L. Giarr e* * DIETT, Universit a di Palermo joint work ... · DIETT 6 Graph theory • It is natural to model information exchange between agents in a cooperative team by directed/undirected

Consensus algorithms

L. Giarre*

* DIETT, Universita di Palermo

joint work with D. Bauso and R. Pesenti

SCUOLA OTTIMIZZAZIONE - TEORIA E APPLICAZIONI

Palermo 2011

DIETT 1

Outline

� Introduction

– Applications

– Why consensus problems

– Graph Theory

– The Consensus Problem

– Reference

• Distributed consensus protocols for coordinating buyers

• Mechanism Design for Optimal Consensus Problems

• Lazy consensus for networks with unknown but bounded

disturbances

• References

SCUOLA DOTTORATO- PALERMO 2011

DIETT 2

Applications:

• Cooperative control for multi-agent systems:

• formation control problems with applications to mobile robots

unmanned air vehicles (UAVs), autonomous underwater vehicles

(AUVs), satellites, aircraft, spacecraft, and automated highway

systems

• nonformation control problems such as task assignment, payload

transport, role assignment, air traffic control, timing, and search,

inventory problems, sensor networks, distributed estimation,

syncronization


DIETT 3

Shared Information

• Definition and management of shared information among a group

of agents to facilitate the coordination among agents

• The shared information may take the form of common objectives,

common control algorithms, relative position information, ...

• Examples of information necessary for cooperation:

– Relative position sensors may enable vehicles to construct

state information for other vehicles (UAV’s formations)

– joint knowledge might be pre-programmed into the vehicles

before a mission begins (robotics team)


DIETT 4

Why Consensus Problems?

• A team of agents must be able to respond to unanticipated

situations sensed as a cooperative task

• As the environment changes, the agents on the team must be in

agreement as to what changes took place.

• A consequence that shared information is assumed necessary for

coordination → cooperation requires that the group of agents

reach consensus on the coordination data.

• Convergence to a common value is called agreement or consensus

problems.

• Consensus problems have a history in computer science. We focus

on applications to cooperative control of multi-agent systems.


DIETT 5

Main aspects

• Challenging aspects:

– decentralized information

– decentralized decision making

– distributed objectives

• Agents may need to be in agreement regarding a certain quantity

of interest

– global state (information sharing)

– future decisions (coordinated decision making)

– common/individual goals


DIETT 6

Graph theory

• It is natural to model information exchange between agents in a

cooperative team by directed/undirected graphs

• A digraph (directed graph) consists of a pair (N,E), where N is

a finite nonempty set of nodes and E is a set of ordered pairs of

nodes, called edges.

• The pairs of nodes in an undirected graph are unordered.

• A directed path is a sequence of ordered edges of the form

(vi1, vi2), (vi2, vi3), , where vij ∈ N , in a digraph.

• An undirected path in an undirected graph is defined

analogously, where (vij , vik) implies (vik, vij).


DIETT 7

Graph theory

• A digraph is called strongly connected if there is a directed path

from every node to every other node.

• An undirected graph is called connected if there is a path

between any distinct pair of nodes.

• A directed tree is a digraph, where every node, except the root,

has exactly one parent.

• A spanning tree of a digraph is a directed tree formed by graph

edges that connect all the nodes of the graph. We say that a

graph has (or contains) a spanning tree if there exists a spanning

tree that is a subset of the graph.

• Note that the condition that a digraph has a spanning tree is

equivalent to the case that there exists a node having a directed

path to all other nodes.


DIETT 8

Graph theory

• The adjacency matrix A = [aij ] of a weighted digraph is defined

as aii = 0 and aij > 0 if(j, i) ∈ E where i = j.

• The Laplacian matrix of the weighted digraph is defined as

L = [ℓij ], where ℓii =∑

j aij and ℓij = −aij where i = j.

• For an undirected graph, the Laplacian matrix is symmetric

positive semi-definite.


DIETT 9

Example

• A team of 4 UAVs in longitudinal flight and initially at different

heights.

