Title: Consensus over matrix-weighted networks with time-delays
Venue: NeurIPS 2024
Accepted: no
Keywords: optimization and control theory; multi-agent systems; matrix-weighted networks; time-delays; network localization
Abstract: This paper studies consensus conditions for leaderless and leader-follower matrix-weighted consensus networks under the presence of constant time-delays. Several delayed consensus algorithms for networks of single- and double-integrators using only the relative positions are considered. Conditions for the networks to asymptotically converge to a consensus or clustering configuration are derived based on direct eigenvalue evaluation or the Lyapunov-Krasovkii theorem. Furthermore, an application of these algorithms in bearing-based network localization is also considered. The theoretical results are supported by numerical simulations.

1

Abstract

This paper studies consensus conditions for leaderless and leader-follower matrixweighted consensus networks under the presence of constant time-delays. Several delayed consensus algorithms for networks of single- and double-integrators using only the relative positions are considered. Conditions for the networks to asymptotically converge to a consensus or clustering configuration are derived based on direct eigenvalue evaluation or the Lyapunov-Krasovkii theorem. Furthermore, an application of these algorithms in bearing-based network localization is also considered. The theoretical results are supported by numerical simulations.

1 Introduction 9

Recently, matrix-weighted consensus, a multi-dimensional extension of the well-known scalar10 weighted consensus algorithm [20], has received a considerable amount of research attention. A 11 matrix-weighted consensus system models diffusion dynamics in a multi-layer system with intra12 and cross-layer interactions between multiple subsystems (or agents). Several applications of matrix13

weighted consensus systems include multi-dimensional opinion dynamics models in [1,33], bearing14

based formation control [7,37], distributed localization of wireless sensor networks [3,4], and network 15 synchronization [31]. 16

17

18

19

20

21

22

23

24

25

26

27

28

29

A matrix-weighted consensus network can be described by a graph with both positive definite and positive semidefinite matrix weights. Associated with the graph, a corresponding Laplacian matrix, whose the kernel (aka the nullspace) may contain further subspaces in addition to the consensus space [2,12,31], can be defined. Necessary and sufficient conditions for a matrix-weighted consensus network to asymptotically achieve consensus or clustering were given in [29,30]. Discrete-time and randomized matrix-weighted consensus were studied in [14,16,28]. The authors in [22] investigated the continuous-time consensus protocol with switching matrix-weighted graphs. A consensus is asymptotically achieved if the weighted integral network over some fixed time period always contains a positive spanning tree, or equivalently, the kernel of the Laplacian matrix of the integrated network contains only the consensus space [22]. The works [15, 16] examined the consensus problems over matrix-weighted networks for double-integrator agents. Controllability of the matrix-weighted consensus network was discussed in [21]. Recent studies on bipartite and multi-partite matrixweighted consensus have been proposed in [10,17,32].

30

31

32

33

34

It practice, time delays are unavoidable if agents communicate their state variables via a wireless

network, especially when the agents are separated by significant distances. When restricted to linear

systems, time delay yields phase lags and alters both the transient and steady-state responses of

the system. If the magnitude of the time delay is sufficiently large, the whole system could be

destabilized. For this reason, it is essential to examine the stability conditions of matrix-weighted

consensus networks under different assumptions on the time delays. It is noteworthy that even 35

with delayed linear differential equations, the exact analysis via characteristic equations will lead to 36

transcendental equations, of which solutions are often complicated [5,23]. An alternative approach 37 for analysing the stability of time-delayed systems is based on Lyapunov-Krasovskii or Lyapunov38 Razumikhin theorems [11,13]. In the literature, a sufficient condition for reaching a consensus in 39 a scalar consensus network with a uniform time delay was given in [20]. Lyapunov-Razumikhin 40 type functional was used for finding sufficient conditions for consensus networks with heterogeneous 41 edge delays and switching interaction topology in [27]. The authors in [25] studied the consensus 42 problem with uniform delay communication and provided consensus conditions by considering 43 some Lyapunov-Razumikhin functionals. The author in [24] considered a consensus problem with 44 heterogeneous communication time delays and introduced a delayed weighted Laplacian for the 45 analysis. The consensus of double-integrator agents with time delay was studied in [34] based on 46 an approximated characteristic equation under the assumption that the delays are sufficiently small. 47 An exact analytic method for a second-order delayed scalar-consensus protocol was proposed in [6]. 48 Stabilization control laws for double- and chain of integrators using delays were proposed in [19], 49 and in the consensus problem over a scalar-weighted graph [26]. 50

51

52

53

54

55

56

57

58

59

60

61

62

63

In this paper, we derive stability conditions of several delayed matrix-weighted consensus models having either a leaderless or a leader-follower topology. A leader-follower network contains several leader agents acting as stationary references during the dynamic process. First, we consider a matrix-weighted consensus network where all the edges have the same constant time delay. For this network, a necessary and sufficient stability condition related to the magnitude of the time delay and the maximum eigenvalue of the matrix-weighted Laplacian is established. Second, we study the matrix-weighted consensus with multiple heterogeneous constant time delays. A stability condition is given in terms of the feasibility of a linear matrix inequality (LMI). Third, we consider a matrixweighted consensus network of double integrators, and show that the network can asymptotically reach the kernel of the matrix-weighted Laplacian by using only the delayed relative positions. As it is assumed that the kernel of the matrix-weighted Laplacian is not restricted to the consensus space, the applicability of the considered models is beyond a consensus problem. In particular, an application of the theoretical results in bearing-based network localization [36] is also discussed.

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

The rest of the paper is organized as follows. In Section 2, the theoretical background is provided and three delayed matrix-weighted consensus models studied in this paper are presented. Sections 3-5 give stability conditions and detailed analysis of each consensus model. An application in bearing-based network localization is discussed in Section 6, and simulation results are provided in Appendix A.5 to support the analysis. Lastly, Section 7 concludes the paper.

Notations: In this paper, R , R + , R d , R m × n respectively denote the sets of real numbers, positive real numbers, d -dimensional vectors with real entries and m × n matrix with real entries. Let 0 d and Θ d respectively denote the zero vector of dimension d and the zero matrix of dimension d × d . For a real m × n matrix A , we use A ⊤ , rank ( A ) , det ( A ) , ker ( A ) , and im ( A ) to denote the transposition, rank, determinant, kernel space and image space of A , respectively. If A is symmetric positive definite (positive semidefinite), we write A > 0 (resp., A ≥ 0 ). Given a vector x ∈ R d , the Euclidean norm of x is denoted by ∥ x ∥ = √ ∑ d i =1 x 2 i .

2 Preliminaries

2.1 Matrix-weighted networks

Consider an undirected, matrix-weighted graph G = ( V , E , A ) with the vertex set V = { 1 , . . . , n } , the edge set E ⊆ V × V of m = |E| edges, and the set of nonnegative definite matrix weights A = { A ij ∈ R d × d } i,j ∈V with A ij = A ⊤ ij ≥ 0 , ∀ i, j and d ≥ 2 . Each edge ( i, j ) ∈ E captures the interactions between two agents i and j , and the existence of ( i, j ) implies the existence of ( j, i ) since the graph is undirected. If ( i, j ) ∈ E , then A ij = 0 ; and if ( i, j ) / ∈ E or i = j , then A ij = Θ d . The neighbor set of a vertex i ∈ V is denoted as N i = { j ∈ V| ( i, j ) ∈ E} . Then, the degree matrix of a vertex i is defined as D i = ∑ j ∈N i A ij .

