Computation of a compact state space model for an adaptive spindle head configuration with piezo actuators using balanced truncation

Abstract

Finite Element models of machine tools or their building blocks are usually very large and thus do not allow for fast simulation or application in controller design. Especially when algebraic constraints come into play the models become differential algebraic equations and therefore are even more difficult to handle in the application. In this contribution we propose a method based on modern system theoretic model order reduction algorithms that allows to generate a first order standard state space reduced order model for a structural model of an adaptive spindle support (ASS) that is of second order index 1 differential algebraic form due to the piezo actuation applied. The accuracy of the method is demonstrated by a numerical frequency domain error analysis

Model reduction for linear and nonlinear magneto‐quasistatic equations

“…Therefore, the preservation of passivity in reduced-order models is crucial for stable simulation of coupled field-circuit problems. In the following, we will show that system (30) (and also (29)) is actually passive and, fortunately, the balanced truncation method applied to the structured system (30), (31) preserves passivity in the reduced-order model (39), (40).…”

“…Recently, some modifications of this approach avoiding the spectral projectors have been presented in [37][38][39] for structured DAE systems including semi-explicit DAEs of index 1, Stokes-type systems of index 2 and mechanical systems of index 1 and 3, see also a recent survey [40]. They all are based on an implicit index reduction and an equivalence between the Schur complement linear systems and systems with the original matrices.…”

“…are the transfer functions of the original and reduced-order models, i = √ −1, and || · || 2 denotes the spectral matrix norm, see [43]. Moreover, the balanced truncation method preserves stability in (39). It should, however, be noticed that the preservation of other properties, in particular, passivity cannot, in general, be guaranteed for this method.…”

“…1287 Finally, we present in Algorithm 3 the computation of the reduced-order model (39), (53) using balanced truncation. The first three steps there are similar to those in Algorithm 1.…”

Model Order Reduction for Magneto-Quasistatic Equations

“…Unfortunately, the model reduction approach from [40] requires the computation of the spectral projectors onto the deflating subspaces of the pencil λE − A corresponding to the finite and infinite eigenvalues. Recently, some modifications of this approach avoiding the spectral projectors have been presented in [13,16,43] for structured DAE systems including semi-explicit DAEs of index 1, Stokes-type systems of index 2 and mechanical systems of index 1 and 3, see also a recent survey [7]. They all are based on an implicit index reduction and an equivalence between the Schur complement linear systems and systems with the original system matrices.…”