Abstract
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 100 × 100 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.
Highlights
The Solution of Coupled Trapezoidal Fully Fuzzy Sylvester Matrix EquationThe solution to the positive CTrFFSME is considered. To get the solution, the positive CTrFFSME is converted to an equivalent system of crisp Sylvester matrix equations (CSMEs), and the solution to this system of CSME is obtained by three different methods
Analyzing the stability of many control systems required solving a couple of crisp Sylvester matrix equations (CSMEs) simultaneously
We presented the solution to the CTrFFSME and its special cases analytically by Fuzzy Matrix Vectorization Method (FMVM) and numerically by Fuzzy Gradient Iterative (FGI) and Fuzzy Least-Square Iterative (FLSI) methods. e FMVM aims to find the exact solution
Summary
The solution to the positive CTrFFSME is considered. To get the solution, the positive CTrFFSME is converted to an equivalent system of CSME, and the solution to this system of CSME is obtained by three different methods. The analytical solution for the CTrFFSME in equation (1) is obtained by extending the method of the fuzzy matrix vectorization method (FMVM) proposed by [48]. E iterative positive solution to the system of equations in equations (36a) and (36b) can be obtained by the GI method in eorem 1 as follows: for 1 ≤ l ≤ 4, we have. By Definition 10, the approximated fuzzy solutions obtained by the previous step to the CTrFFSME in equation (1) can be written as follows:. E iterative positive solution to the system of equations in equations (36a) and (36b) can be obtained by the LSI method in eorem 2 as follows: for 1 ≤ l ≤ 4, we have. E section illustrates the three proposed methods by solving two examples sized 2 × 2 and 100 × 100, followed by the solutions verification
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