Consensus and distributed estimation Sandro Zampieri Universita di - - PowerPoint PPT Presentation

▶
consensus and distributed estimation
SMART_READER_LITE
LIVE PREVIEW

Consensus and distributed estimation Sandro Zampieri Universita di - - PowerPoint PPT Presentation

Aug 24, 2022 138 likes •1.81k views

Consensus and distributed estimation Sandro Zampieri Universita di Padova 5th HYCON2 PhD School Outline Problem description and scientific context Multi-agent systems: a distributed estimation and control architecture Description of the

slide-1
SLIDE 1

5th HYCON2 PhD School

Sandro Zampieri Universita’ di Padova

Consensus and distributed estimation

slide-2
SLIDE 2

5th HYCON2 PhD School

Outline

Problem description and scientific context Multi-agent systems: a distributed estimation and control architecture Description of the linear consensus algorithm Examples of applications Performance analysis Time varying consensus algorithm and its performance analysis Conclusions

slide-3
SLIDE 3

5th HYCON2 PhD School

Networked control systems

Water distribution Smart grids Traffic control Wireless sensor networks Swarm robotics Communication networks

slide-4
SLIDE 4

5th HYCON2 PhD School

Scientific context

pV = nRT

Statistical mechanics how the local interactions of particles may yield simple thermodynamics laws describing the global behavior.

slide-5
SLIDE 5

5th HYCON2 PhD School

Scientific context

Flocking: collective animal behavior given by the motion of a large number of coordinated individuals

COOPERATION: Simple global behavior from local interactions

slide-6
SLIDE 6

5th HYCON2 PhD School

Scientific context

Graph describing friendship relations in an high school

Social and economic networks: individual social and economic interactions produce global phenomena

slide-7
SLIDE 7

5th HYCON2 PhD School

Problem description

The object of our investigations is to study the behavior of “complex” systems constituted by the interconnection of many units which are themselves dynamical systems. The behavior of these systems will depend on the dynamics of the units and on the interconnection topology. We want to understand how these two features produce the global dynamics. A remarkable solution of a question of this type (stability) can be found in JA Fax, RM Murray - IEEE Transactions on Automatic Control, 2004 Information flow and cooperative control of vehicle formations

+ =

slide-8
SLIDE 8

5th HYCON2 PhD School

Multi-agent systems architecture for distributed estimation

ESTIMATOR

y(s)

ˆ x

s = space variable y(s) = spatial data ˆ x = data based decision

slide-9
SLIDE 9

5th HYCON2 PhD School

Multi-agent systems architecture for distributed estimation

is opinion of the node i has of

ˆ xi ˆ xj y(s)

yi

yj sensor sensing link

communication link

i j

ˆ x

ˆ xi

slide-10
SLIDE 10

5th HYCON2 PhD School

Multi-agent systems architecture for distributed estimation

ESTIMATOR

Advantages: intrinsic robustness and adaptivity due to redundancy

y(s) ˆ x ˆ xi ˆ x1

slide-11
SLIDE 11

5th HYCON2 PhD School

Multi-agent systems architecture for distributed estimation

ESTIMATOR

t = time s = space variable y(s, t) = time-varying spatial data ˆ x(t) = time-varying data base decision

ˆ x(t) y(s, t)

slide-12
SLIDE 12

5th HYCON2 PhD School

Multi-agent systems architecture for distributed estimation

y(s, t) u(t)

System Controller Controller

ESTIMATOR

y(s, t)

ˆ x(t)

u(t)

slide-13
SLIDE 13

5th HYCON2 PhD School

The consensus algorithm

Main idea: Having a set of agents to agree upon a certain value (usually

global function) using only local information exchange (local interaction)

x = f(y1, . . . , yN) = F

  • 1

N

N

  • i=1

Gi(yi)

  • yj

xi is the estimates

  • f x of the node i

yi i j xi xj

Distributed computation of general functions

  • Computational efficient (linear &

asynchronous)

  • Independent of graph topology
  • Incremental (i.e. anytime)
  • Robust to failure
slide-14
SLIDE 14

5th HYCON2 PhD School

The consensus algorithm

A distributed algorithm is said to reach the average consensus if xi(t) − → 1 N

N

  • i=1

yi for all i = 1, . . . , N.