̸

100

101

102

103

104

105

106

107

108

109

110

111

block entries 87

̸

Note that L is symmetric, positive semi-definite, and ker ( L ) ⊇ im ( 1 n ⊗ I d ) . A matrix-weighted 88 graph and its corresponding matrix-weighted Laplacian is depicted in Fig. 1 as an example. 89

We order the edges in E such that E = { e 1 , . . . , e m } , and adopt the notation A ij ≡ A k , ∀ e k = 90 ( i, j, k = 1 , . . . , m ) . For each edge ( i, j ) , we specify a vertex to be the starting vertex and the other 91 vertex as the end vertex. The incidence matrix H = [ h ki ] ∈ R m × n is defined as follows 92

Then, L = ¯ H ⊤ blkdiag ( A k ) ¯ H , where ¯ H = H ⊗ I d , and ' ⊗ ' denotes the Kronecker product. 93

Suppose that the matrix-weighted Laplacian L has l ≥ d eigenvalues 0 with l linearly independent 94 eigenvectors v 1 , . . . , v l . This assumption allows the possibilities of achieving a consensus or and 95 clustering when the following consensus algorithm is performed on a matrix-weighted network of 96 single integrators 97

where x i ∈ R d is the state vector of agent i ∈ V . Let x = [ x ⊤ 1 , . . . , x ⊤ n ] ⊤ ∈ R dn , the matrix98 weighted consensus algorithm (3) can be rewritten in matrix form as 99

and it has been shown that x ( t ) → ¯ x = 1 n ∑ n i =1 x i (0) ∈ ker ( L ) , as t → + ∞ [18,29]. Throughout the paper, the shorthand x ij ( t ) = x j ( t ) -x i ( t ) will be used.

From the assumption that zero is a semi-simple eigenvalue of multiplicity l ≥ d , and L is symmetric, positive semi-definite, there exists an orthonormal matrix P = [ p 1 , . . . , p dn ] = [ R , Q ] ∈ R dn × dn such that R = [ p 1 , . . . , p l ] ∈ R dn × l , Q = [ p l +1 , . . . , p dn ] ∈ R dn × ( dn -l ) ,

̸

and ∥ p i ∥ = 1 , ∀ i, j = 1 , . . . , dn so that the matrix-weighted Laplacian is diagonalizable as P ⊤ LP = Λ , where Λ = [ Θ l 0 l × ( dn -l ) 0 ( dn -l ) × l ¯ Λ ] = diag ( λ 1 , . . . , λ dn ) (and ¯ Λ = diag ( λ l +1 , . . . , λ dn ) spectively) the diagonal matrix containing all eigenvalues (all positive eigenvalues) of L . Note that R ⊇ im ( 1 n ⊗ I d ) since the kernel of a matrix-weighted Laplacian always contains the consensus space. Also, Q ⊤ R = 0 , Q ⊤ Q = I dn -l .

Consider a partition of the vertex set into two disjoint subsets V a and V b such that V a ∪ V b = V V a ∩ V b = ∅ , |V a | = n a , |V b | = n b , n a + n b = n . The agents associated with the vertices in

, re-( dn -l ) × l ,

V a and V b are referred to as leaders and followers, respectively. By labeling the vertices such that 112 V a = { 1 , . . . , n a } , V b = { n a +1 , . . . , n } , the matrix-weighted Laplacian is partitioned as 113

where L a = L ⊤ a ∈ R dn a × dn a , L ⊤ ab ∈ R dn a × dn b , and L b = L ⊤ b ∈ R dn b × dn b . Let L ′ denote the matrix-weighted Laplacian corresponding to the subgraph induced by the vertices in V b and edges in G . If n a = 0 , we have a leaderless network while for n a ≥ 1 , we have a leader-follower network. We prove the following lemma on the matrix-weighted Laplacian (5).

Lemma 2.1. Let rank ( L ) = dn -l , rank ( L ′ ) = dn b -l , n a ≥ 1 and l ≥ d +1 . If ∀ ξ ∈ ker ( L ′ ) , [ 0 ⊤ dn a , ξ ⊤ ] ⊤ / ∈ ker ( L ) , then the matrix L b is symmetric positive definite.

Proof. Let B = blkdiag ( L ab ( 1 n a ⊗ I d )) = blkdiag ( B 1 , . . . , B n b ) ∈ R dn b × dn b , we have L b = L ′ -B . Suppose that L b is not positive definite, then there exists ξ = [ ξ ⊤ 1 , . . . , ξ ⊤ n b ] ⊤ ∈ R dn b such that ξ ⊤ L b ξ = ξ ⊤ ( L ′ -B ) ξ = 0 dn b . From the assumption on L ′ , it follows that ξ ∈ ker ( L ′ ) . Furthermore, ξ ⊤ B ξ = ∑ n b k =1 ξ ⊤ k B k ξ k = 0 . Since each matrix weight in B k = ∑ n a j =1 [ L ab ] kj is negative semidefinite, it follows that ξ k ∈ ker ([ L ab ] kj ) , ∀ j = 1 , . . . , n a , or equivalently L ⊤ ab ξ = 0 dn a . Then, we have L [ 0 dn a ξ ] = [ L ⊤ ab ξ L b ξ ] = 0 dn , which shows that [ 0 ⊤ dn a , ξ ⊤ ] ⊤ ∈ ker ( L ) . This contradiction implies that L b must be positive definite.

2.2 Problem formulation

This paper aims to give some conditions for stability and/or reaching a consensus when time delays are present in (4) and its expanded versions. Particularly, the following matrix-weighted consensus models with time delays will be studied.

Model 1 Matrix-weighted consensus of single-integrators with a uniform constant time-delay τ > 0 :

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

∀ i ∈ V b , and ˙ x i ( t ) = 0 d , ∀ i ∈ V a .

Model 2 Matrix-weighted consensus of single-integrators with heterogeneous constant time-delays

where i ∈ V b , τ ij ≥ 0 is the time-delay associated with an edge ( i, j ) ∈ E , and ˙ x i ( t ) = 0 d , ∀ i ∈ V a . 134 Model 3 Matrix-weighted consensus of double-integrators with two constant time-delays: 135

where x k i = [ x k 1 i , . . . , x k di ] ⊤ ∈ R d , i ∈ V b , and ˙ x k i ( t ) = 0 d , ∀ i ∈ V a , k = 1 , 2 . Here, x i = 136 [( x 1 i ) ⊤ , ( x 2 i ) ⊤ ] ⊤ and x 1 1 , x 2 i are referred to as the position and the velocity of agent i , and α > 0 is a 137 control gain. 138

139

140

141

142

For each model, the initial condition is given as x ( θ ) = x (0) , ∀ θ ∈ [ -τ k , 0] .

3 Matrix-weighted consensus of single-integrators with a uniform time-delay

In this section, we give condition on the time-delay to ensure the model (6) to asymptotically achieve a consensus for leaderless and leader-follower matrix-weighted networks.

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

3.1 Leaderless network

The following theorem provides necessary and sufficient consensus condition for a leaderless matrixweighted consensus network.

Theorem 3.1. Consider a leaderless n -agent network with V a = ∅ , and rank ( L ) = dn -l , l ≥ d . Under the consensus algorithm (6) , x ( t ) asymptotically converges to x ∗ = RR ⊤ x (0) ∈ ker ( L ) if and only if τ < π 2 λ dn , where λ dn is the largest eigenvalue of L .

Proof. The proof of this theorem is given in Appendix A.2.

