On Local Input-Output Stability of Nonlinear Feedback Systems via Local Graph Separation

A new type of local input-output stability for nonlinear systems is defined, called M-local boundedness, which can be viewed as a local version of established definitions of global boundedness. This definition states that the system is bounded if the input Lebesgue signal has a norm smaller than M. Using graph separation concepts and a novel topological argument, which partitions the output space of the system into feasible and infeasible regions based on the restriction of the system input, sufficient conditions for M-local boundedness of a nonlinear feedback system are derived. Using this theorem, a new local nonlinear small gain condition is found for a closed-loop system with additive inputs. This small gain condition is then used in a numerical example, in which a differential equation with a quadratic element was partitioned into a feedback system and bounds on the norm of the input were found which ensured the system was M-locally stable.