Reconfigurable cooperative control for extremum seeking Arthur Kahn (ONERA), Julien Marzat (ONERA), Hélène Piet Lahanier (ONERA), Michel Kieffer (L2S) Aerial Robotics Workshop, ENSAM Paris, December 8 2015
Problem statement
Local approach
Global Approach
Context Source location and surveillance missions ◮
Forest fire source location
◮
Chemical or gas leaks
◮
Surveillance of large areas or search and rescue
Interest for multi-vehicle systems (MVS) ◮
Mission repartition
◮
Robustness to faults or agent loss
Conclusions and perspectives
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Field maximization with a MVS
Goal → Find the global maximum of an initially unknown spatial field Means → Multi-vehicle system (MVS) → Each vehicle measures the field value at its position Constraints → Accurately locate field maximum → Take into account vehicle dynamics → Avoid collisions between vehicles → Limit the number of measurements 3/34
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Outline
Problem statement Local approach Global Approach Conclusions and perspectives
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Reconfigurable cooperative control for extremum seeking
Global Approach
Conclusions and perspectives
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Assumptions Consider some unknown, continuous, and time-invariant scalar field φ : x ∈ D ⊂ R2 → φ (x) ∈ R to be maximized using N identical mobile agents with dynamics M¨xi + C (xi , x˙ i ) x˙ i = ui , and measurement equation at xi y (xi ) = φ (xi ) + wi (ηi ), wi measurement noise and ηi the i-th sensor state
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ηi = 0 nominal sensor
◮
ηi = 1 faulty sensor (bias or modified variance)
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Problem statement
◮
N identical vehicles with lossless synchronized communication
◮
Communication radius R defines agent i neighbourhood Ni (t) = {j | kxi (t) − xj (t)k 6 R} .
◮
Available information at time tk for agent i Si (tk ) =
k [
ℓ=0 ◮
{[yj (tℓ ), xj (tℓ )] | j ∈ Ni (tℓ ) ∩ M (tℓ )} .
Define a strategy to find efficiently (time, measurements) xM = arg max {φ (x)} x∈D
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Topics addressed
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Reconfigurable cooperative control for extremum seeking
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Problem statement
Local approach
Proposed solutions
1 Define iteratively vehicle sampling positions 2 Model computation from measurements 3 Move vehicles with collision avoidance
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Reconfigurable cooperative control for extremum seeking
Global Approach
Conclusions and perspectives
Problem statement
Local approach
Proposed solutions
1 Define iteratively vehicle sampling positions 2 Model computation from measurements 3 Move vehicles with collision avoidance
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Reconfigurable cooperative control for extremum seeking
Global Approach
Conclusions and perspectives
Problem statement
Local approach
Proposed solutions
1 Define iteratively vehicle sampling positions 2 Model computation from measurements 3 Move vehicles with collision avoidance
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Reconfigurable cooperative control for extremum seeking
Global Approach
Conclusions and perspectives
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Proposed solutions
Local approach 1 Define iteratively vehicle sampling positions
Optimal sensor placement
2 Model computation from measurements 3 Move vehicles with collision avoidance
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Proposed solutions
Local approach 1 Define iteratively vehicle sampling positions
Optimal sensor placement
2 Model computation from measurements
Local linear model
3 Move vehicles with collision avoidance
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Proposed solutions
Local approach
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1 Define iteratively vehicle sampling positions
Optimal sensor placement
2 Model computation from measurements
Local linear model
3 Move vehicles with collision avoidance
Formation control
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Proposed solutions
Global approach 1 Define iteratively vehicle sampling positions
Constrained sampling criterion
2 Model computation from measurements 3 Move vehicles with collision avoidance
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Proposed solutions
Global approach 1 Define iteratively vehicle sampling positions
Constrained sampling criterion
2 Model computation from measurements
Kriging model of the field
3 Move vehicles with collision avoidance
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Proposed solutions
Global approach
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1 Define iteratively vehicle sampling positions
Constrained sampling criterion
2 Model computation from measurements
Kriging model of the field
3 Move vehicles with collision avoidance
Spread the vehicles in the area
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Section 2 Local approach
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Reconfigurable cooperative control for extremum seeking
Conclusions and perspectives
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Local approach "Gradient climbing" algorithm (Ögren 2004, Cortes 2009) 1. Vehicles are kept in a close formation 2. Vehicles measure the field value at their positions and broadcast 3. Cooperative gradient estimation from measurements 4. Computation of formation motion along gradient direction Contributions
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Cooperative weighted least-square estimation with local model
◮
Outlier detection: adaptive threshold related to cooperative estimation model
◮
Optimal sensor placement with faulty sensors (Fisher information matrix)
◮
Fleet control: vehicle formation motion and reconfiguration
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Field modeling Locally, spatial field φ can be written T T 1 xik . xik ∇φ b x−b xik ∇2 φ (χ i ) x − b xik + x − b xik + φi (x) = φ b 2 k φ b xi k Parameter vector α i = to be estimated xik ∇φ b Local linear model T xik ∇φ b xik + x − b xik , φ i (x) = φ b
with modeling error ei (x) = φi (x) − φ i (x)
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Weighted least-squares Measurement of vehicle j T k k yj (tk ) = 1 xj (tk ) − b α xi i + ei (xj (tk )) + nj (tk ) .
