Consensus of Multi-Integral Fractional-Order Multiagent Systems with Nonuniform Time-Delays

Consensus of fractional-order multiagent systems (FOMASs) with single integral has been wildly studied. However, the dynamics with multiple integral (especially double integral to sextuple integral) also exist in FOMASs, and they are rarely studied at present. In this paper, consensus problems for multi-integral fractional-order multiagent systems (MIFOMASs) with nonuniform timedelays are addressed.The consensus conditions for MIFOMASs are obtained by a novel frequency-domain method which properly eliminates consensus problems of the systems associated with nonuniform time-delays. Besides, the method revealed in this paper is applicable to classical high-ordermultiagent systems which is a special case ofMIFOMASs. Finally, several numerical simulations with different parameters are performed to validate the correctness of the results.


Introduction
The research related to multiagent systems (MASs) has been going on for decades, due to its many meaningful applications, e.g., sweep coverage control of MASs [1], flocking behavior of mobile robots [2], and coordinated attitude control of a formation of satellites [3].Consensus is an agreement on the quality of certain concerns about the specific states of all agents, which is one of the most fundamental requirements for the research on MASs.
Up to now, numerous studies have been conducted to resolve the problems about consensus of MASs with different dynamics.During the past decades, a lot of results have been accomplished about consensus of first-order MASs [4][5][6][7][8][9][10][11].In [4], a simple model was presented for the phase transition of a set of self-driven particle, and it was demonstrated that the headings of all agents in MASs converged to a common value by simulation.In [5], authors provided some theoretical explanations for Vicsek's linearized model and analyzed the alignment of undirected switching topologies of agents that were regularly connected.Based on the research works of [5], more relaxed consensus conditions over dynamic switching topologies were given in [6,7].Authors in [8] put forward a framework about consensus theory of MASs with directed information flow, link/node failures, time-delays, and so on.Robust  ∞ control about consensus problems for MASs with parameter uncertainties, external disturbances, nonidentical state, and time-delays was discussed in [9].In recent years, more and more researchers have paid more attention to consensus of second-order MASs [12][13][14][15][16].For instance, authors in [12] investigated consensus problems for second-order continuous-time MASs in the presence of jointly connected topologies and time-delay.In [13], two types of consensus problems for second-order MASs with and without delay over switching topology and directed topology were studied.In [14], the local consensus problem for secondorder MASs whose dynamics were nonlinear dynamics under directed and switching random topology was discussed, and several sufficient conditions were derived to ensure the MASs reach consensus.Furthermore, taking into account the fact that high-order MASs were widely used, consensus problems for high-order MASs have been studied in [17][18][19][20][21].In particular, the output consensus problem in [17] was addressed for high-order MASs with external disturbances, and some conditions were derived to ensure consensus for the MASs.The consensus problems in [18] were considered for a class of high-order MASs with time-delays and switching networks, and a nearest-neighbour rule was designed and some conditions were derived to guarantee consensus for the systems with time-delays.The conditions of consensus in [21] for high-order MASs with nonuniform time-delays were proposed by a novel frequency-domain approach which properly resolved the challenges associated with multiple time-delays.
It is worth noting that many results above about MASs were based on the integer-order dynamics.In fact, many scholars have declared that the essential characteristic or behavior of an object in the complex environment could be better revealed by adopting fractional-order dynamics.Examples include unmanned aerial vehicles operating in an environment with the impacts of rain and wind [22], food searching with the help of the individual secretions and microbial [23], and submarine robots in the bottom of the sea with large amounts of microorganisms and viscous substances [24].Compared to integer-order dynamics, fractional-order dynamics provided an excellent tool in the description of memory and hereditary properties [25,26].Moreover, authors in [27,28] indicated that the integer-order systems were only the special examples of the fractionalorder systems.Based on these facts, the research results on consensus of FOMASs with single integral in [29][30][31][32][33][34] have been continuously springing up in recent years.As we know, consensus problem of FOMASs was first proposed and investigated by Cao et al. [29].Next, consensus control of FOMASs with time-delays was studied by Yang et al. [30,31], where homogeneous dynamics and heterogeneous dynamics were used to illustrate the agent of system.In [32], consensus problem of linear FOMASs with input timedelay and the consensus problem of nonlinear FOMASs with input time-delay were investigated, respectively.In [33], consensus problems were studied for FOMASs with nonuniform time-delays.Meanwhile, by means of matrix theory tool, Laplace transform and graph theory tool, two delay margins were obtained as the consensus conditions.Lately, consensus of FOMASs with double integral was proposed in [35][36][37][38][39].The consensus problem of FOMASs with double integral over fixed topology was studied in [35].By applying Mittag-Leffler function, Laplace transform, and dwell time technique, consensus for FOMASs with double integral over switching topology was investigated in [36].Based on the sliding mode estimator, consensus problem for FOMASs with double integral was studied in [37].By means of matrix theory tool, Laplace transform, and graph theory tool, consensus problems for a FOMAS with double integral and time-delay were studied in [38].Nevertheless, the above research results on the consensus problems of FOMASs with or without time-delays were based on the single-integral fractionalorder or double-integral fractional-order dynamics.To this day, there is almost no research on consensus problems of MIFOMASs with time-delays, especially nonuniform timedelays.
Motivated by above analysis, we extend FOMASs from single-integral fractional-order dynamics to multi-integral fractional-order ones in this paper.Consensus problems of FOMASs with multiple integral in the presence of nonuniform time-delays are studied.The main idea of this paper is to first obtain the characteristic polynomial of a MIFOMAS with imaginary eigenvalues through the model transformation of the system and then determine the stability conditions of the system according to this characteristic polynomial, so as to determine the consensus conditions of the system according to the stability conditions of the system.The consensus conditions of the MIFOMAS with nonuniform time-delays can be obtained by inequalities.
The remainder of this article is organized as follows.In Section 2, fractional calculus and its Laplace transform are given.In Section 3, the knowledge about graph theory is shown out.In Sections 4 and 5, consensus algorithms for a MIFOMAS in the presence of nonuniform timedelays are studied.In Section 6, some numerical examples with different parameters are simulated to verify the results.Finally, conclusions are drawn out in Section 7.

