Soft Safety Analysis of Interconnected Systems

This article introduces the concept of soft safety for nonlinear dynamical systems, examined from an input–output perspective, and establishes criteria for verifying soft safety in interconnected systems. Diverging from traditional safety approaches that enforce rigid/hard constraints on system states, our proposal involves integrating soft barrier functions with respect to system inputs and outputs to characterize soft safety. Particularly, when the soft barrier function takes a quadratic form with a constant QSR multiplier—later extended to a dynamic multiplier—we further define QSR safety for a class of nonlinear systems subject to quadratic constraints on input–output signals. The article presents methods for analyzing QSR safety in both open-loop systems with state-space representations and interconnected systems, with these methods subsequently employed to design QSR soft safe systems meeting prescribed safety and stability requirements. Numerical and simulation examples are provided to illustrate the rationality of the concept and the efficacy of the results.