Consensus for Mixed-Order Multiagent Systems over Jointly Connected Topologies via Impulse Control

1Key Lab of Intelligent Analysis and Decision on Complex Systems, School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2Key Laboratory of Intelligent Air-Ground Cooperative Control for Universities in Chongqing, College of Automation, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 3School of Automation, China University of Geosciences, Wuhan, Hubei 430074, China


Introduction
Consensus of multiple dynamic agents is an interesting topic [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].Most of the consensus results are on the homogeneous dynamics.However, because of the various restrictions and the complexity of environments or the different task divisions, the dynamics of different agents in the same system may be different.That is, in a system, the dynamics of some agents are first ordered, and the others may be second ordered, even high ordered, which is called a mixed-order multiagent system or a heterogeneous multiagent system [20][21][22][23].In addition, most of the consensus results on multiagent systems required the network to be connected, even undirected connected.However, practically, the communication links may be broken or disconnected because of the obstacle among agents or the change of the agents themselves.That is to say, the topology of the network could not maintain being connected all the time.Just the opposite, it is usually varying along with the time [24][25][26][27][28].To reduce the information exchange capacity of agents, this work adopts the impulse control technique, which only uses the information at the impulse instants [10,[29][30][31][32], that is, exerting control only at the impulse instant and no control at any other time.The main idea of impulsive control is to drive the state variable of the controlled system instantaneously to some value which is determined by an impulsive control law at each impulsive instant.It is more reasonable to perform this state change within a period of time [10].Liu et al. [10] propose the pulsemodulated intermittent control.They found that consensus significantly relies on the sampling period, the control gains, the digraph, and the pulse function and gave some necessary and sufficient conditions to ensure the consensus of the controlled system.The impulse control method not only can avoid the abrupt changes between the agents' states but also can greatly reduce the amount of information transferring.Moreover, the continuous time control protocols may lead to chattering because the neighbor relations might change abruptly with the changing of agents' states.Because of all the above problems, this paper studies the consensus problem of the mixed-order multiagent networks over the jointly connected topologies via impulse control technique.For all we know, there are no related topic's results till now.

Complexity
The rest of this work is organized as follows.The necessary knowledge used in the work is presented in Section 2. The problems and impulse-control protocols for the given multiagent systems are proposed in Section 3. Some simulation results are given in Section 4 to show the feasibility of the control technique.Section 5 summarizes the main ideas and methods of the work.

Preliminaries and Some Necessary Lemmas
. .Preliminaries.In this section, some necessary notations and the knowledge of graph theory are given.
: -dimensional real column vector set;   : -dimensional identity matrix; 1 = [1, 1, . . ., 1]  : a column vector with all elements being 1 and an appropriate dimension; 0: a zero vector or a zero matrix with an appropriate dimension;   : the set of all real -dimensional matrices.For a network, G = {V, E, A} denotes the weighted graph, where V = {V 1 , V 2 , . . ., V  } is the node set,  ∈ Υ is the th node, Υ = {1, 2, . . ., } is the node index set of G, and E ⊆ V × V is the edge set.Throughout this paper, the elements of E denote the communication links between agents.Ordered pair (V  , V  ) represents a edge in G; (V  , V  ) ∈ E if and only if the th agent can directly receive the th agent's information.N  = {V  ∈ V | (V  , V  ) ∈ E} denotes the neighbor agents set of the th agent.For the weighted directed graph G, A = [  ] is the weighted adjacency matrix, and   ≥ 0, ∀,  ∈ Υ.More specifically, if (V  , V  ) ∈ E,   > 0, otherwise   = 0, and   = 0 for all  ∈ Υ.If there is a sequence (V  , V  1 ), (V  1 , V  2 ), . . ., (V   , V  ) from two different nodes V  and V  , it is said there is a path between the nodes V  and V  .If there is a path between any two different nodes, the graph is called connected.D = diag{∑  =1   ,  = 1, . . ., } is the degree matrix of graph G, and L = D − A = [  ] × is the Laplacian matrix of graph G. From the definition of L, one can find that all the row sums of L are zero, and L has a right eigenvector 1  with the zero eigenvalue.If there is a node in a digraph, which satisfies the fact that there is a directed path from this node to any other node, the digraph is called containing a spanning tree.
For graphs G 1 , G 2 , . . ., G  , which have the same node set V, their connection is called the union graph G 1− , whose node set is V, edge set is the union edge sets of all graphs in the collections, and the connected weight between agent  and agent  is the sum of   of the connection graphs G 1 , G 2 , . . ., G  .Graphs G 1 , G 2 , . . ., G  are called jointly connected, if their union graph G 1− contains a spanning tree.
Matrix B = [  ] ∈  × is nonnegative if all its elements   are nonnegative.For a nonnegative matrix B = [  ] ∈  × , if it satisfies B1  = 1  , then it is called (row) stochastic.A stochastic matrix B ∈  × is said to be indecomposable and aperiodic (SIA) if lim →∞ B  = 1    , where  ∈   is a constant vector.
Several lemmas are given in the following for further analysis.
Lemma 1 (see [9]).For a nonnegative matrix H = [ℎ  ] ∈   , if its row sums are the same positive constant, which is given by  > 0, then  is an eigenvalue of H with the eigenvector 1, and  is also the spectral radius of matrix H, i.e., (H) = .Eigenvalue  of matrix H has algebraic multiplicity equal to one, iff the graph of H has a spanning tree.If the graph of matrix H has a spanning tree and all the diagonal element ℎ  > 0,  = 1, 2, . . ., , then  is the unique eigenvalue of matrix H with the maximum modulus.
Lemma 2 (see [9]).For a stochastic matrix S = [  ] ∈   , if S has an eigenvalue  = 1 with the associate algebraic multiplicity equal to one, and all the other eigenvalues satisfy || < 1, then S is SIA. at is, there exists a constant vector  satisfing S   =  and 1   = 1, such that lim →∞ S  = 1  .

