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Consensus-based Source-seeking with a CircularFormation of Agents
Lara Brinon Arranz and Luca Schenato
ISR, Instituto Superior Tecnico, Lisboa & Universita di Padova
GIPSA-lab, Grenoble19th December 2013
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Introduction
About myself:
Sep. 2008: Master degree in Automation and ElectronicsEngineering, 2008, UPM, Spain
Nov. 2011: Ph.D. degree in Automatic Control, University ofGrenoble, France (GIPSA-lab/INRIA)
2011-12: ATER, Grenoble INP, France
Currently: Post-doctoral scholar, Instituto Superior Tecnico, Lisbon,Portugal
Ph.D. thesis
Cooperative control design for a fleet of Autonomous UnderwaterVehicles under communication constraints
- Advisors: Carlos Canudas de Wit and Alexandre Seuret- CONNECT and FeedNetBack projects
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Introduction
About myself:
Sep. 2008: Master degree in Automation and ElectronicsEngineering, 2008, UPM, Spain
Nov. 2011: Ph.D. degree in Automatic Control, University ofGrenoble, France (GIPSA-lab/INRIA)
2011-12: ATER, Grenoble INP, France
Currently: Post-doctoral scholar, Instituto Superior Tecnico, Lisbon,Portugal
Ph.D. thesis
Cooperative control design for a fleet of Autonomous UnderwaterVehicles under communication constraints
- Advisors: Carlos Canudas de Wit and Alexandre Seuret- CONNECT and FeedNetBack projects
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 2 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
FeedNetBack Project
-Networked Control Systems- 7th Framework Programme of the European Commission
- Partners:
Universita di Padova
Universidad de Sevilla
KTH Stockholm
ETH Zurich
INRIA Grenoble
Ifremer, France
Case Study: Autonomous Underwater Vehicles (AUVs)
Source-seeking task: Locating and following the source of ascalar field of interest
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Context and Motivations
Source-seeking problem: To locate and follow the source ofa scalar field of interest (temperature, salinity, pollutant flow).Signal strength σ(z) : R2 → R
Applications: environmental monitoring, rescue operations,pollution sensing, sound source localization
Strategy: Gradient-descent methods[Bachmayer and Leonard 2002]
⇒ the gradient information is usually unknown
⇒ estimation of the gradient by collecting spatiallydistributed measurements
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Context and Motivations
Source-seeking problem: To locate and follow the source ofa scalar field of interest (temperature, salinity, pollutant flow).Signal strength σ(z) : R2 → R
Applications: environmental monitoring, rescue operations,pollution sensing, sound source localization
Strategy: Gradient-descent methods[Bachmayer and Leonard 2002]
⇒ the gradient information is usually unknown
⇒ estimation of the gradient by collecting spatiallydistributed measurements
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 4 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Context and Motivations
Source-seeking problem: To locate and follow the source ofa scalar field of interest (temperature, salinity, pollutant flow).Signal strength σ(z) : R2 → R
Applications: environmental monitoring, rescue operations,pollution sensing, sound source localization
Strategy: Gradient-descent methods[Bachmayer and Leonard 2002]
⇒ the gradient information is usually unknown
⇒ estimation of the gradient by collecting spatiallydistributed measurements
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 4 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Gradient estimation: Single vehicle
Extremum seeking techniques[Cochran and Krstic 2009]
Stochastic extremum seeking[Liu and Krstic 2010]
Stochastic gradient-descent[Atanasov et al. 2012] SOURCE
∇σ(c) c
Main disadvantage
the vehicle may have to travel over large distances
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Gadient estimation: Group of vehicles
Model parameters estimation[Fiorelli et al. 2003, Ogren et al.
