SLIDE 1
Team optimal decentralized state estimation Aditya Mahajan and - - PowerPoint PPT Presentation
Team optimal decentralized state estimation Aditya Mahajan and - - PowerPoint PPT Presentation
Team optimal decentralized state estimation Aditya Mahajan and Mohammad Afshari McGill University IEEE Conference on Decision and Control 19 December 2018 Lets revisit separation of estimation and control in centralized systems
SLIDE 2
SLIDE 3
Let’s revisit separation of estimation and control in centralized systems
SLIDE 4
Decentralized estimation–(Afshari and Mahajan)
1 STANDARD LQG MODEL
x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). Choose u(t) = gt(y(1 : t), u(1 : t − 1)) to min 𝔽[
T
∑
t=1
[x(t)⊺Qx(t) + u(t)⊺Ru(t)]]
Separation in estimation and control
SLIDE 5
Decentralized estimation–(Afshari and Mahajan)
1 STANDARD LQG MODEL
x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). Choose u(t) = gt(y(1 : t), u(1 : t − 1)) to min 𝔽[
T
∑
t=1
[x(t)⊺Qx(t) + u(t)⊺Ru(t)]]
COMPLETION OF SQUARES
Total cost can be written as 𝔽[
T
∑
t=1
(L(t)x(t) + u(t))⊺S(t)(L(t)x(t) + u(t)) + w(t)⊺P(t + 1)w(t)]
Separation in estimation and control
SLIDE 6
Decentralized estimation–(Afshari and Mahajan)
1 STANDARD LQG MODEL
x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). Choose u(t) = gt(y(1 : t), u(1 : t − 1)) to min 𝔽[
T
∑
t=1
[x(t)⊺Qx(t) + u(t)⊺Ru(t)]]
COMPLETION OF SQUARES
Total cost can be written as 𝔽[
T
∑
t=1
(L(t)x(t) + u(t))⊺S(t)(L(t)x(t) + u(t)) + w(t)⊺P(t + 1)w(t)]
Linear System u(t) w(t) y(t)
Separation in estimation and control
SLIDE 7
Decentralized estimation–(Afshari and Mahajan)
1 STANDARD LQG MODEL
x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). Choose u(t) = gt(y(1 : t), u(1 : t − 1)) to min 𝔽[
T
∑
t=1
[x(t)⊺Qx(t) + u(t)⊺Ru(t)]]
COMPLETION OF SQUARES
Total cost can be written as 𝔽[
T
∑
t=1
(L(t)x(t) + u(t))⊺S(t)(L(t)x(t) + u(t)) + w(t)⊺P(t + 1)w(t)]
Linear System u(t) w(t) y(t)
¯ x(t) = part of state depending on u(1 : t). ˜ x(t) = part of state depending on w(1 : t). From linearity, x(t) = ¯ x(t) + ˜ x(t).
Separation in estimation and control
SLIDE 8
Decentralized estimation–(Afshari and Mahajan)
1 STANDARD LQG MODEL
x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). Choose u(t) = gt(y(1 : t), u(1 : t − 1)) to min 𝔽[
T
∑
t=1
[x(t)⊺Qx(t) + u(t)⊺Ru(t)]]
COMPLETION OF SQUARES
Total cost can be written as 𝔽[
T
∑
t=1
(L(t)x(t) + u(t))⊺S(t)(L(t)x(t) + u(t)) + w(t)⊺P(t + 1)w(t)]
Linear System u(t) w(t) y(t)
¯ x(t) = part of state depending on u(1 : t). ˜ x(t) = part of state depending on w(1 : t). From linearity, x(t) = ¯ x(t) + ˜ x(t). Substitute u(t) = ˆ z(t)−L¯ x(t) in expression for total cost
Separation in estimation and control
SLIDE 9
Decentralized estimation–(Afshari and Mahajan)
1 STANDARD LQG MODEL
x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). Choose u(t) = gt(y(1 : t), u(1 : t − 1)) to min 𝔽[
T
∑
t=1
[x(t)⊺Qx(t) + u(t)⊺Ru(t)]]
COMPLETION OF SQUARES
Total cost can be written as 𝔽[
T
∑
t=1
(L(t)x(t) + u(t))⊺S(t)(L(t)x(t) + u(t)) + w(t)⊺P(t + 1)w(t)] = 𝔽[
T
∑
t=1
(L(t)˜ x(t) + ˆ z(t))⊺S(t)(L(t)˜ x(t) + ˆ z(t)) + w(t)⊺P(t + 1)w(t)]
Linear System u(t) w(t) y(t)
¯ x(t) = part of state depending on u(1 : t). ˜ x(t) = part of state depending on w(1 : t). From linearity, x(t) = ¯ x(t) + ˜ x(t). Substitute u(t) = ˆ z(t)−L¯ x(t) in expression for total cost
Separation in estimation and control
SLIDE 10
Decentralized estimation–(Afshari and Mahajan)
1 STANDARD LQG MODEL
x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). Choose u(t) = gt(y(1 : t), u(1 : t − 1)) to min 𝔽[
T
∑
t=1
[x(t)⊺Qx(t) + u(t)⊺Ru(t)]]
COMPLETION OF SQUARES
Total cost can be written as 𝔽[
T
∑
t=1
(L(t)x(t) + u(t))⊺S(t)(L(t)x(t) + u(t)) + w(t)⊺P(t + 1)w(t)] = 𝔽[
T
∑
t=1
(L(t)˜ x(t) + ˆ z(t))⊺S(t)(L(t)˜ x(t) + ˆ z(t)) + w(t)⊺P(t + 1)w(t)]
Linear System u(t) w(t) y(t)
¯ x(t) = part of state depending on u(1 : t). ˜ x(t) = part of state depending on w(1 : t). From linearity, x(t) = ¯ x(t) + ˜ x(t). Substitute u(t) = ˆ z(t)−L¯ x(t) in expression for total cost
STATIC REDUCTION
σ(y(1 : t), u(1 : t − 1)) = σ(˜ y(1 : t − 1)). Thus, wlog, consider ˆ z(t) = ˜ gt(˜ y(1 : t)).
