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A consensus recovery approach to nonlinear multi-agent system under node failure
Accepted Manuscript
A Consensus Recovery Approach to Nonlinear Multi-Agent System under Node Failure Lulu Li, Daniel W.C. Ho, Jianquan Lu PII: DOI: Reference:
S0020-0255(16)30479-0 10.1016/j.ins.2016.06.050 INS 12324
To appear in:
Information Sciences
Received date: Revised date: Accepted date:
29 July 2015 27 April 2016 28 June 2016
Please cite this article as: Lulu Li, Daniel W.C. Ho, Jianquan Lu, A Consensus Recovery Approach to Nonlinear Multi-Agent System under Node Failure, Information Sciences (2016), doi: 10.1016/j.ins.2016.06.050
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A Consensus Recovery Approach to Nonlinear Multi-Agent System under Node Failure
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Lulu Lia,b,∗, Daniel W. C. Hob , Jianquan Luc a School
of Mathematics, Hefei University of Technology, Hefei 230009, China of Mathematics, City University of Hong Kong, Hong Kong c Department of Mathematics, Southeast University, Nanjing 210096, China b Department
Abstract
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In large-scale networks, node failure is unavoidable and it generally brings undesirable effects on the stability and performance of the systems. Then a natural question arises: is it possible to cope with the problem arising from the node failure in the networks, while retaining the corresponding network property (or make the network systems robust against the
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node failure)? In this paper, a consensus recovery approach will be proposed to investigate the consensus of general directed nonlinear multi-agent networks with node failure. The objective of the consensus recovery method is to compen-
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sate for the undesirable effects of the failure nodes, and to retain the consensus property. Numerical examples are provided to demonstrate the effectiveness of the theoretical results obtained, even for large-scale multi-agent systems.
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Keywords: Multi-agent networks, Network reduction approach, Consensus
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recovery approach, Node failure
1. Introduction
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Recently, multi-agent networks, such as distributed robots and mobile sensor
networks, have been widely studied due to their broad applications ([5, 6, 24]).
✩ The work was supported by a GRF grant from HKSAR (CityU 11300415) and the National Natural Science Foundation of China under Grants 11426081, 61503115 and 61573102 ∗ Corresponding author Email addresses: [email protected] (Lulu Li), [email protected] (Daniel W. C. Ho), [email protected] (Jianquan Lu)
Preprint submitted to Information Sciences
June 29, 2016
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The main focus of studying multi-agent networks is to determine how consensus 5
emerges as a result of local interactions among agents, where consensus implies the agreement of a collection of agents on a common state value. Examples of
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typical consensus behavior include cooperation of unmanned air vehicles, flocking of birds, and swarming of fishes ([22, 25]). In the past few years, substantial
consensus or synchronization problems have been studied in much previous lit10
erature ([1, 7, 10, 11, 12, 15, 16, 17, 18, 26, 30, 32, 33, 34]). In [11], under a
new protocol and strong connected network topology, practical consensus can be achieved for multi-agent networks with time delays and quantization. In
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[32], the distributed consensus problem for a class of multiple Euler-Lagrange systems is solved under considering the time-delays and packet dropouts simul15
taneously. Meanwhile, many typical consensus problems were investigated in recent years, such as average consensus [17], cluster consensus [1], second-order consensus [13], leader-follower consensus ([3, 10]), event-based consensus [12] and references therein.
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In many realistic network systems, due to the complexity of systems and undesirable external attacks (disturbance), the occurrence of nodes (or links)
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failure is inevitable ([4, 23, 28, 31]). For instance, a sensor may not be recharged due to battery failure in a sensor network. Subsequently, this failure sensor may not be easily repaired under a hostile environment. This situation can be
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modeled by a networked multi-agent system in which the node failure occur in the process of network states evolving. Some papers have been devoted to
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studying the networked systems with link failure. In [9], an erasure model is studied, that is, the network links fail independently in space (independently of each other) and in time (link failure events are temporarily independent).
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In [31], consensus recovery of linear multi-agent systems under node failure
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is investigated. A recovery approach is proposed for the multi-agent system under external attacks. However, the proposed recovery approach ignores the importance of the theoretical analysis and cannot be applied to maintain the final consensus value. In [28], synchronization problems are investigated for complex dynamical networks subject to recoverable link failures. 2
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Devastating consequence would result due to the failure of the nodes in multi-
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agent systems. In particular, when a noncompensable failure occurs, which means replacement of the failure nodes is not feasible, the failure nodes and the
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connected links have to be isolated (or removed). The noncompensable node failures generally bring undesirable effects on the stability and performance of 40
multi-agent systems. For instance, the consensus property of multi-agent net-
works may be lost under some fatal nodes/link failures. Naturally, an important question is raised Q1: Is it possible to isolate (or remove) the failure nodes of the network and to retain the consensus property? It is a crucial, yet an open
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research problem for many real-life applications.
Moreover, with the proliferation of distributed multi-agent systems interact-
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ing through networks, the increasing complexity and higher safety demands of modern engineering systems highlight the significance of the reliability of the system. In order to improve the reliability, a recovery scheme should be designed to maintain the performance of the system. However, until now, little literature has taken this issue into account. This paper is devoted to studying
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this problem from the aspect of consensus. An efficient consensus recovery al-
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gorithm will be proposed and theoretically proved to deal with the problems arising from the node failure in network systems. Although multi-agent networks have been extensively applied in practical work, the distributed behavior of the networks do bring some essential difficulties
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in theoretical research. In particular, as the multi-agent networks began to be
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implemented for critical tasks such as military or power system applications, the risk of network failures became more of a concern. It is highly desirable to exploit effective protocols to retain the consensus of the multi-agent networks.
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Motivated by the above discussions, we aim to design a consensus recovery
approach to compensate for the undesirable effects of the failure nodes and the consensus property is retained after the recovery process. The main contributions of this paper are presented as follows: • To reduce the size of the networks, while maintaining the consensus prop3
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erty, a novel network reduction approach is proposed in this paper. The
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main advantage of this method is that, for each agent, only local information of its neighboring agents is used in the process of network size
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reduction, and it makes our reduction method practical and easy to implement in real-world networks.
• To improve the reliability of the network system and to retain the consen-
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sus property under node failure, an efficient consensus recovery approach will be provided based on the network reduction method.
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The organization of the remaining part is presented as follows. In Section 2, some basic definitions involving algebraic graph theory are introduced. In 75
Section 3, consensus analyses of nonlinear multi-agent networks are presented in detail by fully utilizing the structure of the network. In Section 4, the network recovery approach is proposed for the consensus problem of the large-size nonlinear networks under general communication topology. In Section 5, a nu-
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merical simulation is given to show the effectiveness of the theoretical results. In Section 6, concluding remarks are drawn.
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N otation : The standard notations will be used in this paper. Throughout this paper, R+ denotes the set of all positive real numbers. Rn and Rm×n represent n-dimensional Euclidean space and the set of m × n real matrices, 85
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respectively; The superscript “T ” represents the transpose. Let C 1 (Rn ; R+ )
denote the family of all continuously differentiable functions V (x) from Rn to R+ . K denotes the family of all continuous nondecreasing functions ϕ : R+ −→
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R+ such that ϕ(r) > 0 if r > 0.
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2. Some Preliminaries 2.1. Algebraic graph theory
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In this subsection, we present some basic knowledge concerning algebraic
graph theory. For more details, please refer to [2].
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Let G(V, E, A) be a weighted directed graph with the set of nodes V = {v1 , v2 , . . . , vN }, the set of edges E ⊂ V × V and a weighted adjacency matrix A = [aij ] with nonnegative adjacency elements aij . The directed graph G is called the interaction graph of the matrix A. An edge of G is denoted
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by eij = (vi , vj ), where vi and vj are called the parent and child vertices, re-
spectively. (vj , vi ) ∈ E means that node vj receives information from node vi .
For adjacency matrix A, (vi , vj ) ∈ E ⇐⇒ aji > 0 and aji is called
the weight of edge (vi , vj ). The set of in-neighbors of vi in G is denoted by 100
Nin (vi ) = {vj : (vj , vi ) ∈ E}. The set of out-neighbors of vi in G is denoted by
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Nout (vi ) = {vj : (vi , vj ) ∈ E}. The Laplacian of the directed graph is defined PN as L = ∆ − A, where ∆ = [∆ij ]N ×N is a diagonal matrix with ∆ii = j=1 aij .
It is easy to show that L has at least one zero eigenvalue with a corresponding eigenvector 1, where 1 = (1, 1, . . . , 1)T .
A path in a digraph is an ordered sequence of vertices such that any two
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consecutive vertices are an directed edge of the digraph. A directed tree is a
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directed graph, where every vertex, except one which is called the root, has exactly one parent. A subgraph Gs of a directed graph is a directed graph 110
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such that the vertex set V(Gs ) ⊂ V(G) and the edge set E(Gs ) ⊂ E(G). If V(Gs ) = V(G), Gs is called the spanning subgraph. A directed tree that is a spanning subgraph of G is called the spanning tree of G. A directed graph G is
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strongly connected if there is a path for any pair of distinct vertices in G. A matrix A is called nonnegative if all its entries are nonnegative, which is
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denoted by A 0. We say nonnegative matrix A is irreducible if the interaction
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graph of A is strongly connected.
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2.2. Basic definition and lemmas Definition 1. (consensus). Consider a multi-agent network A with N agents. Let xi denote the state of agent i. We say that multi-agent network A reaches a consensus if and only if xi = xj for all i, j = 1, 2, . . . , N . If for all xi (0) ∈ R, i =
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1, 2, . . . , N , xi (t) converges to some common equilibrium point x∗ (depending on the initial values of the agents), as t −→ +∞, then we say that multi-agent 5
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network A solves a consensus problem asymptotically. The common value x∗ is called the group decision value.
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of Theorem 1.
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In the following, we give an important lemma, which will be used in the proof
Lemma 1. Consider an n-dimensional ordinary differential equation ˙ x(t) = f (x(t)), x(0) = x0 ,
(1)
x = (x1 , . . . , xn )T , f = (f1 , f2 , . . . , fn )T ∈ C(Rn ; Rn ). Suppose initial value
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problem (1) has an unique solution on [0, +∞). If there exists positive definite function V (x) ∈ C 1 (Rn ; R+ ) satisfying V (0) = 0 such that V˙ (x(t)) |(1) = −W (x(t)) + a(t),
where lim a(t) = 0, W (x(t)) is a positive function, i.e., W (x) ≥ 0 and W (x) = t→+∞
0 if and only if x = 0, then lim x(t) = 0.
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t→+∞
Proof: The proof of Lemma 1 is similar to the proof of Lyapunov stability
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theorem. Thus it is omitted here.
