Matrix iteration algorithms for solving the generalized Lyapunov matrix equation

In this paper, we first recall some well-known results on the solvability of the generalized Lyapunov equation and rewrite this equation into the generalized Stein equation by using Cayley transformation. Then we introduce the matrix versions of biconjugate residual (BICR), biconjugate gradients stabilized (Bi-CGSTAB), and conjugate residual squared (CRS) algorithms. This study’s primary motivation is to avoid the increase of computational complexity by using the Kronecker product and vectorization operation. Finally, we offer several numerical examples to show the effectiveness of the derived algorithms.

Generalized convergence analysis of the fractional order systems

The aim of the present work is to generalize the contraction theory for the analysis of the convergence of fractional order systems for both continuous-time and discrete-time systems. Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. The result of this study is a generalization of the Lyapunov matrix equation and linear eigenvalue analysis. The proposed approach gives a necessary and sufficient condition for exponential and global convergence of nonlinear fractional order systems. The examples elucidate that the theory is very straightforward and exact.

Modified Function Projective Synchronization for a Partially Linear and Fractional-Order Financial Chaotic System with Uncertain Parameters

This paper investigates the modified function projective synchronization between fractional-order chaotic systems, which are partially linear financial systems with uncertain parameters. Based on the stability theory of fractional-order systems and the Lyapunov matrix equation, a controller is obtained for the synchronization between fractional-order financial chaotic systems. Using the controller, the error systems converged to zero as time tends to infinity, and the uncertain parameters were also estimated so that the phenomenon of parameter distortion was effectively avoided. Numerical simulations demonstrate the validity and feasibility of the proposed method.

Solutions to the matrix equation X − AXB = CY+R and its application

The solution of the nonhomogeneous Yakubovich matrix equation [Formula: see text] is important in stability analysis and controller design in linear systems. The nonhomogeneous Yakubovich matrix equation [Formula: see text], which contains the well-known Kalman–Yakubovich matrix equation and the general discrete Lyapunov matrix equation as special cases, is investigated in this paper. Closed-form solutions to the nonhomogeneous Yakubovich matrix equation are presented using the Smith normal form reduction. Its equivalent form is provided. Compared with the existing method, the method presented in this paper has no limit to the dimensions of an unknown matrix. The present method is suitable for any unknown matrix, not only low-dimensional unknown matrices, but also high-dimensional unknown matrices. As an application, parametric pole assignment for descriptor linear systems by PD feedback is considered.