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This paper considers model order reduction based on low-rank decomposition of the cross Gramian for linear systems. The proposed approach uses the low-rank factors of the cross Gramian to generate approximate balanced model for the large-scale system. Then, the reduced-order model is obtained by truncating the states corresponding to the small approximate Hankel singular values. The low-rank factors are directly constructed from the expansion coefficients of impulse responses in the space spanned by Legendre polynomials by solving block tridiagonal linear systems. In contrast to balanced truncation related approaches which require to solve Sylvester equation to compute the full cross Gramian, the proposed method needs to solve sparse linear equations, and only one singular value decomposition is applied to a low-dimensional matrix. The stability preservation of the reduced model is briefly discussed. And in combination with the dominant subspace projection method, the reduction procedure is modified to alleviate the shortcoming, which may unexpectedly lead to unstable systems even though the original one is stable. Finally, numerical experiments are given to demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the topic of model order reduction (MOR) for coupled systems. At first, an approximation via Laguerre polynomials expansions to controllability and observability gramians for such systems are presented, which provides a low-rank decomposition form whose factors are constructed from a recurrence formula instead of Lyapunov equations. Then, in combination of balanced truncation and dominant subspace projection method, a series of MOR algorithms are proposed that preserve the coupled structures. What's more, some main properties of reduced-order models, such as stability preservation, are well discussed. Finally, three numerical simulations are provided to illustrate the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a series of structure-preserving model order reduction algorithms for K-power bilinear systems via Laguerre functions. The method first aims to rewritten the K-power bilinear system as a general bilinear system and calculate the approximate low-rank factors of the cross Gramian of the bilinear system by combining the idea of Laguerre functions expansion of the matrix exponential function. After that, the approximate balanced system of the K-power bilinear system is constructed by the corresponding projection transformation of each subsystem. In order to achieve the purpose of model order reduction, the states with smaller singular values are then truncated, so as to further obtain the reduced order model. For this approach, there is a disadvantage that unstable systems may be generated although the original one is stable. To alleviate the inadequacies of this approach, we have improved the model reduction procedure, which is based upon the dominant subspace projection method. In addition, we also carried out a correlation analysis on the stability of improved algorithms. Finally, numerical experiments are employed to substantiate the effectiveness of the presented algorithms. [ABSTRACT FROM AUTHOR]

The cross Gramian is a useful tool in model order reduction but only applicable to square dynamical systems. Throughout this paper, time-limited cross Gramians is firstly extended to square bilinear systems that satisfies a generalized Sylvester equation, and then concepts from decentralized control are used to approximate a cross Gramian for non-square bilinear systems. In order to illustrate these cross Gramians, they are calculated efficiently based on shifted Legendre polynomials and applied to dimension reduction, which leads to a lower dimensional model by truncating the states that are associated with smaller approximate generalized Hankel singular values. In combination of the dominant subspace projection method, our reduction procedure is modified to produce a bounded-input bounded-output stable-preserved reduced model under some certain conditions. At last, the performance of numerical experiments indicates the validity of our reduction methods. [ABSTRACT FROM AUTHOR]

• A model order reduction algorithm based on low-rank Gramian approximation for linear systems is presented. • The method computes the low-rank factors using a recurrence formula, which makes it computationally efficient. • The above approach is equivalent to the Cholesky factor alternating direction implicit method under certain conditions • An effective strategy is given to modify the above approach to produce a stable reduced model under certain conditions. • The algorithms are extended to bilinear systems successfully and a series of model reduction algorithms are derived. In this paper, we propose a series of model order reduction algorithms based on low-rank Gramian approximation for linear and bilinear systems. The main idea of the approach for linear systems is to use approximate low-rank factors of the controllability and observability Gramians to generate approximate balanced system for the large-scale system. Then, the reduced-order models are obtained by truncating the states corresponding to the smaller approximate Hankel singular values. The low-rank factors are constructed directly from the Laguerre functions expansion coefficient vectors of the matrix exponential functions by solving a recurrence formula instead of Lyapunov equations. In addition, the reduction procedure is modified with the idea of dominant subspace projection method to produce a stable reduced model under certain conditions. Furthermore, our algorithms are extended to bilinear systems successfully, with a series of corresponding algorithms for bilinear systems derived. Finally, numerical experiments are provided to demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

