Leaderless Consensus of Semilinear Hyperbolic Multiagent Systems with Semipositive or Seminegative Definite Convection

Dai, Jiashu; Yang, Chengdong; Xu, Dong; Wen, Shiping; Jian, Muwei; Yang, Dongliang

doi:https://doi.org/10.1155/2022/1976765

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

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Analysis, Control and Applications of Passivity in Complex Networks 2022

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Research Article | Open Access

Volume 2022 | Article ID 1976765 | https://doi.org/10.1155/2022/1976765

Leaderless Consensus of Semilinear Hyperbolic Multiagent Systems with Semipositive or Seminegative Definite Convection

Jiashu Dai,¹Chengdong Yang,²Dong Xu,³Shiping Wen,⁴Muwei Jian,²and Dongliang Yang⁵

Academic Editor: Akbar Ali

Received28 Apr 2022

Accepted06 Jul 2022

Published08 Aug 2022

Abstract

This paper deals with a leaderless consensus of semilinear first-order hyperbolic partial differential equation-based multiagent systems (HPDEMASs). A consensus controller under an undirected graph is designed. Dealing with different convection assumptions, two different boundary conditions are presented, one right endpoint and the other left endpoint. Two sufficient conditions for leaderless consensus of HPDEMAS are presented by giving the gain range in the case of the symmetric seminegative definite convection coefficient and the semipositive definite convection coefficient, respectively. Two examples are presented to show the effectiveness of the control methods.

1. Introduction

As one well-known group of dynamical behavior, multiagent systems (MASs) received many researchers’ attention in the last few decades [1]. There are a number of applications for MASs in engineering fields, for instance, power engineering [2], artificial intelligence [3], energy optimization [4], security [5, 6], traffic decision [7], and precision agriculture [8].

Consensus control of MASs is to derive agents to do a designated task synchronously[9, 10]. Many meaningful control methods have been presented, such as event-triggered control [11, 12], containment control [13], pinning control [14], impulsive control [15], sampled-data control [16], and adaptive control [17].

To put it another way, the mentioned literature assumed dynamics of MASs depending only on time. In practice, dynamics of all processes depends on both time and space. As a consequence, it is necessary to study spatio-temporal MASs [18]. Qi et al. proposed boundary control for PDE-modeled MASs(PDEMASs) under 3-D space with a control delay [19] and formation control for PDEMASs on a cylindrical surface [20]. An iterative learning algorithm was proposed for the consensus of multiagent system PDEMASs [21]. Yang et al. proposed several control methods for the consensus of semilinear PDEMASs or partial integro-differential equation-based MASs without and with time delays [22–24]. Several iterative learning methods were studied for the consensus of PDEMASs[21, 25–27].

Most of the above references are modeled by parabolic PDE-based MASs, while there are few works considering hyperbolic PDE-based MASs (HPDEMASs). The consensus of HPDEMASs is meaningful and significant, as a result of existence of a number of hyperbolic PDE systems in practice [28, 29], including gas dynamics [30], reactor [31], traffic flow [32], and hyperbolic Hopfield neural networks [33]. There are several important studies about consensus of HPDEMASs. For example, Fu et al. proposed the containment control method for the consensus of linear parabolic PDEMASs and second-order HPDEMASs [34]. Wang and Huang proposed the boundary control approach for the finite-time consensus of HPDEMASs by assuming the convection coefficient to be 1 [35]. Zhang et al. proposed boundary control for the leader-following consensus of MASs with input delays by assuming the convection coefficient to be a positive definite diagonal matrix[36]. However, there are still technical difficulties in the consensus of semilinear first-order HPDEMASs for the cases of the convection coefficient to be symmetric seminegative definite or semipositive definite, which are motives of this paper.

This paper mainly studies leaderless consensus control of a semilinear HPDEMAS with two sorts of boundary conditions in one-dimensional space. The contribution of this paper contains (1) a class of HPDEMAS models is given, assuming two sorts of conditions, one symmetric seminegative definite convection coefficient and the other semipositive definite convection coefficient. (2) Dealing with different convection assumptions, two different boundary conditions are presented, one right endpoint and the other left endpoint. (3) A consensus controller based on communication is studied to drive HPDEMAS to reach leaderless consensus. (4) Dealing with two sorts of convection coefficients, two sufficient conditions for the consensus of the leaderless HPDEMAS are, respectively, reached.

