Solving Positive Trapezoidal Fully Fuzzy Sylvester Matrix Equation

Many problems in systems and control theory are related to solvability of Sylvester matrix equations. In many applications, at least some of the parameters of the system should be represented by fuzzy numbers rather than crisp ones. In most of the previous literature, the solutions of fuzzy Sylvester matrix equation are only presented with triangular fuzzy numbers. In this paper, we propose two analytical methods for solving Positive Trapezoidal Fully Fuzzy Sylvester Matrix Equation (PTrFFSME). The PTrFFSME is converted to an equivalent system of crisp Sylvester Matrix Equations (SME) using the existing arithmetic fuzzy multiplication operations. The necessary and sufficient conditions for the existence and uniqueness of the positive fuzzy solutions to the PTrFFSME are investigated. In addition, the equivalency between the solution to the system of SME and the PTrFFSME are discussed. The proposed methods are illustrated by solving one example.


Introduction
Sylvester matrix equation in the form AX + XB = C played a very important role in many areas such as in control systems [1][2][3], reduction of non-linear control systems models [4,5], system design [6], theory of orbits [7] and medical imaging acquisition system [8][9][10][11]. In many scenarios, the classical linear system is not well suited to handle uncertain information ties in real-life problems, since some coefficient values may be vague and imprecise due to incomplete information [12]. Using fuzzy numbers rather than the crisp numbers is a common way of expressing such imprecision [13]. The SME can be extended to a Fuzzy Sylvester Matrix Equation (FSME) in the form AX +XB =C if the solution matrixX and the constant matrixC are in fuzzy form. The FSME was studied in [14,15], where the Kronecker product was applied to convert the FSME to a fuzzy linear system. However, this method can be applied to a FSME with small size only. In order to overcome this shortcoming, authors in [16] applied Dubois and Prade's arithmetic multiplication operations [17] to convert the FSME to a SME and then the fuzzy solution obtained by applying Bartle's Stewart method.
When all the parameters of the SME are in the form positive trapezoidal fuzzy matrices then it is called PTrFFSME.
Several studies were conducted to solve the Triangular Fully Fuzzy Sylvester Matrix Equation (TFFSME), in the formÃX +XB =C. Shang et al. [18] extended the TFFSME into a system of three crisp linear matrix equations, while Malkawi et al. [19] applied the Kronecker product method to convert the TFFSME to a Fully Fuzzy Linear System (FFLS). The FFLS is then converted to a linear system and the solution obtained by direct inversion method. Similarly, Daud et al. [20][21][22] applied Kronecker product and Vec-operator to convert the TFFSME to a system of matrix linear equation. However, in all these methods the size of the obtained linear system equation is much larger than the TFFSME considered. Therefore, these methods are limited to TFFSME with small size. Hou et al. [23] proposed a method for solving TFFSME with parametric fuzzy numbers using α − cut. However, the consistency of the fuzzy solution cannot be checked before applying the method. Therefore, their method needs further modifications.
In a recent study by [24], new analytical methods for solving both positive and negative Trapezoidal Fully Fuzzy Sylvester Matrix Equation (TrFFSME) were introduced. It was the first attempt to extend the concept of Kronecker product and Vec-operator for solving TrFFSME. The arbitrary fuzzy solution to the TrFFSME with arbitrary coefficients was considered by [25] where two-stage algorithm was constructed to find all possible solutions. The main limitation of this method is that it needs along computational time and, therefore, large memory storage in order to find all possible fuzzy solutions.
A coupled of TrFFSME in general form was considered in [26] where the fuzzy solution is obtained analytically by the fuzzy matrix vectorization method and numerically by gradient and least square methods. However, these methods need further modification to be applied for TrFFSME in LR-form. On the other hand, there was a study on the TFFSME in the formÃX −XB =C by [27]. The method obtained maximal and minimal symmetric positive solution for the TFFSME. However, this method is still limited for some TFFSME applications because the method used required long multiplication process and therefore long computational time.
Most of the existing literature for solving TFFSME and TrFFSME are based on Kronecker product and Vec-operator. As the size of the system increases the computational time is also increased due to the complexity of the procedure applied. Although analytical solutions, which can be computed using Vec-operator and Kronecker product, are important, the computational efforts rapidly increase with the dimensions of the matrices to be solved. For example, it required getting the inverse of mn × mn matrix for a system of size m × n which leads to computation complexity. Therefore, this method is limited to systems with small coefficients only. Consequently, the TFFSME and TrFFSME were considered with small size matrices only. Alternative ways exist which transform the matrix equations into forms for which solutions may be readily computed, such as the Jordan canonical form [28] and Hessenberg-Schur form [29]. Therefore, in dealing with this shortcoming, this paper presents two different methods for solving the PTrFFSME; the generalized Bartels Stewart's method and matrix vectorization method. The proposed methods are able to obtain positive fuzzy solutions for large PTrFFSME with short computational timing using Matlab or Mathematica. This paper is organized as follows. In Section 2, the preliminary concepts and arithmetic operations of trapezoidal fuzzy numbers are discussed. In Section 3, two new methods for solving PTrFFSME are developed. In Section 4, numerical example is solved using the proposed methods. In Section 5, conclusion about the proposed methods and achieved results will be drawn.

