A balancing method for reduced order modelling of unstable systems

Prakash, R.; Rao, Shodhan

doi:10.1016/0895-7177(90)90219-d

Cited by 4 publications

(6 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, observability and controllability (characteristics, which describe to what extent the states -natural modes -are relative regarding the out-, inputs of the system, and if they are observable or controllable) of the natural modes are used as reduction criteria [10].…”

Section: State-space Model Developmentmentioning

confidence: 99%

Advanced state space modeling of non-proportional damped machine tool mechanics

Neugebauer

Scheffler

Wabner

et al. 2010

CIRP Journal of Manufacturing Science and Technology

View full text Add to dashboard Cite

Section: State-space Model Developmentmentioning

confidence: 99%

Advanced state space modeling of non-proportional damped machine tool mechanics

Neugebauer

Scheffler

Wabner

et al. 2010

CIRP Journal of Manufacturing Science and Technology

View full text Add to dashboard Cite

“…We set the matrix of the diagonal terms as the unity matrix. Outer diagonal elements subsequently become the zero matrix, if the just made assumption (10) and the conjugate complex characteristic are considered (see [2]).…”

Section: K M Cmentioning

confidence: 99%

State space modeling of non-proportional passive damping in machine tools

Neugebauer

Scheffler

Wabner

et al. 2010

Int J Adv Manuf Technol

View full text Add to dashboard Cite

In the context of virtual development processes of mechatronic systems like machine tools, the preventive assessment of the effects of light-weight structural components on the characteristics of controlled feed systems and therefore on the power- and energy requirement ofmachine tools is of great importance. The modelling of the damping is often limited in practical computation of real machine tools, because they are mostly limited to proportional damping characteristics. Yet the presence of discrete dampers, as for instance, in damping units in guides, or known diverse material damping in construction (components made of glass- or carbon fibre reinforced plastic), demands that the possibility of observance of non-proportional damping, in regard to later model-updating, has to be given. The article demonstrates the theoretical state-space modelling of non-proportional damped mechanics using the example of a biaxial vertical lathe

show abstract

“…Although developed for asymptotically stable systems, balanced truncation and optimal H 2 approximation can be extended to unstable stable systems without poles on the imaginary axis. In particular, a reduced-order model can be obtained by balancing and truncating frequency-domain controllability and observability Gramians [41,51]. By extending the H 2 norm to the L 2 -induced Hilbert-Schmidt norm, an iteratively corrected rational Krylov algorithm was proposed for optimal L 2 model reduction [31].…”

mentioning

confidence: 99%

“…However, the methods in Refs. [41,51,31] cannot be applied to marginally stable systems, as the frequency-domain controllability and observability Gramians as well as the L 2 -induced Hilbert-Schmidt norm are not well defined when there are poles on the imaginary axis.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Structure-preserving model reduction for marginally stable LTI systems

Peng¹,

Carlberg²

2017

Preprint

View full text Add to dashboard Cite

This work proposes a structure-preserving model reduction method for marginally stable linear time-invariant (LTI) systems. In contrast to Lyapunov-stability-based approaches-which ensure the poles of the reduced system remain in the open left-half plane-the proposed method preserves marginal stability by reducing the subsystem with poles on the imaginary axis in a manner that ensures those poles remain purely imaginary. In particular, the proposed method decomposes a marginally stable LTI system into (1) an asymptotically stable subsystem with eigenvalues in the open left-half plane and (2) a pure marginally stable subsystem with a purely imaginary spectrum. We propose a method based on inner-product projection and the Lyapunov inequality to reduce the first subsystem while preserving asymptotic stability. In addition, we demonstrate that the pure marginally stable subsystem is a generalized Hamiltonian system; we then propose a method based on symplectic projection to reduce this subsystem while preserving pure marginal stability. In addition, we propose both inner-product and symplectic balancing methods that balance the operators associated with two quadratic energy functionals while preserving asymptotic and pure marginal stability, respectively. We formulate a geometric perspective that enables a unified comparison of the proposed inner-product and symplectic projection methods. Numerical examples illustrate the ability of the method to reduce the dimensionality of marginally stable LTI systems while retaining accuracy and preserving marginal stability; further, the resulting reduced-order model yields a finite infinite-time energy, which arises from the pure marginally stable subsystem.

show abstract

A balancing method for reduced order modelling of unstable systems

Cited by 4 publications

References 4 publications

Advanced state space modeling of non-proportional damped machine tool mechanics

Advanced state space modeling of non-proportional damped machine tool mechanics

State space modeling of non-proportional passive damping in machine tools

Structure-preserving model reduction for marginally stable LTI systems

Contact Info

Product

Resources

About