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Proportional-Retarded (PR) Protocol for a Large Scale Multi-agent Network with Noisy Measurements; Stability and Performance

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Part of the Advances in Delays and Dynamics book series (ADVSDD,volume 10)

Abstract

A distributed Proportional-Retarded (PR) protocol is studied for leaderless coordination of a large scale multi-agent network with undirected links and noisy measurements. The protocol, already developed for SISO systems, introduces a time delay to mimic a derivative action allowing improvement of agents’ transient responses but being insensitive to high-frequency signals. The main contribution of this chapter is the stability analysis of the closed-loop network system of arbitrarily large scale, in terms of PR protocol parameters. Next, this is connected to our recent results in [28] from which we summarize how PR protocol can be tuned for the network system to achieve certain performance as dictated by its rightmost eigenvalues. Over a case study, the results are demonstrated.

This work was supported in part by the US National Science Foundation Award 1536397.

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Fig. 1
Fig. 2

Notes

  1. 1.

    It is worthy of mention that the radius of convergence of the series is \(e^{-1}\). For practical computation, the reader is referred to [31] where additional asymptotic formulae can be found considering all the branches of the Lambert W function.

  2. 2.

    The scalar Lambert W function is available as embedded function in MATLAB, see the function lambertw.

  3. 3.

    Note that Proposition 6 uses \(\lambda _{\min }\) to guarantee the placement of \(\gamma ^*\) at a desired location \(\gamma _d\) through a stabilizing pair \((k_p,k_r)\). Since \(\gamma _d<0\) is a necessary and sufficient condition for the stability of the consensus network, it can be conjectured that the stabilizing pair \((k_p,k_r)\) must lie within the stability domain associated with \(\lambda _{\max }\). Therefore, one may consider Proposition 6 as a link between two important Laplacian eigenvalues, namely, \(\lambda _{\min }\) and \(\lambda _{\max }\).

References

  1. Werfel, J., Petersen, K., Nagpal, R.: Designing collective behavior in a termite-inspired robot construction team. Science 343, 754–758 (2014)

    CrossRef  Google Scholar 

  2. Helbing, D.: Traffic and related self-driven many-particle systems. Physics 73(4) (2001)

    Google Scholar 

  3. Porfiri, M., Roberson, D.G., Stilwell, D.J.: Tracking and formation control of multiple autonomous agents: a two-level consensus approach. Automatica 43(8), 1318–1328 (2007)

    MathSciNet  CrossRef  Google Scholar 

  4. Koh, M.H., Sipahi, R.: A consensus dynamics with delay-induced instability can self-regulate for stability via agent regrouping. Chaos 26(11) (2016)

    Google Scholar 

  5. Sipahi, R., Niculescu, S.-I., Abdallah, C.T., Michiels, W., Gu, K.: Stability and stabilization of systems with time delay. IEEE Control. Syst. Mag. 31(1), 38–65 (2011)

    MathSciNet  CrossRef  Google Scholar 

  6. Ramírez, A., Garrido, R., Mondié, S.: Integral retarded velocity control of dc servomotors. In: Proceedings of the 11th IFAC Workshop Time Delay Systems, pp. 558–563 (2013)

    Google Scholar 

  7. Ramírez, A., Garrido, R., Sipahi, R., Mondié, S.: On delay-based control of alternative to PI/PID controllers under noisy measurements. In: Proceedings of the 13th IFAC Workshop Time Delay Systems, pp. 188–193 (2016)

    Google Scholar 

  8. Ramírez, A., Mondié, S., Garrido, R.: Proportional integral retarded control of second order linear systems. In: Proceedings of the 52nd IEEE Conference Decision Control, pp. 2239–2244 (2013)

    Google Scholar 

  9. Ramírez, A., Sipahi, R., Mondié, S., Garrido, R.: Design of maximum decay rate for SISO systems with delayed output feedback using elimination theory. In Proceedings of the 12th IFAC Workshop Time Delay Systems, pp. 221–226 (2015)

    Google Scholar 

  10. Ramírez, A., Sipahi, R., Mondié, S., Garrido, R.: An analytical approach to tuning of delay-based controllers for LTI-SISO systems. SIAM J. Control. Optim. 55(1), 397–412 (2017)

    MathSciNet  CrossRef  Google Scholar 

  11. Suh, I.H., Bien, Z.: Proportional minus delay controller. IEEE Trans. Autom. Control. 24(2), 370–372 (1979)

    MathSciNet  CrossRef  Google Scholar 

  12. Olfati-Saber, R.: Ultrafast consensus in small-world networks. In: Proceeding of the American Control Conference, pp. 2371–2378 (2005)

    Google Scholar 

  13. Qiao, W., Atay, F.M., Sipahi, R.: Graph Laplacian design for fast consensus of a LTI system with heterogeneous agent couplings and homogeneous inter-agent delays. In: Proceeding of the ASME Dynamic Systems and Control Conference, pp. 1–8 (2013)

    Google Scholar 

  14. Vyhlídal, T., Zítek, P.: Mapping based algorithm for large-scale computation of quasi-polynomial zeros. IEEE Trans. Autom. Control. 54(1), 171–177 (2009)

    MathSciNet  CrossRef  Google Scholar 

  15. Michiels, W., Vyhlídal, T.: An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type. Automatica 41(6) (2005)

    Google Scholar 

  16. Koh, M., Sipahi, R.: Achieving fast consensus by edge elimination in a class of consensus dynamics with large delays. In: Proceeding of the American Control Conference, pp. 5364–5369 (2016)

