Gramian Matrix

2014 ◽  
Author(s):  
Lisa M. Sullivan
Keyword(s):  
Gramian Matrix

Error-Controlled Model Reduction in Flexible Multibody Dynamics

2010 ◽  
Vol 5 (3) ◽  
Author(s):  
Jörg Fehr ◽  
Peter Eberhard
Keyword(s):  
Model Reduction ◽  
Error Bounds ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Reduction Process ◽  
Flexible Multibody Dynamics ◽  
Scanning Tunneling ◽  
Error Estimator ◽  
Gramian Matrix ◽  
Flexible Multibody

One important issue for the simulation of flexible multibody systems is the quality controlled reduction in the flexible bodies degrees of freedom. In this work, the procedure is based on knowledge about the error induced by model reduction. For modal reduction, no error bound is available. For Gramian matrix based reduction methods, analytical error bounds can be developed. However, due to numerical reasons, the dominant eigenvectors of the Gramian matrix have to be approximated. Within this paper, two different methods are presented for this purpose. For moment matching methods, the development of a priori error bounds is still an active field of research. In this paper, an error estimator based on a new second order adaptive global Arnoldi algorithm is introduced and further assists the user in the reduction process. We evaluate and compare those methods by reducing the flexible degrees of freedom of a rack used for active vibration damping of a scanning tunneling microscope.

scholarly journals Matrix of ℤ-module

2015 ◽  
Vol 23 (1) ◽  
pp. 29-49 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Linear Transformation ◽  
Bilinear Form ◽  
Finite Rank ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Gramian Matrix ◽  
Selection Of

Summary In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module V, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22] and coding theory [14]. Some theorems in this article are described by translating theorems in [24], [26] and [19] into theorems of ℤ-module.

Program for Inverting a Gramian Matrix

1961 ◽  
Vol 21 (3) ◽  
pp. 721-727 ◽  
Author(s):  
Kern Dickman ◽  
Henry F. Kaiser
Keyword(s):  
Gramian Matrix

Controllability criteria of fractional differential dynamical systems with non-instantaneous impulses

2019 ◽  
Vol 37 (3) ◽  
pp. 777-793
Author(s):  
B Sundara Vadivoo ◽  
R Raja ◽  
Jinde Cao ◽  
G Rajchakit ◽  
Aly R Seadawy
Keyword(s):  
Dynamical System ◽  
Algebraic Approach ◽  
Main Section ◽  
Volterra Type ◽  
Differential Systems ◽  
Gramian Matrix ◽  
Fractional Differential Systems ◽  
Fractional Differential ◽  
Fractional Dynamical System ◽  
Sufficiency Conditions

Abstract This manuscript prospects the controllability criteria of non-instantaneous impulsive Volterra type fractional differential systems. By enroling an appropriate Gramian matrix that is often defined by the Mittag-Leffler function and with the assistance of Laplace transform, the necessary and sufficiency conditions for the controllability of non-instantaneous impulsive Volterra-type fractional differential equations are derived by using algebraic approach and Cayley–Hamilton theorem. An important feature present in our paper is that we have taken non-instantaneous impulses into the fractional order dynamical system and studied the controllability analysis, since this do not exist in the available source of literature. Inclusively, we have provided two illustrative examples with the existence of non-instantaneous impulse into the fractional dynamical system. So this demonstrates the validity and efficacy of our obtained criteria of the main section.

scholarly journals The Versatile Cross Gramian for System Theoretic Model Reduction and More

2015 ◽  
Author(s):  
Christian Himpe ◽  
Mario Ohlberger
Keyword(s):  
Model Order Reduction ◽  
Order Reduction ◽  
Theoretic Model ◽  
Model Order ◽  
Gramian Matrix ◽  
The Cross ◽  
Analysis System ◽  
Single Matrix ◽  
Controllability And Observability ◽  
Symmetric Systems

For input-output systems, the cross gramian matrix encodes controllability and observability information into a single matrix, which are essential to system-theoretic applications. This system gramian can be used, in example, for model order reduction, sensitivity analysis, system identification, decentralized control and parameter identification. Beyond linear symmetric systems, the cross gramian is also available for parametric, non-symmetric, non-square and nonlinear systems.

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