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Robust Kalman Filtering under Model Perturbations

Mattia Zorzi

arXiv:1508.01906v3math.OC

TL;DR

The paper addresses robust filtering under model perturbations by formulating incremental uncertainty with a τ-divergence minimax approach. It derives a family of Kalman-like robust estimators and reports that τ tunes their conservativeness, while extending risk-sensitive filtering to a related family.

  • Problem

    The paper addresses robust filtering when model increments may be perturbed, with mismodeling tolerance specified at each increment.

    Source: §I · p. 2, §VIII · p. 16

  • Method

    It bounds incremental model perturbations using the τ-divergence family and derives minimax estimators against least-favorable models.

    Source: §VIII · p. 16

  • Results

    The proposed robust estimators have an iterative structure similar to the Kalman filter, and the simulation study shows that τ tunes how conservative the robust filter is.

    Source: §VIII · p. 16

  • Takeaways & Limitations

    The paper extends the risk-sensitive filter to a family of risk-sensitive-like filters.

    Source: §VIII · p. 16

Abstract

from arXiv · show

We consider a family of divergence-based minimax approaches to perform robust filtering. The mismodeling budget, or tolerance, is specified at each time increment of the model. More precisely, all possible model increments belong to a ball which is formed by placing a bound on the Tau-divergence family between the actual and the nominal model increment. Then, the robust filter is obtained by minimizing the mean square error according to the least favorable model in that ball. It turns out that the solution is a family of Kalman like filters. Their gain matrix is updated according to a risk sensitive like iteration where the risk sensitivity parameter is now time varying. As a consequence, we also extend the risk sensitive filter to a family of risk sensitive like filters according to the Tau-divergence family.

I. INTRODUCTION

The section motivates robust filtering for model mismatch and extends incremental divergence-based minimax filtering to τ-parametrized robust Kalman and risk-sensitive-like filters.

I. INTRODUCTION

  • The introduction notes that a linear state-space model may inadequately describe phenomena and that the resulting Kalman filter does not perform well in practice.This motivates developing robust versions of the standard Kalman filter.Source: §I · p. 1
  • The cited risk-sensitive approach yields a Kalman-like filter, but its sensitivity parameter is not explicitly connected to the discrepancy between actual and nominal models.The paper frames this missing connection as motivation for its divergence-based extension.Source: §I · p. 2
  • Earlier dynamic robust filtering assigns a mismodeling budget to each model time increment and yields a Kalman-like estimator with a time-varying risk-sensitivity parameter.This approach iterates the Bayes estimator using least-favorable statistics.Source: §I · p. 2
  • The paper extends robust Kalman filtering under incremental model perturbations to a τ-divergence-parametrized family, and likewise extends risk-sensitive filtering to τ-risk-sensitive-like filters.The proposed filters have a time-varying risk-sensitivity parameter; the simulation study shows that τ tunes the robust filter's conservativeness.Source: §I · p. 2

II. ROBUST STATIC ESTIMATION

This section formulates robust static estimation as a minimax problem over a τ-divergence uncertainty ball. In the Gaussian case, the robust estimator is the Bayes estimator, with the least-favorable model perturbing covariance.

II. ROBUST STATIC ESTIMATION

  • For Gaussian models, τ-divergence measures deviation between actual and nominal models through their means and covariances.The divergence is described via a normalized prediction error; it is nonnegative and vanishes when the densities match.Source: §II · pp. 3–4
  • The minimax formulation selects an estimator minimizing mean-square error against the worst-case density in the τ-divergence ball around the nominal model.The ball's tolerance represents the allowed mismodeling budget.Source: §II · p. 4
  • Under Gaussian assumptions, the Bayes estimator is robust over the τ-parametrized uncertainty class, with only x's covariance perturbed relative to nominal.The section also identifies the optimal robust estimator as the Bayes estimator.Source: §II · pp. 4–5
  • Corollary 2.1 identifies the least-favorable probability density and states that the optimal estimator is the Bayes estimator (8), for a priori fixed θ > 0 with θ^-1 > (1 − τ)∥P∥.Source: §II · p. 5

III. ROBUST FILTERING PROBLEM

This section defines robust filtering by placing a τ-divergence tolerance on each model increment and minimizing mean-square error against a least-favorable transition model. The formulation assumes Gaussian conditional densities and recovers the cited robust filter when τ = 0.

