The key highlights and insights of this content are:
The authors address the distributed state feedback controller design problem for continuous-time linear time-invariant systems using linear matrix inequalities (LMIs).
They target a class of non-block-diagonal Lyapunov functions that have the same sparsity pattern as the distributed controllers, which can generalize the conventional block-diagonal relaxation.
By leveraging a block-diagonal factorization of sparse matrices and Finsler's lemma, the authors first present a nonlinear matrix inequality for stabilizing distributed controllers with such Lyapunov functions. This inequality becomes necessary and sufficient over chordal communication graphs.
The authors then derive an LMI by relaxing the nonlinear matrix inequality, which completely covers the conventional block-diagonal relaxation. They also provide analogous results for H∞ control.
Numerical examples demonstrate the efficacy of the proposed methods, showing that they outperform the conventional block-diagonal relaxation.
The computation of inverse matrices in the proposed method can be decomposed into smaller ones corresponding to subsystems formed by cliques of the communication graph, which enhances the scalability.
The authors show that the derived conditions provide a necessary and sufficient condition for distributed (H∞) controllers with the class of non-block-diagonal Lyapunov functions under chordal sparsity, allowing for easy evaluation of the conservatism.
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