American Institute of Mathematical Sciences

Advanced
September  2011, 16(2): 489-505. doi: 10.3934/dcdsb.2011.16.489

Consensus of discrete-time linear multi-agent systems with observer-type protocols

Zhongkui Li 1, , Zhisheng Duan 2, and  Guanrong Chen 3,

1. 

School of Automation, Beijing Institute of Technology, Beijing 100081, P. R., China
2. 

State Key Lab for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, P. R., China
3. 

Department of Electronic Engineering, City University of Hong Kong, Hong Kong

Received  December 2009 Revised  October 2010 Published  June 2011

This paper concerns the consensus of discrete-time multi-agent systems with linear or linearized dynamics. An observer-type protocol based on the relative outputs of neighboring agents is proposed. The consensus of such a multi-agent system with a directed communication topology can be cast into the stability of a set of matrices with the same low dimension as that of a single agent. The notion of discrete-time consensus region is then introduced and analyzed. For neurally stable agents, it is shown that there exists an observer-type protocol having a bounded consensus region in the form of an open unit disk, provided that each agent is stabilizable and detectable. An algorithm is further presented to construct a protocol to achieve consensus with respect to all the communication topologies containing a spanning tree. Moreover, for the case where the agents have no poles outside the unit circle, an algorithm is proposed to construct a protocol having an origin-centered disk of radius
$\delta$ ($0<\delta<1$) as its consensus region. Finally, the consensus algorithms are applied to solve formation control problems of multi-agent systems.
Keywords: discrete-time linear system, multi-agent system, formation control., consensus region, observer-type protocol, Consensus.
Mathematics Subject Classification: Primary: 93A14, 93C55; Secondary: 93C0.
Citation: Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489
References:
[1]

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J. Cortés, Distributed algorithms for reaching consensus on general functions, Automatica, 44 (2008), 726-737. doi: 10.1016/j.automatica.2007.07.022.  Google Scholar

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Z. S. Duan, G. R. Chen and L. Huang, Synchronization of weighted networks and complex synchronized regions, Phys. Lett. A, 372 (2008), 3741-3751. doi: 10.1016/j.physleta.2008.02.056.  Google Scholar

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[6]

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[7]

P. Frasca, R. Carli, F. Pagnani and S. Zampieri, Average consensus on networks with quantized communication, Int. J. Robust Nonlinear Control, 19 (2008), 1787-1816. doi: 10.1002/rnc.1396.  Google Scholar

[8]

Y. Hong, J. Hu and L. Gao, Tracking control for multi-agent consensus with an active leader and variable topology, Automatica, 42 (2006), 1177-1182. doi: 10.1016/j.automatica.2006.02.013.  Google Scholar

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Y. Hong, G. R. Chen and L. Bushnell, Distributed observers design for leader-following control of multi-agent, Automatica, 44 (2008), 846-850. doi: 10.1016/j.automatica.2007.07.004.  Google Scholar

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A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonous agents using neareast neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.  Google Scholar

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T. Katayama, On the matrix Riccati equation for linear systems with a random gain, IEEE Trans. Autom. Control, 21 (1976), 770-771. doi: 10.1109/TAC.1976.1101325.  Google Scholar

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Z. K. Li, Z. S. Duan and L. Huang, $H_\infty$ control of networked multi-agent systems, J. Syst. Sci. Complex., 22 (2009), 35-48. doi: 10.1007/s11424-009-9145-y.  Google Scholar

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[16]

P. Lin, Y. M. Jia and L. Li, Distributed robust $H_\infty$ consensus control in directed networks of agents with time-delay, Syst. Control Lett., 57 (2008), 643-653. doi: 10.1016/j.sysconle.2008.01.002.  Google Scholar

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P. Lin and Y. M. Jia, Further results on decentralised coordination in networks of agents with second-order dynamics, IET Contr. Theory Appl., 3 (2009), 957-970. doi: 10.1049/iet-cta.2008.0263.  Google Scholar

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[19]

