Consensus Analysis of High-Order Multiagent Systems with General Topology and Asymmetric Time-Delays

This paper focuses on the consensus problem for high-order multiagent systems (MAS) with directed network and asymmetric time-varying time-delays. It is proved that the high-order multiagent system can reach consensus when the network topology contains a spanning tree and time-delay is bounded.Themain contribution of this paper is that a Lyapunov-like design framework for the explicit selection of protocol parameters is provided.The Lyapunov-like design guarantees the robust consensus of the highorder multiagent system with respect to asymmetric time-delays and is independent of the exact knowledge of the topology when the communication linkages among agents are undirected and connected.


Introduction
In the last few years, substantial research effort from a number of researchers has been poured on the study of consensus problems for multiagent systems (MAS) due to its powerful engineering applications, such as formation control of autonomous vehicles, collective behavior of flocks, and distributed decision making in sensor networks, to name a few.The pioneering contributions in systems and control community have been made by [1,2].Until now, it has been proved that the consensus problem for single-integrator MAS can always be solved under certain mild conditions on the network topology [3][4][5].
Due to the complexity of real systems, the study on single-integrator MAS can not meet the needs of practical applications.Thus, recently, widespread interest in MAS with agents modeled by general dynamics has been excited among researchers, such as double-integrator model [6,7] and highorder-integrator model.Specifically, high-order-integrator (or high-order) MAS have been studied in [8], where the proposed consensus protocol involves the relative information of all-order derivatives of agents's state.Reference [9] has further extended the partial results of [8] and derived a linearmatrix-inequality-based protocol design.Reference [10] has provided a scheme to choose the coupling strength for fixed and connected topology.
In most practical networks, communication time-delays caused by limited transmission speed and distance cannot be neglected.Within the literature on consensus for MAS with time-delays, recent years have witnessed the introduction of numerous distributed protocols.According to the inducements of time-delays, these protocols can be categorized into the ones in which delays only affect the state information of the agents' neighbors [11], the ones in which delays affect both the agents' own state information and their neighbors' state information [4,7,[12][13][14][15][16], the ones containing distributed delays [17], and so forth.In particular, [4] has shown that the consensus of single-integrator MAS might be destabilized by large delays.By using Laplace transform technique, [7] has proved that second-order multiagent systems with bounded and constant time-delays can reach consensus.Based on nonexpansiveness of constant delay operator and Gershgorin's circle theorem, [11] has derived a local controller for high-order MAS with diverse constant time-delays.However, time-varying communication delays are very common in MAS due to the mobility of agents and the disturbance from environment.Hence, it is necessary to study the consensus for 2 Mathematical Problems in Engineering high-order MAS with time-varying communication delays by exploring other method of protocol design.
In this paper, we investigate the consensus problem for high-order MAS with directed interactions and asymmetric time-varying communication delays.By asymmetric communication delays, we mean that the delay in communication channel from the agent to its neighbor differs from the one in the reversed channel if there are bidirectional communication linkages between them.We assume that the agent can only observe the information of the first state variable of its neighbors, and we propose a distributed protocol by applying both the instantaneous information of the agent's own and the time-delayed relative information with respect to its neighbors.Based on a reduced-order time delay system, we derive some sufficient conditions characterized by linear matrix inequalities (LMIs), according to Lyapunov-Krasovskii functional approach.The main contribution of this paper is that we establish a Lyapunovlike design framework for the explicit selection of protocol parameters.The protocol design framework can not only guarantee the solvability of the LMIs aforementioned but also is independent of the exact knowledge of the topology when the communication linkages among agents are undirected and connected.This implies that the Lyapunov-like protocol design guarantees the robust consensus of high-order MAS with respect to asymmetric time-varying communication delays.Compared with the protocol design in [9], our design is not based on LMI but only needs to solve a simple Lyapunov equation and a simple algebraic inequality.In contrast to the literature on single-/double-integrator MAS, the results here are not simple extensions of the results therein since the protocol parameters have important effect on the consensus convergence of high-order MAS.
The remainder of this paper is organized as follows.Section 2 states the problem formulation and Section 3 presents the main results.Section 4 carries out some numerical examples and the last section provides some concluding remarks.
Notations.We let R be the set of real numbers.R  is the -dimensional real vector space.R  is the set of -by- matrices with elements in R.   ∈ R  is an identity matrix.Given a matrix  ∈ R  , () denotes its spectrum (set of eigenvalues);  < 0 means that  is negative definite.diag{ 1 , . . .,   } defines a diagonal matrix with diagonal elements being  1 , . . .,   .Sometimes . ., } and  − 1 = {1, . . .,  − 1} are two index sets.⊗ denotes the Kronecker product.

