Consensus of Delayed Fractional-Order Multiagent Systems Based on State-Derivative Feedback

A state-derivative feedback (SDF) is added into the designed control protocol in the existing paper to enhance the robustness of a fractional-order multiagent system (FMS) against nonuniform time delays in this paper. By applying the graph theory and the frequency-domain analysis theory, consensus conditions are derived to make the delayed FMS based on state-derivative feedback reach consensus. Compared with the consensus control protocol designed in the existing paper, the proposed SDF control protocol with nonuniform time delays can make the FMS with SDF and nonuniform time delays tolerate longer time delays, which means that the convergence speed of states of the delayed FMS with SDF is accelerated indirectly. Finally, the corresponding results of simulation are given to verify the feasibility of our theoretical results.


Introduction
It is well known that the distributed coordination control of multiagent systems has received extensive research attention in various fields including robotics and physics.In the distributed coordination control, it is a critical problem for us to design control laws with the information of states of the agents and their neighbors to insure that multiple agents can agree on certain quantities of interest and this problem is often referred to as the consensus problem [1].With the development of technologies such as computers, networks, and communications, consensus of multiagent systems has gradually shown enormous potential applications in the field of swarming [2], flocking [3], formation control [4], unmanned air vehicles [5], and distributed sensor networks [6].
With the development of traditional integer-order derivatives and integrals, the concept of fractional calculus has long been proposed.The earliest concept of fractional calculus could be probably traced back to the 17th century [7].Generally, different from the integer-order derivatives and integrals, the essential characteristic or behavior of an object could be better revealed by the orders of fractional calculus [8].With the development of fractional-order derivatives and integrals, its applications have been considered by many scholars.The authors in [9] studied the fractionalorder derivatives and integrals to establish the stress-strain relationships of viscoelastic materials.The authors in [10] simulated the fractional-order dynamical characteristics of self-similar protein.In [11,12], the proportional-integral differential (PID) controllers whose dynamics were fractionalorder dynamics were proposed and the performance of the fractional-order PID controllers was superior to that of the classical integer-order ones.Moreover, it has been stated in [13] that fractional derivatives were excellent tools for representing the memories and hereditary effects of all manner of materials and processes.
Although fractional-order derivatives and integrals have been studied for a long time, their applications in multiagent systems have just attracted the attention of researchers in recent years.As far as we know, the consensus problem of FMSs was first investigated in [8].Since then, many research results have been continuously springing up about consensus problems of FMSs [13][14][15][16][17].The consensus problem of FMSs paper mainly studies the consensus of FMSs based on SDF.Finally, although the consensus of delayed FMSs based on SDF in [22,23] was investigated, all the time delays in [22,23] were uniform time-delays, which contain the same value.However, this paper considers nonuniform timedelays, which contain up to n n − 1 different values when the FMS consists of n agents.
The main contents of this paper are as follows.Section 2 introduces some basic preliminaries about graph theory, fractional operator, and its Laplace transform.A control protocol based on delayed SDF is designed and the closed-loop dynamics is built in Section 3. The convergence analysis of consensus and the sufficient conditions are obtained in Section 4. In Section 4, we also study the effect of the designed protocol with delayed SDF on the robustness of the FMS against nonuniform time delays.In Section 5, to verify the theoretical results, some examples are simulated.Finally, the conclusions are presented in Section 6.