Remark 3.2 . Observe that if rank ( L ) = dn -d and the stability condition τ < π 2 λ dn holds, then l = d , ∑ d p =1 p k p ⊤ k = 1 n ( 1 n 1 ⊤ n ⊗ I d ) and the system asymptotically achieves a consensus. A similar consensus condition was given in [20] for scalar-weighted consensus networks but the proof is different from that of Theorem 3.1.

3.2 Leader-follower network

Next, we consider the leader-follower network under the consensus law (6). Let x a = [ x ⊤ 1 , . . . , x ⊤ n a ] ⊤ and x b = [ x ⊤ n a +1 , . . . , x ⊤ n ] ⊤ respectively denote the stacked vectors of the leader and the follower agents. The behaviors of the network is given in the following theorem.

Theorem 3.3. Consider a leader-follower n -agent network with n a ≥ 1 , rank ( L ) = dn -l , l ≥ d , and L b is positive definite. Under the consensus algorithm (6) , x b asymptotically converges to x ∗ b = L -1 b L ab x a if and only if τ < π 2 λ b max , where λ b max is the largest eigenvalue of L b .

Proof. We can write the n -agent network in matrix form as follows

As x a ( t ) = x a (0) , ∀ t ≥ -τ , we consider the variable transformation δ b ( t ) = x b ( t ) + L -1 b L ab x a , and derive the equation

The proof that the delayed system (10) is asymptotically stable if and only if τ < π 2 λ b max is similar to the proof of Thm. 3.1 and will be omitted.

Remark 3.4 . It is remarked that if a consensus algorithm is performed in a leader-follower scalarweighted graph with non-collocated leaders, the followers will asymptotically converge to fixed points inside the convex hull of the leaders' position. In contrast, as shown in Thm. 3.3, for a matrix-weighted consensus, the convergence points of follower agents may lie outside the convex hull of the leaders' positions. This property finds application in the bearing-based network localization problem discussed in Section 6.

4 Matrix-weighted consensus of single integrators with heterogeneous delays

In this section, we study the matrix-weighted consensus algorithms with heterogeneous time delays (7). We first study the problem for a leaderless matrix-weighted network and then consider the problem for a leader-follower network.

4.1 Leaderless network 176

Due to symmetry, we have τ ij = τ ji , ∀ ( i, j ) ∈ E . We can rewrite the dynamics (7) in the matrix form 177 as follows: 178

where r ≤ |E| , τ k = τ ij if e k = ( i, j ) , for k = 1 , . . . , r , and L k = [ L kij ] ∈ R dn × dn is a matrix 179 whose d × d blocks are defined by 180

̸

̸

It is observed that L k is a part of the Laplacian matrix corresponding to an update with time delay τ k , and L = ∑ r k =1 L k . As in the previous section, R ⊤ L k = 0 l × nd , for k = 1 , . . . , r . It follows that x ∗ = RR ⊤ x ( t ) is time-invariant.

Moreover, we have L k = PΛ k P ⊤ , where Λ k = [ Θ l 0 l × ( dn -l ) 0 ( dn -l ) × l ¯ Λ k ] and ¯ Λ k = Q ⊤ L k Q ∈ R ( dn -l ) × ( dn -l ) . Define δ ( t ) = Q ⊤ x ( t ) ∈ R dn -l , then the equation (11) can be rewritten in the following form [13]:

The stability of the system (12) is stated in the following theorem, whose proof can be found in Appendix A.3.

Theorem 4.1. Consider the leaderless matrix-weighted consensus network with time delays (12) , where rank ( L ) = dn -l , n a = 0 and l ≥ d . Suppose that the time delays τ k are sufficient small such that the LMI (13) holds, where τ = ∑ r i =1 τ i . 1 Then, the origin is a globally uniformly asymptotically equilibrium of (12) and x ( t ) → x ∗ ∈ ker ( L ) as t → + ∞ .

181

182

183

184

185

186

187

188

189

190

191

192

4.2 Leader-follower network 193

Next, we consider the leader-follower network under the consensus algorithm (7). Similar to 194 the previous section, we can define δ b ( t ) = x b ( t ) + L -1 b L ab x a , where L ab = ∑ r k =1 L abk and 195 L b = ∑ r k =1 L bbk . That is, each matrix L k contributes a part to the matrices L ab and L b . Then, 196

We can now state a theorem on the delayed-system (14), whose proof is similar to the proof of 197 Theorem 4.1 and will be omitted. 198

199

200

201

202

Theorem 4.2. Suppose that the n -agent network has a leader follower structure, n a ≥ 1 , rank ( L ) = dn -l , l ≥ d , and L b is positive definite. If the time delays τ k are chosen such that the LMI (15) holds and τ = ∑ r i =1 τ i , then δ b = 0 dn b is globally unniformly asymptotically stable, and x ( t ) → L -1 b L ab x a as t → + ∞ .

̸

̸

203

204

205

206

207

208

209

210

211

5 Matrix-weighted consensus of double-integrators without relative velocity measurements using two time delays

5.1 Leaderless network

Consider a leaderless matrix-weighted network. We express the network (8) in the matrix form as follows

First, observe that ( 1 ⊤ n ⊗ I d ) ˙ x 2 ( t ) = -( 1 ⊤ n ⊗ I d ) Lx 1 ( t -τ 1 ) + α ( 1 ⊤ n ⊗ I d ) Lx 1 ( t -τ 2 ) = 0 dn . Hence, ( 1 ⊤ n ⊗ I d ) x 2 ( t ) = ( 1 ⊤ n ⊗ I d ) x 2 (0) = 0 dn . This property will be used in proving the main theorem of this subsection.

we can rewrite the system as 212

Let z 1 = Q ⊤ x 1 , z 2 = Q ⊤ x 2 , and z = [( z 1 ) ⊤ , ( z 2 ) ⊤ ] ⊤ . The differential equation governing the 213 z -system is 214

The eigenvalues of F ( τ 2 ) ∈ R 2( dn -l ) × 2( dn -l ) satisfy the characteristic equation 215

216

217

218

where τ = τ 2 -τ 1 . 219

Finally, we can state the following theorem whose proof can be found in Appendix A.4. 220

where λ i > 0 , i = l + 1 , . . . , dn, are the positive eigenvalues of the matrix-weighted Laplacian matrix L . Thus, for α < 1 and τ 2 > 0 , F ( τ 2 ) is Hurwitz, and we can find a symmetric positive definite matrix Π ∈ R 2( dn -l ) × 2( dn -l ) satisfying the Lyapunov equation

221

222

223

224

225

226

227

228

229

230

231

232

233

(a)

(

G

1

,

x

)

(b)

(

G

1

,

y

)

(c)

(

G

2

,

z

)

(d)

(

G

3

,

x

′

)

(e)

(

G

3

,

y

′

)

(f)

(

G

4

,

z

′

)

g12

P12

g12

P12

P12

g12

g13

8

8

P13

P14841

g23

P23

P14B41

g23

P23

P14S41

g23

P23

5

5

g34

g34

g34

4

4

P34

P34

2

2

2

Theorem 5.1. Consider the leaderless delayed second-order consensus model (8) , where rank ( L ) = dn -l , n a = 0 , α < 1 , x 2 i (0) = 0 d , ∀ i = 1 , . . . , n , and τ 1 > 0 . Suppose that there exist positive definite matrices W ∈ R ( dn -l ) × ( dn -l ) , Z ∈ R ( dn -l ) × ( dn -l ) and Π ∈ R 2( dn -l ) × 2( dn -l ) such that the matrix

is negative definite, where

Then, x 1 ( t ) → ker ( L ) , x 2 ( t ) → 0 dn as t → + ∞ .

