zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Second-order leader-following consensus of nonlinear multi-agent systems via pinning control. (English) Zbl 1213.37131
The authors consider a nonlinear multi-agent system composed of $N$ coupled autonomous agents with second-order dynamics: $$ \dot x_i(t)=v_i(t), \quad \dot v_i(t)=f(t, x_i(t), v_i(t)), \quad i=1,\dots,N, \tag 1$$ where $n$-vectors $x_i$ and $v_i$ are the position and velocity states of agent $i$, $f$ is a nonlinear vector-valued continuous function, $u_i$ is the control input for agent $i$. The virtual leader for multi-agent system (1) is described by $$ \dot x_r(t)=v_r(t), \quad \dot v_r(t)=f(t, x_r(t), v_r(t)). \tag 2$$ The multi-agent system (1) is said to achieve second-order leader-following consensus, if its solution satisfies $\lim_{t\to\infty}\|x_i(t)-x_r(t)\|=0$, $\lim_{t\to\infty}\|v_i(t)-v_r(t)\|=0$, $i=1,\dots,n$, for any initial conditions. With the use of graph theory, matrix theory, and LaSalle’s invariance principle, a consensus algorithm based on pinning control is developed in the paper for second-order multi-agent systems with time-varying and constant reference velocities $v_r$. A pinned-agent selection scheme is provided to determine what kind of followers and how many followers should be informed by the virtual leader, sufficient conditions are derived to guarantee the global asymptotic stability of the second-order leader-following consensus. Numerical simulation results are presented for systems with 10 and 100 agents.

37N35Dynamical systems in control
93C15Control systems governed by ODE
34H05ODE in connection with control problems
Full Text: DOI
[1] Reynolds, C. W.: Flocks, herds, and schools: a distributed behavioral model, Comput. graph. 21, No. 4, 25-34 (1987)
[2] Vicsek, T.; Czirók, A.; Ben-Jacob, E.; Cohen, I.; Shochet, O.: Novel type of phase transition in a system of self-driven particles, Phys. rev. Lett. 75, No. 6, 1226-1229 (1995) · Zbl 0869.92002
[3] Olfati-Saber, R.; Murray, R. M.: Consensus problems in networks of agents with switching topology and time-delays, IEEE trans. Automat. control 49, No. 9, 1520-1533 (2004)
[4] Ren, W.; Beard, R. W.: Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE trans. Automat. control 50, No. 5, 655-661 (2005)
[5] Sun, Y.; Wang, L.; Xie, G.: Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays, Systems control lett. 57, 175-183 (2008) · Zbl 1133.68412 · doi:10.1016/j.sysconle.2007.08.009
[6] Lu, J.; Ho, D. W. C.; Kurths, J.: Consensus over directed static networks with arbitrary finite communication delays, Phys. rev. E 80, 066121 (2009)
[7] Ren, W.; Atkins, E.: Distributed multi-vehicle coordinated control via local information exchange, Int. J. Robust nonlinear control 17, 1002-1033 (2007) · Zbl 1266.93010
[8] Xie, G.; Wang, L.: Consensus control for a class of networks of dynamic agents, Int. J. Robust nonlinear control 17, 941-959 (2007) · Zbl 1266.93013
[9] Hong, Y.; Chen, G.; Bushnell, L.: Distributed observers design for leader-following control of multi-agent networks, Automatica 44, 846-850 (2008) · Zbl 1283.93019
[10] Ren, W.: On consensus algorithms for double-integrator dynamics, IEEE trans. Automat. control 53, No. 6, 1503-1509 (2008)
[11] Sun, Y.; Wang, L.: Consensus problems in networks of agents with double-integrator dynamics and time-varying delays, Int. J. Control 82, No. 10, 1937-1945 (2009) · Zbl 1178.93013 · doi:10.1080/00207170902838269
[12] Tian, Y.; Liu, C.: Robust consensus of multi-agent systems with diverse input delays and asymmetric interconnection perturbations, Automatica 45, 1347-1353 (2009) · Zbl 1162.93027 · doi:10.1016/j.automatica.2009.01.009
[13] Lin, P.; Jia, Y.: Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies, Automatica 45, 2154-2158 (2009) · Zbl 1175.93078 · doi:10.1016/j.automatica.2009.05.002
[14] Wu, C. W.; Chua, L. O.: Synchronization in an array of linearly coupled dynamical systems, IEEE trans. Circuits syst. I 42, No. 8, 430-447 (1995) · Zbl 0867.93042 · doi:10.1109/81.404047