Preliminaries
2.1.Graph Theory.Let G V , ℰ, A be an interacted graph with the node set and a weighted adjacency matrix A = a ik ∈ ℝ n×n .The node indices belong to a finite index set ℐ = 1, 2, … , n .An edge e ik = v k , v i depicts that node v i can receive information from node v k , which means a ik > 0, otherwise a ik = 0. Besides, we assume a ii = 0 for i ∈ ℐ .Define N i = k ∈ ℐ , k ≠ i as the subscript set of neighbours of node v i .If a graph describes all the edges e ik ∈ ℰ to satisfy a ik = a ki ≥ 0, then the graph is called an undirected graph; if there exists any a ik ≠ a ki , then the graph is called a directed graph.A directed path is a sequence of edges in a directed graph with the form v 1 , v 2 , v 2 , v 3 , v 3 , v 4 , … , where v i ∈ V , and if there is a path from every node to every other node, the graph is said to be strongly connected.A spanning tree exists in a directed graph, which means there is a node such that every other node has a directed path to this node.The out-degree of node v i is defined as deg out v i = ∑ M k=1 a ik , and the Laplacian matrix of the interaction graph is For some graphs such as G 1 , G 2 , … , G M , and graph G composed of the same nodes, the L of graph G is the sum of the other graphs' Laplacian matrix if the edge set of graph G is the sum of that of the other graphs Lemma 1 (see [25]).If graph G is an undirected connected graph, then its Laplacian matrix L has a zero eigenvalue and the other eigenvalues are positive real numbers.
Lemma 2 (see [25]).If graph G is a directed graph and has a spanning tree, then its Laplacian matrix L has a zero eigenvalue and the other eigenvalues have a positive real part.

Fractional
Operator.There are several common fractional operator definitions such as the Caputo operator and Grunwald-Letnikov operator.This paper will use the Caputo operator to analyze the asymptotic consensus properties because it is widely used in engineering and its physical meaning is easy to understand.The derivative of the Caputo operator of f t is defined as follows: where α is the order of the derivative of Caputo operator, u − 1 < α ≤ u, u ∈ Z + , and Γ ⋅ is given by where σ is an arbitrary real number.

Laplace Transform.
In order to facilitate the development of the subsequent results, we let f α t replace C 0 D α t f t , and let F s = L f t = ∞ 0 − e −st f t dt, then the Laplace transform of the Caputo derivative is obtained: 3

Problem Formulation
Assume that a FMS is made up of n agents, each of which is considered as a node in graph G. G V , ℰ, A represents the communication topology of the FMS.The dynamic model of agent i is given as follows: where the ith agent's state is denoted by x i t ∈ ℝ, the α order Caputo derivative of x i t is denoted by x α i t α ∈ 0, 1 , and the control input is denoted by u i t ∈ ℝ. Definition 1.The FMS in (4) can achieve consensus when the states of all agents satisfy lim t→+∞ x i t − x k t = 0, 5 for ∀i, k ∈ ℐ.
Authors in [21] studied the consensus problems of a FMS with nonuniform time delays, and the distributed control protocol is designed by where a ik denotes the element of A, N i denotes the subscript set of neighbours of the agent i, and τ ik is the time delay it takes agent i to receive the information of state of the agent k.If τ ik = τ ki holds for all i, k ∈ ℐ, the time delays are said to be symmetric.Otherwise, the time delays are said to be asymmetric.In [21], it has been illustrated that the consensus of the FMS in (4) with nonuniform time delays can be achieved by the protocol in (6) when all the τ ik < τ, which is called the maximum tolerable delay.Moreover, the FMS in (4) cannot achieve consensus by the protocol in (6) when all the τ ik > τ.Motivated by the method in [1,24], we shall use the information x k t − τ ik + γx α k t − τ ik and x i t − τ ik + γx α i t − τ ik , respectively, instead of x k t − τ ik and x i t − τ ik to reduce the impact of time delays on consensus, where γ denotes the intensity of the delayed SDF.In addition, we also assume that Finally, the control protocol in ( 6) can be rewritten to T .By the protocol in (7), the closed-loop dynamics of the FMS in (4) with SDF and nonuniform time delays can be written as where φ α t represents the Caputo derivative of φ t with α order and L m represents the Laplacian matrix of a subgraph, which is associated with the time delay τ m .

