title: Small-gain theorem
text: # 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).
  
  