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Kalman Filtering with Intermittent Observations: Weak Convergence to a Stationary Distribution

Soummya Kar, Bruno Sinopoli, Jose M. F. Moura

arXiv:0903.2890v2cs.ITcs.LGmath.ST

TL;DR

The paper analyzes the evolution of the Random Riccati Equation and its steady-state behavior. Using a random-dynamical-systems approach, it establishes convergence to a unique invariant distribution under specified arrival-probability conditions and characterizes that distribution's fractal support.

  • Problem

    The paper studies the evolution of the Random Riccati Equation and the associated steady-state distribution.

  • Method

    The paper uses a novel approach based on random dynamical systems and analyzes ergodic invariant measures.

  • Results

    The error covariance converges in distribution to a unique steady-state distribution when γ > γsb, while that distribution has finite mean when γ > γbim.

  • Takeaways & Limitations

    The steady-state distribution has fractal characteristics, and its support can be characterized alongside ergodic behavior.

Abstract

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The paper studies the asymptotic behavior of Random Algebraic Riccati Equations (RARE) arising in Kalman filtering when the arrival of the observations is described by a Bernoulli i.i.d. process. We model the RARE as an order-preserving, strongly sublinear random dynamical system (RDS). Under a sufficient condition, stochastic boundedness, and using a limit-set dichotomy result for order-preserving, strongly sublinear RDS, we establish the asymptotic properties of the RARE: the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution, whose support exhibits fractal behavior. In particular, this weak convergence holds under broad conditions and even when the observations arrival rate is below the critical probability for mean stability. We apply the weak-Feller property of the Markov process governing the RARE to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points, which, in general, is not dense in the set of positive semi-definite matrices. We use the explicit characterization of the support of the invariant distribution and the almost sure ergodicity of the sample paths to easily compute the moments of the invariant distribution. A one dimensional example illustrates that the support is a fractured subset of the non-negative reals with self-similarity properties.

I. INTRODUCTION

The paper studies the distributional asymptotics of Kalman-filter error covariances under Bernoulli observation losses, extending analysis beyond mean stability. It establishes weak convergence to a unique invariant distribution and characterizes that distribution's fractured, potentially fractal support.

  • Motivation: The paper targets the asymptotic distribution of Kalman-filter error covariances when observations arrive through a Bernoulli process.Earlier work primarily studied mean covariance stability, while this paper analyzes the covariance distribution and its fluctuations.
  • Main contributions: For stabilizable and detectable systems, any nonzero packet-arrival probability ensures stochastic boundedness and weak convergence, while mean stability may require a much higher probability.The sufficient stochastic-boundedness condition is also necessary under broad assumptions including stabilizability and detectability.
  • Main contributions: Weak convergence to an attracting invariant distribution can hold below the critical probability required for mean stability.Stochastic boundedness implies weak convergence, whereas operating above the mean-stability threshold ensures finite invariant-distribution mean.
  • Support characterization: The weak-Feller Markov process yields the invariant distribution's support as the closure of an explicitly defined countable set.That set is generally not dense in the positive semidefinite matrices.
  • Scalar example: A scalar example shows that the invariant distribution's support is highly fractured and self-similar, exhibiting characteristics of a fractal set.Almost-sure ergodicity and explicit support identification also enable numerical computation of invariant-distribution moments and probabilities.

B. Notation and Preliminaries

This section introduces the mathematical setting for intermittent-observation Kalman filtering and the random covariance process it generates. The conditional prediction-error covariances follow a random algebraic Riccati equation because packet arrivals are random.

  • System model: Under stabilizability of (A,Q) and detectability of (A,C), the deterministic covariance sequence converges to the algebraic Riccati equation's unique fixed point.This fixed-point behavior provides the deterministic reference for the intermittent-observation model.
  • System model: Observation packets are independently received or dropped according to Bernoulli variables with arrival probability γ.γt = 1 denotes packet arrival and γt = 0 denotes dropout.
  • System model: The conditional prediction-error covariance sequence is updated by a random algebraic Riccati equation.Unlike the classical Kalman-filter case, Pt is random because it depends on the packet-arrival sequence.
  • Operator framework: The analysis treats the covariance process as a Markov family and establishes weak convergence of its induced probability measures.The framework uses transition operators and a Markov-Feller structure on the positive semidefinite matrix space.

CXCT + R

The paper distinguishes stochastic boundedness from boundedness in mean and uses these stability notions to identify critical packet-arrival probabilities. Above the stochastic-boundedness threshold, a unique invariant distribution exists.

  • Stability notions: Stochastic boundedness is uniform boundedness in probability, whereas boundedness in mean is a stronger stability requirement.Boundedness in mean implies stochastic boundedness.
  • Critical probabilities: The critical probability γsb separates stochastic boundedness from explosion: above γsb the covariance sequence is stochastically bounded for every initial condition, while below it explodes.For stochastic boundedness, operation must be strictly above γsb because the defining infimum may not be attained.
  • Critical probabilities: The mean-stability threshold γbim separates finite and divergent expected covariance behavior for at least some initial conditions.Prior results state that the expected covariance diverges as time grows when γ ≤ γbim for an appropriate initial condition.

