Small-Gain Theorem for Safety Verification Under High-Relative-Degree Constraints

This article develops a small-gain technique for the safety analysis and verification of interconnected systems with high-relative-degree safety constraints. To this end, a high-relative-degree input-to-state safety (ISSf) approach is proposed to quantify the influence of external inputs on the subsystem safety. With a coordination transformation, the relationship between ISSf barrier functions (ISSf-BFs) and the existing high-relative-degree (or high-order) barrier functions is established to simplify the safety analysis under external inputs. With high-relative-degree ISSf-BFs, a small-gain theorem is proposed for safety verification. It is shown that, under the small-gain condition, the compositional safe set is forward invariant and asymptotically stable. The effectiveness of the proposed small-gain theorem is illustrated on the output-constrained decentralized control of two inverted pendulums connected by a spring mounted on two carts.