Consensus of high-order feed-forward non-linear systems with low gain and communication constraints

Abstract

In this paper, the consensus problem of high-order multi-agent systems with inherent feed-forward delayed non-linear dynamics is investigated. First, a new kind of aperiodically intermittent communication strategy, which can be regarded as an extension of continuous communication, is proposed. The time-varying delay is considered in the multi-agent systems and the delay is not restricted to less than the communication width. Moreover, in order to reach a consensus, the effect of the feed-forward non-linearity and the time-varying delay is eliminated by designing a low-gain control protocol. Based on Lyapunov stability theory, graph theory and inequality techniques, some sufficient conditions related to the input gain and the communication width are obtained for achieving the consensus. Finally, a simulation example is presented to demonstrate the effectiveness of the proposed method.

Keywords

aperiodically intermittent communication consensus multi-agent systems time delay

Introduction

Over the last few years, multi-agent systems have received increasing attention because multiple agents may perform a task more efficiently than a single agent, reduce sensitivity to possible agent faults and offer greater flexibility during mission execution. Cooperative control of multi-agent systems has gained renewed interests due to its broad applications in many areas, including rendezvous (Conte and Pennesi, 2010; Dimarogonas and Kyriakopoulos, 2007), formation control (Defoort et al., 2008; Fax and Murray, 2004), flocking (Cucker and Smale, 2007; Su et al., 2009; Tanner et al., 2007) and distributed sensor networks (Zhang et al., 2013). The objective of distributed cooperative control is to guarantee a group of autonomous agents, via local communication, will achieve some challenging task with such advantages as low costs, high adaptivity and easy maintenance.

As one of the important and fundamental problems in the area of cooperative control, the goal of consensus is to design distributed control schemes that enable all agents to reach an agreement regarding a certain quantity of interest by relying on neighbours’ information. In recent years, consensus problems have been extensively studied and these studies have yielded fruitful results (Chen et al., 2009; Fan et al., 2013; Lin et al., 2017; Meng and Chen, 2013; Olfati-Saber and Murray, 2004; Ren and Beard, 2005; Xu et al., 2017; Yu et al., 2010, 2017a).

All these aforementioned works mainly studied the consensus problems in which the agents are governed by first-order dynamics or second-order dynamics. However, in reality, many classes of multi-agent systems are modelled by high-order non-linear dynamics. In addition, feed-forward systems, which are also called upper triangular systems, have attracted a great deal of attention since they can be used to describe many practical systems, such as the ball and beam system (Barbu et al., 1997), the cart–pendulum system (Mazenc and Bowong, 2003) and the TORA system (Sepulchre et al., 1997), among others. Hence, the consensus of multi-agent systems with high-order feed-forward non-linear dynamics is a desirable question to research. Lee et al. (2017) investigated the consensus of a multi-agent system with feed-forward non-linear dynamics and a continuous communication protocol.

In practical applications, due to the limitation of sensing ranges, the failure of physical devices and external disturbances, the agents may only communicate with their neighbours at discontinuous time intervals. In consideration of these practical situations, some researchers have devoted work to proposing intermittent control protocols to guarantee consensus (Huang et al., 2014, 2015; Li and Su, 2015; Wen et al., 2012, 2013, 2014). In the work of Wen et al. (2012), the consensus of first-order multi-agent systems with non-linear dynamics and intermittent information transmission was investigated and some sufficient conditions were obtained under a fixed strongly connected topology. In the later work of Wen et al. (2013), second-order consensus problem of multi-agent systems with inherent delayed non-linear dynamics and intermittent communication was studied. A new kind of distributed protocol was designed for a consensus of multi-agent systems with general linear node dynamics by Wen et al. (2014), and the consensus could be achieved if each agent was stabilized and the communication rate was larger than a threshold value. The leader-following consensus problems for multi-agent systems with intermittent communication were discussed by Huang et al. (2014). In Huang et al.’s (2015) research, consensus problems for multi-agent systems with second-order dynamics under delayed and intermittent communication were investigated. In addition, an adaptive intermittent control was proposed for achieving consensus by Li and Su (2015). Although these intermittent control protocols were proposed for achieving consensus, most of the results were obtained under periodically intermittent protocols. In fact, the requirement of periodicity is unreasonable and unnecessary, and aperiodically intermittent communication is more relevant in many real scenarios. Hence, Yu et al. (2017b) investigated the second-order consensus of multi-agent systems with aperiodically intermittent communications.

Motivated by these observations, this paper is devoted to investigating distributed consensus of high-order multi-agent systems with feed-forward non-linear dynamics and aperiodically intermittent communications. By using the Lyapunov control approach, some sufficient conditions are derived for achieving consensus. The main contributions of this work can be summarized as follows: (1) the aperiodically intermittent communication protocol is designed for the high-order multi-agent system, which is more relevant than a periodically intermittent communication protocol. In addition, continuous communication can be regarded as a special case of the aperiodically intermittent communication. Hence, the proposed protocol in this paper is more practical than existing ones. (2) In the earlier works discussed, it is always assumed that the time delay is less than the communication width. However, this restriction has been removed in this paper. (3) In order to reach a consensus, a low-gain parameter control scheme is proposed which is used to eliminate the effect of the feed-forward non-linearity and the time-varying delay.

