An optimization framework for resilient batch estimation in Cyber-Physical Systems

TL;DR

A class of resilient state estimators for linear time-varying discrete-time systems that induces an estimation error which, under certain conditions, is independent of the extreme values of the (impulsive) measurement noise.

Abstract

This paper proposes a class of resilient state estimators for LTV 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. The latter in addition to being arbitrary in its form, need not be strictly bounded. In this setting, we construct the estimator as the set-valued map which associates to the measurements, the minimizing set of some appropriate performance functions. We consider a family of such performance functions each of which yielding a specific instance of the general estimator. It is then shown that the proposed class of estimators 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. Hence, the estimation error may be bounded while the measurement noise is virtually unbounded. Moreover, we provide several error bounds (in different configurations) whose expressions depend 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.

Figures (4)

• Figure 1: Probability of exact recovery (expressed in percentage) by the estimator \ref{['eq:Constrained-Estimator']} in the presence of only sparse measurement noise $\left\{s_t\right\}$. The level of sparsity of the noise is expressed in terms of a fraction of nonzero values in the sequence $\left\{s_t: t\in \mathbb{T}\right\}$ with $\left|\mathbb{T}\right|=T=100$.
• Figure 2: Average relative estimation error (in logarithm scale) induced by different estimators versus sparsity level of the sparse noise $\left\{s_t\right\}$. The relative error is expressed here as $\|\hat{X}-X\|_2/\left\|X\right\|_2$ where $X$ and $\hat{X}$ denote the true and estimated state matrices respectively. Parameters of the estimator $\mathcal{E}$ in \ref{['eq:def_est']}: $\lambda=1000$, $W_t=I_2$ and $V_t=1$ for all $t$.
• Figure 3: Average relative estimation error (in log scale) induced by different estimators for different levels of both dense noises $w_t$ and $v_t$. Parameters of the estimator $\mathcal{E}$ in \ref{['eq:def_est']}: $\lambda=1000$, $W_t=I_2$ and $V_t=1$ for all $t$.
• Figure 4: Average relative estimation error induced by $\mathcal{E}_{\ell_2^2,\ell_1}$ and $\mathcal{E}_{\ell_1,\ell_1}$ for the system \ref{['eq:example-matrices']}, SNR$=30$dB and 30$\%$ of non-zero entries in $\{s_t\}$ for different values of the regularization parameter $\lambda$.

Theorems & Definitions (39)

• Definition 1
• Lemma 1: kircher_analysis_2020
• Remark 1
• Lemma 2: Lower Bound of a loss function
• proof
• Proposition 1: Well-definedness of the estimator
• Lemma 3: Equivalent condition of Observability
• Definition 2: Resilience of an estimator
• Lemma 4
• proof