Continuous distributed algorithms for solving linear equations in finite time

Automatica 113 (2020) 108755

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Brief paper

Continuous distributed algorithms for solving linear equations in finite time✩ ∗

Xinli Shi a,b , Xinghuo Yu b , Jinde Cao c , , Guanghui Wen c a b c

School of Cyber Science and Engineering, Southeast University, Nanjing 210096, China School of Engineering, RMIT University, Melbourne VIC 3001, Australia School of Mathematics, Southeast University, Nanjing 210096, China

article

info

Article history: Received 1 December 2018 Received in revised form 6 September 2019 Accepted 24 November 2019 Available online xxxx Keywords: Distributed algorithms Finite/fixed-time consensus Multi-agent networks Linear equations

a b s t r a c t This paper studies the distributed algorithms to obtain a solution of the linear equation Ax = b in finite time (FT) over a multi-agent network. In order to guarantee the settling time without depending on the initial states, the fixed-time (FxT) distributed algorithms are also provided to obtain a solution within a globally bounded time. Specifically, three distributed nonlinear algorithms are developed. The first one is designed to achieve FT/FxT consensus on a solution with special initialization. The second is to obtain the solution in FT/FxT with free initialization by first driving the local states to satisfy the special initialization in FT/FxT time. The last one is to guarantee the FT/FxT convergence to a solution closest to specific points when multiple solutions exist. Finally, three case studies are performed to show the effectiveness of the proposed algorithms. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Nowadays, the distributed algorithms for solving linear equations over a multi-agent network have aroused great interest. A common problem is to design an efficient protocol to solve Ax = b in a distributed way, for which the matrix A = [AT1 , AT2 , . . . , ATn ]T , the vector b = [bT1 , bT2 , . . . , bTn ]T and the ith agent only has access to the information of Ai and bi . In the control process, each agent owns a local state xi as an estimation of the solution and exchanges information with its neighbors to reach an agreement on a solution of Ax = b. This distributed scheme has many advantages. By decomposing the matrix into smaller ones, it can avoid storing large amounts of data on a single memory and make a large-scale linear equation problem tractable. Besides, it can protect the customers’ privacy when some sensitive information exists (Anderson, Mou, Morse, & Helmke, 2016). A natural way to handle the above distributed linear equation problem is to treat it as a distributed optimization problem and solve it with existing distributed algorithms (Chen & Ren, ✩ This work was supported by the Natural Science Foundation of China under Grant Nos. 61833005, 61722303 and 61703095, in part by the Australian Research Council under Grant Nos. DE180101268 and DP120102303. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Julien M. Hendrickx under the direction of Editor Christos G. Cassandras. ∗ Corresponding author. E-mail addresses: [email protected] (X. Shi), [email protected] (X. Yu), [email protected] (J. Cao), [email protected] (G. Wen). https://doi.org/10.1016/j.automatica.2019.108755 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

2016; Gharesifard & Cortés, 2014; Jakovetić, Xavier, & Moura, 2014; Kia, Cortés, & Martínez, 2015; Lin, Ren, & Song, 2016; Nedić, Ozdaglar, & Parrilo, 2010; Shi, Cao, & Huang, 2018; Shi, Johansson, & Hong, 2013; Song & Chen, 2016). However, these methods may not be applicable from various aspects as discussed in Mou, Liu, and Morse (2015) and Shi, Anderson, and Helmke (2017). Specifically, if the equality problem is formulated into a distributed constrained optimization, in order to apply the projection method proposed by Lin et al. (2016) and Nedić et al. (2010), it requires that the intersection of the affine subspaces Mi = {x ∈ Rm : Ai x = bi } has interior point and is compact, which is violated when Ax = b has a unique solution or multiple solutions. For the distributed control method proposed in Shi et al. (2013), the boundedness of the intersection is needed. On the other side, when treating the equality problem as a distributed unconstrained optimization, additional states are introduced and transmitted in the algorithms (Gharesifard & Cortés, 2014; Jakovetić et al., 2014; Kia et al., 2015; Shi et al., 2018), which increase the communication cost of the network. Furthermore, in order to achieve an exponential convergence rate, the strongly convexity assumption is imposed on the local functions (Chen & Ren, 2016; Kia et al., 2015; Shi et al., 2018), which can be only satisfied when each Ai has full column rank. With the strongly convexity assumption, a continuous zero-gradient-sum method is given in Song and Chen (2016) for solving distributed optimization in finite time. In Lin, Ren, and Farrell (2017), a nonsmooth distributed FT optimization algorithm is proposed, which is so complex for applications. In a whole, there are few FT

2

X. Shi, X. Yu, J. Cao et al. / Automatica 113 (2020) 108755

convergent algorithms for distributed optimization with general convex objective functions and constraints. Partially inspired by constrained consensus algorithm proposed in Nedić et al. (2010), a distributed iterative algorithm with exponential convergence rate was designed in Mou et al. (2015) over repeatedly jointly strongly connected networks. An improved version of Mou et al. (2015) was developed in Wang, Mou, and Sun (2017) with free initialization and the ability to achieve a solution closest to a given point. Furthermore, Liu, Morse, Nedić, and Başar (2017) provided the necessary and sufficient conditions on a time-dependent graph for the designed algorithm to achieve an exponential convergence rate. Additionally, two asynchronous distributed algorithms were proposed in Liu, Mou and Morse (2018) for solving Ax = b with at least one solution. A discrete-time distributed algorithm was proposed over jointly strongly connected topologies with M-Fejer mappings in Wang, Ren, and Duan (2019). From a viewpoint of control theory, by mapping the solutions of the linear equations into the equilibria of a interactive dynamical system, a continuous-time linear protocol was proposed in Anderson et al. (2016) to solve Ax = b exponentially when A has full row rank. In Liu, Chen, Başar, and Nedić (2016), a continuous-time distributed algorithm has been provided by assuming Ai has full row rank. In Shi et al. (2017), two continuous-time network flows called ‘‘projection consensus’’ flow and ‘‘consensus + projection’’ flow, respectively, were proposed for solving linear equations asymptotically. When there are no exact solutions for the linear equations, the distributed algorithms to obtain the least squares solutions were investigated (Liu, Lou, Anderson and Shi, 2018; Shi et al., 2017; Yang, George, Qin, Yi, & Wu, 2018). In the aforementioned distributed approaches, the linear distributed consensus algorithms (Olfati-Saber & Murray, 2004) have been embedded to achieve an agreement on the final states asymptotically or exponentially and meanwhile, the private equations are satisfied. However, in some practical systems, the FT convergence of the control or estimation is required (Cortés, 2006; Perruquetti, Floquet, & Moulay, 2008; Wang & Xiao, 2010) especially in a real-time control process when the control accuracy is crucial. Specifically, when the finite settling time is globally bounded, the FxT convergence is investigated (Chen & ` Sánchez-Torres, Li, 2018; Gómez-Gutiérrez, Vázquez, Čelikovsky, & Ruiz-León, 2018; Hong, Yu, Wen, & Yu, 2017; Zuo, Han, Ning, Ge, & Zhang, 2018) since the FxT stability was first studied in Polyakov (2012). Although lots of linear protocols have been proposed for solving linear equations as introduced previously, there are few results on the fully distributed FT/FxT solver with nonlinear continuous-time dynamics. A recent FT solver can be found in Zhou, Wang, Mou, and Anderson (2019), in which a nonsmooth continuous-time dynamics is proposed to solve the linear equations with minimum l1 in view of Filippov solutions. For distributed discrete-time methods, the existing FT algorithms are mostly designed for distributed consensus problems (Yuan, Stan, Shi, Barahona, & Goncalves, 2013). Recently, based on (Yuan et al., 2013), a discrete-time approach is provided in Yang et al. (2018) for achieving the least squares solution within FT iterations. However, global information is needed to determine the algorithm parameter and additional computations are required for calculating the rank and kernel of the expanding square Hankel matrix in each iteration. The fully distributed FT discretetime solver for linear equations is also a challenging problem and beyond the scope of this paper. In this paper, we only focus on the FT continuous-time law in a fully distributed manner since it has simpler structure and easy to analyze by using Lyapunov stability theory. In particular, the main task of this paper is to achieve the FT/FxT consensus on the solution of linear equations when A has full row rank by