Vehicle i collects all measurements from Ni (tk ) yi,k = Ri,k α ki + ni,k + ei,k
Weight matrix W i,k = 2 2 k k −||xN (tk )−b xi ||2 −||x1 (tk )−b xi ||2 −2 , . . . , diag ση−2 exp σ exp kw kw ηN (tk ) 1 (tk ) 10000
k =5 w
9000
kw = 10 kw = 50
8000
k = 100
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Reconfigurable cooperative control for extremum seeking
k = 200 w
kw = 500
6000
Weight
−1 T T b ki = Ri,k Wi,k Ri,k α Ri,k Wi,k yi,k
w
7000
kw = 1000
5000 4000 3000 2000 1000 0 0
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Distance to the estimate
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Problem statement
Local approach
Global Approach
Conclusions and perspectives
Model-based fault detection scheme
System Mission Communication
Faults
Control
Faults
Actuators Sensors
act on measure
Environment
Faults
Fault detection and isolation System values Measure, model, estimation ...
Problem characteristics
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Local model shared by vehicles
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Spatially-varying modeling error
Reconfigurable cooperative control for extremum seeking
Residual construction
Decision
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Fault detection Faulty sensor of vehicle i : yi = φ (xi ) + wi (ηi ) + d Detection residual ri = φˆi (xi ) − yi Adaptive threshold for residual analysis r |ri | < kFDI
T σ02 1 + hi hT − 2h [i] + h i i Ui hi i
Takes into account measurement noise, sensor locations and modeling error For fault isolation:
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For each vehicle, bank of N residuals rij excluding the j-th measurement
◮
Consensus between vehicles to identify the faulty sensors
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Fault detection results d =3
d =5
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Faulse detection rate
Reconfigurable cooperative control for extremum seeking
True detection rate
0.9
Threshold
True detection rate
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0.2
Faulse detection rate
1
N = 15
0
0
Faulse detection rate
0
0
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0.6
Faulse detection rate
0.8
1
Threshold
0
True detection rate
9
Threshold
True detection rate
0.9
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N =7
10
1
9
Threshold
10
1 0.9
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Optimal sensor placement
Find sensor locations (and associated formation shapes) that ◮
minimise estimate variance and modeling error influence
◮
take into account different sensor variances (faults)
Minimise a function of estimation error covariance matrix T −1 b k+1 = R Σ i,k+1 Wi,k+1 Ri,k+1 α i
2 , ∀{i, j}, j > i under collision avoidance constraint kxi − xj k22 > Rsafety
Several optimal design criteria (Walter & Pronzato 1987)
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Optimal sensor placement
T-optimal solution T (x1 (tk+1 ) . . . xN (tk+1 )) = arg max tr Ri,k+1 Wi,k+1 Ri,k+1 (x1 ,...,xN )
2 s.t. kxi − xj k22 > Rsafety , ∀{i, j}, j > i.
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Optimal sensor placement
T-optimal solution T (x1 (tk+1 ) . . . xN (tk+1 )) = arg max tr Ri,k+1 Wi,k+1 Ri,k+1 (x1 ,...,xN )
2 s.t. kxi − xj k22 > Rsafety , ∀{i, j}, j > i.
