1. Convex Reformulation of LMI-Based Distributed Controller Design with a Class of Non-Block-Diagonal Lyapunov Functions
- Author
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Watanabe, Yuto, Fushimi, Sotaro, and Sakurama, Kazunori
- Subjects
Mathematics - Optimization and Control ,Electrical Engineering and Systems Science - Systems and Control - Abstract
This study addresses a distributed state feedback controller design problem for continuous-time linear time-invariant systems by means of linear matrix inequalities (LMI). As the exact convexification is still open, the block-diagonal relaxation of Lyapunov functions has been prevalent despite its conservatism. In this work, we target a class of non-block-diagonal Lyapunov functions that has the same sparsity as distributed controllers. By leveraging a block-diagonal factorization of sparse matrices and Finsler's lemma, we first present a (nonlinear) matrix inequality for stabilizing distributed controllers with such Lyapunov functions, which boils down to a necessary and sufficient condition for such controllers if the sparsity pattern is chordal. As a relaxation of the inequality, we derive an LMI that completely covers the conventional relaxation and then provide analogous results for $H_\infty$ control. Lastly, numerical examples underscore the efficacy of our results., Comment: 8 pages
- Published
- 2024