Advertisement

Semi-global Consensus of Multi-agent Systems with Impulsive Approach

  • Zhen LiEmail author
  • Jian-an Fang
  • Tingwen Huang
  • Wenqing Wang
  • Wenbing Zhang
Living reference work entry

Abstract

Consensus analysis is a basic issue of multi-agent systems. As an important topic of this issue, semi-global consensus problems have aroused interests since the capability of actuator is usually limited in the presence of a finite range in practice. In theory, semi-global consensus problems refer to design a one-parameter family of control protocols whose domain of attraction can tend to the entire state space. To deal with these problems, the low-gain feedback control strategy has been recently extended. The presented chapter offers a short survey of current studies on this topic, and then we develop the basic idea of low-gain feedback control strategy to apply a distributed impulsive strategy. Similarly with the low-gain feedback control, the magnitude of the proposed impulsive protocol can converge to zero as the low-gain parameter tends to zero. By utilizing the Lyapunov function and low-gain theory, a parametric discrete-time Riccati equation is developed for calculating control gain matrix. Then, based on low-and-high-gain feedback control, another distributed impulsive strategy is considered such that this control protocol can be limited in a finite range. Furthermore, two algorithms are proposed to solve the corresponding the control gain matrices. Subsequently, future research topics are discussed.

Keywords

Multi-agent systems Semi-global consensus Low-gain feedback control Impulsive approach 

