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

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.


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][4][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. 6,7Average-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, averagerendezvous 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.][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 averagetracking 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. 12,13However, the PI average-tracking algorithm for the time-varying reference signals leads to estimation errors. 10][16][17][18][19] Connectivity preservation has been an important requirement in flocking model of multiagent systems. 20,21n 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][23][24][25] secondorder multiagent systems, [26][27][28][29][30] multiple Euler-Lagrange systems, 31 linear high-order multiagent systems, 32,33 and so on.Besides, Hui 34 and Dong 35 designed continuous and noncontinuous control algorithm to deal with the finitetime 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 averagerendezvous 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 pÂq denote the set of real numbers, p-dimensional real vectors and p Â q real matrices, respectively.1 n ¼ ½1; 1; Á Á Á ; 1 T denotes the n-dimensional column vector with all elements of 1, and I n denotes an n Â n identity matrix.k Á k is the 2-norm, and denotes the Kronecker product.

Problem description
Consider the first-order multiagent systems composed of n agents as follows where x i 2 R m and u i 2 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 GðtÞ ¼ ðV ; EðtÞÞ that is composed of a set of nodes V ¼ f1;. ..; ng, a set of edges EðtÞ V Â V .Take d > 0 as the radius of agents' sensing region, and the distance between agents i and j is defined as We define EðtÞ ¼ fði; jÞjd i; j ðtÞ < d À s; i 6 ¼ j; i; j 2 V g for t 2 R with s 2 ð0; dÞ.
In addition, N i ðtÞ ¼ f jjði; jÞ 2 EðtÞg 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 2 R m , and the control goal of agents is that all the agents' states converge to the average value r ¼ : Meanwhile, the connectivity of agents ( 1) is preserved all the while, that is, GðtÞ is connected for all t !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. 36ext, we list a critical lemma that will play important role in the proof of main results.

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][23][24][25][26][27][28][29][30]32,33 we design an artificial potential function given by where ðÁÞ is a scaling function, and q > 0 is an assigned positive real number.ðyÞ is nonnegative for y 2 ½0; d.
Calculating the derivative of ðyÞ with respect to y yields which holds with y 2 ½0; d, so ðyÞ is monotonously increasing for y 2 ½0; d.Generally speaking, the chosen potential function for maintaining connectivity should be nonnegative and monotonically increasing.
According to the definition of function ðÁÞ, we obtain where r denotes the gradient, and w ij ðtÞ is expressed as Evidently, !ij ðtÞ is monotonously increasing for d ij 2 ½0; d, so we get ! Now, we propose the average-rendezvous algorithm as follows u i ðtÞ ¼ kðr i À x i ðtÞÞ À a X j2N i ðtÞ r x i ðd i; j ðtÞÞ À z i ðtÞ where k > 0; a > 0; b > 0 and N i ðtÞ is the neighbor set of agent i. Obviously, the algorithm ( 7) is a PI average-tracking algorithm.][12][13][14][15][16][17][18][19] The closed-loop form of agents (1) with algorithm ( 7) is written as Then, the closed-loop system (9) with our proposed algorithm is illustrated in Figure 1.

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!1 x i ðtÞ ¼ c; 8i 2 V , where c 2 R m is a constant vector.holds.Hence, the asymptotic stationary consensus is obviously achieved.
(Sufficiency) Let x Ã ¼ 1 n P n i¼1 x i and z Ã ¼ 1 n P n i¼1 z i , and symmetric weights of undirected topology leads to If the agents reach an asymptotic stationary consensus, lim t!1 x Ã ðtÞ ¼ lim t!1 x i ðtÞ ¼ c; i 2 V holds for some constant vector c 2 R m .Moreover, it is concluded form formulation (11) To continue the convergence analysis of system (8), we make the variable transformations x i ¼ x i À r; z i ¼ z i À kðr i À rÞ and get _ x i ðtÞ ¼ Àk x i ðtÞ À a X j2N i ðtÞ r x i ðd i; j ðtÞÞ À z i ðtÞ Then, we construct the following Lyapunov function for system (13) where V ðÁÞ, V 1 ðÁÞ, and V 2 ðÁÞ are all scaling functions.Now, we analyze the topology connectivity for agents (8) with the following assumptions: Lemma 3.Under Assumptions 1 and 2, the topology connectivity of agents ( 8) is preserved for t 2 ½0; þ1Þ, if Proof.With Assumption 1, suppose that the topology of agents ( 8) is connected over t 2 ½0; t À 1 and unconnected at t 1 , that is, at least one edge is broken at t À 1 .Based on the definition of function ðtÞ in (3), we get ðt and hold, where m i ðtÞ; i ¼ 2; Á Á Á ; n are the nonzero eigenvalues of W ðtÞ.
Proof.Calculating the derivative of V ðtÞ along the agents' trajectories yields

The characteristic equation of
FðtÞ is formulated as detðlI À FðtÞÞ ¼ 0; (19)   which is equivalent to from Lemma 1, where e ¼ ð1 þ ka À bÞ.Furthermore, (20) equals Next, we investigate the following equation When m i ðtÞ ¼ 0, equation ( 21) becomes and it is obvious that two roots of ( 22) are l 11 ¼ 0 and When m i ðtÞ 6 ¼ 0, equation ( 21) turns to be 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 FðtÞ is semi-positive definite, that is, _ V ðtÞ 0. According to Lemma 3, recursively, _ V ðtÞ 0 guarantees that the connectivity of topology is maintained all the time, so W ðtÞ just has a simple eigenvalue at zero.Therefore, the eigenvalues of FðtÞ are all positive real numbers except for one simple eigenvalue at zero.
Based on the definition of FðtÞ, _ V ðtÞ ¼ 0 yields with arbitrary gðtÞ 2 R. Substituting above formulation into the system (12) yields _ gðtÞ ¼ 0; that is, gðtÞ is a time-invariant constant, and we replace it by g.
From the definitions of x T ðtÞ and z T ðtÞ, we obtain 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.
c Remark 1. Evidently, the assumption that V ð0Þ < d q is very important for connectivity preservation.By computation, we get À rÞ T ðxð0Þ À rÞ Thus, V ð0Þ < d q 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 q.Actually, the error xð0Þ À r truly affects the connectivity preservation, since the stabilization part kðr i À x i ðtÞÞ requires the agent to reach the reference signal and does not care about the connectivity maintenance.
Evidently, if 1 þ ka À b > 0, condition (16a) holds for arbitrary positive real numbers m i ðtÞ.Thus, we get the following results: Corollary 1.Under Assumptions 1 and 2, the averagerendezvous of agents ( 8) with connectivity preservation is reached asymptotically, if Remark 2. In fact, the condition ( 25) is relatively conservative, but it is just relevant to the control parameters k; a; b.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.

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][4][5] and nonlinear multiagent systems. 37

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).

Figure 6 .
Figure 6.Evolution of the number of edges.