Convex Reformulation of LMI-Based Distributed Controller Design with a Class of Non-Block-Diagonal Lyapunov Functions
<|reference_start|>Convex Reformulation of LMI-Based Distributed Controller Design with a Class of Non-Block-Diagonal Lyapunov Functions: 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.<|reference_end|>
@article{watanabe2024convex,
    title={Convex Reformulation of LMI-Based Distributed Controller Design with a
  Class of Non-Block-Diagonal Lyapunov Functions},
    author={Yuto Watanabe, Sotaro Fushimi, Kazunori Sakurama},
    journal={arXiv preprint arXiv:2404.04576},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.04576},
    primaryClass={math.OC cs.SY eess.SY}
}