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On the asynchronous bipartite consensus for discrete-time second-order multi-agent systems with switching topologies
Accepted Manuscript
On the asynchronous bipartite consensus for discrete-time second-order multi-agent systems with switching topologies Jinliang Shao, Lei Shi, Yangzhen Zhang, Yuhua Cheng PII: DOI: Reference:
S0925-2312(18)30891-9 https://doi.org/10.1016/j.neucom.2018.07.056 NEUCOM 19809
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
15 May 2018 14 July 2018 24 July 2018
Please cite this article as: Jinliang Shao, Lei Shi, Yangzhen Zhang, Yuhua Cheng, On the asynchronous bipartite consensus for discrete-time second-order multi-agent systems with switching topologies, Neurocomputing (2018), doi: https://doi.org/10.1016/j.neucom.2018.07.056
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Highlights • The asynchronous bipartite consensus for a group of agents with second-order dynamics is examined in this paper. • The properties of the product of infinite time-varying row-stochastic matrices from a noncompact set are explored to analyze this problem.
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• A sufficient condition that depends on switching topologies is established.
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On the asynchronous bipartite consensus for discrete-time second-order multi-agent systems with switching topologies✩ Jinliang Shaoa , Lei Shia , Yangzhen Zhangb , Yuhua Chenga,∗ School of Automation Engineering, University of Electronic Science and Technology of China, Sichuan, 611731 b University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai, 200240
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Abstract
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The asynchronous bipartite consensus for a group of agents with second-order dynamics is examined in this paper, where the asynchrony means that the time instants when each agent receives the neighbors’ data information are completely independent of other agents’. The communication among the agents is described by a time-varying signed and structurally balanced digraph, which is equivalent to assuming that the agents can be divided into two groups without any common agents, in which the agents within the same group are cooperative and the agents between different groups are competitive. An asynchronous distributed control protocol is designed to implement the bipartite consensus. By using the product properties of row-stochastic matrices from a noncompact set, a sufficient condition can be established under a loose assumption that is the union of communication topologies related to any time intervals with given length has a spanning tree. Finally, a simulation instance is provided to verify the reachability of asynchronous bipartite consensus.
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Keywords: Bipartite consensus; Second-order multi-agent systems (MASs); Asynchronous situation.
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Over the past decade, the issue of consensus in multi-agent systems (MASs) has been a topic of intense concern for scientists because of its wide applications in military and civilian fields, such as unmanned air vehicle formation, flocking and rendezvous [1–4]. In the existing related literature [5–16], it is usually assumed that information exchange between the agents is beneficial to their cooperation, which is reflected in the communication topology is that each edge contains positive weights. Such an assumption can ensure the realization of multi-agent consensus, that is, all agents achieve the exact same state in the long run. Unlike the case of fully cooperative agents, in the context of many practical applications, the information exchange between agents is not necessarily completely beneficial to their cooperation. Conversely, agents may also exhibit non-cooperative or confrontational interactions with certain neighbors in situations such as markets or social networks. For example, in a social network described in literature [17], the interpersonal relationship between people may be friendly or hostile, which may lead
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1. Introduction
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✩ This research was supported in part by the National Science Foundation of China (61671109), the China Postdoctoral Science Foundation(2017M612944, 2018T110962), the Special Postdoctoral Foundation of Sichuan Province, and the Fundamental Research Funds for the Central Universities (ZYGX2018J073). ∗ Corresponding author Email addresses: [email protected] (Jinliang Shao), [email protected] (Lei Shi), [email protected] (Yangzhen Zhang), [email protected] (Yuhua Cheng)
Preprint submitted to Neurocomputing
August 3, 2018
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2. Notations and preliminary lemmas
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to the emergence of two completely opposite groups. In a multi-agent network consisting of two different group of agents, if the agents belonging to the same group are in a cooperative relationship and the agents belonging to different groups are in a competitive relationship, then in the long run, the two group of agents will reach the final state of the same size but opposite sign. Consequently, we say that such a consensus issue is called bipartite consensus, in which the communication topology is structurally balanced. Zhang et al. [18] studied the bipartite consensus issue of general linear MASs with a signed digraph based on both dynamic output feedback and state feedback control protocols, and Qin et al. [19] solved the bipartite consensus issue for general linear MASs with input saturation. By using the properties of the Laplacian matrix and rooted cycles of digraph, the bipartite consensus for a team of agents with first-order dynamics is investigated in [20]. A bipartite consensus problem for high-order MASs with unknown disturbances is considered in [21]. In addition, the authors in [22] analyzed the adaptive finite-time bipartite consensus for second-order MASs with unknown external disturbances, and [23] studied the bipartite consensus issue under arbitrary finite communication delays. In practical engineering applications, there may exist differences in the clocks for different agents, and these difference can affect system performance and cause system instability if the sensor is timedriven. Therefore, it is more practical to consider the consensus of MASs under asynchronous situation than synchronous situation in which all agents’ clocks are identical. Asynchrony means that each agent detects the neighbors’ state information at certain discrete time instants determined by its own clock that is independent of the other agents’. In recent years, some valuable results on the consensus of MASs under asynchronous situation have been reported in [24–29]. As we introduced previously, concerning the bipartite consensus of MASs, many valuable works have been emerged in [18–23]. However, it has to be mentioned that the results presented in above works are based on synchronous situation. This consideration shift our attention to the problem of bipartite consensus of discrete-time second-order MASs under asynchronous situation. In addition, the communication topology we consider in this paper is time-varying, this is different from the literatures [18–23] in which the communication topology is assumed to be fixed. In general, time-varying topology is more common than fixed topology in practical applications because of the limited communication range and the reliability of communication links between agents. For solving the bipartite consensus problem of discrete-time second-order MASs with switching topologies under asynchronous situation considered in this paper, we first utilize appropriate model transformation techniques and the parameter selection strategy to ensure that the asynchronous second-order system can be equivalently transformed into a higher-order augmented system with matrix-vector form, in which the coefficient matrices are time-varying row-stochastic matrices from a non-compact set. Then the products properties of an infinite number of time-varying row-stochastic matrices from a non-compact set can be used to solve the asynchronous bipartite consensus problem. The outline of this article is as follows. We first introduce some basic notations about graph in Section 2, and then present the asynchronous bipartite consensus model formulation in Section 3. In Section 4, a sufficiency criterion is established to ensure asynchronous bipartite consensus for secondorder MASs with time-varying topology. The effectiveness of the control protocol is verified by a simulation examples in Section 5. Finally, the conclusion of this article is shown in Section 6.
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Notations: Let Rn and Rn×m denote, respectively, the set of n dimensional real column vectors and the set of n × m-dimensional real matrices. N and Z+ are used to denote the set of natural numbers and 3
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the set of positive integers, respectively. 1n ∈ Rn is the vector of all 1’s, 0n×m ∈ Rn×m denotes a matrix of all 0’s and In ∈ Rn×n is an identity matrix. Matrix Q = [qi j ]n×n is nonnegative, denoted by Q ≥ 0, if qi j ≥ 0 for any i, j = 1, 2, · · · , n. A nonnegative matrix Q is row-stochastic if Q1n = 1n , further the row-stochastic matrix Q is stochastic indecomposable and aperiodic (SIA) if limk→∞ Qk = 1n zT , Q where z ∈ Rn . Let ki=1 Q(i) = Q(k)Q(k − 1) · · · Q(0) denote the left product of matrices. Given a nonnegative matrix Q ∈ Rn×n , let G[Q] denote a digraph with weighted adjacency matrix Q, a node set {1, 2, · · · , n} in which the elements are the indexes of Q’s rows, and a edge set E[Q]. For nonnegative matrices Q = [qi j ]n×n and R = [ri j ]n×n , the digraphs G[Q] and G[R] are said to be of the same type, denoted by G[Q] ∼ G[R], if qi j > 0 ⇔ ri j > 0 and qi j = 0 ⇔ ri j = 0. The notation sgn(c) represents the signum function, and it is defined as follows: 1, c > 0 0, c = 0 sgn(c) = − 1, c < 0.
