Cross Gramian Research Articles

Dimension reduction based on time-limited cross Gramians for bilinear systems

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.

System identification based on the cross Gramian

In order to make mechanical structures more reliable, research is being carried out into methods of detecting and localizing damage to structures, known as Structural Health Monitoring (SHM). One method for continuously monitoring a structure is called State Projection Estimation Error (SP2E) and was developed at the University of Applied Sciences Leipzig. This method is based on state space systems, which are generated by output-only system identification. It is state-of-the-art to create a stabilization diagram to choose the best model order. Since the monitoring process is continuous, a state space model is created every 10 minutes, and therefore it is not possible to look at a stabilization diagram. Thus, the system identification process has to be automated. A novel approach developed in the SPP 100+ research program takes place via the theory of the cross Gramian. The cross Gramian is defined by the Hankel operator, which maps the past input to the future output. This Gramian indicates the interaction between the states and the input/output of a system and can be seen as the energy capacity of a system. By comparing the power of the measurement with the identified energy capacity, the model order selection can be automated. There are limits to the existence of the cross Gramian for multiple-input, multiple-output (MIMO) systems. The system has to be square, stable and symmetric. The covariance driven system identification never provides a symmetric system. A way to create a symmetric system out of the asymmetric measurement data is presented by the use of a symmetrizer, such that the cross Gramian can always be computed. With this symmetrizer a method to estimate the system matrix A by a stable Lyapunov equation, which is classically computed by a numerically unstable pseudo inverse, is presented.

A mixed model reduction method based on the symmetrizer for two classes of dynamical systems

In regard of the cross Gramian and the block Arnoldi process, this paper focuses on mixed model reduction for two classes of dynamical systems. By studying the correlation between the cross Gramian and the Hankel singular values, model reduction via the cross Gramian is investigated for symmetric regular linear systems. Considering the non-symmetric regular linear system, associated with the symmetrizer, a convex optimization problem is established to obtain a symmetric system. Then, the non-symmetric regular linear system is divided into two parts, in which one is determined by the symmetric system and another is a non-symmetric counterpart. Model reduction via the cross Gramian is employed to reduce the first part. According to the structure of the non-symmetric counterpart, a modified version of the block Arnoldi process is developed to reduce the non-symmetric counterpart. The H2 error bound between the original system and the reduced system is discussed. This model reduction approach avoids partitioning the non-symmetric system into several single-input and single-output systems. Moreover, the proposed model reduction method can also be applied to the non-symmetric linear systems. Finally, the proposed model reduction approach is illustrated by a 2D convective thermal flow benchmark and an optimal cooling problem of steel profiles.

Model order reduction based on low-rank decomposition of the cross Gramian

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.

Structure-preserving model order reduction for K-power bilinear systems via Laguerre functions

ABSTRACT 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.

Cross-Gramian-Based Model Reduction for Descriptor Systems

In this paper, we explore model order reduction for large-scale square descriptor systems. A balancing-free square-root method is proposed. The balancing-free square-root method is based on two cross Gramians, one of which is known as the proper cross Gramian and the other as the improper cross Gramian. The proper cross Gramian is the unique solution of a projected generalized continuous-time Sylvester equation, and the improper cross Gramian solves a projected generalized discrete-time Sylvester equation. In order to compute the low-rank factors of these two cross Gramians, we extend the low-rank iteration of the alternating direction implicit method and the Smith method to the projected generalized Sylvester equations. We illustrate the effectiveness of the balance truncation method with one numerical example.

Model order reduction of port-Hamiltonian systems with inhomogeneous initial conditions via approximate finite-time Gramians

Based on the approximate finite-time Gramians, this paper studies model order reduction method of port-Hamiltonian systems with inhomogeneous initial conditions. The approximate controllability and observability Gramians on the finite-time interval [T1,T2](0≤T1<T2<∞) can be obtained by the shifted Legendre polynomials and the reduced port-Hamiltonian system is constructed by the union of dominant eigenspaces. Since the port-Hamiltonian system is square, the cross Gramian on the time interval [T1,T2] can also be approximated by using the shifted Legendre polynomials. Then, the truncated singular value decomposition of the approximate finite-time cross Gramian is carried out to obtain the projection matrix. Finally, the proposed methods are verified by two numerical examples.

