Distributed motion coordination of robotic networks Lecture 5 - - PowerPoint PPT Presentation

Distributed motion coordination

• f robotic networks

Lecture 5 – agreement Jorge Cort´ es

Applied Mathematics and Statistics Baskin School of Engineering University of California at Santa Cruz http://www.ams.ucsc.edu/˜jcortes

Summer School on Geometry Mechanics and Control Centro Internacional de Encuentros Matem´ aticos Castro Urdiales, June 25-29, 2007 ,

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Roadmap

Lecture 1: Introduction, examples, and preliminary notions Lecture 2: Models for cooperative robotic networks Lecture 3: Rendezvous Lecture 4: Deployment Lecture 5: Agreement

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Today

1 Agreement – Basic coordination capability 2 Non-deterministic continuous-time dynamical systems –

nonsmooth stability analysis

3 Algebraic graph theory – interplay between graph theory and

linear algebra

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Outline

1 Agreement

Quick graph theory review

2 Design of distributed algorithms for χ-consensus

Implications on allowable interconnection topologies Systematic design of distributed algorithms for χ-consensus Exponential convergence of power-mean consensus algorithms

3 Further results: switching topologies and finite-time convergence 4 Conclusions

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Reaching agreement is critical

Coordination tasks such as self-organization formation pattern distributed estimation parallel processing require individual agents to agree on the identity of a leader jointly synchronize their operation decide specific pattern to form fuse the information gathered

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Agreement

Objective: given χ : Rn → R, agree on value of χ, i.e., for (p1(0), . . . , pn(0)) ∈ Rn, pi(t) − → χ(p1(0), . . . , pn(0)) Directed graph G as communication topology (–fixed, for simplicity) Traditional topic in computer science In cooperative control, consensus for computing e.g., weighted-least squares estimate of noisy signal product of conditional independent probabilities statistical moments of spatially distributed measurements

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Let’s agree!

We want to compute the average age of the people sitting at the table Each of us can only talk to her/his neighbors to the left, and to the right

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Let’s agree!

We want to compute the average age of the people sitting at the table Each of us can only talk to her/his neighbors to the left, and to the right How could we do it? For instance, by message passing I tell my right-neighbor and left-neighbor my age. They also tell me the same information about themselves When I receive the information about the age of my right-neighbor, I pass it to my left-neighbor, and the other way around We keep doing this message passing until everybody knows everybody’s age, and then each one of us can compute the average

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Message passing

Drawbacks of this approach? With more number of people sitting at the table, algorithm takes longer How many messages do I send/receive, and how much memory do I need to keep track of it? How do I know when I have received everybody’s ages? What if somebody sits at the table while we are running our message passing? Or if somebody leaves?

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Let’s agree – averaging law

Here is an alternative I define a variable p, which I initially set up to my own age At each communication round, I send my current p When I receive pright from my right-neighbor, and pleft from my left-neighbor, I recompute my p as follows pnew = 1 2pold + 1 4(pright + pleft)

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Let’s agree – averaging law

Here is an alternative I define a variable p, which I initially set up to my own age At each communication round, I send my current p When I receive pright from my right-neighbor, and pleft from my left-neighbor, I recompute my p as follows pnew = 1 2pold + 1 4(pright + pleft) Amazingly, this algorithm makes p tend to the average age of everybody sitting at the table! Convergence is exponentially fast I don’t need to know how many we are I need a pretty low memory to run the algorithm

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Outline

1 Agreement

Quick graph theory review

2 Design of distributed algorithms for χ-consensus

Implications on allowable interconnection topologies Systematic design of distributed algorithms for χ-consensus Exponential convergence of power-mean consensus algorithms

3 Further results: switching topologies and finite-time convergence 4 Conclusions

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Graph-theoretic notions

G = (V, E, A) weighted digraph, E ∈ P(V × V) and A = (aij) ∈ Rn×n

≥0

Weighted out-degree and in-degree dout(i) =

n

• j=1

aij and din(i) =

n

• j=1

aji G is weight-balanced if each vertex has equal in- and out-degree G topologically balanced if each vertex has the same number of incoming and outgoing edges