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Consider a team of n agents which are labeled by v1 , v2 , · · · , vn . A digraph G = (V, E, A) is used to describe the information interchange between the agents, here each agent is regarded as a vertex of digraph G. An edge of G is represented by (vi , v j ), where vi and v j are called, respectively, parent vertex and child vertex. The adjacency matrix satisfies (vi , v j )∈ E ⇔ a ji , 0, besides, ai j > 0 if the connection from v j to vi is cooperative; ai j > 0 if the connection from v j to vi is antagonistic. Assume that aii = 0 for any i = 1, 2, · · · , n. The set of neighbors of vertex vi is denoted by Ni = {v j ∈ V : (v j , vi ) ∈ E}. A digraph has a spanning tree if there exists a vertex called root such that there exists a directed path from the root to any other vertices. Digraph G is structural balanced, if all vertices can be divided into two subsets V1 and V2 such that V1 ∪V2 = V, V1 ∩V2 = ∅. If agents vi , v j belong to the same subset, then ai j ≥ 0, and if ai j ≤ 0, then agents vi , v j belong to different subsets. Without loss of generality, we assume that V1 = {v1 , v2 , · · · , vm } and V2 = {vm+1 , vm+2 , · · · , vn }. Corresponding to the division of sets V1 and V2 , partition the adjacency matrix A associated with G as ! A11 A12 A= , A21 A22
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where A12 ∈ Rm×n and A21 ∈ Rn×m are non-positive matrices and show the relationship between the agents in different sets; A11 ∈ Rm×m and A22 ∈ Rn×n are nonnegative matrices and describe the internal communication of the agents in the same set. The in-degree matrix of the agents in V1 and the P P P in-degree matrix of the agents in V2 are defined as D1 = diag nj=1 |a1 j |, nj=1 |a2 j |, · · · , nj=1 |am j | Pn Pn Pn and D2 = diag j=1 |am+1, j |, j=1 |am+2, j |, · · · , j=1 |am+n, j | , respectively. In terms of these constructions, the overall graph Laplacian matrix can be written as ! D1 − A11 −A12 L= . −A21 D2 − A22
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In the remainder of this section, we list several lemmas that will be used to draw the main conclusion of this paper. Lemma 1. [24] Let D1 , D2 , . . . , Dh be nonnegative matrices, D1 D2 · · · Dh 0 0 · · · 0 D = . .. .. , .. .. . . . 0 0 ··· 0 4
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N0 =
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P Let N(k) = D + N0k for any k ∈ Z+ . The digraph G[Nk ] contains a spanning tree if digraph G hi=1 Di contains a spanning tree.
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Lemma 2. [24] Suppose that A ∈ Rn×n is a row stochastic matrix. Then A is SIA (stochastic indecomposable and aperiodic) if the digraph G[A] has a directed spanning tree with the property that the root node of the spanning tree has a self-loop in G[A]. 3. Model formulation
In the continuous-time setting, the agents are modeled by the following second-order dynamics
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x˙i (t) = ϑi (t), ϑ˙ i (t) = ui (t), 68
where xi (t) ∈ R p , ϑi (t) ∈ R p , ui (t) denote, respectively, the position, the velocity and the control input of agent vi . Consider the following consensus protocol: X X ui (t) = −γ1 ϑi (t) + γ2 |ai j | sgn(ai j )x j (t) − xi (t) + γ3 |ai j | sgn(ai j )ϑ j (t) − ϑi (t) , (2) v j ∈Ni
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where γ1 > 0, γ2 > 0 and γ3 > 0 are the fixed gain parameters. In a realistic multi-agent network, continuous data exchange between agents is difficult due to some practical limitations, including environmental interference and communication distance limitations. This fact guides the existing large number of discrete-time consensus models. By referring the discrete-time consensus model in [30], the discrete-time dynamics of agents in this paper are usually described as: xi (tk+1 ) = xi (tk ) + T k ϑi (tk ),
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where xi (tk ) ∈ R p , ϑi (tk ) ∈ R p , ui (tk ) denote, respectively, the position, the velocity and the control input of agent vi at time instant tk ; {tk }∞ 0 is the discrete time instant sequence which satisfies tk+1 − tk = T k > 0 and t0 = 0. Let structural balanced digraph G(tk ) = (V, E(tk ), A(tk )) describe the time-varying data exchange between the agents. An edge (vi , v j ) ∈ E(tk ) if and only if agent v j can detect the information of agent vi at time tk . The set of neighbors of vertex vi at time tk is denoted by Ni (tk ) = {v j ∈ V : (v j , vi ) ∈ E(tk )}. The adjacency matrix A(tk ) = [ai j (tk )]n×n satisfies ai j (tk ) , 0 ⇔ (v j , vi ) ∈ E(tk ), and all positive weighting factors have a common lower and upper bound, i.e. |ai j (tk )| ∈ [α, α] whenever ai j (tk ) , 0, where 0 < α < α. The Laplacian matrix L(tk ) associated with the communication digraph G(tk ) can be partitioned as ! D1 (tk ) − A11 (tk ) −A12 (tk ) L(tk ) = , −A21 (tk ) D2 (tk ) − A22 (tk )
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Figure 1: Time sequences {tk1 } and {tk2 } in asynchronous setting.
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P P P P where D1 (tk ) = diag nj=1 |a1 j (tk )|, nj=1 |a2 j (tk )|, · · · , nj=1 |am j (tk )| and D2 (tk ) = diag nj=1 |am+1, j (tk )|, Pn Pn j=1 |am+2, j (tk )|, · · · , j=1 |an j (tk )| . The situation that all the agents receive the information of the neighbors synchronously is not easy to be achieved in practical application, and this consideration leads us to shift our attention to a more practical case that each agent only detects the neighbors’ state information at certain discrete time determined by its own clock that is independent of the other agents’. Here we assume that agent vi receives its neighbours’ state information at discrete time instants t0i , t1i , · · · , tki , · · · , which i i i i are randomly chosen in the set {tk }∞ k=0 and satisfy t0 = t0 < t1 < · · · < tk < · · · , and let {tk } = j i i i i {t0 , t1 , · · · , tk , · · · } for simplicity. It’s worth noting that {tk } is independent of {tk } for any vi , v j ∈ V, i , j in asynchronous setting, e.g., (See an example with two agents v1 and v2 in Fig. 1). In addition, to ensure the accessibility of the asynchronous group consensus, we let the time sequences {tki }, i = 1, 2, · · · , m + n, satisfy: for any k ∈ N, there exists a positive real number κˆ that meets the condition i 0 < tk+1 − tki ≤ κˆ , which implies 0 < T k ≤ κˆ . By referring protocol (2), we propose the asynchronous bipartite consensus protocol for system (1) as follows X a (t ) hsgn a (k) x (ti ) − x (t )i ui (tk ) = − γ1 ϑi (tk ) + γ2 ij k ij j ω i k
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X a (t ) hsgn a (t )ϑ (ti ) − ϑ (t )i , t ∈ [ti , ti ). ij k ij k j ω i k k ω ω+1
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Definition 1. Using protocol (4), it is said that the asynchronous bipartite consensus for second-order multi-agent system (1) can be achieved if
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i It can be seen that the neighbors’ information received by agent vi at time tk ∈ [tωi , tω+1 ) all comes i from time tω .
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Lemma 3. Consider system (3) with protocol (4). For any i ∈ {1, j 2, k· · · , n} and k ∈ N, the number of i κˆ , where T min = mink∈N {T k } and time instants in the set t s | t s ∈ tki , tk+1 is no more than h = Tmin j k κˆ κˆ T min is the largest integer less than T min . 6
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i Proof. If the single ordered sequence with discrete time in the set t s | t s ∈ tki , tk+1 is evenly spaced i is the and all the time steps are T min , then the number of time instants in the set t s | t s ∈ tki , tk+1 j k κˆ most and it is no more than h = T min .
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In this section, the asynchronous multi-agent systems is transformed into an augmented system with nonnegative coefficient matrices through appropriate model transformation techniques, it means that the asynchronous bipartite consensus problem can be equivalently transformed into a deterministic problem that is the convergence of infinite products of time-varying nonnegative matrices. Then mixed tools from graph theory and SIA matrices, are employed to solve the convergence problem. Following the above guidance, we first present the following model transformations. Let h iT T X1 (tk ) = x1T (tk ), x2T (tk ), · · · , xm (tk ) , iT h V1 (tk ) = ϑT1 (tk ), ϑT2 (tk ), · · · , ϑTm (tk ) , iT h T T T X2 (tk ) = xm+1 (tk ), xm+2 (tk ), · · · , xm+n (tk ) , iT h V2 (tk ) = ϑTm+1 (tk ), ϑTm+2 (tk ), · · · , ϑTm+n (tk ) , " #T γ3 T γ3 T T T T T ξ(tk ) = X1 (tk ), −X2 (tk ), X1 (tk ) + V1 (tk ), −X2 (tk ) − V2 (tk ) . γ2 γ2
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Remark 1. From Lemma 3, it can be seen that the earliest information received by the agent at the time tk comes from tk−h , which is crucial in the process of transforming a heterogeneous multi-agent system into a time-varying augmented system with matrix-vector form under asynchronous situation.
