Solving Arbitrary Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation with Necessary Arithmetic Multiplication Operations

A couple of Sylvester matrix equations (CSME) are required to be solved simultaneously in many applications, especially in analysing 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 stability analysis of control systems. Thus, this paper proposes a new method for solving Arbitrary Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation (ACTrFFSME). New arithmetic fuzzy multiplication operations are developed and applied to convert the ACTrFFSME to a reduced system of non-linear equations. Then it is converted to a system of absolute equations where the arbitrary fuzzy solution is obtained by solving that system. The proposed method can solve many arbitrary fuzzy equations, such as fully fuzzy matrix equations, Sylvester and Lyapunov fully fuzzy matrix equations with triangular and trapezoidal fuzzy numbers without any restrictions. We illustrate the proposed method by solving two numerical examples.


Introduction
Sylvester matrix equation (SME) in the form AX + XB = C has many important applications in the design and analysis of linear control systems [1], observer design [2], reduction of large-scale dynamical systems [3], restoration of noisy images [4,5], medical imaging data acquisition, model reduction [6] and stochastic control, image processing and filtering [5].
However, there are many applications where CSME in the form AX + YB = C DX + YE = F are required to be solved simultaneously. The CSME is important in making the computational process less complicated, especially in analysing the stability of control systems so that the control system always performs well according to its specifications [7]. Researchers for many years have proposed many analytical and numerical methods for solving the CSME with crisp numbers [8][9][10][11][12][13]. However, in many applications, some system parameters are represented by fuzzy numbers rather than crisp numbers due to uncertainty problems such as conflicting requirements during the system process, the distraction of any elements and noise. Fuzzy logic has been studied since the 1920s, as infinite valued logic by Lukasiewicz and Tarski [14], and the fuzzy set theory was introduced by Lotfi Zadeh [15] in 1965 while the set theory was developed by Georg Cantor [16]. Fuzzy Relation Equations (FREs) with the max-min composition was first studied by Sanchez [17]. The theory and applications of FREs can be found in Di Nola et al. [18], which indicated that if the solvability of max-continuous t-norm FREs is assumed, then 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. Since then, FREs based on various compositions have been investigated. Some common compositions include max-min [19][20][21][22][23][24], max-product [25][26][27][28], max-Archimedean t-norm [29,30], u-norm [31], max t-norm [32] and max arithmetic mean [33,34]. The conditions for the existence of a solution to the inverse problem concerned with FRE are investigated in [35], a finite system of FREs with sup-T composition was studied in [36], and a system of FREs was investigated in [37,38].
When all parameters of the CSME are in the fuzzy form, it is called coupled fully fuzzy Sylvester matrix equation (CFFSME). where,Ã = (ã ij ) m×p ,X = (x ij ) p×n andẼ = (ẽ ij ) m×n is called a Fully Fuzzy Matrix Equation (FFME). To the best of our knowledge, the ACFFSME in Equation (1.1) is not solved by any known method. Thus, researchers restrict themselves to some of its special cases. For example, the authors in [39][40][41][42] considered the solvability for a Fully Fuzzy Linear System (FFLS) and authors in [43,44] considered the FFME in Equation (1.4) with triangular fuzzy numbers (TFNs). However, these methods cannot be extended to FFME with trapezoidal fuzzy numbers (TrFNs). In addition, several analytical methods have been proposed for triangular FFSME (TFFSME). The most important of which is the Kronecker product and matrix inversion [45], Kronecker product and associated linear system [46][47][48]. However, the methods were restricted only to positive triangular fuzzy numbers (TFNs) only. Positive and negative solutions for trapezoidal FFSME (TrFFSME) are considered in [49]. However, the method cannot be extended to arbitrary TrFFSME. A few studies have been conducted for solving a pair of fuzzy matrix equations. [50] proposed a method for solving a pair of fuzzy matrix equations in the form: In addition, [51,52] proposed analytical methods for solving a pair of fully fuzzy matrix equations (PFFME) in the form:

