Global quasi-minimal residual method for the Sylvester equations

In this paper, a global quasi-minimal residual （QMR） method was presented for solving the Sylvester equations. Some properties were investigated with a new matrix product for the global QMR method. Numerical results with the global QMR and GMRES methods compared with the block GMRES method were given. The results show that the global QMR method is less time-consuming than the global GMRES （generalized minimal residual） and block GMRES methods in some cases.

ON PMHSS ITERATION METHODS FOR CONTINUOUS SYLVESTER EQUATIONS

The modified Hermitian and skew-Hermitian splitting （MHSS） iteration method and preconditioned MHSS （PMHSS） iteration method were introduced respectively. In the paper, on the basis of the MHSS iteration method, we present a PMHSS iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and complex symmetric positive definite/semi-definite matrices. Under suitable conditions, we prove the convergence of the PMHSS iteration method and discuss the spectral properties of the preconditioned matrix. Moreover, to reduce the computing cost, we establish an inexact variant of the PMHSS iteration method and analyze its convergence property in detail. Numerical results show that the PMHSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.

Distributed solving Sylvester equations with fractional order dynamics

This paper addresses distributed computation Sylvester equations of the form AX+XB=C with fractional order dynamics.By partitioning parameter matrices A,B and C,we transfer the problem of distributed solving Sylvester equations as two distributed optimization models and design two fractional order continuous-time algorithms,which have more design freedom and have potential to obtain better convergence performance than that of existing first order algorithms.Then,rewriting distributed algorithms as corresponding frequency distributed models,we design Lyapunov functions and prove that proposed algorithms asymptotically converge to an exact or least squares solution.Finally,we validate the effectiveness of proposed algorithms by providing a numerical example.

Solutions to the generalized Sylvester matrixequations by a singular value decomposition

In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × （n ＋ pr）. The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair （A, B） and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.

A direct method for solving projected generalized continuous-time Sylvester equations

This article is devoted to the numerical solution of a projected generalized Sylvester equation with relatively small size. Such an equation arises in stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. The algebraic formula of the solution of the projected generalized continuous-time Sylvester equation is presented. A direct method based on the generalized Schur factorization is proposed. Moreover, its low-rank version for problems with low-rank right-hand sides is also proposed. The computational cost of the direct method is estimated. Numerical simulation show that this direct method has high accurncv

Parameterized Solution to a Class of Sylvester Matrix Equations

A class of formulas for converting linear matrix mappings into conventional linear mappings are presented. Using them, an easily computable numerical method for complete parameterized solutions of the Sylvester matrix equation AX - EXF = BY and its dual equation XA - FXE = YC are provided. It is also shown that the results obtained can be used easily for observer design. The method proposed in this paper is universally applicable to linear matrix equations.

A note on combined generalized Sylvester matrix equations

The solution of two combined generalized Sylvester matrix equations is studied. It is first shown that the two combined generalized Sylvester matrix equations can be converted into a normal Sylvester matrix equation through extension, and then with the help of a result for solution to normal Sylvester matrix equations, the complete solution to the two combined generalized Sylvester matrix equations is derived. A demonstrative example shows the effect of the proposed approach.

An iterative algorithm for solving a class of matrix equations

In this paper, an iterative algorithm is presented to solve the Sylvester and Lyapunov matrix equations. By this iterative algorithm, for any initial matrix X1, a solution X＊ can be obtained within finite iteration steps in the absence of roundoff errors. Some examples illustrate that this algorithm is very efficient and better than that of [ 1 ] and [2].

Solving the generalized Sylvester matrix equation AV+BW=VF via Kronecker map

This note considers the solution to the generalized Sylvester matrix equation AV ＋ BW = VF with F being an arbitrary matrix, where V and W are the matrices to be determined. With the help of the Kronecker map, an explicit parametric solution to this matrix equation is established. The proposed solution possesses a very simple and neat form, and allows the matrix F to be undetermined.

Cauchy matrix approach to three non-isospectral nonlinear Schrödinger equations

This paper aims to develop a direct approach,namely,the Cauchy matrix approach,to non-isospectral integrable systems.In the Cauchy matrix approach,the Sylvester equation plays a central role,which defines a dressed Cauchy matrix to provideτfunctions for the investigated equations.In this paper,using the Cauchy matrix approach,we derive three non-isospectral nonlinear Schrödinger equations and their explicit solutions.These equations are generically related to the time-dependent spectral parameter in the Zakharov–Shabat–Ablowitz–Kaup–Newell–Segur spectral problem.Their solutions are obtained from the solutions of unreduced non-isospectral nonlinear Schrödinger equations through complex reduction.These solutions are analyzed and illustrated to show the non-isospectral effects in dynamics of solitons.

A Minimum Residual Based Gradient Iterative Method for a Class of Matrix Equations

In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including Sylvester matrix equations and general coupled matrix equations. The iterative method uses a negative gradient as steepest direction and seeks for an optimal step size to minimize the residual norm of next iterate. It is shown that the iterative sequence converges unconditionally to the exact solution for any initial guess and that the norm of the residual matrix and error matrix decrease monotonically. Numerical tests are presented to show the efficiency of the proposed method and confirm the theoretical results.

