A new tabular form for determining root distribution of a complex polynomial with respect to the imaginary axis

Chen, S.S.; Tsai, Jason Sheng‐Hong

doi:10.1109/9.241571

Cited by 18 publications

(3 citation statements)

References 9 publications

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“…Lev-Ari, Bistritz and Kailath [5] proposed fast triangular factorization to evaluate the inertia of Bezoutian matrices with displacement structure, which leads to the Routh-Hurwitz and Schur-Cohn tests as well as more general test for regions with arbitrary circles and straight lines. Due to the extensive works, Routh test has been extended to solve zero distribution problems for complex polynomials (e.g., [5] and [6]) and subregions enclosed by various type of boundaries (e.g., which are transformed from imaginary axis by any rational function [7], or arbitrary lines, or segments [8]). Recently, Bistritz [9] presented fraction-free forms of the Routh test for complex and real integer polynomials Technical development of Routh test also includes the treatment for singular cases.…”

Section: Introductionmentioning

confidence: 99%

Routh-type table test for zero distribution of polynomials with commensurate fractional and integer degrees

Liang

Wang

Wang³

2017

Journal of the Franklin Institute

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This paper mainly presents Routh-type table test methods for zero distribution of polynomials with commensurate fractional degrees on the left-half plane, right-half plane and imaginary axis in the complex plane. The proposed tabular methods are derived for extension and generalization of the Routh test, which is widely used in controls for zero distribution of polynomials with integer degrees. Singular cases are discussed and handled efficiently and simply. Necessary and sufficient conditions for the second singular case are completely analyzed in terms of symmetric zeros. A particular property is revealed that a polynomial with commensurate fractional degrees without pure imaginary zero may still be stable in the presence of the second singular case, which is impossible for a real polynomial with integer degrees. Furthermore, we present a test to solve the zero distribution problem with respect to general sector region for polynomials with commensurate fractional degrees and real/complex coefficients. Finally, numerical examples are given to illustrate the correctness and effectiveness of the results. The proposed methods have broad application areas, including various systems, circuits and

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Section: Introductionmentioning

confidence: 99%

Routh-type table test for zero distribution of polynomials with commensurate fractional and integer degrees

Liang

Wang

Wang³

2017

Journal of the Franklin Institute

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“…However, the same result, i.e., the confinement of all the roots in the aforementioned minor LHP sector (no roots in the corresponding major sector), had already been obtained well before by means of Routh-Hurwitz arguments (Usher, 1957;Luthi, 1942-43) or could easily have been achieved based on generalisations of the Routh-Hurwitz criteria (Hurwitz, 1895;Routh, 1877) to polynomials with complex coefficients (Frank, 1946;Billarz,1944). New formulations, extensions and improvements of similar algebraic conditions, including • the analysis of the critical cases and different tabular-form presentations, can be found in (Sivanandam & Sreekala, 2012;Chen & Tsai, 1993;Benidir & Picinbono, 1991;Agashe, 1985;Hwang & Tripathi; and, more recently, in (Bistritz, 2013) where numerically very efficient variants are presented . A different approach has been followed in (Kaminski et al, 2015) where, for q > 1, a test based on regular chains for semi-algebraic sets (Chen et al, 2013) has been suggested.…”

Section: Stability Conditionsmentioning

confidence: 99%

Fractional Order System Forced-response Decomposition and Its Application

Casagrande

Krajewski

Viaro

2018

Mathematical Techniques of Fractional Order Systems

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This chapter deals with the additive decomposition of the forced response of a fractionalorder system. Precisely, it is shown how, by solving a simple polynomial Diophantine equation, this response can almost always be decomposed into the sum of a systemdependent component and an input-dependent component. The system-dependent component is formed from the same modes as the system and, assuming stabi li ty, characterises the transient behaviour of the system in the response to sustained inputs. The input-dependent component is formed from the same modes as the input, and accounts for the steady-state or long-term response of a stable system to a persistent input. Simple conditions based on the classical Routh and Mikhai lov criteria are provided to check the system input-output stability. Several examples show that the aforementioned decomposition can profitably be exploited to find simplified models in such a way that the asymptotic response is kept unchanged and, at the same time, the transient behaviour is well approximated. The decomposition proves useful also for solving the so-called model-matching problem that is of particular interest in controller synthesis.

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“…, n) can be assigned such that the roots of the complex-coefficient characteristic equation (13) have negative real parts. According to Chen and Tsai (1993), the roots are in the open left half complex plane if and only if…”

Section: Every Equilibrium State Of the Following Systemmentioning

confidence: 99%