Separation in estimation and control
SLIDE 11
Decentralized estimation–(Afshari and Mahajan)
1 STANDARD LQG MODEL
x(t + 1) = Ax(t) + Bu(t) + w(t), y(t) = Cx(t) + v(t). Choose u(t) = gt(y(1 : t), u(1 : t − 1)) to min 𝔽[
T
∑
t=1
[x(t)⊺Qx(t) + u(t)⊺Ru(t)]]
COMPLETION OF SQUARES
Total cost can be written as 𝔽[
T
∑
t=1
(L(t)x(t) + u(t))⊺S(t)(L(t)x(t) + u(t)) + w(t)⊺P(t + 1)w(t)] = 𝔽[
T
∑
t=1
(L(t)˜ x(t) + ˆ z(t))⊺S(t)(L(t)˜ x(t) + ˆ z(t)) + w(t)⊺P(t + 1)w(t)]
Linear System u(t) w(t) y(t)
¯ x(t) = part of state depending on u(1 : t). ˜ x(t) = part of state depending on w(1 : t). From linearity, x(t) = ¯ x(t) + ˜ x(t). Substitute u(t) = ˆ z(t)−L¯ x(t) in expression for total cost
STATIC REDUCTION
σ(y(1 : t), u(1 : t − 1)) = σ(˜ y(1 : t − 1)). Thus, wlog, consider ˆ z(t) = ˜ gt(˜ y(1 : t)). Thus, ˆ z(t) = −L 𝔽[˜ x(t) | ˜ y(1 : t)]
Separation in estimation and control
SLIDE 12
Decentralized estimation–(Afshari and Mahajan)
2
Separation centralized stochastic control, the
- ptimal control action depends on the solution of an
estimation problem: 𝔽[
T
∑
t=1
(L(t)˜ x(t) + ˆ z(t))⊺S(t)(L(t)˜ x(t) + ˆ z(t))] Does the same happen in decentralized control?
Motivation for current work
SLIDE 13
Decentralized estimation–(Afshari and Mahajan)
2
Separation centralized stochastic control, the
- ptimal control action depends on the solution of an
estimation problem: 𝔽[
T
∑
t=1
(L(t)˜ x(t) + ˆ z(t))⊺S(t)(L(t)˜ x(t) + ˆ z(t))] Does the same happen in decentralized control? In decentralized estimation, is L 𝔽[x(t) | I(t)] the best estimate?
Motivation for current work
SLIDE 14
Decentralized estimation–(Afshari and Mahajan)
2
Separation centralized stochastic control, the
- ptimal control action depends on the solution of an
estimation problem: 𝔽[
T
∑
t=1
(L(t)˜ x(t) + ˆ z(t))⊺S(t)(L(t)˜ x(t) + ˆ z(t))] Does the same happen in decentralized control? In decentralized estimation, is L 𝔽[x(t) | I(t)] the best estimate? There is a long history of duality between estimation and control.
Motivation for current work
SLIDE 15
Decentralized estimation–(Afshari and Mahajan)
2
Separation centralized stochastic control, the
- ptimal control action depends on the solution of an
estimation problem: 𝔽[
T
∑
t=1
(L(t)˜ x(t) + ˆ z(t))⊺S(t)(L(t)˜ x(t) + ˆ z(t))] Does the same happen in decentralized control? In decentralized estimation, is L 𝔽[x(t) | I(t)] the best estimate? There is a long history of duality between estimation and control. Decentralized control is interesting. Ergo, decentralized estimation is interesting. Decentralized estimation is interesting in it’s own right in certain applications.
Motivation for current work
SLIDE 16
DECENTRALIZED state estimation is
fundamentally different from
CENTRALIZED state estimation.
SLIDE 17
Decentralized estimation–(Afshari and Mahajan)
3
x ˆ z y
x ∼ 𝒪(0, Σx), y = Cx + v, v ∼ 𝒪(0, R). Choose ˆ z = g(y) to minimize 𝔽[(Lx − ˆ z)⊺S(Lx − ˆ z)].
Centralized estimation
SLIDE 18
Decentralized estimation–(Afshari and Mahajan)
3
x ˆ z y
x ∼ 𝒪(0, Σx), y = Cx + v, v ∼ 𝒪(0, R). Choose ˆ z = g(y) to minimize 𝔽[(Lx − ˆ z)⊺S(Lx − ˆ z)].
OPTIMAL ESTIMATE: ˆ
z = LKy, where K = ΣxC⊺(CΣxC⊺ + R)−1.
Centralized estimation
SLIDE 19
Decentralized estimation–(Afshari and Mahajan)
3
x ˆ z y
x ∼ 𝒪(0, Σx), y = Cx + v, v ∼ 𝒪(0, R). Choose ˆ z = g(y) to minimize 𝔽[(Lx − ˆ z)⊺S(Lx − ˆ z)].
OPTIMAL ESTIMATE: ˆ
z = LKy, where K = ΣxC⊺(CΣxC⊺ + R)−1. The optimal estimation strategy
DOES NOT depend on the weight S.
Centralized estimation
SLIDE 20
Decentralized estimation–(Afshari and Mahajan)
3
x ˆ z y
x ∼ 𝒪(0, Σx), y = Cx + v, v ∼ 𝒪(0, R). Choose ˆ z = g(y) to minimize 𝔽[(Lx − ˆ z)⊺S(Lx − ˆ z)].
OPTIMAL ESTIMATE: ˆ
z = LKy, where K = ΣxC⊺(CΣxC⊺ + R)−1. The optimal estimation strategy
DOES NOT depend on the weight S.
y1 = C1x + v1, y2 = C2x + v2. Choose ˆ z1 = g1(y1) and ˆ z2 = g2(y2) to minimize 𝔽 ⎡ ⎣[ L1x − ˆ z1 L2x − ˆ z2 ]
⊺
S [ L1x − ˆ z1 L2x − ˆ z2 ] ⎤ ⎦ .
x ˆ z1 ˆ z2 y1 y2
Centralized estimation vs decentralized estimation
SLIDE 21
Decentralized estimation–(Afshari and Mahajan)
3
x ˆ z y
x ∼ 𝒪(0, Σx), y = Cx + v, v ∼ 𝒪(0, R). Choose ˆ z = g(y) to minimize 𝔽[(Lx − ˆ z)⊺S(Lx − ˆ z)].
OPTIMAL ESTIMATE: ˆ
z = LKy, where K = ΣxC⊺(CΣxC⊺ + R)−1. The optimal estimation strategy
DOES NOT depend on the weight S.
y1 = C1x + v1, y2 = C2x + v2. Choose ˆ z1 = g1(y1) and ˆ z2 = g2(y2) to minimize 𝔽 ⎡ ⎣[ L1x − ˆ z1 L2x − ˆ z2 ]
⊺
S [ L1x − ˆ z1 L2x − ˆ z2 ] ⎤ ⎦ .
x ˆ z1 ˆ z2 y1 y2
OPTIMAL ESTIMATE: ˆ
zi = Fiyi, i ∈ {1, 2}, where ∑
j∈{1,2}[SijFjΣji − SijLjΘi] = 0,
i ∈ {1, 2}, and Σij = CiΣxC⊺
j + δijRi and Θi = ΣxC⊺ i .