Problem Formulation
Consider a dynamical system of  autonomous agents, which are labelled 1 through .Each agent is modelled as the following th order integrator: where  ≥ 1 is a positive integer and denotes the order of the differential equations;   ∈ R, and  ()   ,  = 1, . . ., , is the th order derivative of   ;   ∈ R is the control input;

𝑖0
]  is the initial state of agent .The interaction/communication topology among agents can be conveniently modeled by weighted directed graph G(A) = {V, E, A}, where V = {V 1 , . . ., V  } is the vertex set, E ⊂ V × V is the arc set, and A = [  ] ∈ R  is the adjacency matrix with   ≥ 0. An arc of G, denoted by (V  , V  ), is an ordered pair of distinct vertices of V; V  and V  are called the tail and the head of the arc, respectively.An arc In this paper, we assume that (V  , V  ) ∉ E and each element of E is unique.Each vertex in G(A) represents an agent of the dynamical system (1); (V  , V  ) ∈ E indicates that there is a communication linkage from agent  to agent ; the element   in A is the weight of the linkage.
A path from V  to V  means that there is a sequence of distinct arcs in E, (V  , V 1 ), (V 1 , V 2 ), . . ., (V  , V  ).A directed tree is a directed graph, where every vertex has exactly one tail except for one special vertex without any tail.We say a graph contains a spanning tree if there exists a subset of arcs E  ⊂ E such that the graph G  = (V, E  ) is a directed tree.A graph is said to be balanced if for each vertex V  the weights of its linkages satisfy ∑  =1   = ∑  =1   ,  ∈ .A graph is said to be undirected if the associated adjacency matrix A is symmetric.Then it is easy to see that any undirected graph is balanced.A directed graph is called strongly connected if there exists a path between any two distinct vertices of the graph; for undirected graph it is called connected.An undirected graph is called complete if for any  ̸ = , (V  , V  ) ∈ E. The Laplacian matrix L = [  ] ∈ R  of G(A) is defined as Let D = diag{ 1 , . . .,   } with   = ∑ V  ∈N    ,  ∈ .Then D and   are called the in-degree matrix of G(A) and the indegree of vertex V  , respectively.From the definition, it is not hard to obtain that L = D − A and L1  = 0. Spectral properties of the Laplacian matrix can be found in [5,18].Hence the details are omitted.
For the system (1), the consensus protocol is described by where   > 0,  ∈  − 1 and  0 > 0 are, respectively, the feedback gains of absolute and relative information (for convenience, we refer to the gains   ,  ∈  − 1, and  0 , as the protocol parameters); piecewise continuous function   () is the time-varying delay affecting the communication linkage from agent  to agent  at time .Notice that, different from [4,14,19,20], the delays in transmissions from agent  to agent  and from agent  to agent  (if there are bidirectional communication linkages between them) can be asymmetric; that is,   (⋅) ̸ =   (⋅).
Remark 1.In protocol (3), the agent  is able to measure its own instantaneous state information and equipped with memories to store the signals   (⋅) which can be used at some future time and needs only to receive the time-delayed signals   ( −   ()) of its neighbors.The control input   () can be implemented by computing the instantaneous information of all-order derivatives of   () and the time-delayed relative information   ( −   ()) −   ( −   ()).The determination of   () can be carried out by assuming that the stored signals   (⋅) of each agent are time-stamped and neighbor  transmits not only the time-delayed signal   (−  ()) but also the time stamp.The above situation can be found in [13,15] and also satisfied easily in practice.
be the state of agent  and the stacked vector of the agents' initial states, respectively.In this paper, we are devoted to solving the following consensus problem for the MAS (1).Definition 2. Consider the MAS (1) with some given protocol   ().If, for any initial state (0), the states of agents satisfy   () −   () → 0 as  → ∞ for all ,  ∈ , then we say that the system solves a consensus problem asymptotically.In addition, if there exists  * ∈ R  such that, for any initial state (0),   () →  * as  → ∞, for all  ∈ , then we say  * is the consensus state of the system.
In order to develop the main results, some helpful lemmas are introduced as follows.
Suppose that G  is a complete undirected graph of  vertices and L  is the associated Laplacian matrix.From the definition, we have L  1  = 0, 1   L  = 0, and the rank of Then Û Û =  −1 and Û ] 1 = 0. Based on this observation and the property of Laplacian matrix, we can obtain the result below.

