Generalized Sylvester Matrix Equation Research Articles

A System of Four Generalized Sylvester Matrix Equations over the Quaternion Algebra

A System of Four Generalized Sylvester Matrix Equations over the Quaternion Algebra

Tikhonov regularization with conjugate gradient least squares method for large-scale discrete ill-posed problem in image restoration

Image restoration is a large-scale discrete ill-posed problem, which can be transformed into a Tikhonov regularization problem that can approximate the original image. Kronecker product approximation is introduced into the Tikhonov regularization problem to produce an alternative problem of solving the generalized Sylvester matrix equation, reducing the scale of the image restoration problem. This paper considers solving this alternative problem by applying the conjugate gradient least squares (CGLS) method which has been demonstrated to be efficient and concise. The convergence of the CGLS method is analyzed, and it is demonstrated that the CGLS method converges to the least squares solution within the finite number of iteration steps. The effectiveness and superiority of the CGLS method are verified by numerical tests.

Compact formula for skew-symmetric system of matrix equations

In this paper, we consider skew-Hermitian solution of coupled generalized Sylvester matrix equations encompassing *-hermicity over complex field. The compact formula of the general solution of this system is presented in terms of generalized inverses when some necessary and sufficient conditions are fulfilled. An algorithm and a numerical example are provided to validate our findings. A numerical example is carried out using determinantal representations of the Moore–Penrose inverse.

An iterative method based on ADMM for solving generalized Sylvester matrix equations

As is well known, the alternating direction method of multipliers (ADMM) is one of the most famous distributed algorithms for solving the convex optimization problems. In this paper, by applying the hierarchical identification principle, we propose an iterative algorithm based on matrix form of ADMM for finding the least squares solution of the generalized Sylvester matrix equation AXB+CXD=E. We prove that the proposed method is convergent under some common conditions for any initial matrices. The simplicity and effectiveness of the new method are shown by the reported numerical results.

Parametric design of reduced-order functional observers for linear time-varying delay systems

This paper investigates the design of reduced-order functional observer for linear time-invariant systems (LTI) with time-varying delay. Firstly, the sufficient condition for the existence of reduced-order delay-free functional observers is obtained. Secondly, by solving generalized Sylvester matrix equations (GSEs), completely parameterized formulas of the coefficient matrices of reduced-order function observer are established. The free parameter matrices in the formulas provide the design degrees of the freedom and improve the convergence rate of the error system. Finally, two numerical and a practical examples are provided to illustrate the effectiveness of the proposed approach.

Precision matrix estimation using penalized Generalized Sylvester matrix equation

Estimating a precision matrix is an important problem in several research fields when dealing with large-scale data. Under high-dimensional settings, one of the most popular approaches is optimizing a Lasso or ell _1 norm penalized objective loss function. This penalization endorses sparsity in the estimated matrix and improves the accuracy under a proper calibration of the penalty parameter. In this paper, we demonstrate that the problem of minimizing Lasso penalized D-trace loss can be seen as solving a penalized Sylvester matrix equation. Motivated by this method, we propose estimating the precision matrix using penalized generalized Sylvester matrix equations. In our method, we develop a particular estimating equation and a new convex loss function constructed through this equation, which we call the generalized D-trace loss. We assess the performance of the proposed method using detailed numerical analysis, including simulated and real data. Extensive results show the advantage of the proposed method compared to other estimation approaches in the literature.

Trajectory tracking control of failure satellite with actuator jumping fault

This paper studied the trajectory tracking control of a failure satellite that has actuator jumping fault. It proposed a design method based on the jumping control input for the model reference tracking controller. By comb-ining the stochastic stability definition with model reference tracking, it gave the mathematical descriptions of the trajectory tracking control of the failure satellite. It used the linear matrix inequality method and the parametric solution method of the nonhomogeneous generalized Sylvester matrix equation to design the robust H∞ state feedback control law and the complete parametric feed-forward tracking compensator respectively, with the disturbance of the failure satellite and its thrust constraint considered simultaneously. The numerical simulation results on the model of the satellite rendezvous system show that the design method proposed in this paper is effective.

