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

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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

Abstract
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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 and Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489

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.
[2]	S. Bowong and J. L. Dimi, Adaptive synchronization of a class of uncertain chaotic systems, Discret. Contin. Dyn. Syst., 9 (2008), 235-248.
[3]	J. Cortés, Distributed algorithms for reaching consensus on general functions, Automatica, 44 (2008), 726-737. doi: 10.1016/j.automatica.2007.07.022.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	R. Horn and C. Johnson, "Matrix Analysis," Cambridge Univ. Press, New York, 1985.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	K. Ogata, "Modern Control Engineering," 3^rd edition, Prentice Hall: Englewood Cliffs, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control, 53 (2008), 1503-1509.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[40]	S. E. Tuna, Synchronizing linear systems via partial-state coupling, Automatica, 44 (2008), 2179-2184. doi: 10.1016/j.automatica.2008.01.004.
[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.
[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.
[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.
[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.
[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.
[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.
[47]	K. M. Zhou and J. C. Doyle, "Essentials of Robust Control," Prentice-Hall, Englewood Cliffs, 1998.

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.
[2]	S. Bowong and J. L. Dimi, Adaptive synchronization of a class of uncertain chaotic systems, Discret. Contin. Dyn. Syst., 9 (2008), 235-248.
[3]	J. Cortés, Distributed algorithms for reaching consensus on general functions, Automatica, 44 (2008), 726-737. doi: 10.1016/j.automatica.2007.07.022.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	R. Horn and C. Johnson, "Matrix Analysis," Cambridge Univ. Press, New York, 1985.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	K. Ogata, "Modern Control Engineering," 3^rd edition, Prentice Hall: Englewood Cliffs, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	W. Ren, On consensus algorithms for double-integrator dynamics, IEEE Trans. Autom. Control, 53 (2008), 1503-1509.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[40]	S. E. Tuna, Synchronizing linear systems via partial-state coupling, Automatica, 44 (2008), 2179-2184. doi: 10.1016/j.automatica.2008.01.004.
[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.
[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.
[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.
[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.
[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.
[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.
[47]	K. M. Zhou and J. C. Doyle, "Essentials of Robust Control," Prentice-Hall, Englewood Cliffs, 1998.

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