Periodic Matrix Equations Research Articles

Convergence analysis of generalized conjugate direction method to solve general coupled Sylvester discrete‐time periodic matrix equations

SummaryThe discrete‐time periodic matrix equations have been extensively applied as a main tool of analysis and design for periodic systems. This study is intended to provide a generalization of conjugate direction method for solving the general coupled Sylvester discrete‐time periodic matrix equations urn:x-wiley:acs:media:acs2742:acs2742-math-0001 The convergence theorem guarantees that the generalized conjugate direction method can compute the solutions of the general coupled Sylvester discrete‐time periodic matrix equations after a finite number of iterations in the absence of round‐off errors. Finally, numerical results are given to exhibit the accuracy and efficiency of the generalized conjugate direction method. Copyright © 2016 John Wiley & Sons, Ltd.

Finding solutions for periodic discrete-time generalized coupled Sylvester matrix equations via the generalized BCR method

The periodic discrete-time matrix equations have wide applications in stability theory, control theory and perturbation analysis. In this work, the biconjugate residual algorithm is generalized to construct a matrix iterative method to solve the periodic discrete-time generalized coupled Sylvester matrix equations [Formula: see text] The constructed method is shown to be convergent in a finite number of iterations in the absence of round-off errors. By comparing with other similar methods in practical computation, we give numerical results to demonstrate the accuracy and the numerical superiority of the constructed method.

Current‐estimation‐based iterative algorithms for solving periodic Lyapunov matrix equations

In this study, two novel iterative algorithms are presented to solve the Lyapunov matrix equations appearing in discrete-time periodic linear systems. In both algorithms, a weighted combination of the estimation in the last and the current steps is used to update the estimation of the unknown matrices. It is shown that the sequences generated by the proposed algorithms with zero initial conditions monotonically converge to the unique positive definite solution of the periodic Lyapunov matrix equation if the associated system is asymptotically stable. Finally, a numerical example is used to illustrate the effectiveness of the proposed algorithms.

GRADIENT BASED ITERATIVE ALGORITHM TO SOLVE GENERAL COUPLED DISCRETE-TIME PERIODIC MATRIX EQUATIONS OVER GENERALIZED REFLEXIVE MATRICES

The discrete-time periodic matrix equations are encountered in periodic state feedback problems and model reduction of periodic descriptor systems. The aim of this paper is to compute the generalized reflexive solutions of the general coupled discrete-time periodic matrix equations. We introduce a gradient-based iterative (GI) algorithm for finding the generalized reflexive solutions of the general coupled discretetime periodic matrix equations. It is shown that the introduced GI algorithm always converges to the generalized reflexive solutions for any initial generalized reflexive matrices. Finally, two numerical examples are investigated to confirm the efficiency of GI algorithm.

Extending the CGLS method for finding the least squares solutions of general discrete-time periodic matrix equations

The periodic matrix equations are strongly related to analysis of periodic control systems for various engineering and mechanical problems. In this work, a matrix form of the conjugate gradient for least squares (MCGLS) method is constructed for obtaining the least squares solutions of the general discrete-time periodic matrix equations ?t,j=1 (Ai,jXi,jBi,j + Ci,jXi+1,jDi,j)=Mi, i=1,2,.... It is shown that the MCGLS method converges smoothly in a finite number of steps in the absence of round-off errors. Finally two numerical examples show that the MCGLS method is efficient.

Solving the general Sylvester discrete-time periodic matrix equations via the gradient based iterative method

The present work proposes a gradient based iterative method to find the solutions of the general Sylvester discrete-time periodic matrix equations ∑j=1m(AijXiBij+CijXi+1Dij+EijYiFij+GijYi+1Hij)=Mi,i=1,2,…. It is proven that the proposed iterative method can obtain the solutions of the periodic matrix equations for any initial matrices. Finally a numerical example is included to demonstrate the validity and applicability of the iterative method.

Parametric solutions to the discrete periodic regul ator equations

Forward-time discrete periodic regulator matrix equations (FDPRE) and reverse-time discrete periodic regulator matrix equations (RDPRE) are firstly studied. The controllability and observability Gramians related with the lifting time variant system are utilized to constitute solutions. Parametric and explicit solutions to these two equations are presented in the form of iteration. The proposed method can provide sufficient design degree of freedom, since there are infinite solutions to be obtained. A detailed algorithm is also presented and its correctness is checked by a numerical example.

A Finite Iterative Method for Solving the General Coupled Discrete-Time Periodic Matrix Equations

Analysis and design of linear periodic control systems are closely related to the discrete-time periodic matrix equations. In this paper, we propose an iterative algorithm based on the conjugate gradient method on the normal equations (CGNE) for finding the solution group of the general coupled periodic matrix equations $$\begin{aligned} \left\{ \begin{array}{l} A_{1,i}X_iB_{1,i}+C_{1,i}X_{i+1}D_{1,i}=E_{1,i},\\ A_{2,i}X_iB_{2,i}+C_{2,i}X_{i+1}D_{2,i}=E_{2,i}, \end{array} \right. ~~~\mathrm {for}~~~i=1,2,3,\ldots . \end{aligned}$$ A 1 , i X i B 1 , i + C 1 , i X i + 1 D 1 , i = E 1 , i , A 2 , i X i B 2 , i + C 2 , i X i + 1 D 2 , i = E 2 , i , for i = 1 , 2 , 3 , ? . By proving some properties of the algorithm, we show that the solution group of the periodic matrix equations can be obtained within a finite number of iterations in the absence of roundoff errors. Numerical examples are given to illustrate the efficiency and accuracy of the proposed algorithm.

Developing CGNE algorithm for the periodic discrete-time generalized coupled Sylvester matrix equations

The discrete-time periodic matrix equations often arise in analysis and design of linear periodic control systems. This paper studies the problem of finding the solutions of the periodic continuous-time generalized coupled Sylvester matrix equations $$\begin{aligned} \left\{ \begin{array}{ll} A_kX_kB_k+C_kY_kD_k=M_k,&{} \hbox {} \\ E_kX_{k+1}F_k+G_kY_kH_k=N_k,&{} \hbox {} \end{array} \right. \quad {\mathrm {for}} \quad k=1,2,\ldots ,\phi . \end{aligned}$$ We propose an iterative method based on the idea of conjugate gradient method on the normal equations (CGNE) for solving this problem. By proving some properties of the iterative method, it is shown that the iterative method can solve the problem within a finite number of iterations in the absence of roundoff errors. Finally, numerical results are carried out to confirm the theoretical conclusions.

Solving periodic Lyapunov matrix equations via finite steps iteration

Periodic Lyapunov matrix equations play important roles in stability analysis and stabilisation of discrete-time periodic systems. In this paper, an iterative algorithm for solving periodic Lyapunov matrix equations is established. It is shown that the proposed iteration converges to the unique solution of the considered matrix equations at finite steps with arbitrary initial condition despite the stability of the associated discrete-time periodic linear systems. Both the reverse-time and forward-time discrete-time periodic Lyapunov matrix equations are considered. Numerical examples are worked out to illustrate the effectiveness of the proposed algorithm.