Sampled-Data Consensus for High-Order Multiagent Systems under Fixed and Randomly Switching Topology

This paper studies the sampled-data based consensus of multiagent system with general linear time-invariant dynamics. It focuses on looking for a maximum allowable sampling period bound such that as long as the sampling period is less than this bound, there always exist linear consensus protocols solving the consensus problem. Both fixed and randomly switching topologies are considered. For systems under fixed topology, a necessary and sufficient sampling period bound is obtained for single-input multiagent systems, and a sufficient allowable bound is proposed formulti-input systems by solving theH ∞ optimal control problem of certain systemwith uncertainty. For systems under randomly switching topologies, tree-type and complete broadcasting network with Bernoulli packet losses are discussed, and explicit allowable sampling period bounds are, respectively, given based on the unstable eigenvalues of agent’s system matrix and packet loss probability. Numerical examples are given to illustrate the results.


Introduction
In recent years, coordination of distributed dynamic systems operating over relative sensing/communication networks has attracted the attention of researchers in system theory, biology, and statistical physics, and so forth [1][2][3][4][5].Consensus is an important problem in coordination of multiagent systems.
The insertion of the communication network among agents makes the analysis and design of multiagent systems more complex.One of the challenging problems associated with the multiagent systems is the influence of the sampling period.There are many researchers studying sampled-data consensus problem for first and second order multiagent systems.For first-order systems, Olfati-Saber and Murray [3] pointed out that the maximum allowable sampling period is the reciprocal of the maximum out-degree.Xie et al. [6] studied the first-order system via delayed sampled control and gave consensus conditions on delay and sampling period.Liu et al. [7] studied first-order consensus problem with logarithmic quantization and gave sufficient conditions on the sampling interval to ensure the -asymptotic average consensus.For second-order integrator agents, Cao and Ren [8] proposed a necessary and sufficient condition on the sampling period, the control gain, and the communication graph.Gao and Wang [9] proposed an allowable bound of sampling period for a given protocol through solving a set of LMIs.Yu et al. [10] gave a necessary and sufficient consensus condition based on the sampling period, the coupling gains, and the spectra of the Laplacian matrix.Xiao and Chen [11] studied the state consensus of multiple double integrators in a sampled-data setting with the assumption that the position-like states were the only detectable information transmitted over the network and gave a necessary and sufficient condition and a sufficient condition for the uniform and nonuniform datasampling cases, respectively.All the aforementioned literature concentrate on first or second order integrator agents.Nevertheless, the consensus problem of high-order agents is important and has been recently considered by many researchers [12][13][14][15][16][17].Zhang and Tian [14] pointed out that for high-order multiagent systems the sampling period should be bounded and proposed an allowable bound based on given consensus protocol.In [15], Gao et al. studied the consensusability of sampled-data multiagent systems with general linear dynamics and gave some sufficient and necessary conditions for consensusability in the case of state feedback.
The other challenging problem associated with multiagent systems is the influence of packet losses.Consensus of discrete-time first and second-order systems in random networks has been studied in some works [18][19][20][21].They show that for low-order integrator agents the solvability of the consensus problem just depends on the connectivity of the mean topology while it is independent of link weights and link existence probabilities.Zhang and Tian [22] studied the influence of packet loss probabilities on the consensus ability of discrete-time high-order multiagent systems and provided a maximum allowable sampling period bound.Although many efforts have been made on studying the consensus of multiagent systems with switching topologies or sampleddata, there is limited work investigating both the influences of packet losses and sampling period for high-order multiagent systems.Zhang and Tian [14] studied this problem, but the proposed allowable sampling period bound was not explicit and was based on given protocol gain.
This paper studies the consensus problem for high-order multiagent systems with sampled data and packet losses and focuses on looking for a maximum allowable sampling period such that as long as the sampling period is less than this period bound, there always exists a state feedback protocol solving the consensus problem.Firstly, fixed topology is discussed.For single input system, by using algebraic graph theory and nonsingular matrix transformation, the consensus problem is converted to the simultaneous stabilizing problem of subsystems, and a necessary and sufficient sampling period bound is obtained.For multi-input system, by solving the  ∞ optimal control problem of certain system with uncertainty, an allowable bound of sampling period is proposed.Both bounds depend on the eigenvalues of agents' system matrix and the Laplacian matrix and are easily computed.Secondly, randomly switching topology case is studied.When the topology is a rooted directed spanning tree with Bernoulli link losses, a sampling period bound is given by studying the spectral radius of expected value of multiagent system matrix.When the network is a complete topology with broadcasting schemes and Bernoulli packet losses, a sampling period bound is given by applying Lyapunov functional analysis.Both bounds are explicitly composed of unstable eigenvalues of agents' system matrix and packet loss probability.
Notations.  denotes the identity matrix with  dimensions.(⋅) represents the spectral radius of a matrix.Re(⋅) and Im(⋅) represent the real part and imaginary part of a number, respectively.⊗ denotes Kronecker product of matrixes.Pr(⋅) and (⋅) denote the probability and expected value of a random process, respectively.‖‖  is the Euclidian norm

