Bipartite coordination problems on networks of multiple mobile agents

Author's Accepted Manuscript

Bipartite coordination problems on networks of multiple mobile agents Deyuan Meng, Yingmin Jia, Junping Du

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S0016-0032(15)00302-6 http://dx.doi.org/10.1016/j.jfranklin.2015.07.009 FI2399

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Journal of the Franklin Institute

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4 December 2014 2 June 2015 11 July 2015

Cite this article as: Deyuan Meng, Yingmin Jia, Junping Du, Bipartite coordination problems on networks of multiple mobile agents, Journal of the Franklin Institute, http: //dx.doi.org/10.1016/j.jfranklin.2015.07.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Bipartite Coordination Problems on Networks of Multiple Mobile Agents Deyuan Meng, Yingmin Jia, and Junping Du

Abstract Learning via iterative or repeated implementation is an intelligent method which takes full advantage of experience data from previous iterations or repetitions in the control signals computation to improve the current system performance. In this paper, we incorporate the idea of iterative learning to deal with bipartite coordination problems for multiple mobile agents in networked environments that are described by signed directed graphs. We aim at high-precision bipartite coordination tasks for networked mobile agents subject to a time-varying reference whose information is only available to a portion of agents. To achieve this objective, we construct iterative learning algorithms for agents using the nearest neighbor rule and address the related asymptotic stability and monotonic convergence issues for them. We establish convergence conditions and the guarantees to their feasibility. In particular, we develop a class of linear matrix inequality conditions, as well as providing formulas for the design of gain matrices. We perform simulations to illustrate the effectiveness of the proposed algorithms in enabling mobile agents to achieve high-precision bipartite coordination on networks associated with signed directed graphs. Index Terms Iterative learning; Bipartite coordination; Mobile agents; Networked environments; Convergence.

I. I NTRODUCTION Networks of multiple mobile agents as well as their distributed control have attracted considerable research interests due to their wide applications in many fields, such as spacecrafts, Deyuan Meng and Yingmin Jia are with the Seventh Research Division, Beihang University (BUAA), Beijing 100191, China. They are also with the School of Automation Science and Electrical Engineering, Beihang University (BUAA), Beijing 100191, China (e-mail: [email protected], [email protected]). Junping Du is with the Beijing Key Laboratory of Intelligent Telecommunications Software and Multimedia, School of Computer Science and Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: [email protected]).

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urban traffic control and unmanned aerial vehicles. A key feature of multi-agent networks is that the agents’ coordination tasks are realized by employing only local knowledge in the form of the relative state or output information between every agent and its nearest neighbors (see, e.g., [1]-[5]). By benefiting from this feature, they can make contributions to proposing distributed algorithms in [1]-[5]. Moreover, there are some well investigated coordination problems on multiagent networks, such as relative formation and consensus seeking (see, e.g., the surveys [6]-[8] and references therein). In addition to the class of coordination problems considered for multi-agent networks in [1][5], there have been discussed another class of coordination problems that require the agents to complete coordination tasks in a desired manner. The leader-following coordination is one of such considered problems for multi-agent networks, by which the agents follow a prescribed leader to seek coordination (see, e.g., [9]-[14]). Another issue on multi-agent networks of desired coordination requirements is to guarantee the agents to move by following any prescribed reference (see, e.g., [15], [16]). Through the leader or reference designed properly in [9]-[16], the coordination of networked agents can be ensured to behave certain desired performances asymptotically with respect to time. However, these control methods may not work effectively when it comes to the class of desired coordination tasks with high-precision achieving requirements for all the time, e.g., see [17] for trajectory-keeping of formation satellites, [18] for trajectory tracking of running trains and [19] for harmonic currents compensation of power networks. To address the high-precision achieving coordination problems in [17]-[19], iterative learning control (ILC) is found as a good approach mainly due to its ability in refining “perfect tracking” system performances over any fixed time interval (see, e.g., the surveys [20]-[22] for more details). In particular, ILC is a class of data-driven methods which play a practically important role in many industrial areas (see, e.g., [23], [24] for detailed discussions). It has been shown that ILC can be incorporated into the algorithm design of each agent by employing only relative state or output information between it and its neighbors (see, e.g., [25]-[33]). They have contributed to developing ILC-based algorithms for coordination tasks of formation keeping in [25]-[29] and consensus tracking in [30]-[33], respectively. Moreover, ILC-based algorithms not only create a fundamentally two-dimensional structure with respect to independent time and iteration axes but also are in distributed manners with local state or output information only needed in their constructions, which are different from the classical ILC algorithms requiring the global state

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or output information of the controlled system as well as an available reference trajectory as prior knowledge. In particular, it has been verified that the ILC-based algorithms are effective in refining high-precision coordination system performances though there exist no references or leaders prescribed as prior knowledge for any of the agents (see, e.g., [27], [28], [31]). Nevertheless, the aforementioned results ([1]-[5], [9]-[19], [25]-[33]) are all established for multi-agent networks associated with graphs which require the relations between agents to be cooperative (i.e., the adjacency weighted matrices of graphs to be nonnegative). There are practical networked environments of agents that admit both cooperative and antagonistic interactions among them (see, e.g., [34]-[37]). Such networks are a kind of “singed networks” associated with singed graphs that have edges with both positive and negative weights to reflect the cooperative and antagonistic interactions between agents, respectively. In [34], [35], it has extended the idea of consensus to a class of “bipartite consensus” for signed networks, which aims at enabling all agents to reach consensus regarding a certain quantity that is the same in modulus but not in sign. Moreover, the bipartite consensus can be developed to other class of coordination problems on signed networks, e.g., see [36] for bipartite formation and [37] for bipartite flocking. However, there have been no studies presented on applying the ILC-based algorithms to deal with highprecision coordination problems on signed networks. In this paper, we aim at providing design guidelines for ILC-based distributed algorithms to address high-precision bipartite coordination problems on signed networks with a time-varying reference trajectory that is available to only a portion of agents. It is shown that the bipartite coordination tasks can be achieved under the action of ILC-based distributed algorithms which are designed through taking advantage of the topology structure of signed networks and the information of the specified reference. We study two kinds of output coordination problems of agents: formation keeping and consensus tracking. Moreover, we solve both asymptotic stability and monotonic convergence issues on the ILC-based distributed algorithms, and develop convergence conditions for them. In particular, we can give a class of linear matrix inequality (LMI) conditions, together with formulas for the gain matrices design. Our established results can be viewed as a significant extension of the existing ILC results for multi-agent networks of, e.g., [25]-[33] to general signed multi-agent networks. Through the numerical simulations, we illustrate the effectiveness of the proposed ILC-based distributed algorithms in refining outputs of agents to realize both bipartite consensus tracking and bipartite formation keeping objectives

