Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 Note 7 Root-Locu Techique Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862. Vector Repreettio of Coplex Nuer A coplex uer, deoted y z, c e repreeted i the coplex pl y uig vector, how i the followig figure. The legth of the vector i the gitude of the coplex uer d the gle eured couterclockwie fro the rel xi i the gle of the coplex uer. I z θ z Re Suppoe tht oth p d re coplex uer. The utrctio of -p i lo coplex uer, which c e repreeted y vector drw fro the loctio of p to the loctio of, how i the followig figure. If i vried, the vector -p chge i oth gitude d gle. p I -p Re Exple If -34j, repreet the vector of,, d 2 i the coplex ple, d the evlute the gitude d gle of F. 2 Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd 2
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 2. Root-Locu Defiitio: The root locu i the plot of the root of yte chrcteritic equtio or the pole of cloed-loop trfer fuctio yte preter i vried. Coider the followig cloed loop yte: R C _ 2 The cloed loop trfer fuctio i otied : G T G For G, the 2 T 2 2 The chrcteritic equtio i otied : 2 2 0 The pole c e foud to e: 2 ± 4 4 ± 2 Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd 3
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 It c e ee fro the ove equtio tht the pole chge the vlue of i vried. The chge i therefore ffect the tility d the perforce of the yte. For the ove ecod order yte, oe h ω ζ 2 / 2 I ecod order yte, the effect of o the turl frequecy d dpig rtio c e redily deteried. For higher order yte, thi lyi i ot oviou. Thi i where the root locu deig ethod i ued. I the ove exple, the chge of pole with icree i c e otied d plotted : I Pole Pole 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 Root-Locu Re Clcultio of the root, give chge i gi, c e redily ccoplihed y uig deig pckge uch MATLAB. However, the root locu y lo e ketched y hd y followig the procedure tht i decried i the followig ectio. Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd 4
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 3. Propertie of the Root-Locu Coider the followig cloed-loop yte R C _ G H The cloed-loop trfer fuctio i G T G H The chrcteritic equtio: GH0 Or GH- Thi i tur require: Mgitude property: the gitude of the ope-loop trfer fuctio i, i.e., G H or G H 2 Agle property: the gle of the ope-loop trfer fuctio i odd ultiple of 80 o, i.e., G H 80 for ±, ± 3, ± 5,... or G H 80 for ±, ± 3, ± 5,... Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd 5
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 Exple Suppoe the ope-loop trfer fuctio G H 2 Are j d 3 jo the root locu? 2 If o, wht i the correpodig vlue of? Fiite d Ifiite the Ope-Loop Pole d Zero Coider the followig ope-loop trfer fuctio: 2 G H 20 I thi exple, there re 2 fiite pole t - d -20 d oe fiite zero t -2. Beide, fuctio could hve ifiite pole d zero. If the fuctio pproche ifiity goe ifiity, the the fuctio h pole t ifiity. If the fuctio pproche zero pproche ifiity, it h zero t ifiity. For the ove exple, pproche ifiity: Thu, the fuctio h ifiite zero. 2 li G H li 0 20 Plee ot tht every fuctio of h equl uer of pole d zero, icludig oth ifiite d fiite pole d zero. I the ove exple, the fuctio h two fiite pole, oe fiite zero d oe ifiite zero, thu the uer of pole the uer of zero 2 Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd 6
