A new solution of pair matrix equations with arbitrary triangular fuzzy numbers

Wan Suhana Wan Daud,Ghassan Malkawi,Nazihah Ahmad

doi:10.3906/mat-1805-9

Wan Suhana Wan Daud, Ghassan Malkawi + Show 1 more

Open Access

https://doi.org/10.3906/mat-1805-9

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Abstract

Pair matrix equations have numerous applications in control system engineering, such as for stability analysis of linear control systems and also for reduction of nonlinear control system models. There are some situations in which the classical pair matrix equations are not well equipped to deal with the uncertainty problem during the process of stability analysis and reduction in control system engineering. Thus, this study presents a new algorithm for solving fully fuzzy pair matrix equations where the parameters of the equations are arbitrary triangular fuzzy numbers. The fuzzy Kronecker product and fuzzy $Vec$-operator are employed to transform the fully fuzzy pair matrix equations to a fully fuzzy pair linear system. Then a new associated linear system is developed to convert the fully fuzzy pair linear system to a crisp linear system. Finally, the solution is obtained by using a pseudoinverse method. Some related theoretical developments and examples are constructed to illustrate the proposed algorithm. The developed algorithm is also able to solve the fuzzy pair matrix equation.

Highlights

In real world applications, matrix equations play an essential role in several situations
Due to the limitations of these two studies, we aim to provide an algorithm for solving an arbitrary PFFME of
The development of the algorithm presented in this study provides the first investigation on how to solve the PFFME in Eq (1)

Summary

Introduction

Matrix equations play an essential role in several situations. There are a number of studies in which all the parameters of the matrix equations are in the form of fuzzy. In [26], the PME consists of A1X + X B1 = C1 and A2X B2 = C2 , where only some of the parameters of the equation are in the form of arbitrary fuzzy numbers. By Definition 8, the coefficient of the FFLS for this equation is a matrix with size of sr × qp This is proof that the PFFME will yield the same number of columns since they have same solution of. Proof Let (A1)p×pXp×q + Xp×q(B1)q×q = Cp×q, (A1X )p×q + (X B1)p×q = Cp×q, be the fully fuzzy matrix equation as shown in Eq (26), where A1 and B1 are fuzzy coefficients and Xp×q is the fuzzy solution.

M1 M2 N1

Solving the PFME using the proposed method

Conclusion