MIMO Nyquist interpretation of the large gain theorem

Research output: Contribution to journalArticle


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.


  • Ryan James Caverly
  • Richard Pates
  • Leila Jasmine Bridgeman
  • James Richard Forbes
External organisations
  • University of Minnesota system
  • Duke University
  • McGill University
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Control Engineering


  • input-output stability, Large gain theorem, linear systems, minimum gain, Nyquist stability criterion, robust control, stability of feedback interconnections
Original languageEnglish
JournalInternational Journal of Control
Publication statusE-pub ahead of print - 2018 Dec 11
Publication categoryResearch