[PPT] - a geometric approach * * ACC09 paper plus other stuff Andrea Censi PowerPoint Presentation

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Kalman filtering with intermittent observations: a geometric approach *

* ACC’09 paper plus other stuff

Andrea Censi

Ph.D. student, Control & Dynamical Systems, California Institute of Technology Advisor: Richard Murray

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Linear/Gaussian estimation

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Consider the discrete-time linear dynamical system x(k + 1)

=

A x(k) + B ω(k), y(k)

=

C x(k) + v(k), with ω(k) and v(k) white Gaussian sequences with covariance matrix equal to identity.

■ Let Q BB∗ and I C∗C. Then the posterior covariance matrix of the error

P(k) = cov (ˆ x(k) − x(k)|y(1), . . . , y(k)) evolves according to the following deterministic map: g : P →

(APA∗ + Q)−1 + I

−1 (a Riccati iteration “in compact form” for the lazy researcher)

■ If (A, B) controllable and (A, C) detectable, P ∞ = limn→∞ gn(P(0))

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... with intermittent observations

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■ [Sinopoli’04] One can model packet drops as follows:

y′(k)

=

γ(k)y(k) γ(k)

∼

Bernoulli, with probability γ

■ Evolution of P(k) as an Iterated Function System [Barnsley]:

◆ execute g if the packet arrives ◆ execute h if the packet does not arrive

g : P

→

(APA∗ + Q)−1 + I

−1 , pg = γ h : P

→

APA∗ + Q, ph = 1 − γ

■ The iteration of P(k) is not deterministic: rather than P ∞, we must speak of the

stationary distribution of P.

■ What is the behavior as a function of the arrival probability γ?

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Three contributions

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1. Existence of the stationary distribution:

■ Literature: Stationary distribution exists if γ > γs [Kar’09]. ■ Contribution: If A nonsingular, it always exists.

2. Mean of the stationary distribution:

■ Literature:

◆ E{P} exists if and only if γ > γc [Sinopoli’04], ◆ γc not precisely characterized yet. [Mo’08, Plarre’09,...].

■ Contribution: The intrinsic Riemannian mean always exists.

3. CDF (performance bounds):

■ Literature: Upper and lower bounds on P({P ≤ M}). [Epstein’05] ■ Contribution: if a certain non-overlapping condition holds, then:

◆ p(P) has a fractal support. ◆ P({P ≤ M}) can be found in closed form.

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A really useful metric

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Let P be the set of positive definite matrices of order n. Define: d(P1, P2) =

n

∑

i=1

log2(λi(P−1

1 P2))

1/2

1. (P, d) is a complete metric space, with the usual topology.
2. d is invariant to conjugacy. For any invertible matrix A:

d(AP1A∗, AP2A∗) = d(P1, P2)

3. d is invariant to inversion:

d(P−1

1 , P−1 2 ) = d(P1, P2)

4. For any two matrices P1, P2 in P, and for any Q ≥ 0,

d(P1 + Q, P2 + Q) ≤ α α + β d(P1, P2) where α = max{λmax(P1), λmax(P2)} and β = λmin(Q).

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Contraction properties of g and h

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■ The maps g, h are compositions of: conjugations, addition, inversion.

g : P

→

(APA∗ + Q)−1 + I

−1 h : P

→

APA∗ + Q

■ Therefore, they are non-expansive in this metric: (also h!)

d(g(P1), g(P2)) ≤ d(P1, P2) d(h(P1), h(P2)) ≤ d(P1, P2)

■ [Bougerol ’93]: If A nonsingular, (A, C) observable, (A, B) controllable, gn is a strict

contraction: d(gn(P1), gn(P2)) ≤ ρ d(P1, P2), ρ < 1

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Existence of stationary distribution

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■ Lemma [Barnsley’88]. An iterated function system { fi} where all fi are

non-expansive, and at least one is a strict contraction, admits a unique attractive stationary distribution (convergence in distribution).

■ Proposition: A stationary distribution for the covariance always exists if A

nonsingular, (A, C) observable, (A, B) controllable. Proof: After n steps, there are 2n combinations of g and h.

n

hhh · · · h

non-expansive ghh · · · h non-expansive . . . hgg · · · g non-expansive ggg · · · g strict contraction Therefore the system satisfies the average-contractivity condition.

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Existence of Riemannian mean(s)

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■ Does it not bother you that E{·} is not invariant to change of coordinates?

E{P} = E{

√

P}2 = E{P−1}−1 Qualitative/quantitative results should be invariant of parametrizations.

