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Distributed Consensus with Limited Communication Data Rate

ANU RSISE System and Control Group Seminar. Distributed Consensus with Limited Communication Data Rate. Tao Li. Key Laboratory of Systems & Control, AMSS, CAS, China. October, 15 th , 2010. Co-authors. Xie Lihua , School of EEE, Nanyang Technological University, Singapore.

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Distributed Consensus with Limited Communication Data Rate

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  1. ANU RSISE System and Control Group Seminar Distributed Consensus with Limited Communication Data Rate Tao Li Key Laboratory of Systems & Control, AMSS, CAS, China October, 15th, 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: Exponentiallyfast exact average-consensus withfinitedata-rate

  16. Shnayder et al. 2004 Power limitation is a critical constraint Energy to transmit 1 bit Energy for 1000-3000 operations ≈ 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 encoding transmission decoding j States are real-valued，but only finite bits of information are transmitted at each time-step

  24. i encoding transmission decoding 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 φj q Z-1 channel (j,i)

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

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

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

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

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

  32. Distributed protocol • decoder ψji channel (j,i) Z-1

  33. Distributed protocol • decoder ψji channel (j,i) Z-1 Encoder φj channel (j,i) Decoder ψji

  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. Nonlinear coupling between consensus error and quantization error Adaptive control

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

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

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

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