• Each UAV controls the vertical rate without knowing the relative

position of all UAVs but only of neighbors

• The information flow in a network of 4 agents ( the

communication network topology) − > A =? L =?


DIETT 10

Example

v1 v3 v4

v2


DIETT 11

Example

• A =

0 1 1 1

1 0 1 0

1 1 0 0

1 0 0 0

• L =

3 −1 −1 −1

−1 2 −1 0

−1 −1 2 0

−1 0 0 1


DIETT 12

Graph theory

• The incidence matrix C = [cij ] has a row for each node, an a

column for each arc.

• The ordering of the arc is arbitrary.

• Defining a direction (arbitrary if the digraph is undirected, for

example arc ij, consider the direction from Ni to Nj).

• The Laplacian matrix of the weighted digraph is defined as

L = C ∗ CT .


DIETT 13

Example

v2

v3

v1

v4

•C =

1 1 1 0

−1 0 0 1

0 −1 0 −1

0 0 −1 0

e12 e13 e14 e23

• L = C ∗ CT


DIETT 14

Graph theory

• For a graph G and its Laplacian matrix L with eigenvalues

λ0 ≤ λ1 ≤ . . . ≤ λn−1

• L is always positive-semidefinite (∀i, λi ≥ 0).

• The second smallest eigenvalue of graph Laplacians λ1 is called

the algebraic connectivity.


DIETT 15

Example

• L =

3 −1 −1 −1

−1 2 −1 0

−1 −1 2 0

−1 0 0 1

• Eigenvalues: 0, 1, 3, 4


DIETT 16

Networks of Dynamic Agents

),( )(iii xxfx =&

• Consider a system of n dynamic agents Γ = {1, . . . , n}

• Model interaction through connected network (graph) G = (Γ, E)

• let xi be the state of agent i


DIETT 17

Consensus Problem

• Distributed and stationary control policy (protocol )

xi = ui(xi, x(i)) ∀i ∈ Γ.

• Let χ : IRn → IR be the agreement function, usually

– χ(ξ) = ave(ξ1, . . . , ξn)

• Consensus reaching

∥xi − χ(x(0))∥ −→ 0 for t −→ ∞.

• System converges to

χ(x(0))1

.


DIETT 18

A Consensus Algorithm for Averaging

• A simple consensus algorithm to reach an AVERAGING

agreement regarding the state of n integrator agents with

dynamics xi = ui

• can be expressed as an nth-order linear system on a graph:

xi(t) =∑j∈Ni

(xj(t)− xi(t)), xi(0) = zi

• The collective dynamics of the group of agents following protocol

can be written as

x = −Lx

where L = [lij ] is the graph Laplacian of the network.


DIETT 19

Averaging

For a connected network:

• The consensus value is equal to the average of the initial values.

• Irrespective of the initial value of the state of each agent, all

agents reach an asymptotic consensus regarding the value of the

function

f(z) =1

n

∑i

zi


DIETT 20

MIMO system

s

s

s

100

0

01

0

001

�

��

�

�

L

b +

-

y=x u

Input bias

Consensus feedback

output


DIETT 21

Example

v1 v3 v4

v2

x = −Lx

with L =

3 −1 −1 −1

−1 2 −1 0

−1 −1 2 0

−1 0 0 1

• Eigenvalues: 0, 1, 3, 4. The second smallest eigenvalue is −1

• It quantifies the speed of convergence of consensus algorithms.


DIETT 22

Comments on the Linear algorithm

• We recall that the above algorithm is working for linear protocols.

• The linearity can be comprised by many factors: saturations,

disturbances, noisy measurements, etc...

• The convergence is only guaranteed for arithmetic mean.


DIETT 23

Comments on the Topology

• Switching topology: some link can be lost (e.g. a dropping in a

transmission channel)

• If the network is still connected, under certain hypothesis on the

switching time, it is possible to prove convergence.

• The topology is not connected at each instant, but it is on

averaging.


DIETT 24

Applications

• J. Cortes, S. Martinez, and F. Bullo, “Robust rendezvous for

mobile autonomous agents via proximity graphs in arbitrary

dimensions”, IEEE TAC, 2004.