2004]
Distributed estimation[Sahyoun et al. 2010, Li and Guo
2012]
Circular formation of AUVs[Moore and Canudas de Wit 2010]
ω0
r1 − c
r2 − cr4 − c
r3 − c
SOURCE
∇σ(c)
Drawbacks
- a priori model of the signal distribution- assumption about the spatial propagation of the signal
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Problem formulation
Signal strength
Unknown positive spatial mapping σ(z) : R2 → R- the source, located at z∗, is the only maximum of the scalar field
Uniformly distributed circularformation of N agents:
ri (k) = ci (k) + DR(φi )e
with radius D > 0 and where- φi = φ0 + i 2π
N
- R(φ) =
[cosφ − sinφsinφ cosφ
]- e = [1 0]T
SOURCE∇σ(c)
r1 − c
r2 − cr3 − c
r4 − c
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Control Objectives
1 Estimating the gradient- approximation of ∇σ(c) = [∇xσ(c),∇yσ(c)]T
- the gradient direction will be the reference velocity for theformation
2 Keeping the circular formation of agents
ci → c ∀i3 Steering the formation towards the source location
c → z∗
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Control Objectives
1 Estimating the gradient- approximation of ∇σ(c) = [∇xσ(c),∇yσ(c)]T
- the gradient direction will be the reference velocity for theformation
2 Keeping the circular formation of agents
ci → c ∀i
3 Steering the formation towards the source location
c → z∗
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 8 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Control Objectives
1 Estimating the gradient- approximation of ∇σ(c) = [∇xσ(c),∇yσ(c)]T
- the gradient direction will be the reference velocity for theformation
2 Keeping the circular formation of agents
ci → c ∀i3 Steering the formation towards the source location
c → z∗
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Contribution
Previous works: [Brinon-Arranz et al. CDC’11], [Moore and Canudas ACC’10]
Gradient estimation: ∇σ(c) ∝ 1N
∑Ni=1 σ(ri )(ri − c)
Centralized Source Seeking control in continuous time:
c =1
N
N∑i=1
σ(ri )(ri − c)
Novel contribution: [Brinon-Arranz and Schenato ECC’13]
Distributed rotation center control for each agent
Discrete time control
Local communication among agents
ci (k + 1) = ci (k) + ui (k), ui (k) = f (ci−1, ci+1)
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Contribution
Previous works: [Brinon-Arranz et al. CDC’11], [Moore and Canudas ACC’10]
Gradient estimation: ∇σ(c) ∝ 1N
∑Ni=1 σ(ri )(ri − c)
Centralized Source Seeking control in continuous time:
c =1
N
N∑i=1
σ(ri )(ri − c)
Novel contribution: [Brinon-Arranz and Schenato ECC’13]
Distributed rotation center control for each agent
Discrete time control
Local communication among agents
ci (k + 1) = ci (k) + ui (k), ui (k) = f (ci−1, ci+1)
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Gradient Approximation
Approximation of the gradient of the signal distribution
- Considering a fleet of N > 2 agentsuniformly distributed along a circularformation of radius D centered at c .
- Each agent is able to collect sig-nal measurements at its position, asσ(ri ). SOURCE
r1 − c
r2 − cr3 − c
r4 − c
Lemma: Gradient Approximation [Brinon-Arranz et al. CDC’11]
1
N
N∑i=1
σ(ri )(ri − c) =D2
2∇σ(c) + o(D2)
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Gradient Approximation
Approximation of the gradient of the signal distribution
- Considering a fleet of N > 2 agentsuniformly distributed along a circularformation of radius D centered at c .
- Each agent is able to collect sig-nal measurements at its position, asσ(ri ). SOURCE
r1 − c
r2 − cr3 − c
r4 − c
σ(r1)(r1 − c)
Lemma: Gradient Approximation [Brinon-Arranz et al. CDC’11]
1
N
N∑i=1
σ(ri )(ri − c) =D2
2∇σ(c) + o(D2)
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Gradient Approximation
Approximation of the gradient of the signal distribution
- Considering a fleet of N > 2 agentsuniformly distributed along a circularformation of radius D centered at c .
- Each agent is able to collect sig-nal measurements at its position, asσ(ri ). SOURCE
r1 − c
r2 − cr3 − c
r4 − c
σ(r2)(r2 − c)
Lemma: Gradient Approximation [Brinon-Arranz et al. CDC’11]
1
N
N∑i=1
σ(ri )(ri − c) =D2
2∇σ(c) + o(D2)
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 10 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Gradient Approximation
Approximation of the gradient of the signal distribution
- Considering a fleet of N > 2 agentsuniformly distributed along a circularformation of radius D centered at c .
