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Consensus networks with switching topology and time-delays over finite fields
Automatica 68 (2016) 39–43
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Technical communique
Consensus networks with switching topology and time-delays over finite fields✩ Xiuxian Li a , Michael Z.Q. Chen a , Housheng Su b,1 , Chanying Li c a
Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong
b
School of Automation, Image Processing and Intelligent Control Key Laboratory of Education Ministry of China, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan 430074, China c
Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
article
info
Article history: Received 30 September 2015 Accepted 21 December 2015
Keywords: Consensus Networks Switching topology Time-delays Finite fields
abstract The consensus problem in networks with both switching topology and time-delays over finite fields is investigated in this paper. The finite field, which is a kind of finite alphabet, is considered due to the fact that networks often possess limited computation, memory, and capabilities of communication. First, by graph-theoretic method, one necessary and sufficient condition is derived for finite-field consensus of switching networks without time-delays. Subsequently, another necessary and sufficient condition on finite-field consensus without time-delays is provided based on FFC property of matrices associated with switching networks. Moreover, by means of the results on delay-free networks, some necessary and sufficient conditions for finite-field consensus of networks with both switching topology and time-delays are obtained. Additionally, it can be shown that switching networks with time-delays present in each self-transmission cannot achieve consensus. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction A multi-agent system or network consists of a group of agents or nodes that communicate with each other locally, aiming to achieve some goals by designing control strategies, which has attracted an increasing interest from numerous researchers over the past decade. Due to a large volume of potential applications in many areas, multi-agent systems have a number of distinct research directions including consensus (Hu, Lam, & Liang, 2013; Olfati-Saber & Murray, 2004; Su, Chen, Lam, & Lin, 2013; Su, Chen, Wang, & Lam, 2014; Su, Chen, Wang, & Lin, 2011), flocking (Su, Wang, & Lin, 2009a; Tanner, Jadbabaie, & Pappas, 2007), formation control (Meng, Jia, Du, & Zhang, 2014), etc. As an important distributed feature of networks, consensus aims at for all agents reaching an agreement of interest decided by themselves, which has been substantially investigated due to a large number of
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hyeong Soo Chang under the direction of Editor André L. Tits. E-mail addresses: [email protected] (X. Li), [email protected] (M.Z.Q. Chen), [email protected] (H. Su), [email protected] (C. Li). 1 Tel.: +86 27 87540210; fax: +86 27 87540210.
http://dx.doi.org/10.1016/j.automatica.2016.01.033 0005-1098/© 2016 Elsevier Ltd. All rights reserved.
applications in a range of domains, such as spacecrafts and robotics (Ren, Beard, & Atkins, 2007). Up to now, consensus has been deeply studied for various network models, including timeindependent networks, networks with broad and synchronous communication, networks with linear and nonlinear dynamics, networks with infinite communication bandwidth, networks with time-varying topologies, networks with gossip and asynchronous communication, and networks with link failures and so on (Chen, Wang, & Li, 2012; Fagiolini & Bicchi, 2013; Hadjicostis & Charalambous, 2014; Li, Liu, Wang, & Yin, 2014; Lin & Jia, 2009; Moreau, 2005; Ni & Cheng, 2010; Nuño, Ortega, Basanez, & Hill, 2011; Proskurnikov, 2013; Song, Cao, & Liu, 2010; Su, Wang, & Lin, 2009b; Tahbaz-Salehi & Jadbabaie, 2008; Wang & Xiao, 2007; You, Li, & Xie, 2013; Zhang & Tian, 2009; Zhao, Hill, & Liu, 2012; Zhou & Lin, 2014). Recently, instead of real numbers, finite fields have been taken into consideration for the consensus problem in networks for the sake of safety and memory constraints, etc. (Pasqualetti, Borra, & Bullo, 2014; Sundaram & Hadjicostis, 2013; Xu & Hong, 2014). For instance, some necessary and sufficient conditions on finitefield consensus of fixed networks with discrete-time iteration were developed completely in Pasqualetti et al. (2014), in which the authors also provided its applications to average consensus and pose estimation in sensor networks. However, in the context
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X. Li et al. / Automatica 68 (2016) 39–43
of finite fields, there are few research reports focusing on the consensus problem for networks with switching topology and time-delays, which inspire this paper. This paper addresses the consensus problem for networks with switching topology and time-delays over finite fields. The finite field, as a kind of finite alphabet, is taken into account since networks often undertake limited computation, memory, and capabilities of communication. Generally speaking, this paper extends the work in Pasqualetti et al. (2014) to the case with switching topology and time-delays. Provided that switching topology and delays complicate the structure of a network which further leads to the complexity of the union of transition graphs of adjacency matrices, more careful observations on the structure of the union of transition graphs are needed to derive one necessary and sufficient condition for consensus with switching topology by graph-theoretic method. In addition, intrinsically distinct from Pasqualetti et al. (2014), FFC property of finite product of adjacency matrices is raised here to obtain another necessary and sufficient condition for consensus with switching topology. Furthermore, regarding networks with both switching topology and time-delays, several necessary and sufficient conditions are presented to guarantee finite-field consensus using the results on delay-free networks. Moreover, it is shown that switching networks with time-delays present in each self-transmission cannot reach consensus. 2. Preliminaries A finite field F is a finite set of elements with addition and multiplication operations satisfying the following axioms (Lidl, 1997; Pasqualetti et al., 2014; Sundaram & Hadjicostis, 2013; Xu & Hong, 2014):
• Closure under addition and multiplication. a + b ∈ F, a · b ∈ F, ∀a, b ∈ F; • Associativity of addition and multiplication. a + (b + c ) = (a + b) + c , a · (b · c ) = (a · b) · c , ∀a, b, c ∈ F; • Commutativity of addition and multiplication. a + b = b + a, a · b = b · a, ∀a, b ∈ F; • Existence of additive and multiplicative identity elements. ∀a ∈ F, ∃0, 1, such that a + 0 = a, a · 1 = a; • Existence of additive and multiplicative inverse elements. ∀a ∈ F, ∃b, c ∈ F, such that a + b = 0, a · c = 1 (a ̸= 0); • Distributivity of multiplication over addition. a · (b + c ) = (a · b) + (a · c ), ∀a, b, c ∈ F. Denote by GN = (VN , EN ) a graph with N nodes, a set of vertices VN = {v1 , v2 , . . . , vN } and a set of edges EN ⊆ VN × VN . vi → vj (or (vi , vj ) ∈ EN ) denotes an edge in which node i sends information to node j. A graph is called undirected if (vi , vj ) ∈ EN implies (vj , vi ) ∈ EN , and directed otherwise. For a node i, the in-degree and out-degree of vi ∈ VN equal the numbers of the in-neighbor set Ni+ = {vj ∈ VN : (vj , vi ) ∈ EN } and the outneighbor set Ni− = {vj ∈ VN : (vi , vj ) ∈ EN }, respectively. The adjacency matrix AN = (aij ) ∈ FNp ×N is defined as: aij > 0 if
vj ∈ Ni+ , aij = 0 (i ̸= j) otherwise, and self-loops are allowed.
