An STL-based Formulation of Resilience in Cyber-Physical Systems

Abstract

Resiliency is the ability to quickly recover from a violation and avoid future violations for as long as possible. Such a property is of fundamental importance for Cyber-Physical Systems (CPS), and yet, to date, there is no widely agreed-upon formal treatment of CPS resiliency. We present an STL-based framework for reasoning about resiliency in CPS in which resiliency has a syntactic characterization in the form of an STL-based Resiliency Specification (SRS). Given an arbitrary STL formula ϕ, time bounds α and β, the SRS of ϕ, R α,β (ϕ), is the STL formula ¬ϕU [0,α] G [0,β) ϕ, specifying that recovery from a violation of ϕ occur within time α (recoverability), and subsequently that ϕ be maintained for duration β (durability). These R-expressions, which are atoms in our SRS logic, can be combined using STL operators, allowing one to express composite resiliency specifications, e.g., multiple SRSs must hold simultaneously, or the system must eventually be resilient. We define a quantitative semantics for SRSs in the form of a Resilience Satisfaction Value (ReSV) function r and prove its soundness and completeness w.r.t. STL's Boolean semantics. The r-value for R α,β (ϕ) atoms is a singleton set containing a pair quantifying recoverability and durability. The r-value for a composite SRS formula results in a set of non-dominated recoverabilitydurability pairs, given that the ReSVs of subformulas might not be directly comparable (e.g., one subformula has superior durability but worse recoverability than another). To the best of our knowledge, this is the first multi-dimensional quantitative semantics for an STL-based logic. Two case studies demonstrate the practical utility of our approach.

Combined Left and Right Temporal Robustness for Control Under STL Specifications

“…While [24] considers the time sensitive control for a subset of STL specifications, the authors in [25] present the time window temporal logic that is used in [26], [27] to obtain control laws for finding temporal relaxations when the specification is not satisfiable. In [28], the STL-based resiliency for cyber-physical system is presented that can capture temporal violations by recoverability and durability.…”