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Small-gain theorem

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Feedback connection between systems S1 and S2.

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 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 and are connected in a feedback loop, then the closed loop system is input-output stable if and both and are stable by themselves. (This norm is typically the -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).[1][2]

Notes

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  1. ^ Glad, Ljung: Control Theory, Page 19
  2. ^ Glad, Ljung: Control Theory (Edition 2:6), Page 31

References

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  • 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

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