Consensus Analysis for Third-Order Multiagent Systems in Directed Networks

We investigate consensus problem for third-order multiagent dynamical systems in directed graph. Necessary and sufficient conditions to consensus of third-order multiagent systems have been established under three different protocols. Compared with existing results, we focus on the relationship between the scaling strengths and the eigenvalues of the involved Laplacian matrix, which guarantees consensus of third-order multiagent systems. Finally, some simulation examples are given to illustrate the theoretical results.


Introduction
Multiagent system is a set composed of multiple computing intelligent agents, where each agent is a physical or abstract entity, perceiving the surrounding environment through the sensor and communicating with its neighboring agents.Recently, the distributed cooperative control of multiagent systems receives much attention due to its wide application in real world, such as cooperative control of unmanned aerial vehicle [1], formation control of mobile robot [2], autonomous underwater submarine [3], and network control in information.Consensus problem of multiagent systems has been studied extensively in recent years.
In the literature about consensus, most of the papers focus on the first-order multiagent system and the second-order multiagent system.For the first-order system, consensus for networks of multiagent systems with fixed and switching topologies was investigated in [4].In [5], the authors gave a theoretical framework for design and analysis of distributed flocking protocols.A theoretical explanation for observed behavior was presented in [6].Xiao et al. studied the consensus problem for discrete-time multiagent systems and established a necessary and sufficient condition to consensus in [7].
For the second-order multiagent system, necessary and (or) sufficient conditions to consensus under directed graph were given in [8].In [8], the authors considered dimensional states and obtained [9].Xie and Wang extended some results of [9] to second-order discrete-time multiagent systems and gave a necessary and sufficient condition to consensus based on algebraic graph theory and matrix theory in [10].Some other consensus problems for second-order multiagent systems such as couple-group consensus and average consensus criteria were studied in [11][12][13][14][15].
In comparison, consensus of high-order multiagent systems receives less attention.In [16], the authors studied thorder ( ≥ 3) consensus algorithms and presented necessary and sufficient conditions under which each information variable and their higher-order derivatives converge to common values.Consensus of multiagent systems with linear higherorder agents was considered in [17], where the authors answered whether the group converges to consensus and what consensus value it eventually reaches.In [18,19], the authors studied the consensus problem for a group of higherorder dynamical agents with switching topologies and timevarying communication delays.In [20], the authors dealt with the consensus problem for high-order multiagent systems with noises and time delays under undirected topologies, and 2 Mathematical Problems in Engineering derived conditions for high-order consensus by using the Routh-Hurwitz stability criterion and the Lyapunov theorem.
In this paper, we will consider the third-order dynamical multiagent system under three different protocols.The state of the system represents position, velocity, and accelerated speed, respectively.In practice, the derivative of accelerated speed is called Jerk, which is a physical quantity to describe the changing rate of acceleration.A Jerk is often required in engineering, especially in transportation design and materials, and so forth.Hence, the study on thirdorder multiagent systems is significant.What is more, many control systems are almost high-order systems in control engineering.Particularly, third-order systems are very common, whose performance indexes may be very complex.In actual engineering, we need the system to work in a consistent state.Therefore, the results of this paper can be used to sustain agents consensus in engineering.
The main contribution of this paper is to establish the equivalent relationship between the scaling strengths and the eigenvalues of the involved Laplacian matrix, which guarantees consensus of the third-order multiagent system in directed graph when it has a spanning tree.Compared with existing results, some easily verifiable necessary and sufficient conditions to consensus are established.Unlike the case of undirected graph, we have to overcome the difficulty caused by the third-order characteristic equations with complex coefficients since the eigenvalues may be complex.
The rest of this paper is organized as follows.In Section 2, we introduce some preliminaries and lemmas about graph and matrix which play a key role in the consensus analysis.In Section 3, some necessary and sufficient conditions to consensus of third-order multiagent system are established for the case of directed graph.Some numerical simulations are given to illustrate the theoretical results in Section 4. Finally, conclusions are drawn in Section 5.