Remark 1. It should be noted that we do not require the derivative of Lyapunov function (energy function) V (x) to be negative, which is a standard
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requirement for Lyapunov asymptotic stability theory. Generally, in Lemma 1, the term a(t) results from the exterior disturbances. Lemma 1 can be used to
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deal with the consensus and synchronization problem of many dynamical sys135
tems and networks, and to ensure the final states of the whole system converge
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as t −→ +∞.
3. Consensus analysis of general multi-agent networks In this section, we will study the nonlinear consensus protocol for multi-
agent networks under general communication structure, which can also be used
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to illustrate the correctness and efficiency of the network reduction approach. Consider the following nonlinear multi-agent networks model X dxi (t) = aij [f (xj (t)) − f (xi (t))], i = 1, . . . , N, dt
(2)
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j∈Ni
where xi (t) ∈ R is the state of the agent i, f (·) is a nonlinear function with the same dimensions of xi satisfying f (0) = 0.
Assumption 1. Throughout this paper, one requires the nonlinear function f (·) to be continuous and strictly monotonically increasing on R.
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For any communication structure, Laplacian matrix can always be written in the following form after certain permutations [29]. Without loss of generality, we assume the Laplacian matrix of model (2) to be L in the form of ···
0
···
0
···
0
···
0
0
0
···
0
.. .
.. .
.. .
Lkk
0
0
···
0
Lk+1,k
Lk+1,k+1
0
···
0
.. .
.. .
.. .
Lk+m,k
Lk+m,k+1
Lk+m,k+2
0
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L11 0 .. . L = 0 Lk+1,1 .. . Lk+m,1
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··· ···
···
···
.. .
···
.. .
··· ···
Lk+m,k+m
,
(3)
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where Lii are irreducible square matrices and in each line, there exist at least one entry satisfying Lk+i,j 6= 0 (i = 1, 2, . . . , m, j = 1, 2, . . . , k + i − 1). In the following, the nodes, known as leaders, are the strongly connected compo-
nents of the network, which is represented by Lii (i = 1, 2, . . . , k). The other nodes
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are served as followers. It can be seen from the network topology that there is no information channel from the followers to the leader. Hence, the followers cannot af-
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fect the states of the leader in the network. In the following, f (xi (t)) is called the observed states of the agent i. Some detailed explanations will be given in Remark 2. As the fundamental of recovery approach, the following theorem reveals a general 155
consensus result about the nonlinear directed multi-agent networks, which can be seen
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as an extension of the main result of [14].
Theorem 1. Consider multi-agent network (2) with Laplacian matrix L. The following conclusions can be obtained:
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(i) The leaders in Lii will achieve consensus separately;
(ii) the observed states of the followers will asymptotically converge to the convex
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leaders in Lii .
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combination of {f (¯ xξˆi ), i = 1, 2, . . . , k}, where x ¯ξˆi is the consensus value of the
Proof: A comprehensive proof can be found in Appendix A and here we shall give an
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outline of the proof.
Outline of the proof: We take three steps for the remaining part of the proof.
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1. LaSalle’s invariant principle is utilized to obtain the conclusion (i). 2. For the followers in the strongly connected component Lk+l,k+l (l = 1, 2, . . . , m), the coefficients of convex combination in part (ii) of Theorem 1 can be determined.
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3. We will construct appropriate function Vk+l (t) satisfying Lemma 1, and then
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part (ii) of Theorem 1 can be proved.
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Remark 2. An interesting result obtained in Theorem 1 is that the observed states
of the followers converge asymptotically to the convex combination of observed con-
sensus value of the leaders, while different followers may have different combination 175
coefficients. In addition, related to previous results [14], when k = 1, we can obtain
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the condition to guarantee the consensus of the model (2). Moreover, the network
cannot reach consensus if k ≥ 2, i.e., the network does not contain a rooted spanning tree. Then the following corollary can be obtained.
if and only if the directed communication topology of the network contains a rooted spanning tree.
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Corollary 1. Under Assumption 1, multi-agent network (2) can realize consensus
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4. Consensus recovery approach A general consensus result about multi-agent network (2) has been given in Section
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3. However, in many realistic network systems, node failure is a common phenomenon and sometimes the failure nodes are non-repairable. Under this circumstance, these nodes should be removed from the network, but the most important property of the
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185
system (consensus here) should be retained. From this aspect, it is necessary to propose an effective method to deal with such a node failure problem. In other words, an efficient method should be developed to retain the consensus property.
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In the next section, we will firstly present a novel network reduction method to
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reduce the size of the network, such that the consensus property of the large-size
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network can be obtained by studying that of the derived small-size network. This network reduction method can make the consensus analysis much easier and also the computational cost much lower. Based on the network reduction algorithm, we will 195
further discuss a consensus recovery method for multi-agent system with node failure.
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If the node p is removed from the network, one straightforward approach for re-
taining the consensus property is to make the information of the node p retained by its neighboring nodes. For example, consider the network structure in Fig. 1. Clearly, the states of nodes qj (j = 1, 2, 3) is affected by the state of node p directly and the states of nodes li (i = 1, 2) through intermediate node p. If the node p is removed, naturally,
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the connection strength between the node li and node qj should be increased (Fig. 1) and the initial value of node qj should be updated to retain the consensus property of
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the network. Now, the remaining question is how to update the initial value of node qj and the connection strength between the node li and node qj . This problem will be solved based on Theorem 1.
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Next, we propose a network reduction algorithm to reduce the size of the network
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while retaining the consensus property. Algorithm 1:
For the multi-agent network A, suppose its original graph is
G(V, E). The following algorithm can be used to reduce the size of the network and 210
meanwhile retain the consensus property:
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Fig. 1. Remove node p in the network.
Step 1. (Reducing the size of the network)
(a) For a node p in the network A, suppose that there exist m connections to the nodes q1 , q2 , . . . , qm and k connections from nodes l1 , l2 , . . . , lk to the node p. Suppose
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that the connection strength from node p to node qj is aj and the connection strength from node li to node p is bi . Removing node p and its connected edges yields a new
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network Ap . Keep the original initial value and coupling unchanged but increase the initial value of node qj by bi a , β j
bi a , b1 +b2 j
k X
bi . For example,
i=1
i = 1, 2, j = 1, 2, 3.
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in Fig.1, cij =
xp (0) and increase the coupling strength between node li
where i = 1, . . . , k, j = 1, . . . , m, and β =
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and node qj by
aj β
(b) For the node with the self-loop, delete the self-loop.
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(c) Repeat Step 1 (a) and (b) until no node can be reduced. Step 2. (Rescaling the initial value of the reduced network) After Step 1, the nodes in the reduced network can be divided into two classes.
The first class contains nodes with zero in-degree and the second class (maybe empty)
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225
contains nodes with non-zero in-degree but zero out-degree. Suppose that each node in the first class of the reduced network has initial value
k X
ξi xi (0). Then, one needs
i=1
k k X 1X ξi xi (0), where γ = ξi . γ i=1 i=1
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to rescale the initial value of the node into
Remark 3. Following the above mentioned reduction process, it should be emphasized that only local information is used in the process of network size reduction. When the node p is removed, one only needs to update the initial values of the out-
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230
neighboring nodes qj and connection strength between in-neighboring nodes li and out-neighboring nodes qj (i = 1, . . . , k, j = 1, . . . , m).
The logical relationship between the reduced small-size network and the original
ˆ suppose that ri is the zero in-degree Remark 4. For the final reduced network A,
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one is presented in the following remark.
node with initial value x ˆri (0) and node qj is the zero out-degree node with the con-
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nection strength aij from node ri to node qj , where i = 1, . . . , k, j = 1, . . . , m. Then, the following facts about the multi-agent network A can be obtained from the reduced
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ˆ network A:
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1. the set of nodes ∆i = {ri−1 + 1, . . . , ri } (i = 1, . . . , k) are strongly connected components of graph G(V, E);
2. the nodes in these strongly connected components are the leaders of the network A and each leader in the network A must belong to one of the nodes set ∆i (i = 1, . . . , k);
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3. the leaders in the same strongly connected component will reach consensus and
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the consensus value x ¯ξˆi is equal to x ˆri (0);
network A and its final state is f −1 [ Pk
1
i=1
aij
k X i=1
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4. nodes qj , j = 1, . . . , m, in the reduced network Aˆ are the followers of the original aij f (¯ xξˆi )];
5. based on the above two conclusions of Remark 4, one can conclude that:
ˆ the • if k = 1, i.e., there is only one leader in the final reduced network A,
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multi-agent network can achieve consensus;
• if k 6= 1 in the reduced network, the multi-agent network will achieve k different consensus values which are respectively decided by leader sets
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∆i , i = 1, . . . , k. Further explanation will be given in Example 2.
To answer the question Q1, based on the network reduction method discussed
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above, we will propose a network recovery approach to solve the node failure problem.
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Considering that in some real networks, the nodes can only store the current state other than the initial state, the network recovery algorithm in the following will use
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the current state information other than initial value information. Let {ξrl−1 +1 , . . . , ξrl } be the normalized left eigenvector of Lll , l = 1, 2, . . . , k,
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with respect to the zero eigenvalue. Suppose agent p fails at time t0 due to external disturbance (see Fig.1). If node p is the leader to be removed, the network has to
retain the information of node p. Hence, the local information of node p should be
adjusted as that in Step 1 of Algorithm 1. As for the followers, the edge weighting
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information between the neighboring nodes should also be adjusted for retaining the convex combination coefficients in the reduced network. Suppose that (i) there exist m
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connections to the nodes q1 , q2 , . . . , qm and k connections from nodes l1 , l2 , . . . , lk to the node p; (ii) the connection strength from node p to node qj is aj and the connection
strength from node li to node p is bi . We have the following algorithm to remove the 270
failure nodes of the network and retain the consensus property.
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Algorithm 2: For the multi-agent network A, suppose its original graph is G(V, E). Step 1. (Removing the failure nodes of the network)
Suppose the number of failure nodes is s (where s is a positive integer) during the 0
time interval [t0 , t1 ]. For any node p which fails at time t0 , there are two cases we should consider:
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Case 1: (Node p is the follower.) Removing node p and its connected edges yields a new network Ap . The coupling strength between node li and node qj is increased by where i = 1, . . . , k, j = 1, . . . , m, and β =
k X
bi . For the node with the self-loop,
i=1
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bi a , β j
delete the self-loop. For example, in Fig.1, cij =
bi a , b1 +b2 j
i = 1, 2, j = 1, 2, 3.
