Consensus of Multiagent System in the Sense of Curvature and Torsion

Mou, Jinping

doi:https://doi.org/10.1155/2019/8084209

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 8084209 | https://doi.org/10.1155/2019/8084209

Consensus of Multiagent System in the Sense of Curvature and Torsion

Jinping Mou¹

Academic Editor: Carlos-Renato Vázquez

Received13 Oct 2018

Revised28 Dec 2018

Accepted17 Jan 2019

Published28 Feb 2019

Abstract

In this paper, the consensus problem is investigated for a distributed multiagent system (MAS), where the consensus is characterized by curvature function and torsion function. According to the Frenet-Serret formulas, a distributed consensus protocol is designed for the tangent, normal, and binormal unit vectors of trajectory of each agent, and then it gets a closed-loop system. Based on the Lyapunov function, several sufficient conditions for consensus are derived for the closed-loop system. Also the consensus problem of multiagent system on a surface is studied. Finally, the numerical examples show the reliability of the proposed methods.

1. Introduction

In the last decades, curvature and torsion are taken as the efficient tools to solve some problems in different communities, such as fluid dynamics [1, 2] and mathematics physics [3]. In recent years, the further research on curvature and torsion has been investigated in [4–8], and the results provide more ways to solve the consensus problems. However, as the efficient tools, curvature and torsion have not attracted much attention in the control field of multiagent system (MAS).

Consensus of MAS is the basic requirement for some applications, which requires the state of each agent converging to an interesting value. For the past decade, consensus problem of MAS has attracted much attention due to its presence in formation, environmental monitoring, and unmanned aerial vehicles control [9]. Therefore, many results on consensus problems of MAS have been obtained (see, e.g., consensus of the first order MAS [10–19]; consensus of the second order MAS [20, 21]; consensus of the fraction order MAS [22]; and the consensus of the scale free networks [23]).

Notice that, in the above results, consensus refers to the fact that the state of each node converges to a same value, or the errors among all agents are limited for the target tracking system. In [14, 15], the consensus results show that all trajectories of nodes converge to a curve. In [17], the tracking errors among all trajectories are limited in the permitted range. However, in some real applications, we require that the distances among all agents keep nonzero constants; for example, we need all unmanned aerial vehicles to keep the same track with nonzero distances, and we call it the same trajectory except for the different locations here (STEDL). But the vast majority of results require the zero error among all agents, and the target tracking system with limited errors also cannot remain the nonzero distances, so the listed results are not suitable for solving the STEDL problems.

The motivation of this paper is that a trajectory of the moving node is decided by its curvature function and torsion function entirely. That is to say, to solve STEDL problem, each agent should have the same curvature function and torsion function, respectively. However, for the different initial locations, the different agents may have the different curvature and torsion functions. Hence, the main challenge of solving STEDL is how to let the different curvature and torsion functions converge to a curvature function and a torsion function asymptotically.

Generally, most of us believe that, without considering the curvature function and the torsion function, under the traditional average consensus protocols, the connected networks can achieve the average consensus (see, e.g., [14–21], and the references therein). However, under the traditional consensus protocols, all trajectories of agents converge to a curve with zero distance or converge to a line if we apply the protocol to the tangent, normal, and binormal vectors of each agent directly without considering the curvature function and the torsion function, and the results are not what we expected sometimes. For example, we want a group of unmanned aerial vehicles having the same trajectory, and the track may be a curve. In this paper, curvature function and torsion function will be considered when we design the consensus protocol, and the results show that the trajectories of all agents converge to a curve except for the different locations.

The aim of this paper is to describe and analyze the STEDL problem. Our approach begins with some conceptions about the consensus in the sense of the curvature and the torsion, and we call it consensus here. The conception of consensus extends the traditional consensus that is discussed in [18]; when we ignore curvature function and the torsion function, the discussed model is the same as that studied in [18]. Then the consensus protocol about curvature and torsion is proposed, we get a closed-loop system, and the stability of the system is discussed. For updating of all trajectories can be expressed by the unit vectors of tangent, normal, and binormal, hence, to discuss consensus, the system is described by unit vectors of the tangent, normal, binormal and the Frenet-Serret formulas.

As two special cases, we study STEDL problems on a plane and on a spherical surface. For the STEDL problem on a plane, since the torsion of a curve on a plane is zero, the consensus problem of MAS boils down to the consensus of curvature, and, for the spherical surface, the consensus of MAS comes down to the relation between torsion and curvature [24, 25]. Finally, the general STEDL problem of MAS on a surface is investigated.

