Consensus of high-order discrete-time linear networked multi-agent systems with switching topology and time delays

Abstract

In this paper, the consensus problem for high-order discrete-time networked multi-agent systems (D-NMAS) is investigated by distributed feedback protocols. By constructing the self-feedback matrix and the neighbouring feedback matrix for networks, consensus protocols are designed under three different cases and the corresponding convergence analysis is provided. Consensus convergence results of networks are provided by three final consensus values, which are related to self-feedback matrices, initial states of networks and the topology of networks, not related to time delays. In the first case where a directed network with a fixed topology is concerned, the high-order discrete-time consensus problem is studied as an example, and a sufficient and necessary condition is obtained. In the scenario with directed networks with switching topology, a sufficient condition guaranteeing the consensus of high-order D-NMAS is derived, after the consensus analysis is transformed into stability analysis. As for directed networks with switching topology and time delays, the discrete-time stability model with time delays is converted into a general discrete-time stability model by an augmented method and the sufficient condition is provided to achieve consensus for directed networks. Furthermore, the sufficient conditions determining the neighbouring feedback matrix are independent of the number of agents. Two numerical examples are provided to demonstrate the correctness and effectiveness of the theoretical results.

Keywords

Consensus discrete-time multi-agent systems switching topology time delays

Introduction

Recently, the problem of consensus for networked multi-agent systems (NMAS) has received great attention, and has been widely applied in cooperative control, such as formation control (Cao et al., 2011; Sinha and Ghose, 2006), flocking (Olfati-Saber, 2006; Zhang et al., 2011), distributed filtering (Açıkmeşe et al., 2014; Olfati-Saber, 2005), synchronization of coupled chaotic oscillators (Barahona and Pecora, 2002; Hengster-Movric et al., 2013; Wang et al., 2014) and so on. The main work of the consensus problem is to make each agent agree on some common values of interest by designing protocols based on the information of its neighbours.

The theoretical framework for posing and solving the consensus problem for NMAS was first introduced in Olfati-Saber and Murray (2003, 2004) and Olfati-Saber et al. (2007). Their work mainly focused on the first-order consensus. Then, the consensus problems in continuous-time models received a tremendous surge of interest and extensive development. Ren and Beard (2005) showed that the union of interaction topologies must contain a spanning tree if the NMAS is expected to achieve consensus asymptotically. Furthermore, time delays and switching topology of this work were presented in Olfati-Saber and Murray (2004), Sun and Wang (2009), Sun et al. (2008), and Tian and Liu (2008). In addition, a sufficient condition and some necessary and sufficient conditions for second-order consensus in NMAS were given under different protocols in Ren (2006) and Yu et al. (2010), respectively. The second-order consensus problems of NMAS with time delays and switching topology were heavily discussed in Hu and Lin (2010), Lin and Jia (2010), and Zhu and Cheng (2010). A rotating consensus algorithm for second-order dynamics in three dimensions was proposed in Wang and Sun (2015). Along with this direction, a framework of high-dimensional state space for the consensus problems of NMAS was studied in Xiao and Wang (2007). Consensus problems for high-order linear time-invariant swarm systems with time delays were investigated in Xi et al. (2010, 2011, 2012).

Unfortunately, the analysis of the consensus stability of the discrete-time NMAS (D-NMAS) is more difficult than that of the continuous-time NMAS, because the D-NMAS should be Schur-stable, which leads to the solution of the feedback gain matrix being a non-linear process. However, the study of discrete-time consensus problem is widely applied in practice (Açıkmeşe et al., 2014; Xu et al., 2011; Zhang and Leonard, 2010). It is relatively easy to analyse the first-order discrete time consensus problems for D-NMAS, which was investigated by Cao et al. (2009), Olfati-Saber and Murray (2004), Tian and Liu (2008), and Xiao and Wang (2006). Two sampled-data-based discrete-time second-order consensus problems of D-NMAS under switching directed interaction were studied in Cao and Ren (2010). Sufficient conditions were derived for the consensus of second-order D-NMAS with no uniform time delays and dynamically changing topologies in Lin and Jia (2009). Based on the properties of stochastic matrix, the first-order discrete-time consensus with time-varying delays and stationary consensus of heterogeneous with first order and second order were discussed in Xiao and Wang (2008), and Liu and Liu (2011). In recent literature, the problems of consensus ability and communication data of high-order D-NMAS were investigated in You and Xie (2011) and Gu et al. (2011). The leader-following consensus problem and the distributed consensus problem for high-order D-NMAS under switching network topology were studied in Su and Huang (2012). A distributed protocol was proposed to compensate actively for the network delay for consensus problems of high-order D-NMAS in Tan and Liu (2013). The problem of distributed output feedback consensus for D-NMAS both fixed topology and stochastic switching topology were investigated in Zhao et al. (2014). To our knowledge, although the high-order D-NMAS with time delay and switching topology were studied in a few reports, the final convergence values of D-NMAS were not provided. However, the convergence values and the corresponding computing procedure are of great significance in reality. In addition, in the very few studies involving the high-order multi-agent systems with time delay and switching topology, the issue of time delay and the influence of switching topology are generally handled separately, which may be not consistent with industrial fact.

Based on the aforementioned discussion, we are motivated to study the problem of consensus analysis for the high-order D-NMAS systematically. Generally, these cases are considered from simple to more complex. To begin with, we study the high-order consensus problem of D-NMAS with fixed topology and without time delays. A sufficient and necessary condition and a final consensus value are given for it. In addition, the consensus high-order problem of D-NMAS is discussed with switching topology and without time delays. Consensus analysis problems are transformed into stability analysis problems for this via a model transformation method. Furthermore, we study the high-order consensus problem of D-NMAS with switching topology and time delays. The discrete-time stability model with time delay is converted into a generally discrete-time stability model using an augmented method. The contribution of this paper is three-fold. Firstly, we give a final consensus value for the D-NMAS in all three cases, which were not given in the existing literature, such as Tan and Liu (2013), You and Xie (2011), and Zhao et al. (2014). Secondly, we derive a sufficient linear matrix inequality (LMI) condition and for consensus problems of high-order D-NMAS with directed switching topology and time delays. To the best of the authors’ knowledge, there is still no literature regarding directed switching topology and time delays of high-order D-NMAS at the same time. Finally, the LMI conditions of D-NMAS are independent of the number of agents, which is different from the existing work about D-NMAS.

The rest of the paper is organized as follows. The consensus problem of high-order D-NMAS is analysed in the next section. Then, based on the model transformation method, the consensus analysis of high-order D-NMAS with switching topology are transformed into stability analysis, and a sufficient condition with independence of the number of agents on the consensus of proposed protocol is provided by a set of dimension-reduction LMIs. We analyse the consensus problem of high-order D-NMAS with switching topology and time delays through a method of extending dimension, and provide two numerical examples to verify the theoretical analysis. Some conclusions are finally drawn and we propose some possible future directions. The notions of graph theory and Kronecker product that will be used in this paper are summarized in Appendices A and B, respectively.

Consensus analysis for high-order D-NMAS

A high-order NMAS can be described as a linear system (Xiao and Wang, 2007), and thus, consider a high-order D-NMAS consisting of N agents indexed by $1, 2, \dots, N$ . The dynamics of agent i are described by a linear discrete-time system

x_{i} (k + 1) = A x_{i} (k) + B u_{i} (k)

(1)

where, $u_{i} (k) \in R^{m}$ and $x_{i} (k) \in R^{d}$ are the consensus protocols and states of agent i, respectively; $A \in R^{d \times d}$ and $B \in R^{d \times m}$ are known constant matrices. The following consensus protocol is applied for agent i

u_{i} (k) = K_{1} x_{i} (k) + K_{2} \sum_{j \in N_{i}} a_{ij} (x_{j} (k) - x_{i} (k))

(2)

where $K_{1}$ and $K_{2}$ are constant gain matrices with appropriate dimensions.