A Sylvester-Based IMEXMethod via Differentiation Matrices for Solving Nonlinear Parabolic Equations

In this paper we describe a new pseudo-spectral method to solve numerically two and three-dimensional nonlinear diffusion equations over unbounded domains,taking Hermite functions,sinc functions,and rational Chebyshev polynomials as basis functions.The idea is to discretize the equations by means of differentiation matrices and to relate them to Sylvester-type equations by means of a fourth-order implicit-explicit scheme,being of particular interest the treatment of three-dimensional Sylvester equations that we make.The resulting method is easy to understand and express,and can be implemented in a transparent way by means of a few lines of code.We test numerically the three choices of basis functions,showing the convenience of this new approach,especially when rational Chebyshev polynomials are considered.

ON HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR CONTINUOUS SYLVESTER EQUATIONS

We present a Hermitian and skew-Herrnitian splitting （HSS） iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semi- definite matrices. The unconditional convergence of the HSS iteration method is proved and an upper bound on the convergence rate is derived. Moreover, to reduce the computing cost, we establish an inexact variant of the HSS iteration method and analyze its convergence property in detail. Numerical results show that the HSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.

A projection method and Kronecker product preconditioner for solving Sylvester tensor equations

The preconditioned iterative solvers for solving Sylvester tensor equations are considered in this paper.By fully exploiting the structure of the tensor equation,we propose a projection method based on the tensor format,which needs less flops and storage than the standard projection method.The structure of the coefficient matrices of the tensor equation is used to design the nearest Kronecker product(NKP) preconditioner,which is easy to construct and is able to accelerate the convergence of the iterative solver.Numerical experiments are presented to show good performance of the approaches.

An explicit solution to right factorization with applicationin eigenstructure assignment

Based on the well-known Leverrier algorithm, a simple explicit solution to right factorization of a linear system is established. This solution is expressed by the controllability matrix of the given system and a symmetric operator matrix. Applications of this solution to a type of generalized Sylvester matrix equatiorls and the problem of parametric eigenstructure assignment by state feedback are investigated,and general complete parametric solutions to these two problems are deduced. These new solutions are simple, and possess desirable structural properties which render the solutions readily implementable. An example demonstrates the effect of the proposed results.

Developing Bi-CG and Bi-CR Methods to Solve Generalized Sylvester-transpose Matrix Equations

The bi-conjugate gradients(Bi-CG)and bi-conjugate residual(Bi-CR)methods are powerful tools for solving nonsymmetric linear systems Ax=b.By using Kronecker product and vectorization operator,this paper develops the Bi-CG and Bi-CR methods for the solution of the generalized Sylvester-transpose matrix equationp i=1(Ai X Bi+Ci XTDi)=E(including Lyapunov,Sylvester and Sylvester-transpose matrix equations as special cases).Numerical results validate that the proposed algorithms are much more efcient than some existing algorithms.

Parametric Approach for the Normal Luenberger Function Observer Design in Second-order Descriptor Linear Systems

In this paper, the normal Luenberger function observer design for second-order descriptor linear systems is considered. It is shown that the main procedure of the design is to solve a so-called second-order generalized Sylvester-observer matrix equation. Based on an explicit parametric solution to this equation, a parametric solution to the normal Luenberger function observer design problem is given. The design degrees of freedom presented by explicit parameters can be further utilized to achieve some additional design requirements.

A System of Periodic Discrete-time Coupled Sylvester Quaternion Matrix Equations

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations AkXk ＋ YkBk = Mk, CkXk＋l ＋ YkDk ---- Nk （k = 1, 2） over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.

Observer-based event-triggered networked multi-drives speed consensus

The problem of having an identical speed when dealing with multiple motors always exists in the industry.There are several methods to address the problem,but all the methodologies have two common drawbacks.Firstly,the control design requires continuous information on the desired speed and the actual speed of motors;secondly,it is sometimes difficult to directly measure the speed variables.In the proposed study,both of these drawbacks are addressed by designing an observer-based event-triggered networked multi-agent system.The proposed method uses the leader following consensus approach with a centralized event triggering control design so that whenever a follower's speed diverges from that of the leader,an event is triggered,which communicates and resets all the agents to the leader's speed.Moreover,an observer is designed such that the ith agent uses its jth neighbor agent and observer speed information to estimate the leader's speed.The stability of the proposed design is formulated by Lyapunov stability,while the simulation results endorse the design concepts and energy saving.

An explicit solution to the matrix equation AV+BW=EVJ

In this note, the matrix equation AV ＋ BW = EVJ is considered, where E, A and B are given matrices of appropriate dimensions, J is an arbitrarily given Jordan matrix, V and W are the matrices to be determined. Firstly, a right factorization of （sE - A）^-1 B is given based on the Leverriver algorithm for descriptor systems. Then based on this factorization and a proposed parametric solution, an alternative parametric solution to this matrix equation is established in terms of the R-controllability matrix of （E, A, B）, the generalized symmetric operator and the observability matrix associated with the Jordan matrix d and a free parameter matrix. The proposed results provide great convenience for many analysis and design problems. Moreover, some equivalent forms are proposed. A numerical example is employed to illustrate the effect of the proposed approach.