Sylvester Matrix Equations Research Articles

Solvability of the Sylvester equation $ AX-XB = C $ under left semi-tensor product

<abstract><p>This paper investigates the solvability of the Sylvester matrix equation $ AX-XB = C $ with respect to left semi-tensor product. Firstly, we discuss the matrix-vector equation $ AX-XB = C $ under semi-tensor product. A necessary and sufficient condition for the solvability of the matrix-vector equation and specific solving methods are studied and given. Based on this, the solvability of the matrix equation $ AX-XB = C $ under left semi-tensor product is discussed. Finally, several examples are presented to illustrate the efficiency of the results.</p></abstract>

An iterative algorithm for the generalized centro-symmetric solution of the generalized coupled Sylvester matrix equations AV+BW=EVF+C, MV+NW=GVH+D

In this paper, an iterative algorithm for solving of the generalized coupled Sylvester matrix equations AV+BW=EVF+C, MV+NW=GVH+D over the generalized centro-symmetric matrices (V,W ) is proposed. For any initial generalized centro-symmetric matrices V0 and W0, a generalized centro-symmetric solution (V,W ) is obtained within a finite number of iterations in the absence of round-off errors. Two numerical examples are presented to support the theoretical results where the efficiency and accuracy of the suggested algorithm are shown.

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.

Lopsided DSS iteration method for solving complex Sylvester matrix equation

Lopsided DSS iteration method for solving complex Sylvester matrix equation

High-order fully-actuated system approaches: Part IX. Generalised PID control and model reference tracking

Tracking control of general dynamical systems in high-order fully actuated (HOFA) system representation is solved. For the case of tracking a constant or slow time-varying signal in the presence of constant or slow time-varying disturbances, a generalised PID control scheme is proposed, which realises asymptotical tracking to a prescribed signal and also guarantees that the state derivatives of certain orders converge to the origin. For the case that the signal to be tracked is generated by a reference model, a model reference tracking (MRT) controller is presented, which relies on the solution to a type of generalised Sylvester matrix equations and guarantees the desired asymptotical tracking requirement. Due to the full-actuation property of the HOFA models, closed-loop systems under both control schemes are constant and linear. Furthermore, based on a general parametric solution to the type of Sylvester matrix equations and a general parametric eigenstructure assignment result, simple and complete parameterisation of the two types of control designs are provided, and feasibility conditions in terms of the system initial values for sub-fully actuated systems are also derived. An illustrative example is presented to demonstrate the application of the proposed approaches and their effects.

Minimum norm partial eigenstructure assignment problems in high‐order system via feedback control

AbstractIn this article, the minimum norm partial eigenstructure assignment problem (MNPESAP) is considered in a symmetric high‐order system using feedback control law with the highest‐order derivative term. Two new orthogonality relations are constructed using the system matrices and undetermined feedback gains. The parametric expressions of the feedback controller and the necessary and sufficient condition for solving the partial eigenstructure assignment problem are derived. The Sylvester matrix equations and gradient formula are presented for solving MNPESAP. An algorithm is proposed to obtain the minimum norm solutions of feedback gains using gradient‐based optimization method. Numerical examples are provided to verify the effectiveness of given results.

Iterative algorithms for discrete-time periodic Sylvester matrix equations and its application in antilinear periodic system

This paper is dedicated to solving the iterative solution to the discrete-time periodic Sylvester matrix equations. Inspired by Jacobi iterative algorithm and hierarchical identification principle, the Jacobi gradient based iterative (JGI) algorithm and the accelerated Jacobi gradient based iterative (AJGI) algorithm are proposed. It is verified that the proposed algorithms are convergent for any initial matrix when the parameter factor μ satisfies certain condition. The necessary and sufficient conditions are provided for the presented new algorithms. Moreover, we also apply the JGI algorithm and AJGI algorithm to study a more generalized discrete-time periodic matrix equations and give the convergence conditions of the algorithms. Finally, two numerical examples are given to illustrate the effectiveness, accuracy and superiority of the proposed algorithms.

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.

Fast second-order implicit difference schemes for time distributed-order and Riesz space fractional diffusion-wave equations

In this paper, fast numerical methods are established to solve a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by a weighted and shifted Grünwald formula in time and a fractional centered difference formula in space. The unconditional stability and second-order convergence in time, space and distributed-order of the difference schemes are analyzed. In the one-dimensional case, the Gohberg-Semencul formula utilizing a preconditioned Krylov subspace method is developed to solve the Toeplitz linear system derived from the proposed difference scheme. In the two-dimensional case, we also design a global preconditioned conjugate gradient method with a truncated preconditioner to solve the resulting Sylvester matrix equations. We prove that the spectrums of the preconditioned matrices in both cases are clustered around 1, such that the proposed numerical methods with preconditioners converge very quickly. Some numerical experiments are carried out to demonstrate the effectiveness of the proposed difference schemes and show that the performances of the proposed fast solution algorithms are better than other testing methods.

A global variant of the COCR method for the complex symmetric Sylvester matrix equation AX + XB = C

Complex symmetric Sylvester matrix equations appear in many applications, such as the numerical solution of the complex Helmholtz equations. In this paper, by designing a global complex symmetric M-Lanczos process we develop a global variant of the conjugate A-orthogonal conjugate residual method (Gl-COCR) for solving the Sylvester matrix equation AX+XB=C with complex symmetric coefficient matrices. To obtain the smooth and monotone convergence behavior, we also propose a smoothed Gl-COCR method, denoted by SGl-COCR. Finally, numerical examples are given to illustrate the performances of our methods.