• R. Olfati-Saber, J. S. Shamma, “Consensus filters for sensors

networks and distributed sensor fusion”, CDC-ECC 2005.

• L. Xiao, S. Boyd, “ Fast linear iterations for distributed

averaging”, Systems and Control Letters 2004.

• A. Papachristodoulou, A. Jadbabaie, “Synchronization in

oscillator networks: switching topologies and non-homogeneous

delays”, TAC 2003.


DIETT 25

Literature

• N. A. Lynch, Distributed Algorithms. San Francisco, California:

Morgan Kaufmann Publishers, Inc., 1996.

• L. Moreau, “Stability of multi-agent systems with

time-dependent communication links”, TAC 2005.

• D. Angeli, P.-A. Bliman, “Extension of a result by Moreau on

stability of leaderless multi-agent systems”, CDC 2005.

Surveys

• W. Ren, R. Beard, E. M. Atkins, “Information consensus in

multi-vehicle cooperative control”, IEEE Control Systems

Magazine 2007.

• Olfati-Saber, Fax, Murray, “Consensus and Cooperation in

Networked Multi-Agent Systems”, Proc. IEEE 2007.


DIETT 26

Outline

• Introduction

– Applications


– Graph Theory


– References

� Distributed consensus protocols for coordinating buyers



disturbances

• References


DIETT 27

Problem Setting

• Consider a group of buyers (agents or Decision makers, DM )

aiming at coordinating their daily ordering decisions.

• Motivation: sharing fixed transportation costs.

• Coordination requirements for each buyer is expressed in terms of

a minimum threshold l on the number of buyers to coordinate

order with.

• Corresponding to a maximum threshold on the fixed

transportation.


DIETT 28

Problem Setting

• Need of a protocol for information interchange among buyers such

that the number of active buyers is maximized.

• Active buyers = the buyers that eventually place orders, on the

basis of the available information.

• This is a consensus problem.

• The desired distributed protocol should turn out to be a

consensus protocol that allows all the node states to eventually

converge to the exact percentage of active buyers.


DIETT 29

Problem Statement

• Coordination can be achieved in a decentralized setting without

the necessity that each buyer communicates the threshold to a

central Decision Maker.

• The problem is how to reach the maximum coordination without

letting them communicate their threshold to every agents, but

only to the neighbors.

• What to communicate?

• How to find the policies to reach the best equilibrium?

• How to get convergence?


DIETT 30

Problem Solution

• At the beginning of the day each buyer exchanges information

with its neighbors regarding its initial local estimate about the

percentage of active buyers.

• Information propagates in a decentralized setting and converges

to a common decision-value on the estimate within a

pre-specified time interval.

• Once convergence is reached, the current active buyers

synchronize their new decision to give up ordering if the

decision-value is lower than their threshold.

• Consensus on the final number of active buyers is reached

asymptotically and coordination is the best achievable for the

assigned thresholds.


DIETT 31

Final Remarks

• The presented protocol can be seen as an algorithm to find the

best strategies associated to the best Nash equilibrium.

• It guarantees the BEST cooperative solution in a

NONCOOPERATIVE context where each buyers is maximizing

its cost.


DIETT 32

Outline

• Introduction

– Applications


– Graph Theory


– References


� Mechanism Design for Optimal Consensus Problems


disturbances


DIETT 33

Main Problems:

• The consensus protocol answers to the question of determining

the policies that the agents must implement to reach a given

consensus.

• The mechanism design problem answers to the question of which

policy is implemented, and hence which consensus value is

reached, by selfish agents with given individual objectives.


DIETT 34

Networks of Dynamic Agents

),( )(iii xxfx =&

• Consider a system of n dynamic agents Γ = {1, . . . , n}

• Model interaction through connected network (graph) G = (Γ, E)


DIETT 35

• let xi be the state of agent i

• let x(i) collect the states of its neighbors


DIETT 36

Consensus Problem

• Distributed and stationary control policy

xi = ui(xi, x(i)) ∀i ∈ Γ.

