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 $\mathcal$ 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 $\mathcal_\infty$-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 (Edition 2:6), Page 31

Notes

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References

* H. K. Khalil, ''Nonlinear Systems,'' third edition, Prentice Hall, Upper Saddle River, New Jersey, 2002;
* C. A. Desoer, M. Vidyasagar, ''Feedback Systems: Input-Output Properties,'' second edition, SIAM, 2009.

See also

* Input-to-state stability

Nonlinear control