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Laplacian matrix – connecting algebra and graph theory

The graph Laplacian of the weighted digraph G is L(G) = Dout(G) − A(G) Lemma (Properties of the Laplacian matrix) The following statements hold:

1 L(G)1n = 0n 2 G is undirected if and only if L(G) is symmetric 3 if G is undirected, then L(G) is positive semidefinite 4 G contains a globally reachable vertex if and only if

rank L(G) = n − 1

5 G is weight-balanced if and only if 1T nL(G) = 0n if and only if

Sym(L(G)) = 1

2(L(G) + L(G)T ) is positive semi-definite.

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Disagreement function

Disagreement function ΦG(P) = 1 2

n

• i,j=1

aij(pj − pi)2 If G weakly connected, ΦG(P) = 0 iff all in agreement If G weight-balanced, ΦG(P) = P T L(G)P If G weight-balanced and weakly connected, λn(Sym(L)) P − Ave(P)12 ≥ ΦG(P) ≥ λ2(Sym(L)) P − Ave(P)12 For G undirected, λ2 is algebraic connectivity

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Stating the consensus problem

Network of agents ˙ pi = ui, i ∈ {1, . . . , n} Continuous χ : V ⊂ Rn → R Continuous u : V ⊂ Rn → Rn asymptotically achieves χ-consensus if, for any P(0) = (p1(0), . . . , pn(0)) ∈ V, any solution of ˙ pi = ui starting at P(0) stays in V and pi(t) − → χ(P(0)), t → +∞ Problem Given weighted digraph G, design u u is distributed over G, and u asymptotically achieves χ-consensus

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Average consensus via gradient descent

Disagreement function ΦG(P) = 1 2

• (i,j)∈E

(pj − pi)2 Gradient descent of ΦG ˙ pi = −∂ΦG ∂pi =

• j∈NG(i)

(pj − pi) With G connected, LaSalle implies consensus is reached. To what?

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Average consensus via gradient descent

Disagreement function ΦG(P) = 1 2

• (i,j)∈E

(pj − pi)2 Gradient descent of ΦG ˙ pi = −∂ΦG ∂pi =

• j∈NG(i)

(pj − pi) With G connected, LaSalle implies consensus is reached. To what? Note that χave = 1

n

n

i=1 pi is conserved along trajectories

L−∇Φg(χave) =

n

• i=1
• j∈NG(i)

(pj − pi) = 0 Asymptotic consensus + conservation of χave ⇒ Gradient flow achieves average-consensus

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Necessary and sufficient conditions for χ-consensus

Is this only way (consensus + conservation of χ) to solve χ-consensus? Let χ : Rn → R be continuous and surjective Let u : Rn → Rn be continuous with trajectories of ˙ pi = ui bounded Theorem u asymptotically achieves χ-consensus if and only if

1 trajectories converge to diag(Rn) 2 χ is constant along trajectories, and 3 χ(p, . . . , p) = p for all p ∈ R

Result can be stated to account for weaker continuity on χ and u

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Design of distributed algorithms for χ-consensus

From a design perspective, result tells us how to synthesize algorithms for χ-consensus

1 steer the system to agreement while, at the same time, 2 conserve the value of χ

Within the class of (time-independent) feedback laws, there are no more ways to solve the problem Additional constraint for us: do 1 and 2 with only local information

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Example: weighted power means – I

For w ∈ Rn

>0, n i=1 wi = 1 and r ∈ R,

χw,r(p1, . . . , pn) =

• n
• i=1

wipr

i

1

r ,

pi > 0, χw,r(p1, . . . , pn) = 0, otherw. For w = ( 1

n, . . . , 1 n), we just use χr

χ−1 Harmonic Mean

• 1

p1 + · · · + 1 pn

−1 χ0 Geometric Mean

n

√p1 · · · pn χ1 Arithmetic Mean or Average

1 n(p1 + · · · + pn)

χ2 Root-Mean-Square

• p2

1 + · · · + p2 n

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Example: weighted power means – II

Coordination algorithm uw,r over weighted digraph G = (V, E, A) (uw,r)i(p1, . . . , pn) = 1 wi p1−r

i n

• j=1

aij(pj − pi) is distributed over G Does uw,r achieve χw,r-consensus?

