Two consensus problems for discrete-time multi-agent systems with switching network topology

Automatica 48 (2012) 1988–1997

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Two consensus problems for discrete-time multi-agent systems with switching network topology✩ Youfeng Su, Jie Huang 1 Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

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abstract

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Article history: Received 9 August 2011 Received in revised form 22 November 2011 Accepted 19 March 2012 Available online 7 July 2012

In this paper, we study both the leaderless consensus problem and the leader-following consensus problem for linear discrete-time multi-agent systems under switching network topology. Under the assumption that the system matrix is marginally stable, we show that these two consensus problems can be solved via the state feedback protocols, provided that the dynamic graph is jointly connected. Our result will contain several existing results as special cases. The proof is based on the stability analysis of a class of linear discrete-time switched systems which may have some independent interest. © 2012 Elsevier Ltd. All rights reserved.

Keywords: Consensus Switching network Multi-agent systems Discrete-time systems Jointly connected

1. Introduction In this paper, we consider two consensus problems of the following class of linear discrete-time multi-agent systems subject to a switching network topology: xi (k + 1) = Axi (k) + Bui (k),

i = 1, . . . , N , k = 0, 1, . . . ,

(1)

where xi (k) ∈ R and ui (k) ∈ R are the state and control of agent i, respectively, and (A, B) is controllable. Over the past decade, there has been extensive interest in consensus problems of both continuous-time and discrete-time multi-agent systems; see the survey paper Olfati-Saber, Fax, and Murray (2007) and the books Bertsekas and Tsitsiklis (1989), Qu (2009), and Ren and Beard (2008). There are two types of consensus problems: the leaderless consensus problem and the leaderfollowing consensus problem. While the leaderless consensus problem aims to design a distributed controller for a multi-agent system so that the states of all agents asymptotically approach a n

m

✩ This work has been supported in part by the Research Grants Council of the Hong Kong Special Administration Region under grant No. 412810, and in part by National Natural Science Foundation of China under grant No. 61174049. The material in this paper was partially presented at the 30th Chinese Control Conference (CCC’2011), 22–24 July 2011, Yantai, China. This paper was recommended for publication in revised form by Associate Editor Huijun Gao under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (Y. Su), [email protected] (J. Huang). 1 Tel.: +852 2609 8473; fax: +852 2603 6002.

0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.03.029

common value, the leader-following consensus problem further requires that the distributed controller is such that the states of all agents converge to a specified trajectory which is usually produced by another agent called the leader. The distributed controller is defined by a communication network topology. The network topology can be described by either a static or a dynamic graph. The most common dynamic network topology is the so-called switching network topology. For the class of continuous-time systems, the two consensus problems were first studied for some simple systems such as single integrators (Jadbabaie, Lin, & Morse, 2003; Ji, Wang, Lin, & Wang, 2009; Lin, 2005; Olfati-Saber & Murray, 2004; Ren & Beard, 2005), double integrators (Hong, Chen, & Bushnell, 2008; Hong, Gao, Cheng, & Hu, 2007; Qin, Gao, & Zheng, 2011; Qin, Zheng, & Gao, 2011; Ren, 2008a), and harmonic oscillators (Ren, 2008b) under various assumptions on the switching network topology. In particular, both the leaderless consensus problem and the leader-following consensus problem for single-integrator systems were solved under the assumption that the switching network topology satisfies the jointly connected condition. It is worth mentioning that the jointly connected condition is quite weak in the sense that it allows the network to be disconnected at any time instant. The leaderless consensus problem for more general linear systems was studied in Wang, Cheng, and Hu (2008) under the assumption that the dynamic graph is frequently connected. The leader-following consensus problem for more general linear systems was studied in Ni and Cheng (2010) under the assumption that the system matrix simultaneously satisfies two matrix inequalities. More recently, the paper Su and Huang (2012)

Y. Su, J. Huang / Automatica 48 (2012) 1988–1997

further considered the two consensus problems for linear marginally stable multi-agent systems. In comparison with the result in Wang et al. (2008), the result in Su and Huang (2012) has relaxed the frequently connected condition to the jointly connected condition. In comparison with the result in Ni and Cheng (2010), the result in Su and Huang (2012) has replaced the requirement of two matrix inequalities by a requirement for only one matrix inequality. It is worth mentioning that Scardovi and Sepulchre (2009) and Xu, Li, Xie, and Lum (2011) studied the leaderless consensus problem of general linear multi-agent systems using dynamic state feedback or dynamic output feedback control design under switching network topology, as opposed to the static state feedback design studied in Ni and Cheng (2010), Su and Huang (2012) and Wang et al. (2008). For the class of discrete-time systems, both the leaderless consensus problem and the leader-following consensus problem have been mainly studied for multi single-integrator systems (Bertsekas & Tsitsiklis, 1989; Jadbabaie et al., 2003; Nedić, Olshevsky, Ozdaglar, & Tsitsiklis, 2009; Olfati-Saber & Murray, 2004; Ren & Beard, 2005; Tsitsiklis, 1984) under the jointly connected assumption. Tuna (2008) studied the leaderless consensus problem for the same class of systems as (1) for the special case where the network topology is static via state feedback control. Recently, You and Xie (2011a,b) further presented the necessary and sufficient conditions for the consensusability of discrete-time linear multi-agent systems with a static network topology via static state feedback and dynamic output feedback control, respectively. Scardovi and Sepulchre (2009) studied the leaderless synchronization of discrete-time linear multi-agent systems for a time-varying network topology via dynamic state feedback and dynamic output feedback control. In this paper, we will further study both the leaderless consensus problem and the leader-following consensus problem of system (1) under the assumption that the dynamic graph is jointly connected in the sense to be precisely described in Section 4. It is noted that the closed-loop system arising from either the leaderless consensus problem or the leader-following consensus problem is a class of linear discrete-time switched systems. Thus the key for establishing our result is to study the stability of this class of systems. What makes this stability analysis interesting is that, under the jointly connected assumption, the dynamic graph can be disconnected at any time instant, and hence the closed-loop system matrix may not be Schur at any time instant. As a result, the common Lyapunov function approach alone is not adequate to furnish a convergence analysis (Cheng, Guo, & Huang, 2003; Liberzon & Morse, 1999). We have to combine the common Lyapunov function approach with some novel technique to complete the convergence analysis in Lemma 3.1. The result of this paper can be viewed as a discrete-time counterpart of the main result in Su and Huang (2012). For the leaderless consensus problem, it extends the results in Bertsekas and Tsitsiklis (1989), Jadbabaie et al. (2003), Nedić et al. (2009), Olfati-Saber and Murray (2004) and Ren and Beard (2005) where the agents are single integrators, and also extends the result in Tuna (2008) to the case where the network topology is switching. For the leader-following consensus problem, our result can also be seen as an extension of that in Jadbabaie et al. (2003) where leader-following consensus of a group of discrete-time single integrators was considered. The rest of this paper is organized as follows. In Section 2, we introduce the descriptions of both consensus problems. In Section 3, a key lemma (Lemma 3.1) for the stability analysis of a class of linear discrete-time switched systems is established. In Section 4, we present our main result. Using Lemma 3.1, we show that both the leaderless consensus problem and the leader-following consensus problem can be achieved via the state feedback protocols, provided that the dynamic graph is jointly connected. Some examples are provided to illustrate our design in