Case 2: (Node p is the leader.) Adjust the local information of p in the re-
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280
0
duced network, i.e., the current state xqj (t0 ) of node qj is increased by
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and the coupling strength between node li and node qj increased by i = 1, . . . , k, j = 1, . . . , m, and β =
k X
bi a , β j
0
xp (t0 ) where
bi . For the node with the self-loop, delete the
i=1
self-loop. 285
aj β
Step 2. (Rescaling the current state of the reduced network)
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Let {ξrl−1 +1 , . . . , ξrl } be the normalized left eigenvector of Lll , l = 1, 2, . . . , k. Suppose sl nodes are removed from the leader group Lll , l = 1, 2, . . . , k, in Step 1.
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Without loss of generality, these nodes are assumed to be nodes rl−1 +1, · · · , rl−1 +sl .
For each remaining node in leader group Lll , suppose that its current state is xc (t1 ). 290
Then, one needs to rescale the current state of the node into (1 −
sl X
ξrl−1 +j )xc (t1 ).
j=1
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We will prove the correctness of Algorithm 2 in the following theorem.
Theorem 2. Consider multi-agent network (2) with Laplacian matrix L as (3). Suppose the number of failure nodes is s (where s is a positive integer) during the time 0
interval [t0 , t0 ]. Then, the consensus property of the network (2) is retained if Algorithm 2 is implemented when the nodes fail.
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Proof: We firstly consider the failure node p is the leader. Without loss of generality, we assume the label of node p to be node 1, which belongs to the first leader group
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(Laplacian matrix is L11 ) and will be removed in Algorithm 2. Assume also that the
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Laplacian matrix of the reduced network is L011 . Suppose L11
a11 a21 = .. . ar1 1
a12
···
a22
···
···
···
ar1 2
···
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a1r1 a2r1 .. . ar1 r1
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According to the step 1 of Algorithm 2, we can find that
a a − 1ra111 31 . .. . a1r1 ar1 1 − a11
a23 −
a13 a21 a11
···
a2r1 −
a33 −
a13 a31 a11
···
a3r1
··· ar1 3 −
···
a13 ar1 1 a11
···
ar1 r1
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L011
a22 − a12 a21 a11 a32 − a12 a31 a11 = .. . a a ar1 2 − 12a11r1 1
a1r1 a21 a11
Let {ξ1 , . . . , ξr1 } be the normalized left eigenvector of L11 and {ξ20 , . . . , ξr0 1 } be the
Hence, we have
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normalized left eigenvector of L011 . Denote Φ = (ξ1 , . . . , ξr1 ) and Φ0 = (ξ20 , . . . , ξr0 1 ).
Φ · L11 = 0 and Φ0 · L011 = 0, which together with
r1 X
ξi = 1 and
r1 X
ξi0 = 1 imply that
i=2
i=1
1 ξi , i = 2, · · · , r1 . 1 − ξ1
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ξi0 =
r1 X
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It follows from Theorem 1 that the final consensus value of the first leader group is ξi xi (0). Note that rl X
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i=1
ξi x˙ i (t)
=
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i=rl−1 +1
=
−
rl X
i=rl−1 +1
0,
ξi
rl X
aij f (xj (t))
j=rl−1 +1
l = 1, 2, . . . , k.
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Hence, (4) implies that r1 X
ξi xi (t) =
i=1
r1 X i=1
ξi xi (0), ∀t > 0.
If node p fails at time t1 and step 1 of Algorithm 2 is implemented, we have r1 X i=2
ξi0 x0i (t) =
r1 X i=2
ξi0 x0i (t1 ), ∀t > t1 .
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(4)
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Then, we can obtain
=
i=2
r1 X
ξi0 (xi (t1 ) +
i=2
= = = =
r1 r1 X X 1 ai1 ( ξi xi (t1 ) + ξi x1 (t1 )) 1 − ξ1 i=2 a 11 i=2 r1 X 1 ( ξi xi (t1 ) + ξ1 x1 (t1 )) 1 − ξ1 i=2 r1 X 1 ( ξi xi (t1 )) 1 − ξ1 i=1 r1 X 1 ξi xi (0)). ( 1 − ξ1 i=1
Hence, r1 X
ξi xi (0) = (1 − ξ1 )
i=1
r1 X i=2
ξi0 x0i (t1 ) = (1 − ξ1 )
r1 X
ξi0 x0i (t).
(5)
i=2
If s nodes (assume they are 1, · · · , s) fail in the first leader group and their failure
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300
ai1 x1 (t1 )) a11
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ξi0 x0i (t1 )
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r1 X
0
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time is ti , i = 1, · · · , s, satisfying t0 ≤ t1 ≤ t2 ≤ · · · ≤ ts ≤ t0 , then repeat the process s times and we can obtain that r1 X
ξi xi (0)
=
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i=1
(s)
(s)
=
(1 − ξ1 )
r1 X
ξi0 x0i (t1 )
i=2
(1 − ξ1 )(1 − ξ20 ) (1 − ξ1 )(1 −
=
(1 − ξ1 − ξ2 ) ······
=
(1 −
s X i=1
ξi )
r1 X
ξi xi (t2 )
r1 X
ξi xi (t),
(2) (2)
i=3
i=s+1
(s)
(2) (2)
ξi xi (t2 )
i=3
r1 X ξ2 (2) (2) ) ξ x (t2 ) 1 − ξ1 i=3 i i
=
=
r1 X
(s) (s)
∀t ≥ ts ,
where {ξi , ξi+1 . . . , ξr1 }, s = 2, . . . ; i = s + 1, . . . , r1 , is the normalized left eigen-
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(s)
vector of L11 . Therefore, after implementing step 2, the new network will have the 305
same consensus property as the original one.
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Next, we shall continue our proof for the case that the failure node p is the follower. Without loss of generality, we assume the label of node p to be node rk + 1, which belongs to the first follower group Lk+1,k+1 and will be removed in Algorithm 2.
We have proved in Theorem 1 that the observed states of the followers will asymp-
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totically converge to the convex combination of {f (¯ xξˆi ), i = 1, 2, . . . , k}, where x ¯ξˆi is the consensus value of the leaders in Lii (the symbols used in this part are the same as those symbols used in the proof of Theorem 1). Let
k
ark +1,rk+1 ark +2,rk+1 , .. . ark+1 ,rk+1
···
ark +2,rk +2
···
···
···
ark+1 ,rk +2
···
1 ar +2,r +1 k − k ark +1,rk +1 Q1 = .. . ark+1 ,rk +1 − ar +1,r +1
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ark +1,rk +2
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Lk+1,k+1
ar +1,r +1 k k ar +2,r +1 k k = .. . ark+1 ,rk +1
0
0
···
0
1
0
···
0
.. .
.. .
0
0
k
18
··· ···
.. . 0
0 0 , .. . 1
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1 0 . Q2 = .. 0 0
ar +1,r +2 − ark +1,rk +1 k
k
···
ar +1,r −1 − akr +1,rk+1+1
ar +1,r − ark +1,rk+1
0
0
k
···
1 .. .
k
k
.. .
···
0
···
1
0
···
0
k+1
.
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and
.. .
0 1
310
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Let Lk+1,i = Q1 Lk+1,i Q2 , i = 1, · · · , k + 1. It can be seen that the matrices L0k+1,k+1
and L0k+1,i can be obtained by removing the first row and column of the matrices Lk+1,k+1 and Lk+1,i , respectively.
In Theorem 1, we have proved that the observed state of the follower group
=
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F (xξˆk+1 (t))
M
Lk+1,k+1 can be explicitly expressed as
=
(r +1)
[f (xξˆ k
k+1
(r
)
(t)), · · · , f (xξˆ k+1 (t))]T
[−L−1 k+1,k+1
k+1
k X
Lk+1,i F (xξˆi (t))].
i=1
easily see that after implementing step 1 of the recovery algorithm, the final convex
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315
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By some simple computation and the matrix equation Lk+1,i = Q1 Lk+1,i Q2 , we can
coefficients will be the same as the original ones, i.e., the final states of the followers
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are retained.
Remark 5. In [31], consensus recovery of linear multi-agent systems under node failure is investigated. However, in this paper, we extend the results of [31] from the
320
following three aspects:
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• The network topology in this paper is directed.
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• Compared with [31], the network recovery algorithm is proved in this paper. • The consensus recovery algorithm given in [31] cannot retain the final consen-
sus value. In this paper, the proposed recovery method can compensate for the
undesirable effects of the failure nodes, and the final consensus value is retained.
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Remark 6. Suppose the network size is N , i.e., the frequency count in Algorithm 2 is no larger than N . Moreover, in Algorithm 2, it has been shown that the consensus property is retained after the recovery procedure. It can be found that as the failure node
p is deleted, one only needs to update the initial values of the out-neighboring nodes and connection strength between in-neighboring nodes and out-neighboring nodes. Hence,
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330
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the computation complexity of the Algorithm 2 is O(N 2 ∗ N ) = O(N 3 ). Remark 7. In many real-world multi-agent networks, the information flow between
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two neighboring nodes is generally affected by many uncertain factors including network induced time delay, random packet loss, quantization and so on [11, 19, 27]. Since the effect of time delay, packet dropouts and quantization is unknown for a failure
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335
node, the algorithm used in this paper cannot be easily extended. The main difficulty
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for the problem lies in how to estimate the effect of time delay, packet dropouts and quantization for a failure node. Hence, extension of the consensus recovery method to the network with time-delays, packet dropouts, and quantization is an important
340
research topic in the future.
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Remark 8. It has been shown that fuzzy logic control is an effective approach to control many real-world multi-agent networks [20, 21]. It is an interesting problem to
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extend the consensus recovery approach to general nonlinear systems based on fuzzy dynamic models in our future work. Using detailed analysis, it can be observed that 345
the procedures to solve this problem include i) propose a fuzzy communication protocol of the multi-agent system; ii) estimate the effect of failure node based on the informa-
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tion in the fuzzy rule; iii) design the consensus recovery algorithms.
5. Numerical examples
In this section, two examples are given to illustrate the effectiveness of the theoretical results. Firstly, a simple example will be given to illustrate the consensus recovery
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350
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process.
Example 1. Consider a simple sensor network consisting of 8 sensors. Let xi (0), i =
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1, . . . , 8, be the initial states of network. f (x(t)) = [f (x1 (t)), . . . , f (x8 (t))]T with f (xi ) = x3i . Suppose the initial values are set as [−1, 3, 5, 2, 2, −3, 0, −2]T . The network topology is displayed in Fig. 2(a), and the weight of each edge is set as 1.
