Extension of the root-locus method to a certain class of fractional-order systems

Merrikh-Bayat, Farshad; Afshar, Mahdi; Karimi-Ghartemani, Masoud

doi:10.1016/j.isatra.2008.08.001

Cited by 33 publications

(12 citation statements)

References 20 publications

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“…Matignon theorem [27] and its generalization [28] indicate that a linear time invariant fractional order system is BIBO stable if and only if its characteristic function as a polynomial with fractional degrees (PFD) has no closed-RHP zero. Thereafter, the stability and stabilization of fractional order systems have been extensively investigated, e.g., by LMI-based methods [29], [30], root locus method [31], analysis for rational order systems [32], and a review on various systems including linear or nonlinear, distributed, and time delay [33]. The classical Routh-Hurwitz criterion has been directly utilized for checking the stability of fractional order systems with commensurate order less than one [34], whereas that test is only sufficient and therefore conservative.…”

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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“…Proposition Let D ( s ) be a fractional‐order polynomial with FDEG{ D ( s )} = n , then the equation D ( s ) = 0 has exactly n roots on the Riemann surface . For a fractional‐order linear time‐invariant system whose poles are in general complex conjugate, the stability condition can be stated as follows …”

Section: Preliminariesmentioning

confidence: 99%

Robust stability analysis for fractional‐order systems with time delay based on finite spectrum assignment

Liu

Zhang

Xue

et al. 2019

Intl J Robust & Nonlinear

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Summary In this paper, the robust stability of a fractional‐order time‐delay system is analyzed in the frequency domain based on finite spectrum assignment (FSA). The FSA algorithm is essentially an extension of the traditional pole assignment method, which can change the undesirable system characteristic equation into a desirable one. Therefore, the presented analysis scheme can also be used as an alternative time‐delay compensation method. However, it is superior to other time‐delay compensation schemes because it can be applied to open‐loop poorly damped or unstable systems. The FSA algorithm is extended to a fractional‐order version for time‐delay systems at first. Then, the robustness of the proposed algorithm for a fractional‐order delay system is analyzed, and the stability conditions are given. Finally, a simulation example is presented to show the superior robustness and delay compensation performance of the proposed algorithm. Moreover, the robust stability conditions and the time‐delay compensation scheme presented can be applied on both integer‐order and fractional‐order systems.

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“…. The quotient is a (N-2) th order polynomial : (20) Eqns. (18) and (19) give the general models for the division process involving a single real root and a pair of conjugate complex root respectively.…”

Section: Sub-step 22: Root Validation Testmentioning

confidence: 99%

Computer-aided root -locus numerical technique

Zubair¹,

Olatunbosun²

2014

Nig. J. Tech.

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The existing technique of manual calcu The existing technique of manual calcu The existing technique of manual calcu The existing technique of manual calcu studies studies studies studies is laborious and time consuming. is laborious and time consuming. is laborious and time consuming. is laborious and time consuming. simple fast mathematical iteration is simple fast mathematical iteration is simple fast mathematical iteration is simple fast mathematical iteration is were formed and te were formed and te were formed and te were formed and tested for convergence. A pair of complementary formulas with highest rate of sted for convergence. A pair of complementary formulas with highest rate of sted for convergence. A pair of complementary formulas with highest rate of sted for convergence. A pair of complementary formulas with highest rate of convergence was selected. The process of convergence was selected. The process of convergence was selected. The process of convergence was selected. The process of were divided into steps which were were divided into steps which were were divided into steps which were were divided into steps which were g g g geometric properties of the root eometric properties of the root eometric properties of the root eometric properties of the root-de de de described in the literature. Root scribed in the literature. Root scribed in the literature. Root scribed in the literature. Root-loci are drawn to scale instead of rough sketches. loci are drawn to scale instead of rough sketches. loci are drawn to scale instead of rough sketches. loci are drawn to scale instead of rough sketches.

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Extension of the root-locus method to a certain class of fractional-order systems

Cited by 33 publications

References 20 publications

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

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

Robust stability analysis for fractional‐order systems with time delay based on finite spectrum assignment

Computer-aided root -locus numerical technique

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