A Test for the Stability of Networks

• Autorzy: Agnieszka Rowinska-Schwarzweller , Christoph Schwarzweller
• Języki publikacji: EN

Abstrakty

EN

A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an all-dominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwitz by checking whether the rational function e(p)/o(p) can be realized as a reactance of one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here e(p) and o(p) denote the even and the odd part of p [25].

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2013
• Tom: 21
• Numer: 1
• Strony: 47-53

Daty

wydano

2013-01-01

Twórcy

autor

Agnieszka Rowinska-Schwarzweller

• Chair of Display Technology University of Stuttgart Allmandring 3b, 70596 Stuttgart, Germany

autor

Christoph Schwarzweller

• Institute of Computer Science University of Gdansk Wita Stwosza 57, 80-952 Gdansk, Poland

Bibliografia

• [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• [2] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
• [3] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
• [4] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
• [5] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
• [6] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
• [7] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [8] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
• [9] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
• [10] Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265-269, 2001.
• [11] Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.
• [12] Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.
• [13] Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.
• [14] Michał Muzalewski and Wojciech Skaba. From loops to abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833-840, 1990.
• [15] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.
• [16] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.
• [17] Piotr Rudnicki and Andrzej Trybulec. Multivariate polynomials with arbitrary number of variables. Formalized Mathematics, 9(1):95-110, 2001.
• [18] Christoph Schwarzweller. Introduction to rational functions. Formalized Mathematics, 20 (2):181-191, 2012. doi:10.2478/v10037-012-0021-1.[Crossref]
• [19] Christoph Schwarzweller and Agnieszka Rowinska-Schwarzweller. Schur’s theorem on the stability of networks. Formalized Mathematics, 14(4):135-142, 2006. doi:10.2478/v10037-006-0017-9.[Crossref]
• [20] Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. FormalizedMathematics, 1(3):445-449, 1990.
• [21] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
• [22] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
• [23] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
• [24] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
• [25] Rolf Unbehauen. Netzwerk- und Filtersynthese: Grundlagen und Anwendungen. Oldenbourg-Verlag, fourth edition, 1993.
• [26] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
• [27] Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992.

Identyfikatory

DOI

10.2478/forma-2013-0005