September  2019, 9(3): 349-360. doi: 10.3934/naco.2019023

Constrained optimal consensus in dynamical networks

1. 

Electrical Engineering Department, Amirkabir University of Technology, Tehran 15914, Iran

2. 

School of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan, NSW 2308, Australia

* Corresponding author: Mohsen Zamani, mohsen.zamani@newcastle.edu.au

Received  April 2018 Revised  April 2019 Published  May 2019

In this paper, an optimal consensus problem with local inequality constraints is studied for a network of single-integrator agents. The goal is that a group of single-integrator agents rendezvous at the optimal point of the sum of local convex objective functions. The local objective functions are only available to the associated agents that only need to know their own positions and of their neighbors. This point is supposed to be confined by some local inequality constraints. To tackle this problem, we integrate the primal-dual gradient-based optimization algorithm with a consensus protocol to drive the agents toward the agreed point that satisfies KKT conditions. The asymptotic convergence of the solution of the optimization problem is proven with the help of LaSalle's invariance principle for hybrid systems. A numerical example is presented to show the effectiveness of our protocol.

Citation: Amir Adibzadeh, Mohsen Zamani, Amir A. Suratgar, Mohammad B. Menhaj. Constrained optimal consensus in dynamical networks. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 349-360. doi: 10.3934/naco.2019023
References:
[1] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.  doi: 10.1017/CBO9780511804441.
[2]

L. ChengH. WangZ.-G. Hou and M. Tan, Reaching a consensus in networks of high-order integral agents under switching directed topologies, International Journal of Systems Science, 47 (2016), 1966-1981.  doi: 10.1080/00207721.2014.966281.

[3]

M.-C. FanZ. Chen and H.-T. Zhang, Semi-global consensus of nonlinear second-order multi-agent systems with measurement output feedback, IEEE Transactions on Automatic Control, 59 (2014), 2222-2227.  doi: 10.1109/TAC.2014.2299351.

[4]

D. Feijer and F. Paganini, Stability of primal–dual gradient dynamics and applications to network optimization, Automatica, 46 (2010), 1974-1981.  doi: 10.1016/j.automatica.2010.08.011.

[5]

B. Gharesifard and J. Cortés, Distributed continuous-time convex optimization on weight-balanced digraphs, IEEE Transactions on Automatic Control, 59 (2014), 781-786.  doi: 10.1109/TAC.2013.2278132.

[6] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge university press, 2012. 
[7]

H. Khalil, Nonlinear Systems, Prentice Hall, 1996.

[8]

S. S. KiaJ. Cortés and S. Martínez, Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication, Automatica, 55 (2015), 254-264.  doi: 10.1016/j.automatica.2015.03.001.

[9]

P. Lin and W. Ren, Constrained consensus in unbalanced networks with communication delays, IEEE Transactions on Automatic Control, 59 (2014), 775-781.  doi: 10.1109/TAC.2013.2275671.

[10]

J. Lu and C. Y. Tang, Zero-gradient-sum algorithms for distributed convex optimization: The continuous-time case, IEEE Transactions on Automatic Control, 57 (2012), 2348-2354.  doi: 10.1109/TAC.2012.2184199.

[11]

J. LygerosK. H. JohanssonS. N. SimicJ. Zhang and S. S. Sastry, Dynamical properties of hybrid automata, IEEE Transactions on Automatic Control, 48 (2003), 2-17.  doi: 10.1109/TAC.2002.806650.

[12]

A. NedicA. Ozdaglar and P. A. Parrilo, Constrained consensus and optimization in multi-agent networks, IEEE Transactions on Automatic Control, 55 (2010), 922-938.  doi: 10.1109/TAC.2010.2041686.

[13]

S. K. Niederländer and J. Cortés, Distributed coordination for nonsmooth convex optimization via saddle-point dynamics, https://arxiv.org/abs/1606.09298.

[14]

S. Rahili and W. Ren, Distributed continuous-time convex optimization with time-varying cost functions, IEEE Transactions on Automatic Control, 62 (2016), 1590-1605.  doi: 10.1109/TAC.2016.2593899.

[15]

W. Ren and E. Atkins, Distributed multi-vehicle coordinated control via local information exchange, International Journal of Robust and Nonlinear Control, 17 (2007), 1002-1033.  doi: 10.1002/rnc.1147.

[16]

H. Rezaee and F. Abdollahi, Average consensus over high-order multiagent systems, IEEE Transactions on Automatic Control, 60 (2015), 3047-3052.  doi: 10.1109/TAC.2015.2408576.

[17] A. Ruszczynski, Nonlinear Optimization, Princeton University Press, 2011. 
[18]

J. Wang and N. Elia, A control perspective for centralized and distributed convex optimization, in 2011 50th IEEE Conference on Decision and Control and European Control Conference, IEEE, (2011), 3800–3805.

[19]

Y. Xie and Z. Lin, Global optimal consensus for multi-agent systems with bounded controls, Systems & Control Letters, 102 (2017), 104-111.  doi: 10.1016/j.sysconle.2017.02.002.

