Local convex directions for Hurwitz stable polynomials

A new condition for a polynomial p(s) to be a local convex direction for a Hurwitz stable polynomial q(s) is derived. The condition is in terms of polynomials associated with the even and odd parts of p(s) and q(s), and constitutes a generalization of Rantzer's phase-growth condition for global convex directions. It is used to determine convex directions for certain subsets of Hurwitz stable polynomials

Fixed order controller design via parametric methods

Cataloged from PDF version of article.In this thesis, the problem of parameterizing stabilizing fixed-order controllers for linear time-invariant single-input single-output systems is studied. Using a generalization of the Hermite-Biehler theorem, a new algorithm is given for the determination of stabilizing gains for linear time-invariant systems. This algorithm requires a test of the sign pattern of a rational function at the real roots of a polynomial. By applying this constant gain stabilization algorithm to three subsidiary plants, the set of all stabilizing first-order controllers can be determined. The method given is applicable to both continuous and discrete time systems. It is also applicable to plants with interval type uncertainty. Generalization of this method to high-order controller is outlined. The problem of determining all stabilizing first-order controllers that places the poles of the closed-loop system in a desired stability region is then solved. The algorithm given relies on a generalization of the Hermite-Biehler theorem to polynomials with complex coefficients. Finally, the concept of local convex directions is studied. A necessary and sufficient condition for a polynomial to be a local convex direction of another Hurwitz stable polynomial is derived. The condition given constitutes a generalization of Rantzer’s phase growth condition for global convex directions. It is used to determine convex directions for certain subsets of Hurwitz stable polynomials.Saadaoui, KarimPh.D

Continuity argument revisited: geometry of root clustering via symmetric products

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $\Omega$, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in control theory: root clustering ($D$-stability) developed by R.E. Kalman, B.R. Barmish, S. Gutman et al., $D$-decomposition(Yu.I. Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A. Fam, J. Meditch, J.Ackermann). Our approach is based on the interpretation of correspondence between roots and coefficients of a polynomial as a symmetric product morphism. We describe the topology of strata up to homotopy equivalence and, for many important cases, up to homeomorphism. Adjacencies between strata are also described. Moreover, we provide an explanation for the special position of classical stability problems: Hurwitz stability, Schur stability, hyperbolicity.Comment: 45 pages, 4 figure

Constructing convex directions for stable polynomials

Cataloged from PDF version of article.New constructions of convex directions for Hurwitz-stable polynomials are obtained. The technique is based on interpretations of the phase-derivative conditions in terms of the sensitivity of the root-locus associated with the even and odd parts of a polynomial

Help on SOS

In this issue of IEEE Control Systems Magazine, Andy Packard and friends respond to a query on determining the region of attraction using sum-of-squares methods

The Lee-Yang and P\'olya-Schur Programs. I. Linear Operators Preserving Stability

In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre, Hurwitz and P\'olya-Schur on univariate polynomials with such properties.Comment: Final version, to appear in Inventiones Mathematicae; 27 pages, no figures, LaTeX2