Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 Nte 8 Rt-Lcu Techique Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd
Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862. Vectr Repreetti f Cplex Nuer A cplex uer, deted y z, c e repreeted i the cplex pl y uig vectr, hw i the fllwig figure. The legth f the vectr i the gitude f the cplex uer d the gle eured cuterclckwie fr the rel xi i the gle f the cplex uer. I z θ z Re Suppe tht th p d re cplex uer. The utrcti f -p i l cplex uer, which c e repreeted y vectr drw fr the lcti f p t the lcti f, hw i the fllwig figure. If i vried, the vectr -p chge i th gitude d gle. p I -p Re Exple If -34j, repreet the vectr f,, d 2 i the cplex ple, d the evlute the gitude d gle f F. 2 Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 2
Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 2. Rt-Lcu Defiiti: The rt lcu i the plt f the rt f yte chrcteritic equti r the ple f cled-lp trfer fucti yte preter i vried. Cider the fllwig cled lp yte: R C _ 2 The cled lp trfer fucti i tied : G T G Fr G 2, the T 2 2 The chrcteritic equti i tied : 2 2 0 The ple c e fud t e: 2 ± 4 4 ± 2 Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 3
Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 It c e ee fr the ve equti tht the ple chge the vlue f i vried. The chge i therefre ffect the tility d the perfrce f the yte. Fr the ve ecd rder yte, e h ω ζ 2 / 2 I ecd rder yte, the effect f the turl frequecy d dpig rti c e redily deteried. Fr higher rder yte, thi lyi i t viu. Thi i where the rt lcu deig ethd i ued. I the ve exple, the chge f ple with icree i c e tied d pltted : I Ple Ple 2 0 0-2 - - 2 -j.00 -- j.00 3 - j.4 -- j.4 4 - j.73 -- j.73 5 - j 2.00 -- j 2.00 0 - j 3.00 -- j 3.00 20 - j 4.36 -- j 4.36 30 - j 5.39 -- j 5.39 40 - j 6.24 -- j 6.24 50 - j 7.00 -- j 7.00 0 0-2 - 0 Rt-Lcu Re Clculti f the rt, give chge i gi, c e redily ccplihed y uig deig pckge uch MATLAB. Hwever, the rt lcu y l e ketched y hd y fllwig the prcedure tht i decried i the fllwig ecti. Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 4
Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 3. Prpertie f the Rt-Lcu Cider the fllwig cled-lp yte R C G _ H The cled-lp trfer fucti i G T G H The chrcteritic equti: GH0 Or GH- Thi i tur require: Mgitude prperty: the gitude f the pe-lp trfer fucti i, i.e., G H r G H 2 Agle prperty: the gle f the pe-lp trfer fucti i dd ultiple f 80, i.e., G H 80 fr ±, ± 3, ± 5,... r G H 80 fr ±, ± 3, ± 5,... Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 5
Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 Exple Suppe the pe-lp trfer fucti G H 2 Are j d 3 j the rt lcu? 2 If, wht i the crrepdig vlue f? Fiite d Ifiite the Ope-Lp Ple d Zer Cider the fllwig pe-lp trfer fucti: 2 G H 20 I thi exple, there re 2 fiite ple t - d -20 d e fiite zer t -2. Beide, fucti culd hve ifiite ple d zer. If the fucti pprche ifiity ge ifiity, the the fucti h ple t ifiity. If the fucti pprche zer pprche ifiity, it h zer t ifiity. Fr the ve exple, pprche ifiity: Thu, the fucti h ifiite zer. 2 li G H li 0 20 Plee t tht every fucti f h equl uer f ple d zer, icludig th ifiite d fiite ple d zer. I the ve exple, the fucti h tw fiite ple, e fiite zer d e ifiite zer, thu the uer f ple the uer f zer 2 Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 6
Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 4. Sketchig the Rt Lcu Befre digitl cputer, ketchig f the rt lcu w perfred y tkig dvtge f the gle d gitude criteri. We will review thi ethd t gi further iight it rt lcu. Whe wrkig i idutry, yu re trgly dvied t ue deig tl uch Mtl. The fllwig rule fcilitte the ketchig f the rt lcu:. Nuer f Brche: The uer f rche f rt lcu equl t the uer f cled lp ple. There will e e rch fr ech cled lp ple. 2. Syetry: The rt lcu i yetricl with repect t the rel xi. 3. Rel-xi Seget: O the rel-xi the rt lcu exit ly i the eget where the uer f fiite pe-lp ple d zer tht re lcted the righthd ide f the eget i dd. Further t the gle criteri, rt the rel-xi ut tify the fllwig requireet: Σ gle fr fiite zer Σ gle fr fiite ple l80 fr, 3,. where Σ u. If we re ciderig pit the rel xi, t thi pit the fllwig c e erved The gulr ctriuti f pir f pe-lp cplex ple r zer i zer. 2 The gulr ctriuti f the pe-lp ple d zer the rel-xi, which re lcted the left-hd ide f the pit, i l zer. S the ctriuti t the gle ly ce fr the pe-lp ple d zer the rel-xi, which re lcted the right-hd ide f the pit cidered. The gulr ctriuti i 80 fr ech f uch pe-lp ple d zer. If the uer f thee pe-lp ple d zer i dd, the pit i therefre the rt lcu. Otherwie, it i t the rt lcu. 4. Strtig d edig pit: The rt lcu rigite t the ple f GH d terite t the zer f GH, icludig the zer t ifiity. The explti fr thi rule i fllw. The chrcteritic equti fr the yte y e expreed i the fr: Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 7
Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 8 0 j j i i p z H G Thi equti y e rerrged d rewritte : 0 i i j j z p Thu fr 0 the rt f the chrcteritic equti re iply the ple f GH. A pprche ifiity ut rei fiite, the rche f the lcu pprch the zer f the pe-lp fucti. If the pe-lp fucti h zer t ifiity, tht i, if > i the uul ce with the del f phyicl yte the lcu will l pprch thee zer. Thi pect f the rt lcu will e prved i the dicui f e f the dditil rule hrtly. 5. Ayptte: The rt lcu pprche tright lie yptte the lcu pprche ifiity. Let the geerl fr f pe-lp fucti e rewritte i the fr: H G...... Fr t yte cidered, < d the ve equti y e writte : H G...... where -, i the uer f ifiite zer. H G li li li The rt lcu fr lrge vlue f the tifie the reltihip: 0 ] li[ ] [ li H G Thi equti h rt tht re give y
Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 0 r r80 fr r ±, ± 3,... The gitude f thee rt pprche ifiity ecue f the upti tht i pprchig ifiity. The gle θ ctriuted y ech rt i therefre: r80 θ Thee gle re the gle f yptte f the rt lcu, ice geerlly the rt lci pprch thee gle d thu pprche ifiity. If we dete the vlue f t which the yptte iterect the rel xi σ, thi vlue i give y u f fiite ple - u f fiite zer σ uer f fiite ple - uer f fiite zer 6. Brekwy d Brek-i Pit: Thi rule ccer pit t which tw r re rche f the lcu ce tgether r rek wy. Pit f thi type re clled rekwy d rek-i pit. Ituitively, whe tw cled lp ple ve twrd ech ther lg the rel xi, they eet t the xiu vlue f the gi tht wuld led t rel ple. The rekwy pit therefre ccur t pit f xiu gi the rel xi. Siilrly, fr the rek i pit, cplex pir f ple retur t the rel-xi d, the gi icree t ifiity, the ple ve twrd the pe lp zer. Therefre, the gi t the rek-i pit i the iiu gi lg the rel-xi etwee tw zer. We wuld therefre e le t fid thee iiu d xiu pit y uig differetil clculu fllw. Fr pe lp trfer fucti GH, it c e hw tht Philip & Hrr, 2000 the rekwy/rek-i pit rt lcu will pper g the rt f the plyil: d[ G H ] 0 d r equivletly, N D N D 0 where N d D re the uertr d deitr plyil f GH. Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 9
Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 The fllwig exple will e give i cl Exple Sketch the rt lcu fr the yte with pe-lp trfer fucti give y 3 5 G H. 2 Exple 2 Fid the rekwy d rek-i pit fr the yte give i Exple. Exple 3 Sketch the rt lcu fr the yte with pe-lp trfer fucti give y G H. 2 3 Exple 4 Sketch the rt lcu fr the yte with the fllwig pe-lp trfer fucti give y G H. 3 0 Ad the deterie where the rt lcu cre the igiry xi. Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 0