Matrix form of Biconjugate Residual Algorithm to Solve the Discrete‐Time Periodic Sylvester Matrix Equations

Hajarian, Masoud

doi:10.1002/asjc.1528

Cited by 14 publications

(4 citation statements)

References 29 publications

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“…The Lyapunov equations and the related matrix equations play an indispensable role in numerous areas of control theory such as stability analysis, optimal control, and model order reduction [28, 29]. These equations are also an integral part in power systems control and signal processing.…”

Section: Model Reduction Strategiesmentioning

confidence: 99%

Comparative analysis of model reduction strategies for circuit based periodic control problems

Hossain

Tahsin

Omar

et al. 2020

Asian Journal of Control

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This paper is a comparative analysis of two prominent iterative algorithms for model order reduction of linear time‐varying (LTV) periodic systems where the system's matrices are singular. Our proposed method is based on a reformulation of the LTV model to an equivalent linear time‐invariant (LTI) model using a suitable discretization procedure. The resulting LTI model is reduced in two ways, once by applying a balanced truncation method and once by applying a Krylov‐based method known as iterative rational Krylov algorithm (IRKA). During the application of balanced truncation, the low‐rank Cholesky factorized alternating directions implicit (LRCF‐ADI) method is used to estimate the solutions of the corresponding LTI form of Lyapunov equations. Since the system's matrices are singular, the concept of pseudo‐inverse is adopted to compute the shift parameters needed in the LRCF‐ADI iterations. For the Krylov‐based IRKA, our work is twofold. We solve the time‐invariant Lyapunov equation for the observability Gramian and apply a moment‐matching Krylov technique. The accuracy and effectiveness of the two proposed techniques are demonstrated with the help of frequency response graphs, bode plots, and eigenstructure of the main and reduced models.

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Section: Model Reduction Strategiesmentioning

confidence: 99%

Comparative analysis of model reduction strategies for circuit based periodic control problems

Hossain

Tahsin

Omar

et al. 2020

Asian Journal of Control

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“…The extending of nonsymmetric iterative methods has become a popular and powerful method for solving linear matrix equations and inverse eigenvalue problems [27,29,30,36,37,[45][46][47]. In the current study, we propose an extended form of the above algorithm to solve Problem 2.…”

Section: Hs Version Of Bcr Algorithm For Solving MX = B Initial Valuementioning

confidence: 99%

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

Hajarian

2018

Asian Journal of Control

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The inverse eigenvalue problem appears repeatedly in a variety of applications. The aim of this paper is to study a quadratic inverse eigenvalue problem of the form AXΛ2 + BXΛ + CX = 0 where A, B and C should be partially bisymmetric under a prescribed submatrix constraint. We derive an efficient matrix method based on the Hestenes‐Stiefel (HS) version of biconjugate residual (BCR) algorithm for solving this constrained quadratic inverse eigenvalue problem. The theoretical results demonstrate that the matrix method solves the constrained quadratic inverse eigenvalue problem within a finite number of iterations in the absence of round‐off errors. Finally we validate the accuracy and efficiency of the matrix method through the numerical results.

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“…In reference [6], an iterative algorithm based on conjugate gradient was developed to solve the DPLME. Moreover, many iterative algorithms for solving other matrix equations can also be used to solve the DPLME [7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Robust gradient‐based neural networks for solving online the discrete periodic Lyapunov matrix equations

Yin,

Zhang

2023

IET Control Theory & Appl

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Here, a gradient‐based neural network (GNN) model is constructed for solving the discrete periodic Lyapunov matrix equation (DPLME) associated with discrete‐time linear periodic systems. In practical applications, the recurrent neural network model should not only converge rapidly, but also be able to tolerate noise. However, the influence of noise on GNN models was seldom considered in the past. In order to improve the convergence and robustness of the GNN model, a novel type of non‐linear activation function is applied to the GNN model. Compared with the traditional activation functions, the activation function used here makes the GNN model to achieve fixed‐time convergence. Besides, when disturbed by bounded noise, the unique positive definite solution of the DPLME can still be obtained by using the GNN model. Finally, simulation experiment is performed to verify the effectiveness and superiority of the proposed GNN model.

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Matrix form of Biconjugate Residual Algorithm to Solve the Discrete‐Time Periodic Sylvester Matrix Equations

Cited by 14 publications

References 29 publications

Comparative analysis of model reduction strategies for circuit based periodic control problems

Comparative analysis of model reduction strategies for circuit based periodic control problems

BCR Algorithm for Solving Quadratic Inverse Eigenvalue Problems for Partially Bisymmetric Matrices

Robust gradient‐based neural networks for solving online the discrete periodic Lyapunov matrix equations

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