Small-gain theorem

In nonlinear systems, the formalism of input-output stability is an important tool in studying the stability of interconnected systems since the gain of a system directly relates to how the norm of a signal increases or decreases as it passes through the system.

The small-gain theorem gives a sufficient condition for finite-gain

The small gain theorem was proved by George Zames in 1966.

It can be seen as a generalization of the Nyquist criterion to non-linear time-varying MIMO systems (systems with multiple inputs and multiple outputs).

-norm, the size of the largest singular value of the transfer function over all frequencies.

Any induced Norm will also lead to the same results).