[IEEE ICASSP 2014 - 2014 IEEE International Conference on Acoustics, Speech and Signal Processing...

A DISTRIBUTED CONSENSUS PLUS INNOVATION PARTICLE FILTER FOR NETWORKSWITH COMMUNICATION CONSTRAINTS

Arash Mohammadi and Amir Asif

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ABSTRACT

( )

Index Terms—

1. INTRODUCTION

2. PROBLEM FORMULATION AND PARTICLE FILTER

x(k)=f(x(k − 1)) + ξ(k),⎡⎢⎣

z(1)(k)

z(N)(k)

⎤⎥⎦

︸︷︷︸z(k)

=

⎡⎢⎣g(1)(x(k))

g(N)(x(k))

⎤⎥⎦

︸︷︷︸g(x(k))

+

⎡⎢⎣ζ(1)(k)

ζ(N)(k)

⎤⎥⎦,

︸︷︷︸ζ(k)

N nxx = [X1, X2, . . . , Xnx ]

T T

z = [z(1)T , . . . ,z(N)T ]T z(l)(k)l 1 ≤ l ≤ N k

{f(·), g(·)} {ξ(·), ζ(·)}

2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP)

U.S. Government work not protected by U.S. copyright 1105

k

P (x(k)|z(1 :k−1)) =∫P (x(k−1)|z(1 :k−1))f(x(k)|x(k−1))dx(k−1)

P (x(k)|z(1 :k)) =P (z(k)|x(k))P (x(k)|z(1 :k−1))

P (z(k)|z(1 :k−1)).

P (x(k)|z(1 : k))Xi(k) (1 ≤ i ≤ Ns)

q(x(0:k)|z(1:k))

Wi(k) =P (Xi(k)|z(1 : k))

q(Xi(0 : k)|z(1 : k))

{Xi(k),Wi(k)}

Xi(k) ∼ q(Xi(k)|Xi(0 :k−1), z(1 :k))

Wi(k) ∝ Wi(k − 1)P (z(k)|Xi(k))P (Xi(k)|Xi(k−1))

q(Xi(k)|Xi(0 :k−1), z(1 :k)).

3. CONSENSUS PLUS INNOVATION BASED DUPF

Nc

Nc(U)Nc < Nc(U)

3.1. Linear Consensus and Innovation Estimation

x(k) = Fx(k−1) + ξ(k)

z(l)(k) = G(l)x(k) + ζ(l)(k) l

x(l)(k + 1) = F∑

i∈ℵ(l)Ulix(i)(k)︸︷︷︸

+B(l) ∑i∈ℵ(l) [G

(i)]T(z(i)(k + 1)−G(i)x(l)(k)

),︸︷︷︸

ℵ(l) l B(l)

U = {Uli}

Fx(l)(k)

Fx(i)(k) i ∈ ℵ(l)

l l

3.2. The CI/DUPF Implementation

Consensus update

l

kl ∑

i∈ℵ(l) x(i)(k) l

k+1∑i∈ℵ(l) x

(i)(k) z(l)(k+1)

Observation update

l

k

x(l)(k) P (l)(k) k

z(l)(k + 1)l

3.2.1. Consensus Update

Step 1. Local UPF l (2nx+1)

S = {W(l)i ,χ

(l)i (k)}2nx

i=0

χ(l)i (k) = x(l)(k)±

{√(nx+κ)P (l)(k)

}i,

i

χ(l)0 (k) =

x(l)(k) {Wi}2nxi=1

W(l)0 = κ/(nx + κ) W(l)

i = 1/(2(nx + κ))κ

χ(l)i (k + 1|k) = f(χ

(l)i (k))

Z(l)i (k+1|k) = g(l)(χ

(l)i (k+1|k))

l

x(l)

(k+1)=x(l)

(k+1|k)+K(l)(k)(z(l)(k+1)−z(l)

(k+1|k))

P(l)

(k+1)=P(l)

(k+1|k)−K(l)(k)P (l)zz (k+1|k)K(l)T(k)

K(l)(k)=P(l)xz (k+1|k)P (l)

zz (k+1|k)−1

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Step 2. Consensus Step{x(l)

(k+1),P(l)

(k+1)}Nl=1

P(l, )−1

(k+1) =N∑

j=1

[P(j)

(k+1)]−1−

[P

(j)(k+1|k)

]−1

+[P

(l)(k+1|k)

]−1

x(l. )

(k + 1) =[P

(l, )(k+1)

]−1[[P

(l)(k + 1|k)

]−1x

(l)(k + 1|k)

+

N∑j=1

[P

(j)(k+1)

]−1x

(j)(k+1)−

[P

(j)(k+1|k)

]−1x

(j)(k+1|k)

],

x(l)c (t+ 1) = x(l)

c (t) + ε∑

j∈ℵ(l)(x(j)c (t)− x(l)

c (t))

P (l)c (t+ 1) = P (l)

c (t) + ε∑

j∈ℵ(l)(P(j)c (t)−P (l)

c (t)),

ε ∈ (0, 1/ΔG) t

P (l)c (t = 0) = [P

(j)(k + 1)]−1 − [P

(j)(k + 1|k)]−1

x(l)c (t = 0) = [P

(j)(k + 1)]−1x

(j)(k + 1)

− [P(j)

(k + 1|k)]−1x(j)

(k + 1|k).

