It is known that input-output approaches based on scaled small-gain theoremswith constant $D$-scalings and integral linear constraints are non-conservativefor the analysis of some classes of linear positive systems interconnected withuncertain linear operators. This dramatically contrasts with the case ofgeneral linear systems with delays where input-output approaches provide, ingeneral, sufficient conditions only. Using these results we provide simplealternative proofs for many of the existing results on the stability of linearpositive systems with discrete/distributed/neutral time-invariant/-varyingdelays and linear difference equations. In particular, we give a simple prooffor the characterization of diagonal Riccati stability for systems withdiscrete-delays and generalize this equation to other types of delay systems.The fact that all those results can be reproved in a very simple waydemonstrates the importance and the efficiency of the input-output frameworkfor the analysis of linear positive systems. The approach is also used toderive performance results evaluated in terms of the $L_1$-, $L_2$- and$L_\infty$-gains. It is also flexible enough to be used for design purposes.
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