which is spanned by the leaders. Different from general leader-following consensus problems, where there exist multiple leaders and multiple followers in containment problems. The coordinate objective is likely to be one or several target sets in multiagent coordination control. These target sets may be a biological communities, a team of robots, a food enrichment area, and so on. For example, a kind of biological group hunts another kind of biological communities, a team of biological group cooperatively builds their nest, and several agents lead a team of agents avoiding hazardous obstacles. Thus, how to control a multiagent system to achieve a common target becomes an interesting problem.
The idea of fractional calculus has been known since the development of the regular calculus, with the first reference probably being dated back to the seventeenth century 17 , where the meaning of derivative of order one-half was first mentioned. Although it has a long history, the fractional calculus applications to physics and engineering are just a recent focus of interest. Moreover, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. As pointed out by many researchers, many physical systems are more suitable to be modeled by fractional-order dynamic equations 18 . Many systems are known to display fractional-order dynamics, such as viscoelastic systems, electromagnetic waves, and quantum evolution of complex systems. In additional, integer-order systems can be regarded as a special case of fractional-order systems.
More recently, many interesting agent-related consensus problems are under investigation, and fractional-order consensus becomes a hot topic. The consensus problem of fractional-order systems is first proposed and investigated by Cao et al. 19,20 , where three different cases of coordination models are introduced. By employing a varyingorder fractional coordination strategy, a higher convergence performance is obtained. Sun et al. 21 study the consensus problem for fractional-order systems under undirected scale-free networks. They also compared the convergence rate of fractional-order dynamics and the integer-order dynamics. In order to increase the convergence speed and ensure the exponential convergence, a switching order consensus protocol is employed. Shen et al. 22,23 consider the consensus problem of fractional-order systems with nonuniform input and/or communication delays over directed networks. Based on the Nyquist stability criterion and frequency domain approach, some sufficient conditions are obtained to ensure the fractional-order consensus. Formation control problems for fractional-order systems were discussed in 24 . However, little research work has been done toward the problem of containment control of fractional-order system, which is the main focus of this paper.
Motivated by the above discussion, in this paper, we consider the containment problem of networked fractional-order systems over directed topologies. Different from generally leader-following consensus problem, there exist multiple leaders in a containment control problem. The objective is to drive the followers to be in the convex hull formed by the leaders. On the other hand, in practice delays unavoidably exist due to the finite speed of transmission, acquisition, and traffic congestions. Therefore, studying the agents with the form of fractional-order dynamics over delayed communication channel becomes very significant.
The rest of the paper is organized as follows. Section 2 gives some preliminaries on algebraic graph theory and Caputo fractional operator and formulates the problem under investigation. In Section 3, containment control under fixed directed topologies and delayed Journal of Applied Mathematics 3 communication channels is investigated. In Section 4, two examples are provided to verify the theoretical analysis. Finally, some concluding remarks will be drawn in Section 5. Some remarks on the notation are given as follows: α stands for the integer part of α, 1 n is an n-dimension column vector with all ones. I N is the identity matrix with dimension N × N. R N×N and R N , respectively, denote the family of all N × N dimensional matrices and the N dimensional column vector. The notation A > 0 A ≥ 0 means that matrix A is positive definite semidefinite . " x stands for the Euclidean norm of vector x. λ i A and ρ A denote the ith eigenvalue of matrix A and spectral radius of matrix A, respectively. diag {· · · } stands for a block diagonal matrix. A⊗B means the Kronecker product of matrices A and B. A set Θ ∈ R N is convex if the line segment between any two elements in Θ lies in Θ, that is, if for any x, y ∈ Θ and any γ with 0 ≤ γ ≤ 1, we have γx 1 − γ y ∈ Θ. Similarly, a vector sum γ 1 x 1 γ 2 x 2 · · · γ n x n is called a convex combination of x 1 , x 2 , . . . , x n if the coefficients satisfy γ i ≥ 0 and n i 1 γ i 1. The convex hull of Θ denoted by co{Θ} is the intersection of all convex sets containing Θ. For nonempty set E, the Euclid distance between point x and set E is defined as d E x inf y∈E x − y . I {1, 2, . . . , N}.

Preliminary
Before formulating our problem, we introduce some basic concepts in graph theory and the Caputo fractional operator for fractional-order networks.

Algebraic Graph Theory
Algebraic graph theory is a natural framework for analyzing coordination problems. Let the interaction topology of information exchanged between N agents be described by a directed graph G {V, E, A}, where V {1, 2, . . . , N} is the set of vertices, vertex i represents the ith agent, E ⊂ V × V is the set of edge. An edge in G is denoted by an ordered pair j, i , representing that agent i can receive information from agent j. The neighborhood of the ith agent is denoted by N i {j ∈ V | j, i ∈ E}. A a ij N×N ∈ R N is called the weighted adjacency matrix of G with nonnegative elements where a ii 0 and a ij ≥ 0 with a ij > 0 for j ∈ N i . The in-degree of agent i is defined as deg in i N j 1 a ij , and the out-degree of agent i is defined as . . , i k−1 , i k is called a path from agent i 1 to agent i k . A directed tree is a directed graph, where every agent has exactly one neighbor except one agent has no neighbors. A spanning tree of G is a directed tree whose vertex set is V and whose edge set is a subset of E. In undirected graphs, if there is a path between any two vertices of a graph G, then G is connected, otherwise disconnected.

