Average-rendezvous with connectivity preservation for first-order multiagent systems with constant reference signals

Abstract

Average-rendezvous problem with connectivity preservation of constant reference signals is investigated for first-order multiagent systems. According to a constructive potential function, a proportional–integral average-rendezvous algorithm is proposed to make agents converge to the average value of constant reference signals as well as the network connectivity is preserved all the time. Sufficient convergence conditions are obtained for the algorithm by designing the proper Lyapunov functions. Numerical examples illustrate the correctness of theoretical results.

Keywords

Average-rendezvous problem connectivity preservation constant reference signals first-order multiagent systems

Introduction

In the past decades, consensus, which is a fundamental collective behavior of multiagent systems, has attracted enormous attention from various research field, and its related research results have been applied in some engineering fields, for example, batch processes,¹ unmanned aerial and ground vehicles,² surface vessels,^3–5 and so on. Consensus problem means that the agents reach an agreement on their states of interest by exchanging information locally in a distributed way.^6,7 Average-rendezvous problem studied in this article means that the agents get different reference signals, and the whole goal requires the agents to reach the average value of reference signals with network connectivity preservation. Obviously, average-rendezvous problem is a special consensus problem and is a combination of average-tracking problem and rendezvous problem with connectivity preservation.

Average-tracking problem has attracted a lot of researchers’ interest for its application in distributed mapping, distributed estimation, and data fusion of sensor networks. For the first-order multiagent systems with constant reference signals, which is research object in this article, the proportional–integral (PI) average-tracking algorithm has been proved to be effective to solve the average-tracking problem.^8
–10 In addition, Liu et al.¹¹ designed a PI average-tracking algorithm based on disturbance’ observers and obtained the sufficient and necessary convergence conditions for the algorithm with identical communication delay by using matrix theory and frequency-domain analysis. Moreover, the PI average-tracking algorithm can also solve the average-tracking problem of agents with unmatched reference signals that some agents get reference signals, but the left agents have no reference signals.^12,13 However, the PI average-tracking algorithm for the time-varying reference signals leads to estimation errors.¹⁰ Besides, some dynamical average-tracking algorithms have been designed and analyzed for the multiagent systems with distinct time-varying reference signals by using different analysis methods.^14

–19

Connectivity preservation has been an important requirement in flocking model of multiagent systems.^20,21 In recent years, rendezvous problem with connectivity preservation, which requires agents to converge to the same location and the connectivity of multiagent network to be guaranteed all the time, has also stimulated some researchers’ interests. To achieve the connectivity preservation, the negative gradient of the potential function based on the agents’ distance is an important part of the rendezvous control algorithm, so various potential functions have been proposed for first-order multiagent systems,^22

–25 second-order multiagent systems,^26

–30 multiple Euler–Lagrange systems,³¹ linear high-order multiagent systems,^32,33 and so on. Besides, Hui³⁴ and Dong³⁵ designed continuous and noncontinuous control algorithm to deal with the finite-time rendezvous problem of first-order multiagent systems, and connectivity preservation and finite-time convergence have been proved by using semistability theory and nonsmooth stability analysis.

Evidently, there has been little attention paid on the average-rendezvous problem. In this article, we take into consideration the first-order multiagent systems with constant reference signals and aim to solve the average-rendezvous problem with connectivity preservation. Contributions of this article are listed as follows:

Agents are interconnected with the help of the position sensors, that is, the topology structure is time-varying as the agents’ positions change.

Based on the gradient of the potential function, a novel PI average-rendezvous algorithm is proposed to achieve the average-rendezvous convergence with maintaining the topology connectivity.

By using the Lyapunov functions, concise sufficient conditions that depend on the control parameters only are obtained for designing the proposed algorithm.

The remainder of this article is organized as follows. In the second section, we give the problem description. In the third section, the average-rendezvous algorithm is designed, and the convergence analysis of our proposed algorithm is presented in the fourth section. The fifth section provides a numerical example to demonstrate the effectiveness of the theoretical results. Finally, the conclusion is summarized in the sixth section.