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Example: weighted power means – II

Coordination algorithm uw,r over weighted digraph G = (V, E, A) (uw,r)i(p1, . . . , pn) = 1 wi p1−r

i n

• j=1

aij(pj − pi) is distributed over G Does uw,r achieve χw,r-consensus?

1 preserves χw,r if and only if G is weight-balanced

Luw,rχw,r = grad χw,r · uw,r = 1T L

2 for equilibria to be in agreement, G must be weakly connected

Convergence to diag(Rn) established with LaSalle via V = χr+1

w,r+1

Luw,rχr+1

w,r+1 = −(r + 1)P T L(G)P ≤ 0

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Questions from power mean example

1 How do digraphs that are both weight-balanced and weakly

connected look like?

2 Is systematic design of distributed algorithms possible for

general χ?

3 Is convergence of uw,r exponential?

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Outline

1 Agreement

Quick graph theory review

2 Design of distributed algorithms for χ-consensus

Implications on allowable interconnection topologies Systematic design of distributed algorithms for χ-consensus Exponential convergence of power-mean consensus algorithms

3 Further results: switching topologies and finite-time convergence 4 Conclusions

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Weight-balanced and weakly connected digraphs

Theorem (Euler, 1736) Given G weakly connected, G is topologically balanced iff there exists a cycle that visits all edges exactly once Theorem (Hooi-Tong, 70) Given G weakly connected, G is weight-balanced (with integer-valued weights) iff every edge of G lies on at least a cycle

1 1 1 1 2 1 1 1 1 1 1 1 1 2

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Weight-balanced and weakly connected digraphs

Theorem (Euler, 1736) Given G weakly connected, G is topologically balanced iff there exists a cycle that visits all edges exactly once Theorem Given G weakly connected, G is weight-balanced (with arbitrary weights) iff every edge of G lies on at least a cycle

1 1 1 1 2 1 1 1 1 1 1 1 1 2

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Systhematic design of distributed algorithms

Given χ : V ⊂ Rn → R be C1, χ(p, . . . , p) = p for all p, and all partial derivatives { ∂χ

∂p1 , . . . , ∂χ ∂pn } have the same constant sign on V, if

G weakly connected and weight-balanced Exists f : V ⊂ Rn → R such that f · grad χ is distributed over G u : V ⊂ Rn → Rn defined by ui = 1

• f ∂χ

∂pi

• n
• j=1

aij(pj − pi), i ∈ {1, . . . , n} is essentially locally bounded and makes V is strongly invariant Then, u distributed over G & asymptotically achieves χ-consensus

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Exponential convergence of χw,r-consensus algorithms

Theorem Coordination algorithm uw,r : Rn

>0 → Rn over weakly connected,

weight-balanced digraph G asymptotically achieves weighted power mean-consensus with exponential convergence rate rate ≥ c λ2(Sym(L(G))) n max{w1, . . . , wn}, with c =      max{p1(0), . . . , pn(0)}1−r, if r > 1 1, if r = 1 min{p1(0), . . . , pn(0)}1−r, if r < 1