Three special kinds of least squares solutions for the quaternion generalized Sylvester matrix equation

<abstract><p>In this paper, we propose an efficient method for some special solutions of the quaternion matrix equation $ AXB+CYD = E $. By integrating real representation of a quaternion matrix with $ \mathcal{H} $-representation, we investigate the minimal norm least squares solution of the previous quaternion matrix equation over different constrained matrices and obtain their expressions. In this way, we first apply $ \mathcal{H} $-representation to solve quaternion matrix equation with special structure, which not only broadens the application scope of $ \mathcal{H} $-representation, but further expands the research idea of solving quaternion matrix equation. The algorithms only include real operations. Consequently, it is very simple and convenient, and it can be applied to all kinds of quaternion matrix equation with similar problems. The numerical example is provided to illustrate the feasibility of our algorithms.</p></abstract>

The semi-tensor product method for special least squares solutions of the complex generalized Sylvester matrix equation

<abstract><p>In this paper, we are interested in the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution for the complex generalized Sylvester matrix equation $ CXD+EXF = G $. By utilizing of the real vector representations of complex matrices and the semi-tensor product of matrices, we first transform solving special least squares solutions of the above matrix equation into solving the general least squares solutions of the corresponding real matrix equations, and then obtain the expressions of the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution. Further, we give two numerical algorithms and two numerical examples, and numerical examples illustrate that our proposed algorithms are more efficient and accurate.</p></abstract>

Model reference control in quasi-linear systems with a parametric feed-forward compensator and state-feedback stabilization controller

In this paper, a parametric feed-forward compensator and a parametric state-feedback stabilization controller are proposed for the model reference control to a class of quasi-linear systems. Quasi-linear systems are a special type of nonlinear systems whose coefficient matrices contain the state variables and also a time-varying parameter vector. The parametric state-feedback stabilization controller guarantees the stability of the closed-loop system and the parametric feed-forward compensator compensates the effect of the reference model state to the tracking error. The complete parametrization of the parametric feed-forward compensator is established based on a complete parametric solution to a class of generalized Sylvester matrix equations and solution of a coefficient matrix such that two matrix equations are satisfied. The established parametric state-feedback stabilization controller only needs a complete parametric solution to the same generalized Sylvester matrix equations but with different sets of freely designed parameters that represent the degrees of design freedom and may be further utilized to improve the system performance. A linear closed-loop form with the desired eigenstructure can be derived with the proposed parametric feed-forward compensator and parametric state-feedback stabilization controller, and a constant linear can even be obtained in certain cases. A numerical example and the application in spacecraft rendezvous are provided to illustrate the effectiveness of the proposed approach.

Robust model reference control for second-order descriptor linear systems subject to parameter uncertainties

This paper is devoted to designing the robust model reference controller for uncertain second-order descriptor linear systems subject to parameter uncertainties. The parameter uncertainties are assumed to be norm-bounded. The design of a robust controller can be divided into two separate problems: a robust stabilization problem and a robust compensation problem. Based on the solution of generalized Sylvester matrix equations, we obtain some sufficient conditions to guarantee the complete parameterization of the robust controller. The parametric forms are expressed by a group of parameter vectors which reveal the degrees of freedom existing in the design of the compensator and can be utilized to solve the robust compensation problem. In order to reduce the effect of parameter uncertainties on the tracking error vector, the robust compensation problem is converted into a convex optimization problem with a set of linear matrix equation constraints. A simulation example is provided to illustrate the effectiveness of the proposed technique.

Convergence analysis of gradient-based iterative algorithms for a class of rectangular Sylvester matrix equations based on Banach contraction principle

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.

The use of homotopy analysis method for solving generalized Sylvester matrix equation with applications

In this research, we introduce and analyze homotopy analysis method (HAM) for solving approximately linear matrix equation $$ \sum \limits \limits _{i=1}^{s}A_iXB_i+C=\mathbf{0} $$ , where $$ A_i,\;B_i \;(i=1,\ldots ,s), \; C \in \mathbb {C}^{n \times n} $$ and $$X\in \mathbb {C}^{n \times n} $$ must be determined. In this method we consider a convergence control parameter $$ \delta $$ , and then we determine the optimum value of $$ \delta $$ for obtaining fast convergence method. Moreover, we obtain the corresponding spectral radius of convergence factor of HAM method. Finally, we will apply this method to solve some test problems to support the theoretical results.

Robust Parametric Control of Lorenz System via State Feedback

This paper considers the parametric control to the Lorenz system by state feedback. Based on the solutions of the generalized Sylvester matrix equation (GSE), the unified explicit parametric expression of the state feedback gain matrix is proposed. The closed loop of the Lorenz system can be transformed into an arbitrary constant matrix with the desired eigenstructure (eigenvalues and eigenvectors). The freedom provided by the parametric control can be fully used to find a controller to satisfy the robustness criteria. A numerical simulation is developed to illustrate the effectiveness of the proposed approach.