A directed path in a directed graph is a sequence of edges of the form (i1 , i2 ), (i2 , i3 ), . . .. A cycle is a path that shares the same first and last vertex. A directed tree is a directed graph, where every node has exactly one parent except one node called root. A root (resp. globally reachable node) is a node which has a directed path to (resp. from) every node in the graph, including itself. The length of a path (resp. cycle) equals the number of edges in the path (resp. cycle). A directed graph is called strongly (resp. weakly) connected if there is a directed (resp. undirected) path between any two nodes. Two subgraphs of the same graph are called disjoint if they have no common nodes.
3. Problem statement Consider a switching network with N nodes over a finite field. For simplicity, throughout this paper the prime field Fp := {0, 1, 2, . . . , p − 1} is considered, where p is a prime number. However, the results here can be easily extended to general finite fields. The network has the dynamics xi (t + 1) =
s(t )
aij xj (t − τij ),
i = 1, 2, . . . , N
(1)
+ j∈Ni (t )∪{i}
where xi (t ) ∈ Fp is the state of this network, and τij ’s are timedelays experienced by information transmission on the link from node j to node i satisfying 0 ≤ τij ≤ τ for some constant τ > 0, that is, τij ’s are bounded, i, j = 1, 2, . . . , N. Meanwhile, s(t )
the adjacency matrix As(t ) = (aij ) ∈ FNp ×N is row-stochastic (see Definition 2) and s(t ) : {0, 1, 2, . . .} → S is the switching signal with finite index set S := {1, 2, . . . , ν}. Therefore, for any k ∈ S , when s(t ) = k, the subnetwork (1) with As(t ) = Ak is activated. As a delayed network, it is assumed that subnetwork s(t ) takes the initial states x(t − 1), x(t − 2), . . . , x(t − τ ) when subnetwork s(t − 1) is switched to subnetwork s(t ) at time t, where x(t ) := (x1 (t ), x2 (t ), . . . , xN (t ))T ∈ FNp . Note that the addition and multiplication operations in the network (1) are performed by modulo, that is, taking the remainder after divided by p. For some k ∈ S , the transition graph of corresponding fixed network x(t + 1) = Ak x(t ) is defined as G∗Ak = (VA∗k , EA∗k ) with vertex set VA∗k = {v : v ∈ FNp } and edge set EA∗k = {(vi , vj ) :
vj = Ak vi , vi , vj ∈ FNp }. This transition graph comprises of disjoint
weakly-connected subgraphs, and only one cycle, maybe of unit length, is contained in each subgraph which embraces a globally reachable node (Hernández Toledo, 2005; Pasqualetti et al., 2014). As for switching network (1), denote by G∗A = (VA∗ , EA∗ ) the union graph of G∗Ak for all k ∈ S , that is, VA∗ = VA∗k = {v : v ∈ FNp } and ∗ edge set EA∗ = k∈S EAk . To proceed, the notion of consensus of networks over finite fields is defined as follows. Definition 1. The network (1) over Fp can achieve (finite-time) consensus if for any initial states in FNp and any switching signal s(t ), there exist a finite time T ∈ N and some constant η ∈ Fp such that x(T + k) = x(T ) = η1 for all k ∈ N. Note that consensus of networks over finite fields can be always achieved in finite time since there are only finite states in networks over finite fields. For brevity, two concepts are introduced as follows. Definition 2. Over the finite field Fp , (1) a matrix M ∈ Fnp×n is called row-stochastic if each row sums to 1; (2) a row-stochastic matrix M ∈ Fnp×n is called finite-field consensusable (FFC, for short) if M has a simple eigenvalue 1 and all other eigenvalues 0, that is, its characteristic polynomial is PM (λ) = λn−1 (λ − 1). This definition is reasonable due to the following result that states two necessary and sufficient conditions for finite-field consensus, which are conducive to consensus analysis later. Theorem 1 (Pasqualetti et al., 2014). For a fixed network x(t + 1) = Mx(t ) over Fp , the following statements are equivalent: (1) this network can achieve consensus; (2) the transition graph of M contains exactly p cycles and all of them are unit cycles around the vertices η1 for η ∈ Fp ; (3) the matrix M is FFC.
