Consensus of Heterogeneous Multiagent Systems with Switching Dynamics

Heterogeneity is an important feature of multiagent systems. This paper addresses the consensus problem of heterogeneous multiagent systems composed of first-integrator and double-integrator agents. The dynamics of each agent switches between continuous-time and discrete-time dynamics. By using the graph theory and nonnegative matrix theory, we derive that the system can achieve consensus if and only if the fixed interaction topology has a directed spanning tree. For switching topologies, we get that the system can reach consensus if each interaction topology has a directed spanning tree. Simulation examples are provided to demonstrate the effectiveness of our theoretical results.


Introduction
Recently, a well-known fact is that the distributed coordination and control of multiagent systems have attracted a great deal of attention from scientific community.The main reason might be that it can be widely applied in the field of system control, applied mathematics, biology, social science [1][2][3][4][5][6], etc.As a kind of classical multiagent systems, the analysis and control of cyber-physical systems are also considered by scientists and engineers.In general, the study of multiagent system pertains to consensus problems [7][8][9][10][11][12], containment control [13][14][15], controllability analysis [16,17], formation control [18], tracking control [19][20][21], etc.
Among the above topics, consensus problem is a basic but critical problem for the multiagent system.Consensus means that a group of agents reach an agreement.In 1995, Vicsek et al. simulated that the moving direction of a group of selfdriven particles can reach consensus by using local communication [1].By virtue of graph theory, Jadbabaie et al. explained this observed consensus behavior of discrete-time multiagent systems from the perspective of the algebraic graph theory [7].In [22], Olfati-Saber and Murray presented a theoretical framework for consensus problems of continuous-time multiagent systems based on graph theory and obtained the necessary and sufficient conditions for solving the average consensus.In [3], Ren and Beard further investigated the consensus problem for multiagent systems with directed topologies.It is shown in [3] that consensus problem can be solved if the union of the dynamically changing interaction graphs has a directed spanning tree frequently enough as the system evolves.After that, researchers investigated consensus problem from different perspectives.As a result, the fruits for solving consensus problem became more and more.In [23], the author presented the necessary and sufficient conditions of achieving consensus for multiagent systems with noises.Wang et al. studied the state consensus problem for multiagent systems with the switching topologies and bounded time delays [24].Considering that agent dynamics is adopted as a typical point mass model based on the Newton's law, consensus problem for multiagent systems with double-integrators was also investigated [25][26][27][28].Moreover, Guaranteed-cost consensus and group consensus were also studied in [6,29].
Heterogeneity is an important feature of multiagent systems; i.e., the dynamics of agents might be different or changeable.On the one hand, different agents might have different dynamics.For instance, autonomous robots were used to control self-organized behavioral patterns in groupliving cockroaches [30].Therefore, many researches began to focus on the study of heterogeneous multiagent systems with multiple dynamics.Zheng et al. studied consensus problem for heterogeneous multiagent systems consisting of singleand double-integrators under different circumstances [31,32].Liu et al. investigated consensus problem for multiagent systems with heterogeneous nonlinear dynamics in [33].Qin et al. [34] considered group consensus for heterogeneous multiagent systems.On the other hand, the dynamics of agents might change dynamically over time.For example, the dynamics of a motor vehicle might switch between automatic shift gear and manual shift gear.Considering this fact, Zhai studied the stability of switched systems which composed of the continuous-time (CT) and discrete-time (DT) dynamics subsystem and offered some algebraic conditions to solve the stability problem under arbitrary switching in [35].In a CT multiagent system, if we sometimes use a computer to activate all the agents in a DT manner, then the multiagent system switches both CT and DT dynamics.As a result, multiagent systems with switching dynamics were also studied.In [36], Zheng and Wang supposed that the dynamics of agents switch between CT and DT subsystems and derive some sufficient conditions of solving consensus problems.Finitetime consensus problem of switched multiagent system wass also considered in [37].
Motivated by the above results, we consider consensus problems for the heterogeneous multiagent systems with switching dynamics.We assume that the system is consisting of single-and double-integrator agents and each agent can switch its dynamics between CT subsystem and DT subsystem.The main contribution of this paper is twofold.Firstly, we prove the system can reach consensus if the fixed interaction topology has a spanning tree.Secondly, for the system with switching topologies, we obtain that the system can solve consensus if each interaction topology has a spanning tree.
The structure of this paper is organized as follows.In Section 2, we introduce some basic notions about graph theory and present our system model.In Section 3, we give the main results of this paper.Section 4 offers some numerical simulations to illustrate the effectiveness of our theoretical results.Finally, conclusions are given in Section 5.
In this paper, R, Z, and C denote the sets of real number, integer number, and complex number.R  denotes the  dimensional real vector space and R × denotes the set of × matrix. 1  ( 0  ) denotes the column vector with all entries equal to one (zero),   denotes an  dimensional identity matrix, and 0 denotes any dimensional zeros matrix with compatible dimension.diag{ 1 ,  2 , . . .,   } denotes a diagonal matrix with diagonal elements being  1 ,  2 , . . .,   .I  = {1, 2, . . ., }, I  /I  = { + 1,  + 2, . . ., }.For a matrix ,   denotes its transpose.Given a complex number  ∈ C, Re() and || denote the real part and the modulus of , respectively.Matrix  = [  ] × is said to be nonnegative (nonpositive) if all entries   are nonnegative (nonpositive), denoted by  ≥ 0 ( ≤ 0), respectively.Furthermore, if all its row sums are 1,  is said to be a stochastic matrix; a stochastic matrix  is called indecomposable and aperiodic () if lim →∞   = 1    , where  is a column vector.