Notations: Let denotes the identity matrix with proper order, denotes the maximum (minimum) eigenvalue of , denotes the smallest nonzero eigenvalue of , and the superscript denotes the transpose.

2. Problem Formulation

This paper studies a class of semilinear HPDEMASs with time delayswhere are space and time, respectively. are the state and control input, respectively. , , , and are a nonlinear function.

The boundary condition isor

The initial condition is

This paper aims to study one controller driving HPDEMAS (1) to the leaderless consensus. Let consensus error and the controller is designed as follows:where is a control gain to be determined and is symmetric positive definite. Assume that the topological structure is defined ; if the agent connects to , otherwise .

Remark 1. Compared with papers [35, 36] using the information of only one neighbor, this controller (5) considers the whole communication information among all neighbors and takes full advantage of that.

Definition 1. ([35]) HPDEMAS (1) reaches a consensus, ifwhere .

Lemma 1. ([37]) For the Laplacian matrix , symmetric positive definite and such that , the following inequality is satisfied:

Assumption 1. ([23]) For any , there exists satisfying

3. Consensus of HPDEMASs with the Seminegative Definite Convection Coefficient

Assumption 2. Assume is symmetric seminegative definite.
Note that Assumption 2 is sort of classical, which is extensively employed in the practice, see, e.g. [38, 39].
The error system of HPDEMAS (1), (2), and (4) can be obtained as followswhere , , , , , , , , , and so is a Laplace matrix [37].

Theorem 1. Suppose that Assumptions 1 and 2 hold. The leaderless HPDEMAS shown in equations (1), (2), and (4) reaches the consensus under the controller (5), if

Proof. We choose the Lyapunov functional candidate as shown in the following equation:Taking the time derivative of , we obtainSince is a Laplace matrix and is a symmetric positive definite matrix, using Lemma 1, one haswhere [40].
For symmetric seminegative definite , employing integrating by parts, one getswhich impliesSince , under Assumption 1, we can getSubstitution of (12), (14), (15) into (11), we obtainwhere .
It is obvious that (9) impliesSubstitution of (17) into (16), we obtain for all nonzero , implying consensus of HPDEMAS (1).

4. Consensus of HPDEMASs with the Symmetric Semipositive Definite Convection Coefficient

Assumption 3. Assume is symmetric semipositive definite.
Note that Assumption 3 is sort of classical, which is extensively employed in practice, see, e.g. [35, 36].
The error system of the HPDEMAS (1), (3), and (4) can be obtained as follows:

Theorem 2. Suppose that Assumptions 1 and 3 hold. The leaderless HPDEMAS shown in equations (1), (3), and (4) reaches the consensus under controller (5) if (9) holds.

Proof. We choose the same Lyapunov functional candidate as in (10). Taking the time derivative of , we obtain (11). For the symmetric semipositive definite , employing integrating by parts, one haswhich impliesSubstitution of (12), (15), (19) into (11), we obtain (16). It is obvious that (9) impliesThe later part of the proof is similar to that of Theorem 2, and so it is omitted.

Remark 2. Different from the control design for consensus of parabolic PDEMASs in [41, 42], this paper deals with the consensus of a class of HPDEMASs.

Remark 3. Consensus of HPDEMASs has been studied by assuming the convection coefficient to be 1 in [35] and to be a positive definite diagonal matrix in [36]. Different from these results, this paper assumes the convection coefficient to be symmetric seminegative and semipositive definite.

5. Numerical Simulation

Example 1. This example considers one HPDEMAS (1) as follows:with random initial conditions. We get the following parameters:The controller (5) is used with the following parameters:From Figure 1, it can be seen that HPDEMAS (1) cannot reach the consensus without control. With Theorem 1, solving (9) by Matlab, is obtained. Figure 2 shows that the HPDEMAS (1) reaches the consensus under controller (5) with . Controller (5) with the feedback gain is shown in Figure 3.