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by considering a class of six-agent signed networks with a time-varying reference trajectory. The remainder parts of this paper are organized as follows. In Section II, we present the considered coordination problems on signed networks with a time-varying reference trajectory. In Section III, we establish the ILC-based distributed algorithms and study their asymptotic stability, based on which we further develop their monotonic convergence to achieve good learning transients in Section IV. We give simulations and conclusions in Sections V and VI, respectively. Before going to Section II, we end the introduction by introducing some notations and preliminaries related to the graph theory. Notations: ZN = {0, 1, · · · , N}, In = {1, 2, · · · , n}, 1n = [1, 1, · · · , 1]T ∈ Rn , I and 0 denote the identity and null matrices with the required dimensions, respectively, diag{·} denotes a block diagonal matrix with zero off-diagonal elements, and a star  in symmetric block matrices denotes a term induced by the symmetry. For any matrices or vectors A and B, ρ (A) is the spectral radius of a square matrix A, A ≥ 0 is a nonnegative matrix with all nonnegative elements, A ≺ 0 (respectively, A  0) is a negative-definite (respectively, positive-definite) matrix, and A ⊗B is the Kronecker product of A and B. For a discrete-time vector function z(t), q is a shift operator, i.e., qz(t) = z(t + 1) and q−1 z(t) = z(t − 1), and z(t) is the L2 -norm of z(t). For any real number a, sign(a) is its sign function, i.e.,

⎧ ⎪ 1, if a > 0 ⎪ ⎪ ⎪ ⎨ sign(a) = 0, if a = 0 ⎪ ⎪ ⎪ ⎪ ⎩ − 1, if a < 0.

Preliminaries in graph theory: A signed directed graph G (A ) is given by a triple G (A ) = (V , E , A ) that consists of a vertex set V = {vi : i ∈ In }, an edge set E ⊆ {(vi , v j ) : vi , v j ∈ V },   and an adjacency matrix A = ai j ∈ Rn×n for the signed weights of G (A ), i.e., ai j = 0 ⇔ (v j , vi ) ∈ E . Assume that there are no self-loops in G (A ), i.e., a ii = 0, ∀i ∈ In . If there is a directed edge (v j , vi ) ∈ E , then v j is a neighbor of vi . We denote the index set for the neighbors of vi as Ni = { j : (v j , vi ) ∈ E }. A path of G (A ) is a sequence of edges in E : (vil , vil+1 ) ∈ E for l = 1, 2, · · · , j − 1, where vi1 , vi2 , · · · , vi j are distinct vertices. When there exists a special vertex that can be connected to all other vertices through paths, G (A ) is said to have a spanning tree. Let G (|A |) = (V , E , |A |). It shares the same vertices and edges with G (A ) but the weights of its edges are, respectively, the absolute values of the weights of edges in G (A ). Hence, the

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Laplacian matrix L|A | of G (|A |) is given by [38] 

∑

L|A | = diag

j∈N1

|a1 j |,

∑

|a2 j |, · · · ,

j∈N2

II. P LANTS



AND

∑

|an j | − |A |.

j∈Nn

P ROBLEMS

Consider networks with n agents that are labeled 1 through n, where the ith agent is regarded as a corresponding vertex vi in the signed directed graph G (A ). From [25]-[33], we know that the use of iterative learning can help refine the high precision coordination performances for multi-agent networks over any fixed time interval. Let t ∈ Z N and k ∈ Z+ denote the time and learning (via iterative or repeated implementation) indexes, respectively. For i ∈ I n , the agent vi is assumed to have the following dynamics over t ∈ Z N and k ∈ Z+ : ⎧ ⎨ xi,k (t + 1) = Axi,k (t) + Bui,k (t) ⎩

(1)

yi,k (t) = Cxi,k (t), xi,k (0) = xi0

where xi,k (t) ∈ Rnx , ui,k (t) ∈ Rnu and yi,k (t) ∈ Rny are the state, protocol (or input) and output, respectively, and A, B and C are matrices of appropriate dimensions. Without loss of generality, we consider CB of full-row rank to meet the basic requirement of our following established learning algorithms on the system structure of agents (see also [30], [31]). We are interested in high-precision output formation keeping problems. Formally speaking, there exits σi ∈ {1, −1} such that all gents to achieve the bipartite output formation objective for t ∈ ZN /{0}  {1, 2, · · · , N} as   lim σi yi,k (t) − σidi (t) − r(t) = 0, k→∞

∀i ∈ In

(2)

where r(t) is the desired time-varying reference trajectory prescribed for a portion of agents, and di (t) is the desired deviation trajectory between the output of v i and r(t) or −r(t). Remark 1: For the objective (2), there can happen two cases distinguished by the value of parameter σi . Let S1 = {i : σi = 1} and S2 = {i : σi = −1}. For any given i ∈ In , if i ∈ S1 , then (2) is equal to

  lim yi,k (t) − di(t) = r(t),

k→∞

t ∈ ZN /{0}

i.e., the output of vi follows the reference trajectory r(t) with a desired deviation di (t) over t ∈ ZN /{0}; otherwise, if i ∈ S2 , then (2) is equal to   lim yi,k (t) − di(t) = −r(t), k→∞

t ∈ ZN /{0}

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i.e., the vi ’s output follows the negative or symmetric trajectory −r(t) of the prescribed reference r(t) with a desired deviation di (t) over t ∈ ZN /{0}. This implies that the output formations of agents are separated into two symmetric groups and can be regulated by prescribing the reference trajectory. In particular, if S1 = In happens, then (2) will become an output formation keeping problem in the special case where the directed graph G (A ) has a nonnegative adjacency weight matrix, i.e., A ≥ 0. Hence, (2) can be considered as a significant extension of the problems treated in, e.g., [28]-[31] to general signed networks. For the particular case where there exist no desired deviations between the outputs of agents and the reference, i.e., di (t) = 0, ∀i, t, the considered output formation keeping objective (2) collapses into an output consensus tracking problem. That is, there exits σ i ∈ {1, −1} such that all gents achieve the bipartite consensus tracking objective for t ∈ Z N /{0} as   lim σi yi,k (t) − r(t) = 0,

k→∞

∀i ∈ In .

(3)

As stated in Remark 1, if i ∈ S1 , then the output of the agent vi follows r(t); and otherwise, if i ∈ S2 , then vi ’s output follows the symmetric trajectory −r(t) of the prescribed reference r(t) for all t ∈ ZN /{0}. To achieve our objectives, we consider a class of structurally balanced networks. The notion of structural balance is introduced for convenience as follows [34]. Definition 1: Let {V1 , V2 } be a bipartition of V , where V1 ∪ V2 = V and V1 ∩ V2 = Ø. Then a signed directed graph G (A ) is said to be structurally balanced if there exists a bipartition of the vertices such that ai j ≥ 0 for ∀vi , v j ∈ Vl (l ∈ {1, 2}) and ai j ≤ 0 for ∀vi ∈ Vl , v j ∈ Vq , l = q (l, q ∈ {1, 2}). III. A SYMPTOTIC S TABILITY R ESULTS We first consider how to achieve the output formation keeping problem (2). To this end, let us denote δ yi,k (t) = yi,k (t) − di (t) as the output formation error of the agent vi relative to the desired reference trajectory, and let ωi be an index that indicates the accessibility of r(t) by the agent vi (i.e., if r(t) is accessible by vi , then ωi = 0; and otherwise, ωi = 0). With the nearest