The rest of this paper is organized as follows. The next section offers some preliminaries in algebraic graph theory, necessary definitions, lemmas and model formulation. Following this, the main results are derived. Next, a simulation example is presented to show the effectiveness of the theoretical results. A short conclusion is given at the end of the paper.

The notation used here is given at the end of the paper.

Preliminaries

In this section, some preliminaries about algebraic graph theory, definitions, lemmas and model formulation for high-order multi-agent systems with feed-forward non-linear dynamics are briefly introduced.

Graph theory

Let $G = (V, E)$ be a directed graph with the set of nodes $V = {v_{1}, \dots, v_{N}$ } and set of edges $E \subseteq V \times V$ . An edge of G is denoted by $e_{ij} = (v_{i}, v_{j})$ for $i \in I : = {1, \dots, N}$ . The set of neighbours of node $v_{i}$ is denoted by $N_{i} = {v_{j} \in V : (v_{j}, v_{i}) \in E}$ . A weighted adjacency matrix $A = [a_{ij}]_{N \times N}$ of G with non-negative entries is defined as $a_{ii} = 0$ , $a_{ij} > 0$ if $e_{ji} \in E$ and $a_{ij} = 0$ , otherwise. A graph with the property that $e_{ij} \in E$ implies $e_{ji} \in E$ is said to be an undirected graph. For an undirected graph, $A$ is symmetric. The Laplacian matrix $L = [ℓ_{ij}]_{N \times N}$ with respect to the graph is defined as $ℓ_{ii} = \sum_{j \neq i} a_{ij}$ and $ℓ_{ij} = - a_{ij}$ for $i \neq j$ . In cases of both undirected and directed graphs, L has at least one zero eigenvalue (denoted by $λ_{1}$ ) with a corresponding eigenvector $1$ , where $1 = (1, 1, \dots, 1)^{T}$ . A directed path from node $v_{i}$ to $v_{j}$ is a sequence of edges of form $(v_{i}, v_{i_{1}})$ , $(v_{i_{1}}, v_{i_{2}})$ , …, $(v_{i_{k}}, v_{j})$ in the directed graph with distinct node $v_{i_{l}} \in V$ for $l = 1, 2, \dots, k$ . A directed graph is called strongly connected if and only if there is a directed path between any pair of distinct nodes (Godsil and Royle, 2013).

Lemma 1. (Yu et al., 2010) (1) For a undirected graph G, one accesses the following properties: the smallest non-zero eigenvalue $λ_{2}$ of L satisfies

λ_{2} = min_{x^{T} 1 = 0, x \neq 0} \frac{x^{T} Lx}{x^{T} x} .

Under an ascending order setting, the set of eigenvalues of L can be sequentially ordered as: $0 = λ_{1} \leq λ_{2} \leq \dots \leq λ_{N}$ . Furthermore, one has that $0 = λ_{1} < λ_{2} \leq \dots \leq λ_{N}$ if and only if the undirected graph G is connected.

(2) For a strongly connected graph G, the following equality holds:

λ_{2} = min_{x^{T} 1 = 0, x \neq 0} \frac{x^{T} \hat{L} x}{x^{T} Ξ x} > 0,

where $\hat{L} = \frac{Ξ L + L^{T} Ξ}{2}$ , $Ξ = diag {ξ_{1}, \dots, ξ_{N}}$ and $ξ = (ξ_{1}, \dots, ξ_{N})^{T} > 0$ satisfying $ξ^{T} L = 0$ and $\sum_{i = 1}^{N} ξ_{i} = 1$ .

Model formulation

Consider a multi-agent system consisting of N agents with feed-forward non-linear dynamics (Xudong, 2011). The dynamics of each agent is described as follows:

{\overset{\cdot}{x}}_{i} (t) = A x_{i} (t) + f (t, x_{i} (t), x_{i} (t - τ (t))) + B u_{i} (t), i = 1, \dots, N,

(1)

where $x_{i} (t) = (x_{i 1} (t), x_{i 2} (t), \dots, x_{in} (t))^{T} \in R^{n}$ and $u_{i} (t) \in R$ represent the state and the control input of the agent i, respectively, $f (t, x_{i} (t), x_{i} (t - τ (t))) = (f_{1} (t, x_{i} (t), x_{i} (t - τ (t))), \dots, f_{n} (t, x_{i} (t), x_{i} (t - τ (t))))^{T}$ is a continuously differentiable vector-valued function representing the inherent delayed non-linear dynamics of agent i, $τ (t) > 0$ is the time delay, the matrices $A \in R^{n \times n}$ , and $B \in R^{n \times 1}$ are given as follows

A = (\begin{matrix} 0 & 1 & \dots & 0 \\ ⋮ ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \\ 0 & 0 & 0 & 0 \end{matrix}), B = (\begin{matrix} 0 \\ ⋮ \\ 0 \\ 1 \end{matrix}) .

