Decentralized stabilizability of multi-agent systems under fixed and switching topologies

Systems & Control Letters 62 (2013) 438–446

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

Decentralized stabilizability of multi-agent systems under fixed and switching topologies✩ Yongqiang Guan a , Zhijian Ji b , Lin Zhang a , Long Wang c,∗ a

School of Automation Science and Electrical Engineering, Beihang University, Beijing, 100191, China

b

Institute of Automation, Qingdao University, Qingdao, 266071, China

c

Center for Systems and Control, College of Engineering, Peking University, Beijing, 100871, China

article

info

Article history: Received 16 July 2012 Received in revised form 14 January 2013 Accepted 17 February 2013 Available online 28 March 2013 Keywords: Multi-agent system Controllability Stabilizability Switching topology Cooperative control

abstract The paper studies decentralized stabilizability for multi-agent systems with general linear dynamics. The stabilizability problem is formulated in a way that the protocol performance can be evaluated by means of the stabilizability region and the feedback gain. For fixed topology, it is proved that the system is stabilizable if and only if external control inputs are exerted on some indicated agents. The result is further shown to be a prerequisite for subsequent design of the corresponding decentralized external selffeedback control, which is also necessary and sufficient. Based on this, two methods are presented to find the agents under which stabilizability can be reached, and the region of stabilizability is given to evaluate the protocol performance. For switching interaction topology, it is shown that the system is stabilizable even if each of its subsystems is not. Finally, the results are employed to cope with the decentralized setpoint formation control problem, for which some necessary and/or sufficient conditions are developed. Numerical simulations are presented to demonstrate the effectiveness of the proposed results. © 2013 Elsevier B.V. All rights reserved.

1. Introduction In recent years, the distributed cooperation control of multiagent systems has received great attention. Compared with traditional control systems, multi-agent systems have enormous advantages such as reliability, flexibility, and adaptability to uncertain environments. Cooperation control of multi-agent systems has a broad range of applications in the fields of science and engineering including formation control of unmanned air vehicles (UAVs), scheduling of automated highway systems, and distributed estimation over sensor networks. Research hotspots in distributed control and coordination of multi-agent systems include consensus problems [1–5], flocking problems [6–8], formation problems [9–11], and containment problems [12–14]. Controllability is a basic concept in classical control theory. The concept of controllability of multi-agent systems was first formulated by Tanner, who established necessary and sufficient conditions in terms of eigenvalues and eigenvectors of the system matrix corresponding to the follower nodes [15]. Thereafter, more

✩ This work is supported by National 973 Program of China (No. 2012CB821203) and National Science Foundation of China (Grant Nos. 61020106005, 61074144, 61075114, 61203374). ∗ Corresponding author. E-mail addresses: [email protected] (Y. Guan), [email protected] (Z. Ji), [email protected] (L. Zhang), [email protected] (L. Wang).

0167-6911/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sysconle.2013.02.010

and more researchers devoted themselves to the investigation of this problem. In [16], the authors gave a sufficient condition for controllability from the graph-theoretic perspective. The condition relies on the notion of equitable partitions of a graph. In [17], relaxed equitable partition was employed to consider controllability properties for a leader–follower network. In [18,19], the authors investigated the controllability of a single-leader multi-agent system under fixed and switching topologies. Both continuous-time and discrete-time cases were considered therein. Controllability under switching topology and time delay was studied in [20,21], respectively. Uncontrollable topology structures and graph-theoretic properties were given in [22]. In addition, the leaders’ selection problem was investigated in [23], where some necessary and sufficient conditions were proposed in terms of Downer branch and subgraphs to characterize the leaders’ role in controllability. Although the controllability has been extensively studied, examining the stabilizability of multi-agent systems is in its infancy. Recent works in this direction include [24], where the authors proposed a new concept of ‘‘stabilizability’’ for multi-agent systems. The concept is studied for a group of single integrators under a fixed topology. In this paper, we study the stabilizability problem in a more general case, where the dynamics of each agent is an Nth-order linear control system, rather than a single or double integrator as in most existing studies. For fixed topology, it is proved that the system is stabilizable if and only if external control inputs are

Y. Guan et al. / Systems & Control Letters 62 (2013) 438–446

exerted on some indicated agents. For switching interaction topology, it is shown that multi-agent system is stabilizable even if each of its subsystems is not stabilizable, if external control inputs are exerted on some indicated agents in the union graph. This paper is partly motivated by Kim et al. [24]. The major differences between this work and [24] are as follows. (i) Under fixed topology, [24] considered the case of agents with single integrator dynamics, while we consider general linear dynamics instead of a single integrator, which brings new features for the study of the stabilizability problem. (ii) In [24], there is no issue of stabilizability protocol design. In our case, it is shown that the stabilizability admits protocol performance evaluation by means of the stabilizability region and the feedback gain. At the same time, two methods are presented to find the agents under which stabilizability can be reached. (iii) In [24], the stabilizability problem is considered only for fixed topology. Here we consider both fixed and switching topologies. It is shown that stabilizability under switching topology can be achieved even if each of the subsystems is not stabilizable. From an application perspective, those results are employed to cope with the decentralized set-point formation control problem, for which some necessary and/or sufficient conditions are developed. Simulations are performed to validate the theoretical results. The paper is organized as follows. Section 2 contains some preliminaries as well as some definitions and lemmas. Section 3 discusses the stabilizability problem of multi-agent systems under fixed topology. In Section 4, the stabilizability problem is studied under switching topology. In Section 5, the results are applied to the decentralized set-point formation control problem. Simulation results are presented in Section 6. The conclusion is given in Section 7. Notation. Throughout this paper, the following notation is used. Let 0(0m×n ) denote an all-zero vector or matrix with compatible dimension (dimension m × n). In and diag{a1 , . . . , an } represent the n × n identity and diagonal matrices, respectively. Matrix P > 0 (≥0, <0, ≤0) means P is positive definite (positive semidefinite, negative definite, or negative semidefinite). Let 1n denote the all-1 vector with dimension n. j is the imaginary unit. R(λ) represents the real part of a complex number λ. ∧(A) denotes the eigenvalue set of A and ∧+ (A) denotes the eigenvalue set of A which have positive real parts. R and C denote the set of real numbers and the set of complex numbers, respectively. C>0 (C≥0 ) denotes the set of complex numbers possessing positive (nonnegative) real parts. ⊗ denotes the Kronecker product. 2. Preliminaries 2.1. Graph preliminaries In this section, some useful concepts and notation in graph theory are briefly reviewed. In this paper, ‘‘nodes’’ or ‘‘agents’’ are used interchangeably with ‘‘vertices’’, and directed graph will be used to model the interaction topology among agents. A directed (weighted) graph is denoted by G = (N , E , A), where N = {v1 , v2 , . . . , vn } and E ⊆ N × N represent, respectively, the vertex set and the edge set; A = [aij ] ∈ Rn×n is the weighted adjacency matrix with aij > 0 representing the reliability of the interaction from agent j to agent i. An edge of G is denoted by eij = {vj , vi }, where vj is called the parent vertex of vi and vi the child vertex of vj . In this paper, we assume that there are no self-loops, i.e., eii ̸∈ E . The set of neighbors of node vi is denoted by Ni = {vj ∈ N : eij = {vj , vi } ∈ E , j ̸= i}. A directed path in a directed graph G is a sequence vi1 , . . . , vik of distinct vertices with (vis , vis+1 ) ∈ E , for s = 1, . . . , k − 1 and a weak path, with either (vis , vis+1 ) or (vis+1 , vis ) ∈ E . A directed graph G is strongly connected if there is a directed path that starts from vi and