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Using protocol (4), system (3) can be equivalently expressed as
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0m×(n−m) 0m×m 0m×(n−m) 0m×m 0 0 0 0 (n−m)×m (n−m)×(n−m) F[y] (tk ) = (n−m)×m (n−m)×(n−m) 0m×(n−m) γ3 T k A11[y] (tk ) −γ3 T k A12[y] (tk ) 0m×m 0(n−m)×m 0(n−m)×(n−m) −γ3 T k A21[y] (tk ) γ3 T k A22[y] (tk )
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where A11[y] (tk ) ∈ Rm×m , A22[y] (tk ) ∈ R(n−m)×(n−m) , y = 1, 2, · · · , h, are nonnegative matrices, and A12[y] (tk ) ∈ Rm×m , A21[y] (tk ) ∈ R(n−m)×(n−m) , y = 1, 2, · · · , h, are non-positive matrices. If tωi = tk−k0 and vi ∈ V1 , where k0 ∈ {0, 1, 2, · · · , h − 1}, then the ith row of matrices A11[k0 +1] (tk ) is equal to the ith row of matrices A11 (tk ) and the ith row of matrices A12[k0 +1] (tk ) is equal to the ith row of matrices A12 (tk ), while the ith rows of all the other matrices A11[y] (tk ), A12[y] (tk ), y , k0 + 1 are equal to zeros. 7
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where H(tk ) = F(tk ) + Q(tk ), in which E(tk ) − diag{E(tk )} + F[1] (tk ) F[2] (tk ) 0 0 0 0 F(tk ) = .. .. . . 0 0 diag{E(tk )} 0 . . . 0 0 I2n 0 · · · 0 0 0 I2n · · · 0 0 . Q(tk ) = .. .. . . .. .. . . . . . 0 0 · · · I2n 0
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If tωi = tk−k00 and vi ∈ V2 , where k00 ∈ {0, 1, 2, · · · , h−1}, then the (i−m)th row of matrices A21[k00 +1] (tk ) is equal to the (i − m)th row of matrices A21 (tk ) and the (i − m)th row of matrices A22[k00 +1] (tk ) is equal to the (i − m)th row of matrices A22 (tk ), while the (i − m)th rows of all the other matrices A21[y] (tk ), A22[y] (tk ), y , k00 + 1 are equal to zeros. As we have introduced in Remark 1, the earliest neighbors’ state information that one agent can receive at time tk comes from time tk−h due to the existence of asynchronous situation, where h is defined in Lemma 3. Therefore, let θ(tk ) = ξT (tk ), ξT (tk−1 ), · · · , ξT (tk−h+1 ) T , then we further rewrite system (5) into the following time-varying augmented system: θ (tk+1 ) = H(tk ) ⊗ I p θ (tk ) , (6)
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Lemma 4. For any k ∈ N, if the gain parameters γ1 , γ2 and T max satisfy: 1 γ2 < , γ T 3 max (7) γ2 1 0 ≤ γ1 + γ3 ∆ − γ < T , 3 max where T max = maxk∈N T k and ∆ = max |Ni (tk )|¯a : i = 1, 2, · · · , n, k ∈ N in which |Ni (tk )| is the number of neighbors of agent vi at time tk , then H(tk ) is a row stochastic matrix with the property that the block E(tk ) contains positive diagonal elements.
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1 1 Proof. Under the conditions γγ32 < Tmax and 0 ≤ γ1 + γ3 ∆ − γγ32 < Tmax , it can be obtained that E(k) is a nonnegative matrix with positive diagonal elements, which combines the fact that F[y] (tk ), y = 1, 2, · · · , h are nonnegative matrices ensures that matrix H(tk ) is nonnegative. Below we prove that H(tk ) is a row-stochastic matrix. According to the definitions of matrices A11[y] (tk ) ∈ Rm×m , A22[y] (tk ) ∈ R(n−m)×(n−m) , A12[y] (tk ) ∈ Rm×m , A21[y] (tk ) ∈ R(n−m)×(n−m) , y = 1, 2, · · · , h, it can be easily P P derived that all row sums of matrix −γ3 T k D1 (tk ) + hy=1 γ3 T k A11[y] (tk ) − hy=1 γ3 T k A12[y] (tk ) are equal P to 0. It thus follows that all row sums of matrix E(tk ) + hy=1 F[y] (tk ) are equal to 1, which together with the fact that the row sum of jth row matrix H(tk ) is equal to 1, where j = 2n + 1, 2n + 2, · · · , 2nh, guarantee that H(tk ) is a row-stochastic matrix.
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Lemma 5. Suppose that the inequalities in (7) hold, and all the discrete h time instants t0 , t1 , t2 , · · · , are divided into an infinite number of uniformly bounded intervals tk j , tk j+1 , j = 1, 2, · · · , starting at h k1 = 0. If the union of digraphs G tk j , G tk j +1 , · · · , G tk j+1 −1 in each time interval tk j , tk j+1 contains Qk j+1 −1 a spanning tree, then matrices i=k H(ti ), j = 1, 2, · · · , are SIA. j
P Pk j+1 −1 Proof. Firstly, it is necessary to prove that digraph G i=k E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) conj Pk j+1 −1 P tains a directed spanning tree. Matrix i=k E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) can be written in detail j as: kX j+1 −1 i=k j
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limtk →∞ θ(tk ) = 12hn c. Therefore, under condition (7), what needs to be shown below in this article is to prove the products convergence of an infinite number of time-varying row stochastic matrices H(tk ), k ∈ N, such as limtk →∞ H(tk )H(tk−1 ) · · · H(t0 ) = 12n yT , where y ∈ R2hn×1 . Before moving on, the following lemmas are presented.
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It is known from Lemma 4 that the block matrix E(tk ) in Q(tk ) contain positive diagonal elements under condition (7), then we have G[Q(tk ) + F(tk )] ∼ G[N(tk )]. Through a simple calculation, one can
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Remark 2. Note that the set Ξ = H(tk )| T k ∈ (0, T max ], ∀k ∈ N is not a compact set because ˜ T k ∈ (0, T max ], k ∈ N may be not compact. Let Ξ˜ = H(tk ) : T k ∈ [0, T max ], ∀k ∈ N , clearly, Ξ ⊂ Ξ. ˜ This is reasonable since there exists a To proceed further we need to exploit the compactness of Ξ. Q constraint that all H(tk ), k ∈ N must belong to a compact set if the result limk→∞ ki=0 H(ti ) = 12nh yT holds, where y is a column vector (Detailed information is displayed in [31]). In the following, we will prove this conclusion.
Lemma 6. Suppose that the conditions in (7) hold, then Ξ˜ is a compact set. Proof. Let F = F[y] (tk ) : y = 1, 2, · · · , h; tk ∈ [0, T max ] . Denote by F (tk ) the set consisting of all matrices F[1] (tk ), F[2] (tk ), · · · , F[h] (tk ) for each T k ∈ [0, T max ]. It is clear that F is a compact set if each F (tk ) is a compact set. Note that when T k = 0, F (tk ) is a compact set because only an element 02n×2n exists in F (tk ). In the following, we consider the case that T k ∈ (0, T max ]. According to the assumption that all the non-zero weighting factors in adjacency matrix A(tk ) = [ai j (tk )]n×n are uniformly lower and upper bounded, i.e. |ai j (tk )| ∈ [α, α] whenever ai j (tk ) , 0, where 0 < α < α, it is easy to obtain that all the possible choices of F[y] (tk ), y = 1, 2, · · · , h are finite for each T k ∈ (0, T max ], that is, F (tk ) is a compact for each T k ∈ (0, T max ]. As a result, F is a compact set. Let S = E(tk ) : T k ∈ [0, T max ] . It is clear that S is a compact set if E(tk ) is compact for each T k ∈ [0, T max ]. Note that when T k = 0, E(tk ) = I2n×2n is compact. When T k ∈ (0, T max ], the fact that |ai j (tk )| ∈ [α, α] for all i, j = 1, 2, · · · , n and k ∈ N implies that all the possible choices of D1 (tk ) and D2 (tk ) are infinite for each T k ∈ (0, T max ]. It follows that E(tk ) is compact for each T k ∈ (0, T max ], further S is a compact set. By the above analysis, it is clear that the following set n H˜ 1 = H(tk )(1, :), H(tk )(2, :), · · · , H(tk )(2n, :) H(tk )(i, :) is i − th o row of matrix H(tk ), i = 1, 2, · · · , 2n, T k ∈ [0, T max ] ,
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derive that there exists ˜ ˜ ˜ F1 F2 · · · Fh 0 0 · · · 0 F˜ = . , .. . . . .. . .. . 0 0 · · · 0 2nh×2nh Qk j+1 P Pk j+1 −1 P k −k such that i=k N(ti ) ≥ F˜ + N0 j+1 j , where hi=1 F˜ i = i=k E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) . Thus, j j P P Pk j+1 −1 G hi=1 F˜ i contains a spanning tree because G i=k E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) contains a j k −k spanning tree, then it follows from Lemma 1 that G F˜ + N0 j+1 j contains a spanning tree, and so is Qk j+1 Qk j+1 k −k G i=k N(ti ) because of i=k N(ti ) ≥ F˜ + N0 j+1 j . Furthermore, G[Q(tk ) + F(tk )] ∼ G[N(tk )] j j Qk j+1 −1 implies that G i=k H(ti ) contains a spanning tree. Assume that vi is the root vertex of the j union of digraphs G tk j , G tk j +1 , · · · , G tk j+1 −1 , it follows that i + n is the root vertex of digraph Qk j+1 −1 G i=k H(ti ) . By Lemma 2 and the fact that i + n has self-loop, it can be obtained that matrices j Qk j+1 −1 i=k j H(ti ), j = 1, 2, · · · , are SIA.
