Distributed estimation and consensus. Luca Schenato University of Padova WIDE 09 7 July 2009, Siena

Transcription

1 Distributed estimation and consensus Luca Schenato University of Padova WIDE 09 7 July 2009, Siena

2 Joint work w/

3 Outline Motivations and target applications Overview of consensus algorithms Application of consensus to WSN: Sensor calibration Least-square parameter identification Time-synchronization Distributed Kalman filtering Open problems Identification Estimation Control

4 Outline Motivations and target applications Overview of consensus algorithms Application of consensus to WSN: Sensor calibration Least-square parameter identification Time-synchronization Distributed Kalman filtering Open problems Identification Estimation Control

5 Networked Control Systems Drive-by-wire systems Swarm robotics Smart structures: space telescope & satellites mesh Wireless Sensor Networks Traffic Control: Internet and transportation Smart materials & MEMS: sheets of sensors and actuators NCSs: physically distributed dynamical systems interconnected by a communication network

6 Wireless Sensor Actuator Networks (WSANs) Small devices μcontroller, Memory Wireless radio Sensors & Actuators Batteries Inexpensive Multi-hop communication Programmable (micro-pc) sensor node BASE STATION

7 Applications: Smart Greenhouse Distributed estimation Distributed control Control under packet loss & random delay Sensor fusion Distributed time synchronization

8 Applications: ThermoEfficiency Labeling Building thermodynamics model identification Sensor selection for identification Optimal sensor placement

9 Applications: Distributed Localization&Tracking Indoor radio signal modeling Real-time localization Distributed tracking Coordination

10 NCSs: what s new for control? Classical architecture: Centralized structure Actuators Plant Sensors Controller

11 NCSs: what s new for control? NCSs: Large scale distributed structure A A Plant S S A S Interference Packet loss Connectivity COMMUNICATION Random delay Limited capacity NETWORK Quantization Congestion C C C C C

12 Outline Motivations and target applications Overview of consensus algorithms Application of consensus to WSN: Sensor calibration Least-square parameter identification Time-synchronization Distributed Kalman filtering Open problems Identification Estimation Control

13 Main idea The consensus problem Having a set of agents to agree upon a certain value (usually global function) using only local information exchange (local interaction) Also known as: Agreement problem (economics, signal processing, social networks) Gossip algorithms (CS & communications) Synchronization ( statistical mechanics) Rendezvous (robotics) Suitable for (noisy) sensor networks

14 Main features Distributed computation of general functions Computational efficient (linear & asynchronous) Independent of graph topology Incremental (i.e. anytime) Robust to failure Global Decision Maker

15 Some history (in control) Convergence of Markov Chains (60 s) and Parallel Computation Alg.(70 s) John Tsitsiklis Problems in Decentralized Decision Making and Computation, Ph.D thesis, MIT 1984 A. Jadbabaie, J. Lin, and A. S. Morse Coordination of groups of mobile autonomous agents using nearest neighbor rules, CDC 02 (Axelby Best Paper Award TAC) Time-varying topologies (worst-case) L. Moreau, Consensus seeking in multi-agent systems using dynamically changing interaction topologies, IEEE, Transactions on Automatic Control, vol 50, No. 2, 2005 M. Cao, A. S. Morse, and B. D. O. Anderson. "Reaching a Consensus in a Dynamically Changing Environment: A Graphical Approach." SIAM Journal on Control and Optimization, Feb 2008 Randomized topologies S. Boyd, A. Ghosh, B. Prabhakar, D. Shah Randomized Gossip Algorithms, TIT 2006 F. Fagnani, S. Zampieri, Randomized consensus algorithms over large scale networks, JSAC 08 Applications: Vehicle coordination: Jadbabaie, Francis s group, Tanner, Kalman Filtering: Olfati Saber-Murray, Alighanbari-How, Carli-Chiuso-Schenato-Zampieri Generalized means: Giarre,Cortes Time-synchronization: Solis-P.R. Kumar,Osvlado-Spagnolini, Carli-Chiuso-Schenato-Zampieri WSN sensor calibration and parameter identification: Bolognani-DelFavero-Schenato-Varagnolo

16 Consensus formulation

17 Consensus formulation (cont )

18 Consensus definition

19 Linear consensus

20 A robotics example: the rendezvous problem Convex hull always shrinks. If communication graph sufficiently connected, then shrinks to a point

21 Stochastic matrix

22 P= Constant matrix P

23 Convergence results P= x x x λ x x 1 x

24 Time varying P(t): broadcast

25 Time varying P(t): symmetric gossip

26 Consensus strategies for WSN

27 Convergence results: P=P(t) deterministic L. Moreau, Consensus seeking in multi-agent systems using dynamically changing interaction topologies, IEEE, Transactions on Automatic Control, vol 50, No. 2, 2005 M. Cao, A. S. Morse, and B. D. O. Anderson. "Reaching a Consensus in a Dynamically Changing Environment: A Graphical Approach." SIAM Journal on Control and Optimization, Feb 2008