Notation. R, R^p , and $R^{p \times q}$ denote the set of real numbers, p-dimensional real vectors and $p \times q$ real matrices, respectively. $1_{n} = [1, 1, \dots {,1]}^{T}$ denotes the n-dimensional column vector with all elements of 1, and I_n denotes an $n \times n$ identity matrix. $∥ \cdot ∥$ is the 2-norm, and $\otimes$ denotes the Kronecker product.

Problem description

Consider the first-order multiagent systems composed of n agents as follows

{\dot{x}}_{i} (t) = u_{i} (t), i = 1, \dots, n

where $x_{i} \in R^{m}$ and $u_{i} \in R^{m}$ denote the position and the control input of agent i.

The interconnection topology of agents is formed by agents sensing each other, that is, each agent can obtain its neighbors’ relative positions by sensing equipment. Hence, the interconnection topology depends on the agents’ states and is described as a time-varying graph $G (t) = (V, E (t))$ that is composed of a set of nodes $V = {1, …, n}$ , a set of edges $E (t) \subseteq V \times V$ . Take $d > 0$ as the radius of agents’ sensing region, and the distance between agents i and j is defined as

d_{i, j} (t) ≐ ∥ x_{i} (t) - x_{j} (t) ∥

We define $E (t) = {(i, j) | d_{i, j} (t) < d - σ, i \neq j, i, j \in V}$ for $t \in R$ with $σ \in (0, d)$ .

If $d_{i, j} (t) \geq d$ , then $(i, j) \in E (t)$ ;

if $(i, j) \in E (t^{-})$ and $d_{i, j} (t) < d - σ$ with $0 < σ < d$ , then $(i, j) \in E (t)$ ;

if $(i, j) \in E (t^{-})$ and $d_{i, j} (t) < d$ , then $(i, j) \in E (t)$ .

In addition, $N_{i} (t) = {j | (i, j) \in E (t)}$ denotes the neighbor set of agent i at time t.

In this article, we study the average-rendezvous problem defined as follow.

Average-rendezvous Problem. Each agent (1) accesses a constant reference signal $r_{i} \in R^{m}$ , and the control goal of agents is that all the agents’ states converge to the average value $\bar{r} ≐ \frac{1}{n} \sum_{i = 1}^{n} r_{i}$ of reference signals asymptotically, that is,

lim_{t \to \infty} x_{i} (t) = \bar{r}

Meanwhile, the connectivity of agents (1) is preserved all the while, that is, $G (t)$ is connected for all $t \geq 0$ .

Apparently, the average-rendezvous problem is equivalent to average-tracking with connectivity preservation, and its results can be applied in many engineering applications, for example, target localization of multiple autonomous underwater vehicles.³⁶

Next, we list a critical lemma that will play important role in the proof of main results.

Lemma 1

If P ₄ is invertible, $det (\begin{matrix} P_{1} & P_{2} \\ P_{3} & P_{4} \end{matrix}) = det (P_{4}) det (P_{1} - P_{2} P_{4}^{- 1} P_{3})$ .¹²

Distributed algorithms

To preserve the connectivity all the time, we adopt the negative gradient of a potential function. By referring existing potential functions for connectivity preservation,^{22

–30,32,33} we design an artificial potential function given by

φ (y) = \frac{y}{d^{2} - y^{2} + θ},0 \leq y \leq d

where $φ (\cdot)$ is a scaling function, and $θ > 0$ is an assigned positive real number. $φ (y)$ is nonnegative for $y \in [0, d]$ . Calculating the derivative of $φ (y)$ with respect to y yields

\frac{d φ (y)}{d y} = \frac{d^{2} + y^{2} + θ}{{(d^{2} - y^{2} + θ)}^{2}} > 0

which holds with $y \in [0, d]$ , so $φ (y)$ is monotonously increasing for $y \in [0, d]$ . Generally speaking, the chosen potential function for maintaining connectivity should be nonnegative and monotonically increasing.