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It is clear that H(tk )(i, :) only includes a non-zero element 1 and other elements are 0 for each i ∈ {2n + 1, 2n + 2, · · · , 2nh}. It means that H(tk )(i, :) is compact for each i ∈ {2n + 1, 2n + 2, · · · , 2nh}. It follows that H˜ 2 is a compact set. By the above analysis, Ξ˜ is a compact set.
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Based on the above lemmas, we now present our main result for asynchronous bipartite consensus of system (1). Theorem 1. Divide all the discrete time into an infinite number of uniformly bounded intervals tk j , tk j+1 , j = 1, 2, · · · , starting at k0 = 0. Under the conditions in (7), the asynchronous bipartite consensus for system (1) with protocol (3) can be achieved if all digraphs G(tk ), k ∈ N are structural balanced and the union of digraphs G(tk j ), G(tk j +1 ), · · · , G(tk j+1 −1 ) associated with each time interval tk j , tk j+1 contains a directed spanning tree.
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In the following, we consider the set n H˜ 2 = H(tk )(2n + 1, :), H(tk )(2n + 2, :), · · · , H(tk )(2nh, :) H(tk )(i, :) is i − th o row of matrix H(tk ), i = 2n + 1, 2n + 2, · · · , 2nh, T k ∈ [0, T max ] .
Qk j+1 −1 Proof. By Lemma 6, we know that matrix G i=k H(ti ) is SIA if the union of digraphs G(tk j ), j G(tk j +1 ), · · · , G(tk j+1 −1 ) contains a directed spanning tree. And we also know that the set Ξ consisting of all matrices H(tk ) is compact. Then, it can be obtained from [31] that the product of infinite SIA matrices H(tk ), k ∈ N that from a compact set Ξ˜ can be written as j+1 −1 ∞ kY Y
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Below we present a numerical example to demonstrate the effectiveness of our result. Under protocol (4), we consider a system consisting six agents. At the same time, randomly choose T k from [0.05, 0.2]. All discrete time instants in {tki }, i = 1, 2, · · · , 6 are randomly chosen in the time sequence j k κˆ i i ≤ κ {tk }∞ under the constraint t − t ˆ = 0.6. Then, we have h = T min = 12. Assume that the 0 k+1 k interaction topology at each time tk can be randomly chosen in Fig. 2, and each interaction topology 11
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of Fig. 2 is chosen at least once in each time interval [t jh , t( j+1)h ), j ∈ N, where the absolute value of weight of each edge is 1. This implies that ∆ = 2 and the union of digraphs Ga , Gb and Gc in each time interval [t jh , t( j+1)h ) has a spanning tree. In addition, let the system evolve in the following asynchronous way: 1 , k ∈ N. 1) agent v1 can receive the state information of agent v6 at update time t4k+1 2 2) agent v2 can receive the state information of agent v1 at update time t4k+1 , k ∈ N, and receive the 2 , k ∈ N. state information of agent v5 at update time t4k+3 3 3) agent v3 can receive the state information of agent v2 at update time t4k+3 , k ∈ N. 4 , k ∈ N. 4) agent v4 can receive the state information of agent v3 at update time t4k+2 5 5) agent v5 can receive the state information of agent v2 at update time t4k+1 , k ∈ N, and receive the 5 state information of agent v4 at update time t4k+3 , k ∈ N. 6 6) agent v6 can receive the state information of agent v5 at update time t4k+2 , k ∈ N.
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Choose γ1 = 2, γ2 = 5 and γ3 = 1.5 which satisfy the inequalities in (7). The trajectories of position and velocity of all the agents are shown in Fig. 3 and Fig. 4. From the results of the simulation shown in Fig. 3 and Fig. 4, we can see that the asynchronous bipartite consensus of system (3) can be reached asymptotically. It verifies the correctness of our theoretical analysis.
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This paper has studied the bipartite consensus for a group of agents with second-order dynamics in an asynchronous situation, where the time-varying communication topology among the agents is a structurally balanced digraph. Based on the proposed asynchronous distributed control protocol, the asynchronous bipartite consensus issue has been equivalently transformed into a product issue of time-varying row-stochastic matrices from a non-compact set. Then, by using the product properties of row-stochastic matrices, a sufficient condition has been established to guarantee the achievement of asynchronous bipartite consensus for second-order MASs with time-varying topology. Finally, a simulation instance has also be provided to verify the effectiveness of our theoretical result.
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[24] F. Xiao, L. Wang, Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays, IEEE Transactions on Automatic Control 53 (8) (2008) 1804–1816. [25] J. Qin, C. Yu, S. Hirche, Stationary consensus of asynchronous discrete-time second-order multiagent systems under switching topology, IEEE Transactions on Industrial Informatics 8 (4) (2012) 986–994. [26] J. Zhan, X. Li, Asynchronous consensus of multiple double-integrator agents with arbitrary sampling intervals and communication delays, IEEE Transactions on Circuits & Systems I Regular Papers 62 (9) (2015) 2301–2311. [27] L. Ding, W. X. Zheng, Consensus tracking in heterogeneous nonlinear multi-agent networks with asynchronous sampled-data communication, Systems & Control Letters 96 (2016) 151–157. [28] L. Shi, J. Shao, W.X. Zheng, T.Z. Huang, Asynchronous containment control for discrete-time second-order multi-agent systems with time-varying delays, Journal of the Franklin Institute 354 (18) (2017) 8552–8569. [29] L. Shi, J. Shao, M. Cao, H. Xia, Asynchronous group consensus for discrete-time heterogeneous multi-agent systems under dynamically changing interaction topologies, Information Sciences 463–464 (2018) 282–293. [30] Y. Chen, J. Lv, X. Yu, Z. Lin, Consensus of discrete-time second-order multiagent systems based on infinite products of general stochastic matrices, SIAM Journal on Control & Optimization 51 (4) (2013) 3274–3301. [31] F. Xiao, L. Wang, Consensus protocols for discrete-time multi-agent systems with time, Automatica 44 (10) (2008) 2577–2582.
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Jinliang Shao received the BSc and PhD degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2003 and 16
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2009, respectively. During 2014, he was a visiting scholar in Australian National University, Australia. He is currently an associate professor in the School of Automation Engineering, UESTC. His research interests include multi-agent system, robust control, and matrix analysis with applications in control theory.
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Lei Shi received the B.S. degree in mathematics and applied mathematics from Shanxi Datong University in 2014. He is currently working toward the Ph.D. degree in the School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu. Her research interests include networked control systems and multiagent systems.
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Yangzhen Zhang is an undergraduate student at University of Michigan-Shanghai Jiao Tong University Joint Institute. Her major is Electrical and Computer Engineering, and her research interests include the consensus and tracking consensus of multi-agent systems.
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Yuhua Cheng received the Ph.D. degree in measurement techniques and automation devices from Sichuan University, Chengdu, China, in 2007. He is currently a Professor and the Dean in the School of Automation Engineering, University of Electronic 18
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On the asynchronous bipartite consensus for discrete-time second-order multi-agent systems with switching topologies Jinliang Shao, Lei Shi, Yangzhen Zhang, Yuhua Cheng PII: DOI: Reference:
S0925-2312(18)30891-9 https://doi.org/10.1016/j.neucom.2018.07.056 NEUCOM 19809
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
15 May 2018 14 July 2018 24 July 2018
Please cite this article as: Jinliang Shao, Lei Shi, Yangzhen Zhang, Yuhua Cheng, On the asynchronous bipartite consensus for discrete-time second-order multi-agent systems with switching topologies, Neurocomputing (2018), doi: https://doi.org/10.1016/j.neucom.2018.07.056
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights • The asynchronous bipartite consensus for a group of agents with second-order dynamics is examined in this paper. • The properties of the product of infinite time-varying row-stochastic matrices from a noncompact set are explored to analyze this problem.
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• A sufficient condition that depends on switching topologies is established.