combining the ‘‘projection consensus’’ flow with a general version of the existing nonlinear FT/FxT consensus protocols (Shi, Cao, Wen, & Yu, 2019; Wang & Xiao, 2010). In Gómez-Gutiérrez et al. (2018), equipped with specific homogeneous functions defined on the sum of the errors of all neighboring nodes, a kind of FT/FxT consensus algorithms are investigated by using homogeneity theory. However, the homogeneity theory does not provided a bound for the convergence time and cannot be applied to the systems with heterogeneous coefficients directly. Differently, in this work, the FT/FxT consensus protocol has heterogeneous exponents and is designed based on the error described by neighboring node. The bound estimation of the finite/fixed settling time is also given correspondingly. Concretely, to achieve FT/FxT convergence to a solution of Ax = b, three nonlinear algorithms are developed to deal with three cases: (a) with initial condition Ai xi (0) = bi , i = 1, 2, . . . , n; (b) with free initialization; (c) obtain a solution closest to specific points when multiple solutions exist. To the best of the authors’ knowledge, there are no existing fully distributed FT/FxT convergent protocols addressing any one of the above cases. The remainder of this paper is structured as follows. In Section 2, some basic notations and concepts are introduced. The problem formulation and the primal methodology are provided in Section 3. In Section 4, the main results for the FT/FxT convergence of the proposed protocols to the solution of the linear equations are shown for three scenarios. Finally, two numerical examples are given in Section 5 to show the effectiveness of the proposed methods. Conclusions are included in Section 6. 2. Preliminaries 2.1. Basic notations and graph representation

Rn0 (Rn+ ) represents the set of n-dimensional vectors with nonnegative (positive) entries. Let 1n ∈ Rn be the vector with all entries being one, and In be an n-dimensional identity matrix. We denote ⟨n⟩ = {1, 2, . . . , n}. For α = [α1 , . . . , αn ]T ∈ Rn+ and x = [x1 , . . . , xn ]T ∈ Rn , the √ Euclidean norm and infinity norm of x are denoted by ∥x∥ = xT x and ∥x∥∞ = maxi∈⟨n⟩ |xi |, respectively, and sgn[α] (x) = [sign(x1 )|x1 |α1 , . . . , sign(xn )|xn |αn ]T with sign(·) being the signum function. Let σ = [σ1 , . . . , σn ]T ∈ Rn , diag(σ ) is a diagonal matrix with σi being the ith diagonal entry. N (L) and I (L) represent the null space and the image space of the linear map L, respectively. When L is positive semidefinite, λ2 (L) is the smallest positive eigenvalue of L. In this paper, the term w.r.t. is an abbreviation for ‘‘with respect to’’. An undirected network is represented by G (V , E , A), for which ¯ The the node set V = ⟨n⟩ and edge set E ⊆ V × V with |E | = m. weighted adjacency matrix A = [aij ]n×n is defined as: aij > 0 if and only if (j, i) or (i, j) ∈ E and aii = 0, ∀i ∈ ⟨n⟩. The set Ni = {j : (j, i) ∈ E } represents the neighbors of agent i. Let N i = Ni ∪ {i} be the closed neighbor set of agent i. The graph G is connected if there exists an undirected path connecting any pair of distinct agents i, j ∈ V . 2.2. Finite-time and fixed-time stability Consider the following autonomous system x˙ (t) = f (x(t)),

(1) n

for which x(t) ∈ X0 and the map f : X0 → R is a continuous vector function and f (0) = 0. For the system (1), the concept of FT/FxT stability is introduced in Definition 1.

X. Shi, X. Yu, J. Cao et al. / Automatica 113 (2020) 108755

Definition 1. The origin is called finite-time stable for system (1) if it is Lyapunov stable and there exists a finite settling time T (x0 ) such that

Table 1 ¯ Basic mathematical symbols with l = mm. ¯

aˆ = δ[aij

for any solution x(t ; x0 ) starting from x0 ∈ X0 . Furthermore, if there exists a bounded Tmax < +∞ such that the above settling time T (x0 ) ≤ Tmax for any x0 ∈ X0 , then it is called fixed-time stable. When X0 = Rn , it is called globally finite-/fixed-time stable. Lemma 1. Let x(t , x0 ) be any solution of the system (1) starting from x0 ∈ X0 . Suppose that there exists a continuous radially unbounded function such that V (x) = 0 ⇒ x = 0. (1) (Bhat & Bernstein, 2000, Theorem 4.2) If V˙ (x(t)) ≤ −cV (x(t)) with c > 0, r ∈ (0, 1), the origin is FT stable with V 1−r (x(0)) settling time T (x0 ) ≤ c(1−r) . (2) (Polyakov, 2012, Lemma 1) If V (x) satisfies that (2)

for some positive scalars c1 , c2 , r1 , r2 : 0 < r1 k < 1, r2 k > 1, the origin is FxT stable with settling time estimated by 1 c1k (1

− r1 k)

+

1 c2k (r2 k

− 1)

.

(3)

3. Problem formulation and methodology 3.1. Problem formulation

¯

](i,j)∈E˜ ⊗ 1m ∈ Rl0

L = H T PH, aˆ min = min{ˆal : l ∈ ⟨¯l⟩}, bˆ min = min{bˆ l : l ∈ ⟨¯l⟩}

3.2. Distributed finite-time algorithm For the purpose of the FT/FxT convergence to X ∗ , the local nonlinear protocol ui is designed as u¯ i (xN i ) = −Pi

∑

aij (δ sgn[αij ] (xi − xj )

j∈Ni

− ηsgn[βij ] (xi − xj )),

(6)

in which δ > 0, η ≥ 0, 0 < αij = αji < 1 < βij = βji , ∀i, j ∈ ⟨n⟩, and Pi = I − ATi (Ai ATi )−1 Ai is the orthogonal projection matrix on N (Ai ). Note that the protocol (6) is continuous over the time when αij > 0 since sgn[αij ] (xi − xj ) = sign(xi − xj )|xi − xj |αij . For each edge ek = (i, j) of the undirected graph G (⟨n⟩, E , A), one can ˜(⟨n⟩, ˜ define a digraph G E ) by assigning an orientation from j to i if i < j, and define the weighted incidence matrix B = [bik ]n×m¯ of ˜ as: for any ek = (i, j) ∈ ˜ G E , bik = aij and bjk = −aij . Before moving on, some basic mathematical symbols frequently used throughout this paper are collected in Table 1. ˆ we can arrange them as α = By the formation of α, β, aˆ , b, ¯ Let [αl ]l∈⟨¯l⟩ , β = [βl ]l∈⟨¯l⟩ , aˆ = [ˆal ]l∈⟨¯l⟩ , bˆ = [bˆ l ]l∈⟨¯l⟩ with ¯l = mm. x = [xi ]i∈⟨n⟩ ∈ Rnm . With the nonlinear protocol (6), the whole system (5) can be written in a compact form (7)

