A Polynomial-Time Algorithm for the Secure State Estimation Problem Under Sparse Sensor Attacks via State Decomposition Technique

Abstract

This paper investigates the secure state estimation problem for cyber-physical systems (CPSs) under sparse sensor attacks. In the existing results, the secure state estimation is usually established as an NP-hard problem where combinatorial candidates should be checked since the set of attacked channels is unknown. For avoiding brute force search, a novel state decomposition technique is proposed such that the state can be reconstructed by a simple majority vote. Necessary and sufficient conditions for the observability of the decomposition elements are given, and based on the obtained conditions, an effective algorithm for designing the decomposition matrix is also proposed. Then, a polynomial-time secure state estimation strategy is constructed based on the proposed state decomposition technique. It is shown that besides <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{2}~s$</tex-math></inline-formula> -sparse eigenvalue observable systems, the secure state estimation problem can be solved in polynomial time for more general cases where each decomposition element is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {B}$</tex-math></inline-formula> -observable for at least <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2s+1$</tex-math></inline-formula> sensors. Finally, the effectiveness of the proposed methods is demonstrated by two simulations showing the decrease of computational complexity and the effectiveness under different cases.