A distributed algorithm is said to reach the consensus if xi(t) − → α for all i = 1, . . . , N.

slide-15
SLIDE 15

5th HYCON2 PhD School

The consensus algorithm

, . . . , ()

  • ( + ) =
  • =

() () = () =

slide-16
SLIDE 16

5th HYCON2 PhD School

The consensus algorithm

( + ) = () () =

i j

> G () − →

  • =

µ() µ

slide-17
SLIDE 17

5th HYCON2 PhD School

The consensus algorithm

µ = / () − →

  • =

µ()

  • (, . . . , ) =
  • =

()

slide-18
SLIDE 18

5th HYCON2 PhD School

Some literature (limited to the control field)

Convergence of Markov Chains (60’s) and Parallel Computation Alg.(70’s) John Tsitsiklis “Problems in Decentralized Decision Making and Computation ”, Ph.D thesis, 1984 Time-varying topologies (deterministic worst-case analysis)

Convergence: Moreau (2005), Jadbabaie, Lin, Morse (2002), Olfati Saber, Murray (2004), Cao, Morse, Anderson (2008), . . . .

Randomized consensus

Convergence: Y. Hatano and M. Mesbahi,(2005), Wu (2006), Boyd, Ghosh, Prabhakar, Sha (2006), Alireza Tahbaz- Salehi, Ali Jadbabaie (2006), Performance: Boyd, Ghosh, Prabhakar, Sha (2006), Patterson, Bamieh, Abbadi (2007), Fagnani, Zampieri, (2007)

Applications

Vehicle coordination: many contributions Distributed Kalman Filtering: Xiao, S. Boyd, and S. Lall. (2005), Olfati Saber (2005), Alighanbari, How (2006), Carli, Chiuso, Schenato, Zampieri (2008), Alriksson, Rantzer (2006), Spanos, R. Olfati-Saber, and R. M. Murray.(2005), I.D. Schizas, G.B. Giannakis, S. I. Roumeliotis, and A Ribeiro.(2007), A. Speranzon, C. Fischione, and K. Johansson (2006) Generalized means: Bauso, L. Giarre’, and R. Pesenti (2006),Cortes (2008) Time-synchronization: Solis, Borkar, Kumar (2006), Simeone, Spagnolini (2007), Carli, Chiuso, Schenato, Zampieri (2008), Schenato, Fiorentin (2009) Sensor and camera network calibration: Barooah, Hespanha (2005) ,Bolognani, DelFavero, Schenato, Varagnolo (2008), Tron, Vidal (2009)

slide-19
SLIDE 19

5th HYCON2 PhD School

Some literature on performance metrics

Rate of convergence, mixing rate of a Markov chain, spectral gap and essential spectral radius of a stochastic matrix (from the 70’s)

Cayley graphs: Diaconis (1990-2000), Carli, Fagnani, Speranzon, Zampieri (2008). Random geometric graph: Boyd, Ghosh, Prabhakar, Sha (2006). Performance Classical literature of Markov chains (Diaconis, Stroock), Xiao, Boyd (2006), B. Bamieh, M. Jovanovic, P . Mitra, and S. Patterson. (2010)

L2 performance metrics

General considerations: Xiao, Boyd, Lall, Diacomis, Kim (2004-2009). Cayley graphs: Bamieh, Javonovic, Mitra, Patterson (2009), Carli, Z. (2008), Garin, Zampieri (2009). Random geometric graph: Barooah, Hespanha (2004-2009), Carli, Lovisari, Zampieri (2010).

slide-20
SLIDE 20

5th HYCON2 PhD School

The consensus algorithm

Idea for the proof of convergence: Convex hull always shrinks. If communication graph sufficiently connected, then shrinks to a point

slide-21
SLIDE 21

5th HYCON2 PhD School

The consensus algorithm

Idea for the proof of convergence: Convex hull always shrinks. If communication graph sufficiently connected, then shrinks to a point

slide-22
SLIDE 22

5th HYCON2 PhD School

The consensus algorithm

Idea for the proof of convergence: Convex hull always shrinks. If communication graph sufficiently connected, then shrinks to a point

slide-23
SLIDE 23

5th HYCON2 PhD School

Variations of the consensus algorithm

˙ () = () − ( + ) = ()() ( + ) = ()() ()

slide-24
SLIDE 24

5th HYCON2 PhD School

Variations of the consensus algorithm

() ∈ R ( + ) = () + () () = ()

  • () =
  • =

() ∈ R× () − → α

slide-25
SLIDE 25

5th HYCON2 PhD School

Variations of the consensus algorithm

() ∈ R ( + ) = (()) + (())() () = (())

  • () =
  • =

() ∈ R× () − → α

slide-26
SLIDE 26

5th HYCON2 PhD School

Example: vehicle formation

() = ((), ())

  • ( + ) =
  • =

()

slide-27
SLIDE 27

5th HYCON2 PhD School

Example: distributed estimation

∈ R = + ˆ :=

  • Sensor

Communication link Sensing link

slide-28
SLIDE 28

5th HYCON2 PhD School

Example: distributed least square

x y

,

  • () =
  • =

θ() () θ () = ()Θ

  • () = [() · · · ()]