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355
We can see that the network is balanced and the final consensus value is the average
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state of the network (see Fig. 3(a)). Suppose node p fails at time t0 = 5 (see Fig. 2(b)). By using the consensus recovery
approach, removing node p gives the reduced graph (Fig. 2(c)). The corresponding 360
state trajectories of system (2) are shown in Fig. 3(a). It can be seen from Fig. 3(b)
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q5
q1
q4
q2
q3 q6
q7
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p
Fig. 2(a). Original network structure.
q5
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q1
p
q2
q4
q3
q7
q6
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Fig. 2(b). Node p has failure due to external attack.
that the network cannot achieve consensus since the the connectivity of the network
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is destroyed by the failure node p. Fig 3(c) shows the evolution of the network when
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the recovery procedure is implemented for the failure node p. The efficiency of the
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proposed network recovery method is well illustrated in this example.
q5
q1
q3 q4
q2
q6
Fig. 2(c). Remove node p in the network.
22
q7
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AC
CE
PT
ED
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Fig. 3(a). The state trajectories of the original network.
Fig. 3(b). The state trajectories of the network when the node p fails at time t0 = 5.
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Fig. 3(c). The state trajectories of the network when the recovery procedure is implemented.
network with only 5 nodes.
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In the following example, a network with 30 nodes will be reduced to a smaller
365
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Example 2. Consider the multi-agent system (2) with 30 nodes (Fig. 4), and the nonlinear function f is chosen as f (x) = tanh(x).
one. The reduced network contains only 5 nodes and the corresponding structure can
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370
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Firstly, based on Algorithm 1, we can reduce the original network into a smaller
be seen in Fig. 5. It can be observed that the network is not strongly connected,
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and hence consensus cannot be achieved. Moreover, according to Remark 4, we can conclude that:
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1. The nodes sets ∆1 = {1, 4, 6, 8, 10, 16, 21, 23, 25, 29}, ∆2 = {7, 12, 13, 15, 17, 18, 26, 27}, and ∆3 = {5, 9, 19, 20, 22} are the leaders in the original network.
24
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AC
CE
PT
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Fig. 4. The original graph of network (2) with 30 nodes.
Fig. 5. The reduced graph of network (2) with 5 nodes.
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After reduction, only leader nodes 1 ∈ ∆1 , 18 ∈ ∆2 and 19 ∈ ∆3 are still active in the reduced network;
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2. The final consensus states of the nodes in the set ∆i , i = 1, 2, 3, are respectively x ˆ1 (0), x ˆ18 (0) and x ˆ19 (0), where x ˆi (0), i = 1, 18, 19, are the initial values of nodes i in the reduced network. For instance, x ˆ19 (0) =
380
x20 (0) + x22 (0)) + 31 x19 (0);
1 (x5 (0) 6
+ x9 (0) +
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3. The nodes set ∆ = {2, 3, 11, 14, 24, 28, 30} are the followers in the original network;
4. The final observed state x(t) of the nodes 24 and 30 are tanh−1 ( 23 tanh(x10 (0))+ 1 tanh(x180 (0))) 3
and tanh−1 ( 51 tanh(x190 (0)) + 45 tanh(x180 (0))), respectively.
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385
The state trajectories of the original network and the reduced network are given
ED
in Fig. 6 and Fig. 7, respectively, which illustrate the effectiveness of Algorithm 1. To demonstrate the effectiveness of the consensus recovery approach, we assume that the
If the consensus recovery is not applied, we can see from Fig 8(a) that the network will
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390
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nodes 1, 2, 10 fail at time t0 = 5 and nodes 5, 22, 28 fail at time t1 = 10, respectively.
not converge to the same final state as the original network in Fig. 6. Fig 8(b) shows
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the evolution of the same network when the proposed consensus recovery algorithm is implemented. It can be found that the network has the same final state as the original network.
26
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5
final states of leaders group 4
3
network state
2
1
0
−2
−3
−4
−5
0
10
20
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−1
30 t
40
50
60
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Fig. 6. The states of the original network. 2
ED
1.5
0.5
0
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reduced network state
1
−0.5
AC
CE
−1
−1.5
−2
0
1
2
3
4
5 t
6
7
8
9
Fig. 7. The states of the reduced network.
27
10
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5 node failure time t=5 node failure time t=10 final states of leaders group in original network
4
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3
network state
2
1
0
−1
−2
−3
−5
0
10
20
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−4
30 t
40
50
60
Fig. 8(a). The states of the network without
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implementing recovery procedure.
node failure time t=5 node failure time t=10 final states of leaders group
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5
4
3
2
network state
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1
0
AC
CE
−1
−2
−3
−4
−5
0
10
20
30 t
40
50
Fig. 8(b). The states of the network when recovery procedure is implemented.
28
60
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395
6. Conclusion and discussions
In this paper, a novel consensus recovery approach has been introduced to analyze
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the consensus of nonlinear coupled multi-agent networks with node failure. The node
removing process consists of one important operation, that is to update the edge weighing and initial values. In the process of removing nodes, only local information
is used, and hence our proposed network reduction method is quite practical and easy
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400
to implement. After the consensus recovery operation, the consensus property is well retained. Theoretical consensus analysis has also been presented in this paper by fully utilizing the network structure. The theoretical results have been well illustrated by
M
using two numerical examples.
There are some difficult problems which remain unsolved for the multi-agent net-
405
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work under node failure. The consensus recovery approaches need to be extended to the network-based environment with time-delays, packet dropouts, and quantization.
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In addition, the consensus recovery method can be applied to sensor locations and
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state estimation for further investigation.
Appendix A: Proof of Theorem 1
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410
Proof: Let x = [x1 , · · · , xN ]T , x1 = [x1 , · · · , xr1 ]T , · · · , xk+m = [xrk+m−1 +1 , · · · ,
xN ]T . F (x1 ) = [f (x1 ), · · · , f (xr1 )]T , F (x2 ) = [f (xr1 +1 ), · · · , f (xr2 )]T , . . . , F (xk+m ) = [f (xrk+m−1 +1 ), · · · , f (xN )]T . The dynamics of the multi-agents network (2) can be
29
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written as: x˙ i = −Lii F (xi ), i = 1, 2, . . . , k.
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(6)
x˙ k+j = −Lk+j,1 F (x1 ) − Lk+j,2 F (x2 ) − . . . − Lk+j,k+j F (xk+j ), j = 1, 2, . . . , m. (7) Firstly, we know that the solution to system (2) exists and is unique on [0, ∞) according to the properties of f in Assumption 1. Let {ξrl−1 +1 , · · · , ξrl } be the normal-
415
xξˆl (t) =
Prl
i=rl−1 +1
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ized left eigenvector of Lll , l = 1, 2, · · · , k, with respect to the zero eigenvalue. Define ξi xi (t), xξˆl (t) = 1 ⊗ (xξˆl (t)) and xξˆl =
Prl
i=rl−1 +1
ξi xi (0). It fol-
lows from Theorem 2 of [14] that the leaders in the same strongly connected component of the network will achieve consensus and final consensus value is xξˆl , l = 1, 2, . . . , k.
M
Next, we will prove that the observed states of the followers will converge asymp-
420
ED
totically to the convex combination of {f (¯ xξˆi ), i = 1, 2, . . . , k}. Hence, it can be seen that
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−L−1 k+h,k+h
k X
Lk+h,i
L−1 k+h,k+h [Lk+h,k+h · 1] = 1.
=
i=1
(8)
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Define
AC
xξˆk+h
(r
+1)
[xξˆ k+h−1
=
F −1 [−L−1 k+h,k+h
k+h
(r
)
, · · · , xξˆ k+h ]T
=
k+h
k+h−1 X
Lk+h,i F (xξˆi )],
h = 1, 2, . . . , m.
i=1
In the following, we will prove that xξˆk+h is the final state of xk+h (t), h =
1, 2, . . . m.
30
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Since there exists at least one entry satisfying Lk+h,j 6= 0 (h = 1, 2, . . . , m, j = 425
1, 2, . . . , k + i − 1), one can easily verify that Lk+h,k+h is a M -matrix. By the property
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of M -matrix, there exists a positive definite diagonal matrix D =diag(drk+h−1 +1 , · · · , ˆ k+h,k+h = 1 (DLk+h,k+h +LTk+h,k+h D) is positive definite drk+h ) such that the matrix L 2 (see [8]). Define the Lyapunov functional as rk+h
X
=
di
Z
xi (t)
(f (s) − f (xξˆk+h ))ds,
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Vk+h (t)
i=rk+h−1 +1
(i) ξk+h
xˆ
h = 1, 2, . . . , m.
430
(9)
Obviously, Vk+h (t) ≥ 0 and Vk+h (t) = 0 if and only if xk+h (t) = xξˆk+h . Choose suffi-
1 2
k+h−1 X
(h)
ci
M
(h) (h) (h) ˆ k+h,k+h )− ciently small positive constants c1 , c2 , . . . , ck+h−1 , such that πh = λmin (L
> 0. Then, we can obtain
i=1
rk+h
=
X
i=rk+h−1 +1
CE
=
AC
=
di (f (xi (t)) − f (xξˆ
k+h
−(F (xk+h (t)) − F (xξˆk+h ))T D(
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=
rk+h (i)
ED
V˙ k+h (t)
k+h X
X
aij f (xj (t))
j=1
Lk+h,i F (xi (t)))
i=1
−(F (xk+h (t)) − F (xξˆk+h ))T DLk+h,k+h [F (xk+h (t)) − (−L−1 k+h,k+h
k+h−1 X
Lk+h,i F (xi (t)))]
i=1
−(F (xk+h (t)) − F (xξˆk+h ))T DLk+h,k+h (F (xk+h (t)) − F (xξˆk+h )) − (F (xk+h (t)) − F (xξˆk+h ))T D[
≤
))
k+h−1 X i=1
Lk+h,i (F (xi (t)) − F (xξˆi (t))]
−Ah (t) + a(h) (t),
(10)
where Ah (t) = πh (F (xk+h (t))−F (xξˆk+h ))T (F (xk+h (t))−F (xξˆk+h )), a(h) (t) =
k+h−1 X i=1
31
(h)
ai (t),
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(h)
and ai (t) = 435
1 λ (LTk+h,i DT DLk+h,i )(F (xi (t))−F (xξˆi (t)))T · 2ci max
(F (xi (t))−F (xξˆi (t))),
h = 1, 2, . . . , m.
(1)
Pk
i=1
(1)
ai (t).
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Clearly, Ah (t) is a positive definite function. For h=1, we have a(h) (t) =
According to the proof of the first part, lim ai (t) = 0. Therefore, lim a(1) (t) = 0. t→+∞
By Lemma 1, one can obtain that
lim xk+1 (t) = xξˆk+1 .
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t→+∞
t→+∞
Similarly, we can prove that
lim xk+h (t) = xξˆk+h , h = 2, . . . , m.
t→+∞
The proof is thus completed.