Fractional Calculus
In [40], several different definitions of fractional calculus have been proposed, in which the Caputo fractional derivative played an important role in fractional-order systems.Because the initial value of Caputo fractional derivative has practical signification in many problems, which is commonly used in the variety of physical fields.Ergo, this paper will model the system dynamical characteristics by using Caputo derivative which is defined by where  ∈ R denotes the initial value,  represents the order of the Caputo derivative, and  − 1 <  ≤ ( ∈  + ).Γ(⋅) is given by where (0 − ) = lim →0 − () and   (0 − ) = lim →0 −   ().

Graph Theory
For a MAS with  agents, the network topology can be denoted by a graph G = (V, E), where V = {

Problem Statement
There are two lemmas [41] for the later analysis.
In this paper, the control protocol for MIFOMAS (4) will be given by where   denotes the neighbors index collection of the agent ,   is the (, )-th element of A,   > 0 is the timedelay which is from the -th agent to the -th agent, and  +1 > 0 are scale coefficients.If all   =   , then the timedelays are symmetrical; else the time-delays are asymmetrical.The symmetric time-delays and the asymmetric timedelays are two different forms of nonuniform time-delays.For ease of analysis, we define that   denote  different time-delays of MIFOMAS (4); i.e.,   ∈ {  : , ∈ I} ( = 1, 2, . . ., ).Then the following control protocol is provided to resolve consensus problems of MIFOMAS (4): and then under the control protocol given by (7), the closed-loop dynamics of MIFOMAS (4) can be described as 1

Main Results
Theorem 4. Suppose that a FOMAS with multiple integral is given by MIFOMAS ( ) whose corresponding network topology G satisfies Lemma .Define the following functions: where