Centralized estimation vs decentralized estimation
SLIDE 22
Decentralized estimation–(Afshari and Mahajan)
3
x ˆ z y
x ∼ 𝒪(0, Σx), y = Cx + v, v ∼ 𝒪(0, R). Choose ˆ z = g(y) to minimize 𝔽[(Lx − ˆ z)⊺S(Lx − ˆ z)].
OPTIMAL ESTIMATE: ˆ
z = LKy, where K = ΣxC⊺(CΣxC⊺ + R)−1. The optimal estimation strategy
DOES NOT depend on the weight S.
y1 = C1x + v1, y2 = C2x + v2. Choose ˆ z1 = g1(y1) and ˆ z2 = g2(y2) to minimize 𝔽 ⎡ ⎣[ L1x − ˆ z1 L2x − ˆ z2 ]
⊺
S [ L1x − ˆ z1 L2x − ˆ z2 ] ⎤ ⎦ .
x ˆ z1 ˆ z2 y1 y2
OPTIMAL ESTIMATE: ˆ
zi = Fiyi, i ∈ {1, 2}, where ∑
j∈{1,2}[SijFjΣji − SijLjΘi] = 0,
i ∈ {1, 2}, and Σij = CiΣxC⊺
j + δijRi and Θi = ΣxC⊺ i .
The optimal estimation strategy
DOES depend on the weight S.
Centralized estimation vs decentralized estimation
SLIDE 23
Decentralized estimation–(Afshari and Mahajan)
4 DYNAMICS
x(t + 1) = Ax(t) + w(t), w(t) ∼ 𝒪(0, Q).
System model
SLIDE 24
Decentralized estimation–(Afshari and Mahajan)
4 DYNAMICS
x(t + 1) = Ax(t) + w(t), w(t) ∼ 𝒪(0, Q).
OBSERVATIONS
The system consists of n agents. yi(t) = Cix(t) + vi(t), vi(t) ∼ 𝒪(0, Ri).
System model
SLIDE 25
Decentralized estimation–(Afshari and Mahajan)
4
dij
DYNAMICS
x(t + 1) = Ax(t) + w(t), w(t) ∼ 𝒪(0, Q).
OBSERVATIONS
The system consists of n agents. yi(t) = Cix(t) + vi(t), vi(t) ∼ 𝒪(0, Ri).
INFO STRUCTURE
Agents communicate over a strongly connected weighted directed graph. Edge weight dij corresponds to link delay. Ii(t) = {yi(1 : t)} ∪ ( ∪
j∈N−
i
Ij(t − dji)
System model
SLIDE 26
Decentralized estimation–(Afshari and Mahajan)
4
dij
DYNAMICS
x(t + 1) = Ax(t) + w(t), w(t) ∼ 𝒪(0, Q).
OBSERVATIONS
The system consists of n agents. yi(t) = Cix(t) + vi(t), vi(t) ∼ 𝒪(0, Ri).
INFO STRUCTURE
Agents communicate over a strongly connected weighted directed graph. Edge weight dij corresponds to link delay. Ii(t) = {yi(1 : t)} ∪ ( ∪
j∈N−
i
Ij(t − dji) d-STEP DELAY SHARING Ii(t) = {y(1 : t − d), yi(t − d + 1 : t)}.
System model
SLIDE 27
Decentralized estimation–(Afshari and Mahajan)
4
dij
DYNAMICS
x(t + 1) = Ax(t) + w(t), w(t) ∼ 𝒪(0, Q).
OBSERVATIONS
The system consists of n agents. yi(t) = Cix(t) + vi(t), vi(t) ∼ 𝒪(0, Ri).
INFO STRUCTURE
Agents communicate over a strongly connected weighted directed graph. Edge weight dij corresponds to link delay. Ii(t) = {yi(1 : t)} ∪ ( ∪
j∈N−
i
Ij(t − dji) d-STEP DELAY SHARING Ii(t) = {y(1 : t − d), yi(t − d + 1 : t)}.
NEIGHBORHOOD SHARING
Ii(t) =
d∗
∪
k=0 ∪ j∈Nk
i
{yj(1 : t − k)}.
System model
SLIDE 28
Decentralized estimation–(Afshari and Mahajan)
5 ESTIMATES
Each agent generates an estimate ˆ zi(t) = gi,t(Ii(t))
System model (continued)
SLIDE 29
Decentralized estimation–(Afshari and Mahajan)
5 ESTIMATES
Each agent generates an estimate ˆ zi(t) = gi,t(Ii(t))
PER-STEP ERROR
Let ˆ z(t) = vec(ˆ z1(t), … , ˆ zn(t)). Then, c(x(t), ˆ z(t)) = (Lx(t) − ˆ z(t))
⊺S(Lx(t) − ˆ
z(t)). where S = ⎡ ⎢ ⎣ S11 ⋅ ⋅ ⋅ S1n ⋮ ⋱ ⋮ Sn1 ⋅ ⋅ ⋅ Snn ⎤ ⎥ ⎦ and L = ⎡ ⎢ ⎣ L1 ⋮ Ln ⎤ ⎥ ⎦
System model (continued)
SLIDE 30
Decentralized estimation–(Afshari and Mahajan)
5 ESTIMATES
Each agent generates an estimate ˆ zi(t) = gi,t(Ii(t))
PER-STEP ERROR
Let ˆ z(t) = vec(ˆ z1(t), … , ˆ zn(t)). Then, c(x(t), ˆ z(t)) = (Lx(t) − ˆ z(t))
⊺S(Lx(t) − ˆ
z(t)). where S = ⎡ ⎢ ⎣ S11 ⋅ ⋅ ⋅ S1n ⋮ ⋱ ⋮ Sn1 ⋅ ⋅ ⋅ Snn ⎤ ⎥ ⎦ and L = ⎡ ⎢ ⎣ L1 ⋮ Ln ⎤ ⎥ ⎦ ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +λ‖¯ x(t) − ¯ z(t)‖2
System model (continued)
SLIDE 31
Decentralized estimation–(Afshari and Mahajan)
5 ESTIMATES
Each agent generates an estimate ˆ zi(t) = gi,t(Ii(t))
PER-STEP ERROR