Preliminaries
Let G = (V, E, A) be an undirected graph of order  with the set of nodes V = {1, 2, . .

Problem Formulation
Consider there are  agents in the communication network.Each agent in the network has identical continuous-time linear dynamics: where   ∈   is the state of agent ,   ∈   is the consensus protocol, and (, ) are constant matrices with appropriate dimensions and are completely controllable.By consensus, we mean a scenario where all states of agents in the network agree on a particular value; that is,   =   , for all , .Consensus is an important even fundamental problem in multiple agent coordination.It is of interest in studying flocking, swarming, and attitude alignment.
Since state information can be used, here we apply a state feedback consensus protocol: where  is the protocol parameter to be designed.In practical applications, the topology is often varying due to communication packet losses and communication range constraints, and thus   is varying.Suppose all agents are clock synchronized and each agent transfers its state information periodically.Let  denote the sampling period, then at each transfer instant ,  = 0, 1, 2, . .., each agent sends its sampled information   () by network.If agent  receives 's information   () at the th sampling period, then   () > 0 and agent  updates its control input   .ZOH is applied in the actuator of agents.Thus for  ∈ [( + 1)), And then Denote  = [  ] as the Laplacian matrix of the topology; Discretizing the above system we obtain that This paper studies the consensus problem defined as follows.
Zhang and Tian [14] have revealed that for high-order multiagent systems, the sampling period should be bounded.In the following sections, we investigate the maximum allowable sampling periods bound (MASPB) for systems under fixed topology and randomly switching topologies such that as long as the sampling period is less than this period bound, there exists a state feedback protocol solving the consensus problem.

MASPB of Systems under Fixed Undirected Topology
This section will look for a MASPB for multiagent systems under fixed undirected topology.To study the system, firstly perform some system transformations.For Laplacian matrix , there exists a nonsingular matrix Applying nonsingular transformation  −1 ⊗  to both sides of (6) we have that system (6) achieves consensus, if and only if Thus the following lemma can be obtained.

Lemma 3. The sampled-data multiagent system (1), (3) under fixed and undirected connected topology converges to consensus, if and only if
holds for , where   are the eigenvalues of  except the zero eigenvalue corresponding to the eigenvector 1  ,  = 1, 2, . . .,  − 1.

MASPB for Single Input Systems.
To provide the maximum allowable sampling period for single input sampleddata multiagent systems, an important lemma is given firstly.
From Lemma 4 we can obtain the following theorem.Here we ignore the case when the discretized system Theorem 5.For single input sampled-data multiagent systems (1) in fixed and connected undirected communication topology, there exists a linear consensus protocol asymptotically solving the consensus problem of the multiagent system, if and only if where 0 <  1 ≤  2 ≤ ⋅ ⋅ ⋅ ≤  −1 are the nonzero eigenvalues of Laplacian matrix and    are the unstable eigenvalues of continuous-time system matrix  satisfying Re(   ) ≥ 0.
Here we ignore the case when the discretized system is uncontrollable.Since (, ) is completely controllable, the discretized system (, ) is also completely controllable.
Obviously, let From Lemma 4, Theorem 5 has been proved.Remark 6. Theorem 5 shows that for first-order, secondorder, even high-order integrator multiagent systems, the allowable sampling period can be arbitrarily large but bounded.It conforms to the results in [8].