Remark 5.2 . The condition α < 1 is only sufficient for our analysis, which is based on (16) to held. Indeed, for certain choices of τ 1 and τ 2 , α = 1 may still make the system achieve asymptotic consensus.

5.2 Leader-follower network

We now consider the consensus algorithm (8) when the matrix-weighted graph has a leader-follower structure. The leaders' positions are time-invariant, thus x 1 a ( t ) = x 1 a (0) := x 1 a , x 2 a ( t ) = 0 dn a , ∀ t ≥ -τ . The equations governs followers' dynamics are given as follows

Using the variable transformation δ 1 b ( t ) = x 1 b ( t ) + L -1 b L ab x 1 and δ 2 b ( t ) = x 2 b ( t ) , we have the 234 equations with the transformed variables 235

Defining E ( τ 2 ) = [ Θ dn b I dn b -(1 -α ) L b -ατ 2 L dn b ] , then E ( τ 2 ) is Hurwitz for α < 1 and τ 2 > 0 . Thus, 236 there exists a symmetric positive definite matrix Π b satisfying the following equation 237

where τ = τ 2 -τ 1 . Similar to the proof of Theorem 5.1, the following theorem can be proved. 238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

Theorem 5.3. Consider the delayed second-order consensus model (8) in a leader-follower network with rank ( L ) = dn -l , n a ≥ 1 , L b > 0 , α < 1 and τ 1 > 0 . Suppose that there exist positive definite matrices W b ∈ R ( dn -l ) × ( dn b ) , Z b ∈ R dn b × dn b , and Π b ∈ R 2 dn b × 2 dn b such that the matrix

is negative definite, where

. Then, x 1 b ( t ) →-L -1 b L ab x a , and x 2 b ( t ) → 0 dn b .

6 Bearing-based network localization under time delays

We consider a wireless sensor network of n nodes in the d ≥ 2 dimensional space. Consider a global coordinate system g Σ , and let the position of the i -th sensor in the network referred in g Σ be denoted as x i ∈ R d .

The network is characterized by ( G , x ) , where G is the interaction graph and x = [ x ⊤ 1 , . . . , x ⊤ n ] ⊤ ∈ R dn , the stacked vector of the global positions of n nodes, is referred to as a realization. Each node (or agent), located at x i ∈ R d , can measure the bearing vector g ij = x j -x i ∥ x j -x i ∥ , which contains the directional information from node i to a neighboring node j ∈ N i . The global position x i is unknown to each agent i , so it needs to update an estimate ˆ x i ( t ) ∈ R d of x i and exchange this information with its neighbors. The process of determining the positions of the network's nodes is called network localization . We assume that the information about the origin of the global coordinate system is unavailable to each agent and each agent maintains a local coordinate systems i Σ , whose axes are aligned with g Σ . This assumption is feasible since we can firstly conduct an orientation alignment algorithm before performing the network localization process.

For each bearing vector g ij , there is a corresponding symmetric positive semidefinite matrix P g ij = I d -g ij g ⊤ ij ∈ R d × d satisfying ker ( P g ij ) = im ( g ij ) and P g ij = P ⊤ g ij = P 2 g ij . Observe that P g ij is an orthogonal projection onto ker ( g ij ) . The bearing-based network localization algorithm [35,36]

can be considered as a matrix-weighted consensus algorithm (3). The network localization algorithm (22) induces the bearing Laplacian L with the ij -th off-diagonal block matrix -P g ij . It has been shown that the necessary and sufficient condition for the network under the update law (22) to be determined up to a translation and a scaling is rank ( L ) = dn -d -1 [37]. Thus, the bearing Laplacian corresponds to l = d +1 , and all theoretical results in Sections 3-5 are applicable for the bearing-based network localization problem with time delays.

7 Conclusions

In this paper, three leaderless and leader-follower matrix-weighted consensus models with constant time-delays were studied. The stability of the considered models was analysed and several conditions for the system to asymptotically converge to a point in the kernel of the matrix-weighted Laplacian were provided. An application in bearing-based network localization with time-delays was also given. Since the current work only focuses on constant time delay, for further studies, it will be interesting to consider time-varying time-delays or adaptive algorithms for stabilizing the matrix-weighted consensus network with time-delays.

275

276

277

278

279

280

281

282

283

References

H.-S. Ahn, Q. V. Tran, M. H. Trinh, M. Ye, J. Liu, and K. L. Moore. Opinion dynamics with cross-coupling topics: Modeling and analysis. IEEE Transactions on Computational Social Systems , 7(3):632-647, 2020.

F Atik, R B Bapat, and M R Kannan. Resistance matrices of graphs with matrix weights. Linear Algebra and its Applications , 571:41-57, 2019.

P. Barooah and J. P. Hespanha. Distributed estimation from relative measurements in sensor networks. In 3rd International Conference on Intelligent Sensing and Information Processing , pages 226-231. IEEE, 2005.

P. Barooah and J. P. Hespanha. Estimation on graphs from relative measurements. IEEE Control 284 Systems Magazine , 27(4):57-74, 2007. 285

Kenneth L. C. and Zvi G. Discrete delay, distributed delay and stability switches. Journal of 286 Mathematical Analysis and Applications , 86(2):592-627, 1982. 287

R Cepeda-Gomez and N Olgac. An exact method for the stability analysis of linear consensus 288 protocols with time delay. IEEE Transactions on Automatic Control , 56(7):1734-1740, 2011. 289

T.-F. Ding, M.-F. Ge, Z.-W. Liu, Y.-W. Wang, and H. R. Karimi. Lag-bipartite formation 290 tracking of networked robotic systems over directed matrix-weighted signed graphs. IEEE 291 Transactions on Cybernetics , 52(7):6759-6770, 2020. 292

E. Fridman. Tutorial on lyapunov-based methods for time-delay systems. European Journal of 293 Control , 20(6):271-283, 2014. 294

G. Goodwin, S. Grabe, and M. Salgado. Control System Design . Pearson, 2000. 295

R Gopika, V Resmi, and Rakesh R Warier. Cluster consensus in multi-partitioned matrix 296 weighted graphs. In Proc. of the 13th Asian Control Conference (ASCC) , pages 1184-1189. 297 IEEE, 2022. 298

K. Gu. An integral inequality in the stability problem of time-delay systems. In Proc. of the 299 39th IEEE Conference on Decision and Control, Sydney, NSW, Australia , volume 3, pages 300 2805-2810, 2000. 301

J Hansen. Expansion in matrix-weighted graphs. Linear Algebra and its Applications , 630:252302 273, 2021. 303

V. B. Kolmanovskii and J-P Richard. Stability of some linear systems with delays. IEEE 304 Transactions on Automatic Control , 44(5):984-989, 1999. 305

306

307

N.-M. Le-Phan, M. H. Trinh, and P. D. Nguyen. Randomized matrix weighted consensus. IEEE Transactions on Network Science and Engineering , 2024.

308

309

310

311

S. Miao and H. Su. Second-order consensus of multiagent systems with matrix-weighted network. Neurocomputing , 433:1-9, 2021.

S Miao and H Su. Second-order hybrid consensus of multi-agent systems with matrix-weighted networks. IEEE Transactions on Network Science and Engineering , 9(6):4338-4348, 2022.