1989

Section 5. Finally, we conclude the paper in Section 6. The graph notation that will be extensively used throughout this paper are summarized in the Appendix. The following notation will be used throughout this paper. ⊗ denotes the Kronecker product of matrices. Some properties of Kronecker product are useful in this paper: (A ⊗ B)T = AT ⊗ BT , (A ⊗ B)(C ⊗ D) = (AC ) ⊗ (BD), (A + B) ⊗ C = A ⊗ C + B ⊗ C , A ⊗ (B + C ) = A ⊗ B + A ⊗ C . span{A} denotes the linear subspace 1

spanned by the columns of A. A 2 denotes the square root of a positive semi-definite matrix A. Z+ denotes the set of nonnegative integers. 1N denotes an N × 1 column vector whose elements are all 1. Given the matrices Ai ∈ Rni ×m , i = 1, . . . , N, we denote col(A1 , . . . , AN ) = [AT1 , . . . , ATN ]T . 2. Problem statement To introduce our problems, let σ : Z+ → P , where P = {1, 2, . . . , ρ}. Throughout this paper, we assume that σ (·) is a piecewise constant switching signal in the sense that there exists a subsequence ki of k, called switching instants, such that σ (k) is a constant for ki ≤ k < ki+1 for any ki ≥ 0. Associated with system (1) and the given piecewise constant switching signal σ (·), we can define a dynamic graph2 Gσ (k) = (V , Eσ (k) ), where V = {1, . . . , N } and (j, i) ∈ Eσ (k) if and only if the control ui of agent i can make use of xi − xj for feedback at time k. Let Lσ (k) = [lij (k)] ∈ RN ×N be any time-varying Laplacian matrix of the graph Gσ (k) . Then we can define the following state feedback protocol: ui (k) = K

N 

lij (k)(xi (k) − xj (k)),

i = 1, . . . , N ,

(2)

j=1

where K ∈ Rm×n is the gain matrix to be designed. Definition 1 (Leaderless Consensus Problem). Given plant (1) and the associated dynamic graph Gσ (k) , find the gain matrix K such that, for i, j = 1, . . . , N, the closed-loop system composed of plant (1) and state feedback protocol (2) has the property that xi (k) − xj (k) → 0 as k → ∞. For the leaderless consensus problem described above, the steady-state behavior of the solution of each subsystem is immaterial. Another type of consensus problem, called the leaderfollowing consensus problem, requires the solution of each subsystem to approach some prescribed discrete-time signal x0 (k) asymptotically. In this paper, we assume that the signal x0 (k) is generated by a linear autonomous system of the form x0 (k + 1) = Ax0 (k),

(3)

with an arbitrary initial state x0 (0) ∈ R . System (3) and system (1) are called the leader system and the follower system, respectively. Associated with system (1), system (3), and the given piecewise constant switching signal σ (·), we can define another dynamic graph G¯ σ (k) = (V¯ , E¯σ (k) ), where V¯ = {0, 1, . . . , N } and (j, i) ∈ E¯σ (k) , i = 1, . . . , N , j = 0, 1, . . . , N, if and only if control ui can make use of xi − xj for feedback at time k. Clearly, Gσ (k) is a subgraph of G¯ σ (k) , and it can be obtained from G¯ σ (k) by removing node 0 from V¯ and all edges incident on node 0 at time k from E¯σ (k) . To introduce our control law for the leader-following case, for any k ≥ 0 and i = 1, 2, . . . , N, let li0 (k) < 0 if (0, i) ∈ E¯σ (k) and li0 (k) = 0 otherwise. Then n

ui (k) = K

N 

lij (k)(xi (k) − xj (k)),

i = 1, . . . , N ,

j =0

where K ∈ Rm×n is the gain matrix to be defined later.

2 See Appendix for a summary of the graph.

(4)

1990

Y. Su, J. Huang / Automatica 48 (2012) 1988–1997

Definition 2 (Leader-Following Consensus Problem). Given leader system (3), follower system (1), and the associated dynamic graph G¯ σ (k) , find the gain matrix K such that the closed-loop system composed of plant (1) and state feedback protocol (4) has the property that xi (k) − x0 (k) → 0 as k → ∞. Remark 2.1. Let ∆σ (k) be an N × N diagonal matrix with −li0 (k) as its ith diagonal element, and let Lσ (k) be a Laplacian of Gσ (k) . Then it can be verified that the matrix

L¯ σ (k) =



01×1

−∆σ (k) 1N



01×N

Lσ (k) + ∆σ (k)

By Remark 2.3, there exists a nonsingular real matrix P such that

¯ . A = P −1 AP

(8)

Under the transformation x¯ i = Pxi , system (1) is transformed to

¯ i (k), x¯ i (k + 1) = A¯ x¯ i (k) + Bu

i = 1, . . . , N ,

(9)

where B¯ = PB. 3. A stability result As will be seen later, our results on both consensus problems are based on the stability analysis of some linear discrete-time switched system. We will first introduce the following assumption.

is a Laplacian of G¯ σ (k) . In this paper, we make the following assumption on the system matrix A. Assumption 1. The system matrix A is neutrally stable, i.e., all eigenvalues of A are semi-simple3 with modulus 1.

