-Gain Analysis for Positive Linear Systems With Unbounded Time-Varying Delays

This technical note addresses the ℓ_∞-gain (respectively, L_∞-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_∞-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_∞-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.