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neighbor rule, we present a distributed iterative learning algorithm as 

 ui,k+1 (t) = ui,k (t) + K ∑ ai j δ y j,k (t + 1) − sign ai j δ yi,k (t + 1) j∈Ni



+ ωi r(t + 1) − sign (ωi ) δ yi,k (t + 1) ,

(4) ∀i ∈ In

where K is a unified nu × ny gain matrix for all agents, sign(a i j ) is the sign of ai j , and ui,0 (t) is an initial input for the agent vi that can be prescribed for ∀t ∈ ZN−1 . For the multi-agent system (1) under the algorithm (4), we denote η i,k (t) = xi,k+1 (t) − xi,k (t) as the state error of the agent vi over two sequential iterations. Then from (1) and (4), we can obtain that ηi,k (t) satisfies

ηi,k (t + 1) = A[xi,k+1 (t) − xi,k (t)] + B[ui,k+1(t) − ui,k (t)] 

 = Aηi,k (t) + BK ∑ ai j δ y j,k (t + 1) − sign ai j δ yi,k (t + 1) j∈Ni

+ ωi r(t + 1) − sign (ωi ) δ yi,k (t + 1)



(5)

which can be combined to deduce that δ yi,k (t) satisfies

δ yi,k+1 (t + 1) = δ yi,k (t + 1) +C[xi,k+1 (t + 1) − xi,k (t + 1)] = δ yi,k (t + 1) +C ηi,k (t + 1) = δ yi,k (t + 1) +CAηi,k (t) +CBK



∑



 ai j δ y j,k (t + 1) − sign ai j δ yi,k (t + 1)

j∈Ni



+ ωi r(t + 1) − sign (ωi ) δ yi,k (t + 1) . (6)

T

 T T (t), · · · , η T (t) and δ yk (t) = δ yT1,k (t), δ yT2,k (t), · · · , For (5) and (6), let ηk (t) = η1,k (t), η2,k n,k

T δ yTn,k (t) . Hence, (5) can be rewritten in a compact form as

ηk (t + 1) = (I ⊗ A)ηk (t) − [(L + |Ω|) ⊗ BK]δ yk(t + 1) + (Ω ⊗ BK)[1n ⊗ r(t + 1)]

(7)

where Ω = diag{ω1 , ω2 , · · · , ωn } denotes the matrix associated with the accessibility of r(t) by the agents in G (A ), and L = [li j ] ∈ Rn×n has elements in the form of ⎧   ⎪ ⎪ ⎨ ∑ ai j  , j = i li j = j∈Ni ⎪ ⎪ ⎩ − ai j , j = i.

(8)

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       In fact, if we denote Δ = ∑ j∈N1 a1 j  , ∑ j∈N2 a2 j  , · · · , ∑ j∈Nn an j  , then L = Δ − A is the Laplacian matrix of the signed directed graph G (A ) (see [34] for more details). Also, we can rewrite (5) as

δ yk+1 (t + 1) = (I ⊗CA)ηk (t) + [I − (L + |Ω|) ⊗CBK]δ yk (t + 1) + (Ω ⊗CBK)[1n ⊗ r(t + 1)]. (9) For (7) and (9), we can easily derive ηk (0) = 0 for all k ∈ Z+ . Note, however, that there exist terms of r(t) in (7) and (9), in addition to those of δ y k (t) and ηk (t). To overcome this problem, we view the desired reference trajectory r(t) as a special vertex that does not get information from any agents in G (A ) but has accessibility associated   with Ω by the agents in G (A ). We define a signed directed graph G A that contains G (A ) ⎤ ⎡ A Ω1n ⎦. A= ⎣ 0 0   For the structure balance of G A , we have the following equivalent result.    Lemma 1: A signed directed graph G A is structurally balanced if and only if there exists a

and this special vertex, where

diagonal matrix D = diag{σ1 , σ2 , · · · , σn } with σi ∈ {1, −1} for all i ∈ In such that DA D = |A | and DΩ = |Ω|.

  By following [34], we can easily validate that a signed directed graph G A is structurally

 = diag{σ1 , σ2 , · · · , σn+1 } with σi ∈ balanced if and only if there exists a diagonal matrix D    AD  = A. With this equivalent result, the sufficiency is {1, −1} for all i ∈ In+1 such that D  = diag{D, 1}, since it leads to immediate if we take D ⎤ ⎡ ⎤ ⎡   DA D DΩ1n |A | |Ω| 1n AD =⎣ ⎦=⎣ ⎦ = A . D 0 0 0 0    AD  = A as For the necessity, we can rewrite D ⎤ ⎡ ⎤ ⎡ |A | |Ω| 1n D A D1 D1 Ω1n σn+1 ⎦=⎣ ⎦ ⎣ 1 0 0 0 0

(10)

where D1 = diag{σ1 , σ2 , · · · , σn }. If σn+1 = 1, we take D = D1 , and can thus obtain from (10) that DA D = |A | and DΩ = |Ω|. Otherwise, if σn+1 = −1, then we take D = −D1 , and the use

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of (10) can also yield DA D = (−D1 )A (−D1 ) = D1 A D1 = |A | DΩ = −D1 Ω = D1 Ωσn+1 = |Ω| . The proof is complete.

  Remark 2: Based on Lemma 1, we can validate that G A is structurally balanced if and    only if G (A ) is structurally balanced, and one partition of G (A ) is a partition of G A and the other partition of G (A ) together with the special vertex r(t) forms another partition of   G A , where the bipartition of graphs is the same as defined in Definition 1.   With Lemma 1, we can give the structure balance of G A by considering Laplacian matrices of signed directed graphs.

   Lemma 2: A signed directed graph G A is structurally balanced if and only if there exists a

diagonal matrix D = diag{σ1 , σ2 , · · · , σn } with σi ∈ {1, −1} for all i ∈ In such that DLD = L|A | and DΩ = |Ω|, where L|A | is the Laplacian matrix of the directed graph G (|A |). Based on Lemma 1, we only need to prove DLD = L|A | ⇔ DA D = |A |. Note that DΔD = Δ, L = Δ − A by (8), and L|A | = Δ − |A |. We have DLD = L|A | ⇔ DΔD − DA D = L|A | ⇔ DΔD − DA D = Δ − |A | ⇔ DA D = |A | . The proof can be completed.  be the Laplacian matrix of the signed Remark 3: In the same way as defined in (8), let L     directed graph G A . We can also obtain that a signed directed graph G A is structurally  = diag{σ1 , σ2 , · · · , σn+1 } with σi ∈ balanced if and only if there exists a diagonal matrix D   = L {1, −1} for all i ∈ In+1 such that D LD

.  A

In the remainder analysis, we consider such a diagonal matrix D determined in Lemma 1 or   2 for the signed directed graph G A when there can occur no confusions. Now, we can present the following formation results for the objective (2). Theorem 1: For the multi-agent system (1) with a desired reference trajectory r(t), let the   signed directed graph G A be structurally balanced. If the algorithm (4) is applied, then the output formation keeping objective (2) holds if and only if there exits a gain matrix K satisfying  ρ I − (L|A | + |Ω|) ⊗CBK < 1. (11)

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To make the proof of Theorem 1 clear to follow, we propose another useful lemma. Lemma 3: Let ek (t) = 1n ⊗ r(t + 1) − (D ⊗ I)δ yk (t + 1). Then the formation keeping objective (2) holds if and only if limk→∞ ek (t) = 0 for t ∈ ZN−1 . By the definitions of ek (t) and δ yk (t), we can see that limk→∞ ek (t) = 0 for t ∈ ZN−1 holds if and only if     lim σi δ yi,k (t + 1) − r(t + 1) = lim σi yi,k (t + 1) − σi di (t + 1) − r(t + 1)

k→∞

k→∞

= 0,

∀i ∈ In ,t ∈ ZN−1

i.e., (2) holds. This completes the proof. Using Lemmas 1-3, we can prove Theorem 4 as follows.

