# Practical bipartite synchronization via pinning control on a network of nonlinear agents with antagonistic interactions

## Abstract

This paper studies the synchronization phenomena in a network of heterogeneous nonlinear systems over signed graph, which can be considered as the perturbation of a network of homogeneous nonlinear systems. Assume that the signed graph is structurally balanced, the nonlinear system satisfies one-sided Lipschitz condition, and a leader pins a subset of agents. Under some proper initial conditions of the leader, we derive some conditions under which bipartite synchronization error can be kept arbitrary small by choosing a proper pining scheme. This property is called practical pinning bipartite synchronization as an alternative synchronization notion for network of heterogeneous nonlinear systems over signed graph. Finally, we present a numerical example to illustrate the effectiveness of the obtained results.

## Keywords

Bipartite synchronization Heterogeneous nonlinear systems Pinning control## References

- 1.Dörfler, F., Chertkov, M., Bullo, F.: Synchronization in complex oscillator networks and smart grids. Proc. Natl. Acad. Sci.
**110**(6), 2005–2010 (2013)MathSciNetCrossRefMATHGoogle Scholar - 2.Strogatz, S.H., Stewart, I., et al.: Coupled oscillators and biological synchronization. Sci. Am.
**269**(6), 102–109 (1993)CrossRefGoogle Scholar - 3.Acebrón, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys.
**77**(1), 137 (2005)CrossRefGoogle Scholar - 4.Ma, J., Qin, H., Song, X., Chu, R.: Pattern selection in neuronal network driven by electric autapses with diversity in time delays. Int. J. Mod. Phys. B
**29**(01), 1450239 (2015)CrossRefGoogle Scholar - 5.Qin, H., Ma, J., Wang, C., Chu, R.: Autapse-induced target wave, spiral wave in regular network of neurons. Sci. China Phys. Mech. Astron.
**57**(10), 1918–1926 (2014)CrossRefGoogle Scholar - 6.Wang, C., He, Y., Ma, J., Huang, L.: Parameters estimation, mixed synchronization, and antisynchronization in chaotic systems. Complexity
**20**(1), 64–73 (2014)MathSciNetCrossRefGoogle Scholar - 7.Altafini, C.: Consensus problems on networks with antagonistic interactions. IEEE Trans. Autom. Control
**58**(4), 935–946 (2013)MathSciNetCrossRefGoogle Scholar - 8.Hu, J., Zheng, W.X.: Bipartite consensus for multi-agent systems on directed signed networks. In: 2013 IEEE 52nd Annual Conference on IEEE Decision and Control (CDC), pp. 3451–3456 (2013)Google Scholar
- 9.Valcher, M.E., Misra, P.: On the consensus and bipartite consensus in high-order multi-agent dynamical systems with antagonistic interactions. Syst. Control Lett.
**66**, 94–103 (2014)MathSciNetCrossRefMATHGoogle Scholar - 10.Zhang, H., Chen, J.: Bipartite consensus of general linear multi-agent systems. In: American Control Conference (ACC), pp. 808–812. IEEE (2014)Google Scholar
- 11.Zhang, H., Chen, J.: Bipartite consensus of linear multi-agent systems over signed digraphs: an output feedback control approach. In: Proceedings on 19th IFAC World Congress, pp. 4681–4686 (2014)Google Scholar
- 12.Fan, M.-C., Zhang, H.-T., Wang, M.: Bipartite flocking for multi-agent systems. Commun. Nonlinear Sci. Numer. Simul.
**19**(9), 3313–3322 (2014)MathSciNetCrossRefGoogle Scholar - 13.Hu, J., Zheng, W.X.: Emergent collective behaviors on coopetition networks. Phys. Lett. A
**378**(26), 1787–1796 (2014)MathSciNetCrossRefMATHGoogle Scholar - 14.Wasserman, S.: Social Network Analysis: Methods and Applications, vol. 8. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
- 15.Altafini, C.: Dynamics of opinion forming in structurally balanced social networks. PloS ONE
**7**(6), e38135 (2012)CrossRefGoogle Scholar - 16.Lee, S.-H.: Predator’s attack-induced phase-like transition in prey flock. Phys. Lett. A
**357**(4), 270–274 (2006)CrossRefGoogle Scholar - 17.Grossberg, S.: Competition, decision, and consensus. J. Math. Anal. Appl.
**66**(2), 470–493 (1978)MathSciNetCrossRefMATHGoogle Scholar - 18.Grossberg, S.: Biological competition: Decision rules, pattern formation, and oscillations. In: Proceedings of the National Academy of Sciences of the United States of America, pp. 2338–2342 (1980)Google Scholar
- 19.Chen, C.C., Wan, Y.-H., Chung, M.-C., Sun, Y.-C.: An effective recommendation method for cold start new users using trust and distrust networks. Inf. Sci.
**224**, 19–36 (2013)MathSciNetCrossRefGoogle Scholar - 20.Victor, P., Cornelis, C., Cock, M., Teredesai, A.: Trust-and distrust-based recommendations for controversial reviews. IEEE Intell. Syst.
**26**(1), 48–55 (2011)CrossRefGoogle Scholar - 21.Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control
**49**(9), 1520–1533 (2004)MathSciNetCrossRefGoogle Scholar - 22.Ren, W., Beard, R.W., et al.: Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control
**50**(5), 655–661 (2005)MathSciNetCrossRefGoogle Scholar - 23.Zhai, S., Yang, X.-S.: Consensus of second-order multi-agent systems with nonlinear dynamics and switching topology. Nonlinear Dyn.
**77**(4), 1667–1675 (2014)MathSciNetCrossRefMATHGoogle Scholar - 24.Zhai, S., Yang, X.S.: Contraction analysis of synchronization of complex switched networks with different inner coupling matrices. J. Franklin Inst.
**350**(10), 3116–3127 (2013)MathSciNetCrossRefMATHGoogle Scholar - 25.Feng, Y., Xu, S., Lewis, F.L., Zhang, B.: Consensus of heterogeneous first-and second-order multi-agent systems with directed communication topologies. Int. J. Robust Nonlinear Control
**25**(3), 362–375 (2015)MathSciNetCrossRefMATHGoogle Scholar - 26.Chen, Y., Sun, J.: Distributed optimal control for multi-agent systems with obstacle avoidance. Neurocomputing
**173**(3), 2014–2021 (2016)CrossRefGoogle Scholar - 27.Wan, X., Sun, J.: Adaptiveimpulsive synchronization of chaotic systems. Math. Comput. Simul.
**81**(8), 1609–1617 (2011)MathSciNetCrossRefMATHGoogle Scholar - 28.Zhai, S., Li, Q.: Pinning bipartite synchronization for coupled nonlinear systems with antagonistic interactions and switching topologies. Syst. Control Lett.
**94**, 127–132 (2016)MathSciNetCrossRefMATHGoogle Scholar - 29.Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci.
**20**(2), 130–141 (1963)CrossRefGoogle Scholar - 30.Yu, W., Cao, J., Chen, G.: Stability and hopf bifurcation of a general delayed recurrent neural network. IEEE Trans. Neural Netw.
**19**(5), 845–854 (2008)CrossRefGoogle Scholar - 31.Montenbruck, J.M., Bürger, M., Allgöwer, F.: Practical synchronization with diffusive couplings. Automatica
**53**, 235–243 (2015)MathSciNetCrossRefGoogle Scholar - 32.Chen, T., Liu, X., Lu, W.: Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I Regul. Pap.
**54**(6), 1317–1326 (2007)Google Scholar - 33.Khalil, H.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2002)MATHGoogle Scholar
- 34.Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar