Optimal stealthy attacks on actuators for strictly proper systems

In this paper, we consider stealthy data injection attacks against control systems, and develop security sensitivity metrics to quantify their impact on the system. The final objective of this work is to use such metrics as objective functions in the design of optimal resilient controllers against stealthy attacks, akin to the classical design of optimal ℋ ∞ robust controllers. As a first metric, the recently proposed ℓ 2 output to output gain is first examined, and fundamental limitations of this gain for systems with strictly proper dynamics are uncovered and characterized. To circumvent such limitations, a new security sensitivity metric is proposed, namely the truncated ℓ 2 gain. Necessary and sufficient conditions for this gain to be finite are derived, which we show can cope with strictly proper systems. Finally, we report preliminary investigations on the design of optimal resilient controllers, which are supported and illustrated through numerical examples.