Let ˆ z(t) = vec(ˆ z1(t), … , ˆ zn(t)). Then, c(x(t), ˆ z(t)) = (Lx(t) − ˆ z(t))
⊺S(Lx(t) − ˆ
z(t)). where S = ⎡ ⎢ ⎣ S11 ⋅ ⋅ ⋅ S1n ⋮ ⋱ ⋮ Sn1 ⋅ ⋅ ⋅ Snn ⎤ ⎥ ⎦ and L = ⎡ ⎢ ⎣ L1 ⋮ Ln ⎤ ⎥ ⎦ ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +λ‖¯ x(t) − ¯ z(t)‖2 ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +
n−1
∑
i=1
λ‖di(t) − ˆ di(t)‖2
System model (continued)
SLIDE 32
Decentralized estimation–(Afshari and Mahajan)
5 ESTIMATES
Each agent generates an estimate ˆ zi(t) = gi,t(Ii(t))
PER-STEP ERROR
Let ˆ z(t) = vec(ˆ z1(t), … , ˆ zn(t)). Then, c(x(t), ˆ z(t)) = (Lx(t) − ˆ z(t))
⊺S(Lx(t) − ˆ
z(t)). where S = ⎡ ⎢ ⎣ S11 ⋅ ⋅ ⋅ S1n ⋮ ⋱ ⋮ Sn1 ⋅ ⋅ ⋅ Snn ⎤ ⎥ ⎦ and L = ⎡ ⎢ ⎣ L1 ⋮ Ln ⎤ ⎥ ⎦
OBJECTIVE
Choose a team estimation problem g to min 𝔽g [
T
∑
t=1
c(x(t), ˆ z(t)) ] ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +λ‖¯ x(t) − ¯ z(t)‖2 ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +
n−1
∑
i=1
λ‖di(t) − ˆ di(t)‖2
System model (continued)
SLIDE 33
Decentralized estimation–(Afshari and Mahajan)
6
Fusion Center
Literature Overview
SLIDE 34
Decentralized estimation–(Afshari and Mahajan)
6
Fusion Center Conensus based methods
Literature Overview
SLIDE 35
Decentralized estimation–(Afshari and Mahajan)
6
Fusion Center Conensus based methods
TEAM OPTIMAL DECENTRALIZED ESTIMATION
Barta, PhD Thesis (1978) Castanon, LIDS Tech Report (1981) Andersland and Teneketzis, JOTA (1996)
Literature Overview
SLIDE 36
Decentralized estimation–(Afshari and Mahajan)
7
1.1 1.2 ⋅ ⋅ ⋅ 1.t ⋅ ⋅ ⋅ ⋮ ⋮ ⋅ ⋅ ⋅ ⋮ ⋮ n.1 n.2 ⋅ ⋅ ⋅ n.t ⋅ ⋅ ⋅
Solution approach: Witsenhausen’s intrinsic model
SLIDE 37
Decentralized estimation–(Afshari and Mahajan)
7
Decentralized estimation is a STATIC TEAM problem.
1.1 1.2 ⋅ ⋅ ⋅ 1.t ⋅ ⋅ ⋅ ⋮ ⋮ ⋅ ⋅ ⋅ ⋮ ⋮ n.1 n.2 ⋅ ⋅ ⋅ n.t ⋅ ⋅ ⋅
Solution approach: Witsenhausen’s intrinsic model
SLIDE 38
Decentralized estimation–(Afshari and Mahajan)
7
Decentralized estimation is a STATIC TEAM problem.
1.1 1.2 ⋅ ⋅ ⋅ 1.t ⋅ ⋅ ⋅ ⋮ ⋮ ⋅ ⋅ ⋅ ⋮ ⋮ n.1 n.2 ⋅ ⋅ ⋅ n.t ⋅ ⋅ ⋅
Stand-alone optimization problem
Solution approach: Witsenhausen’s intrinsic model
SLIDE 39
Decentralized estimation–(Afshari and Mahajan)
7
Decentralized estimation is a STATIC TEAM problem.
1.1 1.2 ⋅ ⋅ ⋅ 1.t ⋅ ⋅ ⋅ ⋮ ⋮ ⋅ ⋅ ⋅ ⋮ ⋮ n.1 n.2 ⋅ ⋅ ⋅ n.t ⋅ ⋅ ⋅
Stand-alone optimization problem
Solution approach: Witsenhausen’s intrinsic model
SLIDE 40
Decentralized estimation–(Afshari and Mahajan)
7
Decentralized estimation is a STATIC TEAM problem.
1.1 1.2 ⋅ ⋅ ⋅ 1.t ⋅ ⋅ ⋅ ⋮ ⋮ ⋅ ⋅ ⋅ ⋮ ⋮ n.1 n.2 ⋅ ⋅ ⋅ n.t ⋅ ⋅ ⋅
Stand-alone optimization problem
Solution approach: Witsenhausen’s intrinsic model
SLIDE 41
Decentralized estimation–(Afshari and Mahajan)
7
Decentralized estimation is a STATIC TEAM problem.
1.1 1.2 ⋅ ⋅ ⋅ 1.t ⋅ ⋅ ⋅ ⋮ ⋮ ⋅ ⋅ ⋅ ⋮ ⋮ n.1 n.2 ⋅ ⋅ ⋅ n.t ⋅ ⋅ ⋅
Stand-alone optimization problem Instead of solving min 𝔽[
T
∑
t=1
c(x(t), ˆ z(t))] we solve min 𝔽[c(x(t), ˆ z(t))] at each t
Solution approach: Witsenhausen’s intrinsic model
SLIDE 42
Decentralized estimation–(Afshari and Mahajan)
7
Decentralized estimation is a STATIC TEAM problem.
1.1 1.2 ⋅ ⋅ ⋅ 1.t ⋅ ⋅ ⋅ ⋮ ⋮ ⋅ ⋅ ⋅ ⋮ ⋮ n.1 n.2 ⋅ ⋅ ⋅ n.t ⋅ ⋅ ⋅
Stand-alone optimization problem Instead of solving min 𝔽[
T
∑
t=1
c(x(t), ˆ z(t))] we solve min 𝔽[c(x(t), ˆ z(t))] at each t Decentralized estimation is a SEQUENCE of static team problems.
Solution approach: Witsenhausen’s intrinsic model
SLIDE 43
A naive application of Radnar’s result does not work.
SLIDE 44
Decentralized estimation–(Afshari and Mahajan)
8 OPTIMAL STRATEGY:
ˆ zi(t) = Fi(t)Ii(t) {Fi(t)}i∈N given by the solution of a system of matrix equations.
