Bipartite consensus-based formation control of high-order multi-robot systems with time-varying delays

Abstract

This paper considers a multi-robot formation control scheme for high-order bipartite consensus under time-varying delays. First, a third-order bipartite consensus protocol under time-varying delays is presented, and then a high-order bipartite consensus protocol is extended from it. Then, necessary and sufficient conditions for third-order and high-order multi-agent systems without time-delays are given to attain bipartite consensus following the protocols. Communication time-delays are added to the protocols, and the theory of Lyapunov asymptotic stability is used to obtain sufficient conditions for the multi-agent systems with communication delays to ultimately attain a bipartite consensus. Furthermore, the given protocol is employed for the formation control of multi-robot systems, and a formation control scheme for multi-robot systems with time-varying delays is proposed. Finally, a few simulations are implemented to demonstrate the rationality and availability of the method.

Keywords

Multi-robot system high-order bipartite consensus formation control time-varying delays

Introduction

Formation control is a research hotspot of multi-robot systems. Multi-robot formation coordinates the movement of the system and completes tasks reliably and efficiently. The multi-robot formation has been widely used in the areas of joint search and rescue, mine clearance, surveillance, and geographic survey, among others. Many scholars have studied the multi-robot formation and designed various formation control strategies. Chen et al. (2010) designed a receding-horizon leader-follower method to dominate non-holonomic mobile robot formation and achieve rapid convergence. Fidan et al. (2013) designed a single-view distance-estimation method to control the formation motion of robots. Guillet et al. (2014) proposed a formation control scheme based on adaptive and predictive algorithms to solve formation control from the perspective of path tracking and to obtain high-precision relative positioning. For the multi-robot formation problem of undirected communication graph, Lee et al. (2014) designed a distributed model predictive control framework to solve it. Alonso-Mora et al. (2019) proposed a distributed method for robots to form an ideal formation by constraint optimization and consensus of multiple robots in the presence of obstacles. Xiao and Chen (2018) used a non-linear model predictive control as the controller to design an incremental centralized formation control system. For the situation where the global pose information of the incomplete mobile robot cannot be obtained, the visual-based flexible formation problem was discussed by Wang et al. (2019), and the operation of the robot in an unknown obstacle environment was also considered.

With the rapid development of consensus theory and its successful application in formation control, the consensus method has shown its potential in formation control. Compared with other formation control schemes, consensus-based formation control has advantages such as high reliability, strong self-healing, and good scalability. This has also allowed the application of consensus methods in formation control issues to be extensively studied (Chen et al., 2019; Fu et al., 2018; Han et al., 2018; Wang et al., 2020; Zheng and Wang, 2012). Peng et al. (2016) proposed a consensus-based adaptive neural network strategy, which converges the non-holonomic mobile robot into the desired formation. In practice, there are not only cooperative relationships but also antagonistic relationships between agents. Altafini (2013) proposed the definition of bipartite consensus and provided the conditions for achieving bipartite consensus. Zhang and Chen (2017) studied the equivalence relationship between bipartite consensus and conventional consensus of linear multi-agent systems. Wen et al. (2017) analyzed the distributed bipartite tracking consensus problem of linear multi-agent systems with a dynamic leader. Zong et al. (2019) analyzed the distributed formation control of the multi-robot system based on the bipartite consensus.

In many practical situations, the research on general high-order system consensus is more meaningful than simple first-order system consensus and second-order system consensus. Many systems have high-order dynamics characteristics, and high-order systems can also describe the dynamic characteristics of the system more accurately. The study of the consistency of high-order systems has important practical significance and theoretical value, which has also inspired many scholars to discuss and study the consensus of high-order systems (Gao et al., 2020; Hu et al., 2018; Hua et al., 2016; You et al., 2019). Abdessameud and Tayebi (2018) designed a fully distributed consensus algorithm by employing the static state feedback approach to study the consensus issue under general directed graphs. An adjustable fractional power feedback method was used by Wang et al. (2016) to discuss the adaptive coordination consensus. Valcher and Misra (2014) analyzed the consensus and bipartite consensus of the antagonistic high-order multi-agent dynamic system based on the structural balance communication diagram. Based on linear matrix inequation (LMI), Dou et al. (2015) advanced a devise approach to explore the bipartite consensus for a high-order system with leaders.

The influence of communication delay on consensus control is an important point that cannot be ignored. In practical application, the phenomenon of communication delay inevitably appears because of the overload of communication lines between agents, the asymmetry of information interaction, and the limitation of the physical characteristics of communication equipment on the speed of information transmission. It affects stability and reduces the overall performance of the system. This problem has drawn the interest of many experts. For the second-order multi-agent system, Zhu and Cheng (2010), Qin et al. (2011), Liu et al. (2014), and Shi and Xie (2020) have all discussed the consensus problem with time-varying delays. The consensus problems for heterogeneous systems, linear time-delay systems, and non-linear higher-order systems with reference trajectories have been analyzed by Tian and Zhang (2012), Zhou and Lin (2014), and Huang et al. (2017), respectively. Some scholars—including Guo et al. (2018), Tian et al. (2018), Zhang et al. (2019), and Ren et al. (2020)—have researched the issue of bipartite consensus.

Based on the above work, this paper investigates the multi-robot formation control problem under the high-order bipartite consensus protocol. Taking into account the stability of the system with cooperation and confrontation between agents, a bipartite consensus protocol for the third-order multi-agent system is proposed and extended to high-order bipartite consensus protocol. The bipartite consensus problem of the multi-agent system is studied, which is then turned into a stability problem by gauge transformation and matrix transformation. In the actual multi-agent network, there are different losses and time-delays in information transfer. Based on the above-mentioned bipartite consensus, uneven communication delays are further added to the protocol. For the solution of the bipartite consensus issue with time-delay systems, the theory of Lyapunov asymptotic stability based on the above-mentioned stability problems is used to construct a Lyapunov–Krasovskii functional with time-delay information to analyze it. Different from Liu et al. (2014) and Gao et al. (2020), the bipartite consensus of high-order systems in coopetition networks is studied. Different from Zhang and Chen (2017) and Wen et al. (2017), we consider the bipartite consensus problem of high-order systems with non-uniform time-varying delays. Different from Chen et al. (2010), Guillet et al. (2014), and Alonso Mora et al. (2019), the problem of bipartite consensus-based formation control of high-order multi-robot systems with communication delays is studied.

The remainder of this paper is arranged as follows. The main issues of this paper and the bipartite consensus protocol are presented in section “Preliminaries and problem formation.” The major results and detailed proofs are provided in section “Main results.” In section “Formation control and simulation of mobile robot,” the multi-robot formation control is proposed and the simulation is performed. In section “Conclusion,” the work of this paper is summarized.

Preliminaries and problem formation

Graph theory

The authors consider a signed graph $G = {V, E, A}$ with mixed weights of positive and negative, where $V = {1, 2, \dots, n}$ is the set of nodes, $E \subseteq V \times V$ is the set of edges, $A = [a_{ij}] \in R^{n \times n}$ is the weighted adjacency matrix, and $a_{ij} \neq 0$ iff $(j, i) \in E$ . The relationship between j to i is cooperative if $a_{ij} > 0$ and antagonistic if $a_{ij} < 0$ . Suppose that the self-loop is not considered, that is, $a_{ii} = 0$ , $i = 1, 2, \dots, n$ , when the situation of $a_{ij} a_{ji} \geq 0$ occurs, the signed graph is referred to as digon sign-symmetric. A directed path of the signed graph G(A) is a concatenation of the edges in E, which can be described by $S = {(i_{1}, i_{2}), (i_{2}, i_{3}), \dots, (i_{s - 1}, i_{s})} \subset E$ .