DXẼ =F
In that study, a direct method was proposed to solve the positive PFFME with arbitrary coefficients by applying the Kronecker product and Vec-operator. However, both methods obtained a positive fuzzy solution only. Recently, The positive fuzzy solution to the CTrFF-SME was obtained by El Sayed, Malkawi and Ahmad [53]. However, this method cannot be applied to ACTrFFSME. Therefore, limitations of the existing methods are pointed out.
(I) Existing methods for solving TFFSME and TrFFSME can only obtain positive or negative fuzzy solutions and cannot be extended to fuzzy systems with arbitrary solutions. Consequently, the existing methods are limited to some real-life situations. However, there exist several instances in real scenarios where the coefficients may not be entirely positive or nonnegative. (II) Analytical methods proposed for solving different arbitrary systems are based on Vecoperator and Kronecker product. It is worth mentioning that the Vec-operator and Kronecker product approach is applicable for fuzzy systems with positive or negative fuzzy numbers only and cannot be applied to fuzzy systems with near-zero fuzzy numbers. In addition, the Kronecker product method for m × n fuzzy system requires getting the inverse of mn × mn matrix, which is impossible for large systems. (III) Using the existing methods, it is not possible to check whether the obtained solution is unique or not. There may be several cases where the fuzzy systems generate unique or infinitely many solutions.
(IV) The existing methods can only be applied to a certain fuzzy number and cannot be extended to other types of fuzzy numbers.
Consequently, the developed methods in the literature limit the size and sign of fuzzy systems. Therefore, it is important to develop a new method without considering Vecoperator and Kronecker product for solving the CTrFFSME and its special cases. Therefore, to deal with this shortcoming, in this paper, reduced fuzzy multiplication operations are applied to convert the arbitrary CTrFFSME into a system of non-linear equations. The reduced system is converted to an equivalent system of absolute equations where the fuzzy solutions are obtained by solving that system.
The proposed method is applicable for solving ACTrFFSME with large-size matrices. In addition, it can also be applied to its special cases in Equations (1.2)-(1.4) with both TFNs and TrFNs without any restriction. This paper is organised as follows: Section 2 introduces preliminary arithmetic operations of trapezoidal fuzzy numbers. In Section 3, a proposed method for solving ACTrFFSME is developed, along with a presentation of its algorithm. In Section 4, a numerical example is presented to illustrate the proposed method. Section 5 is dedicated to the conclusion.

Preliminaries
The following are basic definitions and results related to TrFNs in fuzzy theory [54][55][56] and fuzzy matrix [57,58]. Definition 2.1: Let X be a universal set. Then, the fuzzy subsetÃ of X is defined by its membership function μÃ : X → [0, 1] which assigns to each element x ∈ X a real number μÃ(x) in the interval [0, 1], where the function value of μÃ(x) represents the grade of membership of x inÃ. A fuzzy setÃ is written asÃ = {(x, μÃ(x)), x ∈ X, μÃ(x) ∈ [0, 1]}.

Definition 2.2:
A fuzzy setÃ, defined on the universal set of real number R, is said to be a fuzzy number if its membership function has the following characteristics: (III) μÃ is piecewise continuous.

Definition 2.3:
A fuzzy numberÃ = (a 1 , a 2 , a 3 , a 4 ) is a TrFN if its membership function is: x > a 4 Figure 1 representation of TrFN in the form (a 1 , a 2 , a 3 , a 4 ).

Definition 2.4:
The sign of the TrFNÃ = (a 1 , a 2 , a 3 , a 4 ) can be classified as:

Definition 2.12:
The solution set of a system is called a finite solution or alternative solution, whereby the number of solutions is more than one and not infinite solutions. 2) can find the product of arbitrary TrFNs. However, when applying these operations to solve arbitrary fuzzy systems, the fuzzy systems are converted to a min-max non-linear system which is very challenging to be solved. Therefore, reducing the min-max non-linear system from four terms to two terms makes the solution to the system much easier in terms of computational timing and memory usage. Thus, a further modification to the AMO in Definition 2.5, Equation (2.2) need to be done to make it more practical in solving arbitrary fuzzy systems. Therefore, in the next Section 3, AMO is reduced to what so-called Reduced Arithmetic Multiplication Operations (RAMO) based on the sign of the TrFNs that are positive, negative or near-zero.