Directly applying Radnar’s result
SLIDE 45
Decentralized estimation–(Afshari and Mahajan)
8 OPTIMAL STRATEGY:
ˆ zi(t) = Fi(t)Ii(t) {Fi(t)}i∈N given by the solution of a system of matrix equations. Ii(t) increases with time; so does the dimension of Fi(t). Complexity of fjnding the optimal solution increases with time.
Directly applying Radnar’s result
SLIDE 46
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t) = Ii(t) ∖ Icom(t),
Alternative idea: Common information approach
SLIDE 47
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t)
= Ii(t) ∖ Icom(t), State estimate ˆ xcom(t) = 𝔽[x(t) | Icom(t)]. Local Innovation ˜ Iloc
i (t)
= Ii(t) − 𝔽[Iloc
i (t) | Icom(t)].
Alternative idea: Common information approach
SLIDE 48
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t)
= Ii(t) ∖ Icom(t), State estimate ˆ xcom(t) = 𝔽[x(t) | Icom(t)]. Local Innovation ˜ Iloc
i (t)
= Ii(t) − 𝔽[Iloc
i (t) | Icom(t)].
Let ˆ Σij(t) = cov(˜ Ii(t), ˜ Ij(t)). and ˆ Θi(t) = cov(x(t), ˜ Ii(t))
Alternative idea: Common information approach
SLIDE 49
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t)
= Ii(t) ∖ Icom(t), State estimate ˆ xcom(t) = 𝔽[x(t) | Icom(t)]. Local Innovation ˜ Iloc
i (t)
= Ii(t) − 𝔽[Iloc
i (t) | Icom(t)].
Let ˆ Σij(t) = cov(˜ Ii(t), ˜ Ij(t)). and ˆ Θi(t) = cov(x(t), ˜ Ii(t))
STRUCTURE OF OPTIMAL ESTIMATORS
ˆ zi(t) = Li ˆ xcom(t) + Fi(t) ˜ Iloc
i (t)
1st term: Common info based estimate 2nd term: Local innovation based correction (depends on weight matrix)
Alternative idea: Common information approach
SLIDE 50
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t)
= Ii(t) ∖ Icom(t), State estimate ˆ xcom(t) = 𝔽[x(t) | Icom(t)]. Local Innovation ˜ Iloc
i (t)
= Ii(t) − 𝔽[Iloc
i (t) | Icom(t)].
Let ˆ Σij(t) = cov(˜ Ii(t), ˜ Ij(t)). and ˆ Θi(t) = cov(x(t), ˜ Ii(t))
STRUCTURE OF OPTIMAL ESTIMATORS
ˆ zi(t) = Li ˆ xcom(t) + Fi(t) ˜ Iloc
i (t)
1st term: Common info based estimate 2nd term: Local innovation based correction (depends on weight matrix)
COMPUTING OPTIMAL GAINS
System of matrix equations: for all i ∈ N,
∑
j∈N
[SijFj(t)ˆ Σji(t) − SijLj ˆ Θj(t)] = 0.
Alternative idea: Common information approach
SLIDE 51
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t)
= Ii(t) ∖ Icom(t), State estimate ˆ xcom(t) = 𝔽[x(t) | Icom(t)]. Local Innovation ˜ Iloc
i (t)
= Ii(t) − 𝔽[Iloc
i (t) | Icom(t)].
Let ˆ Σij(t) = cov(˜ Ii(t), ˜ Ij(t)). and ˆ Θi(t) = cov(x(t), ˜ Ii(t))
STRUCTURE OF OPTIMAL ESTIMATORS
ˆ zi(t) = Li ˆ xcom(t) + Fi(t) ˜ Iloc
i (t)
1st term: Common info based estimate 2nd term: Local innovation based correction (depends on weight matrix)
COMPUTING OPTIMAL GAINS
System of matrix equations: for all i ∈ N,
∑
j∈N
[SijFj(t)ˆ Σji(t) − SijLj ˆ Θj(t)] = 0.
PROOF IDEA
Same as Radnar. Show that the proposed strategy is PBPO For convex static teams: PBPO ⟹ global optimal.
Alternative idea: Common information approach
SLIDE 52
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t)
= Ii(t) ∖ Icom(t), State estimate ˆ xcom(t) = 𝔽[x(t) | Icom(t)]. Local Innovation ˜ Iloc
i (t)
= Ii(t) − 𝔽[Iloc
i (t) | Icom(t)].
Let ˆ Σij(t) = cov(˜ Ii(t), ˜ Ij(t)). and ˆ Θi(t) = cov(x(t), ˜ Ii(t))
STRUCTURE OF OPTIMAL ESTIMATORS
ˆ zi(t) = Li ˆ xcom(t) + Fi(t) ˜ Iloc
i (t)
1st term: Common info based estimate 2nd term: Local innovation based correction (depends on weight matrix)
COMPUTING OPTIMAL GAINS
System of matrix equations: for all i ∈ N,
∑
j∈N
[SijFj(t)ˆ Σji(t) − SijLj ˆ Θj(t)] = 0.
PROOF IDEA
Same as Radnar. Show that the proposed strategy is PBPO For convex static teams: PBPO ⟹ global optimal.
VECTORIZED SOLUTION
Equivalen to
F(t) = Γ(t)−1η(t)
where F(t) = vec(F1(t), … , Fn(t)) and Γ(t) and η(t) depends on Sij, ˆ Σij(t), and ˆ Θi(t).
Alternative idea: Common information approach
SLIDE 53
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t)
= Ii(t) ∖ Icom(t), State estimate ˆ xcom(t) = 𝔽[x(t) | Icom(t)]. Local Innovation ˜ Iloc
i (t)
= Ii(t) − 𝔽[Iloc
i (t) | Icom(t)].
Let ˆ Σij(t) = cov(˜ Ii(t), ˜ Ij(t)). and ˆ Θi(t) = cov(x(t), ˜ Ii(t))
STRUCTURE OF OPTIMAL ESTIMATORS
ˆ zi(t) = Li ˆ xcom(t) + Fi(t) ˜ Iloc
i (t)
1st term: Common info based estimate 2nd term: Local innovation based correction (depends on weight matrix)
COMPUTING OPTIMAL GAINS
System of matrix equations: for all i ∈ N,
∑
j∈N
[SijFj(t)ˆ Σji(t) − SijLj ˆ Θj(t)] = 0.
PROOF IDEA
Same as Radnar. Show that the proposed strategy is PBPO For convex static teams: PBPO ⟹ global optimal.