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A Consensus Recovery Approach to Nonlinear Multi-Agent System under Node Failure Lulu Li, Daniel W.C. Ho, Jianquan Lu PII: DOI: Reference:
S0020-0255(16)30479-0 10.1016/j.ins.2016.06.050 INS 12324
To appear in:
Information Sciences
Received date: Revised date: Accepted date:
29 July 2015 27 April 2016 28 June 2016
Please cite this article as: Lulu Li, Daniel W.C. Ho, Jianquan Lu, A Consensus Recovery Approach to Nonlinear Multi-Agent System under Node Failure, Information Sciences (2016), doi: 10.1016/j.ins.2016.06.050
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A Consensus Recovery Approach to Nonlinear Multi-Agent System under Node Failure
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Lulu Lia,b,∗, Daniel W. C. Hob , Jianquan Luc a School
of Mathematics, Hefei University of Technology, Hefei 230009, China of Mathematics, City University of Hong Kong, Hong Kong c Department of Mathematics, Southeast University, Nanjing 210096, China b Department
Abstract
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In large-scale networks, node failure is unavoidable and it generally brings undesirable effects on the stability and performance of the systems. Then a natural question arises: is it possible to cope with the problem arising from the node failure in the networks, while retaining the corresponding network property (or make the network systems robust against the
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node failure)? In this paper, a consensus recovery approach will be proposed to investigate the consensus of general directed nonlinear multi-agent networks with node failure. The objective of the consensus recovery method is to compen-
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sate for the undesirable effects of the failure nodes, and to retain the consensus property. Numerical examples are provided to demonstrate the effectiveness of the theoretical results obtained, even for large-scale multi-agent systems.
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Keywords: Multi-agent networks, Network reduction approach, Consensus
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recovery approach, Node failure
1. Introduction
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Recently, multi-agent networks, such as distributed robots and mobile sensor
networks, have been widely studied due to their broad applications ([5, 6, 24]).
✩ The work was supported by a GRF grant from HKSAR (CityU 11300415) and the National Natural Science Foundation of China under Grants 11426081, 61503115 and 61573102 ∗ Corresponding author Email addresses: [email protected] (Lulu Li), [email protected] (Daniel W. C. Ho), [email protected] (Jianquan Lu)
Preprint submitted to Information Sciences
June 29, 2016
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The main focus of studying multi-agent networks is to determine how consensus 5
emerges as a result of local interactions among agents, where consensus implies the agreement of a collection of agents on a common state value. Examples of
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typical consensus behavior include cooperation of unmanned air vehicles, flocking of birds, and swarming of fishes ([22, 25]). In the past few years, substantial
consensus or synchronization problems have been studied in much previous lit10
erature ([1, 7, 10, 11, 12, 15, 16, 17, 18, 26, 30, 32, 33, 34]). In [11], under a
new protocol and strong connected network topology, practical consensus can be achieved for multi-agent networks with time delays and quantization. In
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[32], the distributed consensus problem for a class of multiple Euler-Lagrange systems is solved under considering the time-delays and packet dropouts simul15
taneously. Meanwhile, many typical consensus problems were investigated in recent years, such as average consensus [17], cluster consensus [1], second-order consensus [13], leader-follower consensus ([3, 10]), event-based consensus [12] and references therein.
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In many realistic network systems, due to the complexity of systems and undesirable external attacks (disturbance), the occurrence of nodes (or links)
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failure is inevitable ([4, 23, 28, 31]). For instance, a sensor may not be recharged due to battery failure in a sensor network. Subsequently, this failure sensor may not be easily repaired under a hostile environment. This situation can be
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modeled by a networked multi-agent system in which the node failure occur in the process of network states evolving. Some papers have been devoted to
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studying the networked systems with link failure. In [9], an erasure model is studied, that is, the network links fail independently in space (independently of each other) and in time (link failure events are temporarily independent).
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In [31], consensus recovery of linear multi-agent systems under node failure
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is investigated. A recovery approach is proposed for the multi-agent system under external attacks. However, the proposed recovery approach ignores the importance of the theoretical analysis and cannot be applied to maintain the final consensus value. In [28], synchronization problems are investigated for complex dynamical networks subject to recoverable link failures. 2
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Devastating consequence would result due to the failure of the nodes in multi-
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agent systems. In particular, when a noncompensable failure occurs, which means replacement of the failure nodes is not feasible, the failure nodes and the
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connected links have to be isolated (or removed). The noncompensable node failures generally bring undesirable effects on the stability and performance of 40
multi-agent systems. For instance, the consensus property of multi-agent net-
works may be lost under some fatal nodes/link failures. Naturally, an important question is raised Q1: Is it possible to isolate (or remove) the failure nodes of the network and to retain the consensus property? It is a crucial, yet an open
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research problem for many real-life applications.
Moreover, with the proliferation of distributed multi-agent systems interact-
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ing through networks, the increasing complexity and higher safety demands of modern engineering systems highlight the significance of the reliability of the system. In order to improve the reliability, a recovery scheme should be designed to maintain the performance of the system. However, until now, little literature has taken this issue into account. This paper is devoted to studying
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this problem from the aspect of consensus. An efficient consensus recovery al-
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gorithm will be proposed and theoretically proved to deal with the problems arising from the node failure in network systems. Although multi-agent networks have been extensively applied in practical work, the distributed behavior of the networks do bring some essential difficulties
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in theoretical research. In particular, as the multi-agent networks began to be
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implemented for critical tasks such as military or power system applications, the risk of network failures became more of a concern. It is highly desirable to exploit effective protocols to retain the consensus of the multi-agent networks.
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Motivated by the above discussions, we aim to design a consensus recovery
approach to compensate for the undesirable effects of the failure nodes and the consensus property is retained after the recovery process. The main contributions of this paper are presented as follows: • To reduce the size of the networks, while maintaining the consensus prop3
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erty, a novel network reduction approach is proposed in this paper. The
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main advantage of this method is that, for each agent, only local information of its neighboring agents is used in the process of network size
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reduction, and it makes our reduction method practical and easy to implement in real-world networks.
• To improve the reliability of the network system and to retain the consen-
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sus property under node failure, an efficient consensus recovery approach will be provided based on the network reduction method.
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The organization of the remaining part is presented as follows. In Section 2, some basic definitions involving algebraic graph theory are introduced. In 75
Section 3, consensus analyses of nonlinear multi-agent networks are presented in detail by fully utilizing the structure of the network. In Section 4, the network recovery approach is proposed for the consensus problem of the large-size nonlinear networks under general communication topology. In Section 5, a nu-
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merical simulation is given to show the effectiveness of the theoretical results. In Section 6, concluding remarks are drawn.
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N otation : The standard notations will be used in this paper. Throughout this paper, R+ denotes the set of all positive real numbers. Rn and Rm×n represent n-dimensional Euclidean space and the set of m × n real matrices, 85
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respectively; The superscript “T ” represents the transpose. Let C 1 (Rn ; R+ )
denote the family of all continuously differentiable functions V (x) from Rn to R+ . K denotes the family of all continuous nondecreasing functions ϕ : R+ −→
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R+ such that ϕ(r) > 0 if r > 0.
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2. Some Preliminaries 2.1. Algebraic graph theory
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In this subsection, we present some basic knowledge concerning algebraic
graph theory. For more details, please refer to [2].
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Let G(V, E, A) be a weighted directed graph with the set of nodes V = {v1 , v2 , . . . , vN }, the set of edges E ⊂ V × V and a weighted adjacency matrix A = [aij ] with nonnegative adjacency elements aij . The directed graph G is called the interaction graph of the matrix A. An edge of G is denoted
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by eij = (vi , vj ), where vi and vj are called the parent and child vertices, re-
spectively. (vj , vi ) ∈ E means that node vj receives information from node vi .
For adjacency matrix A, (vi , vj ) ∈ E ⇐⇒ aji > 0 and aji is called
the weight of edge (vi , vj ). The set of in-neighbors of vi in G is denoted by 100
Nin (vi ) = {vj : (vj , vi ) ∈ E}. The set of out-neighbors of vi in G is denoted by
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Nout (vi ) = {vj : (vi , vj ) ∈ E}. The Laplacian of the directed graph is defined PN as L = ∆ − A, where ∆ = [∆ij ]N ×N is a diagonal matrix with ∆ii = j=1 aij .
It is easy to show that L has at least one zero eigenvalue with a corresponding eigenvector 1, where 1 = (1, 1, . . . , 1)T .
A path in a digraph is an ordered sequence of vertices such that any two
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consecutive vertices are an directed edge of the digraph. A directed tree is a
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directed graph, where every vertex, except one which is called the root, has exactly one parent. A subgraph Gs of a directed graph is a directed graph 110
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such that the vertex set V(Gs ) ⊂ V(G) and the edge set E(Gs ) ⊂ E(G). If V(Gs ) = V(G), Gs is called the spanning subgraph. A directed tree that is a spanning subgraph of G is called the spanning tree of G. A directed graph G is
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strongly connected if there is a path for any pair of distinct vertices in G. A matrix A is called nonnegative if all its entries are nonnegative, which is
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denoted by A 0. We say nonnegative matrix A is irreducible if the interaction
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graph of A is strongly connected.
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2.2. Basic definition and lemmas Definition 1. (consensus). Consider a multi-agent network A with N agents. Let xi denote the state of agent i. We say that multi-agent network A reaches a consensus if and only if xi = xj for all i, j = 1, 2, . . . , N . If for all xi (0) ∈ R, i =
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1, 2, . . . , N , xi (t) converges to some common equilibrium point x∗ (depending on the initial values of the agents), as t −→ +∞, then we say that multi-agent 5
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network A solves a consensus problem asymptotically. The common value x∗ is called the group decision value.
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of Theorem 1.
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In the following, we give an important lemma, which will be used in the proof
Lemma 1. Consider an n-dimensional ordinary differential equation ˙ x(t) = f (x(t)), x(0) = x0 ,
(1)
x = (x1 , . . . , xn )T , f = (f1 , f2 , . . . , fn )T ∈ C(Rn ; Rn ). Suppose initial value
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problem (1) has an unique solution on [0, +∞). If there exists positive definite function V (x) ∈ C 1 (Rn ; R+ ) satisfying V (0) = 0 such that V˙ (x(t)) |(1) = −W (x(t)) + a(t),
where lim a(t) = 0, W (x(t)) is a positive function, i.e., W (x) ≥ 0 and W (x) = t→+∞
0 if and only if x = 0, then lim x(t) = 0.
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t→+∞
Proof: The proof of Lemma 1 is similar to the proof of Lyapunov stability
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theorem. Thus it is omitted here.