Assumption 2. The switching instants satisfy ki+1 − ki ≥ τ for some positive integer τ > 1 (called the dwell time (Liberzon & Morse, 1999)) for all ki . Lemma 3.1. Consider the linear discrete-time switched system

Remark 2.2. The system matrix A is said to be marginally stable if all the eigenvalues of A are inside the unit circle, and those eigenvalues with modulus 1 are semi-simple3 . Assumption 1 can be relaxed to the case where the matrix A is marginally stable. In fact, if the matrix A is marginally stable but not neutrally stable, then, as indicated in You and Xie  (2011b),  there is a nonsingular  matrix T such that TAT −1 =

Au 0

0 As

and TB =

Bu Bs

, where

(n−nu )×(n−nu )

nu ×nu

ξ (k + 1) = (IN ⊗ A¯ − µFσ (k) ⊗ (B¯ B¯ T A¯ ))ξ (k),

(10)

where the pair (A¯ , B¯ ) is as defined in (7) and (9), Fσ (k) is symmetric and positive semi-definite for any k ≥ 0, σ (k) is a piecewise constant switching signal with dwell time τ that satisfies Assumption 2, and µ satisfies





1

Au ∈ R is neutrally stable, As ∈ R is Schur, Bu ∈ nu ×m (n−nu )×m . Under the coordinate transformation R  , and Bs ∈ R

0 < µ ≤ min

system (1) is equivalent to

(1) If a solution ξ (k) of (10) has the property that there exists an infinite subsequence {tj } of {k} satisfying tj+1 − tj < ν for some positive ν and any j ≥ 0 such that ξ (tj ) is orthogonal to the null

x¯ 1i x¯ 2i

= Txi , i = 0, 1, . . . , N, where x¯ 1i ∈ Rnu and x¯ 2i ∈ Rn−nu ,

x¯ 1i (k + 1) = Au x¯ 1i (k) + Bu ui (k) x¯ 2i (k + 1) = As x¯ 2i (k) + Bs ui (k),

(5) i = 1, . . . , N ,

(6)

x¯ 20 (k + 1) = As x¯ 20 (k). Now, by the same argument as in You and Xie (2011b), if there exists a gain matrix K1 ∈ Rm×nu such that the controller ui (k) = N K1 j=1 lij (k)(¯x1i (k) − x¯ 1j (k)), i = 1, . . . , N, solves the leaderless consensus problem of system (5), this controller, which takes the form (2) with K = [K1 , 0]T , also solves the leaderless consensus problem of system (1). Similarly, if there exists a gain matrix N K1 ∈ Rm×nu such that ui (k) = K1 j=0 lij (k)(¯x1i (k) − x¯ 1j (k)), i = 1, . . . , N, solves the leader-following consensus problem of follower system (5), this controller, which takes the form (4) with K = [K1 , 0]T , also solves the leader-following consensus problem of system (1). Thus, in what follows, we only focus on the neutrally stable A. Remark 2.3. Under Assumption 1, we can assume that the Jordan form A¯ of A takes the following form.

  α1 −β1 ,..., β1 α1   −βr , Is−2r , −In−s , αr

 αr βr

.

Then the following hold.

tj+1 −1 k=tj

Fσ (k) ) ⊗ In , then

lim ξ (k) = 0.

k→∞

(11)

(2) If there exists an infinite subsequence {tj } of {k} satisfying tj+1 − tj < ν for some positive ν and any j ≥ 0 such that the matrix

 t −1 ( kj=+1tj Fσ (k) ) ⊗ In is nonsingular, then the origin of system (10) is asymptotically stable. To present the proof of Lemma 3.1, we need the following lemma. Lemma 3.2. Given φ(k) : Z+ → Rn , suppose that there exists a subsequence {ki } of {k} satisfying ki+1 − ki ≥ 2 such that, for all ki ≤ k < ki+1 − 1,

φ(k + 1) = (M − Q (k))φ(k),

(12)

where M is any square matrix and Q (k) is such that limk→∞ Q (k)φ(k) = 0. If limk→∞ F (k + s)φ(k) = 0 for some bounded matrix F (k) and some nonnegative integer s, then limki →∞ F (ki + s + 1)M φ(ki ) = 0. Proof. By direct calculation, we have lim F (ki + s + 1)M φ(ki )

A¯ = blockdiag

×

∥Fσ (k) ⊗ (A¯ T B¯ B¯ T A¯ )∥

space of the matrix (

and leader system (3) is equivalent to x¯ 10 (k + 1) = Au x¯ 10 (k)

σ (k)∈P

ki →∞

(7)

where 0 ≤ 2r ≤ s ≤ n, and, for k = 1, . . . , r, αk2 + βk2 = 1. Clearly, A¯ is an orthogonal matrix.

3 An eigenvalue of a matrix is semi-simple if all the Jordan blocks associated with this eigenvalue have dimension 1.

= lim F (ki + s + 1) (φ(ki + 1) − Q (ki )φ(ki )) ki →∞

= lim F (ki + s + 1)φ(ki + 1) − 0 ki →∞

= lim F (ki + s)φ(ki ) = 0.  ki →∞

Remark 3.1. From the proof of Lemma 3.2, if (12) holds for all k ≥ 0 and the other conditions remain unchanged, then the result of Lemma 3.2 can be strengthened to limk→∞ F (k + s + 1)M φ(k) = 0.

Y. Su, J. Huang / Automatica 48 (2012) 1988–1997

Now we give the proof of Lemma 3.1. Proof. Part (1): Let V (ξ (k)) =

1 2

ξ T (k)ξ (k).