Reduced Multiplication Operations for Semi-Restricted TrFNs
In this section, the fuzzy arithmetic multiplication operations for arbitrary TrFNs in Equation (2.2) are reduced for semi-restricted TrFNs. In the following corollaries, the multiplication between semi-restricted TrFNs is reduced from four terms into two terms only. This reduction contributes significantly to the solution of the arbitrary CTrFFSME in Equation (1.1) in Section 4.
Proof: Straightforward similar to Corollary 3.1 and 3.2.
The following Figure 2 summarises the reduced fuzzy multiplication operations for TrFNs.

Proposed Method
In the following section, a new method is proposed for solving the CTrFFSME in Equation (1.1).

Arbitrary Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation
In this section, the arbitrary CTrFFSME in Equation (1.1) is converted to an equivalent reduced system of non-linear equations based on the RAMO in Section 3. In addition, the obtained non-linear system is converted to a system of absolute equations. Solving the system of absolute equations will give the fuzzy solution to the arbitrary CTrFFSME in Equation (1.1). In the following Definition 6.1.2, the system of non-linear equations is introduced.

Definition 4.1:
The system of equations in the form, ij e (2) ij , y ij e (3) ij , y ij e (2) ij , y ij e ij e (3) ij , y ij e (2) ij , y ij e is called a system of non-linear equations. In the following Theorem 4.1, the arbitrary CTrFFSME in Equation (1.1) is converted to an equivalent system of non-linear equations.

Remark 4.1:
The number of non-linear equations obtained from m × n ACTrFFSME is equal to 8 × m × n equations. For example, 2 × 2 CTrFFSME can be converted to a system of 32 non-linear equations.
Authors in [41] showed that the reduced non-linear system of equations can be converted to an equivalent system of absolute equations. Therefore, the obtained non-linear system in Equation (3.2) can be converted to an equivalent system of absolute equations where the nature of the fuzzy solutions of the ACTrFFSME depends upon the nature of the solutions of the absolute system which may be no solution, unique, many definite or infinitely many solutions.
The following algorithm can be used to obtain the fuzzy solution to the ACTrFFSME in Equation (1.1) as follows: Algorithm 1 Solving arbitrary CTrFFSME Step 1: Convert the CTrFFSME in Equation (1.1) to the reduced non-linear system in Equation (4.2) using Theorem 4.1.
Step 2: Convert the reduced non-linear system to an absolute system using Equation (2.6) in Definition 2.10.
Step 3: Solve the system of absolute equations.

Feasibility of the arbitrary fuzzy solution to the CTrFFSME
The obtained arbitrary fuzzy solution in Equation (4.3) to the CTrFFSME in Equation (1.1) is feasible (strong arbitrary fuzzy solution) if the following conditions are satisfied: (III) At least one element ofX is near-zero TrFN. (IV) At least one element ofỸ is near-zero TrFN.

Remark 4.2:
If the solution fails to satisfy the feasibility conditions, then it is an infeasible fuzzy solution (weak and also known as a non-fuzzy solution). Algorithm 1 can be applied to ACTrFFSME with different sizes. In the following Section 4.2, the proposed method is explicitly applied to a 2 × 2 arbitrary CTrFFSME.