439

ends at vj between every pair of distinct vertices vi , vj in G, and is weakly connected if any two vertices can be jointed by a weak path. A strong component of a directed graph is an induced subgraph that is maximal, and subject to being strongly connected. Since any subgraph consisting of only a vertex is strongly connected, it follows that each vertex lies in a strong component. Two vertices in the same strong component have an equivalence relation. For a directed graph G, the in-degree and out-degree of node   vi are defined as degin (vi ) = vj ∈Ni aij and degout (vi ) = vj ∈Ni aji , respectively. The degree matrix of G is a diagonal matrix defined as

△ = [△ij ], where △ij = degin (vi ) for i = j; otherwise, △ij = 0. The Laplacian matrix L(G) = [lij ] ∈ Rn×n of a graph  G, abbreviated as L, is defined by lij = −aij if i ̸= j and lij = vj ∈Ni aij if i = j. It is obvious that L = △ − A. Definition 1 ([25]). An independent strongly connected component (iSCC) of a digraph G = (N , E , A) is an induced subgraph G¯ = (N¯ , E¯ , A¯ ) which is maximal, subject to being strongly connected, and satisfies (vj , vi ) ̸∈ E for any vj ∈ N \ N¯ and vi ∈ N¯ . That is, G¯ is strongly connected, and the unweighted digraph induced by any set N˜ with N¯ ⊆ N˜ ⊆ N is strongly connected if and only if N˜ = N¯ . Furthermore, there is no edge eij = {vj , vi } ∈ E with parent vertex vj ∈ N \ N¯ and child vertex vi ∈ N¯ . Remark 1. Since a single vertex of a directed graph constitutes a strongly connected component, any directed graph contains up to m (1 ≤ m ≤ n) iSCCs. The method of finding all the iSCCs for any directed graph will be shown in Section 3.4. 2.2. Basic lemmas The following three lemmas play a basic role for further analysis of stabilizability in subsequent sections. Lemma 1 ([26]). Let A ∈ Cn×n and A ≥ 0. Then A is irreducible if and only if the directed graph G is strongly connected. n×n Lemma 2 ([26]). A matrix is nonsingular if A is  A = [aij ] ∈ C irreducible and |aii | ≥ | a | for all i, with the inequality being ij j̸=i strict for at least one i.

Lemma 3 ([14]). Suppose that directed graph G = (N , E , A) is weakly connected and that L is the Laplacian matrix of G. Then Rank(L) = n − m if and only if G = (N , E , A) contains m iSCCs. 3. Stabilizability under fixed topology In this section, the multi-agent system has general linear dynamics. The stabilizability results are first derived with respect to fixed topology. Then, the design method is proposed for the feedback gain matrix. Finally, two methods are given to find the external control input vertices. 3.1. Problem formulation Consider a group of n identical agents with general continuoustime linear dynamics. The dynamics of each agent is described by x˙ i (t ) = Axi (t ) + Bui (t ),

i = 1 , . . . , n,

N

(1) P

where xi ∈ R is the state of agent i, and ui ∈ R is the control input. A ∈ RN ×N and B ∈ RN ×P are the system matrix and the input matrix, respectively. Assumption 1. The pair (A, B) is stabilizable.

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Y. Guan et al. / Systems & Control Letters 62 (2013) 438–446

Consider the following control protocol: ui (t ) = ui0 ( t ) + ui1 (t ) ui0 (t ) = K aij (xj (t ) − xi (t ))

(2)

vj ∈Ni

ui1 (t ) = K · di · ri (t ), where ui0 and ui1 (t ) are the interagent control and the external control, respectively. K ∈ RP ×N is a feedback gain matrix to be designed, and ri ∈ RN is the external control input of the agent i. di ∈ R is a constant, indicating whether agent i is affected by the external input ri . That is, di > 0 if and only if external input ri is applied to the agent i; otherwise, di = 0. The weights aij for i, j = 1, . . . , n are assumed to be given by the interaction topology G. System (1) together with protocol (2) can be written in matrix form:

 x˙ (t ) = In ⊗ A − L ⊗ (BK ) x(t ) + D ⊗ (BK ) r (t ), 





(3)

where x(t ) = [xT1 (t ), . . . , xTn (t )]T , r (t ) = [r1T (t ), . . . , rnT (t )]T , and D = diag(d1 , . . . , dn ). L is the Laplacian matrix of G. Definition 2. For a given gain matrix K , the control protocol (2) is said to be able to solve the stabilizability problem of multi-agent system (1), if there exists an external control input r (t ) = −Kf x(t ), such that the closed-loop system (3) is stable. Here, Kf is called the feedback matrix. 3.2. Stabilizability analysis In subsequent arguments, we assume that the topology is weakly connected. Otherwise, the topology can be divided into several weakly connected subgraphs, and there will be no information transmission in these subgraphs. In this case, the subgraphs can be considered independently. Without loss of generality, we suppose that the directed graph G has m (m ≥ 1) iSCCs Gl1 , . . . , Glm . Let N (Gl1 ) = {v1 , v2 , . . . , vk1 }, N (Gl2 ) = {vk1 +1 , vk1 +2 , . . . , vk2 }, . . . , N (Glm ) = {vkm−1 +1 , vkm−1 +2 , . . . , vkm } and

N (Glm+1 ) ∪ · · · ∪ N (Gls ) = N (G) \ N (∪m i=1 Gli ) = {vkm +1 ,

vkm +2 , ..., vkm+1 }, where k1 = n1 , k2 = k1 + n2 , . . . , km = km−1 + nm , km+1 = km + nm+1 + · · · + ns and ni is the number of nodes in Gli . Then the Laplacian matrix L and the diagonal matrix D read L11  0



 .  .  .  0

Lm+1,1 D1



0 . . . 0 0

0 D2

0 L22

.. .