(13)

1 2

(21)

k¯ i →∞

1

Let A¯ = blockdiag{A1 , . . . , An−r }, where A¯ l , l = 1, . . . , n − r, are the components of the lth Jordan block as described in (7), and let B¯ = col(B¯ 1 , . . . , B¯ n−r ), where the row dimension of B¯ l is the same as the dimension of A¯ l . Let η = col(η1 , . . . , ηN ) with ηs ∈ Rn , s = 1, . . . , N, and let ηs = col(η1s , . . . , η(n−r )s ), where the row dimension of ηls is the same as the dimension of A¯ l . Then (18) and (21) imply that

ξ T (k + 1)ξ (k + 1) − ξ T (k)ξ (k)

2 = −µξ T (k) (Fσ (k) ⊗ (A¯ T B¯ B¯ T A¯ ) 1 − µFσ2(k) ⊗ (A¯ T B¯ B¯ T A¯ )2 )ξ (k) 2

µ ≤ − ξ T (k) (Fσ (k) ⊗ (A¯ T B¯ B¯ T A¯ ))ξ (k) 2

≤ 0.

¯ Q (k) = µFσ (k) ⊗ (B¯ B¯ T A¯ ) Let F (k) = IN ⊗ (B¯ T A¯ ), M = IN ⊗ A, ¯ = µ(Fσ (k) ⊗ B)F (k), and φ(k) = η(k). Then, (18) implies that limk→∞ F (k)η(k) = 0 and limk→∞ Q (k)η(k) = 0. Thus, applying Lemma 3.2 to (20) with s = 0 gives lim (IN ⊗ (B¯ T A¯ )) (IN ⊗ A¯ )η(k¯ i ) = 0.

Then the difference of (13) along system (10) satisfies

1V (ξ (k))|(10) =

1991

(14)

lim B¯ Tl A¯ l ηls (k¯ i ) = 0

(22)

lim B¯ Tl A¯ 2l ηls (k¯ i ) = 0.

(23)

k¯ i →∞

That is, V (ξ (k)) is non-increasing and lower bounded. Thus, k k)−V (k−1))+V (0), limk→∞ V (ξ (k)) exists. Since V (k) = i=1 (V ( ∞ the existence of limk→∞ V (k) implies that of i=1 (V (k) − V (k − 1)). Therefore, limk→∞ ∆V (k) = limk→∞ (V (k + 1) − V (k)) = 0. Thus, by (14), limk→∞ ξ T (k) (Fσ (k) ⊗ (A¯ T B¯ B¯ T A¯ ))ξ (k) = 0,

Since the controllability of (A¯ , B¯ ) implies that of (A¯ l , B¯ l ), (B¯ Tl , A¯ Tl ) is observable. We further claim that (B¯ Tl , A¯ l ) is observable. In fact, the

i.e. limk→∞ (Fσ2(k) ⊗ (B¯ T A¯ ))ξ (k) = 0. Then

claim is obvious when A¯ l = 1 or −1. When A¯ l =

1

lim (Fσ (k) ⊗ (B¯ T A¯ ))ξ (k) = 0.

k→∞

(15)

lim ξ (k) = 0

(16)

(17)

Let η(k) = (Fσ (k) ⊗ In )ξ (k). Since Fσ (k) ⊗ (B¯ T A¯ ) = (IN ⊗ (B¯ T A¯ )) (Fσ (k) ⊗ In ), (15) implies that lim (IN ⊗ (B¯ T A¯ ))η(k) = 0.

k→∞

(1) k¯ 0 = k0 . (2) If q0 = 1, then k¯ 1 = k1 ; otherwise, if q0 ≥ 2, then k¯ j =

k0 + jτ , k1 ,

j = 1, . . . , q0 − 1; j = q0 .

(3) For i = 2, 3 . . ., suppose that k¯ p = ki−1 for some p > 0. Then, if qi = 1, k¯ p+1 = ki ; otherwise, if qi ≥ 2, k¯ j =



ki−1 + (j − p)τ , ki ,

j = p + 1, . . . , p + qi − 1; j = p + qi .

(24)

¯ η(k) = (IN ⊗ A¯ − µFσ (k¯ i ) ⊗ (B¯ B¯ T A¯ ))k−ki η(k¯ i ).

Then ¯

∥η(k)∥ ≤ ∥IN ⊗ A¯ − µFσ (k¯ i ) ⊗ (B¯ B¯ T A¯ )∥k−ki ∥η(k¯ i )∥ ¯ ≤ M k−ki ∥η(k¯ i )∥

≤ max{M 2τ , 1}∥η(k¯ i )∥,

(25)

where M , maxσ (k)∈P ∥IN ⊗ A¯ − µFσ (k) ⊗ (B¯ B¯ T A¯ )∥. The fact that P is a finite set implies M < +∞. By (24) and (25), we can see that η(k) is a Cauchy sequence that contains a subsequence η(k¯ i ) which converges to zero. Thus, limk→∞ η(k) = 0, i.e. (17) holds. Step 2. We will show that (17) implies (16). In fact, (10) is in the ¯ Q (k) = µFσ (k) ⊗ (B¯ B¯ T A¯ ), and form of (12) with M = IN ⊗ A, φ(k) = ξ (k). Let F (k) = Fσ (k) ⊗ IN . Then, (15) and (17) imply that limk→∞ Q (k)ξ (k) = 0 and limk→∞ F (k)ξ (k) = 0, respectively. Thus, applying Lemma 3.2 with s = 0 to (10) and noting Remark 3.1 gives lim (Fσ (k+1) ⊗ In ) (IN ⊗ A¯ )ξ (k) = 0.

Since (IN ⊗ A¯ ) (Fσ (k) ⊗ In ) = (Fσ (k) ⊗ In ) (IN ⊗ A¯ ) and IN ⊗ A¯ is nonsingular, (25) implies that (19)

lim (Fσ (k+1) ⊗ In )ξ (k) = 0.

k→∞

That is, limk→∞ F (k + 1)ξ (k) = 0. Thus, applying Lemma 3.2 with s = 1 to (10) and noting Remark 3.1 gives

Therefore, by (19), for any k¯ i ≤ k < k¯ i+1 − 1,

η(k + 1) = (Fσ (k+1) ⊗ In ) (IN ⊗ A¯ − µFσ (k) ⊗ (B¯ B¯ T A¯ ))ξ (k) = (Fσ (k) ⊗ In ) (IN ⊗ A¯ − µFσ (k) ⊗ (B¯ B¯ T A¯ ))ξ (k)

lim (Fσ (k+2) ⊗ In ) (IN ⊗ A¯ )ξ (k) = 0,

k→∞

= (IN ⊗ A¯ − µFσ (k) ⊗ (B¯ B¯ T A¯ )) (Fσ (k) ⊗ In )ξ (k) = (IN ⊗ A¯ − µFσ (k) ⊗ (B¯ B¯ T A¯ ))η(k).



k→∞

By this construction, it can be easily verified that k¯ i satisfies

τ ≤ k¯ i+1 − k¯ i < 2τ .

lim η(k¯ i ) = 0.