S Miao and H Su. Behaviors of matrix-weighted networks with antagonistic interactions. 312 Applied Mathematics and Computation , 467:128490, 2024. 313

H. M. Nguyen and M. H. Trinh. Leaderless- and leader-follower matrix-weighted consensus 314 with uncertainties. Measurements, Control and Automation , 3(2):33-41, 2022. 315

S-I Niculescu and W Michiels. Stabilizing a chain of integrators using multiple delays. IEEE 316 Transactions on Automatic Control , 49(5):802-807, 2004. 317

318

319

R. Olfati-Saber and R. M. Murray. Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control , 49(9):1520-1533, 2004.

L. Pan, H. Shao, M. Mesbahi, Y. Xi, and D. Li. On the controllability of matrix-weighted 320 networks. IEEE Control Systems Letters , 4(3):572-577, 2020. 321

L. Pan, H. Shao, M. Mesbahi, Y. Xi, and D. Li. Consensus on matrix-weighted switching 322 networks. IEEE Transactions on Automatic Control , 66:5990-5996, 2021. 323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

S. Ruan and J. Wei. On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis , 10:863-874, 2003.

K Sakurama and K Nakano. Average-consensus problem for networked multi-agent systems with heterogeneous time-delays. IFAC Proceedings Volumes , 44(1):2368-2375, 2011.

A. Seuret, D. V. Dimarogonas, and K. H. Johansson. Consensus under communication delays. In Proc. of the 47th IEEE Conference on Decision and Control (CDC) , pages 4922-4927. IEEE, 2008.

S K Soni, Xiaogang Xiong, A Sachan, S Kamal, and S Ghosh. Delayed output feedback based leader-follower and leaderless consensus control of uncertain multiagent systems. IET Control Theory & Applications , 15(15):1956-1970, 2021.

Y. G. Sun, L. Wang, and G. Xie. Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays. Systems & Control Letters , 57(2):175183, 2008.

Q. V. Tran, M. H. Trinh, and H.-S. Ahn. Discrete-time matrix-weighted consensus. IEEE Transactions on Control of Network Systems , 8(4):1568-1578, 2021.

M. H. Trinh, C. V. Nguyen, Y.-H. Lim, and H.-S. Ahn. Matrix-weighted consensus and its applications. Automatica , 89:415-419, 2018.

M. H. Trinh, M. Ye, H.-S. Ahn, and B D O Anderson. Matrix-weighted consensus with leaderfollowing topologies. In Proc. of the 11th Asian Control Conference (ASCC) , pages 1795-1800. IEEE, 2017.

S. E. Tuna. Synchronization under matrix-weighted Laplacian. Automatica , 73:76-81, 2016.

C. Wang, L. Pan, H. Shao, D. Li, and Y. Xi. Characterizing bipartite consensus on signed matrix-weighted networks via balancing set. Automatica , 141:110237, 2022.

M. Ye, M. H. Trinh, Y.-H. Lim, B. D. O. Anderson, and H.-S. Ahn. Continuous-time opinion dynamics on multiple interdependent topics. Automatica , 115(108884), 2020.

WYu, G Chen, M Cao, and W Ren. Delay-induced consensus and quasi-consensus in multiagent dynamical systems. IEEE Transactions on Circuits and Systems I: Regular Papers , 60(10):2679-2687, 2013.

S. Zhao and D. Zelazo. Bearing-based distributed control and estimation of multi-agent systems. In Proc. of the European Control Conference, Zürich, Switzerland , pages 2202-2207, 2015.

S. Zhao and D. Zelazo. Localizability and distributed protocols for bearing-based network localization in arbitrary dimensions. Automatica , 69:334-341, 2016.

S. Zhao and D. Zelazo. Bearing rigidity theory and its applications for control and estimation of network systems: Life beyond distance rigidity. IEEE Control Systems Magazine , 2018.

A Appendix / supplemental material

A.1 Time-delay systems and the Lyapunov-Krasovskii theorem

Consider the functional differential equation

where x ( t ) ∈ R n , and the notation x t = x ( t + θ ) , ∀ θ ∈ [ -τ, 0] is adopted. The function f : R ×C n,τ → R n is continuous in both arguments and is locally Lipschitz in the second argument. 2 Furthermore, it is assumed that f ( t, 0 n ) = 0 n , ∀ t ∈ R so that x ≡ 0 n is a solution of the system.

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

386

387

388

Lemma A.1 (Lyapunov-Krasovskii Theorem) . [8] Suppose that f maps R × (bounded sets of C n,τ ) into bounded sets of R n , and there exist functions u, v, w : R + → R + which are continuous, nondecreasing functions, u ( s ) > 0 , v ( s ) > 0 , w ( s ) > 0 , ∀ s > 0 , u (0) = v (0) = 0 . If there exists a continuous function V : R ×C n × L 2 ( -h, 0) → R + , such that

then, the solution x ( t ) ≡ 0 n is uniformly asymptotically stable. If in addition,

then the solution x ( t ) ≡ 0 n is globally uniformly asymptotically stable.

The following lemmas are useful for analysing the stability of time-delay systems. A short introduction to time-delay systems and the Lyapunov-Krasovskii method are given in Appendix A.1 for quick reference, while we refer the reader to [8] for a tutorial on the topic.

Lemma A.2 (Jensen's inequality) . Denote

where a ≤ b , f : [ a, b ] → [0 , ∞ ) , x ( s ) ∈ R n . Then, for any positive definite matrix K ∈ R n × n , there holds

Lemma A.3 (Wirtinger's Inequality) . Let z ( t ) : ( a, b ) → R n be absolutely continuous with ˙ z ∈ 380 L 2 ( a, b ) and z ( a ) = 0 n or z ( b ) = 0 n . Then, for any positive definite matrix W ∈ R n × n , there 381 holds 382

A.2 Proof of Theorem 3.1 383

We rewrite the consensus system (6) in the following matrix form 384

Consider the variable transformation δ ( t ) = Q ⊤ x ( t ) . By expressing L = PΛP ⊤ , we have 385

As R ⊤ ˙ x ( t ) = 0 l , R ⊤ x ( t ) , which shows that R ⊤ x ( t ) = R ⊤ x (0) = ∑ l i =1 p ⊤ i x (0) is time invariant. The n -agent system (25) asymptotically converges to a point in ker ( L ) if and only if δ ( t ) → 0 dn -l , as t → + ∞ , or all roots of the characteristic equation

must have negative real parts. Equation (26) is equivalent to s + λ k e -τs = 0 , ∀ k = l +1 , . . . , dn . 389 Let s = σ + ȷω , where σ, ω ∈ R , we have 390

Thus, the roots of (26) satisfy σ = -λ k e -τσ cos( ωτ ) , ω = λ k e -τσ sin( ωτ ) . 391

(Necessity) If σ < 0 , ∀ ω , it follows that cos( ωτ ) = cos( λ k ωe -τσ τ sin( τω )) > 0 , ∀ ω . This implies 392 that | λ k τe -τσ sin( τω ) | ≤ λ k τe -τσ < π 2 , ∀ k = l +1 , . . . , dn , or τ < πe τσ 2 λ dn ≤ π 2 λ dn . 393

(Sufficiency) if τ < π 2 λ dn , because σ 2 + ω 2 = λ 2 k e -2 τσ , it follows that | ω | ≤ λ k e -τσ , and 394 τ | ω | ≤ πe -τσ 2 . If σ ≥ 0 , then e -τσ ≤ 1 . It follows that cos( τω ) ≥ cos( π 2 ) ≥ 0 , and σ = 395 -λ k e -τσ cos( π 2 ) ≤ 0 . This contradiction implies that σ < 0 . 396

Therefore, we conclude that σ < 0 if and only if τ < π 2 λ dn . 397

Next, let the condition τ < π 2 λ dn be satisfied, and x ( t ) = Φ( t ) , ∀ t ∈ [ -τ, 0] , and Φ( t ) = x (0) , the 398 Laplace transform of (25) gives 399

Using the final value theorem [9], we have 400

401

402

which completes the proof.