The self-feedback matrix $K_{1}$ , which is defined beforehand by pole assignment method, is designed for assigning the final consensus value $c (k)$ by changing the dynamics of each agent. The neighbouring feedback matrix $K_{2}$ is used to act on the interaction between any two neighbouring agents. The definition of consensus for the high-order D-NMAS (1) with consensus protocol (2) is given as follows.

Definition 1. For given gain matrices $K_{1}$ and $K_{2}$ , system (1) is said to achieve consensus if for any given bounded initial condition, there exists a vector-valued $c (k)$ , which is dependent on the initial condition such that $li m_{k \to \infty} (x (k) - 1_{N} \otimes c (k)) = 0$ , where $c (k)$ is called a final consensus value.

Let $x (k) = [x_{1}^{T} (k), x_{2}^{T} (k), \dots, x_{N}^{T} (k)]^{T}$ , and the dynamics of the high-order D-NMAS (1) with the protocol (2) can be described by the following closed-loop discrete-time networked dynamics

x (k + 1) = (I_{N} \otimes (A + B K_{1}) - L \otimes B K_{2}) x (k) .

(3)

In order to analyse the consensus problem of the closed-loop D-NMAS (3), we provide the following lemma about graph theory.

Lemma 1 (Ren and Beard, 2005). Let L be the Laplacian matrix of a digraph G. Zero is a simple eigenvalue of L, and all the other eigenvalues of L have positive real parts if and only if G has a spanning tree.

Let $λ_{i} (i = 1, 2, \dots, N)$ be eigenvalues of the Laplacian matrix $L \in R^{N \times N}$ for a directed topology G, where $λ_{1} = 0$ with the associated eigenvector ${\bar{u}}_{1} = 1_{N}$ , and $Re (λ_{1}) \leq Re (λ_{2}) \leq \dots \leq Re (λ_{N})$ . Denote by $U = [{\bar{u}}_{1}, {\bar{u}}_{2}, \dots, {\bar{u}}_{N}] \in C^{N \times N}$ a non-singular matrix satisfying $U^{- 1} LU = J_{L}$ , where $J_{L}$ is the Jordan canonical form of L.

Let $H = I_{N} \otimes (A + B K_{1}) - L \otimes B K_{2}$ , and suppose that $p_{j} \in C^{Nd} (j = 1, 2, \dots, Nd)$ are a group of linear independent eigenvectors or generalized eigenvectors of $H .$ $p_{j} (j = 1, 2, \dots, d)$ and $p_{j} (j = d + 1, d + 2, \dots, Nd)$ are eigenvectors and generalized eigenvectors of H corresponding to eigenvalues of $A + B K_{1}$ and $A + B K_{1} - λ_{i} B K_{2} (i = 1, 2, \dots, N),$ respectively. The structure of $p_{j} (j = 1, 2, \dots, d)$ is given by the following lemma.

Lemma 2 (Xi et al., 2010). Let $c_{1}, c_{2}, \dots, c_{d} \in C^{d}$ be eigenvectors or generalized eigenvectors of H corresponding to eigenvalues of $A + B K_{1}$ , then $p_{j} = 1_{N} \otimes c_{j} (j = 1, 2, \dots, d)$ are eigenvectors or generalized eigenvectors of H.

Definition 2. Let $p_{j} = u_{i} \otimes c_{l} (j = (i - 1) d + l; i = 1, 2, \dots, N; l = 1, 2, \dots, d)$ . A consensus subspace (CS) $C (H)$ is spanned by $p_{l} = {\bar{u}}_{1} \otimes c_{l} = 1_{d} \otimes c_{l} (l = 1, 2, \dots, d)$ , and a complement consensus subspace (CCS) is defined as the subspace $\bar{C} (H)$ spanned by $p_{d + 1}, p_{d + 2}, \dots, p_{Nd}$ .

Notice that any vector in $C (H)$ has a form $1_{d} \otimes c_{l} (l = 1, 2, \dots, d)$ , this is the reason of $C (H)$ called a consensus subspace. Meanwhile, it is not difficult to find that $p_{l} (l = 1, 2, \dots, d)$ are independent, so one can obtain the following lemma.

Lemma 3 (Xi et al., 2010). $C (H) \oplus \bar{C} (H) = C^{Nd}$ , and $C (H)$ and $\bar{C} (H)$ are invariant subspaces of H.

As stability analysis of a discrete-time system usually uses Schur-stable theory, which needs applying to the concept of spectral radius for matrices, then Definition 3 is given as follows.

Definition 3. For any square matrix M, the spectral radius $ρ (M) = max {| eig (M) |}$ is defined as the maximal magnitude of the eigenvalues of M.

Theorem 1. Assume that the interaction topology Ghas a spanning tree. Then system (3) can achieve consensus if and only if $ρ (A + B K_{1} - λ_{i} B K_{2}) < 1 (i = 2, 3, \dots, N)$ .

Proof. Let $\tilde{x} (k) = (U^{- 1} \otimes I_{d}) x (k) = [{\tilde{x}}_{1}^{T} (k), {\tilde{x}}_{2}^{T} (k), \dots, {\tilde{x}}_{N}^{T} (k)]^{T}$ . Then the closed-loop discrete-time system (3) is equivalent to

\tilde{x} (k + 1) = (I_{N} \otimes (A + B K_{1}) - J_{L} \otimes B K_{2}) \tilde{x} (k),

(4)

where $J_{L}$ is the Jordan canonical form of L.

By Lemma 3, one has $x (k) = x_{C} (k) + x_{\bar{C}} (k)$ . For an arbitrary initial state $x (0)$ , there exists $α_{j} (0) (j = d + 1, d + 2, \dots, Nd)$ satisfying $x (0) = x_{C} (0) + x_{\bar{C}} (0),$ where $x_{C} (0) = \sum_{j = 1}^{d} α_{j} (0) p_{j}$ , and $x_{\bar{C}} (0) = \sum_{j = d + 1}^{Nd} α_{j} (0) p_{j}$ . The response of system (3) due to $x (0)$ is

\begin{matrix} x_{\bar{C}} (k) = {PJ}_{H}^{k} P^{- 1} x_{\bar{C}} (0) = P [\begin{matrix} J_{C}^{k} & 0 \\ 0 & J_{\bar{C}}^{k} \end{matrix}] \\ {[0, \dots, 0, α_{d + 1}^{H} (0), \dots, α_{Nd}^{H} (0)]}^{H} . \end{matrix}

(5)

where $J_{C} = (A + B K_{1})$ , and $J_{\bar{C}} = I_{N - 1} \otimes (A + B K_{1}) - J_{\bar{L}} \otimes B K_{2}$ are upper triangular matrices.

Suppose that $J_{\bar{C}}$ has s Jordan blocks. It follows that

J_{\bar{C}} = [\begin{matrix} J_{1} (λ_{\bar{C}, 1}) \\ J_{2} (λ_{\bar{C}, 2}) \\ ⋱ \\ J_{s} (λ_{\bar{C}, s}) \end{matrix}],

where

\begin{matrix} J_{i} (λ_{\bar{C}, i}) = [\begin{matrix} λ_{\bar{C}, i} & 1 \\ λ_{\bar{C}, i} & 1 \\ λ_{\bar{C}, i} & ⋱ \\ ⋱ & 1 \\ λ_{\bar{C}, i} \end{matrix}] & (i = 1, 2, \dots, s) \end{matrix} .