Lemma 3. Suppose that G is a graph with the associated
Proof.See the Appendix: Proofs of Lemmas.
The following result can be considered as a special case of Jensen's integral inequality given in [21].

Main Results
In this section, we first provide an equivalent condition for the consensus convergence of the systems (1) and (3) based on an orthogonal state transformation and a reduced-order time delay system.Then we give a Lyapunov-like parameter design for the protocol and prove that the maximum allowable upper bounds of time-varying delays can be determined by solving some optimization problems.
Suppose that the interaction topology of the system (1) is modelled by G(A); the associated Laplacian matrix is L. Then the dynamics of agent  can be written as where Denote () = [  1 () ⋅ ⋅ ⋅    ()]  by the stacked vector of the agents' states.The closed-loop dynamics of the systems (1) and ( 3) are in the form of where  denotes the number of different time-delays over the communication channels of the system (it is easy to get that  ≤ ( − 1));   () ∈ {  () : ,  ∈ ,  ̸ = } for  = 1, . . ., ; L  = [   ] ∈ R  is defined as Mathematical Problems in Engineering Next, we assume that the time-delays in (7) satisfy where   > 0. Define  0 = max{  :  = 1, . . ., }.The initial state of the system ( 7) is assumed to be () = (0),  ∈ [− 0 , 0].Remark 6.The matrix L  in (7) amounts to consider a Laplacian matrix that is associated with a subgraph of G(A) with all linkages affected by the same time-delay.Hence, where   is given as therein.Then ∑  =1 L = L. Furthermore, it is important to point out that even when the topology G(A) is undirected, L  is unnecessarily symmetric due to   (⋅) ̸ =   (⋅).Consequently, the results of [4,14,19,20] are invalid to deal with the consensus problem for MAS with asymmetric communication delays.
Applying the orthogonal linear transformation () = (  ⊗   )() to the system (7) and denoting where   and L are defined as in Remark 6.It can be seen that the system (11) is independent of the dynamics of  1 () and has the order (−1); the system (10) can be regarded as a forced system with the forced motion caused by the delayed state of  2 ().From   = [(1/ √ )1  Û] and some direct computation, it follows that In addition, Proposition 7 will indicate that  2 () can be considered as the disagreement state of the system (7).Note that, for the time delay system in ( 10) and ( 11), we assume that the initial state is where  10 ∈ R  ,  20 ∈ R (−1) , and  0 is given in (7).
We are now in a position to present an equivalent condition for the consensus convergence of the system (7).Proposition 7. Consider the system (7) with topology G(A).
Remark 8.It should be pointed out that the orthogonal linear transformation () = (  ⊗   )() is not uniquely defined by   .Actually, any orthogonal matrix with the first column being (1/ √ )1  also can derive the result of Proposition 7. In addition, the above orthogonal linear transformation can be seen as an improvement of the transformation on disagreement space which was displayed in Lemma 5.2 of [20].
Remark 9.In contrast to the transformation (7) of [15] and the tree-type transformation of [16], the order of the reduced-order system (11) induced by the orthogonal linear transformation () = (  ⊗   )() is lower than those of the reduced-order systems induced by them (in the context that all of the three transformations are applied to the system (7)).It is easy to see that the order of the reduced-order system (11) is  − , whereas the order of the reducedorder system which is derived from the transformation (7) of [15] (or the tree-type transformation of [16]) is  − 1. Seen from the linear-matrix-inequality-based sufficient conditions which will be given in Theorem 12, our orthogonal linear transformation can derive lower-order linear matrix inequalities.This will reduce the computation cost to some extent when estimating the maximum allowable upper bound  0 of time-varying delays.
Proof.See the Appendix: Proofs of Lemmas.
By choosing the parameters   ,  ∈  − 1 and  0 as in Lemma 11 and making use of Proposition 7, we can obtain the following theorem.

