A Limiting Property of the Matrix Exponential
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Date
2014-04
Publication Type
Journal Article
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Abstract
A limiting property of the matrix exponential is proven: if the (1,1)-block of a 2-by-2 block matrix becomes 'arbitrarily small' in a limiting process, the matrix exponential of that matrix tends to zero in the (1,1)-, (1,2)-, and (2,1)-blocks. The limiting process is such that either the log norm of the (1,1)-block goes to negative infinity, or, for a certain polynomial dependency, the matrix associated with the largest power of the variable that tends to infinity is stable. The limiting property is useful for simplification of dynamic systems that exhibit modes with sufficiently different time scales. The obtained limit then implies the decoupling of the corresponding dynamics. © 1963-2012 IEEE.
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published
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Journal / series
Volume
59 (4)
Pages / Article No.
1105 - 1110
Publisher
IEEE
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Subject
Limiting property; Logarithmic norm; Matrix exponential; Time-scale separation
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Notes
Manuscript received 15 October 2012, Revised 7 June 2013, Accepted 15 October 2013, Date of publication 24 October 2013, Date of current Version 20 March 2014.