AXAT + Q

Under stochastic boundedness, the random prediction covariance sequence converges weakly to a unique invariant distribution, including regimes below the critical probability for mean stability. The distribution’s support is characterized as the closure of a countable Riccati-generated set and can be fractured and self-similar.

  • Operating above γsb guarantees weak convergence of {Pt} to a unique invariant distribution independent of the initial condition.
  • Weak convergence can hold below γbim, although finite mean requires γ > γbim.
  • For stable systems, stochastic boundedness and the theorem’s conclusions also hold at γ = 0, whereas unstable systems require γ > 0.
  • Stochastic boundedness is a sufficient condition for existence and uniqueness of an attracting invariant measure in general systems, beyond stabilizability and detectability assumptions.
  • The invariant measure’s support is the topological closure of the countable set S generated by compositions of the Riccati maps applied to P∗.
  • The support is generally not dense in [P∗, ∞), is unbounded and fractured, and exhibits self-similarity in the scalar example.

IV. RANDOM DYNAMICAL SYSTEM FORMULATION

The paper models the Riccati covariance sequence as a random dynamical system driven by a two-sided stationary i.i.d. Bernoulli observation process. This RDS is distributionally equivalent to the original random Riccati iterates.

  • RDS construction: The Riccati recursion is embedded as an RDS whose state space is the cone of positive semidefinite matrices.The construction uses the random observation indicator to select the Riccati map at each iterate.
  • RDS construction: The observation process is represented on a canonical two-sided binary path space with an ergodic left-shift dynamical system.The binary projections are i.i.d., with probability γ of an observation indicator equal to one.
  • RDS construction: The RDS satisfies the cocycle framework through measurable transformations and compositions of the one-step Riccati maps.The paper verifies joint measurability and the cocycle construction needed for the RDS representation.
  • Distributional equivalence: For any deterministic initial covariance, the RDS iterates and the original random Riccati sequence are equivalent in distribution.This follows because each selected map is driven by an i.i.d. Bernoulli coordinate with success probability γ.

V. PROPERTIES OF THE RDS (θ, ϕ)

The constructed RDS is analyzed using order, sublinearity, equilibria, and boundedness properties. Under the stated positivity and ergodicity assumptions, a limit-set dichotomy supplies a unique positive almost equilibrium in the relevant regime.

  • Equilibria and orbits: An equilibrium is a random state invariant under the RDS, so its iterates have the same distribution as the equilibrium itself.The paper distinguishes an everywhere-valid equilibrium from an almost equilibrium holding outside a null set.
  • Equilibria and orbits: Forward and pull-back orbits are equivalent in distribution because the underlying transformations preserve the probability measure.This lets pull-back asymptotic results yield distributional conclusions for forward trajectories.
  • Limit-set dichotomy: For an ergodic, conditionally compact, strongly sublinear, order-preserving RDS with positive forcing, the limit-set dichotomy gives precisely one of two alternatives.One alternative includes a unique positive almost equilibrium on an invariant full-measure set.
  • Order and sublinearity: The Riccati RDS is order-preserving, and when Q is positive definite it is also strongly sublinear.Order preservation follows from the corresponding property of the one-step maps; strong sublinearity is extended from one step to arbitrary times.
  • Order and sublinearity: The positive definite process noise condition Q ≫ 0 is used to establish strong sublinearity of the Riccati RDS.The proof establishes the property for one step and then extends it inductively to all times.

VI. PROOFS OF THEOREMS 9,10

The proofs establish weak convergence of the covariance sequence to a unique stationary distribution by combining the RDS dichotomy with boundedness and distributional equivalence. They also connect the result to unique ergodicity of the associated Markov-Feller process.

  • Theorem 9: The proof verifies the hypotheses of the limit-set theorem using finite dimensionality, cone properties, and the positivity assumptions on the Riccati model.These conditions rule out the alternative to the almost-equilibrium case in the regime considered.
  • Theorem 9: For γ above the stochastic-boundedness threshold, the RDS has a unique almost equilibrium and the associated Markov-Feller process is uniquely ergodic.The proof first obtains convergence for initial conditions dominated by a multiple of the equilibrium, then extends it to arbitrary initial states.
  • Theorem 9: A modified initial condition equal to the original covariance almost surely is constructed so that it is dominated by the almost equilibrium.This permits application of the restricted convergence result without changing the distribution of the desired process.
  • Theorem 9: The random prediction-error covariance sequence converges weakly to the unique stationary distribution.The conclusion follows from almost-sure convergence for the modified initial condition, distributional equivalence of forward and pull-back orbits, and the RDS representation.
  • Theorem 9: The proof also derives the theorem’s additional boundedness conclusion using an existing mean-stability result and Fatou’s lemma.The cited argument combines the established distributional convergence with standard expectation bounds.

B. Proof of Theorem 10

The proof characterizes the invariant measure’s support using the weak-Feller property and the points reachable by finite compositions of the two Riccati maps. The resulting support is the closure of a countable, generally fractured set.

