Complex Engineering Systems

Open Access Research Article

Correspondence to: Dr. Yurong Liu, Department of Mathematics, Yangzhou University, No. 180 Siwangting Road, Hanjiang Dsitrict, Yangzhou 225002, Jiangsu, China. E-mail: yrliu@yzu.edu.cn

© The Author(s) 2022. **Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, sharing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

This paper is concerned with the sampled-data bipartite tracking consensus problem for a class of nonlinear multiagent systems (MASs) with input saturation. Both competitive and cooperative interactions coexist among agents in the concerned network. By resorting to Lyapunov stability theory and the linear matrix inequality (LMI) technique, several criteria are obtained to ensure that the considered MASs can achieve the bipartite tracking consensus. Besides, with the help of the decoupled method, the dimensions of LMIs are reduced for mitigation of the computation complexity so that the obtained results can be applied to large-scaled MASs. Furthermore, the controller gain matrix is explicitly expressed in terms of solutions to a set of LMIs. We also provide an estimate of elliptical attraction domain of bipartite tracking consensus. Finally, numerical simulation is exploited to support our theoretical analysis.

Bipartite tracking consensus, sampled-data control, nonlinear multiagent systems, input saturation

Over the last decade, adequate attention has been gained to various research on multiagent systems (MASs) due to pervasive applications in a variety of areas including, but are not limited to, coordination control of unmanned aerial vehicles ^{[1]}, formation control of multiple robot systems ^{[2]}, and dynamics of opinion forming ^{[3]}. Generally speaking, a MAS is composed of a group of agents, and these agents share local information from their neighbors through communication channels. The main objective of consensus problem for MASs is to construct an appropriate protocol so that agents can approach an agreement. Such a problem is one of the hottest topics in analysis and synthesis of cooperative behaviors. Thus far, most existing literature on MASs is concerned with consensus problem (see ^{[4-15]}). For example, the tracking consensus issue of single-integrator MASs was discussed in ^{[10]}, where a network-based consensus control protocol was designed to ensure that the followers' states reach an agreement on the leader's state. In ^{[12]}, the authors designed the consensus protocol for heterogeneous second-order nonlinear MASs with uniformly connected topologies in the presence of both uncertainties and disturbances.

It should be pointed out that communication linkages in the above-mentioned MASs are mainly concentrated on a cooperative network. However, in the real world, only cooperative linkages are insufficient to describe the intricate interactions among individuals. For example, in social networks, there are both competitive and cooperative relationships during the process of communication. In this case, the bipartite consensus entered the researchers' field of vision. The so-called bipartite consensus for MASs means that all agents reach a final state with an identical magnitude but opposite sign ^{[16-20]}. Recently, some scholars commit themselves to studying bipartite consensus problem of MASs with signed graphs, and there are some results on this topic scattered in the literature. For instance, based on state feedback and output feedback, the authors of ^{[21]} designed two distributed protocols to solve the bipartite output consensus problem for heterogeneous MASs with structurally balanced graphs. In ^{[22]}, the bipartite tracking consensus problem for linear MASs was investigated, in which the dynamic leader's control input was nonzero and unknown.

Notice that continuous information transmission among agents may cause a heavy burden and congestion in communication networks. Thus, it would lead to some difficulties in practical applications for some continuous-type consensus protocols. For this reason, the sampled-data control approach has been used to design various consensus protocols for MASs. The sampled-data control approach captures inherent properties of digital control, where the control input signal can be kept constant via a zero-order holder until the next sampling instant. Recently, many results have been reported concerning the sampled-data consensus of MASs ^{[23-25]}. Particularly, in ^{[26]}, with the aid of the input delay technique and decoupled method, the sampled-data consensus problem was investigated for nonlinear MASs with randomly occurring deception attacks. For a class of second-order MASs, an improved aperiodic sampled-data consensus protocol was designed in ^{[27]}, where only the sampled position data were exchanged among neighboring agents. Nevertheless, under competitive and cooperative communications, sampled-based bipartite tracking consensus problems are far from being adequately investigated due mainly to some difficulties aroused by a signed communication topology, which is one of the main motivations of this paper.