Preliminaries
In this section, we give the basic concept and theory of algebraic graph, problem formulation, and some definitions and lemmas.For readers who want to know the details about algebraic graph theory, please refer to [21][22][23][24].
Let  = (], , ) be a weighted directed graph of order  with the finite set of vertexes ] = { Consider third-order dynamic systems of the form where  ≥ 0,   , V  ,   ∈   represent the position, velocity, and accelerated speed statement, respectively, and   ∈   is the control input.
Throughout this paper, we are concerned with the following three protocols: where  > 0,  > 0, and  > 0 are scaling parameters (strengths).3), (4), and ( 5) can be rewritten into the following systems, respectively: where The following definitions and lemmas are required in this paper.
Definition 1. Assume the third-order multiagent system (6 Definition 2. Assume the third-order system (7 Lemma 4 (see [8]). has a simple eigenvalue 0 and all the other eigenvalues have positive real parts if and only if the directed graph has a directed spanning tree.
Lemma 5 (see [16]).L1 has three zero-eigenvalues at least.L1 has only three zero-eigenvalues if and only if  has a simple zero-eigenvalue.Moreover, the geometric multiplicity of zeroeigenvalue of L1 equals one if  has a simple zero-eigenvalue.
Lemma 6 (see [16]).The third-order multiagent system ( 6) can achieve consensus if and only if L1 has only a three-multiplicity zero-eigenvalue and all the other eigenvalues have negative real parts.Moreover, if system ( 6) achieves consensus, one has for  = 1, 2, . . .,  as  → ∞, where  satisfying   1 = 1 is the unique nonnegative left eigenvector of  associated with zero-eigenvalue.

Main Results
We first extend Lemmas 5 and 6 to protocols (4) and ( 5) which are useful to the proof of the main results in this paper.The following two lemmas extend Lemma 5 to the case of matrices L2 and L3 , respectively.
On the other hand, based on the above analysis,  has a simple zero-eigenvalue if and only if L2 has only two zero-eigenvalues and the other eigenvalue is −.What is more, if  has only one simple zero-eigenvalue,  has only a linear independent eigenvector  corresponding to zeroeigenvalue, which in turn implies that  = [  , 0  , 0  ]  is the unique linear independent eigenvector of L2 corresponding to zero-eigenvalue.Therefore, the geometric multiplicity of zero-eigenvalues of L2 equals one.This completes the proof of Lemma 8.

Lemma 9. L3 has a zero-eigenvalue at least. L3 has only a zero-eigenvalue if and only if 𝐿 has a simple zero-eigenvalue.
Moreover, the geometric multiplicity of zero-eigenvalue of L3 equals one if  has a simple zero-eigenvalue.
Proof.Let ,  be eigenvalues of L3 and , respectively.By the character equation of L3 , we have  3 +  2 +  = 0 when  = 0.It implies that L3 has at least a zero-eigenvalue by Lemma 4. The remaining proof can be similarly done as Lemma 8.This completes the proof of Lemma 9.

Lemma 10. The third-order multiagent system (7) achieves consensus if and only if L2 has only a two-multiplicity zeroeigenvalue and all the other eigenvalues have negative real parts. Moreover, if system (7) achieves consensus, one has
() → 0, (19) for  = 1, 2, . . .,  as  → ∞, where  satisfying   1 = 1 is the unique nonnegative left eigenvector of  associated with zero-eigenvalue.
Proof.(⇐) By the proof of Lemmas 6 and 8 and the properties of Kronecker product ( [23,25]), there exists a nonsingular matrix  which is composed of eigenvector and generalized eigenvector such that where  ∈  3−3 is a column vector whose elements are 0,  is the Jordan standard form, and J is the upper diagonal Jordan block matrix corresponding to the nonzero eigenvalues of L2 .Note that  J →  (3−3)×(3−3) as  → ∞. where It implies that (19) holds.Therefore, system (7) achieves consensus.
(⇒) Otherwise, the matrix L2 does not have exactly a zero-eigenvalue of multiplicity two, or the other eigenvalues have positive real parts.Then, similar to the proof of Lemma 6, the rank of lim  → ∞  L2  is greater than 2, which contradicts the fact that third-order consensus is achieved.This completes the proof of Lemma 10.

