1. Distributed Consensus with Limited Communication Data Rate Tao Li Key Laboratory of Systems & Control, AMSS, CAS, China ANU RSISE System and Control Group Seminar October, 15 th, 2010

2. Co-authors  Xie Lihua, School of EEE, Nanyang Technological University, Singapore.  Fu Minyue, School of EE&CS, University of Newcastle, Australia.  Zhang Ji-feng, AMSS, Chinese Academy of Sciences, China

3. Outline Background and motivation

4. Outline Background and motivation Case with time-invariant topology  protocol design and closed-loop analysis  parameter selection algorithm  asymptotic convergence rate

5. Outline Background and motivation Case with time-invariant topology  protocol design and closed-loop analysis  parameter selection algorithm  asymptotic convergence rate Communication energy minimization

6. Outline Background and motivation Case with time-invariant topology  protocol design and closed-loop analysis  parameter selection algorithm  asymptotic convergence rate Communication energy minimization Case with time-varying topologies  protocol design and closed-loop analysis  finite duration of link failures

7. Outline Background and motivation Case with time-invariant topology  protocol design and closed-loop analysis  parameter selection algorithm  asymptotic convergence rate Communication energy minimization Case with time-varying topologies  protocol design and closed-loop analysis  finite duration of link failures Concluding remarks

8. Background and Motivation

9. Distributed estimation over sensor networks Maximum likelihood estimate ： Xiao, Boyd & Lall, ISIPSN, 2005

10. Central strategy ： remote fusion center drawbacks ： high communication cost low robustness

11. Distributed consensus: to achieve agreement by distributed information exchange average consensus

12. Literature: precise information exchange  Olfati-Saber & Murray TAC 2004  Beard, Boyd, Kingston, Ren, Xiao, et al. Features exact information exchange exponentially fast convergence zero steady-state error

13. The communication channels between agents have limited capacity. Quantization plays an important role.

14. Literature: quantized information exchange  Kashyap et al. Automatica 2007  Carli et al. NeCST 2007  Frasca et al. IJNRC 2009  Kar & Moura arXiv:0712 2009 Features infinite levels nonzero steady-state error

15. Theoretic motivation: Exponentially fast exact average-consensus with finite data-rate

16. Power limitation is a critical constraint Energy to transmit 1 bit Energy for 1000-3000 operations ≈ Shnayder et al. 2004 How to select the number of quantization levels (data rate) and convergence rate to minimize the communication energy

17. Channel uncertainties ： Link failures Packet dropouts Channel noises How to design robust encoding-decoding schemes and control protocols against channel uncertainties

18. Problem 1: How many bits does each pair of neighbors need to exchange at each time step to achieve consensus of the whole network ?

19. Problem 2: What is the relationship between the consensus convergence rate and the number of quantization levels (data rate)?

20. Problem 3: What is the relationship of the communication energy cost, the convergence rate and the data rate ?

21. Problem 4: How to design encoders and decoders when the communication topology is time- varying or the transmission is unreliable ?

22. Case with time- invariant topology

23. i encodingtransmissiondecoding j States are real-valued ， but only finite bits of information are transmitted at each time-step

24. i encodingtransmissiondecoding j States are real-valued ， but only finite bits of information are transmitted at each time-step

25. Difficulties  quantization error divergence of the states  finite-level quantizer nonlinearity, unbounded quantization error Key points  design proper encoder, decoder and protocol stabilize the whole network eliminate the effect of quantization error on achieving exact consensus

26. Distributed protocol encoder φ j Z -1 q channel (j,i)

27. Distributed protocol encoder φ j Z -1 q channel (j,i)

28. Distributed protocol encoder φ j Z -1 q finite-level symmetric uniform quantizer channel (j,i)

29. Distributed protocol encoder φ j Z -1 q innovation channel (j,i)

30. Distributed protocol encoder φ j Z -1 q scaling function channel (j,i)

31. Distributed protocol Symmetric uniform quantizer (2K+1) levels bits

32. Distributed protocol decoder ψ ji Z -1 channel (j,i)

33. Distributed protocol decoder ψ ji Z -1 Encoder φ j channel (j,i) Decoder ψ ji channel (j,i)

34. Distributed protocol protocol

35. Distributed protocol protocol

36. Distributed protocol protocol error compensation

37. Distributed protocol protocol average preserved

38. control gain: h quantization levels: 2K+1 scaling fucntion: g(t)

