Noncommutative convexity vs LMI'S

J. William Helton, Scott McCullough, Victor Vinnikov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


Most linear control problems convert directly to matrix inequalities, MIs. Many of these are badly behaved but a classical core of problems convert to linear matrix inequalities (LMIs). in many engineering systems problems convexity has all of the advantages of a LMI. Since LMIs have a structure which is seemingly much more ridged than convexity, there is the hope that a convexity based theory will be less restrictive than LMIs. A dimensionless MI is a MI where the unknowns are matrices and appear in the formula in a manner which respects matrix multiplication. This holds for most of the classic MIs of control theory. The results presented here suggest the surprising conclusion that for dimensionless MIs convexity offers no greater generality than LMIs. in fact, we prove, for a class of model situations, that a convex dimensionless MI is equivalent to an LMI.

Original languageEnglish
Title of host publicationProceedings of the 16th IFAC World Congress, IFAC 2005
PublisherIFAC Secretariat
Number of pages6
ISBN (Print)008045108X, 9780080451084
StatePublished - 1 Jan 2005

Publication series

NameIFAC Proceedings Volumes (IFAC-PapersOnline)
ISSN (Print)1474-6670


  • Algebraic approaches
  • Convex optimization
  • LMI
  • Linear control systems
  • Linear matrix inequality

ASJC Scopus subject areas

  • Control and Systems Engineering


Dive into the research topics of 'Noncommutative convexity vs LMI'S'. Together they form a unique fingerprint.

Cite this