X. Li et al. / Automatica 68 (2016) 39–43
4. Finite-field consensus without time-delays In this section, the network (1) is addressed without timedelays, i.e., τij = 0, i, j = 1, 2, . . . , N. Hence, the network (1) can be rewritten as x(t + 1) = As(t ) x(t ),
(2)
where x(t ) = (x1 (t ), x2 (t ), . . . , xN (t )) ∈ is the state of this network and the other notations are defined in (1). It is known that finite-field consensus requires more stringent conditions than that for consensus over real numbers (Pasqualetti et al., 2014). More precisely, two necessary and sufficient conditions based on the transition graph and the characteristic polynomial are presented for fixed networks in Theorem 1. Consequently, a natural question is if it is sufficient to ensure finite-time consensus for the network (2) when row-stochastic matrices A1 , A2 , . . . , Aν are all FFC. The answer is negative which can be seen from the following example. T
FNp
Example 1. Consider the network (2) over the finite field F3 =
{0, 1, 2}, let ν = 2 and A1 =
1 1 0
0 0 1
0 0 , 0
A2 =
1 0 1
0 0 0
0 1 . 0
It is easy to verify that both A1 and A2 are FFC, which, by Theorem 1, follows that both fixed network x(t + 1) = A1 x(t ) and x(t + 1) = A2 x(t ) can reach consensus. However, when subnetworks x(t + 1) = A2 x(t ) and x(t + 1) = A1 x(t ) are activated alternately every unit time with initial state x(0) = (0, 0, 1)T , i.e., x(1) = A2 x(0), x(2) = A1 x(1), x(3) = A2 x(2), x(4) = A1 x(3), . . . , it is straightforward to see that x(t ) oscillates between (0, 0, 1)T and (0, 1, 0)T . Hence, the switching network cannot achieve consensus. Therefore, to guarantee consensus it is indispensable to impose other conditions on the network (2) besides FFC property of all Ai ’s, i ∈ S . In the sequel, a necessary and sufficient condition is provided for consensus of the network (2) based on the characterization of the transition graph. Theorem 2. For the switching network (2) over Fp , it can achieve consensus if and only if the graph G∗A contains exactly p cycles and all of them are unit cycles around the vertices η1 for η ∈ Fp . Proof. The argument for sufficiency is a variation of the proof of Theorem 4.1 in Pasqualetti et al. (2014), which is omitted here. In what follows, the necessity is proved in detail. Since the network (2) can achieve consensus, fixed network x(t + 1) = Ak x(t ) can achieve consensus as well for any k ∈ S because As(t ) may be equal to Ak for all time t ≥ 0. As a result, by Theorem 1, one concludes that the transition graph G∗Ak has exactly p cycles of unit length around the vertices η1 for η ∈ Fp , and then combining the arbitrariness of k ∈ S with the definition of G∗A yields that G∗A contains at least p cycles of unit length. In the following, it only needs to show that G∗A does not contain other cycles. By contradiction argument, if there is another cycle C0 in G∗A and note that for any η ∈ Fp , vertex η1 has only one edge pointed out to itself, then η1 is not contained in cycle C0 . Suppose that C0 consists of m vertices v1 , v2 , . . . , vm with v1 → v2 → · · · → vm → v1 , that is, there are some matrices Ak1 , Ak2 , . . . , Akm satisfying v2 = Ak1 v1 , v3 = Ak2 v2 , . . . , vkm = Akm−1 vm−1 , v1 = Akm vm . Notice that Aki ’s, i = 1, 2, . . . , m, are not necessary to be distinct. Consequently, consider a network x(t + 1) = As(t ) x(t ) with initial value x(0) = v1 and As(t ) equaling Ak1 , Ak2 , . . . , Akm periodically in ascending order of unit time, that is, As(0) = Ak1 , As(1) = Ak2 , . . . , As(m−1) = Akm , As(m) = Ak1 , . . .. It is straightforward to see that this network cannot achieve consensus which is a contradiction. This ends the proof.
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The result of Theorem 2 can be carefully observed from the structure of the graph G∗A , since the state can oscillate periodically along a circle of length l ≥ 2 if the graph contains other cycles besides these p unit cycles, which can also be seen in above proof. Note that the condition in this theorem will involve an increasing amount of computation when the number of the index set S and the number of agents of this network becomes large, which induces linear and exponential growth on edges (ν pN , due to the unit outdegree of each vertex for each subnetwork) of the graph G∗A . In view of this, the following theorem presents another necessary and sufficient condition for consensus of the network (2) based on the FFC property of Ak ’s, k ∈ S . Theorem 3. The switching network (2) can achieve consensus if and only if any finite product of Ak ’s is FFC, k ∈ S . Proof. (Sufficiency). Since any finite product of Ak ’s, k ∈ S , is FFC, then Ak is FFC for any k ∈ S , which, by Theorem 1, follows that every transition graph associated with Ak for k ∈ S contains exactly p cycles of unit length around the vertices η1 for η ∈ Fp . As a consequence, the graph G∗A contains at least p unit cycles around these p vertices. To proceed, it will be proved by contradiction that G∗A does not contain any other cycles. If there is an additional cycle C0 of length m: v1 → v2 → · · · → vm → v1 , where vi , i = 1, 2, . . . , m, is a vertex in G∗A (note that the length of all cycles in G∗A must be finite because of the finiteness of vertices in this graph). As a result, there exist matrices Ak1 , Ak2 , . . . , Akm , ki ∈ S , i = 1, 2, . . . , m, such that v2 = Ak1 v1 , v3 = Ak2 v2 , . . . , vm = Akm−1 vm−1 , v1 = Akm vm . Similarly to that in proof of Theorem 2, η1 cannot be contained in cycle C0 for any η ∈ S . Thus, the transition graph G∗M of matrix M, where M := Akm Akm−1 · · · Ak1 , contains a unit cycle around the vertex v1 that is different from the unit cycles around the vertices η1 for η ∈ Fp , which, together with Theorem 1, follows that row-stochastic matrix M is not FFC. This contradicts that any finite product of Ak ’s, k ∈ S , is FFC. Therefore, G∗A contains exactly p cycles of unit length around the vertices η1 for η ∈ Fp , and hence, invoking Theorem 2, the switching network (2) can achieve consensus. (Necessity). Because the switching network can reach consensus, by Theorem 2 the graph G∗A contains exactly p unit cycles around the vertices η1 for η ∈ Fp , and consensus will be achieved at most time T := pN − p since there are pN vertices in graph G∗A . Therefore, invoking Theorem 1, any product of Ak ’s with length not less than T is indeed FFC. Subsequently, it still needs to prove that any Ak ’s product with length less than T is FFC. Let X be any finite product of Ak ’s with length less than T , k ∈ S . It is easy to verify that transition graph G∗X associated with X contains exactly p unit cycles around the vertices η1 for η ∈ Fp (if not, G∗X will contain other cycles that are different from these p unit cycles, and thus the graph G∗A will contain the same other cycles except these p unit cycles by the definition of G∗A , which is a contradiction by Theorem 2). In view of Theorem 1, it is straightforward to obtain that X is FFC. Summarizing the above analysis completes the proof. In Theorem 3, since it may involve complex computations, it is not necessary to verify the FFC property of all finite products of Ak ’s, k ∈ S . Actually, the graph G∗A associated with the network (2) contains precisely pN vertices, which results in that the switching network that can achieve consensus will indeed reach consensus at most time T := pN − p. Therefore, by similar proof as in Theorem 3, the following result is obtained. Theorem 4. The network (2) can achieve consensus if and only if any finite product of length up to T of Ak ’s is FFC, k ∈ S .