(2)

Assumption 2. There exist non-negative constant $ρ_{1}$ and $ρ_{2}$ , such that

{\begin{matrix} ∥ f_{j} (t, x_{i} (t), x_{i} (t - τ (t))) - f_{j} (t, x_{k} (t), x_{k} (t - τ (t))) ∥ \\ \leq ρ_{1} ∥ \sum_{l = j + 2}^{n} (x_{il} (t) - x_{kl} (t)) ∥ + ρ_{2} ∥ \sum_{l = j + 2}^{n} (x_{il} (t - τ (t)) - x_{kl} (t - τ (t))) ∥, \\ j = 1, \dots, n - 2 \\ f_{j} (x_{i} (t)) = 0, j = n - 1, n, \end{matrix}

(3)

where $x_{i}, y_{i} \in R^{n}, t \geq 0$ .

Assumption 3. There exists a known constant $τ > 0$ such that $0 \leq τ (t) \leq τ$ .

Lemma 4. (Lee, 2017) For any constant $λ_{2} (\hat{L}) > 0$ , there always exists a unique matrix $P = P^{T} > 0$ that satisfies the algebraic Riccati equation:

A^{T} P + PA - 2 λ_{2} (\hat{L}) PB B^{T} P + I = 0,

(4)

where $I \in R^{n \times n}$ is an identity matrix.

In this paper, the main goal is investigating the consensus of the multi-agent system (1) by designing a distributed control protocol. Continuous control protocols were designed in the works cited earlier (Chen et al., 2009; Olfati-Saber and Murray, 2004; Ren and Beard, 2005; Xu et al., 2017; Yu et al., 2010, 2017a). However, in practical applications, due to the limitation of sensing ranges, the failure of physical devices and external disturbances, the agents may only communicate with their neighbours at some discontinuous time intervals. However, the input range of the controller is always bounded and restricted. Hence, the following consensus protocol with low gain and aperiodically intermittent communication is considered:

{\begin{matrix} {\overset{\cdot}{x}}_{i} (t) = A x_{i} (t) + f (t, x_{i} (t), x_{i} (t - τ (t))) - B B^{T} P Θ \sum_{j = 1}^{N} l_{ij} x_{j} (t), t \in [t_{m}, s_{m}], \\ {\overset{\cdot}{x}}_{i} (t) = A x_{i} (t) + f (t, x_{i} (t), x_{i} (t - τ (t))), t \in (s_{m}, t_{m + 1}), \\ m = 0, 1, 2, \dots, i = 1, 2, \dots, N, \end{matrix}

(5)

where $Θ = diag (1 / θ^{n}, 1 / θ^{n - 1}, \dots, 1 / θ)$ is a gain matrix with a design parameter $θ \geq 1$ to be determined. $τ (t) > 0$ is the time-varying delay. $L = [l_{ij}]_{N \times N}$ is the Laplacian matrix of the fixed communication topology $G (A)$ . Furthermore, the initial functions $x_{i} (t) = ϕ_{i} (t)$ are continuous for all $t \in [- τ, 0]$ and $i = 1, 2, \dots, N$ .

Remark 5. In the work of Wen et al. (2013), the requirement of periodicity of intermittent communication is quite restrictive. Consequently, in this paper, a more general control scheme – aperiodically intermittent communication – is proposed. Periodically intermittent communication is just a special case of aperiodically intermittent communication, which can also be included in the aperiodically intermittent communication, as shown in Figures 1 and 2. For any time span, $[t_{m}, t_{m + 1})$ , $t_{0} = 0, m = 0, 1, 2, \dots,$ $[t_{m}, s_{m}]$ represents the time interval over which each agent could interact with its neighbours, called the communication time, and $s_{m} - t_{m} = δ_{m}$ is the mth communication width, where $t_{m}$ and $s_{m}$ denote the start time and the end time of mth, respectively. $(s_{m}, t_{m + 1})$ represents the time interval over which each pair of neighbouring agents is no longer in force, called the rest time, and $t_{m + 1} - s_{m}$ is the mth rest width. $t_{m + 1} - t_{m} = ω_{m}$ is the sum of the communication width $s_{m} - t_{m}$ and rest width $t_{m + 1} - s_{m}$ . Obviously, when $s_{m} - t_{m} \equiv δ$ and $t_{m + 1} - t_{m} \equiv ω$ , the aperiodically intermittent communication becomes periodically intermittent communication.

Definition 6. The consensus in multi-agent system (1) is said to be achieved if for any initial conditions,

lim_{t \to \infty} ∥ x_{i} (t) - x_{j} (t) ∥ = 0, i, j = 1, 2, \dots, N .

(6)

Now we give some notations for aperiodically intermittent communication strategy.

Assumption 7. For the aperiodically intermittent communication strategy, there exist two positive scalars $0 < δ < ω < + \infty$ , such that for $m = 0, 1, 2, \dots$

{\begin{matrix} inf_{m} (s_{m} - t_{m}) = δ \\ sup_{m} (t_{m + 1} - t_{m}) = ω . \end{matrix}

(7)

In this assumption, the time span of each communication width is less than $δ$ , while the sum span of the communication and rest width is no larger than $ω$ . Then the span of rest width is no larger than $ω - δ$ .

Definition 8. For the aperiodically intermittent communication, define

σ = lim_{m \to \infty} sup \frac{t_{m + 1} - s_{m}}{t_{m + 1} - t_{m}} .

(8)

The physical meaning of $σ$ is maximum proportion of rest width $t_{m + 1} - s_{m}$ in the time span $t_{m + 1} - t_{m}$ . Obviously, $σ \in [0, 1)$ . When $σ = 0$ , the aperiodically intermittent communication becomes a continuous communication. Without loss of generality, it is supposed that $σ \in (0, 1)$ .

