American Institute of Mathematical Sciences

Advanced
July  2014, 10(3): 929-943. doi: 10.3934/jimo.2014.10.929

Finite-time optimal consensus control for second-order multi-agent systems

Rui Li 1, and  Yingjing Shi 2,

1. 

School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, China
2. 

School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu, China

Received  June 2012 Revised  July 2013 Published  November 2013

We propose an optimal consensus design method for solving a finite-time optimal control problem involving a second-order multi-agent system. With this method, the optimal consensus problem can be modeled as an optimal parameter selection problem with continuous state inequality constraints and free terminal time. By virtue of the constraint transcription method and a time scaling transform method, a gradient-based optimization algorithm is developed to solve this optimal parameter selection problem. Furthermore, a new consensus protocol is designed, by which the consensus value of the system velocity can be chosen to be an arbitrary value. For illustration, simulation studies are carried out to demonstrate the proposed method.
Keywords: multi-agent system, optimal parameter selection problem, Finite time consensus, gradient-based optimization algorithm..
Mathematics Subject Classification: Primary: 58E25; Secondary: 49N9.
Citation: Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929
References:
[1]

E. A. Blanchard, R. C. Loxton and V. Rehbock, A computational algorithm for a class of non-smooth optimal control problems arising in aquaculture operations, Applied Mathematics and Computation, 219 (2013), 8738-8746. doi: 10.1016/j.amc.2013.02.070.

[2]

Y. C. Cao and W. Ren, Optimal linear-consensus algorithms: An LQR perspective, IEEE Transactions on Systems, Man, and Cybernetics Part B: Cybernetics, 40 (2010), 810-830.

[3]

R. Carli, G. Como, P. Frasca and F. Garin, Distributed averaging on digital erasure networks, Automatica J. IFAC, 47 (2011), 115-121. doi: 10.1016/j.automatica.2010.10.015.

[4]

T. Dierks, Formation Control of Mobile Robots and Unmanmed Aerial Vehicles, Ph.D thesis, Missouri University of Science and Technology, 2009.

[5]

J. Fax and M. Murray, Information flow and cooperative control of vehicle formations, IEEE Transactions on Automatic Control, 49 (2004), 1465-1476. doi: 10.1109/TAC.2004.834433.

[6]

V. Gazi and K. M. Passino, Stability analysis of social foraging swarms, IEEE Transactions on Systems Man Cybernet., 34 (2004), 539-557. doi: 10.1109/TSMCB.2003.817077.

[7]

V. Gupta, V. Hassibi and R. M. Murray, On sensor fusion in the presence of packet-dropping communication channels, in 44th IEEE Conference on Decision and Control, 2005, 3547-3552. doi: 10.1109/CDC.2005.1582712.

[8]

D. Jakovetic, J. Xavier and J. M. F. Moura, Cooperative convex optimization in networked systems: Augmented Lagrangian algorithms with directed Gossip communication, IEEE Transactions on Signal Processing, 59 (2011), 3889-3902. doi: 10.1109/TSP.2011.2146776.

[9]

C. H. Jiang, Q. Lin, C. J. Yu, K. L. Teo and G.-R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, J. Optim. Theory Appl., 154 (2012), 30-53. doi: 10.1007/s10957-012-0006-9.

[10]

P. Lin and Y. M. Jia, Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies, Automatica J. IFAC, 45 (2009), 2154-2158. doi: 10.1016/j.automatica.2009.05.002.

[11]

Y. Liu and K. M. Passino, Stable social foraging swarms in a noisy environment, IEEE Transactions on Automatic Control, 49 (2004), 30-44. doi: 10.1109/TAC.2003.821416.

[12]

R. C. Loxton, K. L. Teo and V. Rehbock, Computational method for a class of switched system optimal control problems, IEEE Transactions on Automatic Control, 54 (2009), 2455-2460. doi: 10.1109/TAC.2009.2029310.

[13]

R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and control, Automatica J. IFAC, 45 (2009), 2250-2257. doi: 10.1016/j.automatica.2009.05.029.

[14]

R. C. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica J. IFAC, 44 (2008), 2923-2929. doi: 10.1016/j.automatica.2008.04.011.

[15]

R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of IEEE, 95 (2007), 215-233. doi: 10.1109/JPROC.2006.887293.

[16]

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

[17]

R. Olfati-Saber and J. S. Shamma, Consensus filters for sensor networks and distributed sensor fusion, in 44th IEEE Conference on Decision and Control, 2005, 6698-6703. doi: 10.1109/CDC.2005.1583238.