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X. Li et al. / Automatica 68 (2016) 39–43
5. Finite-field consensus with time-delays In this section, we mainly present some results on consensus of the switching network (1) with bounded delays for two cases: (C1) τii = 0, 0 ≤ τij ≤ τ , and (C2) 1 ≤ τii ≤ τ , 0 ≤ τij ≤ τ for i ̸= j, i, j = 1, 2, . . . , N, where τ > 0 is a constant integer. Note that τij ∈ {0, 1, . . . , τ }. Then the network (1) can be rewritten in a loose form:
6. Conclusion
x(t + 1) = C0,s(t ) x(t ) + C1,s(t ) x(t − 1) + C2,s(t ) x(t − 2)
+ · · · + Cτ ,s(t ) x(t − τ ), where Ck,s(t ) , k
(3)
= 0, 1, . . . , τ , is an appropriate nonnegative s(t )
matrix with entries 0 or aij determined by delay k that is underwent by information transmission on the link received at time t. That is, the messages available for x(t ) to update its state to x(t + 1) come from its own state and all delayed states received by that time t. Accordingly, for any time instance t it always holds τ
Proof. By contradiction, if this network can achieve consensus, then by Theorem 3 each matrix Dk , k ∈ S , is FFC, which means that Dk has a simple eigenvalue 1 and all other eigenvalues 0 with multiplicity N (τ + 1) − 1. Therefore, by linear algebra, the trace of matrix Dk must equal 1, which contradicts the fact that Tr (Dk ) = 0 since all diagonal elements of C0,s(t ) are zero. This ends the proof.
Ck,s(t ) = As(t ) ,
(4)
The finite-field consensus problem in networks with both switching topology and time-delays has been investigated in this paper. For switching networks without time-delays, two necessary and sufficient conditions for finite-field consensus have been developed by graph-theoretic approach and the FFC property of corresponding matrices, respectively. Regarding switching networks with time-delays, some necessary and sufficient conditions have been derived by making use of the results on delay-free switching networks. In addition, for networks with time-delays present in each self-transmission, it has been shown that finite-field consensus cannot be reached.
k=0
where all diagonal entries of C0,s(t ) are zeros for case (C2). Note that for each fixed k, Ck,s(t ) switches every unit time as switching signal s(t ) ∈ S evolves over time. In order to study the switching network (3), an equivalent augmented system is introduced as follows (Wang & Xiao, 2007):
x(t + τ ) = C0,s(t +τ −1) x(t + τ − 1) + C1,s(t +τ −1) · x(t + τ − 2) + · · · + Cτ ,s(t +τ −1) x(t − 1), x(t + τ − 1) = x(t + τ − 1), .. . x(t ) = x(t ).
(5)
Define y(t ) := (xT (t +τ ), xT (t +τ − 1), . . . , xT (t ))T ∈ result, the network (3) can be equivalently rewritten in a compact form: y(t + 1) = Ds(t ) y(t ),
(6)
where C0,s(t +τ ) I 0 Ds(t ) =
.. .
0
.. .
··· ··· ··· .. .
Cτ −1,s(t +τ ) 0 0
0
···
I
C1,s(t +τ ) 0 I
.. .
Cτ ,s(t +τ ) 0 0 .
.. .