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On the asynchronous bipartite consensus for discrete-time second-order multi-agent systems with switching topologies✩ Jinliang Shaoa , Lei Shia , Yangzhen Zhangb , Yuhua Chenga,∗ School of Automation Engineering, University of Electronic Science and Technology of China, Sichuan, 611731 b University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai, 200240
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Abstract
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The asynchronous bipartite consensus for a group of agents with second-order dynamics is examined in this paper, where the asynchrony means that the time instants when each agent receives the neighbors’ data information are completely independent of other agents’. The communication among the agents is described by a time-varying signed and structurally balanced digraph, which is equivalent to assuming that the agents can be divided into two groups without any common agents, in which the agents within the same group are cooperative and the agents between different groups are competitive. An asynchronous distributed control protocol is designed to implement the bipartite consensus. By using the product properties of row-stochastic matrices from a noncompact set, a sufficient condition can be established under a loose assumption that is the union of communication topologies related to any time intervals with given length has a spanning tree. Finally, a simulation instance is provided to verify the reachability of asynchronous bipartite consensus.
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Keywords: Bipartite consensus; Second-order multi-agent systems (MASs); Asynchronous situation.
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Over the past decade, the issue of consensus in multi-agent systems (MASs) has been a topic of intense concern for scientists because of its wide applications in military and civilian fields, such as unmanned air vehicle formation, flocking and rendezvous [1–4]. In the existing related literature [5–16], it is usually assumed that information exchange between the agents is beneficial to their cooperation, which is reflected in the communication topology is that each edge contains positive weights. Such an assumption can ensure the realization of multi-agent consensus, that is, all agents achieve the exact same state in the long run. Unlike the case of fully cooperative agents, in the context of many practical applications, the information exchange between agents is not necessarily completely beneficial to their cooperation. Conversely, agents may also exhibit non-cooperative or confrontational interactions with certain neighbors in situations such as markets or social networks. For example, in a social network described in literature [17], the interpersonal relationship between people may be friendly or hostile, which may lead
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1. Introduction
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✩ This research was supported in part by the National Science Foundation of China (61671109), the China Postdoctoral Science Foundation(2017M612944, 2018T110962), the Special Postdoctoral Foundation of Sichuan Province, and the Fundamental Research Funds for the Central Universities (ZYGX2018J073). ∗ Corresponding author Email addresses: [email protected] (Jinliang Shao), [email protected] (Lei Shi), [email protected] (Yangzhen Zhang), [email protected] (Yuhua Cheng)
Preprint submitted to Neurocomputing
August 3, 2018
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2. Notations and preliminary lemmas
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to the emergence of two completely opposite groups. In a multi-agent network consisting of two different group of agents, if the agents belonging to the same group are in a cooperative relationship and the agents belonging to different groups are in a competitive relationship, then in the long run, the two group of agents will reach the final state of the same size but opposite sign. Consequently, we say that such a consensus issue is called bipartite consensus, in which the communication topology is structurally balanced. Zhang et al. [18] studied the bipartite consensus issue of general linear MASs with a signed digraph based on both dynamic output feedback and state feedback control protocols, and Qin et al. [19] solved the bipartite consensus issue for general linear MASs with input saturation. By using the properties of the Laplacian matrix and rooted cycles of digraph, the bipartite consensus for a team of agents with first-order dynamics is investigated in [20]. A bipartite consensus problem for high-order MASs with unknown disturbances is considered in [21]. In addition, the authors in [22] analyzed the adaptive finite-time bipartite consensus for second-order MASs with unknown external disturbances, and [23] studied the bipartite consensus issue under arbitrary finite communication delays. In practical engineering applications, there may exist differences in the clocks for different agents, and these difference can affect system performance and cause system instability if the sensor is timedriven. Therefore, it is more practical to consider the consensus of MASs under asynchronous situation than synchronous situation in which all agents’ clocks are identical. Asynchrony means that each agent detects the neighbors’ state information at certain discrete time instants determined by its own clock that is independent of the other agents’. In recent years, some valuable results on the consensus of MASs under asynchronous situation have been reported in [24–29]. As we introduced previously, concerning the bipartite consensus of MASs, many valuable works have been emerged in [18–23]. However, it has to be mentioned that the results presented in above works are based on synchronous situation. This consideration shift our attention to the problem of bipartite consensus of discrete-time second-order MASs under asynchronous situation. In addition, the communication topology we consider in this paper is time-varying, this is different from the literatures [18–23] in which the communication topology is assumed to be fixed. In general, time-varying topology is more common than fixed topology in practical applications because of the limited communication range and the reliability of communication links between agents. For solving the bipartite consensus problem of discrete-time second-order MASs with switching topologies under asynchronous situation considered in this paper, we first utilize appropriate model transformation techniques and the parameter selection strategy to ensure that the asynchronous second-order system can be equivalently transformed into a higher-order augmented system with matrix-vector form, in which the coefficient matrices are time-varying row-stochastic matrices from a non-compact set. Then the products properties of an infinite number of time-varying row-stochastic matrices from a non-compact set can be used to solve the asynchronous bipartite consensus problem. The outline of this article is as follows. We first introduce some basic notations about graph in Section 2, and then present the asynchronous bipartite consensus model formulation in Section 3. In Section 4, a sufficiency criterion is established to ensure asynchronous bipartite consensus for secondorder MASs with time-varying topology. The effectiveness of the control protocol is verified by a simulation examples in Section 5. Finally, the conclusion of this article is shown in Section 6.
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Notations: Let Rn and Rn×m denote, respectively, the set of n dimensional real column vectors and the set of n × m-dimensional real matrices. N and Z+ are used to denote the set of natural numbers and 3
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the set of positive integers, respectively. 1n ∈ Rn is the vector of all 1’s, 0n×m ∈ Rn×m denotes a matrix of all 0’s and In ∈ Rn×n is an identity matrix. Matrix Q = [qi j ]n×n is nonnegative, denoted by Q ≥ 0, if qi j ≥ 0 for any i, j = 1, 2, · · · , n. A nonnegative matrix Q is row-stochastic if Q1n = 1n , further the row-stochastic matrix Q is stochastic indecomposable and aperiodic (SIA) if limk→∞ Qk = 1n zT , Q where z ∈ Rn . Let ki=1 Q(i) = Q(k)Q(k − 1) · · · Q(0) denote the left product of matrices. Given a nonnegative matrix Q ∈ Rn×n , let G[Q] denote a digraph with weighted adjacency matrix Q, a node set {1, 2, · · · , n} in which the elements are the indexes of Q’s rows, and a edge set E[Q]. For nonnegative matrices Q = [qi j ]n×n and R = [ri j ]n×n , the digraphs G[Q] and G[R] are said to be of the same type, denoted by G[Q] ∼ G[R], if qi j > 0 ⇔ ri j > 0 and qi j = 0 ⇔ ri j = 0. The notation sgn(c) represents the signum function, and it is defined as follows: 1, c > 0 0, c = 0 sgn(c) = − 1, c < 0.
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Consider a team of n agents which are labeled by v1 , v2 , · · · , vn . A digraph G = (V, E, A) is used to describe the information interchange between the agents, here each agent is regarded as a vertex of digraph G. An edge of G is represented by (vi , v j ), where vi and v j are called, respectively, parent vertex and child vertex. The adjacency matrix satisfies (vi , v j )∈ E ⇔ a ji , 0, besides, ai j > 0 if the connection from v j to vi is cooperative; ai j > 0 if the connection from v j to vi is antagonistic. Assume that aii = 0 for any i = 1, 2, · · · , n. The set of neighbors of vertex vi is denoted by Ni = {v j ∈ V : (v j , vi ) ∈ E}. A digraph has a spanning tree if there exists a vertex called root such that there exists a directed path from the root to any other vertices. Digraph G is structural balanced, if all vertices can be divided into two subsets V1 and V2 such that V1 ∪V2 = V, V1 ∩V2 = ∅. If agents vi , v j belong to the same subset, then ai j ≥ 0, and if ai j ≤ 0, then agents vi , v j belong to different subsets. Without loss of generality, we assume that V1 = {v1 , v2 , · · · , vm } and V2 = {vm+1 , vm+2 , · · · , vn }. Corresponding to the division of sets V1 and V2 , partition the adjacency matrix A associated with G as ! A11 A12 A= , A21 A22
AC
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PT
where A12 ∈ Rm×n and A21 ∈ Rn×m are non-positive matrices and show the relationship between the agents in different sets; A11 ∈ Rm×m and A22 ∈ Rn×n are nonnegative matrices and describe the internal communication of the agents in the same set. The in-degree matrix of the agents in V1 and the P P P in-degree matrix of the agents in V2 are defined as D1 = diag nj=1 |a1 j |, nj=1 |a2 j |, · · · , nj=1 |am j | Pn Pn Pn and D2 = diag j=1 |am+1, j |, j=1 |am+2, j |, · · · , j=1 |am+n, j | , respectively. In terms of these constructions, the overall graph Laplacian matrix can be written as ! D1 − A11 −A12 L= . −A21 D2 − A22
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In the remainder of this section, we list several lemmas that will be used to draw the main conclusion of this paper. Lemma 1. [24] Let D1 , D2 , . . . , Dh be nonnegative matrices, D1 D2 · · · Dh 0 0 · · · 0 D = . .. .. , .. .. . . . 0 0 ··· 0 4
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N0 =
I 0 ··· I 0 ··· 0 I ··· .. .. . . . . . 0 0 ···
0 0 0 0 0 0 . .. .. . . I 0
P Let N(k) = D + N0k for any k ∈ Z+ . The digraph G[Nk ] contains a spanning tree if digraph G hi=1 Di contains a spanning tree.