MASPB for Multi-Input Systems.
For multi-input systems, the degree of instability of the system matrix cannot be presented by the unstable eigenvalues directly.Here, we look for the allowable sampling period bound by studying the robust control of uncertain system.
From Lemma 3, the consensus problem is simplified to the simultaneous stabilization of  − 1 specific systems like Define   =  * ; then the simultaneous stabilization problem of systems (11) can be solved by studying the robust control of the following system with uncertain parameters where the uncertainty Δ satisfies that |Δ| ≤   .Denote system ( 13) by a linear model as follows: where Δ is the uncertain parameter in the model and satisfies that |Δ| ≤   .
The uncertain system ( 14) is quadratic stable, if and only if where () is the transfer function of system ( 14) when Δ = 0; that is, Next, we study the condition of sampling period by solving the optimal  ∞ function problem.
Theorem 7.For multi-input sampled-data multiagent systems (1) in a fixed communication topology, the topology is undirected and connected.If  <  * , where where  * () is obtained by solving the following optimal problem  * () = min  (18) subject to the following inequality then there exists a linear consensus protocol asymptotically solving the consensus problem of the multiagent system.
Proof.From Bounded Real Lemma, ‖()‖ ∞ <  if and only if there exists symmetric positive definite matrix  > 0 such that Left-and right-multiplying the both sides of by  −1 , we obtain that Define  =  −1 and  −1 = ; then by applying Schur Complement Lemma, the above inequality is equivalent to (19).Solving the optimal  ∞ control problem, we obtain the minimum  ∞ spectrum  * and optimal control gain .Therefore, if  * <  −1  , the optimal control gain  can always simultaneously stabilize systems (11).
For a given communication topology,   is fixed.If a sampling period  guarantees that the obtained minimum  ∞ spectrum  * () for this sampled-data system matrix (, ) is less than  −1  , then under this sampling period there exists control gain  simultaneously stabilizing systems (11).We look for the maximum allowable sampling period by searching the maximum  under which there holds  * () <  −1  .So, as long as the sampling period  <  * , there exists a common control gain  simultaneously stabilizing systems (11), and hence from Lemma 1 there exists a linear consensus protocol solving the consensus problem of the sampled-data multiagent system.This theorem has been proved.Remark 8. Zhang and Tian [14] studied the consensus of general linear dynamical multiagent systems and gave allowable sampling period bounds based on given protocol gain.Comparing with their results, the bound in Theorem 7 is less conservative since the bound is obtained by finding optimal protocol gain.

MASPB of Systems under Randomly Switching Topologies
This section focuses on looking for MASPBs for multiagent systems under randomly switching topologies.Two types of topology are discussed.

Rooted Directed Spanning Tree with Bernoulli Link Losses.
Firstly we consider systems under tree-type topology.Due to communication constraints, the links between agents are time-varying and driven by a Bernoulli process.Assume all links in the network are independent.For an edge (, ) ∈ E in the tree,   () is varying between 0 and 1. Define  (0 <  < 1) as the packet loss probability of the network; then Pr(  () = 0) = .For a tree-type graph, number the agents such that each agent's parent node in the graph is lower numbered than itself.Then the Laplacian matrix is a lower triangular matrix with diagonal elements  11 () ≡ 0 and   () ( > 1) is switching among 0 and 1 with probability Pr(  () = 0) = .() can be denoted as [ 0 0 * () ], where () is a ( − 1)-dimensional lower triangular matrix.
Lemma 9. Consider a controllable system with packet loss where () ∈ {0, 1} denotes packet loss process and is driven by an i.i.d.process with loss probability .
Theorem 10.For multiagent systems (1)-( 3) under tree-type lossy network, there exists a linear consensus protocol solving the mean square consensus problem, if where    are the unstable eigenvalues of .

