MIMO Nyquist interpretation of the large gain theorem

Abstract

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.

Keywords:

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

Acknowledgements

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.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

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.