Numerical Solutions for Coupled Trapezoidal Fully Fuzzy Sylvester Matrix Equations

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations (AXB − CYD, EXF − GYH) = (M, N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices X and Y. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [X1, Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round‐off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair to a given matrix pair [X0, Y0] in Frobenius norm can be derived by finding the least‐norm generalized reflexive solution pair of a new corresponding generalized coupled Sylvester matrix equation pair , where . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices

It is known that solving coupled matrix equations with complex matrices can be very difficult and it is sufficiently complicated. In this work, we propose two iterative algorithms based on the Conjugate Gradient method (CG) for finding the reflexive and Hermitian reflexive solutions of the coupled Sylvester-conjugate matrix equations(including Sylvester and Lyapunov matrix equations as special cases). The iterative algorithms can automatically judge the solvability of the matrix equations over the reflexive and Hermitian reflexive matrices, respectively. When the matrix equations are consistent over reflexive and Hermitian reflexive matrices, for any initial reflexive and Hermitian reflexive matrices, the iterative algorithms can obtain reflexive and Hermitian reflexive solutions within a finite number of iterations in the absence of roundoff errors, respectively. Finally, two numerical examples are presented to illustrate the proposed algorithms.

On the generalized Sylvester mapping and matrix equations

Roth's similarity theorem on the consistency of Sylvester's matrix equation AX − XA = C can be extended to a theorem on rank minimization if the common eigenvalues of A and B are nonderogatory or semisimple.

We derive an iterative procedure for solving a generalized Sylvester matrix equation AXB+CXD = E, where A,B,C,D,E are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation AXB=C, the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.

Stability is one of the most important concepts of analysis and design of control systems. Stability analysis of fuzzy control systems has been difficult since fuzzy systems are essentially nonlinear systems. Recently, stability analysis techniques [l]-[6] [13]-[16] based on nonlinear stability theory have been reported. Langari and Tomizuka [2, 6] analyzed stability of fuzzy linguistic control systems. Kitamura [3] realized stability analysis of fuzzy control systems by using extended circle criterion. Farinwata [5] and Kato [14] utilized phase plane analysis type of techniques, tanaka and co-authors [1,3,4] applied Lyapunov approach to fuzzy control systems and recently developed LMI-based designs [10, 17, 28-37], where LMI stands for nonlinear matrix inequality.

In this article, we investigate a variant of the biconjugate residual (BCR) algorithm to solve the generalized Sylvester matrix equation which includes the well‐known Lyapunov, Stein and Sylvester matrix equations. We show that the BCR algorithm with any (special) initial matrix pair can smoothly compute the (least Frobenius norm) solution pair of the generalized Sylvester matrix equation within a finite number of iterations in the absence of round‐off errors. Finally the accuracy and effectiveness of the BCR algorithm in comparison to some existing algorithms are demonstrated by two numerical examples.

In this article, we present two new algorithms referred to as the improved modified gradient-based iterative (IMGI) algorithm and its relaxed version (IMRGI) for solving the complex conjugate and transpose (CCT) Sylvester matrix equations, which often arise from control theory, system theory, and so forth. Compared with the gradient-based iterative (GI) (A.-G. Wu, L.-L. Lv, and G.-R. Duan, Iterative algorithms for solving a class of complex conjugate and transpose matrix equations, Appl. Math. Comput. 217 (2011), 8343–8353) and the relaxed GI (RGI) (W.-L. Wang, C.-Q. Song, and S.-P. Ji, Iterative solution to a class of complex matrix equations and its application in time-varying linear system, J. Appl. Math. Comput. 67 (2021), 317–341) algorithms, the proposed ones can make full use of the latest information and need less computations, which leads to higher computational efficiency. With the real representation of a complex matrix as a tool, we establish sufficient and necessary conditions for the convergence of the IMGI and the IMRGI algorithms. Finally, some numerical examples are given to illustrate the effectiveness and advantages of the proposed algorithms.

The general coupled matrix equations over generalized bisymmetric matrices

In this paper, an iterative algorithm is established for finding the generalized bisymmetric solution group to the coupled matrix equations (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases). It is proved that proposed algorithm consistently converges to the generalized bisymmetric solution group for any initial generalized bisymmetric matrix group. Finally a numerical example indicates that proposed algorithm works quite effectively in practice.

The problem of state feedback optimal pole assignment is to design a feedback gain such that the closed-loop system has desired eigenvalues and such that certain quadratic performance index is minimized. Optimal pole assignment controller can guarantee both good dynamic response and well robustness properties of the closed-loop system. With the help of a class of linear matrix equations, necessary and sufficient conditions for the existence of a solution to the optimal pole assignment problem are proposed in this paper. By properly choosing the free parameters in the parametric solutions to this class of linear matrix equations, complete solutions to the optimal pole assignment problem can be obtained. A numerical example is used to illustrate the effectiveness of the proposed approach.

By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations∑k=1qAikXkBik=Ci,i=1,2,…,p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups(X1,X2,…,Xq), such as symmetric, generalized bisymmetric, and(R,S)-symmetric matrix groups. By this iterative method, for any initial matrix group(X1(0),X2(0),…,Xq(0)), a solution group(X1*,X2*,…,Xq*)can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group(X¯1,X¯2,…,X¯q)in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method.

Keeping in view that a lot of physical systems with inverse problems can be written by matrix equations, the least-norm of the solution to a general Sylvester matrix equation with restrictions A1X1=C1,X1B1=C2,A2X2=C3,X2B2=C4,A3X1B3+A4X2B4=Cc, is researched in this chapter. A novel expression of the general solution to this system is established and necessary and sufficient conditions for its existence are constituted. The novelty of the proposed results is not only obtaining a formal representation of the solution in terms of generalized inverses but the construction of an algorithm to find its explicit expression as well. To conduct an algorithm and numerical example, it is used the determinantal representations of the Moore–Penrose inverse previously obtained by one of the authors.

Bivariate matrix functions provide a unified framework for various tasks in numerical linear algebra, including the solution of linear matrix equations and the application of the Frechet derivative. In this work, we propose a novel tensorized Krylov subspace method for approximating such bivariate matrix functions and analyze its convergence. While this method is already known for some instances, our analysis appears to result in new convergence estimates and insights for all but one instance, Sylvester matrix equations.