- Each agent is able to collect sig-nal measurements at its position, asσ(ri ). SOURCE
r1 − c
r2 − cr3 − c
r4 − c
σ(r3)(r3 − c)
Lemma: Gradient Approximation [Brinon-Arranz et al. CDC’11]
1
N
N∑i=1
σ(ri )(ri − c) =D2
2∇σ(c) + o(D2)
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 10 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Gradient Approximation
Approximation of the gradient of the signal distribution
- Considering a fleet of N > 2 agentsuniformly distributed along a circularformation of radius D centered at c .
- Each agent is able to collect sig-nal measurements at its position, asσ(ri ). SOURCE
r1 − c
r2 − cr3 − c
r4 − c
σ(r4)(r4 − c)
Lemma: Gradient Approximation [Brinon-Arranz et al. CDC’11]
1
N
N∑i=1
σ(ri )(ri − c) =D2
2∇σ(c) + o(D2)
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Gradient Approximation
Approximation of the gradient of the signal distribution
- Considering a fleet of N > 2 agentsuniformly distributed along a circularformation of radius D centered at c .
- Each agent is able to collect sig-nal measurements at its position, asσ(ri ). SOURCE
∇σ(c)
r1 − c
r2 − cr3 − c
r4 − c
Lemma: Gradient Approximation [Brinon-Arranz et al. CDC’11]
1
N
N∑i=1
σ(ri )(ri − c) =D2
2∇σ(c) + o(D2)
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Gradient Approximation
Proof:Multi-variable Taylor series expansion of σ at c :
σ(ri )− σ(c) = ∇σ(c)T (ri − c) + o(D)
By multiplying by the relative vector (ri − c) and summing over i :
1
N
N∑i=1
σ(ri )(ri−c)+σ(c)1
N
N∑i=1
(ri−c) =1
N
N∑i=1
(ri−c)(ri−c)T∇σ(c)+o(D2)
thanks to the uniform distribution then∑N
i=1(ri − c) = 0, and usingtrigonometric properties:
N∑i=1
(ri − c)(ri − c)T = D2N∑i=1
R(φi )eeTR(φi )
T =ND2
2I2
and then previous equation holds.
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Distributed solution
All-to-all communicationThanks to previous Lemma, at each instant k each agent cancompute the gradient estimation
1
N
N∑i=1
σ(ri )(ri (k)− c(k))
Limited communication- Undirected communication graph G = (V ,E )- At each instant k each agent computes its position ri (k), itscenter ci (k) and its estimated gradient vector
fi (k) = σ(ri )(ri (k)− ci (k))
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Algorithm
Distributed source-seeking algorithm
for i = 1, . . . ,N dohi (0) = gi (0) = gi (−1) = ci (0) + σ(ri (0))(ri (0)− ci (0)) initialization
for k = 1, 2, . . . dofor i = 1, . . . ,N dogi (k) = ci (k) + σ(ri (k))(ri (k)− ci (k)) gradient estimationgi (k) = (1− α)gi (k − 1) + αgi (k) low-pass filterhi (k) = hi (k − 1) + gi (k − 1)− gi (k − 2) local estimate of z∗
h(k) = (P ⊗ I2)h(k) consensus
for i = 1, . . . ,N doci (k) = (1− ε)ci (k − 1) + εhi (k)
- P ∈ RN×N is a doubly stochastic matrix consistent with graph G- Separation of time scales is regulated by ε ∈ (0, 1]- Low-pass filter is regulated by α ∈ (0, 1]
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
Distributed Source-seeking
Theorem
Let consider previous algorithm and ‖ci (0)− z∗‖ < r for somearbitrary r > 0, then there exists ε such that for all ε ∈ (0, ε):
limk→∞
ci (k)− cj(k) = 0, ∀i , j
Moreover, all the centers ci converge asymptotically to theneighborhood of the maximum of the signal distribution σ(z)located at z∗, s.t. when k →∞:
‖ci (k)− z∗‖ ≤ β(D), ∀i
limD→0
β(D) = 0
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
Sketch of the proof: singular perturbation model analysis
Distributed source-seeking algorithm
gi (k) = ci (k) + σ(ri (k))(ri (k)− ci (k)) fastgi (k) = (1− α)gi (k − 1) + αgi (k) dynamicshi (k) = hi (k − 1) + gi (k − 1)− gi (k − 2)
h(k) = (P ⊗ I2)h(k)
ci (k) = (1− ε)ci (k − 1) + εhi (k) slow dynamics
Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci
=⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )
limk→∞
hi (k) =1
N
N∑i=1
(ci + σi (ri )(ri − ci )) = h(c)
exponentially fast with rate given by esr(P).