Θ = [θ · · · θ]

  • ˆ

Θ := argminΘ

  • =

( − ()Θ)

slide-29
SLIDE 29

5th HYCON2 PhD School

Example: distributed least square

x y

  • ˆ

Θ =

  • =

()() −

  • =

()

  • () = ()() ∈ R×

(∞) =

  • =

()() () = () ∈ R (∞) =

  • =

()

average consensus average consensus

slide-30
SLIDE 30

5th HYCON2 PhD School

Example: distributed least square

x y

  • ˆ

Θ =

  • =

()() −

  • =

()

  • () = ()() ∈ R×

(∞) =

  • =

()() () = () ∈ R (∞) =

  • =

()

average consensus average consensus

Initial knowledge

  • f the node i

Final knowledge

  • f the node i
slide-31
SLIDE 31

5th HYCON2 PhD School

Example: distributed least square

x y

  • ˆ

Θ =

  • =

()() −

  • =

()

  • () = ()() ∈ R×

(∞) =

  • =

()() () = () ∈ R (∞) =

  • =

()

ˆ Θ = ((∞))− (∞)

average consensus average consensus

Initial knowledge

  • f the node i

Final knowledge

  • f the node i
slide-32
SLIDE 32

5th HYCON2 PhD School

Example: distributed calibration

  • = (, ) = {, . . . , }

⊆ × (,) (, ) ∈ min

,...,

  

  • (,)∈

(,)( − − (,))    (,)

  • min
  • || − ||
  • ||×|| , , −

= diag{ : ∈ }

slide-33
SLIDE 33

5th HYCON2 PhD School

Example: distributed calibration

  • min

| = || − ||

  • ( + ) = () + α

() = = − α α () := − () ( + ) = () () =

  • (∞) = − (∞) = −
  • =
slide-34
SLIDE 34

5th HYCON2 PhD School

Example: distributed decision making

( = ) = ( = ) = /

  • ( = | = ) = ( = | = ) =

( = | = ) = ( = | = ) = −

  • L(, . . . , ) =

log (|, · · · , ) (|, · · · , ) =

  • ( − ) log −
  • ˆ

= ⇐ ⇒ L(, . . . , ) >

  • −
slide-35
SLIDE 35

5th HYCON2 PhD School

Pros and cons

Advantages Disadvantages

slide-36
SLIDE 36

5th HYCON2 PhD School

Performance metrics

( + ) = () () = () − →

  • =

µ(),

  • =

µ =

  • µ /

() (∞)

slide-37
SLIDE 37

5th HYCON2 PhD School

ρ(P)

Performance metrics

() − →

  • =

µ()

  • (µ, . . . , µ)

µ = / , ρ() ρ() = max

λ∈Λ()\{} |λ|

Λ()

slide-38
SLIDE 38

5th HYCON2 PhD School

  • Performance metrics
slide-39
SLIDE 39

5th HYCON2 PhD School

Performance metrics

  • ρ()
slide-40
SLIDE 40

5th HYCON2 PhD School

We consider here network topologies coming from wireless sensor networks applications, namely the geometric graphs

Network topologies

slide-41
SLIDE 41

5th HYCON2 PhD School

Network topologies

R

  • γ

ρ

slide-42
SLIDE 42

5th HYCON2 PhD School

Network topologies

slide-43
SLIDE 43

5th HYCON2 PhD School

Network topologies

slide-44
SLIDE 44

5th HYCON2 PhD School

Network topologies

slide-45
SLIDE 45

5th HYCON2 PhD School

Network topologies

slide-46
SLIDE 46

5th HYCON2 PhD School

Network topologies

γ

slide-47
SLIDE 47

5th HYCON2 PhD School

Network topologies

(, )

slide-48
SLIDE 48

5th HYCON2 PhD School

Network topologies

  • (, )
slide-49
SLIDE 49

5th HYCON2 PhD School

Network topologies

  • (, )

ρ = min

  • (,)

(,)

  • (, ) ≤ (,)

ρ

slide-50
SLIDE 50

5th HYCON2 PhD School

In certain cases we need to restrict to lines and grids

Network topologies

slide-51
SLIDE 51

5th HYCON2 PhD School

The simplest “geometric” graphs are circles and toruses (no boundary effects)