Linear consensus protocols

The third-order system is given by

{\overset{\cdot}{x}}_{i} = v_{i}, {\overset{\cdot}{v}}_{i} = a_{i}, {\overset{\cdot}{a}}_{i} = u_{i}

(1)

where $i = 1, 2, \dots, n$ , $x_{i} \in R^{m}$ denotes the position, $v_{i} \in R^{m}$ denotes the velocity, $a_{i} \in R^{m}$ denotes the acceleration, and $u_{i} \in R^{m}$ denotes the control input (or protocol). For the convenience of research, m is set to 1.

The consensus protocol for system (1) can be depicted by

\begin{matrix} u_{i} = & - γ_{1} v_{i} (t) - γ_{2} a_{i} (t) \\ - \sum_{j \in N_{i}} {β_{0} | a_{ij} | [x_{i} (t - τ_{ij} (t)) - sgn (a_{ij}) x_{j} (t - τ_{ij} (t))] \\ + β_{1} | a_{ij} | [v_{i} (t - τ_{ij} (t)) - sgn (a_{ij}) v_{j} (t - τ_{ij} (t))] \\ + β_{2} | a_{ij} | [a_{i} (t - τ_{ij} (t)) - sgn (a_{ij}) a_{j} (t - τ_{ij} (t))]} \end{matrix}

(2)

where γ₁, γ₂, β₀, β₁, and β₂ are the positive gains, and, τ_ij(t) are the unknown time-varying delays and satisfy the conditions of 0 ≤ τ_ij(t) ≤ h and h > 0.

Concerning the presented protocol equation (2), the bipartite consensus can be reached if $\lim_{t \to \infty} [| x_{i} (t) | - | x_{j} (t) |] = 0$ , $\lim_{t \to \infty} [| v_{i} (t) | - | v_{j} (t) |] = 0$ , $\lim_{t \to \infty} [| a_{i} (t) | - | a_{j} (t) |] = 0$ , $\forall x_{i} (0)$ , $v_{i} (0)$ , and $a_{i} (0) (i, j = 1, 2, \dots, n)$ .

If $τ_{ij} (t) = 0$ , system (1) is simplified as

\overset{\cdot}{x} = v, \overset{\cdot}{v} = a, \overset{\cdot}{a} = - γ_{1} v - γ_{2} a - β_{0} Lx - β_{1} Lv - β_{2} La

(3)

where $v = [v_{1}, \dots, v_{n}]^{T}$ , $x = [x_{1}, \dots, x_{n}]^{T}$ , and $a = [a_{1}, \dots, a_{n}]^{T}$ . The Laplacian matrix L can be given as

l_{iq} = {\begin{matrix} \sum_{j \in N_{i}} | a_{ij} |, & q = i \\ - a_{ik}, & q \neq i \end{matrix}

(4)

Consider the high-order system is given by

{\overset{\cdot}{x}}_{i}^{(0)} = x_{i}^{(1)}, {\overset{\cdot}{x}}_{i}^{(1)} = x_{i}^{(2)}, \dots, {\overset{\cdot}{x}}_{i}^{(e - 2)} = x_{i}^{(e - 1)}, {\overset{\cdot}{x}}_{i}^{(e - 1)} = u_{i},

(5)

where ${\overset{\cdot}{x}}_{i}^{(l)}, l \in [0, e - 1]$ denotes the states.

The high-order control system protocol for system (5) is described as follows

\begin{matrix} u_{i} = & - \sum_{l = 1}^{e - 1} γ_{l} x_{i}^{(l)} (t) \\ - \sum_{l = 0}^{e - 1} \sum_{j \in N_{i}} {β_{l} | a_{ij} | [x_{i}^{(l)} (t - τ_{ij} (t)) - sgn (a_{ij}) x_{j}^{(l)} (t - τ_{ij} (t))]} \end{matrix}

(6)

where $γ_{l}, β_{l}, l \in [0, e - 1]$ are the positive gains. τ_ij(t) are the unknown time-varying delays and satisfy the conditions of 0 ≤ τ_ij(t) ≤ h and h > 0.

The bipartite consensus of protocol equation (6) can be reached if $\lim_{t \to \infty} [| x_{i}^{(l)} (t) | - | x_{j}^{(l)} (t) |] = 0, l = 0, \dots, e - 1$ .

If $τ_{ij} (t) = 0$ , system (5) is rewritten by

\begin{matrix} {\overset{\cdot}{x}}^{(0)} = & x^{(1)}, {\overset{\cdot}{x}}^{(1)} = x^{(2)}, \dots, {\overset{\cdot}{x}}^{(e - 2)} = x^{(e - 1)}, \\ {\overset{\cdot}{x}}^{(e - 1)} = - \sum_{l = 1}^{e - 1} γ_{l} x^{(l)} - \sum_{l = 0}^{e - 1} β_{l} L x^{(l)} \end{matrix}

(7)

where $x^{(l)} = [x_{1}^{(l)}, \dots, x_{n}^{(l)}]^{T}, l \in [0, e - 1]$ .

Gauge transformation

The signed graph G(A) is structurally balanced. If its nodes are divided into V₁ and V₂, $V_{1} \overset{}{\cup} V_{2} = V$ , $V_{1} \overset{}{\cap} V_{2} = \emptyset$ , and $a_{ij} \geq 0$ , $\forall i, j \in V_{p}$ , $(p \in {1, 2})$ , $a_{ij} \leq 0$ , $\forall i \in V_{p}$ , $j \in V_{b}$ , $(p, b \in (1, 2), p \neq b)$ , and G(A) is structurally unbalanced otherwise.

$B = {B | B = diag {ξ_{1}, \dots, ξ_{n}}, ξ_{i} = \pm 1, i = 1, 2, \dots, n}$ is provided as a set of canonical transformations. The transformation of the system (3) is as follows

ϑ = Bx, ζ = Bv, ε = Ba, B \in B

(8)

Since $B^{- 1} = B$ , then $x = B ϑ$ , $v = B ζ$ , $a = B ε$ , $\overset{\cdot}{ϑ} = B \overset{\cdot}{x} = ζ$ , and $\overset{\cdot}{ζ} = B \overset{\cdot}{v} = ε$ .

For system (3), by $\overset{\cdot}{a} = - γ_{1} v - γ_{2} a - β_{0} LB ϑ - β_{1} LB ζ - β_{2} LB ε$ , one can obtain that

\overset{\cdot}{ε} = B \overset{\cdot}{a} = - γ_{1} ζ - γ_{2} ε - β_{0} L_{B} ϑ - β_{1} L_{B} ζ - β_{2} L_{B} ε

where $L_{B} = BLB$ is described as a new Laplacian matrix, which can be shown as follows

l_{B, iq} = {\begin{matrix} \sum_{j \in N_{i}} | a_{ij} |, & q = i \\ - ξ_{i} ξ_{q} a_{iq}, & q \neq i \end{matrix}

Lemma 1

If there is a structurally balanced directed graph with a spanning tree is represented by $G (A_{g})$ (Tian et al., 2018), then

$\exists B \in B$ , and any elements contained in BAB are on-negative.

0 is a simple eigenvalue of L.

The transformation is applied to equation (7) as

ϑ^{(l)} = B x^{(l)}, l \in [0, e - 1], B \in B

(9)

Since $B^{- 1} = B$ , then $x^{(l)} = B ϑ^{(l)}$ .

