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Runtime Enforcement of CPS against Signal Temporal Logic

Han Su, Saumya Shankar, Srinivas Pinisetty, Partha S. Roop, Naijun Zhan

TL;DR

This work addresses enforcing dense-time CPS properties specified in STL by translating STL formulas into Timed Transducers (TTs) and encoding continuous signals into timed words. It presents a formal pipeline: encode signals as timed words, translate STL into TT via a compositional construction that handles $p_1\mathcal{U}_{I}p_2$ and $p_1\mathcal{R}_{I}p_2$, and enforce the property with an online enforcer that minimally modifies the signal using a tractable optimization. The contributions include a sound, transparent, and minimally invasive enforcer, a compositional TT framework with $\land$/$\lor$-products, and empirical case studies on autonomous-vehicle scenarios demonstrating efficiency and scalability. This framework enables real-time safety guarantees for CPS with continuous-time dynamics and lays groundwork for extending to bidirectional enforcement and more general STL specifications.

Abstract

Cyber-Physical Systems (CPSs), especially those involving autonomy, need guarantees of their safety. Runtime Enforcement (RE) is a lightweight method to formally ensure that some specified properties are satisfied over the executions of the system. Hence, there is recent interest in the RE of CPS. However, existing methods are not designed to tackle specifications suitable for the hybrid dynamics of CPS. With this in mind, we develop runtime enforcement of CPS using properties defined in Signal Temporal Logic (STL). In this work, we aim to construct a runtime enforcer for a given STL formula to minimally modify a signal to satisfy the formula. To achieve this, the STL formula to be enforced is first translated into a timed transducer, while the signal to be corrected is encoded as timed words. We provide timed transducers for the temporal operators \emph{until} and \emph{release} noting that other temporal operators can be expressed using these two. Our approach enables effective enforcement of STL properties for CPS. A case study is provided to illustrate the approach and generate empirical evidence of its suitability for CPS.

Runtime Enforcement of CPS against Signal Temporal Logic

TL;DR

This work addresses enforcing dense-time CPS properties specified in STL by translating STL formulas into Timed Transducers (TTs) and encoding continuous signals into timed words. It presents a formal pipeline: encode signals as timed words, translate STL into TT via a compositional construction that handles and , and enforce the property with an online enforcer that minimally modifies the signal using a tractable optimization. The contributions include a sound, transparent, and minimally invasive enforcer, a compositional TT framework with /-products, and empirical case studies on autonomous-vehicle scenarios demonstrating efficiency and scalability. This framework enables real-time safety guarantees for CPS with continuous-time dynamics and lays groundwork for extending to bidirectional enforcement and more general STL specifications.

Abstract

Cyber-Physical Systems (CPSs), especially those involving autonomy, need guarantees of their safety. Runtime Enforcement (RE) is a lightweight method to formally ensure that some specified properties are satisfied over the executions of the system. Hence, there is recent interest in the RE of CPS. However, existing methods are not designed to tackle specifications suitable for the hybrid dynamics of CPS. With this in mind, we develop runtime enforcement of CPS using properties defined in Signal Temporal Logic (STL). In this work, we aim to construct a runtime enforcer for a given STL formula to minimally modify a signal to satisfy the formula. To achieve this, the STL formula to be enforced is first translated into a timed transducer, while the signal to be corrected is encoded as timed words. We provide timed transducers for the temporal operators \emph{until} and \emph{release} noting that other temporal operators can be expressed using these two. Our approach enables effective enforcement of STL properties for CPS. A case study is provided to illustrate the approach and generate empirical evidence of its suitability for CPS.

Paper Structure

This paper contains 36 sections, 8 theorems, 17 equations, 11 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Organization
  3. Background and Problem Formulation
  4. Timed Transducer
  5. Timed Language
  6. Timed Transducer
  7. Signal Temporal Logic
  8. Runtime Enforcement
  9. Constraints on an Enforcer.
  10. Problem Formulation
  11. Signal Encoding
  12. Variable Points
  13. Relevant Points
  14. Signal encoding
  15. Constructing Timed Transducer from STL
  16. ...and 21 more sections

Key Result

Proposition 7

Given a signal $\bm{x}\xspace$ and a predicate $p(\bm{x}\xspace)$, assume $t_0 < t_1 < \cdots < t_k$ denote all the variable points of $\bm{x}\xspace$ with respect to $p(\bm{x}\xspace)$. It is then established that the truth value of $p(\bm{x}\xspace)$ remains constant within each open interval $(t_

Figures (11)

  • Figure 1: Overview
  • Figure 1: Transitions in TT of $p_1\mathcal{U}\xspace_{[4,5]} p_2$
  • Figure 2: Timed Transducer $\mathcal{A}_P$
  • Figure 3: Signal Encoding against Formula $p_1\mathcal{U}\xspace_{[4,5]}p_2$
  • Figure 4: Timed Transducer $\mathcal{A}\xspace_{\mathcal{U}\xspace}$
  • ...and 6 more figures

Theorems & Definitions (27)

  • Definition 1: Timed transducer
  • Example 2
  • Remark 1
  • Definition 3: STL Semantics
  • Example 4
  • Definition 5: Constraints on an Enforcer
  • Definition 6: Variable Point [bae2019bounded, Def. 2.8]
  • Proposition 7
  • Example 8
  • Definition 9: Relevant Point
  • ...and 17 more