Abstract
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.
Highlights
Sylvester Matrix Equations (SME) in the form AX + XB = C plays a vital role in many fields such as in control systems [1], medical imaging data acquisition, model reduction [2] and stochastic control, in addition to image processing and filtering [3]
The theory and applications of Fuzzy Relation Equations (FREs) can be found in Di Nola et al [9], which indicated that if the solvability of max-continuous t-norm FREs is assumed, the solution set for the FREs can be fully determined from a unique greatest solution and all minimal solutions, and the number of minimal solutions is always finite
Most of the analytical methods proposed for solving Triangular fully fuzzy Sylvester matrix equation (TFFSME) and Trapezoidal FFSME (TrFFSME) in the literature are based on Dubois and Prade’s arithmetic operator for multiplication, restricted only to positive fuzzy numbers with very small fuzziness [41]
Summary
Most of the analytical methods proposed for solving TFFSME and TrFFSME in the literature are based on Dubois and Prade’s arithmetic operator for multiplication, restricted only to positive fuzzy numbers with very small fuzziness [41]. These methods are limited to positive coefficients and positive fuzzy solutions only. Many researchers have applied Kaufmann and Gupta’s arithmetic multiplication operator for solving TFFSME with arbitrary coefficients; their methods are limited to positive fuzzy solutions only, and these methods cannot detect all possible fuzzy solutions.
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