One has

\begin{matrix} lim_{k \to \infty} x_{\bar{C}} (k) = P [\begin{matrix} lim_{k \to \infty} J_{C}^{k} & 0 \\ 0 & lim_{k \to \infty} J_{\bar{C}}^{k} \end{matrix}] \\ {[0, \dots, 0, α_{d + 1}^{H} (0), \dots, α_{Nd}^{H} (0)]}^{H} \\ = P [\begin{matrix} lim_{k \to \infty} J_{C}^{k} \\ lim_{k \to \infty} J_{1} (λ_{\bar{C}, 1}) \\ lim_{k \to \infty} J_{2}^{k} (λ_{\bar{C}, 2}) \\ ⋱ \\ lim_{k \to \infty} J_{s}^{k} (λ_{\bar{C}, s}) \end{matrix}] \\ {[0, \dots, 0, α_{d + 1}^{H} (0), \dots, α_{Nd}^{H} (0)]}^{H}, \end{matrix}

(6)

where

\begin{matrix} J_{i}^{k} (λ_{\bar{C}, i}) & = {[\begin{matrix} λ_{\bar{C}, i} & 1 \\ λ_{\bar{C}, i} & ⋱ \\ ⋱ & 1 \\ λ_{\bar{C}, i} \end{matrix}]}^{k} \\ = [\begin{matrix} λ_{\bar{C}, i}^{k} & * & \dots & * \\ λ_{\bar{C}, i}^{k} & ⋱ & ⋮ \\ ⋱ & * \\ λ_{\bar{C}, i}^{k} \end{matrix}] (i = 1, 2, \dots, s) . \end{matrix}

Sufficiency. Due to $ρ (A + B K_{1} - λ_{i} B K_{2}) < 1 (i = 2, 3, \dots, N)$ , then $| λ_{\bar{C}, i} | < 1 (i = 1, 2, \dots, s)$ . One obtains

lim_{k \to \infty} λ_{\bar{C}, i}^{k} = 0 \Rightarrow lim_{k \to \infty} J_{i}^{k} (λ_{\bar{C}, i}) = 0 (i = 1, 2, \dots, s) .

(7)

It follows that $lim_{k \to \infty} x_{\bar{C}} (k) = 0$ .

The response of system (3) due to $x (0)$ is

\begin{matrix} x_{C} (k) = {PJ}_{H}^{k} P^{- 1} x_{C} (0) = P [\begin{matrix} J_{C}^{k} & 0 \\ 0 & J_{\bar{C}}^{k} \end{matrix}] \\ {[α_{1}^{H} (0), \dots, α_{d}^{H} (0), 0, \dots, 0]}^{H} . \end{matrix}

(8)

Let ${[β_{1}^{H} (k), β_{2}^{H} (k), \dots, β_{d}^{H} (k)]}^{H} = J_{C}^{k} {[α_{1}^{H} (0), α_{2}^{H} (0), \dots, α_{d}^{H} (0)]}^{H},$ then one can have

x_{C} (k) = P {[β_{1}^{H} (k), β_{2}^{H} (k), \dots, β_{d}^{H} (k), 0, \dots, 0]}^{H} .

(9)

As the first d column vectors of P are $p_{j} = 1_{N} \otimes c_{j} (j = 1, 2, \dots, d)$ by Lemma 2, one has

x_{C} (k) = 1_{N} \otimes \sum_{j = 1}^{d} β_{j} (k) c_{j} .

(10)

From (7–10), one obtains

lim_{k \to \infty} x (k) = x_{C} (k) = 1_{N} \otimes \sum_{j = 1}^{d} β_{j} (k) c_{j} .

(11)

Hence, there exists a vector $\tilde{c} (k) = \sum_{j = 1}^{d} β_{j} (k) c_{j} \in R^{d}$ satisfying $li m_{k \to \infty} (x (k) - 1_{N} \otimes \tilde{c} (k)) = 0$ . Then the proof of sufficiency of Theorem 1 is completed.

Necessity. We prove the conclusion by contradiction. If the system (3) achieve consensus and $ρ (A + B K_{1} - λ_{i} B K_{2}) \geq 1 (i = 2, 3, \dots, N)$ , there must exist an eigenvalue $λ_{m}$ such that $| λ_{m} | \geq 1$ . Consequently, one has $lim_{k \to \infty} λ_{\bar{C}, m}^{k} \neq 0 \Rightarrow lim_{k \to \infty} J_{m}^{k} (λ_{\bar{C}, m}) \neq 0$ . Then $lim_{k \to \infty} x_{\bar{C}} (k) \neq 0$ is obtained.

Hence, $lim_{k \to \infty} x (k) = lim_{k \to \infty} x_{C} (k) + lim_{k \to \infty} x_{\bar{C}} (k) \neq x_{C} (k)$ . This is contradicted with the hypothesis that system (3) achieve consensus. Then the proof of necessity of Theorem 1 is completed.

The proof of Theorem 1 is completed.▪

Remark 1. A sufficient and necessary condition is given in Theorem 1 to guarantee the consensus of closed-loop D-NMAS (3). This condition is similar to that of the high-order continue-time NMAS in Xi et al. (2010). The difference between them is that for D-NMAS, $A + B K_{1} - λ_{i} B K_{2}$ should be Schur-stable, but for continuous-time NMAS, $A + B K_{1} - λ_{i} B K_{2}$ is Hurwitz stable. It should be specially explained that if the interaction topology G has a spanning tree, the convergence of consensus of system (3) is only dependent on the neighbouring feedback matrix $K_{2}$ .

Corollary 1. When the interaction topology G does not have a spanning tree, system (3) can achieve consensus if and only if $ρ (A + B K_{1}) < 1$ and $ρ (A + B K_{1} - λ_{i} B K_{2}) < 1 (i = 2, 3, \dots, N)$ .

Proof. If the interaction topology G does not have a spanning tree, one can determine that the Laplacian matrix L of G has at least two zero eigenvalues by Lemma 1. Thus, there exist at least two diagonal blocks of $A + B K_{1}$ in $I_{N} \otimes (A + B K_{1}) - J_{L} \otimes B K_{2}$ . Hence, it is necessary that $ρ (A + B K_{1}) < 1$ and $ρ (A + B K_{1} - λ_{i} B K_{2}) < 1 (i = 2, 3, \dots, N)$ are simultaneously satisfied, and system (3) can asymptotically achieve consensus with a final consensus value 0. Therefore, system (3) can achieve consensus if and only if $ρ (A + B K_{1}) < 1$ and $ρ (A + B K_{1} - λ_{i} B K_{2}) < 1 (i = 2, 3, \dots, N)$ .

This completes the proof of Corollary 1.▪

Remark 2. If interaction topology G does not have a spanning tree, by Corollary 1, consensus problems of the D-NMAS (3) are equal to asymptotic stability problems in $R^{Nd}$ . Due to the interactions of agents, the D-NMAS (3) may not achieve consensus even if every agent has been stabilized.

Theorem 2. If system (3) achieves consensus, then the final consensus value $c (k)$ satisfies

\begin{matrix} lim_{k \to \infty} (c (k) - {(A + B K_{1})}^{k} (e_{1}^{T} U^{- 1} \otimes I_{d}) x (0)) = 0 & k = 0, 1, 2, \dots \end{matrix},

(12)

where $e_{1}^{T} \in R^{N}$ denotes a N dimensions unit row vector with its first entry of 1 and 0 elsewhere.

Proof. Let $x_{C} (k) = (U \otimes I_{d}) [{\tilde{x}}_{1}^{H} (k), 0]^{H}$ , $\tilde{x}' (k) = [{\tilde{x}}_{2}^{H} (k), \dots, {\tilde{x}}_{N}^{H} (k)]^{H}$ and $x_{\bar{C}} (k) = (U \otimes I_{d}) [0, \tilde{x}' (k)]^{H}$ . Therefore, $x (k)$ can be uniquely decomposed as $x (k) = x_{C} (k) + x_{\bar{C}} (k)$ , From Theorem 1, we know that if system (3) achieves consensus, the response of system (3) due to $x_{\bar{C}} (0)$ will satisfy $lim_{k \to \infty} x_{\bar{C}} (k) = 0$ . Hence, the final consensus value $c (k)$ is determined solely upon $x_{C} (k)$ . Hence the final consensus value $c (k)$ is determined solely upon $x_{C} (k)$ . Substituting $\tilde{x} (0) = (U^{- 1} \otimes I_{d}) x (0)$ into $x_{C} (0) = {\bar{u}}_{1} \otimes {\tilde{x}}_{1} (0)$ yields $x_{C} (0) = (e_{1}^{T} U^{- 1} \otimes I_{d}) x (0)$ . Hence,

x_{C} (k) = {(A + B K_{1})}^{k} (e_{1}^{T} U^{- 1} \otimes I_{d}) x (0)

(13)

Though Definition 1, we have $\begin{matrix} {lim}_{k \to \infty} (c (k) - {(A + B K_{1})}^{k} (e_{1}^{T} U^{- 1} \otimes I_{d}) x (0)) = 0 k = 0, 1, 2, \dots \end{matrix}$ .

