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 \mathcal{L} stability of the feedback connection. 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).

Theorem. Assume two stable systems S_1 and S_2 are connected in a feedback loop, then the closed loop system is input-output stable if |S_1| \cdot |S_2| and both S_1 and S_2 are stable by themselves. (This norm is typically the \mathcal{H}_\infty-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).

A complementing result due to Georgiou, Khammash and Megretski (1997), referred to as the large-gain theorem, quantifies the minimum loop-gain needed to stabilize an unstable, possibly nonlinear and time-varying, plant; the minimum loop-gain being 1.