ROOT-LOCUS ANALYSIS. Lecture 11: Root Locus Plot. Consider a general feedback control system with a variable gain K. Y ( s ) ( ) K

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1 ROOT-LOCUS ANALYSIS Coder a geeral feedback cotrol yte wth a varable ga. R( Y( G( + H( Root-Locu a plot of the loc of the pole of the cloed-loop trafer fucto whe oe of the yte paraeter ( vared. Root locu allow u to vualze the chage the cloed-loop pole a the paraeter creae fro to fty. A tcto uually ade betwee the followg categore of root-locu (a Root-Locu Plot The plot of the root loc whe oe paraeter vare potve value; th paraeter uually the forward ga,.e. < (b Copleetary Root-Locu Plot The plot obtaed for egatve value of,.e. < (c Root Cotour Plot The plot obtaed whe ore tha oe paraeter vare. Exaple Coder the yte. R( + G( Y( The effect of o the root ad how by the table below Reark Suppoe that G (, fd the root of the C.E. for ( + < ad plot thee root the -plae breakaway pot Y ( ( G R ( + G ( + + The C.E The root of the C.E. are + for for ; + for > for >

2 The root-locu plotted a how OLTF pole root-locu a creaed + ω breakaway pot -plae We hall develop a procedure to ketch the root-locu wthout coputg the root of the C.E. pot by pot. 5 ( + The root loc plot of + Gp ( + ad Gp ( ( ( + ( + are gve below. Take ote of the ayptote ad break-away ( pot. ω break- pot break-away ω pot OLTF pole OLTF zero ayptote root loc cetrod of ayptote break-away pot -plae 6 Matheatcal Defto of Root-Locu Coder the cloed-loop trafer fucto Y ( ( G R ( + G ( H ( The Charactertc Equato (C.E. + G(H(.e. G(H( Root-locu defed by the codto (I Magtude Codto G ( H ( (II Agle Codto G( H( ± ( + π where,,, (a teger 7 The ope-loop trafer fucto ay be expreed a ( + z( + z ( + z G( H( ( + p ( + p ( + p where -z are the zero of OLTF -p are the pole of OLTF Hece, the C.E. + G( H ( gve by ( + p ( + p ( + p + ( + z ( + z ( + z Whe, the root of the C.E. are p, p,, p Whe, the root of the C.E. are z, z,, z I other wor, a vare fro to, the root of the C.E. travere fro the pole of OLTF to the zero of OLTF. 8

3 Wth the OLTF G(H( gve by ( + z( + z ( + z G( H ( ( + p ( + p ( + p Magtude Codto G( H ( Agle codto + z + p G( H ( ( + z ( + p ± ( + π ; a teger ( Equato ( ad ( defe the root-locu. The root-locu of a yte a plot of all the value of whch atfy eq ( ad (. ( 9 Thu, gve the pole-zero plot of G(H(, the root-locu plot of the cloed-loop yte volve (a A earch for pot the -plae whch atfy equato (. (b The deterato of the value of o the root-locu ug equato (. Exaple Coder G H The pole ad zero are how the dagra, ad we wat to check whether o the root locu. ( + z ( + p( + p p p C ( ( ; z z A p D p ω B p p a arbtrary pot ope loop pole ope loop zero + z + p + p ( ( + z + ( + p + ( + p ± ( + π ( + + ± ( + π z p p p If a pot o the Root-Locu, t ut atfy the followg two codto Magtude Codto Agle Codto If atfe equato (4, the the value of at that pot obtaed fro eq (,.e. B C D A Next, repeat for ew value of ad o o. ( (4 Rule for Cotructg the Root-Locu The OLTF gve by ( + z( + z ( + z G( H ( ; ( ( + p ( + p ( + p The cloed-loop C.E..e. ( + p ( + p ( + p + uber of fte pole of G(H( p ;,, uber of fte zero of G(H( z ;,, Rule, pot o the root-locu are at the fte pole of G(H(. Rule +, pot o the root-locu are at the fte zero of G(H( ad fty. Rule The uber of brache of the root loc equal to. Rule 4 The root loc are yetrcal wth repect to the real-ax of the -plae. ( + z ( + z ( + z