Complete Network with Broadcasting Schemes and
Bernoulli Link Losses.This subsection considers a network of agents with complete graph and broadcasting schemes.At each sampling instant, all agents compete for the communication channel with the same opportunity.Just one of the agents can succeed and broadcast its information to all other agents.Due to communication constraints, when the agent broadcasts its information, the links between itself and other agents may be lost.The process is driven by a Bernoulli process.Assume all links in the network are independent.Thus at each sampling instant , agent  broadcasts its information to other  − 1 agents with probability 1/.When agent  sends its information to agent , the edge (, ) may be lost due to packet losses.Then   () is varying between 0 and 1. Define  (0 <  < 1) as the packet loss probability of the network, then Pr(  () = 0) = .
Proof.Firstly, we will show that for agents' system matrices , just unstable parts should be considered.For  and , there always exists a nonsingular matrix  such that ], where   ,   are Hurwitz stable and unstable matrices, respectively, and   ,   are Schur stable and unstable matrices, respectively.Define  −1 = [0   ], then system matrix in ( 6) is similar to Obviously, the system achieves consensus if and only if the part x(( + 1)) = (  ⊗   − () ⊗  2   ) x() achieves consensus.Therefore, just unstable part of  should be controlled.For depiction simplicity, in the following we just consider the unstable matrix .

Simulation Examples
6.1.Fixed Topology Cases.Firstly consider a network of 5 agents.The topology is undirected and cyclic as given in Figure 1.Then the eigenvalues of Laplacian matrix are  4 = 3.618,  1 = 1.382.
Consider a network of agents with single input.The system matrices are From Theorem 5, the maximum allowable sampling period bound is 0.4024 s.Choose  = 0.35 s, then by solving LMIs ( −   )  ( −   ) <  we obtain a control gain  = [0.2183−0.2537 1.5837].The trajectories of agents are given in Figure 2. Obviously, the system reaches consensus asymptotically.
If  = 0.45 s, then the LMIs (−  )  (−  ) <  are infeasible.For a control gain  = [0 −0.6643 1.2524], the trajectory of average consensus errors is given in Figure 3. Obviously, the system does not achieve consensus.
Next consider a network of agents with multi-input.The system matrices are From Theorem 7, the maximum allowable sampling period bound is 0.85 s.Choose  = 0.75 s, then by solving LMI (19) we obtain a control gain  = [ 0 0 −0.9836 0.4841 0.7337 0.2499 ].The trajectories of agents are given in Figure 4. Obviously, the system reaches consensus asymptotically.

Tree-Type
Network with Bernoulli Link Losses.Consider a network of 5 agents with system matrix (36).The topology is a rooted spanning tree as shown in Figure 5.The packet loss probability is 0.1.Then by applying Theorem 10, the maximum allowable sampling period bound is 0.5756 s.Choose  = 0.

Complete Network with Broadcasting Schemes and
Bernoulli Link Losses.Consider a network of 5 agents with system matrix (36).The network is a complete broadcasting graph.The packet loss probability is 0.1.Then by applying Theorem 11, the maximum allowable sampling period bound is 0.4164 s.Choose  = 0.4 s, then by solving LMIs ( − )  ( − ) +    <  we obtain a control gain  = [0.3999−0.6330 3.6687].The trajectories of agents are given in Figure 7. Obviously, the system reaches consensus.

Conclusion
This paper focuses on looking for an allowable sampling period bound such that as long as the sampling period is less than this period bound, there exists a state feedback consensus algorithm solving the consensus problem.The allowable sampling period bounds for sampled-data multiagent systems under fixed topology and two specific Bernoulli lossy networks are provided.Comparing with existing results, the proposed MASPDs are explicitly related to unstable eigenvalues of agents' system matrix and packet loss probability and can be directly computed.

Figure 2 :
Figure 2: Trajectories of single input agents under fixed topology.

Figure 3 :
Figure 3: Average consensus errors with unallowable sampling period.
. , }, edge set E ⊆ V × V, and adjacency matrix A = [  ] × which describes the linkages of nodes.If the edge (, ) ∈ E, that is, vehicle  can obtain information from vehicle , then   > 0, otherwise   = 0.If for any  and ,   =   , we say the graph is undirected.The Laplacian matrix of graph G is denoted by  = [  ] × with   = ∑  =1   and   = − 1 ≤  2 ≤ ⋅ ⋅ ⋅ ≤   be given.There exists a common control gain  ∈  1× such that ( −   ) < 1 hold for all , if and only if The degree of instability of  is ∏  | Re (   ) ) .