A.3 Proof of Theorem 4.1

Consider the functional V ( t, δ ( t ) , ˙ δ t ) = V 1 ( δ ( t )) + V 2 ( ˙ δ t ) , where V 1 = δ ( t ) ⊤ δ ( t ) and V 2 = 403 ∑ r k =1 ∫ τ k 0 ds ∫ t t -s ˙ δ ( h ) ⊤ ˙ δ ( h ) dh. The derivatives of V 1 and V 2 along a trajectory of (12) are given 404 by 405

and 406

where τ = ∑ r i =1 τ i , and in (29) we have used the Jensen's inequality in Lemma A.2. Define the 407 ( r +1)( dn -l ) vector 408

from Eqs. (28) and (29), one gets

409

410

411

412

where M is given in (13). From the assumption that M < 0 , there exists γ > 0 such that

or the origin is a globally uniformly asymptotically stable equilibrium of the system (12) (Appendix A.1). Thus, x ( t ) → x ∗ ∈ ker ( L ) , as t → + ∞ .

The matrix M is a summation of two matrices, the first one is positive definite when τ k are small, 413 and the second one can be made arbitrarily small by choosing τ k small. This implies that the LMI 414 (13) is feasible if τ k , k = 1 , . . . , r , are sufficiently small. 415

A.4 Proof of Theorem 5.1 416

Consider the following functionals 417

where Z , W ∈ R ( dn -l ) × ( dn -l ) are positive definite matrices. Denoting β 1 ( t ) = ∫ t t -τ 1 z 2 ( s ) ds, and 418 β 2 ( t ) = τ 2 z 2 ( t ) -( z 1 ( t ) -z 1 ( t -τ 2 )) , and taking the time derivatives of V j , j = 1 , 2 , 3 , we have 419

Based on Jensen's inequality, the second term in ˙ V 2 can be evaluated as follows 420

Thus, 421

Next, based on Wirtinger's and Jensen's inequalities, we have 422

Thus, ˙ V 3 ≤ τ 4 2 (˙ z 2 ( t )) ⊤ ˆ W ˙ z 2 ( t ) -π 2 4 ( β 2 ( t )) ⊤ ˆ W β 2 ( t ) . Choosing the Lyapunov functional 423 V ( z ( t ) , ˙ z t ) = V 1 ( z ( t )) + V 2 ( z t ) + V 3 (˙ z t ) , and let η = [ z ( t ) ⊤ , ( ¯ Λ β 1 ( t )) ⊤ , ( ¯ Λ β 2 ( t )) ⊤ ] ⊤ , we 424 can compute 425

where X , Y are defined as in (18). Since 426

we have the following equation (˙ z 2 ( t )) ⊤ ¯ ΛW ¯ Λ ˙ z 2 ( t ) = 427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

Thus, if the LMI Ξ 1 + τ 4 2 Ξ 2 < 0 is feasible, the z -system is globally uniformly asymptotically stable. By Schur's complement, this condition is equivalent to

Thus, ˙ V ( z t , ˙ z t ) ≤ -c ∥ z ∥ 2 for some c > 0 , or equivalently, z = 0 is globally uniformly asymptotically stable (Appendix A.1) and x k ( t ) → ker ( L ) , k = 1 , 2 , if the LMI (39) is satisfied. Since x 2 i (0) = 0 d , ∀ i = 1 , . . . , n , due to the observation at the beginning of the Subsection 5.1, we conclude that x 2 i ( t ) → 0 d , ∀ i = 1 , . . . , n .

Finally, we consider the feasibility of the LMI (39). As F ( τ 2 ) is affinely dependent on τ 2 , let Π be a solution of the Lyapunov equation (16), then Π does not have any term that is affine dependent on τ 2 , i.e., Π = O (1) . Let τ 1 be selected such that τ 1 = O ( τ 2 2 ) ,

Choose R = τ -1 1 I dn , W = τ -2 2 I dn , by Schur complement, the LMI Ξ ( τ 2 ) < 0 gives the approximated evaluation

which is satisfied for small positive τ 2 .

A.5 Simulation results

A.5.1 Matrix-weighted consensus models with time delays

In this subsection, we consider a matrix-weighted network of 10 agents in R 3 with the interaction graph as depicted in Fig. 4(a). The edge weights are selected so that rank ( L ) = 3 n -3 = 27 . We will below simulate the network of 10 agents under different assumptions of the time-delays.

y

(a)

x

z

(b)

The network has a uniform time-delay : We consider the consensus network with a uniform constant delay. The maximum eigenvalue of the matrix-weighted Laplacian is calculated to be 10.9235, and thus, the upper bound of the delay is τ max ≈ 0 . 1438 (seconds). For τ = 0 . 1 < τ max , Fig. 4(b) shows that the n -agent system asymptotically consents on a common vector. However, for τ = 0 . 25 > τ max , simulation result in Fig. 4(c) shows that the consensus system becomes unstable.

The network has heterogeneous time delays : Next, let the matrix-weighted network has heterogeneous edge time delays as given in Table 1. In Simulation 1, the time delays are τ 1 = 0 . 05 , τ 2 = 0 . 10 , τ 3 = 0 . 15 . The system asymptotically achieves a consensus. As shown in Figs. 5(a), heterogeneous delays cause significant fluctuations on the process of reaching an agreement.

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

(a)

τ

= 0

.

1

s

(b)

τ

= 0

.

25

s

C1i

0.8

T2i

0.6

33i

2

0.4

0.2

Q

10

15

[8]

×108

0

2

Z1i

C2i

1

1

33i

0

1

-1

2ki

-2

0

5

10

15

(a)

(b)

T,=0.05s,7,=0.10,r,=0.15

0.9

0

0.7

0.8

0.6

=！

0.5

=1,2,3,

0.4

E0

0.1

0

L

5

Time [s]

10

15

T=0.05,2=0.10,T=0.20

4

X

1i

2

1

-2

2

Time [s]

3

4

5

For Simulation 2, the time delays are changed to τ 1 = 0 . 05 , τ 2 = 0 . 10 , τ 3 = 0 . 20 . In this case, the system becomes unstable as shown in Fig. 5(c).

Consensus of double integrators without velocity measurements : We consider the same matrix weighted graphs and conduct simulations for different values of the time delays τ 1 , τ 2 and the control gain k to demonstrate the continuous dependencies of the MWC algorithm (8) with regard to the design parameters.

We first fix the time delays τ 1 = 0 . 05 , τ 2 = 0 . 25 and vary the control gain k from 1 . 1 to 0 . 2 . It can be seen that if k = 1 . 1 (exceeding 1) and k = 0 . 2 (being too small so that the LMI does not hold), the system becomes unstable (see Figs. 6(a)- (f)). For k = 0 . 3 , 0 . 5 , 0 . 85 , 1 , the agents asymptotically achieve a consensus. It can be observed from Figs. 6(b)-6(e) that when k is smaller, the interaction between agents becomes weaker and thus, more fluctuations are exhibited during the process of reaching a consensus.

Second, we fix k = 0 . 85 , τ 1 = 0 . 05 (sec), and vary τ 2 . Simulation results corresponding to τ 2 = 0 . 25 , 0 . 6 , and 0 . 66 are shown in Figs. 6(c), (g), (h), respectively. Clearly, after τ 2 exceeds the limit (about 0.658 (sec)), the network becomes unstable.