The proof of Theorem 2 is completed.▪

Remark 3. The final consensus value $c (k)$ of system (3) is given by Theorem 2. That is, the final consensus value, which is related to the self-feedback matrix $K_{1}$ and U determined by the interaction topology G, can be represented in a simple transformation of $x (0)$ . This indicates that the final consensus value $c (k)$ may change under switching topology, and the resulting details will be discussed below.

Consensus analysis for high-order D-NMAS with switching topology

Now, we consider an interaction switching topology $G_{δ (k)}$ for a D-NMAS at time k. Denoted by $Γ_{N} = {G_{δ (k)}}$ a finite collection of N order digraphs of all possible topologies for $G_{δ (k)}$ that can be analytically expressed as

Γ_{N} = {G_{δ (k)} = (V_{δ (k)}, ℰ_{δ (k)}, A_{δ (k)}) : r a n k (ℒ (G_{δ (k)})) = N - 1, 1^{T} ℒ (G_{δ (k)}) = 0},

where $G_{δ (k)} \in Γ_{N}$ , and $δ (k) \in I_{Γ_{N}} \subset N$ denotes a switching signal that is the index set associated with the elements of $Γ_{N}$ . Let $L (δ (k)) = L (G_{δ (k)})$ denote the Laplacian matrix of the interaction topology $G_{δ (k)}$ , and $λ_{1} (δ (k)), λ_{2} (δ (k)), \dots, λ_{N} (δ (k))$ be all the eigenvalues of the matrix $L (δ (k))$ , which satisfy

0 = | λ_{1} (δ (k)) | \leq | λ_{2} (δ (k)) | \leq \dots | λ_{N} (δ (k)) | .

Assumption 1. Every interaction topology $G_{δ (k)} \in Γ_{N}$ has a spanning tree.

Remark 4. In the existing literature, there have two main methods to investigate the consensus problem of NMAS with switching topologies, one is Assumption 1, e.g. Pei and Sun (2015) and Wen et al. (2013); another is joint-contained spanning tree topologies, e.g. Su and Huang (2012). In this paper, we adopt the former option.

For convenience, we define the following notations,

\underline{λ} = \underset{λ_{i}}{\arg \min} {Re {λ_{2} (δ (k))} | G_{δ (k)} \in Γ_{N}, G_{δ (k)} have a spanning tree},

\begin{matrix} \bar{λ} = \underset{λ_{i}}{\arg max} \\ {Re {λ_{N} (δ (k))} | G_{δ (k)} \in Γ_{N}, G_{δ (k)} have a spanning tree}, \end{matrix}

\begin{matrix} σ_{M} = max \\ {Im {λ_{i} (δ (k))} | G_{δ (k)} \in Γ_{N}, G_{δ (k)} have a spanning tree, i = 1, 2, \dots, N} . \end{matrix}

Let ${\tilde{λ}}_{1, 2} = Re (\underline{λ}) \pm j σ_{M}$ , ${\tilde{λ}}_{3, 4} = Re (\bar{λ}) \pm j σ_{M}$ , $Φ_{0}$ , $Φ_{1}$ and $Φ_{2}$ be real symmetric matrices independent of $λ_{i} (δ (k))$ and ${\tilde{λ}}_{i}, (i = 1, 2, 3, 4)$ . Then we have the following lemma.

Lemma 4 (Xi et al., 2011). If $Φ_{0} + Re ({\tilde{λ}}_{i}) Φ_{1} + Im ({\tilde{λ}}_{i}) Φ_{2} < 0$ $(i = 1, 2, 3, 4)$ , then $Φ_{0} + Re (λ_{i}) Φ_{1} + Im (λ_{i}) Φ_{2} < 0,$ $(i = 2, 3, \dots, N)$ .

The following consensus protocol with switching topology is applied for agent i

u_{i} (k) = K_{1} x_{i} (k) + K_{2} \sum_{j \in N_{i} (k)} a_{ij} (δ (k)) (x_{j} (k) - x_{i} (k),

(14)

where $K_{1}$ and $K_{2}$ are constant gain matrices with appropriate dimensions.

Let $x (k) = [x_{1}^{T} (k), x_{2}^{T} (k), \dots, x_{N}^{T} (k)]^{T}$ , then the dynamics of the high-order D-NMAS (1) with the protocol (14) can be described by a closed-loop discrete-time networked dynamics as

x (k + 1) = (I_{N} \otimes (A + B K_{1}) - L (δ (k) \otimes B K_{2}) x (k)

(15)

From Theorem 2, one know that $x (k)$ can be decomposed as the consensus dynamics $x_{C} (k)$ and the disagreement dynamics $x_{\bar{C}} (k)$ . The discrete-time system (15) can achieve consensus if only $x_{\bar{C}} (k)$ is asymptotically converge to zero vector, which has not to do with $x_{C} (k)$ . So, one can use a method to convert the consensus problem into a stable problem of the discrete-time system (15).

Therefore, set $e_{i} (k) = x_{i + 1} (k) - x_{1} (k) (i = 1, 2, \dots, N - 1)$ and $e (k) = [e_{1}^{T} (k), e_{2}^{T} (k), \dots, e_{N - 1}^{T} (k)]^{T}$ . The discrete-time networked dynamics (15) can be rewritten as

e (k + 1) = (I_{N - 1} \otimes (A + B K_{1}) - \tilde{L} (δ (k)) \otimes B K_{2}) e (k),

(16)

where

\tilde{L} (δ (k)) = [\begin{matrix} l_{22} (δ (k)) - l_{12} (δ (k)) & \dots & l_{2 N} (δ (k)) - l_{1 N} (δ (k)) \\ ⋮ & ⋱ & ⋮ \\ l_{N 2} (δ (k)) - l_{12} (δ (k)) & \dots & l_{NN} (δ (k)) - l_{1 N} (δ (k)) \end{matrix}]

(17)

Obviously, the closed-loop discrete-time system (15) can achieve consensus if and only if the error discrete-time system (16) is asymptotically stable. Then, the consensus problem of the closed-loop discrete-time system (15) is transformed into the stability problem of the error system (16) by a model transformation. Thus, we can use the discrete-time stability theorems to analyse the high-order discrete-time system (16) for the consensus analysis of the high-order discrete-time system (15).

Lemma 5 (Guan et al., 2012). Let $λ_{1}, λ_{2}, \dots, λ_{N}$ where $0 = | λ_{1} | \leq | λ_{2} | \leq \dots | λ_{N} |$ , and $μ_{1}, μ_{2}, \dots, μ_{N - 1}$ where $| μ_{1} | \leq | μ_{2} | \leq \dots | μ_{N - 1} |$ denote the eigenvalues of Laplacian matrix L and matrix $\tilde{L}$ , respectively, then one has $λ_{2} = μ_{1}, λ_{3} = μ_{2}, \dots, λ_{N} = μ_{N - 1}$ . The matrix $\tilde{L}$ is defined as

\tilde{L} = [\begin{matrix} l_{22} - l_{12} & \dots & l_{2 N} - l_{1 N} \\ ⋮ & ⋱ & ⋮ \\ l_{N 2} - l_{12} & \dots & l_{NN} - l_{1 N} \end{matrix}] .