The remainder of this paper is organized as follows. In Section 2, it introduces some concepts, and the main notations are shown in Table 1. Section 3 proposes the STEDL problem. In Section 4, several consensus criteria are proposed for MAS. Section 5 provides some numerical examples to illustrate the reliability of the proposed methods. In Section 6, we get several conclusions.

2. Preliminaries

Let be the agent set, where , . Let be a weighted symmetric adjacent matrix, where is the weight of the edge , takes value in or , and . shows that there exists information flow between nodes and , or else . refers to the edge set, where denotes an edge between and . denotes the neighbor set of , where . Hence, the topology of the multiagent system (MAS) can be denoted by an undirected weighted graph .

, let be the state of agent , where , and are the functions, so the state of MAS can be denoted by

According to the theory of the differential geometry, the trajectory of is decided entirely by the curvature function and the torsion function. For convenience, let and be the curvature function and torsion function of , respectively. Therefore, and , if then and have the same trajectory except for the difference of their locations. Consequently, based on the curvature function and torsion function, a consensus definition is proposed in the following.

Definition 1. , , if then MAS is the consensus in the sense of curvature and torsion. For short, we call it consensus here.

Remark 2. In Definition 1, consensus is different from the consensus mentioned in the recent literatures (e.g., see [18–23]). In these literatures, the consensus requires that, for every two vectors and , they satisfy and all trajectories of them converge to interest curve; the state error between every two agents converges to zero vector, and However, the results cannot ensure that every two agents keep the nonzero distance. Similarly, for the second-order MAS [21], based on the different initial values, consensus of the velocity and acceleration leads that trajectories of all agents to converge to a curve; however, every two agents cannot keep the nonzero distance.

Definition 3 (see [26]). The tangent unit vector is denoted by , the normal unit vector is denoted by , and the binormal unit vector is denoted by , where

Remark 4. In Definition 1, consensus is decided by the curvature function and the torsion function. On the other hand, the curvature function and torsion function are influenced by the updating of the tangent vector , normal vector , and binormal vector of each agent , which satisfy the following Frenet-Serret formulas:According to (8), if then we have Hence, we get the following result.

Proposition 5. , if (9) holds, then MAS achieves the consensus.

3. Problem Statement

In this section, the dynamics of tangent vector, normal vector, and binormal vector are displayed for each agent firstly and then the dynamics of tangent vectors, normal vectors, and binormal vectors are described for MAS.

, the dynamics of are described by where , , and are the controls of agent , and they are defined by

From (11) and (12), we obtain the dynamics of vectors , , and below: where is the Laplacian matrix:

For convenience, denote ; then (13) can be represented aswhere

In the following, we are going to discuss the consensus of system (15).

4. The Main Results

In this section, the stability of the error system of system (15) is studied firstly. Then the consensus problem is investigated for system (15) in space, where the ranges of the curvature functions and torsion functions are gotten, and the criteria of consensus are proposed. Finally, the CT consensus problem is studied for system (15) in some surfaces.

Denote , where ,and it obtains thatand the error system

Notice that , and hence, , and the error system (19) can be shown by

In order to study the stability of (19), we should discuss the relation between the matrices and . Denotewhereand then it gets thatwhere

To study the stability, we need the following.

Assumption 6. Suppose that , , satisfy where Then, we discuss the stability of system (15).
Let the Lyapunov candidate function be presented as it follows thatBased on (22), it gets

Theorem 7. If is connected, and if , then system (19) is the asymptotically stable, where is the second largest eigenvalue of .

Proof. Following inequality (31), it holds that since it holds thatsimilarly, we have hence, System (19) is asymptotically stable.

Theorem 8. Under (11), if is connected, then system (15) achieves consensus.

Proof. From the proof of Theorem 7, if , then . Hence, so system (15) achieves the consensus.

Remark 9. Theorem 8 shows the sufficient conditions of system (15) about the consensus. However, it does not provide the values about and ; the following results show the values.

Corollary 10. If system (15) achieves the consensus, then and satisfy the following equations:

Proof. from (11) and (12), we haveOn the one hand, following the result of Theorem 7, it is obtained that and , then and On the other hand, and, hence, Similarly, we have Denote sowhere , and we haveSubstitute (49) into (46); it holds that similarly, we get (38).

Following Theorem 7, we obtain the following result.