Example 2. This example considers one HPDEMAS (1) as follows:with random initial conditions.
We get , and the other parameters are the same as (22). The parameters and of the controller (5) are the same as (23).
From Figure 4, it can be seen that HPDEMAS (1) cannot reach the consensus without control. With Theorem 2, solving (9) by Matlab, is obtained. Figure 5 shows that the HPDEMAS (1) reaches the consensus under controller (5) with . The controller (5) with the feedback gain is shown in Figure 6.

6. Conclusion

This paper has dealt with leaderless consensus control of a class of semilinear HPDEMASs. One consensus controller of HPDEMASs under the structure of undirected graphs, making use of communication among agents, was established. Firstly, for the case of the symmetric seminegative definite convection coefficient, the boundary condition of the right endpoint was given. For the case of the symmetric semipositive definite convection coefficient, the boundary condition of the left endpoint was given. Two sufficient conditions for the consensus of HPDEMASs were obtained. Two examples illustrated the effectiveness of developed theoretical results. In future work, containment control, event-triggered control, and many other factors will be studied.

Data Availability

All the data in the simulation are included within this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China (Grant no. 61976123), the Natural Science Research in Colleges and Universities of Anhui Province of China (Grant nos. KJ2020A0362, KJ2020A0361, KJ2019ZD15), the Anhui Natural Science Foundation (Grant no. 2108085MF213), the Taishan Young Scholars Program of Shandong Province, the Key Development Program for Basic Research of Shandong Province (Grant No.ZR2020ZD44), and the Natural Science Foundation of Heilongjiang Province (Grant no. LH2020F049).

