THE CONCEPT OF THE ROOT LOCUS. H(s) THE CONCEPT OF THE ROOT LOCUS

Transcription

1 So far i the tudie of cotrol yte the role of the characteritic equatio polyoial i deteriig the behavior of the yte ha bee highlighted. The root of that polyoial are the pole of the cotrol yte, ad their locatio i the coplex -plae reveal iforatio about the tability ad the perforace of the yte. Coider the yte how below: R() + - G () C() H() It trafer fuctio i give by: C( ) R( ) G ( ) + G ( ) H( ) G( ) + G( ) H( ) ' ' The loop gai of the yte ca be expreed a a uerator polyoial over a deoiator polyoial a follow: 2 G H a 0 + a + a 2 + LL + a + ( ) ( ) a 2 b + b + b + LL + b + b 0 2 The characteritic equatio i the give by: G H a a a LL ( ) ( ) a a 2 b + b + b + L L+ b + b 0 2 A a exaple coider that the yte i the previou figure have the followig loop gai: G( ) H ( ) ( + )( + 2)

2 The characteritic equatio for the yte i give by: + G( ) H( ) ( 2 + ) The uerator of the above equatio i i fact the deoiator of the yte trafer fuctio. The root of that polyoial are the pole of the yte. A varie betwee zero ad ifiity the pole of the yte are calculated a follow: j j j j j j j j j j j j j j 3.688

3 The plot of thee pole a they igrate throughout the zero ad ifiity are a how below: -doai whe varie betwee -2 - The root locu i othig but the path of the yte pole a the gai of the yte varie betwee zero ad ifiity. Sice the root locu repreet the path of the root of the characteritic equatio a the gai varie fro zero to ifiity, it follow that every poit o the root locu ut atify the characteritic equatio, aely; + G( ) H( ) 0 Rearragig give; G( ) H( ) Both G() ad H() are coplex variable. The above equatio ca hece be cat i polar or vector for a follow: G( ) H( ) G( ) H( ) G( ) H( )

4 Thi lat relatiohip pecifie the coditio that ut prevail for ay poit o the root locu. The relatiohip i geerally cat i the for of two coditio a follow: Magitude Coditio: G( ) H( ) ( + z ) ( + z ) LL ( + z ) 2 ( + p ) ( + p ) LL ( + p ) 2 ( p ) ( p ) ( p ) LL + ( + z ) ( + z ) LL ( + z ) 2 Thi equatio tate that the agitude of all the vector fro a poit o the root locu to the pole over the agitude of all the vector fro the ae poit to the zero ut equal the gai that reult i that poit o the locu. Agle Coditio: G( ) H( ) ± 80 ( 2k + ) or k 0,, 2, L Thi equatio ca be rewritte a: ( ) ( ) LL LL G H + z + + z + + z z + p + p + p ± 80 ( 2k + ) Thi equatio tate that the u of all the agle of vector fro a poit o the root locu to the zero of the yte iu the u of all the agle of vector fro the ae poit to the pole of the yte ut be equal to 80E or ultiple thereof.

5 or the yte aalized before ad how below at the poit give by -.5 ± agle of the vector fro that poit to the two pole are calculated a follow: j.3229, the θ θ ta ta The u of thee two agle i 80E, ad hece the agle criteria i atified The agitude of the two vector are calculated a follow: l l 2 2 ( ) + ( 5. + ) ( ) + ( ) The agitude criteria tate that; ( + p ) ( + p2 ) LL ( + p) ( + z ) ( + z ) LL ( + z ) or Thi reult i a gai equal to that which produced the poit o the root locu.

6 Digital coputer ca be ued to draw the root locu. The proce ued etail icreetig the gai ad calculatig the root of the characteritic equatio. The poit are the joit together to create the path of the root. While thi proce i uitable for aalizig yte, a good udertadig of why ad how the root locu portrate chage a varie, a well a a reult of addig pole ad zero i vital for ucceful yte deig. I what follow, a iple yet very effective ad rapid ethod for ketchig a approxiate root locu will be outlied with the reao for each tep. The Ed Poit of a Root Locu The characteritic equatio ca be rearraiged a follow: ( + p )( + p ) LL ( + p ) ( + z )( + z ) LL( + z ) 2 2 Whe 0 the above equatio becoe; ( + p )( + p ) LL( + p ) 2 0 Thi idicate that the root locii tart at the pole of the yte whe 0.

