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 tradeoff can be exploited by the designer depending on the particular application at hand. We conclude the article with a variety of case studies.

We study the temporal robustness of stochastic signals. This topic is of particular interest in interleaving processes such as multi-agent systems where communication and individual agents induce timing uncertainty. For a deterministic signal and a given specification, we first introduce the synchronous and the asynchronous temporal robustness to quantify the signal’s robustness with respect to synchronous and asynchronous time shifts in its sub-signals. We then define the temporal robustness risk by investigating the temporal robustness of the realizations of a stochastic signal. This definition can be interpreted as the risk associated with a stochastic signal to not satisfy a specification robustly in time. In this definition, general forms of specifications such as signal temporal logic specifications are permitted. We show how the temporal robustness risk is estimated from data for the value-at-risk. The usefulness of the temporal robustness risk is underlined by both theoretical and empirical evidence. In particular, we provide various numerical case studies including a T-intersection scenario in autonomous driving.

Urban Air Mobility, the scenario where hundreds of manned and Unmanned Aircraft Systems (UASs) carry out a wide variety of missions (e.g., moving humans and goods within the city), is gaining acceptance as a transportation solution of the future. One of the key requirements for this to happen is safely managing the air traffic in these urban airspaces. Due to the expected density of the airspace, this requires fast autonomous solutions that can be deployed online. We propose Learning-‘N-Flying (LNF), a multi-UAS Collision Avoidance (CA) framework. It is decentralized, works on the fly, and allows autonomous Unmanned Aircraft System (UAS)s managed by different operators to safely carry out complex missions, represented using Signal Temporal Logic, in a shared airspace. We initially formulate the problem of predictive collision avoidance for two UASs as a mixed-integer linear program, and show that it is intractable to solve online. Instead, we first develop Learning-to-Fly (L2F) by combining (1) learning-based decision-making and (2) decentralized convex optimization-based control. LNF extends L2F to cases where there are more than two UASs on a collision path. Through extensive simulations, we show that our method can run online (computation time in the order of milliseconds) and under certain assumptions has failure rates of less than 1% in the worst case, improving to near 0% in more relaxed operations. We show the applicability of our scheme to a wide variety of settings through multiple case studies.

The determination of safety assurances for automated driving vehicles is one of the most critical challenges in the industry today. Several behavioral safety models for automated driving have been proposed recently and standards discussions are on the way. In this paper we present a method to automatically explore the performance of automated vehicle (AV) safety models utilizing robustness of Metric Temporal Logic (MTL) specifications as a continuous metric of safety. We present a case study of the Responsibility Sensitive Safety model (RSS), introducing a safety evaluation pipeline based on the CARLA driving simulator, RSS and a set of safety-critical driving scenarios. Our method automatically extracts safety relevant profiles for these scenarios providing practical parametric boundaries for implementation. Furthermore, we evaluate the trade-offs between safety and utility within the safe RSS parameter space through a proposed naturalistic benchmark challenge that we open-sourced. We analyze different RSS parameter configurations including assertive and more conservative settings, extracted by our specification-driven framework. Our results show that while maintaining the safety boundaries, the extracted RSS configuration for assertive driving behavior achieves the highest utility.

Implantable medical devices are safety-critical systems whose incorrect operation can jeopardize a patient's health, and whose algorithms must meet tight platform constraints like memory consumption and runtime. In particular, we consider here the case of implantable cardioverter defibrillators, where peak detection algorithms and various others discrimination algorithms serve to distinguish fatal from non-fatal arrhythmias in a cardiac signal. Motivated by the need for powerful formal methods to reason about the performance of arrhythmia detection algorithms, we show how to specify all these algorithms using Quantitative Regular Expressions (QREs). QRE is a formal language to express complex numerical queries over data streams, with provable runtime and memory consumption guarantees. We show that QREs are more suitable than classical temporal logics to express in a concise and easy way a range of peak detectors (in both the time and wavelet domains) and various discriminators at the heart of today's arrhythmia detection devices. The proposed formalization also opens the way to formal analysis and rigorous testing of these detectors' correctness and performance, alleviating the regulatory burden on device developers when modifying their algorithms. We demonstrate the effectiveness of our approach by executing QRE-based monitors on real patient data on which they yield results on par with the results reported in the medical literature.

As methods and tools for cyber-physical systems (CPS) grow in capabilities and use, one-size-fits-all solutions start to show their limitations. In particular, tools and languages for programming an algorithm or modeling a CPS that are specific to the application domain are typically more usable, and yield better performance, than general-purpose languages and tools. In the domain of cardiac arrhythmia monitoring, a small, implantable medical device continuously monitors the patient's cardiac rhythm and delivers electrical therapy when needed. The algorithms executed by these devices are streaming algorithms, so they are best programmed in a streaming language that allows the programmer to reason about the incoming data stream as the basic object, rather than force her to think about lower-level details like state maintenance and minimization. Because these devices are resource-constrained, it is useful if the programming language allowed predictable performance in terms of processing runtime and energy consumption, or more general costs. StreamQRE is a declarative streaming programming language, with an efficient and portable implementation and strong theoretical guarantees. In particular, its evaluation algorithm guarantees constant cost (runtime, memory, energy) per data item and also calculates upper bounds on the per-item cost. Such an estimate of the cost allows early exploration of the algorithmic possibilities, while maintaining a handle on worst case performance, on the basis of which hardware can be designed and algorithms can be tuned.

We show that metric temporal logic (MTL) the extension of linear temporal logic to real time, can be viewed as linear time-invariant filtering, by interpreting addition, multiplication, and their neutral elements, over the idempotent dioid (max,min,0,1). Moreover, by interpreting these operators over the field of reals (+,x,0,1), one can associate various quantitative semantics to a metric-temporal-logic formula, depending on the filter's kernel used: square, rounded-square, Gaussian, low-pass, band-pass, or high-pass. This remarkable connection between filtering and metric temporal logic allows us to freely navigate between the two, and to regard signal-feature detection as logical inference. To the best of our knowledge, this connection has not been established before. We prove that our qualitative, filtering semantics is identical to the classical MTL semantics. We also provide a quantitative semantics for MTL, which measures the normalized, maximum number of times a formula is satisfied within its associated kernel, by a given signal. We show that this semantics is sound, in the sense that, if its measure is 0, then the formula is not satisfied, and it is satisfied otherwise. We have implemented both of our semantics in Matlab, and illustrate their properties on various formulas and signals, by plotting their computed measures.