References

A. Dorri, S. Kanhere, and R. Jurdak, “Multi-agent systems: A survey,” IEEE Access, vol. 6, Article ID 28573, 2018.
View at: Publisher Site | Google Scholar
D. J. McArthur, E. M. Davidson, V. M. Catterson et al., “Multi-agent systems for power engineering applications-Part I: Concepts, approaches, and technical challenges,” IEEE Transactions on Power Systems, vol. 22, no. 4, pp. 1743–1752, 2007.
View at: Publisher Site | Google Scholar
J. Ferber and G. Weiss, Multi-agent Systems: An Introduction to Distributed Artificial Intelligence, Addison-Wesley, Boston, Massachusetts, United States, vol. 1, 1999.
A. Briones, F. Prieta, M. Mohamad, S. Mohamad, M. Omatu, and J. Corchado, “Multi-agent systems applications in energy optimization problems: A state-of-the-art review,” Energies, vol. 11, no. 8, p. 1928, 2018.
View at: Publisher Site | Google Scholar
W. He, W. Xu, X. Ge, Q.-L. Han, W. Du, and F. Qian, “Secure Control of Multiagent Systems Against Malicious Attacks: A Brief Survey,” IEEE Transactions on Industrial Informatics, vol. 18, no. 6, pp. 3595–3608, 2022.
View at: Publisher Site | Google Scholar
W. He, Z. Mo, Q.-L. Han, and F. Qian, “Secure impulsive synchronization in Lipschitz-type multi-agent systems subject to deception attacks,” IEEE/CAA Journal of Automatica Sinica, vol. 7, no. 5, pp. 1–9, 2020.
View at: Publisher Site | Google Scholar
M. Zouari, N. Baklouti, M. H. Kammoun, M. Ben Ayed, A. M Alimi, and J. Sanchez-Medina, “A multi-agent system for road traffic decision making based on hierarchical interval type-2 fuzzy knowledge representation system,” in Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 1–6, IEEE, Luxembourg, Luxembourg, July 11th -14th, 2021.
View at: Publisher Site | Google Scholar
M. Davoodi, S. Faryadi, and J. M. Velni, “A graph theoretic-based approach for deploying heterogeneous multi-agent systems with application in precision agriculture,” Journal of Intelligent and Robotic Systems, vol. 101, no. 1, pp. 10–15, 2021.
View at: Publisher Site | Google Scholar
W. Ren, R. W. Beard, and E. M. Atkins, “A survey of consensus problems in multi-agent coordination,” in Proceedings of the American Control Conference, pp. 1859–1864, IEEE, Atlanta Ga, June 8-10 2005.
View at: Google Scholar
X. Tan, M. Cao, and J. Cao, “Distributed dynamic event-based control for nonlinear multi-agent systems,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 68, no. 2, pp. 687–691, 2021.
View at: Publisher Site | Google Scholar
Z. Liu, A. Zhang, J. Qiu, and Z. Li, “Event-triggered control of second-order nonlinear multi-agent systems with directed topology,” Neurocomputing, vol. 452, pp. 820–826, 2021.
View at: Publisher Site | Google Scholar
W. Xu, W. He, D. W. C. Ho, and J. . Kurths, “Fully distributed observer-based consensus protocol: Adaptive dynamic event-triggered schemes,” Automatica, vol. 139, Article ID 110188, 2022.
View at: Publisher Site | Google Scholar
D. Wang and W. Wang, “Necessary and sufficient conditions for containment control of multi-agent systems with time delay,” Automatica, vol. 103, pp. 418–423, 2019.
View at: Publisher Site | Google Scholar
X. Li, Z. Yu, Z. Li, and N. Wu, “Group consensus via pinning control for a class of heterogeneous multi-agent systems with input constraints,” Information Sciences, vol. 542, pp. 247–262, 2021.
View at: Publisher Site | Google Scholar
W. He, X. Gao, W. Zhong, and F. Qian, “Secure impulsive synchronization control of multi-agent systems under deception attacks,” Information Sciences, vol. 459, pp. 354–368, 2018.
View at: Publisher Site | Google Scholar
X. Ge, Q.-L. Han, D. Ding, X.-M. Zhang, and B. Ning, “A survey on recent advances in distributed sampled-data cooperative control of multi-agent systems,” Neurocomputing, vol. 275, pp. 1684–1701, 2018.
View at: Publisher Site | Google Scholar
X. Jin, S. Lü, C. Deng, and M. Chadli, “Distributed adaptive security consensus control for a class of multi-agent systems under network decay and intermittent attacks,” Information Sciences, vol. 547, pp. 88–102, 2021.
View at: Publisher Site | Google Scholar
G. Ferrari-Trecate, A. Buffa, and M. Gati, “Analysis of coordination in multi-agent systems through partial difference equations,” IEEE Transactions on Automatic Control, vol. 51, no. 6, pp. 1058–1063, 2006.
View at: Publisher Site | Google Scholar
J. Qi, S. Wang, J.-an Fang, and M. Diagne, “Control of multi-agent systems with input delay via PDE-based method,” Automatica, vol. 106, pp. 91–100, 2019.
View at: Publisher Site | Google Scholar
J. Qi, S.-X. Tang, and C. Wang, “Parabolic PDE-based multi-agent formation control on a cylindrical surface,” International Journal of Control, vol. 92, no. 1, pp. 77–99, 2019.
View at: Publisher Site | Google Scholar
Q. Fu, L. Du, G. Xu, and J. Wu, “Consensus control for multi-agent systems with distributed parameter models via iterative learning algorithm,” Journal of the Franklin Institute, vol. 355, no. 10, pp. 4453–4472, 2018.
View at: Publisher Site | Google Scholar