VECTORIZED SOLUTION
Equivalen to
F(t) = Γ(t)−1η(t)
where F(t) = vec(F1(t), … , Fn(t)) and Γ(t) and η(t) depends on Sij, ˆ Σij(t), and ˆ Θi(t).
Alternative idea: Common information approach
WHAT ELSE IS NEEDED?
Iteratively compute ˆ xcom(t) and ˜ Iloc
i (t).
Iteratively update ˆ Σij(t) and ˆ Θi(t).
Follow Witsenhausen’s idea!
SLIDE 54
Decentralized estimation–(Afshari and Mahajan)
10 SYSTEM IN TERMS OF DELAYED STATE
Defjne w(k)(ℓ, t) =
t−ℓ−1
∑
τ=t−k
At−ℓ−τ−1w(τ)
Iterative update of estimates and covariances
SLIDE 55
Decentralized estimation–(Afshari and Mahajan)
10 SYSTEM IN TERMS OF DELAYED STATE
Defjne w(k)(ℓ, t) =
t−ℓ−1
∑
τ=t−k
At−ℓ−τ−1w(τ) Then, x(t) = Akx(t − k) + w(k)(0, t) + vi(t) yi(t) = Ci Ak x(t − k) + Ci w(k)(0, t) + vi(t).
Iterative update of estimates and covariances
SLIDE 56
Decentralized estimation–(Afshari and Mahajan)
10 SYSTEM IN TERMS OF DELAYED STATE
Defjne w(k)(ℓ, t) =
t−ℓ−1
∑
τ=t−k
At−ℓ−τ−1w(τ) Then, x(t) = Akx(t − k) + w(k)(0, t) + vi(t) yi(t) = Ci Ak x(t − k) + Ci w(k)(0, t) + vi(t). Let d∗ be the diameter of the graph. Then, Icom(t) = y(1 : t − d∗) Iloc
i (t) ⊆ y(t − d∗ + 1 : t).
We can fjnd a matrix Cloc
i
and vectors wloc
i (t) and vloc i (t) such that
Iloc
i (t) = Cloc i
x(t−d∗ +1)+wloc
i (t)+vloc i (t)
Iterative update of estimates and covariances
SLIDE 57
Decentralized estimation–(Afshari and Mahajan)
10 SYSTEM IN TERMS OF DELAYED STATE
Defjne w(k)(ℓ, t) =
t−ℓ−1
∑
τ=t−k
At−ℓ−τ−1w(τ) Then, x(t) = Akx(t − k) + w(k)(0, t) + vi(t) yi(t) = Ci Ak x(t − k) + Ci w(k)(0, t) + vi(t). Let d∗ be the diameter of the graph. Then, Icom(t) = y(1 : t − d∗) Iloc
i (t) ⊆ y(t − d∗ + 1 : t).
We can fjnd a matrix Cloc
i
and vectors wloc
i (t) and vloc i (t) such that
Iloc
i (t) = Cloc i
x(t−d∗ +1)+wloc
i (t)+vloc i (t)
COMPUTING ESTIMATES AND INNOVATION
Defjne ˆ x(t − d∗ + 1) = 𝔽[x(t − d∗ + 1) | Icom(t)]. Then, ˆ xcom(t) = Ad∗−1 ˆ x(t − d∗ + 1) ˜ Iloc
i (t) = Iloc i (t) − Cloc i
ˆ x(t − d∗ + 1)
Iterative update of estimates and covariances
SLIDE 58
Decentralized estimation–(Afshari and Mahajan)
10 SYSTEM IN TERMS OF DELAYED STATE
Defjne w(k)(ℓ, t) =
t−ℓ−1
∑
τ=t−k
At−ℓ−τ−1w(τ) Then, x(t) = Akx(t − k) + w(k)(0, t) + vi(t) yi(t) = Ci Ak x(t − k) + Ci w(k)(0, t) + vi(t). Let d∗ be the diameter of the graph. Then, Icom(t) = y(1 : t − d∗) Iloc
i (t) ⊆ y(t − d∗ + 1 : t).
We can fjnd a matrix Cloc
i
and vectors wloc
i (t) and vloc i (t) such that
Iloc
i (t) = Cloc i
x(t−d∗ +1)+wloc
i (t)+vloc i (t)
COMPUTING ESTIMATES AND INNOVATION
Defjne ˆ x(t − d∗ + 1) = 𝔽[x(t − d∗ + 1) | Icom(t)]. Then, ˆ xcom(t) = Ad∗−1 ˆ x(t − d∗ + 1) ˜ Iloc
i (t) = Iloc i (t) − Cloc i
ˆ x(t − d∗ + 1)
KEEPING TRACK OF COVARIANCES
ˆ Σij(t) = Cloc
i
P(t − d∗ + 1) Cloc
j ⊺
+ Pw
ij (t) + Pv ij(t).
ˆ Θi(t) = Ad∗−1 P(t − d∗ + 1)Cloc
j ⊺ + Pσ i (t).
Iterative update of estimates and covariances
SLIDE 59
Decentralized estimation–(Afshari and Mahajan)
11 ASSUMPTIONS
(A, √Q) is stabilizable and (A, C) is detectable
OBJECTIVE
min lim sup
T→∞
1 T
T
∑
t=1
c(x(t), ˆ z(t))
Extension to infinite horizon setup
SLIDE 60
Decentralized estimation–(Afshari and Mahajan)
11 ASSUMPTIONS
(A, √Q) is stabilizable and (A, C) is detectable
OBJECTIVE
min lim sup
T→∞
1 T
T
∑
t=1
c(x(t), ˆ z(t))
STRUCTURE OF OPTIMAL ESTIMATORS
ˆ zi(t) = Li ˆ xcom(t) + Fi ˜ Iloc
i (t)
Note: Fi, Σij and Θi are time-homogeneous
COMPUTING OPTIMAL GAINS
System of matrix equations: for all i ∈ N,
∑
j∈N
[SijFjˆ Σji − SijLj ˆ Θj] = 0.