For the system (7), it is required that the initial states satisfy (4)

where A = col{A1 , A2 , . . . , An } and b = col{b1 , b2 , . . . , bn } with Ai ∈ Rni ×m , bi ∈ Rni , i ∈ ⟨n⟩. It is assumed Ai and bi are only known by the ith agent. The objective of this work is to design a distributed continuous-time algorithm in order to solve Ax = b in finite time under some assumptions. Denote the solution set of (4) as X ∗ . Let the local protocol of the ith agent be formulated as x˙ i (t) = ui (xN i ), ∀i ∈ ⟨n⟩

bˆ = η[aij

x˙ (t) = −PHAd sgn[α] (H T x) − PHBd sgn[β] (H T x).

Consider the following linear equation: Ax = b,

−βij

Pi = I − ATi (Ai ATi )−1 Ai , P = diag([Pi ]i∈⟨n⟩ ), H = B ⊗ Im

r

T (x0 ) ≤

](i,j)∈E˜ ⊗ 1m ∈

¯ Rl+ ,

ˆ A = [Ai ]i∈⟨n⟩ , b = [bi ]i∈⟨n⟩ , Ad = diag(aˆ ), Bd = diag(b)

t →T (x0 )

V˙ (x(t)) ≤ −(c1 V r1 (x(t)) + c2 V r2 (x(t)))k

¯

α = [αij ](i,j)∈E˜ ⊗ 1m ∈ Rl+ , β = [βij ](i,j)∈E˜ ⊗ 1m ∈ Rl+ , −αij

lim x(t ; x0 ) = 0 and x(t) = 0, ∀t ≥ T (x0 )

3

(5)

in which xi ∈ Rm is the local estimation of the solution and xN i = col{xj : j ∈ N i } is the state collection of the agents in N i . Let x = col{xi : i ∈ ⟨n⟩} be the state collection of all the agents. Then, this paper aims to design ui such that there exists a bounded settling time T (x0 ) and x∗ ∈ X ∗ satisfying that lim xi (t ; x0 ) = x∗ and xi (t) = x∗ , ∀t ≥ T (x0 )

t →T (x0 )

for any given initial state x0 = [xi (0)]i∈⟨n⟩ ∈ X0 ⊆ Rnm . If there exists a global bounded settling time T0 such that T (x0 ) ≤ T0 for any x0 ∈ X0 , the dynamics (5) is called a fixed-time distributed algorithm for solving (4). In this paper, the following assumption is imposed for the matrix A and communication network G . When A has full row rank, Eq. (4) has at least one solution, i.e., X ∗ ̸ = ∅. Assumption 1. The matrix A has full row rank and the graph G (⟨n⟩, E , A) is undirected and connected, i.e., A = AT .

Ai xi (0) = bi , ∀i ∈ ⟨n⟩.

(8)

Then, it can be verified that the manifold M = {x = [xi ]i∈⟨n⟩ : Ai xi = bi , i ∈ ⟨n⟩} is forward invariant. By letting z = H T x, the system (7) is transformed into

Sz : z˙ (t) = −LAd sgn[α] (z) − LBd sgn[β] (z).

(9) T

For the system Sz , it is obvious that I (H ) is a forward invariant subspace. The following lemma is useful in the later part, the proof of which is similar to that of Zhou et al. (2019, Lemma 4) and is omitted here. Lemma 2.

Under Assumption 1, one has that

I (H) ∩ N (P) = 0, I (H T ) ∩ N (L) = 0.

Corollary 1.

(10)

Under Assumption 1, it holds that I (H T ) ⊥ N (L).

Proof. For any y ∈ N (L), it holds that Ly = H T PHy = 0, which indicates that PHy = 0 and Hy = 0 by Lemma 2. Therefore, z T y = xT Hy = 0 for any z = H T x ∈ I (H T ). □ Corollary 2. For the system Sz , z = 0 is the unique equilibrium point in the subspace I (H T ). Proof. Let f = Ad sgn[α] + Bd sgn[β] . Suppose that z e = H T xe ∈ I (H T ) is an equilibrium point of Sz . Then, it holds that Lf (z e ) = H T PHf (z e ) = 0, which indicates that PHf (z e ) = 0. With Lemma 2, it follows that Hf (z e ) = 0 and thus (xe )T Hf (z e ) = z e f (z e ) = 0, which implies that z e = 0 since f is sign preserving. □

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X. Shi, X. Yu, J. Cao et al. / Automatica 113 (2020) 108755

The concept of sign preserving function is introduced in Definition 2, and one main property is provided in Lemma 3, which is essential for stability analysis of the system Sz .

Specially, by applying Lemma 6 with homogeneous coefficients αl = α0 , βl = β0 , l ∈ ⟨¯l⟩, one can easily obtain the following corollary (Wang, Yu, Wen, & Chen, 2018).

Definition 2. A continuous function φ : R ↦ → R is sign preserving if it satisfies: (1) φ (0) = 0; (2) yφ (y) > 0, ∀y ̸ = 0. A continuous vector field f : Rn ↦ → Rn represented by f (z) = [fi (zi )], ∀z ∈ Rn is sign preserving if each component fi , i ∈ ⟨n⟩ is sign preserving.

Corollary 3. Suppose that zl ≥ 0, 0 < α0 < 1 < β0 , l ∈ ⟨¯l⟩ for any positive integer ¯l. Then, it holds that

Lemma 3 (Shi et al., 2019, Lemma 4). Given a nonzero positive semidefinite matrix L ∈ Rn×n and a positive diagonal matrix D = diag([di ]) ∈ Rn×n . Suppose that f : Rn ↦ → Rn is sign preserving and define

λf (L, D) ≜

f T (z)Lf (z)

inf

0̸ =Dz ⊥N (L)

.

f T (z)f (z)

(11)

Then, it holds that λf (L, D) > 0. If D = In , λf (L, D) is simply written as λf (L). Particularly, when L is positive definite, λf (L, D) = λf (L) = λ2 (L). Remark 1. By the definition (11), when f (z) = z, λf (L) = λ2 (L). Specifically, when L is the Laplacian matrix of a connected graph G , λ2 (L) is the algebraic connectivity of G , which is used to measure the convergence rate of a linear Laplacian-flow dynamics for achieving the multi-agent consensus asymptotically (Olfati-Saber & Murray, 2004).