(18)

Under Assumption 2, for any ki , there exists qi ≥ 1 such that ki < ki + τ < ki + 2τ < · · · < ki + qi τ ≤ ki+1 < ki + (qi + 1)τ . Now, we construct a new sequence {k¯ i }, which contains {ki } as a subsequence, as follows.



βl αl , the

On the other hand, by (20), for any k¯ i ≤ k < k¯ i+1 ,

Step 1. We first show that (15) implies that lim (Fσ (k) ⊗ In )ξ (k) = 0.

αl −βl

claim is also clear by noting that A¯ Tl = −A¯ l . As the dimension of ηls is at most 2, (22) and (23) imply that limk¯ i →∞ A¯ l ηls (k¯ i ) = 0. Since

k¯ i →∞

by the following two steps.

k→∞



A¯ l is nonsingular, limk¯ i →∞ ηls (k¯ i ) = 0, i.e.,

Now we will show that (15) implies that k→∞

k¯ i →∞

which also implies that (20)

lim (Fσ (k+2) ⊗ In )ξ (k) = 0.

k→∞

1992

Y. Su, J. Huang / Automatica 48 (2012) 1988–1997

Thus, Lemma 3.2 and Remark 3.1 can be applied to (10) with s = 2, 3, . . .. As a result, for any s = 0, 1, . . ., lim (Fσ (k+s) ⊗ In )ξ (k) = 0.

−

tj+1 −1 k=tj

Lσ (k) is a Metzler matrix with zero row sum. Moreover,

tj+1 −1

it can be seen that Γ (

k=tj

Lσ (k) ) =

tj+1 −1

Gσ (k) . Thus, by

k=tj

tj+1 −1

k→∞

Remark A.1, under Assumption 4, the matrix

Since {tj } is a subsequence of {k}, we have

exactly one zero eigenvalue, and its null space is span{1N }.

lim (Fσ (tj +s) ⊗ In )ξ (tj ) = 0.

(26)

tj →∞

Let Jj = (

tj+1 −1 k=tj

Fσ (k) ) ⊗ In and ζj = Jj ξ (tj ). Then (26) leads to

lim ζj = 0.

(27)

j→∞

where

Ď Jj

denotes the Moore–Penrose inverse of Jj (Ben-Israel & Ď

Greville, 2003). Thus, ∥ξ (tj )∥ ≤ ∥Jj ∥ · ∥ζj ∥. Since P is a finite set

0 < µ ≤ min



lim ξ (tj ) = 0.

N

By (14), ∥ξ (k)∥ is non-increasing as k → ∞. This fact together with (28) lets us conclude that (16) holds. Part (2): Since Jj is nonsingular, the Moore–Penrose inverse of Jj becomes the inverse of Jj . We still have (28), and hence any solution of system (10) will approach the origin asymptotically. Therefore, the origin of system (10) is asymptotically stable.  Remark 3.2. If B¯ is of full row rank, both conclusions of the above lemma can be obtained without Assumption 2. In fact, from the above proof, one can see that Assumption 2 is only used in Step 1 to guarantee that η(k) satisfies (20). From (20), one can further prove that limk→∞ η(k) = 0. However, if B¯ and hence B¯ l are of full row rank, one can directly obtain the following from (18): lim B¯ Tl A¯ l ηls (k) = 0.

(29)

k→∞

Since A¯ l is nonsingular, B¯ Tl A¯ l is also of full column rank. Thus, we have BTl Al Ď

(¯ ¯ ) (¯ ¯ ) = BTl Al

 

I2 ,



1,



4. Solvability of the two consensus problems

N 

xc (k + 1) =

4.1. Leaderless consensus The dynamic graph Gσ (k) is bidirected for any

Assumption 4 (Joint Connectedness Jadbabaie et al., 2003). There exists an infinite subsequence {tj } of {k} such that tj+1 − tj is uniformly bounded for any j ≥ 0 and

k=tj

Gσ (k) is connected.

Remark 4.1. Under Assumption 3, the matrix

xi (k + 1)

i=1

N





N 

 xi (k)   i=1  = A   N 

= Axc (k).

(31)

As in Olfati-Saber and Murray (2004) and Wang et al. (2008), define a decomposition of xi (k) as xi (k) = xc (k) + wi (k),

i = 1, . . . , N .

(32)

Eq. (32) can be put in the following compact form: x(k) = 1N ⊗ xc (k) + w(k), where x(k) = col(x1 (k), . . . , xN (k)) and w(k) = col(w1 (k), . . . , wN (k)). We call w(k) the (group) disagreement N N vector. Since i=1 wi (k) = i=1 xi (k) − Nxc (k) = 0, w(k) is orthogonal to span{1N ⊗ In } for any k ≥ 0. On the other hand, by Remark 4.1, the null space of (

tj+1 −1 k=tj

Lσ (k) ) ⊗ In is span{1N ⊗ In }.

Thus, for any k ≥ 0, w(k) is orthogonal to the null space of

(

tj+1 −1

Lσ (k) ) ⊗ In . k=tj Under (2), the closed-loop system of agent i is

xi (k + 1) = Axi (k) + µBBT P T PA

N 

lij (k) (xi (k) − xj (k)).

(33)

j =1

N 

tj+1 −1 k=tj

Lσ (k) is

symmetric and positive semi-definite, and hence the matrix

lij (k) (¯xi (k) − x¯ j (k)).