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 15 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
Sketch of the proof: singular perturbation model analysis
Distributed source-seeking algorithm
gi (k) = ci (k) + σ(ri (k))(ri (k)− ci (k)) fastgi (k) = (1− α)gi (k − 1) + αgi (k) dynamicshi (k) = hi (k − 1) + gi (k − 1)− gi (k − 2)
h(k) = (P ⊗ I2)h(k)
ci (k) = (1− ε)ci (k − 1) + εhi (k) slow dynamics
Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.
=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )
limk→∞
hi (k) =1
N
N∑i=1
(ci + σi (ri )(ri − ci )) = h(c)
exponentially fast with rate given by esr(P).
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 15 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
Sketch of the proof: singular perturbation model analysis
Distributed source-seeking algorithm
gi (k) = ci (k) + σ(ri (k))(ri (k)− ci (k)) fastgi (k) = (1− α)gi (k − 1) + αgi (k) dynamicshi (k) = hi (k − 1) + gi (k − 1)− gi (k − 2)
h(k) = (P ⊗ I2)h(k)
ci (k) = (1− ε)ci (k − 1) + εhi (k) slow dynamics
Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci )
=⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )
limk→∞
hi (k) =1
N
N∑i=1
(ci + σi (ri )(ri − ci )) = h(c)
exponentially fast with rate given by esr(P).
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 15 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
Sketch of the proof: singular perturbation model analysis
Distributed source-seeking algorithm
gi (k) = ci (k) + σ(ri (k))(ri (k)− ci (k)) fastgi (k) = (1− α)gi (k − 1) + αgi (k) dynamicshi (k) = hi (k − 1) + gi (k − 1)− gi (k − 2)
h(k) = (P ⊗ I2)h(k)
ci (k) = (1− ε)ci (k − 1) + εhi (k) slow dynamics
Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]
=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )
limk→∞
hi (k) =1
N
N∑i=1
(ci + σi (ri )(ri − ci )) = h(c)
exponentially fast with rate given by esr(P).
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 15 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
Sketch of the proof: singular perturbation model analysis
Distributed source-seeking algorithm
gi (k) = ci (k) + σ(ri (k))(ri (k)− ci (k)) fastgi (k) = (1− α)gi (k − 1) + αgi (k) dynamicshi (k) = hi (k − 1) + gi (k − 1)− gi (k − 2)
h(k) = (P ⊗ I2)h(k)
ci (k) = (1− ε)ci (k − 1) + εhi (k) slow dynamics
Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)
=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )
limk→∞
hi (k) =1
N
N∑i=1
(ci + σi (ri )(ri − ci )) = h(c)
exponentially fast with rate given by esr(P).
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 15 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
Sketch of the proof: singular perturbation model analysis
Distributed source-seeking algorithm
gi (k) = ci (k) + σ(ri (k))(ri (k)− ci (k)) fastgi (k) = (1− α)gi (k − 1) + αgi (k) dynamicshi (k) = hi (k − 1) + gi (k − 1)− gi (k − 2)
h(k) = (P ⊗ I2)h(k)
ci (k) = (1− ε)ci (k − 1) + εhi (k) slow dynamics
Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )
limk→∞
hi (k) =1
N
N∑i=1
(ci + σi (ri )(ri − ci )) = h(c)
exponentially fast with rate given by esr(P).
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 15 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
Sketch of the proof: singular perturbation model analysis
Distributed source-seeking algorithm
gi (k) = ci (k) + σ(ri (k))(ri (k)− ci (k)) fastgi (k) = (1− α)gi (k − 1) + αgi (k) dynamicshi (k) = hi (k − 1) + gi (k − 1)− gi (k − 2)
h(k) = (P ⊗ I2)h(k)
ci (k) = (1− ε)ci (k − 1) + εhi (k) slow dynamics
Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )
limk→∞
hi (k) =1
N
N∑i=1
(ci + σi (ri )(ri − ci )) = h(c)
exponentially fast with rate given by esr(P).GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 15 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
Slow Dynamics:If we insert the fast dynamics hi (k) = h(c) into the slow dynamics we get
ci (k) = (1− ε)ci (k − 1) + εh(c(k − 1))
=⇒ limk→∞
ci (k)− cj(k) = 0
Therefore we restrict our attention to the scenario whereci (k) = c(k),∀i :
h(c) = c +1
N
N∑i=1
σ(ri (c))(ri (c)− c) ≈ c(k) +D2
2∇σ(c)
And thus the dynamics of c are given by
c(k + 1) = (1− ε)c(k) + ε(c(k) + D2
2 ∇σ(c) + o(D2))
= c(k) + εD2
2 ∇σ(c) + εo(D2)
which is the standard gradient-ascent update away from the error o(D2).