for the MIFOMAS ( ) with symmetric time-delays, then the control protocol ( ) can resolve the consensus problem of the MIFOMAS ( ) with symmetric time-delays, and on the contrary, then the control protocol ( ) can not resolve the consensus problem of the MIFOMAS ( ) with symmetric time-delays. e value of
corresponding to  in   () is determined by the following equation: where |  ()| is the modulus of   () and   is the maximum eigenvalue of .
Proof.We shall apply the frequency-domain method to analyze the MIFOMAS (10) with symmetric time-delays, and we can get where Ψ() is the Laplace transform of (), (0 − ) is the initial value of (), and Motivated by the stability analysis of a fractional-order system in [42], we can study consensus of the MIFOMAS (10) with symmetric time-delays by analyzing the characteristic eigenvalues' position of the characteristic polynomial det[   ()] of the MIFOMAS (10) with symmetric timedelays.Specifically, consensus of the MIFOMAS (10) without delays (all   = 0) is necessary for consensus of the MIFOMAS (10) with symmetric time-delays; that is to say, in this case the characteristic eigenvalues of det[   ()] of the MIFOMAS (10) with symmetric time-delays are all situated in the left half plane (LHP) of the complex plane, and as   increases continuously from zero, the characteristic eigenvalue of det[   ()] of the MIFOMAS (10) with symmetric time-delays will change continuously from the LHP to the right half plane (RHP) of the complex plane.Once the characteristic eigenvalue of det[   ()] of the MIFOMAS (10) with symmetric time-delays reaches the RHP of the complex plane through the imaginary axis, the MIFOMAS (10) with symmetric time-delays will be unstable and can not achieve consensus.Ergo, we only need to consider the critical time-delay when the nonzero characteristic eigenvalue of det[   ()] of the MIFOMAS (10) with symmetric timedelays is just situated on the imaginary axis for the first time as   increases continuously from zero, and the corresponding time-delay is just the delay margin  of the MIFOMAS (10) with symmetric time-delays.
According to (24), we can get where   () is the principal value of the argument of   (), and Re[  ()] and Im[  ()], respectively, denote the real part and the imaginary part of   ().
According to (26), it is easy to obtain that Consider a FOMAS with single integral; we have and  1 () =   1  2 ( 2 = 1).According to (25), we should only consider  ≤  =  (1/ 1 )  , and if all (27).Therefore, when all   < , the characteristic eigenvalues of det[   ()] of the MIFOMAS (10) with symmetric time-delays are all situated in the LHP and the FOMAS with single integral will remain stable and can achieve consensus.On the contrary, the FOMAS with single integral will not remain stable and can not achieve consensus.Theorem 4 is proven for  = 1.
In the following, the FOMAS with multiple integral (double integral to sextuple integral) shall be analyzed step by step.For convenience of analysis, we first need to define some symbolic parameters: and For the FOMAS with double integral, we can get the first derivative of  2 (): For the FOMAS with triple integral, Because of  3 () = Im[ 3 ()]/Re[ 3 ()], we can get the first derivative of  3 (): In a similar way, the [  ()] 2 and    () of the FOMASs with quadruple integral to sextuple integral can be, respectively, calculated under the appropriate parameters, and they are as follows: In summary, we have found that the first derivatives of   () (2 ≤  ≤ 6) listed above are negative values, and    () < 0 means that   () = tan[  ()] are monotonically decreasing with the growth of .Then it can be deduced that the arguments   () also decrease monotonically and continuously about  because the values of   () vary smoothly.Evidently, we can analyze the features of   () (2 ≤  ≤ 6) together.
Remark .Consensus of the MIFOMAS (10) without symmetric time-delays is necessary for consensus of this system with symmetric time-delays.

𝜔), then the control protocol ( ) can resolve the consensus problem for the highorder MAS with symmetric time-delays, and on the contrary, then the control protocol ( ) can not resolve the consensus problem for the high-order MAS with symmetric time-delays.
e value of  corresponding to  in   () is determined by the following equation: where |  ()| is the modulus of   () and   is the maximum eigenvalue of .
Remark .The dynamic model and the control protocol in Corollary 7 were discussed in [21], and the conclusion in Corollary 7 is less conservative than that in [21].The proof about Corollary 7 is the same as that of Theorem 4.

Theorem 9. Suppose that a FOMAS with multiple integral is given by MIFOMAS ( ) whose corresponding network topology
G satisfies Lemma .Define the following functions:

the MIFOMAS ( ) with asymmetric time-delays, then the control protocol ( ) can resolve the consensus problem of the MIFO-MAS ( ) with asymmetric time-delays, and on the contrary, then the control protocol ( ) can not resolve the consensus problem of the MIFOMAS ( ) with asymmetric time-delays.
e value of   corresponding to  in Γ  (  ) is determined by the following equation: where |  (  )| is the modulus of   (  ).
Calculate the principal value of the argument of (46); we have Inequality ( 50) is in contradiction with inequality (49).Therefore, as long as all   < , the characteristic eigenvalues of det[   ()] of the MIFOMAS (10) with asymmetric timedelays can not reach or pass through the imaginary axis, then the MIFOMAS (10) with asymmetric time-delays will remain stable and can achieve consensus.On the contrary, the MIFOMAS (10) with asymmetric time-delays will not be stable and can not achieve consensus.Theorem 9 is proven.
Remark .Consensus of the MIFOMAS (10) without asymmetric time-delays is necessary for consensus of this system with asymmetric time-delays.
Corollary 12.If we suppose that a FOMAS with multiple integral is given by MIFOMAS ( ) whose corresponding network topology G satisfies Lemma and  1 =  2 = . . .=   = 1, then the MIFOMAS ( ) with asymmetric time-delays can be transformed into high-order MAS with asymmetric time-delays whose dynamic model is an integer-order dynamic model and the following functions can be obtained: = arctan   () , where  ∈ {1, and Im[  ()] and Re[  ()] denote the imaginary part and real part of   (), respectively.
For the high-order MAS with asymmetric time-delays, if all   satisfy   <  = min |  | ̸ =0 {Γ  (  )}, then the control protocol ( ) can resolve the consensus problem for the high-order MAS with asymmetric time-delays, and on the contrary, then the control protocol ( ) can not resolve the consensus problem for the high-order MAS with asymmetric time-delays.e value of   corresponding to  in Γ  (  ) is determined by the following equation: where |  (  )| is the modulus of   (  ).

Simulation Results
The correctness and validity of the theoretical results for Theorems 4 and 9 will be verified by some numerical simulations in this section.Under different network topologies, the FOMAS with different multiple integral will be considered.First of all, to validate Theorem 4, we consider a FOMAS composed of 4 agents.Figure 1 shows the network topology depicted with a connected and undirected graph G, and Figure 1 has five different time-delays which are symmetric time-delays and it shows full connectivity.All the delays are marked with   , where  and  are the indexes, which are used to represent the connected agents  and .If we suppose the weight of each edge of graph G in Figure 1 is 1, then the adjacency matrix and Complexity 13 the corresponding Laplacian matrix of G are, respectively, where   = 4 is the maximum eigenvalue of .  2 and 3: the two subfigures in Figure 2 show the trajectories of all agents' states when all symmetric time-delays are less than the delay margin , which indicates that the FOMAS with double integral and symmetric time-delays is stable and consensus  of the FOMAS with double integral and symmetric timedelays can be reached; the two subfigures in Figure 3 show the trajectories of all agents' states when all symmetric timedelays exceed the delay margin , which indicates that the FOMAS with double integral and symmetric time-delays is unstable and consensus of the FOMAS with double integral and symmetric time-delays can not be reached.and 5: the three subfigures in Figure 4 show the trajectories of all agents' states when all symmetric time-delays are less than the delay margin , which indicates that the FOMAS with triple integral and symmetric time-delays is stable and consensus of the FOMAS with triple integral and symmetric time-delays can be reached; the three subfigures in Figure 5 show the trajectories of all agents' states when all symmetric time-delays exceed the delay margin , which indicates that the FOMAS with triple integral and symmetric time-delays is unstable and consensus of the FOMAS with triple integral and symmetric time-delays can not be reached.
Example .For a FOMAS with sextuple integral and symmetric time-delays under the undirected graph, let us set  1 = 0.9,  2 = 0.8,  3 = 0.7,  4 = 0.6,  5 = 0.5,  6 = 0.4, and    6 and 7: the six subfigures in Figure 6 show the trajectories of all agents' states when all symmetric time-delays are less than the delay margin , which indicates that the FOMAS with sextuple integral and symmetric time-delays is stable and consensus of the FOMAS with sextuple integral and symmetric timedelays can be reached; the six subfigures in Figure 7 show the trajectories of all agents' states when all symmetric timedelays exceed the delay margin , which indicates that the FOMAS with sextuple integral and symmetric time-delays is unstable and consensus of the FOMAS with sextuple integral and symmetric time-delays can not be reached.Next, to examine Theorem 9, we give a network topology described in Figure 8, which is a directed graph G with a spanning tree.It also contains five different time-delays which are asymmetric time-delays and displays full connectivity.If we suppose the weight of each edge of graph G in Figure 8
Example .For a FOMAS with double integral and asymmetric time-delays under the directed graph, let us set  1 = 0.9,  2 = 0.   9 show the trajectories of all agents' states when all asymmetric time-delays are less than the delay margin , which indicates that consensus of the FOMAS with double integral and asymmetric time-delays can be reached; the two subfigures in Figure 10 show the trajectories of all agents' states when all asymmetric time-delays exceed the     11 show the trajectories of all agents' states when all asymmetric time-delays are less than the delay margin , which indicates that consensus of the FOMAS with triple integral and asymmetric time-delays can be reached; the three subfigures in Figure 12 show the trajectories of all agents' states when all asymmetric time-delays exceed the delay margin , which indicates that consensus of the FOMAS with triple integral and asymmetric time-delays can not be reached.
Example .For a FOMAS with sextuple integral and asymmetric time-delays under the directed graph, let us set  1 = 0.9,  2 = 0.8,  3 = 0.7,  4 = 0.6,  5 = 0.5,  6 = 0.   displayed in Figures 13 and 14: the six subfigures in Figure 13 show the trajectories of all agents' states when all asymmetric time-delays are less than the delay margin , which indicates that consensus of the FOMAS with sextuple integral and asymmetric time-delays can be reached; the six subfigures in Figure 14 show the trajectories of all agents' states when all asymmetric time-delays exceed the delay margin , which indicates that consensus of the FOMAS with sextuple integral and asymmetric time-delays can not be reached.