T 2 Lagrangian L = tr Ri,k+1 Wi,k+1 Ri,k+1 + ∑ µij kxi − xj k22 − Rsafety j>i
Two solutions for µij = 0 (inactive constraints),
2
k+1 k+1 b x (t ) − x
= kw − 1
i k+1 i xi (tk+1 ) = b xi 2 16/34
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Optimal sensor placement
D-optimal solution (x1 (tk+1 ) . . . xN (tk+1 )) = arg max det (x1 ,...,xN )
T Ri,k+1 Wi,k+1 Ri,k+1
2 s.t. kxi − xj k22 > Rsafety , ∀{i, j}, j > i.
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Optimal sensor placement
D-optimal solution (x1 (tk+1 ) . . . xN (tk+1 )) = arg max det (x1 ,...,xN )
T Ri,k+1 Wi,k+1 Ri,k+1
2 s.t. kxi − xj k22 > Rsafety , ∀{i, j}, j > i.
T 2 Lagrangian L = det Ri,k+1 Wi,k+1 Ri,k+1 + ∑ µij kxi − xj k22 − Rsafety j>i
Two solutions for µij = 0 (inactive constraints),
2 2k
w k+1 k+1 b = x (t ) − x
i k+1 i xi (tk+1 ) = b xi 3 2 16/34
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Numerical solutions N = 3 agents, no faulty agent D-optimal
T-optimal 8 8
6
6 4 4 2
y
y
2 0 -2
0 -2
-4 -4 -6 -6
-8
-8 -10
-5
0
5
10
x
√ radius = kw − 1 17/34
Reconfigurable cooperative control for extremum seeking
-10
-8
-6
-4
-2
0
x
radius =
2
r
4
2kw 3
6
8
10
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Numerical solutions N = 5 agents, D-optimal placement 1 faulty agent
No fault 8 10 6
8 6
4
4 2
y
y
2 0
0 -2
-2
-4 -4
-6 -8
-6
-10 -8 -10
-8
-6
-4
-2
0
2
4
6
8
10
x
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Reconfigurable cooperative control for extremum seeking
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-5
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Problem statement
Local approach
Global Approach
Conclusions and perspectives
Numerical solutions Conclusion on T-optimal and D-optimal sensor placement ◮
All vehicles should be located on a circle with inactive constraints
◮
A faulty agent is placed further from the fleet, due to estimation weight
Sensor placement to minimize modeling error ◮
Be as close as possible to estimation position
Formation characteristics ◮ ◮
T-optimal → concentric circles
D-optimal → compact formation around estimation position with active constraints A. Kahn, J. Marzat, H. Piet-Lahanier, M. Kieffer, Cooperative estimation with outlier detection and fleet
reconfiguration for multi-agent systems, IFAC Workshop on Multi-Vehicule Systems 2015
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Cooperative guidance law
◮
Manage vehicle motions to respect sensor placement
◮
Locate field maximum
A virtual point b xk is used in a two-layer control law ◮
High-level control ◮
◮
Low-level control ◮ ◮
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Move the virtual point to track the field maximum Keep the agents in formation around the virtual point Avoid collisions between vehicles
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
High-level control
Gradient climbing of estimation position b xk
c k c b b xk x . xk+1 = b xk + λ k ∇φ
∇φ b 2
b xk can be proven to converge to maximum for concave fields
Decentralized computation of estimation position is possible with incomplete communication graph J. Marzat, A. Kahn, H. Piet-Lahanier Cooperative guidance of Lego Mindstorms NXT mobile robots, 11th International Conference on Informatics in Control, Automation and Robotics, Vienne Autriche, 2014
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Reconfigurable cooperative control for extremum seeking
Problem statement
Experiment
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Local approach
Global Approach
h
Reconfigurable cooperative control for extremum seeking
