Numerical Solutions for Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equations
Analyzing the stability of many control systems required solving a couple of crisp Sylvester matrix equations (CSMEs) simultaneously. However, there are some situations in which the crisp Sylvester matrix equations are not well equipped to deal with the uncertainty problem during the stability analysis of control systems. This paper constructs analytical and numerical methods for solving a couple of trapezoidal fully fuzzy Sylvester matrix equations (CTrFFSMEs) to overcome the drawbacks of the existing crisp methods. In developing these new methods, fuzzy arithmetic multiplication is applied on the CTrFFSME to transform it into an equivalent system of four CSMEs. Then, the fuzzy solution is obtained analytically by the fuzzy matrix vectorization method and numerically by gradient and least square methods. The analytical method can obtain the exact solution; however, it is limited to small-sized systems while the numerical methods can approximate the solution for large dimensional systems up to with a very small error bound for any initial value. In addition, the proposed methods are applied to other fuzzy systems such as Sylvester and Lyapunov matrix equations. The proposed methods are illustrated by solving numerical examples with different size systems.
The Sylvester matrix equation (SME) has massive applications in control theory [1, 2], system theory , optima control , linear descriptor systems , sensitivity analysis , perturbation theory , system design , theory of orbits , design and analysis of linear control systems , reduction of large-scale dynamical systems , restoration of noisy images [12, 13], medical imaging data acquisition and model reduction , and stochastic control, image processing, and filtering . CSME must be solved simultaneously in many applications, such as analyzing the stability of control systems . Researchers for many years have proposed many analytical and numerical methods for solving CSME with crisp numbers.
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 matrix for a system of size which leads to computation complexity. Therefore, this method is limited to systems with small coefficients only. In addition, for some applications such as stability analysis, it is often not necessary to compute analytical solutions; approximate solutions or bounds of solutions are sufficient. Also, if the parameters in system matrices are uncertain, it is not possible to obtain analytical solutions for robust stability results [16, 17]. Alternative ways exist which transform the matrix equations into forms for which solutions may be readily computed, such as the Jordan canonical form  and Hessenberg–Schur form . However, these methods are computationally expensive for large systems. In the field of matrix algebra and system identification, iterative algorithms for large systems have received much attention . Starke and Niethammer  presented an iterative method for solutions of the SME by using the SOR technique while Jonsson and Kägström proposed recursive block algorithms for solving the coupled Sylvester matrix equations . Kägström derived an approximate solution of the coupled Sylvester equation .
Many authors studied the least square solutions of CSME [24–31] while authors in  discussed the solvability conditions and general solutions for mixed Sylvester equations. Recently, a relaxed gradient-based algorithm for solving generalized CSME was introduced by  in addition to the conjugate gradient least square algorithm  and gradient-based approach  and the BCR algorithm proposed by [36–38]. However, in many applications, some of the system parameters are represented by fuzzy numbers rather than crisp numbers due to uncertainty problems such as conflicting requirements during the system process and the distraction of any elements and noise. When all parameters of the CSME are in the fuzzy form, then it is called the coupled fully fuzzy Sylvester matrix equation (CFFSME).
Definition 1. The couple fully fuzzy matrix equation can be written aswhere , , , , , , , and .
Equation (1) is of interest in many different applications. However, until now, there are fewer studies for the solution of this equation. In the fuzzy literature, most of the solution methods are proposed for its special cases, such as fully fuzzy Sylvester matrix equations (FFSMEs), fully fuzzy matrix equations (FFMEs), fully fuzzy linear systems (FFLSs), and fuzzy linear system (FLS).
The first approach of solving FLS was accomplished by , which proposed a general model for solving a FLS by transferring FLS to a linear system. Allahviranloo et al.  proposed a method to obtain symmetric solutions of the FLS based on a 1-cut expansion. They extended the same method in  to obtain symmetric solutions of the FFLS. Sufficient conditions needed for getting positive solutions of the FFLS were discussed by Malkawi and his colleagues [42, 43]. Otadi and Mosleh extended the FFLS to FFME in . Several analytical methods have been proposed for solving the triangular fully fuzzy Sylvester matrix equation (TFFSME) [45–47]. However, these methods are restricted only for positive triangular fuzzy numbers and require a long multiplication process and consequently long computational timing. Consequently, researchers limit the sizes of the TFFSME to . Recently, authors in  considered solution of the trapezoidal fully fuzzy Sylvester matrix equation (TrFFSME) by transforming the TrFFSME to a system of crisp linear matrix equations where the positive and negative fuzzy solutions are obtained by applying Kronecker product and Vec-operator method. However, these algorithms are not suitable for TrFFSME with large sizes.
