<inline-formula> <tex-math notation="TeX">$\ell_{\infty}/L_{\infty} $</tex-math></inline-formula>-Gain Analysis for Positive Linear Systems With Unbounded Time-Varying Delays

Abstract

This technical note addresses the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -gain (respectively, L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -gain) analysis problem for discrete-time (respectively, continuous-time) positive linear systems with unbounded time-varying delays. For the discrete-time case, by virtue of the monotonicity of an auxiliary system and the corresponding delay-free system, it is shown that the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -gain of discrete-time positive systems with unbounded delays is insensitive of the magnitude of delays. For the continuous-time case, we first construct a sampled-data positive system and analyze its monotonicity and asymptotic behavior. Then, by comparing the delayed positive system with the proposed sampled-data positive system and the corresponding delay-free system, it is further proved that the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> -gain of continuous-time positive systems with unbounded delays is also independent of the delays and fully determined by the system matrices. The application of the results is illustrated by a sampled-data system with unbounded sampling intervals.