   [Proof of Theorem 1] Since the signed directed graph G A is structurally balanced, it follows from Lemma 1 that we have a diagonal matrix D = diag{σ 1 , σ2 , · · · , σn } with σi ∈ {1, −1} for all i ∈ In such that DA D = |A | and DΩ = |Ω|. Hence, if we denote ξk (t) = (D ⊗ I)ηk (t), then we can obtain with (7) that

ξk (t + 1) = (I ⊗ A)ξk (t) − [(DL + D|Ω|) ⊗ BK]δ yk(t + 1) + (DΩ ⊗ BK)[1n ⊗ r(t + 1)] = (I ⊗ A)ξk (t) − [(DLD + D|Ω|D) ⊗ BK](D ⊗ I)δ yk(t + 1) + [(L|A | + DΩ) ⊗ BK][1n ⊗ r(t + 1)]

(12)

= (I ⊗ A)ξk (t) + [(L|A | + |Ω|) ⊗ BK][1n ⊗ r(t + 1) − (D ⊗ I)δ yk (t + 1)] = (I ⊗ A)ξk (t) + [(L|A | + |Ω|) ⊗ BK]ek (t) where we also use DLD = L|A | by Lemma 2, L|A | 1n = 0 by [38], D|Ω|D = |Ω|, and DΩ = |Ω|. With these facts, we can deduce from (9) that ek+1 (t) = 1n ⊗ r(t + 1) − (D ⊗ I)δ yk+1 (t + 1) = −(D ⊗ I)(I ⊗CA)ηk (t) + 1n ⊗ r(t + 1) − [D ⊗ I − (DL + D|Ω|) ⊗CBK]δ yk (t + 1) − (DΩ ⊗CBK)[1n ⊗ r(t + 1)] = −(I ⊗CA)(D ⊗ I)ηk (t) + 1n ⊗ r(t + 1) − [DD ⊗ I − (DLD + D|Ω|D) ⊗CBK] × (D ⊗ I)δ yk (t + 1) − [(L|A | + DΩ) ⊗CBK][1n ⊗ r(t + 1)] = −(I ⊗CA)ξk (t) + [I − (L|A | + |Ω|) ⊗CBK][1n ⊗ r(t + 1) − (D ⊗ I)δ yk (t + 1)] = −(I ⊗CA)ξk (t) + [I − (L|A | + |Ω|) ⊗CBK]ek(t). (13)

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By following [20], [21], we can use the lifting technique to obtain e k = eTk (0), eTk (1), · · · ,

T eTk (N − 1) , and thus can develop from (12) and (13) that ek+1 = Hek , where H is a block lower triangular matrix with block diagonal entries in terms of I − (L |A | + |Ω|) ⊗CBK. Based on the standard matrix analysis theory [39], we know that lim k→∞ ek = 0, which is equivalent to limk→∞ ek (t) = 0 for t ∈ ZN−1 , holds if and only if ρ (H) < 1, which is equivalent to the condition (11). With this fact and based on Lemma 3, the objective (2) can be achieved if and only if the condition (11) can be guaranteed. This proof can be completed. For the output consensus tracking problem (3), we can directly obtain a distributed iterative learning algorithm from (4) as ui,k+1 (t) = ui,k (t) + K



∑



 ai j y j,k (t + 1) − sign ai j yi,k (t + 1)

j∈Ni



+ ωi r(t + 1) − sign (ωi ) yi,k (t + 1) ,

(14) ∀i ∈ In .

It is not difficult to see that if A ≥ 0 particularly holds, then (4) and (14) collapse into classical distributed iterative learning algorithms similar to those of [25]-[33]. Thus, in contrast to them, (4) and (14) broaden the application of distributed iterative learning algorithms to general signed multi-agent networks. Similar to Theorem 1, the following theorem gives consensus results for the objective (3). Theorem 2: For the multi-agent system (1) with a desired reference trajectory r(t), let the   signed directed graph G A be structurally balanced. If the algorithm (14) is applied, then the output consensus tracking objective (3) holds if and only if there exits a gain matrix K satisfying (11). In the same way as the derivations of (7) and (9), we can deduce that if the algorithm (14) is applied to the multi-agent system (1), then we have

ηk (t + 1) = (I ⊗ A)ηk (t) − [(L + |Ω|) ⊗ BK]yk(t + 1) + (Ω ⊗ BK)[1n ⊗ r(t + 1)]

(15)

yk+1 (t + 1) = (I ⊗CA)ηk (t) + [I − (L + |Ω|) ⊗CBK]yk (t + 1) + (Ω ⊗CBK)[1n ⊗ r(t + 1)] (16) where yk (t) is defined in the same way as ηk (t). Based on (15) and (16), we can obtain the objective (3) if and only if the condition (11) holds, which can be proved by following the same lines as the proof of Theorem 1 and thus is omitted here.

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For the convergence condition (11) in Theorems 1 and 2, we can present a necessary and sufficient guarantee to its satisfaction.

  Theorem 3: There exits a gain matrix K satisfying (11) if and only if G A has a spanning

tree.

              Since G A  and G A share the same vertices and edges, we know that G A has   a spanning tree if and only if G A has a spanning tree. By noting ⎤ ⎡ L|A | + |Ω| −|Ω|1n ⎦ L  = ⎣ A  0 0 we know that all eigenvalues of L|A | + |Ω| (i.e., the eigenvalues of L  except the only zero A      eigenvalue) have positive real-parts if and only if G A has a spanning tree [38]. With this fact, we can develop the remainder of this proof by following the same way as used in the proof of [30, Theorem 12], which is thus not detailed here. Remark 4: From Theorems 1–3, it is clear that we can achieve bipartite coordination results under the structurally balanced networks which are only required to have a spanning tree. It is interesting that the considered agents cooperate and/or compete to work through networks associated with signed graphs, whereas the given convergence condition depends on graphs whose edge weights are nonnegative. This makes it possible to design the developed algorithm in the same way as in the popular but particular case that multi-agent networks are associated with graphs with all nonnegative edge weights (see, e.g., [28]-[31]). Thus, the results of Theorems 1–3 are much more general, especially in comparison with those of [28]-[31]. IV. M ONOTONIC C ONVERGENCE R ESULTS In Theorems 1 and 2, we discuss how to achieve the formation keeping objective (2) and the consensus tracking objective (3), respectively. However, the convergence performance related to their processes is not considered, which plays an important role in all kinds of iterative learning algorithms [20]-[22]. In general, monotonic convergence is one of the most desirable objectives to guarantee good learning behavior and prevent high-overshoot or large transient growth. This observation motivates us to further realize the monotonic convergence for the resulting iterative learning processes of the multi-agent system (1) under the distributed algorithms (4) and (14), respectively.

