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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. [ABSTRACT FROM AUTHOR]

The conic sector theorem is a powerful input–output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis. Without means to systematically identify and impose exterior conic properties, existing analysis and synthesis methods are limited to employing a special case of the conic sector theorem which may prove overly conservative when considering very strictly passive plants. This paper removes the current limitation by establishing the exterior conic sector lemma, which provides matrix inequalities equivalent to exterior conic bounds for linear time-invariant systems. [ABSTRACT FROM PUBLISHER]

This paper focuses on the newly developed notion of minimum gain and the corresponding Large Gain Theorem. The Large Gain Theorem is an input-output stability result particularly well suited to unstable plants connected in feedback with stable or unstable controllers. This paper aims to facilitate the practical application of these results. An altered definition of minimum gain broadens the applicability of the Large Gain Theorem, and the novel Minimum Gain Lemma provides LMI conditions that imply and are often equivalent to a minimum gain for LTI systems. Numerical examples are provided to clarify the differences between the existing and proposed definitions of minimum gain, highlight the utility of the newly established Minimum Gain Lemma, and demonstrate how the paper's contributions may be employed in practice. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]