Remark 11. If the consensus protocol is chosen asthe method is widely used in many literatures (see, e.g., [13, 23] et al.).
If we apply (51) to vectors , and directly, then we get that all trajectories of agents become some parallel lines finally (see the simulation results of numerical Example 2). This is because , and satisfy (51); on the other hand, , and satisfy (8). Thereby, , , while . Thus, the protocol (51) is a special case of protocol (11).

Definition 12. STEDL problem on a curved surface refers to all agents moving on a curved surface and, under protocol (51), system (15) achieves CT consensus.

In the following, STEDL problems on a curved surface are investigated. First, as the special cases, STEDL problems on a plane and on a spherical surface are studied.

Corollary 13. If is connected, and all agents are moving in a plane, then, under protocol (11), system (15) achieves the consensus if

Proof. If all agents are moving in a plane, then , . From Assumption 6 and (6), we have and (52).

Corollary 14. Under the protocol (12), if is connected, all agents are moving on a spherical surface, and system (15) achieves the consensus; then the following inequality holds:where is the length of arc, and and are the constants.

Proof. According to literatures [27, 28], the following holds:Hence, from (54), we obtain that for each agent , if its satisfies that This completes the proof.

Remark 15. System (15) achieves the consensus if and only if , Since And, hence, if and only ifwhere is a constant vector.
According to Remark 15, it is not difficult to get the following equivalent statement. , if and only if (9) hold. So we get another equivalent statement of Definition 1.

Proposition 16. , if then MAS (15) achieves the consensus.

In the following, CT-consensus problem of MAS on curved surfaces is studied. To solve the problem, we should discuss the relations among the curvature function, torsion function, surface, and curves.

Let be the function, where , and is an open region. Suppose is the trajectory of agent on the surface , where is the length of arc of , satisfyand are the constants. Hence, where

Then we obtain that

Since denote

we get

Notice that ; the following holds:

On the other hand, since , the following hold:and

Following inequalities (70) and (72), we have the following corollary.

Corollary 17. If is connected, and all agents are moving on a surface , and satisfy (62), then, under protocol (11), system (15) achieves CT consensus if for each agent ,

Proof. From Assumption 6, (70), and inequality (72), it holds that , , and ; then this completes the proof.

In the next section, several numerical examples show reliability of the proposed methods.

5. Numerical Examples

Comparing with the traditional consensus protocol, the methods proposed in this paper get the better effect.

Example 1. Consider a connected MAS (15) with three agents; the topology of (15) is shown as Figure 1, where , and sthe curvature functions and torsion functions of the system are listed as follows. and According to Corollary 13, under the control of protocol (11), MAS (15) achieves the consensus. The simulation result is shown as Figure 2, where three agents begin from axis, and their trajectories become the same shape; namely, all agents achieve the consensus.

Example 2. Following Example 1, if we ignore the curvature function and torsion function and apply the general consensus protocol to the tangent, normal, and binormal unit vectors of each agent, then we get (51). Under protocol (51), we get another simulation result that is shown as Figure 3, where the trajectory of each agent converges to straight line, and all trajectories are parallel lines finally.

Example 3. Following Example 1, if we ignore the curvature function and torsion function and apply the general consensus protocol of literatures [18, 19] to directly, namely, then we get where The simulation result is shown as Figure 4, where all agents cannot keep nonzero distance.

The above simulation results show that, under protocol (11), all agents achieve the consensus, where, among all agents, they keep the nonzero constant distances, and all trajectories are the curves with the same shape; if we ignore the curvature function and torsion function, the trajectories of all agents are parallel lines asymptotically; if we apply the general consensus protocol to , all agents achieve the consensus with zero distance. In fact, in some applications, we would prefer to choose the result of Example 1.

6. Conclusion

In this paper, we have discussed the STEDL problem which is the same trajectory except for the different locations. We have offered the concept of consensus firstly; then a kind of protocol of the consensus has been proposed; finally, several consensus criteria have been obtained. However, there are several problems worth considering. One is that the criteria of the consensus are too strong, there might exist others ways to solve STEDL problems, and some weaker conditions could be found. Another is that if there are some disturbances among the agents, the STEDL problem is very difficult to investigate, and the related consensus problems will be considered by me in the future.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This work has been supported by the Fund of Zhejiang Provincial Education Department (Y201533479), Research Initiation Fund of Taizhou University (0104010004), and the Cultivation Funds of Taizhou University (2015PY009; 2016PY017).

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Copyright © 2019 Jinping Mou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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