For system (7), by ${\overset{\cdot}{x}}^{(e - 1)} = - \sum_{l = 1}^{e - 1} γ_{l} x^{(l)} - \sum_{l = 0}^{e - 1} β_{l} LB ϑ^{(l)}$ , one can obtain that

{\overset{\cdot}{ϑ}}^{(e - 1)} = B {\overset{\cdot}{x}}^{(e - 1)} = - \sum_{l = 1}^{e - 1} γ_{l} ϑ^{(l)} - \sum_{l = 0}^{e - 1} β_{l} L_{B} ϑ^{(l)}

Main results

Networks without time-delays

By equation (8), the third-order system (1) with the presented protocol equation (2) $(τ_{ij} (t) = 0)$ is converted into

\overset{\cdot}{ϑ} = ζ, \overset{\cdot}{ζ} = ε, \overset{\cdot}{ε} = - γ_{1} ζ - γ_{2} ε - β_{0} L_{B} ϑ - β_{1} L_{B} ζ - β_{2} L_{B} ε

(10)

Denote $ϑ = [ϑ_{1}, \dots, ϑ_{n}]^{T}$ , $ζ = [ζ_{1}, \dots, ζ_{n}]^{T}$ , $ε = [ε_{1}, \dots, ε_{n}]^{T}$ , and $Γ = [ϑ^{T}, ζ^{T}, ε^{T}]^{T}$ . Equation (10) can be rewritten as

\overset{\cdot}{Γ} (t) = [\begin{matrix} 0 & I_{n} & 0 \\ 0 & 0 & I_{n} \\ - β_{0} L_{B} & - γ_{1} I_{n} - β_{1} L_{B} & - γ_{2} I_{n} - β_{2} L_{B} \end{matrix}] Γ (t)

(11)

Remark 1

Owing to $G (A_{g})$ , $\exists B \in B$ causes all elements contained in BAB to be non-negative, and 0 is a simple eigenvalue of $L_{B} = BLB$ . Through equation (8), system (3) can be transformed into equation (11). Based on the definition of B, $\lim_{t \to \infty} [| a_{i} (t) | - | a_{j} (t) |] = 0$ iff $\lim_{t \to \infty} ε_{i} (t) = \lim_{t \to \infty} ε_{j} (t)$ . In this situation, $\lim_{t \to \infty} [| v_{i} (t) | - | v_{j} (t) |] = 0$ iff $\lim_{t \to \infty} ζ_{i} (t) = \lim_{t \to \infty} ζ_{j} (t)$ ; $\lim_{t \to \infty} [| x_{i} (t) | - | x_{j} (t) |] = 0$ iff $\lim_{t \to \infty} ϑ_{i} (t) = \lim_{t \to \infty} ϑ_{j} (t)$ , $\forall x_{i} (0)$ , $v_{i} (0)$ , and $a_{i} (0) (i, j = 1, 2, \dots, n)$ . Thus, the third-order system (1) will asymptotically reach bipartite consensus iff $\lim_{t \to \infty} [ε_{i} (t) - ε_{j} (t)] = 0$ , $\lim_{t \to \infty} [ζ_{i} (t) - ζ_{j} (t)] = 0$ , and $\lim_{t \to \infty} [ϑ_{i} (t) - ϑ_{j} (t)] = 0$ .

Proposition 1

In the presence of $G (A_{g})$ , the bipartite consensus can be asymptotically reached for the third-order system (1) of application presented protocol equation (2) iff a positive definite matrix $K \in R^{3 (n - 1) \times 3 (n - 1)}$ makes the following inequality hold

\begin{matrix} [\begin{matrix} 0 & {I_{n}}_{- 1} & 0 \\ 0 & 0 & I_{n - 1} \\ - β_{0} M L_{B} N & - γ_{1} I_{n - 1} - M β_{1} L_{B} N & - γ_{2} I_{n - 1} - β_{2} M L_{B} N \end{matrix}] K \\ + K [\begin{matrix} 0 & {I_{n}}_{- 1} & 0 \\ 0 & 0 & I_{n - 1} \\ - β_{0} M L_{B} N & - γ_{1} I_{n - 1} - M β_{1} L_{B} N & - γ_{2} I_{n - 1} - β_{2} M L_{B} N \end{matrix}] < 0 \end{matrix}

(12)

where $L_{B} = BLB$ , $M = [\begin{matrix} 1 & - I_{n - 1} \end{matrix}]$ , and $N = [\begin{matrix} 0 & - I_{n - 1} \end{matrix}]^{T}$ .

Proof

Suppose

{\tilde{ε}}_{i} = ε_{1} - ε_{i + 1}, \tilde{ε} = {[{\tilde{ε}}_{1}, \dots, {\tilde{ε}}_{n - 1}]}^{T}

(13)

\tilde{ζ} = {[{\tilde{ζ}}_{1}, \dots, {\tilde{ζ}}_{n - 1}]}^{T}

(14)

and

\tilde{ϑ} = {[{\tilde{ϑ}}_{1}, \dots, {\tilde{ϑ}}_{n - 1}]}^{T}

(15)

Then, $\tilde{ϑ} = M ϑ$ , $\tilde{ζ} = M ζ$ , and $\tilde{ε} = M ε$ were obtained. Therefore

\dot{\tilde{ϑ}} = \tilde{ζ}

(16)

\dot{\tilde{ζ}} = \tilde{ε}

(17)

and

\dot{\tilde{ε}} = M \overset{\cdot}{ε} = - γ_{1} \tilde{ζ} - γ_{2} \tilde{ε} - β_{0} M L_{B} N \tilde{ϑ} - β_{1} M L_{B} N \tilde{ζ} - β_{2} M L_{B} N \tilde{ε}

(18)

It is not difficult to see that $\lim_{t \to \infty} [ϑ_{i} (t) - ϑ_{j} (t)] = 0$ and $\lim_{t \to \infty} {\tilde{ϑ}}_{i} (t) = 0$ are equivalent. Therefore, the bipartite consensus turns into a stability problem.

Let $\tilde{Γ} = [{\tilde{ϑ}}^{T}, {\tilde{ζ}}^{T}, {\tilde{ε}}^{T}]^{T}$ , then

\dot{\tilde{Γ}} (t) = [\begin{matrix} 0 & {I_{n}}_{- 1} & 0 \\ 0 & 0 & I_{n - 1} \\ - β_{0} M L_{B} N & - γ_{1} I_{n - 1} - β_{1} M L_{B} N & - γ_{2} I_{n - 1} - β_{2} M L_{B} N \end{matrix}] \tilde{Γ} (t)

(19)

and $\tilde{Γ} (t) = ϖ Γ (t)$ , where $ϖ = diag {M, M, M}$ and $rank (ϖ) = 3 (n - 1)$ . For equation (11), in case that $Γ_{i} (t)$ , $i = 1, 2, \dots, 3 n$ is its solution, then $ϖ Γ_{i}$ is the solution of equation (19).

Assuming that the bipartite consensus can be asymptotically reached for the third-order system (1) under the presented protocol equation (2), that is, $\lim_{t \to \infty} [ϑ_{i} (t) - ϑ_{j} (t)] = 0$ , $\lim_{t \to \infty} [ζ_{i} (t) - ζ_{j} (t)] = 0$ , and $\lim_{t \to \infty} [ε_{i} (t) - ε_{j} (t)] = 0$ . Then, $\lim_{t \to \infty} ϖ Γ_{i} (t) = 0$ ; therefore, equation (12) holds.

