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The present work provides two alternatives to formulate time-discountedincremental input/output-to-state stability (i-IOSS) as a suitabledetectability notion for general nonlinear systems with non-additivedisturbances. Both formulations are related to existing i-IOSS notions whichresult as special cases. Previous results that provide a sufficientLyapunov-function condition for i-IOSS and that prove i-IOSS to be necessaryfor the existence of robustly stable full-order state observers are generalizedto the presented time-discounted i-IOSS definition for general nonlinearsystems. For linear systems, explicit i-IOSS bounds are presented.

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