··· ··· .. .

0 0

0 0

0 Lm+1,2

··· ···

Lmm Lm+1,m

0 Lm+1,m+1

.. .

··· ··· .. .

0 0

0 0

··· ···

Dm 0

.. .

.. .

0 0 



.. .

Proof (Sufficiency). Since Ljj ∈ Rnj ×nj , j = 1, . . . , m, is the Laplacian matrix associated with the iSCC Gj = (Nj , Ej , Aj ), Ljj is irreducible diagonally dominant by Lemma 1. For each j = 1, . . . , m, there exists a node qj ∈ N such that dqj > 0, and thus Dj ̸= 0. From Lemma 2, we know that Ljj + Dj , j = 1, . . . , m, is nonsingular. Therefore, all the eigenvalues of Ljj + Dj , j = 1, . . . , m, have positive real parts. Applying Lemma 3 yields that ∧(Lm+1,m+1 ) ⊆ C>0 . This further implies that all the eigenvalues of L + D have positive real parts. (Necessary) Suppose that, for some j with 1 ≤ j ≤ m, there is no such a vertex qj . We assume j = 1 without loss of generality; then D1 = 0. It follows that λ1 = 0 is an eigenvalue of L + D. This is a contradiction.  Let x¯ 1 = [xT1 , . . . , xTk1 ]T , x¯ 2 = [xTk1 +1 , . . . , xTk2 ]T , . . . , x¯ m+1 =

[ , . . . , xTkm+1 ]T , r¯1 = [r1T , . . . , rkT1 ]T , r¯2 = [rkT1 +1 , . . . , rkT2 ]T , . . . , r¯m+1 = [rkTm +1 , . . . , rkTm+1 ]T . Then the dynamics (3) can be xTkm +1

rewritten as follows: x˙¯ 1 (t )  x˙¯ 2 (t )   





L¯ 11  0 



0 L¯ 22

··· ··· .. .

0 0

0 0



    ..   .. . . . .. .. ..   . = .       x˙¯ m (t )   0 ¯ 0 ··· Lmm 0 ¯Lm+1,1 L¯ m+1,2 · · · L¯ m+1,m L¯ m+1,m+1 ˙x¯ m+1 (t )   D¯  0 ··· 0 0 x¯ 1 (t ) 1 ¯2 ··· 0 0 D 0   x¯ 2 (t )     .    . . . ..  .   .. .. .. ×  ..  +  .. .    x¯ (t )    ¯ 0 0 · · · Dm 0  m x¯ m+1 (t ) ¯ m+1 0 0 ··· 0 D   r¯1 (t )  r¯2 (t )   .   ×  ..  ,  r¯ (t )  m r¯m+1 (t )

(5)

¯i = where L¯ ii = Ini ⊗ A − Lii ⊗ (BK ), L¯ m+1,i = −Lm+1,i ⊗ (BK ), D ¯ Di ⊗(BK ), i = 1, . . . , m, and Lm+1,m+1 = Ikm+1 −km ⊗ A − Lm+1,m+1 ⊗ (BK ). The following assumption is assumed throughout this section.

 Assumption 2. The directed graph G = (N , E , A) is weakly connected and contains m(m ≥ 1) distinct iSCCs Gj = (Nj , Ej , Aj ), j = 1, . . . , m.

  ,   (4)

..   . ,  0 Dm+1

where Ljj ∈ Rnj ×nj , j = 1, . . . , m, is the Laplacian matrix associated with the iSCC Gj = (Nj , Ej , Aj ), and Di ∈ Rni ×ni , i = 1, . . . , m + 1, is a diagonal matrix with all entries nonnegative. The following lemma serves for the derivation of main results. Lemma 4. Suppose that the directed graph G = (N , E , A) is weakly connected and contains m (m ≥ 1) distinct iSCCs Gj = (Nj , Ej , Aj ), j = 1, . . . , m. Then all the eigenvalues of L + D have positive real parts if and only if, for each j = 1, . . . , m, there exists a node qj ∈ Nj such that dqj > 0.

Now, we can present the main result of this section. The result establishes a necessary and sufficient condition for stabilizability of multi-agent system (3). Theorem 1. Under Assumptions 1 and 2, consider multi-agent system (1) with control protocol (2). Suppose that the agent’s dynamics (1) is not asymptotically stable; then there exists K such that multi-agent system (3) is stabilizable if and only if, for each j = 1, . . . , m, there exists a node qj ∈ Nj such that dqj > 0. Proof (Sufficiency). Without loss of generality, assume that the j  node in the theorem is qj = h=1 kh − kj + 1, j = 1, . . . , m, and dqj = 1. Then Dj = diag(1, ∗, . . . , ∗) for all j = 1, . . . , m and Dj ∈ Rnj ×nj , where ‘‘ ∗ ’’ stand for the numbers of no interest. Now suppose by contradiction that (3) is not stabilizable; that is, there exists a complex number s ∈ C≥0 and a nonzero row vector w ¯ T ∈ CnN such that w ¯ T [sInN − In ⊗ A + L ⊗(BK ) D ⊗(BK )] = 0. SupT pose that w ¯ is partitioned as w ¯ T = [w ¯ 1T , . . . , w ¯ mT , w ¯ mT +1 ], where

Y. Guan et al. / Systems & Control Letters 62 (2013) 438–446

w ¯ jT ∈ Cnj N for j = 1, . . . , m + 1. Then we get that   w ¯ jT sInj N − Inj ⊗ A + Ljj ⊗ (BK )   +w ¯ mT +1 Lm+1,j ⊗ (BK ) = 0,   w ¯ mT +1 sInm+1 N − (Ikm+1 −km ) ⊗ A + Lm+1,m+1 ⊗ (BK ) = 0,   w ¯ jT Dj ⊗ (BK ) = 0,

(6) (7) (8)

for j = 1, . . . , m. As all the eigenvalues of Lm+1,m+1 have positive real parts and the pair (A, B) is stabilizable, there exists K such that A − λi BK is Hurwitz for all λi ∈ ∧(Lm+1,m+1 ) (the existence of K is given in Section 3.3), which implies that (Ikm+1 −km )⊗A−Lm+1,m+1 ⊗ (BK ) is Hurwitz. Therefore, we claim that w ¯ mT +1 = 0 from (7). If not, by (7), it is a left eigenvector associated to the eigenvalues s ∈ C≥0 of (Ikm+1 −km ) ⊗ A − Lm+1,m+1 ⊗ (BK ). This is not possible, since (Ikm+1 −km )⊗ A − Lm+1,m+1 ⊗(BK ) is Hurwitz. Therefore, (6) becomes

  w ¯ jT sInj N − Inj ⊗ A + Ljj ⊗ (BK ) = 0,

j = 1, . . . , m.

