Advertisement

Control Theory and Technology

, Volume 17, Issue 1, pp 85–98 | Cite as

Distributed optimal consensus of multiple double integrators under bounded velocity and acceleration

  • Zhirong Qiu
  • Lihua XieEmail author
  • Yiguang Hong
Article
  • 16 Downloads

Abstract

This paper studies a distributed optimal consensus problem for multiple double integrators under bounded velocity and acceleration. Assigned with an individual and private convex cost which is dependent on the position, each agent needs to achieve consensus at the optimum of the aggregate cost under bounded velocity and acceleration. Based on relative positions and velocities to neighbor agents, we design a distributed control law by including the integration feedback of position and velocity errors. By employing quadratic Lyapunov functions, we solve the optimal consensus problem of double-integrators when the fixed topology is strongly connected and weight-balanced. Furthermore, if an initial estimate of the optimum can be known, then control gains can be properly selected to achieve an exponentially fast convergence under bounded velocity and acceleration. The result still holds when the relative velocity is not available, and we also discuss an extension for heterogeneous Euler-Lagrange systems by inverse dynamics control. A numeric example is provided to illustrate the result.

Keywords

Distributed optimization double integrators bounded velocity bounded input 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Jakovetic, J. Xavier, J. M. F. Moura. Fast distributed gradient methods. IEEE Transactions on Automatic Control, 2014, 59(5): 1131–1146.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Nedić, A. Olshevsky. Distributed optimization over timevarying directed graphs. IEEE Transactions on Automatic Control, 2015, 60(3): 601–615.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    W. Shi, Q. Ling, K. Yuan, et al. On the linear convergence of the ADMM in decentralized consensus optimization. IEEE Transactions Signal Processing, 2014, 62(7): 1750–1761.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Mokhtari, Q. Ling, A. Ribeiro. Network Newton distributed optimization methods. IEEE Transactions Signal Processing, 2017, 65(1): 146–161.MathSciNetCrossRefGoogle Scholar
  5. [5]
    V. Cevher, S. Becker, M. Schmidt. Convex optimization for big data: scalable, randomized, parallel algorithms for big data analytics. IEEE Signal Processing Magazine, 2014, 31(5): 32–43.CrossRefGoogle Scholar
  6. [6]
    A. H. Sayed. Adaptation, learning, and optimization over networks. MAL, 2014, 7(4/5): 311–801.zbMATHGoogle Scholar
  7. [7]
    J. Wang, N. Elia. Control approach to distributed optimization. Annual Allerton Conference on Communication, Control, Computing, Allerton: IEEE, 2010: 557–561.Google Scholar
  8. [8]
    B. Gharesifard, J. Cortes. Distributed continuous-time convex optimization on weight-balanced digraphs. IEEE Transactions Automatica Control, 2014, 59(3): 781–786.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    S. S. Kia, J. Cortés, S. Martínez. Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication. Automatica, 2015, 55: 254–264.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Z. Qiu, S. Liu, L. Xie. Distributed constrained optimal consensus of multi-agent systems. Automatica, 2016, 68: 209–215.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    X. Zeng, P. Yi, Y. Hong. Distributed continuous-time algorithm for constrained convex optimizations via nonsmooth analysis approach. IEEE Transactions on Automatic Control, 2017, 62(10): 5227–5233.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    S. Yang, Q. Liu, J. Wang. A multi-agent system with a proportionalintegral protocol for distributed constrained optimization. IEEE Transactions on Automatic Control, 2017, 62(7): 3461–3467.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    D. Mateos-Nunez, J. Cortes. Distributed saddle-point subgradient algorithms with laplacian averaging. IEEE Transactions on Automatic Control, 2017, 62(6): 2720–2735.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    P. Yi, Y. Hong, F. Liu. Initialization-free distributed algorithms for optimal resource allocation with feasibility constraints application to economic dispatch of power systems. Automatica, 2016, 74: 259–269.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Y. Zhang, Z. Deng, Y. Hong. Distributed optimal coordination for multiple heterogeneous Euler{Lagrangian systems. Automatica, 2017, 79: 207–213.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Z. Deng, Y. Hong. Multi-agent optimization design for autonomous Lagrangian systems. Unmanned Systems, 2016, 4(1): 5–13.CrossRefGoogle Scholar
  17. [17]
    Y. Zhang Y. Hong. Distributed optimization design for high-order multi-agent systems. Chinese Control Conference, Hangzhou: IEEE, 2015: 7251–7256.Google Scholar
  18. [18]
    Y. Xie, Z. Lin. Global optimal consensus for multi-agent systems with bounded controls. Systems & Control Letters, 2017, 102: 104–111.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    H. Su, M. Z. Q. Chen, J. Lam, et al. Semi-global leader-following consensus of linear multi-agent systems with input saturation via low gain feedback. IEEE Transactions Circuits Systems: Regular Papers, 2013, 60(7): 1881–1889.MathSciNetCrossRefGoogle Scholar
  20. [20]
    Z. Zhao, Z. Lin. Semi-global leader-following consensus of multiple linear systems with position rate limited actuators. International Journal of Robust Nonlinear Control, 2015, 25(13): 2083–2100.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A. Abdessameud, A. Tayebi. On consensus algorithms design for double integrator dynamics. Automatica, 2013, 49(1): 253–260.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    A. Abdessameud, A. Tayebi. Synchronization of networked Lagrangian systems with input constraints. IFAC Proceedings Volumes, 2011, 44(1): 2382–2387.CrossRefGoogle Scholar
  23. [23]
    Q. Wang, H. Gao. Global consensus of multiple integrator agents via saturated controls. Journal of the Franklin Institute, 2013, 350(8): 2261–2276.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Z. Zhao, Z. Lin. Global leader-following consensus of a group of general linear systems using bounded controls. Automatica, 2016, 68: 294–304.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Z. Qiu, Y. Hong, L. Xie. Optimal consensus of Euler-Lagrangian systems with kinematic constraints. IFAC PapersOnline, 2016, 49(22): 327–332.MathSciNetCrossRefGoogle Scholar
  26. [26]
    M. S. Bazaraa, H. D. Sherali, C. M. Shetty. Nonlinear Programming: Theory and Algorithms. 3rd ed. Hoboken: John Wiley & Sons, Inc., 2006.CrossRefzbMATHGoogle Scholar
  27. [27]
    M. W. Spong, M. Vidyasagar. Robot Dynamics and Control. Delhi: Wiley India Pvt. Limited, 2008.Google Scholar

Copyright information

© Editorial Board of Control Theory & Applications, South China University of Technology and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Electrical and Electronic EngineeringNanyang Technological UniversitySingaporeSingapore
  2. 2.Key Laboratory of Systems and Control, Institute of Systems ScienceChinese Academy of SciencesBeijingChina

Personalised recommendations