Extension to infinite horizon setup
SLIDE 61
Decentralized estimation–(Afshari and Mahajan)
12
x(t) ∈ ℝ4, n = 4 and agent i observes xi(t). Per-step cost: ∑
i∈N
‖xi(t) − ˆ zi(t)‖2 + λ‖¯ x(t) − ¯ z(t)‖2 2-step delay sharing information structure
Example
SLIDE 62
Decentralized estimation–(Afshari and Mahajan)
12
x(t) ∈ ℝ4, n = 4 and agent i observes xi(t). Per-step cost: ∑
i∈N
‖xi(t) − ˆ zi(t)‖2 + λ‖¯ x(t) − ¯ z(t)‖2 2-step delay sharing information structure
Example
SLIDE 63
Decentralized estimation–(Afshari and Mahajan)
12
x(t) ∈ ℝ4, n = 4 and agent i observes xi(t). Per-step cost: ∑
i∈N
‖xi(t) − ˆ zi(t)‖2 + λ‖¯ x(t) − ¯ z(t)‖2 2-step delay sharing information structure
BASELINE STRATEGY
ˆ zi(t) = Li 𝔽[x(t) | Ii(t)]
OPTIMAL STRATEGY
ˆ zi(t) = Lixcom(t) + Fi(t) ˜ Iloc
i (t)
Example
SLIDE 64
Decentralized estimation–(Afshari and Mahajan)
12
x(t) ∈ ℝ4, n = 4 and agent i observes xi(t). Per-step cost: ∑
i∈N
‖xi(t) − ˆ zi(t)‖2 + λ‖¯ x(t) − ¯ z(t)‖2 2-step delay sharing information structure
BASELINE STRATEGY
ˆ zi(t) = Li 𝔽[x(t) | Ii(t)] 17.67
OPTIMAL STRATEGY
ˆ zi(t) = Lixcom(t) + Fi(t) ˜ Iloc
i (t)
14.54
17% better
Example
SLIDE 65
Decentralized estimation–(Afshari and Mahajan)
13
Summary
SLIDE 66
Decentralized estimation–(Afshari and Mahajan)
13
Summary
Decentralized estimation–(Afshari and Mahajan)
4
dij
DYNAMICS
x(t + 1) = Ax(t) + w(t), w(t) ∼ 𝒪(0, Q).
OBSERVATIONS
The system consists of n agents. yi(t) = Cix(t) + vi(t), vi(t) ∼ 𝒪(0, Ri).
INFO STRUCTURE
Agents communicate over a strongly connected weighted directed graph. Edge weight dij corresponds to link delay. Ii(t) = {yi(1 : t)} ∪ ( ∪
j∈N−
i
Ij(t − dji)
System model
SLIDE 67
Decentralized estimation–(Afshari and Mahajan)
13
Summary
Decentralized estimation–(Afshari and Mahajan)
4
dij
DYNAMICS
x(t + 1) = Ax(t) + w(t), w(t) ∼ 𝒪(0, Q).
OBSERVATIONS
The system consists of n agents. yi(t) = Cix(t) + vi(t), vi(t) ∼ 𝒪(0, Ri).
INFO STRUCTURE
Agents communicate over a strongly connected weighted directed graph. Edge weight dij corresponds to link delay. Ii(t) = {yi(1 : t)} ∪ ( ∪
j∈N−
i
Ij(t − dji)
System model
Decentralized estimation–(Afshari and Mahajan)
5 ESTIMATES
Each agent generates an estimate ˆ zi(t) = gi,t(Ii(t))
PER-STEP ERROR
Let ˆ z(t) = vec(ˆ z1(t), … , ˆ zn(t)). Then, c(x(t), ˆ z(t)) = (Lx(t) − ˆ z(t))
⊺S(Lx(t) − ˆ
z(t)). where S = ⎡ ⎢ ⎣ S11 ⋅ ⋅ ⋅ S1n ⋮ ⋱ ⋮ Sn1 ⋅ ⋅ ⋅ Snn ⎤ ⎥ ⎦ and L = ⎡ ⎢ ⎣ L1 ⋮ Ln ⎤ ⎥ ⎦
OBJECTIVE
Choose a team estimation problem g to min 𝔽g [
T
∑
t=1
c(x(t), ˆ z(t)) ] ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +λ‖¯ x(t) − ¯ z(t)‖2 ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +
n−1
∑
i=1
λ‖di(t) − ˆ di(t)‖2
System model (continued)
SLIDE 68
Decentralized estimation–(Afshari and Mahajan)
13
Summary
Decentralized estimation–(Afshari and Mahajan)
4
dij
DYNAMICS
x(t + 1) = Ax(t) + w(t), w(t) ∼ 𝒪(0, Q).
OBSERVATIONS
The system consists of n agents. yi(t) = Cix(t) + vi(t), vi(t) ∼ 𝒪(0, Ri).
INFO STRUCTURE
Agents communicate over a strongly connected weighted directed graph. Edge weight dij corresponds to link delay. Ii(t) = {yi(1 : t)} ∪ ( ∪
j∈N−
i
Ij(t − dji)
System model
Decentralized estimation–(Afshari and Mahajan)
5 ESTIMATES
Each agent generates an estimate ˆ zi(t) = gi,t(Ii(t))
PER-STEP ERROR
Let ˆ z(t) = vec(ˆ z1(t), … , ˆ zn(t)). Then, c(x(t), ˆ z(t)) = (Lx(t) − ˆ z(t))
⊺S(Lx(t) − ˆ
z(t)). where S = ⎡ ⎢ ⎣ S11 ⋅ ⋅ ⋅ S1n ⋮ ⋱ ⋮ Sn1 ⋅ ⋅ ⋅ Snn ⎤ ⎥ ⎦ and L = ⎡ ⎢ ⎣ L1 ⋮ Ln ⎤ ⎥ ⎦
OBJECTIVE
Choose a team estimation problem g to min 𝔽g [
T
∑
t=1
c(x(t), ˆ z(t)) ] ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +λ‖¯ x(t) − ¯ z(t)‖2 ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +
n−1
∑
i=1
λ‖di(t) − ˆ di(t)‖2
System model (continued)
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t)
= Ii(t) ∖ Icom(t), State estimate ˆ xcom(t) = 𝔽[x(t) | Icom(t)]. Local Innovation ˜ Iloc
i (t)
= Ii(t) − 𝔽[Iloc
i (t) | Icom(t)].
Let ˆ Σij(t) = cov(˜ Ii(t), ˜ Ij(t)). and ˆ Θi(t) = cov(x(t), ˜ Ii(t))
STRUCTURE OF OPTIMAL ESTIMATORS
ˆ zi(t) = Li ˆ xcom(t) + Fi(t) ˜ Iloc
i (t)
1st term: Common info based estimate 2nd term: Local innovation based correction (depends on weight matrix)
Alternative idea: Common information approach
SLIDE 69
Decentralized estimation–(Afshari and Mahajan)
13
Summary
Decentralized estimation–(Afshari and Mahajan)
4
dij
DYNAMICS
x(t + 1) = Ax(t) + w(t), w(t) ∼ 𝒪(0, Q).
OBSERVATIONS
The system consists of n agents. yi(t) = Cix(t) + vi(t), vi(t) ∼ 𝒪(0, Ri).