C. Yang, H. He, T. Huang et al., “Consensus for non‐linear multi‐agent systems modelled by PDEs based on spatial boundary communication,” IET Control Theory & Applications, vol. 11, no. 17, pp. 3196–3200, 2017.
View at: Publisher Site | Google Scholar
C. Yang, T. Huang, A. Zhang, J. Qiu, J. Cao, and F. E. Alsaadi, “Output consensus of multiagent systems based on pdes with input constraint: A boundary control approach,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 51, no. 1, pp. 370–377, 2021.
View at: Publisher Site | Google Scholar
J. Dai, C. Yang, X. Yan, J. Wang, K. Zhu, and C. Yang, “Leaderless consensus control of nonlinear PIDE-type multi-agent systems with time delays,” IEEE Access, vol. 10, pp. 21211–21218, 2022.
View at: Publisher Site | Google Scholar
X. Dai, C. Wang, S. Tian, and Q. Huang, “Consensus control via iterative learning for distributed parameter models multi-agent systems with time-delay,” Journal of the Franklin Institute, vol. 356, no. 10, pp. 5240–5259, 2019.
View at: Publisher Site | Google Scholar
Y.-H. Lan, B. Wu, Y.-X. Shi, and Yi-P. Luo, “Iterative learning based consensus control for distributed parameter multi-agent systems with time-delay,” Neurocomputing, vol. 357, pp. 77–85, 2019.
View at: Publisher Site | Google Scholar
Y.-H. Lan, W. Bin, and Y. Zhou, “Iterative learning consensus control with initial state learning for fractional order distributed parameter models multi‐agent systems,” Mathematical Methods in the Applied Sciences, vol. 45, no. 1, pp. 5–20, 2022.
View at: Publisher Site | Google Scholar
S.-H. Tsai, J.-W. Wang, En-S. Song, and H.-K. Lam, “Robust control for nonlinear hyperbolic PDE systems based on the polynomial fuzzy model,” IEEE Transactions on Cybernetics, vol. 51, no. 7, pp. 3789–3801, 2019.
View at: Google Scholar
Ji Krstic and M. Krstic, “Event-triggered output-feedback backstepping control of sandwich hyperbolic PDE systems,” IEEE Transactions on Automatic Control, vol. 67, no. 1, pp. 220–235, 2022.
View at: Publisher Site | Google Scholar
S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications, OUP Oxford, England, 2006.
I. Aksikas, “Optimal control and duality-based observer design for a hyperbolic PDEs system with application to fixed-bed reactor,” International Journal of Systems Science, vol. 52, no. 12, pp. 2493–2504, 2021.
View at: Publisher Site | Google Scholar
I. Karafyllis, N. Bekiaris-Liberis, and M. Papageorgiou, “Feedback control of nonlinear hyperbolic PDE systems inspired by traffic flow models,” IEEE Transactions on Automatic Control, vol. 64, no. 9, pp. 3647–3662, 2019.
View at: Publisher Site | Google Scholar
M. Kobayashi, “Storage capacity of hyperbolic Hopfield neural networks,” Neurocomputing, vol. 369, pp. 185–190, 2019.
View at: Publisher Site | Google Scholar
Q. Fu, P. Yu, G. Xu, and J. Wu, “Containment control for partial differential multi-agent systems,” Physica A: Statistical Mechanics and its Applications, vol. 529, Article ID 121549, 2019.
View at: Publisher Site | Google Scholar
X. Huang and N. Huang, “Finite-time consensus of multi-agent systems driven by hyperbolic partial differential equations via boundary control,” Applied Mathematics and Mechanics, vol. 42, no. 12, pp. 1799–1816, 2021.
View at: Publisher Site | Google Scholar
H. Zhang, T. Wang, and J. Qiu, “PDE-based leader-following consensus of multi-agent systems with input delay under spatial boundary communication,” IFAC-PapersOnLine, vol. 54, no. 18, pp. 181–185, 2021.
View at: Publisher Site | Google Scholar
J. Qin, H. Gao, and W. X. Zheng, “Exponential synchronization of complex networks of linear systems and nonlinear oscillators: A unified analysis,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 3, pp. 510–521, 2015.
View at: Publisher Site | Google Scholar
J.-W. Wang, H.-N. Wu, and H.-X. Li, “Distributed fuzzy control design of nonlinear hyperbolic PDE systems with application to nonisothermal plug-flow reactor,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 3, pp. 514–526, 2011.
View at: Publisher Site | Google Scholar
I. Aksikas, A. Fuxman, J. F. Forbes, and J. J. Winkin, “LQ control design of a class of hyperbolic PDE systems: Application to fixed-bed reactor,” Automatica, vol. 45, no. 6, pp. 1542–1548, 2009.
View at: Publisher Site | Google Scholar
A. Pilloni, A. Pisano, Y. Orlov, and E. Usai, “Consensus-based control for a network of diffusion PDEs with boundary local interaction,” IEEE Transactions on Automatic Control, vol. 61, no. 9, pp. 2708–2713, 2016.
View at: Publisher Site | Google Scholar
D. Bahuguna, R. Sakthivel, and A. Chadha, “Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with infinite delay,” Stochastic Analysis and Applications, vol. 35, no. 1, pp. 63–88, 2017.
View at: Publisher Site | Google Scholar
H. Long and H. T. P. Thao, “Hyers-Ulam stability for nonlocal fractional partial integro-differential equation with uncertainty,” Journal of Intelligent and Fuzzy Systems, vol. 34, no. 1, pp. 233–244, 2018.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Jiashu Dai et al. 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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