In addition, a few studies have been conducted for solving a pair of fuzzy matrix equations. Sadeghi, Abbasbandy, and Abbasnejad  proposed a method for solving a pair of fuzzy matrix equations in the form as follows:Moreover, Daud, Ahmad, and Malkawi [50–53] proposed analytical methods for solving FFSME and a pair of fully fuzzy matrix equations (PFFME) in the form as follows:In that study, a direct method was proposed to solve the PFFME by applying the Kronecker product and Vec-operator. However, both methods required a long multiplication process and were consequently limited to small-sized systems. In general, the existing methods proposed for solving PFFME, TFFSME, and TrFFSME are based on Kronecker product and Vec-operator and therefore limited to small systems or . Only a few researchers considered fuzzy systems with sizes . Fuzzy systems with sizes greater than are not investigated till now. In addition, the CFFSME is not investigated in the fuzzy literature.
To deal with this shortcoming, in this paper, three different methods are proposed for solving CFFSME with trapezoidal fuzzy numbers (CTrFFSME) and its special cases. The fuzzy solution to the CTrFFSME is obtained analytically by the fuzzy matrix vectorization method and numerically by gradient and least square methods. The fuzzy matrix vectorization method can obtain the exact solution; however, it is restricted to small systems. Therefore, it is important to develop mathematical models and numerical procedures that solve the CFFSME and special cases with big sizes while the numerical methods can obtain the solution for large dimensional systems up to with a very small error bound compared with the existing numerical approaches, which were applied up to fuzzy systems [54–63]. Moreover, the proposed methods can also be applied to other fuzzy systems such as Sylvester and Lyapunov matrix equations with triangular fuzzy numbers (TFNs) and trapezoidal (TrFNs) fuzzy numbers.
To illustrate the effectiveness of the proposed methods for solving the CTrFFSME in equation (1), we consider various sizes of fuzzy systems, namely, small and large . In addition, we compare the performance of the proposed methods by calculating the number of iterations , convergence factor , error , error bound , convergence rate, CPU time, real-time, and memory usage. In addition to the graphical representation of the relative error when the number of iterations () increases.
This paper is organized as follows. Section 2 introduces preliminary arithmetic operations of trapezoidal fuzzy numbers. In Section 3, three proposed methods for solving CTrFFSME are developed along with a presentation of its algorithms. In Section 4, numerical examples are presented to illustrate the proposed methods. Section 5 is dedicated to the conclusion.
The following are the basic definitions and results related to TrFNs in fuzzy theory [64–66] and matrix theory [67–69].
Definition 2. Let be a universal set. Then, the fuzzy subset of is defined by its membership function which assigns to each element a real number in the interval , where the function value of represents the grade of membership of in . A fuzzy set is written as .
Definition 3. A fuzzy set , defined on the universal set of real number , is said to be a fuzzy number if its membership function has the following characteristics:(i) is convex, i.e.,(ii) is normal, i.e., such that .(iii) is piecewise continuous.
Definition 4. A fuzzy number is a TrFN in the general form if its membership function is as follows:In Figure 1, the TrFN in general form is presented.
Definition 5. The sign of the TrFN can be classified as follows: is positive (negative) iff is zero iff is near zero iff
Definition 6. Operations of TrFNs.
The arithmetic operations of TrFNs are presented as follows: let and be two TrFNs, then(i)Addition:(ii)Subtraction:(iii)Symmetric image:(iv)Scalar multiplication: let , then(v)Multiplication: the multiplication between fuzzy numbers is neither commutative nor associative. Thus, TrFNs multiplication operations can be classified as follows: Case I. If and be two arbitrary TrFNs, then where Case II. If , then Case III. If , then Case IV. If and , then Case V. If and , then(vi)Equality: the fuzzy numbers and are equal iff
Definition 7. A matrix is called a trapezoidal fuzzy matrix if each element of is a TrFN.
