  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.