[20]

S. YangQ. Liu and J. Wang, A multi-agent system with a proportional-integral protocol for distributed constrained optimization, IEEE Transactions on Automatic Control, 62 (2017), 3461-3467.  doi: 10.1109/TAC.2016.2610945.

[21]

P. YiY. Hong and F. Liu, Distributed gradient algorithm for constrained optimization with application to load sharing in power systems, Systems & Control Letters, 83 (2015), 45-52.  doi: 10.1016/j.sysconle.2015.06.006.

show all references

References:
[1] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.  doi: 10.1017/CBO9780511804441.
[2]

L. ChengH. WangZ.-G. Hou and M. Tan, Reaching a consensus in networks of high-order integral agents under switching directed topologies, International Journal of Systems Science, 47 (2016), 1966-1981.  doi: 10.1080/00207721.2014.966281.

[3]

M.-C. FanZ. Chen and H.-T. Zhang, Semi-global consensus of nonlinear second-order multi-agent systems with measurement output feedback, IEEE Transactions on Automatic Control, 59 (2014), 2222-2227.  doi: 10.1109/TAC.2014.2299351.

[4]

D. Feijer and F. Paganini, Stability of primal–dual gradient dynamics and applications to network optimization, Automatica, 46 (2010), 1974-1981.  doi: 10.1016/j.automatica.2010.08.011.

[5]

B. Gharesifard and J. Cortés, Distributed continuous-time convex optimization on weight-balanced digraphs, IEEE Transactions on Automatic Control, 59 (2014), 781-786.  doi: 10.1109/TAC.2013.2278132.

[6] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge university press, 2012. 
[7]

H. Khalil, Nonlinear Systems, Prentice Hall, 1996.

[8]

S. S. KiaJ. Cortés and S. Martínez, Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication, Automatica, 55 (2015), 254-264.  doi: 10.1016/j.automatica.2015.03.001.

[9]

P. Lin and W. Ren, Constrained consensus in unbalanced networks with communication delays, IEEE Transactions on Automatic Control, 59 (2014), 775-781.  doi: 10.1109/TAC.2013.2275671.

[10]

J. Lu and C. Y. Tang, Zero-gradient-sum algorithms for distributed convex optimization: The continuous-time case, IEEE Transactions on Automatic Control, 57 (2012), 2348-2354.  doi: 10.1109/TAC.2012.2184199.

[11]

J. LygerosK. H. JohanssonS. N. SimicJ. Zhang and S. S. Sastry, Dynamical properties of hybrid automata, IEEE Transactions on Automatic Control, 48 (2003), 2-17.  doi: 10.1109/TAC.2002.806650.

[12]

A. NedicA. Ozdaglar and P. A. Parrilo, Constrained consensus and optimization in multi-agent networks, IEEE Transactions on Automatic Control, 55 (2010), 922-938.  doi: 10.1109/TAC.2010.2041686.

[13]

S. K. Niederländer and J. Cortés, Distributed coordination for nonsmooth convex optimization via saddle-point dynamics, https://arxiv.org/abs/1606.09298.

[14]

S. Rahili and W. Ren, Distributed continuous-time convex optimization with time-varying cost functions, IEEE Transactions on Automatic Control, 62 (2016), 1590-1605.  doi: 10.1109/TAC.2016.2593899.

[15]

W. Ren and E. Atkins, Distributed multi-vehicle coordinated control via local information exchange, International Journal of Robust and Nonlinear Control, 17 (2007), 1002-1033.  doi: 10.1002/rnc.1147.

[16]

H. Rezaee and F. Abdollahi, Average consensus over high-order multiagent systems, IEEE Transactions on Automatic Control, 60 (2015), 3047-3052.  doi: 10.1109/TAC.2015.2408576.

[17] A. Ruszczynski, Nonlinear Optimization, Princeton University Press, 2011. 
[18]

J. Wang and N. Elia, A control perspective for centralized and distributed convex optimization, in 2011 50th IEEE Conference on Decision and Control and European Control Conference, IEEE, (2011), 3800–3805.

[19]

Y. Xie and Z. Lin, Global optimal consensus for multi-agent systems with bounded controls, Systems & Control Letters, 102 (2017), 104-111.  doi: 10.1016/j.sysconle.2017.02.002.

[20]

S. YangQ. Liu and J. Wang, A multi-agent system with a proportional-integral protocol for distributed constrained optimization, IEEE Transactions on Automatic Control, 62 (2017), 3461-3467.  doi: 10.1109/TAC.2016.2610945.

[21]

P. YiY. Hong and F. Liu, Distributed gradient algorithm for constrained optimization with application to load sharing in power systems, Systems & Control Letters, 83 (2015), 45-52.  doi: 10.1016/j.sysconle.2015.06.006.

Figure 1.  States' trajectories for a ring network of single-integrator agents under the control law (15)
Figure 2.  Simulation results of [21]: States' trajectories for a ring network of single-integrator agents
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