Lemma 11. The third-order multiagent system (8) achieves consensus if and only if L3 has only a simple zero-eigenvalue and all the other eigenvalues have negative real parts. Moreover, if system (8) achieves consensus, one has
() → 0, (23) for  = 1, 2, . . .,  as  → ∞, where  satisfying   1 = 1 is the unique nonnegative left eigenvector of  associated with zero-eigenvalue.
Proof.The proof is similar to that of Lemma 10 and hence is omitted.
Remark 12.Although Lemmas 6, 10, and 11 have presented necessary and sufficient conditions guaranteeing the thirdorder consensus, it is still unclear whether the eigenvalues of Laplacian matrix  and the scaling parameters , , and  have effect on the third-order consensus.In the following theorems, we will answer this problem certainly.
Remark 13.By Lemmas 4-6 and 8-11, we have that a necessary condition to consensus of third-order multiagent system (2) under protocols (3)-( 5) is that the directed graph  has a spanning tree.Without loss of generality, we assume throughout this paper that the directed graph  has a spanning tree.Theorem 14. Assume that the directed graph  has a spanning tree.The third-order multiagent system (6) achieves consensus if and only if the scaling parameters , , and  satisfy where   ( = 2, . . ., ) is the nonzero eigenvalue of the Laplacian matrix .
Proof.Let  be the eigenvalue of L1 .We have We first have that the simple zero-eigenvalue of the Laplacian matrix  yields a three-multiplicity zero-eigenvalue of the matrix L1 .Set Denote   = R(  ) + ℓ(  )j.Then (25) yields that By Lemma 6, it is sufficient to show that all eigenvalues of (27) have negative real part, which is equivalent to the following inequalities by Lemma 7: Note that  > 0 and the directed graph  has a spanning tree.By Lemma 4, we have that  1 > 0 is obvious.Therefore, (28) is equivalent to where By substituting (31) into (29), we get (24).Therefore, by Lemma 6, we have that system (6) achieves consensus if and only if the scaling parameters , , and  satisfy (24).The proof of Theorem 14 is completed.
Theorem 15.Assume that the directed graph  has a spanning tree.The third-order multiagent system (7) achieves consensus if and only if the scaling parameters , , and  satisfy where   ( = 2, . . ., ) is the nonzero eigenvalue of the Laplacian matrix .
Proof.Let  denote the eigenvalue of L2 .We have Firstly, we have that the simple zero-eigenvalue of the Laplacian matrix  yields a two-multiplicity zero-eigenvalue of L2 by Lemma 8. Set Denote   = R(  ) + ℓ(  )j.Then (33) yields that By Lemma 10, it is sufficient to show that all eigenvalues of (33) have negative real part.Similar to the proof of Theorem 14, we have that (35) is equivalent to (29) where ,, =  ,, = 0,  +  > 3, By substituting (36) into (29), we can obtain (32).Therefore, by Lemma 10, we have that system (7) By substituting (41) into (29), we get (37).Therefore, by Lemma 11, we have that system (8) achieves consensus if and only if the scaling parameters , , and  satisfy (37).This completes the proof of Theorem 16.
Remark 17.For the particular case when  is an undirected connected graph, it is easy to see that all the other eigenvalues of the Laplacian matrix  are positive real number except a zero-eigenvalue.In this case, conditions ( 24), (32), and (37) in Theorems 14-16 can be simplified as follows: where   > 0 is the eigenvalue of the Laplacian matrix  for  = 2, . . ., .It is easy to see that the above inequalities reduce to   > ,  > , and  >   , respectively.
Based on Remark 17, we obtain the following corollaries by Theorems 14-16 immediately.