Network topologies

slide-52
SLIDE 52

5th HYCON2 PhD School

Rate of convergence

  • G

G −

  • / ≤ ρ() ≤ −

/ ,

(Boyd, Ghosh, Prabhakar, Sha 2006, Lovisari, Zampieri 2011)

slide-53
SLIDE 53

5th HYCON2 PhD School

Rate of convergence

slide-54
SLIDE 54

5th HYCON2 PhD School

Rate of convergence

slide-55
SLIDE 55

5th HYCON2 PhD School

Rate of convergence

slide-56
SLIDE 56

5th HYCON2 PhD School

Rate of convergence

slide-57
SLIDE 57

5th HYCON2 PhD School

Rate of convergence

Circle

slide-58
SLIDE 58

5th HYCON2 PhD School

Rate of convergence

Torus

slide-59
SLIDE 59

5th HYCON2 PhD School

Rate of convergence

Circle and torus

slide-60
SLIDE 60

5th HYCON2 PhD School

Rate of convergence

Geometric graph

slide-61
SLIDE 61

5th HYCON2 PhD School

Rate of convergence

Random graph

slide-62
SLIDE 62

5th HYCON2 PhD School

Rate of convergence

Random graph and circle

slide-63
SLIDE 63

5th HYCON2 PhD School

H2 performance

()

  • () :=
  • ∞
  • =

||() − (∞)||

  • || · ||
slide-64
SLIDE 64

5th HYCON2 PhD School

H2 performance

  • ( + 1) = () + ()
  • () := lim sup

→∞

1 {||() − ()1 1||} () = 1/ () () 1 1 1

slide-65
SLIDE 65

5th HYCON2 PhD School

Consensus with noise

Circle

slide-66
SLIDE 66

5th HYCON2 PhD School

Consensus with noise

Geometric graph

slide-67
SLIDE 67

5th HYCON2 PhD School

H2 performance

() = 1 tr

∞

  • =0
  • 2 − 1

1 11 1

  • =

1

  • λ∈Λ()\{1}

1 1 − λ2 1 1 1

slide-68
SLIDE 68

5th HYCON2 PhD School

L2 performance

  • G

G ≤ () ≤ = log() ≤ () ≤ log() = ≤ () ≤ ≥ ,

slide-69
SLIDE 69

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

slide-70
SLIDE 70

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

:=

slide-71
SLIDE 71

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

:= / /

slide-72
SLIDE 72

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

:= R / /

slide-73
SLIDE 73

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

:= R / /

  • () =
  • =

R

slide-74
SLIDE 74

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

slide-75
SLIDE 75

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

/

slide-76
SLIDE 76

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

  • R
slide-77
SLIDE 77

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

slide-78
SLIDE 78

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

slide-79
SLIDE 79

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

slide-80
SLIDE 80

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

slide-81
SLIDE 81

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

Current Kirchhoff’s current law Ohm’s law Condition to get uniqueness

   = C = 1 1 = 0 C /2

  • C

1 1 1 1

  • =
  • =
  • C

1 1 1 1 −1

slide-82
SLIDE 82

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

Green function

  • =
  • C

1 1 1 1 −1

  • C = −
  • −

1 1 1 1 −1 =

  • ()

−11 1 −11 1

  • () :=

∞

  • =0
  • − 1

1 11 1

  • = ()
slide-83
SLIDE 83

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

  • −
slide-84
SLIDE 84

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

() = tr ()

= () C = −

  • −

1 1 1 1 −1 =

  • ()

−11 1 −11 1

  • C

1 1 1 1

  • =
  • =
  • C

1 1 1 1 −1

  • C

1 1 1 1 −1 =

  • ()

−11 1 −11 1

slide-85
SLIDE 85

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

= − R = − = ( − ) = ( − )() = ( − )()( − )

  • =

R = tr ()

slide-86
SLIDE 86

5th HYCON2 PhD School

Electrical network (Doyle, Snell)

() = tr

∞

  • =
  • −
  • =

R = tr () () =

  • =

R () :=

∞

  • =0
  • − 1

1 11 1

slide-87
SLIDE 87

5th HYCON2 PhD School

Example: distributed estimation

∈ R = + ˆ :=

  • Sensor

Communication link Sensing link

slide-88
SLIDE 88

5th HYCON2 PhD School

Example: distributed estimation

  • ( + ) = ()

() = () :=

  • (, ) :=
  • E[(() − )]
slide-89
SLIDE 89

5th HYCON2 PhD School

Performance in distributed estimation

(, ) = tr =

  • λ∈Λ()