Lemma 9. (Yu et al., 2017b) Assume function $y (t)$ is continuous and non-negative when $t \in [- τ, + \infty)$ and satisfies the following condition:

{\begin{matrix} \overset{\cdot}{y} (t) \leq - γ_{1} y (t) + γ_{2} y (t - τ (t)), t \in [t_{m}, s_{m}], m = 0, 1, 2, \dots \\ \overset{\cdot}{y} (t) \leq γ_{3} y (t) + γ_{4} y (t - τ (t)), t \in (s_{m}, t_{m + 1}), m = 0, 1, 2, \dots \end{matrix}

(9)

where $γ_{1}$ , $γ_{2}$ , $γ_{3}$ , $γ_{4}$ are constants and $t_{0} = 0$ . Suppose for the aperiodically intermittent communication, there exists a constant $σ \in (0, 1)$ , where $σ$ is defined in Definition 8. If $0 \leq τ (t) \leq τ$ holds and

γ_{1} > \bar{γ} \geq 0, η = γ_{1} + γ_{3} > 0,

(10)

ϱ = λ^{*} - η σ > 0,

(11)

then

y (t) \leq sup_{- τ \leq s \leq 0} y (s) \exp {- ϱ t}, t \geq 0,

(12)

where $\bar{γ} = max {γ_{2}, γ_{4}}$ , $λ^{*} > 0$ is the unique positive solution of the equation

λ^{*} - γ_{1} + \bar{γ} \exp {λ^{*} τ} = 0 .

(13)

Main results

In this section, the consensus problem of multi-agent system (5) in the presence of a time-varying delay is considered. Let $e_{i} (t) = x_{i} (t) - \sum_{k = 1}^{N} ξ_{k} x_{k} (t)$ . The following error dynamical system can be given by:

{\begin{matrix} {\overset{\cdot}{e}}_{i} (t) = A e_{i} (t) + f (t, x_{i} (t), x_{i} (t - τ (t))) - \sum_{k = 1}^{N} ξ_{k} f (t, x_{k} (t), x_{k} (t - τ (t))) \\ - B B^{T} P Θ \sum_{j = 1}^{N} l_{ij} e_{j} (t), t \in [t_{m}, s_{m}], m = 0, 1, 2, \dots \\ {\overset{\cdot}{e}}_{i} (t) = A e_{i} (t) + f (t, x_{i} (t), x_{i} (t - τ (t))) - \sum_{k = 1}^{N} ξ_{k} f (t, x_{k} (t), x_{k} (t - τ (t))) \\ t \in (s_{m}, t_{m + 1}), m = 0, 1, 2, \dots \end{matrix}

(14)

where $i = 1, 2, \dots, N$ .

Let ${\bar{f}}_{i} (t) = f (t, x_{i} (t), x_{i} (t - τ (t))) - \sum_{k = 1}^{N} ξ_{k} f (t, x_{k} (t), x_{k} (t - τ (t)))$ , $e (t) = (e_{1}^{T} (t), e_{2}^{T} (t),$ $\dots, e_{N}^{T} (t))^{T}$ , and $\bar{F} (t) = ({\bar{f}}_{1}^{T} (t), \dots, {\bar{f}}_{N}^{T} (t))^{T}$ . Then system (14) can be written as

{\begin{matrix} \overset{\cdot}{e} (t) = (I_{N} \otimes A) e (t) + \bar{F} (t) - (L \otimes B B^{T} P Θ) e (t), t \in [t_{m}, s_{m}], m = 0, 1, \dots \\ \overset{\cdot}{e} (t) = (I_{N} \otimes A) e (t) + \bar{F} (t), t \in (s_{m}, t_{m + 1}), m = 0, 1, \dots \end{matrix}

(15)

Considering the state transformation,

\begin{matrix} y (t) = (y_{1}^{T} (t), y_{1}^{T} (t), \dots, y_{N}^{T} (t {))}^{T} = (I_{N} \otimes Ω) e (t), \\ Ω = diag (θ, θ^{2}, \dots, θ^{n}) . \end{matrix}

(16)

Then, equation (15) can be transformed into

{\begin{matrix} \overset{\cdot}{y} (t) = (I_{N} \otimes Ω A Ω^{- 1}) y (t) + (I_{N} \otimes Ω) \bar{F} (t) - (L \otimes Ω B B^{T} P Θ Ω^{- 1}) y (t), \\ t \in [t_{m}, s_{m}], m = 0, 1, \dots \\ \overset{\cdot}{y} (t) = (I_{N} \otimes Ω A Ω^{- 1}) y (t) + (I_{N} \otimes Ω) \bar{F} (t), t \in (s_{m}, t_{m + 1}), m = 0, 1, \dots \end{matrix}

(17)

By simple calculation, one has

Ω A Ω^{- 1} = \frac{1}{θ} A, Ω B B^{T} P Θ Ω^{- 1} = \frac{1}{θ} B B^{T} P .