Thus, from (8) and (9), we get that

  w ¯ jT sInj N − Inj ⊗ A + (Ljj + Dj ) ⊗ (BK ) = 0, j = 1, . . . , m.

Next, a necessary and sufficient condition will be derived to design a decentralized self-feedback control r (t ) = −Kf x(t ) to stabilize multi-agent system (3). The control provides considerable flexibility to the designer since the result indicates that any negative selffeedback to a vertex in each iSCC stabilizes system (3). Theorem 2. Under Assumptions 1 and 2, let ri (t ) = −ki xi (t ),

i ∈ N,

(10)

By Lemma 4 and Assumption 1, there exists K such that A − λBK , λ ∈ ∧(Ljj +Dj ) is Hurwitz (the existence of K is given in Section 3.3). This implies that (Ikm+1 −km )⊗A−Lm+1,m+1 ⊗(BK ) is Hurwitz. Therefore, from (10), we obtain that w ¯ jT = 0, j = 1, . . . , m. Accordingly,

w ¯ jT = 0 for all j = 1, . . . , m + 1, which contradicts the assumption that w ¯ T ̸= 0. (Necessity) Suppose that, for some j with 1 ≤ j ≤ m, there is no such node qj . We assume that j = 1; then D1 = 0. By (5), we obtain   that x˙¯ 1 (t ) = L¯ 11 x¯ 1 (t ); that is, x˙¯ 1 (t ) = In1 ⊗ A − L11 ⊗ (BK ) x¯ 1 (t ). As λ1 = 0 is the eigenvalue of L11 , A − λ1 BK = A. From the assumption that the agent’s dynamics is not asymptotically stable, we can obtain that the whole systems is not stabilizable. This is a contradiction.  Corollary 1. Let G = (N , E , A) be any connected graph. Under Assumption 1, suppose that the agent’s dynamics (1) is not asymptotically stable; then there exists K such that system (3) is stabilizable if and only if there exists a vertex q ∈ N such that dq > 0. Proof. Note that any connected graph contains exactly one iSCC. Let the iSCC be G1 = (N1 , E1 , A1 ). Then by Theorem 1, system (3) is stabilizable if and only if there exists a vertex q ∈ N1 such that dq > 0.  Remark 2. In [24], the stabilizability was studied for a group of single integrators. The differences between the stabilizability problem studied in [24] and this paper are as follows. (i) In this paper, the results are derived for a system with general linear dynamics instead of a single integrator. Subsequent arguments show that general linear dynamics brings new features for the study of the stabilizability problem. (ii) For general linear dynamics, Theorem 1 implies that the feedback gain matrix K is crucial to the stabilizability. However, for a single integrator, there is no issue of the feedback gain matrix K . Naturally there arises a question: How does one design the feedback gain matrix K to guarantee the stabilizability of multi-agent systems with general linear dynamics? In subsequent arguments, it will be shown that the design of K relates to not only the stabilizable pair (A, B), but also the structure of interconnection topology. (iii) For general linear dynamics, the concept of a stabilizability region can be defined, which can be employed to evaluate the performance of the designed protocol.

(11)

in which ki is a nonnegative number. Suppose that the agent’s dynamics (1) is not asymptotically stable; then there exists K such that (3) is exponentially stable if and only if, for each j = 1, . . . , m, there exists a node qj ∈ Nj such that dqj > 0 and kqj > 0. Proof (Sufficiency). Under the external control of (11), the closedloop system of (3) can be summarized as follows: x˙ (t ) = In ⊗ A − (L + DK1 ) ⊗ (BK ) x(t ),



(9)

441



(12)

where K1 = diag(k1 , k2 . . . , kn ). Without loss of generality, it is assumed that the matrices L and D are of the form in (4) and K1 = diag(K11 , . . . , Kmm , Km+1m+1 ), where the Kii are ni × ni diagonal matrices. From Lemma 4, we know that all the eigenvalues of L + DK1 have positive real parts. Therefore, there exists K such that all matrices A − λBK , λ ∈ ∧(L + DK1 ) are Hurwitz (the existence of K is given in Section 3.3), which implies that In ⊗ A −(L + DK1 )⊗(BK ) is Hurwitz. Thus system (12) is exponentially stable. (Necessity) Suppose that, for some j with 1 ≤ j ≤ m, there is no such node qj . Without loss of generality, we assume that j = 1; then D1 = 0. From (5), we obtain that L + DK1 has at least a zero eigenvalue; i.e., λ1 = 0 is the eigenvalue of L11 + D1 K11 . Therefore, A − λ1 BK = A. From the assumption that the agent’s dynamics is not asymptotically stable, we obtain that the whole system (12) is not exponentially stable. This is a contradiction.  Corollary 2. Let G = (N , E , A) be any connected graph. Under Assumption 1, let the external controls be as shown in (11). Suppose that the agent’s dynamics (1) is not asymptotically stable; then there exists K such that (3) is exponentially stable if and only if there exists a vertex q ∈ N such that dq > 0. Proof. Let G1 = (N1 , E1 , A1 ) be the iSCC. Then, by Theorem 2, system (3) is exponentially stable if and only if there exists a vertex q ∈ N1 such that dq > 0.  Remark 3. From Theorem 2 and Corollary 2, we know that only one external control input injected to a vertex of each iSCC is enough for decentralized stabilizability when the graph has m distinct iSCCs. Moreover, Theorem 2 implies that, for any arbitrarily small gains kqi > 0, any negative self-feedback control to a vertex in each iSCC stabilizes system (3). 3.3. Stabilizability protocol design Theorems 1 and 2 show that the design of a stabilizability protocol involves the feedback gain matrix K . In particular, Theorem 2 indicates that, under decentralized self-feedback control r (t ) = −(K1 ⊗ IN )x(t ), the stabilizability problem can be cast into the stability of system (12). Note that the stability of system (12) is equivalent to A − λBK being Hurwitz for all λ ∈ ∧(L + DK1 ). This observation inspires the investigation of the following two questions. (i) How does one design feedback gain matrix K such that A − λBK is Hurwitz? (ii) What is the relationship between stabilizability region and λ?