Furthermore, the bipartite consensus is asymptotically reached for system (3) if inequality equation (12) holds; in other words, if $\lim_{t \to \infty} {\tilde{Γ}}_{i} (t) = \lim_{t \to \infty} ϖ Γ_{i} (t) = 0$ . Another description can be given as follows

\begin{matrix} \lim_{t \to \infty} [ϑ_{1} (t) - ϑ_{i} (t)] = 0, \lim_{t \to \infty} [ζ_{i} (t) - ζ_{j} (t)] = 0, \\ \lim_{t \to \infty} [ε_{i} (t) - ε_{j} (t)] = 0 \end{matrix}

Therefore, $\lim_{t \to \infty} [ϑ_{i} (t) - ϑ_{j} (t)] = 0 = \lim_{t \to \infty} [ϑ_{i} (t) - ϑ_{1} (t)] + \lim_{t \to \infty} [ϑ_{1} (t) - ϑ_{j} (t)] = 0$ , $\lim_{t \to \infty} [ζ_{i} (t) - ζ_{j} (t)] = 0 = \lim_{t \to \infty} [ζ_{i} (t) - ζ_{1} (t)] + \lim_{t \to \infty} [ζ_{1} (t) - ζ_{j} (t)] = 0$ , and $\lim_{t \to \infty} [ε_{i} (t) - ε_{j} (t)] = 0 = \lim_{t \to \infty} [ε_{i} (t) - ε_{1} (t)] + \lim_{t \to \infty} [ε_{1} (t) - ε_{j} (t)] = 0$ , that is, $\lim_{t \to \infty} [| x_{i} (t) | - | x_{j} (t) |] = 0$ , $\lim_{t \to \infty} [| v_{i} (t) | - | v_{j} (t) |] = 0$ , and $\lim_{t \to \infty} [| a_{i} (t) | - | a_{j} (t) |] = 0$ , $\forall x_{i} (0)$ , $v_{i} (0)$ , $a_{i} (0) (i, j = 1, 2, \dots, n)$ . The bipartite consensus for the protocol equation (2) can therefore be asymptotically reached.

With equation (9), the system (5) with the protocol equation (6) $(τ_{ij} (t) = 0)$ is converted into

{\overset{\cdot}{ξ}}^{(e - 1)} = B {\overset{\cdot}{x}}^{(e - 1)} = - \sum_{l = 1}^{e - 1} γ_{l} ϑ^{(l)} - \sum_{l = 0}^{e - 1} β_{l} L_{B} ϑ^{(l)}

(20)

Denote $Γ = [ϑ^{(0) T}, ϑ^{(1) T}, \dots, ϑ^{(e - 1) T}]^{T}$ , then equation (20) can be rewritten as

\overset{\cdot}{Γ} (t) = [\begin{matrix} 0 & I_{n} & 0 & \dots & 0 \\ 0 & 0 & I_{n} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & I_{n} \\ - β_{0} L_{B} & - γ_{1} I_{n} - β_{1} L_{B} & - γ_{2} I_{n} - β_{2} L_{B} & \dots & - γ_{e - 1} I_{n} - β_{e - 1} L_{D} \end{matrix}] Γ (t)

(21)

Remark 2

The system (5) asymptotically reaches bipartite consensus iff $\lim_{t \to \infty} ϑ_{i}^{l} (t) = ϑ_{j}^{l} (t)$ , that is, $\lim_{t \to \infty} [| x_{i}^{(l)} (t) | - | x_{j}^{(l)} (t) |] = 0, l = 0, \dots, e - 1$ .

Proposition 2

In the presence of $G (A_{g})$ , the bipartite consensus of the system (5) with the protocol equation (6) $(τ_{ij} (t) = 0)$ can be asymptotically reached iff a positive definite matrix $S \in R^{e (n - 1) \times e (n - 1)}$ makes the following inequality hold

\begin{matrix} [\begin{matrix} 0 & I_{n - 1} & 0 & \dots & 0 \\ 0 & 0 & I_{n - 1} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & I_{n - 1} \\ - β_{0} L_{B} & - γ_{1} I_{n - 1} - β_{1} L_{B} & - γ_{2} I_{n - 1} - β_{2} L_{B} & \dots & - γ_{e - 1} I_{n - 1} - β_{e - 1} L_{B} \end{matrix}] S \\ + S [\begin{matrix} 0 & I_{n - 1} & 0 & \dots & 0 \\ 0 & 0 & I_{n - 1} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & I_{n - 1} \\ - β_{0} L_{B} & - γ_{1} I_{n - 1} - β_{1} L_{B} & - γ_{2} I_{n - 1} - β_{2} L_{B} & \dots & - γ_{e - 1} I_{n - 1} - β_{e - 1} L_{B} \end{matrix}] < 0 \end{matrix}

(22)

Proof

Suppose

{\tilde{ϑ}}_{i}^{l} = ϑ_{1}^{l} - ϑ_{i + 1}^{l}, {\tilde{ϑ}}^{l} = [{\tilde{ϑ}}_{1}^{l}, \dots, {\tilde{ϑ}}_{n - 1}^{l}]^{T}

(23)

Then, ${\tilde{ϑ}}^{l} = M ϑ^{l}$ was obtained. Therefore

{\dot{\tilde{ϑ}}}^{e - 1} = M {\overset{\cdot}{ϑ}}^{e - 1} = - \sum_{l = 1}^{e - 1} γ_{l} {\tilde{ϑ}}^{(l)} - \sum_{l = 0}^{e - 1} β_{l} M L_{B} N {\tilde{ϑ}}^{(l)}

(24)

Let $\bar{Γ} = [{\tilde{ϑ}}^{(0) T}, {\tilde{ϑ}}^{(1) T}, \dots, {\tilde{ϑ}}^{(e - 1) T}]^{T}$ , then

\overset{\cdot}{\bar{Γ}} (t) = [\begin{matrix} 0 & I_{n - 1} & 0 & \dots & 0 \\ 0 & 0 & I_{n - 1} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & I_{n - 1} \\ - β_{0} L_{B} & - γ_{1} I_{n - 1} - β_{1} L_{B} & - γ_{2} I_{n - 1} - β_{2} L_{B} & \dots & - γ_{e - 1} I_{n - 1} - β_{e - 1} L_{B} \end{matrix}] \bar{Γ} (t)

(25)

and $\bar{Γ} (t) = \overset{⌢}{ϖ} Γ (t)$ , where $\overset{⌢}{ϖ} = diag {M_{1}, \dots, M_{e}}$ and $rank (\overset{⌢}{ϖ}) = e (n - 1)$ . For equation (21), in case that $Γ_{i} (t), i = 1, 2, \dots, en$ is its solution, then $ϖ Γ_{i}$ is the solution of equation (25).

Assuming that the bipartite consensus can be asymptotically reached for the high-order system (5) employing the high-order protocol equation (6), that is, $\lim_{t \to \infty} [ϑ_{i}^{l} (t) - ϑ_{j}^{l} (t)] = 0$ . Then, $\lim_{t \to \infty} \overset{⌢}{ϖ} Γ_{i} (t) = 0$ ; hence, equation (22) holds.

Furthermore, the bipartite consensus is asymptotically reached for system (25) if inequality equation (22) holds; in other words, if $\lim_{t \to \infty} {\bar{Γ}}_{i} (t) = \lim_{t \to \infty} \overset{⌢}{ϖ} Γ_{i} (t) = 0$ . Another description can be given as $\lim_{t \to \infty} [ϑ_{i}^{l} (t) - ϑ_{j}^{l} (t)] = 0$ .

Therefore, $\lim_{t \to \infty} [ϑ_{i}^{l} (t) - ϑ_{j}^{l} (t)] = 0 = \lim_{t \to \infty} [ϑ_{i}^{l} (t) - ϑ_{1}^{l} (t)] + \lim_{t \to \infty} [ϑ_{1}^{l} (t) - ϑ_{j}^{l} (t)] = 0$ , that is, $\lim_{t \to \infty} [| x_{i}^{l} (t) | - | x_{j}^{l} (t) |] = 0, (i, j = 1, 2, \dots, n)$ . The bipartite consensus for the protocol equation (6) can therefore be asymptotically reached.