Therefore, it follows that

{\begin{cases} \overset{\cdot}{y} (t) = \frac{1}{θ} (I_{N} \otimes A) y (t) + (I_{N} \otimes Ω) \bar{F} (t) - \frac{1}{θ} (L \otimes B B^{T} P) y (t), \\ t \in [t_{m}, s_{m}], m = 0, 1, \dots \\ \overset{\cdot}{y} (t) = \frac{1}{θ} (I_{N} \otimes A) y (t) + (I_{N} \otimes Ω) \bar{F} (t), t \in (s_{m}, t_{m + 1}), m = 0, 1, \dots \end{cases}

(18)

Theorem 10. Suppose that Assumptions 2, 3 and 7 hold, and the communication topology $G (A)$ is strongly connected. Then the consensus in system (5) is achieved if the following conditions are satisfied:

θ > max {1, 2 (\frac{ξ_{max} λ_{max} (P) ρ_{2}}{ξ_{min} λ_{min} (P)} + ρ_{1} + ρ_{2}) \sqrt{n} ∥ Ξ \otimes P ∥},

(19)

ϱ = λ^{*} - η σ > 0, η = β_{1} + β_{3} > 0,

(20)

where $ξ_{max} = max_{1 \leq i \leq N} {ξ_{i}}$ , $ξ_{min} = min_{1 \leq i \leq N} {ξ_{i}}$ , $P > 0$ and satisfies equation (4), $Ξ = diag {ξ_{1}, ξ_{2},$ $\dots, ξ_{N}}$ , $β_{1} = \frac{α_{1}}{ξ_{max} λ_{max} (P)}$ , $β_{2} = \frac{α_{2}}{ξ_{min} λ_{min} (P)}$ , $β_{3} = \frac{α_{3}}{ξ_{min} λ_{min} (P)}$ , $λ^{*}$ is the unique positive solution of $λ - β_{1} + β_{2} \exp {λ \bar{τ}} = 0$ , $α_{1} = \frac{1}{θ} - \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥$ , $α_{2} = \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥$ , $α_{3} = \frac{ξ_{max}}{θ} λ_{max} (PA + A^{T} P) + \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥$ .

Proof. Construct the following Lyapunov function candidate:

V (t) = y^{T} (t) (Ξ \otimes P) y (t),

(21)

where $Ξ = diag (ξ_{1}, ξ_{2}, \dots, ξ_{N})$ , and $P > 0$ is a positive definite matrix.

For $t \in [t_{m}, s_{m}]$ , $m = 0, 1, 2, \dots$ , taking the time derivative of $V (t)$ along the trajectories of equation (18), one obtains

\begin{matrix} \overset{\cdot}{V} (t) = 2 y^{T} (t) (Ξ \otimes P) [\frac{1}{θ} (I_{N} \otimes A) y (t) + (I_{N} \otimes Ω) \bar{F} (t) \\ - \frac{1}{θ} (L \otimes B B^{T} P) y (t)] \\ = \frac{1}{θ} y^{T} (t) (Ξ \otimes (PA + A^{T} P)) y (t) \\ - \frac{1}{θ} y^{T} (t) ((Ξ L + L^{T}) Ξ \otimes PB B^{T} P) y (t) \\ + 2 y^{T} (t) (Ξ \otimes P Ω) \bar{F} (t) \\ = \frac{1}{θ} y^{T} (t) [(Ξ \otimes (PA + A^{T} P)) - ((Ξ L + L^{T} Ξ) \otimes PB B^{T} P)] y (t) \\ + 2 y^{T} (t) (Ξ \otimes P Ω) \bar{F} (t) . \end{matrix}

(22)

Combined with the definition of $\bar{F} (t)$ , it follows that

\begin{matrix} 2 y^{T} (t) (Ξ \otimes P Ω) \bar{F} (t) \\ = 2 \sum_{j = 1}^{N} ξ_{i} y_{i}^{T} (t) P Ω {\bar{f}}_{i} (t) \\ = 2 \sum_{j = 1}^{N} ξ_{i} y_{i}^{T} (t) P Ω [f (t, x_{i} (t), x_{i} (t - τ (t))) \\ - \sum_{k = 1}^{N} ξ_{k} f (t, x_{k} (t), x_{k} (t - τ (t)))] \\ = 2 \sum_{j = 1}^{N} ξ_{i} y_{i}^{T} (t) P Ω [f (t, x_{i} (t), x_{i} (t - τ (t))) - f (t, \sum_{k = 1}^{N} ξ_{k} x_{k} (t), \\ \sum_{k = 1}^{N} ξ_{k} x_{k} (t - τ (t)))] \\ + 2 \sum_{j = 1}^{N} ξ_{i} y_{i}^{T} (t) P Ω [f (t, \sum_{k = 1}^{N} ξ_{k} x_{k} (t), \sum_{k = 1}^{N} ξ_{k} x_{k} (t - τ (t))) \\ - \sum_{k = 1}^{N} ξ_{k} f (t, x_{k} (t), x_{k} (t - τ (t)))] . \end{matrix}

(23)

According to the definition of $e_{i} (t)$ , one has that $\sum_{i = 1}^{N} ξ_{i} e_{i} (t) = 0$ and $\sum_{i = 1}^{N} ξ_{i} y_{i}^{T} (t) = 0$ . As a result, it has

\begin{matrix} 2 \sum_{j = 1}^{N} ξ_{i} y_{i} (t) P Ω [f (t, \sum_{k = 1}^{N} ξ_{k} x_{k} (t), \sum_{k = 1}^{N} ξ_{k} x_{k} (t - τ (t))) \\ - \sum_{k = 1}^{N} ξ_{k} f (t, x_{k} (t), x_{k} (t - τ (t)))] = 0 . \end{matrix}