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Lemma 2. [24] Suppose that A ∈ Rn×n is a row stochastic matrix. Then A is SIA (stochastic indecomposable and aperiodic) if the digraph G[A] has a directed spanning tree with the property that the root node of the spanning tree has a self-loop in G[A]. 3. Model formulation
In the continuous-time setting, the agents are modeled by the following second-order dynamics
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x˙i (t) = ϑi (t), ϑ˙ i (t) = ui (t), 68
where xi (t) ∈ R p , ϑi (t) ∈ R p , ui (t) denote, respectively, the position, the velocity and the control input of agent vi . Consider the following consensus protocol: X X ui (t) = −γ1 ϑi (t) + γ2 |ai j | sgn(ai j )x j (t) − xi (t) + γ3 |ai j | sgn(ai j )ϑ j (t) − ϑi (t) , (2) v j ∈Ni
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where γ1 > 0, γ2 > 0 and γ3 > 0 are the fixed gain parameters. In a realistic multi-agent network, continuous data exchange between agents is difficult due to some practical limitations, including environmental interference and communication distance limitations. This fact guides the existing large number of discrete-time consensus models. By referring the discrete-time consensus model in [30], the discrete-time dynamics of agents in this paper are usually described as: xi (tk+1 ) = xi (tk ) + T k ϑi (tk ),
(3)
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where xi (tk ) ∈ R p , ϑi (tk ) ∈ R p , ui (tk ) denote, respectively, the position, the velocity and the control input of agent vi at time instant tk ; {tk }∞ 0 is the discrete time instant sequence which satisfies tk+1 − tk = T k > 0 and t0 = 0. Let structural balanced digraph G(tk ) = (V, E(tk ), A(tk )) describe the time-varying data exchange between the agents. An edge (vi , v j ) ∈ E(tk ) if and only if agent v j can detect the information of agent vi at time tk . The set of neighbors of vertex vi at time tk is denoted by Ni (tk ) = {v j ∈ V : (v j , vi ) ∈ E(tk )}. The adjacency matrix A(tk ) = [ai j (tk )]n×n satisfies ai j (tk ) , 0 ⇔ (v j , vi ) ∈ E(tk ), and all positive weighting factors have a common lower and upper bound, i.e. |ai j (tk )| ∈ [α, α] whenever ai j (tk ) , 0, where 0 < α < α. The Laplacian matrix L(tk ) associated with the communication digraph G(tk ) can be partitioned as ! D1 (tk ) − A11 (tk ) −A12 (tk ) L(tk ) = , −A21 (tk ) D2 (tk ) − A22 (tk )
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CE
ϑi (tk+1 ) = ϑi (tk ) + T k ui (tk ), 75
v j ∈Ni
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(1)
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Figure 1: Time sequences {tk1 } and {tk2 } in asynchronous setting.
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P P P P where D1 (tk ) = diag nj=1 |a1 j (tk )|, nj=1 |a2 j (tk )|, · · · , nj=1 |am j (tk )| and D2 (tk ) = diag nj=1 |am+1, j (tk )|, Pn Pn j=1 |am+2, j (tk )|, · · · , j=1 |an j (tk )| . The situation that all the agents receive the information of the neighbors synchronously is not easy to be achieved in practical application, and this consideration leads us to shift our attention to a more practical case that each agent only detects the neighbors’ state information at certain discrete time determined by its own clock that is independent of the other agents’. Here we assume that agent vi receives its neighbours’ state information at discrete time instants t0i , t1i , · · · , tki , · · · , which i i i i are randomly chosen in the set {tk }∞ k=0 and satisfy t0 = t0 < t1 < · · · < tk < · · · , and let {tk } = j i i i i {t0 , t1 , · · · , tk , · · · } for simplicity. It’s worth noting that {tk } is independent of {tk } for any vi , v j ∈ V, i , j in asynchronous setting, e.g., (See an example with two agents v1 and v2 in Fig. 1). In addition, to ensure the accessibility of the asynchronous group consensus, we let the time sequences {tki }, i = 1, 2, · · · , m + n, satisfy: for any k ∈ N, there exists a positive real number κˆ that meets the condition i 0 < tk+1 − tki ≤ κˆ , which implies 0 < T k ≤ κˆ . By referring protocol (2), we propose the asynchronous bipartite consensus protocol for system (1) as follows X a (t ) hsgn a (k) x (ti ) − x (t )i ui (tk ) = − γ1 ϑi (tk ) + γ2 ij k ij j ω i k
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v j ∈Ni (tk )
+ γ3
X a (t ) hsgn a (t )ϑ (ti ) − ϑ (t )i , t ∈ [ti , ti ). ij k ij k j ω i k k ω ω+1
(4)
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Definition 1. Using protocol (4), it is said that the asynchronous bipartite consensus for second-order multi-agent system (1) can be achieved if
lim
xi (tk ) − x j (tk )
= 0, ∀vi , v j ∈ Vq , q ∈ {1, 2}, tk →∞
lim
xi (tk ) + x j (tk )
= 0, ∀vi ∈ Vq1 , v j ∈ Vq2 , q1 , q2 ∈ {1, 2} and q1 , q2 ,
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95
i It can be seen that the neighbors’ information received by agent vi at time tk ∈ [tωi , tω+1 ) all comes i from time tω .
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PT
v j ∈Ni (tk )
tk →∞
lim kϑi (tk )k = 0, ∀vi ∈ V.
tk →∞
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Lemma 3. Consider system (3) with protocol (4). For any i ∈ {1, j 2, k· · · , n} and k ∈ N, the number of i κˆ , where T min = mink∈N {T k } and time instants in the set t s | t s ∈ tki , tk+1 is no more than h = Tmin j k κˆ κˆ T min is the largest integer less than T min . 6
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i Proof. If the single ordered sequence with discrete time in the set t s | t s ∈ tki , tk+1 is evenly spaced i is the and all the time steps are T min , then the number of time instants in the set t s | t s ∈ tki , tk+1 j k κˆ most and it is no more than h = T min .
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4. Main result
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In this section, the asynchronous multi-agent systems is transformed into an augmented system with nonnegative coefficient matrices through appropriate model transformation techniques, it means that the asynchronous bipartite consensus problem can be equivalently transformed into a deterministic problem that is the convergence of infinite products of time-varying nonnegative matrices. Then mixed tools from graph theory and SIA matrices, are employed to solve the convergence problem. Following the above guidance, we first present the following model transformations. Let h iT T X1 (tk ) = x1T (tk ), x2T (tk ), · · · , xm (tk ) , iT h V1 (tk ) = ϑT1 (tk ), ϑT2 (tk ), · · · , ϑTm (tk ) , iT h T T T X2 (tk ) = xm+1 (tk ), xm+2 (tk ), · · · , xm+n (tk ) , iT h V2 (tk ) = ϑTm+1 (tk ), ϑTm+2 (tk ), · · · , ϑTm+n (tk ) , " #T γ3 T γ3 T T T T T ξ(tk ) = X1 (tk ), −X2 (tk ), X1 (tk ) + V1 (tk ), −X2 (tk ) − V2 (tk ) . γ2 γ2
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Remark 1. From Lemma 3, it can be seen that the earliest information received by the agent at the time tk comes from tk−h , which is crucial in the process of transforming a heterogeneous multi-agent system into a time-varying augmented system with matrix-vector form under asynchronous situation.