¯l ∑ (

α0

zl

β

+ zl 0

)2

≥(

l=1

¯l ∑

α0 +1

(zl

β +1

+ zl 0

))r1 ,

l=1

¯l ∑ (

α0

zl

β

+ zl 0

)2

¯l 1−β0 ∑ α +1 β +1 ≥ (2¯l) β0 +1 ( (zl 0 + zl 0 ))r2 l=1

l=1

with r1 = (α0 + β0 )/(1 + β0 ) and r2 = 2β0 /(1 + β0 ). 4. Main results In this section, three types of nonlinear algorithms are investigated for different considerations. In Section 4.1, the FT/FxT convergence of the system (5) with protocol u¯ i to a solution of the linear equations with initialization (8) is shown. In Section 4.2, a novel distributed algorithm is developed to achieve the FT/FxT convergence to a solution with free initialization. In Section 4.3, the previous algorithms are applied to obtain a solution closest to specific points with modified initializations when multiple solutions exist. 4.1. Case 1: FT/FxT convergence with initialization (8)

In order to show that each state xi of the system (7) converges to the same point in X ∗ in FT/FxT, it can be achieved by showing the FT/FxT stability of Sz at the origin w.r.t. I (H T ).

In the following, the FT/FxT stability of Sz at the origin w.r.t. I (H T ) is shown with the help of Lemmas 5 and 6.

Lemma 4. If the system Sz is FT/FxT stable at the origin w.r.t. the subspace I (H T ), then the system (7) with condition (8) will converge to 1n ⊗ x∗ with x∗ ∈ X ∗ in FT/FxT.

Theorem 1. Let Assumption 1 hold. Consider the system (5) with protocol (6), i.e., ui = u¯ i , and the transformed system Sz with z0 ∈ I (H T ). Then,

Proof. Let x(t) be any solution of the system (7) with initial point x(0) satisfying (8) and z(t) = H T x(t). Then, z(t) is a solution to Sz with initial point z(0) = H T x(0). Since the system Sz is FT/FxT stable at the origin w.r.t. the subspace I (H T ), it indicates that z(t) = H T x(t) will converge to the origin in FT/FxT T0 . Since x(t) ∈ M, then the state x(t) will converge to (1n ⊗ Rm ) ∩ M. That is to say, x(t) will converge to 1n ⊗ x∗ with x∗ ∈ X ∗ . □ In order to derive the main results of this part, several key inequalities are presented in advance in Lemmas 5 and 6. Lemma 5 (Goldberg, 1987, Theorem 1). If η1 , η2 , . . . , ηk ≥ 0 and µ > ν ≥ 1, then (

k ∑

µ

1

ηi ) µ ≤ (

k ∑

1

1

1

ηiν ) ν ≤ k ν − µ (

k ∑

i=1

i=1

µ

1

ηi ) µ .

(12)

i=1

Lemma 6 (Shi et al., 2019, Lemma 7). Suppose that zl ≥ 0, 0 < αl < α +β 1 < βl , l ∈ ⟨¯l⟩ for any positive integer ¯l. Let κ = maxl∈⟨¯l⟩ { βl +1l } and l

βl χ χ = minl∈⟨¯l⟩ { β2+ }. For any α0 , β0 satisfying β0 ∈ (1, 2−χ ] and l 1 α0 ∈ [κ (β0 + 1) − β0 , 1), it holds that ¯l ∑ (

αl

βl )2

zl + zl

l=1

αl +1

(zl

βl +1

+ zl

Proof. Denote z = [zk ]ek ∈E˜ with zk ∈ Rm . Since z ∈ I (H T ), there exists x = [xi ]i∈⟨n⟩ with xi ∈ Rm such that zk = aij (xi − xj ) when i < j for each ek = (i, j) ∈ ˜ E . For the system Sz defined on the subspace I (H T ), consider the following Lyapunov function V =

−β ij ∑ ( δ a−α ηaij ij T ) ij 1Tm |zk |αij +1 + 1m |zk |βij +1 αij + 1 βij + 1 ˜

ek =(i,j)∈E

=

¯l ∑ ( l=1

aˆ l

αl + 1

∑(

αl

βl )2

zl + zl

≥ (2¯l)

1−β0 β0 +1

¯l

(

∑

αl +1

(zl

+ zl

bˆ l

βl + 1

) |zl |βl +1 .

V˙ = f T (z)z˙ = −f T (z)Lf (z). Since f is sign preserving and I (H T ) ⊥ N (L) by Corollary 1, with Lemma 3, V˙ can be further estimated as V˙ = −f T (z)Lf (z) ≤ −λf (L)f T (z)f (z).

(13)

(1) If η = 0, Eq. (13) can be further relaxed as V˙ ≤ −ˆa2min λf (L)(sgn[α] (z))T sgn[α] (z)

)) , r1

βl +1

|zl |αl +1 +

Define f ≜ Ad sgn[α] + Bd sgn[β] . Taking the derivative of V along the solution of Sz (L) gives that

l=1

¯l

l=1

≥(

¯l ∑

(1) if η = 0, Sz is FT stable at the origin w.r.t. I (H T ); (2) if η > 0, Sz is FxT stable at the origin w.r.t. I (H T ).

r2

))

l=1

with r1 = (α0 + β0 )/(1 + β0 ) and r2 = 2β0 /(1 + β0 ).

= −ˆa2min λf (L)

¯l ∑

|zl |2αl

(14)

l=1

To obtain the differential inequality in the statement (1) of Lemma 1, we consider two cases for different possible values of

X. Shi, X. Yu, J. Cao et al. / Automatica 113 (2020) 108755

z. Let α0 = ∥α∥∞ . If ∥z ∥∞ ≤ 1, by applying Lemma 5 with ν = 1, µ = α20α+1 and ηi = |zl |2αl , one has that 0

¯l

∑

|zl |2αl ≥ (

l=1

¯l ∑

(|zl |2αl )

1+α0 2α0

2α0

) 1+α0 ≥ (

l=1

¯l ∑

2α0

Ax=b

l=1

¯l ∑

Consider the Lagrange function as L(x, λ) = ∥ˆx∗ − x∥2 +λT (Ax − b). Let the optimal primal–dual solution be (x∗ , λ∗ ). Then, by the optimal conditions, it is necessary that

2α0

|zl |αl +1 ) 1+α0 ≤ −ˆa2min λf (L)ρ −r V r (t)

2(xˆ ∗ − x∗ ) + AT λ∗ = 0,

l=1

with r = ¯l ∑

2α0 1+α0

According to Corollary 4, all the agents will reach consensus on one solution xˆ ∗ satisfying Axˆ ∗ = b + ∆ in FT/FxT. Then, it remains to solve that d2 (xˆ ∗ , X ∗ ) ≜ min ∥ˆx∗ − x∥2 .

|zl |αl +1 ) 1+α0 ,

which implies that V˙ ≤ −ˆa2min λf (L)(

5

aˆ

and ρ = maxl∈⟨¯l⟩ { α +l 1 }. If ∥z ∥∞ > 1, then l

|zl |2αl > 1 ≥ V r (z(t))/V r (z0 )

which gives that λ∗ √ ∆T (AAT )−1 ∆. □

2(AAT )−1 ∆ and d(xˆ ∗ , X ∗ )