Third, we fix k = 0 . 85 , τ 2 = 0 . 25 (sec), and vary τ 1 . Simulation results are depicted in Figs. 6 (a), (c), (j)-(l), corresponding to τ 1 = 0 , 0 . 05 , 0 . 1 , 0 . 2 , 0 . 22 , respectively. As τ 1 gradually reaches to τ 2 , the network tends to be less stable, and when τ 2 = 0 . 22 (sec), the network becomes unstable. Thus, simulation results are consistent with the analysis.

A.5.2 Bearing-based network localization with time delays

Below, we give simulations of the bearing-based network localization laws with time delays to 474 reinforce our analysis. Specifically, in all simulations in this subsection, a 10-agent network will be 475

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

×1049

,= 0.05s, 2= 0.25s,k=1.1

T,= 0.05s,2=0.25s,k=1

T,= 0.05s,T2=0.25s,k=0.85

10

cii

10

0.8

C

10

0.8

C

5

0.6

0.6

2

0

2

0.4

2

0.4

-5

0.2

0.2

0

0

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

Time [s]

Time [s]

Time [s]

T,=0.05s,T2=0.25s,k=0.5

T,=0.05s,2=0.25s，k=0.3

T=0.05s,T2=0.25s,k=0.2

100

10

0.8

0.8

50

0.6

一

0.6

1,2,3,

N

0.4

2,3,

0.4

2

0

=

0.2

0.2

-50

0

H

0

0

50

100

150

200

250

300

0

50

100

150

200

250

300

-100

0

50

100

150

200

250

300

Time [s]

Time [s]

Time [s]

T，=0.05s,T2=0.6s,k=0.85

T,=0.05s，T2=0.66s,k=0.85

T,= 0.0s,2=0.25s,k=0.85

600

10

0.8

10

400

T2i

Tii

1

10

0.8

1I

0.6

I

200

23i

1

0.6

3

2

0.4

2

3

0

2

0.4

-200

0.2

-400

0.2

0

-600

0

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

Time [s]

Time [s]

Time [s]

T=0.1s,T2=0.25s,k=0.85

T= 0.2s,T2=0.25s,k=0.85

X104

10

0.8

Zli

0

0.8

C1i

C1i

C2i

32i

32i

11

0.6

C3i

0.6

T3i

2

C3i

1,2,

'ee

=

0.4

0.2

2

*ee

0.4

0.2

'ee

2i

-2

0

0

0

0

50

100

150

200

250

300

0

50

100

150

200

250

300

4

0

50

100

150

200

250

300

Time [s]

Time [s]

Time [s]

considered. The graph G and the true position of the nodes are given as follows. It can be checked 476 that the bearing Laplacian satisfies rank ( L ) = 26 . 477

478

479

480

481

482

483

Bearing-based network localization with uniform constant time delays : Consider the bearing-based network localization (6) with a constant time delay. The simulation results are depicted in Figs. 7(a)-(b) for τ = 0 . 1 , and Figs. 7(c)-(d). For τ = 0 . 1 , the estimate ˆ x asymptotically converges to an x ∗ , which differs from the correct position x by a translation and a scaling. For τ = 0 . 2 , after 20 seconds of simulation, it can be observed that ˆ x tends to grow unbounded (instability).

Bearing-based network localization with heterogeneous time-delays : Next, we simulate the 484 network localization algorithm (7) with parameters given in Table 2. For τ 3 = 0 . 2 (sec), it is 485 observed from Figures 8(a)-(b) that ˆ x converges to a configuration x ∗ , and the sum of squared 486 bearing errors ∑ ( i,j ) ∈E ∥ g ij -g ∗ ij ∥ 2 asymptotically converges to zero. Thus, x ∗ is a configuration 487

489

Bearing-based network localization of double-integrators with two constant time delays: 490

Finally, we conduct simulations of the network localization algorithms for double-integrator agents 491 with two time-delays. The results are depicted as in Fig. 9. We can observe that

(a)

(b)

(c)

(d)

, =0.05s, , = 0.25s, k = 0.7

14

18

16

（j） -9

Init Est

Traj.

, = 0.2, r, = 0.87, k = 0.7

25

30

(） I  9

-Init. Est.

Est. at t=50

Traj.

Bearing Errors

—Est at t=10

Est at t=20

1510

20

6

-0.5

0.5

-0.5

-1

-1.5

10

20

Time

30

40

50

10

20

Time

30

40

50

492

(a)

(b)

(c)

(d)

r = 0.2

Init. Est.

Traj.

Est. at t=10

20

18

16

（） 9i9i

= 0.3

30

25

（99

Bearing Errors

14

suoug Suueeg

15

6

Init. Est.

Traj.

10

Est. at t=8

Est. at t=20

Z

Time

10

2

10

Time

15

20

25

satisfying all the sensed bearing vectors. As τ 3 changes from 0.2 (sec) to 0.3 (sec), the network 488 becomes unstable, as shown in Figs. 8(c)-(d).

(a)

(b)

(c)

(d)

,=0.1,,=0.15,,=0.2

16

18

（e9 -9

, = 0.1, r, =0.2, r, =0.33

Init. Est.

40

35

（j）99

Bearing Errors

14

10

Traj.

Est. at t=10

Est. at t=20

Bearing Errors

25

30

05

2

0

15

-0.5

Init. Est.

Fin. Est.

Traj.

4

10

10

Q

5

Time

10

15

Time

10

15

20

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The paper is the first one studying effects of time-delay in matrix-weighted consensus networks. Our analytical tool is control theory for linear systems and the Lyapunov-Krasovskii theorem. Application of the considered consensus algorithms in network localization is also discussed and supported by simulations.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes] .

Justification: The analysis is restricted to constant time-delay, which is the fundamental case for studies any delayed system. A sentence in the conclusion has been stated to address this case.

3. Theory Assumptions and Proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [No]

Justification: All mathematical proofs are provided for leaderless networks. The proofs for leader-follower networks are similar and thus, have been omitted in the submission.

4. Experimental Result Reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [No]

Justification: the paper does not contain any experiment. The results are theoretical and only numerical simulations are provided.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [No]

Justification: The paper does not produce any data. Simulation codes are available and can be shared after the paper is published.

6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [No]

Justification: The paper does not include tuning of hyperparameters.

7. Experiment Statistical Significance

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [No]

Justification: The paper does not contain any experiment, so no information about the statistical significance of the experiments is needed.

8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [No]

Justification: the result in the paper is theoretical and no experiments are reported. Simulations are given to illustrate the theoretical results, and thus, does not require any special hardware/computer.

9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: We claim that the research conducted in the paper conform, in every respect, with NeurIPS Code of Ethics.

10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [No]

Justification: this research mainly concerns on matrix-weighted consensus algorithm - a generalized model of the consensus algorithm. Currently, no negative potential negative societal impacts of the algorithm have been known.

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [No]

Justification: The paper mainly focuses on theory. Simulation results are given to support the theoretical analysis.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [No]

Justification: The authors of the paper possess rights on any algorithms and numerical simulations reported in this submission.

13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

Answer: [No]

Justification: The paper does provided descriptions of all theoretical results and numerical simulations of the paper (in the main body of the paper and in the appendix/supplementary files).

14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [No]

Justification: There is no experiments and research with human subjects reported in this paper.