    Google Scholar 

  17. Koh, M., Sipahi, R.: Optimizing agent coupling strengths in a network dynamics with inter-agent delays for achieving fast consensus. In: Proceeding of the American Control Conference, pp. 5358–5363 (2016)

    Google Scholar 

  18. Ramírez, A., Garrido, R., Mondié, S.: Velocity control of servo systems using an integral retarded algorithm. ISA Trans. 58, 357–366 (2015)

    CrossRef  Google Scholar 

  19. Ramírez, A., Mondié, S., Garrido, R., Sipahi, R.: Design of proportional-integral-retarded ( PIR) controllers for second-order LTI systems. IEEE Trans. Autom. Control. 61(6), 1688–1693 (2016)

    MathSciNet  CrossRef  Google Scholar 

  20. Cao, Y., Ren, W.: Multi-agent consensus using both current and outdated states with fixed and undirected interaction. J. Intell. Robot. Syst. 58(1), 95–106 (2010)

    CrossRef  Google Scholar 

  21. Huang, N., Duan, Z., Chen, G.: Some necessary and sufficient conditions for order multi-agent systems with sampled position data. Automatica 63 (2016)

    Google Scholar 

  22. Li, J., Xu, S., Chu, Y., Wang, H.: Distributed average consensus control in networks of agents using outdated states. IET Control. Theory Appl. 4(5), 746–758 (2010)

    MathSciNet  CrossRef  Google Scholar 

  23. Meng, Z., Cao, Y., Ren, W.: Stability and convergence analysis of multi-agent consensus with information reuse. Int. J. Control. 83(5), 1081–1092 (2010)

    MathSciNet  CrossRef  Google Scholar 

  24. Meng, Z., Li, Z., Vasilakos, A.V., Chen, S.: Delay-induced synchronization of identical linear multiagent systems. IEEE Trans. Cybern. 43(2), 476–489 (2013)

    CrossRef  Google Scholar 

  25. Song, Q., Yu, W., Cao, J., Liu, F.: Reaching synchronization in networked harmonic oscillators with outdated position data. IEEE Trans. Cybern. 46(7), 1566–1578 (2016)

    CrossRef  Google Scholar 

  26. Yu, W., Chen, G., Cao, M., Ren, W.: Delay-induced consensus and quasi-consensus in multi-agent dynamical systems. IEEE Trans. Circuits Syst. I: Regul. Pap. 60(10), 2679–2687 (2013)

    Google Scholar 

  27. Yu, W., Zheng, W.X., Chen, G., Ren, W., Cao, J.: Second-order consensus in multi-agent dynamical systems with sampled position data. Automatica 47(7), 1496–1503 (2011)

    MathSciNet  CrossRef  Google Scholar 

  28. Ramírez, A., Sipahi, R.: Single-delay and multiple-delay proportional-retarded protocols for fast consensus in a large-scale network. Submitted (2017)

    Google Scholar 

  29. Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, USA (1988)

    Google Scholar 

  30. Hale, J.K., Sjoerd, M., Verduyn, L.: Introduction to Functional Differential Equations. Springer, New York (1993)

    CrossRef  Google Scholar 

  31. Corless, R., Gonnet, G., Hare, D., Jeffrey, D., Knuth, D.: On the Lambert W function. Adv. Comput. Math. 5(1), 329–359 (1996)

    MathSciNet  CrossRef  Google Scholar 

  32. Yi, S., Nelson, P.W., Ulsoy, A.G.: Time-delay Systems: Analysis and Control Using the Lambert W Function. World Scientific, Singapore (2010)

    CrossRef  Google Scholar 

  33. Yi, S., Nelson, W., Ulsoy, A.G.: Proportional-integral control of first-order time-delay systems via eigenvalue assignment. IEEE Trans. Control. Syst. Technol. 21(5), 1586–1594 (2013)

    CrossRef  Google Scholar 

  34. Datko, R.: A procedure for determination of the exponential stability of certain differential-difference equations. Q. Appl. Math. 36(3), 279–292 (1978)

    MathSciNet  CrossRef  Google Scholar 

  35. Breda, D.: On characteristic roots and stability charts of delay differential equations. Int. J. Robust Nonlinear Control 22(8), 892–917 (2012)

    MathSciNet  CrossRef  Google Scholar 

  36. Cepeda-Gomez, R., Olgac, N.: An exact method for the stability analysis of linear consensus protocols with time delay. IEEE Trans. Autom. Control 56(7), 1734–1740 (2011)

    MathSciNet  CrossRef  Google Scholar 

  37. Ghadami, R., Shafai, B.: Decomposition-based distributed control for continuous-time multi-agent systems. IEEE Trans. Autom. Control 58(1), 258–264 (2013)

    MathSciNet  CrossRef  Google Scholar 

  38. Sipahi, R., Lämmer, S., Helbing, D., Niculescu, S.-I.: On stability problems of supply networks constrained with transport delay. ASME J. Dyn. Syst. Meas. Control 131(2) (2009)

    Google Scholar 

  39. Sipahi, R., Qiao, W.: Responsible eigenvalue concept for the stability of a class of single-delay consensus dynamics with fixed topology. IET Control. Theory Appl. 5(4), 622–629 (2011)

    MathSciNet  CrossRef  Google Scholar 

  40. Gu, K., Niculescu, S.I., Chen, J.: On stability crossing curves for general systems with two delays. J. Math. Anal. Appl. 1(311), 231–253 (2005)

    MathSciNet  CrossRef  Google Scholar 

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Ramírez, A., Sipahi, R. (2019). Proportional-Retarded (PR) Protocol for a Large Scale Multi-agent Network with Noisy Measurements; Stability and Performance. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_16

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