=

=

4.2. Case 2: Distributed algorithm with free initialization

l=1

since V (t) is decreasing along the time. Thus, we have that V˙ ≤ −ˆa2min λf (L) min{ρ −r , V −r (z0 )}V r (t) ≜ −ϖ0 V r (t). Then, by the statement (1) of Lemma 1, the origin is FT stable V 1−r (z(0)) with settling time T (z0 ) ≤ ϖ (1−r) for any z0 ∈ I (H T ). 0 (2) If η > 0, Eq. (13) can be further relaxed as V˙ ≤ −λf (L)f T (z)f (z)

= −λf (L)

¯l ∑ (

aˆ 2l |zl |2αl + 2aˆ l bˆ l |zl |αl +βl + bˆ 2l |zl |2βl

≤ −γ λf (L)

¯l ∑ (

αl

|zl | + |zl |

ui = u¯ i − ATi (δ˜ i sgn[α˜ i ] (Ai xi − bi ) + η˜ i sgn[βi ] (Ai xi − bi )) ˜

f

for i ∈ ⟨n⟩, in which δ˜ i > 0, η˜ i ≥ 0 and 0 < α˜ i < 1 < β˜ i . The second term is to drive the state xi to the manifold Mi = {xi ∈ Rm : Ai xi = bi } in FT/FxT, as stated in the following result. f

Theorem 3. For the local dynamics x˙ i = ui starting for any xi (0) ∈ Rm , one has that

)

l=1 2

To eliminate the negative effects cased by the initialization errors, a novel protocol with free initialization is designed as below

(1) if η˜ i = 0, xi converges to Mi in finite time; (2) if η˜ i > 0, xi converges to Mi in fixed time.

) βl 2

l=1

Proof. Let yi = Ai xi − bi . Since Ai Pi = 0, it can be derived that

≤ −c1 V r1 − c2 V r2 ,

y˙ i = Ai x˙ i = −Li (δ˜ i sgn[α˜ i ] (yi ) + η˜ i sgn[βi ] (yi )). ˜

for which the last inequality is implied by Lemma 6, the parameters c1 , c2 are given as follows c1 =

1 2

γ λf (L)ρ˜ 2

−r1

, c2 =

1 2

γ λf (L)(2¯l) 2

1−β0 1+β0

ρ˜

−r2 aˆ l

bˆ l

with γ = min{ˆamin , bˆ min }, ρ˜ = maxl∈⟨¯l⟩ {min{ α +1 , β +1 }} and l l r1 , r2 , β0 are provided in Lemma 6. From Lemma 1, the origin is FxT stable with settling time T0 ≤ c (11−r ) + c (r1−1) for any z0 ∈ I (H T ). □

1

1

2 2

With Lemma 4 and Theorem 1, one can easily obtain the following result. Corollary 4. Let Assumption 1 hold. With the nonlinear protocol ui = u¯ i and the initial condition (8), the interactive system (5) converges to the same solution of Ax = b in finite (resp. fixed) time with η = 0 (resp. η > 0). With the initialization (8), the possible round-off errors may make system (7) fail to converge to the manifold M, as indicated in Lemma 2. It can be seen that the network dimension has less impact on the final solution, which is mainly affected by the initialization errors. Lemma 2. Suppose that there exist errors in the initialization (8), i.e., Ai xˆ i (0) = bi + ∆i , ∀i ∈ ⟨n⟩. With the FT/FxT protocol ui = u¯ i , the system (5) will converge to the same solution xˆ ∗ of Ax = b + ∆ and the deviation of xˆ ∗ from X ∗ is d(xˆ ∗ , X ∗ ) = which ∆ = [∆i ]i∈⟨n⟩ .

√ ∆T (AAT )−1 ∆, in

Proof. With protocol (6), it holds that Ai x˙ i (t) = 0, ∀i ∈ ⟨n⟩. Then, with the perturbed initialization, i.e., Ai xˆ i (0) = bi + ∆i , ∀i ∈ ⟨n⟩, one can always get that Ai xi (t) = Ai xˆ i (0) = bi + ∆i , ∀i ∈ ⟨n⟩.

(15)

Ai ATi .

with Li = Since Ai has full row rank, then Li is positive definite. Consider the following Lyapunov function Vi =

δ˜i η˜ i ˜ 1T |yi |α˜ i +1 + 1Tm |yi |βi +1 . α˜ i + 1 m β˜ i + 1

(16)

Let f˜i ≜ δ˜ i sgn[α˜ i ] (yi ) + η˜ i sgn[βi ] (yi ). Taking the derivative of Vi along (15) gives that ˜

V˙ i = −f˜iT Li f˜i ≤ −λ2 (Li )f˜iT f˜i .

(17)

(1) If η˜ i = 0, it can be derived from Eq. (17) that V˙ i ≤ −δ˜ i2 λ2 (Li )(sgn[α˜ i ] (yi ))T (sgn[α˜ i ] (yi )) m ∑ = −δ˜i2 λ2 (Li ) |yik |2α˜ i k=1 m

≤ −δ˜i2 λ2 (Li )(

∑

2α˜ i

|yik |α˜ i +1 ) α˜ i +1

k=1 r

≜ −ϖi Vi 1 2α˜

2−r

with ri = α˜ +i1 and ϖi = δ˜ i i (α˜ i + 1)ri λ2 (Li ). The last second i inequality holds according to Lemma 5 with µ = 1/ri and ν = 1. V 1−r (yi (0))

i By Lemma 1, yi (t) converges to zero in finite time T0 ≤ ϖ i (1−ri ) for any yi (0) = Ai xi (0), which indicates that xi (t) converges to Mi in finite time T0 . (2) If η˜ i > 0, from Eq. (13), one has that

V˙ i ≤ −λ2 (L)f˜iT (yi )f˜i (yi )

= −λ2 (L)

m ∑ ( 2 ˜ ˜ ) δ˜i |yik |2α˜ i + 2δ˜i η˜ i |yik |α˜ i +βi + β˜ i2 |yik |2βi k=1

6

X. Shi, X. Yu, J. Cao et al. / Automatica 113 (2020) 108755

≤ −γi2 λ2 (L)

m ∑ (

|yik |α˜ i + |yik |βi ˜

Since N (PA ) = I (AT ), (19) is equivalent to PA x = PA q¯ and Ax∗ = b. □

)2

k=1 r˜i,1 ci,1 Vi

≤ −˜

In order to solve (18) in a distributed way, a new initial condition is imposed as below

r˜

− c˜i,2 Vi i,2 ,

for which the last inequality is implied by Corollary 3 with α0 = α˜ i and β0 = β˜ i , the parameters c˜1 , c˜2 are given below c˜1 =

1 2

−˜ri,1

γi2 λ2 (L)ρ˜ i

, c˜2 =

1 2

1−β˜ i

−˜ri,2

γi2 λ2 (L)(2m) 1+β˜ i ρ˜ i δ˜

with γi = min{δ˜ i , η˜ i }, ρ˜ i = min{ α˜ +i 1 ,

η˜ i } β˜ i +1

and r˜i,1 = (α˜ i + ˜βi )/(1 + β˜ i ), r˜i,2 = 2β˜ i /(1 + β˜ i ). From Lemma 1, yi (t) converges to zero in fixed settling time T0 ≤ c˜ (11−˜r ) + c˜ (r˜1 −1) for any i,1 i,1 i,2 i,2 yi (0) = Ai xi (0), which indicates that xi (t) converges to Mi in fixed i

time T0 .