Preliminaries
In this section, we present some basic concepts and results of algebra graph theory [38] and give the system model.

Graph Theory.
Let  = {, , } be a weighted directed graph with a vertex set  = {1, 2, . . ., }, an edge set  = {(, ) ∈  × }.The adjacency matrix  = [  ] is defined as   > 0 if and only if (, ) ∈ , and   = 0 otherwise.(, ) ∈  is an edge of the graph , where  is called the parent vertex and  is called the child vertex.A directed tree is the directed graph, where every vertex has exactly one parent except one special vertex called the root vertex.A directed spanning tree is a directed tree, which consists of all vertices and some edges of .The degree matrix  = diag{ 1 ,  2 , . . .,   } is the diagonal matrix with   = ∑  =1   .The Laplacian matrix of the graph  is defined as Lemma 1 (see [3]).The directed graph  associated with the Laplacian matrix  has a directed spanning tree if and only if  has only one zero eigenvalue with algebraic multiplicity 1.
(2)   has a directed spanning tree if and only if  has a directed spanning tree.
(2) (Sufficiency) If the graph  has a directed spanning tree, then matrix  has exactly one zero eigenvalue; i.e., rank(L)= − 1. and and we have   = , where Since  and  are the nonsingular matrix, we have (  ) = () =  + () =  +  − 1, which implies that the matrix   has exactly one zero eigenvalue.Therefore, the directed graph   associated with   has a directed spanning tree.
(Necessity) If the graph  does not have a spanning tree, then the Laplacian matrix  of the graph  has at least two zero eigenvalues.Thus, we know that (  ) <  +  − 1.It follows from Lemma 1 that the graph   does not have a spanning tree.
Lemma 4 (see [3]).Let  1 ,  2 , . . .,   be a finite set of SIA matrices with the property that for each sequences for each    , The dynamics of agent  ∈ I  switch between a CT model and a DT model.Specifically, the dynamics of the doubleintegrator agent  ∈ I  can be modeled as V  ( + 1) = V  () +   () , for DT subsystem, (7) and the dynamics of the first-integrator agent  ∈ I  /I  can be described as ẋ  () =   () , for CT subsystem,   ( + 1) =   () +   () , for DT subsystem, (8) where   ∈ R, V  ∈ R, and   ∈ R are the position, the velocity, and the control input of the agent , respectively.The initial states are   (0) =  0 ,  ∈ I  , and Definition 5.The heterogeneous multiagent system ( 7)-( 8 holds for any initial states (0) ∈ R  , V(0) ∈ R  .

