Paper Info
Title
Sparse solution of the Lyapunov equation for large-scale interconnected systems
Abstract
We consider the problem of computing an approximate banded solution of the continuous-time Lyapunov equation A ¿ X ¿ + X ¿ A ¿ T = P ¿ , where the coefficient matrices A ¿ and P ¿ are large, symmetric banded matrices. The (sparsity) pattern of A ¿ describes the interconnection structure of a large-scale interconnected system. Recently, it has been shown that the entries of the solution X ¿ are spatially localized or decaying away from a banded pattern. We show that the decay of the entries of X ¿ is faster if the condition number of A ¿ is smaller. By exploiting the decay of entries of X ¿ , we develop two computationally efficient methods for approximating X ¿ by a banded matrix. For a well-conditioned and sparse banded A ¿ , the computational and memory complexities of the methods scale linearly with the state dimension. We perform extensive numerical experiments that confirm this, and that demonstrate the effectiveness of the developed methods. The methods proposed in this paper can be generalized to (sparsity) patterns of A ¿ and P ¿ that are more general than banded matrices. The results of this paper open the possibility for developing computationally efficient methods for approximating the solution of the large-scale Riccati equation by a sparse matrix.
Year
DOI
Venue
2016
10.1016/j.automatica.2016.06.002
Automatica (Journal of IFAC)
Keywords
Field
DocType
Complex systems,Large-scale Lyapunov equation,Large-scale optimization problems and methods,Remote and distributed control
Complex system,Mathematical optimization,Condition number,Lyapunov equation,Matrix (mathematics),Riccati equation,Interconnection,Band matrix,Mathematics,Sparse matrix
Journal
Volume
Issue
ISSN
73
C
0005-1098
Citations 
PageRank 
References 
3
0.39
31
Authors
2
Name
Order
Citations
PageRank
Aleksandar Haber1524.67
Michel Verhaegen21074140.85