Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases

Many real problems in control systems are related to the solvability of the generalized Sylvester matrix equation either using analytical or numerical methods. However, in many applications, the classical generalized Sylvester matrix equation are not well equipped to handle uncertainty in real-life...

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Main Author:	Ahmed Abdelaziz Elsayed, Ahmed
Format:	Thesis
Language:	eng eng
Published:	2022
Subjects:	QA Mathematics QA76.76 Fuzzy System.
Online Access:	https://etd.uum.edu.my/9715/1/permission%20to%20deposit-903671.pdf https://etd.uum.edu.my/9715/2/s903671_01.pdf
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spelling	my-uum-etd.97152022-08-04T02:33:32Z Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases 2022 Ahmed Abdelaziz Elsayed, Ahmed Ahmad, Nazihah Malkawi, Ghassan Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Art & Sciences QA Mathematics QA76.76 Fuzzy System. Many real problems in control systems are related to the solvability of the generalized Sylvester matrix equation either using analytical or numerical methods. However, in many applications, the classical generalized Sylvester matrix equation are not well equipped to handle uncertainty in real-life problems such as conflicting requirements during the system process, the distraction of any elements and noise. Thus, crisp number in this matrix equation is replaced by fuzzy numbers and called generalized fully fuzzy Sylvester matrix equation when all parameters are in fuzzy form. The existing fuzzy analytical methods have four main drawbacks, the avoidance of using near-zero fuzzy numbers, the lack of accurate solutions, the limitation of the size of the systems, and the positive sign restriction of the fuzzy matrix coefficients and fuzzy solutions. Meanwhile, the convergence, feasibility, existence and uniqueness of the fuzzy solution are not examined in many fuzzy numerical methods. In addition, many studies are limited to positive fuzzy systems only due to the limitation of fuzzy arithmetic operation, especially for multiplication between trapezoidal fuzzy numbers.Therefore, this study aims to construct new analytical and numerical methods, namely fuzzy matrix vectorization, fuzzy absolute value, fuzzy Bartle’s Stewart, fuzzy gradient iterative and fuzzy least-squares iterative for solving arbitrary generalized Sylvester matrix equation for special cases and couple Sylvester matrix equations. In constructing these methods, new fuzzy arithmetic multiplication operators for trapezoidal fuzzy numbers are developed. The constructed methods overcome the positive restriction by allowing the negative, near-zero fuzzy numbers as the coefficients and fuzzy solutions. The necessary and sufficient conditions for the existence, uniqueness, and convergence of the fuzzy solutions are discussed, and a complete analysis of the fuzzy solution is provided. Some numerical examples and the verification of the solutions are presented to demonstrate the constructed methods. As a result, the constructed methods have successfully demonstrated the solutions for the arbitrary generalized Sylvester matrix equation for special and general cases based on the new fuzzy arithmetic operations, with minimum complexity fuzzy operations. The constructed methods are applicable to either square or non-square coefficient matrices up to 100 × 100. In conclusion, the constructed methods have significant contribution to the application of control system theory without any restriction on the system. 2022 Thesis https://etd.uum.edu.my/9715/ https://etd.uum.edu.my/9715/1/permission%20to%20deposit-903671.pdf text eng staffonly https://etd.uum.edu.my/9715/2/s903671_01.pdf text eng public other doctoral Universiti Utara Malaysia
institution	Universiti Utara Malaysia
collection	UUM ETD
language	eng eng
advisor	Ahmad, Nazihah Malkawi, Ghassan
topic	QA Mathematics QA76.76 Fuzzy System.
spellingShingle	QA Mathematics QA76.76 Fuzzy System. Ahmed Abdelaziz Elsayed, Ahmed Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases
description	Many real problems in control systems are related to the solvability of the generalized Sylvester matrix equation either using analytical or numerical methods. However, in many applications, the classical generalized Sylvester matrix equation are not well equipped to handle uncertainty in real-life problems such as conflicting requirements during the system process, the distraction of any elements and noise. Thus, crisp number in this matrix equation is replaced by fuzzy numbers and called generalized fully fuzzy Sylvester matrix equation when all parameters are in fuzzy form. The existing fuzzy analytical methods have four main drawbacks, the avoidance of using near-zero fuzzy numbers, the lack of accurate solutions, the limitation of the size of the systems, and the positive sign restriction of the fuzzy matrix coefficients and fuzzy solutions. Meanwhile, the convergence, feasibility, existence and uniqueness of the fuzzy solution are not examined in many fuzzy numerical methods. In addition, many studies are limited to positive fuzzy systems only due to the limitation of fuzzy arithmetic operation, especially for multiplication between trapezoidal fuzzy numbers.Therefore, this study aims to construct new analytical and numerical methods, namely fuzzy matrix vectorization, fuzzy absolute value, fuzzy Bartle’s Stewart, fuzzy gradient iterative and fuzzy least-squares iterative for solving arbitrary generalized Sylvester matrix equation for special cases and couple Sylvester matrix equations. In constructing these methods, new fuzzy arithmetic multiplication operators for trapezoidal fuzzy numbers are developed. The constructed methods overcome the positive restriction by allowing the negative, near-zero fuzzy numbers as the coefficients and fuzzy solutions. The necessary and sufficient conditions for the existence, uniqueness, and convergence of the fuzzy solutions are discussed, and a complete analysis of the fuzzy solution is provided. Some numerical examples and the verification of the solutions are presented to demonstrate the constructed methods. As a result, the constructed methods have successfully demonstrated the solutions for the arbitrary generalized Sylvester matrix equation for special and general cases based on the new fuzzy arithmetic operations, with minimum complexity fuzzy operations. The constructed methods are applicable to either square or non-square coefficient matrices up to 100 × 100. In conclusion, the constructed methods have significant contribution to the application of control system theory without any restriction on the system.
format	Thesis
qualification_name	other
qualification_level	Doctorate
author	Ahmed Abdelaziz Elsayed, Ahmed
author_facet	Ahmed Abdelaziz Elsayed, Ahmed
author_sort	Ahmed Abdelaziz Elsayed, Ahmed
title	Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases
title_short	Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases
title_full	Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases
title_fullStr	Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases
title_full_unstemmed	Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases
title_sort	arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases
granting_institution	Universiti Utara Malaysia
granting_department	Awang Had Salleh Graduate School of Arts & Sciences
publishDate	2022
url	https://etd.uum.edu.my/9715/1/permission%20to%20deposit-903671.pdf https://etd.uum.edu.my/9715/2/s903671_01.pdf
_version_	1747828656701440000

Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases

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