On ℓ∞ and L∞ gains for positive systems with bounded time-varying delays

This paper is devoted to the analysis of the ℓ∞ and L∞ gains for positive linear systems with interval time-varying delays. Through exploiting the monotonicity of the state trajectory, we first prove that for positive systems with constant delays, the ℓ∞ and L∞ gains are fully governed by the system matrices but independent of the delay size. Moreover, for positive systems with bounded time-varying delays, by comparing with the nominal systems with constant delays, it turns out that the ℓ∞ and L∞ gains are exactly the same as that of the constant delay systems. The results in this paper reveal that the ℓ∞ and L∞ gains of positive linear systems are not sensitive to the magnitude of time delays and hence the computation of ℓ∞ and L∞ gains of positive systems with bounded time-varying delays can be reduced to computing the ℓ∞ and L∞ gains of the corresponding delay-free systems. Both continuous-time and discrete-time cases are considered in this paper.

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