Corrections and Comments to “Model Reduction of Discrete Linear Systems Via Frequency-Domain Balanced Structure”

Abstract

This note gives the correction to a few mistakes in the aforementioned paper. It is also found that the derived reduced model (fourthorder) is unstable.Index Terms-Discrete linear system, model reduction.A sixth-order Chebyshev Type-I filter, shown in (1) at the bottom of the page, is given in [1]. This filter is unstable (see Fig. 1). Therefore, the simulations results shown in [1, are irrelevant.The filter, shown in (2) at the bottom of the page, considered in [2] is stable (see Fig. 1). The reduced-order models of [2] are the same as in [1]. Thus, the filter shown in [1] appears to be the same as the one shown in [2].Therefore, it is evident that the equation showing sixth-order Chebyshev Type-I filter [1], contains a few typographical errors. The errors are as follows.i) The incorrect coefficient of z in the numerator. The correct coefficient of z is 0.06417. Fig. 1. Time response of original filters [1], [2]. ii) Missing coefficient of z in the denominator. The correct missing coefficient of z is 00.6859. iii) The incorrect coefficient of z 0 in the denominator. The correct coefficient of z 0 is 0.1962. Thus, the correct form of filter T (z) in [1] is as shown in (2). The reduced third-order [1], [2] and the fourth-order [1], [2] models are A rf = 0:6320 00:4430 0:0438 0:4430 0:2518 0:1919 00:0438 0:1919 0:6390 B rf = [0:5853 0 0:5870 0:0986] 3 C rf = [0:5853 0:5870 0 0:0986] G3(z) = C rf (e jw I 0 A rf ) 01 B rf = 00:0117z 2 + 0:4730z 0 0:3239 z 3 0 1:5228z 2 + 0:8852z 0 0:2117T (z) = 0:01069z 6 + 0:06417z 5 + 0:1604z 4 + 0:2139z 3 + 0:1604z 2 + 0:6417z + 0:01069 z 6 0 1:464z 5 + 2:291z 4 0 2:06z 3 + 1:464z 2 0 0:685 = 0:01069 + 0:0799z 5 + 0:1359z 4 + 0:2359z 3 + 0:1447z 2 + 0:6417z + 0:0180 z 6 0 1:464z 5 + 2:291z 4 0 2:06z 3 + 1:464z 2 0 0:685(1) T1(z) = 10 02 1:069z 6 + 6:417z 5 + 16:04z 4 + 21:39z 3 + 16:04z 2 + 6:417z + 1:069 z 6 0 1:464z 5 + 2:291z 4 0 2:06z 3 + 1:464z 2 0 0:6859z + 0:1962 = 0:01069 + 0:0799z 5 + 0:1359z 4 + 0:2359z 3 + 0:1447z 2 + 0:0715z + 0:0086 z 6 0 1:464z 5 + 2:291z 4 0 2:06z 3 + 1:464z 2 0 0:6859z + 0:1962(2) 1549-8328/$25.00

Reduction of discrete interval system using clustering of poles with Padé approximation: a computer-aided approach

“…reduced computational effort in simulation, simplified understanding of system, simpler control laws etc., model reduction has been ample area of research. Several methods have been proposed for reduction of continuous-time and discrete-time systems Aoki, 1968;Glover, 1984;Sinha and Kuszta, 1983;Huttan and Friedland, 1975;Shamash, 1975;Rao el al., 1978;Singh el al., 2004;Prajapati et al, 2007). Among them, Padé approximation method has found to be very useful in theoretical physics research (Baker and Graves-Morris, 1981;Baker, 1975) due to being computationally simple but the reduced-order model obtained using Padé approximation method often leads to be unstable even though the high-order system is stable.…”

“…Among them, Padé approximation method has found to be very useful in theoretical physics research (Baker and Graves-Morris, 1981;Baker, 1975) due to being computationally simple but the reduced-order model obtained using Padé approximation method often leads to be unstable even though the high-order system is stable. To overcome limitaion, many improvements have been proposed (Huttan and Friedland, 1975;Shamash, 1975;Rao el al., 1978;Singh el al., 2004;Prajapati et al, 2007) in literature.…”

Model Reduction Via Limited Frequency Interval Gramians

“…Inspired by Wang et al [18], Gugercin and Antoulas [5] modified the Gawronski and Juang's method [4] to obtain stable reduced-order models and error bounds. In [20], Wang and Zilouchian have extended the technique [1] (which is similar to [4]) for discrete-time sytems (Please also see [9] and [17] for corrections and authors' reply to the paper [20]). In the paper [20], the authors provide a proof of stability of reduced-order models and derivation of error bounds.…”