Conclusions and perspectives
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Low-level control
Vehicle dynamics M¨xi (t) + C (xi (t), x˙ i (t)) x˙ i (t) = ui (t) Proposed control law (similar to Cheah, 2009) x˙ i (t) ui (t) =M b x¨i (t) + C (xi (t), x˙ i (t))x˙ i (t) − k1 x˙ i (t) − b N T (xi (t) − xj (t)) (xi (t) − xj (t)) + 2k2 ∑ (xi (t) − xj (t)) exp − q j=1 xi (t)) , − k3i (ηi , t)(xi (t) − b 21/34
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Control stability Candidate Lyapunov function V (X(t)) " 1 N V (X(t)) = ∑ (x˙ i (t) − b x˙ (t))T M(x˙ i (t) − b x˙ (t)) 2 i=1
x(t)) + (xi (t) − b x(t))T k3i (xi (t) − b # N (xi (t) − xj (t))T (xi (t) − xj (t) + k2 ∑ exp − q j=1
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Control stability Candidate Lyapunov function V (X(t)) " 1 N V (X(t)) = ∑ (x˙ i (t) − b x˙ (t))T M(x˙ i (t) − b x˙ (t)) 2 i=1
x(t)) + (xi (t) − b x(t))T k3i (xi (t) − b # N (xi (t) − xj (t))T (xi (t) − xj (t) + k2 ∑ exp − q j=1
Speed control term
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Control stability Candidate Lyapunov function V (X(t)) " 1 N V (X(t)) = ∑ (x˙ i (t) − b x˙ (t))T M(x˙ i (t) − b x˙ (t)) 2 i=1
+ (xi (t) − b x(t)) x(t))T k3i (xi (t) − b # N (xi (t) − xj (t))T (xi (t) − xj (t) + k2 ∑ exp − q j=1
Position control term
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Control stability Candidate Lyapunov function V (X(t)) " 1 N V (X(t)) = ∑ (x˙ i (t) − b x˙ (t))T M(x˙ i (t) − b x˙ (t)) 2 i=1
x(t)) + (xi (t) − b x(t))T k3i (xi (t) − b # N (xi (t) − xj (t))T (xi (t) − xj (t) + k2 ∑ exp − q j=1
Collision avoidance term
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Control stability Candidate Lyapunov function V (X(t)) " 1 N V (X(t)) = ∑ (x˙ i (t) − b x˙ (t))T M(x˙ i (t) − b x˙ (t)) 2 i=1
x(t)) + (xi (t) − b x(t))T k3i (xi (t) − b # N (xi (t) − xj (t))T (xi (t) − xj (t) + k2 ∑ exp − q j=1
Control law can be proven to be Lyapunov stable A. Kahn, J. Marzat, H. Piet-Lahanier, M. Kieffer, Cooperative estimation with outlier detection and fleet reconfiguration for multi-agent systems, IFAC Workshop on Multi-Vehicule Systems, Gênes Italie, 2015
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Reconfiguration 10 8 6 4 2
y
Optimal sensor placement → desired formation shape
0 -2 -4 -6 -8 -10 -10
-5
0
x
Faulty agent i → modified control law k3i (ηi = 0) > k3i (ηi = 1)
Faulty agents are "pushed" far from the formation center
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Reconfigurable cooperative control for extremum seeking
5
10
Problem statement
Local approach
Reconfiguration
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Reconfigurable cooperative control for extremum seeking
Global Approach
Conclusions and perspectives
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Local approach: complete loop simulation
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Local approach: complete loop simulation
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Section 3 Global Approach
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Reconfigurable cooperative control for extremum seeking
Conclusions and perspectives
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Kriging Unknown field φ (x) modeled as Y (x) = r(x)T β + Z (x) ◮
r(x) regression vector
◮
β parameter vector
◮
Z (x) zero-mean Gaussian process with covariance C (Z (x1 ), Z (x2 ))
Kriging provides a Gaussian distribution for each x with ◮
a mean value µ(x)
◮
a prediction variance σ 2 (x)
How to choose sampling points ?
