MIMO Nyquist interpretation of the large gain theorem

Ryan James Caverly, Richard Pates, Leila Jasmine Bridgeman, James Richard Forbes

Research output: Contribution to journalArticlepeer-review


The Large Gain Theorem is an input-output stability result with intriguing applications in the field of control systems. This paper aims to increase understanding and appreciation of the Large Gain Theorem by presenting an interpretation of it for linear time-invariant systems using the well-known Nyquist stability criterion and illustrative examples of its use. The Large Gain Theorem is complementary in nature to the Small Gain Theorem, as it uses a lower bound on the gain of the open-loop system to guarantee closed-loop stability, rather than an upper bound on the gain of the open-loop system. It is shown that the stipulations of the Large Gain Theorem ensure that the multi-input multi-output Nyquist stability criterion is satisfied. Numerical examples of minimum gain and systems that satisfy the Large Gain Theorem are presented, along with examples that make use of the Large Gain Theorem to guarantee robust closed-loop stability.

Original languageEnglish (US)
Pages (from-to)2326-2335
Number of pages10
JournalInternational Journal of Control
Issue number10
StatePublished - Oct 2 2020

Bibliographical note

Funding Information:
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada's Postgraduate Scholarship program PGSD3-471184-2015, as well as the Swedish Foundation for Strategic Research and the Swedish Research Council through the LCCC Linnaeus Center. The authors would like to thank Mr Anthony Maalouly for his preliminary contributions to this work, particularly, introducing Minkowski's inequality for determinants, and Dr Alex Walsh for discussions that helped clarify the proof of Theorem 3.1.

Publisher Copyright:
© 2018 Informa UK Limited, trading as Taylor & Francis Group.


  • Large gain theorem
  • Nyquist stability criterion
  • input-output stability
  • linear systems
  • minimum gain
  • robust control
  • stability of feedback interconnections


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