Model Order Reduction Based on Approximate Cross Gramian and Laguerre Series for Linear Input-Output Systems

In this paper, we first focus on the topic of calculation of cross gramians for linear square systems, which are constructed approximately from the Laguerre series expansion coefficient vectors that are obtained by solving a recurrence formula instead of solving Sylvester equations directly. Then based on such approximate cross gramian, the reduced-order models (ROMs) are produced by truncating the states that are associated with the smaller approximate Hankel singular values (HSVs). In addition, combining with the idea of dominant subspace projection method, we modify our proposed algorithm to obtain a ROM that preserves the stability. What's more, our algorithms are extended to non-square case successfully. The main properties of ROMs are discussed as well. Finally, some numerical simulations are provided to illustrate the effectiveness of our proposed algorithms in the views of accuracy and computational cost.

Structure‐preserving interval‐limited balanced truncation reduced models for port‐Hamiltonian systems

In this study, the authors propose structure-preserving balanced truncation methods of port-Hamiltonian systems over the frequency and time intervals. First, the port-Hamiltonian system is reduced based on the frequency-interval controllability and observability Gramians. In order to reduce the computational cost, the frequency-interval cross Gramian is utilised to yield the frequency-interval balanced truncation method. For the symmetric port-Hamiltonian systems, the authors prove that these two methods can generate equivalent reduced systems. Additionally, they are also devoted in exploring two structure-preserving balanced truncation methods over time intervals, where one is based on the time interval controllability and observability Gramians and the other is based on the time interval cross Gramian. All the resulting reduced systems have port-Hamiltonian structure, and as a consequence, they are passive. Two numerical examples are simulated to demonstrate the efficiency of the proposed methods.

Cross-Gramian-based dominant subspaces

A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work, we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings.

Balanced Model Reduction for Linear Time-Varying Symmetric Systems

The goal of this paper is to develop balancing theory for the linear time-varying (LTV) symmetric systems. To this end, first, we extend the concept of symmetry in terms of the dual system. Then, we define the cross Gramian for the LTV systems. For LTV symmetric systems, we establish a connection among the controllability, observability, and cross Gramians. In particular, if one of these three Gramians is obtained, the other two Gramians can be constructed. Based on this fact, we show that the symmetry structure is preserved under balanced truncation if the Hankel singular values are pointwise distinct.

Model Order Reduction of Linear Systems via the Cross Gramian and SVD

On the basis of the cross Gramian and the singular value decomposition, this brief investigates model order reduction of linear time-invariant systems in the view of the symmetric case and the nonsymmetric case. It guarantees the reduced system for the nonsymmetric system to be data-sparse. The asymptotical stability of the reduced system is discussed and an output error bound is provided. The correlation of the Hankel singular values in the proposed method and the balanced truncation method is presented. Finally, numerical results indicate that the reduced system of the symmetric system is equivalent to that resulting from balanced truncation, while the reduced system of the nonsymmetric system has a good performance.

H2 optimal model order reduction of the discrete system on the product manifold

• The cross Gramian-based reduction for the discrete systems is proposed. • The H 2 norm of the discrete system is expressed by the cross Gramian. • The constraint conditions of the H 2 optimal problem is further relaxed. • The product manifold is used to reduce the discrete system. This paper studies the H 2 optimal model order reduction of the discrete system based on the cross Gramian, which equips both the controllability and observability information. The cross Gramian is first employed to obtain the H 2 norm of the discrete systems. Then, the constrained conditions of the H 2 optimal model order reduction problem of the discrete systems are relaxed, and a new H 2 optimal model order reduction problem on the product manifold is reformulated. Finally, geometry of the resultant optimization problem is investigated and an iterative algorithm is presented. The efficiency of the proposed algorithm is demonstrated by a practical example.

Frequency interval model reduction of complex fir digital filters

In this paper, a model reduction method for FIR filters with complex coefficients based on frequency interval impulse response Gramians is developed. The advantage of the proposed method is that only one Lyapunov equation needs to be solved in order to obtain the information regarding the frequency interval controllability and observability of the system. In addition this method overcomes the limitations of using cross Gramians which are not applicable for filters with complex coefficients. The effectiveness of the proposed method is demonstrated by a numerical example.