4 (The explaato for the followg rule 5-8, ad are gve the Appedx. Rule 5 There are ayptote. A the root loc are ayptotc to traght le (ayptote wth agle wth the real-ax gve by ( + π ;,,,( Rule 6 The ayptote terect o the real ax at the pot gve by (fte_pole_of_ G( H ( (fte_zero_of_ G( H ( c.e. c ( p ( z ( ( Rule 7 Root loc are foud o a ecto of the real ax oly f the total uber of real pole ad zero of G(H( to the rght of the ecto ODD. Rule 8 (a The agle of departure fro coplex pole are gve by d 8 + where the agle of G(H( at that pole wth the agle cotrbuto fro the pole telf gored. (b The agle of arrval at coplex zero are gve by a 8 where the agle of G(H( at that zero wth the agle cotrbuto fro the zero telf gored. 4 Rule 9 The pot where the root loc terect the agary ax are obtaed ug the Routh-Hurwtz crtero. Rule The breakaway/break pot are pot where two or ore brache of the root locu depart fro or arrve at a a brach. The agle of departure/arrval of breakaway/break pot gve by ( + 8 ba ;,,, k k where k uber of loc leavg or approachg the pot. 5 Suppoe that the C.E. expreed a + G( H ( + A ( A( The, d A ( A( B ( B ( The breakaway/break pot are gve by d (7.e. the breakaway/break pot are gve by the root of A( B ( A ( (6 NB NOT all root of equato (8 correpod to the actual breakaway/break pot. (5 (8 6

5 Wth codto (7, we ca ue eq. (6 to derve a uercal ethod to obta the breakaway/break pot. For breakaway( pot o the real ax,, o we have A( (9 We plot a graph of for varou value of betwee the elected pot where the breakaway( pot expected. The at whch axu (or u the breakaway (or break pot. the breakaway pot. the break pot. 7 Rule The value of at a pot o the root locu obtaed by applyg the Magtude Codto ad obtaed fro G( H (.e. + p + z product_of_vector_legth_fro_oltf_pole product_of_vector_legth_fro_oltf_zero 8 Appedx Explaato of rule 5-8, ad. Rule 5 A +, loc wll approach fty..e. There are ayptote. If By the Agle Codto G( H ( ( ( + π ( + π ;,,,( ω 9 Rule 6 Fro the OLTF G( H ( ( + z ( + z ( + z ( + p ( + p ( + p + z + + z + p + + p Dvde the deoator ad the uerator by the uerator ter, G( H ( (A + p z Coder the followg fucto P( ( c ( + c (A

6 ( + π ;,,,( The C.E. + P( ha ( root locu brache whch are traght le pag through the pot c ad havg agle, The cae of 4 llutrated below. It ca be readly verfed that every pot o the 4 traght le atfe the agle codto. ω That, the root locu of + P( a et of ( le draw fro c at agle ( + π ;,,,(. 45 o c 45 o (4 pole here The +G(H( behave the ae aer a +P( for value of, becaue the frt two hgher order ter of ther deoator are detcal. +G(H( approache +P( for value of. Therefore, the brache of + G(H( whch ted to fty approach the traght le root locu brache of + P(. The traght le root loc of + P( act a ayptote to the ( brache of +G(H(. Fro eq (A ad (A.e. ( p z c c ( p ( z ( - the cetrod of the ayptote. Rule 7 Coder the followg cae - ω So, G( H ( ( 8 ± ( + 8 R R ( o + p p z R Nuber of zero o the rght of o. R Nuber of pole o the rght of o. ( o + z o ( o + p ( o + z ( o + p p z The pole ad zero o the real ax to the rght of th pot o cotrbute a agle of 8 each. The pole ad zero to the left of th pot o cotrbute a agle of each. The et agle cotrbuto of a coplex cougate pole or zero par alway zero. p.e. R R (ad hece R R Rule 8 Coder the followg cae + ut be a odd uber. o p d ( o p ω p 5 4 4

7 At pot o, the et agle cotrbuto of all other pole ad zero at th pot ( I the lt a the pot o o the root locu approache p, P equal the agle of departure of the root locu fro the pole p, ( P d. Fro the Agle Codto or d ( d 8 +. Slar explaato apple to agle of arrval. 5 Rule The Charactertc Equato d + G(H( If C.E. ha a ultple root at b of ultplcty r,.e. + G(H( ( + b r A ( where A ( doe ot cota the factor ( + b, the [ ] r + G( H ( ( + b A ( + A ( r( + b d ' r r ' ( + b ra ( + ( + b A ( At b, RHS (oly f r [ G H ] + ( ( (A 6 The C.E. ca be expreed a So, + G( H ( + A ( d ( ( ( ( [ ( ( ] A B A + G H B A ( A( B ( A ( A( Fro eq (A, the root are the breakaway pot, therefore the breakaway pot are alo gve by the root of Fro eq (A4, (A4 (A5 (A6 7 Hece d A ( A( B ( B ( A ( B ( A ( B ( Wth eq (A5, we have d (A7 That, the breakaway( pot ca alo be coputed fro the root of (A7, or through a uercal ethod. Warg NOT all root of equato (A5 (or (A7 correpod to the actual breakaway pot. (Verfy wth Agle Codto!!! 8

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