Applications of the Proposed Method to Other Fuzzy Systems and Fuzzy Numbers.
The proposed method is able to solve different arbitrary fuzzy systems with triangular and trapezoidal fuzzy numbers, without any restrictions. It can be directly applied to the TrFFME in Equation (1.4). In the following Theorem 4.2 the TrFFME in Equation (1.4) is converted to an equivalent system of non-linear matrix equations. Theorem 4.2: IfÃ = (ã ij ) m×n , ∀1 ≤ i, j ≤ m, n andX = (x ij ) n×r , ∀1 ≤ i, j ≤ n, r. Then the arbitrary TrFFME in Equation (1.4) can be written as follows: ij , x (4) ij ) and E = (ẽ ij ) m×r = (e (1) ij , e (2) ij , e (3) ij , e (4) ij ) arbitrary TrFNs. By Equation (2.2) in Definition 2.5, we have, N, P, Q) where, By the definition of arbitrary TrFFME in Eq. (1.4), it can be written as: ij x (4) ij ) = e (4) ij and the proof is completed. Based on algorithm 1, the following algorithm is used to obtain the arbitrary fuzzy solution to the TrFFME in Equation (1.4) as follows: Algorithm 2 Solving arbitrary TrFFME.
Step 2: Convert the reduced non-linear system to an absolute system using Definition 2.6.
Step 3: Solve the system of absolute equations and check which solution(s) satisfy the following.
Step 4: By solving the system of absolute equations and eliminating the non-fuzzy solutions, the following arbitrary fuzzy solution(s) is obtained: np , x np , x Similarly, the proposed method is able to solve different arbitrary fuzzy systems with triangular and trapezoidal fuzzy numbers without any restrictions. It can be directly applied to the following: Fully fuzzy Sylvester matrix equationÃX +XD =Ẽ. Fully fuzzy continuous-time Lyapunov matrix equationÃX +XÃ T =Ẽ.
To illustrate the proposed methods, two examples are solved in the following Section 5. The first example is for ACTrFFSME, and the second example is for FFME with TFNs.

Numerical Example
In this section, the proposed method in Section 3 is illustrated by solving two examples.

Solution
The solution to the given CTrFFSME is obtained by Algorithm 1 as follows: Step 1: Converting the given arbitrary 2 × 2 CTrFFSME to a reduced system of non-linear equations using RAMO in Section 3 as follows:

, 5y
(2) 21 ) + min(2y (2) 22 , −3y Step 2: The reduced non-linear system in step 1 is reduced to an absolute system using Equation (2.6) in Definition 2.10; see Appendix A for the obtained absolute system of equations.
Step 3: Solve the system of absolute equations in step 2 and choose the solution(s) that satisfy the following.
Step 4: By solving the system of absolute equations and eliminating the non-fuzzy solutions, the following arbitrary fuzzy solution is obtained: .  Adding,ÃX +ỸB andDX +ỸẼ gives: The obtained arbitrary fuzzy solution satisfies the given PCTrFFSME and it is clearly feasible.
The following Example 5.2, was first solved by Kumar et al. [39] and obtained only one fuzzy solution. However, Malkawi et al. [41] considered the same example and obtained two fuzzy solutions. To support the developed method, the same example is considered.

Remark 5.1:
The methods used in solving the example considered by Kumar et al. [39] and Malkawi et al. [41] can only be applied to fuzzy systems with TFNs. Therefore, in the following example, we extend the TFNs to TrFNs by equating the mean values in the TrFNs used, to apply the developed method in this section.

Example 5.2:
Consider the following arbitrary TrFFME: 11 , x 11 , x The fuzzy solutions to the given TrFFME can be obtained by Algorithm 2 as follows: Step 1: Convert the arbitrary TrFFME in Equation (1.4) to a reduced non-linear system in Equation (3.5) as follows:

, 2x
(1) min(2x Subject to: x Step 2: Convert the reduced non-linear system to an absolute system using Definition 2.6. Subject to: Step 3: Solve the system of absolute equations and check which solution(s) satisfy the following.
Step 4: By solving the system of absolute equations and eliminating the non-fuzzy solutions, the following arbitrary fuzzy solutions are obtained: For future work, the proposed method will be extended to a couple of arbitrary generalised trapezoidal fully fuzzy Sylvester matrix equations with arbitrary coefficients.

Human and animal rights
This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent
Informed consent was obtained from all individual participants included in the study.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This research received no external funding. . 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. (2x (2)