Studies On Interval Polynomial Stability

Based on the stability criteria for polynomials with fixed coefficients, The Hurwitz stability problems of interval polynomials and the convex combinations of polynomials are discussed in this thesis. In detail, the contents contained in this thesis are as follows:⑴Some significant stability criteria for polynomials, which are foundations for stability analysis of interval polynomials and the convex combinations of polynomials, are summarized at the beginning. These results, such as Routh theorem, Hurwitz theorem, Hermite-Biehler theorem and the stability criteria based on phase properties of Hurwitz polynomials, describe the stability properties of polynomials from different aspects. They make it easy to analyze the stability of polynomials, using different criteria under different conditions.⑵Kharitonov's theorem is an important stability criterion for interval polynomial. This paper gives a new simple proof of Kharitonov's theorem by using the classical zero exclusion principle without invoking the Hermite-Biehler Theorem. A simplification form of Kharitonov's theorem for degree less than 6 is analyzed, and a necessary condition for the stability of the interval polynomials that can reduce the degree is given. Limitations of Kharitonov's theorem are discussed, and two important extensions of the theorem, the Edge Theorem and the Box Theorem, are also discussed. The perturbation ranges of the coefficients of Hurwitz polynomials, which is related to interval polynomial stability, are discussed, and three analysis methods are proposed.⑶The Hurwitz stability criteria for the convex combinations of polynomials are studied. Three basic methods---the positive polynomial pairs, the Routh table and the resultant---are discussed. Moreover, based on the symmetric Rouche's theorem, a necessary and sufficient condition for the Hurwitz stability of the convex combinations of polynomials is proposed. This condition is simple and can be easily applied to the Hurwitz stability analysis of the convex combinations of polynomials.