Ai xi (0) = bi , Pi xi (0) = Pi qi .

Similar to Case 1, the initialization (8) is included in (20) to ensure that the state xi always remains on the manifold Mi and the final consensus state satisfies Ax = b. Since N (Ai ) = I (Pi ), there exists a unique solution x∗qi of (20). Similar to the proof of Lemma 7, it can be shown that the x∗qi is the unique solution of the following local problem min x

□

Combining with Theorem 3, it can be shown that the equation f Ax = b can be solved by x˙ i = ui (x), i ∈ ⟨n⟩ in FT/FxT with free initialization, as summarized in the following main result. Corollary 5. Let Assumption 1 hold. With the nonlinear protocol f ui = ui , the interactive system (5) converges to the same solution of Ax = b in finite (resp. fixed) time when η = η˜ i = 0 (resp. η, η˜ i > 0) for any xi (0) ∈ Rm , i ∈ ⟨n⟩. Proof. When η = η˜ i = 0, from Theorem 3, the system state x f converges to M in finite time. Then, ui = u¯ i and the dynamics f x˙ i = ui reduces to x˙ i = u¯ i . After that, by Corollary 4, Ax = b will be solved in finite time. Similarly, one can show that Ax = b will be solved in fixed time by applying Theorem 3 and Corollary 4 when η, η˜ i > 0. □

1 2

∥x − qi ∥2 , s.t ., Ai x = bi .

Theorem 4. Let Assumption 1 hold. With the nonlinear protocol ui = u¯ i and the initial condition (20), the interactive system (5) converges to x∗q¯ in finite (resp. fixed) time with η = 0 (resp. η > 0). Proof. By Corollary 4, all xi converge to the same solution x∗ ∈ X ∗ in finite (fixed) time with η = 0 (η > 0). With Lemma 7, it is sufficient to show that PA x∗ = PA q¯ . According to the proof of Theorem∑2 in Wang et al. (2017), it holds that PA Pi = PA , i ∈ ⟨n⟩. n Let x¯ = i=1 xi /n. Then, one can derive that PA x˙¯ =

n 1∑

n

=−

u¯ i = u¯ i − ATi (Ai ATi )−1 (δ˜ i sgn[α˜ i ] (Ai xi − bi )

=0

+ η˜ i sgn[βi ] (Ai xi − bi )) ˜

Similarly, one can show that Theorem 3 and Corollary 5 hold by f f replacing ui with u¯ i . 4.3. Case 3: Achieve the solution closest to specified points Since the solutions of the linear equation Ax = b can be multiple, the property of the final solution obtained by (7) with the initial condition (8) is not clear, which depends on the initial states chosen to satisfy (8). In this part, the initial condition (8) will be modified to solve the following problem min x

n 1 ∑

2n

∥x − qi ∥ , s.t ., Ax = b, 2

(18)

i=1

in which qi ∈ Rm is the given point preferred by the ith agent. Specially, when the preferred specific points are zero, it reduces to obtain a solution with∑the least Euclidean norm. Denote the n averaged point as q¯ = 1n i=1 qi and the unique solution of (18) ∗ as xq¯ , for which one can obtain the following result. Lemma 7. x = x∗ is the unique solution of (18) if and only if it satisfies that Ax = b and PA x = PA q¯ with PA = I − AT (AAT )−1 A being the orthogonal projection matrix on N (A).

(21)

With the help of Lemma 7, the main result of this section is shown as follows.

Remark 2. Another FT/FxT solver for linear equation Ax = b with free initialization can be designed as f

(20)

PA u¯ i (xN i )

i=1

PA ∑ n

aij (δ sgn[αij ] (xi − xj ) − ηsgn[βij ] (xi − xj ))

(i,j)∈E

since G is undirected and αij = αji , βij = βji . With the initial condition (20), it can be seen that PA x¯ ≡ PA q¯ , which indicates that PA x∗ ≡ PA q¯ , i.e., x∗ = x∗q¯ . □ f

By replacing u¯ i with ui , the initial condition (8) can be dropped by applying Corollary 5. Corollary 6. Let Assumption 1 hold and the initial states satisfy f Pi xi (0) = Pi qi , i ∈ ⟨n⟩. With the nonlinear protocol ui = ui , all state ∗ xi will converge xq¯ in finite (resp. fixed) time when η = η˜ i = 0 (resp. η, η˜ i > 0). Proof. By Corollary 5, all xi converges to the same solution x∗ ∈ X ∗ in finite (resp. fixed) time with η = η˜ i = 0 (resp. η, η˜ i > 0). ∗ With Lemma 7, it is sufficient to show that ∑n PA x = PA q¯ . Since T Pi Ai = 0 and PA Pi = PA , i ∈ ⟨n⟩, with x¯ = i=1 xi /n, one has that PA x˙¯ =

n 1∑

n

f

PA ui (xN i ) =

i=1

n 1∑

n

PA u¯ i (xN i ) = 0

i=1

according to the proof of Theorem 4. With the initial condition Pi xi (0) = Pi q¯ , i ∈ ⟨n⟩, it holds that PA x¯ ≡ PA q¯ , which implies that PA x∗ ≡ PA q¯ , i.e., x∗ = x∗q¯ . □ 5. Numerical examples

∗

Proof. By the Karush–Kuhn–Tucker (KKT) conditions, x is the unique solution of (18) if and only if there exists a Lagrange multiplier λ such that x − q¯ = AT λ, Ax∗ = b.

(19)

In order to testify the effectiveness of the proposed FT/FxT nonlinear protocols for solving linear equations in a distributed way, two numerical examples are provided in this section over a connected communication network composed of five agents,

X. Shi, X. Yu, J. Cao et al. / Automatica 113 (2020) 108755

7

Table 2 Performance comparison (V (t)) among three protocols. Case studies (t)

LP

FTP

FxTP

Case 1 (100 s) Case 2 (100 s) Case 3 (30 s)

1.41 2.10 6.05 × 10−4

2.78 × 10−7 1.75 × 10−7 6.67 × 10−7

1.73 × 10−7 1.10 × 10−7 5.94 × 10−7

Fig. 1. The communication network in the simulation. f

f

as shown in Fig. 1, for which the edge weights are all ones. For both examples, the heterogeneous coefficients of the protocol u¯ i f or ui are chosen as αij = αji = 0.1 ∗ min{i, j}, βij = βji = f 1 + 0.1 ∗ min{i, j}. For both protocol u¯ i and ui , we set δ = δ˜ i = 20. f ˜ For protocol ui , α˜ i = 0.1 ∗ i and βi = 1 + 0.1 ∗ i. In the simulations, to achieve higher accuracy, all the continuous-time protocols are implemented by an embedded explicit Runge–Kutta methods of orders 2 and 3 (Bogacki & Shampine, 1989), like ODE solver ode23 in MATLAB. As an alternative, one can also use the Euler’s integration method with small step-size dt = 0.0001 s as in Gómez-Gutiérrez et al. (2018).

protocol (ui with η˜ i = η = 0) and FxT nonlinear protocol (ui with η˜ i = η = δ ). Then, the values of V (t) with three protocols are illustrated in Fig. 2(b). The performance comparisons among three algorithms for both case studies at the end of simulation (t = 100 s) are provided in Table 2, which indicates that the FT/FxT protocol can achieve the solution with higher accuracy in a faster way compared with the linear one. From Fig. 2, one can see that there are slight differences between FxT and FT algorithms since the initial conditions are chosen close to the optimal solution. In the case that the initial conditions are unknown and unbounded, the FxT protocol has a great advantage, because it guarantees the existence of a bound for the convergence time (Gómez-Gutiérrez et al., 2018).