According to the definition of function $φ (\cdot)$ , we obtain

\nabla_{x_{i}} φ (d_{i, j} (t)) = ω_{i j} (t) (x_{i} (t) - x_{j} (t))

where $\nabla$ denotes the gradient, and $w_{i j} (t)$ is expressed as

ω_{i j} (t) = {\begin{matrix} \frac{d^{2} + d_{i, j} {(t)}^{2} + θ}{{(d^{2} - d_{i, j} {(t)}^{2} + θ)}^{2}}, & e_{i j} \in E (t), \\ 0, & otherwise \end{matrix}

Evidently, $ω_{i j} (t)$ is monotonously increasing for $d_{i j} \in [0, d]$ , so we get

ω_{i j} (t) \in [\frac{1}{d^{2} + θ}, \frac{2 d^{2} + θ}{θ^{2}}]

Now, we propose the average-rendezvous algorithm as follows

\begin{matrix} u_{i} (t) = κ (r_{i} - x_{i} (t)) - α \sum_{j \in N_{i} (t)} \nabla_{x_{i}} φ (d_{i, j} (t)) - z_{i} (t) \\ {\dot{z}}_{i} (t) = β \sum_{j \in N_{i} (t)} \nabla_{x_{i}} φ (d_{i, j} (t)) \end{matrix}

where $κ > 0, α > 0, β > 0$ and $N_{i} (t)$ is the neighbor set of agent i. Obviously, the algorithm (7) is a PI average-tracking algorithm. However, it follows from (6) that the algorithm (7) is nonlinear and the coupled gains between neighboring agents depends on the states, so the convergence analysis of algorithm (7) is more complicated than the existing average-tracking algorithms.^{10

–19}

The closed-loop form of agents (1) with algorithm (7) is written as

\begin{matrix} {\dot{x}}_{i} (t) = κ (r_{i} - x_{i} (t)) - α \sum_{j \in N_{i} (t)} \nabla_{x_{i}} φ (d_{i, j} (t)) - z_{i} (t) \\ {\dot{z}}_{i} (t) = β \sum_{j \in N_{i} (t)} \nabla_{x_{i}} φ (d_{i, j} (t)) \end{matrix}

Define $x = [x_{1}^{T}, \dots, x_{n}^{T}]^{T}, z = [z_{1}^{T}, \dots, z_{n}^{T}]^{T}, r = [r_{1}^{T}, \dots, r_{n}^{T}]^{T}$ , and the compact-form of above system (8) is

\begin{matrix} \dot{x} (t) = - κ (r - x (t)) - α W (t) \otimes I_{m} x (t) - z (t) \\ \dot{z} (t) = β W (t) \otimes I_{m} x (t) \end{matrix}

where $W (t) = {w_{i j} (t)}$ is defined as

w_{i j} (t) = {\begin{matrix} - ω_{i j} (t), & i \neq j, (i, j) \in E (t); \\ - \sum_{j = 1}^{n} ω_{i j} (t), & i = j; \\ 0, & otherwise \end{matrix}

Then, the closed-loop system (9) with our proposed algorithm is illustrated in Figure 1.

Figure 1.

Block diagram of PI average-rendezvous algorithm. PI: proportional–integral.

Convergence analysis

In this section, we proceed the convergence analysis of the closed-loop multiagent system (8).

First of all, we claim that the average-tracking is equivalent to the stationary consensus seeking of agents (8).

Lemma 2

The agents (8) reach the average-consensus tracking of constant reference signals asymptotically, if and only if the agents (8) achieve a stationary consensus asymptotically, that is, ${lim}_{t \to \infty} x_{i} (t) = c, \forall i \in V$ , where $c \in R^{m}$ is a constant vector.

Proof

(Necessity) If the agents (8) reach the average-consensus tracking of constant reference signals asymptotically, (2) holds. Hence, the asymptotic stationary consensus is obviously achieved.

(Sufficiency) Let $x^{*} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}$ and $z^{*} = \frac{1}{n} \sum_{i = 1}^{n} z_{i}$ , and symmetric weights of undirected topology leads to

\begin{matrix} {\dot{x}}^{*} (t) = κ (\bar{r} - x^{*} (t)) \\ {\dot{z}}^{*} (t) = 0 \end{matrix}

If the agents reach an asymptotic stationary consensus, ${lim}_{t \to \infty} x^{*} (t) = {lim}_{t \to \infty} x_{i} (t) = c, i \in V$ holds for some constant vector $c \in R^{m}$ . Moreover, it is concluded form formulation (11) that

lim_{t \to \infty} x_{i} (t) = lim_{t \to \infty} x^{*} (t) = \bar{r}, i \in V .