λ

slide-90
SLIDE 90

5th HYCON2 PhD School

/

  • ρ() =

(, ) = /, ≥

Performance in distributed estimation

slide-91
SLIDE 91

5th HYCON2 PhD School

× =          / /

  • · · ·

· · · / / / / · · · · · ·

  • /

/ / · · ·

  • · · ·

· · · / / / / · · · · · ·

  • /

/          ρ() − − →

Performance in distributed estimation

slide-92
SLIDE 92

5th HYCON2 PhD School

20 40 60 80 100 120 140 160 180 200 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32

time t N=24 N=30 N=6 N=12 N=18 N=

J(PN,t)

8

Performance in distributed estimation

(, ) < / (, ) = (∞, )

  • ,

(, ) max , √

slide-93
SLIDE 93

5th HYCON2 PhD School

Performance in distributed estimation

slide-94
SLIDE 94

5th HYCON2 PhD School

Time varying consensus algorithm

( + 1) = ()() G ()

() > 0 , G := G ∪ G+1 ∪ · · · G(−1)−1

  • () −

→ α () () − → 1

  • (0)
slide-95
SLIDE 95

5th HYCON2 PhD School

Time varying consensus algorithm

( + 1) = ()() G ()

() > 0 , G := G ∪ G+1 ∪ · · · G(−1)−1

  • () −

→ α () () − → 1

  • (0)
slide-96
SLIDE 96

5th HYCON2 PhD School

Performance of randomized consensus algorithms

  • Hatano Mesbahi
  • Boyd, Ghosh, Prabhakar, Shah
  • Tahbaz-Salehi, Jadbabaie
  • Porfiri, Stilwell
  • Kar, Moura
  • Patterson, Bamieh, Abbadi
  • Wu
  • Fagnani, Zampieri
  • () () > 0

, ( + 1) = ()() () →

  • = µ(0) µ

( − 1) · · · (0) → 1 1µ µ := (µ1, . . . , µ)

slide-97
SLIDE 97

5th HYCON2 PhD School

Performance of randomized consensus algorithms

  • Hatano Mesbahi
  • Boyd, Ghosh, Prabhakar, Shah
  • Tahbaz-Salehi, Jadbabaie
  • Porfiri, Stilwell
  • Kar, Moura
  • Patterson, Bamieh, Abbadi
  • Wu
  • Fagnani, Zampieri
  • () () > 0

, ( + 1) = ()() () →

  • = µ(0) µ

( − 1) · · · (0) → 1 1µ µ := (µ1, . . . , µ)

slide-98
SLIDE 98

5th HYCON2 PhD School xi(t) xj(t) i j

Gossip algorithm (Boyd et al. 2006)

G (, ) G > 0 ( + 1) = 1/2() + 1/2() ( + 1) = 1/2() + 1/2() ( + 1) = () = ,

slide-99
SLIDE 99

5th HYCON2 PhD School xi(t) xj(t) i j

Gossip algorithm (Boyd et al. 2006)

G (, ) G > 0 ( + 1) = 1/2() + 1/2() ( + 1) = 1/2() + 1/2() ( + 1) = () = ,

slide-100
SLIDE 100

5th HYCON2 PhD School xi(t) xj(t) i j

Gossip algorithm (Boyd et al. 2006)

G (, ) G > 0 ( + 1) = 1/2() + 1/2() ( + 1) = 1/2() + 1/2() ( + 1) = () = ,

slide-101
SLIDE 101

5th HYCON2 PhD School

i j i j P(t) = 1 ! 1 1/2 1/2 1 ! 1 1/2 1/2 1 ! 1 " # $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

Gossip algorithm (Boyd et al. 2006)

slide-102
SLIDE 102

5th HYCON2 PhD School

Asymmetric gossip algorithm

xj(t) i j

G (, ) G > 0 ( + 1) = 1/2() + 1/2() ( + 1) = () =

slide-103
SLIDE 103

5th HYCON2 PhD School

Asymmetric gossip algorithm

xj(t) i j

G (, ) G > 0 ( + 1) = 1/2() + 1/2() ( + 1) = () =

slide-104
SLIDE 104

5th HYCON2 PhD School

Asymmetric gossip algorithm

xj(t) i j

G (, ) G > 0 ( + 1) = 1/2() + 1/2() ( + 1) = () =

slide-105
SLIDE 105

5th HYCON2 PhD School

i j

P(t) = 1 ! 1 1/2 1/2 1 ! 1 1 1 ! 1 " # $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

Asymmetric gossip algorithm

slide-106
SLIDE 106

5th HYCON2 PhD School i j

G ( + 1) = 1/2() + 1/2() ( + 1) = ()

  • Broadcast communication
slide-107
SLIDE 107

5th HYCON2 PhD School i j

G ( + 1) = 1/2() + 1/2() ( + 1) = ()