C. Q. Ma and J. F. Zhang, Necessary and sufficient conditions for consensusability of linear multi-agent systems, IEEE Trans. Autom. Control, 55 (2010), 1263-1268. doi: 10.1109/TAC.2010.2042764.  Google Scholar

[20]

K. Ogata, "Modern Control Engineering," 3rd edition, Prentice Hall: Englewood Cliffs, 1996. Google Scholar

[21]

R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.  Google Scholar

[22]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 51 (2006), 401-420. doi: 10.1109/TAC.2005.864190.  Google Scholar

[23]

R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Pro. IEEE, 97 (2007), 215-233. doi: 10.1109/JPROC.2006.887293.  Google Scholar

[24]

L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109-2112. doi: 10.1103/PhysRevLett.80.2109.  Google Scholar

[25]

W. Ren and R. W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topogies, IEEE Trans. Autom. Control, 50 (2005), 655-661. doi: 10.1109/TAC.2005.846556.  Google Scholar

[26]

W. Ren, R. W. Beard and E. M. Atkins, Information consensus in multivehicle cooperative control, IEEE Control Syst. Mag., 27 (2007), 71-82. doi: 10.1109/MCS.2007.338264.  Google Scholar

[27]

W. Ren, K. L. Moore and Y. Q. Chen, High-order and model reference consensus algorithms in cooperative control of multi-vehicle systems, J. Dyn. Syst. Meas. Control-Trans. ASME, 129 (2007), 678-688. Google Scholar

[28]

W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control, 53 (2008), 1503-1509.  Google Scholar

[29]

W. Ren and N. Sorensen, Distributed coordination architecture for multi-robot formation control, Robot. Auton. Syst., 56 (2008), 324-333. doi: 10.1016/j.robot.2007.08.005.  Google Scholar

[30]

A. Rahmani, M. Ji, M. Mesbahi and M. Egerstedt, Controllability of multi-agent systems from a graph-theorectic perspective, SIAM J. Control Optim., 48 (2009), 162-186. doi: 10.1137/060674909.  Google Scholar

[31]

L. Scardavi and S. Sepulchre, Synchronization in networks of identical linear systems, Automatica, 45 (2009), 2557-2562. doi: 10.1016/j.automatica.2009.07.006.  Google Scholar

[32]

L. Schenato, B. Sinopoli, M. Franceschetti, K. Poolla, M. I. Jordan and S. S. Sastry, Foundations of control and estimation over lossy networks, Proc. IEEE, 95 (2007), 163-187. doi: 10.1109/JPROC.2006.887306.  Google Scholar

[33]

J. H. Seo, H. Shim and J. Back, Consensus of high-order linear systems using dynamic output feedback compensator: Low gain approach, Automatica, 45 (2009), 2659-2664. doi: 10.1016/j.automatica.2009.07.022.  Google Scholar

[34]

B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. I. Jordan and S. S. Sastry, Kalman filtering with intermittent observations, IEEE Trans. Autom. Control, 49 (2004), 1453-1464. doi: 10.1109/TAC.2004.834121.  Google Scholar

[35]

Y. G. Sun and W. Long, Consensus problems in networks of agents with double-integrator dynamics and time-varying delays, Int. J. Control, 82 (2009), 1937-1945. doi: 10.1080/00207170902838269.  Google Scholar

[36]

R. S. Smith and F. Y. Hadaegh, Control of deep-space formation-flying spacecraft; Relative sensing and switched information, J. Guid. Control Dyn., 28 (2005), 106-114. doi: 10.2514/1.6165.  Google Scholar

[37]

H. S. Su, X. F. Wang and Z. L. Lin, Flocking of multi-agents with a virtual leader, IEEE Trans. Autom. Control, 54 (2009), 293-307. doi: 10.1109/TAC.2008.2010897.  Google Scholar

[38]

H. G. Tanner, A. Jadbabaie and G. J. Pappas, Flocking in fixed and switching networks, IEEE Trans. Autom. Control, 52 (2007), 863-868. doi: 10.1109/TAC.2007.895948.  Google Scholar

[39]

