A Note on the Connection Between Incremental Input-to-State Stability and Fading Memory in Nonlinear Systems

Recently, Angeli1 has proposed the concept of incremental input-to-state stability, which can be viewed as an extension of the well-known input-to-state stability to the context of incremental stability. In this note, we show how the concept is connected with fading memory that has played a central role in the problem of uniformly approximating input-output nonlinear maps. Specifically, we show that the input-output map of an incrementally input-to-state stable nonlinear system has fading memory. Using this connection, we are able to establish a body of results on uniform approximation of incrementally input-to-state stabile systems by simple nonlinear structures such as finite Volterra series, bilinear systems, and Laguerre systems. We also show that the output responses of those systems to asymptotically almost periodic inputs are also asymptotically almost periodic.