  • Support characterization: The weak-Feller Markov process makes the invariant measure attractive and allows its support to be identified from reachable finite-time states.Starting from any state, the only points reached with nonzero probability are obtained by applying finite compositions of f0 and f1.
  • Support characterization: The support of the invariant measure is the topological closure of the set generated by finite compositions of f0 and f1 applied to the deterministic Riccati fixed point.The reachable set is explicitly defined as S = {f_i1 ◦ f_i2 ◦ ··· ◦ f_is(P*)} over all finite binary compositions.
  • Support geometry: The generated support lies above the deterministic fixed point P* and is not generally dense in the interval [P*, ∞).The lower-bound property is established inductively using order preservation of f0 and f1.

VII. A SCALAR EXAMPLE AND NUMERICAL STUDIES

The scalar example shows that the invariant distribution’s support is highly fractured, self-similar, and generally not dense in its ambient interval.

  • A. Scalar Example: The scalar support is a highly fractured subset of [P ∗, ∞) with self-similar, fractal-like structure.The paper distinguishes internal fractures from holes between consecutive sets in the support construction.
  • A. Scalar Example: For n ≥ 1, the recursive sets satisfy Sn = {2Y + 1, Y ∈ Sn−1}, producing self-similar support segments.The first recursive set occupies a later interval, and subsequent sets repeat the scaled structure.
  • A. Scalar Example: The support is built from a base set S0 whose stretched copies Sn recur across successive intervals, with stretching factor 2^n.Each Sn is obtained recursively, and the pattern repeats over the real line.
  • A. Scalar Example: Open intervals between consecutive Sn sets are absent from the support, creating large-scale holes in addition to internal fractures.The construction therefore yields both inter-set gaps and finer disconnected components within S0.
  • A. Scalar Example: The authors call the structure “fractal like” because they do not establish self-similarity at every scale.They note that a rigorous fractal characterization is beyond the scope of the analysis.

B. Numerical Studies

Numerical studies examine eigenvalue and trace distributions of the invariant measure and show concentration near the deterministic Riccati fixed point as the arrival probability approaches one.

  • B. Numerical Studies: The simulations use a 10-dimensional system to study invariant-measure eigenvalue distributions for γsb < γ < 1.The system matrices are generated randomly, with dimensions 10 × 10 and 5 × 10.
  • B. Numerical Studies: As γ increases to 1, the largest-eigenvalue distributions approach δλ10(P ∗), concentrated at the deterministic fixed point’s largest eigenvalue.For γsb < γ < 1, the eigenvalue support lies in [λ10(P ∗), ∞).
  • B. Numerical Studies: As γ increases to 1, the trace distributions approach δTr(P ∗), concentrating the conditional mean-squared error at the deterministic fixed point.The trace is interpreted as the conditional mean-squared error.
  • B. Numerical Studies: Because the Riccati sequence converges in distribution to µγ, designers can tune γ to target satisfactory eigenvalue and trace behavior.The paper connects these distributions to control and estimation system design.
  • B. Numerical Studies: Almost-sure ergodicity means a single sample path suffices to compute invariant-distribution moments under the stated assumptions.This avoids empirically generating the full invariant distribution, though integrability assumptions remain relevant.

VIII. CONCLUSIONS AND FUTURE WORK

The conclusions present a random-dynamical-systems analysis of Kalman filtering with Bernoulli observation losses and summarize convergence, support, and computational consequences.

  • VIII. CONCLUSIONS AND FUTURE WORK: The approach analyzes the random Riccati equation through random dynamical systems to characterize the filter’s steady-state behavior.The paper presents this as a novel framework for the Bernoulli-loss observation model.
  • VIII. CONCLUSIONS AND FUTURE WORK: The error covariance converges in distribution to a unique steady-state distribution when the arrival probability satisfies γ > γsb.The steady-state distribution has finite mean only when γ > γbim.
  • VIII. CONCLUSIONS AND FUTURE WORK: The steady-state distribution has fractal support, and ergodicity enables numerical evaluation of the error-covariance distribution.The support characterization and ergodic arguments jointly provide the computational route.
  • VIII. CONCLUSIONS AND FUTURE WORK: The analysis and results are stated to extend to control over erasure channels.The authors identify more complex communication-control tradeoffs as future work.

APPENDIX A

The appendix establishes stochastic boundedness and weak-Feller properties for the Markov process governing the random Riccati recursion.

  • APPENDIX A: The transition operator is weak-Feller because the Riccati process switches between continuous Lyapunov and Riccati maps.The transition probability is expressed as a γ-weighted combination of the two map outcomes.
  • APPENDIX A: The boundedness argument covers unstable A when C is invertible, while stable A admits boundedness through the state variance.The appendix notes that the general non-invertible case is treated elsewhere.
  • APPENDIX A: The tail bound is obtained by tracking the most recent observation arrival and bounding the intervening Lyapunov growth.The probability of a gap of length k is bounded by (1 − γ)^k.
  • APPENDIX A: For every γ > 0, the prediction-covariance sequence is stochastically bounded from every initial condition under the stated assumptions.The proof uses increasingly likely observation arrivals and an upper bound on the covariance tail probability.
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