Problem 10.1
Root-clustering for multivariate
polynomials and robust stability analysis
Pierre-Alexandre Bliman
INRIA
Rocquencourt BP 105
78153 Le Chesnay cedex
France
1 DESCRIPTION OF THE PROBLEM
Problem (2, 3) is a linear matrix inequality in the m + 1 unknown matrices P, Q1, . . . , Qm, a convex optimization problem.
The LMIs (2, 3) obtained for increasing values of k constitute indeed a family of weaker and weaker sufficient conditions for (1). Conversely, property (1) necessarily implies solvability of the LMIs for a certain rank k and beyond.
See [1] for details.
Numerical experimentations have shown that the precision of the criteria obtained for small values of k (2 or 3) may be remarkably good already, but rational use of this result requires to have a priori information on the size of the least k, if any, for which the LMIs are solvable. Bounds, especially upper bound, on this quantity are thus highly desirable, and they should be computed with low complexity algorithms.
2 MOTIVATIONS AND COMMENTS
We expose here some problems related to property (1).
Robust stability
Property (1) is equivalent to asymptotic stability of the uncertain system