Actuator saturation is ubiquitous and unavoidable in practical engineering owing to various physical restrictions, such as power amplifiers and proportional valves ^{[28]}. The saturation phenomenon usually causes performance degradation, undesirable oscillatory behavior, and even instability ^{[29, 30]}. Hence, it is critically important to take into account input constraints when designing distributed consensus protocols for MASs. Recent years have witnessed much research on consensus problems for MASs subject to input saturation ^{[31-36]}. For example, the containment control issue for MASs with bounded actuation was investigated in ^{[33]}, where an anti-windup compensation was designed by using convex conditions to improve the performance in the presence of actuator saturation. By considering the one-sided Lipschitz condition and input saturation, a new region of stability was provided in ^{[37]} to ensure the consensus error of one-sided Lipschitz nonlinear MASs was asymptotically stable. Based on the low-gain feedback technique, the authors of ^{[38]} studied the semi-global bipartite consensus problem for MASs with input saturation. However, a thorough literature search found that the research on the bipartite consensus problem for MASs is still in infancy, especially for the case when the MASs are involved in both signal-sampling and input saturation.

Inspired by the above discussions, we aim to further investigate the sampled-based bipartite tracking consensus of nonlinear MASs with input saturation. By resorting to Lyapunov stability theory and LMI technique, some criteria are established to ensure that the considered MASs can achieve the bipartite tracking consensus. The main contributions of this paper can be highlighted as follows.

**Notation**: In this paper,

A directed signed graph is denoted by

**Definition 1**^{[16]} A directed signed graph

Let

**Lemma 1**^{[16]} A signed graph

**Lemma 2**^{[39]} Given a matrix

Consider a group of agents with

and the leader is modeled by

where

**Assumption 1***Assume that the communication graph *

**Definition 2*** ^{[22]} The MAS in Equations* (1)

*for any initial conditions *

In this paper, we put forward to the following sampled-based controller (control scheme) of the

where

According to Assumption 1 and Lemma 1, there exists

Set

From Equations (1) and (2) and the distributed control protocol in Equation (4), one can get that

where

where

Since the leader has no neighbors, the Laplacian matrix of the MAS in Equations (1) and (2) can be partitioned as

where

In this paper, we aim to design the sampled-based distributed protocol for the MAS in Equations (1) and (2) subject to input saturation and derive some conditions to ensure that the bipartite tracking consensus can be achieved.

Our main aim in this section is to establish some criteria to ensure that the considered MASs in Equations (1) and (2) can achieve the bipartite tracking consensus. Before proceeding, we give the following assumption and lemmas.

**Assumption 2***The odd nonlinear vector-valued function *

*for all *

**Lemma 3**^{[40]} Let

*where *

**Lemma 4**^{[41]} Let

**Lemma 5**^{[42]} Let matrices

**Lemma 6**^{[26]} Let

In what follows, we always suppose

The following theorem gives a condition to ensure the bipartite tracking consensus can be achieved for the MAS in Equations (1) and (2) with distributed protocol Equation (4).

**Theorem 1***Suppose Assumptions 1 and 2 hold. Let gain matrix *

*with * (1)

*with *

**Proof 1***Let *

*Equation* (7) *can be rewritten as*

*where *

*Construct the following Lyapunov function:*

*Denote * (11)

*According to Assumption 2 and *

*For *

*Noting that *

*One the other hand, if *

*From Equations* (15)–(17)*, it follows that*

*which implies that*

*It can also be validated that*

*Note that*

*we obtain*

*From Lemmas 4 and 5, it follows*

*and*

*Substituting Equations* (14), (20)*, and* (23)–(25) *into Equation* (13)*, one has*

*Without loss of generality, we assume * (8)

*which implies that*

*Furthermore, it follows from Equation* (28) *that*

*By Kronecker product, we can rewrite Equation* (29) *as*

*that is,*

*where *

*Hence, according to Lemma 3, we have*

*where *

*Subsequently, the inequality in Equation* (26) *in combination with Equation* (32) *indicates that*

*where *

*According to Equation* (9)*, one has*

*Next, we need to show that*

*In fact, suppose that Equation* (35) *is not true, then there exists *

*In light of Equations* (10) *and* (34)*, one has that*

*which means that there exists *

*Then, we get*

*Let *

*which further implies*

*In view of Equation* (34)*, one obtains that*

*which contradicts with Equation* (41)*. Therefore,*

*Furthermore, the inequality in Equation* (34) *in combination with Equation* (43) *indicates that*

*where *

*From the inequality in Equation* (44)*, one gets*

*Integrating both sides of the inequality in Equation* (45) *from *

*After a simple calculation, it is easy to see that*

*Setting * (10)

*Hence,*

*It is easy to see that *

*For any *

*Since *

*Consequently, in view of Equation* (49)*, we obtain that * (1)