Remark 1. It should be noted that we do not require the derivative of Lyapunov function (energy function) V (x) to be negative, which is a standard
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requirement for Lyapunov asymptotic stability theory. Generally, in Lemma 1, the term a(t) results from the exterior disturbances. Lemma 1 can be used to
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deal with the consensus and synchronization problem of many dynamical sys135
tems and networks, and to ensure the final states of the whole system converge
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as t −→ +∞.
3. Consensus analysis of general multi-agent networks In this section, we will study the nonlinear consensus protocol for multi-
agent networks under general communication structure, which can also be used
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140
to illustrate the correctness and efficiency of the network reduction approach. Consider the following nonlinear multi-agent networks model X dxi (t) = aij [f (xj (t)) − f (xi (t))], i = 1, . . . , N, dt
(2)
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j∈Ni
where xi (t) ∈ R is the state of the agent i, f (·) is a nonlinear function with the same dimensions of xi satisfying f (0) = 0.
Assumption 1. Throughout this paper, one requires the nonlinear function f (·) to be continuous and strictly monotonically increasing on R.
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For any communication structure, Laplacian matrix can always be written in the following form after certain permutations [29]. Without loss of generality, we assume the Laplacian matrix of model (2) to be L in the form of ···
0
···
0
···
0
···
0
0
0
···
0
.. .
.. .
.. .
Lkk
0
0
···
0
Lk+1,k
Lk+1,k+1
0
···
0
.. .
.. .
.. .
Lk+m,k
Lk+m,k+1
Lk+m,k+2
0
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L11 0 .. . L = 0 Lk+1,1 .. . Lk+m,1
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··· ···
···
···
.. .
···
.. .
··· ···
Lk+m,k+m
,
(3)
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where Lii are irreducible square matrices and in each line, there exist at least one entry satisfying Lk+i,j 6= 0 (i = 1, 2, . . . , m, j = 1, 2, . . . , k + i − 1). In the following, the nodes, known as leaders, are the strongly connected compo-
nents of the network, which is represented by Lii (i = 1, 2, . . . , k). The other nodes
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are served as followers. It can be seen from the network topology that there is no information channel from the followers to the leader. Hence, the followers cannot af-
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fect the states of the leader in the network. In the following, f (xi (t)) is called the observed states of the agent i. Some detailed explanations will be given in Remark 2. As the fundamental of recovery approach, the following theorem reveals a general 155
consensus result about the nonlinear directed multi-agent networks, which can be seen
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as an extension of the main result of [14].
Theorem 1. Consider multi-agent network (2) with Laplacian matrix L. The following conclusions can be obtained:
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(i) The leaders in Lii will achieve consensus separately;
(ii) the observed states of the followers will asymptotically converge to the convex
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leaders in Lii .
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combination of {f (¯ xξˆi ), i = 1, 2, . . . , k}, where x ¯ξˆi is the consensus value of the
Proof: A comprehensive proof can be found in Appendix A and here we shall give an
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outline of the proof.
Outline of the proof: We take three steps for the remaining part of the proof.
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1. LaSalle’s invariant principle is utilized to obtain the conclusion (i). 2. For the followers in the strongly connected component Lk+l,k+l (l = 1, 2, . . . , m), the coefficients of convex combination in part (ii) of Theorem 1 can be determined.
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3. We will construct appropriate function Vk+l (t) satisfying Lemma 1, and then
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part (ii) of Theorem 1 can be proved.
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Remark 2. An interesting result obtained in Theorem 1 is that the observed states
of the followers converge asymptotically to the convex combination of observed con-
sensus value of the leaders, while different followers may have different combination 175
coefficients. In addition, related to previous results [14], when k = 1, we can obtain
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the condition to guarantee the consensus of the model (2). Moreover, the network
cannot reach consensus if k ≥ 2, i.e., the network does not contain a rooted spanning tree. Then the following corollary can be obtained.
if and only if the directed communication topology of the network contains a rooted spanning tree.
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Corollary 1. Under Assumption 1, multi-agent network (2) can realize consensus
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4. Consensus recovery approach A general consensus result about multi-agent network (2) has been given in Section
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3. However, in many realistic network systems, node failure is a common phenomenon and sometimes the failure nodes are non-repairable. Under this circumstance, these nodes should be removed from the network, but the most important property of the
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185
system (consensus here) should be retained. From this aspect, it is necessary to propose an effective method to deal with such a node failure problem. In other words, an efficient method should be developed to retain the consensus property.
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In the next section, we will firstly present a novel network reduction method to
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reduce the size of the network, such that the consensus property of the large-size
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network can be obtained by studying that of the derived small-size network. This network reduction method can make the consensus analysis much easier and also the computational cost much lower. Based on the network reduction algorithm, we will 195
further discuss a consensus recovery method for multi-agent system with node failure.
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If the node p is removed from the network, one straightforward approach for re-
taining the consensus property is to make the information of the node p retained by its neighboring nodes. For example, consider the network structure in Fig. 1. Clearly, the states of nodes qj (j = 1, 2, 3) is affected by the state of node p directly and the states of nodes li (i = 1, 2) through intermediate node p. If the node p is removed, naturally,
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the connection strength between the node li and node qj should be increased (Fig. 1) and the initial value of node qj should be updated to retain the consensus property of
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the network. Now, the remaining question is how to update the initial value of node qj and the connection strength between the node li and node qj . This problem will be solved based on Theorem 1.
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Next, we propose a network reduction algorithm to reduce the size of the network
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while retaining the consensus property. Algorithm 1:
For the multi-agent network A, suppose its original graph is
G(V, E). The following algorithm can be used to reduce the size of the network and 210
meanwhile retain the consensus property:
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Fig. 1. Remove node p in the network.
Step 1. (Reducing the size of the network)
(a) For a node p in the network A, suppose that there exist m connections to the nodes q1 , q2 , . . . , qm and k connections from nodes l1 , l2 , . . . , lk to the node p. Suppose
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that the connection strength from node p to node qj is aj and the connection strength from node li to node p is bi . Removing node p and its connected edges yields a new
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network Ap . Keep the original initial value and coupling unchanged but increase the initial value of node qj by bi a , β j
bi a , b1 +b2 j
k X
bi . For example,
i=1
i = 1, 2, j = 1, 2, 3.
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in Fig.1, cij =
xp (0) and increase the coupling strength between node li
where i = 1, . . . , k, j = 1, . . . , m, and β =
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and node qj by
aj β
(b) For the node with the self-loop, delete the self-loop.
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(c) Repeat Step 1 (a) and (b) until no node can be reduced. Step 2. (Rescaling the initial value of the reduced network) After Step 1, the nodes in the reduced network can be divided into two classes.
The first class contains nodes with zero in-degree and the second class (maybe empty)
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225
contains nodes with non-zero in-degree but zero out-degree. Suppose that each node in the first class of the reduced network has initial value
k X
ξi xi (0). Then, one needs
i=1
k k X 1X ξi xi (0), where γ = ξi . γ i=1 i=1
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to rescale the initial value of the node into
Remark 3. Following the above mentioned reduction process, it should be emphasized that only local information is used in the process of network size reduction. When the node p is removed, one only needs to update the initial values of the out-
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230
neighboring nodes qj and connection strength between in-neighboring nodes li and out-neighboring nodes qj (i = 1, . . . , k, j = 1, . . . , m).
The logical relationship between the reduced small-size network and the original
ˆ suppose that ri is the zero in-degree Remark 4. For the final reduced network A,
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one is presented in the following remark.
node with initial value x ˆri (0) and node qj is the zero out-degree node with the con-
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nection strength aij from node ri to node qj , where i = 1, . . . , k, j = 1, . . . , m. Then, the following facts about the multi-agent network A can be obtained from the reduced
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ˆ network A:
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1. the set of nodes ∆i = {ri−1 + 1, . . . , ri } (i = 1, . . . , k) are strongly connected components of graph G(V, E);
2. the nodes in these strongly connected components are the leaders of the network A and each leader in the network A must belong to one of the nodes set ∆i (i = 1, . . . , k);
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3. the leaders in the same strongly connected component will reach consensus and
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the consensus value x ¯ξˆi is equal to x ˆri (0);
network A and its final state is f −1 [ Pk
1
i=1
aij
k X i=1
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4. nodes qj , j = 1, . . . , m, in the reduced network Aˆ are the followers of the original aij f (¯ xξˆi )];
5. based on the above two conclusions of Remark 4, one can conclude that:
ˆ the • if k = 1, i.e., there is only one leader in the final reduced network A,
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multi-agent network can achieve consensus;
• if k 6= 1 in the reduced network, the multi-agent network will achieve k different consensus values which are respectively decided by leader sets
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∆i , i = 1, . . . , k. Further explanation will be given in Example 2.
To answer the question Q1, based on the network reduction method discussed
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above, we will propose a network recovery approach to solve the node failure problem.
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Considering that in some real networks, the nodes can only store the current state other than the initial state, the network recovery algorithm in the following will use
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the current state information other than initial value information. Let {ξrl−1 +1 , . . . , ξrl } be the normalized left eigenvector of Lll , l = 1, 2, . . . , k,
260
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with respect to the zero eigenvalue. Suppose agent p fails at time t0 due to external disturbance (see Fig.1). If node p is the leader to be removed, the network has to
retain the information of node p. Hence, the local information of node p should be
adjusted as that in Step 1 of Algorithm 1. As for the followers, the edge weighting
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information between the neighboring nodes should also be adjusted for retaining the convex combination coefficients in the reduced network. Suppose that (i) there exist m
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connections to the nodes q1 , q2 , . . . , qm and k connections from nodes l1 , l2 , . . . , lk to the node p; (ii) the connection strength from node p to node qj is aj and the connection
strength from node li to node p is bi . We have the following algorithm to remove the 270
failure nodes of the network and retain the consensus property.
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Algorithm 2: For the multi-agent network A, suppose its original graph is G(V, E). Step 1. (Removing the failure nodes of the network)
Suppose the number of failure nodes is s (where s is a positive integer) during the 0
time interval [t0 , t1 ]. For any node p which fails at time t0 , there are two cases we should consider:
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Case 1: (Node p is the follower.) Removing node p and its connected edges yields a new network Ap . The coupling strength between node li and node qj is increased by where i = 1, . . . , k, j = 1, . . . , m, and β =
k X
bi . For the node with the self-loop,
i=1
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bi a , β j
delete the self-loop. For example, in Fig.1, cij =
bi a , b1 +b2 j
i = 1, 2, j = 1, 2, 3.