Consensus Convergence Analysis
where and λ n is the maximum eigenvalue of L.
3 Complexity Proof 1.Here, the dynamic performance of the FMS in (8) with SDF and symmetric time delays is studied, so it is not necessary for us to consider the impact of the initial state.Taking the Laplace transform to the FMS in (8) with SDF and symmetric time delays, we have where Ψ s is the Laplace transform of φ t , φ 0 − is the initial value of φ t , and I n ∈ ℝ n×n is the unit matrix.
From (11), we have the characteristic equation of the FMS in (8) with SDF and symmetric time delays: The roots of ( 12) are called the eigenvalues of the FMS in (8) with SDF and symmetric time delays.First of all, we assume that the FMS in (8) with SDF and symmetric time delays is stable and can reach consensus when τ m = 0.Then, it is easy to obtain that as τ m increases continuously from zero, the eigenvalues of the FMS in (8) with SDF and symmetric time delays in the complex plane will change continuously from the LH (left half-plane) to the RH (right half-plane).Once the trajectories of these eigenvalues reach the RH through the imaginary axis, the FMS in (8) with SDF and symmetric time delays will no longer be stable, which results in the failure of the consensus condition.So, it is essential for us to consider the time delay τ when the nonzero eigenvalues of the FMS in (8) with SDF and symmetric time delays appear on the imaginary axis for the first time, and the time delay τ, which is known as maximum tolerable delay, will become the critical point of stability of the FMS in (8) with SDF and symmetric time delays.Now set s = −jω and it is the imaginary eigenvalue of the FMS in (8) with SDF and symmetric time delays, u ∈ ℂ n is the corresponding eigenvector, u = 1, and let u H be the conjugate transpose of u, then the following equation can be obtained: L m e jωτ m u = 0 13 Since all the roots of ( 12) appear in the form of conjugate pairs, it is only necessary to study the case ω > 0. Let the left side of (13) be multiplied by u H , then we have the following series of equations: Then, we define 1 + γω α cos πα/2 2 + γω α sin πα/2 2 e −j arctan γω α sin πα/2 /1+γω α cos πα/2 14 a m e jωτ m = ω α e j π 2−α /2 1 + γω α cos πα/2 2 + γω α sin πα/2 2 e −j arctan γω α sin πα/2 /1+γω α cos πα/2 , 15 4 Complexity where a m = u H L m u/u H u.
If we suppose that δ = sin πα/2 and σ = cos πα/2 , then it is easy to arrive at δ 2 + σ 2 = 1 and we get According to (18), we can calculate the first derivative of τ ω about ω: where and If we assume that there is a function Z ω established by then the first derivative of Z ω is as follows: τ ω is a decreasing function of ω, and when ω ≤ ω, there is On the other hand, when all τ m < τ, the following inequality can be obtained: The contradiction between the inequality in (25) and the inequality in ( 24) is obvious.Accordingly, when all τ m are less than τ, we can avoid the eigenvalues of the FMS in (8) with SDF and symmetric time delays crossing the imaginary axis to reach the unstable RH, and the FMS in (8) with SDF and symmetric time delays can reach consensus; when all τ m are equal to τ, s = −jω is an imaginary eigenvalue of the FMS in (8) with SDF and symmetric time delays, whose corresponding eigenvector u ω makes ∑ M m=1 a m = λ n hold; when all τ m are more than τ, there must exist at least one eigenvalue of the FMS in (8) with SDF and symmetric time delays in the RH, and the states of the FMS in (8) with SDF and symmetric time delays will no longer converge and the FMS in (8) with SDF and symmetric time delays cannot reach consensus.
Corollary 1.Consider a FMS with SDF and symmetric time delays over a connected and undirected graph G.When α = 1, by the distributed control protocol in (7), the FMS in (8) with SDF and symmetric time delays can asymptotically achieve consensus if all τ m < τ, and the FMS in (8) and λ n is the maximum eigenvalue of L.