Networks with time-varying delays

If $τ_{ij} (t) \neq 0$ , let $τ_{ij} (t) \in {τ_{q} (t) : q = 1, 2, \dots, m}$ , and the piecewise continuous time-varying delays $τ_{q} (t), q = 1, 2, \dots, m$ satisfy $0 \leq τ_{q} (t) \leq h$ and ${\overset{\cdot}{τ}}_{q} (t) \leq λ$ . Then, under the combination of conversion equation (8) and protocol equation (2), system (1) is described by

\overset{\cdot}{Γ} (t) = Ψ Γ (t) + \sum_{q = 1}^{m} L_{q} Γ (t - τ_{q} (t))

(26)

where

\begin{matrix} Ψ = [\begin{matrix} 0 & I_{n} & 0 \\ 0 & 0 & I_{n} \\ 0 & - γ_{1} I_{n} & - γ_{2} I_{n} \end{matrix}], \\ L_{q} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ - β_{0} L_{q} & - β_{1} L_{q} & - β_{2} L_{q} \end{matrix}], L_{q} = [l_{ij}^{(q)}] \end{matrix}

are defined as follows

l_{ij}^{(q)} = {\begin{matrix} \sum_{j = 1, j \neq i} | a_{ij} |, j = i \\ 0, j \neq i, τ_{q} (\cdot) \neq τ_{ij} (\cdot) \\ - ξ_{i} ξ_{j} a_{ij}, j \neq i, τ_{q} (\cdot) = τ_{ij} (\cdot) \end{matrix}

Note that $L_{q} 1 = 0, q = 1, 2, \dots, m$ and $\sum_{q = 1}^{m} L_{q} = L_{B}$ . According to equations (13)–(15), a reduced-order system description of equation (26) can be obtained as follows

\dot{\tilde{Γ}} (t) = \tilde{Ψ} \tilde{Γ} (t) + \sum_{q = 1}^{m} {\tilde{L}}_{q} \tilde{Γ} (t - τ_{q} (t))

(27)

where

\begin{matrix} \tilde{Ψ} = [\begin{matrix} 0 & I_{n - 1} & 0 \\ 0 & 0 & I_{n - 1} \\ 0 & - γ_{1} I_{n - 1} & - γ_{2} I_{n - 1} \end{matrix}], \\ {\tilde{L}}_{q} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ - β_{0} M L_{q} N & - β_{1} M L_{q} N & - β_{2} M L_{q} N \end{matrix}] \end{matrix}

Lemma 2

When and only when all solutions tend to zero for system (27), the third-order system (1) employing equation (2) reaches the bipartite consensus asymptotically.

Proof

It is easy to observe the sufficiency.

Necessity

Suppose that in the case of $t \to \infty$ , a solution $\tilde{Γ} (t)$ that does not tend to 0 exists, and ${\tilde{Γ}}^{T} (t) = [{\tilde{ϑ}}_{1} (t), \dots, {\tilde{ϑ}}_{n - 1} (t) {\tilde{ζ}}_{1} (t), \dots, {\tilde{ζ}}_{n - 1} (t) {\tilde{ε}}_{1} (t), \dots, {\tilde{ε}}_{n - 1} (t)]$ . Let $ϑ_{1} (t)$ , $ζ_{1} (t)$ , and $ε_{1} (t)$ conform to ${\overset{\cdot}{ϑ}}_{1} (t) = ζ_{1} (t)$ and ${\overset{\cdot}{ζ}}_{1} (t) = ε_{1} (t)$ . Let $ζ_{i + 1} (t) = ζ_{1} (t) - {\tilde{ζ}}_{i} (t), i = 1, 2, \dots, n - 1$ . One can obtain from equation (27) that

{\begin{matrix} {\overset{\cdot}{ϑ}}_{1} (t) = ζ_{1} (t) \\ {\overset{\cdot}{ζ}}_{1} (t) = ε_{1} (t) \\ {\overset{\cdot}{ζ}}_{i + 1} (t) = {\overset{\cdot}{ζ}}_{1} (t) - {\dot{\tilde{ζ}}}_{i} (t) = ε_{i + 1} (t) \end{matrix}

(28)

From equation (26) and $L_{q} 1 = 0$ , the following can be obtained

\begin{matrix} \dot{\tilde{ε}} (t) = - γ_{1} \tilde{ζ} (t) - γ_{2} \tilde{ε} (t) - \sum_{q = 1}^{m} β_{0} M L_{q} N \tilde{ϑ} (t - τ_{q} (t)) \\ - \sum_{q = 1}^{m} β_{1} M L_{q} N \tilde{ζ} (t - τ_{q} (t)) - \sum_{q = 1}^{m} β_{2} M L_{q} N \tilde{ε} (t - τ_{q} (t)) \\ = - γ_{1} M ζ (t) - γ_{2} M ε (t) \\ - \sum_{q = 1}^{m} β_{0} M L_{q} [ϑ_{1} (t - τ_{q} (t)) 1 + N \tilde{ϑ} (t - τ_{q} (t))] \\ - \sum_{q = 1}^{m} β_{1} M L_{q} [ζ_{1} (t - τ_{q} (t)) 1 + N \tilde{ζ} (t - τ_{q} (t))] \\ - \sum_{q = 1}^{m} β_{2} M L_{q} [ε_{1} (t - τ_{q} (t)) 1 + N \tilde{ε} (t - τ_{q} (t))] \\ = - γ_{1} M ζ (t) - γ_{2} M ε (t) - \sum_{q = 1}^{m} β_{0} M L_{q} ϑ (t - τ_{q} (t)) \\ - \sum_{q = 1}^{m} β_{1} M L_{q} ζ (t - τ_{q} (t)) - \sum_{q = 1}^{m} β_{2} M L_{q} ε (t - τ_{q} (t)) \end{matrix}

(29)

From equations (28) and (29), $Γ (t)$ is a solution of system (26) that can be obtained.

Since $\lim_{t \to \infty} \tilde{Γ} (t) \neq 0$ , $\exists i, j$ , such that $\lim_{t \to \infty} ζ_{i} \neq 0$ , $\lim_{t \to \infty} ε_{i} \neq 0$ or

$\lim_{t \to \infty} [ϑ_{i} (t) - ϑ_{j} (t)] = \lim_{t \to \infty} [ϑ_{i} (t) - ϑ_{1} (t)] + \lim_{t \to \infty} [ϑ_{1} (t) - ϑ_{j} (t)] \neq 0$ ,

$\lim_{t \to \infty} [ζ_{i} (t) - ζ_{j} (t)] = \lim_{t \to \infty} [ζ_{i} (t) - ζ_{1} (t)] + \lim_{t \to \infty} [ζ_{1} (t) - ζ_{j} (t)] \neq 0$ , and

$\lim_{t \to \infty} [ε_{i} (t) - ε_{j} (t)] = \lim_{t \to \infty} [ε_{i} (t) - ε_{1} (t)] + \lim_{t \to \infty} [ε_{1} (t) - ε_{j} (t)] \neq 0$ .

It can be concluded that the solution $Γ (t)$ cannot make the bipartite consensus asymptotically reached for the system (1). In other words, the analysis result is contradictory to the initial hypothesis.

Lemma 3

Denote any real differentiable vector function and any positive definite constant matrix by $x (t) \in R^{n}$ and $D = D^{T} \in R^{n \times n}$ , respectively (Sun et al., 2008), one has

h \int_{t - τ_{q} (t)}^{t} {\overset{\cdot}{x}}^{T} (s) D \overset{\cdot}{x} (s) ds \geq {[x (t) - x (t - τ_{q} (t))]}^{T} D [x (t) - x (t - τ_{q} (t))]

where $0 \leq τ_{q} (t) \leq h$ .