5.1. Example 1

5.2. Example 2

The equation matrix A and vector b are given below

⎡

7

⎢4 ⎢ ⎢10 ⎢ A=⎢1 ⎢5 ⎢ ⎣5 5

8 4 8 5 1 2 8

5 2 4 7 2 8 3

10 3 8 2 3 1 6

7 6 5 7 7 7 7

10 3 8 3 2 7 5

⎤

⎡

The third case study is performed to verify Theorem 4. It aims to achieve the solution of Ax = b closest to a common specific point q = [5 4 1 3 4 5]T , i.e., qi = q, ∀i ∈ ⟨n⟩. The matrix A and vector b are provided as follows

⎤

25 5 ⎢8⎥ 7⎥ ⎢ ⎥ ⎥ 8⎥ ⎢17⎥ ⎢ ⎥ ⎥ 4⎥ , b = ⎢18⎥ . ⎢17⎥ ⎥ 7⎥ ⎢ ⎥ ⎣27⎦ 5⎦ 9 8

(22)

For the linear equation Ax = b, there exists a unique solution x∗ = [1.12 − 2.03 2.59 1.86 1.43 − 0.34 − 0.96]T . In the follows, two case studies will be conducted with and without considering the initial condition (8), respectively, in order to validate Corollaries 4 and 5. For the first case study, the local states xi are initialized to satisfy the condition Ai xi (0) = bi . Three continuous-time protocols are performed, i.e., linear protocol (Anderson et al., 2016), FT nonlinear protocol (u¯ i with η = 0) and FxT nonlinear protocol (u¯ i with η = δ ), abbreviated as LP, FTP and FxTP, respectively. Note that the linear protocol proposed in Anderson et al. (2016) is corresponding u¯ i with αij = 1, (i, j) ∈ E and η = 0. ∑to n ∗ 2 Define V (t) = i=1 ∥xi (t) − x ∥ as the system error function. The values of V (t) with three protocols are presented in Fig. 2(a), from which one can see that the FxT protocol exhibits the fastest convergence rate. For the linear protocol, all states xi converge to x∗ asymptotically. For the second case study, all the local states xi are freely initialized, e.g., xi (0) = 0. Three continuoustime protocols are performed, i.e., linear protocol, FT nonlinear

2 ⎢6 ⎢ A = ⎢1 ⎣1 3

⎡

1 8 7 4 2

4 4 6 4 8

8 8 3 3 8

2 2 5 8 4

10 2 1⎥ ⎢12⎥ ⎢ ⎥ ⎥ 8⎥ , b = ⎢ 6 ⎥ . ⎣11⎦ ⎦ 5 4 4

⎤

⎡

⎤

(23)

By solving (18), the optimal solution is x∗q = [2.19 − 0.62 − ∑n ∗ 2 3.64 1.73 1.06 2.44]. Let V (t) = i=1 ∥xi (t) − xq ∥ be the system error function. Suppose that the initial states are chosen to satisfy the condition (20). For the purpose of comparison, three protocols are performed, i.e., linear protocol, FT nonlinear protocol (u¯ i with η = 0) and FxT nonlinear protocol (u¯ i with η = δ ). Then, the values of V (t) are plotted in Fig. 2(c). By the simulation results, V (t) with FT/FxT protocol converges to zero with error less than 10−4 in 3 s, which takes the linear protocol more than 30 s. The performance comparisons among three algorithms at the end of simulation (t = 30 s) are given in Table 2. 6. Conclusion In this paper, three FT/FxT distributed algorithms are developed over a multi-agent network for linear equations Ax = b when A has full row rank. With specific initializations, a nonlinear

Fig. 2. The values of V (t) by three protocols: (a) Case 1 with initial condition (8); (b) Case 2 with free initialization; (c) Case 3 with modified initialization (20).

8

X. Shi, X. Yu, J. Cao et al. / Automatica 113 (2020) 108755

distributed protocol is provided by combining the projectionconsensus flow (Shi et al., 2017) and a general version of the existing FT/FxT consensus algorithms (Shi et al., 2019; Wang & Xiao, 2010), which can obtain a solution in FT/FxT time over an undirected and fixed network. To eliminate the initial conditions, a nonlinear term is added to drive the local states to satisfy the specific initialization. When multiple solutions emerge, a modified initialization is given for the predesigned distributed algorithms to obtain one solution closest to specific points in FT/FxT. For the purpose of practical applications, the command signals deigned from the continuous-time protocols can be converted to digital forms by using sampling technique. Moreover, the delay and asynchronous communication may occur in the network control systems. Future work will concern the design and stability analysis of robust protocols under the uncertain environment in presence of signal delay and noises as well as the asynchronous implementation of its discrete version for solving equation problems. References Anderson, B., Mou, S., Morse, A. S., & Helmke, U. (2016). Decentralized gradient algorithm for solution of a linear equation. Numerical Algebra, Control & Optimization, 6(3), 319–328. Bhat, S. P., & Bernstein, D. S. (2000). Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization, 38(3), 751–766. Bogacki, P., & Shampine, L. (1989). A 3(2) pair of runge–kutta formulas. Applied Mathematics Letters, 2(4), 321–325. Chen, G., & Li, Z. (2018). A fixed-time convergent algorithm for distributed convex optimization in multi-agent systems. Automatica, 95, 539–543. Chen, W., & Ren, W. (2016). Event-triggered zero-gradient-sum distributed consensus optimization over directed networks. Automatica, 65, 90–97. Cortés, J. (2006). Finite-time convergent gradient flows with applications to network consensus. Automatica, 42(11), 1993–2000. Gharesifard, B., & Cortés, J. (2014). Distributed continuous-time convex optimization on weight-balanced digraphs. IEEE Transactions on Automatic Control, 59(3), 781–786. Goldberg, M. (1987). Equivalence constants for lp norms of matrices. Linear and Multilinear Algebra, 21(2), 173–179. ` S., Sánchez-Torres, J. D., & Gómez-Gutiérrez, D., Vázquez, C. R., Čelikovsky, Ruiz-León, J. (2018). On finite-time and fixed-time consensus algorithms for dynamic networks switching among disconnected digraphs. International Journal of Control, http://dx.doi.org/10.1080/00207179.2018.1543896, in press. Hong, H., Yu, W., Wen, G., & Yu, X. (2017). Distributed robust fixed-time consensus for nonlinear and disturbed multiagent systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47(7), 1464–1473. Jakovetić, D., Xavier, J., & Moura, J. M. (2014). Fast distributed gradient methods. IEEE Transactions on Automatic Control, 59(5), 1131–1146. Kia, S. S., Cortés, J., & Martínez, S. (2015). Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication. Automatica, 55, 254–264. Lin, P., Ren, W., & Farrell, J. A. (2017). Distributed continuous-time optimization: nonuniform gradient gains, finite-time convergence, and convex constraint set. IEEE Transactions on Automatic Control, 62(5), 2239–2253. Lin, P., Ren, W., & Song, Y. (2016). Distributed multi-agent optimization subject to nonidentical constraints and communication delays. Automatica, 65, 120–131. Liu, J., Chen, X., Başar, T., & Nedić, A. (2016). A continuous-time distributed algorithm for solving linear equations. In Proceedings of the american control conference, pp. 5551–5556. Liu, Y., Lou, Y., Anderson, B., & Shi, G. (2018). Network flows that solve least squares for linear equations. arXiv preprint arXiv:1808.04140. Liu, J., Morse, A., Nedić, A., & Başar, T. (2017). Exponential convergence of a distributed algorithm for solving linear algebraic equations. Automatica, 83, 37–46. Liu, J., Mou, S., & Morse, A. S. (2018). Asynchronous distributed algorithms for solving linear algebraic equations. IEEE Transactions on Automatic Control, 63(2), 372–385. Mou, S., Liu, J., & Morse, A. (2015). A distributed algorithm for solving a linear algebraic equation. IEEE Transactions on Automatic Control, 60(11), 2863–2878. Nedić, A., Ozdaglar, A., & Parrilo, P. A. (2010). Constrained consensus and optimization in multi-agent networks. IEEE Transactions on Automatic Control, 55(4), 922–938. http://dx.doi.org/10.1109/TAC.2010.2041686.

