{"title":"具有L∞误差界的线性多变量系统的乘法逼近","authors":"K. Glover","doi":"10.23919/ACC.1986.4789204","DOIUrl":null,"url":null,"abstract":"It is shown that a p<sup>x</sup>m transfer function G(s) with p⩾m can be decomposed as G = (I-ν<sub>N</sub>Δ<sub>N</sub>)<sup>-1</sup>...(I-ν<sub>r+2</sub>Δ<sub>r+2</sub>)<sup>-1</sup> (I-ν<sub>r=1</sub>Δ<sub>r=1</sub>)<sup>-1</sup>G<sub>O</sub> where Δ<sub>i</sub> are stable all-pass transfer functions, 1=ν<sub>1</sub>..=ν<sub>r</sub>≫ν<sub>r=1</sub>..≫ν<sub>N</sub>≫0 are the Hankel-singular-values of GW*<sup>-1</sup> where G*G=WW* with W stable and minimum phase. Results on the McMillan degree of G<sup>Λ</sup>:=(I-ν<sub>i</sub>Δ<sub>i</sub>)..(I-ν<sub>N</sub>Δ<sub>N</sub>)G then show that G<sup>Λ</sup> gives a good low order approximation to G in the sense of relative error.","PeriodicalId":266163,"journal":{"name":"1986 American Control Conference","volume":"216 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1986-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"40","resultStr":"{\"title\":\"Multiplicative approximation of linear multivariable systems with L∞ error bounds\",\"authors\":\"K. Glover\",\"doi\":\"10.23919/ACC.1986.4789204\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"It is shown that a p<sup>x</sup>m transfer function G(s) with p⩾m can be decomposed as G = (I-ν<sub>N</sub>Δ<sub>N</sub>)<sup>-1</sup>...(I-ν<sub>r+2</sub>Δ<sub>r+2</sub>)<sup>-1</sup> (I-ν<sub>r=1</sub>Δ<sub>r=1</sub>)<sup>-1</sup>G<sub>O</sub> where Δ<sub>i</sub> are stable all-pass transfer functions, 1=ν<sub>1</sub>..=ν<sub>r</sub>≫ν<sub>r=1</sub>..≫ν<sub>N</sub>≫0 are the Hankel-singular-values of GW*<sup>-1</sup> where G*G=WW* with W stable and minimum phase. Results on the McMillan degree of G<sup>Λ</sup>:=(I-ν<sub>i</sub>Δ<sub>i</sub>)..(I-ν<sub>N</sub>Δ<sub>N</sub>)G then show that G<sup>Λ</sup> gives a good low order approximation to G in the sense of relative error.\",\"PeriodicalId\":266163,\"journal\":{\"name\":\"1986 American Control Conference\",\"volume\":\"216 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1986-06-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"40\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"1986 American Control Conference\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.23919/ACC.1986.4789204\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"1986 American Control Conference","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.23919/ACC.1986.4789204","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Multiplicative approximation of linear multivariable systems with L∞ error bounds
It is shown that a pxm transfer function G(s) with p⩾m can be decomposed as G = (I-νNΔN)-1...(I-νr+2Δr+2)-1 (I-νr=1Δr=1)-1GO where Δi are stable all-pass transfer functions, 1=ν1..=νr≫νr=1..≫νN≫0 are the Hankel-singular-values of GW*-1 where G*G=WW* with W stable and minimum phase. Results on the McMillan degree of GΛ:=(I-νiΔi)..(I-νNΔN)G then show that GΛ gives a good low order approximation to G in the sense of relative error.
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