A new structure-preserving model order reduction technique based on Laguerre-Gramian for second-order form systems is presented in this article. The main task of the proposed approach is to use the Laguerre polynomial expansion of the matrix exponential function to obtain the approximate low-rank decomposition of the Gramians for the equivalent first-order representation of the original second-order form system. The approximate balanced system is generated by a balancing transformation which is directly computed from the expansion coefficients of impulse responses in the space spanned by Laguerre polynomials, without computing the full Gramians for the first-order representation. Then, the reduced second-order model is constructed by truncating the states with small approximate Hankel singular values (HSVs). The above method has a disadvantage that it may unexpectedly result in unstable systems although the original one is stable. Therefore, modified reduction procedure combined with the dominant subspace projection method is presented to alleviate the limitation. Finally, two numerical experiments are provided to demonstrate the effectiveness of the algorithms. [ABSTRACT FROM AUTHOR]

This study focuses on the topic of model order reduction (MOR) for coupled systems with inhomogeneous initial conditions and presents an MOR method by general orthogonal polynomials with Arnoldi algorithm. The main procedure is to use a series of expansion coefficients vectors in the space spanned by orthogonal polynomials that satisfy a recursive formula to generate a projection based on the multiorder Arnoldi algorithm. The resulting model not only match desired number of expansion coefficients but also has the same coupled structure as the original system. Moreover, the stability is preserved as well. The error bound between the outputs is well‐discussed. Finally, numerical results show that the authors' method can deal well with those systems with inhomogeneous initial conditions in the views of accuracy and computational cost. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a finite-time model order reduction method for linear systems via shifted Legendre polynomials. The main idea of the approach is to use finite-time empirical Gramians, which are constructed from impulse responses by solving block tridiagonal linear systems, to generate approximate balanced system for the large-scale system. The balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted Legendre polynomials, without individual reduction of the Gramians and a separate eigenvector solve. Then, the reduced-order model is constructed by truncating the states corresponding to the small approximate Hankel singular values (HSVs). The stability preservation of the reduced model is briefly discussed. And in combination with the dominant subspace projection method, we modify the reduction procedure to alleviate the shortcomings of the above method, which may unexpectedly lead to unstable systems even though the original one is stable. Furthermore, the properties of the resulting reduced models are considered. Finally, numerical experiments are given to demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the Krylov subspace-based model order reduction methods of second-order systems with time delays, and present two structure-preserving methods for model order reduction of these second-order systems, which avoid to convert the second-order systems into first-order ones. One method is based on a Krylov subspace by using the Taylor series expansion, the other method is based on the Laguerre series expansion. These two methods are used in the multi-order Arnoldi algorithm to construct the projection matrices. The resulting reduced models can not only preserve the structure of the original systems, but also can match a certain number of approximate moments or Laguerre expansion coefficients. The effectiveness of the proposed methods is demonstrated by two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we present a time domain model order reduction method for multi-input multi-output (MIMO) bilinear systems by general orthogonal polynomials. The proposed method is based on a multi-order Arnoldi algorithm applied to construct the projection matrix. The resulting reduced model can match a desired number of expansion coefficient terms of the original system. The approximate error estimate of the reduced model is given. And we also briefly discuss the stability preservation of the reduced model in some cases. Additionally, in combination with Krylov subspace methods, we propose a two-sided projection method to generate reduced models which capture properties of the original system in the time and frequency domain simultaneously. The effectiveness of the proposed methods is demonstrated by two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose a model order reduction (MOR) method based on general orthogonal polynomials for K-power bilinear systems in the time domain. Constructing proper projection matrices by solving a series of linear equations, a reduced K-power bilinear system is produced, which preserves the original coupled structure. It can match several expansion coefficients of the original output. Then the error bound of our algorithm is also investigated. Moreover, the stability of the reduced system is discussed as well. Finally, two numerical examples are provided to illustrate the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we present a model order reduction (MOR) method based on Laguerre polynomials and singular value decomposition (SVD) for coupled systems in the frequency domain. By constructing projection matrices from the global perspective and then blockdiagonalizing them, the reduced system is produced, which not only retains the structure of the original system, but also matches the first several Laguerre coefficients. In addition, the connection between our algorithm and the moment matching approximation is also discussed. The error estimation of our method is given as well. Besides, the stability of the reduced system is also studied. Finally, two numerical examples are provided to verify the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]