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Kriging 4 Real function Sampling position Estimated mean Confidence bound
3
Amplitude
2 1 0 -1 -2 -3 -4 0
1
2
3
4
5
x
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Reconfigurable cooperative control for extremum seeking
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Problem statement
Local approach
Global Approach
Conclusions and perspectives
Kriging 4 Real function Sampling position Estimated mean Confidence bound
3
Amplitude
2 1 0 -1 -2 -3 -4 0
1
2
3
4
5
x
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Reconfigurable cooperative control for extremum seeking
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Problem statement
Local approach
Global Approach
Conclusions and perspectives
Kriging-based existing sampling criteria
Kriging-based sampling criterion for seeking xM = arg maxφ (x) x∈D
For optimizing costly-to-evaluate functions Based on n measurements, choose the n + 1-th
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◮
Kushner, 1962
◮
Expected improvement (Jones, 1998)
◮
Confidence bound (Cox, 1997)
CKushner (x) = P(µ(x) > fmax + ε)
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Kriging-based existing sampling criteria
Kriging-based sampling criterion for seeking xM = arg maxφ (x) x∈D
For optimizing costly-to-evaluate functions Based on n measurements, choose the n + 1-th
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◮
Kushner, 1962
◮
Expected improvement (Jones, 1998)
◮
Confidence bound (Cox, 1997)
CEI (x) = (µ(x)−fmax )Ψ(z)+ σˆ (x)ψ(z)
Reconfigurable cooperative control for extremum seeking
µ(x) − fmax z= σˆ (x)
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Kriging-based existing sampling criteria
Kriging-based sampling criterion for seeking xM = arg maxφ (x) x∈D
For optimizing costly-to-evaluate functions Based on n measurements, choose the n + 1-th
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◮
Kushner, 1962
◮
Expected improvement (Jones, 1998)
◮
Confidence bound (Cox, 1997)
Reconfigurable cooperative control for extremum seeking
Clcb (x) = µ(x) + blcb σˆ (x)
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Existing Kriging-based sampling criteria for MVS
Now looking for all vehicle positions X ◮
Choi, 2008
◮
Xu & Choi, 2011
∑4p=1 λp (t)Ξp (X(t), t) CChoi (X(t)) = ∑4p=1 λp (t)
Ξ1 = µ, Ξ2 = −µ, Ξ3 = σ 2 , Ξ4 = ln(2πσ 2 ) Minimise uncertainy mean on J , grid of interest points.
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◮
More exploration criteria than global optimization criteria
◮
Do not take into account vehicle dynamics explicitely
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Existing Kriging-based sampling criteria for MVS
Now looking for all vehicle positions X ◮
Choi, 2008
◮
Xu & Choi, 2011
1 2 CXu (X(t)) = σ ∑ zj (X(t)) |J | j∈J
Ξ1 = µ, Ξ2 = −µ, Ξ3 = σ 2 , Ξ4 = ln(2πσ 2 ) Minimise uncertainy mean on J , grid of interest points.
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◮
More exploration criteria than global optimization criteria
◮
Do not take into account vehicle dynamics explicitely
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Existing Kriging-based sampling criteria for MVS
Now looking for all vehicle positions X ◮
Choi, 2008
◮
Xu & Choi, 2011
Ξ1 = µ, Ξ2 = −µ, Ξ3 = σ 2 , Ξ4 = ln(2πσ 2 ) Minimise uncertainy mean on J , grid of interest points.
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◮
More exploration criteria than global optimization criteria
◮
Do not take into account vehicle dynamics explicitely
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Proposed criterion
Goals ◮
Locate global maximum
◮
Limit exploration to areas of interest
◮
Take into account vehicle dynamics (k)
Ji (x) = kxi (tk ) − xk2 −
∑ j∈Ni (tk )
αkxj (tk ) − xk2 ,
n o (k) xdi (tk ) = arg min Ji (x) x∈D
i (tk ) s.t. φˆi,k (x) + bσφ ,i,k (x) > fmax
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Proposed criterion
Goals ◮
Locate global maximum
◮
Limit exploration to areas of interest
◮
Take into account vehicle dynamics (k)
Ji (x) = kxi (tk ) − xk2 −
∑ j∈Ni (tk )
αkxj (tk ) − xk2 ,
n o (k) xdi (tk ) = arg min Ji (x) x∈D
i (tk ) s.t. φˆi,k (x) + bσφ ,i,k (x) > fmax
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Proposed criterion
Goals ◮
Locate global maximum
◮
Limit exploration to areas of interest
◮
Take into account vehicle dynamics (k)
Ji (x) = kxi (tk ) − xk2 −
∑ j∈Ni (tk )