(34)

j=1

Using (31)–(34) shows that w ¯ i (k) = P wi (k) satisfies

w ¯ i (k + 1) = A¯ w ¯ i (k) + µB¯ B¯ T A¯

Now we are ready to present the solvability conditions of the leaderless consensus problem and the leader-following consensus problem, respectively.

tj+1 −1

x (k)

x¯ i (k + 1) = A¯ x¯ i (k) + µB¯ B¯ T A¯

Premultiplying both sides of (29) by (IN ⊗ (B¯ Tl A¯ l )Ď ) gives limk→∞ ηls (k) = 0, and hence limk→∞ η(k) = 0.

Assumption 3. k ≥ 0.

(30)

Let x¯ i = Pxi , i = 1, . . . , N. Then

 αl βl ; −βl αl when A¯ l = 1 or − 1.

when A¯ l =

.

Proof. Let xc (k) = i=N1 i , where xc (k) is called the center of all agents at time k. Since the dynamic graph is bidirected at any time, applying protocol (2) to (1) gives

(28)

tj →∞



1

∥Lσ (k) ⊗ (A¯ T B¯ B¯ T A¯ )∥

σ (k)∈P

Ď

and tj+1 − tj is bounded, the set {∥Jj ∥, k = 1, 2, . . . , } contains only finitely many distinct real numbers, and thus there exists a Ď finite real number J such that, for all k = 1, 2, . . ., ∥Jj ∥ ≤ J. Then ∥ξ (tj )∥ ≤ J ∥ζj ∥. Therefore, (27) implies that

Lσ (k) has

Theorem 4.1. Under Assumptions 1–4, the leaderless consensus problem is solvable by distributed state feedback protocol (2) with gain matrix K = µBT P T PA, where P is defined in (8) and µ satisfies

Ď

Since ξ (tj ) is orthogonal to the null space of Jj , we have ξ (tj ) = Jj ζj ,

k=tj

N 

lij (t ) (w ¯ i (k) − w ¯ j (k)),

j =1

which can be put into the following compact form:

w( ¯ k + 1) = (IN ⊗ A¯ − µLσ (k) ⊗ (B¯ B¯ T A¯ ))w( ¯ k).

(35)

System (35) is in the form of (10) with Fσ (k) = Lσ (k) and ξ (k) = w( ¯ k). It is also noted that w(k) (and hence w( ¯ k)) is orthogonal to tj+1 −1 the null space of ( k=tj Lσ (k) )⊗ In . Then, by Lemma 3.1, we have limk→∞ w( ¯ k) = 0. Hence lim w(k) = lim (IN ⊗ P −1 )w( ¯ k) = 0.

k→∞

k→∞

Therefore, all the states xi (k) asymptotically converge to xc (k). The proof is thus completed.  Remark 4.2. If Assumption 2 is not satisfied, the trajectory of (Lσ (k) ⊗ In )w( ¯ k) may fall into the null space of IN ⊗ (B¯ T A¯ ). Then

Y. Su, J. Huang / Automatica 48 (2012) 1988–1997

(Lσ (k) ⊗ In )w( ¯ k) may not approach zero asymptotically unless B is of full row rank. Thus the consensus may not be achieved if Assumption 2 is not satisfied and B is not of full row rank. Examples will be presented in Section 5 to illustrate this point. Remark 4.3. From the proof of Theorem 4.1, we can see that the stability analysis of closed-loop system (39) is based on Lemma 3.1. By Remark 3.2, when B (and hence B¯ l ) is of full row rank, Lemma 3.1 does not need Assumption 2. Therefore, we have the following corollary. Corollary 4.1. Suppose that B is of full row rank. Under Assumptions 1, 3 and 4, distributed state feedback protocol (2) with gain matrix K = µBT P T PA, where P is chosen as in (8) and µ satisfies (30), solves the leaderless consensus problem. Remark 4.4. The static network topology can be viewed as a special case of switching network topology when the switching index set contains only one element. Thus, Theorem 4.1 and Corollary 4.1 extend the results in Tuna (2008), which only considered the static network topology. Remark 4.5. When A = 1 and B = 1, plant (1) consists of discretetime single integrators, and the consensus problem under switching network topology has been studied in Bertsekas and Tsitsiklis (1989), Jadbabaie et al. (2003), Nedić et al. (2009), Olfati-Saber and Murray (2004), and Ren and Beard (2005). Thus, Theorem 4.1 and Corollary 4.1 can be seen as an extension of the results of these papers. 4.2. Leader-following consensus We need the following assumption. Assumption 5. There exists an infinite subsequence {tj } of {k} such that tj+1 − tj is uniformly bounded for any j ≥ 0, and every node is reachable from node 0 in the union graph

tj+1 −1 k=tj

G¯ σ (k) .

Remark 4.6. When the dynamic graph G¯ σ (k) satisfies Assumption 5, for convenience, we also say that G¯ σ (k) is jointly connected. Remark 4.7. Let Hσ (k) = Lσ (k) + ∆σ (k) . Then, under Assumption 3, Hσ (k) is symmetric and  for any k ≥ 0. By  positive semi-definite 01×N 01×1 is a Laplacian of G¯ σ (k) , Remark 2.1, L¯ σ (k) =

−∆σ (k) 1N

Hσ (k)

and is such that Γ (L¯ σ (k) ) = G¯ σ (k) . Let

M¯ j =

=

L¯ σ (k)



1

∥Hσ (k) ⊗ (A¯ T B¯ B¯ T A¯ )∥

σ (k)∈P

.

(36)

Proof. Under (4), the closed-loop system of agent i is xi (k + 1) = Axi (k) + µBBT P T PA

N 

lij (k) (xi (k) − xj (k)).