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Proof sketch

Take V =

χr+1

w,r+1(P )−χr+1 w,r(P )

r+1

as Lyapunov function Luw,rV = −P(t)T L(G)P(t) ≤ −λ2(Sym(L(G))) P(t) − χ1(P(t))12

2

To prove exponential convergence, we need a

relationship between

V (P(t)) and P(t) − χ1(P(t))12

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Proof sketch

Take V =

χr+1

w,r+1(P )−χr+1 w,r(P )

r+1

as Lyapunov function Luw,rV = −P(t)T L(G)P(t) ≤ −λ2(Sym(L(G))) P(t) − χ1(P(t))12

2

≤ −λ2(Sym(L(G))) n max{wi} χ2

w,2

• P(t) − χw,1(P(t))1
• To prove exponential convergence, we need a

relationship between

V (P(t)) and P(t) − χ1(P(t))12

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Proof sketch

Take V =

χr+1

w,r+1(P )−χr+1 w,r(P )

r+1

as Lyapunov function Luw,rV = −P(t)T L(G)P(t) ≤ −λ2(Sym(L(G))) P(t) − χ1(P(t))12

2

≤ −λ2(Sym(L(G))) n max{wi} χ2

w,2

• P(t) − χw,1(P(t))1
• To prove exponential convergence, we need a

relationship between

V (P(t)) and P(t) − χ1(P(t))12 Luw,rV ≤ −2cλ2(Sym(L(G))) n max{wi} V (t) ⇒ V (t) ≤ V (0)e−2 c λ2(Sym(L(G)))

n max{wi} t

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Some simulations

Plot of network evolution

0.1 0.2 0.3 0.4 0.5Time 2 4 6 8 10 0.2 0.4 0.6 0.8 1 Time 2 4 6 8 10

Harmonic mean Geometric mean

0.5 1 1.5 2 2.5 3 Time 2 4 6 8 10 2 4 6 8 10Time 2 4 6 8 10

Average mean Two-norm mean

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Some simulations

Log plot of disagreement

0.1 0.2 0.3 0.4 0.5 0.0001 0.001 0.01 0.1 1 10 0.2 0.4 0.6 0.8 1 0.001 0.01 0.1 1

Harmonic mean Geometric mean

0.5 1 1.5 2 2.5 3 0.01 0.1 1 2 4 6 8 10 0.05 0.1 0.5 1 5

Average mean Two-norm mean

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Outline

1 Agreement

Quick graph theory review

2 Design of distributed algorithms for χ-consensus

Implications on allowable interconnection topologies Systematic design of distributed algorithms for χ-consensus Exponential convergence of power-mean consensus algorithms

3 Further results: switching topologies and finite-time convergence 4 Conclusions

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Further results: switching interconnection topologies

For Γ = {G1, . . . , Gm} finite collection of weakly connected, weight-balanced digraphs, switching signal σ : R≥0 → {1, . . . , m} establishes network graph Gσ(t) ∈ Γ at time t ∈ R≥0 ˙ pi(t) = ui(Gσ(t)), i ∈ {1, . . . , n}

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Further results: switching interconnection topologies

For Γ = {G1, . . . , Gm} finite collection of weakly connected, weight-balanced digraphs, switching signal σ : R≥0 → {1, . . . , m} establishes network graph Gσ(t) ∈ Γ at time t ∈ R≥0 ˙ pi(t) = ui(Gσ(t)), i ∈ {1, . . . , n} Theorem Let χ : V ⊂ Rn → R be continuously differentiable and surjective. Let Γ = {G1, . . . , Gm} be a finite collection of weakly connected, weight-balanced digraphs. Assume

1 grad χ is out-distributed over Gk, for k ∈ {1, . . . , m}; 2 { ∂χ ∂p1 , . . . , ∂χ ∂pn } have same constant sign on V; 3 coordination algorithm u(Gk) : V ⊂ Rn → Rn associated with Gk is

essentially locally bounded and such that V is strongly invariant. Then, switching system asymptotically achieves χ-consensus for any signal σ : R≥0 → {1, . . . , m}

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Further results: finite-time consensus

Previous algorithms achieve consensus exponentially fast, but asymptotically What if we want/need to achieve consensus in finite time? E.g., agree on the value of a variable

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Motivation from gradient dynamical systems