28 Convergence results: P=P(t) randomized F. Fagnani, S. Zampieri, Randomized consensus algorithms over large scale networks, IEEE Journal on Selected Areas in Communications, 2008

29 Generalized mean D. Bauso, L. Giarre and R. Pesenti, "Nonlinear protocols for Optimal Distributed Consensus in Networks of Dynamic Agents, Systems and Control Letters, 2006 J. Cortés, Distributed algorithms for reaching consensus on general functions, Automatica 44 (3) (2008),

30 leader Node counting

31 Outline Motivations and target applications Overview of consensus algorithms Application of consensus to WSN: Sensor calibration Least-square parameter identification Time-synchronization Distributed Kalman filtering Open problems Identification Estimation Control

32 Localization with WSN

33 Localization with WSN Transmission power Model parameter Receiver parameter

34 Offset effect

35 WSN sensor calibration

36 Calibration as consensus problem update equation Steady state

37 Experimental Testbed

38 Experimental results

39

40 Parameter identification

41 For each link: Modeling

42 Modeling (cont d)

43 Least-square Identification

44 Consensus-based Identification

45 Communication schemes

46 Experimental results

47 Tracking results

48 Time synchronization in sensor networks sensor node BASE STATION ON OFF Node i transmission Node j

49 Clock characteristics & standard clock pair synch Node 1 skew Offset: instantaneous time difference Skew: clock speed Drift: derivative of clock speed offset Node 2 synchronizing node Offset synch: periodically remove offset with respect to reference clock Skew compensation: estimate relative speed with respect to reference clock synch. period Reference node

50 State-of-the-art Tree-based sync root Cluster-based sync single-hop clusters gateways nodes comm. links Distributed nodes i j

51 Modeling

52 P-control DT integrator

53 PI-control

54 C(z)-control

55 Parameter design (undirected graphs) Suboptimal design (no topology needed): Optimal design: almost convex problem (SDP + 1D non-convex search)

56 Model w/ noise Suboptimal design still OK Optimal design: almost convex problem (Semidefinite programming in K+ 1D non-convex search in ff)

57 Simulations

58 Outline Motivations and target applications Overview of consensus algorithms Application of consensus to WSN: Sensor calibration Least-square parameter identification Time-synchronization Distributed Kalman Filtering Open problems Identification Estimation Control

59 Estimation framework Static estimation Dynamic estimation Hierarchical estimation Distributed estimation All-to-all communication Multi-hop communication

60 Estimation framework Static estimation Dynamic estimation Hierarchical estimation Distributed estimation All-to-all communication Multi-hop communication

61 Problem setup Sensor j Sensor i Sensor Communication links yi Sensing links x

62 Desired solution: centralized Kalman filter

63 Distributed Kalman Filter [Olfati-Saber,Spanos,Murray,Alriksson,Rantzer] j i measurement consensus step communication measurement

64 Steady-state performance parameters COST FUNCTION optimization variables

65 Objective

66 Assumption: Q normal

67 Convexity for fixed l or Q

68 Joint optimization: special cases

69 Theorem Fast communication

70 Theorem Small measurement noise

71 Theorem High measurement noise

72 = k k k k k k k k k k k k k k Q k L M M O M M M O L L We assume that N=100, q=1 and r=1. Simulation results: circulant graph

73 Simulation results: Circulant graph J 1 (Q 1 opt,l1 opt ) J 1 r J m

74 Simulation results: Circulant graph k(ρ opt ) K opt opt k 1 k r opt k( Q ) F m

75 Simulation results: Random geometric graph J 1 r J 1 (Q 1 opt,l 1 opt ) J m

76 Takeaways points Consensus algorithms fit naturally in distributed estimation problems Some analytical results for scalar dynamics under special regimes Optimizing second λ 2 (Q) is not necessarily optimal strategy

77 Outline Motivations and target applications Overview of consensus algorithms Application of consensus to WSN: Sensor calibration Least-square parameter identification Time-synchronization Open problems Identification Estimation Control

78 Identification: large scale structured systems Correlation graph comm. links nodes Communication graph correlation. links nodes Carlos Carvalho " Structure and Sparsity in High-Dimensional Multivariate Analysis, Ph.D. Theis, Duke Univ., 2007 A. P. Dempster, Covariance Selection, Biometrics, Vol. 28, No. 1, Special Multivariate Issue (Mar., 1972), pp

79 Non-parametric estimation comm. links nodes kernel Predd, J.B.; Kulkarni, S.B.; Poor, H.V. Distributed learning in wireless sensor networks Signal Proce Magazine,2006

80 Soft Hierarchical Control Time synchronization example: comm. links nodes root i j