Definition 8. A fuzzy matrix will be as follows:(i)Positive (negative) and denoted by , if each element of is positive (negative) TrFN(ii)Nonnegative (nonpositive) and denoted by , if each element of is nonnegative (nonpositive) TrFNs(iii)Arbitrary if at least one element of is near zero TrFNsIn Remark 1, the positive trapezoidal fuzzy matrix is written as four separated crisp matrices.
Remark 1. The positive trapezoidal fuzzy matrices can be written as four separated crisp matrices as follows:where are four crisp matrices sized .
In Remark 2, the multiplication of positive trapezoidal fuzzy matrices is introduced.
Remark 2. The product of the two positive trapezoidal fuzzy matrices , , and , , can be represented as follows:where represent the multiplication of the fuzzy number of of matrix with of matrix . In addition, this product is equivalent to the product of the following crisp matrices:
Definition 9. The Vec-operator generates a column vector from a matrix by stacking the column vectors of as . In addition, if , then .
Theorem 1 (see ). If the crisp linear matrix equation has a unique solution , then the gradient iterative solution given by converges to or for any initial value .
Theorem 2 (see ). If the crisp linear matrix equation has a unique solution , then the gradient iterative solution given by converges to or for any initial value .
In Section 3, the solution to the CTrFFSME in equation (1) is discussed.
3. The Solution of Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equation
In this section, the solution to the positive CTrFFSME is considered. To get the solution, the positive CTrFFSME is converted to an equivalent system of CSME, and then the solution to this system of CSME is obtained by three different methods. In Section 3.1, the positive CTrFFSME in equation (1) is converted to an equivalent system CSME based on the arithmetic multiplication operation in Definition 6.
3.1. Systems of CSME
In this section, the positive CTrFFSME in equation (1) is converted to four systems of CSME. The next theorem shows that the CTrFFSME can be written as four systems of CSME.
Theorem 3. Fundamental theorem of the coupled trapezoidal fully fuzzy Sylvester matrix equation.
In the CTrFFSME in equation (1), if and , , and , , and , , , and , , then the positive CTrFFSME is equivalent to the following systems of CSME:
Proof. Since ,,,,,,, and in equation (1) are positive trapezoidal fully fuzzy matrices, respectively, equation (12) in Definition 6 can be used to find , ,, and in equation (1) as follows:s.t all i = 1, …, m and j = 1, …, n. Combining and , and , we getBy Definition 1, the positive CTrFFSME is equivalent to the following systems of CSME:
Remark 3. The equivalent systems of CSME in equation (20) to the CTrFFSME in equation (1) can also be written as follows:To solve the CTrFFSME in equation (1), we consider the corresponding systems of CSME in equation (24).
Remark 4. The nature of the solutions of the CTrFFSME in equation (1) depends upon the nature of the solutions of the system of CSME in equation (24), which may be unique, trivial, or infinitely many solutions ; that is, the CTrFFSME may yield no solution, unique solution, or infinitely many solutions.
Since the systems of CSME obtained in equation (24) are similar, in Remark 5, the systems of CSME are represented in a more general form.
Remark 5. Based on equation (24), the CTrFFSME in equation (1) can be written as follows: for , we haveBased on Theorem 3, the CTrFFSME in equation (1) is transferred to an equivalent linear system of CSME in crisp form, which can be solved analytically and numerically by many classical methods in linear algebra like the matrix inversion method or Gaussian method. The main advantage of the analytical methods is that the exact fuzzy solution to the CTrFFSME in equation (1) can be obtained.
However, since most of the analytical methods are based on Vec-operator and Kronecker products, the system’s size in equation (1) is limited to small sizes . For CTrFFSME with large dimensions , iterative algorithms to find an approximated solution are more practical . Therefore, in the following section, three different methods are proposed for solving the CTrFFSME in equation (1). The first method aims to find the exact fuzzy solution by extending the concept of matrix vectorization and the Kronecker product. In addition, two iterative methods are applied to approximate the fuzzy solution of the CTrFFSME with large dimensions. In Definition 10, the positive trapezoidal fuzzy solution is defined.
Definition 10. Trapezoidal positive fuzzy solution matrix in a general form.
Let be a trapezoidal fuzzy matrix. If is an exact solution of equation (20) such that , , then is called a positive fuzzy solution of equation (1).
3.2. Proposed Analytical Method for Solving CTrFFSME
In the following method, the analytical solution for the CTrFFSME in equation (1) is obtained by extending the method of the fuzzy matrix vectorization method (FMVM) proposed by .