αkxj (tk ) − xk2 ,
n o (k) xdi (tk ) = arg min Ji (x) x∈D
i (tk ) s.t. φˆi,k (x) + bσφ ,i,k (x) > fmax
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Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Criterion illustration
Instant 1 30/34
Reconfigurable cooperative control for extremum seeking
Conclusions and perspectives
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Criterion illustration
4 3
Amplitude
2 1 0 -1 -2 -3 -4 0
1
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Instant 2 30/34
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Local approach
Global Approach
Conclusions and perspectives
Criterion illustration
4 3
Amplitude
2 1 0 -1 -2 -3 -4 0
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Instant 3 30/34
Reconfigurable cooperative control for extremum seeking
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Local approach
Global Approach
Conclusions and perspectives
Criterion illustration
4 3
Amplitude
2 1 0 -1 -2 -3 -4 0
1
2
3
4
5
x
Instant 4 30/34
Reconfigurable cooperative control for extremum seeking
6
7
8
9
10
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Criterion illustration
4 3
Amplitude
2 1 0 -1 -2 -3 -4 0
1
2
3
4
5
x
Instant 5 30/34
Reconfigurable cooperative control for extremum seeking
6
7
8
9
10
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Criterion illustration
4 3
Amplitude
2 1 0 -1 -2 -3 -4 0
1
2
3
4
5
x
Instant 6 30/34
Reconfigurable cooperative control for extremum seeking
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Problem statement
Local approach
Global Approach
Conclusions and perspectives
Global approach : full simulation
A. Kahn, J. Marzat, H. Piet-Lahanier, M. Kieffer, Global extremum seeking by Kriging with a multi-agent system, 17th IFAC SYSID, Beijing China, 2015 31/34
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Comparison with reference method 30 Proposed criterion Xu criterion, period=5 Xu criterion, period=20
30 25 20 15 10 5 0 0
200
400
600
800
distance to the maximum (m)
distance to the maximum
35
25 20 15 10 5 0 0
1000
k
Contribution ◮
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Proposed criterion Xu criterion, period=5 Xu criterion, period=20
100 200 300 400 number of measurements
500
Limitations
Quick convergence to a small error with limited measurements
Reconfigurable cooperative control for extremum seeking
◮
True covariance parameters usually unknown
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Conclusions and perspectives Two approaches for maximum location with a multi-vehicle system
Local approach ◮
Cooperative estimation with associated optimal placement
◮
Fault detection and identification
◮
Formation control with reconfiguration
Global approach ◮
◮
Kriging-based criterion for global optimization to limit search area Perspectives ◮ ◮
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Fault diagnosis and reconfiguration with Kriging model Incorporate communication constraints in criterion
Reconfigurable cooperative control for extremum seeking
Problem statement
Local approach
Global Approach
Conclusions and perspectives
Publications ◮ Kahn, A., Marzat, J., Piet Lahanier, H. Formation flying control via elliptical virtual structure, IEEE International Conference on Networking, Sensing and Control, 158-163, Evry, France, (2013) ◮ Piet Lahanier, H., Kahn, A., Marzat, J. Cooperative guidance laws for maneuvering target interception, IFAC Symposium on Automatic Control in Aerospace, 296-301, Würzburg, Germany, (2013) ◮ Marzat, J., Piet Lahanier, H., Kahn, A. Cooperative guidance of Lego Mindstorms NXT mobile robots, 11th International Conference on Informatics in Control, Automation and Robotics, 605-610, Vienne, Austria, (2014) ◮ Bertrand S., Marzat J., Piet-Lahanier H., Kahn A., Rochefort Y., MPC Strategies for Cooperative Guidance of Autonomous Vehicles, Aerospace Lab Journal, 8, 1-18, (2014) ◮ Kahn, A., Marzat, J., Piet Lahanier, H., Kieffer, M. Cooperative estimation with outlier detection and fleet reconfiguration for multi-agent systems, IFAC Workshop on Multi-Vehicule Systems, 11-16, Genova, Italy, (2015) ◮ Kahn, A., Marzat, J., Piet Lahanier, H., Kieffer, M. Global extremum seeking by Kriging with a multi-agent system, 17th IFAC Symposium on System Identification, Bejing, China (2015)
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Reconfigurable cooperative control for extremum seeking
