Time-Discounted Incremental Input/Output-to-State Stability

The present work provides two alternatives to formulate time-discounted incremental input/output-to-state stability (i-IOSS) as a suitable detectability notion for general nonlinear systems with non-additive disturbances. Both formulations are related to existing i-IOSS notions which result as special cases. Previous results that provide a sufficient Lyapunov-function condition for i-IOSS and that prove i-IOSS to be necessary for the existence of robustly stable full-order state observers are generalized to the presented time-discounted i-IOSS definition for general nonlinear systems. For linear systems, explicit i-IOSS bounds are presented.