13

For the formation keeping objective (2), we consider its monotonic convergence in the sense that ek+1 (t) < ek (t), ∀k ∈ Z+ by noting Lemma 3, where ek (t) is the L2 -norm of ek (t). We again consider (12) and (13) to arrive at   ek+1 (t) = C(qI − A)−1 B + D ek (t)

(17)

where

 B = L|A | + |Ω| ⊗ BK  C = −I ⊗CA, D = I − L|A | + |Ω| ⊗CBK.   If the H∞ -norm condition C(qI − A)−1 B + D∞ < 1 holds, then we can obtain from (17) that   ek+1 (t) ≤ C(qI − A)−1 B + D∞ ek (t) (18) < ek (t), ∀k ∈ Z+ A = I ⊗ A,

and

 k ek (t) ≤ C(qI − A)−1 B + D∞ e0 (t) →0

as k → ∞.

(19)

  In addition, the following lemma to achieve the H∞ -norm condition C(qI − A)−1 B + D∞ < 1 can be presented. Lemma 4: For the state-space system (A, B, C, D), if there exists a positive-definite matrix Q  0 such that

⎡

−Q







⎤

⎢ ⎥ ⎢QAT −Q   ⎥ ⎢ ⎥ ⎢ ⎥≺0 T ⎢B ⎥ 0 −I  ⎣ ⎦ 0 CQ D −I

  then C(qI − A)−1 B + D∞ < 1 holds.

A consequence of bounded real lemma for discrete systems (see, e.g., [31]). With the above development, we give the following theorem for the objective (2) and its monotonic convergence. Theorem 4: For the multi-agent system (1) with a desired reference trajectory r(t), let the   signed directed graph G A be structurally balanced. With the algorithm (4) being applied, if

14

there exist a positive-definite matrix Q  0 and a matrix X that satisfy the following LMI: ⎤ ⎡ −Q    ⎥ ⎢  ⎥ ⎢Q I ⊗ AT −Q   ⎥ ⎢ (20) ⎥≺0 ⎢ ⎥ ⎢ Ψ 0 −I  1 ⎦ ⎣ 0 − (I ⊗CA) Q Ψ2 −I 

 T  Ψ1 = − I ⊗ X T L|A | + |Ω| ⊗ BT   Ψ2 = I + L|A | + |Ω| ⊗CB (I ⊗ X )

where

(21)

then the output formation keeping objective (2) holds with its limit being approached monotonically in the sense that ek+1 (t) < ek (t), ∀k ∈ Z+ . If the LMI (20) is feasible, then the gain matrix can be computed by K = −X .

(22)

  By (21) and (22), we can use Lemma 4 to obtain for (17) that C(qI − A)−1 B + D∞ < 1 holds under the LMI condition (20). By noting (18) and (19), we can deduce based on Lemma 3 that we not only can accomplish (2) but also can guarantee its monotonic convergence in the sense that ek+1 (t) < ek (t), ∀k ∈ Z+ . The proof is complete. For the consensus tracking objective (3), let εk (t) = 1n ⊗ r(t + 1) − (D ⊗ I)yk (t + 1), and we consider its monotonic convergence in the sense that ε k+1 (t) < εk (t), ∀k ∈ Z+ . This can work since we can easily validate that limk→∞ εk (t) = 0 for t ∈ ZN−1 if and only if the consensus tracking objective (3) holds. Toward this end, we employ (15) and (16) to obtain

ξk (t + 1) = (I ⊗ A)ξk (t) + [(L|A | + |Ω|) ⊗ BK]εk(t) εk+1 (t) = −(I ⊗CA)ξk (t) + [I − (L|A | + |Ω|) ⊗CBK]εk(t) which implies

  εk+1 (t) = C(qI − A)−1 B + D εk (t)

(23)

where A, B, C and D are the same matrices as defined in (17), and ξk (t) = (D ⊗ I)ηk (t) as used in (12) and (13). With (23) and in the same way as used in the establishment of Theorem 4, we can develop the following theorem for the monotonic convergence of the objective (3). Theorem 5: For the multi-agent system (1) with a desired reference trajectory r(t), let the    signed directed graph G A be structurally balanced. With the algorithm (14) being applied,

15

if there exist a positive-definite matrix Q  0 and a matrix X that satisfy the LMI (20), then the output consensus tracking objective (3) holds with its limit being approached monotonically in the sense that εk+1 (t) < εk (t), ∀k ∈ Z+ , where the gain matrix K can be computed by (22). By the Schur’s complement formula, we can easily validate that (11) is a necessary condition for the LMI condition (20) in Theorems 4 and 5. This, together with Theorem 3, leads to the following necessity result immediately.

   Corollary 1: The LMI (20) is feasible only if G A has a spanning tree.

In addition to Corollary 1, we know also that the LMI (20) requires the stability of the agents’ system matrix A (i.e., all its eigenvalues lie strictly within the unit circle). If this condition on A can not be met, then the result of Theorem 4 can not be achieved. An alternative way to solve this problem is to make the protocol of each agent have two parts: a pure feedback to stabilize its plant and a distributed learning controller to better improve its system performance iteratively through interacting with other agents. Here, we assume that (A, B) is stabilizable, and consider an iterative learning algorithm as fb

ui,k (t) = ui,k (t) + udl i,k (t),

∀i ∈ In

(24)

where for the output formation keeping objective (2), we apply fb

ui,k (t) = K1 xi,k (t) dl udl i,k+1 (t) = ui,k (t) + K2



∑



 ai j δ y j,k (t + 1) − sign ai j δ yi,k (t + 1)

j∈Ni



+ ωi r(t + 1) − sign (ωi ) δ yi,k (t + 1) ,

(25)

∀i ∈ In .

