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 $$\|S_1\| \cdot \|S_2\| < 1$$ 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).