7 The Ed Poit of a Root Locu The characteritic equatio ca alo be arraiged a follow: ( + p)( + p2) LL( + p ) ( + z )( + z ) LL ( + z ) 2 Whe 4 the above equatio becoe; 0 2 ( + z )( + z ) LL ( + z ) Thi idicate that the root locii ed at the zero of the yte whe 4. Root locii tart at the yte pole (whe 0) ad ed at the yte zero (whe 4). The Nuber of Seget of a Root Locu Sice root locii ut tart at pole ad ed at zero, the it i fair to aue that the uber ditict eget of root locii i equal to the uber of pole or the uber of zero. If the uber of pole i the loop gai equatio i larger tha the uber of zero, thi ea that a uber of root locii eget equal to the differece betwee the uber of pole ad zero will be edig at ifiity. If the uber of zero i the loop gai equatio i larger tha the uber of pole, thi ea that a uber of root locii eget equal to the differece betwee the uber of zero ad pole will be tartig fro ifiity.

8 Shape of Coplex Seget of a Root Locu Sice pole ad root that occur off the real axi ut be i coplex cojugate pair, it follow that coplex portio of the root locii alway occur a coplex cojugate portio. Thee portio appear a irror iage of oe aother with the partig lie beig the real axi. Root Locii Seget o the Real Axi Root locii o the real axi ca be foud by applyig the agle criteria. Coider the cae how i the figure, ad aue a root idicated by the black dot. Applyig the agle criteria give: ( + z ) ( + p ) ( + p ) ( + p ) 2 3 ( θ θ θ ) Thi idicate that the aued poit violate the agle criteria ad hece caot be o a eget of the root locu Vector agle 80E Vector agle 0E

9 Root Locii Seget o the Real Axi or the cae whe the aued root locatio i betwee the firt ad ecod pole a how i the figure. Applyig the agle criteria give: ( + z ) ( + p ) ( + p ) ( + p ) 2 3 ( θ θ θ ) Thi idicate that the aued poit cofor to the agle criteria ad hece i o a eget of the root locu Vector agle 80E Vector agle 0E Root Locii Seget o the Real Axi or the cae whe the aued root locatio i betwee the ecod ad the third pole a how i the figure. Applyig the agle criteria give: ( + z ) ( + p ) ( + p ) ( + p ) 2 3 ( θ θ θ ) Thi idicate that the aued poit violate the agle criteria ad hece caot be o a eget of the root locu. Vector agle 80E Vector agle 0E

10 Root Locii Seget o the Real Axi or the cae whe the aued root locatio i beyod the third pole a how i the figure. Applyig the agle criteria give: ( + z ) ( + p ) ( + p ) ( + p ) 2 3 ( θ θ2 θ3 ) ( 2k + ) k 2 Thi idicate that the aued poit cofor to the agle criteria ad hece i o a eget of the root locu Vector agle 80E Vector agle 0E Root Locii Seget o the Real Axi It hould be oted at thi poit that coplex cojugate pole or zero do ot eed to be icluded ice the pair would cotribute ± 360 E a ca be ee below: With referece to the figure, auig a tet poit o the real axi, the agle of vector fro the coplex cojugate pole how to the tet poit are 2 ad 2 2. Their relatiohip are give a follow: θ θ θ 2 + θ θ + θ

11 Root Locii Seget o the Real Axi By ipectio of the previou reult, the followig rule ca be iferred: Root locii o the real axi will occur to the left of odd uber of pole ad zero. Agle of Ayptote of Root Locii Aue the yte how with three pole ad two coplex zero. urther aue a tet poit oewhere i the coplex doai. The agle that the vector fro the pole ad zero to that tet poit are how. If that poit lie o a eget of a root locu, the: 2 4 θ θ2 θ 3 + θ4 + θ 5 ± ( 2k + ) π A the tet poit i oved away fro the real axi ad toward ifiity, the agitude of the agle ted toward each other. 2 5

