Implicit iterative algorithms with a tuning parameter for discrete stochastic Lyapunov matrix equations

Parametric Smith iterative algorithms for discrete Lyapunov matrix equations

“…For the discrete Lyapunov matrix equation (1), denote the eigenvalue set of A⊗A as in (18). If the matrix A is Schur stable, then |a i ± ib i | < 1 for any i ∈ I [1, s].…”

“…In [16], the matrices A and B were firstly written as the sum of diagonal matrices and other matrices, and then a Jacobi-like algorithm was proposed. Recently, the Lyapunov matrix equations appearing in stochastic linear systems were investigated in [18] and [19], and iterative algorithms were presented to solve this kind of matrix equations in the case where the associated systems are stochastically stable. In addition, a kind of coupled matrix equations in a general form was investigated in [3], and an iterative method was developed to solve this kind of matrices by extending the idea of conjugate gradient methods.…”

The filtering‐based maximum likelihood iterative estimation algorithms for a special class of nonlinear systems with autoregressive moving average noise using the hierarchical identification principle

“…Due to their good statistical properties of consistency, asymptotic normality, and availability, numerous likelihood estimation methods are developed for different models. 29 In the fields of parameter estimation and filtering identification, the iterative estimation methods can give more accurate parameter estimates than the recursive estimation methods. 18 In the fields of bilinear systems identification, Gibson et al proposed an expectation maximization algorithm for the maximum likelihood estimation of multivariable bilinear systems 19 ; a maximum likelihood least squares-based iterative algorithm and a filtering-based maximum likelihood gradient iterative algorithm are derived for parameter estimation of bilinear systems with colored noise.…”

“…28 For example, Zhang et al established an implicit iterative algorithm for solving a class of Lyapunov matrix equations. 29 In the fields of parameter estimation and filtering identification, the iterative estimation methods can give more accurate parameter estimates than the recursive estimation methods. 30 After decades of development, some iterative estimation methods have been used to identify different systems.…”

Spectral decomposition based solutions to the matrix equation

“…There also exist plenty of works that devote to compute the solutions to equation (1) numerically, generally under condition (2). Since in this paper we are not concerned with numerical solutions, readers are suggested to see [31–39] and the references therein for detailed information. Some other topics for this matrix equations have also been studied in the literature.…”