(24)

Based on equations (23) and (24), one gets

\begin{matrix} 2 y^{T} (t) (Ξ \otimes P Ω) \bar{F} (t) = 2 \sum_{j = 1}^{N} ξ_{i} y_{i} (t) P Ω \\ \times [f (t, x_{i} (t), x_{i} (t - τ (t))) - f \\ (t, \sum_{k = 1}^{N} ξ_{k} x_{k} (t), \sum_{k = 1}^{N} ξ_{k} x_{k} (t - τ (t)))] \end{matrix}

(25)

Considering Assumption 2, and $θ \geq 1$ , it has

\begin{matrix} | θ^{j} (f_{j} (t, x_{i} (t), x_{i} (t - τ (t))) - f_{j} (t, \sum_{k = 1}^{N} ξ_{k} x_{k} (t), \sum_{k = 1}^{N} ξ_{k} x_{k} (t - τ (t)))) | \\ \leq θ^{j} (ρ_{1} \sqrt{\sum_{k = j + 2}^{n} e_{ik}^{2} (t)} + ρ_{2} \sqrt{\sum_{k = j + 2}^{n} e_{ik}^{2} (t - τ (t))}) \\ = ρ_{1} \sqrt{\sum_{k = j + 2}^{n} {(θ^{j} e_{ik} (t))}^{2}} + ρ_{2} \sqrt{\sum_{k = j + 2}^{n} {(θ^{j} e_{ik} (t - τ (t)))}^{2}} \\ = \frac{ρ_{1}}{θ^{2}} \sqrt{{(θ^{j + 2} e_{i (j + 2)} (t))}^{2} + \dots + {(θ^{j + 2} e_{in} (t))}^{2}} \\ + \frac{ρ_{2}}{θ^{2}} \sqrt{{(θ^{j + 2} e_{i (j + 2)} (t - τ (t)))}^{2} + \dots + {(θ^{j + 2} e_{in} (t - τ (t)))}^{2}} \\ \leq \frac{ρ_{1}}{θ^{2}} \sqrt{{(θ^{j + 2} e_{i (j + 2)} (t))}^{2} + \dots + {(θ^{n} e_{in} (t))}^{2}} \\ + \frac{ρ_{2}}{θ^{2}} \sqrt{{(θ^{j + 2} e_{i (j + 2)} (t - τ (t)))}^{2} + \dots + {(θ^{n} e_{in} (t - τ (t)))}^{2}} \\ \leq \frac{ρ_{1}}{θ^{2}} ∥ y_{i} (t) ∥ + \frac{ρ_{2}}{θ^{2}} ∥ y_{i} (t - τ (t)) ∥ \end{matrix}

(26)

Using inequality (26), it follows that

\begin{matrix} 2 y^{T} (t) (Ξ \otimes P Ω) \bar{F} (t) \\ \leq 2 ∥ y (t) ∥ ∥ Ξ \otimes P ∥ ∥ (I_{N} \otimes Ω) \bar{F} (t) ∥ \\ \leq \frac{2 ρ_{1} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t) ∥^{2} \\ + \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t) ∥ ∥ y (t - τ (t)) ∥ \\ \leq \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t) ∥^{2} \\ + \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t - τ (t)) ∥^{2} . \end{matrix}

(27)

Based on inequality (27), one has

\begin{matrix} \overset{\cdot}{V} (t) \leq \frac{1}{θ} y^{T} (t) [(Ξ \otimes (PA + A^{T} P)) - ((Ξ L + L^{T} Ξ) \otimes PB B^{T} P)] y (t) \\ + \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t) ∥^{2} \\ + \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t - τ (t)) ∥^{2} \end{matrix}

(28)

According to Lemma 1, one has

\begin{matrix} \overset{\cdot}{V} (t) \leq \frac{1}{θ} y^{T} (t) [(Ξ \otimes (PA + A^{T} P)) - 2 λ_{2} (\hat{L}) (Ξ \otimes PB B^{T} P)] y (t) \\ + \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t) ∥^{2} \\ + \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t - τ (t)) ∥^{2} \\ \leq \frac{1}{θ} y^{T} (t) [Ξ \otimes (PA + A^{T} P - 2 λ_{2} (\hat{L}) PB B^{T} P)] y (t) \\ + \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t) ∥^{2} \\ + \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t - τ (t)) ∥^{2} . \end{matrix}

(29)

By using Lemma 4, it has

\begin{matrix} \overset{\cdot}{V} (t) \leq - \frac{ξ_{min}}{θ} y^{T} (t) y (t) + \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t) ∥^{2} \\ + \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t - τ (t)) ∥^{2} \\ \leq - \frac{α_{1}}{ξ_{max} λ_{max} (P)} V (t) + \frac{α_{2}}{ξ_{min} λ_{min} (P)} V (t - τ (t)) \\ = - β_{1} V (t) + β_{2} V (t - τ (t)), \end{matrix}

(30)

where $ξ_{min} = min_{1 \leq i \leq N} {ξ_{i}}$ and $ξ_{max} = max_{1 \leq i \leq N} {ξ_{i}}$ , $α_{1} = \frac{1}{θ} - \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥$ , $α_{2} = \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥$ , $β_{1} = \frac{α_{1}}{ξ_{max} λ_{max} (P)}$ , $β_{2} = \frac{α_{2}}{ξ_{min} λ_{min} (P)}$ .