Lemma 6 (Ghaoui et al., 1994). The following is an LMI

(\begin{matrix} Q (x) & S (x) \\ S {(x)}^{T} & R (x) \end{matrix}) > 0

where $Q (x) = Q {(x)}^{T}$ and $R (x) = R {(x)}^{T}$ are equivalent to one of the following conditions,

1) $Q (x) > 0$ , $R (x) - S {(x)}^{T} Q {(x)}^{- 1} S (x) > 0$

2) $R (x) > 0$ , $Q (x) - S (x) R {(x)}^{- 1} S {(x)}^{T} > 0$

Theorem 3. The error discrete-time system (16) is asymptotically stable, if there exist two positive definite matrices P and Q with $2 d \times 2 d$ dimensions, such that P and Q are a solution for the following minimization problem

\begin{matrix} min & tr [PQ] \end{matrix}

(18)

subject to

[\begin{matrix} P & {\tilde{Λ}}_{i} (k) \\ {\tilde{Λ}}_{i}^{T} (k) & Q \end{matrix}] > 0, i = 1, 2, 3, 4,

(19)

[\begin{matrix} P & I \\ I & Q \end{matrix}] > 0, P > 0, Q > 0

(20)

where tr [•] denotes the trace of a given matrix, ${\tilde{Λ}}_{i} (k) = Λ_{A} + Λ_{B} Λ_{K_{1}} - ϒ_{{\tilde{λ}}_{i}} Λ_{B} Λ_{K_{2}}$ , $i = 1, \dots, 4$ , where $Λ_{X} = [\begin{matrix} X & 0 \\ 0 & X \end{matrix}]$ with $X = A, B, K_{1}, K_{2}$ , and $ϒ_{{\tilde{λ}}_{i}} = [\begin{matrix} Re ({\tilde{λ}}_{i} I & - Im ({\tilde{λ}}_{i}) I \\ Im ({\tilde{λ}}_{i}) I & Re ({\tilde{λ}}_{i}) I \end{matrix}]$ .

Proof. Let $\tilde{U} (k) = [{\tilde{u}}_{1} (k), {\tilde{u}}_{2} (k), \dots, {\tilde{u}}_{N - 1} (k),] \in C^{N \times N}$ be a non-singular matrix satisfying ${\tilde{U}}^{- 1} (k) \tilde{L} (δ (k)) \tilde{U} (k) = {\tilde{J}}_{L} (δ (k))$ , where ${\tilde{J}}_{L} (δ (k))$ is the Jordan canonical form of matrix $\tilde{L} (δ (k))$ . Let $\tilde{e} (k) = ({\tilde{U}}^{- 1} (k) \otimes I_{m}) e (k) = [{\tilde{e}}_{1}^{T} (k), {\tilde{e}}_{2}^{T} (k), \dots, {\tilde{e}}_{N - 1}^{T} (k)]^{T}$ . The discrete-time system (16) can be converted to

\tilde{e} (k + 1) = (I_{N - 1} \otimes (A + B K_{1}) - J_{\tilde{L} (δ (k))} \otimes B K_{2}) \tilde{e} (k) .

(21)

By Assumption 1, we can obtain that $J_{\tilde{L} (δ (k))}$ is a triangular matrix. From the proof of Theorem 1, one can obtain that the asymptotical stability of discrete-time system (21) is equivalent to that of the following discrete-time subsystem

ξ_{i} (k + 1) = (A + B K_{1} - μ_{i} (δ (k)) B K_{2}) ξ_{i} (k) .

(22)

By decomposing of real and imaginary parts, system (22) can be rewritten as

{\hat{ξ}}_{i} (k + 1) = (Λ_{A} + Λ_{B} Λ_{K_{1}} - ϒ_{μ_{i} (δ (k))} Λ_{B} Λ_{K_{2}}) {\hat{ξ}}_{i} (k),

(23)

where, $i = 1, \dots, N - 1$ and ${\hat{ξ}}_{i} (k) = {[Re {(ξ_{i} (k))}^{T}, Im {(ξ_{i} (k))}^{T}]}^{T}$ .

By Lemma 5, the discrete-time system (23) is equivalent to

{\hat{ξ}}_{i} (k + 1) = Λ_{i} (k) {\hat{ξ}}_{i} (k)

(24)

where $Λ_{i} (k) = Λ_{A} + Λ_{B} Λ_{K_{1}} - ϒ_{λ_{i} (δ (k))} Λ_{B} Λ_{K_{2}}, (i = 2, \dots, N)$ and $ϒ_{λ_{i} (δ (k))} = [\begin{matrix} Re (λ_{i} (δ (k))) I & - Im (λ_{i} (δ (k))) I \\ Im (λ_{i} (δ (k))) I & Re (λ_{i} (δ (k))) I \end{matrix}]$

Define a Lyapunov function for system (24) as

V_{i} (k) = {\hat{ξ}}_{i}^{T} (k) P {\hat{ξ}}_{i} (k),

(25)

where P is a positive definite matrix with $2 md \times 2 md$ dimensions. Taking derivative of $V_{i} (k)$ with respect to time along the trajectory of (24), one has

\begin{matrix} Δ V_{i} (k) & = {\hat{ξ}}_{i}^{T} (k + 1) P {\hat{ξ}}_{i} (k + 1) - {\hat{ξ}}_{i}^{T} (k) P {\hat{ξ}}_{i} (k) \\ = {\hat{ξ}}_{i}^{T} (k) (Λ_{i}^{T} (k) P Λ_{i} (k) - P) {\hat{ξ}}_{i} (k) . \end{matrix}

(26)

Clearly, if $Λ_{i}^{T} (k) P Λ_{i} (k) - P < 0$ , the discrete time system (16) is asymptotically stable.

By the Schur complement of Lemma 6, $P - Λ_{i}^{T} (k) P Λ_{i} (k) > 0$ is equivalent to

[\begin{matrix} P & Λ_{i} (k) \\ Λ_{i}^{T} (k) & P^{- 1} \end{matrix}] > 0

(27)

By setting $Q = P^{- 1}$ , through the Cone Complementarity Linearization (CCL) algorithm in Ghaoui et al. (1997), the non-linear matrix inequality (NMI) (27) is transformed to solve the following optimization problem

\begin{matrix} min & tr [PQ] \end{matrix}

(28)

subject to

[\begin{matrix} P & Λ_{i} (k) \\ Λ_{i}^{T} (k) & Q \end{matrix}] > 0, i = 2, 3, \dots, N,

(29)

[\begin{matrix} P & I \\ I & Q \end{matrix}] > 0, P > 0, Q > 0

(30)

By Lemma 4, the above expressions (28)–(30) are equivalent to (18)–(20), which can be solved by Algorithm 1.

Algorithm 1	A possible way to the optimization problem (18)–(20)
Step 1	Find a feasible point $P_{0}$ , $Q_{0}$ for LMI (18)–(20), if there are none, exit. Set $k = 0$ .
Step 2	Set $P_{k} = P$ , $Q_{k} = Q$ , and find $P_{k + 1}$ , $Q_{k + 1}$ that solve the LMI problem $P_{k}$ : min $tr (P_{k} Q + Q_{k} P)$ subject to LMI (18)–(20).
Step 3	If $tr (P_{k} Q + Q_{k} P) < 4 d + ε$ , exit. Otherwise, set $k = k + 1$ and go to step 2.

Thus, the proof of Theorem 3 is completed.▪

Remark 5. From Theorem 3, it can be noted that a sufficient LMI stability condition is established for the discrete-time system (21) by a Lyapunov function approach, then the high-order D-NMAS system (1) with the protocol (14) achieves consensus. Meanwhile, one can also get the matrix $Λ_{K_{2}}$ , and thereby the neighbouring feedback matrix $K_{2}$ can be obtained.