Lemma 2 is proved. $□$

To continue the convergence analysis of system (8), we make the variable transformations ${\bar{x}}_{i} = x_{i} - \bar{r}, {\bar{z}}_{i} = z_{i} - κ (r_{i} - \bar{r})$ and get

\begin{matrix} {\dot{\bar{x}}}_{i} (t) = - κ {\bar{x}}_{i} (t) - α \sum_{j \in N_{i} (t)} \nabla_{{\bar{x}}_{i}} φ (d_{i, j} (t)) - {\bar{z}}_{i} (t) \\ {\dot{\bar{z}}}_{i} (t) = β \sum_{j \in N_{i} (t)} \nabla_{{\bar{x}}_{i}} φ (d_{i, j} (t)) \end{matrix}

Let $\bar{x} = [{\bar{x}}_{1}^{T}, \dots, {\bar{x}}_{n}^{T}]^{T}, \bar{z} = [{\bar{z}}_{1}^{T}, \dots, {\bar{z}}_{n}^{T}]^{T}$ , and the compact-form of above system (12) is

\begin{matrix} \dot{\bar{x}} (t) = - κ \bar{x} (t) - α W (t) \otimes I_{m} \bar{x} (t) - \bar{z} (t) \\ \dot{\bar{z}} (t) = β W (t) \otimes I_{m} \bar{x} (t) \end{matrix}

Then, we construct the following Lyapunov function for system (13)

\begin{matrix} V (t) = V_{1} (t) + V_{2} (t) \\ V_{1} (t) = \frac{1}{2} \sum_{i = 1}^{n} \sum_{j \in N_{i} (t)} φ (d_{i, j} (t)) \\ V_{2} (t) = \frac{1}{2} (κ {\bar{x}}^{T} (t) + {\bar{z}}^{T} (t)) (κ \bar{x} (t) + \bar{z} (t)) \end{matrix}

where $V (\cdot)$ , $V_{1} (\cdot)$ , and $V_{2} (\cdot)$ are all scaling functions.

Now, we analyze the topology connectivity for agents (8) with the following assumptions:

Assumption 1

$G (0)$ is connected.

Assumption 2

$V (0) < \frac{d}{θ}$ .

Lemma 3

Under Assumptions 1 and 2, the topology connectivity of agents (8) is preserved for $t \in [0, + \infty)$ , if

V (t) \leq V (0)

Proof

With Assumption 1, suppose that the topology of agents (8) is connected over $t \in [0, t_{1}^{-}]$ and unconnected at t ₁, that is, at least one edge is broken at $t_{1}^{-}$ . Based on the definition of function $φ (t)$ in (3), we get $φ (t_{1}^{-}) \geq \frac{d}{θ}$ and $V (t_{1}^{-}) \geq V (0)$ , which contradicts the condition (15) and Assumption 2, so the topology connectivity is preserved for $t \in [0, + \infty)$ . Lemma 3 is proved. $□$

Theorem 1

Under Assumptions 1 and 2, the average-rendezvous of agents (8) with connectivity preservation is reached asymptotically, if

κ + κ (1 + κ α - β) μ_{i} (t) + α μ_{i}^{2} (t) + κ^{3} > 0, i = 2, \dots, n

16a

and

κ α > \frac{1}{4} {(1 + κ α - β)}^{2}

16b

hold, where $μ_{i} (t), i = 2, \dots, n$ are the nonzero eigenvalues of $W (t)$ .

Proof

Calculating the derivative of $V (t)$ along the agents’ trajectories yields

\begin{matrix} \dot{V} (t) = \sum_{i = 1}^{n} {(\sum_{j \in N_{i} (t)} \nabla_{{\bar{x}}_{i}} φ (∥ {\bar{x}}_{i} (t) - {\bar{x}}_{j} (t) ∥))}^{T} {\dot{\bar{x}}}_{i} (t) \\ + (κ {\bar{x}}^{T} (t) + {\bar{z}}^{T} (t)) (κ \dot{\bar{x}} (t) + \dot{\bar{z}} (t)) \\ = - [\begin{matrix} {\bar{x}}^{T} & {\bar{z}}^{T} \end{matrix}] (Φ (t) \otimes I_{m}) [\begin{matrix} \bar{x} \\ \bar{z} \end{matrix}], \end{matrix}

