On the Computational Complexity of the Secure State-Reconstruction Problem

TL;DR

This paper explores the computational difficulty of reconstructing linear system states from corrupted sensor data, establishing NP-hardness, and identifying conditions for polynomial-time solutions based on eigenvalue observability.

Contribution

It introduces eigenvalue observability as a new concept and characterizes when state reconstruction is computationally feasible under adversarial sensor attacks.

Findings

Reconstruction is NP-hard in general.

Eigenvalue observability allows polynomial-time reconstruction under certain conditions.

The gap between eigenvalue and sparse observability disappears for matrices with unitary geometric multiplicity.

Abstract

In this paper, we discuss the computational complexity of reconstructing the state of a linear system from sensor measurements that have been corrupted by an adversary. The first result establishes that the problem is, in general, NP-hard. We then introduce the notion of eigenvalue observability and show that the state can be reconstructed in polynomial time when each eigenvalue is observable by at least $2s+1$ sensors and at most $s$ sensors are corrupted by an adversary. However, there is a gap between eigenvalue observability and the possibility of reconstructing the state despite attacks - this gap has been characterized in the literature by the notion of sparse observability. To better understand this, we show that when the $\mathbf{A}$ matrix of the linear system has unitary geometric multiplicity, the gap disappears, i.e., eigenvalue observability coincides with sparse observability, and there exists a polynomial time algorithm to reconstruct the state provided the state can be reconstructed.