• Let χ : IRn → IR be the agreement function, usually

– χ(ξ) = ave(ξ1, . . . , ξn)

– χ(ξ) = min(ξ1, . . . , ξn)

– χ(ξ) = max(ξ1, . . . , ξn)

• Consensus reaching

∥xi − χ(x(0))∥ −→ 0 for t −→ ∞.


DIETT 37

Consensus problem

Problem 1 (Consensus Problem) Consider a network G = (Γ, E) of

dynamic agents with first-order dynamics. For any function χ

determine a (distributed stationary) protocol, whose components have

the feedback form xi = ui(xi, x(i)), that makes the agents

asymptotically reach consensus on χ(x(0)) for any initial state x(0).


DIETT 38

Agreement on any mean of order p

• χ is any mean of order p not only ave/min/max.

• The group decision value is permutation invariant function of the

agents’ initial states: the value is independent of the agents

indexes: χ(ξ1, ξ2, . . . , ξn) = χ(ξσ(1), ξσ(2), . . . , ξσ(n)) (Perm. Inv.)

• The agreement function is confinated between:

mini∈Γ{ξi} ≤ χ(ξ) ≤ maxi∈Γ{ξi}, for all ξ ∈ IRn.


DIETT 39

Structure

mean χ(x) f(y) g(z)

arithmetic∑

i∈Γ1nxi

1ny z

geometric n√∏

i∈Γ xi e1ny log z

harmonic 1∑i∈Γ

nxi

ny

1z

mean of order p p

√∑i∈Γ

1nx

pi

q

√1ny zp

• Structure χ(ξ) = f(∑

i∈Γ g(ξi)),

• f, g : IR → IR with dg(ξi)dξi

= 0 and for all ξi.


DIETT 40

Consensus protocol

• A protocol solving the consensus problem that is stationary and

distributed is such that the system state trajectory enjoys the

property that χ is time-invariant.

• We find a family of protocols guaranteeing this property.

• We prove that such protocols are consensus protocols.


DIETT 41

Time Invariancy of χ(x(t))

• Basic result: stationary means

limt−→∞

∥xi − χ(x(0))∥ = 0 =⇒ χ(x) time-invariant

• (Protocol design rule) For any χ(.), the protocol

ui(xi, x(i)) =

1dg(xi)dxi

∑j∈Ni

ϕ(xj , xi), for all i ∈ Γ

lets the value χ(x(t)) be time-invariant if ϕ(xj , xi) = −ϕ(xi, xj).


DIETT 42

• Consider the linear function ϕ(xj , xi) = α(xj − xi) and the

different means.

• The arithmetic mean is time-invariant under protocol

u(xi, x(i)) = α

∑j∈Ni

(xj − xi)

• the geometric mean under protocol

u(xi, x(i)) = αxi

∑j∈Ni

(xj − xi)

• the harmonic mean under protocol

u(xi, x(i)) = −αx2

i

∑j∈Ni

(xj − xi)

• the mean of order p under protocol

u(xi, x(i)) = α

x1−pi

p

∑j∈Ni

(xj − xi).


DIETT 43

Sufficient conditions for convergence• g(.) strictly increasing

• ϕ(xj , xi) = αϕ(ϑ(xj)− ϑ(xi)),

– α > 0

– ϕ : IR → IR continuous, locally Lipschitz, odd and strictly

increasing,

– ϑ : IR → IR differentiable with dϑ(xi)dxi

locally Lipschitz and

strictly positive.

• For any χ(.), the protocol

ui(xi, x(i)) = α

1dgdxi

∑j∈Ni

ϕ(ϑ(xj)− ϑ(xi)), for all i ∈ Γ.

lets the agents reach consensus on χ(x(0)1 (Liapunov Approach).


DIETT 44

Lyapunov Approach

• define a new variables ηi = g(xi)− g(χ(x(0))).

• η = 0 corresponds to x = χ(x(0))1

• candidate Lyapunov function

V (η) =1

2

∑i∈Γ

η2i

• η = 0 is asymptotically stable equilibrium (in the quotient space

IRn/span{1}).


DIETT 45

Switching Topology

• E finite set of all possible edgesets connecting the agents in Γ.