In (25), K1 ∈ Rnu ×nx and K2 ∈ Rnu ×ny are the gain matrices, and sign(ai j ) and ui,0 (t) are the same as in (4). Thus, in addition to Theorem 4, we present the following output formation keeping results by applying (24) and (25). Theorem 6: For the multi-agent system (1) with a desired reference trajectory r(t), let the   signed directed graph G A be structurally balanced. With the algorithm (24) and (25) being applied, if there exist a positive-definite matrix Q  0 and matrices X and Y that satisfy the

16

following LMI:

⎤ −(I ⊗ Q)    ⎥ ⎢ ⎥ ⎢(I ⊗ Q) I ⊗ AT + Ψ −(I ⊗ Q)   3 ⎥ ⎢ ⎥≺0 ⎢ ⎥ ⎢ 0 −I  Ψ 1 ⎦ ⎣ 0 − (I ⊗CA) (I ⊗ Q) + Ψ4 Ψ2 −I ⎡



 T  Ψ1 = − I ⊗ X T L|A | + |Ω| ⊗ BT   Ψ2 = I + L|A | + |Ω| ⊗CB (I ⊗ X )   Ψ3 = I ⊗Y T I ⊗ BT

where

(26)

(27)

Ψ4 = − (I ⊗CB) (I ⊗Y ) then the output formation keeping objective (2) holds with its limit being approached monotonically in the sense that ek+1 (t) < ek (t), ∀k ∈ Z+ . If the LMI (26) is feasible, then the gain matrices can be computed by K1 = Y Q−1 , K2 = −X .

(28)

By replacing A with A + BK1 , we can establish this theorem based on Lemma 4 in the same way as performed in the derivation of Theorem 4, which is omitted here. If we consider the output consensus tracking objective (3), then we apply the iterative learning algorithm (24) with fb ui,k (t) = K1 xi,k (t) dl udl i,k+1 (t) = ui,k (t) + K2



∑



 ai j y j,k (t + 1) − sign ai j yi,k (t + 1)

j∈Ni



+ ωi r(t + 1) − sign (ωi ) yi,k (t + 1) ,

(29)

∀i ∈ In

and can thus present the following consensus tracking results. Theorem 7: For the multi-agent system (1) with a desired reference trajectory r(t), let the   signed directed graph G A be structurally balanced. With the algorithm (24) and (29) being applied, if there exist a positive-definite matrix Q  0 and matrices X and Y that satisfy the LMI (26), then the output consensus tracking objective (3) holds with its limit being approached monotonically in the sense that ε k+1 (t) < εk (t), ∀k ∈ Z+ , where the gain matrices K1 and K2 can be computed by (28).

17

Similar to Corollary 1, we can give the following necessity guarantee to the LMI condition (26) in Theorems 6 and 7.

  Corollary 2: The LMI (26) is feasible only if G A has a spanning tree.

Remark 5: By Theorems 4–7, we not only can achieve the bipartite coordination objectives for multi-agent networks but also can guarantee monotonic convergence for their resulting iterative learning processes. This makes the system performance of all agents become better and better from iteration to iteration and also improves the basic results of Theorem 1. However, it is worth pointing out that more information of the agents’ plant is required to establish Theorems 4–7. This coincides with the fact that the more prior knowledge of plants is known, the better system performance can be refined by iterative learning methods (see also [20], [21]). In contrast to the existing results (see, e.g., [29], [31]), Theorems 4–7 demonstrate that monotonically convergent ILC design can be extended to deal with coordination tasks of signed networks. To make the proposed algorithm (24) and (25) (or (29)) be effectively implemented, we finally introduce the following implementation procedures. 1) Let the tolerance ε > 0, the reference trajectory r(t), ∀t ∈ ZN , and the initial control input ui,0 (t), ∀t ∈ ZN , ∀i ∈ In be given. 2) Let k = 0, and go to the step 3) to start iteration. 3) Apply ui,k (t) to the multi-agent system (1), and measure the output y i,k (t), ∀t ∈ ZN , ∀i ∈ In . 4) Compute the error ek (t) (or εk (t)), ∀t ∈ ZN−1 . If maxt∈ZN−1 ek (t) > ε (or maxt∈ZN−1 εk (t) >

ε ), then go to the next step 5). Otherwise, go to the final step 8). 5) Use the conditions (26)–(28) to calculate gain matrices K1 and K2 .   6) Apply the algorithm (24) and (25) (or (29)) in the directed graph G A to calculate the control input ui,k+1 (t), ∀t ∈ ZN , ∀i ∈ In . 7) Let k = k + 1, and return to the step 3). 8) Stop the iteration.

18

-1

v1

1 -1

r(t)

1

v6

1

v2 -1 v3

1 v5

1 1

v4

Fig. 1: A signed directed graph with six vertices vi , i ∈ I6 and a special vertex generating the desired reference trajectory r(t).

V. S IMULATION R ESULTS Next, we give simulations to illustrate the derived results by considering networks associated with a signed directed graph, where each agent is described by the system (1) with ⎤ ⎡ ⎤ ⎡ 1.2 0.3 0 0.72 0 0 ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0.72 0.60 −1.20 1 0 1 0 2 ⎥ 0 ⎥ ⎢−0.5 ⎢ ⎦. A=⎢ ⎥ ,C = ⎣ ⎥,B = ⎢ ⎢ 0.4 −0.8⎥ ⎢1.00 0 −1.04 −0.81⎥ 0 1 0 1 ⎦ ⎣ ⎦ ⎣ 0.7 1 0 0.70 0.81 0 Clearly, CB has the full-row rank. We consider the initial states of agents as x10 = [0.6, 0.6, 0.6, 0.6]T,

x20 = [0.9, 0.9, 0.9, 0.9]T

x30 = −[0.9, 0.9, 0.9, 0.9]T, x40 = [0.3, 0.3, 0.3, 0.3]T x50 = −[0.6, 0.6, 0.6, 0.6]T, x60 = −[0.3, 0.3, 0.3, 0.3]T and apply the zero initial inputs of agents, i.e., u i,0 (t) = 0, ∀i,t. We perform simulations under the signed directed graph in Fig. 1. From this figure, we can    see that G A is structurally balanced and there is a bipartition: V 1 = {v1 , v2 } and V2 = {v3 , v4 , v5 , v6 }. Though all agents can access the desired reference trajectory r(t), the agents in V2 ‘trust’ its information, whereas the agents in V 1 ‘distrust’. This implies that v 3 , v4 , v5 and v6 use the information of r(t), while v1 and v2 do not, but employ its opposite information, i.e., −r(t). In addition, with regard to Fig. 1, we can determine a diagonal matrix D used in Lemmas 1-3 as D = diag{−1, −1, 1, 1, 1, 1}.

19

2

3 t=1

1

2 t=1

Outputs of Agents yi

Outputs of Agents yi

1.5 y1 y2 y3 y4 y5 y6

0.5 0 −0.5 −1

t=1

1 0 −1 −2

−1.5

t=1 −2 0

5

10

15 20 Time Step t

25

30

−3 0

5

10

15 20 Time Step t

25

30

Fig. 2: The output consensus tracking performance of all agents after the algorithm (14) is applied for k = 150 iterations. Left: the first coordinate. Right: the second coordinate.

6

10

4

Error Norm k (t)

10

2

10

0

10

−2

10

−4

10

−6

10

0

100

200 300 Iteration Number k

400

500

Fig. 3: The learning performance of the algorithm (14) evaluated in the sense of the error norm εk (t) for the first 500 iterations (asymptotic stability).

