Conic stability of polynomials and positive maps (original) (raw)
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The paper conside& robust stability properties for Schur polynomia~~ of the f o g By ploning coefficient variations in planes defined by variable pairs ai, an-i for eachi and requiring in each such 'plane the region of obtained coefficients to be bounded by lines of slope 4S0, 90° a 4 135". we ah& that stability for. all polynomials defined by comer points is necessary and sufficient for stabiity of all polynomials defmed by any' points in the region. Using this idea, one can construct several necessity and differing sufficiency conditions for the stability of polynomials where each a i can vary independently in an i n t e r v a l~~i ]. As the sufficiency condi-The "weak" version of Kharitonov's theorem states that a. necessary and sufficient condition for robust stability is that all'comer polynomials &-stable (comerpolynomi& are obtained when ai {a&) V'i). .The "strong" version states that a necessary and suffi&ent condition foistability is that a particular four of the 2n+l comer polynomials a r i stable. Actually, when n < 6, stability of fewer than four comer polynomials constitutes anecessary and sufficient condition for robust stability, see [2], ' '. .. . ~i v e n t h i s time-continuous stability result, an. obvious conjecture that can be made concerning the discrete-time problem (when the region of interest is lzl < 1) is rhat stability of all comer polynomials is necessary and sufficient for robust stabiity. This is not wrr&t, and there exist countetexamp1es demonstrating thefalsity of this conjecture, see 13-41. Nevertheless, certain rather more specialtions become closer to necessity cohditions the number of ized results available. Let us now recall some. If% = distinct polynomials for which stabiity has to b-S tested infor i = O,l,.. ..,(n-1) L 2 (see. footnote for the meaning of creases.