Small Gain Theorems on Input-to-Output Stability

Abstract — This paper presents nonlinear small gain theo-rems for both continuous time systems and discrete time systems in a general setting of input-to-output stability. Most small gain theorems in the past literature were stated for interconnected systems of which the subsystems are either input-to-state sta-ble, or are input-to-output stable and satisfy an additional con-dition of input/output-to-state stability. In this work we show that when the input-to-output stability is strengthened with an output Lagrange stability property, a small gain theorem can be obtained without the input/output-to-state stability condition. I