Stein Matrix Equation Research Articles

Low‐rank approximations of frequency‐limited discrete‐time Gramians

AbstractWe consider the approximate computation of Gramians of large‐scale linear discrete‐time state‐space systems. Of particular interest are the Gramians which arise in frequency‐limited balanced truncation model order reduction. While there are many results for the continuous‐time situation, the discrete‐time scenario is usually considered to a much lesser extent. The Gramians are solutions of discrete‐time Lyapunov matrix equations (also called Stein matrix equations) which, if the reduction is restricted to smaller frequency intervals, also incorporate matrix functions in the form of matrix logarithms. We discuss the efficient numerical handling of these matrix function as well as the computation of low‐rank Gramian approximation by rational Krylov subspace and related iterative methods.

Global quaternion generalized minimal residual method for generalized coupled Sylvester quaternion matrix equations with application to colour image encryption and decryption

ABSTRACT Sylvester matrix equations play important roles in control and system theory. We consider a class of the generalized coupled Sylvester quaternion matrix equations, which includes a lot of special matrix equations such as Stein matrix equation, Lyapunov matrix equation and Sylvester matrix equation on the quaternion skew field. First, the quaternion Krylov subspace and its F-orthogonal basis are constructed via a new quaternion matrix product and the global quaternion Arnoldi process. Then a global quaternion generalized minimal residual (Gl-QGMRES) method is proposed for solving the generalized coupled Sylvester quaternion matrix equations. The convergence of the proposed method is theoretically analysed. Finally, the effectiveness and feasibility of the proposal are verified by some numerical experiments. As an application of Gl-QGMRES method, we establish a kind of colour image encryption and decryption by embedding watermark images.

A Low-Rank Global Krylov Squared Smith Method for Solving Large-Scale Stein Matrix Equation

A Low-Rank Global Krylov Squared Smith Method for Solving Large-Scale Stein Matrix Equation

The general algebraic solution of dual fuzzy linear systems and fuzzy Stein matrix equations

In this paper, we present a new method for solving a dual fuzzy linear system (DFLS), AX˜+C˜=BX˜+D˜, where the coefficient matrices A and B are arbitrary real m×n matrices and C˜ and D˜ are given fuzzy number vectors. A necessary and sufficient condition for the R-consistency of the associated system of linear equations is obtained. The straightforward method for solving m×n DFLS based on an arbitrary {1}-inverse of A−B is introduced. Also, as an application, we present the first algorithm for solving the fuzzy Stein matrix equations, based on {1}-inverses. Finally, these results are illustrated by examples.

Stein-based preconditioners for weak-constraint 4D-var

Algorithms for data assimilation try to predict the most likely state of a dynamical system by combining information from observations and prior models. Variational approaches, such as the weak-constraint four-dimensional variational data assimilation formulation considered in this problem, can ultimately be interpreted as a minimization problem. One of the main challenges of such a formulation is the solution of large linear systems of equations which arise within the inner linear step of the adopted nonlinear solver. Depending on the adopted approach, these linear algebraic problems amount to either a saddle point linear system or a symmetric positive definite (SPD) one. Both formulations can be solved by means of a Krylov method, like GMRES or CG, that needs to be preconditioned to ensure fast convergence in terms of the number of iterations. In this paper we illustrate novel, efficient preconditioning operators which involve the solution of certain Stein matrix equations. In addition to achieving better computational performance, the latter machinery allows us to derive tighter bounds for the eigenvalue distribution of the preconditioned linear system for certain problem settings. A panel of diverse numerical results displays the effectiveness of the proposed methodology compared to current state-of-the-art approaches.

Shifted Global BiCG-Type Methods for Solving the Large Stein Equation X + AXB = C

The Stein matrix equation plays an essential role in control and communications theory, linear algebra, image restoration, and so on. In this paper, we propose two shifted variants of global BiCG-type methods to solve the large Stein matrix equation which make full use of the shifted structure of the matrix equation. These modifications will not add more matrix-matrix multiplications and inner products. Finally, numerical examples are given to illustrate the effectiveness of the presented methods.

An intelligent fuzzy robustness ZNN model with fixed‐time convergence for time‐variant Stein matrix equation

On account of the rapid progress of zeroing neural network (ZNN) and the extensive use of fuzzy logic system (FLS), this article proposes an intelligent fuzzy robustness ZNN (IFR-ZNN) model and applies it to solving the time-variant Stein matrix equation (TVSME) problem. Be different from ZNN models before, the IFR-ZNN model uses a fuzzy parameter as the design parameter and adopts a first proposed improved nonlinear piecewise activation function. Particularly, the FLS that generates the fuzzy parameter utilizes an improved membership function of nonuniform distribution which can improve the adaptability and robustness of the IFR-ZNN model. Based on the above two optimizations, the proposed IFR-ZNN model possesses three significant advantages: (1) fixed-time convergence independent of initial states; (2) superior robustness to tolerate two kinds of noises simultaneously; and (3) better adaptiveness based on computational error. Besides, the upper bounds of fixed-time convergence of the IFR-ZNN model under noisy or non-noisy situations are calculated theoretically, and the stability as well as the excellent adaptability are analyzed in detail. Finally, simulation comparison results manifest the availability and meliority of the proposed IFR-ZNN model in solving the TVSME problem.

