Small-gain theorem

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Theorem used for studying closed-loop stability

title: "Small-gain theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["nonlinear-control"] description: "Theorem used for studying closed-loop stability" topic_path: "general/nonlinear-control" source: "https://en.wikipedia.org/wiki/Small-gain_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Theorem used for studying closed-loop stability ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/a7/Feedback_connection_between_two_systems..svg" caption="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 \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| 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).

A complementing result due to Georgiou, Khammash and Megretski (1997), referred to as the large-gain theorem, quantifies the minimum loop-gain needed to stabilize an unstable, possibly nonlinear and time-varying, plant; the minimum loop-gain being 1.

Notes

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.

References

  1. G. Zames, "On the input-output stability of time-varying nonlinear feedback systems Part one: Conditions derived using concepts of loop gain, conicity, and positivity," in IEEE Transactions on Automatic Control, vol. 11, no. 2, pp. 228-238, April 1966.
  2. Glad, Ljung: Control Theory (Edition 2:6), Page 31
  3. Andrew R. Teel, Tryphon T. Georgiou, Laurent Praly, Eduardo S. Sontag, ''Input-output stability'', The Control Handbook. 1996; 4250, section 44.1.
  4. Georgiou, Tryphon T., Mustafa Khammash, and Alexander Megretski. "On a large-gain theorem." Systems & control letters 32.4 (1997): 231-234.

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