For $t \in (s_{m}, t_{m + 1})$ , $m = 0, 1, 2, \dots$ , taking the time derivative of $V (t)$ along the trajectories of (18), one obtains

\begin{matrix} \overset{\cdot}{V} (t) = 2 y^{T} (t) (Ξ \otimes P) [\frac{1}{θ} (I_{N} \otimes A) y (t) + (I_{N} \otimes Ω) \bar{F} (t)] \\ = \frac{1}{θ} y^{T} (t) (Ξ \otimes (PA + A^{T} P)) y (t) + 2 y^{T} (t) (Ξ \otimes P Ω) \bar{F} (t) . \end{matrix}

(31)

By using a similar analysis method, one gets

\begin{matrix} \overset{\cdot}{V} (t) \leq \frac{1}{θ} y^{T} (t) (Ξ \otimes (PA + A^{T} P)) y (t) \\ + \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t) ∥^{2} \\ + \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t - τ (t)) ∥^{2} \\ \leq \frac{ξ_{max}}{θ} λ_{max} (PA + A^{T} P) y^{T} (t) y (t) \\ + \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t) ∥^{2} \\ + \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥ ∥ y (t - τ (t)) ∥^{2} \\ \leq \frac{α_{3}}{ξ_{min} λ_{min} (P)} V (t) + \frac{α_{2}}{ξ_{min} λ_{min} (P)} V (t - τ (t)) \\ = β_{3} V (t) + β_{2} V (t - τ (t)) . \end{matrix}

(32)

where $ξ_{min} = min_{1 \leq i \leq N} {ξ_{i}}$ and $ξ_{max} = max_{1 \leq i \leq N} {ξ_{i}}$ , $α_{3} = \frac{ξ_{max}}{θ} λ_{max} (PA + A^{T} P) + \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥$ , $α_{2} = \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥$ , $β_{3} = \frac{α_{3}}{ξ_{min} λ_{min} (P)}$ , $β_{2} = \frac{α_{2}}{ξ_{min} λ_{min} (P)}$ .

According to equation (19), one can obtain that

\frac{θ - 2 (ρ_{1} + ρ_{2}) \sqrt{n} ∥ Ξ \otimes P ∥}{θ^{2}} \geq \frac{ξ_{max} λ_{max} (P)}{ξ_{min} λ_{min} (P)} \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} ∥ Ξ \otimes P ∥,

that is:

\frac{θ - 2 (ρ_{1} + ρ_{2}) \sqrt{n} ∥ Ξ \otimes P ∥}{θ^{2} ξ_{max} λ_{max} (P)} \geq \frac{2 ρ_{2} \sqrt{n} ∥ Ξ \otimes P ∥ ξ_{min} λ_{min} (P)}{θ^{2}} .

Then, $β_{1} > β_{2}$ . Combining with equation (20) and using Lemma 9, one has

V (t) \leq sup_{- τ \leq s \leq 0} V (s) \exp {- ϱ t},

(33)

where $ϱ = λ^{*} - η σ > 0$ , $η = β_{1} + β_{2}$ , $λ^{*}$ is the unique positive solution of the equation

λ^{*} - β_{1} + β_{2} \exp {λ^{*} τ} = 0 .

Based on inequality (33), one has $lim_{t \to \infty} V (t) = 0$ , that is $lim_{t \to \infty} y (t) = 0$ . Then, the consensus can be achieved. The proof is completed.

Remark 11. In this paper, it is only assumed that the time-varying delay is a bounded function. Hence, the restriction on the time-varying delay $τ (t)$ is less conservative and more practically applicable. In addition, the results are dependent on the parameter $θ$ in Definition 8, and do not directly rely on the communication width or rest width. Consequently, the communication width and the rest width can be small enough.

Corollary 12. Suppose that Assumptions 2 and 7 hold, and the communication topology $G (A)$ is undirected and connected. Then the consensus in system (5) is achieved if the following conditions are satisfied:

θ > max {1, 2 (\frac{λ_{max} (P) ρ_{2}}{λ_{min} (P)} + ρ_{1} + ρ_{2}) \sqrt{n} λ_{max} (P)},

(34)

\tilde{ϱ} = {\tilde{λ}}^{*} - \tilde{η} σ > 0, \tilde{η} = {\tilde{β}}_{1} + {\tilde{β}}_{3} > 0,

(35)

where $P > 0$ and satisfies $A^{T} P + PA - 2 λ_{2} (L) PB B^{T} P + I = 0$ , ${\tilde{β}}_{1} = \frac{{\tilde{α}}_{1}}{λ_{max} (P)}$ , ${\tilde{β}}_{2} = \frac{{\tilde{α}}_{2}}{λ_{min} (P)}$ , ${\tilde{β}}_{3} = \frac{{\tilde{α}}_{3}}{λ_{min} (P)}$ , ${\tilde{λ}}^{*}$ is the unique positive solution of $\tilde{λ} - {\tilde{β}}_{1} + {\tilde{β}}_{2} \exp {\tilde{λ} \bar{τ}} = 0$ , ${\tilde{α}}_{1} = \frac{1}{θ} - \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} λ_{max} (P)$ , ${\tilde{α}}_{2} = \frac{2 ρ_{2} \sqrt{n}}{θ^{2}} λ_{max} (P)$ , ${\tilde{α}}_{3} = \frac{1}{θ} λ_{max} (PA + A^{T} P) + \frac{2 (ρ_{1} + ρ_{2}) \sqrt{n}}{θ^{2}} λ_{max} (P)$ .