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Using protocol (4), system (3) can be equivalently expressed as
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ξ(tk+1 ) = E(tk )ξ(tk ) +
h X
F[y] (tk )ξ(tk−y+1 ),
(5)
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PT
j=1
where
CE
γ T Im − 2γ3 k Im 0(n−m)×m E(tk ) = (γ1 γ3 −γ2 )Tk Im γ3 0(n−m)×m
γ2 T k γ3 Im
0m×(n−m)
In−m −
γ2 T k γ3 In−m
0m×(n−m) (γ1 γ3 −γ2 )T k In−m γ3
0(n−m)×m Im −
(γ1 γ3 −γ2 )T k Im −γ3 T k D1 (tk ) γ3
0(n−m)×m
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γ2 T k γ3 In−m
0m×(n−m) (γ γ −γ )T In−m − 1 3γ 2 k In−m −γ3 T k D2 (tk ) 3
0m×(n−m) 0m×m 0m×(n−m) 0m×m 0 0 0 0 (n−m)×m (n−m)×(n−m) F[y] (tk ) = (n−m)×m (n−m)×(n−m) 0m×(n−m) γ3 T k A11[y] (tk ) −γ3 T k A12[y] (tk ) 0m×m 0(n−m)×m 0(n−m)×(n−m) −γ3 T k A21[y] (tk ) γ3 T k A22[y] (tk )
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0m×(n−m)
,
, y = 1, 2, · · · , h,
where A11[y] (tk ) ∈ Rm×m , A22[y] (tk ) ∈ R(n−m)×(n−m) , y = 1, 2, · · · , h, are nonnegative matrices, and A12[y] (tk ) ∈ Rm×m , A21[y] (tk ) ∈ R(n−m)×(n−m) , y = 1, 2, · · · , h, are non-positive matrices. If tωi = tk−k0 and vi ∈ V1 , where k0 ∈ {0, 1, 2, · · · , h − 1}, then the ith row of matrices A11[k0 +1] (tk ) is equal to the ith row of matrices A11 (tk ) and the ith row of matrices A12[k0 +1] (tk ) is equal to the ith row of matrices A12 (tk ), while the ith rows of all the other matrices A11[y] (tk ), A12[y] (tk ), y , k0 + 1 are equal to zeros. 7
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where H(tk ) = F(tk ) + Q(tk ), in which E(tk ) − diag{E(tk )} + F[1] (tk ) F[2] (tk ) 0 0 0 0 F(tk ) = .. .. . . 0 0 diag{E(tk )} 0 . . . 0 0 I2n 0 · · · 0 0 0 I2n · · · 0 0 . Q(tk ) = .. .. . . .. .. . . . . . 0 0 · · · I2n 0
... ··· ··· .. .
F[h−1] (tk ) F[h] (tk ) 0 0 0 0 .. .. . . 0 0
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If tωi = tk−k00 and vi ∈ V2 , where k00 ∈ {0, 1, 2, · · · , h−1}, then the (i−m)th row of matrices A21[k00 +1] (tk ) is equal to the (i − m)th row of matrices A21 (tk ) and the (i − m)th row of matrices A22[k00 +1] (tk ) is equal to the (i − m)th row of matrices A22 (tk ), while the (i − m)th rows of all the other matrices A21[y] (tk ), A22[y] (tk ), y , k00 + 1 are equal to zeros. As we have introduced in Remark 1, the earliest neighbors’ state information that one agent can receive at time tk comes from time tk−h due to the existence of asynchronous situation, where h is defined in Lemma 3. Therefore, let θ(tk ) = ξT (tk ), ξT (tk−1 ), · · · , ξT (tk−h+1 ) T , then we further rewrite system (5) into the following time-varying augmented system: θ (tk+1 ) = H(tk ) ⊗ I p θ (tk ) , (6)
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Lemma 4. For any k ∈ N, if the gain parameters γ1 , γ2 and T max satisfy: 1 γ2 < , γ T 3 max (7) γ2 1 0 ≤ γ1 + γ3 ∆ − γ < T , 3 max where T max = maxk∈N T k and ∆ = max |Ni (tk )|¯a : i = 1, 2, · · · , n, k ∈ N in which |Ni (tk )| is the number of neighbors of agent vi at time tk , then H(tk ) is a row stochastic matrix with the property that the block E(tk ) contains positive diagonal elements.
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1 1 Proof. Under the conditions γγ32 < Tmax and 0 ≤ γ1 + γ3 ∆ − γγ32 < Tmax , it can be obtained that E(k) is a nonnegative matrix with positive diagonal elements, which combines the fact that F[y] (tk ), y = 1, 2, · · · , h are nonnegative matrices ensures that matrix H(tk ) is nonnegative. Below we prove that H(tk ) is a row-stochastic matrix. According to the definitions of matrices A11[y] (tk ) ∈ Rm×m , A22[y] (tk ) ∈ R(n−m)×(n−m) , A12[y] (tk ) ∈ Rm×m , A21[y] (tk ) ∈ R(n−m)×(n−m) , y = 1, 2, · · · , h, it can be easily P P derived that all row sums of matrix −γ3 T k D1 (tk ) + hy=1 γ3 T k A11[y] (tk ) − hy=1 γ3 T k A12[y] (tk ) are equal P to 0. It thus follows that all row sums of matrix E(tk ) + hy=1 F[y] (tk ) are equal to 1, which together with the fact that the row sum of jth row matrix H(tk ) is equal to 1, where j = 2n + 1, 2n + 2, · · · , 2nh, guarantee that H(tk ) is a row-stochastic matrix.
CE
129
PT
ED
128
···
,
136
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From the structure of state vector ξ(tk ), it is clear that the realization of asynchronous bipartite consensus is equivalent to limtk →∞ ξ(tk ) = 12n c, where c is a constant, further equivalent to 8
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Lemma 5. Suppose that the inequalities in (7) hold, and all the discrete h time instants t0 , t1 , t2 , · · · , are divided into an infinite number of uniformly bounded intervals tk j , tk j+1 , j = 1, 2, · · · , starting at h k1 = 0. If the union of digraphs G tk j , G tk j +1 , · · · , G tk j+1 −1 in each time interval tk j , tk j+1 contains Qk j+1 −1 a spanning tree, then matrices i=k H(ti ), j = 1, 2, · · · , are SIA. j
P Pk j+1 −1 Proof. Firstly, it is necessary to prove that digraph G i=k E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) conj Pk j+1 −1 P tains a directed spanning tree. Matrix i=k E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) can be written in detail j as: kX j+1 −1 i=k j
E(ti ) − diag{E(ti )} +
γ T Im − 2γ3 k Im 0(n−m)×m = (γ1 γ3 −γ2 )Tk Im γ3 0(n−m)×m
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In−m −
γ2 T k γ3 In−m
0m×(n−m) (γ1 γ3 −γ2 )T k In−m γ3
γ2 T k γ3 Im
0(n−m)×m
(γ γ −γ )T Im − 1 3γ 2 k Im −γ3 T k D1 (tk )+γ3 T k A11[y] (tk ) 3
−γ3 T k A21[y] (tk )
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165
0m×(n−m) γ2 T k γ3 In−m
−γ3 T k A12[y] (tk )
(γ γ −γ )T In−m − 1 3γ 2 k In−m −γ3 T k D2 (tk )+γ3 T k A22[y] (tk ) 3
Without loss of generality, let vu1 be the root node of directed spanning tree in the union of digraphs G tk j , G tk j +1 , · · · , G tk j+1 −1 , then there exist a directed path vu1 → vu2 → · · · → vuz for any vuz ∈ V, P Pk j+1 −1 E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) , where u1 , u2 , u3 · · · , uz ∈ {1, 2, · · · , n}. By the structure of i=k j Pk j+1 −1 E(ti ) − it is obvious that there exists a directed path u1 + n → u2 + n → · · · → uz + n in G i=k j Ph diag{E(ti )} + y=1 Fy (ti ) for any uz + n ∈ {n + 1, n + 2, · · · , 2n}. According to the fact that (uz + n, uz ) ∈ P Pk j+1 −1 E i=k E(ti )−diag{E(ti )}+ hy=1 Fy (ti ) , uz = 1, 2, · · · , n, one can obtain that there exists a directed j P Pk j+1 −1 path u1 + n → u2 + n → · · · → uz + n → uz in G i=k E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) for any j P Pk j+1 −1 uz ∈ {1, 2, · · · , n}. This implies that digraph G i=k E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) contains a j directed spanning tree. Let N(tk ) = N(0) + F(tk ), where I2n 0 · · · 0 0 I2n 0 · · · 0 0 N0 = 0 I2n · · · 0 0 . .. .. . . . . . .. .. . . 0 0 · · · I2n 0
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0m×(n−m)
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y=1
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Fy (ti )
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h X
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limtk →∞ θ(tk ) = 12hn c. Therefore, under condition (7), what needs to be shown below in this article is to prove the products convergence of an infinite number of time-varying row stochastic matrices H(tk ), k ∈ N, such as limtk →∞ H(tk )H(tk−1 ) · · · H(t0 ) = 12n yT , where y ∈ R2hn×1 . Before moving on, the following lemmas are presented.
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It is known from Lemma 4 that the block matrix E(tk ) in Q(tk ) contain positive diagonal elements under condition (7), then we have G[Q(tk ) + F(tk )] ∼ G[N(tk )]. Through a simple calculation, one can
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Remark 2. Note that the set Ξ = H(tk )| T k ∈ (0, T max ], ∀k ∈ N is not a compact set because ˜ T k ∈ (0, T max ], k ∈ N may be not compact. Let Ξ˜ = H(tk ) : T k ∈ [0, T max ], ∀k ∈ N , clearly, Ξ ⊂ Ξ. ˜ This is reasonable since there exists a To proceed further we need to exploit the compactness of Ξ. Q constraint that all H(tk ), k ∈ N must belong to a compact set if the result limk→∞ ki=0 H(ti ) = 12nh yT holds, where y is a column vector (Detailed information is displayed in [31]). In the following, we will prove this conclusion.
