Sampled-based bipartite tracking consensus of nonlinear multiagents subject to input saturation
Correspondence to: Dr. Yurong Liu, Department of Mathematics, Yangzhou University, No. 180 Siwangting Road, Hanjiang Dsitrict, Yangzhou 225002, Jiangsu, China. E-mail:
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.
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 , formation control of multiple robot systems , and dynamics of opinion forming . 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 , where a network-based consensus control protocol was designed to ensure that the followers' states reach an agreement on the leader's state. In , 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  designed two distributed protocols to solve the bipartite output consensus problem for heterogeneous MASs with structurally balanced graphs. In , 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 , 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 , 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 . 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 , 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  to ensure the consensus error of one-sided Lipschitz nonlinear MASs was asymptotically stable. Based on the low-gain feedback technique, the authors of  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,
2. PROBLEM FORMULATION
A directed signed graph is denoted by
Definition 1 A directed signed graph
Lemma 1 A signed graph
Lemma 2 Given a matrix
Consider a group of agents with
and the leader is modeled by
Assumption 1Assume that the communication graph
Definition 2 The MAS in Equations (1) and (2) is said to achieve the bipartite tracking consensus, if there exists an appropriate distributed control scheme such that
for any initial conditions
In this paper, we put forward to the following sampled-based controller (control scheme) of the
According to Assumption 1 and Lemma 1, there exists
From Equations (1) and (2) and the distributed control protocol in Equation (4), one can get that
Since the leader has no neighbors, the Laplacian matrix of the MAS in Equations (1) and (2) can be partitioned as
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.
3. MAIN RESULTS
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 2The odd nonlinear vector-valued function
Lemma 3 Let
Lemma 4 Let
Lemma 5 Let matrices
Lemma 6 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 1Suppose Assumptions 1 and 2 hold. Let gain matrix
Equation (7) can be rewritten as
Construct the following Lyapunov function:
According to Assumption 2 and
One the other hand, if
From Equations (15)–(17), it follows that
which implies that
It can also be validated that
From Lemmas 4 and 5, it follows
Substituting Equations (14), (20), and (23)–(25) into Equation (13), one has
Without loss of generality, we assume
which implies that
Furthermore, it follows from Equation (28) that
By Kronecker product, we can rewrite Equation (29) as
Hence, according to Lemma 3, we have
Subsequently, the inequality in Equation (26) in combination with Equation (32) indicates that
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
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
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
It is easy to see that
Consequently, in view of Equation (49), we obtain that
Remark 1Theorem 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 2Let gain matrix
Proof 2Clearly, it suffices to prove that the LMIs in Equation (52) imply the LMIs in Equation (9).
In light of
Remark 2Similar to the approach discussed in  for computational complexity, the computational complexity of the LMIs in Equation (9) in Theorem 1 can be represented as
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
and a upper bound of the sampling period
Proof 3Pre- and post-multiplying Equation (51) by
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 1For the ellipsoidal set
Proof 4The proof can be obtained directly from Theorem 2 and Schur complement, and is therefore omitted here.
Remark 3Since the maximization problem of ellipsoidal set
4. SIMULATION STUDY
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
Availability of data and materials
Financial support and sponsorship
The work was supported by the National Natural Science Foundation of China under Grants 62173292, 62003090, 61873230 and 61773017.
Conflicts of interest
All authors declared that there are no conflicts of interest.
Ethical approval and consent to participate
Consent for publication
The Author(s) 2022.
1. Peng Z, Wang D, Wang W, Liu L. Containment control of networked autonomous underwater vehicles: A predictor-based neural DSC design. ISA Trans 2015;59:160-71.
2. Abdollahi F, Khorasani K. A decentralized markovian jump H∞ control routing strategy for mobile multi-agent networked systems. IEEE Trans Contr Syst Technol 2011;19:269-83.
3. Guan ZH, Hu B, Shen XS. Introduction to hybrid intelligent networks: Modeling, communication, and control. Springer; 2019.
4. Yuan Y, Wang Z, Zhang P, Dong H. Nonfragile near-optimal control of stochastic time-varying multiagent systems with control-and state-dependent noises. IEEE Trans Cybern 2019;49:2605-17.
5. Li H, Liao X, Huang T, Zhu W, Liu Y. Second-order global consensus in multiagent networks with random directional link failure. IEEE Trans Neural Netw Learn Syst 2015;26:565-75.
6. Ma L, Wang Z, Han QL, Liu Y. Consensus control of stochastic multi-agent systems: A survey. Sci China Inf Sci 2017;60.
7. Olfati-Saber R, Murray RM. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans Automat Contr 2004;49:1520-33.
8. Li Z, Ren W, Liu X, Fu M. Consensus of multi-agent systems with general linear and lipschitz nonlinear dynamics using distributed adaptive protocols. IEEE Trans Automat Contr 2013;58:1786-91.
9. Zhang W, Ho DWC, Tang Y, Liu Y. Quasi-consensus of heterogeneous-switched nonlinear multiagent systems. IEEE Trans Cybern 2020;50:3136-46.
10. Ding L, Han QL, Guo G. Network-based leader-following consensus for distributed multi-agent systems. Automatica 2013;49:2281-86.
11. Ding L, Zheng WX. Network-based practical consensus of heterogeneous nonlinear multiagent systems. IEEE Trans Cybern 2017;47:1841-51.
12. Lu M, Liu L. Consensus of heterogeneous second-order nonlinear uncertain multiagent systems under switching networks. IEEE Trans Automat Contr 2021;66:3331-38.
13. Zou L, Wang Z, Hu J, Liu Y, Liu X. Communication-protocol-based analysis and synthesis of networked systems: Progress, prospects and challenges. Inter J Syst Sci 2021;52:3013-34.
