Linear-Programming-Based Decentralized Stabilizing Controller Synthesis for Interconnected Positive Systems and its Optimality Property

This study is concerned with decentralized stabilizing controller synthesis problem for interconnected systems constructed from positive subsystems. The main issue is how to design a controller for each subsystem locally so that positivity and stability of the overall interconnected system can be attained. Under a specific interconnection structure, the author has already shown that such a controller can be designed locally without any information about the rest of the positive subsystems. Namely, under the specific interconnection structure, we can reduce the original problem into a set of $L_{1^{-}}$ induced-norm optimal controller synthesis problems for subsystems each of which can be solved indeed locally via linear programming. On the basis of these preceding results, in this study, we show an optimality property of such local $L_{1}$ -induced-norm optimal controllers in a global sense. More precisely, we prove that the set of $L_{1}$ -induced-norm optimal controllers is indeed globally optimal in the sense that it minimizes the abscissa of the coefficient matrix of the overall interconnected closed-loop system.