Extended Bernstein-Kantorovich-Stancu Operators with Multiple Parameters and Approximation Properties

Kadak, Uğur; Özlük, Muharrem

doi:10.1080/01630563.2021.1895833

Cited by 4 publications

(7 citation statements)

References 40 publications

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“…The modified BKS operator is defined in [ 25 ] as

where

and

represent a continuous function defined on the interval

.…”

Section: Model Derivation and Control Objectivesmentioning

confidence: 99%

“…

where

denotes the floor function, and

for

. The associated shape parameters

are constrained to be

[ 25 ].…”

Section: Model Derivation and Control Objectivesmentioning

confidence: 99%

“…By derivation, it follows that in case λ i ,

, and

are any real numbers satisfying the above conditions, the following moments were obtained in [ 25 ]

where

…”

Section: Model Derivation and Control Objectivesmentioning

confidence: 99%

“…Moreover, with the development of the BK operator, some improved BK operators, such as λ -BK [ 17 , 18 ],

-BK [ 19 ],

-BK [ 20 ], and other methods, have also been developed [ 21 , 22 , 23 , 24 ]. In this paper, a new BKS operator, which is introduced in [ 25 ], is implemented in the manipulator controller to optimize the parameter update process of the disturbance observer and improve the trajectory tracking accuracy of the robotic manipulator controller.…”

Section: Introductionmentioning

confidence: 99%

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A Modified Robotic Manipulator Controller Based on Bernstein-Kantorovich-Stancu Operator

Zhang

Wang

2022

Micromachines

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With the development of intelligent manufacturing and mechatronics, robotic manipulators are used more widely. There are complex noises and external disturbances in many application cases that affect the control accuracy of the manipulator servo system. On the basis of previous research, this paper improves the manipulator controller, introduces the Bernstein–Kantorovich–Stancu (BKS) operator, and proposes a modified robotic manipulator controller to improve the error tracking accuracy of the manipulator controller when observing complex disturbances and noises. In addition, in order to solve the problem that the coupling between the external disturbances of each axis of the manipulator leads to a large amount of computation when observing disturbances, an improved full-order observer is designed, which simplifies the parameters of the controller combined with the BKS operator and reduces the complexity of the algorithm. Through a theoretical analysis and a simulation test, it was verified that the proposed manipulator controller could effectively suppress external disturbance and noise, and the application of the BKS operator in the manipulator servo system control is feasible and effective.

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“…The modified BKS operator is defined in [ 25 ] as

where

and

represent a continuous function defined on the interval

.…”

Section: Model Derivation and Control Objectivesmentioning

confidence: 99%

“…

where

denotes the floor function, and

for

. The associated shape parameters

are constrained to be

[ 25 ].…”

Section: Model Derivation and Control Objectivesmentioning

confidence: 99%

“…By derivation, it follows that in case λ i ,

, and

are any real numbers satisfying the above conditions, the following moments were obtained in [ 25 ]

where

…”

Section: Model Derivation and Control Objectivesmentioning

confidence: 99%

“…Moreover, with the development of the BK operator, some improved BK operators, such as λ -BK [ 17 , 18 ],

-BK [ 19 ],

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Modified Robotic Manipulator Controller Based on Bernstein-Kantorovich-Stancu Operator

Zhang

Wang

2022

Micromachines

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“…In recent years, statistical convergence properties for some specific linear positive operators and their various modifications have been studied by many researchers (e.g. see [11,[25][26][27][28][29] for more details).…”

Section: Let Us Considermentioning

confidence: 99%

A numerical comparative study of generalized Bernstein-Kantorovich operators

Kadak¹,

Özger²

2021

MFC

Self Cite

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<p style='text-indent:20px;'>In this paper, a new generalization of the Bernstein-Kantorovich type operators involving multiple shape parameters is introduced. Certain Voronovskaja and Grüss-Voronovskaya type approximation results, statistical convergence and statistical rate of convergence of proposed operators are obtained by means of a regular summability matrix. Some illustrative graphics that demonstrate the convergence behavior, accuracy and consistency of the operators are given via Maple algorithms. The proposed operators are comprehensively compared with classical Bernstein, Bernstein-Kantorovich and other new modifications of Bernstein operators such as <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein, <inline-formula><tex-math id="M2">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich, <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Bernstein and <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich operators.</p>

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Approximation by Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators

Berwal,

Mohiuddine,

Kajla

et al. 2024

Math Methods in App Sciences

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In this research paper, we construct a new sequence of Riemann–Liouville type fractional ‐Bernstein–Kantorovich operators. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first order modulus of continuity of these operators. Further, we study Voronovskaja type theorem, quantitative Voronovskaya type theorem, Chebyshev–Grüss inequality and Grüss–Voronovskaya type theorem.

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Extended Bernstein-Kantorovich-Stancu Operators with Multiple Parameters and Approximation Properties

Cited by 4 publications

References 40 publications

A Modified Robotic Manipulator Controller Based on Bernstein-Kantorovich-Stancu Operator

A Modified Robotic Manipulator Controller Based on Bernstein-Kantorovich-Stancu Operator

A numerical comparative study of generalized Bernstein-Kantorovich operators

Approximation by Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators

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