Gradient descent flow of V : R → R, V (x) = x2/2 ˙ x = − grad V (x) = −x has

• minima of V as stable equilibria,
• trajectories converge (exponentially fast) to critical points

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Motivation from gradient dynamical systems

Gradient descent flow of V : R → R, V (x) = x2/2 ˙ x = − grad V (x) = −x has

• minima of V as stable equilibria,
• trajectories converge (exponentially fast) to critical points

However, they never get there! Trajectories of ˙ x = − grad V (x) | grad V (x)| = − sgn(grad V (x)) = − sgn(x) get to critical points of V in finite time Is this always the case?

1 2 3 4 5

• 2
• 1.5
• 1
• 0.5

0.5 1

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Finite-time convergence

Flow of locally Lipschitz vector field cannot achieve task in finite time! For finite-time convergence, we must go by non-autonomous flows (e.g., Coron 95) continuous (non-Lipschitz) flows (Bhat& Bernstein 00) discontinuous flows (e.g., Ryan 91 for particular planar systems) Our approach study of discontinuous flows via nonsmooth stability analysis

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Nonsmooth LaSalle Invariance Principle

(1st-order) Evolution of V along Filippov solution t → V (x(t)) is differentiable a.e., d dtV (x(t)) ∈ LXV (x(t)) = {a ∈ R | ∃v ∈ K[X](x) s.t. ζ · v = a , ∀ζ ∈ ∂V (x)}

• set-valued Lie derivative

Theorem (LaSalle Invariance Principle) For S compact and strongly invariant with max LXV(x) ≤ 0 Any Filippov solution starting in S converges to largest weakly invariant set contained in S ∩ ZX,V = S ∩

• x | 0 ∈

LXV(x)

• E.g., nonsmooth gradient flow ˙

x = − Ln[∂V ](x) converges to critical set

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Finite-time convergence

First-order information

Theorem LaSalle’s theses + ∃ neighborhood U of ZX,V ∩ S in S with max LXV < −ǫ < 0 a.e. on U \ (ZX,V ∩ S). Then any Filippov solution starting in S attains ZX,V ∩ S in finite time

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Finite-time convergence

First-order information

Theorem LaSalle’s theses + ∃ neighborhood U of ZX,V ∩ S in S with max LXV < −ǫ < 0 a.e. on U \ (ZX,V ∩ S). Then any Filippov solution starting in S attains ZX,V ∩ S in finite time E.g., nonsmooth flow ˙ x = − sgn(x) with V (x) = x2/2 has Differential inclusion: K[− sgn(x)] =   

1, x < 0, [−1, 1], x = 0, −1, x > 0,

Set-valued Lie derivative: L− sgn(x)V =   

x, x < 0, 0, x = 0, −x x > 0,

= −|x| By LaSalle, solutions starting in S converge to S ∩ Z− sgn(x),V = {0} However, above theorem does not help to establish finiteness

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Finite-time convergence

Second-order information

Theorem LaSalle’s theses +

1 x ∈ Rd →

LXV (x) is single-valued, Lipschitz and regular d2 dt2 (V (x(t))) exists a.e. and d2 dt2 (V (x(t))) ∈ LX( LXV )(x(t)) a.e.

2 ∃U of ZX,V ∩ S in S with max

LX( LXV ) > ǫ > 0 a.e. on U \ (ZX,V ∩ S) Then any Filippov solution starting in S attains ZX,V ∩ S in finite time

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Finite-time convergence

Second-order information

Theorem LaSalle’s theses +

1 x ∈ Rd →

LXV (x) is single-valued, Lipschitz and regular d2 dt2 (V (x(t))) exists a.e. and d2 dt2 (V (x(t))) ∈ LX( LXV )(x(t)) a.e.