Y. P. Tian and C. L. Liu, Robust consensus of multi-agent systems with diverse input delays and asymmetric interconnection perturbations, Automatica, 45 (2009), 1347-1353. doi: 10.1016/j.automatica.2009.01.009.  Google Scholar

[40]

S. E. Tuna, Synchronizing linear systems via partial-state coupling, Automatica, 44 (2008), 2179-2184. doi: 10.1016/j.automatica.2008.01.004.  Google Scholar

[41]

S. E. Tuna, Conditions for synchronizability in arrays of coupled linear systems, IEEE Trans. Autom. Control, 54 (2009), 2416-2420. doi: 10.1109/TAC.2009.2029296.  Google Scholar

[42]

T. Vicsek, A. Cziroók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transitions in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[43]

J. H. Wang, D. Z. Cheng and X. M. Hu, Consensus of multi-agent linear dynamic systems, Asian J. Control, 10 (2008), 144-155. doi: 10.1002/asjc.15.  Google Scholar

[44]

G. Xie and L. Wang, Consensus control for a class of networks of dynamic agents, Int. J. Robust Nonlinear Control, 17 (2007), 941-959. doi: 10.1002/rnc.1144.  Google Scholar

[45]

R. Yamapi and R. S. Mackay, Stability of synchronization in a shift-invariant ring of mutually coupled oscillators, Discret. Contin. Dyn. Syst., 10 (2008), 973-996. doi: 10.3934/dcdsb.2008.10.973.  Google Scholar

[46]

H. T. Zhang, M. Z. Q. Chen, T. Zhou and G. B. Stan, Ultrafast consensus via predictive mechanisms, Europhysics Letters, 83 (2008), 40003. doi: 10.1209/0295-5075/83/40003.  Google Scholar

[47]

K. M. Zhou and J. C. Doyle, "Essentials of Robust Control," Prentice-Hall, Englewood Cliffs, 1998. Google Scholar

show all references

References:
[1]

D. Bauso, L. Giarré and R. Pesenti, Consensus for netowrks with unknown but bounded disturbances, SIAM J. Control Optim., 48 (2009), 1756-1770. doi: 10.1137/060678786.  Google Scholar

[2]

S. Bowong and J. L. Dimi, Adaptive synchronization of a class of uncertain chaotic systems, Discret. Contin. Dyn. Syst., 9 (2008), 235-248.  Google Scholar

[3]

J. Cortés, Distributed algorithms for reaching consensus on general functions, Automatica, 44 (2008), 726-737. doi: 10.1016/j.automatica.2007.07.022.  Google Scholar

[4]

Z. S. Duan, G. R. Chen and L. Huang, Synchronization of weighted networks and complex synchronized regions, Phys. Lett. A, 372 (2008), 3741-3751. doi: 10.1016/j.physleta.2008.02.056.  Google Scholar

[5]

Z. S. Duan, G. R. Chen and L. Huang, Disconnected synchronized regions of complex dynamical networks, IEEE Trans. Autom. Control, 54 (2009), 845-849. doi: 10.1109/TAC.2008.2009690.  Google Scholar

[6]

J. A. Fax and R. M. Murray, Information flow and cooperative control of vehicle formations, IEEE Trans. Automat. Control, 49 (2004), 1465-1476. doi: 10.1109/TAC.2004.834433.  Google Scholar

[7]

P. Frasca, R. Carli, F. Pagnani and S. Zampieri, Average consensus on networks with quantized communication, Int. J. Robust Nonlinear Control, 19 (2008), 1787-1816. doi: 10.1002/rnc.1396.  Google Scholar

[8]

Y. Hong, J. Hu and L. Gao, Tracking control for multi-agent consensus with an active leader and variable topology, Automatica, 42 (2006), 1177-1182. doi: 10.1016/j.automatica.2006.02.013.  Google Scholar

[9]

Y. Hong, G. R. Chen and L. Bushnell, Distributed observers design for leader-following control of multi-agent, Automatica, 44 (2008), 846-850. doi: 10.1016/j.automatica.2007.07.004.  Google Scholar

[10]

R. Horn and C. Johnson, "Matrix Analysis," Cambridge Univ. Press, New York, 1985.  Google Scholar

[11]