3.2.1. Fuzzy Matrix Vectorization Method (FMVM) for Solving CTrFFSME
In this method, the CTrFFSME in equation (1) is solved analytically using Vec-Operator and Kronecker product. The following steps summarize the methods: Step 1. Decompose , , , , , , , and into , , , , , , , and where , respectively, and convert the CTrFFSME in equation (1) to the system of linear matrix equations in equation (24) using Theorem 3. Step 2. Applying Vec-operator and Kronecker product on equation (24) gives the following: Step 3. By equation (25), the systems of coupled linear matrix equations in equation (27) can be combined and written as follows: where where the crisp matrix is , and the crisp vectors are . Step 4. Multiplying the system of equation in equation (27) by matrix multiplicative inverse gives Step 5: By Definitions 9 and 10, the obtained solution in equation (29) can be written as follows:
In the following remark, the system of equations in equation (26) is written in a general form.
Remark 6. Based on equations (25) and (27), the CTrFFSME in equation (1) is equivalent to the following system:for , we haveThe sufficient conditions to have a unique fuzzy solution to the CTrFFSME are discussed in Corollary 1.
Corollary 1. For , the CTrFFSME in equation (1) has a unique solution if and only if the matrix is nonsingular. Then, this solution is obtained byand can be written as
3.2.2. Feasibility of the Positive Fuzzy Solution to the CTrFFSME
The obtained positive fuzzy solution in equation (30) to the CTrFFSME in equation (1) is feasible (strong fuzzy solution) if the following conditions are satisfied: for ,(i), (ii), (iii), (iv),
Remark 7. If the solution fails to satisfy the feasibility conditions, it is infeasible (weak fuzzy solution).
The algorithm of the FMVM for solving the CTrFFMSE in equation (1) is given in the following five steps.In Section 3.3, the positive solution to the CTrFFSME is approximated numerically by extending the GI method in Theorem 1 and LSI method in Theorem 2.
3.3. Proposed Numerical Methods for Solving CTrFFSME
In this section, two numerical methods are developed: the fuzzy gradient iterative (FGI) method and the fuzzy least square iterative (FLSI) method. The details of the FGI methods are discussed as follows.
3.3.1. Fuzzy Gradient Iterative (FGI) Method for CTrFFSME
In this method, the CTrFFSME in equation (1) is solved numerically by extending the GI method for solving the crisp linear matrix equation in Theorem 1 to the CTrFFSME in equation (1). The following steps summarize the methods: Step 1. Decompose , , , , , , , and into , , , , , , , and where , respectively, and convert the CTrFFSME in equation (1) to the system of linear matrix equations in equation (24) using Theorem 3. Step 2. Using the hierarchical identification principle and Remark 5, the system of CSME in equation (25) can be decomposed into two subsystems: for , where the iterative positive solution to the system of CSME in equation (25) is the average of the iterative solution for the subsystems. Let and . From equations (25) and (35), the following can be obtained: for , Step 3. The iterative positive solution to the system of equations in equations (36a) and (36b) can be obtained by the GI method in Theorem 1 as follows: for , we have Step 4. Substitute equations (35) into (37) as follows: The obtained algorithm in equation (38) can be written as follows: for , we have Let and , then for , the approximated solution in equation (39) can be written in reduced form as follows: where the convergence factor (step size) is given by It can also be obtained as follows: where . At step of the iteration, the following relative error is considered: Step 5. By Definition 10, the approximated fuzzy solutions obtained by the previous step to the CTrFFSME in equation (1) can be written as follows:
It can also be written in matrix form as
In Theorem 4, we prove that the iterative solution obtained by the FGI method converges to the positive solution of the positive CTrFFSME for any initial value.
Theorem 4. If the system of CSME in equation (25) has a unique positive solution , then the iterative solution in equation (43) converges to for any initial values for (i.e., if , then and ).
Proof. Let be the error at each , for and .From equations (24), (39), (46a), and (46b), the following is obtained:Taking to both sides of equation (47) gives the following:
Remark 8. The following steps in the proof are long; therefore, the system in equation (48) needs to be separated into two equations in the following steps of the proof.
It is not hard to prove that . Thus, equation (48) can be written asApplying norm properties and matrix multiplication givesApplying norm properties on equations (50a) and (50b) gives