where

Φ (t) = [\begin{matrix} Φ_{11} (t) & Φ_{21} (t) \\ Φ_{21} (t) & κ I_{n} \end{matrix}]

with $Φ_{11} (t) = κ (1 + κ α) W (t) + α W {(t)}^{2} - κ β W (t) + κ^{3} I_{n}$ and $Φ_{21} (t) = \frac{1}{2} ((1 + κ α) W (t) - β W (t) + 2 κ^{2} I_{n})$ .

The characteristic equation of $\bar{Φ} (t)$ is formulated as

det (λ I - Φ (t)) = 0,

which is equivalent to

\begin{matrix} det ((λ - κ) (λ I - (κ ε W (t) + α W {(t)}^{2} + κ^{3} I)) \\ - \frac{1}{4} {(ε W (t) + 2 κ^{2} I)}^{2}) = 0, \end{matrix}

from Lemma 1, where $ε = (1 + κ α - β)$ . Furthermore, (20) equals

\prod_{i = 1}^{n} ((λ - κ) (λ - (κ ε μ_{i} (t) + α μ_{i}^{2} (t) + κ^{3})) - \frac{1}{4} {(ε μ_{i} (t) + 2 κ^{2})}^{2}) = 0

Next, we investigate the following equation

(λ - κ) (λ - (κ ε μ_{i} (t) + α μ_{i}^{2} (t) + κ^{3})) - \frac{1}{4} {(ε μ_{i} (t) + 2 κ^{2})}^{2} = 0

When $μ_{i} (t) = 0$ , equation (21) becomes

λ (λ - κ - κ^{3}) = 0

and it is obvious that two roots of (22) are $λ_{11} = 0$ and $λ_{12} = κ + κ^{3}$ . When $μ_{i} (t) \neq 0$ , equation (21) turns to be

\begin{matrix} λ^{2} - (κ + κ ε μ_{i} (t) + α μ_{i}^{2} (t) + κ^{3}) λ \\ + κ (κ ε μ_{i} (t) + α μ_{i}^{2} (t)) - \frac{1}{4} ε^{2} μ_{i}^{2} (t) - ε μ_{i} (t) κ^{2} = 0 \end{matrix}

and it follows from condition (16) that the roots of equation (23) are all positive real numbers.

Hence, the roots of equation (19) are positive real numbers and zero, and we conclude that $Φ (t)$ is semi-positive definite, that is, $\dot{V} (t) \leq 0$ . According to Lemma 3, recursively, $\dot{V} (t) \leq 0$ guarantees that the connectivity of topology is maintained all the time, so $W (t)$ just has a simple eigenvalue at zero. Therefore, the eigenvalues of $Φ (t)$ are all positive real numbers except for one simple eigenvalue at zero.

Based on the definition of $Φ (t)$ , $\dot{V} (t) = 0$ yields

[{\bar{x}}^{T} (t), {\bar{z}}^{T} (t)] = γ (t {) [1}_{n m}^{T}, κ 1_{m n}^{T}]

with arbitrary $γ (t) \in R$ . Substituting above formulation into the system (12) yields $\dot{γ} (t) = 0,$ that is, $γ (t)$ is a time-invariant constant, and we replace it by $γ$ .

From the definitions of ${\bar{x}}^{T} (t)$ and ${\bar{z}}^{T} (t)$ , we obtain

x (t {) = 1}_{n} \otimes \bar{r} + γ 1_{n m}

z (t) = κ (r - 1_{n} \otimes \bar{r}) + γ κ 1_{n m}

which implies that the agents (8) reach a stationary consensus, and it is concluded from Lemma 2 that the agents converge to the average value of reference signals. Theorem 1 is proved. $□$

Remark 1

Evidently, the assumption that $V (0) < \frac{d}{θ}$ is very important for connectivity preservation. By computation, we get

\begin{matrix} V (0) = V_{1} (0) + V_{2} (0) \\ V_{1} (0) = \frac{1}{2} \sum_{i = 1}^{n} \sum_{j \in N_{i} (t)} φ (d_{i, j} (0)) \\ V_{2} (0) = \frac{1}{2} κ^{2} {(x (0) - r)}^{T} (x (0) - r) \end{matrix}