[15] Pecora, L. M.; Carroll, T. L.: Master stability functions for synchronized coupled systems, Phys. rev. Lett. 80, No. 10, 2109-2112 (1998)
[16] Lu, W.; Chen, T.: New approach to synchronization analysis of linearly coupled ordinary differential systems, Physica D 213, 214-230 (2006) · Zbl 1105.34031 · doi:10.1016/j.physd.2005.11.009
[17] Ren, W.: Synchronization of coupled harmonic oscillators with local interaction, Automatica 44, 3195-3200 (2008) · Zbl 1153.93421 · doi:10.1016/j.automatica.2008.05.027
[18] Su, H.; Wang, X. F.; Lin, Z.: Synchronization of coupled harmonic oscillators in a dynamic proximity network, Automatica 45, 2286-2291 (2009) · Zbl 1179.93102 · doi:10.1016/j.automatica.2009.05.026
[19] D’humieres, D.; Beasley, M. R.; Huberman, B. A.; Libchaber, A.: Chaotic states and routes to chaos in the forced pendulum, Phys. rev. A 26, No. 6, 3483-3496 (1982)
[20] Amster, P.; Mariani, M. C.: Some results on the forced pendulum equation, Nonlinear anal. 68, 1874-1880 (2008) · Zbl 1148.34313 · doi:10.1016/j.na.2007.01.018
[21] Khalil, H. K.: Nonlinear systems, (2002) · Zbl 1003.34002
[22] Yu, W.; Chen, G.; Cao, M.; Kurths, J.: Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics, IEEE trans. Syst. man cybern. Part B 40, No. 3, 881-891 (2010)
[23] Chen, F.; Chen, Z.; Xiang, L.; Liu, Z.; Yuan, Z.: Reaching a consensus via pinning control, Automatica 45, 1215-1220 (2009) · Zbl 1162.93305 · doi:10.1016/j.automatica.2008.12.027
[24] Liu, X.; Chen, T.; Lu, W.: Consensus problem in directed networks of multi-agents via nonlinear protocols, Phys. lett. A 373, 3122-3127 (2009) · Zbl 1233.34012 · doi:10.1016/j.physleta.2009.06.054
[25] Ren, W.: Multi-vehicle consensus with a time-varying reference state, Systems control lett. 56, 474-483 (2007) · Zbl 1157.90459 · doi:10.1016/j.sysconle.2007.01.002
[26] Wang, X. F.; Chen, G.: Pinning control of scale-free dynamical networks, Physica A 310, 521-531 (2002) · Zbl 0995.90008 · doi:10.1016/S0378-4371(02)00772-0
[27] Li, X.; Wang, X. F.; Chen, G.: Pinning a complex dynamical network to its equilibrium, IEEE trans. Circuits syst. I 51, No. 10, 2074-2087 (2004)
[28] Yu, W.; Chen, G.; Lü, J.: On pinning synchronization of complex dynamical networks, Automatica 45, 429-435 (2009) · Zbl 1158.93308 · doi:10.1016/j.automatica.2008.07.016
[29] Xiang, J.; Chen, G.: On the V-stability of complex dynamical networks, Auomatica 43, 1049-1057 (2007) · Zbl 05246818
[30] Xiang, J.; Chen, G.: Analysis of pinning-controlled networks: a renormalization approach, IEEE trans. Automat. control 54, No. 8, 1869-1875 (2009)
[31] Chen, T.; Liu, X.; Lu, W.: Pinning complex networks by a single controller, IEEE trans. Circuits syst. I 54, No. 6, 1317-1326 (2007)
[32] Lu, J.; Ho, D. W. C.; Wang, Z.: Pinning stabilization of linearly coupled stochastic neural networks via minimum number of controllers, IEEE trans. Neural netw. 20, No. 10, 1617-1629 (2009)
[33] Xiong, W.; Ho, D. W. C.; Huang, C.: Pinning synchronization of time-varying polytopic directed stochastic networks, Phys. lett. A 374, 439-447 (2010) · Zbl 1235.34160
[34] Lu, W.; Li, X.; Rong, Z.: Global stabilization of complex networks with diagraph topologies via a local pinning algorithm, Automatica 46, 116-121 (2010) · Zbl 1214.93090 · doi:10.1016/j.automatica.2009.10.006
[35] Song, Q.; Cao, J.: On pinning synchronization of directed and undirected complex dynamical networks, IEEE trans. Circuits syst. I 57, No. 3, 672-680 (2010)
[36] Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in system and control theory, (1994) · Zbl 0816.93004
[37] Wilkinson, J. H.: The algebraic eigenvalue problem, (1965) · Zbl 0258.65037
[38] Langville, A. N.; Stewart, W. J.: The Kronecker product and stochastic automata networks, J. comput. Appl. math. 167, 429-447 (2004) · Zbl 1104.68062 · doi:10.1016/j.cam.2003.10.010
[39] Huijberts, H.; Nijmeijer, H.; Oguchi, T.: Anticipating synchronization of chaotic Lur’e systems, Chaos 17, 013117 (2007) · Zbl 1159.37356 · doi:10.1063/1.2710964
[40] Park, S. M.; Kim, B. J.: Dynamic behaviors in directed networks, Phys. rev. E 74, 026114 (2006)