Temporal Robustness of Temporal Logic Specifications: Analysis and Control Design

Abstract

We study the temporal robustness of temporal logic specifications and show how to design temporally robust control laws for time-critical control systems. This topic is of particular interest in connected systems and interleaving processes such as multi-robot and human-robot systems where uncertainty in the behavior of individual agents and humans can induce timing uncertainty. Despite the importance of time-critical systems, temporal robustness of temporal logic specifications has not been studied, especially from a control design point of view. We define synchronous and asynchronous temporal robustness and show that these notions quantify the robustness with respect to synchronous and asynchronous time shifts in the predicates of the temporal logic specification. It is further shown that the synchronous temporal robustness upper bounds the asynchronous temporal robustness. We then study the control design problem in which we aim to design a control law that maximizes the temporal robustness of a dynamical system. Our solution consists of a Mixed-Integer Linear Programming (MILP) encoding that can be used to obtain a sequence of optimal control inputs. While asynchronous temporal robustness is arguably more nuanced than synchronous temporal robustness, we show that control design using synchronous temporal robustness is computationally more efficient. This trade-off can be exploited by the designer depending on the particular application at hand. We conclude the paper with a variety of case studies.

Combined Left and Right Temporal Robustness for Control Under STL Specifications

“…Formally, if θ ± ϕ (x,t) > 0 then χ ϕ (x,t) = 1 and if θ ± ϕ (x,t) < 0 then χ ϕ (x,t) = −1. In [15], we also showed that the absolute value of the left (right) time robustness measures how robustly a signal x satisfies a formula ϕ at time t with respect to time shifts in the predicates of formula ϕ. In fact, one can asynchronously shift predicates in time to the left by up to |θ + ϕ (x,t)| and the specification will not change its satisfaction.…”

“…and then, to obtain the θ ± ϕ (x,t), one needs to apply the standard recursive inf/sup rules to each θ ± p (x,t), similarly to the characteristic function χ ϕ (x,t), see [15] for details.…”

“…The authors define the left (right) time robustness by quantifying the maximal permissible left (right) time shifts in the predicates of the STL specification that do not result in a violation of the specification. In our previous works [14], [15], we analyze various properties of left (right) time robustness and propose an MILP encoding to control † The authors are with the Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia PA, USA. {nellro, larsl, morari, pappasg}@seas.upenn.edu.…”

“…The signal x ← τ is called a τ-early signal if ∀p k ∈ AP, ∀t ∈ T, χ p k (x ← τ ,t) = χ p k (x,t + τ k ). The signal x → τ is called a τ-late signal if ∀p k ∈ AP, ∀t ∈ T, χ p k (x → τ ,t) = χ p k (x,t − τ k ), see[15].…”