Case 2: (Node p is the leader.) Adjust the local information of p in the re-
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0
duced network, i.e., the current state xqj (t0 ) of node qj is increased by
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and the coupling strength between node li and node qj increased by i = 1, . . . , k, j = 1, . . . , m, and β =
k X
bi a , β j
0
xp (t0 ) where
bi . For the node with the self-loop, delete the
i=1
self-loop. 285
aj β
Step 2. (Rescaling the current state of the reduced network)
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Let {ξrl−1 +1 , . . . , ξrl } be the normalized left eigenvector of Lll , l = 1, 2, . . . , k. Suppose sl nodes are removed from the leader group Lll , l = 1, 2, . . . , k, in Step 1.
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Without loss of generality, these nodes are assumed to be nodes rl−1 +1, · · · , rl−1 +sl .
For each remaining node in leader group Lll , suppose that its current state is xc (t1 ). 290
Then, one needs to rescale the current state of the node into (1 −
sl X
ξrl−1 +j )xc (t1 ).
j=1
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We will prove the correctness of Algorithm 2 in the following theorem.
Theorem 2. Consider multi-agent network (2) with Laplacian matrix L as (3). Suppose the number of failure nodes is s (where s is a positive integer) during the time 0
interval [t0 , t0 ]. Then, the consensus property of the network (2) is retained if Algorithm 2 is implemented when the nodes fail.
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Proof: We firstly consider the failure node p is the leader. Without loss of generality, we assume the label of node p to be node 1, which belongs to the first leader group
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(Laplacian matrix is L11 ) and will be removed in Algorithm 2. Assume also that the
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Laplacian matrix of the reduced network is L011 . Suppose L11
a11 a21 = .. . ar1 1
a12
···
a22
···
···
···
ar1 2
···
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a1r1 a2r1 .. . ar1 r1
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According to the step 1 of Algorithm 2, we can find that
a a − 1ra111 31 . .. . a1r1 ar1 1 − a11
a23 −
a13 a21 a11
···
a2r1 −
a33 −
a13 a31 a11
···
a3r1
··· ar1 3 −
···
a13 ar1 1 a11
···
ar1 r1
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L011
a22 − a12 a21 a11 a32 − a12 a31 a11 = .. . a a ar1 2 − 12a11r1 1
a1r1 a21 a11
Let {ξ1 , . . . , ξr1 } be the normalized left eigenvector of L11 and {ξ20 , . . . , ξr0 1 } be the
Hence, we have
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normalized left eigenvector of L011 . Denote Φ = (ξ1 , . . . , ξr1 ) and Φ0 = (ξ20 , . . . , ξr0 1 ).
Φ · L11 = 0 and Φ0 · L011 = 0, which together with
r1 X
ξi = 1 and
r1 X
ξi0 = 1 imply that
i=2
i=1
1 ξi , i = 2, · · · , r1 . 1 − ξ1
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ξi0 =
r1 X
ED
It follows from Theorem 1 that the final consensus value of the first leader group is ξi xi (0). Note that rl X
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i=1
ξi x˙ i (t)
=
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i=rl−1 +1
=
−
rl X
i=rl−1 +1
0,
ξi
rl X
aij f (xj (t))
j=rl−1 +1
l = 1, 2, . . . , k.
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Hence, (4) implies that r1 X
ξi xi (t) =
i=1
r1 X i=1
ξi xi (0), ∀t > 0.
If node p fails at time t1 and step 1 of Algorithm 2 is implemented, we have r1 X i=2
ξi0 x0i (t) =
r1 X i=2
ξi0 x0i (t1 ), ∀t > t1 .
16
(4)
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Then, we can obtain
=
i=2
r1 X
ξi0 (xi (t1 ) +
i=2
= = = =
r1 r1 X X 1 ai1 ( ξi xi (t1 ) + ξi x1 (t1 )) 1 − ξ1 i=2 a 11 i=2 r1 X 1 ( ξi xi (t1 ) + ξ1 x1 (t1 )) 1 − ξ1 i=2 r1 X 1 ( ξi xi (t1 )) 1 − ξ1 i=1 r1 X 1 ξi xi (0)). ( 1 − ξ1 i=1
Hence, r1 X
ξi xi (0) = (1 − ξ1 )
i=1
r1 X i=2
ξi0 x0i (t1 ) = (1 − ξ1 )
r1 X
ξi0 x0i (t).
(5)
i=2
If s nodes (assume they are 1, · · · , s) fail in the first leader group and their failure
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300
ai1 x1 (t1 )) a11
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ξi0 x0i (t1 )
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r1 X
0
ED
time is ti , i = 1, · · · , s, satisfying t0 ≤ t1 ≤ t2 ≤ · · · ≤ ts ≤ t0 , then repeat the process s times and we can obtain that r1 X
ξi xi (0)
=
AC
CE
PT
i=1
(s)
(s)
=
(1 − ξ1 )
r1 X
ξi0 x0i (t1 )
i=2
(1 − ξ1 )(1 − ξ20 ) (1 − ξ1 )(1 −
=
(1 − ξ1 − ξ2 ) ······
=
(1 −
s X i=1
ξi )
r1 X
ξi xi (t2 )
r1 X
ξi xi (t),
(2) (2)
i=3
i=s+1
(s)
(2) (2)
ξi xi (t2 )
i=3
r1 X ξ2 (2) (2) ) ξ x (t2 ) 1 − ξ1 i=3 i i
=
=
r1 X
(s) (s)
∀t ≥ ts ,
where {ξi , ξi+1 . . . , ξr1 }, s = 2, . . . ; i = s + 1, . . . , r1 , is the normalized left eigen-
17
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(s)
vector of L11 . Therefore, after implementing step 2, the new network will have the 305
same consensus property as the original one.
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Next, we shall continue our proof for the case that the failure node p is the follower. Without loss of generality, we assume the label of node p to be node rk + 1, which belongs to the first follower group Lk+1,k+1 and will be removed in Algorithm 2.
We have proved in Theorem 1 that the observed states of the followers will asymp-
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totically converge to the convex combination of {f (¯ xξˆi ), i = 1, 2, . . . , k}, where x ¯ξˆi is the consensus value of the leaders in Lii (the symbols used in this part are the same as those symbols used in the proof of Theorem 1). Let
k
ark +1,rk+1 ark +2,rk+1 , .. . ark+1 ,rk+1
···
ark +2,rk +2
···
···
···
ark+1 ,rk +2
···
1 ar +2,r +1 k − k ark +1,rk +1 Q1 = .. . ark+1 ,rk +1 − ar +1,r +1
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ark +1,rk +2
M
ED
Lk+1,k+1
ar +1,r +1 k k ar +2,r +1 k k = .. . ark+1 ,rk +1
0
0
···
0
1
0
···
0
.. .
.. .
0
0
k
18
··· ···
.. . 0
0 0 , .. . 1
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1 0 . Q2 = .. 0 0
ar +1,r +2 − ark +1,rk +1 k
k
···
ar +1,r −1 − akr +1,rk+1+1
ar +1,r − ark +1,rk+1
0
0
k
···
1 .. .
k
k
.. .
···
0
···
1
0
···
0
k+1
.
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and
.. .
0 1
310
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Let Lk+1,i = Q1 Lk+1,i Q2 , i = 1, · · · , k + 1. It can be seen that the matrices L0k+1,k+1
and L0k+1,i can be obtained by removing the first row and column of the matrices Lk+1,k+1 and Lk+1,i , respectively.
In Theorem 1, we have proved that the observed state of the follower group
=
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F (xξˆk+1 (t))
M
Lk+1,k+1 can be explicitly expressed as
=
(r +1)
[f (xξˆ k
k+1
(r
)
(t)), · · · , f (xξˆ k+1 (t))]T
[−L−1 k+1,k+1
k+1
k X
Lk+1,i F (xξˆi (t))].
i=1
easily see that after implementing step 1 of the recovery algorithm, the final convex
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315
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By some simple computation and the matrix equation Lk+1,i = Q1 Lk+1,i Q2 , we can
coefficients will be the same as the original ones, i.e., the final states of the followers
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are retained.
Remark 5. In [31], consensus recovery of linear multi-agent systems under node failure is investigated. However, in this paper, we extend the results of [31] from the
320
following three aspects:
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• The network topology in this paper is directed.
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• Compared with [31], the network recovery algorithm is proved in this paper. • The consensus recovery algorithm given in [31] cannot retain the final consen-
sus value. In this paper, the proposed recovery method can compensate for the
undesirable effects of the failure nodes, and the final consensus value is retained.
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Remark 6. Suppose the network size is N , i.e., the frequency count in Algorithm 2 is no larger than N . Moreover, in Algorithm 2, it has been shown that the consensus property is retained after the recovery procedure. It can be found that as the failure node
p is deleted, one only needs to update the initial values of the out-neighboring nodes and connection strength between in-neighboring nodes and out-neighboring nodes. Hence,
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330
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the computation complexity of the Algorithm 2 is O(N 2 ∗ N ) = O(N 3 ). Remark 7. In many real-world multi-agent networks, the information flow between
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two neighboring nodes is generally affected by many uncertain factors including network induced time delay, random packet loss, quantization and so on [11, 19, 27]. Since the effect of time delay, packet dropouts and quantization is unknown for a failure
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335
node, the algorithm used in this paper cannot be easily extended. The main difficulty
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for the problem lies in how to estimate the effect of time delay, packet dropouts and quantization for a failure node. Hence, extension of the consensus recovery method to the network with time-delays, packet dropouts, and quantization is an important
340
research topic in the future.
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Remark 8. It has been shown that fuzzy logic control is an effective approach to control many real-world multi-agent networks [20, 21]. It is an interesting problem to
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extend the consensus recovery approach to general nonlinear systems based on fuzzy dynamic models in our future work. Using detailed analysis, it can be observed that 345
the procedures to solve this problem include i) propose a fuzzy communication protocol of the multi-agent system; ii) estimate the effect of failure node based on the informa-
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tion in the fuzzy rule; iii) design the consensus recovery algorithms.
5. Numerical examples
In this section, two examples are given to illustrate the effectiveness of the theoretical results. Firstly, a simple example will be given to illustrate the consensus recovery
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350
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process.
Example 1. Consider a simple sensor network consisting of 8 sensors. Let xi (0), i =
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1, . . . , 8, be the initial states of network. f (x(t)) = [f (x1 (t)), . . . , f (x8 (t))]T with f (xi ) = x3i . Suppose the initial values are set as [−1, 3, 5, 2, 2, −3, 0, −2]T . The network topology is displayed in Fig. 2(a), and the weight of each edge is set as 1.