Conclusion
The consensus problems of a FOMAS with multiple integral under nonuniform time-delays are studied in this paper.Taking into account two kinds of nonuniform time-delays, the sufficient conditions have been derived in the form of inequalities for the MIFOMAS with nonuniform time-delays.Numerical simulations of the MIFOMAS with nonuniform time-delays over undirected topology and directed topology are performed to verify these theorems.Finally, the simulation results show that the selected examples have achieved the desired results: the MIFOMAS with nonuniform time-delays under given conditions can achieve the consensus.With the help of the above research of this paper, distributed formation control of the MIFOMAS with nonuniform time-delays will be one of the most significant topics, which will be one of our future research tasks.

Figure 2 :Figure 3 :
Figure 2: The trajectories of all agents' states in the FOMAS with double integral and symmetric time-delays when all   <  in Example 1.

Figure 4 :
Figure 4: The trajectories of all agents' states in the FOMAS with triple integral and symmetric time-delays when all   <  in Example 2.

Figure 9 :
Figure 9: The trajectories of all agents' states in the FOMAS with double integral and asymmetric time-delays when all   <  in Example 4.

Figure 12 :
Figure 12: The trajectories of all agents' states in the FOMAS with triple integral and asymmetric time-delays when all   >  in Example 5.

Figure 13 :
Figure 13:  The trajectories of all agents' states in the FOMAS with sextuple integral and asymmetric time-delays when all   <  in Example 6.
If nodes   and   are connected and   =   , then G is an undirected graph; otherwise the G is a directed graph.In a directed graph, a directed path is a sequence of edges by ( 1 ,  2 ), ( 2 ,  3 ), . .., where (  ,   ) ∈ E. The directed graph has a directed spanning tree if all other nodes have directional paths from the same node.The Laplacian matrix of the graph G is defined by  = Δ − A ∈ R × , where Δ ≜ diag{  ( 1 ),   ( 2 ), . . .,   (  ), . . .,   (  )} is a diagonal matrix with   (  ) = ∑  =1   .Supposing some graphs G 1 , G 2 , . . ., G  and graph G consist of the same nodes, and the edge set of graph G is the sum of the edge sets of other graphs G 1 , G 2 , . . ., G  , then there is  = ∑  =1   , which means the Laplacian matrix of graph G is the sum of other graphs' Laplacian matrix.
1 , . . .,   } and E ⊆ V 2 , respectively, represent the set of nodes and the set of edges.The node indices belong to a finite index set I = {1, 2, . . ., }.The weighted adjacency matrix is denoted by A = [  ] × .The element of the -th row and the -th column in matrix A indicates the connection state between agents   and   .If nodes   and   are connected, i.e.,   ∈ E, then   > 0, and   is called a neighbor of node   .  = { ∈ I,  ̸ = } denotes the index set of all neighbors of agent .
Figure 6: The trajectories of all agents' states in the FOMAS with sextuple integral and symmetric time-delays when all   <  in Example 3.
Figure 7: The trajectories of all agents' states in the FOMAS with sextuple integral and symmetric time-delays when all   >  in Example 3.
Figure 14:The trajectories of all agents' states in the FOMAS with sextuple integral and asymmetric time-delays when all   >  in Example 6.