III. ROBUST FILTERING PROBLEM

  • The filter chooses an estimator to minimize mean-square error when evaluated under a least-favorable transition density in the allowed ball.The estimator depends on the observation history Y_t, although that dependence is suppressed in the notation.Source: §III · pp. 6–7
  • The formulation assumes the conditional density of z_t given past observations under the least-favorable model is Gaussian.The authors additionally assume the least-favorable transition density yields that Gaussian conditional density.Source: §III · p. 7
  • The robust filtering problem bounds τ-divergence from a pseudo-nominal transition model at each time step, with a separate tolerance c_t.The resulting minimax estimator has finite second-order moments over the allowed model class.Source: §III · p. 7
  • For τ = 0, the solution to the minimax problem coincides with the earlier robust-filtering solution in.The section notes that the corresponding divergence is Kullback-Leibler when the relevant densities are Gaussian.Source: §III · p. 7

IV. ROBUST KALMAN FILTERS

The optimal robust estimator solves a minimax problem and has a Kalman-like form. Its model uncertainty is represented by a τ-divergence ball, yielding a parametric family of robust Kalman filters.

IV. ROBUST KALMAN FILTERS

  • The minimax solution is a Kalman-like estimator, forming a parametric family of robust Kalman filters.The family is characterized by τ.Source: §IV · p. 8
  • The uncertainty set bounds the τ-divergence between candidate and nominal conditional densities by c_t.The minimax problem is reformulated as a static minimax problem over this ball.Source: §IV · p. 8
  • The derivation uses a Gaussian pseudo-nominal density and characterizes the least-favorable conditional covariance.The passages identify this covariance as part of the least-favorable conditional model.Source: §IV · p. 8

yt . From (17), we obtain

The robust filter follows an iterative scheme with Kalman-filter structure, but distorts the covariance through V_t and updates its gain through a Riccati-like equation. The τ=0 case has radial distortion as a function of estimation error.

yt . From (17), we obtain

  • The robust filter retains Kalman-filter iterative structure, with covariance distortion through V_t and a Riccati-like gain update.Algorithm 1 gives the iterative scheme for the robust estimator.Source: yt . From (17), we obtain · p. 9
  • For τ=0, the case considered in, the distortion is a radial function of the estimation.Source: yt . From (17), we obtain · p. 10

V. LEAST-FAVORABLE MODEL

The least-favorable model is characterized by moving the distortion into the noise, whose conditional distribution depends on the estimation error. The section also describes its state-space representation and a procedure for evaluating estimators under that model.

V. LEAST-FAVORABLE MODEL

  • The least-favorable noise has a Gaussian conditional distribution that depends on the estimation error.The distortion is moved into noise v_t to characterize the least-favorable model.Source: §V · p. 11
  • The least-favorable model admits a state-space representation with an input that is white Gaussian noise of unit variance.The passages introduce the representation's matrices but do not specify their values here.Source: §V · p. 11
  • Constructing the least-favorable model uses a forward sweep for gains and a backward sweep for Ω_t over [0, T].Extending the simulation interval requires a new backward sweep.Source: §V · p. 11
  • The error-covariance quantity Π_t obeys a Lyapunov equation with Π_0 = I_2 ⊗ V_0.The passage attributes the Lyapunov result to.Source: §V · p. 12

VI. RISK SENSITIVE FILTERING

The paper extends risk-sensitive filtering to the τ-divergence family. The resulting estimator has the same form as the robust filter, while its risk-sensitivity parameter is constant rather than time-varying.

