Gradient-based neural networks for solving periodic Sylvester matrix equations

This paper considers neural network solutions of a category of matrix equation called periodic Sylvester matrix equation (PSME), which appear in the process of periodic system analysis and design. A linear gradient-based neural network (GNN) model aimed at solving the PSME is constructed, whose state is able to converge to the unknown matrix of the equation. In order to obtain a better convergence effect, the linear GNN model is extended to a nonlinear form through the intervention of appropriate activation functions, and its convergence is proved through theoretical derivation. Furthermore, the different convergence effects presented by the model with various activation functions are also explored and analyzed, for instance, the global exponential convergence and the global finite time convergence can be realized. Finally, the numerical examples are used to confirm the validity of the proposed GNN model for solving the PSME considered in this paper as well as the superiority in terms of the convergence effect presented by the model with different activation functions.     

Factor gradient iterative algorithm for solving a class of discrete periodic Sylvester matrix equations

At present, gradient iteration methods have been used to solve various Sylvester matrix equations and proved effective. Based on this method, we generalize the factor gradient iterative method (FGI) for solving forward periodic Sylvester matrix equations (FPSME) and backward periodic Sylvester matrix equations (BPSME). To accelerate the convergence of the iterative method, we refer to Gauss-Seidel and Jacobi iterative construction ideas and use the latest matrix information in the FGI iterative method to obtain the modified factor gradient iterative (MFGI) method. Then, the convergence of the proposed methods and the selection of optimal factors are proved. The last numerical examples illustrate the effectiveness and applicability of the iterative methods.     

Conjugate gradient-based iterative algorithm for solving generalized periodic coupled Sylvester matrix equations

This paper focuses on constructing a conjugate gradient-based (CGB) method to solve the generalized periodic coupled Sylvester matrix equations in complex space. The presented method is developed from a point of conjugate gradient methods. It is proved that the presented method can find the solution of the considered matrix equations within finite iteration steps in the absence of round-off errors by theoretical derivation. Some numerical examples are provided to verify the convergence performance of the presented method, which is superior to some existing numerical algorithms both in iteration steps and computation time.     

A relaxed gradient based algorithm for solving generalized coupled Sylvester matrix equations

The present work proposes a relaxed gradient based iterative (RGI) algorithm to find the solutions of coupled Sylvester matrix equations $AX+YB=C,DX+YE=F$. It is proved that the proposed iterative method can obtain the solutions of the coupled Sylvester matrix equations for any initial matrices X₀ and Y₀. Next the RGI algorithm is extended to the generalized coupled Sylvester matrix equations of the form $A_{i1}{X}_1{B}_{i1}+A_{i2}{X}_2{B}_{i2}+?+A_{ip}{X}_p{B}_{ip}=C_i,(i=1,2,\dots ,p)$. Then, we compare their convergence rate and find RGI is faster than GI, which has maximum convergence rate, under an appropriative positive number ω and the same convergence factor µ₁ and µ₂. Finally, a numerical example is included to demonstrate that the introduced iterative algorithm is more efficient than the gradient based iterative (GI) algorithm of (Ding and Chen 2006) in speed, elapsed time and iterative steps.     

An iterative algorithm for generalized periodic multiple coupled Sylvester matrix equations

The paper is indicated to constructing a modified conjugate gradient iterative (MCG) algorithm to solve the generalized periodic multiple coupled Sylvester matrix equations. It can be proved that the proposed approach can find the solution within finite iteration steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrices to obtain the least Frobenius norm solution of the system. Some numerical examples are illustrated to show the performance of the proposed approach and its superiority over the existing method CG.     

Gradient based approach for generalized discrete-time periodic coupled Sylvester matrix equations

The paper is dedicated to solving the generalized periodic discrete-time coupled Sylvester matrix equation, which is frequently encountered in control theory and applied mathematics. The solvable condition and a iterative algorithm for this equation are presented. The proposed method is developed from a point of least squares method. The rationality of the method is testified by theoretical analysis, which shows that the algorithm can solve the problem within finite number of iterations. The presented approach is numerically reliable and requires less computation. A numerical example illustrates the effectiveness of the raised result.     

A finite iterative algorithm for the general discrete-time periodic Sylvester matrix equations

The purpose of this paper is to present an iterative algorithm for solving the general discrete-time periodic Sylvester matrix equations. It is proved by theoretical analysis that this algorithm can get the exact solutions of the periodic Sylvester matrix equations in a finite number of steps in the absence of round-off errors. Furthermore, when the discrete-time periodic Sylvester matrix equations are consistent, we can obtain its unique minimal Frobenius norm solution by choosing appropriate initial periodic matrices. Finally, we use some numerical examples to illustrate the effectiveness of the proposed algorithm.     