References

  1. D.B. Arieh, T. Easton, B. Evans, Minimum cost consensus with quadratic cost functions. IEEE Trans. Syst. Man Cybern. A Syst. Humans 1(39), 210–217 (2009)CrossRefGoogle Scholar
  2. C. Belta, V. Kumar, Abstraction and control for groups of robots. IEEE Trans. Robot. 20(5), 865–875 (2004)CrossRefGoogle Scholar
  3. C.L.P. Chen, Y. Liu, G. Wen, Fuzzy neural network-based adaptive control for a class of uncertain nonlinear stochastic systems. IEEE Trans. Cybern. 44(5), 583–593 (2014)CrossRefGoogle Scholar
  4. M.Z.Q. Chen, L. Zhang, H. Su, G. Chen, Stabilizing solution and parameter dependence of modified algebraic Riccati equation with application to discrete-time network synchronization. IEEE Trans. Autom. Control 61(1), 228–233 (2016a)MathSciNetCrossRefGoogle Scholar
  5. C.L.P. Chen, G. Wen, Y. Liu, Z. Liu, Observer-based adaptive backstepping consensus tracking control for high-order nonlinear semi-strict-feedback multiagent systems. IEEE Trans. Cybern. 46(7), 1591–1601 (2016b)CrossRefGoogle Scholar
  6. Z. Guan, Y. Wu, G. Feng, Consensus analysis based on impulsive systems in multiagent networks. IEEE Trans. Circuits Syst. I Regul. Pap. 59(1), 170–178 (2012)MathSciNetCrossRefGoogle Scholar
  7. Z. Guan, Z. Liu, G. Feng, M. Jian, Impulsive consensus algorithms for second-order multi-agent networks with sampled information. Automatica 48(7), 1397–1404 (2013)MathSciNetCrossRefGoogle Scholar
  8. K.M. Hengster, K. You, F.L. Lewis, L. Xie, Synchronization of discrete-time multi-agent systems on graphs using Riccati design. Automatica 49(2), 414–423 (2013)MathSciNetCrossRefGoogle Scholar
  9. R.A. Horn, C.R. Johnson, Martix Analysis (Springer, New York, 2001)Google Scholar
  10. P. Hou, A. Saberi, Z. Lin, P. Sannuti, Simultaneous external and internal stabilization of linear systems with input saturation and non-input-additive sustained disturbances. Automatica 34(12), 1547–1557 (1998)CrossRefGoogle Scholar
  11. T. Kailath, Linear Systems. (Prentice Hall, Englewood Cliffs, 1980)Google Scholar
  12. H. Li, X. Liao, T. Huang, Second-order locally dynamical consensus of multiagent systems with arbitrarily fast switching directed topologies. IEEE Trans. Syst. Man Cybern. Syst. 4345(6), 1343–1353 (2013)CrossRefGoogle Scholar
  13. Y. Li, S. Tong, T. Li, Hybrid fuzzy adaptive output feedback control design for uncertain MIMO nonlinear systems with time-varying delays and input saturation. IEEE Trans. Fuzzy Syst. 24(1), 841–853 (2016a)CrossRefGoogle Scholar
  14. H. Li, J. Wang, P. Shi, Output-feedback based sliding mode control for fuzzy systems with actuator saturation. IEEE Trans. Fuzzy Syst. 24(6), 1282–1293 (2016b)CrossRefGoogle Scholar
  15. P. Lin, Y. Jia, Consensus of a class of second-order multi-agent systems with time-delay and jointly-connected topologies. IEEE Trans. Autom. Control 55(3), 778–784 (2010)MathSciNetCrossRefGoogle Scholar
  16. Z. Lin, A. Saberi, Semi-global exponential stabilization of linear systems subject to input saturation via linear feedbacks. Syst. Control Lett. 21(3), 225–239 (1993)MathSciNetCrossRefGoogle Scholar
  17. Z. Lin, A. Saberi, A.A. Stoorvogel, R. Mantri, An improvement to the low gain design for discrete-time linear systems in the presence of actuator saturation nonlinearity. Int. J. Robust Nonlinear Control 10(3), 117–135 (2000)MathSciNetCrossRefGoogle Scholar
  18. X. Liu, Impulsive control and optimization. Appl. Math. Comput. 73(1), 77–98 (1995)MathSciNetzbMATHGoogle Scholar
  19. Z. Liu, Z. Guan, X. Shen, G. Feng, Consensus of multi-agent networks with aperiodic sampled communication via impulsive algorithms using position-only measurements. IEEE Trans. Autom. Control 57(10), 2639–2643 (2012)MathSciNetCrossRefGoogle Scholar
  20. B. Liu, W. Lu, T. Chen, Pinning consensus in networks of multiagents via a single impulsive controller. IEEE Trans. Neural Netw. Learn. Syst. 24(7), 1141–1149 (2013)CrossRefGoogle Scholar
  21. J. Lu, Z. Wang, J. Cao, D.W. Ho, J. Kurths, Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay. Int. J. Bifurcat. Chaos 12(7), 1250176 (2012)Google Scholar
  22. J. Lu, C. Ding, J. Lou, J. Cao, Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers. J. Franklin I. 352(11), 5024–5041 (2015)MathSciNetCrossRefGoogle Scholar
  23. I. Palomares, L. Martínez, A semisupervised multiagent system model to support consensus-reaching processes. IEEE Trans. Fuzzy Syst. 4(22), 762–777 (2014)CrossRefGoogle Scholar
  24. I.J. Pérez, F.J. Cabrerizo, S. Alonso, E.H. Viedma, A new consensus model for group decision making problems with non-homogeneous experts. IEEE Trans. Syst. Man Cybern. Syst. 4(44), 494–498 (2014)CrossRefGoogle Scholar
  25. M. Prüfer, Turbulence in multistep methods for initial value problems. SIAM J. Appl. Math. 45(1), 32–69 (1985)MathSciNetCrossRefGoogle Scholar
  26. J. Qin, H. Gao, C. Yu, On discrete-time convergence for general linear multi-agent systems under dynamic topology. IEEE Trans. Autom. Control 59(4), 1054–1059 (2014)MathSciNetCrossRefGoogle Scholar
  27. A. Saberi, P. Sannuti, B.M. Chen, \({\mathcal {H}_{2}}\) Optimal Control (Prentice Hall, Englewood Cliffs, 1995)Google Scholar
