Optimal Distributed State Estimation with Communication
Cost: A Majorization Theory Approach
Gabriel M. Lipsa and Nuno C. Martins
Majorization Theory:
xk+1 = axk + wk
Problem Formulation:
Estimator
Pre-processor
2 2 )
wk ∼ N 0, σW
(NσW
x0 = 0
vk = Pk {xt }kt=0
k
x̂k = Ek {{vt }t=0
Pk : Rk+1 → R {∅}
Ek : (R {∅}
{∅})k+1 → R
The symmetric nonincreasing rearrangement of the function f is given by:
σ
f (x) =
rk =
0, vk = ∅
∅
1, vk =
Proof of Theorem 2:
∞
0
I(z∈Rn :f (z)>ρ)σ (x)dρ
Let {τk }Tk=1 be a sequence of positive real numbers. We define the policy
P o as it follows:
• if vt = ∅ for all t ∈ {{1, . . . , k − 1},
} let vk = ∅ if xk ∈ [−τ
− k , τk ], otherwise
let vk = xk ;
• after a perfect sample was sent, adopt the optimal threshold policy for the
remaining time.
T
2
2
Jβ α, σW
, C, P, E = lim
β k E (xk − x̂k ) + Crk
T →∞
2
Jβ α, σW
,C
=
k=0
2
min Jβ α, σW
, C, P, E
P,E
f
Optimal solution:
f g if:
k
E(α),k {vt }t=0 = x̂k =
P(τ ),k
k
{ t }t=0
{x
=
αx̂k−1 , vk = ∅
, k≥1
vk ,
ow
∅
∅,
xk
fK (x) =
−τk < xk − x̂k < τk
−
,k ≥ 1
ow
Theorem 1: Let the following parameters be given: the variance of the
2
process noise σW
, the system’s dynamic constant α, the communication cost C,
and the discount factor β. There exist a positive real constant ν, such that P(τ )
∞
2
and E(α) are an optimal solution to min Jβ α, σW , C, P, E , where τ = {τk }k=0
P ,E
is a sequence of positive real numbers such that τk = ν for all positive integers k.
JT,β
2
α, σW
, C, P, E
2
JT,β α, σW
,C
=
=
T
k=0
k
2
β E (xk − x̂k ) + Crk
2
min JT,β α, σW
, C, P, E
P,E
Theorem 2: Let the following parameters parameters be given: the variance
2
, the system’s dynamic constant α, the communication
of the process noise σW
cost C, the discount factor β and the time horizon T . There exists a sequence
T
of positive real numbers τ = {τk }k=0 , such that P(τ ) and E(α) are an optimal
2
solution to min JT,β α, σW , C, P, E .
P,E
Remark: The optimal cost JT,β
condition x0 .
Theorem 2 can be proved using an induction argument. Choose an arbitrary
pre-processor policy P, this policy will define ω k , γk|k−1 and γk for all k ∈
{1, . . . , T }
2
α, σW
,C
does not depend on the initial
fσ
σ
||x||≤ρ
f (x)dx ≥
K
f (x)
,x
f (x)dx
||x||≤ρ
g σ (x)dx, for all ρ ≥ 0
The cost obtained by adopting the policy P can be lower bounded as follows:
JT,β
∈K
0, x ∈
/K
A nonnegative function f : Rn → R+ is symmetric and nonincreasing if
it can be expressed via a nonincreasing function g : R+ → R+ such that
f (x) = g(||x||).
Lemma: Let f, g : R → R be two probability distribution functions, such
that f is symmetric and nonincreasing and f g. Let κ be a real number in
interval κ ∈ (0, 1). Let K = [−τ, τ ] be the symmetric interval,
such that
the
τ
f (x)dx = 1 − κ. For any function λ : R → [0, 1] satisfying R g(x)λ(x)dx =
−τ
1 − κ, the following holds:
fK g·λ
1−κ
f ∗bg∗b
Lemma: Let f be a neat and even probability distribution function on the
real line. Let µ, be a probability
distribution function on the real line, such that
µ ≺ f . If we define x̂µ = R xµ(x)dx, then the following holds:
2
x
f (x)dx ≤
R
2
α, σW
, C, P
P, E
T
2
(P)
P ≥ k=1 Eω k (xk − x̂k ) γk
2
+ C + JT −k,β α, σW , C (1 − γk|k−1 )γk−1
The cost obtained by adopting the policy P o is given by:
JT,β
2
o
(P ) = k=1 Eωok (xk − x̂k ) γk
2
+ C + JT −k,β α, σW , C (1 − γk|k−1 )γk−1
2
α, σW
, C, P o , E
o
T
It follows then that:
2
JT,β α, σW
, C, P, E
Proof of Theorem 1:
(P) ≥
2
JT,β α, σW
, C, P o , E
o
(P )
Let the time horizon T go to infinity and the results follow.
Lemma [Hajek]: Let f and g be two probability distribution functions
on Rn , with f is symmetric and nonincreasing and f g. For a symmetric
nonincreasing probability distribution function b the following holds:
Preliminary notations:
Remark:For a given
pre-processor policy P, the estimtor which minimizes
2
, C is the conditional expectation of the state xk given the
the cost JT,β α, σW
entire past, i.e. the estimator which computes x̂k = E [xk |vt , 1 ≤ t ≤ k].
o
The policy P o will define ω ok , γk|k−1
and γko for all k ∈ {1, . . . , T }. Choose
o
the thresholds τk such that γk|k−1 = γk|k−1
for all k ∈ {1, . . . , T }. It can be
shown that ω ok ω k for all k ∈ {1, . . . , T }, and moreover ω ok is symmetric and
nonincreasing.
2
(x
−
x̂
)
µ(x)dx
µ
R
ω k = f (xk |vt = ∅, 1 ≤ t ≤ k, x0 )
γk = P (vt = ∅, 1 ≤ t ≤ k, x0 )
γk|k−1 = P (vk = ∅|vt = ∅, 1 ≤ t ≤ k − 1, x0 )
Conclusions:
We have shown that for the given problem, the decision to send or not to send
a sample of the underlying stochastic process depends only on the estimation
error and that there exists an optimal threshold policy.
References:
Bruce Hajek, Kevin Mitzel and Sichao Yang, “Paging and Registration in
Cellular Networks: Jointly Optimal Policies and an Iterative Algorithm,” IEEE
Transactions on Information Theory, Feb 2008
O. C. Imer and T. Basar. “Optimal estimation with limited measurements,”
in Proc. IEEE CDC/ECC 2005, Seville, Spain submitted to IEEE Transactions
on Automatic Control
Albert W. Marshall, Ingram Olkin, Barry Arnold, ”Inequalities: Theory of
Majorization and Its Applications (Springer Series in Statistics),” Academic
Press, New York