Input Delay Tolerance of Nonlinear Systems Under Smooth Feedback: A Semiglobal Control Framework

Abstract

This article studies the problem of when a globally asymptotically stabilizable nonlinear system by smooth feedback is tolerable with respect to input delay. We illustrate, by means of theoretic and practical examples, some fundamental limitations including: 1) A globally exponentially stabilizable (GES) nonlinear system may not guarantee global asymptotic stabilizability (GAS) of the nonlinear system even with an arbitrarily small input delay; 2) GAS cannot even ensure semiglobal asymptotic stabilizability of the nonlinear system with a small input delay. To overcome these obstacles, we introduce the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">semiglobal input delay tolerance</i> (SGIDT) and present a semiglobal control framework for the asymptotic analysis/synthesis of input delay tolerance of multi-input–multi-output (MIMO) nonlinear systems under smooth feedback. With the aid of Razumikhin theorem and the converse Lyapunov theorem on global asymptotic local exponential stability (GALES), it is proved that in the case of state feedback, GALES does ensure the SGIDT of a MIMO nonlinear system. In the case of output feedback, it is further proved that GALES and uniform observability imply the SGIDT of the nonlinear system.