Remark 13. In the work of Lee (2017), the consensus of the multi-agent system with feed-forward non-linear dynamics and time-varying communication was considered. In that paper, the authors assumed that the communication among all agents is continuous and the communication graph is undirected and connected. However, in our paper aperiodically intermittent communication is considered. The continuous communication can be regarded as a special case of intermittent communication. The restriction on the communication graph is relaxed and it is only assumed that the communication graph is strongly connected, which is more common than an undirected network.

Numerical example

In this section a numerical example is given to demonstrate the effectiveness of the theoretical results.

Consider a three-order multi-agent system with feed-forward non-linear dynamics in system (1), where

A = (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}), B = (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}), f = (\begin{matrix} f_{1} (t, x_{i} (t), x_{i} (t - τ (t))) \\ 0 \\ 0 \end{matrix}),

(35)

$f_{1} (t, x_{i} (t), x_{i} (t - τ (t))) = 0.1 \sin x_{i 3} (t) + 0.1 \cos x_{i 3} (t - 0.5 | \sin t |)$ , $i = 1, 2, 3, 4$ . The communication topology among the multi-agent system is presented in Figure 3 with weights on each edge being 0.5. According to Assumptions 2 and 3, one obtains that $ρ_{1} = ρ_{2} = 0.1$ , $τ (t) \leq 0.5$ . $λ_{2} (L_{s}) = 0.5$ . By condition (19) of Theorem 10, one has

P = (\begin{matrix} 2.4142 & 2.4142 & 1 \\ 2.4142 & 4.8284 & 2.4142 \\ 1 & 2.4142 & 2.4142 \end{matrix}),

$θ > 9.2153$ . Choosing $θ = 100$ , it has $λ^{*} = 0.517$ , $η = 0.2357$ , $λ = 0.293$ , $σ < 0.2193$ . We choose $δ / ω = 0.8$ . Then the state trajectories of all agents are shown in Figures 4 –6 with the initial conditions $x_{1} (0) = [5.2, 0.5, - 0.3]^{T}$ , $x_{2} (0) = [- 4.5, - 0.25, 0.5]^{T}$ , $x_{3} (0) = [- 8, 0.45, - 0.5]^{T}$ , $x_{4} (0) = [10, - 0.5, 0.3]^{T}$ . It can be seen that the consensus problem in system (1) is indeed solved and the simulation results demonstrate the theoretical analysis very well.

In addition, when $σ = 0$ , the aperiodically intermittent communication becomes a continuous communication. Condition (20) is naturally satisfied. Therefore, choosing $θ = 50$ , the state trajectories of all agents are shown in Figures 7 –9.

Remark 14. Based on the simulation example, it is found that the parameter $θ$ plays a critical role in the convergence speed of the consensus. The greater the parameter $θ$ , the slower the convergence speed. Moreover, the communication width is also affected by the parameter $θ$ , which can be seen in condition (20). Hence, in practical applications, consensus can be achieved by adjusting the parameter and the communication width to achieve a trade-off.

Conclusion

In this paper the distributed consensus for multi-agent systems with high-order feed-forward non-linear dynamics is considered. In consideration of practical situations, a novel kind of aperiodically intermittent consensus protocol is proposed. The main difference between existing works is a low-gain matrix introduced in the distributed protocol, which can eliminate the effect of the feed-forward non-linearity. In combination with Lyapunov stability theory, graph theory and inequality techniques, some sufficient conditions related to the input gain and the communication width are obtained for achieving consensus. It is found that consensus can be achieved by selecting an appropriate parameter for the low-gain matrix. The communication delay is not considered in the design process. In future work the extension of our results to general directed network with communication delay will be investigated.

Footnotes

Notation

In this paper, R denotes the set of real numbers. $R^{n}$ denotes the n-dimensional Euclidean space. Let $I_{N}$ be the identity matrix with dimension N, and $1_{N}$ be the N-dimensional column vector, with each entry being 1. For a square matrix A, let $A^{T}$ represent its transpose, $λ_{2} (A)$ , $λ_{max} (A)$ and $λ_{min} (A)$ be the second smallest eigenvalue of A, the maximum and minimum eigenvalues of A, respectively. For a real symmetric matrix B, $B > 0$ $(B < 0)$ if B is positive (negative) definite. ∥·∥ denotes the Euclidean norm. ⊗ denotes the Kronecker product of matrices.

Conflict of interest

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 by the National Natural Science Foundation of the People’s Republic of China under grant nos. 61473244 and 11402223, and by the Scientific Research Program of the Higher Education Institution of Xinjiang under grant no. XJEDU2017T001.

ORCID iD

Haijun Jiang

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