Decentralized control of interconnected positive systems using L1-induced norm characterization

This study concerns decentralized control of interconnected positive systems. 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, we will show that such a controller can be designed optimally without any information about the rest of the positive subsystems. This achievement is based on our recent characterization of stability of interconnected positive systems in terms of the L₁-induced norm of each subsystem, where the L₁-induced norm is evaluated with positive weighting vectors that are coupled mutually over the subsystems. The key observation in this study is that, under the specific interconnection structure, we can decouple the L₁-induced norm conditions completely. We thus reduce the original stabilization problem into a set of L₁-induced norm optimal controller synthesis problems each of which can be solved indeed locally. In particular, the L₁-induced norm optimal controller synthesis can be done efficiently by solving a semidefinite programming problem.