Consensus with Fixed Topology.
In this subsection, we discuss the consensus of the heterogeneous multiagent with switched dynamics ( 7)-( 8) under directed fixed topology.At the time , we assume the switching rule is arbitrary.First, we present the linear consensus protocol for system (7)- (8) with the fixed topology.For the first-integrator agent  ∈ I  /I  , we let and for the double-integrator agent  ∈ I  , we let where   is the (, ) entry of the adjacency matrix  of , ℎ > 0 is the sampling period, and  > 0 is the feedback gain.
(Sufficiency).Since  > 1 + 2 max ∈I    and we have   > 0 and It follows that  > 0 and Θ 11 −  < 0. By Lemma 3, it is easy to find that   is the Laplacian matrix of the directed graph   .Recalling ( 16), we know that   is the interaction graph of the first-integrator multiagent system ( 14)- (15). has a spanning tree, we have that   has a spanning tree by Lemma 3.Moreover, it follows from Lemma 2 that lim and lim For any  > 0, we have  =   +   , where   ∈ R is the total duration time on CT subsystem and   ∈ Z is the total duration time on DT subsystem.Due to we have When  → ∞, there are three cases as follows.

Consensus with Switching Topology.
In this subsection, we discuss the consensus problem of system ( 7)-( 8) under directed switching topology.The interaction topology is dynamically changing; we assume that the switching rule is followed CT and DT subsystems.
we present the linear consensus protocol for system ( 7)-( 8) with the switching topology.For the first-integrator agent  ∈ I  /I  , we let where   () is the (, ) entry of (), ℎ > 0 is the sampling period, and  > 0 is the feedback gain.Likewise, for the double-integrator agent  ∈ I  , we let Let   =   ()V  +   ( ∈ I  ), where and   () is the degree of vertex  in ().
When  → ∞, there are two cases as follows.

Simulations
In this section, some simulations are presented to verify the efficiency of the theoretical results.Let ℎ = 0.8,  = 2,  = 1, and the other initial conditions of all the agents are generated randomly.
Example 11.The interaction graph  is shown in Figure 1; it is obvious that the interaction graph in Figure 1 has a directed spanning tree.The heterogeneous multiagent system consists of three single-integrator agents 1-3 and three doubleintegrator agents 4-6.The switching law of the heterogeneous multiagent system (7)-( 8) and simulation results are shown in Figure 2, respectively.We can find that the linear consensus protocols ( 10)-( 11) can solve consensus problem for the heterogeneous multiagent system (7)-( 8) which is consistent with the result of Theorem 6.
Example 12. Figure 3 shows the communication topologies  1 ,  2 ,  3 .It is obvious that  1 ,  2 ,  3 have the directed spanning tree.The heterogeneous multiagent system consists of six vertices with a directed switching topology.Vertices 1 − 3 denote the single-integrator agents and vertices 4 − 6 denote the double-integrator agents.The switching law of topology, system, the trajectories of agents 1 − 6, and the velocity of agents 4 − 6 are shown in (a), (b), (c), and (d) in Figure 4, respectively.It is shown that linear consensus protocols ( 29)-( 30) can solve consensus problem for system (7)-( 8) under the directed switching topology in Figure 4.This result is consistent with the theoretical result in Theorem 10.

Conclusions
This paper studied the heterogeneous multiagent system composed of single-integrators and double-integrators whose dynamics can switch between DT and CT dynamics arbitrarily.Linear consensus protocols are given to solve the  consensus problem of the system.The necessary and sufficient condition is established for solving the consensus of the heterogeneous multiagent system with switching dynamics.Moreover, under switching topologies, we also proved that the system can solve consensus problems when all interaction topologies have spanning tree.Finally, some examples are offered to illustrate the effectiveness of theoretical results.