Theorem 1

In the presence of $G (A_{g})$ , the bipartite consensus can be asymptotically reached for third-order system (1) employing equation (2), if there exist positive definite matrices $ρ, ι_{q}, υ_{q} \in R^{3 (n - 1) \times 3 (n - 1)}$ , such that

\begin{matrix} [\begin{matrix} {\tilde{Ψ}}^{T} ρ + ρ \tilde{Ψ} + \sum_{q = 1}^{m} ι_{q} - \sum_{q = 1}^{m} υ_{q} & ρ \tilde{L} + \tilde{υ} \\ {\tilde{L}}^{T} ρ + {\tilde{υ}}^{T} & - (1 - λ) \hat{ι} - \hat{υ} \end{matrix}] \\ + h^{2} [\begin{matrix} {\tilde{Ψ}}^{T} \\ {\tilde{L}}^{T} \end{matrix}] (\sum_{q = 1}^{m} υ_{q}) [\begin{matrix} \tilde{Ψ} & \tilde{L} \end{matrix}] < 0 \end{matrix}

(30)

where $\tilde{L} = [{\tilde{L}}_{1}, {\tilde{L}}_{2} \dots, {\tilde{L}}_{m}]$ , $\tilde{υ} = [{\tilde{υ}}_{1}, {\tilde{υ}}_{2} \dots, {\tilde{υ}}_{3}]$ , $\hat{ι} = diag {ι_{1}, ι_{2} \dots, ι_{m}}$ , $\hat{υ} = diag {υ_{1}, υ_{2} \dots, υ_{m}}$ , and $\tilde{Ψ}$ , ${\tilde{L}}_{q}$ are the same as before.

Proof

Introduce the following Lyapunov–Krasovskii function

V (t) = V_{1} (t) + V_{2} (t) + V_{3} (t)

where $V_{1} (t) = {\tilde{Γ}}^{T} (t) ρ \tilde{Γ} (t)$ , $V_{2} (t) = \sum_{q = 1}^{m} \int_{t - τ_{q} (t)}^{t} {\tilde{Γ}}^{T} (s) ι_{q} \tilde{Γ} (s) ds$ , and $V_{3} (t) = h \sum_{q = 1}^{m} \int_{- h}^{0} \int_{t + θ}^{t} {\dot{\tilde{Γ}}}^{T} (s) υ_{q} \dot{\tilde{Γ}} (s) dsd θ$ .

With the solution of equation (27) and Lemma 3, one can obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} (t) & = \overset{T}{\tilde{Γ}} (t) ρ \tilde{Γ} (t) + {\tilde{Γ}}^{T} (t) ρ \dot{\tilde{Γ}} (t) \\ = {\tilde{Γ}}^{T} (t) [{\tilde{Ψ}}^{T} ρ + ρ \tilde{Ψ}] \tilde{Γ} (t) + 2 \sum_{q = 1}^{m} {\tilde{Γ}}^{T} (t) ρ {\tilde{L}}_{q} \tilde{Γ} (t - τ_{q} (t)) \end{matrix}

(31)

\begin{matrix} {\overset{\cdot}{V}}_{2} (t) & = \sum_{q = 1}^{m} [{\tilde{Γ}}^{T} (t) ι_{q} \tilde{Γ} (t) - (1 - {\overset{\cdot}{τ}}_{q} (t)) {\tilde{Γ}}^{T} (t - τ_{q} (t)) ι_{q} \tilde{Γ} (t - τ_{q} (t))] \\ \leq \sum_{q = 1}^{m} {\tilde{Γ}}^{T} (t) ι_{q} \tilde{Γ} (t) - (1 - λ) \sum_{q = 1}^{m} {\tilde{Γ}}^{T} (t - τ_{q} (t)) ι_{q} \tilde{Γ} (t - τ_{q} (t)) \end{matrix}

(32)

\begin{matrix} {\overset{\cdot}{V}}_{3} (t) & = h \sum_{q = 1}^{m} [h {\dot{\tilde{Γ}}}^{T} (t) υ_{q} \dot{\tilde{Γ}} (t)] - \int_{t - h}^{t} {\dot{\tilde{Γ}}}^{T} (s) υ_{q} \dot{\tilde{Γ}} (s) ds \\ \leq h^{2} \sum_{q = 1}^{m} \overset{T}{\tilde{Γ}} (t) υ_{q} \dot{\tilde{Γ}} (t) - h \sum_{q = 1}^{m} \int_{t - τ_{q} (t)}^{t} \overset{T}{\tilde{Γ}} (s) υ_{q} \dot{\tilde{Γ}} (s) ds \\ \leq h^{2} \sum_{q = 1}^{m} {\dot{\tilde{Γ}}}^{T} (t) υ_{q} \dot{\tilde{Γ}} (t) \\ - \sum_{q = 1}^{m} {[\tilde{Γ} (t) - \tilde{Γ} (t - τ_{q} (t))]}^{T} υ_{q} [\tilde{Γ} (t) - \tilde{Γ} (t - τ_{q} (t))] \end{matrix}

(33)

By combining equations (31)–(33), the following can be obtained

\begin{matrix} \overset{\cdot}{V} (t) & \leq h^{2} \sum_{q = 1}^{m} \overset{T}{\tilde{Γ}} (t) υ_{q} \dot{\tilde{Γ}} (t) \\ + \sum_{q = 1}^{m} {\tilde{Γ}}^{T} (t) ι_{q} \tilde{Γ} (t) + {\tilde{Γ}}^{T} (t) [{\tilde{Ψ}}^{T} ρ + ρ \tilde{Ψ}] \tilde{Γ} (t) \\ + 2 \sum_{q = 1}^{m} {\tilde{Γ}}^{T} (t) ρ {\tilde{L}}_{q} \tilde{Γ} (t - τ_{q} (t)) \\ - (1 - λ) \sum_{q = 1}^{m} {\tilde{Γ}}^{T} (t - τ_{q} (t)) ι_{q} \tilde{Γ} (t - τ_{q} (t)) \\ - \sum_{q = 1}^{m} {[\tilde{Γ} (t) - \tilde{Γ} (t - τ_{q} (t))]}^{T} υ_{q} [\tilde{Γ} (t) - \tilde{Γ} (t - τ_{q} (t))] \end{matrix}

From equations (27) and (30), $\overset{\cdot}{V} (t) < 0$ can be obtained. Hence, system (27) achieves asymptotic stability.

Given the above analysis, it can be understood that the bipartite consensus can be asymptotically reached for third-order system (1) employing equation (2).

If $τ_{ij} (t) \neq 0$ , under the combination of protocol equation (6) and conversion equation (9), system (5) is described by

\overset{\cdot}{Γ} (t) = Ψ Γ (t) + \sum_{q = 1}^{m} L_{q} Γ (t - τ_{q} (t))

(34)

where

\begin{matrix} Ψ = [\begin{matrix} 0 & I_{n} & 0 & \dots & 0 \\ 0 & 0 & I_{n} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & I_{n} \\ 0 & - γ_{1} I_{n} & - γ_{2} I_{n} & \dots & - γ_{e - 1} I_{n} \end{matrix}], \\ L_{q} = [\begin{matrix} 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 0 \\ - β_{0} L_{q} & - β_{1} L_{q} & - β_{2} L_{q} & \dots & - β_{e - 1} L_{q} \end{matrix}] \end{matrix}

According to equation (23), a reduced-order system description of equation (34) can be achieved as follows

\overset{\cdot}{\bar{Γ}} (t) = \bar{Ψ} \bar{Γ} (t) + \sum_{q = 1}^{m} {\bar{L}}_{q} \bar{Γ} (t - τ_{q} (t))

(35)

where

\begin{matrix} \bar{Ψ} = [\begin{matrix} 0 & I_{n - 1} & 0 & \dots & 0 \\ 0 & 0 & I_{n - 1} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & I_{n - 1} \\ 0 & - γ_{1} I_{n - 1} & - γ_{2} I_{n - 1} & \dots & - γ_{e - 1} I_{n - 1} \end{matrix}], \\ {\bar{L}}_{q} = [\begin{matrix} 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 0 \\ - β_{0} M L_{q} N & - β_{1} M L_{q} N & - β_{2} M L_{q} N & \dots & - β_{e - 1} M L_{q} N \end{matrix}] \end{matrix}

Proof

The sufficiency is obvious.

