# Observability Robustness under Sensor Failures: a Computational Perspective

## TL;DR

This paper investigates the computational complexity of maintaining system observability under sensor failures, proving NP-hardness of related problems and proposing polynomial-time solutions under certain practical assumptions.

## Contribution

It introduces the first complexity analysis of sensor removal impacts on observability and offers efficient algorithms for specific cases with bounded eigenvalue multiplicities.

## Key findings

- All problems are NP-hard, even in special cases.
- Certain problems exhibit a submodular structure, limiting approximation guarantees.
- Polynomial-time solutions are possible under practical eigenvalue multiplicity bounds.

## Abstract

This paper studies the robustness of observability of a linear time-invariant system under sensor failures from a computational perspective. To be precise, the problem of determining the minimum number of sensors whose removal can destroy system observability, as well as the problem of determining the minimum number of state variables that need to be prevented from being directly measured by the existing sensors to destroy observability, is investigated. The first one is closely related to the ability of unique state reconstruction of a system under adversarial sensor attacks, and the dual of both problems are in the opposite direction of the well-studied minimal controllability problems. It is proven that all these problems are NP-hard, both for a numerical system and a structured system, even restricted to some special cases. It is also shown that the first problems both for a numerical system and a structured one share a cardinality-constrained submodular minimization structure, for which there is no known constant or logarithmic factor approximation in general. On the other hand, for the first two problems, under a reasonable assumption often met by practical systems, that the eigenvalue geometric multiplicities of the numerical systems or the matching deficiencies of the structured systems are bounded by a constant, by levering the rank-one update property of the involved rank function, it is possible to obtain the corresponding optimal solutions by traversing a subset of the feasible solutions. We show such a method has polynomial time complexity in the system dimensions and the number of sensors.