Thus, $V (0) < \frac{d}{θ}$ depends not only on the initial distances between neighboring agents but also on the initial distances between agents’ states and corresponding reference signals. However, Assumption 2 can be reached by adopting sufficient small $θ$ . Actually, the error $x (0) - r$ truly affects the connectivity preservation, since the stabilization part $κ (r_{i} - x_{i} (t))$ requires the agent to reach the reference signal and does not care about the connectivity maintenance.

Evidently, if $1 + κ α - β > 0$ , condition (16a) holds for arbitrary positive real numbers $μ_{i} (t)$ . Thus, we get the following results:

Corollary 1

Under Assumptions 1 and 2, the average-rendezvous of agents (8) with connectivity preservation is reached asymptotically, if

\sqrt{κ α} > \frac{1}{2} (1 + κ α - β) > 0

holds.

Remark 2

In fact, the condition (25) is relatively conservative, but it is just relevant to the control parameters $κ, α, β$ . Notably, not only the conditions (16) and (25) in Theorem 1 and Corollary 1 but also Assumption 2 determine the choice of the control parameters. In short, the control algorithm (7) can be designed conveniently with Assumption 2 and condition (25) in advance.

Numerical example

Investigate a multiagent system composed of six agents (5) moving in the two-dimensional (2D) space, and some parameters are set as $d = 3$ , $θ = 0.1$ , and $δ = 0.5$ . Then, the potential function is

φ (y) = \frac{y}{9 - y^{2} + 0.1},0 \leq y \leq 3.

The initial states of agents are chosen as $x_{1} {(0) = [1, 2]}^{T}, x_{2} {(0) = [1, 3]}^{T}, x_{3} {(0) = [3, 4]}^{T}, x_{4} {(0) = [5, 4]}^{T}, x_{5} {(0) = [3, 1]}^{T}, x_{6} {(0) = [4, 1]}^{T}$ , and it follows from the definition of edges that the initial topology of agents is shown in Figure 2 that is undirected and connected and satisfies Assumption 1. The constant reference signals of agents are chosen as $r_{1} {= [10, 12]}^{T}, r_{2} {= [2, 3]}^{T}, r_{3} {= [5, 4]}^{T}, r_{4} {= [14, 15]}^{T}, r_{5} {= [5, 6]}^{T}, r_{6} {= [6, 8]}^{T}$ , so the average value is ${[7, 8]}^{T}$ .

Figure 2.

Initial topology of agents.

Besides, choose the control parameters α and $β$ firstly as $α = 1, β = 0.3$ , and we obtain from $V (0) < \frac{d}{θ} = 30$ in Assumption 2 and the condition (25) in Corollary 1 that

κ \in (0.2046, 0.3468) .

Then, we select $κ = 0.3$ .

With above assigned parameters, thus, the agents reach the average value of reference signals asymptotically. Figure 3 shows that agents’ positions converge to the average-point of reference signals in the 2D space, and the convergence of each component of state variable is presented in Figure 4, in which all the agents’ states reach a consensus. In addition, we illustrate the distances among the agents in Figure 5, but we just provide the distances between agent 1 and others for simplicity in the figure, where $d_{1 j} = \sqrt{{(x_{1} - x_{j})}^{T} * (x_{1} - x_{j})}, j = 2, 3, \dots,6$ . Figure 5 also shows that all the six agents converge to a rendezvous. Finally, the number of edges linking agents is also shown in Figure 6. From Figure 6, the edges of the agents’ topology increase gradually to the largest number of edges, so the connectivity of the multiagent network topology is preserved. Furthermore, it is found that increasing the sensing radius can speed up the convergence rate, since the larger sensing radius brings faster connectivity.

Figure 3.

Average-rendezvous convergence.

Figure 4.

Evolution of agents’ states.

Figure 5.

Distances between agents.

Figure 6.

Evolution of the number of edges.