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Y. Guan et al. / Systems & Control Letters 62 (2013) 438–446

(i) Design of the feedback gain matrix K Given a stabilizable pair (A, B), where A ∈ RN ×N and B ∈ RN ×P , the following algebraic Riccati equation AT P + PA + IN − PBBT P = 0

(13)

has a unique solution P = P > 0 [27]. We can rewrite (13) as (A − BBT P )T P + P (A − BBT P ) + (IN + PBBT P ) = 0, whereby we infer that A − BBT P is Hurwitz. T

Lemma 5 ([28]). Let A ∈ RN ×N and B ∈ RN ×P satisfy (13) for some symmetric positive definite P. Then, for all x ≥ 1 and y ∈ R, matrix A − (x + jy)BBT P is Hurwitz. Proposition 1. Under Assumption 1, consider system (1) with 1 1 protocol (2). Let K = max{1, min , min , R(λ) R(λ) 1 minλ∈∧(L+DK ) 1

λ∈∧+ (L)

λ∈∧(L+D)

}BT P, where P is the solution to (13). Then, for all R(λ)

λ ∈ ∧+ (L), λ ∈ ∧(L + D) or λ ∈ ∧(L + DK1 ), the matrices A − λBK are Hurwitz. Proof. The proof can be easily derived from Lemma 5.



Remark 4. When applying Proposition 1 to construct feedback gain matrix K , we do not need to know the exact eigenvalues of L + D in (3) and L + DK1 in (12). Instead, we can use a lower bound for the real parts of the eigenvalues of L + D and L + DK1 . However, it is well known that adding or deleting some communication links of topology may not change the lower bound for the real parts of the eigenvalues of L+D and L+DK1 . Therefore, by choosing a reasonably small lower bound, the stabilizability of multi-agent systems given by Theorem 1 or Theorem 2 will maintain a certain robustness margin with respect to modifications of the communication topology, like adding or deleting some communication links. (ii) Region of stabilizability Definition 3. Consider the control protocol (2). The stabilizability region is a complex region defined as S , {s ∈ C|A − sBK is Hurwitz}. By Theorem 2 and Definition 3, stabilizability is achieved if λi ∈

S for all λi ∈ ∧(L + DK1 ), i = 1, . . . , n. The stabilizability region is

used to evaluate the performance of stabilizability protocol (2). The following result shows a desired property of the proposed control protocol. Proposition 2. For protocol (2) with a linear quadratic regulator (LQR) based control gain K = BT P, where P is satisfied with (13), the stabilizability region is unbounded. Specifically, the stabilizability region is S = {x + jy|x ∈ [1/2, ∞), y ∈ (−∞, ∞)}. Proof. Let s = x + jy. Straightforward computation gives the Lyapunov equation

(A − sBK )∗ P + P (A − sBK ) = −IN − (2x − 1)PBBT P . Since P > 0, A − sBK is Hurwitz if and only if −IN −(2x − 1)PBBT P < 0. A sufficient condition is that x ≥ 21 . This completes the proof.  Remark 5. The stabilizability region issue considered in this paper is similar to the consensus (synchronization) region issue studied in [29]. The stabilizability region is employed to evaluate the performance of stabilizability protocols. In a certain sense, the larger the stabilizability region the better the performance of the protocol. In addition, a larger stabilizability region implies that the stabilizability control protocol is more robust to the graph topology. (Details can be found in Remark 4.)

3.4. Finding the external control vertices Since system (3) is stabilizable as long as the external control inputs are exerted on any vertex of each iSCC, we concentrate on finding these nodes. Two methods are proposed for this problem, in which the key point is to obtain the independent strongly connected components. To this end, method one is to find an orthogonal and nonnegative left eigenvector basis corresponding to eigenvalue 0 of L, and method two is to use graph algorithms. Method one (1) For a group of agents with a directed topology G = (N , E , A), calculate the weighted adjacency matrix A and the Laplacian matrix L. (2) Find an orthogonal and nonnegative left eigenvector basis corresponding to eigenvalue 0 of L. Denote the number of these left eigenvectors as m. (3) Choose m vertices, with each of them corresponding to any nonzero component of each of the m basis vectors. Then, make the external control input acted to them. Method two (1) Use Algorithms 8.5.4 and 8.5.5 of [30] to get all strongly connected components of G = (N , E , A). (2) Choose such strongly connected components which have no edges starting from the other strongly connected component while ending at it, and denote the number of them as m. (3) Select m vertices such that the external control input is applied to them, with each of the vertices corresponding to each of such strongly connected components. Remark 6. Although the two methods are equally effective from the viewpoint of iSSCs, their complexities are different in practical applications. The complexity of method one is O (N 2 ) when QR decomposition is used to calculate the left eigenvector basis associated with the eigenvalue 0 of L, while the other is O (|E |) when the Jungnickel algorithm is employed. 4. Stabilizability under switching topology For switching topology, the control protocol is given as follows: ui (t ) = ui0 (t  ) + ui1 (t ) ui0 (t ) = K aij (t )(xj (t ) − xi (t )) vj ∈Ni (t )

(14)

ui1 (t ) = K · di (t ) · ri (t ) i = 1, . . . , n. The difference from the fixed graph is that both the neighbors Ni (t ) of each agent and the connection weight aij , di (i, j = 1, . . . , n) are time varying, and, moreover, Laplacian L(t ) associated with the switching interconnection graph is also time varying. Denote G¯ = {G1 , . . . , GM } as the set of graphs with all possible topologies, and M , {1, . . . , M } as its index set. Let σ (t ) : [0, ∞) −→ M be a piecewise constant switching signal with successive times to describe the topology switches between subintervals. For convenience, we denote Lp (p ∈ M) as the Laplacian of the digraph p p Gp (p ∈ M), Dp = diag(d1 , . . . , dn )(p ∈ M). As in [5], consider an infinite sequence of nonempty, bounded, and contiguous time intervals [tk , tk+1 ), k = 0, 1, . . ., with t0 and tk+1 − tk ≤ T for some constant T > 0. In each interval [tk , tk+1 ) there is a sequence of nonoverlapping subintervals

[tk0 , tk1 ), . . . , [tkl , tkl+1 ), . . . , [tkmk −1 , tkmk ), tk = tk0 ,

tk+1 = tkmk

(15)

satisfying tkl+1 − tkl ≥ τ , 0 ≤ l ≤ mk for some integer mk ≥ 0 and given constant τ ≥ 0, such that, during each time interval [tkl , tkl+1 ), the graph Gσ (t ) is fixed; we denote it by Gtkl . Now one

Y. Guan et al. / Systems & Control Letters 62 (2013) 438–446

can more reasonably view (1) as a system in the form of a switching network:

where

   x˙ (t ) = In ⊗ A − Lσ (t ) ⊗ (BK ) x(t ) + Dσ (t ) ⊗ (BK ) r (t ).