Theorem 4. If system (15) achieves consensus, then the final consensus value $c (k)$ satisfies

{\begin{matrix} \begin{matrix} lim_{{\tilde{k}}_{i} \to \infty} (c ({\tilde{k}}_{i}) - (e_{1}^{T} U {(δ ({\tilde{k}}_{i}))}^{- 1} \otimes I_{d}) x ({\tilde{k}}_{i})) = 0, & where {\tilde{k}}_{i} is the i_{th} time of switching topology i = 1, 2, \dots, \end{matrix} \\ \begin{matrix} lim_{k \to \infty} (c (k) - {(A + B K_{1})}^{k - \tilde{k}} (e_{1}^{T} U {(δ (k))}^{- 1} \otimes I_{d}) x ({\tilde{k}}_{i})) = 0, & where k = {\tilde{k}}_{i} + 1, {\tilde{k}}_{i} + 2, \dots {\tilde{k}}_{i + 1} - 1, i = 1, 2, \dots, \end{matrix} \end{matrix}

(31)

Proof. By the proof of Theorem 2, one know that if the interaction topology of the D-NMAS does not switch, the final consensus value is $\begin{matrix} c (k) = {(A + B K_{1})}^{k} (e_{1}^{T} U^{- 1} \otimes I_{d}) x (0) k = 0, 1, 2, \dots \end{matrix}$ . For this reason, one can treat the state of the D-NMAS at the switching time $\tilde{k}$ as an initial time, which means that $c (\tilde{k}) = (e_{1}^{T} U {(δ ({\tilde{k}}_{i}))}^{- 1} \otimes I_{d}) x ({\tilde{k}}_{i})$ , where ${\tilde{k}}_{i}$ denotes the $i_{th}$ time of switching topology. Therefore, before the arrival of next switching signal at time ${\tilde{k}}_{i + 1}$ , the final consensus value can be constructed as $c (k) = {(A + B K_{1})}^{k - \tilde{k}} (e_{1}^{T} U {(δ (k))}^{- 1} \otimes I_{d}) x (\tilde{k})$ .

The proof of Theorem 4 is completed.▪

Remark 6. The final consensus value $c (k)$ of the system (15) is given by Theorem 4. That is, the final consensus value $c (k)$ will be changed by switching topology. However, it does not alter during two neighbour switching signals. This is discussed under two different cases including the switching time $\tilde{k}$ and the other times.

Consensus analysis for high-order D-NMAS with switching topology and time delays

First, the following assumption is given.

Assumption 2. Network time-delay $τ$ is a constant and known positive integer.

The following consensus protocol with switching topology and time delays is applied for agent i

u_{i} (k) = K_{1} x_{i} (k) + K_{2} \sum_{j \in N_{i} (k)} a_{ij} (δ (k)) (x_{j} (k - τ) - x_{i} (k - τ)

(32)

where $K_{1}$ and $K_{2}$ are constant gain matrices with appropriate dimensions, and $τ \in N^{*}$ denotes time of delay.

Let $x (k) = [x_{1}^{T} (k), x_{2}^{T} (k), \dots, x_{N}^{T} (k)]^{T}$ . The dynamics of the high-order D-NMAS (1) with the protocol (32) can be described by a closed-loop discrete-time networked dynamics through

{\begin{matrix} x (k + 1) = (I_{N} \otimes (A + B K_{1})) x (k) - (L (δ (k)) \otimes B K_{2}) x (k - τ) \\ x (k) = φ (k), k \in {- τ, - τ + 1, \dots, 0} . \end{matrix}

(33)

By setting $e_{i} (k) = x_{i + 1} (k) - x_{1} (k) (i = 1, 2, \dots, N - 1)$ and $e (k) = [e_{1}^{T} (k), e_{2}^{T} (k), \dots, e_{N - 1}^{T} (k)]^{T}$ , the discrete-time networked dynamics (33) can be rewritten as

{\begin{matrix} e (k + 1) = (I_{N - 1} \otimes (A + B K_{1})) e (k) - (\tilde{L} (δ (k)) \otimes B K_{2}) e (k - τ) \\ e (k) = φ_{e} (k), k \in {- τ, - τ + 1, \dots, 0}, \end{matrix}

(34)

where $\tilde{L} (δ (k))$ is given by (17)

The closed-loop discrete-time system (33) can achieve consensus if and only if the error discrete-time system (34) is asymptotically stable. Hence, we can also use the discrete-time stability theorems to analyse the consensus problems of the discrete-time system (33).

Theorem 5. The error discrete-time system (34) is asymptotically stable, if there exist two positive definite matrices R and S with $2 d τ \times 2 d τ$ dimensions, such that Rand S are a solution to the following minimization problem:

\begin{matrix} min & tr [RS] \end{matrix}

(35)

subject to

[\begin{matrix} R & {\tilde{F}}_{i} (k) \\ {\tilde{F}}_{i}^{T} (k) & S \end{matrix}] > 0

(36)

[\begin{matrix} R & I \\ I & S \end{matrix}] > 0, R > 0, S > 0

(37)

where

{\tilde{F}}_{i} (k) = [\begin{matrix} Λ_{A} + Λ_{B} Λ_{K_{1}} & 0 & \dots & - ϒ_{{\tilde{λ}}_{i}} Λ_{B} Λ_{K_{2}} \\ I_{2 m} & 0 & 0 & 0 \\ 0 & ⋱ & ⋱ & ⋮ \\ 0 & 0 & I_{2 m} & 0 \end{matrix}], i = 1, 2, 3, 4 .

(38)

Proof. Let $\tilde{U} (k) = [{\tilde{u}}_{1} (k), {\tilde{u}}_{2} (k), \dots, {\tilde{u}}_{N - 1} (k),] \in C^{N \times N}$ is a non-singular matrix satisfying ${\tilde{U}}^{- 1} (k) \tilde{L} (δ (k)) \tilde{U} (k) = {\tilde{J}}_{L} (δ (k))$ , where ${\tilde{J}}_{L} (δ (k))$ is the Jordan canonical form of matrix $\tilde{L} (δ (k))$ .

Let $\tilde{e} (k) = ({\tilde{U}}^{- 1} (k) \otimes I_{m}) e (k) = [{\tilde{e}}_{1}^{T} (k), {\tilde{e}}_{2}^{T} (k), \dots, {\tilde{e}}_{N - 1}^{T} (k)]^{T}$ , then the discrete-time system (34) can be converted to

\tilde{e} (k + 1) = (I_{N - 1} \otimes (A + B K_{1})) \tilde{e} (k) - (J_{\tilde{L} (δ (k))} \otimes B K_{2}) \tilde{e} (k - τ)

(39)

Because $\tilde{L} (δ (k))$ is a triangular matrix, it can be shown that asymptotical stability of the discrete-time system (39) is equivalent to that of the following discrete-time subsystem

ξ_{i} (k + 1) = (A + B K_{1}) ξ_{i} (k) - μ_{i} (δ (k)) B K_{2} ξ_{i} (k - τ)

(40)

Though decomposing of real and imaginary parts, system (40) can be rewritten as

{\hat{ξ}}_{i} (k + 1) = (Λ_{A} + Λ_{B} Λ_{K_{1}}) {\hat{ξ}}_{i} (k) - ϒ_{μ_{i} (δ (k))} Λ_{B} Λ_{K_{2}} {\hat{ξ}}_{i} (k - τ)

(41)

where $i = 1, \dots, N - 1$ and ${\hat{ξ}}_{i} (k) = {[Re {(ξ_{i} (k))}^{T}, Im {(ξ_{i} (k))}^{T}]}^{T}$ .