• The elements E ∈ E define family of dynamical systems

xi = uiE(xi, x(i)) ∀i ∈ Γ,

• We define a switching function and we assume dwell time to

avoid Zeno behaviors [Jadbabaie et al.2003]

• To extend the results and prove convergence we must prove that

V (η) = 12

∑i∈Γ η

2i is a common Lyapunov function

i) positive definite

ii) continuously differentiable and

V (η) ≤ −W (η), ∀η ∈ IRn, E ∈ E with W (η) independent

from E.


DIETT 46

Common Lyapunov Function

• time varying edge costs

cij = α(g(xj)− g(xi))ϕ(ϑ(xj)− ϑ(xi)), ∀(i, j) ∈ E.

• subset Q ⊆ E of edgsets defining (connected) trees (Γ, Q).

• For each Q ∈ Q define continuous and positive definite function

WQ(η) =∑

(i,j)∈Q

cij .

• find minimum spanning tree

W (η) = minQ∈Q

{WQ(η)}

• W (η), continuous, positive definite and

V (η) ≤ −W (η), ∀η ∈ IRn, E ∈ E


DIETT 47

Mechanism design

• The distributed/individual optimality of the considered protocols

allows interpreting our consensus problem as a non cooperative

differential game

• A supervisor entails the agents to reach a consensus by imposing

individual objectives.

• The main topic of the mechanism design is the definition of game

rules or incentive schemes that induce self-interested players to

cooperate and reach Pareto optimal solution.


DIETT 48

Mechanism Design

• Recast as noncooperative differential game

Ji(xi, x(i), ui) = lim

T−→∞

∫ T

0

(F (xi, x

(i)) + ρu2i

)dt

– ρ > 0 weights control energy

– F : IR× IRn → IR nonnegative penalty function that measures

the deviation of xi from neighbors’ states.

• Mechanism design problem: Design F (.) such that the

agents reach consensus.

• Difficult problem: reformulate as Receding Horizon problem.


DIETT 49

No mechanism Design

Consider the case when it is not employed.

• The distributed protocols are generally planned at high level by a

supervisor.

• The supervisor communicates the planned control policies to the

agents that are in charge to implement them at a low level.

• The agents are only said what to do by the supervisor and do not

exhibit any decision capability.

• A drawback in systems with large dimensions and complex

structures (too onerous)

• if agents are not motivated to behave in the desired way, then a

costly monitoring system may be necessary.


DIETT 50

Mechanism Design

• The advantage of the mechanism design is that intelligence as

well as implementation capabilities are distributed.

• Reviewing the asymptotically consensus reaching as a team

objective

• The supervisor decomposes this team objective in n individual

objectives.

• the agents are said what to aim at instead of “what to do”, and

are free to find the best solution to their subtasks.


DIETT 51

Assumptions

• We need to make some assumptions on the information available

to each agent and its computational capabilities.

• At each time instant, the generic agent can observe the state of

its neighbor agents

• but cannot predict their future behavior

• Hence uses a naive expectation of the state of the neighbor agents

in evaluating its individual objective function.

• naive expectation means that the agents choose their optimal

control policy assuming the neighbors’ state variables as

constants.


DIETT 52

Naive Expectation

• The naive expectation is a major assumption, but it is reasonable

when agents:

– cannot keep record of the past behavior of their neighbors

– are so numerous that they can infer little or none on the

group decision value from the punctual observation of their

neighbors’ state

– are organized in a network with a switching topology whose

switching times and pattern are unpredictable.


DIETT 53

Why Naive Expectation?

• Reduces the RH problem to 1-dimension

Ji(xi(tk), x(i)(tk), ui(τ, tk)) = lim

T−→∞

∫ T

tk

(F(xi(τ, tk), x

(i)(τ, tk)) + ρu2i (τ, tk)

)dτ

⇓

Ji(xi(tk), ui(τ, tk)) = limT−→∞

∫ T

tk

(F(xi(τ, tk)) + ρu2

i (τ, tk))dτ.


DIETT 54

Optimality results

• If the supervisor imposes convex penalty functions, rational

agents have a unique optimal protocol.