Necessity

Suppose that in the case of $t \to \infty$ , a solution $\bar{Γ} (t)$ that does not tend to 0 exists, and ${\bar{Γ}}^{T} (t) = [{\tilde{ϑ}}^{(0)} (t), {\tilde{ϑ}}^{(1)} (t), \dots, {\tilde{ϑ}}^{(e - 1)} (t)]$ . Let $ϑ_{i + 1}^{l} (t) = ϑ_{1}^{l} (t) - {\tilde{ϑ}}_{i}^{l} (t), i = 1, 2, \dots, n - 1$ . It can be seen from equation (35) that

{\begin{matrix} {\overset{\cdot}{ϑ}}_{1}^{(0)} (t) = ϑ_{1}^{(1)} (t) \\ {\overset{\cdot}{ϑ}}_{1}^{(1)} (t) = ϑ_{1}^{(2)} (t) \\ ⋮ \\ {\overset{\cdot}{ϑ}}_{i + 1}^{(e - 2)} (t) = {\overset{\cdot}{ϑ}}_{1}^{(e - 2)} (t) - {\overset{\cdot}{\tilde{}} ϑ}_{1}^{(e - 2)} (t) = ϑ_{i + 1}^{(e - 1)} (t) \end{matrix}

(36)

From equation (34) and $L_{q} 1 = 0$ , the following can be obtained

\begin{matrix} {\overset{\cdot}{\tilde{ϑ}}}^{e - 1} (t) & = - \sum_{l = 1}^{e - 1} γ_{l} {\tilde{ϑ}}^{(l)} (t) - \sum_{l = 0}^{e - 1} \sum_{q = 1}^{m} β_{l} M L_{q} N {\tilde{ϑ}}^{(l)} (t - τ_{q} (t)) \\ = - \sum_{l = 1}^{e - 1} γ_{l} M ϑ^{(l)} (t) - \sum_{l = 0}^{e - 1} \sum_{q = 1}^{m} β_{l} M L_{q} \\ [ϑ_{1}^{(l)} (t - τ_{q} (t)) 1 + N {\tilde{ϑ}}^{(l)} (t - τ_{q} (t))] \\ = - \sum_{l = 1}^{e - 1} γ_{l} M ϑ^{(l)} (t) - \sum_{l = 0}^{e - 1} \sum_{q = 1}^{m} β_{l} M L_{q} ϑ^{(l)} (t - τ_{q} (t)) \end{matrix}

(37)

From equations (36) and (37), $Γ (t)$ is a solution of system (34) that can be obtained. Since $\lim_{t \to \infty} \bar{Γ} (t) \neq 0$ , $\exists i, j$ , such that $\lim_{t \to \infty} ϑ_{i}^{l} \neq 0$ or $\lim_{t \to \infty} [ϑ_{i}^{l} (t) - ϑ_{j}^{l} (t)] = \lim_{t \to \infty} [ϑ_{i}^{l} (t) - ϑ_{1}^{l} (t)] + \lim_{t \to \infty} [ϑ_{1}^{l} (t) - ϑ_{j}^{l} (t)] \neq 0$ .

It can be concluded that the solution $Γ (t)$ cannot make the bipartite consensus of the system (5) be asymptotically reached. Thus, the analysis result is contradictory to the initial hypothesis.

Theorem 2

In the presence of $G (A_{g})$ , the bipartite consensus can be asymptotically reached for system (5) employing equation (6), if there exist positive definite matrices $\overset{⌢}{ρ}, {\overset{⌢}{ι}}_{q}, {\overset{⌢}{υ}}_{q} \in R^{e (n - 1) \times e (n - 1)}$ , such that

\begin{matrix} [\begin{matrix} {\bar{Ψ}}^{T} \overset{⌢}{ρ} + \overset{⌢}{ρ} \bar{Ψ} + \sum_{q = 1}^{m} {\overset{⌢}{ι}}_{q} - \sum_{q = 1}^{m} {\overset{⌢}{υ}}_{q} & \overset{⌢}{ρ} \bar{L} + \tilde{υ} \\ {\bar{L}}^{T} ρ + {\tilde{υ}}^{T} & - (1 - λ) \hat{ι} - \hat{υ} \end{matrix}] \\ + h^{2} [\begin{matrix} {\bar{Ψ}}^{T} \\ {\bar{L}}^{T} \end{matrix}] (\sum_{q = 1}^{m} {\overset{⌢}{υ}}_{q}) [\begin{matrix} \bar{Ψ} & \bar{L} \end{matrix}] < 0 \end{matrix}

(38)

where $\bar{L} = [{\bar{L}}_{1}, {\bar{L}}_{2}, \dots, {\bar{L}}_{m}]$ , $\tilde{υ}$ , $\hat{ι}$ , $\hat{υ}$ , $\bar{Ψ}$ , and ${\bar{L}}_{q}$ are the same as before.

Proof

A proof similar to Theorem 1 is used, which is not described to avoid repetition.

Given the above analysis, it can be understood that the bipartite consensus can be asymptotically reached for system (5) employing equation (6).

Formation control and simulation of mobile robot

Formation control of mobile robot

The kinematic equations of the mobile robot are written by

{\begin{matrix} \overset{\cdot}{x} & = v \cos θ \\ \overset{\cdot}{y} & = v \sin θ \\ \overset{\cdot}{z} & = z \\ m \overset{\cdot}{v} & = f \\ J \overset{\cdot}{ω} & = T \\ \overset{\cdot}{θ} & = ω \end{matrix}

(39)

where $(x, y, z)$ represents the position coordinates of the robot; $θ$ and $J$ denote the orientation and the mass moments of inertia, respectively; $v$ , $m$ , and $f$ , respectively, refer to the mass, the angular velocity, and the force of the robot; and $ω$ and $T$ mean the linear velocity and the torque, respectively. Here, friction factors are not taken into account.

A reference not at the center of rotation of the robot is defined as

{\begin{matrix} x_{i} = x + l \cos θ \\ y_{i} = y + l \sin θ \\ z_{i} = z \end{matrix}

where $(x_{i}, y_{i}, z_{i})$ is the center of gravity coordinate of the robot platform, and $l$ represents the distance between $(x, y, z)$ and $(x_{i}, y_{i}, z_{i})$ .

Based upon the previous discussion, the equation of motion can be described as follows

{\begin{matrix} {\overset{\cdot}{x}}_{i} = v_{xi} {\overset{\cdot}{v}}_{xi} = a_{xi} {\overset{\cdot}{a}}_{xi} = u_{xi} \\ {\overset{\cdot}{y}}_{i} = v_{yi} {\overset{\cdot}{v}}_{yi} = a_{yi} {\overset{\cdot}{a}}_{yi} = u_{yi} \\ {\overset{\cdot}{z}}_{i} = v_{zi} {\overset{\cdot}{v}}_{zi} = a_{zi} {\overset{\cdot}{a}}_{zi} = u_{zi .} \end{matrix}

(40)

The motion equation of a mobile robot subject to holonomic constraints is represented by equation (40). Next, the designs of $u_{xi}$ , $u_{yi}$ , and $u_{zi}$ are discussed.

