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Matrix Concentration Inequalities for Sensor Selection

Christopher I. Calle, Shaunak D. Bopardikar

Abstract

In this work, we address the problem of sensor selection for state estimation via Kalman filtering. We consider a linear time-invariant (LTI) dynamical system subject to process and measurement noise, where the sensors we use to perform state estimation are randomly drawn according to a sampling with replacement policy. Since our selection of sensors is randomly chosen, the estimation error covariance of the Kalman filter is also a stochastic quantity. Fortunately, concentration inequalities (CIs) for the estimation error covariance allow us to gauge the estimation performance we intend to achieve when our sensors are randomly drawn with replacement. To obtain a non-trivial improvement on existing CIs for the estimation error covariance, we first present novel matrix CIs for a sum of independent and identically-distributed (i.i.d.) and positive semi-definite (p.s.d.) random matrices, whose support is finite. Next, we show that our guarantees generalize an existing matrix CI. Also, we show that our generalized guarantees require significantly fewer number of sampled sensors to be applicable. Lastly, we show through a numerical study that our guarantees significantly outperform existing ones in terms of their ability to bound (in the semi-definite sense) the steady-state estimation error covariance of the Kalman filter.

Matrix Concentration Inequalities for Sensor Selection

Abstract

In this work, we address the problem of sensor selection for state estimation via Kalman filtering. We consider a linear time-invariant (LTI) dynamical system subject to process and measurement noise, where the sensors we use to perform state estimation are randomly drawn according to a sampling with replacement policy. Since our selection of sensors is randomly chosen, the estimation error covariance of the Kalman filter is also a stochastic quantity. Fortunately, concentration inequalities (CIs) for the estimation error covariance allow us to gauge the estimation performance we intend to achieve when our sensors are randomly drawn with replacement. To obtain a non-trivial improvement on existing CIs for the estimation error covariance, we first present novel matrix CIs for a sum of independent and identically-distributed (i.i.d.) and positive semi-definite (p.s.d.) random matrices, whose support is finite. Next, we show that our guarantees generalize an existing matrix CI. Also, we show that our generalized guarantees require significantly fewer number of sampled sensors to be applicable. Lastly, we show through a numerical study that our guarantees significantly outperform existing ones in terms of their ability to bound (in the semi-definite sense) the steady-state estimation error covariance of the Kalman filter.
Paper Structure (13 sections, 10 theorems, 32 equations, 1 figure)

This paper contains 13 sections, 10 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Statistical Properties
  5. Sampling Policy
  6. Dynamical System Model
  7. Kalman filtering
  8. Problem Statement
  9. Main Results
  10. Matrix Concentration Inequalities
  11. Sensor Selection for State Estimation
  12. Numerics
  13. Conclusion

Key Result

Lemma 1

Let $( \boldsymbol{Z}_i )_{i \in [\gamma]}$ denote a sequence of $\gamma$ i.i.d. and p.s.d. random matrices. Let the tuple $\bar{\mathcal{T}} \space := \space ( d, \bar{\delta}, \gamma, \bar{\rho}, \bar{\epsilon}, p )$ of parameters satisfy the inequality almost surely for the scalar $\bar{\rho} \geq 1$ and the equality for the scalars $\bar{\delta}, \bar{\epsilon} \in (0,1)$. Then, the event o

Figures (1)

  • Figure 1: (a) Plot of the upper bounds on the worst-case estimation performance for varying values of the refinement factor $\zeta$. First, observe that $\overline{\lambda}( U_S )$ is a function of $\zeta \in [0,1]$. Also, observe that $\overline{\lambda}( \bar{U}_S )$ and $\overline{\lambda}( P_{\mathcal{S}} )$ are constant since they are not functions of $\zeta$. We remind the reader that $\overline{\lambda}( P_\mathcal{S} )$ is a random variable since it depends on a selection $\mathcal{S}$ that is randomly drawn via the sampling policy in Section \ref{['subsection:sampling_policy']}. The average value of $\overline{\lambda}( P_{\mathcal{S}} )$ is indicated by the black curve and the variability of $\overline{\lambda}( P_\mathcal{S} )$ is captured by the standard deviation, where the error bars indicate $\pm$ one standard deviation. (b) Plot of the upper bounds on the worst-case estimation performance for varying number of sampled sensors. The quantity $\overline{\lambda}( U_S )$ is plotted for varying values of the refinement factor $\zeta$. Comments in subplot (a), regarding the quantities $\overline{\lambda}( U_S )$ and $\overline{\lambda}( P_\mathcal{S} )$, also apply to subplot (b).

Theorems & Definitions (15)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Lemma 2
  • Corollary 2
  • proof : Sketch
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 5 more