Olfati-Saber, R., & Murray, R. M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), 1520–1533. Perruquetti, W., Floquet, T., & Moulay, E. (2008). Finite-time observers: application to secure communication. IEEE Transactions on Automatic Control, 53(1), 356–360. Polyakov, A. (2012). Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Transactions on Automatic Control, 57(8), 2106–2110. Shi, G., Anderson, B. D., & Helmke, U. (2017). Network flows that solve linear equations. IEEE Transactions on Automatic Control, 62(6), 2659–2674. Shi, X., Cao, J., & Huang, W. (2018). Distributed parametric consensus optimization with an application to model predictive consensus problem. IEEE Transactions on Cybernetics, 48(7), 2024–2035. Shi, X., Cao, J., Wen, G., & Yu, X. (2019). Finite-time stability for network systems with nonlinear protocols over signed digraphs. IEEE Transactions on Network Science and Engineering, http://dx.doi.org/10.1109/TNSE.2019.2941553, in press. Shi, G., Johansson, K. H., & Hong, Y. (2013). Reaching an optimal consensus: Dynamical systems that compute intersections of convex sets. IEEE Transactions on Automatic Control, 58(3), 610–622. Song, Y., & Chen, W. (2016). Finite-time convergent distributed consensus optimisation over networks. IET Control Theory & Applications, 10(11), 1314–1318. Wang, X., Mou, S., & Sun, D. (2017). Improvement of a distributed algorithm for solving linear equations. IEEE Transactions on Industrial Electronics, 64(4), 3113–3117. Wang, P., Ren, W., & Duan, Z. (2019). Distributed algorithm to solve a system of linear equations with unique or multiple solutions from arbitrary initializations. IEEE Transactions on Control of Network Systems, 6(1), 82–93. Wang, L., & Xiao, F. (2010). Finite-time consensus problems for networks of dynamic agents. IEEE Transactions on Automatic Control, 55(4), 950–955. Wang, H., Yu, W., Wen, G., & Chen, G. (2018). Finite-time bipartite consensus for multi-agent systems on directed signed networks. IEEE Transactions on Circuits and Systems. I. Regular Papers, 65(12), 4336–4348. Yang, T., George, J., Qin, J., Yi, X., & Wu, J. (2018). Distributed finite-time least squares solver for network linear equations. arXiv preprint arXiv:1810.00156. Yuan, Y., Stan, G.-B., Shi, L., Barahona, M., & Goncalves, J. (2013). Decentralised minimum-time consensus. Automatica, 49(5), 1227–1235. Zhou, J., Wang, X., Mou, S., & Anderson, B. D. (2019). Finite-time distributed linear equation solver for solutions with minimum l1 norm. IEEE Transactions on Automatic Control, http://dx.doi.org/10.1109/TAC.2019.2932031, in press. Zuo, Z., Han, Q.-L., Ning, B., Ge, X., & Zhang, X.-M. (2018). An overview of recent advances in fixed-time cooperative control of multiagent systems. IEEE Transactions on Industrial Informatics, 14(6), 2322–2334.

Xinli Shi received the B.S. degree in software engineering, the M.S. degree in applied mathematics and the Ph.D. degree in control science and engineering from Southeast University, Nanjing, China, in 2013, 2016 and 2019, respectively. From 2018 to 2019, he was a visiting Ph.D. student at RMIT University, Melbourne, Australia. He is currently a lecture with the School of Cyber Science and Engineering, Southeast University, Nanjing, China. His current research interests include model predictive control, distributed optimization, and network control systems.

Xinghuo Yu received the B.Eng and M.Eng degrees in electrical engineering from the University of Science and Technology of China, Hefei, China, in 1982 and 1984, and the Ph.D. degree from Southeast University, Nanjing, China, in 1988, respectively. He is Associate Deputy Vice-Chancellor and Distinguished Professor of RMIT University (Royal Melbourne Institute of Technology), Melbourne, Victoria, Australia. His current research interests include variable structure and nonlinear control, complex and intelligent systems, and smart energy systems. He was a recipient of the 2018 M. A. Sargent Medal of Engineers Australia, the 2013 Dr.-Ing. Eugene Mittelmann Achievement Award of the IEEE Industrial Electronics Society, and a number of awards and honors for his contributions. He is the President (2018-2019) of the IEEE Industrial Electronics Society.

X. Shi, X. Yu, J. Cao et al. / Automatica 113 (2020) 108755 Jinde Cao received the B.S. degree from Anhui Normal University, Wuhu, China, the M.S. degree from Yunnan University, Kunming, China, and the Ph.D. degree from Sichuan University, Chengdu, China, all in mathematics/applied mathematics, in 1986, 1989, and 1998, respectively. He is an Endowed Chair Professor, the Dean of the School of Mathematics, the Director of the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence of China at Southeast University. He was a recipient of the National Innovation Award of China, Obada Prize and the Highly Cited Researcher Award in Engineering, Computer Science, and Mathematics by Thomson Reuters/Clarivate Analytics. He is elected as a member of the Academy of Europe, a member of the European Academy of Sciences and Arts, a fellow of Pakistan Academy of Sciences, and an IASCYS academician.

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Guanghui Wen received the Ph.D. degree in mechanical systems and control from Peking University, China, in 2012. He is a Professor with the School of Mathematics, Southeast University, Nanjing, China. His current research interests include cooperative control of multi-agent systems, analysis and synthesis of complex networks, cyber–physical systems, and resilient control. He was the recipient of the Best Student Paper Award in the 6th Chinese Conference on Complex Networks in 2010, the Second Prize of Natural Science Award from Ministry of Education of China in 2016, and the First Prize of Scientific and Technological Progress Award of Chinese Institute of Command and Control in 2016. He was awarded a National Natural Science Fund for Excellent Young Scholars in 2017.