  • Broadcast communication
slide-108
SLIDE 108

5th HYCON2 PhD School

Consensus with packet drops

Patterson, Bamieh, Abbadi Fagnani, Zampieri Preciado, Tahbaz-Salehi, Jadbabaie

  • ( + 1) =
  • =1

()

  • (),

∈ N, , = 1, . . . , = () = 0

  • [() = 1] =

[() = 0] = 1 − = 1, . . . , () 1 1

slide-109
SLIDE 109

5th HYCON2 PhD School

Consensus with packet drops

  • ( + 1) =

1

  • ∈ ()

 

∈

()()  

  • ( + 1) = (1 −
  • )() +
  • ∈\{}

()()

slide-110
SLIDE 110

5th HYCON2 PhD School

Performance metrics

() = 1 ||() − 1 1()||2 = 1

  • =1

|() − ()|2 () = 1/ () () β() = |() − (0)|2

slide-111
SLIDE 111

5th HYCON2 PhD School

Convergence

() () > 0 := E[()]

slide-112
SLIDE 112

5th HYCON2 PhD School

Examples

  • [() = − 1/2( − )( − )] =
  • = ,

∀ = G = G () ≥ 1/2

slide-113
SLIDE 113

5th HYCON2 PhD School

Examples

  • () ≥ 1/

= (, )

  • =

E

  • ()
  • ∈ ()
  • =

E

  • ()
  • ∈ ()
  • () = 0
  • P[() = 0]

+ E

  • ()
  • ∈ ()
  • () = 1
  • P[() = 1]

= E

  • ()
  • ∈ ()
  • () = 1
  • ≥ /||
slide-114
SLIDE 114

5th HYCON2 PhD School

Rate of convergence

() () > 0 lim

→∞

1 log ||() − (∞)|| =

  • ||() − (∞)||
slide-115
SLIDE 115

5th HYCON2 PhD School

200 400 600 800 1000 1200

  • 60
  • 50
  • 40
  • 30
  • 20
  • 10

Rate of convergence

= 2ν = 1/2 = 0 >>

  • = 8
slide-116
SLIDE 116

5th HYCON2 PhD School

Rate of convergence

  • ¯

() := E[()]

  • () = 1

||() − (∞)||2 () ¯ () ()1/2

  • ¯

()1/2

  • =

() P[|() − ¯ ()| ≥ δ] ≤ exp

  • − δ2α()

||(0)||4

  • α() ()

α() = 2 2/ P[|()−¯ ()| ≥ δ]

  • ()1/2 ¯

()1/2

slide-117
SLIDE 117

5th HYCON2 PhD School

Mean square analysis

  • Ω := − 1

11

E[()] = 1 E[()Ω()] = 1 (0)∆()(0)

  • ∆() := E[(0)(1) · · · ( − 1)Ω( − 1) · · · (1)(0)]

∆(0) := Ω ∆( + 1) = L(∆()) L : R× → R× L() = E[(0)(0)]

slide-118
SLIDE 118

5th HYCON2 PhD School

  ∆( + 1) = L(∆()) ∆(0) := Ω E[()] = 1

(0)∆()(0)

   δ( + 1) = Lδ() E[()] = ((0))δ() L := E[(0) ⊗ (0)] δ() = vect(∆()) L

Mean square analysis

slide-119
SLIDE 119

5th HYCON2 PhD School

  • E[()]

:= lim

→∞

1 log E[()] Sym ×

  • L|Sym

L|Sym

  • esr
  • 2 ≤ ≤ sr(L(Ω)) esr(·)

sr(·)

Mean square analysis

slide-120
SLIDE 120

5th HYCON2 PhD School

  • = 1/2

2

= 1 − 1

  • 2

= (1 − β)(1 − β)

β := E

  • 1

2+−2

  • {0, 1, . . . , }

P[ = ] =

  • (1 − )−

= 0, 1, . . . ,

Examples

slide-121
SLIDE 121

5th HYCON2 PhD School

Distance of the consensus value from the average

β(∞) = |(∞) − (0)|2 = |(µ − −11)(0)|2 E[β(∞)] = (0)(0) E[()] = E[µµ] − −211

  • := E[µµ] = 1

lim

→∞ L()

L() = 11 = 1

slide-122
SLIDE 122

5th HYCON2 PhD School

Distance of the consensus value from the average

β(∞) = |(∞) − (0)|2 = |(µ − −11)(0)|2 E[β(∞)] = (0)(0)

  • = E[µµ] − 1

E[µ]1 − 1 1E[µ] + −211 = E[µµ] − −211

  • := E[µµ] = 1

lim

→∞ L()

L() = 11 = 1

slide-123
SLIDE 123

5th HYCON2 PhD School

  • =1

=

  • =1
  • L

= 1 ( + 1)

