Solving sylvester matrix equations with LR bipolar triangular fuzzy numbers in electric circuits problems

Bipolar crisp numbers refer to two different functions and information in a given system, namely positive and negative components. Likelihood and unlikelihood information can be simultaneously represented by bipolar crisp numbers rather than classical crisp numbers. However, since bipolar crisp numbers are inadequate in dealing with uncertainty problem, bipolar fuzzy numbers (BFN) are used instead. BFN in Sylvester matrix equations (SME) plays an essential role in the control system such as in electrical controller. An electrical controller of RLC circuit consisting of resistor (R), inductor (L), and capacitor (C), is used to control the amount of electric currents flowing across the electric circuits. Besides, complex numbers which consist of real and imaginary parts are used in solving RLC circuit, where real numbers denote resistance, while imaginary numbers denote inductance or capacitance. To the best of our knowledge, the integration of SME with either BFN or complex BFN is not yet explored. Therefore, this study aims to construct analytical approaches in solving bipolar fuzzy Sylvester matrix equation (FSME), complex bipolar FSME, bipolar fully fuzzy Sylvester matrix equation (FFSME), and complex bipolar fully fuzzy linear system (FFLS) in left-right (LR) bipolar triangular fuzzy numbers. In order to obtain the solutions, bipolar FSME, complex bipolar FSME, and bipolar FFSME are converted into the bipolar linear system by utilizing Kronecker product and Vecoperator. Next, an equivalent bipolar linear system (EBLS), equivalent complex bipolar linear system (ECBLS), associated bipolar linear system (ABLS), and associated complex bipolar linear system (ACBLS) are established. Then, the final solutions of the constructed methods are obtained using inverse method. Therefore, four analytical approaches have been constructed in solving bipolar FSME, complex bipolar FSME, bipolar FFSME, and complex bipolar FFLS in LR forms. Several examples are presented to illustrate the constructed methods. Moreover, the application of RLC circuits with complex bipolar FSME and complex bipolar FFLS are also carried out. In conclusion, the new findings of analytical approaches add to the fuzzy equations body of knowledge with significant applications in bipolar electrical controllers

Unrestricted solutions of arbitrary linear fuzzy systems

Solving linear fuzzy system has intrigued many researchers due to its ability to handle imprecise information of real problems. However, there are several weaknesses of the existing methods. Among the drawbacks are heavy dependence on linear programing, avoidance of near zero fuzzy numbers, lack of accurate solutions, focus on limited size of the systems, and restriction to the matrix coefficients and solutions. Therefore, this study aims to construct new methods which are associated linear systems, min-max system and absolute systems in matrix theory with triangular fuzzy numbers to solve linear fuzzy systems with respect to the aforementioned drawbacks. It is proven that the new constructed associated linear systems are equivalent to linear fuzzy systems without involving any fuzzy operation. Furthermore, the new constructed associated linear systems are effective in providing exact solution as compared to linear programming, which is subjected to a number of constraints. These methods are also able to provide accurate solutions for large systems. Moreover, the existence of fuzzy solutions and classification of possible solutions are being checked by these associated linear systems. In case of near zero fully fuzzy linear system, fuzzy operations are required to determine the nature of solution of fuzzy system and to ensure the fuzziness of the solution. Finite solutions which are new concept of consistency in linear systems are obtained by the constructed min-max and absolute systems. These developed methods can also be modified to solve advanced fuzzy systems such as fully fuzzy matrix equation and fully fuzzy Sylvester equation, and can be employed for other types of fuzzy numbers such as trapezoidal fuzzy number. The study contributes to the methods to solve arbitrary linear fuzzy systems without any restriction on the system

Approximate Solution of LR Fuzzy Sylvester Matrix Equations

The fuzzy Sylvester matrix equation AX~+X~B=C~ in which A,B are m×m and n×n crisp matrices, respectively, and C~ is an m×n LR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method

Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases

Many real problems in control systems are related to the solvability of the generalized Sylvester matrix equation either using analytical or numerical methods. However, in many applications, the classical generalized Sylvester matrix equation are not well equipped to handle uncertainty in real-life problems such as conflicting requirements during the system process, the distraction of any elements and noise. Thus, crisp number in this matrix equation is replaced by fuzzy numbers and called generalized fully fuzzy Sylvester matrix equation when all parameters are in fuzzy form. The existing fuzzy analytical methods have four main drawbacks, the avoidance of using near-zero fuzzy numbers, the lack of accurate solutions, the limitation of the size of the systems, and the positive sign restriction of the fuzzy matrix coefficients and fuzzy solutions. Meanwhile, the convergence, feasibility, existence and uniqueness of the fuzzy solution are not examined in many fuzzy numerical methods. In addition, many studies are limited to positive fuzzy systems only due to the limitation of fuzzy arithmetic operation, especially for multiplication between trapezoidal fuzzy numbers.Therefore, this study aims to construct new analytical and numerical methods, namely fuzzy matrix vectorization, fuzzy absolute value, fuzzy Bartle’s Stewart, fuzzy gradient iterative and fuzzy least-squares iterative for solving arbitrary generalized Sylvester matrix equation for special cases and couple Sylvester matrix equations. In constructing these methods, new fuzzy arithmetic multiplication operators for trapezoidal fuzzy numbers are developed. The constructed methods overcome the positive restriction by allowing the negative, near-zero fuzzy numbers as the coefficients and fuzzy solutions. The necessary and sufficient conditions for the existence, uniqueness, and convergence of the fuzzy solutions are discussed, and a complete analysis of the fuzzy solution is provided. Some numerical examples and the verification of the solutions are presented to demonstrate the constructed methods. As a result, the constructed methods have successfully demonstrated the solutions for the arbitrary generalized Sylvester matrix equation for special and general cases based on the new fuzzy arithmetic operations, with minimum complexity fuzzy operations. The constructed methods are applicable to either square or non-square coefficient matrices up to 100 × 100. In conclusion, the constructed methods have significant contribution to the application of control system theory without any restriction on the system