(37)

j =0

Let wi (k) = P (xi (k) − x0 (k)). Then, by (3) and (37), we have

w ¯ i (k + 1) = A¯ w ¯ i (k) + µB¯ B¯ T A¯

N 

lij (k) (w ¯ i (k) − w ¯ j (k)),

(38)

j =0

where w ¯ 0 (k) , 0. Since Lσ (k) 1N = 0, we have Hσ (k) 1N = ∆σ (k) 1N for any k ≥ 0. Then system (38) can be put into the compact form

w( ¯ k + 1) = (IN ⊗ A¯ − µHσ (k) ⊗ (B¯ B¯ T A¯ ))w( ¯ k),

(39)

where w( ¯ k) = col(w ¯ 1 (k), . . . , w ¯ N (k)). System (39) is in the form of (10) with Fσ (k) = Hσ (k) and ξ (k) = w( ¯ k). By Remark 4.7, under Assumption 5, (

tj+1 −1 k=tj

Hσ (k) ) ⊗ In is nonsingular. Therefore,

the conditions of part (2) of Lemma 3.1 are all satisfied. Then, by Lemma 3.1, limk→∞ w( ¯ k) = 0. Thus, all the states xi (t ) asymptotically converge to x0 (t ).  Similarly, we have the following corollary. Corollary 4.2. Suppose that B is of full row rank. Under Assumptions 1, 3 and 5, distributed state feedback protocol (4) with gain matrix K = µBT P T PA, where P is defined in (8) and µ satisfies (36), is such that the leader-following consensus is achieved. Remark 4.8. When A = 1 and B = 1, plant (1) reduces to discretetime single integrators, and the leader-following consensus problem under switching network topology has been studied in Jadbabaie et al. (2003). Thus Theorem 4.2 and Corollary 4.2 have extended the result of Jadbabaie et al. (2003) to more general systems. Remark 4.9. It may appear that the leader-following consensus problem could have been treated as a special case of the leaderless consensus problem. But the two problems have a subtle difference. In fact, for the leader-following case, the control law relies on the leader state x0 (k), and the trajectories of all followers will asymptotically approach x0 (k) = Ak−1 x0 (0), where x0 (0) is an arbitrary initial condition of the leader. In contrast, for the leaderless case, the trajectories of all agents can only

N

i=1 xi (0)

N

t −011×1  j+1 ∆σ (k) 1N k=tj

Ak−1 x (0)

01×N

−

tj+1 −1 k=tj

, where xi (0) is the initial condition of the ith agent.

 Hσ (k)

.

5. Examples

¯ j is a Metzler matrix with zero Then it can be verified that M ¯ j) = row sum, and Γ (M

0 < µ ≤ min



N

k=tj



and µ satisfies

i i=1 asymptotically track a specific trajectory xc (k) = N of the center system whose initial condition has to be xc (0) =

tj+1 −1



1993

tj+1 −1

G¯ σ (k) . By Remark A.1, under k=tj ¯ Assumption 5, the matrix Mj has exactly one zero eigenvalue, and all the nonzero eigenvalues have negative real parts. Since the first

¯ j is a zero row vector, we can conclude that row of M

tj+1 −1 k=tj

Hσ (k)

tj+1 −1 is nonsingular. Thus, ( k=tj Hσ (k) ) ⊗ In is nonsingular. Theorem 4.2. Under Assumptions 1–3 and 5, the leader-following consensus problem is solved by distributed state feedback protocol (4) with gain matrix K = µBT P T PA, where P is defined in (8)

In this section, we provide two examples to illustrate our design. Example 1. Consider the leaderless consensus problem of system (1) with N = 4 and

 √

2

 2 A=  √2 − 2

√  2

√2  , 2 

  B=

0 . 1

(40)

2

Since A is orthogonal, Assumption 1 is satisfied. We will consider two cases of switching network topologies.

1994

Y. Su, J. Huang / Automatica 48 (2012) 1988–1997

Fig. 2. Lyapunov function V (k) and its difference 1V (k) for case 1. Fig. 1. The switching network topology Gσ (k) with P = {1, 2, 3, 4}.

Case 1. For any k ∈ Z+ ,

 1 if k = 4s + 1   2 if k = 4s + 2 σ (k) =  3 if k = 4s + 3 4 if k = 4s + 4,

(41)

where s ∈ Z+ . Since σ (k) is a periodic signal with period 4, the graph Gσ (k) is defined by four graphs Gi , i = 1, 2, 3, 4, which are shown in Fig. 1. Let the Laplacian Li corresponding to each Gi , i = 1, 2, 3, 4, be given as follows:



−1

1  −1 L1 =  0 0



0 0 L3 =  0 0

1 0 0 0 0 0 0



0 0 0 0



0 0 1 −1



0 1 −1 0



0 0 0 0

0 0 L2 =  0 0

0 0 , 0 0

1 0 L4 =  0 −1

0 0  , −1 1

0 −1 1 0 0 0 0 0



0 0 , 0 0

−1



0  . (42) 0  1

It can be seen that Gσ (k) is bidirected and jointly connected, i.e., Assumptions 3 and 4 are also satisfied. However, Assumption 2 is not satisfied, since τ = 1. Under the control gain

 √

K = µB A = µ − T

2

2

√  2

2

,

(43)

where µ = 0.5, the simulation result of the closed-loop system is shown in Figs. 2 and 3. It can be seen from Fig. 2 that limk→∞ V (k) ̸= 0, even though limk→∞ 1V (k) = 0. As a result, none of the states of the four agents approaches the center, as can be seen from Fig. 3. Fig. 3. The errors between states of the agents and their center for case 1.

Case 2. For any k ∈ Z+ ,

 1 if k = 8s + 1 or 8s + 2   2 if k = 8s + 3 or 8s + 4 σ (k) =  3 if k = 8s + 5 or 8s + 6 4 if k = 8s + 7 or 8s + 8,

(44)

where s ∈ Z+ , and the possible switching graphs are also shown in Fig. 1. It can be verified that Assumptions 2–4 are satisfied with τ = 2. Under the same control gain as (43) with µ = 0.5, the simulation result of the closed-loop system is shown in Fig. 4. It can be seen that all the states of the agents converge to their center.

Example 2. Consider the leader-following consensus problem of system (1) with N = 4 and A, B as defined in (40). Again, we consider two cases of switching network topologies. Case 1. Let σ (k) be as defined in (41). Let the four graphs G¯ i , i = 1, 2, 3, 4, that define the graph G¯ σ (k) be as shown in Fig. 5. Since, for i = 1, 2, 3, 4, the subgraphs Gi of the graphs G¯ i are the same as those in Example 1, we can let the Laplacian Li of Gi be given by (42). Let ∆1 = diag {1, 0, 0, 0}, ∆2 = diag {0, 0, 0, 1}, ∆3 = diag {0, 0, 1, 0}, ∆4 = diag {0, 0, 0, 0}. We can see that

Y. Su, J. Huang / Automatica 48 (2012) 1988–1997

1995

Fig. 6. Lyapunov function V (k) and its difference 1V (k) for case 1.