Moreover, to illustrate efficiency of our proposed algorithm for complex multiagent network, we consider the average-rendezvous problem of $20$ agents with initial states set as $x_{1} {(0) = [1, 2]}^{T}$ , $x_{2} {(0) = [1, 3]}^{T}$ , $x_{3} {(0) = [3, 4]}^{T}$ , $x_{4} {(0) = [5, 4]}^{T}$ , $x_{5} {(0) = [3, 1]}^{T}$ , $x_{6} {(0) = [4, 1]}^{T}$ , $x_{7} {(0) = [1.1, 2.1]}^{T}$ , $x_{8} {(0) = [1.1, 3.1]}^{T}$ , $x_{9} {(0) = [3.1, 4.1]}^{T}$ , $x_{10} {(0) = [5.1, 4.1]}^{T}$ , $x_{11} {(0) = [3.1, 1.1]}^{T}$ , $x_{12} {(0) = [4.1, 1.1]}^{T}$ , $x_{13} {(0) = [1.2, 2.2]}^{T}$ , $x_{14} {(0) = [1.2, 3.2]}^{T}$ , $x_{15} {(0) = [3.2, 4.2]}^{T}$ , $x_{16} {(0) = [5.2, 4.2]}^{T}$ , $x_{17} {(0) = [3.2, 1.2]}^{T}$ , $x_{18} {(0) = [4.2, 1.2]}^{T}$ , $x_{19} {(0) = [2.2, 1.3]}^{T}$ , $x_{20} {(0) = [3, 1.3]}^{T}$ . Choosing the same parameters $d, θ$ , and $δ$ as above, the initial topology of $20$ agents is shown in Figure 7. Then, the agents’ reference signals of agents are set as $r_{1} {= [2.5, 2.6]}^{T}$ , $r_{2} {= [2, 3]}^{T}$ , $r_{3} {= [2, 3.2]}^{T}$ , $r_{4} {= [2.5, 3.0]}^{T}$ , $r_{5} {= [3.5, 2.6]}^{T}$ , $r_{6} {= [2.6, 3.8]}^{T}$ , $r_{7} {= [3, 2]}^{T}$ , $r_{8} {= [2.3, 3]}^{T}$ , $r_{9} {= [2.5, 3.1]}^{T}$ , $r_{10} {= [3.4, 2.5]}^{T}$ , $r_{11} {= [2.5, 3.1]}^{T}$ , $r_{12} {= [2.6, 3.4]}^{T}$ , $r_{13} {= [2.8, 2]}^{T}$ , $r_{14} {= [1.2, 3]}^{T}$ , $r_{15} {= [3.5, 2.7]}^{T}$ , $r_{16} {= [2, 5]}^{T}$ , $r_{17} {= [3.1, 2.9]}^{T}$ , $r_{18} {= [2, 2.8]}^{T}$ , $r_{19} {= [2, 4]}^{T}$ , $r_{20} {= [2.7, 3]}^{T}$ , and the average value is $\bar{r} {= [2.535, 3.035]}^{T}$ . With the same control parameters $κ, α,$ and $β$ as above, the agents’ trajectories reach the average-value of reference signals, see Figures 8 and 9.

Figure 7.

Initial topology of $20$ agents.

Figure 8.

Average-rendezvous convergence of $20$ agents.

Figure 9.

Evolution of $20$ agents’ states.

Conclusion

In this article, we study the average-rendezvous problem for the first-order multiagent systems with constant reference signals. Based on the normal PI average-tracking algorithm, we propose two PI average-rendezvous algorithms. One is obtained by replacing the proportional part by the gradient of the potential function, and another one is obtained by replacing the proportional and integral parts both by the gradient of the potential function. With the help of the Lyapunov functions, we get the sufficient rendezvous convergence conditions of two algorithms, respectively. Moreover, a simple condition, which is convenient for setting parameters, is presented for the second algorithm. However, this article just considers the first-order agents with constant reference signals, and the analysis and synthesis of the average-rendezvous problem for complex multiagent systems with time-varying reference signals will be investigated in our future work, for example, surface vessels^3–5 and nonlinear multiagent systems.³⁷

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 project was supported by the “Qinglan project” of Jiangsu Province and the National Natural Science Foundation of China (Grant No.61973139, 61473138).

ORCID iD

Cheng-Lin Liu

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