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355
We can see that the network is balanced and the final consensus value is the average
AC
state of the network (see Fig. 3(a)). Suppose node p fails at time t0 = 5 (see Fig. 2(b)). By using the consensus recovery
approach, removing node p gives the reduced graph (Fig. 2(c)). The corresponding 360
state trajectories of system (2) are shown in Fig. 3(a). It can be seen from Fig. 3(b)
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q5
q1
q4
q2
q3 q6
q7
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p
Fig. 2(a). Original network structure.
q5
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q1
p
q2
q4
q3
q7
q6
M
Fig. 2(b). Node p has failure due to external attack.
that the network cannot achieve consensus since the the connectivity of the network
ED
is destroyed by the failure node p. Fig 3(c) shows the evolution of the network when
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the recovery procedure is implemented for the failure node p. The efficiency of the
AC
CE
proposed network recovery method is well illustrated in this example.
q5
q1
q3 q4
q2
q6
Fig. 2(c). Remove node p in the network.
22
q7
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AC
CE
PT
ED
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Fig. 3(a). The state trajectories of the original network.
Fig. 3(b). The state trajectories of the network when the node p fails at time t0 = 5.
23
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Fig. 3(c). The state trajectories of the network when the recovery procedure is implemented.
network with only 5 nodes.
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In the following example, a network with 30 nodes will be reduced to a smaller
365
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Example 2. Consider the multi-agent system (2) with 30 nodes (Fig. 4), and the nonlinear function f is chosen as f (x) = tanh(x).
one. The reduced network contains only 5 nodes and the corresponding structure can
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370
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Firstly, based on Algorithm 1, we can reduce the original network into a smaller
be seen in Fig. 5. It can be observed that the network is not strongly connected,
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and hence consensus cannot be achieved. Moreover, according to Remark 4, we can conclude that:
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1. The nodes sets ∆1 = {1, 4, 6, 8, 10, 16, 21, 23, 25, 29}, ∆2 = {7, 12, 13, 15, 17, 18, 26, 27}, and ∆3 = {5, 9, 19, 20, 22} are the leaders in the original network.
24
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AC
CE
PT
ED
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Fig. 4. The original graph of network (2) with 30 nodes.
Fig. 5. The reduced graph of network (2) with 5 nodes.
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After reduction, only leader nodes 1 ∈ ∆1 , 18 ∈ ∆2 and 19 ∈ ∆3 are still active in the reduced network;
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2. The final consensus states of the nodes in the set ∆i , i = 1, 2, 3, are respectively x ˆ1 (0), x ˆ18 (0) and x ˆ19 (0), where x ˆi (0), i = 1, 18, 19, are the initial values of nodes i in the reduced network. For instance, x ˆ19 (0) =
380
x20 (0) + x22 (0)) + 31 x19 (0);
1 (x5 (0) 6
+ x9 (0) +
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3. The nodes set ∆ = {2, 3, 11, 14, 24, 28, 30} are the followers in the original network;
4. The final observed state x(t) of the nodes 24 and 30 are tanh−1 ( 23 tanh(x10 (0))+ 1 tanh(x180 (0))) 3
and tanh−1 ( 51 tanh(x190 (0)) + 45 tanh(x180 (0))), respectively.
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385
The state trajectories of the original network and the reduced network are given
ED
in Fig. 6 and Fig. 7, respectively, which illustrate the effectiveness of Algorithm 1. To demonstrate the effectiveness of the consensus recovery approach, we assume that the
If the consensus recovery is not applied, we can see from Fig 8(a) that the network will
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390
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nodes 1, 2, 10 fail at time t0 = 5 and nodes 5, 22, 28 fail at time t1 = 10, respectively.
not converge to the same final state as the original network in Fig. 6. Fig 8(b) shows
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the evolution of the same network when the proposed consensus recovery algorithm is implemented. It can be found that the network has the same final state as the original network.
26
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5
final states of leaders group 4
3
network state
2
1
0
−2
−3
−4
−5
0
10
20
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−1
30 t
40
50
60
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Fig. 6. The states of the original network. 2
ED
1.5
0.5
0
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reduced network state
1
−0.5
AC
CE
−1
−1.5
−2
0
1
2
3
4
5 t
6
7
8
9
Fig. 7. The states of the reduced network.
27
10
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5 node failure time t=5 node failure time t=10 final states of leaders group in original network
4
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3
network state
2
1
0
−1
−2
−3
−5
0
10
20
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−4
30 t
40
50
60
Fig. 8(a). The states of the network without
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implementing recovery procedure.
node failure time t=5 node failure time t=10 final states of leaders group
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5
4
3
2
network state
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1
0
AC
CE
−1
−2
−3
−4
−5
0
10
20
30 t
40
50
Fig. 8(b). The states of the network when recovery procedure is implemented.
28
60
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395
6. Conclusion and discussions
In this paper, a novel consensus recovery approach has been introduced to analyze
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the consensus of nonlinear coupled multi-agent networks with node failure. The node
removing process consists of one important operation, that is to update the edge weighing and initial values. In the process of removing nodes, only local information
is used, and hence our proposed network reduction method is quite practical and easy
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400
to implement. After the consensus recovery operation, the consensus property is well retained. Theoretical consensus analysis has also been presented in this paper by fully utilizing the network structure. The theoretical results have been well illustrated by
M
using two numerical examples.
There are some difficult problems which remain unsolved for the multi-agent net-
405
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work under node failure. The consensus recovery approaches need to be extended to the network-based environment with time-delays, packet dropouts, and quantization.
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In addition, the consensus recovery method can be applied to sensor locations and
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state estimation for further investigation.
Appendix A: Proof of Theorem 1
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410
Proof: Let x = [x1 , · · · , xN ]T , x1 = [x1 , · · · , xr1 ]T , · · · , xk+m = [xrk+m−1 +1 , · · · ,
xN ]T . F (x1 ) = [f (x1 ), · · · , f (xr1 )]T , F (x2 ) = [f (xr1 +1 ), · · · , f (xr2 )]T , . . . , F (xk+m ) = [f (xrk+m−1 +1 ), · · · , f (xN )]T . The dynamics of the multi-agents network (2) can be
29
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written as: x˙ i = −Lii F (xi ), i = 1, 2, . . . , k.
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(6)
x˙ k+j = −Lk+j,1 F (x1 ) − Lk+j,2 F (x2 ) − . . . − Lk+j,k+j F (xk+j ), j = 1, 2, . . . , m. (7) Firstly, we know that the solution to system (2) exists and is unique on [0, ∞) according to the properties of f in Assumption 1. Let {ξrl−1 +1 , · · · , ξrl } be the normal-
415
xξˆl (t) =
Prl
i=rl−1 +1
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ized left eigenvector of Lll , l = 1, 2, · · · , k, with respect to the zero eigenvalue. Define ξi xi (t), xξˆl (t) = 1 ⊗ (xξˆl (t)) and xξˆl =
Prl
i=rl−1 +1
ξi xi (0). It fol-
lows from Theorem 2 of [14] that the leaders in the same strongly connected component of the network will achieve consensus and final consensus value is xξˆl , l = 1, 2, . . . , k.
M
Next, we will prove that the observed states of the followers will converge asymp-
420
ED
totically to the convex combination of {f (¯ xξˆi ), i = 1, 2, . . . , k}. Hence, it can be seen that
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−L−1 k+h,k+h
k X
Lk+h,i
L−1 k+h,k+h [Lk+h,k+h · 1] = 1.
=
i=1
(8)
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Define
AC
xξˆk+h
(r
+1)
[xξˆ k+h−1
=
F −1 [−L−1 k+h,k+h
k+h
(r
)
, · · · , xξˆ k+h ]T
=
k+h
k+h−1 X
Lk+h,i F (xξˆi )],
h = 1, 2, . . . , m.
i=1
In the following, we will prove that xξˆk+h is the final state of xk+h (t), h =
1, 2, . . . m.
30
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Since there exists at least one entry satisfying Lk+h,j 6= 0 (h = 1, 2, . . . , m, j = 425
1, 2, . . . , k + i − 1), one can easily verify that Lk+h,k+h is a M -matrix. By the property
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of M -matrix, there exists a positive definite diagonal matrix D =diag(drk+h−1 +1 , · · · , ˆ k+h,k+h = 1 (DLk+h,k+h +LTk+h,k+h D) is positive definite drk+h ) such that the matrix L 2 (see [8]). Define the Lyapunov functional as rk+h
X
=
di
Z
xi (t)
(f (s) − f (xξˆk+h ))ds,
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Vk+h (t)
i=rk+h−1 +1
(i) ξk+h
xˆ
h = 1, 2, . . . , m.
430
(9)
Obviously, Vk+h (t) ≥ 0 and Vk+h (t) = 0 if and only if xk+h (t) = xξˆk+h . Choose suffi-
1 2
k+h−1 X
(h)
ci
M
(h) (h) (h) ˆ k+h,k+h )− ciently small positive constants c1 , c2 , . . . , ck+h−1 , such that πh = λmin (L
> 0. Then, we can obtain
i=1
rk+h
=
X
i=rk+h−1 +1
CE
=
AC
=
di (f (xi (t)) − f (xξˆ
k+h
−(F (xk+h (t)) − F (xξˆk+h ))T D(
PT
=
rk+h (i)
ED
V˙ k+h (t)
k+h X
X
aij f (xj (t))
j=1
Lk+h,i F (xi (t)))
i=1
−(F (xk+h (t)) − F (xξˆk+h ))T DLk+h,k+h [F (xk+h (t)) − (−L−1 k+h,k+h
k+h−1 X
Lk+h,i F (xi (t)))]
i=1
−(F (xk+h (t)) − F (xξˆk+h ))T DLk+h,k+h (F (xk+h (t)) − F (xξˆk+h )) − (F (xk+h (t)) − F (xξˆk+h ))T D[
≤
))
k+h−1 X i=1
Lk+h,i (F (xi (t)) − F (xξˆi (t))]
−Ah (t) + a(h) (t),
(10)
where Ah (t) = πh (F (xk+h (t))−F (xξˆk+h ))T (F (xk+h (t))−F (xξˆk+h )), a(h) (t) =
k+h−1 X i=1
31
(h)
ai (t),
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(h)
and ai (t) = 435
1 λ (LTk+h,i DT DLk+h,i )(F (xi (t))−F (xξˆi (t)))T · 2ci max
(F (xi (t))−F (xξˆi (t))),
h = 1, 2, . . . , m.
(1)
Pk
i=1
(1)
ai (t).
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Clearly, Ah (t) is a positive definite function. For h=1, we have a(h) (t) =
According to the proof of the first part, lim ai (t) = 0. Therefore, lim a(1) (t) = 0. t→+∞
By Lemma 1, one can obtain that
lim xk+1 (t) = xξˆk+1 .
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t→+∞
t→+∞
Similarly, we can prove that
lim xk+h (t) = xξˆk+h , h = 2, . . . , m.
t→+∞
The proof is thus completed.
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