Robustness Analysis for Case 1.
According to Theorem 1, for the given FMS in (4), if applying the control protocol in (6), that is, the control protocol in (7) with γ = 0, we obtain If applying the control protocol in (7), we obtain In order to show the effect of the protocol in (7) on the robustness against symmetric time delays, we are supposed to further determine the range of α to insure τ up2 > τ up1 , which means Obviously, the inequality in (29) contains multiple parameters.However, for a fix undirected interconnection topology, we can determine the range of α and find a proper value of γ to improve the robustness of the FMS with symmetric time delays.Since it is very difficult for us to solve the inequality in (29) by the analytic method, we shall analyze it by the graphical method and let λ n = 3 5229.Figure 1 is the three-dimensional diagram of τ up with respect to parameters α and γ, and it is easy for us to find proper parameters α, γ to 6 Complexity ensure τ up > 0 by the graphical method.According to Figure 1, we can find that α ∈ α * , 1 when γ changes from 0 to 1/λ n and τ up > 0, and α * which is decided by the inequality in (29) is easy to obtain in Figure 1 when the value of γ is determined.
In particular, if we assume γ = 0 1, then the relationship between τ up and α can be shown in Figure 2. According to Figure 2, it is obvious that α ∈ 0 854,1 α * ≜ 0 854 when τ up > 0 and γ = 0 1.Theorem 2. Consider a FMS with SDF and asymmetric time delays over a directed interconnection graph G that has a spanning tree.By the distributed control protocol in (7), the FMS in (8) with SDF and asymmetric time delays can asymptotically achieve consensus if all τ m < τ, and the FMS in (8) with SDF and asymmetric time delays cannot achieve consensus if all τ m > τ. where λ i which makes τ minimized, and λ i is the ith eigenvalue of L.
Proof 2. Let one apply the above frequency-domain proof method, which has been used in proving Theorem 1. Suppose that s = −jω ≠ 0 is the eigenvalue of the FMS in (8) with SDF and asymmetric time delays on the imaginary axis, u ∈ ℂ n is the corresponding eigenvector, and u = 1.According to Lemma 2, one can get Taking the modulus of both sides of (31) and regarding ω as the function of B a , we can get where ω B a is an increasing function of B a .
Calculating the principal value of the argument of (31) on both sides separately, and we have According to the definition of B a in (31), we have and it yields that The contradiction between the inequality in (36) and the inequality in (35) is obvious.Accordingly, when all τ m < τ, the eigenvalues of the FMS in (8) with SDF and asymmetric time delays cannot reach or cross the imaginary axis, then the FMS in (8) with SDF and asymmetric time delays will remain stable and the FMS in (8) with SDF and asymmetric time delays can reach consensus.On the other hand, when all τ m > τ, there must exist at least one eigenvalue of the FMS in (8) with SDF and asymmetric time delays in the RH, then the states of the FMS in (8) with SDF and asymmetric time delays will no longer converge and the FMS in (8) with SDF and asymmetric time delays cannot reach consensus.
Corollary 2. Consider a FMS with SDF and asymmetric time delays over a directed interconnection graph G with a spanning tree.When α = 1, by the distributed control protocol in (7), the FMS in (8) with SDF and asymmetric time delays can asymptotically achieve consensus if all τ m < τ, and the FMS in (8) with SDF and asymmetric time delays cannot achieve consensus if all τ m > τ. τ = min where 1/ λ i , λ i is the λ i which makes τ minimized, and λ i is the ith eigenvalue of L. 4.2.2.Robustness Analysis for Case 2. According to Theorem 2, for the given FMS in (4), if applying the protocol in (6), that is, the control protocol in (7) with γ = 0, we obtain When we apply the control protocol in (7), we obtain where In order to show the effect of the protocol in (7) on the robustness against asymmetric time delays, we are supposed to further determine the range of α to insure up2 > τ up1 , which means Obviously, the inequality in (40) contains multiple parameters.However, for a fix directed interconnection graph G that has a spanning tree, we can determine the range of α and find a proper value of γ to improve the robustness of the FMS with asymmetric time delays.Since it is very difficult for us to solve the inequality in (40) by the analytic method, we shall analyze it by the graphical method and let γ * = 0 6967.Figure 3 is the three-dimensional diagram of τ up with respect to parameters α and γ, it is easy for us to find proper parameters α, γ to ensure τ up > 0 by the graphical method.According to Figure 3, we can find that α ∈ α * , 1 when γ changes from 0 to γ * and τ up > 0, and α * which is decided by the inequality in (40) is easy to obtain in Figure 3 when the value of γ is determined.
It is obvious that all symmetric time delays τ m are greater than τ up1 and less than τ up2 .
Figures 6 and 7 show the trajectories of x i t with the symmetric time delays by applying the different control protocols when τ up1 < all τ m < τ up2 .From these simulation results, it is obvious that the given FMS in (4) by the control protocol in (6) diverges and cannot reach consensus, whereas the FMS in (4) applying the SDF control protocol in (7) converges to the same states and can reach consensus.Hence, x 3 (t) x 4 (t) x 1 (t) x 2 (t) x i (t) x 3 (t) x 4 (t) x 1 (t) x 2 (t) 9 Complexity the introduced SDF control protocol can enhance the robustness of the FMS in (4) to symmetric time delays.
It is obvious that all asymmetric time delays τ m are greater than τ up1 and less than τ up2 .
Figures 10 and 11 show the trajectories of x i t with the asymmetric time delays by applying the different control protocols when τ up1 < all τ m < τ up2 .From these simulation results, it is obvious that the given FMS in (4) by the control protocol in (6) diverges and cannot reach consensus, whereas the FMS in (4) applying the SDF control protocol in (7) converges to the same states and can reach consensus.Hence, the introduced SDF control protocol can enhance the robustness of the FMS in (4) to asymmetric time delays.
On the other hand, under the same conditions, we suppose that τ 14 = 0 88 s, τ 21 = 0 89 s, τ 31 = 0 90 s, and τ 42 = 0 91 s. Figure 12 shows the trajectories of x i t applying the protocol in (7) under asymmetric time delays when all τ m > τ up2 .It is clear that the FMS in (4) cannot reach consensus.x 3 (t) x 4 (t) x 1 (t) x 2 (t) Figure 8: The trajectories of x i t applying the protocol in (7) under symmetric time delays when all τ m > τ up2 .x 3 (t) x 4 (t) x 1 (t) x 2 (t) Figure 10: The trajectories of x i t applying the protocol in (6) under asymmetric time delays when all τ m > τ up1 .