  28. A. Saberi, P. Hou, A.A. Stoorvogel, On simultaneous global external and global internal stabilization of critically unstable linear systems with saturating actuators. IEEE Trans. Autom. Control 45(6), 1042–1052 (2000)MathSciNetCrossRefGoogle Scholar
  29. M. Samejima, R. Sasaki, Chance-constrained programming method of it risk countermeasures for social consensus making. IEEE Trans. Syst. Man Cybern. Syst. 5(45), 725–733 (2015)CrossRefGoogle Scholar
  30. F. Sivrikaya, B. Yener, Time synchronization in sensor networks: a survey. IEEE Netw. 18(4), 45–50 (2004)CrossRefGoogle Scholar
  31. S. Strogatz, Exploring complex networks. Nature 410(6825), 268–276 (2001)CrossRefGoogle Scholar
  32. H. Su, M.Z.Q. Chen, J. Lam, Z. Lin, Semi-global leader-following consensus of linear multi-agent systems with input saturation via low gain feedback. IEEE Trans. Circuits Syst. I Regul. Pap. 60(7), 1881–1889 (2013)MathSciNetCrossRefGoogle Scholar
  33. H.J. Sussmann, E.D. Sontag, Y. Yang, A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Autom. Control 39(12), 2411–2425 (1994)MathSciNetCrossRefGoogle Scholar
  34. Y. Tang, H. Gao, J. Lu, J. Kurths, Pinning distributed synchronization of stochastic dynamical networks: a mixed optimization approach. IEEE Trans. Neural Netw. Learn. Syst. 25(10), 1804–1815 (2014)CrossRefGoogle Scholar
  35. Y. Tang, H. Gao, W. Zhang, J. Kurths, Leader-following consensus of a class of stochastic delayed multi-agent systems with partial mixed impulses. Automatica 53, 346–354 (2015)MathSciNetCrossRefGoogle Scholar
  36. A.R. Teel, Semi-global stabilization of linear controllable systems with input nonlinearities. IEEE Trans. Autom. Control 40(1), 96–100 (1995)MathSciNetCrossRefGoogle Scholar
  37. Y. Wang, M. Yang, H.O. Wang, Z. Guan, Robust stabilization of complex switched networks with parametric uncertainties and delays via impulsive control. IEEE Trans. Circuits Syst. I Regul. Pap. 56(9), 2100–2108 (2009)MathSciNetCrossRefGoogle Scholar
  38. X. Wang, A. Saberi, H.F. Grip, A.A. Stoorvogel, Simultaneous external and internal stabilization of linear systems with input saturation and non-input-additive sustained disturbances. Automatica 48(10), 2633–2639 (2012)MathSciNetCrossRefGoogle Scholar
  39. C. Wang, X. Yu, W. Lan, Semi-global output regulation for linear systems with input saturation by composite nonlinear feedback control. Int. J. Control 87(10), 1985–1997 (2014)MathSciNetzbMATHGoogle Scholar
  40. J. Wang, H. Wu, T. Huang, S. Ren, Passivity and synchronization of linearly coupled reaction-diffusion neural networks with adaptive coupling. IEEE Trans. Cybern. 45(9), 1942–1952 (2015)CrossRefGoogle Scholar
  41. X. Wang, H. Su, X. Wang, W.G. Chen, An overview of coordinated control for multi-agent systems subject to input saturation. Perspect. Sci. 7(4), 133–139 (2016)CrossRefGoogle Scholar
  42. G. Wen, C.P. Chen, Y. Liu, Z. Liu, Neural-network-based adaptive leader-following consensus control for second-order nonlinear multi-agent systems. IET Control Theory Appl. 9(13), 1927–1934 (2015)MathSciNetCrossRefGoogle Scholar
  43. G. Yang, J. Wang, Y.C. Soh, Guaranteed cost control for discrete-time linear systems under controller gain perturbations. Linear Algebra Appl. 312(1–3), 161–180 (2000)MathSciNetCrossRefGoogle Scholar
  44. T. Yang, Z. Meng, D.V. Dimarogonas, K.H. Johansson, Global consensus for discrete-time multi-agent systems with input saturation constraints. Automatica 50(2), 499–506 (2014)MathSciNetCrossRefGoogle Scholar
  45. K. You, L. Xie, Network topology and communication data rate for consensusability of discrete-time multi-agent systems. IEEE Trans. Autom. Control 56(10), 2262–2275 (2011)MathSciNetCrossRefGoogle Scholar
  46. W. Zhang, Y. Tang, Q. Miao, J.a. Fang, Synchronization of stochastic dynamical networks under impulsive control with time delays. IEEE Trans. Neural Netw. Learn. Syst. 25(10), 1758–1768 (2014)Google Scholar
  47. B. Zhou, G. Duan, Z. Lin, A parametric Lyapunov equation approach to the design of low gain feedback. IEEE Trans. Autom. Control 53(6), 1548–1554 (2008)MathSciNetCrossRefGoogle Scholar
  48. B. Zhou, Z. Lin, G. Duan, A parametric Lyapunov equation approach to low gain feedback design for discrete-time systems. Automatica 45(1), 238–244 (2009)MathSciNetCrossRefGoogle Scholar
  49. B. Zhou, G. Duan, Z. Lin, Approximation and monotonicity of the maximal invariant ellipsoid for discrete-time systems by bounded controls. IEEE Trans. Autom. Control 55(2), 440–447 (2010)MathSciNetCrossRefGoogle Scholar
  50. L. Zhou, X. Xiao, G. Lu, Simultaneous semi-global Lp-stabilization and asymptotical stabilization for singular systems subject to input saturation. Syst. Control Lett. 61(3), 403–411 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Zhen Li
    • 1
    Email author
  • Jian-an Fang
    • 2
  • Tingwen Huang
    • 3
  • Wenqing Wang
    • 1
  • Wenbing Zhang
    • 4
  1. 1.School of AutomationXi-an University of Posts & TelecommunicationsXi-anChina
  2. 2.School of Information Science and TechnologyDonghua UniversityShanghaiChina
  3. 3.The Science ProgramTexas A&M UniversityDohaQatar
  4. 4.Department of MathematicsYangzhou UniversityJiangsuChina

Section editors and affiliations

  • Yang Tang
    • 1
  1. 1.Department of AutomationEast China University of Science and TechnologyShanghaiChina

Personalised recommendations