Hyperbolic tangent variant-parameter robust ZNN schemes for solving time-varying control equations and tracking of mobile robot

Time-varying Lyapunov and Stein matrix equations (TVLSMEs) are very ubiquitous in engineering applications. To solve TVLSMEs, many zeroing neural networks (ZNNs) activated by nonlinear activation functions have been proposed. In the conventional ZNN schemes, the designed convergence parameters (DCPs) before the nonlinear activation functions are extraordinarily important because DCPs determine their convergent rates basically. However, the DCPs are usually stated to be constant, which is not practical because the DCPs are usually time-varying in realistic hardware situations, especially the external noises injected. Hence, many varying-parameter ZNNs (VP-ZNNs) with time-varying DCPs have been proposed previously. Compared with finite-parameter ZNNs, these traditional VP-ZNNs are proved to have better convergence, however their downside is that DCPs usually increases over time, and becomes even infinite eventually. Evidently, infinity large DCPs would be lack of robustness are unacceptable in practice, especially the external noises injected. Additionally, even if VP-ZNNs converge over time, the growth of DCPs will result in a huge waste of computing resources. Inspired by that, a new hyperbolic tangent varying-parameter ZNNs (HTVP-ZNNs) with time-varying DCPs are proposed in this paper. Considering the noisy environment, we also develop the robust HTVP-ZNNs (HTVPR-ZNNs). Both of them are able to solve the time-variant Lyapunov and Stein matrix equations in prescribed-time. Theoretically the convergent prescribed-time of the HTVPR-ZNN and the upper time threshold of the DCPs are analyzed mathematically. And numerical experiments and trajectory tracking tasks of the mobile robot substantiate the outstanding convergence of the HTVPR-ZNN schemes.

On Some Numerical Methods for Solving Large Differential Nonsymmetric Stein Matrix Equations

In this paper, we propose a new numerical method based on the extended block Arnoldi algorithm for solving large-scale differential nonsymmetric Stein matrix equations with low-rank right-hand sides. This algorithm is based on projecting the initial problem on the extended block Krylov subspace to obtain a low-dimensional differential Stein matrix equation. The obtained reduced-order problem is solved by the backward differentiation formula (BDF) method or the Rosenbrock (Ros) method, the obtained solution is used to build the low-rank approximate solution of the original problem. We give some theoretical results and report some numerical experiments.

Two algebraic methods for least squares L-structured and generalized L-structured problems of the commutative quaternion Stein matrix equation

Two algebraic methods for least squares L-structured and generalized L-structured problems of the commutative quaternion Stein matrix equation

Design and Analysis of Two Prescribed-Time and Robust ZNN Models With Application to Time-Variant Stein Matrix Equation.

The zeroing neural network (ZNN) activated by nonlinear activation functions plays an important role in many fields. However, conventional ZNN can only realize finite-time convergence, which greatly limits the application of ZNN in a noisy environment. Generally, finite-time convergence depends on the original state of ZNN, but the original state is often unknown in advance. In addition, when meeting with different noises, the applied nonlinear activation functions cannot tolerate external disturbances. In this article, on the strength of this idea, two prescribed-time and robust ZNN (PTR-ZNN) models activated by two nonlinear activation functions are put forward to address the time-variant Stein matrix equation. The proposed two PTR-ZNN models own two remarkable advantages simultaneously: 1) prescribed-time convergence that does not rely on original states and 2) superior noise-tolerance performance that can tolerate time-variant bounded vanishing and nonvanishing noises. Furthermore, the detailed theoretical analysis is provided to guarantee the prescribed-time convergence and noise-tolerance performance, with the convergence upper bounds of steady-state residual errors calculated. Finally, simulative comparison results indicate the effectiveness and the superiority of the proposed two PTR-ZNN models for the time-variant Stein matrix equation solving.

On the semigroup property for some structured iterations

Nonlinear matrix equations play a crucial role in science and engineering problems. However, solutions of nonlinear matrix equations cannot, in general, be given analytically. One standard way of solving nonlinear matrix equations is to apply the fixed-point iteration with usually only the linear convergence rate. To advance the existing methods, we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order. We realize our way by starting with examples of solving the scalar equations and, also, connect this method with some well-known equations including, but not limited to, the Stein matrix equation, the generalized eigenvalue problem, the generalized nonlinear matrix equation, the discrete-time algebraic Riccati equations to express the capacity of this method.