Conclusion
In order to enhance the robustness of a FMS against nonuniform time delays, a control protocol based on SDF and nonuniform time delays is introduced in this paper.First of all, the consensus problem is investigated for the FMS with SDF and symmetric time delays over undirected topology.Then, the consensus problem is investigated for the FMS with SDF and asymmetric time delays over directed topology.By the robustness analysis, it is obvious that the control protocol-based on SDF with the appropriate intensity can enhance the robustness for the FMS to nonuniform time delays.Finally, the validity of the theoretical analysis is verified by the corresponding simulation results.In addition, inspired by [26,27], the consensus problems or formation control problems of delayed double-integrator FMSs based on round-robin protocols or attacks will be one of the most interesting topics of our future research work.x 3 (t) x 4 (t) x 1 (t) x 2 (t) x i (t) x 3 (t) x 4 (t) x 1 (t) x 2 (t) Figure 11: The trajectories of x i t applying the protocol in (7) under asymmetric time delays when all τ m < τ up2 .

Figure 1 :Figure 2 :
Figure 1: The three-dimensional diagram of τ up with respect to parameters α and γ in case 1.

Figure 3 : 8 Complexity 5 . 1 .
Figure 3: The three-dimensional diagram of τ up with respect to parameters α and γ in case 2.

Figure 5 :
Figure 5: The communication topology in example 1.

Figure 6 :
Figure 6: The trajectories of x i t applying the protocol in (6) under symmetric time delays when all τ m > τ up1 .

Figure 7 :Figure 4 :
Figure 7: The trajectories of x i t applying the protocol in (7) under symmetric time delays when all τ m < τ up2 .

Figure 9 :
Figure 9: The connected interaction topology in example 2.

Figure 12 :
Figure12:The trajectories of x i t applying the protocol in(7) under asymmetric time delays when all τ m > τ up2 .