A Vector Small-Gain Theorem for General Nonlinear Control Systems

TL;DR

This paper introduces a new vector small-gain theorem for nonlinear control systems using vector Lyapunov functions, enabling stability analysis of complex, large-scale, and delay systems with broad applicability.

Contribution

It develops a novel vector small-gain theorem employing vector Lyapunov functions, extending stability analysis to diverse nonlinear control systems.

Findings

Recovers several recent stability results as special cases

Applicable to large-scale, sampled-data, and time-delay systems

Demonstrated on a biochemical circuit model

Abstract

A new Small-Gain Theorem is presented for general nonlinear control systems. The novelty of this research work is that vector Lyapunov functions and functionals are utilized to derive various input-to-output stability and input-to-state stability results. It is shown that the proposed approach recovers several recent results as special instances and is extendible to several important classes of control systems such as large-scale complex systems, nonlinear sampled-data systems and nonlinear time-delay systems. An application to a biochemical circuit model illustrates the generality and power of the proposed vector small-gain theorem.