• We prove this result through the Pontryagin Minimum Principle.


DIETT 55

Main Result:

• If update time interval δ −→ 0 then

i) the penalty function

F(xi(τ, tk)) −→ F (xi, x(i)) = ρ

1dgdxi

∑j∈Ni

(ϑ(xj)− ϑ(xi))

2

ii) the applied receding horizon control law

u⋆iRH

(τ) −→ ui(xi, x(i)) =

1dgdxi

∑j∈Ni

(ϑ(xj)− ϑ(xi)).

• Fact: for any χ and x(0), with the penalty F (xi, x(i)) as above,

selfish agents reach consensus on χ(x(0)).


DIETT 56

Vertical alignement Maneuver for UAVs

• A team of 4 UAVs in longitudinal flight and initially at different

heights.

• Each UAV controls the vertical rate without knowing the relative

position of all UAVs but only of neighbors according to the

communication network topology:

v1 v3 v4

v2


DIETT 57

Arithmetic Mean

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 55

10

15

20

time

heig

ht

F (xi, x(i)) =

(∑j∈Ni

(xj − xi))2

; u(xi, x(i)) =

∑j∈Ni

(xj − xi).


DIETT 58

Geometric Mean

0 0.1 0.2 0.3 0.4 0.5 0.6 0.75

10

15

20

time

heig

ht

F (xi, x(i)) =

(xi

∑j∈Ni

(xj − xi))2

; u(xi, x(i)) = xi

∑j∈Ni

(xj − xi).


DIETT 59

Harmonic Mean

0 0.02 0.04 0.06 0.08 0.1 0.125

10

15

20

time

heig

ht

F (xi, x(i)) =

(x2i

∑j∈Ni

(xj − xi))2

; u(xi, x(i)) = −x2

i

∑j∈Ni

(xj −xi)


DIETT 60

Mean of order 2

0 20 40 60 80 100 120 140 160 180 2005

10

15

20

time

heig

ht

F (xi, x(i)) =

(1

2xi

∑j∈Ni

(xj − xi))2

; u(xi, x(i)) = 1

2xi

∑j∈Ni

(xj −xi)


DIETT 61

Concluding Remarks

• AIM:To make the agents’ states reach consensus on a group

decision value with general structure

• RESULT: the agents can reach consensus with a distributed and

stationary linear or non-linear protocol, provided that the

networks defined by the communication links between agents is

connected though possibly time variant.

• SOLUTION: Consensus is the result of a mechanism design

(game theoretic approach).

• A supervisor imposes individual objectives.

• The agents reach asymptotically consensus as a side effect of the

optimization of their own individual objectives on a local basis.


DIETT 62

Outline

• Introduction

– Applications


– Graph Theory


– References



� Lazy consensus for networks with unknown but bounded

disturbances

• References


DIETT 63

Illustrative example 1/2

1 2 3

x1 x2 x3

x1

x2

x3

= −

1 −1 0

−1 2 −1

0 −1 1

︸︷︷︸

L

x1

x2

x3

,

x1 = x2 − x1

x2 = (x1 − x2) + (x3 − x2)

x3 = x2 − x3

• Find equilibria by imposing x = 0 and obtain

x1 = x2 = x3 =x1(0) + x2(0) + x3(0)

3


DIETT 64

Illustrative example 2/2

0 10 20 30 40 50 600

5

10

15

20

25

t

x i


DIETT 65

UBB disturbances

1 2 3

x1+d21 x2 x3+d23

• neighbors’ measures y(i)

• closed-loop dynamics under protocol ui(xi, y(i))

x1 = (x2 + d12)− x1

x2 = [(x1 + d21)︸︷︷︸y21

−x2] + [(x3 + d23)︸︷︷︸y23

−x2]

x3 = (x2 + d32)− x3

• Find equilibria by imposing x = 0 and obtain

x1 = x2 + d12, 0 = d12 + d21 + d23 + d32, x3 = x2 + d32

• No equilibria unless d12 + d21 + d23 + d32 = 0.