  • − 1

11

  • Examples
slide-124
SLIDE 124

5th HYCON2 PhD School

  • = 1

2 1 − β 1 − β + 1/

  • − 1

11

  • β := E
  • 1

2+−2

  • {0, 1, . . . , } P[ = ] =
  • (1 − )−

= 0, 1, . . . ,

1−β 1−β+1/ ≤ 1

≤ 1 2

  • − 1

11

  • −2 ∞

Examples

slide-125
SLIDE 125

5th HYCON2 PhD School

Higher order consensus

( + ) = () + () () = () () =

  • =

() (() − ())

slide-126
SLIDE 126

5th HYCON2 PhD School

+()

Higher order consensus

( + ) = () + () () = () () =

  • =

() (() − ())

slide-127
SLIDE 127

5th HYCON2 PhD School

+()

Higher order consensus

( + ) = () + () () = () () =

  • =

() (() − ()) () , |() − ()| → ∀,

slide-128
SLIDE 128

5th HYCON2 PhD School

Higher order consensus: another example

  • ( + ) = β() + ( − β)( − )

β

  • ( + ) = β()() + ( − β)( − )
slide-129
SLIDE 129

5th HYCON2 PhD School

Higher order consensus: another example

  • ( + )

= () + () () = () () =

  • =

() (() − ()) + () () := − () =

  • ,

=

  • ,

=

  • =

−β

  • ,

=

  • − β

β −

slide-130
SLIDE 130

5th HYCON2 PhD School

Clock synchronization

time time estimate

slide-131
SLIDE 131

5th HYCON2 PhD School

Clock synchronization

clock time profile time time estimate

slide-132
SLIDE 132

5th HYCON2 PhD School

Reference time profile

Clock synchronization

clock time profile time time estimate

slide-133
SLIDE 133

5th HYCON2 PhD School

Reference time profile

Clock synchronization

time time estimate

slide-134
SLIDE 134

5th HYCON2 PhD School

Reference time profile

Clock synchronization

clock time profile with time corrections time time estimate

slide-135
SLIDE 135

5th HYCON2 PhD School

Reference time profile

Clock synchronization

time time estimate

slide-136
SLIDE 136

5th HYCON2 PhD School

Reference time profile

Clock synchronization

clock time profile with time and slope corrections time time estimate

slide-137
SLIDE 137

5th HYCON2 PhD School

Architectures for clock synchronization

root

Master slave architecture (NTP time-synchronization protocol)

j i

slide-138
SLIDE 138

5th HYCON2 PhD School

Leaderless distributed architecture

j i

Architectures for clock synchronization

slide-139
SLIDE 139

5th HYCON2 PhD School

Clock synchronization

Clock synchronization with no reference time

x2(t) x3(t) x1(t) ()

slide-140
SLIDE 140

5th HYCON2 PhD School

Mathematical description of a clock

∆

slide-141
SLIDE 141

5th HYCON2 PhD School

Mathematical description of a clock

We neglect the clock quantization

  • () = −

∆ + () ∆

si(t) t

slide-142
SLIDE 142

5th HYCON2 PhD School

Mathematical description of a clock

We neglect the clock quantization

  • () = −

∆ + ()

  • ()

= () + ˆ ∆(() − ()) ˆ ∆ ∆ ∆

si(t) t

slide-143
SLIDE 143

5th HYCON2 PhD School

Mathematical description of a clock

We neglect the clock quantization

ˆ ∆()

We can have time dependent estimation

  • f the oscillator period
  • () = −

∆ + ()

  • ()

= () + ˆ ∆(() − ()) ˆ ∆ ∆ ∆

si(t) t

slide-144
SLIDE 144

5th HYCON2 PhD School

Mathematical description of a clock

We neglect the clock quantization

ˆ ∆()

We can have time dependent estimation

  • f the oscillator period

= () + ˆ ∆ − ∆

  • () = −

∆ + ()

  • ()

= () + ˆ ∆(() − ()) ˆ ∆ ∆ ∆

si(t) t

slide-145
SLIDE 145

5th HYCON2 PhD School

Mathematical description of a clock

si(t) t

We neglect the clock quantization

= () + ˆ ∆ − ∆

  • () = −

∆ + ()

  • ()

= () + ˆ ∆(() − ()) ∆

yi(t) t t0 yi(t0)

slide-146
SLIDE 146

5th HYCON2 PhD School

Mathematical description of a clock

yi(t) t th

, , , . . .    (+

) = (− ) + ()

