The projection lemma (often also referred to as the elimination lemma) is one of the most powerful and useful tools in the context of linear matrix inequalities for system analysis and control. In its traditional formulation, the projection lemma only applies to strict inequalities, however, in many applications we naturally encounter non-strict inequalities. As such, we present, in this note, a non-strict projection lemma that generalizes both its original strict formulation as well as an earlier non-strict version. We demonstrate several applications of our result in robust linear-matrix-inequality-based marginal stability analysis and stabilization, a matrix S-lemma, which is useful in (direct) data-driven control applications, and matrix dilation.
%0 Generic
%1 meijer2023nonstrict
%A Meijer, T. J.
%A Holicki, T.
%A van den Eijnden, S. J. A. M.
%A Scherer, C. W.
%A Heemels, W. P. M. H.
%D 2023
%K PN4-3(II) PN4 curated EXC2075
%R 10.48550/arXiv.2305.08735
%T The Non-Strict Projection Lemma
%X The projection lemma (often also referred to as the elimination lemma) is one of the most powerful and useful tools in the context of linear matrix inequalities for system analysis and control. In its traditional formulation, the projection lemma only applies to strict inequalities, however, in many applications we naturally encounter non-strict inequalities. As such, we present, in this note, a non-strict projection lemma that generalizes both its original strict formulation as well as an earlier non-strict version. We demonstrate several applications of our result in robust linear-matrix-inequality-based marginal stability analysis and stabilization, a matrix S-lemma, which is useful in (direct) data-driven control applications, and matrix dilation.
@misc{meijer2023nonstrict,
abstract = {The projection lemma (often also referred to as the elimination lemma) is one of the most powerful and useful tools in the context of linear matrix inequalities for system analysis and control. In its traditional formulation, the projection lemma only applies to strict inequalities, however, in many applications we naturally encounter non-strict inequalities. As such, we present, in this note, a non-strict projection lemma that generalizes both its original strict formulation as well as an earlier non-strict version. We demonstrate several applications of our result in robust linear-matrix-inequality-based marginal stability analysis and stabilization, a matrix S-lemma, which is useful in (direct) data-driven control applications, and matrix dilation.},
added-at = {2023-10-31T14:48:39.000+0100},
author = {Meijer, T. J. and Holicki, T. and van den Eijnden, S. J. A. M. and Scherer, C. W. and Heemels, W. P. M. H.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/222db4863f73728775fd234a44d37d9d8/simtech},
doi = {10.48550/arXiv.2305.08735},
howpublished = {ArXiV},
interhash = {e0f5e2044efad9ec8b8bc283392ee13e},
intrahash = {22db4863f73728775fd234a44d37d9d8},
keywords = {PN4-3(II) PN4 curated EXC2075},
timestamp = {2024-03-12T10:23:33.000+0100},
title = {The Non-Strict Projection Lemma},
year = 2023
}