[18]

W. Ren and R. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Transactions on Automatic Control, 50 (2005), 655-661. doi: 10.1109/TAC.2005.846556.

[19]

W. Ren, R. W. Beard and E. M. Atkins, Information consensus in multivehicle cooperative control: Collective group behavior through local interaction, IEEE Control System Magazine, 27 (2007), 71-82.

[20]

H. Sayyaadi and M. R. Doostmohammadian, Finite-time consensus in directed switching network topologies and time-delayed communications, Scientia Iranica, 18 (2011), 75-85. doi: 10.1016/j.scient.2011.03.010.

[21]

E. Semsar and K. Khorasani, Optimal control and game theoretic approaches to cooperative control of a team of multi-vehicle unmanned systems, in Proceedings of the 2007 IEEE International Conference on, Networking, Sensing and Control, London, 2007, 628-633. doi: 10.1109/ICNSC.2007.372852.

[22]

A. Tahbaz-Salehi and A. Jadbabaie, A necessary and sufficient condition for consensus over random networks, IEEE Transactions on Automatic Control, 53 (2008), 791-795. doi: 10.1109/TAC.2008.917743.

[23]

K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems, Journal of the Australian Mathematical Society, Series B, 40 (1999), 314-335. doi: 10.1017/S0334270000010936.

[24]

K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach to optimal control problems, in World Congress of Nonlinear Analysts '92, De Gruyter, 1996, 2763-2774. doi: 10.1515/9783110883237.2763.

[25]

L. Wang and F. Xiao, Finite-time consensus problems for networks of dynamic agents, IEEE Transactions on Automatic Control, 55 (2010), 950-955. doi: 10.1109/TAC.2010.2041610.

[26]

X. L. Wang and Y. G. Hong, Finite-time consensus for multi-agent networks with second-order agent dynamics, in Proceedings of the 17th IFAC World Congress, The International Federation of Automatic Control, 2008, 15185-15190.

[27]

F. Xiao and L. Wang, Consensus protocols for discrete-time multi-agent systems with time-varying delays, Automatica J. IFAC, 44 (2008), 2577-2582. doi: 10.1016/j.automatica.2008.02.017.

[28]

L. Xiao and S. Boyd, Fast linear iterations for distributed averaging, Systems Control Letters, 53 (2004), 65-78. doi: 10.1016/j.sysconle.2004.02.022.

[29]

W. Zhang and J. H. Hu, Optimal multi-agent coordination under tree formation constraints, IEEE Transactions on Automatic Control, 53 (2008), 692-705. doi: 10.1109/TAC.2008.919855.

show all references

References:
[1]

E. A. Blanchard, R. C. Loxton and V. Rehbock, A computational algorithm for a class of non-smooth optimal control problems arising in aquaculture operations, Applied Mathematics and Computation, 219 (2013), 8738-8746. doi: 10.1016/j.amc.2013.02.070.

[2]

Y. C. Cao and W. Ren, Optimal linear-consensus algorithms: An LQR perspective, IEEE Transactions on Systems, Man, and Cybernetics Part B: Cybernetics, 40 (2010), 810-830.

[3]

R. Carli, G. Como, P. Frasca and F. Garin, Distributed averaging on digital erasure networks, Automatica J. IFAC, 47 (2011), 115-121. doi: 10.1016/j.automatica.2010.10.015.

[4]

T. Dierks, Formation Control of Mobile Robots and Unmanmed Aerial Vehicles, Ph.D thesis, Missouri University of Science and Technology, 2009.

[5]

J. Fax and M. Murray, Information flow and cooperative control of vehicle formations, IEEE Transactions on Automatic Control, 49 (2004), 1465-1476. doi: 10.1109/TAC.2004.834433.

[6]

V. Gazi and K. M. Passino, Stability analysis of social foraging swarms, IEEE Transactions on Systems Man Cybernet., 34 (2004), 539-557. doi: 10.1109/TSMCB.2003.817077.

[7]

V. Gupta, V. Hassibi and R. M. Murray, On sensor fusion in the presence of packet-dropping communication channels, in 44th IEEE Conference on Decision and Control, 2005, 3547-3552. doi: 10.1109/CDC.2005.1582712.

[8]

D. Jakovetic, J. Xavier and J. M. F. Moura, Cooperative convex optimization in networked systems: Augmented Lagrangian algorithms with directed Gossip communication, IEEE Transactions on Signal Processing, 59 (2011), 3889-3902. doi: 10.1109/TSP.2011.2146776.

[9]

C. H. Jiang, Q. Lin, C. J. Yu, K. L. Teo and G.-R. Duan, An exact penalty method for free terminal time optimal control problem with continuous inequality constraints, J. Optim. Theory Appl., 154 (2012), 30-53. doi: 10.1007/s10957-012-0006-9.