DIETT 66

UBB disturbances

0 10 20 30 40 50 600

5

10

15

20

25

t

x i

• take dij = 1 for all i, j then x = −Lx+ (1, 2, 1)T has a drift


DIETT 67

UBB disturbances

0 10 20 30 40 50 600

5

10

15

20

25

t

x i

• take d12 = d23 = 1, d32 = d21 = −1, then x = −Lx+ (1, 0,−1)T


DIETT 68

ϵ-consensus

• bounded tube of radius ϵ,

T = {x ∈ Rn : |xi − xj | ≤ 2ϵ, ∀ i, j ∈ Γ} .

• given equilibrium x∗ ∈ T

• asymptotically ϵ-consensus

x(t) → x∗ for t → ∞

• for ϵ = 0 we have the standard consensus definition.


DIETT 69

Problem formulation

• find a rule to estimate yij ∈ [yij − ξ, yij + ξ] such that u(xi, y(i))

is ϵ-consensus protocol.

• study dependence of T on

– the topology E,

– disturbances D

• hypercube D = {d : −ξ ≤ dij ≤ ξ, ∀(i, j) ∈ Γ2}


DIETT 70

Lazy rule

j=1 j=2 j=3

10

4

6

8

2

9

6

3

4

2

5

8

3

5

7

cg

ỹ2j

y2j


DIETT 71

Lazy rule

j=1 j=2 j=3

10

4

6

8

2

9

6

3

4

2

5

8

3

5

7

cg

pick

4 4 4

ỹ2j


DIETT 72

Lazy rule

• pick the solution of

y(i) = arg minyij∈[yij−ξ,yij+ξ], j∈Ni

|∑j∈Ni

(yij − xi)|.

• it maintains the sign: for each i ∈ Γ, for each t ≥ 0 either

sign(ui(xi, y(i))) = sign(

∑j∈Ni

(xj − xi)),

or

sign(ui(xi, y(i))) = 0.


DIETT 73

Equilibria

• all equilibria are such that

ui(xi, y(i)) = sign(ui(xi, y

(i))) = 0, for all i

• assume d(t) unknown but constant

• any solution x is an equilibrium if and only if it belongs to

P (d,E) =

{x : −

∑j∈Ni

dij

|Ni|− ξ ≤

∑j∈Ni

xj

|Ni|− xi ≤ −

∑j∈Ni

dij

|Ni|+ ξ, ∀i ∈ Γ

};


DIETT 74

Polyhedron P (0, E)

x1 x2

x3

{π1}


DIETT 75

Equilibria on ∂P (0, E)

j=1 j=2 j=3

10

4

6

8

2

cg

ỹij

-ξ


DIETT 76

Equilibria on ∂P (0, E)

j=1 j=2 j=3

10

4

6

8

2

cg

ỹij ξ


DIETT 77

Generic polyhedron P (D,E)

• For a generic {d(t) ∈ D, t ≥ 0}, define

P (D,E) :=∪d∈D

P (d,E) =

{−2ξ ≤

∑j∈Ni

xj

|Ni|− xi ≤ 2ξ, ∀i ∈ Γ

}.

• ∀x ∈ P (D,E), there exists a {d(t) ∈ D, t ≥ 0} s.t. x is equ.;

• Given a {d(t) ∈ D, t ≥ 0} all its equilibria are in P (D,E).


DIETT 78

Stability

• Thm.: all system trajectories converges to equilibria in P (D,E)

• Proof: consider Lyapunov function V (x) = 12

∑(i,j)∈E(xj − xi)

2.

– V (x) = 0 if and only if x ∈ {π1};– V (x) > 0 for all x ∈ {π1}.– V (x) < 0 for all x ∈ P (D,E). Actually

V (x) =∑

(i,j)∈E

(xj − xi)(uj − ui) = −∑i∈Γ

ui

∑j∈Ni

(xj − xi) =

= −∑i∈Γ

sign(ui) sign

∑j∈Ni

(xj − xi)

|ui|

∣∣∣∣∣∣∑j∈Ni

(xj − xi)

∣∣∣∣∣∣SCUOLA DOTTORATO- PALERMO 2011

DIETT 79

Stability

• assume d bang-bang

• Thm.: all system trajectories converges to {π1}

• extensions to switching topology


DIETT 80

Conclusions

• Despite the literature on consensus is now becoming extensive,

only few approaches have considered a disturbance affecting the

measurements.

