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.
Sylvester Matrix Equations (SME) play a central role in applied mathematics, particularly in systems and control theory. A fuzzy theory is normally applied to represent the uncertainty of real problems where the classical SME is extended to Fully Fuzzy Sylvester Matrix Equation (FFSME). The existing analytical methods for solving FFSME are based on Vec-operator and Kronecker product. Nevertheless, these methods are only applicable for nonnegative fuzzy numbers, which limits the applications of the existing methods. Thus, this paper proposes a new numerical method for solving arbitrary Trapezoidal FFSME (TrFFSME), which includes near-zero trapezoidal fuzzy numbers to overcome this limitation. The TrFFSME is converted to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication operations. Then the non-linear system is solved using a newly developed two-stage algorithm. In the first stage algorithm, initial values are determined. Subsequently, the second stage algorithm obtains all possible finite fuzzy solutions. A numerical example is solved to illustrate the proposed method. Besides, this proposed method can solve other forms of fuzzy matrix equations and produces finite fuzzy and non-fuzzy solutions compared to the existing methods.
In the fuzzy literature, researchers have applied the concept of Vec-operator and Kronecker product for solving arbitrary Fuzzy Matrix Equations (FME). However, this approach is limited to positive or negative FMEs and cannot be applied to FMEs with near-zero fuzzy numbers. Therefore, this paper proposes a new analytical method for solving a family of arbitrary FMEs. The proposed method is able to solve Arbitrary Generalized Trapezoidal Fully Fuzzy Sylvester Matrix Equations (AGTrFFSME), in addition to many unrestricted FMEs such as Sylvester, Lyapunov and Stein fully fuzzy matrix equations with arbitrary triangular or trapezoidal fuzzy numbers. The proposed method thus fruitfully removes the sign restriction imposed by researchers and is, therefore, better to use in several engineering and scientific applications. The AGTrFFSME is converted to a system of non-linear equations, which is reduced using new multiplication operations between trapezoidal fuzzy numbers. The feasibility conditions are introduced to distinguish between fuzzy and non-fuzzy solutions to the AGTrFFSME.
Many authors proposed analytical methods for solving fully fuzzy Sylvester matrix equation (FFSME) based on Vec-operator and Kronecker product. However, these methods are restricted to nonnegative fuzzy numbers and cannot be extended to FFSME with near-zero fuzzy numbers. The main intention of this paper is to develop a new numerical method for solving FFSME with near-zero trapezoidal fuzzy numbers that provides a wider scope of trapezoidal fully fuzzy Sylvester matrix equation (TrFFSME) in scientific applications. This numerical method can solve the trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients and find all possible finite arbitrary solutions for the system. In order to obtain all possible fuzzy solutions, the TrFFSME is transferred to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication between trapezoidal fuzzy numbers. The fuzzy solutions to the TrFFSME are obtained by developing a new two-stage algorithm. To illustrate the proposed method numerical example is solved.
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