Generalized Mean Robustness for Signal Temporal Logic

Robustness functions provide quantitative scores to measure the satisfaction of temporal logic formulas. We introduce a general class of parameterized robustness functions for signal temporal logic (STL), and demonstrate how it can be used for control problems involving STL specifications. We employ power means and generalized functional means to capture robust satisfaction over space and time. We show that our general definition encompasses many of the STL robustness functions in the literature. Most importantly, we show how that our notion of robustness addresses the two main limitations of the the traditional robustness (masking and locality), which currently limit using robustness-based approaches for control. The proposed robustness function parameters affect the conservativeness of the score, and can be chosen based on desired performance. We show how the proposed robustness can be used for control.