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 4. Sketchig the Root Locu Before digitl coputer, ketchig of the root locu w perfored y tkig dvtge of the gle d gitude criteri. We will review thi ethod to gi further iight ito root locu. Whe workig i idutry, you re trogly dvied to ue deig tool uch Mtl. The followig rule fcilitte the ketchig of the root locu:. Nuer of Brche: The uer of rche of root locu equl to the uer of cloed loop pole. There will e oe rch for ech cloed loop pole. 2. Syetry: The root locu i yetricl with repect to the rel xi. 3. Rel-xi Seget: O the rel-xi the root locu exit oly i the eget where the uer of fiite ope-loop pole d zero tht re locted o the righthd ide of the eget i odd. Further to the gle criterio, root o the rel-xi ut tify the followig requireet: Σ gle fro fiite zero Σ gle fro fiite pole l80 for, 3,. where Σ u. If we re coiderig poit o the rel xi, t thi poit the followig c e oerved The gulr cotriutio of pir of ope-loop coplex pole or zero i zero. 2 The gulr cotriutio of the ope-loop pole d zero o the rel-xi, which re locted o the left-hd ide of the poit, i lo zero. So the cotriutio to the gle oly coe fro the ope-loop pole d zero o the rel-xi, which re locted o the right-hd ide of the poit coidered. The gulr cotriutio i 80 o fro ech of uch ope-loop pole d zero. If the uer of thee ope-loop pole d zero i odd, the poit i therefore o the root locu. Otherwie, it i ot o the root locu. 4. Strtig d edig poit: The root locu origite t the pole of GH d terite t the zero of GH, icludig thoe zero t ifiity. The expltio for thi rule i follow. The chrcteritic equtio for the yte y e expreed i the for: Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd 7
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd 8 0 j j i i p z H G Thi equtio y e rerrged d rewritte : 0 i i j j z p Thu for 0 the root of the chrcteritic equtio re iply the pole of GH. A pproche ifiity ut rei fiite, the rche of the locu pproch the zero of the ope-loop fuctio. If the ope-loop fuctio h zero t ifiity, tht i, if > i the uul ce with the odel of phyicl yte the locu will lo pproch thee zero. Thi pect of the root locu will e proved i the dicuio of oe of the dditiol rule hortly. 5. Ayptote: The root locu pproche tright lie yptote the locu pproche ifiity. Let the geerl for of ope-loop fuctio e rewritte i the for: o o H G...... For ot yte coidered, < d the ove equtio y e writte : o o H G...... where -, i the uer of ifiite zero. H G li li li The root locu for lrge vlue of the tifie the reltiohip: 0 ] li[ ] [ li H G Thi equtio h root tht re give y
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 0 or r80 for r ±, ± 3,... The gitude of thee root pproche ifiity ecue of the uptio tht i pprochig ifiity. The gle θ cotriuted y ech root i therefore: r80 θ Thee gle re the gle of yptote of the root locu, ice geerlly the root loci pproch thee gle d thu pproche ifiity. If we deote the vlue of t which the yptote iterect the rel xi σ, thi vlue i give y u of fiite pole - u of fiite zero σ uer of fiite pole - uer of fiite zero 6. Brekwy d Brek-i Poit: Thi rule cocer poit t which two or ore rche of the locu coe together or rek wy. Poit of thi type re clled rekwy d rek-i poit. Ituitively, whe two cloed loop pole ove towrd ech other log the rel xi, they eet t the xiu vlue of the gi tht would led to rel pole. The rekwy poit therefore occur t poit of xiu gi o the rel xi. Siilrly, for the rek i poit, coplex pir of pole retur to the rel-xi d, the gi icree to ifiity, the pole ove towrd the ope loop zero. Therefore, the gi t the rek-i poit i the iiu gi log the rel-xi etwee two zero. We would therefore e le to fid thee iiu d xiu poit y uig differetil clculu follow. For ope loop trfer fuctio GH, it c e how tht Philip & Hror, 2000 the rekwy/rek-i poit o root locu will pper og the root of the polyoil: d[ G H ] 0 d or equivletly, Nʹ D N Dʹ 0 where N d D re the uertor d deoitor polyoil of GH. Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd 9
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 The followig exple will e give i cl Exple Sketch the root locu for the yte with ope-loop trfer fuctio give y 3 5 G H. 2 Exple 2 Fid the rekwy d rek-i poit for the yte give i Exple. Exple 3 Sketch the root locu for the yte with ope-loop trfer fuctio give y G H. 2 3 Exple 4 Sketch the root locu for the yte with the followig ope-loop trfer fuctio give y G H. 3 0 Ad the deterie where the root locu croe the igiry xi, d the rge of gi i.e., i order tht the yte i tle. Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd 0