39. Main assumptions A1) G is a connected graph A2) Denote

40. Nonlinear coupling between consensus error and quantization error

41. Adaptive control

42. Selecting proper h, K, g(t)

43. Lemma. Suppose A1)-A2) hold. For any given and by taking with

44. Lemma. Suppose A1)-A2) hold. For any given and by taking with

45. where convergence rate:

46. Lemma. Suppose A1)-A2) hold. For any given and by taking with

47. Lemma. Suppose A1)-A2) hold. For any given and by taking with Convergence rate for perfect communication

49. To save communication bandwidth Minimize the number of quantization levels 2K+1

50. Theorem. Suppose A1)-A2) hold. For any given let Then (i) is not empty (ii) for any given, let

51. For a connected network, average- consensus can be achieved with exponential convergence rate base on a single-bit exchange between each pair of neighbors at each time step

52. Parameter selection algorithm Selecting a constant Selecting the control gain Choose Nonlinear coupled inequalities

53. Asymptotic convergence rate Theorem. Suppose G is connected, then for any given, we have where

54. Asymptotic convergence rate Theorem. Suppose G is connected, then for any given, we have where Synchronizability Barahona & Pecora PRL 2002 Donetti et al. PRL 2005

55. The highest asymptotic convergence rate increases as the number of quantization levels and the synchronizability increase and decreases as the network expands

56. selection of edges: initial states: a network with 30 nodes

57. 1 bit quantizer convergence factor ： 0.95

58. 1 bit quantizer convergence factor ： 0.975

59. Communication energy minimization

60. convergence time constant Xiao & Boyd 2004 Consensus error decreasing by 1/e Model of communication energy

61. Energy for transmitting a single bit Energy for receiving a single bit The faster, the larger The faster, the smaller

62. Fast convergence does not imply power saving. The optimization of the total communication cost leads to tradeoff between number of quantization levels and convergence rate.

63. Energy for transmitting a single bit Energy for receiving a single bit Optimization w.r.t. Suboptimal solution

64. Network with 30 nodes and 0-1 weights

66. Case with time- varying topologies

67. Network model

68. Distributed protocol encoder φ ji Z -1 channel (j,i)

69. Distributed protocol encoder φ ji Z -1 channel (j,i)

70. Distributed protocol decoder ψ ji Z -1 channel (j,i)

71. Distributed protocol protocol

72. Main assumptions A3) A4) A5) There is a constant, such that A6) There exist an integer T>0 and a real constant such that

73. Theorem. Suppose A3)-A6) hold. If where, and the numbers of quantization levels satisfy

74. where then

75. Finite duration of link failures NCS with finite dropouts  Xiao, et al., IJNRC, 2009  Xiong & Lam, Automatica, 2007  Yu, et al., CDC, 2004 A7) There is an integer, such that

76. Theorem. Suppose A3)-A7) hold. For any given let Then is nonempty, and if

77. selecting h and g(t ), such that, and Note that

78. If the duration of link failures in the network is bounded, then control gain and scaling function can be selected properly ， s. t. 3-bit quantizers suffice for asymptotic average-consensus with an exponential convergence rate.

79. Concluding remarks

80. Fixed topology: protocol based on difference encoder and decoder Asymptotic average-consensus with a single-bit exchange Relationship between asymptotic convergence rate and system parameters Algorithm for selecting the control parameters and energy minimization

81. Time-varying topology: different quantizers for different channels Adaptive algorithm for selecting the numbers of quantization levels Special case with jointly-connected graph flow and finite duration of link failures: 3-bit quantizer

82. Future topics  Noisy channels  Random link failures  Model uncertainty  …