• we have assumed an UBB noise in the neighbors’ state feedback

as it requires the least amount of a-priori knowledge on the

disturbance.

• Because of the presence of the disturbances convergence to

equilibria with all equal components is not possible.

• Main contribution: the introduction and solution of the

ϵ-consensus problem, where the states converge in a tube of ray ϵ

asymptotically or in finite time.


DIETT 81

Outline

� Introduction

– Applications


– Graph Theory


– References




disturbances

� References


DIETT 82

References

• D. Bauso, L. Giarre, R. Pesenti, Consensus for networks with

unknown but bounded disturbances. SIAM JOURNAL ON

CONTROL AND OPTIMIZATION, 48(3); 1756-1770,2009,

• D. Bauso, L. Giarre, R. Pesenti. Distributed Consensus in

Noncooperative Inventory Games. EUROPEAN JOURNAL OF

OPERATIONAL RESEARCH; 192(3), 866-878, 2009;

• D. Bauso, L. Giarre, R. Pesenti. Consensus in Noncooperative

Dynamic Games: a Multi-Retailer Inventory Application. IEEE

TRANSACTIONS ON AUTOMATIC CONTROL, 53(4),

998-1003, 2008.

• D. Bauso, L. Giarre, R. Pesenti. Nonlinear protocols for optimal

distributed consensus in networks of dynamic agents. Systems

and Control Letters, 915-928, 55(11),2006.


DIETT 83

Conference papers

• D. Bauso, L. Giarre, R. Pesenti. Quantized Dissensus in

switching networks with nodes death and duplication, NecSys,

Venice, 2009.

• D. Bauso, L. Giarre, R. Pesenti. “Dissensus, Death and

Division”, IEEE ACC, 2307-2312, 2009.

• D. Bauso, L. Giarre, R. Pesenti. “Challenging aspects in

Consensus protocols for Networks”, Proceedings of the 3rd IEEE

ISCCSP, 2008.

• D. Bauso, L. Giarre, R. Pesenti. “Noncooperative Dynamic

games for Inventory Applications: a consensus approach ”, IEEE

CDC, p. 4819-4824,2008.


DIETT 84

Conference papers

• D. Bauso, L. Giarre, R. Pesenti. “Lazy consensus for network

with unknown but bounded noise”, IEEE CDC, new orleans,

december 12-14,2007, p. 2284-2290, 2007.

• D. Bauso, L. Giarre, R. Pesenti. “Distributed consensus for

switched networks with unknown but bounded noise”, 3rd

International workshop on : Networked Control Systems :

Tolerant to faults, 2007.

• D. Bauso, L. Giarre, R. Pesenti, Mechanism Design for Optimal

Consensus Problems, CDC- 2006.

• D. Bauso, L. Giarre, R. Pesenti, Distributed consensus in

Networks of Dynamic Agents, CDC-ECC, 2005.


DIETT 85

• D. Bauso, L. Giarre, R. Pesenti, Existence and Optimality of

Nash Equilibria in Inventory Games, 16th IFAC World Congress,

2005.

• D. Bauso, L. Giarre, R. Pesenti, Multiple UAV Cooperative Path

Planning via Neuro-Dynamic Programming, CDC 2004.

• D. Bauso and L. Giarre and R. Pesenti, Neuro-Dynamic

Programming for Cooperative Inventory Control, ACC 2004.

• D. Bauso, L. Giarre, R. Pesenti, Distributed Consensus Protocols

for Coordinating Buyers, CDC 2003.

• D. Bauso, L. Giarre, R. Pesenti, Attitude Alignment for a Team

of UAVs under decentralized Information Structure, CCA 2003.


L. Giarr e* * DIETT, Universit a di Palermo joint work ... · DIETT 6 Graph theory • It is natural to model information exchange between agents in a cooperative team by directed/undirected

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