This research was partially supported by the Research Grants Council, Hong Kong, through the General Research Fund under Grant 106140120, the HKU CRCG Seed Funding Programme for Basic Research 201411159037, the Natural Science Foundation of China under Grants 61374053 and 61473129, the Program for New Century Excellent Talents in University from Chinese Ministry of Education under Grant NCET-12-0215, and the National Natural Science Foundation of China under Grant 61422308. References
N (τ +1) Fp . As a
Acknowledgments
(7)
0
Along this line, the consensus problem for the switching network (3) with time-delays is equivalently transformed into the consensus problem for the switching network (6) without time-delays. In what follows, let G∗D denote the union graph of transition graphs G∗Dk ’s, k ∈ S . Therefore, in view of results on delay-free switching networks, the following results are obtained. Theorem 5. The network (3) with delays in (C1) over Fp can achieve consensus if and only if G∗D contains exactly p cycles and all of them are unit cycles around the vertices η1 for η ∈ Fp . Theorem 6. The network (3) with delays in (C1) over Fp can achieve consensus if and only if any finite product of length up to pN (τ +1) − p of Dk ’s is FFC, k ∈ S . Theorem 7. The network (3) with multiple delays in (C2) over Fp cannot achieve consensus.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
Consensus networks with switching topology and time-delays over finite fields✩ Xiuxian Li a , Michael Z.Q. Chen a , Housheng Su b,1 , Chanying Li c a
Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong
b
School of Automation, Image Processing and Intelligent Control Key Laboratory of Education Ministry of China, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan 430074, China c
Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
article
info
Article history: Received 30 September 2015 Accepted 21 December 2015
Keywords: Consensus Networks Switching topology Time-delays Finite fields
abstract The consensus problem in networks with both switching topology and time-delays over finite fields is investigated in this paper. The finite field, which is a kind of finite alphabet, is considered due to the fact that networks often possess limited computation, memory, and capabilities of communication. First, by graph-theoretic method, one necessary and sufficient condition is derived for finite-field consensus of switching networks without time-delays. Subsequently, another necessary and sufficient condition on finite-field consensus without time-delays is provided based on FFC property of matrices associated with switching networks. Moreover, by means of the results on delay-free networks, some necessary and sufficient conditions for finite-field consensus of networks with both switching topology and time-delays are obtained. Additionally, it can be shown that switching networks with time-delays present in each self-transmission cannot achieve consensus. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction A multi-agent system or network consists of a group of agents or nodes that communicate with each other locally, aiming to achieve some goals by designing control strategies, which has attracted an increasing interest from numerous researchers over the past decade. Due to a large volume of potential applications in many areas, multi-agent systems have a number of distinct research directions including consensus (Hu, Lam, & Liang, 2013; Olfati-Saber & Murray, 2004; Su, Chen, Lam, & Lin, 2013; Su, Chen, Wang, & Lam, 2014; Su, Chen, Wang, & Lin, 2011), flocking (Su, Wang, & Lin, 2009a; Tanner, Jadbabaie, & Pappas, 2007), formation control (Meng, Jia, Du, & Zhang, 2014), etc. As an important distributed feature of networks, consensus aims at for all agents reaching an agreement of interest decided by themselves, which has been substantially investigated due to a large number of
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hyeong Soo Chang under the direction of Editor André L. Tits. E-mail addresses: [email protected] (X. Li), [email protected] (M.Z.Q. Chen), [email protected] (H. Su), [email protected] (C. Li). 1 Tel.: +86 27 87540210; fax: +86 27 87540210.
http://dx.doi.org/10.1016/j.automatica.2016.01.033 0005-1098/© 2016 Elsevier Ltd. All rights reserved.
applications in a range of domains, such as spacecrafts and robotics (Ren, Beard, & Atkins, 2007). Up to now, consensus has been deeply studied for various network models, including timeindependent networks, networks with broad and synchronous communication, networks with linear and nonlinear dynamics, networks with infinite communication bandwidth, networks with time-varying topologies, networks with gossip and asynchronous communication, and networks with link failures and so on (Chen, Wang, & Li, 2012; Fagiolini & Bicchi, 2013; Hadjicostis & Charalambous, 2014; Li, Liu, Wang, & Yin, 2014; Lin & Jia, 2009; Moreau, 2005; Ni & Cheng, 2010; Nuño, Ortega, Basanez, & Hill, 2011; Proskurnikov, 2013; Song, Cao, & Liu, 2010; Su, Wang, & Lin, 2009b; Tahbaz-Salehi & Jadbabaie, 2008; Wang & Xiao, 2007; You, Li, & Xie, 2013; Zhang & Tian, 2009; Zhao, Hill, & Liu, 2012; Zhou & Lin, 2014). Recently, instead of real numbers, finite fields have been taken into consideration for the consensus problem in networks for the sake of safety and memory constraints, etc. (Pasqualetti, Borra, & Bullo, 2014; Sundaram & Hadjicostis, 2013; Xu & Hong, 2014). For instance, some necessary and sufficient conditions on finitefield consensus of fixed networks with discrete-time iteration were developed completely in Pasqualetti et al. (2014), in which the authors also provided its applications to average consensus and pose estimation in sensor networks. However, in the context
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of finite fields, there are few research reports focusing on the consensus problem for networks with switching topology and time-delays, which inspire this paper. This paper addresses the consensus problem for networks with switching topology and time-delays over finite fields. The finite field, as a kind of finite alphabet, is taken into account since networks often undertake limited computation, memory, and capabilities of communication. Generally speaking, this paper extends the work in Pasqualetti et al. (2014) to the case with switching topology and time-delays. Provided that switching topology and delays complicate the structure of a network which further leads to the complexity of the union of transition graphs of adjacency matrices, more careful observations on the structure of the union of transition graphs are needed to derive one necessary and sufficient condition for consensus with switching topology by graph-theoretic method. In addition, intrinsically distinct from Pasqualetti et al. (2014), FFC property of finite product of adjacency matrices is raised here to obtain another necessary and sufficient condition for consensus with switching topology. Furthermore, regarding networks with both switching topology and time-delays, several necessary and sufficient conditions are presented to guarantee finite-field consensus using the results on delay-free networks. Moreover, it is shown that switching networks with time-delays present in each self-transmission cannot reach consensus. 2. Preliminaries A finite field F is a finite set of elements with addition and multiplication operations satisfying the following axioms (Lidl, 1997; Pasqualetti et al., 2014; Sundaram & Hadjicostis, 2013; Xu & Hong, 2014):
• Closure under addition and multiplication. a + b ∈ F, a · b ∈ F, ∀a, b ∈ F; • Associativity of addition and multiplication. a + (b + c ) = (a + b) + c , a · (b · c ) = (a · b) · c , ∀a, b, c ∈ F; • Commutativity of addition and multiplication. a + b = b + a, a · b = b · a, ∀a, b ∈ F; • Existence of additive and multiplicative identity elements. ∀a ∈ F, ∃0, 1, such that a + 0 = a, a · 1 = a; • Existence of additive and multiplicative inverse elements. ∀a ∈ F, ∃b, c ∈ F, such that a + b = 0, a · c = 1 (a ̸= 0); • Distributivity of multiplication over addition. a · (b + c ) = (a · b) + (a · c ), ∀a, b, c ∈ F. Denote by GN = (VN , EN ) a graph with N nodes, a set of vertices VN = {v1 , v2 , . . . , vN } and a set of edges EN ⊆ VN × VN . vi → vj (or (vi , vj ) ∈ EN ) denotes an edge in which node i sends information to node j. A graph is called undirected if (vi , vj ) ∈ EN implies (vj , vi ) ∈ EN , and directed otherwise. For a node i, the in-degree and out-degree of vi ∈ VN equal the numbers of the in-neighbor set Ni+ = {vj ∈ VN : (vj , vi ) ∈ EN } and the outneighbor set Ni− = {vj ∈ VN : (vi , vj ) ∈ EN }, respectively. The adjacency matrix AN = (aij ) ∈ FNp ×N is defined as: aij > 0 if
vj ∈ Ni+ , aij = 0 (i ̸= j) otherwise, and self-loops are allowed.