A¯ k =



(16)

Definition 4. Suppose that we have a set of M graphs Gk = (N , E k , Ak ), k = 1, . . . , M, with the same node set N = {v1 , . . . , vn }. M k ˜ The graph G˜ = (N˜ , E˜ , A˜ ) where N˜ = N , E˜ = k=1 E , A =

M

M

Ak is called the union of Gk and is denoted by k=1 Gk . Given a piecewise constant switching signal σ (t ), Gσ (t ) = (N , E σ (t ) , Aσ (t ) ) denotes a time-varying graph, where E σ (t ) ⊆ N × N , σ (t ) Aσ (t ) = [aij ] ∈ Rn×n for all t ≥ 0. For any s ≥ 0, S > 0, let  σ (t ) G˜ ([s, s + S )) = ; we call G˜ ([s, s + S )) the union t ∈[s,s+S ) G σ (t ) graph of G over the time interval [s, s + S ). k=1

Assumption 3. There exists an infinite sequence of time intervals [tk , tk+1 ), k = 0, 1, . . ., with t0 = 0 and tk+1 − tk ≤ T (k ≥ 0) for T > 0, and a sequence of nonoverlapping subintervals in the form of (15) with tkl+1 − tkl ≥ τ in each interval for τ > 0. The union t

t

t

t

graph G˜ ([tk , tk+1 )) contains m distinct iSCCs G˜ j k = (N˜j k , E˜j k , A˜ j k ) with j = 1, . . . , m.

M

k Lemma 6. Suppose that the union graph G˜ = k=1 G contains m distinct iSCCs G˜ j = (N˜j , E˜j , A˜ j ) with j = 1, . . . , m. Let matrices L1 , . . . , LM and D1 , . . . , DM be associated with the graphs G1 , . . . , GM , respectively. If, for each j = 1, . . . , m, there exists a M node q ∈ N˜j such that d˜ qj > 0, then (i) all the eigenvalues of k=1

( Lk + Dk ) have positive real parts, and (ii) all the eigenvalues of M τ (Lk + Dk ) have positive real parts, where τk > 0 and kM=1 k k=1 τk = 1. M Proof. (i) Note that, for G˜ = k=1 Gk , its weighted Laplacian L˜ U = M M k k ˜ k=1 L , and the corresponding matrix DU = k=1 D . Then the proof can be implemented in the same way as the sufficiency proof ˜ U instead of L and D, respectively. of Lemma 4 by using L˜ U and D M (ii) A similar discussion for the weighted matrices L˜ U = k=1

τk Lk and D˜ U =

M

k=1

τk Dk yields the conclusion.



Inspired by [5], the averaging method is employed here to derive the following stabilizability result under switching topology. Theorem 3. Under Assumptions 1 and 3, consider multi-agent system (1) with control protocol (14) and the associated switched graph Gσ (t ) with T sufficiently small. There exists a positive number δ such that P > 0 is a solution to the Riccati inequality PA + AT P + IN − 2δ PBBT P < 0.

(17)

T

Let K = B P and ri (t ) = −ki xi (t ),

(18)

Proof. Under external control (18), the closed-loop system (16) reads x˙ (t ) = In ⊗ A − H σ (t ) ⊗ (BK ) x(t ),





σ (t )

(19)

σ (t )

where H =L + D K1 , K1 = diag(k1 , k2 . . . , kn ). During each [tk , tk+1 ), the average system of (19) is x˙¯ = A¯ k x¯ ,

tk



tk+1 − tk [ t ¯ k ,tk+1 )

= In ⊗ A − H ⊗ (BK ),  [tk ,tk+1 ) ¯ H = t ∈[tk ,tk+1 ) τσ (t ) H σ (t ) , and τl = (tkl+1 − tkl )/(tk+1 − tk ), 0 ≤ l ≤ mk . Define   δ = min

k = 0, 1, . . . ,

(20)

min

inf

(τ0 ,...,τmk )∈Γk λ∈∧(H¯ [tk ,tk+1 ) )

R(λ) | k = 0, 1, . . . ,

(21)

m

k where Γk = {(τ0 , . . . , τmk ) | l=0 τl , τ ≤ τl < 1}. Noting that minλ∈∧(H¯ [tk ,tk+1 ) ) R(λ) depends continuously on τ0 , . . . , τmk , and that the set Γk is compact, by Lemma 6, we have

min

inf

(τ0 ,...,τmk )∈Γk λ∈∧(H¯ [tk ,tk+1 ) )

=

R(λ)

min

∗ H km ) λ∈∧(τ0∗ H k0 +···+τm k

R(λ) > 0,

which implies that δ in (21) is a positive number since set (21) is finite due to the finiteness of all possible topologies. Next, we show that, for each k, system (20) is asymptotically stable. Let Uk ∈ Rn×n be a unitary matrix such that Uk−1 H¯ [tk ,tk+1 ) Uk = ¯ k is the Schur upper triangular matrix with diagonal elements ∧ λ¯ k1 , . . . , λ¯ kn being the eigenvalues of H¯ [tk ,tk+1 ) . Taking xˆ = (U ⊗ In )−1 x¯ , (20) becomes

¯ k ⊗ BK )ˆx, x˙ˆ = (In ⊗ A − ∧ whose stability is equivalent to that of the subsystems,

¯ ki BBT P )ˆxi , x˙ˆ i = (A − λ

i = 1 , . . . , n.

¯ = ¯ + j¯ then Denoting λ k i

xki

yki ,

¯ ki BBT P ) + (A − λ¯ ki BBT P )∗ P = PA + AT P − 2x¯ ki PBBT P P (A − λ ≤ PA + AT P − 2δ PBBT P ≤ −IN < 0. Therefore system (20) is globally asymptotically stable for each k = 0, 1, . . .. Now we can obtain from Lemma 2.4 of [5] that there exists a positive α ∗ dependent on T , such that, for ∀α > α ∗ , the switching system x˙ (t ) = [In ⊗ A − H σ (α t ) ⊗(BK )]x(t ) is asymptotically stable. By Remark 2.5 of [5], α ∗ can be made smaller than 1 if T is chosen sufficiently small. Since α > α ∗ is arbitrary, just pick α = 1. That is, system (19) is asymptotically stable.  Corollary 3. Suppose that the union graph G˜ ([tk , tk+1 )) is connected. Under Assumption 1, let K = BT P, where P > 0 is a solution of (17), and let the external controls be as shown in (18). Then, multi-agent system (16) is asymptotically stable if, in G˜ ([tk , tk+1 )), there exists a t node q ∈ N˜ tk such that d˜ qk > 0 and kq > 0. t

t

t

Proof. Let G˜ 1k = (N˜1 k , E˜1k , A˜ 1k ) be the iSCC in G˜ ([tk , tk+1 )). Then, by Theorem 3, system (16) is asymptotically stable if there exists a t

where ki is a nonnegative number. Then system (16) is asymptotically stable if in G˜ ([tk , tk+1 )), for each j = 1, . . . , m, there exists a node t t qj ∈ N˜j k such that d˜ qkj > 0 and kqj > 0.

