Stochastic Models to Mitigate Sparse Sensor Attacks in Continuous-Time Non-Linear Cyber-Physical Systems

Cyber-Physical Systems are very vulnerable to sparse sensor attacks. But current protection mechanisms employ linear and deterministic models which cannot detect attacks precisely. Therefore, in this paper, we propose a new non-linear generalized model to describe Cyber-Physical Systems. This model includes unknown multivariable discrete and continuous-time functions and different multiplicative noises to represent the evolution of physical processes and random effects in the physical and computational worlds. Besides, the digitalization stage in hardware devices is represented too. Attackers and most critical sparse sensor attacks are described through a stochastic process. The reconstruction and protection mechanisms are based on a weighted stochastic model. Error probability in data samples is estimated through different indicators commonly employed in non-linear dynamics (such as the Fourier transform, first-return maps, or the probability density function). A decision algorithm calculates the final reconstructed value considering the previous error probability. An experimental validation based on simulation tools and real deployments is also carried out. Both, the new technology performance and scalability are studied. Results prove that the proposed solution protects Cyber-Physical Systems against up to 92% of attacks and perturbations, with a computational delay below 2.5 s. The proposed model shows a linear complexity, as recursive or iterative structures are not employed, just algebraic and probabilistic functions. In conclusion, the new model and reconstruction mechanism can protect successfully Cyber-Physical Systems against sparse sensor attacks, even in dense or pervasive deployments and scenarios.