Investigation of robust stability of fractional order multilinear affine systems: -convex parpolygon approach

Yeroğlu, Celaleddin; Senol, Bilal

doi:10.1016/j.sysconle.2013.06.005

Cited by 24 publications

(33 citation statements)

References 21 publications

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“…To begin with, we give some definitions. Definition Let

{bold-italic \bar{K}}_{i} (s) = ({}, {\overset{K}{true}}_{i}^{1} (s), {\overset{K}{true}}_{i}^{2} (s), {\overset{K}{true}}_{i}^{3} (s), \dots)

and

{bold-italic \bar{S}}_{i} (s) = ({}, {\overset{S}{true}}_{i}^{1} (s), {\overset{S}{true}}_{i}^{2} (s), {\overset{S}{true}}_{i}^{3} (s), \dots)

denote all the outer vertices and outer segments of P i ( s ) in Equation when s traverses on

\partial scriptD

(see References , for more information about the outer vertices and segments). Definition Define the outer generalized fractional order segment polynomials

{bold-italic \bar{Δ}}_{E} (s) = \cup_{l = 1}^{m} {bold-italic \bar{Δ}}_{E}^{l} (s)

where

\begin{matrix} {\overset{normalΔ}{true}}_{E}^{l} (s) & = F_{1} (s) {\overset{K}{true}}_{1} (s) + \dots + F_{l - 1} (s) {\overset{K}{true}}_{l - 1} (s) + F_{l <...>} \end{matrix}

…”

Section: Main Results For Fractional Order Parametric Controlmentioning

confidence: 99%

“…In Reference , a simple and efficient graphical procedure was developed for testing the stability of interval fractional order systems. Yeroglu et al discussed robust stability problem of fractional order systems with multi‐linear affine uncertainty structure…”

Section: Introductionmentioning

confidence: 99%

“…Section 4 will explain the method to determine the D-stabilizing region of FOPID controller for a fixed plant. By combining the results in Sections 3 and 4, we will present the graphical tuning method of FOPID controllers for According to References [16,17], the outer vertices and outer segments of P i .s/ are always no more than 2 nC1 and .n C 1/2 n , respectively. Therefore, if one uses Theorem 3.2 instead of Theorem 3.1, the computational burden to check the robust D-stability can be reduced.…”

mentioning

confidence: 99%

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Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust ‐stability

Zheng

Tang

Song

2015

Intl J Robust & Nonlinear

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SUMMARYThis paper focuses on the graphical tuning method of fractional order proportional integral derivative (FOPID) controllers for fractional order uncertain system achieving robust D-stability. Firstly, general result is presented to check the robust D-stability of the linear fractional order interval polynomial. Then some alternative algorithms and results are proposed to reduce the computational effort of the general result. Secondly, a general graphical tuning method together with some computational efficient algorithms are proposed to determine the complete set of FOPID controllers that provides D-stability for interval fractional order plant. These methods will combine the results for fractional order parametric robust control with the method of FOPID D-stabilization for a fixed plant. At last, two important extensions will be given to the proposed graphical tuning methods: determine the D-stabilizing region for fractional order systems with two kinds of more general and complex uncertainty structures: multi-linear interval uncertainty and mixed-type uncertainties. Numerical examples are followed to illustrate the effectiveness of the method.

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“…To begin with, we give some definitions. Definition Let

{bold-italic \bar{K}}_{i} (s) = ({}, {\overset{K}{true}}_{i}^{1} (s), {\overset{K}{true}}_{i}^{2} (s), {\overset{K}{true}}_{i}^{3} (s), \dots)

and

{bold-italic \bar{S}}_{i} (s) = ({}, {\overset{S}{true}}_{i}^{1} (s), {\overset{S}{true}}_{i}^{2} (s), {\overset{S}{true}}_{i}^{3} (s), \dots)

denote all the outer vertices and outer segments of P i ( s ) in Equation when s traverses on

\partial scriptD

(see References , for more information about the outer vertices and segments). Definition Define the outer generalized fractional order segment polynomials

{bold-italic \bar{Δ}}_{E} (s) = \cup_{l = 1}^{m} {bold-italic \bar{Δ}}_{E}^{l} (s)

where

\begin{matrix} {\overset{normalΔ}{true}}_{E}^{l} (s) & = F_{1} (s) {\overset{K}{true}}_{1} (s) + \dots + F_{l - 1} (s) {\overset{K}{true}}_{l - 1} (s) + F_{l <...>} \end{matrix}

…”

Section: Main Results For Fractional Order Parametric Controlmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust ‐stability

Zheng

Tang

Song

2015

Intl J Robust & Nonlinear

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show abstract

“…for the closed-loop uncertain quasipolynomials in which the plant numerator contains the time-delay term). Besides, the works [10][11][12][13] extended the idea of the value set concept also to fractional order uncertain polynomials.…”

Section: Robust Stability Analysis For (Fractional Order Time-delay) mentioning

confidence: 99%

“…the stability under all possible variations of uncertain parameters has to be verified. Quite naturally, the combination of the robust stability problem under parametric uncertainty with fractional order systems has become also the important research subject in the recent years [10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Robust Stabilization of a Main Irrigation Canal Pool via Fractional Order PI Controller: A Graphical Investigation

Matušů¹

2016

DAAAM Proceedings

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This contribution is focused on graphical investigation of robust stability for a feedback control loop containing the parametrically uncertain second order time-delay model of a main irrigation canal pool and a fractional order PI controller. The robust stability of the family of fractional order closed-loop characteristic quasipolynomials is analyzed by means of the sampled value sets and application of the zero exclusion condition.

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Robust stabilization of fractional‐order plant with general interval uncertainties based on a graphical method

Zheng

2017

Intl J Robust & Nonlinear

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This paper concentrates on the robust stabilization of fractional-order plant suffering from general interval uncertainties by using fractional-order controllers.General interval uncertainties mean that the coefficients and orders of the denominator and nominator of the fractional-order plant are all uncertain and lie in specified intervals. Two main contributions are presented, as follows.(i) Based on a graphical method, a necessary and sufficient criterion is proposed for the stabilization of the general interval fractional-order plant. By adopting some well-defined multivalue functions, the proposed method can explicitly construct the nonconvex boundary of the value set of the fractional system.(ii) Two alternative methods are presented to improve the computational efficiency of the stabilization test. Based on a newly developed redundancy elimination technique, the first method can avoid computing and plotting many segments, which are in the interior of the value set of the fractional system. The second method utilizes a novel convex polygon bounding approach. It can efficiently construct a convex polygon to bound the nonconvex value set of the general interval fractional system. Examples are followed to illustrate the effectiveness of the proposed methods.

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Investigation of robust stability of fractional order multilinear affine systems: -convex parpolygon approach

Cited by 24 publications

References 21 publications

Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust ‐stability

Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust ‐stability

Robust Stabilization of a Main Irrigation Canal Pool via Fractional Order PI Controller: A Graphical Investigation

Robust stabilization of fractional‐order plant with general interval uncertainties based on a graphical method

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