Numerical Simulations
In this section, several simulation results are presented to illustrate the theoretical results.The topology graph in the simulation is shown in Figure 1 which has four agents and 0-1 weight.It is not difficult to see that the corresponding Laplacian matrix  has four eigenvalues:  1 = 0,  2 = 1,  3 = 1.5 + 0.866, and  4 = 1.5 − 0.866.
If  =  =  = 1, condition (24) is satisfied.By Theorem 14, consensus of system (6) is achieved.The position, velocity, and accelerated speed states of agents are shown in Figure 2.
If  = 0.1,  =  = 1, condition (32) is satisfied.By Theorem 15, consensus of system ( 7) is achieved.The position, velocity, and accelerated speed states of agents are shown in Figure 3.If  = 0.1,  = 2, and  = 1, condition (37) is satisfied.By Theorem 16, consensus of system (8) is achieved.The position, velocity, and accelerated speed states of agents are shown in Figure 4.

Conclusion
This paper studies the consensus problem of third-order dynamic multiagent systems in directed graph.Under three different protocols, necessary and sufficient conditions are established by analyzing the roots of the involved characteristic equations with complex coefficients.Compared with some existing results in the literature, our results illustrate the equivalent relationship between the scaling strengths and the eigenvalues of the involved Laplacian matrix when the third-order multiagent system achieves consensus.In the forthcoming study, we will further study the consensus of higher-order multiagent systems with delay and switching topologies.

Figure 1 :
Figure 1: A digraph with a spanning tree.
1 ,  2 , . . .,   }, set of edges  = {(  ,   ) :   ,   ∈ ]} ⊆ ] × ], and a weighted adjacent matrix  = (  ) × with nonnegative entries   .An edge of  is denoted by   = (  ,   ) and   ∈  if and only if   > 0; otherwise,   = 0. We assume   = 0; namely, the graph has no self-loops.A directed path is a sequence of different ordered boundaries of form (  ,  1 ), ( 1 ,  2 ), . . ., (  ,   ) in a directed graph where   ∈ ].A directed graph has a spanning tree if there exists at least one agent that has a directed path to all other agents.The Laplacian matrix  = [  ] which is the graph Laplacian induced by information flow is defined by   = − ∑   ̸ =   ,   = −  ,  ̸ = .Obviously, the matrix  satisfies Suppose that  is an eigenvalue of L2 and  = [  ,   ,   ]  is the corresponding characteristic vector, where , , and  are -dimensional column vector.Noting that Lemma 8. L2 has two zero-eigenvalues at least.L2 has only two zero-eigenvalues if and only if  has a simple zero-eigenvalue.Moreover, the geometric multiplicity of zeroeigenvalue of L2 equals one if  has a simple zero-eigenvalue.Proof.
(8)ieves consensus if and only if the scaling parameters , , and  satisfy (32).This completes the proof of Theorem 15.Assume that the directed graph  has a spanning tree.The third-order multiagent system(8)achieves consensus if and only if the scaling parameters , , and  satisfy achieves consensus if and only if the scaling parameters , , and  satisfy (7)re   > 0 ( = 2, 3, ..., ) is the nonzero eigenvalue of the Laplacian matrix .Corollary 19.Assume that  is an undirected connected graph.The third-order multiagent system(7)achieves consensus if and only if the scaling parameters , , and  satisfy  > , (44) where   > 0 ( = 2, 3, . . ., ) is the nonzero eigenvalue of the Laplacian matrix .Corollary 20.Assume that  is an undirected connected graph.The third-order multiagent system (8) achieves consensus if and only if the scaling parameters , , and  satisfy  >   ,