Small-gain theorem

{{Short description|Theorem used for studying closed-loop stability}}

File:Feedback connection between two systems..svg

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 $\backslash mathcal\{L\}$ 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 and both $S\_1$ and $S\_2$ are stable by themselves. (This norm is typically the $\backslash mathcal\{H\}\_\backslash infty$-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).Glad, Ljung: Control Theory, Page 19Glad, Ljung: Control Theory (Edition 2:6), Page 31