Construction of Polynomial Polytopes by Using Segment Lemma

Abstract

A polynomial with all roots lying in the open left half plane of the complex plane is called Hurwitz stable. The convex hull of a finite number of polynomials of the same order is called a polynomial polytope. By the Edge theorem, a polynomial polytope with the invariant degree is Hurwitz stable if and only if all edges are Hurwitz stable. Due to the Edge Theorem, the segment stability criterions are of great importance. In this paper, we consider a construction of stable polytopes by using the Segment Lemma. It is constructed stable polytopes with nonzero volumes. The obtained results can be used, for example, in the stabilization problems of unstable transfer functions by lower-order controllers.

Keywords

• Hurwitz stability
• Polynomial
• Segment Lemma
• Polynomial polytope

References

1. [1] Bartlett, A. C., Hollot, C. V., Lin, H., “Root locations of an entire polytope of polynomials: It suffices to check the edges”, Mathematics of Control, Signals, and Systems, 1(1) (1988) : 61-71.
2. [2] Bhattacharyya S. P., Chapellat H. ve Keel L. H., “Robust control: The parametric approach”, New Jersey: Prentice-Hall, (1995).
3. [3] Barmish B. R., “New Tools for robustness of linear systems”, New York: Macmillan Publishing Company, (1994).
4. [4] Bialas, S., “A necessary and sufficient condition for the stability of convex combinations of stable polynomials or matrices”, Bulletin of The Polish Academy of Sciences Technical Sciences, 33(9-10) (1985) : 473-480.
5. [5] Aguirre, B. and Suárez, R., “Algebraic test for the Hurwitz stability of a given segment of polynomials”, Boletin De La Sociedad Matematica Mexicana, 12 (2006) : 261-275.
6. [6] Aguirre-Hernández, B., García-Sosa, F. R., Loredo-Villalobos, C. A., Villafuerte-Segura, R., Campos-Cantón, E., “Open problems related to the Hurwitz stability of polynomials segments”, Mathematical Problems in Engineering, Volume 2018, Article ID2075903 (2018) : 8 pages.
7. [7] Kalinina, E. A., Smol’kin Y. A., Uteshev, A. Yu., “Stability and distance to instability for polynomial matrix families. Complex perturbations”, Linear and Multilinear Algebra, (2020) DOI: 10.1080/03081087.2020.1759500.
8. [8] Gayvoronskiy, S. A., Ezangina, T., Puskarev, M., Khozhaev, I., “IE control theory & applications, 14(18) (2020) : 2825-2835.

9. [9] Hinrichsen, D., Pritchard, A. J., “Text in applied mathematics: vol. 48, Mathematical systems theory I. Modelling, state space analysis, stability and robustness”, Berlin, Springer-Verlag, (2005).
10. [10] Aguirre-Hernández, B., Frías-Armenta, M. E., Verduzco, F., “Smooth trivial vector bundle structure of the space of Hurwitz polynomials”, Automatica, 45 (2009) : 2864-2868.
11. [11] Dzhafarov, V., Esen, Ö., Büyükköroğlu, T., “Infinite polytopes in Hurwitz stability region”, Automatica, 106 (2019) : 301-305.
12. [12] Dzhafarov, V., Esen, Ö., Büyükköroğlu, T., “On polytopes in Hurwitz region”, Systems & Control Letters, 141 (2020) : 1-5.
13. [13] Aguirre-Hernández, B.; Frías-Armenta, M.E. and Verduzco, F., “On differential structures of polynomial spaces in Control Theory”, Journal of Systems Science and Systems Engineering, 21(3) (2012) : 372-382.