Robustness analysis of systems' safety through a new notion of input‐to‐state safety

Abstract

Summary In this paper, we propose a new robustness notion that is applicable for certifying systems' safety with respect to external disturbance signals. The proposed input‐to‐state safety notion allows us to certify systems' safety in the presence of the disturbances, which is analogous to the notion of input‐to‐state stability for analyzing systems' stability.

Data-Driven Safety Verification of Discrete-Time Networks: A Compositional Approach

“…, q ir i ) ∈ R r i are determined later. Given the basis functions p j , to enforce conditions (4) on B i in (7), we reformulate the search for B i into the following RCP.…”

“…Assumption 1: Functions c j : R n i ×R p i ×R r i +1 ×R (p i +n i ) 2 → R, j = 1, 2, 3, are locally Lipschitz with respect to the first two arguments, uniformly in the others, with the corresponding Lipschitz constants L j > 0, j = 1, 2, 3. 1 Remark 2: As shown in [16,Lemma 5.4], one can compute Lipschitz constants L j , j = 1, 2, 3, (or their upper bounds) by assuming continuous differentiability of the system dynamics g i and bounding the coefficients q i in (7). The latter can be ensured by enforcing extra constraints over q i in SCP-i.…”

“…Problem 3: Consider network in (1). Given sets X i , X 0i , X ui , data sets D i , and the structure of B i as in (7), solve the following problem:…”

“…Let the network be composed of = 100 subsystems. Taking the basis functions p j (x i ) = x j−1 i , j ∈ {1, 2, 3}, in (7), the sub-barrier candidates are expressed as B i (x i ) = This figure illustrates that for all possible values of states within X 15 = [19, 28] and of internal inputs within W 15 = [38, 56], the decay condition (4c) is satisfied. The other matrices X i are not presented here due to the space constraints.…”

“…Barrier functions have been introduced as a tool for safety verification of dynamical systems [5], [6]. They are also used to verify robustness and/or design controllers enforcing safety, e.g., [7], [8]. Most of these results, however, rely on having a sufficiently accurate knowledge of models of the systems.…”