Formation tracking control for multi-agent systems: A wave-equation based approach

Abstract

This paper considers the formation tracking control problem of large-scaled Multi-Agent Systems (MAS) for which the model is based on a system of mutually independent wave Partial Differential Equations (PDEs). The spatial derivatives in the equation correspond to the underlying communication topology of the agents. A leader-follower mode is employed in the control algorithm, with which the agents on the boundary of PDEs are chosen as leaders knowing the tracking trajectory and all the other agents are followers. Each follower has only the information of its own relative position and velocity to its topological neighbors. With a designed distributed controller, the formation tracking error is bounded by a constant proportional to the acceleration of the desired trajectory. Robustness of the control law to a perturbation in the velocity measurement is also discussed. Furthermore, some simulation studies are provided to show the effectiveness of the control algorithm.

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References

  1. [1]

    L. Fang and P. J. Antsaklis, “Decentralized formation tracking of multi-vehicle systems with nonlinear dynamics,” Proc. of the 14th Mediterranean Conf. on Control and Automation, MED’06, pp. 1–6, 2006.

    Google Scholar 

  2. [2]

    K. K. Oh, M. C. Park, and H. S. Ahn, “A survey of multiagent formation control,” Automatica, vol. 53, pp. 424–440, March 2015. [click]

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    Y. Cao, W. Yu, W. Ren, and G. Chen, “An overview of recent progress in the study of distributed multi-agent coordination,” IEEE Transactions on Industrial Informatics, vol. 9, no. 1, pp. 427–438, February 2013. [click]

    Article  Google Scholar 

  4. [4]

    G. Antonelli, F. Arrichiello, F. Caccavale, and A. Marino, “Decentralized time-varying formation control for multirobot systems,” The International Journal of Robotics Research, vol. 33, no. 7, pp. 1029–1043, May 2014.

    Article  Google Scholar 

  5. [5]

    R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. of the IEEE, vol. 95, no. 1, pp. 215–233, January 2007. [click]

    Article  MATH  Google Scholar 

  6. [6]

    M. Rubenstein, A. Cornejo, and R. Nagpal, “Programmable self-assembly in a thousand-robot swarm,” Science, vol. 345, no. 6198, pp. 795–799, August 2014. [click]

    Article  Google Scholar 

  7. [7]

    F. Y. Hadaegh, S. J. Chung, and H. M. Manohara, “On development of 100-gram-class spacecraft for swarm applications,” IEEE Systems Journal, vol. 10, no. 2, pp. 673–684, June 2016. [click]

    Article  Google Scholar 

  8. [8]

    T. Vicsek, A. Zafeiris, “Collective motion,”. Physics Reports, vol. 517, no. 3, pp. 71–140, 2012. [click]

    Article  Google Scholar 

  9. [9]

    A. Sarlette, and R. Sepulchre, “A PDE viewpoint on basic properties of coordination algorithms with symmetries,” Proc. of the 48th IEEE Conf. on Decision and Control and 28th Chinese Control Conf., pp. 5139–5144, 2009.

    Google Scholar 

  10. [10]

    G. Ferrari-Trecate, A. Buffa, and M. Gati, “Analysis of coordination in multi-agent systems through partial difference equations,” IEEE Transactions on Automatic Control, vol. 51, no. 6, pp. 1058–1063, June 2006. [click]

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    T. Meurer, and M. Krstic, “Finite-time multi-agent deployment: A nonlinear PDE motion planning approach,” Automatica, vol. 47, no. 11, pp. 2534–2542, November 2011. [click]

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    P. Frihauf, and M. Krstic, “Leader-enabled deployment onto planar curves: A PDE-based approach,” IEEE Transactions on Automatic Control, vol. 56, no. 8, pp. 1791-1806, August 2011. [click]

    Google Scholar 

  13. [13]

    J. Qi, R. Vazquez, and M. Krstic, “Multi-agent deployment in 3-D via PDE control,” IEEE Transactions on Automatic Control, vol. 60, no. 4, pp. 891-906, April 2015. [click]

    Google Scholar 

  14. [14]

    N. Ghods, and M. Krstic, “Multi-agent deployment over a source,” IEEE Transactions on Control Systems Technology, vol. 20, no. 1, pp. 277–285, January 2012. [click]

    Google Scholar 

  15. [15]

    H. Hao, P. Barooah, and P. G. Mehta, “Stability margin scaling laws for distributed formation control as a function of network structure,” IEEE Transactions on Automatic Control, vol. 56, no. 4, pp. 923–929, April 2011. [click]

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    C. Xu, Y. Dong, Z. Ren, H. Jiang, and X. Yu, “Sensor deployment for pipeline leakage detection via optimal boundary control strategies,” Journal of Industrial and Management Optimization, vol. 11, no. 1, pp. 199–216, January 2015.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    V. D. Blondel, J. M. Hendrickx, and J.N. Tsitsiklis, “Continuous-time average-preserving opinion dynamics with opinion-dependent communications,” SIAM Journal on Control and Optimization, vol. 48, no. 8, pp. 5214–5240, October 2010.