Main Results
Consider a system with  agents.Suppose that the system consists of  1 (0 <  1 < ) first-order agents and ( −  1 ) second-order agents.In general, assume the first  1 agents are first ordered and the other (− 1 ) agents are second ordered.For the first-order agents, their dynamics are described as And the dynamics for the second-order agents are given as where In the robot fault-tolerant control and hybrid robot formation environment, the stationary consensus in Definition 3, that is, lim →∞ ‖,   ()−  ()‖ = 0, ∀,  = 1, 2, . . ., , and lim →∞   () = 0, ∀ =  1 + 1,  1 + 2, . . ., , means that the position states of the robots tend to be the same and the velocity states tend to be zero with the development of the time.
From the above analysis, systems (1)-( 2) can achieve consensus asymptotically if and only if the union graph of the jointed networks contains a spanning tree.This completes the proof.
Remark .The advantage of the method in this work is adopting the impulse control method to solve the continuous time consensus problems.The impulse control technique requires much less information of the multiple agents than the usual method [29][30][31][32].Accordingly, it greatly reduces the control cost in the engineering applications.
Remark .Note that when  1 = , systems (1)-( 2) reduce to the single-order multiagent systems.And when  1 = 0, systems (1)-( 2) reduce to the general second-order systems.Hence the first-order and second-order multiagent systems could be regarded as the special cases of the considered mixed-order systems in this work.That is, the results in this paper are the generalization of the existing consensus results of the first-order and second-order multiagent systems.

Numerical Simulation Examples
To verify the correctness of the main results, some numerical examples are given in the following.
From Figures 1 and 2 one can find that graphs G 1 and G 2 contain spanning trees, and the union of graphs G 3 , G 4 , and G 5 contains a spanning tree, while the union of graphs G 3 , G 4 , and G 6 contains no spanning tree.Note that, under protocols (4) and ( 5) with  1 = 0.5,  2 = 0.8, ℎ = 0.5, conditions (i), (ii), and (iii) in Theorems 4 and 5 hold.The state trajectories of the agents over networks G 1 and G 2 are given in Figures 3 and 4, respectively, which show that all the position state trajectories converge together and velocity state trajectories converge to zero.The state trajectories over networks switching among G 3 , G 4 , and G 5 are given in Figure 5, and the state trajectories over networks switching among G 3 , G 4 , and G 6 are given in Figure 6. Figure 5 illustrates that the position and velocity state trajectories can converge together when the union network graph contains a spanning tree, while Figure 6 illustrates that the state trajectories cannot converge together when the union network graph contains no spanning tree.The simulation results verify the correctness of the main results in this work.

Remark .
From the numerical simulation results, which have not been given in the paper because of the limited space, under the conditions in Theorems 4 and 5, the control  gains  1 and  2 affect the convergence speed, but it is not monotonically increasing or decreasing as the control gains increase.

Conclusions
This work studies the consensus problem for the mixed-order multiagent systems over the jointly connected topologies.By adopting the graph theory, matrix theory, and control theory, impulse consensus protocols are designed and analyzed for the mixed-order multiagent systems.Some simulation examples are given to verify the correctness of the main results and the effectiveness of the control method.However, the related topics of the systems in the turbulent, noisy, or some other uncertainty case have not been considered.The related problems will be studied in the future work.