H2 optimal model order reduction on the Stiefel manifold for the MIMO discrete system by the cross Gramian

ABSTRACTIn this paper, the H2 optimal model order reduction method for the large-scale multiple-input multiple-output (MIMO) discrete system is investigated. First, the MIMO discrete system is resolved into a number of single-input single-output (SISO) subsystems, and the H2 norm of the original MIMO discrete system is expressed by the cross Gramian of each subsystem. Then, the retraction and the vector transport on the Stiefel manifold are introduced, and the geometric conjugate gradient model order reduction method is proposed. The reduced system of the original MIMO discrete system is generated by using the proposed method. Finally, two numerical examples show the efficiency of the proposed method.

New integer-order approximations of discrete-time non-commensurate fractional-order systems using the cross Gramian

This paper presents a new approach to modeling of linear time-invariant discrete-time non-commensurate fractional-order single-input single-output state space systems by means of the Balanced Truncation and Frequency Weighted model order reduction methods based on the cross Gramian. These reduction methods are applied to the specific rational (integer-order) FIR-based approximation to the fractional-order system, which enables to introduce simple, analytical formulae for determination of the cross Gramian of the system. This leads to significant decrease of computational burden in the reduction algorithm. As a result, a rational and relatively low-order state space approximator for the fractional-order system is obtained. A simulation experiment illustrates an efficiency of the introduced methodology in terms of high approximation accuracy and low time complexity of the proposed method.

$H_{2}$ model order reduction based on gramians of discrete-time bilinear systems

Abstract In this paper, on the basis of gramians, $H_{2}$-optimal model order reduction of discrete-time bilinear dynamic systems is investigated. First, based on the reachability gramian and the observability gramian, the $H_{2}$-optimal necessary conditions of the discrete-time bilinear system are derived and the corresponding algorithm is proposed. When the algorithm is convergent, the resulting reduced system satisfies the necessary conditions. Then, for the single-input and single-output system, the $H_2$-optimal necessary conditions based on the generalized cross gramian are presented and the related algorithm is accordingly generated. Similarly, the convergent algorithm can make the reduced system satisfy the necessary conditions. Finally, two numerical examples are employed to illustrate the effectiveness of the proposed methods.

Model Reduction via Generalized Frequency Interval Cross Gramian

In this paper, a new model reduction technique based on the generalized limited frequency interval cross Gramian is developed for continuous linear time invariant systems. The proposed technique is not only applicable to unstable systems but can also be applied to the systems which contain eigenvalues of opposite polarities and equal magnitude. The standard cross Gramian is generalized for such systems by transforming the system matrix of the original system using a Riccati equation.

Fast Low‐Rank Empirical Cross Gramians

AbstractThe cross Gramian matrix encodes the input‐output coherence of linear control systems and is used in projection‐based model reduction. The empirical cross Gramian is a data‐driven variant of the cross Gramian which also extends to nonlinear systems. A drawback of the empirical cross Gramian for large‐scale systems is its full order and dense structure; yet, it may be computed column‐wise. Using the hierarchical approximate proper orthogonal decomposition (HAPOD), this partial computability can be exploited to obtain a truncated projection for model order reduction without assembling a full cross Gramian. (© 2017 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Reduced optimal models via cross Gramian for continuous linear time‐invariant systems

In this study, the authors focus on the problem of model order reduction (MOR) suitable for continuous linear time-invariant (LTI) systems. Specifically, for single-input single-output (SISO) LTI systems, a couple of MOR algorithms via the cross Gramian are presented. The authors first give a cross Gramian-based optimal MOR iterative algorithm to reduce the SISO system. Further, the authors explore the optimisation problem on the Stiefel manifold. Based on the geometric notions on this manifold, a conjugate gradient iterative algorithm (CGIAHM) is proposed. The conjugate gradient direction used to search for the minimiser is derived by applying the notion of vector transport. It is worth mentioning that the conjugate gradient used in the CGIAHM algorithm is a decent direction for the cost function due to the ingenious construction of this algorithm. In addition, the authors’ algorithms are extended to solve the optimal MOR problem for general LTI systems. Finally, two numerical examples demonstrate the effectiveness of their algorithms.