A. Consensus Tracking We first perform simulations to illustrate the output consensus tracking objective (3) with ⎤ ⎡   t 1− t 2+1 30   ⎦ for t ∈ Z30 . r(t) = ⎣ 6 πt 2 cos 10 To apply the consensus algorithm (14), we choose ⎡ ⎤ 0.1531 0.0255 ⎦ K=⎣ −0.0102 0.0816

2

2

1.5

1.5

1

t=1

Outputs of Agents yi

Outputs of Agents yi

20

y1 y2 y3 y4 y5 y6

0.5 0 −0.5 −1

t=1

−1.5 −2 0

t=1

1 0.5 0 −0.5 −1 −1.5

5

10

15 20 Time Step t

25

30

−2 0

t=1 5

10

15 20 Time Step t

25

30

Fig. 4: The output consensus tracking performance of all agents after the algorithm (24) and (29) is implemented for only k = 50 iterations. Left: the first coordinate. Right: the second coordinate.

 and thus ρ I − (L|A | + |Ω|) ⊗CBK = 0.9387 follows, which implies that the convergence condition (11) is satisfied. Fig. 2 depicts the output trajectories of all agents when the algorithm (14) is applied after k = 150 iterations. Obviously, we can see from this figure that all agents achieve consensus tracking with the desired reference in two groups, i.e., the outputs of v 3 , v4 , v5 and v6 following r(t) and those of v1 and v2 tracking −r(t). That is, the bipartite consensus tracking is achieved for all time steps t ∈ Z30 /{0}, as described by the objective (3). Note that the bipartite consensus tracking objective (3) can be reflected by the stability of ε k (t)  N−1 T as k → ∞. Thus, εk (t) = ∑t=0 εk (t)εk (t) is used to evaluate the learning performance of the algorithm (14) when it is considered for the multi-agent networks under signed graphs. In Fig. 3, we depict the iteration evolution of ε k (t) for the first 500 iterations. We can clearly see that the error εk (t) can be enabled to converge to zero with increasing iterations. This guarantees that the control input can be refined to gradually achieve the bipartite consensus tracking of agents along the iteration axis. However, it is obvious from Fig. 3 that there occurs a high overshoot (even greater than 10 5 ) before the error decreases. We can easily validate that A is an unstable matrix (there exists an eigenvalue of A equal to 1.0353), and thus we may not avoid high overshoots through achieving the monotonic convergence of εk (t) by Theorem 5. In view of this observation, we consider the consensus algorithm given by (24) and (29), under which the monotonic convergence of ε k (t) can be guaranteed with the LMI condition (26) as shown by Theorem 7. Hence, we solve this

21

2

10

0

Error Norm k (t)

10

−2

10

−4

10

−6

10

−8

10

0

50

100 Iteration Number k

150

200

Fig. 5: The learning performance of the algorithm (24) and (29) evaluated in the sense of the error norm εk (t) for the first 200 iterations (monotonic convergence).

LMI and then apply (28) to compute the gain matrices of (29) as ⎤ ⎤ ⎡ ⎡ −0.6740 −0.5727 0.6712 0.4871 0.2379 0.0393 ⎦ , K2 = ⎣ ⎦. K1 = ⎣ −0.1905 −0.3948 0.0832 −0.0360 −0.0163 0.1296 When using the algorithm (24) and (29), Figs. 4 and 5 plot the simulation results. We can see from Fig. 4 that the agents can achieve bipartite consensus tracking with the desired reference trajectory only after k = 50 iterations. Moreover, Fig. 5 shows that not only can the error ε k (t) be driven to converge to zero with the increase of iterations but also the convergence can be achieved in a monotonic manner from iteration to iteration. B. Formation Keeping Next, we perform simulations to demonstrate the formation keeping algorithms. Note that in Fig. 1, we have a bipartition: V1 = {v1 , v2 } and V2 = {v3 , v4 , v5 , v6 }. Therefore, we consider our formation keeping objectives as shown in Fig. 6 when the agents move by following a desired reference as

⎡

⎤

2 ⎦ r(t) = ⎣ 0.5t

for t ∈ Z30 .

By Fig. 6, we attempt to guarantee the two agents in V 1 to keep the same vertical level with the desired horizontal distance and the four agents in V 2 to compose the desired relative formation

22

y-axis (second coordinate)

v3 v1

v4

v2

s(t)

s(t) v6

v5

x-axis (first coordinate)

Fig. 6: The desired formations between agents when they move by following the desired reference.

y-axis (second coordinate)

20 15

y1 y2 y3 y4 y5 y6

10 5 0

t=0

−5 −10 −10

0 10 x-axis (first coordinate)

Fig. 7: The formation keeping performance achieved by the outputs of all agents after the algorithm (4) is applied for k = 150 iterations.

in the form of a square, respectively. However, the edge of the formations in Fig. 6 varies with respect to the time step, which is revealed by s(t). Here, we consider s(t) = 2 cos(π t/15) with ⎤ ⎡ ⎡ ⎤   πt + t + 1 t +1 2 cos 15 5 ⎦, ⎦ d2 (t) = ⎣ d1 (t) = ⎣ 5 t t 6 6 ⎤ ⎤ ⎡ ⎡   πt + t + 1 t +1 2 cos 15 5 ⎦ ⎦, d3 (t) = ⎣ 5 d4 (t) = ⎣ t t 6 6 ⎤ ⎤ ⎡   ⎡ πt + t + 1 t +1 2 cos 15 5  ⎦ .  5  ⎦ , d6 (t) = ⎣ d5 (t) = ⎣ t − 2 cos π t t − 2 cos π t 6 15 6 15

23

6

10

4

Error Norm ek (t)

10

2

10

0

10

−2

10

−4

10

−6

10

0

100

200 300 Iteration Number k

400

500

Fig. 8: The learning performance of the algorithm (4) evaluated in the sense of the error norm ek (t) for the first 500 iterations (asymptotic stability).

We first implement the formation algorithm (4), and use the same gain matrix K as adopted by (14) in the above subsection. Figs. 7 and 8 describe the simulation results. It is obvious from Fig. 7 that all agents can form and keep the desired formations as the same as those given in Fig. 6 for all the time except t = 0 after the algorithm (4) is performed for k = 150 iterations, and the formation shapes can also vary with respect to the changing of time. Moreover, we can see from Fig. 8 that ek (t) converges to zero with increasing iterations. This clearly implies that the bipartite formation keeping objective (2) is well accomplished by noting the equivalent result of Lemma 3. However, the high overshoot phenomenon observed by Fig. 3 also happens in Fig. 8 for the same reason. To avoid causing such bad learning transients, we employ the algorithm (24) and (25) with the same gain matrices K1 and K2 as those computed from the LMI condition (26) for the consensus algorithm (24) and (29) in the above subsection. We depict the corresponding simulation results for the bipartite formations of agents in Figs. 9 and 10. From Figs. 9 and 10, it is clear that we not only can guarantee all agents to achieve the bipartite formation keeping objective (2) with desired formations but also can obtain good learning transients and gain a faster convergence speed.

24

y-axis (second coordinate)

20

y1 y2 y3 y4 y5 y6

15 10 5 0

t=0

−5 −10 −10

0 10 x-axis (first coordinate)

Fig. 9: The formation keeping performance achieved by the outputs of all agents after the algorithm (24) and (25) is performed for only k = 50 iterations.

