Control Barrier Functions for Safe CPS Under Sensor Faults and Attacks

Control barrier function based attack-recovery with provable guarantees

“…where η is the bound on the second time-derivative B and τ > 0. We have the following result stating that the attack detection mechanism in (10) detects the attack before the system trajectories leave the safe set.…”

“…where x : R + → R n is the solution of (1) resulting from applying the input u : R + → U and η is the bound on the second time-derivative B. Then, for each τ ≥ 0, it holds that td ≤ T , where td is given in (10).…”

“…Lemma 2 implies that the attack-detection mechanism in (10) raises an alert on or before the system trajectories reach the boundary of the set ∂S under an attack. In other words, while the detection-mechanism (10) can have false positives (i.e., raise an alert when there is no attack), it will never have a false negative (i.e., it will not miss any attack).…”

“…Given system (1) with F ∈ C 1 , B ∈ C 2 and the attack model (2), suppose that Assumption 1 holds, and Assumptions 2-3 hold for some c ∈ (0, c M ). Then, there exist feedback laws λ : R n → U and k s : R n → U s such that under the effect of the input u in (17) with tj d is defined in (10), the system trajectories of (1) resulting from applying (17) satisfy x(t) ∈ S for all t ≥ 0 and for all x(0) ∈ X 0 = int(S).…”

“…In the absence of knowledge of actual system input under an attack, we utilize an approximation scheme and show that the attackdetection mechanism is sound, i.e., it does not generate any false negatives. While there is work on CBF-based safety of CPS under faults and attacks [10], [11], to the best of the authors' knowledge, this is the first work utilizing CBF conditions for attack detection; 2) Based on the zeroing-CBF condition [8], we propose an adaptation scheme to minimize the false-positive rate of the attack-detection mechanism; 3) Finally, we use a switching law for input assignment and a QP formulation for online feedback synthesis for both nominal and safe feedback. We illustrate the efficacy of the proposed method in a case study involving an attack on a motor of a quadrotor and show how the proposed framework can recover the quadrotor from an attack.…”