A novel finite-time complex-valued zeoring neural network for solving time-varying complex-valued Sylvester equation

A novel finite-time complex-valued zeroing neural network (FTCVZNN) for solving time-varying Sylvester equation is proposed and investigated. Asymptotic stability analysis of this network is examined with any general activation function satisfying a condition or with an odd monotonically increasing activation function. So far, finite-time model studies have been investigated for the upper bound time of convergence using a linear activation function with design formula for the derivative of the error or with variations of sign-bi-power activation functions to zeroing neural networks. A function adaptive coefficient for sign-bi-power activation function (FA-CSBP) is introduced and examined for faster convergence. An upper bound on convergence time is derived with the two components in the function adaptive coefficients of sign-bi-power activation function. Numerical simulation results demonstrate that the FTCVZNN with function adaptive coefficient for sign-bi-power activation function is faster than applying a sign-bi-power activation function to the zeroing neural network (ZNN) and the other finite-time complex-valued models for the selected example problems.     

Finite-time convergent zeroing neural network for solving time-varying algebraic Riccati equations

Various forms of the algebraic Riccati equation (ARE) have been widely used to investigate the stability of nonlinear systems in the control field. In this paper, the time-varying ARE (TV-ARE) and linear time-varying (LTV) systems stabilization problems are investigated by employing the zeroing neural networks (ZNNs). In order to solve the TV-ARE problem, two models are developed, the ZNNTV-ARE model which follows the principles of the original ZNN method, and the FTZNNTV-ARE model which follows the finite-time ZNN (FTZNN) dynamical evolution. In addition, two hybrid ZNN models are proposed for the LTV systems stabilization, which combines the ZNNTV-ARE and FTZNNTV-ARE design rules. Note that instead of the infinite exponential convergence specific to the ZNNTV-ARE design, the structure of the proposed FTZNNTV-ARE dynamic is based on a new evolution formula which is able to converge to a theoretical solution in finite time. Furthermore, we are only interested in real symmetric solutions of TV-ARE, so the ZNNTV-ARE and FTZNNTV-ARE models are designed to produce such solutions. Numerical findings, one of which includes an application to LTV systems stabilization, confirm the effectiveness of the introduced dynamical evolutions.     

Tensor form of GPBiCG algorithm for solving the generalized Sylvester quaternion tensor equations

Sylvester quaternion tensor equations have a wide range of applications in image processing and system and control theory. In this paper, by the Kronecker product and vectorization operator and the properties of quaternion tensors, we focus mainly on proposing the tensor form of the generalized product-type biconjugate gradient method for solving generalized Sylvester quaternion tensor equations. As an application, we apply the proposed method to restore a blurred and noisy-free color video. The obtained numerical results illustrate the effectiveness of our method compared with some existing methods.     

A robust fast convergence zeroing neural network and its applications to dynamic Sylvester equation solving and robot trajectory tracking

In order to find the theoretical solution of a dynamic Sylvester equation (DSE) in noisy environment, a robust fast convergence zeroing neural network (RFCZNN) is proposed in this paper. Unlike the original zeroing neural network (ZNN) model with existing activation functions (AF), by introducing a new AF, the proposed RFCZNN model guarantees fixed-time convergence to theoretical solution of DSE and robustness against noise simultaneously. The effectiveness and robustness of the proposed RFCZNN model are investigated in theory and demonstrated through simulation results. In addition, its effectiveness and robustness are further verified by a successful robotic trajectory tracking application in noisy environment.     

A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations

In this paper, we discuss the properties of the eigenvalues related to the symmetric positive definite matrices. Several new results are established to express the structures and bounds of the eigenvalues. Using these results, a family of iterative algorithms are presented for the matrix equation AX=F and the coupled Sylvester matrix equations. The analysis shows that the iterative solutions given by the least squares based iterative algorithms converge to their true values for any initial conditions. The effectiveness of the proposed iterative algorithm is illustrated by a numerical example.     

Generalized conjugate direction algorithm for solving generalized coupled Sylvester transpose matrix equations over reflexive or anti-reflexive matrices

The paper studies the iterative solutions of the generalized coupled Sylvester transpose matrix equations over the reflexive (anti-reflexive) matrix group by the generalized conjugate direction algorithm. The convergence analysis shows that the solution group can be obtained within finite iterative steps in the absence of round-off errors for any initial given reflexive (anti-reflexive) matrix group. Furthermore, we can get the minimum-norm solution group by choosing special kinds of initial matrix group. Finally, some numerical examples are given to demonstrate the algorithm considered is quite effective in actual computation.     

Symmetric least squares solution of a class of Sylvester matrix equations via MINIRES algorithm

The purpose of this paper is deriving the minimal residual (MINIRES) algorithm for finding the symmetric least squares solution on a class of Sylvester matrix equations. We prove that if the system is inconsistent, the symmetric least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrix to obtain the minimum norm least squares symmetric solution of the problem. Finally, we give some numerical examples to illustrate the performance of MINIRES algorithm.