A. Jadbabaie, J. Lin and A. S. Morse, Coordination of groups of mobile autonous agents using neareast neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001. doi: 10.1109/TAC.2003.812781.  Google Scholar

[12]

T. Katayama, On the matrix Riccati equation for linear systems with a random gain, IEEE Trans. Autom. Control, 21 (1976), 770-771. doi: 10.1109/TAC.1976.1101325.  Google Scholar

[13]

G. Lafferriere, A. Williams, J. Caughman and J. J. P. Veerman, Decentralized control of vehicle formations, Syst. Control Lett., 54 (2005), 899-910. doi: 10.1016/j.sysconle.2005.02.004.  Google Scholar

[14]

Z. K. Li, Z. S. Duan and L. Huang, $H_\infty$ control of networked multi-agent systems, J. Syst. Sci. Complex., 22 (2009), 35-48. doi: 10.1007/s11424-009-9145-y.  Google Scholar

[15]

Z. K. Li, Z. S. Duan, G. R. Chen and L. Huang, Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint, IEEE Trans. Circuits Syst. I-Regul. Pap., 51 (2010), 213-224.  Google Scholar

[16]

P. Lin, Y. M. Jia and L. Li, Distributed robust $H_\infty$ consensus control in directed networks of agents with time-delay, Syst. Control Lett., 57 (2008), 643-653. doi: 10.1016/j.sysconle.2008.01.002.  Google Scholar

[17]

P. Lin and Y. M. Jia, Further results on decentralised coordination in networks of agents with second-order dynamics, IET Contr. Theory Appl., 3 (2009), 957-970. doi: 10.1049/iet-cta.2008.0263.  Google Scholar

[18]

C. Liu, Z. S. Duan, G. R. Chen and L. Huang, Analyzing and controlling the network synchronization regions, Physica A, 386 (2007), 531-542. doi: 10.1016/j.physa.2007.08.006.  Google Scholar

[19]

C. Q. Ma and J. F. Zhang, Necessary and sufficient conditions for consensusability of linear multi-agent systems, IEEE Trans. Autom. Control, 55 (2010), 1263-1268. doi: 10.1109/TAC.2010.2042764.  Google Scholar

[20]

K. Ogata, "Modern Control Engineering," 3rd edition, Prentice Hall: Englewood Cliffs, 1996. Google Scholar

[21]

R. Olfati-Saber and R. M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Autom. Control, 49 (2004), 1520-1533. doi: 10.1109/TAC.2004.834113.  Google Scholar

[22]

R. Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans. Autom. Control, 51 (2006), 401-420. doi: 10.1109/TAC.2005.864190.  Google Scholar

[23]

R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Pro. IEEE, 97 (2007), 215-233. doi: 10.1109/JPROC.2006.887293.  Google Scholar

[24]

L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109-2112. doi: 10.1103/PhysRevLett.80.2109.  Google Scholar

[25]

W. Ren and R. W. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topogies, IEEE Trans. Autom. Control, 50 (2005), 655-661. doi: 10.1109/TAC.2005.846556.  Google Scholar

[26]

W. Ren, R. W. Beard and E. M. Atkins, Information consensus in multivehicle cooperative control, IEEE Control Syst. Mag., 27 (2007), 71-82. doi: 10.1109/MCS.2007.338264.  Google Scholar

[27]

W. Ren, K. L. Moore and Y. Q. Chen, High-order and model reference consensus algorithms in cooperative control of multi-vehicle systems, J. Dyn. Syst. Meas. Control-Trans. ASME, 129 (2007), 678-688. Google Scholar

[28]

W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control, 53 (2008), 1503-1509.  Google Scholar

[29]

W. Ren and N. Sorensen, Distributed coordination architecture for multi-robot formation control, Robot. Auton. Syst., 56 (2008), 324-333. doi: 10.1016/j.robot.2007.08.005.  Google Scholar

[30]

A. Rahmani, M. Ji, M. Mesbahi and M. Egerstedt, Controllability of multi-agent systems from a graph-theorectic perspective, SIAM J. Control Optim., 48 (2009), 162-186. doi: 10.1137/060674909.  Google Scholar