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    H. Rastgoftar and S. Jayasuriya, “Evolution of multi-agent systems as continua,” Journal of Dynamic Systems, Measurement, and Control, vol. 136, no. 4, pp. 041014, April 2014. [click]

    Article  Google Scholar 

  19. [19]

    Y. Zhao, Z. Duan, G. Wen, and Y. Zhang, “Distributed finite-time tracking control for multi-agent systems: an observer-based approach,” Systems & Control Letters, vol. 62, no. 1, pp. 22–28, January 2013.

    MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    Y. Cao and W. Ren, “Multi-vehicle coordination for double-integrator dynamics under fixed undirected/directed interaction in a sampled-data setting,” International Journal of Robust and Nonlinear Control, vol. 20, no. 9, pp. 987–1000, May 2010.

    MathSciNet  MATH  Google Scholar 

  21. [21]

    J. Qi, F. Pan, and J.-P. Qi, “A PDE approach to formation tracking control for multi-agent systems,” Proc. of the 34th Chinese Control Conference, pp. 7136–7141, 2015.

    Google Scholar 

  22. [22]

    X. Dong, B. Yu, Z. Shi, and Y. Zhong, “Time-varying formation control for unmanned aerial vehicles: Theories and applications,” IEEE Transactions on Control Systems Technology, vol. 23, no. 1, pp. 340–348, January 2015.

    Article  Google Scholar 

  23. [23]

    T. Meurer, Control of Higher-dimensional PDEs: Flatness and Backstepping Designs, Springer Science & Business Media, 2012. [click]

    Google Scholar 

  24. [24]

    M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, 2008.

    Google Scholar 

  25. [25]

    S. Kawashima, and Y. Shibata, “Global existence and exponential stability of small solutions to nonlinear viscoelasticity,” Communications in Mathematical Physics, vol. 148, no. 1, pp. 189–208, August 1992.

    MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge University Press, 1996. [click]

    Google Scholar 

  27. [27]

    J, Qi, “Simulation movie of a 3-D formation tracking example 2016,” https://www.dropbox.com/s/ipnan4c1b6x478r/formationtrackingwave.mp4?dl=0; or http://pan.baidu.com/s/1bbOrHc

  28. [28]

    A. R. Mitchell and D. F. Griffiths, The Finite Difference Method in Partial Differential Equations, JohnWiley, 1980.

    Google Scholar 

  29. [29]

    N. Sun, Y. Fang, H Chen, and L Biao, “Amplitude-saturated nonlinear output feedback antiswing control for underactuated cranes with double-pendulum cargo dynamics,” IEEE Transactions on Industrial Electronics, vol. 64, no. 3, pp. 2135–2146, March 2017. [click]

    Article  Google Scholar 

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Authors

Corresponding author

Correspondence to Jie Qi.

Additional information

Recommended by Associate Editor Hyo-Sung Ahn under the direction of Editor Duk-Sun Shim. This work was partially supported by National Natural Science Foundation of China 61773112, 61473265, Natural Science Foundation of Shanghai 16ZR1401200 and the Fundamental Research Funds for the Central Universities 2232015D3-24.

Shu-Xia Tang received her Ph.D. in Mechanical Engineering in 2016 from the Department of Mechanical & Aerospace Engineering, University of California, San Diego, USA. She is currently a postdoctoral research fellow and lecturer at the Department of Applied Mathematics, University of Waterloo, Canada. Her main research interests are control and estimation in distributed parameter systems. Recent research also includes optimal actuator and sensor design problems.

Jie Qi received the Ph.D. degree in Systems Engineering (2005) and the B.S. degree in Automation (2000) from Northeastern University in Shenyang, China. She is currently a Professor in Automation Department, Donghua University, China. Her research interests include multi-agent cooperative control, the control of distributed parameter systems, complex system modeling and intelligent optimization.

Jing Zhang is currently pursuing the Ph.D. degree with the College of Information Science and Technology, Donghua University, Shanghai, China. She received the M.S. degree in Control Science and Engineering (2017) from Donghua University and the B.S. degree in Automation (2013) from Shanxi University, Taiyuan, China. Her research interests include boundary control and multi-agent cooperative control.

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Tang, SX., Qi, J. & Zhang, J. Formation tracking control for multi-agent systems: A wave-equation based approach. Int. J. Control Autom. Syst. 15, 2704–2713 (2017). https://doi.org/10.1007/s12555-016-0562-0

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Keywords

  • Distributed control
  • formation tracking
  • MAS
  • robustness
  • wave PDE