Preliminaries
In this section some basic arithmetic operations of fuzzy numbers are introduced [17,30]. Definition 2.1: A fuzzy numberÃ = (m, p, α, β) is said to be LR-TrFN, if its membership function is given by
(2.5) Scalar multiplication: Let λ ∈ R then: Definition 2.6: Let the two matrices A and X be n × n and m × m respectively. Then the matrix, is called the Kronecker product of A and X, it is also called the direct product or tensor product. Definition 2.8: [33,34]: q×p and I p , I q are identity matrices of order p and q respectively, then:

Definition 2.9:
The Kronecker difference of two matrices can be considered as a matrix difference defined by where A is a square matrix of order p and B is a square matrix of order q and I p , I q are identity matrices of order p and q respectively and ⊗ represents the Kronecker product.
For example, the Kronecker difference of two 2 × 2 matrices a ij and b ij is given by: Definition 2.10: The trapezoidal fully fuzzy matrix equation that can be written as is called a TrFFSME and it can also be written in the form: Definition 2.11: [35]: The Schur factorization of a matrix A is the factorization: where, • R is upper triangular matrix which is called a Schur form of A.
• Q is a unitary matrix (QQ T = I).

Proposed Method
In this section, the solution to the PTrFFSME is discussed. The PTrFFSME in Equation (2.10) is extended to a system of four SME using arithmetic fuzzy multiplication operations. The positive fuzzy solution afterwards is obtained by developing two different methods. In the following Theorem 3.1, the PTrFFSME is converted to an equivalent system of matrix equations.
, then the PTrFFSME iñ AX −XD =Ẽ is equivalent to the following SME: We have from Definition 2.5 and by Equation (2.3), Therefore, Which can be written as SinceÃX −XB =C can be written as Therefore, the PTrFFSMEÃX −XD =Ẽ is equivalent to the following system of SME: To solve the PTrFFSME in Equation (2.10), we consider the corresponding SME in Equation (3.1). The analytical solution of the system of SME in Equation (3.1) can be obtained by many classical methods. In the following, the Bartels Stewart Method (BSM) [36] is generalized in the fuzzy environment.

Method 1:
Generalized BSM for solving PTrFFSME: Step 1: Suppose m ij , a ij , n ij and b ij are real and have real Schur decompositions m ij = where U and V are orthogonal and R and S are upper quasi-triangular. Then the first two equations in Equation (3.1) can be transformed to: Consequently, they can be written as Then, this system can be written as Gaussian elimination and back substitution are applied to obtain w 1 .
Step 2: The values of x ij and y ij can be computed as follows: Step 3: The third and fourth equations in Equation (3.1) can be written as follows: If we let, Then Equation (3.2) can be written as Since Equation (3.3) has the same structure as the first two equations in Equation (3.1), it can be transformed to: that is, or equivalently Gaussian elimination and back substitution are applied to obtain w 2 .
Step 4: The values of z ij and q ij can be computed as follows: Step 5: Combining the values of x ij , y ij , z ij and q ij which obtained in step 2 and step 4. The solution of TrFFSME is represented by: The following Matrix Vectorization Method (MVM) is based on the concept of Kronecker product, Kronecker difference and Vec-operator. It is an extension of the method proposed for solving PTrFFSME in the formÃX +XB =C by Elsayed et al. [24].
In this method we apply the concept of Kronecker product and Vec-operator to Equation (3.1) as follows: Step1: Applying subtraction property of equality on the third and fourth equations in Equation (3.1) we get: Vec(g ij ) Step 2: Let, . .
Step 3: With the assumption that R 1 and R 2 are non-singular, the system of equations in Equation (3.6) can be written as follows: By the multiplicative inverse of R 1 and R 2 we obtain the following: Step 4: The solution of the TrFFSME is represented by: In the following Definition 3.1, the positive fuzzy solution to the PTrFFSME in LR form is defined.

