Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Resilient State Estimation for Discrete-Time Linear Systems

Alexandre Kircher, Laurent Bako, Eric Blanco, Mohamed Benallouch

TL;DR

It is shown that the proposed resilient state estimator enjoys the property of resilience, that is, it induces an estimation error which, under certain conditions, is independent of the extreme values of the (impulsive) measurement noise.

Abstract

This paper proposes a resilient state estimator for LTI discrete-time systems. The dynamic equation of the system is assumed to be affected by a bounded process noise. As to the available measurements, they are potentially corrupted by a noise of both dense and impulsive natures. In this setting, we construct the estimator as the map which associates to the measurements, the minimizing set of an appropriate (convex) performance function. It is then shown that the proposed estimator enjoys the property of resilience, that is, it induces an estimation error which, under certain conditions, is independent of the extreme values of the (impulsive) measurement noise. Therefore, the estimation error may be bounded while the measurement noise is virtually unbounded. Moreover, the expression of the bound depends explicitly on the degree of observability of the system being observed and on the considered performance function. Finally, a few simulation results are provided to illustrate the resilience property.

Resilient State Estimation for Discrete-Time Linear Systems

TL;DR

It is shown that the proposed resilient state estimator enjoys the property of resilience, that is, it induces an estimation error which, under certain conditions, is independent of the extreme values of the (impulsive) measurement noise.

Abstract

This paper proposes a resilient state estimator for LTI discrete-time systems. The dynamic equation of the system is assumed to be affected by a bounded process noise. As to the available measurements, they are potentially corrupted by a noise of both dense and impulsive natures. In this setting, we construct the estimator as the map which associates to the measurements, the minimizing set of an appropriate (convex) performance function. It is then shown that the proposed estimator enjoys the property of resilience, that is, it induces an estimation error which, under certain conditions, is independent of the extreme values of the (impulsive) measurement noise. Therefore, the estimation error may be bounded while the measurement noise is virtually unbounded. Moreover, the expression of the bound depends explicitly on the degree of observability of the system being observed and on the considered performance function. Finally, a few simulation results are provided to illustrate the resilience property.

Paper Structure

This paper contains 11 sections, 5 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Notations
  4. The estimation Problem
  5. Problem
  6. Resilient optimization-based estimator
  7. Preliminaries
  8. Characterization of the resilience property
  9. Simulation Results
  10. Conclusion
  11. Additional elements to the proof of Theorem 1

Key Result

Lemma 1

Let $G:\mathbb{R}^{n\times m}\rightarrow\mathbb{R}_{\geq 0}$ be a nonnegative continuous function satisfying the following properties: Then for any norm $\left\|\cdot\right\|$ on $\mathbb{R}^{n\times m}$, there exists $d>0$ such that for all $S\in\mathbb{R}^{n\times m}$, $G(S)\geq d \sigma(\left\|S\right\|)$.

Figures (3)

  • Figure 1: State of the system and its estimates (resilient estimator and smoother) in absence of sparse noise
  • Figure 2: State, estimated states (through resilient estimation and smoothing) and output of the system in presence of sparse noise
  • Figure 3: State of the system and its estimate (resilient estimator) in presence of sparse noise

Theorems & Definitions (12)

  • Definition 1: class-$\mathcal{K_\infty}$ functions
  • Lemma 1
  • proof
  • Lemma 2: Lower Bound on $H$
  • proof
  • Definition 2: $r$-Resilience index $p_r$
  • Theorem 1: Upper bound on the estimation error
  • proof
  • Corollary 1
  • proof
  • ...and 2 more