VI. RISK SENSITIVE FILTERING

  • A larger θ penalizes estimators with large errors more strongly in the risk-sensitive objective.The classic filter is linked to the minimax formulation with constant θ.Source: §VI · p. 12
  • The τ-risk-sensitive estimator takes the same form as the robust filter in equations (19)–(20).The extension applies the τ-divergence family to the minimax problem.Source: §VI · p. 13
  • For 0 ≤ τ < 1, V_{t+1} is defined when 0 < P_{t+1} < (θ(1 − τ))^-1 I_n; for τ=1, positive definiteness of P_{t+1} suffices.These are the stated conditions for the covariance update to be well-defined.Source: §VI · p. 13
  • The extension uses a constant risk-sensitivity parameter, unlike the robust filter's time-varying parameter governed by c_t.The algorithmic scheme is similar, with Step 4 removed.Source: §VI · p. 13

VII. SIMULATION RESULTS

The simulations compare the standard Kalman filter with two robust Kalman filters under nominal and perturbed models. The robust filters’ steady-state risk sensitivity depends on τ.

VII. SIMULATION RESULTS

  • The experiment considers three filters: the standard Kalman filter (KF), robust Kalman filter RKF0 with τ = 0, and robust Kalman filter RKF1.The model is time-invariant, and x0 ∼ N(0, V0) with V0 = 0.01·I2.Source: §VII · p. 13
  • RKF0 and RKF1 have steady-state risk sensitivities of approximately 0.19 and 0.23, respectively.These values correspond to the risk-sensitive filters with τ = 0 and τ = 1.Source: §VII · p. 14
  • The study compares filter error variances under nominal and least-favorable models, with large and small differences between those models.The stated tolerance settings distinguish large discrepancy (c large) from similar models (c small).Source: §VII · p. 14

A. Large tolerance

With large tolerance, the robust filters outperform the standard Kalman filter even when the least-favorable model comes from a different τ class. Which filter performs best depends on the model used for evaluation.

A. Large tolerance

  • On the nominal model, the standard Kalman filter performs best, while RKF1’s error variances are slightly smaller than RKF0’s.The nominal-model results are shown in Figure 2’s first row.Source: A. Large tolerance · pp. 14, 16, Figure 2 · p. 15
  • For model (23) with τ = 0 and c = 10−1, RKF0 is the best estimator because it was designed with respect to this model.Figure 2’s second row depicts the variances for this condition.Source: A. Large tolerance · p. 14, Figure 2 · p. 15
  • On the τ = 1 least-favorable model, RKF1 is best because it is optimal for the underlying model, and the standard Kalman filter is worst.This comparison is shown in Figure 2’s third row.Source: A. Large tolerance · pp. 15–16
  • Smaller τ makes the filter more conservative by admitting models with larger mean-square error into the uncertainty class.The paper notes this property was also observed for static estimation.Source: A. Large tolerance · p. 16
  • The robust Kalman filters outperform the standard Kalman filter even when the least-favorable model belongs to an uncertainty class with a different τ.The experiments also report that the best robust filter is the one designed for the underlying model.Source: A. Large tolerance · pp. 14, 16

B. Small tolerance

With small tolerance, the robust filters perform comparably to the standard Kalman filter; when model discrepancy is limited, the standard filter’s performance does not deteriorate much.

B. Small tolerance

  • At tolerance c = 5 · 10−3 (Figure 3), RKF0 and RKF1 provide the same performance, comparable with that of the standard Kalman filter.Source: B. Small tolerance · p. 16
  • When the nominal and least-favorable models are not too different, the standard Kalman filter applied to the least-favorable model does not deteriorate much.Source: B. Small tolerance · p. 16

VIII. CONCLUSIONS

The paper formulates robust filtering under incremental model perturbations using the τ-divergence family. Its minimax estimators retain a Kalman-like iterative structure, and simulations show τ tunes conservatism.

VIII. CONCLUSIONS

  • The proposed robust estimators solve a minimax problem for incremental model perturbations characterized by the τ-divergence family.Source: §VIII · p. 16
  • The estimators have an iterative structure similar to the Kalman filter, and the corresponding least-favorable models are derived.Source: §VIII · p. 16
  • The paper extends the risk-sensitive filter to a family of risk-sensitive-like filters.Source: §VIII · p. 16
  • The simulation study shows that τ tunes how conservative the robust filter is.Source: §VIII · p. 16
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