Distributed L₂-gain control of large-scale systems: A space construction approach

This note studies the distributed L₂-gain control problem for discrete-time large-scale systems. The considered system is transformed equivalently to a high dimensional system. The classical controller design methods are not suitable to be applied directly due to the matrix inversion terms in the transformed system. Using a space construction approach, this paper derives necessary and sufficient stability conditions in the form of linear matrix inequalities for large-scale systems. In addition, under given L₂-gain γ, necessary and sufficient conditions are established such that the considered system is asymptotically stable and the prescribed L₂-gain is satisfied. Then, these results are used to design distributed controllers for large-scale systems. Based on a matrix construction method, distributed controllers can be constructed via the solutions of a set of linear matrix inequalities to guarantee that the closed-loop system is stable with the given L₂-gain. Finally, the advantages of the proposed theoretical results are demonstrated by two examples.