12 Agle of Ayptote of Root Locii I the liit whe the tet poit i at ifiity, the agitude of all the agle becoe the ae, i.e., θ θ2 θ 3 θ4 θ 5 α k The agle criterio i thi cae becoe: θ θ2 θ 3 + θ4 + θ 5 ± ( 2k + ) π ( P Z) α ( 2k + ) π k 2 4 The agle of the ayptote ca be calculated a follow: α k k + π ( 2 ) ( P Z) 2 5 Iterectio of the Ayptote with the Real Axi A the root locu head toward ifiity, ad auig a poit o the root locu at ifiity, the yte characteritic equatio i give by: ( + zi ) + i 0 ( + p ) j j The legth of the vector fro the tet poit to the fiite pole ad zero ted toward each other a the tet poit ted toward ifiity.

13 Iterectio of the Ayptote with the Real Axi I thi cae all the vector ca be aued to origiate fro a igle poit a. The characteritic equatio i thi cae i give by: ( + σ ) ( + σ ) ( + σ ) a a a Iterectio of the Ayptote with the Real Axi Expadig the previou expreio uig a bioial expaio give: ( )( ) σ a + σa + L! 2! A the tet poit,, head toward ifiity the above expreio ay be trucated after the the firt two ter i the deoiator, i.e., +! σ a

14 Iterectio of the Ayptote with the Real Axi Expadig the exact characteritic equatio give: zi + LL + zi i j j i + p + LL + p j j Agai a the tet poit head toward ifiity both the uerator ad deoiator of the above expreio ay be trucated after the the firt two ter a follow: i j z p i j Iterectio of the Ayptote with the Real Axi Carryig out log diviio of the deoiator by the uerator of the above expreio give: + z + p + ( ) z ( p z) ( p z) p z + + L L Agai a the tet poit head toward ifiity the characteritic equatio reultig fro the above expreio ay be trucated after the the firt two ter a follow: 0 + ( ) + p z

15 Iterectio of the Ayptote with the Real Axi The two for of the trucated characteritic equatio are: 0 + +! σa 0 + ( ) + p z Copairig thee two for reult i the followig cocluio: ( ) ( ) p z σ a Iterectio of the Ayptote with the Real Axi The iterectio of the ayptote with the real axi i give by the quatiet of the differece betwee the u of the pole ad the zero ad the differece betwee the uber of pole ad the zero, i.e., p - z a -

16 Break Away ad Break I Poit o the Real Axi A ha bee deduced before, root locii eget o the real axi exit to the left of odd uber of pole ad zero. I the evet that two pole exit ext to each other, the root locii origiatig fro the pair whe 0 have to leave the real axi at oe poit. A we travere the real axi over a root locu fro oe pole to the other the agitude of icreae till the ukow breakaway poit i reached where the value of peak. A we ove further alog the real axi i the ae directio we jup to aother root locu ad the value of tart decreaig till it reache zero agai at the other pole. Break Away ad Break I Poit o the Real Axi If there i root locii eget betwee two zero o the real axi the ae arguet ca be ade with the exceptio that a iiu of would be looked for itead of a axiu. The break away or break i to the real axi ca be foud by lookig for the axiu value of. The characteritic equatio ca be writte a: - j i ( + pj) ( + z ) i Differetiatig with repect to ad equatig the reult to zero the required brakaway or break i poit b ca be olved for.

17 Iterectio of the Root Locii with the Iagiary Axi The iterectio of the root locii with the iagiary axi ca be foud uig the Routh-Hurwitz criteria by lookig for the liitig value of ad olvig for the poit of cro-over ito the right half of the coplex plae fro the auxilliary equatio. The iterectio of the root locii with the iagiary axi ay alo be foud by ubtitutig i the characteritic equatio ad olvig for k ad the iterectio. Exit or Etry Agle fro Coplex Pole ad Zero The agle ade by a root locu leavig a coplex pole or arivig at a coplex zero ca be evaluated by auig a tet pole or zero i cloe proxiity of the coplex pole or zero, the applyig the agle criteria.