15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [No]

Justification: The studies of matrix-weighted consensus have not known to cause any potential risks.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the conditions that lead to consensus in matrix-weighted consensus networks when constant time delays are present. The analysis considers both leaderless and leader-follower settings. The paper considers single integrators with uniform time delays, heterogeneous time delays, and double integrators with two constant time delays. The paper derives the conditions for asymptotic convergence to a consensus or clustering configuration. The mathematical techniques include direct eigenvalue evaluation and application of the Lyapunov-Krasovkii theorem. The paper explains how the derived results can be applied to the problem of bearing-based network localization.\n\nSTRENGTHS:\nS1. 1The paper provides a novel theoretical characterization of the asymptotic consensus for matrix-weighted consensus networks in several settings. In multi-agent networks  \n\nS2.\tThe analysis is clearly presented and the arguments are logical and well-structured. The employed techniques are innovative.\n\nWEAKNESSES:\nW1.\tAlthough the scope of NeurIPS is broad, and papers are encouraged from diverse fields, this paper makes little attempt to connect to any learning-based problem or application. The paper seems to be a poor match for a machine learning conference. There is not a single citation of a paper from one of the leading machine learning conferences or journals. 24 of the 37 references are associated with papers in control journals and conferences. The paper would be of much more interest to the control community. If the authors consider this to be a significant research contribution that furthers the understanding of matrix-weighted consensus, then why not submit it to Automatica (this is the forum for several other cited matrix-weighted consensus papers), or IEEE Trans. Automatic Control, or IEEE Trans. Control of Networked Systems. If there is a belief that the paper exposes a problem or a technique to the machine learning community, then there must be a much more convincing effort to highlight the connections – where/how would the machine learning community find the presented results useful? \n\nW2.\tThe main paper shows how the theoretical results are applicable to the bearing-based network localization problem. There is very little explanation of whether the proposed approach to localization is advantageous, and how the theoretical results are useful – whether it is for analysis of a network, or for design of a network. The appendix provides the results of simulations. But even there, the discussion is limited to simple observations regarding the behaviour of the simulated network (consensus/instability). The simulation analysis needs to be more convincing and explain in detail how the theoretical results are useful for this problem. Alternatively, additional theory could be provided that pertains to the bearings-based network localization task. \n\nW3.\tThe presentation of the paper should be improved. In particular, the figures on pages 16-18 are far too small. The text in these figures is illegible. It becomes almost impossible to understand what information the figures are supposed to convey.\n\nQUESTIONS:\nQ1.\tWhat is the motivation for submission of this paper to NeurIPS? What is the relevance/connection to the machine learning community? \n\nQ2.\tCan the authors provide more information about the bearing-based network localization problem and the application of the presented theoretical results? What insights do the theoretical results provide? \n\nQ3.\tThe restriction to constant time delays is undesirable. Can the authors discuss how the work might be extended to a more realistic setting of random time delays? How would the results change if the constant time delay were considered to be the maximum delay?\n\nLIMITATIONS:\nThe paper includes one or two sentences in the conclusion to discuss the limitations. The only acknowledged limitation is the restriction to the constant time delay setting. The paper would be strengthened by a much more thorough discussion of the limitations. For example, it would be helpful to understand whether the authors consider the constant time restriction to simply lead to conservative bounds, or whether the results would be completely inapplicable in a variable time delay setting.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nPaper under review analyzes the consensus of agents over a network. The agents have arbitrary but identical state space dimension, so not just scalar dynamics. The communication between agents is delayed and can be heterogeneous. Lyapunov–Razumikhin functionals with an LMI (that grows with the size of the network) are the main analysis tools. The literature in this area is vast. The results may already be contained as special cases of more general results from the control literature that was not referenced in the original submission\n\nSTRENGTHS:\n- originality: very crowded area of research\n- quality: high quality except for not being very precise about what is different from other work, and lacks a detailed discussion about what impacts the LMI bound\n- clarity: paper is well written\n- significance: more general results on the same topic appear to already exist\n\nWEAKNESSES:\n- Authors discuss some of the relevant literature but are never explicit about what actually is different. This makes it very challenging to understand what is different and new about this contribution.\n- Along the same lines as the above statement. Results and analysis techniques may be already present in the papers referenced below.   \n\t- Jiang, W., Liu, K., & Charalambous, T. (2022) in particular solves the more general problem of consensus with heterogenous delays where each agent is an arbitrary linear dynamical system (A,B,C). The results in this paper appear to be a subset of that class of systems.\n- LMI grows with the size of the network making it not scale well\n- No interpretation of the LMI once it is derived (but they are huge matrices which kind of make them hard to interpret)\n\nQUESTIONS:\n- How are Theorem 3.1 and its proof different from (Olfati-Saber and Murray, 2004), **explicity**?\n- You may want to check these references (and many more can be found that you did not reference that seem even more relevant than the ones you originally referenced)\n\t1. Y. G. Sun and L. Wang, “Consensus of multi-agent systems in directed networks with nonuniform time-varying delays,” IEEE Trans. Autom. Control, vol. 54, no. 7, pp. 1607–1613, Jul. 2009.\n\t2. C.-L. Liu and F. Liu, “Consensus problem of second-order dynamic agents with heterogeneous input and communication delays,” Int. J. Comput. Commun., vol. 5, no. 3, pp. 325–335, 2010.\n\t3. Jiang, W., Chen, Y., & Charalambous, T. (2021). Consensus of general linear multi-agent systems with heterogeneous input and communication delays. _IEEE Control Systems Letters_, _5_(3), 851-856.\n\t4. Jiang, W., Liu, K., & Charalambous, T. (2022). Multi-agent consensus with heterogeneous time-varying input and communication delays in digraphs. _Automatica_, _135_, 109950.\n- How does toplogy or degree of heterogeneity affect the LMI\n- Is your LMI construction similar to Jiang, W., Liu, K., & Charalambous, T. (2022)?\n- Is NeurIPS and appropriate venue for consensus theory papers?\n\nLIMITATIONS:\nlimitations discussion is not really sufficient, they only discuss future directions", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-17"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper investigates consensus conditions for matrix-weighted consensus networks, both leaderless and leader-follower, in the presence of constant time-delays. It explores delayed consensus algorithms for networks of single- and double-integrators using relative positions. The study derives conditions for networks to achieve consensus or clustering using eigenvalue evaluation and the Lyapunov-Krasovkii theorem. It also discusses an application in bearing-based network localization. Some numerical simulations are also provided to demonstrate effectiveness of the results.\n\nSTRENGTHS:\nThis paper studied an important problem in the control theory, proposing novel algorithms and extending existing work to matrix-weighted networks with time-delays. They provided structured and rigorous mathematical analysis/proof of their results and provide helpful numerical simulations to validate their theoretical results.\n\nWEAKNESSES:\nIn my opinion, the major weakness (in terms of publication at NeurIPS) of this paper is below. This paper has strong focus on control theory and consensus algorithms, which makes it less relevant to the core interests of the NeurIPS audience, since NeurIPS emphasizes more on machine learning methodologies and applications. The paper does not clearly establish connections to machine learning problems or provide good experimental results involving machine learning tasks.\n\nThis paper contains quite some valuable novel research results, but it seems to me it is more appropriate for publication on a traditional control journal or conference, rather than such a top tier machine learning conference like NeurIPS. It is just not a good fit, and it might be better to reserve the space (which is quite limited) for some other good candidate paper submissions more relevant to Machine Learning, which are more aligned with the interests of core audiences of NeurIPS.\n\nQUESTIONS:\nSee my above comment about the weakness.\n\nLIMITATIONS:\nN/A.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-18"}