Low-Complexity Quickest Change Detection in Linear Systems With Unknown Time-Varying Pre- and Post-Change Distributions

Abstract

Motivated by the sequential detection of false data injection attacks (FDIAs) in a dynamic smart grid, we consider a more general problem of sequentially detecting a time-varying change in a dynamic linear regression model. To be specific, when the change occurs, a time-varying unknown vector is added in the linear regression model. The parameter vector of the linear regression model is also assumed to be unknown and time-varying. Thus, the pre- and post-change distributions are both unknown and time-varying. This imposes a significant challenge for designing a computationally efficient sequential detector. We first propose two Cumulative-Sum-type algorithms to address this challenge. One is called generalized Cumulative-Sum (GCUSUM) algorithm, and the other one is called relaxed generalized Cumulative-Sum (RGCUSUM) algorithm, which is a modified version of the GCUSUM. It can be shown that the computational complexity of the proposed RGCUSUM algorithm scales linearly with the number of observations. Next, considering Lordon’s setup, for any given constraint on the expected false alarm period, a lower bound on the threshold employed in the proposed RGCUSUM algorithm is derived, which provides a useful guideline for the design of the proposed RGCUSUM algorithm to achieve any prescribed performance requirement in practice. In addition, for any given threshold employed in the proposed RGCUSUM algorithm, an upper bound on the expected detection delay is also provided. The performance of the proposed RGCUSUM algorithm is numerically studied in the context of an IEEE standard power system under FDIAs. Moreover, the numerical results demonstrate the superiority of the proposed RGCUSUM in computational efficiency.