Fig. 4. The errors between states of the agents and their center for case 2.

Fig. 7. The errors between states of the agents and the leader for case 1.

in (43) with µ = 0.7, the simulation result of the closed-loop system is shown in Figs. 6 and 7. It can be seen from Fig. 6 that limk→∞ V (k) ̸= 0, even though limk→∞ 1V (k) = 0. As a result, the states of the agents 2 and 3 do not approach the center, as can be seen from Fig. 7. Fig. 5. The switching network topology G¯ σ (k) with P = {1, 2, 3, 4}.

Assumptions 3 and 5 are also satisfied. However, Assumption 2 is not satisfied, since τ = 1. Under the same control gain as

Case 2. Let σ (k) be as defined in (44). It can be verified that Assumptions 2 to 3, and 5 are all satisfied with τ = 2. Under the same control gain as (43) with µ = 0.7, the simulation result of the closed-loop system is shown in Fig. 8. It can be seen that all the states of the agents converge to their leader.

1996

Y. Su, J. Huang / Automatica 48 (2012) 1988–1997

V } ⊆ V × V . If an edge (i, j) ∈ E , then node i is called a neighbor of node j. Ni denotes the subset of V composed of those nodes

that are neighbors of the node i. Edge (i, j) is called bidirected if (i, j) ∈ E implies that (j, i) ∈ E . The graph is called bidirected if every edge in E is bidirected. If E contains a subset of the form {(i1 , i2 ), (i2 , i3 ), . . . , (ik , ik+1 )}, then this subset is called a path of G from i1 to ik+1 , and node ik+1 is said to be reachable from node i1 . A graph is called connected if there exists a node i such that any other node is reachable from node i. Node i is called a root of the graph. A graph Gs = (Vs , Es ) is a subgraph of G = (V , E ) if Vs ⊆ V and Es ⊆ E ∩ (Vs × Vs ). Given a set of r graphs {Gi = (V , Ei ), i = 1, . . . , r } with the same node set, the graph r G = (V , E ), where E = i=1 Ei , is called the union graph of Gi . + Given a map σ : Z → P , where P = {1, 2, . . . , ρ}, we can define a dynamic graph Gσ (k) = (V , Eσ (k) ), where V = {1, . . . , N } and Eσ (k) ⊆ V × V for all k ≥ 0. A matrix L = [lij ] ∈ RN ×N with zero row sum is said to be a Laplacian matrix of a graph G if, for i, j = 1, . . . , N , i ̸= j, lij < 0 iff ⇔ (j, i) ∈ E , and lij = lji if (j, i) is a bidirected edge of E . Clearly, L1N = 0. A matrix whose off-diagonal elements are nonnegative is called a Metzler matrix. Thus, −L is a Metzler matrix with zero row sum. Given any matrix M = [mij ] ∈ RN ×N , one can define a graph denoted by Γ (M ) = (V , E ), where V = {1, . . . , N }, and (i, j) ∈ E , i ̸= j, i, j = 1, . . . , N, if and only if mji ̸= 0. Clearly, if L is any Laplacian matrix of a graph G, then Γ (L) = G. Remark A.1. It is shown in Lin (2005) that a Metzler matrix M ∈ RN ×N with zero row sum has at least one zero eigenvalue and all the nonzero eigenvalues have negative real parts. Furthermore, M has exactly one zero eigenvalue, and its null space is span{1N }, if and only if Γ (M ) is connected. A symmetric Metzler matrix with zero row sum is negative semi-definite.

Fig. 8. The errors between states of the agents and the leader for case 2.

6. Conclusion In this paper, we have studied both the leaderless consensus problem and the leader-following consensus problem for discretetime linear neutrally stable systems under switching network topology. By establishing a stability result on a class of linear discrete-time switched systems, we have shown that both consensus problems can be solved via state feedback protocols, provided that the dynamic graph is jointly connected. The result of this paper can be viewed as a discrete-time counterpart of the main result in Su and Huang (2012). For the leaderless consensus, the result of this paper contains the results in Bertsekas and Tsitsiklis (1989), Jadbabaie et al. (2003), Nedić et al. (2009), OlfatiSaber and Murray (2004), and Ren and Beard (2005), where the agents are single integrators as special cases. On the other hand, it also extends the result in Tuna (2008) to the case where the network topology is dynamic. For the leader-following consensus, our result also contains the leader-following consensus of single integrators (Jadbabaie et al., 2003) as a special case. It is of interest to extend our result to the directed network topology. Another possible extension of our work is to consider the H∞ consensus control problem (Shen, Wang, & Hung, 2010). The consensus problem for systems with time delay (Qin, Gao et al., 2011) is also of interest for further study. Appendix. Graph Most of the graph notation in this appendix can be found in Godsil and Royle (2001). A graph G = (V , E ) consists of a finite node set V = {1, . . . , N } and an edge set E = {(i, j) : i ̸= j, i, j ∈

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1997

You, K., & Xie, L. (2011a). Coordination of discrete-time multi-agent systems via relative output feedback. Internal Journal of Robust and Nonlinear Control, 21(13), 1587–1605. You, K., & Xie, L. (2011b). Network topology and communication data rate for consensusability of discrete-time multi-agent systems. IEEE Transactions on Automatic Control, 56(10), 2262–2275. Youfeng Su received his B.S. degree in 2005 and his M.S. degree in 2008, both from East China Normal University, Shanghai, China, and his Ph.D. degree in 2012 from The Chinese University of Hong Kong, Hong Kong, China. He is currently a postdoctoral fellow in the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong, China. His research interests include cooperative control, multiagent systems, output regulation, and switched systems.

Jie Huang is a Professor with the Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong, China. His research interests include nonlinear control theory and applications, multiagent systems, and flight guidance and control. Dr. Huang is a Fellow of IEEE, and a Fellow of IFAC.