σ (t )

In ⊗ A − H σ (t ) ⊗ (BK ) dt

 tk+1 

t

i ∈ N,

443

t k node q ∈ N˜1 such that d˜ qk > 0 and kq > 0.



Remark 7. Different from Proposition 1, the feedback gain K = BT P in Theorem 3 can be calculated in a distributed way since P > 0 can be computed with a sufficiently small parameter δ1 which is independent of the global information. In addition, we see that the graph conditions given in Theorem 3 and Corollary 3 only involve the topology structure of the union graph and do not require stabilizability of the subsystems. This is not only desirable in applications, as it can provide more freedom to the design of a multi-agent system, but also convenient for the design of a switching path to guarantee the switching stabilizability of the system.

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Y. Guan et al. / Systems & Control Letters 62 (2013) 438–446

5. Application to formation control In this section, the obtained results are applied to the formation control of multi-agent systems. A pertinent work is [24], which addresses the decentralized set-point formation control problem with agent dynamics modeled as single integrators under fixed topology. Consider multi-agent system (1) with interaction topology modeled by a graph G = (N , E , A). A set-point h0 ∈ RN , H = [hT1 , hT2 , . . . , hTn ]T ∈ RnN describes a constant formation structure of the agent network in a reference coordinate frame, where hi ∈ RN is the formation variable corresponding to agent i. Then, variable hi − hj can be used to denote the relative formation vector between agents i and j, which is independent of the reference coordinate. Fig. 1. A directed graph with 13 nodes, where r serves as the external control input.

20 15 10 5

x(t)

Definition 5 ([24]). The decentralized set-point control problem with formation H = [hT1 , hT2 , . . . , hTn ]T for multi-agent system (1) can be solved if there exist controls ui and rqj such that (a) each agent is allowed to use its own state information and that of its neighboring agents for feedback, as well as the desired displacements hi ; (b) the information on the set-point h0 is available only for the nodes qj ; and (c) under the controls, the multi-agent system satisfies that limt →∞ ∥ xi (t ) − (hi + h0 ) ∥= 0 for each i ∈ N . First, for multi-agent system (1), we propose the following interagent and external controls: ui0 (t ) = K



−10

vj ∈Nj σ (t )

−15

(22)

−20 0

2

6

4

8

10

t

where K ∈ RP ×N is a feedback matrix to be designed, and ki is nonnegative number. σ (t ) is a switching signal. Next, we give the following theorem to show that a multiagent system under decentralized control protocol (22) solves the problem of decentralized set-point control with formation under fixed and switching topologies, respectively. Theorem 4. Under Assumptions 1 and 2, suppose that the agent’s dynamics is not asymptotically stable. Then, for multi-agent system (1) with control protocol (22), there exists K such that the decentralized set-point control problem with formation H = [hT1 , hT2 , . . . , hTn ]T can be achieved if and only if, for each j = 1, . . . , m, there exists a node qj ∈ Nj such that dqj > 0 and kqj > 0. Proof. Let x˜ i = xi − hi − h0 , i = 1, . . . , n and x˜ = [˜xT1 , . . . , x˜ Tn ]T . Then system (1) under protocol (22) can be written as x˙˜ (t ) = In ⊗ A − (L + DK1 ) ⊗ (BK ) x˜ (t ),



Fig. 2. The state trajectories of all the agents with external control input.

where Lσ (t ) is the Laplacian matrix of Gσ (t ) = (N , E σ (t ) , Aσ (t ) ), K = BT P is the feedback gain matrix, and the diagonal matrices Dσ (t ) and K1 are defined in (19). The rest of the proof is similar to the proof of Theorem 3, and is thus omitted for brevity.  Remark 8. The difference between the formation control given by Definition 5 and the one based on consensus [10,11] lies in the fact that the former formation eventually reached by agents does not depend on the initial conditions xi (0). In addition, the formation can be put in any location by setting h0 . The necessary and/or sufficient graphical conditions presented in Theorems 4 and 5 expand the results given in [24], where it is assumed that the agent dynamics is a single integrator and the communication topology is fixed.

(23)

where L is the Laplacian matrix of G = (N , E , A), , K is the feedback gain matrix (the existence of K can be obtained by Proposition 1), and the diagonal matrices D and K1 are defined in (12). The rest of the proof is similar to the proof of Theorem 2, and is thus omitted for brevity.  Theorem 5. Under Assumptions 1 and 3, let K = B P, where P > 0 is a solution of (17). Then, for system (1), the decentralized setpoint control problem with formation H = [hT1 , hT2 , . . . , hTn ]T can be achieved under decentralized control protocol (22) if in G˜ ([tk , tk+1 )), t t for each j = 1, . . . , m, there exists a node qj ∈ N˜j k such that d˜ qkj > 0 and kqj > 0. T

Proof. Similarly, let x˜ i = xi − hi − h0 , i = 1, . . . , n and x˜ = [˜xT1 , . . . , x˜ Tn ]T ; then system (1) under protocol (22) can be written as x˙˜ (t ) = In ⊗ A − (Lσ (t ) + Dσ (t ) K1 ) ⊗ (BK ) x˜ (t ),



−5

 σ (t )  aij (xj (t ) − hi ) − (xi (t ) − hj )

ui1 (t ) = K ·di · ri ( t )  ri (t ) = −ki xi (t ) − hi − h0 ,



0



(24)

6. Simulations Example 1. The first example aims at demonstrating the effectiveness of Theorem 2. Consider multi-agent system (1)  with xi =

[xi1 xi2 ]T ∈ R2 , which consists of 13 agents and A =  1 0

−4 4

1 1

,B =

.Suppose that the interacting topology of multi-agent system

(1) is described as in Fig. 1. The external control input, namely r, is assumed to be applied to vertices 1, 3, and 6. Suppose that ri (t ) = −5xi (t ), i = 1, . . . , 13. From Proposition 1, we can obtain that K = [9.5110 13.5034], such that all matrices A − λi BK , λi ∈ ∧(L + DK1 ), i = 1, . . . , 13 are Hurwitz, where DK1 = diag{5 0 5 0 0 5 0 0 0 0 0 0 0}. Assume that all the initial conditions of the agents are randomly chosen. The simulation result is shown in Fig. 2. We see that the state trajectories of all the agents converge to zero, which is in agreement with the theoretical analysis results.