{P(l)c (t),x

(l)c (t)}

N

t

Step 3. Generating the ParticlesX

(l)i (k+1) ∼ N

(x

(l, )(k+1), P

(l, )(k+1)

)l

N(l)p X

(l)i (k + 1)

3.2.2. Innovation Update

W(l)i (k+1)

X(l)i (k+1)

W(l)i (k +1)∝P

(z(k+1)|X(l)

i (k+1))P(X

(l)i (k+1)|X(l)

i (k))

N(x

(l, )(k + 1),P

(l, )(k + 1)

) ,

P(z(k+1)|X(l)

i (k+1)) X(l)i (k+1)

z(k+1)

Case 1 – Single Consensus Iteration:

t ≤ 1 lz(j)(k+1)

j ∈ ℵ(l)

W(l)i (k+1)∝P

(z(j∈ℵ(l))(k+1)|X(l)

i (k+1))P(X

(l)i (k+1)|X(l)

i (k))

N(x

(l, )(k + 1),P

(l, )(k + 1)

) .

l

g(j)(·) j ∈ ℵ(l)

Case 2 – Multiple Consensus Iterations:

tt = 2 t = 3

P(z(k)|x(k)

)� T1(z(k))T2

(S(z(k)),x(k)

),

T1(z(k)) x(k)T2(S(z(k)),x(k))

x(k)

S(z(k)) = ∑Nl=1 Y(l)(k) Y(l)(k)

S(z(k))P(z(k+1)|X(l)

i (k+1))

X(l)i (k+1)

t

Example:

Z(l)(k) = g(x(k)) + ζ(l)(k),

1 ≤ l ≤ N ζ(l)(k) ∼ N (0, σ(l)2(k))

P (z(k)|x(k)) ∝ exp{−

N∑l=1

Z(l)2(k)

2σ(l)2(k)−

g2(x(k))

N∑l=1

1

2σ(l)2(k)+ g(x(k))

N∑l=1

Z(l)(k)

σ(l)2(k)

},

1107

Fig. 1

et al.

l

G1(k) =∑N

l=1

Z(l)2(k)

2σ(l)2(k)︸︷︷︸Y(l)1 (k)

, G2(k) =∑N

l=1

1

2σ(l)2(k)︸︷︷︸Y(l)2 (k)

G3(k) =∑N

l=1

Z(l)(k)

2σ(l)2(k)︸︷︷︸Y(l)3 (k)

.

4. EXPERIMENTAL RESULTS

N 15 × 15 2

√2 log(N)/N

f(x(k))

{10, 10} −110◦

σξ(k) = 1.6 × 10−2

σ2ζ(l)

(k) l

D(l)(x(k))

σ2ζ(l)

(k) = Bm[D(l)(x(k))]2 + 0.11D(l)(x(k))

σθ = 2.5◦

Ns = 1000

(k)=

√√√√ 1

n

1

N

n∑j=1

N∑l=1

(Xj(k)−X(l)

j (k))2+(Yj(k)−Y (l)

j (k))2

Fig. 2

k = 20 t = 1, 2, 3

nScenario 1:

1629

Scenario 2{2, 3}

tScenario 3

N = {10, 20, 30, 40, 50}

5. SUMMARY

1108

6. REFERENCES

IEEE Tran. Sig. Proc.

IEEETransactions on Robotics

CRCPress

Artech House

IEEE Transactions on Signal Processing

IEEE Signal Processing Magazine

IEEE Inter.Con. on Acoustics, Speech, & Signal Processing (ICASSP)

Nonlin-ear Estimation and Applications to Industrial Systems Control

IEEE Workshop on Statistical SignalProcessing (SSP)

IEEE Int. Con. on Informa-tion and Automation

Proc. IEEE Int. Con. on Info. Fusion

IEEE Inter. Con. on Robotics and Automation



IEEE Int. Con. on Communications (ICC)

IEEE Int. Con. onAcoustics, Speech, & Signal Processing (ICASSP)

IEEE Int. Con. on Acoustics, Speech, & SignalProcessing (ICASSP)

IEEE Int.Con. on Acoustics, Speech, & Signal Processing (ICASSP)

IEEE Int. Con. on Acoustics, Speech, & Signal Pro-cessing (ICASSP)

IEEEInt. Conference on Robotics and Automation

Proceedingsof the IEEE

IEEE Journalon Selected Areas in Communications

Proceedings of the IEEE

IEEE Transactions on Infor-mation Theory

IEEE Signal Processing Magazine

arXiv preprint arXiv:1111.4580

EURASIP App. Sig. Proc.

IEEE Transac-tions on Robotics

IEEE SignalProcessing Magazine


Multi-target Multi-sensor Tracking

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