Caputo Fractional Operator
For an arbitrary real number α, the Riemann-Liouville and Caputo fractional derivatives are defined, respectively, as Journal of Applied Mathematics where m α 1 is the first integer which is not less than α and Γ · is the Euler's gamma function.
It is worth pointing out that the advantage of Caputo approach is that the initial conditions for fractional-order differential equations with Caputo derivatives take on the same form as that for integer-order differential equations. For details, please refer to 17 . Therefore, we will only use the Caputo fractional operator in this paper to model the system dynamics. For notational simplicity, we rewrite C a D α t f t as f α t in the rest of the paper.

Laplace Transform
In the following, we will introduce the Laplace transform of the caputo fractional derivative which will fascinate the development of the subsequent results:

Problem Formulation
Consider a networked fractional-order system consisting of N m agents, where N agents labeled by 1, 2, . . . , N are referred to as the follower agents and the other agents labeled by N 1, . . . , N m act as leaders of the team. The information interaction topology among N following-agents is described by the graph G V, E, A , and the whole system involving N m agents is conveniently modeled by a weighted directed graph G V, E, A with V {1, 2, . . . , N m} and A a ij ∈ R N m × N m , i, j 1, 2, . . . , N m, where the lower block submatrix of order N can be regarded as A. In this paper, we regard the convex hull spanned by multiple leaders as a virtual leader. The graph G has a spanning tree meaning that there exits a path from the virtual leader to every follower agent. In general, the dynamic of each leader is independent of the follower agents. x k represents the position state of the leader k and keeps being a constant.
The dynamic of follower agent i takes the following form: where x i t ∈ R n is the position state, u i t ∈ R n is the control input of agent i, and x α i t ∈ R n is the αth derivative of x i t . In practice, the fractional order α often lies in 0, 1 , so we assume that the order α is a positive real number but not more than 1 in this paper.
For the aforementioned fractional-order dynamics, the following control rule will be used for follower agent i: Inserting the control rule 2.5 into each follower agent dynamic 2.4 , the dynamics of agent i becomes Our objective is to let the N follower agents move into the polytope region formed by the leaders; that is, for any x i t i 1, 2, . . . , N it can be represented as a convex hull of x k t k N 1, N 2, . . . , N m when t → ∞, namely:

Convergence Analysis
Before starting our main results, we begin with the following lemmas which will play an important role in the proof of main results.

Lemma 3.1. If graph G has a spanning tree, then the matrix H associated with G is a positive stable matrix; that is, all the eigenvalues of H lie in the open right hand plane.
Proof. This lemma follows from Lemma 4 in 9 by considering the convex region formed by the leaders as a virtual leader.
Lemma 3.2 see 25 . The following autonomous system:

and A ∈ R n×n , is asymptotically stable if and only if | arg ρ A | > απ/2 is satisfied for all eigenvalues of matrix A. Also, this system is stable if and only if | arg ρ A | ≥ απ/2 is satisfied for all eigenvalues of matrix A with those critical eigenvalues satisfying | arg ρ A | απ/2 having geometric multiplicity of one. The geometric multiplicity of an eigenvalue λ of the matrix A is the dimension of the subspace of vectors v for which Av λv.
When the time delay τ 0, the dynamic network degenerates to the delay-free case, and the dynamic 2.6 can be rewritten as Then, we have the following theorem.

Theorem 3.3.
For the fractional-order dynamic system 3.2 , the follower agents can enter the region spanned by the leaders if the fixed interaction graph G has a directed spanning tree.
Proof . By Lemma 3.1, we learn that matrix H is invertible. Let δ t x t − H −1 B I m ⊗1 N ⊗ I n x t . Then, we get the following error system: Since all the eigenvalues of matrix H lie in the open right hand plane, then, | arg −H | ∈ π/2, π and 2 arg ρ H /π > 1 hold. It follows from Lemma 3.2 that system 3.3 is asymptotically stable for any α ∈ 0, 1 .
In the sequel, we will prove that all the follower agents can be aggregated in a polytope region formed by the leaders. In other words, we need only to prove that for any vector x * i t ∈ R n i 1, 2, . . . , N , the region can be expressed as a convex hull of x k t ∈ R n k N 1, N 2, . . . , N m . This problem can be transformed to prove that matrix H −1 B I m ⊗1 N ⊗I n is a row stochastic matrix that is, it is a nonnegative matrix, and the sum of the items in every row is 1.
Since all the eigenvalues of H have positive real parts, there exist a positive scalar κ > 0 and nonnegative matrix Z such that H κI − Z holds. Obviously, κ > ρ Z and λ i H κ − λ i Z , ∀i 1, 2, . . . , N. Therefore, It can be seen that matrix H −1 ⊗ I n is a nonnegative matrix and so is Notice that H L B 1 m ⊗ I N and L1 N 0; we obtain