[31]

L. Scardavi and S. Sepulchre, Synchronization in networks of identical linear systems, Automatica, 45 (2009), 2557-2562. doi: 10.1016/j.automatica.2009.07.006.  Google Scholar

[32]

L. Schenato, B. Sinopoli, M. Franceschetti, K. Poolla, M. I. Jordan and S. S. Sastry, Foundations of control and estimation over lossy networks, Proc. IEEE, 95 (2007), 163-187. doi: 10.1109/JPROC.2006.887306.  Google Scholar

[33]

J. H. Seo, H. Shim and J. Back, Consensus of high-order linear systems using dynamic output feedback compensator: Low gain approach, Automatica, 45 (2009), 2659-2664. doi: 10.1016/j.automatica.2009.07.022.  Google Scholar

[34]

B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. I. Jordan and S. S. Sastry, Kalman filtering with intermittent observations, IEEE Trans. Autom. Control, 49 (2004), 1453-1464. doi: 10.1109/TAC.2004.834121.  Google Scholar

[35]

Y. G. Sun and W. Long, Consensus problems in networks of agents with double-integrator dynamics and time-varying delays, Int. J. Control, 82 (2009), 1937-1945. doi: 10.1080/00207170902838269.  Google Scholar

[36]

R. S. Smith and F. Y. Hadaegh, Control of deep-space formation-flying spacecraft; Relative sensing and switched information, J. Guid. Control Dyn., 28 (2005), 106-114. doi: 10.2514/1.6165.  Google Scholar

[37]

H. S. Su, X. F. Wang and Z. L. Lin, Flocking of multi-agents with a virtual leader, IEEE Trans. Autom. Control, 54 (2009), 293-307. doi: 10.1109/TAC.2008.2010897.  Google Scholar

[38]

H. G. Tanner, A. Jadbabaie and G. J. Pappas, Flocking in fixed and switching networks, IEEE Trans. Autom. Control, 52 (2007), 863-868. doi: 10.1109/TAC.2007.895948.  Google Scholar

[39]

Y. P. Tian and C. L. Liu, Robust consensus of multi-agent systems with diverse input delays and asymmetric interconnection perturbations, Automatica, 45 (2009), 1347-1353. doi: 10.1016/j.automatica.2009.01.009.  Google Scholar

[40]

S. E. Tuna, Synchronizing linear systems via partial-state coupling, Automatica, 44 (2008), 2179-2184. doi: 10.1016/j.automatica.2008.01.004.  Google Scholar

[41]

S. E. Tuna, Conditions for synchronizability in arrays of coupled linear systems, IEEE Trans. Autom. Control, 54 (2009), 2416-2420. doi: 10.1109/TAC.2009.2029296.  Google Scholar

[42]

T. Vicsek, A. Cziroók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transitions in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[43]

J. H. Wang, D. Z. Cheng and X. M. Hu, Consensus of multi-agent linear dynamic systems, Asian J. Control, 10 (2008), 144-155. doi: 10.1002/asjc.15.  Google Scholar

[44]

G. Xie and L. Wang, Consensus control for a class of networks of dynamic agents, Int. J. Robust Nonlinear Control, 17 (2007), 941-959. doi: 10.1002/rnc.1144.  Google Scholar

[45]

R. Yamapi and R. S. Mackay, Stability of synchronization in a shift-invariant ring of mutually coupled oscillators, Discret. Contin. Dyn. Syst., 10 (2008), 973-996. doi: 10.3934/dcdsb.2008.10.973.  Google Scholar

[46]

H. T. Zhang, M. Z. Q. Chen, T. Zhou and G. B. Stan, Ultrafast consensus via predictive mechanisms, Europhysics Letters, 83 (2008), 40003. doi: 10.1209/0295-5075/83/40003.  Google Scholar

[47]

K. M. Zhou and J. C. Doyle, "Essentials of Robust Control," Prentice-Hall, Englewood Cliffs, 1998. Google Scholar

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