Transfer Function Reduction

SYNOPSIS A method of generating reduced order models from the Laurent series expansion of a transfer function is examined by means of the Hankel Matrix and its correspondence to the field of rational functions. The approach enables particularly simple results to be derived regarding the composition of the reduced form and the avoidance of non minimum phase characteristics therein.

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On Minimum Phase

at - Automatisierungstechnik ◽

10.1524/auto.2013.1002 ◽

2013 ◽

Vol 61 (12) ◽

pp. 805-817

Author(s):

Achim Ilchmann ◽

Fabian Wirth

Keyword(s):

Transfer Function ◽

Transfer Functions ◽

Main Tool ◽

Rational Functions ◽

Phase System ◽

Zero Dynamics ◽

Minimal Realization ◽

Minimum Phase ◽

Denominator Polynomial ◽

The Time Domain

Abstract We discuss the concept of `minimum phase' for scalar semi Hurwitz transfer functions. The latter are rational functions where the denominator polynomial has its roots in the closed left half complex plane. In the present note, minimum phase is defined in terms of the derivative of the argument function of the transfer function. The main tool to characterize minimum phase is the Hurwitz reflection. The factorization of a weakly stable transfer function into an all-pass and a minimum phase system leads to the result that any semi Hurwitz transfer function is minimum phase if, and only if, its numerator polynomial is semi Hurwitz. To characterize the zero dynamics, we use the Byrnes-Isidori form in the time domain and the internal loop form in the frequency domain. The uniqueness of both forms is shown. This is used to show in particular that asymptotic stable zero dynamics of a minimal realization of a transfer function yields minimum phase, but not vice versa.

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Constrained Optimal Pade´ Model Reduction

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2802378 ◽

1997 ◽

Vol 119 (4) ◽

pp. 685-690 ◽

Cited By ~ 5

Author(s):

T. N. Lucas

Keyword(s):

Transfer Function ◽

Iterative Process ◽

Natural Extension ◽

Initial Time ◽

Reduced Order Models ◽

Optimal Method ◽

Reduced Order ◽

Numerical Examples ◽

Linear Sets ◽

Multipoint Padé Approximation

A frequency-domain multipoint Pade´ approximation method is given that produces optimal reduced order models, in the least integral square error sense, which are constrained to match the initial time response values of the full and reduced systems for impulse or step inputs. It is seen to overcome a perceived drawback of the unconstrained optimal models, i.e., that they do not guarantee a proper rational reduced order transfer function for a step input. The method is easy to implement when compared to existing constrained optimal methods, and consists of solving only linear sets of equations in an iterative process. It is also seen to be a natural extension of an existing optimal method. Numerical examples are given to illustrate its application.

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Order Reduction of Discrete-Time System Via Bilinear Routh Approximation

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2896138 ◽

1990 ◽

Vol 112 (2) ◽

pp. 292-297 ◽

Cited By ~ 16

Author(s):

Chyi Hwang ◽

Ching-Shieh Hsieh

Keyword(s):

Transfer Function ◽

Approximation Method ◽

Order Reduction ◽

Reduced Order Models ◽

Discrete Time System ◽

Initial Value ◽

Time System ◽

Bilinear Transformation ◽

Reduced Order ◽

Canonical Expansion

In this paper, a method of combining the Routh approximation method with the bilinear transformation is presented for deriving stable reduced-order models of a strictly proper z-transfer function Gn(z). It is based on applying the bilinear transformation to the (z+1)Gn(z), and then deriving a new bilinear Routh γ − δ canonical expansion for Gn(z). The proposed bilinear Routh approximation method has all the advantages of the Routh approximation method [5] while without having initial-value problem caused by the bilinear transformation. A numerical example is included to illustrate the procedure.

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