**Remark 1***Theorem 1 establishes a sufficient condition to ensure the bipartite tracking consensus for the concerned network. The condition is rather general, but it might have heavy computation burden for large-scale MASs. To reduce such computation burden, based on Theorem 1 and by utilizing the matrix decomposition technique, we derive the following theorem, giving a low-dimensional condition for the bipartite tracking consensus.*

**Theorem 2***Let gain matrix *

*where * (1)

**Proof 2***Clearly, it suffices to prove that the LMIs in Equation* (52) *imply the LMIs in Equation* (9).

*Since matrix * (33)

*where *

*with *

*Denoting*

*one has*

*where *

*In light of * (52)

**Remark 2***Similar to the approach discussed in ^{[15]} for computational complexity, the computational complexity of the LMIs in Equation* (9)

We establish above sufficient conditions to ensure the MAS in Equations (1) and (2) can achieve the bipartite tracking consensus. Next, we consider the design problem of controller.

**Theorem 3.***Under Assumptions 1 and 2, for given positive scalars *

*with * (1)

*and a upper bound of the sampling period *

*with *

**Proof 3***Pre- and post-multiplying Equation* (51) *by *

*Selecting * (56)

*Using the Schur complement, the inequalities in Equation* (61) *are equivalent to the inequalities in Equation* (57). *Consequently, by Theorem 2, the bipartite tracking consensus is reached for the MAS in Equations* (1) *and* (2).

In what follows, a corollary is presented to maximize an estimate of elliptical attraction domain of bipartite tracking consensus.

**Corollary 1***For the ellipsoidal set *

**Proof 4***The proof can be obtained directly from Theorem 2 and Schur complement, and is therefore omitted here.*

**Remark 3***Since the maximization problem of ellipsoidal set * (62)

A simulation example is provided in this section to confirm the theoretical results.

Consider the MASs consisting of six agents, and the corresponding parameters are listed as follows:

Obviously, the nonlinear function

It is easy to verify that the six agents can be divided into two clusters:

According to Theorem 3, the MAS in Equations (1) and (2) achieves the bipartite tracking consensus for any initial conditions

In this paper, we have investigated the sampled-data tracking consensus problem for a class of nonlinear MASs subjected to input saturation over cooperation–competition networks. Based on the Lyapunov stability theory and some analysis tips, some LMI-based criteria are derived to guarantee the concerned MASs can reach the bipartite tracking consensus. Besides, by utilizing matrix decoupling method, the dimensions of LMIs are reduced to avoid a heavy computational burden. Moreover, an optimization problem is presented to maximize an estimate of ellipsoidal attraction domain of bipartite tracking consensus. Finally, a simulation example is provided to verify our main theoretical results. For the sampled-data-based bipartite tracking consensus of nonlinear MASs subject to input saturation, there are still some topics worthy of being investigated in the future, including the extension of our results to more general MASs with mixed time delays and other network-induced phenomena.

The authors would like to thank the reviewers for their valuable comments.

Conceptualization, Writing - original draft, Writing - review & editing: Yu L

Writing - original draft: Cui Y

Writing - review & editing: Lu Z

Methodology, Supervision: Liu Y

Not applicable.

The work was supported by the National Natural Science Foundation of China under Grants 62173292, 62003090, 61873230 and 61773017.

All authors declared that there are no conflicts of interest.

Not applicable.

Not applicable.

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**OAE Style**

Yu L, Cui Y, Lu Z, Liu Y. Sampled-based bipartite tracking consensus of nonlinear multiagents subject to input saturation.
* Complex Eng Syst* 2022;2:6. http://dx.doi.org/10.20517/ces.2022.08

**AMA Style**

Yu L, Cui Y, Lu Z, Liu Y. Sampled-based bipartite tracking consensus of nonlinear multiagents subject to input saturation.
*Complex Engineering Systems.*
2022; 2(2):6. http://dx.doi.org/10.20517/ces.2022.08

**Chicago/Turabian Style**

Yu, Luyang, Ying Cui, Zongya Lu, Yurong Liu. 2022. "Sampled-based bipartite tracking consensus of nonlinear multiagents subject to input saturation"
*Complex Engineering Systems.*
2, no.2: 6. http://dx.doi.org/10.20517/ces.2022.08

**ACS Style**

Yu, L.; Cui Y.; Lu Z.; Liu Y. Sampled-based bipartite tracking consensus of nonlinear multiagents subject to input saturation.
*Complex. Eng. Syst.*
**2022**, *2*, 6. http://dx.doi.org/10.20517/ces.2022.08

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