Stability Analysis of Discrete-time Interconnected Positive Systems using Weighted <i>l</i>₁-induced Norm

In this paper, we study the stability analysis of discrete-time interconnected positive systems by means of a weighted <i>l</i>₁-induced norm, where the weighted <i>l</i>₁-induced norm is computed from the <i>l</i>₁norm of the input and output signals evaluated with given weighting vectors. In the literature,it was shown that the weighted <i>L</i>₁-induced norm of continuous-time LTI positive systems can be characterized by linear programming problem (LP). Moreover, the weighted <i>L</i>₁-induced norm was proved to be useful for the stability analysis of interconnected positive systems. On the basis of these results, we first show that the weighted <i>l</i>₁-induced norm of discrete-time LTI positive systems can be again characterized by LP. From this preliminary result, we can construct a transformation from a given discrete-time positive system to a continuous-time positive system, where the weighted <i>L</i>₁-induced norm of the resulting system is equivalent to the weighted <i>l</i>₁-induced norm of the original system. By means of this key transformation, we can readily extend the continuous-time case results to the stability analysis of discrete-time interconnected positive systems.