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.

中文翻译：

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具有未知时变变化前和变化后分布的线性系统中的低复杂度最快变化检测

由于动态智能电网中错误数据注入攻击（FDIA）的顺序检测的推动，我们考虑了在动态线性回归模型中顺序检测时变变化的更普遍的问题。具体而言，当发生更改时，会在线性回归模型中添加随时间变化的未知向量。线性回归模型的参数向量也被假定为未知且随时间变化。因此，变化前和变化后的分布都是未知的并且随时间变化。这对设计计算有效的顺序检测器提出了重大挑战。我们首先提出两种累积和类型算法来解决这一挑战。一种称为广义累积和（GCUSUM）算法，另一种称为松弛广义累积和（RGCUSUM）算法，这是GCUSUM的修改版本。可以看出，所提出的RGCUSUM算法的计算复杂度与观察次数成线性比例关系。接下来，考虑Lordon的设置，对于预期的虚假警报周期的任何给定约束，都将推导所提出的RGCUSUM算法中采用的阈值下限，这将为设计所提出的RGCUSUM算法以实现任何规定的性能提供有用的指导在实践中的要求。另外，对于所提出的RGCUSUM算法中采用的任何给定阈值，还提供了预期检测延迟的上限。在FDIAs下，在IEEE标准电源系统的背景下，对所提出的RGCUSUM算法的性能进行了数值研究。而且，