Definition 3.1: Positive Fuzzy Solution to PTrFFSME in LR Form
A trapezoidal fuzzy solution matrixX = (x ij ) n×m = (x ij , y ij , z ij , q ij ) where x ij > 0, y ij > 0, z ij > 0, q ij > 0, x ij ≤ y ij and x ij − z ij > 0 is called a positive fuzzy solution of the PTrFFSME in LR form.
The existence and uniqueness of the positive fuzzy solution to the PTrFFSME are discussed in the following Corollary 3.1.

three positive trapezoidal fuzzy matrices. Then the positive fuzzy solution to the PTrFFSMẼ AX −XB =C in LR form exists and it is unique if the following conditions are satisfied:
Proof: I) LetÃ,B andC are non-negative matrices. The PTrFFSME can be written as a system of SME in Equation (3.1) by Theorem 3.1. By applying the Vec-operator and Kronecker product to the system of SME, the following system is obtained , then > 0 and therefore, III) So far, the solution to the SME in Equation In order for this solution to be equivalent to the positive fuzzy solution to the PTrFFSME the following conditions must be satisfied y ij − x ij ≥ 0 and x ij − z ij ≥ 0. Now, the feasibility of the positive solution to the PTrFFSME in LR form is discussed.

Numerical Examples
In this section, the proposed methods are illustrated by solving the following example. where,X The three proposed methods are applied to obtain the fuzzy solutionX = x 11x12 x 21x22 as follows: Method 1: Generalized BSM for solving PTrFFSME.
Step 1: We decompose the following matrices by applying Definition 2.11 as follows: We get: This will be followed by obtaining P 1 and P 2 by the definition of Kronecker difference Definition 2.9 as follows: Also, D 1 and D 2 can be computed as follows: ⎞ ⎠ . Applying Definition 2.7 on W 1 and W 2 gives: Now we can solve for w 1 and w 2 as follows: Gaussian elimination and back substitution are applied to obtain W 1 and W 2 . Step 2: We compute x ij and y ij as follows: Step 3: The values of x ij and y ij are used to compute h α 1 and f α 1 as follows: The values of h α 1 and f α 1 are substituted in the following equations.
Step 4: Since the obtained equations have exactly the same structure as the first two equations in Equation (3.1), z ij and q ij can be computed similar to x ij and y ij . Thus, z ij = 2 3 2 2 and q ij = 1 1 2 1 .
Step 1: We can obtain the following: Step 2: Since R 1 · S 1 = T 1 , we compute R 1 , T 1 and solve for S 1 as follows: Multiplying both sides of Equation (4.1) by R 1 −1 we get: Thus, x ij = 44 33 and y ij = 55 44 .
The values of Vec(x ij ) and Vec(y ij ) is substituted in Equation (3.13) to compute Vec(z ij ) and Vec(q ij ) as follows: Step 3: We also compute, And since, Multiplying both sides of Equation (4.2) by R 2 −1 we get: Thus, z ij = 2 3 2 2 and q ij = 1 1 2 1 .

Verification of the solution
To verify the obtained fuzzy solution, we first multiplyÃ andX as follows: The value ofÃX −XB is exactly equal to the constant matrixC.

Conclusion
In this paper, the solution to the PTrFFSMEÃX −XB =C is obtained analytically. The positive fuzzy solution is obtained by two analytical methods, the generalized BSM, in addition to MVM. The existence and uniqueness of the positive fuzzy are discussed. In terms of accuracy, the two methods are obtained the positive fuzzy solution to the PTrFFSME. In addition, the methods can also be applied to TFFSME. Both methods can be applied for large TrFFSME using Matlab or Mathematica. The main limitation of the MVM is that it needs a long computational time and, therefore, large memory storage in order to find the fuzzy solution.
In future work, a further modification to the existing arithmetic fuzzy operations is needed in order to reduce the complexity of the obtained system of SME. In addition, optimization techniques need to be developed to overcome the limitation of the developed methods. As future research, the ideas presented in this paper will be modified and applied to TrFFSME with near zero trapezoidal fuzzy numbers. . Her current research interests include fuzzy mathematics, topology, and mathematical modelling in medicine, healthcare, finance, and psychology. She has been awarded several research grants by the Ministry of Education Malaysia and UUM as a principal and co-investigator. The quality of her work has been published in international journals (indexed by SCOPUS or Web of Science), conference proceedings, books and book chapter. She has been teaching mathematics for more than 20 years at the undergraduate and master's level. She has also supervised master and PhD students. Her expertise was acknowledged by her academic peers when she was appointed as an external/internal examiner, article reviewer and external assessor. At national level, she has involved in revising the mathematics syllabus based on the Secondary School Standard Curriculum and in organizing STEM service-learning activities to inspire and strengthen STEM education among young people.