ˆ ∆(+

) = ˆ

∆(−

) + ()

slide-147
SLIDE 147

5th HYCON2 PhD School

Mathematical description of a clock

yi(t) t th th+1

  • (−

+) = (+ ) + ˆ

∆(+

)+ −

∆

slide-148
SLIDE 148

5th HYCON2 PhD School

Mathematical description of a clock

yi(t) t th th+1

(−

+)

= (−

) + + −

∆ ˆ ∆(+

) + ()

  • (−

+) = (+ ) + ˆ

∆(+

)+ −

∆

slide-149
SLIDE 149

5th HYCON2 PhD School

Mathematical description of a clock

yi(t) t th th+1

= (−

) + + −

∆ ( ˆ ∆(−

) + ()) + ()

(−

+)

= (−

) + + −

∆ ˆ ∆(+

) + ()

  • (−

+) = (+ ) + ˆ

∆(+

)+ −

∆

slide-150
SLIDE 150

5th HYCON2 PhD School

Mathematical description of a clock

yi(t) t th th+1

  • ˆ

∆(−

+) = ˆ

∆(−

) + ()

= (−

) + + −

∆ ( ˆ ∆(−

) + ()) + ()

(−

+)

= (−

) + + −

∆ ˆ ∆(+

) + ()

  • (−

+) = (+ ) + ˆ

∆(+

)+ −

∆

slide-151
SLIDE 151

5th HYCON2 PhD School

Mathematical description of a clock

  • () :=

(−

)

ˆ ∆(−

)

  • () :=
  • ()
  • ()

      ( + ) =

  • +−

∆

  • (() + ())

() =

  • ()
slide-152
SLIDE 152

5th HYCON2 PhD School

Consensus based clock synch

  • () =
  • =

() (() − ())

  • =
  • () ∈ R

() ∈ R×

slide-153
SLIDE 153

5th HYCON2 PhD School

Consensus based clock synch

() |() − ()| → ∀,

  • () =
  • =

() (() − ())

  • =
  • () ∈ R

() ∈ R×

slide-154
SLIDE 154

5th HYCON2 PhD School

Communication graph

i j

G()

() () =

slide-155
SLIDE 155

5th HYCON2 PhD School

Consensus for higher order systems

slide-156
SLIDE 156

5th HYCON2 PhD School

Ideal (unrealistic) time-invariant case

  • = +

() =

  • ( + ) =
  • /∆
  • (() + ())

() =

  • =

(() − ())

slide-157
SLIDE 157

5th HYCON2 PhD School

Ideal (unrealistic) time-invariant case

() () () () := () −

  • ()
slide-158
SLIDE 158

5th HYCON2 PhD School

Ideal (unrealistic) time-invariant case

() () () () := () −

  • ()
  • Synchronization error
slide-159
SLIDE 159

5th HYCON2 PhD School

=

  • =
  • Ideal (unrealistic) time-invariant case
slide-160
SLIDE 160

5th HYCON2 PhD School

=

  • =
  • Ideal (unrealistic) time-invariant case
slide-161
SLIDE 161

5th HYCON2 PhD School

=

  • =
  • =
  • Ideal (unrealistic) time-invariant case
slide-162
SLIDE 162

5th HYCON2 PhD School

=

  • =
  • =
  • Ideal (unrealistic) time-invariant case

() =

  • =
  • (() − ())
slide-163
SLIDE 163

5th HYCON2 PhD School

Ideal (unrealistic) time-invariant case

  • ∆

= /, ≤ ∆/ =    −

  • max{,}

(, ) ∈ E = −

=

=

  • ∆ ∆

∞ ∆

slide-164
SLIDE 164

5th HYCON2 PhD School

Ideal (unrealistic) time-invariant case

Metropolis weights

  • ∆

= /, ≤ ∆/ =    −

  • max{,}

(, ) ∈ E = −

=

=

  • ∆ ∆

∞ ∆

slide-165
SLIDE 165

5th HYCON2 PhD School

  • Clock synchronization
slide-166
SLIDE 166

5th HYCON2 PhD School

Conclusions

The consensus algorithm is an instance of a completely distributed

  • design. This is an extreme design paradigm.

It is intrinsically robust to external changes and highly self-adaptive so that a limited initial configuration and tuning effort is necessary. None or limited information about the global structure of the system is necessary to the units. Graceful performance degradation. Importance of the interaction network topology.

slide-167
SLIDE 167

5th HYCON2 PhD School

Conclusions

There are two important messages: The consensus algorithm should be analyzed in the context of the applications in which it is used. This yields different performance indices with different relations with network topology. In large scale networks both time and the number of agents may be

  • large. Therefore there might emerge several asymptotic regimes in

relation to how these two quantities grow with respect to each other.