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