■ The manifold of positive definite matrices is not flat; we must be careful. ■ The expectation can be generalized to a Riemannian manifolds M with distance d as

follows: M{X} arg inf

y∈M E

d2(X, y)
■ In our case, different distances give different “critical probabilities”:

covariances std devs information Riemannian d =

||P1−P2||F || √

P1−

√

P2||F

||P−1

1 −P−1 2 ||F

∑ log2(λi(P−1

1 P2)

1

2

γd

c =

γc

< γc

none none

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Which is more natural?

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covariances std devs information Riemannian d =

||P1−P2||F || √

P1−

√

P2||F

||P−1

1 −P−1 2 ||F

∑ log2(λi(P−1

1 P2)

1

2

γd

c =

γc

< γc

none none

■ Covariances: Corresponds to the average squared error E{||e||2}. ■ Standard deviations

◆ Corresponds to the average error E{||e||} (physical meaning).

■ Information matrices

◆ P−1 is the natural parametrization for Gaussians.

■ Riemannian distance

◆ Information Geometry interpretation: the natural distance from the Fisher

Information Metric on the manifodl of Gaussian distributions.

◆ d(P1, P2) can be linked to the probability of distinguishing the two

distributions from the samples [Amari’00].

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Finally the fractals

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Cantor set and Cantor function

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■ Instructions: take a segment, remove the middle third, repeat.

↑ Cantor Set

The CDF is the Cantor Function →

■ The Cantor set is a “fractal”:

◆ it is totally disconnected ◆ it is self-similar ◆ it has non-integer dimension

■ The Cantor function is a singular function (“devil’s staircase”)

◆ continuous ◆ differentiable almost everywhere, with derivative 0

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Cantor set

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■ Proposition: For some value of the parameters, the distribution of P−1 is exactly the

(scaled/translated) Cantor set.

■ Black plot: nominal system A = 1/

√

3, Q = 0, I = 1 and packet dropping governed by a Markov chain with transition matrix T = [α, 1 − α; 1 − β, β], α = β = 0.5.

Varying Q: Q = 0; Q = 1; Q = 2; Q = 3. Varying the parameters of the Markov Chain: α = 0.5, β = 0.5; α = 0.3, β = 0.3; α = 0.1, β = 0.1; α = 0.9, β = 0.9.

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Fractal properties

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■ Proposition: If h and g have disjoint range, then:

◆ the stationary distribution is homeomorphic to the Cantor set, ◆ it is a compact, totally disconnected set ◆ the cumulative distribution function is a singular function

■ If h and g have non-disjoint range, then it might be still fractal.

◆ This is an open problem in number theory.

■ Proposition: If h and g satisfy the stronger condition:

h(P

∞) ≥ g(0)

then one can find a closed form solution for P({P ≤ M}) = "ugly" formula involving the "digit representation" of M

◆ This implies that C is invertible (strong condition). ◆ This is also valid for Markov Chain driving the packet drops.

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Conclusions

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Three contributions:

■ The stationary distribution always exists. ■ The intrinsic Riemannian mean always exists. ■ P({P ≤ M}) in closed form with a (strong) non-overlapping condition.

Main ideas:

■ Theory of Iterated Function Systems — many ready-to-use results in books with

very nice fractal illustrations [Barnsley].

■ A useful Riemannian metric for positive definite matrices — natural Information

Geometry metric for manifold of Gaussian distributions

■ Contraction properties of Riccati recursions in this metric.

Open problems:

■ Case of a singular A. ■ Computing the Riemannian mean (only proved existence). ■ Computing the CDF P({P ≤ M}) without non-overlapping condition.

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Metric spaces properties

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■ (P, d) is a complete metric space: every Cauchy sequence has a limit in P. ■ Fixed Point Theorem: If f strict contraction mapping:

sup

x,y

d( f (x), f (y)) d(x, y)

= q < 1

and complete metric space, then limn→∞ f n(x) = x0.

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Cantor set, more formally

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■ Let {0, 1}N the Cantor space with the following metric:

d(x, y) = 2−k; k first digit for which xk = yk Note that with this metric, 0.11111 · · · = 1.0000 . . . because d(0.1, 1.0) = 0.

■ Any set is called “Cantor set” if it is homeomorphic to the Cantor space.

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Cantor set, more formally

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■ Let {0, 1}N represent the arrival sequence. You can write the final covariance as:

ϕ : {0, 1}N

→

P arrival sequence

→

final covariance

■ If you can prove that ϕ is invertible, then the range of ϕ is a Cantor set. ■ Non-overlapping condition: If the ranges of h and g are non-overlapping, then ϕ is