Gradient Iterative Method with Optimal Convergent Factor for Solving a Generalized Sylvester Matrix Equation with Applications to Diffusion Equations

We introduce a gradient iterative scheme with an optimal convergent factor for solving a generalized Sylvester matrix equation ∑i=1pAiXBi=F, where Ai,Bi and F are conformable rectangular matrices. The iterative scheme is derived from the gradients of the squared norm-errors of the associated subsystems for the equation. The convergence analysis reveals that the sequence of approximated solutions converge to the exact solution for any initial value if and only if the convergent factor is chosen properly in terms of the spectral radius of the associated iteration matrix. We also discuss the convergent rate and error estimations. Moreover, we determine the fastest convergent factor so that the associated iteration matrix has the smallest spectral radius. Furthermore, we provide numerical examples to illustrate the capability and efficiency of this method. Finally, we apply the proposed scheme to discretized equations for boundary value problems involving convection and diffusion.

Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form sum_{t=1}^{p}A_{t}XB_{t}=C which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson’s equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.

Parametric control to a type of descriptor quasi‐linear systems based on dynamic compensator and multi‐objective optimisation

This study investigates the parametric approach for a type of descriptor quasi-linear systems by utilising dynamic compensator and multi-objective optimisation. Based on the solutions of generalised Sylvester matrix equation, the generally parameterised expressions of dynamic compensator and the left and right eigenvector matrices are both established, meanwhile, a group of arbitrary parameters are obtained. With the parametric approach, the closed-loop system can be transformed into a linear time-invariant one with an expected eigenstructure by using a group of canonical matrix pairs. Simultaneously, it also presents a novel technique to design multi-objective optimisation for descriptor quasi-linear systems. Multiple performance indexes such as low sensitivity, disturbance attenuation, robustness degree, and low gains are formulated by arbitrary parameters. Based on the above indexes, robustness criteria and low gain criteria can be expressed by a synthetic objective function which includes each performance index weighted. By utilising the degrees of freedom in arbitrary parameters, a dynamic compensator can be obtained by solving a multi-objective optimisation problem. Finally, two examples are proposed to prove that the parametric approach is effective.

Algorithms for Solving Nonhomogeneous Generalized Sylvester Matrix Equations

This paper considers a new method to solve the first-order and second-order nonhomogeneous generalized Sylvester matrix equations AV+BW= EVF+R and MVF2+DV F+KV=BW+R, respectively, where A,E,M,D,K,B, and F are the arbitrary real known matrices and V and W are the matrices to be determined. An explicit solution for these equations is proposed, based on the orthogonal reduction of the matrix F to an upper Hessenberg form H. The technique is very simple and does not require the eigenvalues of matrix F to be known. The proposed method is illustrated by numerical examples.

Robust model reference control for uncertain second-order system subject to parameter uncertainties

This paper is devoted to designing a robust model reference controller for uncertain second-order systems subject to parameter uncertainties. The system matrix of the first-order reference model is more general and the parameter uncertainties are assumed to be norm-bounded. The design of robust controller can be devided into two separate problems: problem robust stabilization and problem robust compensation. Based on the solution of generalized Sylvester matrix equations, we obtain some sufficient conditions to guarantee the complete parameterization of the controller. Then, the problem robust compensation of the closed-loop system is estimated by solving a convex optimisation problem with a set of linear matrix equations constraints. Two simulation examples are provided to illustrate the effectiveness of the proposed technique.

Model Reference Control for Linear Time-Varying Systems: A Direct Parametric Approach

This paper investigates model reference control for linear time-varying (LTV) systems. It is shown that the problem can be decomposed into two sub-problems: a state feedback stabilization problem of LTV systems and a feed-forward compensation problem. Firstly, the sufficient condition for the existence of the model reference control is deduced. The condition concerns two coefficient matrices such that two matrix equations are met simultaneously, and the state feedback gain matrix stabilizes LTV systems according to the unified spectral theory. Secondly, generally complete parametrization of the model reference tracking controller including a state feedback controller and a feed-forward compensator is proposed based on the solution to a type of generalized Sylvester matrix equations. Furthermore, a general procedure for solving the model reference control problem is also provided. Finally, a numerical example shows the feasibility and effectiveness of the parametric design approach to the model reference control in LTV systems.