Conjugate gradient-like algorithms for constrained operator equation related to quadratic inverse eigenvalue problems

As is well known, linear operator equations have wide applications in many areas of engineering and applied mathematics. In the present paper, we are interested in solving the linear operator equation $$\begin{aligned} {\mathscr {F}}(J)+{\mathscr {G}}(K)+{\mathscr {H}}(L)=N, \end{aligned}$$ where J, K and L should be partially bisymmetric under a prescribed submatrix constraint. Three conjugate gradient-like algorithms are derived for solving this constrained operator equation including the Lyapunov, Stein and Sylvester matrix equations and the quadratic inverse eigenvalue problem as special cases. The algorithms converge to the solutions of the linear operator equation within a finite number of iterations in the absence of round-off errors. At the end, the accuracy and efficiency of the introduced algorithms are demonstrated numerically with three examples.

Sign function and ANN based pole placement for computing interval controls

This paper deals with estimation of interval controls for interval Linear Time Invariant (LTI) plants using pole placement technique. Generally, the parameters associated with LTI plant system are assumed to be crisp values, but due to the occurrence of error in experimentation, measurement or due to maintenance induced errors, the parameters are imprecise. As such, the LTI problem reduces to interval LTI. The pole placement problem is then reduced to interval algebraic equation viz. interval Diophantine equation using proper transfer function. Further, the interval Diophantine equation is transformed to Interval System of Linear Equations (ISLE) viz. interval Sylvester matrix equation. Initially, using interval arithmetic an interval algebraic approach has been applied to solve the ISLE for computing non-negative or non-positive controls. Then, another approach using Artificial Neural Network (ANN) is proposed for computing the interval controls. Further, algorithms based on sign function and ANN procedures have been discussed. Finally, the efficiency of the proposed algorithms has been verified using different order interval plants.

A relaxed MSIO iteration algorithm for solving coupled discrete Markovian jump Lyapunov equations

In this paper, combining the multi-step Smith-inner-outer (MSIO) iteration framework with some tunable parameters, a relaxed MSIO iteration method is proposed for solving the Sylvester matrix equation and coupled Lyapunov matrix equations (CLMEs) in the discrete-time jump linear systems with Markovian transitions. The convergence properties of the relaxed MSIO iteration method are investigated, and the choices of the parameters are also discussed. In order to accelerate the convergence rate of the relaxed MSIO iteration method for solving the CLMEs, a current-estimation-based and a weighted relaxed MSIO iteration algorithms are presented, respectively. Finally, several numerical examples are given to verify the superiorities of the proposed relaxed algorithms.

An extension method for fully fuzzy Sylvester matrix equation

An extension method is proposed to solve a class of fully fuzzy Sylvester matrix equation (FFSME) under some mild assumptions. This method consists of two steps. Firstly, the fully fuzzy system is transferred into a series of interval Sylvester matrix equations through $$\alpha $$ -cut. Secondly, these interval systems are extended into crisp systems which is easy to be solved. The solutions of FFSME are not presumed to be triangular-type fuzzy numbers. Moreover, some examples are presented to show the validity of our method.

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.

An SOR-type algorithm based on IO iteration for solving coupled discrete Markovian jump Lyapunov equations

In this paper, based on the inner-outer (IO) iteration framework [17], by introducing some tunable parameters, an SOR-type IO (SIO) iteration method is proposed for solving the Sylvester matrix equation and coupled Lyapunov matrix equations (CLMEs) in the discrete-time jump linear systems with Markovian transitions. Fisrtly, the SIO iteration algorithm for solving the discrete Sylvester matrix equation is developed, its convergence property is analyzed and the choices of the parameters are also discussed. Next, the SIO iteration algorithm is used to solve the CLMEs. Moreover, by using the latest estimations, a current-estimation-based SIO (CSIO) iteration algorithms are also constructed for solving the CLMEs, respectively. The boundedness and monotonicity of the iteration sequence derived from the proposed algorithm with zero initial conditions are established. Finally, several numerical examples are implemented to illustrate the superiorities of the proposed iteration algorithms.

Constrainted least squares solution of Sylvester equation

<abstract> In this paper, we study several constrainted least squares solutions of quaternion Sylvester matrix equation. We first propose a real vector representation of quaternion matrix and study its properties. By using this real vector representation, semi-tensor product of matrices, swap matrix and Moore-Penrose inverse, we derive compatible conditions and the expressions of several constrainted least squares solutions of quaternion Sylvester equation. </abstract>

On the CRI method for solving Sylvester equation with complex symmetric positive semi-definite coefficient matrices

Combination of real and imaginary parts (CRI) method is an efficient method for solving a class of large sparse linear systems with complex symmetric positive semi-definite coefficient matrices. In this work we will extend CRI approach to determine the approximate solution of Sylvester equation with complex symmetric semi-definite positive coefficient matrices. We show that the new algorithm converges unconditionally to the unique exact solution of the Sylvester matrix equation. In the end we test the new scheme by solving some numerical examples.