Seven robots are required to effectively move from the initial position to the predetermined destination and achieve bipartite consensus in the X-, Y-, and Z-directions. Furthermore, the information exchange topology of seven robots is shown in Figure 1, and the directed edge (i → j robot) denotes that the jth robot can obtain $x_{i} - x_{im}$ , $y_{i} - y_{im}$ , and $z_{i} - z_{im}$ . To make $(x_{im}, y_{im}, z_{im})$ become the ideal destination of the ith robot, the control law among $u_{xi}$ , $u_{yi}$ , and $u_{zi}$ is proposed as follows

{\begin{matrix} u_{xi} = & - γ_{1} v_{xi} - γ_{2} a_{xi} \\ - \sum_{j \in N_{i}} {β_{0} | a_{ij} | [(x_{i} - x_{im}) (t - τ_{ij} (t)) - sgn (a_{ij}) (x_{j} - x_{jm}) (t - τ_{ij} (t))]} \\ - \sum_{j \in N_{i}} {β_{1} | a_{ij} | [v_{xi} (t - τ_{ij} (t)) - sgn (a_{ij}) v_{xj} (t - τ_{ij} (t))]} \\ - \sum_{j \in N_{i}} {β_{2} | a_{ij} | [a_{xi} (t - τ_{ij} (t)) - sgn (a_{ij}) a_{xj} (t - τ_{ij} (t))]} \\ u_{yi} = & - γ_{1} v_{yi} - γ_{2} a_{yi} \\ - \sum_{j \in N_{i}} {β_{0} | a_{ij} | [(y_{i} - y_{im}) (t - τ_{ij} (t)) - sgn (a_{ij}) (y_{j} - y_{jm}) (t - τ_{ij} (t))]} \\ - \sum_{j \in N_{i}} {β_{1} | a_{ij} | [v_{yi} (t - τ_{ij} (t)) - sgn (a_{ij}) v_{yj} (t - τ_{ij} (t))]} \\ - \sum_{j \in N_{i}} {β_{2} | a_{ij} | [a_{yi} (t - τ_{ij} (t)) - sgn (a_{ij}) a_{yj} (t - τ_{ij} (t))]} \\ u_{zi} = & - γ_{1} v_{zi} - γ_{1} a_{zi} \\ - \sum_{j \in N_{i}} {β_{0} | a_{ij} | [(z_{i} - z_{im}) (t - τ_{ij} (t)) - sgn (a_{ij}) (z_{j} - z_{jm}) (t - τ_{ij} (t))]} \\ - \sum_{j \in N_{i}} {β_{1} | a_{ij} | [v_{zi} (t - τ_{ij} (t)) - sgn (a_{ij}) v_{zj} (t - τ_{ij} (t))]} \\ - \sum_{j \in N_{i}} {β_{2} | a_{ij} | [a_{zi} (t - τ_{ij} (t)) - sgn (a_{ij}) a_{zj} (t - τ_{ij} (t))]} \end{matrix}

(41)

Note that equation (41) is used to ensure that the seven robots maintain an ideal formation during the transition.

Simulation results

The cooperation and competition among robots are represented by solid lines and dotted lines with weights of 1 and −1 between nodes in Figure 1, respectively. Let $x_{e} = x_{i} - x_{im}$ , $y_{e} = y_{i} - y_{im}$ , $z_{e} = z_{i} - z_{im}$ , $γ_{1} = 1$ , $γ_{2} = 1$ , $β_{0} = 2$ , $β_{1} = 3$ , $β_{2} = 2$ , $x_{f} (0) = [1.6 - 0.5 - 1.8 1 2 - 1.5 - 1]$ , $v_{x} (0) = [- 2.7 - 1 2 - 2 - 1.5 3 2.5]$ , $a_{x} (0) = [1.5 2 - 0.5 2.5 1.8 - 1 - 2]$ , $y_{f} (0) = [1.4 - 3 - 1.8 3 - 2 - 1 2.6]$ , $v_{y} (0) = [- 2.5 - 1.5 - 3 - 3.5 2.5 3.5 1.8]$ , $a_{y} (0) = [- 3 - 0.5 1 2.5 - 2.5 - 3.5 - 2]$ , $z_{f} (0) = [- 1.5 1.5 - 2.5 1 2.5 - 1 0.5]$ , $v_{z} (0) = [- 2 - 0.5 1 0.7 - 2.5 2 1.2]$ , and $a_{z} (0) = [3 0.5 - 2 - 1 1 - 2.5 1.5]$ .

When $τ_{ij} = 0$ , Figure 2 shows the position change trail, velocity change trail, and acceleration change trail of the seven robots in the X-direction under the control law equation (41). All of the robots that have reached the desired consensus can be observed in Figure 2.

The position variety trajectories, velocity variety trajectories, and acceleration variety trajectories for the seven robots in the Y-direction with time under the control law equation (41) are described in Figure 3. The position, velocity, and acceleration of the seven robots converge to the desired value and achieve consensus.

From the position motion curve, velocity motion curve, and acceleration motion curve in the Z-direction described in Figure 4, it can also be understood that each robot achieves the desired consensus. All the robots can effectively move from the initial position to the predetermined position.

If $τ_{ij} \neq 0$ , let $γ_{1} = 1$ , $γ_{2} = 1$ , $β_{0} = 2$ , $β_{1} = 3$ , and $β_{2} = 2$ , by solving equation (30), h = 0.1526 can be obtained, where h is the upper bound of τ_ij(t). Figure 5 describes the position motion curve, velocity motion curve, and acceleration motion curve of each robot in X-direction with time when h = 0.1526, and it can be seen that each robot can achieve consensus.

The position variety trajectories, velocity variety trajectories, and acceleration variety trajectories of the seven robots in the Y-direction under the control law equation (41) at h = 0.1526 are shown in Figure 6. The position, velocity, and acceleration for the seven robots converged to the expected value and reached consensus.

The position change trail, velocity change trail, and acceleration change trail of the seven robots in the Z-direction under the control law equation (41) at h = 0.1526 are shown in Figure 7, and the seven robots can also reach a consensus. All the robots can effectively move from the initial position to the predetermined position.

If $τ_{ij} \neq 0$ , let $γ_{1} = 2$ , $γ_{2} = 2$ , $β_{0} = 1$ , $β_{1} = 3$ , and $β_{2} = 2$ . By solving equation (30), h = 0.1634 can be obtained The position trajectories of all robots are drawn as shown in Figure 8, in which each robot of the same weight can reach its destination.

In the other case, if $τ_{ij} = 0$ , let $γ_{1} = 0.1$ , $γ_{2} = 0.2$ , $β_{0} = 1.2$ , $β_{1} = 2.8$ , and $β_{2} = 1$ . The position trajectories of all robots are drawn as shown in Figure 9. It can be determined that even if the information exchange topology changes over time, the seven robots still maintain the expected formation between them.

Conclusion

In this paper, the issue of multi-robot formation control in a high-order system subject to time-delays based on bipartite consensus was studied. The bipartite consensus of multi-agent systems with third-order dynamics and higher-order dynamics under the directed graph was proposed. The analysis and proof for the third-order and high-order systems were given. Furthermore, the necessary and sufficient conditions for the system to reach bipartite consensus without time-delays were obtained. Considering the case with time-delays, the Lyapunov–Krasovskii functional with time-delay information was used to analyze this problem. According to the given bipartite consensus protocol, a multi-robot formation control protocol was designed to solve the control issue of multi-robot formation. The future work will include the discussion of the relationship between the values of the parameters $γ_{1}$ , $γ_{2}$ , $β_{0}$ , $β_{1}$ , and $β_{2}$ , and the multi-robot formation and the study of the formation control of the general linear multi-agent system based on bipartite consensus.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported, in part, by the National Nature Science Foundation of China under grants 61873136, 61374062, and 61603288, and, in part, by the Shandong Provincial Natural Science Foundation under grant ZR201709260010.

ORCID iDs

Wei Lu

Chengguo Zong

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