Lemma 6. Suppose that the conditions in (7) hold, then Ξ˜ is a compact set. Proof. Let F = F[y] (tk ) : y = 1, 2, · · · , h; tk ∈ [0, T max ] . Denote by F (tk ) the set consisting of all matrices F[1] (tk ), F[2] (tk ), · · · , F[h] (tk ) for each T k ∈ [0, T max ]. It is clear that F is a compact set if each F (tk ) is a compact set. Note that when T k = 0, F (tk ) is a compact set because only an element 02n×2n exists in F (tk ). In the following, we consider the case that T k ∈ (0, T max ]. According to the assumption that all the non-zero weighting factors in adjacency matrix A(tk ) = [ai j (tk )]n×n are uniformly lower and upper bounded, i.e. |ai j (tk )| ∈ [α, α] whenever ai j (tk ) , 0, where 0 < α < α, it is easy to obtain that all the possible choices of F[y] (tk ), y = 1, 2, · · · , h are finite for each T k ∈ (0, T max ], that is, F (tk ) is a compact for each T k ∈ (0, T max ]. As a result, F is a compact set. Let S = E(tk ) : T k ∈ [0, T max ] . It is clear that S is a compact set if E(tk ) is compact for each T k ∈ [0, T max ]. Note that when T k = 0, E(tk ) = I2n×2n is compact. When T k ∈ (0, T max ], the fact that |ai j (tk )| ∈ [α, α] for all i, j = 1, 2, · · · , n and k ∈ N implies that all the possible choices of D1 (tk ) and D2 (tk ) are infinite for each T k ∈ (0, T max ]. It follows that E(tk ) is compact for each T k ∈ (0, T max ], further S is a compact set. By the above analysis, it is clear that the following set n H˜ 1 = H(tk )(1, :), H(tk )(2, :), · · · , H(tk )(2n, :) H(tk )(i, :) is i − th o row of matrix H(tk ), i = 1, 2, · · · , 2n, T k ∈ [0, T max ] ,
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derive that there exists ˜ ˜ ˜ F1 F2 · · · Fh 0 0 · · · 0 F˜ = . , .. . . . .. . .. . 0 0 · · · 0 2nh×2nh Qk j+1 P Pk j+1 −1 P k −k such that i=k N(ti ) ≥ F˜ + N0 j+1 j , where hi=1 F˜ i = i=k E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) . Thus, j j P P Pk j+1 −1 G hi=1 F˜ i contains a spanning tree because G i=k E(ti ) − diag{E(ti )} + hy=1 Fy (ti ) contains a j k −k spanning tree, then it follows from Lemma 1 that G F˜ + N0 j+1 j contains a spanning tree, and so is Qk j+1 Qk j+1 k −k G i=k N(ti ) because of i=k N(ti ) ≥ F˜ + N0 j+1 j . Furthermore, G[Q(tk ) + F(tk )] ∼ G[N(tk )] j j Qk j+1 −1 implies that G i=k H(ti ) contains a spanning tree. Assume that vi is the root vertex of the j union of digraphs G tk j , G tk j +1 , · · · , G tk j+1 −1 , it follows that i + n is the root vertex of digraph Qk j+1 −1 G i=k H(ti ) . By Lemma 2 and the fact that i + n has self-loop, it can be obtained that matrices j Qk j+1 −1 i=k j H(ti ), j = 1, 2, · · · , are SIA.
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It is clear that H(tk )(i, :) only includes a non-zero element 1 and other elements are 0 for each i ∈ {2n + 1, 2n + 2, · · · , 2nh}. It means that H(tk )(i, :) is compact for each i ∈ {2n + 1, 2n + 2, · · · , 2nh}. It follows that H˜ 2 is a compact set. By the above analysis, Ξ˜ is a compact set.
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Based on the above lemmas, we now present our main result for asynchronous bipartite consensus of system (1). Theorem 1. Divide all the discrete time into an infinite number of uniformly bounded intervals tk j , tk j+1 , j = 1, 2, · · · , starting at k0 = 0. Under the conditions in (7), the asynchronous bipartite consensus for system (1) with protocol (3) can be achieved if all digraphs G(tk ), k ∈ N are structural balanced and the union of digraphs G(tk j ), G(tk j +1 ), · · · , G(tk j+1 −1 ) associated with each time interval tk j , tk j+1 contains a directed spanning tree.
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In the following, we consider the set n H˜ 2 = H(tk )(2n + 1, :), H(tk )(2n + 2, :), · · · , H(tk )(2nh, :) H(tk )(i, :) is i − th o row of matrix H(tk ), i = 2n + 1, 2n + 2, · · · , 2nh, T k ∈ [0, T max ] .
Qk j+1 −1 Proof. By Lemma 6, we know that matrix G i=k H(ti ) is SIA if the union of digraphs G(tk j ), j G(tk j +1 ), · · · , G(tk j+1 −1 ) contains a directed spanning tree. And we also know that the set Ξ consisting of all matrices H(tk ) is compact. Then, it can be obtained from [31] that the product of infinite SIA matrices H(tk ), k ∈ N that from a compact set Ξ˜ can be written as j+1 −1 ∞ kY Y
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5. Simulation results
Below we present a numerical example to demonstrate the effectiveness of our result. Under protocol (4), we consider a system consisting six agents. At the same time, randomly choose T k from [0.05, 0.2]. All discrete time instants in {tki }, i = 1, 2, · · · , 6 are randomly chosen in the time sequence j k κˆ i i ≤ κ {tk }∞ under the constraint t − t ˆ = 0.6. Then, we have h = T min = 12. Assume that the 0 k+1 k interaction topology at each time tk can be randomly chosen in Fig. 2, and each interaction topology 11
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of Fig. 2 is chosen at least once in each time interval [t jh , t( j+1)h ), j ∈ N, where the absolute value of weight of each edge is 1. This implies that ∆ = 2 and the union of digraphs Ga , Gb and Gc in each time interval [t jh , t( j+1)h ) has a spanning tree. In addition, let the system evolve in the following asynchronous way: 1 , k ∈ N. 1) agent v1 can receive the state information of agent v6 at update time t4k+1 2 2) agent v2 can receive the state information of agent v1 at update time t4k+1 , k ∈ N, and receive the 2 , k ∈ N. state information of agent v5 at update time t4k+3 3 3) agent v3 can receive the state information of agent v2 at update time t4k+3 , k ∈ N. 4 , k ∈ N. 4) agent v4 can receive the state information of agent v3 at update time t4k+2 5 5) agent v5 can receive the state information of agent v2 at update time t4k+1 , k ∈ N, and receive the 5 state information of agent v4 at update time t4k+3 , k ∈ N. 6 6) agent v6 can receive the state information of agent v5 at update time t4k+2 , k ∈ N.
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Choose γ1 = 2, γ2 = 5 and γ3 = 1.5 which satisfy the inequalities in (7). The trajectories of position and velocity of all the agents are shown in Fig. 3 and Fig. 4. From the results of the simulation shown in Fig. 3 and Fig. 4, we can see that the asynchronous bipartite consensus of system (3) can be reached asymptotically. It verifies the correctness of our theoretical analysis.
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This paper has studied the bipartite consensus for a group of agents with second-order dynamics in an asynchronous situation, where the time-varying communication topology among the agents is a structurally balanced digraph. Based on the proposed asynchronous distributed control protocol, the asynchronous bipartite consensus issue has been equivalently transformed into a product issue of time-varying row-stochastic matrices from a non-compact set. Then, by using the product properties of row-stochastic matrices, a sufficient condition has been established to guarantee the achievement of asynchronous bipartite consensus for second-order MASs with time-varying topology. Finally, a simulation instance has also be provided to verify the effectiveness of our theoretical result.
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Jinliang Shao received the BSc and PhD degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 2003 and 16
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2009, respectively. During 2014, he was a visiting scholar in Australian National University, Australia. He is currently an associate professor in the School of Automation Engineering, UESTC. His research interests include multi-agent system, robust control, and matrix analysis with applications in control theory.
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Lei Shi received the B.S. degree in mathematics and applied mathematics from Shanxi Datong University in 2014. He is currently working toward the Ph.D. degree in the School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu. Her research interests include networked control systems and multiagent systems.
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Yangzhen Zhang is an undergraduate student at University of Michigan-Shanghai Jiao Tong University Joint Institute. Her major is Electrical and Computer Engineering, and her research interests include the consensus and tracking consensus of multi-agent systems.
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Yuhua Cheng received the Ph.D. degree in measurement techniques and automation devices from Sichuan University, Chengdu, China, in 2007. He is currently a Professor and the Dean in the School of Automation Engineering, University of Electronic 18
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Science and Technology, Chengdu. His current research interests include nondestructive evaluation, structural health monitoring, complex system fault diagnosis, DPHM, and the application of power electronics.
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