By Lemma 5, the discrete-time system (41) is equivalent to

{\hat{ξ}}_{i} (k + 1) = (Λ_{A} + Λ_{B} Λ_{K_{1}}) {\hat{ξ}}_{i} (k) - ϒ_{λ_{i} (δ (k))} Λ_{B} Λ_{K_{2}} {\hat{ξ}}_{i} (k - τ)

(42)

where $i = 2, 3, \dots, N$ . By extending the dimensions of the state of the discrete-time system (42), it can be rewritten that

\begin{matrix} [\begin{matrix} {\hat{ξ}}_{i} (k) \\ {\hat{ξ}}_{i} (k - 1) \\ ⋮ \\ {\hat{ξ}}_{i} (k - τ) \end{matrix}] = [\begin{matrix} Λ_{A} + Λ_{B} Λ_{K_{1}} & 0 & \dots & - ϒ_{λ_{i} (k)} Λ_{B} Λ_{K_{2}} \\ I_{2 d} & 0 & 0 & 0 \\ 0 & ⋱ & ⋱ & ⋮ \\ 0 & 0 & I_{2 d} & 0 \end{matrix}] \\ [\begin{matrix} {\hat{ξ}}_{i} (k - 1) \\ {\hat{ξ}}_{i} (k - 2) \\ ⋮ \\ {\hat{ξ}}_{i} (k - τ - 1) \end{matrix}] \end{matrix}

(43)

Let $ς_{i} (k) = {[{\hat{ξ}}_{i}^{T} (k), {\hat{ξ}}_{i}^{T} (k - 1), \dots, {\hat{ξ}}_{i}^{T} (k - τ)]}^{T}$ . Then we have

ς_{i} (k) = F_{i} (k) ς_{i} (k - 1)

(44)

where

F_{i} (k) = [\begin{matrix} Λ_{A} + Λ_{B} Λ_{K_{1}} & 0 & \dots & - ϒ_{λ_{i} (k)} Λ_{B} Λ_{K_{2}} \\ I_{2 d} & 0 & 0 & 0 \\ 0 & ⋱ & ⋱ & ⋮ \\ 0 & 0 & I_{2 d} & 0 \end{matrix}]

(45)

Define a Lyapunov function for the discrete-time system (44) as

V_{i} (k) = ς_{i}^{T} (k) R ς_{i} (k)

(46)

where R is a $2 d τ \times 2 d τ$ dimensional positive definite matrix.

Taking the time derivative of $V_{i} (k)$ along the trajectory of (44), one has

\begin{matrix} Δ V_{i} (k) & = ς_{i}^{T} (k + 1) R ς (k + 1) - ς_{i}^{T} (k) R ς_{i} (k) \\ = ς_{i}^{T} (k) (F_{i}^{T} (k) R F_{i} (k) - R) ς_{i} (k) . \end{matrix}

(47)

Clearly, if $F_{i}^{T} (k) R F_{i} (k) - R < 0$ , the discrete time system (44) is asymptotically stable.

By the Schur complement of Lemma 6, $R - F_{i}^{T} (k) R F_{i} (k) > 0$ is equivalent to

[\begin{matrix} R & F_{i} (k) \\ F_{i}^{T} (k) & R^{- 1} \end{matrix}] > 0

(48)

By setting $S = R^{- 1}$ , through the CCL algorithm (Ghaoui et al., 1997), the NMI (48) is transformed to solving the following optimization problem

\begin{matrix} min & tr [RS] \end{matrix}

(49)

subject to

[\begin{matrix} R & F_{i} (k) \\ F_{i}^{T} (k) & S \end{matrix}] > 0, where i = 2, \dots, N

(50)

[\begin{matrix} R & I \\ I & S \end{matrix}] > 0, R > 0, S > 0

(51)

By Lemma 4, the above expression (49)–(51) are equivalent to (35)–(38), which can be solved by Algorithm 1.

Thus, the proof of Theorem 5 is completed.

Remark 7. From Theorem 5, it can be noted that the high-order consensus problem of D-NMAS with switching topology and time delays, is converted into a generally discrete-time stability model by an augmented method. A neighbouring feedback matrix $K_{2}$ can be calculated to insure the stability of the discrete-time system (34). Then the high-order closed-loop D-NMAS system (33) achieves consensus. It is worth mentioning that, if LMI (35)–(38) have no solution, the time delay $τ$ would be not acceptable.

Theorem 6. The time-delay $τ$ , which only work on whether the system (33) achieving consensus, has no influence on the final consensus value $c (k)$ . If system (33) achieves consensus, then the final consensus value $c (k)$ satisfies (31).

Proof. Let $\tilde{x} (k) = (U^{- 1} \otimes I_{d}) x (k) = [{\tilde{x}}_{1}^{T} (k), {\tilde{x}}_{2}^{T} (k), \dots, {\tilde{x}}_{N}^{T} (k)]^{T}$ . By Lemma 1, the closed-loop discrete-time system (33) is equivalent to

{\tilde{x}}_{1} (k + 1) = (A + B K_{1}) {\tilde{x}}_{1} (k),

(52)

\begin{matrix} {\tilde{x}}_{i} (k + 1) = (A + B K_{1}) {\tilde{x}}_{i} (k) - λ_{i} B K_{2} {\tilde{x}}_{i} (k - τ) . (i = 2, 3, \dots, N) \end{matrix}

(53)

From the proof of Theorem 1, one know that the subsystem (52) determines the consensus value of system (33). If subsystem (53) is asymptotically stable, system (33) can reach consensus. Namely, the time-delay $τ$ does not directly change the consensus value of system (33), but they determine whether system (33) can achieves consensus by destroying the stability of the system (34).

This completes the proof of Theorem 6.▪

Remark 8. From Theorem 6, we know that the time delay $τ$ does not directly change the consensus value, but it may destroy the convergence ability of consensus of a high-order D-NMAS system. There exists a trade-off between the constringency speed and time delays when designing the neighbouring feedback matrix $K_{2}$ .

Simulation studies

In this section, three numerical examples are given to illustrate the effectiveness of the proposed theoretical results. The proposed consensus protocols are applied to achieve state alignment among six agents. The dynamics of them are described by (1) with

A = [\begin{matrix} 1.01 & - 0.02 & 0.0 \\ 0.01 & 1.02 & 0.01 \\ 0 & 0.01 & 0.85 \end{matrix}], B = [\begin{matrix} 0.01 & - 0.02 \\ 0 & 0.01 \\ 0.02 & 0.01 \end{matrix}]

Example 1. We apply the consensus protocol (2) to achieve consensus among those six agents (indexed by 1–6) with a fixed topology. The interconnection topology of D-NMAS (1) is shown in Figure 1, and the graph is connected with containing a spanning tree.

Figure 1.

A fixed interaction topology of six agents.

The state initial values of all the agents 1–6 are randomly produced as $x_{1} (0) = {[- 4, 3 - 1]}^{T}$ , $x_{2} (0) = {[- 2, - 3, - 2]}^{T}$ , $x_{3} (0) = {[4, 1, 2]}^{T}$ , $x_{4} (0) = {[6, 9, 6]}^{T}$ , $x_{5} (0) = {[10, 7, 4]}^{T}$ , and $x_{6} (0) = {[7, - 6, 2]}^{T}$ . We choose that

\begin{array}{l} K_{1} = [\begin{matrix} 0.3952 & 0.5128 & 0.2761 \\ 0.3016 & - 0.1447 & 0.1127 \end{matrix}], \\ K_{2} = [\begin{matrix} 16.9131 & 23.8223 & 8.6727 \\ - 9.5176 & 12.0880 & 4.2149 \end{matrix}], \end{array}

which satisfy the conditions in Theorem 1. The simulation results are shown in Figures 2 –4.

Figure 2.

The trajectories of state 1 for discrete-time networked multi-agent systems (D-NMAS) (1).

Figure 3.

The trajectories of state 2 for discrete-time networked multi-agent systems (D-NMAS) (1)

Figure 4.

The trajectories of state 3 for discrete-time networked multi-agent systems (D-NMAS) (1).

In Figures 2 –4, the state trajectories of D-NMAS (1) are given. It can be seen that the state trajectories of D-NMAS (1) are asymptotically coverage to consensus values $c (k)$ , which are produced by Theorem 1 and marked by a red asterisk.

Example 2. We apply the consensus protocol (14) to achieve consensus among the six agents under switching topologies. Four interconnection topologies of $Γ_{N}$ are shown in Figure 5, and the graphs are connected with containing a spanning tree. The switching signal $δ (k)$ is given as shown in Figure 6.

Figure 5.

Four interaction topologies of six agents.

Figure 6.

The switching signal $δ (k)$ of discrete-time networked multi-agent systems (D-NMAS) (1).