Y. Guan et al. / Systems & Control Letters 62 (2013) 438–446

445

Example 2. The second example aims at demonstrating the effectiveness of Theorem 3. Consider multi-agent system (1) with xi = [xi1 xi2 ]T ∈ R2 , which consists of six agents and A = 0

−1

1 0

,B =

0 1

.The switching topologies of system (1) are

illustrated in Fig. 3. Gi , i = 1, 2, 3, 4 are four possible topologies which are switched as G1 → G2 → G3 → G4 → G1 → . . ., and each graph is activated for 0.5s. The node r serves as the external control input, and D1 = diag(0, 0, 0, 0, 0, 0), D2 = diag(0, 0, 0, 0, 1, 0), D3 = diag(0, 0, 0, 0, 0, 0) and D4 = diag (1, 0, 0, 0, 0, 0). The Laplacian matrices L1 , L2 , L3 and L4 are as follows:

0 0

0 0 0 L = 0  2

0 0

0 0 0 L = 0  3

0 0

0 0 0 4 L = −2  0 0

0 2 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

0 −2 2 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 , 0  0 0



0 0 0 , 0  0 0

Fig. 3. Four possible network topologies G1 , G2 , G3 , and G4 .

6



0 0 0 −4 0 0

0 0 0 4 0 0

0 0 0 0 0 −1.5

0 0 0 0 0 0

0 0 0 2 0 0

0 0 0 0 0 0

4 2

0 0  0  , 0   0 1.5



0 0 0 . 0  0 0

x(t)

0 3.5 0 0 0 0

−4 −6



By Theorem 1, it is easy to check that the subsystems of system (16) with (−Li , Di ) (i = 1, 2, 3, 4) are all unstabilizable. From 2s+1.5 Fig. 3, we find that the union graph k=2s Gk contains two distinct iSCCs. They are N˜1k = {1} and N˜2k = {5}. The external control, namely r, is applied to agent 1 and agent 5 for any s = 0, 1, . . .. We choose a small parameter δ = 0.0257; the matrix K in Theorem 3 is calculated as K = [0.5237 6.8926]. Assume that all the initial conditions of the agents are randomly chosen. Fig. 4 shows that, under external control ri (t ) = −5xi (t ), i = 1, . . . , 6, the state trajectories of all the closed-loop system (19) asymptotically converge to zero. This is in agreement with the theoretical analysis results. Example 3. The third example is to validate Theorem 4. Consider the interacting topology depicted by Fig. 1. The external control input, namely r, is assumed to be applied to the vertices 1, 3, and 6. Let A, B, and the feedback gain K be   the same as in Example 1. Assume that ri (t ) = −5 xi (t )−hi −h0 , i = 1, . . . , 13, where h0 = [60 60]T , the agents starting from an arbitrary configuration (all the initial conditions of the agents are randomly chosen) and H = [hT1 , hT2 , . . . , hT13 ]T , where h1 = [0 0]T , h2 = [−10 − 10]T , h3 = [0 − 10]T , h4 = [10 − 10]T , h5 = [−20 − 20]T , h6 = [−10 − 20]T , h7 = [0 − 20]T , h8 = [10 − 20]T , h9 = [20 − 20]T , h10 = [−10 − 30]T , h11 = [0 − 30]T , h12 = [10 − 30]T , h13 = [0 − 40]T . Fig. 5 shows the simulation result. Example 4. The fourth example is to validate Theorem 5. Consider the interacting topology depicted in Fig. 3. Let A, B, and the

0 −2

0

10

20

30

40

50

t Fig. 4. The state trajectories of all the agents with external control input.

70 60

t=0 t=10

50 40

xi2 axis

 0 −3.5  0 1 L =  0 

30 20 10 0 −10 −20 −20

0

20

40 xi1 axis

60

80

100

Fig. 5. Thirteen agents form a diamond. The asterisk and circle denote the configuration of the multi-agent system at t = 0 and 10, respectively.

feedback gain  K be the same  as in Example 2. We assume that ri (t ) = −5 xi (t ) − hi − h0 , i = 1, . . . , 6, where h0 = [30 30]T , the agents starting from an arbitrary configuration (all the initial conditions of the agents are randomly chosen) and H = [hT1 , hT2 , . . . , hT6 ]T , where h1 = [0 0]T , h2 = [−5 − 5.5]T , h3 = [5 − 5.5]T , h4 = [−5 − 12.5]T , h5 = [5 − 12.5]T , h6 = [0 − 17.5]T . Fig. 6 shows the simulation result. 7. Conclusion In this paper, the stabilizability problem of multi-agent systems with agents modeled by general linear dynamics was formulated

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Y. Guan et al. / Systems & Control Letters 62 (2013) 438–446

30 25

t=0 t=50

xi2 axis

20 15 10 5 0 −5 −5

0

5

10

15 20 xi1 axis

25

30

35

40

Fig. 6. Six agents form a diamond. The asterisk and circle denote the configuration of the multi-agent system at t = 0 and 50, respectively.

and considered. Two important issues of stabilizability protocol design with respect to the stabilizability region and the feedback gain have been addressed. In addition, two methods were presented to find the agents under which stabilizability can be reached. Some necessary and/or sufficient conditions were proposed to stabilize the system through a decentralized external self-feedback control input in scenarios with fixed and switching topologies. Finally, the results were employed to cope with the decentralized set-point formation control problem, for which some necessary and/or sufficient conditions were developed. Consideration of agents with nonidentical or discrete-time linear dynamics leaves an interesting topic for future research. References [1] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control 49 (9) (2004) 1520–1533. [2] Y. Sun, L. Wang, G. Xie, Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays, Systems & Control Letters 57 (2008) 175–183. [3] Y. Zheng, W. Chen, L. Wang, Finite-time consensus for stochastic multi-agent systems, International Journal of Control 84 (10) (2011) 1644–1652. [4] Y. Zheng, L. Wang, Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements, Systems & Control Letters 61 (2012) 871–878. [5] W. Ni, X. Wang, C. Xiong, Leader-following consensus of multiple linear systems under switching topologies: an averaging method. Available at: http://arxiv.org/abs/1211.1438. [6] H. Shi, L. Wang, T. Chu, Virtual leader approach to coordinated control of multiple mobile agents with asymmetric interactions, Physica D 231 (1) (2006) 51–65. [7] R. Olfati-Saber, Flocking for multi-agent dynamic systems: algorithms and theory, IEEE Transactions on Automatic Control 51 (3) (2006) 401–420.

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