Sensor Fault and Delay Tolerant Control for Networked Control Systems Subject to External Disturbances.

In this paper, the problem of sensor fault and delay tolerant control problem for a class of networked control systems under external disturbances is investigated. More precisely, the dynamic characteristics of the external disturbance and sensor fault are described as the output of exogenous systems first. The original sensor fault and delay tolerant control problem is reformulated as an equivalence problem with designed available system output and reformed performance index. The feedforward and feedback sensor fault tolerant controller (FFSFTC) can be obtained by utilizing the solutions of Riccati matrix equation and Stein matrix equation. Based on the designed fault diagnoser, the proposed FFSFTC is further reconstructed to compensate for the sensor fault and delayed measurement effects. Finally, numerical examples are provided to illustrate the effectiveness of our proposed FFSFTC with different cases with various types of sensor faults, measurement delays and external disturbances.

On Some Extended Block Krylov Based Methods for Large Scale Nonsymmetric Stein Matrix Equations

In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term A X B − X + E F T = 0 . These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments.

Lanczos version of BCR algorithm for solving the generalised second‐order Sylvester matrix equation

It is well known that the biconjugate residual (BCR) algorithm and its variants are powerful procedures to find the solution of large sparse non-symmetric systems equation A x = b . In this study, the authors develop the Lanczos version of BCR algorithm for computing the solution pair ( V , W ) of the generalised second-order Sylvester matrix equation E V F + G V H + B V C = D W E + M , which includes the second-order Sylvester, Lyapunov and Stein matrix equations as special cases. The convergence results show that the algorithm with any initial matrices converges to the solutions within a finite number of iterations in the absence of round-off errors. Finally, two numerical examples are provided to support the theoretical findings and to testify the effectiveness and usefulness of the algorithm.

Normality conditions for semilinear matrix operators of the Stein type

Normality conditions are found for the operators associated with the semilinear analogs of the Stein matrix equation, namely, with the equations xxxxxxxx and X − AX*B = C.

On the solution of the linear matrix equation [formula omitted

In this paper, we derive a formula to compute the solution of the linear matrix equation X=Af(X)B+C via finding any solution of a specific Stein matrix equation X=AXB+C, where the linear (or anti-linear) matrix operator f is period-n. According to this formula, we should pay much attention to solve the Stein matrix equation from recently famous numerical methods. For instance, Smith-type iterations, Bartels–Stewart algorithm, and etc. Moreover, this transformation is used to provide necessary and sufficient conditions of the solvable of the linear matrix equation. On the other hand, it can be proven that the general solution of the linear matrix equation can be presented by the general solution of the Stein matrix equation. The necessary condition of the uniquely solvable of the linear matrix equation is developed. It is shown that several representations of this formula are coincident. Some examples are presented to illustrate and explain our results.

Erratum to “A Note on the ⊤-Stein Matrix Equation”

Erratum to “A Note on the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>⊤</mml:mi></mml:mrow></mml:math>-Stein Matrix Equation”

On the convergence of inexact Newton methods for discrete-time algebraic Riccati equations

In this paper, we present a convergence analysis of the inexact Newton method for solving Discrete-time algebraic Riccati equations (DAREs) for large and sparse systems. The inexact Newton method requires, at each iteration, the solution of a symmetric Stein matrix equation. These linear matrix equations are solved approximatively by the alternating directions implicit (ADI) or Smithʼs methods. We give some new matrix identities that will allow us to derive new theoretical convergence results for the obtained inexact Newton sequences. We show that under some necessary conditions the approximate solutions satisfy some desired properties such as the d-stability. The theoretical results developed in this paper are an extension to the discrete case of the analysis performed by Feitzinger et al. (2009) [8] for the continuous-time algebraic Riccati equations. In the last section, we give some numerical experiments.

A Note on the⊤-Stein Matrix Equation

This note is concerned with the linear matrix equationX=AX⊤B + C, where the operator(·)⊤denotes the transpose (⊤) of a matrix. The first part of this paper sets forth the necessary and sufficient conditions for the unique solvability of the solutionX. The second part of this paper aims to provide a comprehensive treatment of the relationship between the theory of the generalized eigenvalue problem and the theory of the linear matrix equation. The final part of this paper starts with a brief review of numerical methods for solving the linear matrix equation. In relation to the computed methods, knowledge of the residual is discussed. An expression related to the backward error of an approximate solution is obtained; it shows that a small backward error implies a small residual. Just like the discussion of linear matrix equations, perturbation bounds for solving the linear matrix equation are also proposed in this work.Erratum to “A Note on the⊤-Stein Matrix Equation”