Theorem 12. Consider the system (7) with fixed topology G(A). Suppose that G(A) contains a spanning tree; the associated Laplacian matrix L has the form of 𝑈
where   is given as in Lemma 3; the parameters   ,  ∈  − 1 and  0 are chosen as in Lemma 11.Then the system solves a consensus problem asymptotically if the time-varying delays satisfy   () ≤  0 , where  0 is obtained from the following optimization problem: where and " * " represents the elements below the main diagonal of a symmetric matrix.
Proof.We first show that there must exist some positive definite matrices  and   such that the first LMI in ( 15) is solvable.From the selection of protocol parameters and Lemma 11, we know that  is Hurwitz stable.Hence there is a positive definite matrix P such that   P + P < 0. By Lemma 5, the first LMI in ( 15) is equivalent to and Σ 1 < 0 is equivalent to    +  +  0 Φ −1 0    < 0. Consequently, if we choose  = P,   =  (−1) ,  = 1, . . ., , then for some sufficiently small  0 ,   P + P +  0 P  P < 0 holds.This means that  = P and   =  (−1) satisfy the LMIs in (15).
Remark 13.From the proof of Theorem 12, it can be seen that the parameter design of protocol (3) in the theorem (i.e., the selection of   ,  ∈  − 1, and  0 ) guarantees the solvability of the LMIs in (15).According to the nature of the parameter design (see Lemma 11), we refer to it as a Lyapunov-like parameter design.
Remark 15.Corollary 14 indicates that the selection of parameters   ,  ∈  − 1, and  0 is independent of the eigenvalues of Laplacian matrix but only depends on the maximum vertex in-degree of the graph.It will reduce greatly the computation and storage costs for the protocol design of practical MAS.Therefore, we can say that the parameter design in Theorem 12 is independent of the precise interaction topology when the underlying graph is connected and guarantees the robust consensus with respect to asymmetric time-varying delays for the high-order MAS (7).
Remark 16.It is worth pointing out that the results of [19] can not be applied to the high-order MAS straightforwardly, since the parameters   ,  ∈  − 1, and  0 have important effect on the consensus of the system.Whereas the Lyapunovlike parameter design given in Theorem 12 can solve the consensus problem for the high-order MAS (7) very well.
Remark 17.Compared with the existing results, the main contribution of this paper is giving the Lyapunov-like parameter design which is easy to implement, independent of the precise interaction topology for the case of connected graphs, and robust with respect to asymmetric time-varying delays.Moreover, the parameter design can guarantee the existence of solution of the linear matrix inequalities given in Theorem 12, although it seems that the estimations of  0 are conservative.More excellent estimation on the maximum allowable upper bound of time-varying delays is a commonly unsolved problem.This requires us to explore other analysis techniques which could reduce the dependence of that estimation on the knowledge of network topology.

Numerical Examples
Consider the system (7) of six agents with dynamics described by a triple-order integrator.The interaction topology among agents is depicted by a cycle with the arcs 2 )}.Thus when  1 = 24 and  2 = 8, the constraint on  0 can be calculated as 0 <  0 < 576/35 ≃ 16.4571.We choose  1 = 24,  2 = 8, and  0 = 5 for the simulation.

Conclusions
This paper has studied the consensus problem for high-order MAS with directed network and asymmetric time-varying communication delays.An equivalent condition for the consensus convergence has been established based on an orthogonal linear transformation.A Lyapunov-like protocol design has been given, which is robust with respect to asymmetric time-varying communication delays and directed interactions.The maximum allowable upper bounds of delays have been determined by solving some optimization problems.Finally, numerical examples have been worked out to demonstrate the effectiveness of the theoretical results.

Proofs of Lemmas
Proof of Lemma 3. From the property of   , let If G is strongly connected and balanced, according to Theorem 7 in [4], we obtain that (L  + L)/2 is a valid Laplacian matrix of an undirected graph.Then the strong connectivity of G implies that the corresponding graph of (L  + L)/2 is connected.Hence (L  + L)/2 is positive semidefinite and its rank is  − 1.This proves that L + L is positive definite.
Denote the -by- leading principle minor of Q2 by Q2 ().