TY - JOUR
T1 - Convex matrix inequalities versus linear matrix inequalities
AU - Helton, J. William
AU - McCullough, Scott
AU - Putinar, Mihai
AU - Vinnikov, Victor
N1 - Funding Information:
Manuscript received June 30, 2007; revised April 04, 2008. Current version published May 13, 2009. This work was supported in part by the National Science Foundation (NSF) and the Ford Motor Company, by NSF Grant DMS-0140112, by NSF Grant DMS-0350911, and by the Israel Science Foundation under Grant 322/00. Recommended by Guest Editors G. Chesi and D. Henrion. J. W. Helton is with the Department of Mathematics, University of California San Diego, La Jolla, CA 92093-0112 USA (e-mail: [email protected]). S. McCullough is with the Department of Mathematics, University of Florida, Gainesville, FL 32611-8105 USA (e-mail: [email protected]). M. Putinar is with the Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA 93106-3080 USA. V. Vinnikov is with the Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2009.2017087
PY - 2009/5/8
Y1 - 2009/5/8
N2 - Most linear control problems lead directly to matrix inequalities (MIs). Many of these are badly behaved but a classical core of problems are expressible as 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 rigid than convex MIs, there is the hope that a convexity based theory will be less restrictive than LMIs. How much more restrictive are LMIs than convex MIs?. There are two fundamentally different classes of linear systems problems: dimension free and dimension dependent. A dimension free MI is a MI where the unknowns are -tuples of matrices and appear in the formulas in a manner which respects matrix multiplication. Most of the classic MIs of control theory are dimension free. Dimension dependent MIs have unknowns which are tuples of numbers. The two classes behave very differently and this survey describes what is known in each case about the relation between convex MIs and LMIs. The proof techniques involve and necessitate new developments in the field of semialgebraic geometry.
AB - Most linear control problems lead directly to matrix inequalities (MIs). Many of these are badly behaved but a classical core of problems are expressible as 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 rigid than convex MIs, there is the hope that a convexity based theory will be less restrictive than LMIs. How much more restrictive are LMIs than convex MIs?. There are two fundamentally different classes of linear systems problems: dimension free and dimension dependent. A dimension free MI is a MI where the unknowns are -tuples of matrices and appear in the formulas in a manner which respects matrix multiplication. Most of the classic MIs of control theory are dimension free. Dimension dependent MIs have unknowns which are tuples of numbers. The two classes behave very differently and this survey describes what is known in each case about the relation between convex MIs and LMIs. The proof techniques involve and necessitate new developments in the field of semialgebraic geometry.
KW - Algebraic approaches
KW - Convex optimization
KW - Linear control systems
KW - Linear matrix inequality (LMI)
UR - http://www.scopus.com/inward/record.url?scp=67349160558&partnerID=8YFLogxK
U2 - 10.1109/TAC.2009.2017087
DO - 10.1109/TAC.2009.2017087
M3 - Article
AN - SCOPUS:67349160558
SN - 0018-9286
VL - 54
SP - 952
EP - 964
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
IS - 5
ER -