A directed path in a directed graph is a sequence of edges of the form (i1 , i2 ), (i2 , i3 ), . . .. A cycle is a path that shares the same first and last vertex. A directed tree is a directed graph, where every node has exactly one parent except one node called root. A root (resp. globally reachable node) is a node which has a directed path to (resp. from) every node in the graph, including itself. The length of a path (resp. cycle) equals the number of edges in the path (resp. cycle). A directed graph is called strongly (resp. weakly) connected if there is a directed (resp. undirected) path between any two nodes. Two subgraphs of the same graph are called disjoint if they have no common nodes.
3. Problem statement Consider a switching network with N nodes over a finite field. For simplicity, throughout this paper the prime field Fp := {0, 1, 2, . . . , p − 1} is considered, where p is a prime number. However, the results here can be easily extended to general finite fields. The network has the dynamics xi (t + 1) =
s(t )
aij xj (t − τij ),
i = 1, 2, . . . , N
(1)
+ j∈Ni (t )∪{i}
where xi (t ) ∈ Fp is the state of this network, and τij ’s are timedelays experienced by information transmission on the link from node j to node i satisfying 0 ≤ τij ≤ τ for some constant τ > 0, that is, τij ’s are bounded, i, j = 1, 2, . . . , N. Meanwhile, s(t )
the adjacency matrix As(t ) = (aij ) ∈ FNp ×N is row-stochastic (see Definition 2) and s(t ) : {0, 1, 2, . . .} → S is the switching signal with finite index set S := {1, 2, . . . , ν}. Therefore, for any k ∈ S , when s(t ) = k, the subnetwork (1) with As(t ) = Ak is activated. As a delayed network, it is assumed that subnetwork s(t ) takes the initial states x(t − 1), x(t − 2), . . . , x(t − τ ) when subnetwork s(t − 1) is switched to subnetwork s(t ) at time t, where x(t ) := (x1 (t ), x2 (t ), . . . , xN (t ))T ∈ FNp . Note that the addition and multiplication operations in the network (1) are performed by modulo, that is, taking the remainder after divided by p. For some k ∈ S , the transition graph of corresponding fixed network x(t + 1) = Ak x(t ) is defined as G∗Ak = (VA∗k , EA∗k ) with vertex set VA∗k = {v : v ∈ FNp } and edge set EA∗k = {(vi , vj ) :
vj = Ak vi , vi , vj ∈ FNp }. This transition graph comprises of disjoint
weakly-connected subgraphs, and only one cycle, maybe of unit length, is contained in each subgraph which embraces a globally reachable node (Hernández Toledo, 2005; Pasqualetti et al., 2014). As for switching network (1), denote by G∗A = (VA∗ , EA∗ ) the union graph of G∗Ak for all k ∈ S , that is, VA∗ = VA∗k = {v : v ∈ FNp } and ∗ edge set EA∗ = k∈S EAk . To proceed, the notion of consensus of networks over finite fields is defined as follows. Definition 1. The network (1) over Fp can achieve (finite-time) consensus if for any initial states in FNp and any switching signal s(t ), there exist a finite time T ∈ N and some constant η ∈ Fp such that x(T + k) = x(T ) = η1 for all k ∈ N. Note that consensus of networks over finite fields can be always achieved in finite time since there are only finite states in networks over finite fields. For brevity, two concepts are introduced as follows. Definition 2. Over the finite field Fp , (1) a matrix M ∈ Fnp×n is called row-stochastic if each row sums to 1; (2) a row-stochastic matrix M ∈ Fnp×n is called finite-field consensusable (FFC, for short) if M has a simple eigenvalue 1 and all other eigenvalues 0, that is, its characteristic polynomial is PM (λ) = λn−1 (λ − 1). This definition is reasonable due to the following result that states two necessary and sufficient conditions for finite-field consensus, which are conducive to consensus analysis later. Theorem 1 (Pasqualetti et al., 2014). For a fixed network x(t + 1) = Mx(t ) over Fp , the following statements are equivalent: (1) this network can achieve consensus; (2) the transition graph of M contains exactly p cycles and all of them are unit cycles around the vertices η1 for η ∈ Fp ; (3) the matrix M is FFC.
