Noncommutative Convexity vs. LMI's
| Authors: | Helton J. William, UC San Diego, United States
McCullough Scott, University of Florida, United States
Vinnikov Victor, Ben Gurion of the Negev, Israel |
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| Topic: | 2.5 Robust Control |
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| Session: | Positive Polynomials and LMI Optimization with Applications to Robust Control |
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| Keywords: | Algebraic approaches, Convex optimization,Linear control systems,LMI,Linear matrix Inequality |
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Abstract
Most linear control problems convertdirectly to matrix inequalities, MIs.Many of theseare badly behaved but a classical core of problems convert tolinear matrix inequalities (LMIs).In many engineering systems problems convexityhas all of the advantages of a LMI.Since LMIs have a structure which is seemingly much more ridgedthan convexity, there is the hope that a convexitybased theory will be less restrictive than LMIs.A dimensionless MI is a MI where the unknowns are matricesand appear in the formula in a manner whichrespects matrix multiplication. This holdsfor most of the classic MIs of control theory.The results presented heresuggest the surprising conclusion that for dimensionless MIsconvexity offers no greater generality than LMIs.In fact, we prove, for a class ofmodel situations, that a convex dimensionlessMI is equivalent to an LMI. Also we give a discussion of dimension dependent results.
