Solving the Fully Fuzzy Sylvester Matrix Equation With Triangular Fuzzy Number

Malkawi, Ghassan; Ahmad, Nazihah; Ibrahim, Haslinda

doi:10.17654/fjmssep2015_037_055

Cited by 24 publications

(26 citation statements)

References 12 publications

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“…The definitions and theorems of the fuzzy Kronecker products and fuzzy V ec-operator, are provided as follows: Definition 8. [17] LetÃ = (ã ij ) m×n andB = (b ij ) p×q be fuzzy matrices. Fuzzy Kronecker product is represented asÃ ⊗ kB , wherẽ…”

Section: Fundamental Concepts Of Fuzzy Kronecker Products and Fuzzy V Ec-operatormentioning

confidence: 99%

“…Subsequently, in 2015, Shang et al [15] proposed their algorithm in solving fully fuzzy Sylvester matrix equation (FFSE), AX +XB =C by applying the arithmetic multiplication operator, which has been previously proposed in Dehghan et al [16]. On the other hand, Malkawi et al [17] have proposed an algorithm which offers faster computational compared to Shang et al [15]. While in 2020, Elsayed et al [18] carried out a study in solving the FFME ofÃX +XB =C, which considering the entries of the equation are in trapezoidal fuzzy numbers.…”

Section: Introductionmentioning

confidence: 99%

“…In developing the algorithm, the existing fuzzy multiplication arithmetic operators are modified as the existing operators introduced by [17] and [20] are not applicable to perform the multiplication involving near-zero fuzzy numbers. Besides that, a new fuzzy subtraction operation is also developed in solving the FFME, since the existing operator is inadequated to subtract a near-zero fuzzy number to a positive fuzzy number.…”

Section: Introductionmentioning

confidence: 99%

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A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation

Daud

Ahmad

Malkawi³

2021

TELKOMNIKA

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Matrix equations have its own important in the field of control system engineering particularly in the stability analysis of linear control systems and the reduction of nonlinear control system models. There are certain conditions where the classical matrix equation are not well equipped to handle the uncertainty problems such as during the process of stability analysis and reduction in control system engineering. In this study, an algorithm is developed for solving fully fuzzy matrix equation particularly forÃXB −X =C, where the coefficients of the equation are in near-zero fuzzy numbers. By modifying the existing fuzzy multiplication arithmetic operators, the proposed algorithm exceeds the positive restriction to allow the near-zero fuzzy numbers as the coefficients. Besides that, a new fuzzy subtraction arithmetic operator has also been proposed as the existing operator failed to satisfy the both sides of the near-zero fully fuzzy matrix equation. Subsequently, Kronecker product and V ec-operator are adapted with the modified fuzzy arithmetic operator in order to transform the fully fuzzy matrix equation to a fully fuzzy linear system. On top of that, a new associated linear system is developed to obtain the final solution. A numerical example and the verification of the solution are presented to demonstrate the proposed algorithm.This is an open access article under the CC BY-SA license.

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Section: Fundamental Concepts Of Fuzzy Kronecker Products and Fuzzy V Ec-operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation

Daud

Ahmad

Malkawi³

2021

TELKOMNIKA

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“…In addition, several analytical methods have been proposed to triangular FFSME (TFFSME). The most important of which are Kronecker product and matrix inversion [20], Kronecker product and associated linear system [21,22]. However, the methods were restricted only for positive triangular fuzzy numbers (TFNs) only.…”

Section: Definition 14mentioning

confidence: 99%

Arbitrary Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation

Elsayed¹,

Ahmad

Malkawi

2021

Preprint

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There are many applications where couple of Sylvester matrix equations (CSME) are required to be solved simultaneously, especially in analyzing the stability of control systems. However, there are some situations in which the crisp CSME are not well equipped to deal with the uncertainty problem during the process of stability analysis in control system engineering. Thus, in this paper a new method for solving a coupled trapezoidal fully fuzzy Sylvester matrix equation (CTrFFSME) with arbitrary coefficients is proposed. The arithmetic fuzzy multiplication operation is applied to convert the CTrFFSME into a system of non-linear equations. Then the obtained non-linear system is reduced and converted to a system of absolute equations where the fuzzy solution is obtained by solving that system. The proposed method can solve many unrestricted fuzzy systems such as Sylvester and Lyapunov fully fuzzy matrix equations with triangular and trapezoidal fuzzy numbers. We illustrate the proposed methods by solving numerical example.

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“…In addition, alternative methods for solving FFSME have been proposed by (Malkawi et al 2015c). Contrary to (Guo and Shang 2013), their research shows that FFSME is converted to FFLS using the Kronecker product and applying their proposed linear system method.…”

Section: Introductionmentioning

confidence: 99%

On the solution of fully fuzzy Sylvester matrix equation with trapezoidal fuzzy numbers

2020

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Sylvester matrix equations play a prominent role in various areas such as control theory, medical imaging acquisition systems, model reduction, and stochastic control. Considering any uncertainty problems such as conflicting requirements during system process, instability of environmental conditions, distraction of any elements and noise, all for which the classical matrix equation is sometimes ill-equipped, fuzzy numbers represent the most effective tool that can be used to model matrix equations in the form of fuzzy equations. In most of the previous literature, the solutions of fuzzy systems are only presented with triangular fuzzy numbers. In this paper, we discuss fully fuzzy Sylvester matrix equation with positive and negative trapezoidal fuzzy numbers. An analytical approach for solving a fully fuzzy Sylvester matrix equation is proposed by transforming the fully fuzzy matrix equation into a system of four crisp Sylvester linear matrix equation. In obtaining the solution the Kronecker product and Vec-operator are used. Numerical examples are solved to illustrate the proposed method.

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Solving the Fully Fuzzy Sylvester Matrix Equation With Triangular Fuzzy Number

Cited by 24 publications

References 12 publications

A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation

A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation

Arbitrary Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation

On the solution of fully fuzzy Sylvester matrix equation with trapezoidal fuzzy numbers

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