3.6
It is obvious that H −1 B I m ⊗ 1 N ⊗ I n is a row stochastic matrix. Thus, the conclusion of Theorem 3.3 holds. In what follows, we will focus on the convergence analysis of 2.6 under fixed and directed interconnection topologies in delayed communication channels.
Journal of Applied Mathematics 7 Theorem 3.5. For the fractional-order dynamic system 2.6 , the follower agents can enter the region spanned by the leaders if the fixed interaction graph G has a directed spanning tree and τ < τ * , where Proof. Since the communication topology G has a spanning tree, all the eigenvalues of H have positive real parts by Lemma 3.1. Therefore, matrix H is invertible.
Then, we get the following error system Taking Laplace transform of system 3.8 , it yields that where ξ s is the Laplace transforms of position states δ t . After some simple manipulation we obtain where Δ s s α I e −τs H⊗I n , which is called a characteristic matrix in 27 . The distribution of det Δ s 's eigenvalues totally determines the stability of system 3.8 . Thus, a sufficient condition for the error dynamics 3.8 which converges to zero is that all the poles of Δ s are located at the open left half plane or s 0.
Then, the characteristic equation of 3.8 is where μ i is the ith eigenvalue of matrix H. Obviously, s / 0. Now, we turn to prove that the poles of 3.11 are located at the open left half plane. Since the directed graph G has a spanning tree, all the eigenvalues of H have positive real parts. Without loss of generality, we assume that the eigenvalues satisfy 0 < Re μ 1 ≤ Re μ 2 ≤ · · · ≤ Re μ n . Based on the Nyquist stability theorem, all the roots of 3.11 lie in the open left half complex plane, if and only if the Nyquist curve e −τs μ i /s α does not enclose the point −1, 0i for any w ∈ R, where i is the imaginary unit.
Therefore, assume that s iw w cos π/2 i sin π/2 w > 0 is a root of 3.11 ; we have Journal of Applied Mathematics where η i arctan Im μ i / Re μ i . Separating the real and imaginary parts of 3.12 yields

3.13
From the above two equations, one gets Obviously, the two terms in the left-hand side of 3.15 are nonnegative the equality holds if and only if both the two terms are zero, namely: Thus, combining 3.16 and 3.17 , we can easily obtain that π 2kπ μ i 1/α τ απ 2 − η i , k 0, 1, 2, . . . .

3.18
Therefore, the smallest time-delay τ > 0 occurs at k 0 and satisfies

3.19
Similarly, one can repeat the very argument for the case that w < 0 and get similar conclusion. Thus, the Nyquist plot of e −τs μ i /s α does not enclose the point −1, 0i for all i > 1 if

3.20
Then all the roots of det Δ s lie in the open left hand plan. Therefore, the error system 3.8 is asymptotically stable, that is, x → x * H −1 B I m ⊗ 1 N ⊗ I n x, as t → ∞. The rest of the proof is similar to that of Theorem 3.3 and hence is omitted.
For the case of the considered undirected graph, that is, agent i and agent j can receive information from each other when there exists an edge between i and j; we can get the following result.
Proof . The proof of Theorem 3.6 is similar to that of the Theorem 3.5 by noting that the eigenvalues of H are nonnegative real number, and hence omitted.

Numerical Examples
In this section, two numerical simulations will be presented to illustrate the effectiveness of the theoretical results obtained in the previous sections. In all the simulations, all dynamics of agent are integrated with a fixed time step 0.05. The following two directed graphs with 0-1 weights will be needed in the analysis of this section. Circle and triangle stand for follower agent and leader, respectively.  Figure 3. It can be easily seen that as time goes on, the whole follower agents will be flocked in the segment −0.5, 0.5 , which is formed by the two leaders. Leader 2 t (s) Example 4.2. Consider a dynamic fractional-order network of five follower agents and four leaders moving in a plane. The interconnection topology among the above agents is given in Figure 2. Clearly, G 2 is strongly connected. The matrix H is and its five eigenvalues are 0.6972, 1.3820, 2.0000, 3.6180, 4.3028, respectively. The initial position states of agents and leaders are given as follows:

4.2
Take the fractional order α 0.96 in 2.1 . Then, it can be seen from Theorem 3.6 that all the followers will enter into the region formed by the leaders if τ < 0.3573.

Conclusion
In this paper, the containment control problem of multiple leaders has been considered for networks of fractional-order dynamics with delay-dependent communication channels. Utilizing algebraic graph theory, matrix theory, Nyquist stability theorem, and frequency domain method, some sufficient conditions are obtained. It is shown that all the follower agents will ultimately move into the convex hull which is spanned by the leaders, for appropriate communication time delay if the topology of weighted network has a spanning tree. Moreover, two numerical simulations are provided to validate the effectiveness of our theoretical analysis.