Let $K_{1} = [\begin{matrix} 0.1924 & 0.4068 & 0.3784 \\ 0.2003 & - 0.1585 & 0.0168 \end{matrix}]$ . From Theorem 3, one can obtain that

K_{2} = [\begin{matrix} 19.2329 & 31.7110 & 6.1623 \\ - 7.8526 & 12.7210 & 3.5049 \end{matrix}]

P = [\begin{matrix} 0.9278 & - 0.2867 & 0.2296 & 0 & 0 & 0 \\ - 0.2867 & 0.6235 & - 0.6211 & 0 & 0 & 0 \\ 0.2296 & - 6211 & 3.4878 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.9278 & - 0.2867 & 0.2296 \\ 0 & 0 & 0 & - 0.2867 & 0.6235 & - 0.6211 \\ 0 & 0 & 0 & 0.2296 & - 0.6211 & 3.4878 \end{matrix}]

Q = [\begin{matrix} 1.2633 & 0.6119 & 0.0248 & 0 & 0 & 0 \\ 0.6119 & 2.2615 & 0.3600 & 0 & 0 & 0 \\ 0.0248 & 0.3600 & 0.3496 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1.2633 & 0.6119 & 0.0248 \\ 0 & 0 & 0 & 0.6119 & 2.2615 & 0.3600 \\ 0 & 0 & 0 & 0.0248 & 0.3600 & 0.3496 \end{matrix}]

The state initial values of all the agents 1–6 are randomly produced as $x_{1} (0) = {[- 10, 4, - 3]}^{T}$ , $x_{2} (0) = {[- 5, - 6, - 7]}^{T}$ , $x_{3} (0) = {[4, 5, 7]}^{T}$ , $x_{4} (0) = {[6, 9, 6]}^{T}$ , $x_{5} (0) = {[10, 7, 4]}^{T}$ , and $x_{6} (0) = {[7, - 6, 2]}^{T}$ . Each agent uses the protocol (14). The simulation results are shown in Figures 7 –9.

Figure 7.

The trajectories of state 1 for discrete-time networked multi-agent systems (D-NMAS) (1).

Figure 8.

The trajectories of state 2 for discrete-time networked multi-agent systems (D-NMAS) (1).

Figure 9.

The trajectories of state 3 for discrete-time networked multi-agent systems (D-NMAS) (1).

From Figures 7 –9, it is clear that the states of six agents all asymptotically converge to the actual $c (k)$ produced by Theorem 3 and marked by the red asterisk, rather than the initial consensus value $c (k)$ marked by the black dashed line obtained by Theorem 2 with fixed topology as shown in Figure 1(a). This demonstrates the correctness of Theorems 3 and 4. The final consensus value is related to the self-feedback matrix $K_{1}$ and interaction topologies.

Remark 9. Through the Example 2, we know that there all exist increasing errors between the initial $c (k)$ and the actual $c (k)$ in Figures 7 –9. The increasing errors are caused by switching topologies. The actual $c (k)$ should be altered when the interaction topology of system (15) changes. However, the increasing errors are not significant, especially in Figure 8, which even seems to be a constant error.

Example 3. We apply the consensus protocol (32) to achieve state alignment among those six agents under above switching topologies with time delay $τ = 2$ . In order to analyse the impacts of the time delays on the consensus of the D-NMAS, we let the initial state, $K_{1}$ and the switching signal $δ (k)$ of D-NMAS (1) remain the same to example 2. According to Theorem 5, we can obtain $K_{2} = [\begin{matrix} 8.0778 & 13.3186 & 2.5882 \\ - 3.2981 & 5.3428 & 1.4721 \end{matrix}]$ . Then, the corresponding simulation results are shown in Figures 10 –12.

Figure 10.

The trajectories of state 1 for discrete-time networked multi-agent systems (D-NMAS) (1) ( $τ = 2$ ).

Figure 11.

The trajectories of state 2 for discrete-time networked multi-agent systems (D-NMAS) (1) ( $τ = 2$ ).

Figure 12.

The trajectories of state 3 for discrete-time networked multi-agent systems (D-NMAS) (1) ( $τ = 2$ ).

From Figures 10 –12, we can see that the all states of six agents can also asymptotically converge to actual $c (k)$ marked by a red symbol asterisk and produced by Theorem 6. Therefore, the correctness of Theorems 5 and 6 are also demonstrated.

Remark 10. Through the above three examples, the correctness and validity of the proposed protocols and theorems are verified. From the simulation results, we can find that the actual $c (k)$ and the initial $c (k)$ are very close, indicating that the self-feedback matrix $K_{1}$ is the dominant factor influencing the final consensus value $c (k)$ while the interaction topologies are less important. At the same time, the convergence rate in example 3 is slower than that in example 2, which shows that the time delay $τ$ does not change the final consensus value of D-NMAS (1) though it may destroy the stability of it. The convergence rate of D-NMAS (1) will slow down, and the time delay $τ$ affects the final consensus value indirectly.

Conclusions

In this paper, we have addressed the consensus problem of high-order D-NMAS in three cases. A novel distributed feedback protocol has been proposed to solve the consensus problem. Final consensus values are given for the networks with proposed protocols. A sufficient and necessary condition is given for directed network achieving consensus with fixed topology and without time delays. Consensus problems are transformed into stability problems for the high-order D-NMAS via a model transformation method. Through stability analysis, sufficient LMI conditions are derived to ensure that the high-order D-NMAS can achieve consensus in the scenarios with and without time delays and switching topology, and which are independent of the number of agents. Further research will be conducted to more general consensus problems of high-order D-NMAS with joint-contained spanning tree topologies and time-varying delays.

Footnotes

Appendix A: Graph

Let a weighted digraph (or directed graph) $G = (V, E, A)$ of order N represents an interaction topology of a network of agents, with the set of nodes $V = {v_{1}, \dots, v_{N}}$ , set of edges $E \subseteq V \times V$ , and a weighted adjacency matrix $A = [a_{ij}]$ with non-negative adjacency elements $a_{ij}$ .

The node indexes belong to a finite index set $I = {1, 2, \dots, N}$ . An edge of G is denoted by $e_{ij} = (v_{i}, v_{j})$ , where $v_{i}$ and $v_{j}$ are called the initial and terminal nodes. It implies that node $v_{j}$ can receive information from node $v_{i}$ , but not necessarily vice versa. The adjacency elements associated with the edges of the graph are positive if $e_{ij} \in E$ while $a_{ij} = 0$ if $e_{ij} \notin E$ . Furthermore, we assume $a_{ii} = 0$ for all $i \in I$ . The set of neighbours of node $v_{i}$ is denoted by $N_{i} = {v_{j} \in V : (v_{i}, v_{j}) \in ε}$ . A cluster is any subset $J \subseteq V$ of the nodes of the graph. The set of neighbours of a cluster $N_{J}$ is defined by $N_{J} = \cup_{v_{i} \in J} N_{i} = {v_{j} \in V : v_{i} \in J, (v_{i}, v_{j}) \in ε}$ , The in-degree and out-degree of node $v_{i}$ are defined as $\deg_{in} (v_{i}) = \sum_{j = 1}^{n} a_{ji}$ and $\deg_{out} (v_{i}) = \sum_{j = 1}^{n} a_{ij}$ respectively, The degree matrix of the digraph G is a diagonal matrix $Δ = [Δ_{ij}]$ , where

Δ_{ij} = {\begin{matrix} 0 & i \neq j \\ \deg_{out} (v_{i}) & i = j \end{matrix}

The graph Laplacian matrix associated with the digraph G is defined as $L (G) = L = Δ - A$ .

Appendix B: Kronecker product

Given matrices $P = {(p_{ij})}_{n \times n} \in R^{m \times n}$ and $Q = {(q_{ij})}_{n \times n} \in R^{p \times q}$ , their Kronecker product (Hom and Johnson, 1999) is defined as

P \otimes Q = [p_{ij} Q] \in R^{mp \times nq} .

For matrices A, B, C and D, with appropriate dimensions, we have the following conditions.

Declaration of conflicting interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grants 61374054 and by Province Natural Science Foundation Research Projection of Shaanxi under Grants 2013JQ8038.

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