INFO STRUCTURE
Agents communicate over a strongly connected weighted directed graph. Edge weight dij corresponds to link delay. Ii(t) = {yi(1 : t)} ∪ ( ∪
j∈N−
i
Ij(t − dji)
System model
Decentralized estimation–(Afshari and Mahajan)
5 ESTIMATES
Each agent generates an estimate ˆ zi(t) = gi,t(Ii(t))
PER-STEP ERROR
Let ˆ z(t) = vec(ˆ z1(t), … , ˆ zn(t)). Then, c(x(t), ˆ z(t)) = (Lx(t) − ˆ z(t))
⊺S(Lx(t) − ˆ
z(t)). where S = ⎡ ⎢ ⎣ S11 ⋅ ⋅ ⋅ S1n ⋮ ⋱ ⋮ Sn1 ⋅ ⋅ ⋅ Snn ⎤ ⎥ ⎦ and L = ⎡ ⎢ ⎣ L1 ⋮ Ln ⎤ ⎥ ⎦
OBJECTIVE
Choose a team estimation problem g to min 𝔽g [
T
∑
t=1
c(x(t), ˆ z(t)) ] ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +λ‖¯ x(t) − ¯ z(t)‖2 ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +
n−1
∑
i=1
λ‖di(t) − ˆ di(t)‖2
System model (continued)
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t)
= Ii(t) ∖ Icom(t), State estimate ˆ xcom(t) = 𝔽[x(t) | Icom(t)]. Local Innovation ˜ Iloc
i (t)
= Ii(t) − 𝔽[Iloc
i (t) | Icom(t)].
Let ˆ Σij(t) = cov(˜ Ii(t), ˜ Ij(t)). and ˆ Θi(t) = cov(x(t), ˜ Ii(t))
STRUCTURE OF OPTIMAL ESTIMATORS
ˆ zi(t) = Li ˆ xcom(t) + Fi(t) ˜ Iloc
i (t)
1st term: Common info based estimate 2nd term: Local innovation based correction (depends on weight matrix)
Alternative idea: Common information approach
Decentralized estimation–(Afshari and Mahajan)
12
x(t) ∈ ℝ4, n = 4 and agent i observes xi(t). Per-step cost: ∑
i∈N
‖xi(t) − ˆ zi(t)‖2 + λ‖¯ x(t) − ¯ z(t)‖2 2-step delay sharing information structure
BASELINE STRATEGY
ˆ zi(t) = Li 𝔽[x(t) | Ii(t)] 17.67
OPTIMAL STRATEGY
ˆ zi(t) = Lixcom(t) + Fi(t) ˜ Iloc
i (t)
14.54
17% better
Example
SLIDE 70
Decentralized estimation–(Afshari and Mahajan)
13
Summary
Decentralized estimation–(Afshari and Mahajan)
4
dij
DYNAMICS
x(t + 1) = Ax(t) + w(t), w(t) ∼ 𝒪(0, Q).
OBSERVATIONS
The system consists of n agents. yi(t) = Cix(t) + vi(t), vi(t) ∼ 𝒪(0, Ri).
INFO STRUCTURE
Agents communicate over a strongly connected weighted directed graph. Edge weight dij corresponds to link delay. Ii(t) = {yi(1 : t)} ∪ ( ∪
j∈N−
i
Ij(t − dji)
System model
Decentralized estimation–(Afshari and Mahajan)
5 ESTIMATES
Each agent generates an estimate ˆ zi(t) = gi,t(Ii(t))
PER-STEP ERROR
Let ˆ z(t) = vec(ˆ z1(t), … , ˆ zn(t)). Then, c(x(t), ˆ z(t)) = (Lx(t) − ˆ z(t))
⊺S(Lx(t) − ˆ
z(t)). where S = ⎡ ⎢ ⎣ S11 ⋅ ⋅ ⋅ S1n ⋮ ⋱ ⋮ Sn1 ⋅ ⋅ ⋅ Snn ⎤ ⎥ ⎦ and L = ⎡ ⎢ ⎣ L1 ⋮ Ln ⎤ ⎥ ⎦
OBJECTIVE
Choose a team estimation problem g to min 𝔽g [
T
∑
t=1
c(x(t), ˆ z(t)) ] ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +λ‖¯ x(t) − ¯ z(t)‖2 ∑
i∈N
‖xi(t)−ˆ zi(t)‖2 +
n−1
∑
i=1
λ‖di(t) − ˆ di(t)‖2
System model (continued)
Decentralized estimation–(Afshari and Mahajan)
9
Common Information Icom(t) = ∩
i∈N
Ii(t), Local Information Iloc
i (t)
= Ii(t) ∖ Icom(t), State estimate ˆ xcom(t) = 𝔽[x(t) | Icom(t)]. Local Innovation ˜ Iloc
i (t)
= Ii(t) − 𝔽[Iloc
i (t) | Icom(t)].
Let ˆ Σij(t) = cov(˜ Ii(t), ˜ Ij(t)). and ˆ Θi(t) = cov(x(t), ˜ Ii(t))
STRUCTURE OF OPTIMAL ESTIMATORS
ˆ zi(t) = Li ˆ xcom(t) + Fi(t) ˜ Iloc
i (t)
1st term: Common info based estimate 2nd term: Local innovation based correction (depends on weight matrix)
Alternative idea: Common information approach
Decentralized estimation–(Afshari and Mahajan)
12
x(t) ∈ ℝ4, n = 4 and agent i observes xi(t). Per-step cost: ∑
i∈N
‖xi(t) − ˆ zi(t)‖2 + λ‖¯ x(t) − ¯ z(t)‖2 2-step delay sharing information structure
BASELINE STRATEGY
ˆ zi(t) = Li 𝔽[x(t) | Ii(t)] 17.67
OPTIMAL STRATEGY
ˆ zi(t) = Lixcom(t) + Fi(t) ˜ Iloc
i (t)
14.54
17% better
Example
Decentralized estimation–(Afshari and Mahajan)
2
Separation centralized stochastic control, the
- ptimal control action depends on the solution of an
estimation problem: 𝔽[
T
∑
t=1
(L(t)˜ x(t) + ˆ z(t))⊺S(t)(L(t)˜ x(t) + ˆ z(t))] Does the same happen in decentralized control? In decentralized estimation, is L 𝔽[x(t) | I(t)] the best estimate? There is a long history of duality between estimation and control. Decentralized control is interesting. Ergo, decentralized estimation is interesting. Decentralized estimation is interesting in it’s own right in certain applications.