14. Liu Y, Ho DWC, Wang Z. A new framework for consensus for discrete-time directed networks of multi-agents with distributed delays. Inter J Contr 2012;85:1755-65.
15. Liu Y, Wang Z, Yuan Y, Liu W. Event-Triggered Partial-Nodes-Based State Estimation for Delayed Complex Networks With Bounded Distributed Delays. IEEE Trans Syst Man Cybern, Syst 2019;49:1088-98.
16. Altafini C. Consensus problems on networks with antagonistic interactions. IEEE Trans Automat Contr 2013;58:935-46.
17. Yang Y, Liu Q, Yue D, Han QL. Predictor-based neural dynamic surface control for bipartite tracking of a class of nonlinear multiagent systems. IEEE Trans Neural Netw Learn Syst 2021:1-12.
18. Zhao G, Hua C. Leaderless and leader-following bipartite consensus of multiagent systems with sampled and delayed information. IEEE Trans Neural Netw Learn Syst 2021:1-14.
19. Lu J, Wang Y, Shi X, Cao J. Finite-time bipartite consensus for multiagent systems under detail-balanced antagonistic interactions. IEEE Trans Syst Man Cybern, Syst 2021;51:3867-75.
20. Wang D, Ma H, Liu D. Distributed control algorithm for bipartite consensus of the nonlinear time-delayed multi-agent systems with neural networks. Neurocomputing 2016;174:928-36.
21. Han T, Zheng WX. Bipartite output consensus for heterogeneous multi-agent systems via output regulation approach. IEEE Trans Circuits Syst II 2021;68:281-85.
22. Wen G, Wang H, Yu X, Yu W. Bipartite tracking consensus of linear multi-agent systems with a dynamic leader. IEEE Trans Circuits Syst II 2018;65:1204-8.
23. Liu W, Huang J. Leader-following consensus for linear multiagent systems via asynchronous sampled-data control. IEEE Trans Automat Contr 2020;65:3215-22.
24. Gao C, Wang Z, He X, Yue D. Sampled-data-based fault-tolerant consensus control for multi-agent systems: A data privacy preserving scheme. Automatica 2021;133:109847.
25. Ge C, Park JH, Hua C, Guan X. Nonfragile consensus of multiagent systems based on memory sampled-data control. IEEE Trans Syst Man Cybern, Syst 2021;51:391-99.
26. Cui Y, Liu Y, Zhang W, Alsaadi FE. Sampled-based consensus for nonlinear multiagent systems with deception attacks: The decoupled method. IEEE Trans Syst Man Cybern, Syst 2021;51:561-73.
27. Ménard T, Moulay E, Coirault P, Defoort M. Observer-based consensus for second-order multi-agent systems with arbitrary asynchronous and aperiodic sampling periods. Automatica 2019;99:237-45.
28. Tarbouriech S, Garcia G, da Silva JMG, Queinnec I. Stability and stabilization of linear systems with saturating actuators London: Springer; 2011.
29. Kan X, Wang Z, Shu H. State estimation for discrete-time delayed neural networks with fractional uncertainties and sensor saturations. Neurocomputing 2013;117:64-71.
30. Chen Y, Wang Z, Alsaadi FE, Liu H. Dynamic output-feedback H∞ control for discrete time-delayed systems with actuator saturations under round-robin communication protocol. Inter J Robust Nonlin Contr 2022;32:1703-20.
31. Gao C, Wang Z, He X, Han QL. On consensus of second-order multiagent systems with actuator saturations: A generalized-nyquist-criterion-based approach. IEEE Trans Cybern 2020;PP:1-11.
32. Chen Y, Wang Z, Shen B, Dong H. Exponential synchronization for delayed dynamical networks via intermittent control: Dealing with actuator saturations. IEEE Trans Neural Netw Learn Syst 2019;30:1000-1012.
33. Li P, Jabbari F, Sun XM. Containment control of multi-agent systems with input saturation and unknown leader inputs. Automatica 2021;130:109677.
34. Chen Y, Wang Z, Hu J, Han QL. Synchronization control for discrete-time-delayed dynamical networks with switching topology under actuator saturations. IEEE Trans Neural Netw Learn Syst 2021;32:2040-53.
35. Wang B, Chen W, Wang J, Zhang B, Shi P. Semiglobal tracking cooperative control for multiagent systems with input saturation: A multiple saturation levels framework. IEEE Trans Automat Contr 2021;66:1215-22.
36. Wu H, Su H. Positive edge consensus of networked systems with input saturation. ISA Trans 2020;96:210-17.
37. Rehan M, Ahn CK, Chadli M. Consensus of one-sided lipschitz multi-agents under input saturation. IEEE Trans Circuits Syst II 2020;67:745-49.
38. Qin J, Fu W, Zheng WX, Gao H. On the bipartite consensus for generic linear multiagent systems with input saturation. IEEE Trans Cybern 2017;47:1948-58.
39. Wen G, Duan Z, Chen G, Yu W. Consensus tracking of multi-agent systems with Lipschitz-type node dynamics and switching topologies. IEEE Trans Circuits Syst I 2014;61:499-511.
40. Tarbouriech S, Prieur C, Silva JMGD. Stability analysis and stabilization of systems presenting nested saturations. IEEE Trans Automat Contr 2006;51:1364-71.
41. Zhang W, Wang Z, Liu Y, Ding D, Alsaadi FE. Sampled-data consensus of nonlinear multiagent systems subject to cyber attacks. Int J Robust Nonlinear Contr 2017;28:53-67.
Cite This Article
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
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
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
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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