2 ∃U of ZX,V ∩ S in S with max

LX( LXV ) > ǫ > 0 a.e. on U \ (ZX,V ∩ S) Then any Filippov solution starting in S attains ZX,V ∩ S in finite time E.g., nonsmooth flow ˙ x = − sgn(x) has L− sgn(x)V = −|x| and ∂(−|x|) =   

1, x < 0, 0, x = 0, −1 x > 0,

L− sgn(x)(L− sgn(x)V ) =   

1, x < 0, 0, x = 0, 1 x > 0.

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Consensus in finite time

For G undirected, grad ΦG = LP and Hess(ΦG) = L. Right assumptions,

L symmetric, positive-semidefinite 0 is eigenvalue of L with eigenvector (1, . . . , 1) Graph G is connected if and only if rank(L) = n − 1

Norm-gradient descent ˙ pi =

• (i,j)∈E(pj − pi)

LP2 Sign-gradient descent ˙ pi = sgn

(i,j)∈E

(pj − pi)

• Norm-gradient flow achieves average consensus in finite time — but

not distributed Sign-gradient flow achieves average max-min consensus in finite time

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Max-consensus and min-consensus

Define sgn+, sgn− : R → R by sgn+(x) =

• 0,

x ≤ 0, 1, x > 0, sgn−(x) =

• 0,

x ≥ 0, −1, x < 0. Theorem: For G connected, ˙ pi = sgn+ n

• j=1

aij(pj − pi)

• ,

˙ pi = sgn− n

• j=1

aij(pj − pi)

• achieve max-consensus and min-consensus in finite time, respectively

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Summary and conclusions

Agreement

1 Analysis and design of distributed algorithms for consensus on

arbitrary functions

2 Studied structure of allowable interconnection topologies 3 Established exponential rate of convergence of weighted

power-mean consensus algorithms Technical tools

1 Dynamical systems and stability analysis 2 Algebraic graph theory 3 Analytic number theory

Applications to synthesis of cooperative strategies for distributed estimation and fusion problems

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References

Algebraic graph theory:

• C. D. Godsil and G. F. Royle. Algebraic Graph Theory, volume 207 of

Graduate Texts in Mathematics. Springer Verlag, New York, 2001

Consensus algorithms:

• R. Olfati-Saber and R. M. Murray. Consensus problems in networks of

agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9):1520--1533, 2004

• D. Bauso, L. Giarr´

e, and R. Pesenti. Nonlinear protocols for optimal distributed consensus in networks of dynamic agents. Systems & Control Letters, 55(11):918--928, 2006

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As promised: objective for the end of the course

Emerging motion coordination discipline Network modeling, algorithm design and validation Network modeling network, ctrl+comm algorithm, task, complexity Coordination algorithms rendezvous, deployment, consensus Systematic algorithm design

1 geometric structures 2 aggregate objective functions 3 class of (gradient) algorithms 4 distributed information processing 5 non-deterministic dynamical systems 6 invariance principles and stability

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Acknowledgments

Collaborators in all the material presented in the course Francesco Bullo Sonia Mart´ ınez

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Acknowledgments

Collaborators in modeling and in nonconvex problems (respectively) Emilio Frazzoli Anurag Ganguli

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Acknowledgments

Collaborators in modeling and in nonconvex problems (respectively) Emilio Frazzoli Anurag Ganguli Plenty of open problems! If you’re interested, check out http://www.soe.ucsc.edu/~jcortes and/or contact me at jcortes@ucsc.edu

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Differences of weighted power means

Beautiful fact from analytic number theory says exactly what we need Theorem (Gao, 03) For w ∈ Rn

>0, n i=1 wi = 1, r ∈ R, and any P = (p1, . . . , pn) ∈ Rn >0

1 2 min

• 1

min{pi}1−r , 1 max{pi}1−r

• χ2

w,2

• P − χw,1(P)1
• ≤

χr+1

w,r+1(P) − χr+1 w,r (P)

r + 1 ≤ 1 2 max

• 1

min{pi}1−r , 1 max{pi}1−r

• χ2

w,2

• P − χw,1(P)1
• Return