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 16 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
Slow Dynamics:If we insert the fast dynamics hi (k) = h(c) into the slow dynamics we get
ci (k) = (1− ε)ci (k − 1) + εh(c(k − 1))
=⇒ limk→∞
ci (k)− cj(k) = 0
Therefore we restrict our attention to the scenario whereci (k) = c(k),∀i :
h(c) = c +1
N
N∑i=1
σ(ri (c))(ri (c)− c) ≈ c(k) +D2
2∇σ(c)
And thus the dynamics of c are given by
c(k + 1) = (1− ε)c(k) + ε(c(k) + D2
2 ∇σ(c) + o(D2))
= c(k) + εD2
2 ∇σ(c) + εo(D2)
which is the standard gradient-ascent update away from the error o(D2).GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 16 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Theorem
- Fast dynamics (consensus):
limk→∞
hi (k) =1
N
N∑i=1
(ci + σi (ri )(ri − ci )) = h(c)
- Slow dynamics (gradient ascent):
limk→∞
ci (k) = c ⇒ h(c) = c + f (c)
c(k + 1) = c(k) + εD2
2∇σ(c) + εo(D2)
In conclusion:
If ε is sufficiently small, then the separation of time-scale holds andlimk→∞ ci (k) = limk→∞ c(k) which converges to the neigborhoodof z∗.
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Simulation results
Case without noise:The scalar field is a combination of two ellipsis and thus with non convexlevel curves whose maximum is z∗ = [0, 0]T .
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−5
0
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ε=0.5
ε=0.8
ε=0.1
α = 1
−5 0 5 10 15 20 25 30 35 40 45 50−5
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ε = 0.5, α = 1
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Simulation results
Case with noise:The signal measurements are corrupted by zero-mean Gaussian noiseN (0, 0.2)
0 2000 4000 6000 8000 10000 12000−5
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α=0.5α=0.1
α=0.8
ε = 0.5
−5 0 5 10 15 20 25 30 35 40 45 50−5
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ε = 0.5, α = 0.5
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Conclusions and Future works
Conclusions
Gradient approximation via a circular formation of agents
No previous acknowledgement of the signal is assumed.
Distributed source-seeking algorithm based on DistributedGradient Descent Consensus
Two tunable parameters ε, α: tradeoff rate of convergence,robustness to noisy measurements and formation stability.
Detailed analysis of the proof and asynchronouscommunication in a new paper submitted to TNCS.
Perspectives
Extension to the 3-dimensional case
Control of the radius Di (k)
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Conclusions and Future works
Conclusions
Gradient approximation via a circular formation of agents
No previous acknowledgement of the signal is assumed.
Distributed source-seeking algorithm based on DistributedGradient Descent Consensus
Two tunable parameters ε, α: tradeoff rate of convergence,robustness to noisy measurements and formation stability.
Detailed analysis of the proof and asynchronouscommunication in a new paper submitted to TNCS.
Perspectives
Extension to the 3-dimensional case
Control of the radius Di (k)
GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 20 / 23
Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
MORPH project (http://morph-project.eu/)
-Marine robotic system of self-organizing, logically linkedphysical nodes- 7th Framework Programme of the European Commission- Partners from Germany, Italy, Portugal, France and Spain
Underwater Robotic System
-Combination of different mobilerobot-modules with distinct andcomplementary resources.- To develop methods to map theunderwater environment withgreat accuracy in complexsituations
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
IST Lisboa
Range-Only Formation Control (ROF)
- Multiple vehicle cooperation is required- Low communication throughput (bandwidth constraints)
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Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions
Thank you for your attention
e-mail: [email protected]: https://sites.google.com/site/lbrinonarranz/
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