X. Li et al. / Automatica 68 (2016) 39–43
4. Finite-field consensus without time-delays In this section, the network (1) is addressed without timedelays, i.e., τij = 0, i, j = 1, 2, . . . , N. Hence, the network (1) can be rewritten as x(t + 1) = As(t ) x(t ),
(2)
where x(t ) = (x1 (t ), x2 (t ), . . . , xN (t )) ∈ is the state of this network and the other notations are defined in (1). It is known that finite-field consensus requires more stringent conditions than that for consensus over real numbers (Pasqualetti et al., 2014). More precisely, two necessary and sufficient conditions based on the transition graph and the characteristic polynomial are presented for fixed networks in Theorem 1. Consequently, a natural question is if it is sufficient to ensure finite-time consensus for the network (2) when row-stochastic matrices A1 , A2 , . . . , Aν are all FFC. The answer is negative which can be seen from the following example. T
FNp
Example 1. Consider the network (2) over the finite field F3 =
{0, 1, 2}, let ν = 2 and A1 =
1 1 0
0 0 1
0 0 , 0
A2 =
1 0 1
0 0 0
0 1 . 0
It is easy to verify that both A1 and A2 are FFC, which, by Theorem 1, follows that both fixed network x(t + 1) = A1 x(t ) and x(t + 1) = A2 x(t ) can reach consensus. However, when subnetworks x(t + 1) = A2 x(t ) and x(t + 1) = A1 x(t ) are activated alternately every unit time with initial state x(0) = (0, 0, 1)T , i.e., x(1) = A2 x(0), x(2) = A1 x(1), x(3) = A2 x(2), x(4) = A1 x(3), . . . , it is straightforward to see that x(t ) oscillates between (0, 0, 1)T and (0, 1, 0)T . Hence, the switching network cannot achieve consensus. Therefore, to guarantee consensus it is indispensable to impose other conditions on the network (2) besides FFC property of all Ai ’s, i ∈ S . In the sequel, a necessary and sufficient condition is provided for consensus of the network (2) based on the characterization of the transition graph. Theorem 2. For the switching network (2) over Fp , it can achieve consensus if and only if the graph G∗A contains exactly p cycles and all of them are unit cycles around the vertices η1 for η ∈ Fp . Proof. The argument for sufficiency is a variation of the proof of Theorem 4.1 in Pasqualetti et al. (2014), which is omitted here. In what follows, the necessity is proved in detail. Since the network (2) can achieve consensus, fixed network x(t + 1) = Ak x(t ) can achieve consensus as well for any k ∈ S because As(t ) may be equal to Ak for all time t ≥ 0. As a result, by Theorem 1, one concludes that the transition graph G∗Ak has exactly p cycles of unit length around the vertices η1 for η ∈ Fp , and then combining the arbitrariness of k ∈ S with the definition of G∗A yields that G∗A contains at least p cycles of unit length. In the following, it only needs to show that G∗A does not contain other cycles. By contradiction argument, if there is another cycle C0 in G∗A and note that for any η ∈ Fp , vertex η1 has only one edge pointed out to itself, then η1 is not contained in cycle C0 . Suppose that C0 consists of m vertices v1 , v2 , . . . , vm with v1 → v2 → · · · → vm → v1 , that is, there are some matrices Ak1 , Ak2 , . . . , Akm satisfying v2 = Ak1 v1 , v3 = Ak2 v2 , . . . , vkm = Akm−1 vm−1 , v1 = Akm vm . Notice that Aki ’s, i = 1, 2, . . . , m, are not necessary to be distinct. Consequently, consider a network x(t + 1) = As(t ) x(t ) with initial value x(0) = v1 and As(t ) equaling Ak1 , Ak2 , . . . , Akm periodically in ascending order of unit time, that is, As(0) = Ak1 , As(1) = Ak2 , . . . , As(m−1) = Akm , As(m) = Ak1 , . . .. It is straightforward to see that this network cannot achieve consensus which is a contradiction. This ends the proof.
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The result of Theorem 2 can be carefully observed from the structure of the graph G∗A , since the state can oscillate periodically along a circle of length l ≥ 2 if the graph contains other cycles besides these p unit cycles, which can also be seen in above proof. Note that the condition in this theorem will involve an increasing amount of computation when the number of the index set S and the number of agents of this network becomes large, which induces linear and exponential growth on edges (ν pN , due to the unit outdegree of each vertex for each subnetwork) of the graph G∗A . In view of this, the following theorem presents another necessary and sufficient condition for consensus of the network (2) based on the FFC property of Ak ’s, k ∈ S . Theorem 3. The switching network (2) can achieve consensus if and only if any finite product of Ak ’s is FFC, k ∈ S . Proof. (Sufficiency). Since any finite product of Ak ’s, k ∈ S , is FFC, then Ak is FFC for any k ∈ S , which, by Theorem 1, follows that every transition graph associated with Ak for k ∈ S contains exactly p cycles of unit length around the vertices η1 for η ∈ Fp . As a consequence, the graph G∗A contains at least p unit cycles around these p vertices. To proceed, it will be proved by contradiction that G∗A does not contain any other cycles. If there is an additional cycle C0 of length m: v1 → v2 → · · · → vm → v1 , where vi , i = 1, 2, . . . , m, is a vertex in G∗A (note that the length of all cycles in G∗A must be finite because of the finiteness of vertices in this graph). As a result, there exist matrices Ak1 , Ak2 , . . . , Akm , ki ∈ S , i = 1, 2, . . . , m, such that v2 = Ak1 v1 , v3 = Ak2 v2 , . . . , vm = Akm−1 vm−1 , v1 = Akm vm . Similarly to that in proof of Theorem 2, η1 cannot be contained in cycle C0 for any η ∈ S . Thus, the transition graph G∗M of matrix M, where M := Akm Akm−1 · · · Ak1 , contains a unit cycle around the vertex v1 that is different from the unit cycles around the vertices η1 for η ∈ Fp , which, together with Theorem 1, follows that row-stochastic matrix M is not FFC. This contradicts that any finite product of Ak ’s, k ∈ S , is FFC. Therefore, G∗A contains exactly p cycles of unit length around the vertices η1 for η ∈ Fp , and hence, invoking Theorem 2, the switching network (2) can achieve consensus. (Necessity). Because the switching network can reach consensus, by Theorem 2 the graph G∗A contains exactly p unit cycles around the vertices η1 for η ∈ Fp , and consensus will be achieved at most time T := pN − p since there are pN vertices in graph G∗A . Therefore, invoking Theorem 1, any product of Ak ’s with length not less than T is indeed FFC. Subsequently, it still needs to prove that any Ak ’s product with length less than T is FFC. Let X be any finite product of Ak ’s with length less than T , k ∈ S . It is easy to verify that transition graph G∗X associated with X contains exactly p unit cycles around the vertices η1 for η ∈ Fp (if not, G∗X will contain other cycles that are different from these p unit cycles, and thus the graph G∗A will contain the same other cycles except these p unit cycles by the definition of G∗A , which is a contradiction by Theorem 2). In view of Theorem 1, it is straightforward to obtain that X is FFC. Summarizing the above analysis completes the proof. In Theorem 3, since it may involve complex computations, it is not necessary to verify the FFC property of all finite products of Ak ’s, k ∈ S . Actually, the graph G∗A associated with the network (2) contains precisely pN vertices, which results in that the switching network that can achieve consensus will indeed reach consensus at most time T := pN − p. Therefore, by similar proof as in Theorem 3, the following result is obtained. Theorem 4. The network (2) can achieve consensus if and only if any finite product of length up to T of Ak ’s is FFC, k ∈ S .