4

10

2

Error Norm ek (t)

10

0

10

−2

10

−4

10

−6

10

−8

10

0

50

100 Iteration Number k

150

200

Fig. 10: The learning performance of the algorithm (24) and (25) evaluated in the sense of the error norm ek (t) for the first 200 iterations (monotonic convergence).

C. Discussions We can see from Figs. 2–5 and Figs. 7–10 that although the agents are subject to antagonistic interactions, the consensus tracking and the formation keeping objectives of them can be achieved through distributed iterative learning algorithms, respectively. Due to antagonistic interactions in Fig. 1, the existing ILC-based algorithms for multi-agent systems are no longer applicable (see, e.g., [25]-[33]). Note also that the multi-agent systems considered in [25]-[33] can be viewed as a special case of those associated with signed graphs (see discussions of [34]). It is not difficult

25

to conclude that our proposed algorithms are effective for more general networks in comparison with those of [25]-[33]. In addition, it is obvious from Figs. 2–5 and Figs. 7–10 that the bipartite coordination tasks of agents can be achieved with high precision over any specified time interval of interest. The existing bipartite coordination algorithms of signed networks can only guarantee agents to complete the required tasks asymptotically with increasing time (see, e.g., [34]-[37]). In contrast, our algorithms make an improvement in achieving high-precision bipartite coordination tasks by taking advantage of iterative learning methods. VI. C ONCLUSIONS In this paper, we have considered two bipartite coordination problems on signed networks: bipartite formation keeping and bipartite consensus tracking. We have given a class of iterative learning algorithms to guarantee all agents to achieve bipartite coordination objectives with arbitrary high realizing precision. Moreover, we have established conditions for both asymptotic stability and monotonic convergence of the associated learning processes, where in particular, a class of LMI conditions have been derived such that good learning transients can be ensured together with formulas of the gain matrices design being given. The established bipartite coordination results have generalized the developed iterative learning results of multi-agent networks in, e.g., [25]-[33] to much general signed networks. This brings new insights into the bipartite coordination of signed networks, which also renders the existing bipartite coordination results of signed networks in, e.g., [34]-[37] to be achieved with arbitrary high precision. With numerical simulations, we have illustrated the proposed bipartite coordination results. We have only considered the linear multi-agent networks without uncertainties. Future studies of this work include developments of the established distributed iterative learning algorithms for nonlinear networked systems (see, e.g., [40]). In practice, the agents as well as interactions between them may be subject to uncertainties, such as stochastic disturbances [41] and information loss [40]. To make the underlying networked systems reliable and safe, a promising future topic is to improve the proposed algorithms by incorporating the effective fault-tolerant and fault diagnosis methods (see, e.g., [41]-[43]). ACKNOWLEDGEMENTS The authors would like to thank the associate editor and anonymous reviewers for their insightful comments and suggestions which greatly improved the quality and presentation of

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our paper. This work was supported by the National Basic Research Program of China (973 Program: 2012CB821200, 2012CB821201), the National Natural Science Foundation of China (NSFC: 61473010, 61134005, 61221061, 61327807, 61320106006), and the Fundamental Research Funds for the Central Universities. R EFERENCES [1] A. Jadbabaie, J. Lin, and A. S. Morse, “Coordination of groups of mobile autonomous agents using nearest neighbor rules,” IEEE Transactions on Automatic Control, vol. 48, no. 6, pp. 998–1001, Jun. 2003. [2] C. Tan and G.-P. Liu, “Consensus of networked multi-agent systems via the networked predictive control and relative outputs,” Journal of the Franklin Institute, vol. 349, no. 7, pp. 2343–2356, Sept. 2012. [3] Y. Gao, J. Ma, M. Zuo, T. Jiang, and J. Du, “Consensus of discrete-time second-order agents with time-varying topology and time-varying delays,” Journal of the Franklin Institute, vol. 349, no. 8, pp. 2598–2608, Oct. 2012. [4] H.-X. Hu, L. Yu, W.-A. Zhang, and H. Song, “Group consensus in multi-agent systems with hybrid protocol,” Journal of the Franklin Institute, vol. 350, no. 3, pp. 575–597, Apr. 2013. [5] Y. Hu, J. Lam, and J. Liang, “Consensus of multi-agent systems with Luenberger observers,” Journal of the Franklin Institute, vol. 350, no. 9, pp. 2769–2790, Nov. 2013. [6] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proceedings of the IEEE, vol. 95, no. 1, pp. 215–233, Jan. 2007. [7] W. Ren, R. W. Beard, and E. Atkins, “Information consensus in multivehicle cooperative control,” IEEE Control Systems Magazine, vol. 27, no. 2, pp. 71–82, Apr. 2007. [8] Y. Cao, W. Yu, W. Ren, and G. Chen, “An overview of recent progress in the study of distributed multi-agent coordination,” IEEE Transactions on Industrial Informatics, vol. 9, no. 1, pp. 427–438, Feb. 2013. [9] Y. Hong, G. Chen, and L. Bushnell, “Distributed observers design for leader-following control of multi-agent networks,” Automatica, vol. 44, no. 3, pp. 846–850, Mar. 2008. [10] W. Ni and D. Cheng, “Leader-following consensus of multi-agent systems under fixed and switching topologies,” Systems and Control Letters, vol. 59, nos. 3-4, pp. 209–217, Mar.-Apr. 2010. [11] G. Miao, S. Xu, and Y. Zuo, “Consentability for high-order multi-agent systems under noise environment and time delays,” Journal of the Franklin Institute, vol. 350, no. 2, pp. 244–257, Mar. 2013. [12] Q. Wang and H. Gao, “Global consensus of multiple integrator agents via saturated controls,” Journal of the Franklin Institute, vol. 350, no. 8, pp. 2261–2276, Oct. 2013. [13] Q. Ma, Z. Wang, and G. Miao, “Second-order group consensus for multi-agent systems via pinning leader-following approach,” Journal of the Franklin Institute, vol. 351, no. 3, pp. 1288–1300, Mar. 2014. [14] L. Cheng, Z.-G. Hou, M. Tan, Y. Lin, and W. Zhang, “Neural-network-based adaptive leader-following control for multiagent systems with uncertainties,” IEEE Transactions on Neural Networks, vol. 21, no. 8, pp. 1351–1358, Aug. 2010. [15] W. Ren, “Multi-vehicle consensus with a time-varying reference state”, Systems and Control Letters, vol. 56, nos. 7-8, pp. 474–483, Jul. 2007. [16] S. Liu, L. Xie, and F. L. Lewis, “Synchronization of multi-agent systems with delayed control input information from neighbors”, Automatica, vol. 47, no. 10, pp. 2152–2164, Oct. 2011. [17] H.-S. Ahn, K. L. Moore, and Y. Chen, “Trajectory-keeping in satellite formation flying via robust periodic learning control,” International Journal of Robust and Nonlinear Control, vol. 20, no. 14, pp. 1655–1666, Sept. 2010.

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