[10]

P. Lin and Y. M. Jia, Consensus of second-order discrete-time multi-agent systems with nonuniform time-delays and dynamically changing topologies, Automatica J. IFAC, 45 (2009), 2154-2158. doi: 10.1016/j.automatica.2009.05.002.

[11]

Y. Liu and K. M. Passino, Stable social foraging swarms in a noisy environment, IEEE Transactions on Automatic Control, 49 (2004), 30-44. doi: 10.1109/TAC.2003.821416.

[12]

R. C. Loxton, K. L. Teo and V. Rehbock, Computational method for a class of switched system optimal control problems, IEEE Transactions on Automatic Control, 54 (2009), 2455-2460. doi: 10.1109/TAC.2009.2029310.

[13]

R. C. Loxton, K. L. Teo, V. Rehbock and K. F. C. Yiu, Optimal control problems with a continuous inequality constraint on the state and control, Automatica J. IFAC, 45 (2009), 2250-2257. doi: 10.1016/j.automatica.2009.05.029.

[14]

R. C. Loxton, K. L. Teo and V. Rehbock, Optimal control problems with multiple characteristic time points in the objective and constraints, Automatica J. IFAC, 44 (2008), 2923-2929. doi: 10.1016/j.automatica.2008.04.011.

[15]

R. Olfati-Saber, J. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of IEEE, 95 (2007), 215-233. doi: 10.1109/JPROC.2006.887293.

[16]

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

[17]

R. Olfati-Saber and J. S. Shamma, Consensus filters for sensor networks and distributed sensor fusion, in 44th IEEE Conference on Decision and Control, 2005, 6698-6703. doi: 10.1109/CDC.2005.1583238.

[18]

W. Ren and R. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Transactions on Automatic Control, 50 (2005), 655-661. doi: 10.1109/TAC.2005.846556.

[19]

W. Ren, R. W. Beard and E. M. Atkins, Information consensus in multivehicle cooperative control: Collective group behavior through local interaction, IEEE Control System Magazine, 27 (2007), 71-82.

[20]

H. Sayyaadi and M. R. Doostmohammadian, Finite-time consensus in directed switching network topologies and time-delayed communications, Scientia Iranica, 18 (2011), 75-85. doi: 10.1016/j.scient.2011.03.010.

[21]

E. Semsar and K. Khorasani, Optimal control and game theoretic approaches to cooperative control of a team of multi-vehicle unmanned systems, in Proceedings of the 2007 IEEE International Conference on, Networking, Sensing and Control, London, 2007, 628-633. doi: 10.1109/ICNSC.2007.372852.

[22]

A. Tahbaz-Salehi and A. Jadbabaie, A necessary and sufficient condition for consensus over random networks, IEEE Transactions on Automatic Control, 53 (2008), 791-795. doi: 10.1109/TAC.2008.917743.

[23]

K. L. Teo, L. S. Jennings, H. W. J. Lee and V. Rehbock, The control parameterization enhancing transform for constrained optimal control problems, Journal of the Australian Mathematical Society, Series B, 40 (1999), 314-335. doi: 10.1017/S0334270000010936.

[24]

K. L. Teo, C. J. Goh and K. H. Wong, A unified computational approach to optimal control problems, in World Congress of Nonlinear Analysts '92, De Gruyter, 1996, 2763-2774. doi: 10.1515/9783110883237.2763.

[25]

L. Wang and F. Xiao, Finite-time consensus problems for networks of dynamic agents, IEEE Transactions on Automatic Control, 55 (2010), 950-955. doi: 10.1109/TAC.2010.2041610.

[26]

X. L. Wang and Y. G. Hong, Finite-time consensus for multi-agent networks with second-order agent dynamics, in Proceedings of the 17th IFAC World Congress, The International Federation of Automatic Control, 2008, 15185-15190.

[27]

F. Xiao and L. Wang, Consensus protocols for discrete-time multi-agent systems with time-varying delays, Automatica J. IFAC, 44 (2008), 2577-2582. doi: 10.1016/j.automatica.2008.02.017.

[28]

L. Xiao and S. Boyd, Fast linear iterations for distributed averaging, Systems Control Letters, 53 (2004), 65-78. doi: 10.1016/j.sysconle.2004.02.022.

[29]

W. Zhang and J. H. Hu, Optimal multi-agent coordination under tree formation constraints, IEEE Transactions on Automatic Control, 53 (2008), 692-705. doi: 10.1109/TAC.2008.919855.

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