Based on the traditional linear algebra theory, this paper propose the attack detection schemes for networked control systems (NCSs) under single stochastic cyber-attack and multiple stochastic cyber-attacks aiming at multiple communication channels of NCSs, respectively. The focus is on designing an anomaly detector for NCSs under cyber-attacks. First, we construct a model of stochastic NCSs with stochastic cyber-attacks which satisfy the Markovian stochastic process. And we also introduced the stochastic attack models that NCSs are possibly exposed to. Next, based on the frequency-domain transformation technique and linear algebra theory, we propose the algebraic detection schemes for possible stochastic cyber-attacks. We transform the detector error dynamics into algebraic equations. By applying the presented approaches, residual information that is caused by different attacks is, respectively, obtained and anomalies on the control system are detected. In addition, sufficient and necessary conditions guaranteeing the detectability of the stochastic cyber-attacks are obtained. The presented detection approaches in this paper are simple, straightforward and easy to implement. The aim of this work is to use traditionalmathematics tools to solve new problems that arise from the complex NCSs. Finally, two simulation examples are provided. The simulation results underline that the detection approaches are effective and feasible in practical application.
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