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X. Li et al. / Automatica 68 (2016) 39–43
5. Finite-field consensus with time-delays In this section, we mainly present some results on consensus of the switching network (1) with bounded delays for two cases: (C1) τii = 0, 0 ≤ τij ≤ τ , and (C2) 1 ≤ τii ≤ τ , 0 ≤ τij ≤ τ for i ̸= j, i, j = 1, 2, . . . , N, where τ > 0 is a constant integer. Note that τij ∈ {0, 1, . . . , τ }. Then the network (1) can be rewritten in a loose form:
6. Conclusion
x(t + 1) = C0,s(t ) x(t ) + C1,s(t ) x(t − 1) + C2,s(t ) x(t − 2)
+ · · · + Cτ ,s(t ) x(t − τ ), where Ck,s(t ) , k
(3)
= 0, 1, . . . , τ , is an appropriate nonnegative s(t )
matrix with entries 0 or aij determined by delay k that is underwent by information transmission on the link received at time t. That is, the messages available for x(t ) to update its state to x(t + 1) come from its own state and all delayed states received by that time t. Accordingly, for any time instance t it always holds τ
Proof. By contradiction, if this network can achieve consensus, then by Theorem 3 each matrix Dk , k ∈ S , is FFC, which means that Dk has a simple eigenvalue 1 and all other eigenvalues 0 with multiplicity N (τ + 1) − 1. Therefore, by linear algebra, the trace of matrix Dk must equal 1, which contradicts the fact that Tr (Dk ) = 0 since all diagonal elements of C0,s(t ) are zero. This ends the proof.
Ck,s(t ) = As(t ) ,
(4)
The finite-field consensus problem in networks with both switching topology and time-delays has been investigated in this paper. For switching networks without time-delays, two necessary and sufficient conditions for finite-field consensus have been developed by graph-theoretic approach and the FFC property of corresponding matrices, respectively. Regarding switching networks with time-delays, some necessary and sufficient conditions have been derived by making use of the results on delay-free switching networks. In addition, for networks with time-delays present in each self-transmission, it has been shown that finite-field consensus cannot be reached.
k=0
where all diagonal entries of C0,s(t ) are zeros for case (C2). Note that for each fixed k, Ck,s(t ) switches every unit time as switching signal s(t ) ∈ S evolves over time. In order to study the switching network (3), an equivalent augmented system is introduced as follows (Wang & Xiao, 2007):
x(t + τ ) = C0,s(t +τ −1) x(t + τ − 1) + C1,s(t +τ −1) · x(t + τ − 2) + · · · + Cτ ,s(t +τ −1) x(t − 1), x(t + τ − 1) = x(t + τ − 1), .. . x(t ) = x(t ).
(5)
Define y(t ) := (xT (t +τ ), xT (t +τ − 1), . . . , xT (t ))T ∈ result, the network (3) can be equivalently rewritten in a compact form: y(t + 1) = Ds(t ) y(t ),
(6)
where C0,s(t +τ ) I 0 Ds(t ) =
.. .
0
.. .
··· ··· ··· .. .
Cτ −1,s(t +τ ) 0 0
0
···
I
C1,s(t +τ ) 0 I
.. .
Cτ ,s(t +τ ) 0 0 .
.. .
This research was partially supported by the Research Grants Council, Hong Kong, through the General Research Fund under Grant 106140120, the HKU CRCG Seed Funding Programme for Basic Research 201411159037, the Natural Science Foundation of China under Grants 61374053 and 61473129, the Program for New Century Excellent Talents in University from Chinese Ministry of Education under Grant NCET-12-0215, and the National Natural Science Foundation of China under Grant 61422308. References
N (τ +1) Fp . As a
Acknowledgments
(7)
0
Along this line, the consensus problem for the switching network (3) with time-delays is equivalently transformed into the consensus problem for the switching network (6) without time-delays. In what follows, let G∗D denote the union graph of transition graphs G∗Dk ’s, k ∈ S . Therefore, in view of results on delay-free switching networks, the following results are obtained. Theorem 5. The network (3) with delays in (C1) over Fp can achieve consensus if and only if G∗D contains exactly p cycles and all of them are unit cycles around the vertices η1 for η ∈ Fp . Theorem 6. The network (3) with delays in (C1) over Fp can achieve consensus if and only if any finite product of length up to pN (τ +1) − p of Dk ’s is FFC, k ∈ S . Theorem 7. The network (3) with multiple delays in (C2) over Fp cannot achieve consensus.
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