ROOT LOCUS TECHNIQUE 93 should be desiged differetly to eet differet specificatios depedig o its area of applicatio. We have observed i Sectio 6.4 of Chapter 6, how the variatio of a sigle paraeter like the forward path gai iflueces the trasiet respose- obviously due to variatio of the closed loop poles. The root locus ethod is a graphical tool to study the variatio of closed loop poles of a syste whe oly oe paraeter, orally the forward path gai, is allowed to vary. However, the locus of roots of the characteristic equatio of a ulti-loop syste ay also be ivestigated as i the case of a sigle-loop syste. Sice the root locus ethod provides graphical iforatio, it ay be used, specially i desig stage of a cotroller, to obtai qualitative iforatio regardig the stability ad perforace of the syste. We preset below a step by step procedure for the root locus ethod for cotiuous syste. We shall cosider typical trasfer fuctios of closed loop systes to illustrate the root locus techiques. 7.3 ROOT LOCUS TECHNIQUE The root-locus ethod has bee established as a useful tool for the aalysis ad desig of liear tie-ivariat cotrol systes. The root-locus diagra is essetially a graphical plot of the loci of the roots of the characteristic equatio of a syste as a fuctio of a real paraeter which varies fro to +. It gives a idicatio of the absolute stability ad, to soe extet, the relative stability of a cotrol syste with respect to the variatio of the syste paraeter. We shall cofie our discussios to oly positive values of, sice it ca be readily exteded to the case of egative values of. Cosider a closed loop syste C() s G () s = R() s + G () s H() s where the forward path gai is kept as a separate desig paraeter. The forward path gai ay iclude the process gai together with the error aplifier gai. Let the loop trasfer fuctio G(s)H(s) for a secod order syste be give by, G(s)H(s) = G (s)h(s) = ( s + 8 ) (7.2) ss ( + 5) Therefore, the characteristic equatio is obtaied as + G (s)h(s) = 0 s(s + 5) + (s + 8) = 0 (7.3) The two ope loop poles of the syste are located at p = 0 ad p 2 = 5 ad oe fiite zero is located at z = 8 whereas the other zero is at. The closed loop poles of the syste are the roots of the characteristic equatio (7.3) ad depeds o the value of. Whe = 0, the closed loop poles are idetical with the ope loop poles, but they are differet for 0. The roots of the above characteristic equatio are calculated for various values of ad are show i Table 7.. Fig. 7. shows the loci of the roots as is varied fro = 0 to =. We observe fro the figure that the secod order equatio (7.3) gives rise to two braches of the loci-oe beig p BEDz ad the other is p 2 BCDF. It is to be oted that as the value of is icreased, the two braches, origiatig at the fiite poles p ad p 2 approach each other ad eets at the poit B producig a double root there ad breaks away to follow separate paths BE ad BC. As is further icreased, the two braches approach each other ad eets at the poit D producig a double root there ad oves o to follow separate paths Dz ad DF (7.)
94 INTRODUCTION TO CONTROL ENGINEERING towards zeros located at z ad ifiity respectively. The poits B ad D are referred to as, respectively, breakaway ad breaki poits. Table 7. Value of Root located at s Root located at s 2 0 0 5 2 4.2020 3.0868 3.52 2.000 3.5 + j.9365 3.5 j.9365 5.0000 5 + j3.873 5 j3.873 0.0000 7.5 + j4.8734 7.5 j4.8734 20.0000 2.5 + j.9365 2.5 j.9365 20.7980 2.885 2.93 25 0 20 8 Fig. 7. Root loci of G (s)h(s) = ( s + 8) ss ( + 5) 7.3. Properties of Root Loci Fro the above observatios ad subsequet aalysis we record below soe iportat properties of the root loci of the characteristic equatio for a cotrol syste with the loop trasfer fuctio G (s)h(s) ad closed loop trasfer fuctio give by Equatio (7.). The roots of the characteristic equatio ust satisfy the equatio + G (s) H(s) = 0 (7.4)
ROOT LOCUS TECHNIQUE 95 or G (s)h(s) = (7.5) I order to satisfy the Equatio (7.5), the followig two coditios ust be satisfied siultaeously (A) Magitude Coditio (B) Phase Agle Coditios G (s)h(s) = 0 G (s)h(s) = (2k + )80 0 G (s)h(s) = 2k 80 where k is ay iteger i.e., k = 0, ±, ± 2,. (7.6) = odd ultiples of 80 (7.7) = eve ultiplies of 80 (7.8) It is to be oted that the phase agles are all easured i a couterclockwise directio fro a horizotal lie. So, ay poit i the plae of the root that satisfy the coditios (7.6) ad (7.7) or (7.8) siultaeously will be a poit o the root loci of the characteristic equatio (7.5). Of the above two coditios, the coditios o phase agle are ore critical tha that of the agitude coditio. We ca always fid a value of (ot ecessarily a iteger) that will satisfy the agitude coditio of (7.6). But the coditio o phase, deads that ay search poit s = s s will be a poit o the root loci if the phase agle is odd ( > 0) or eve ( < 0) ultiple of 80. The costructio of root loci, therefore, is essetially to search for a trial poit s = s s that satisfy the phase coditios (7.7), for > 0 or (7.8), for < 0 ad the fid the value of fro the aplitude coditio (7.6). Based o these observatios, the properties for root loci i the plae of the roots are recorded below for positive values of. The relevat chages i the properties ca be readily icorporated for egative values of. CAD tools like MATLAB ay be used for drawig root loci. However, the egieer ust be failiar with the rules i order to iterpret the plots.. Poits of the loci correspodig to = 0 The poits o the root loci correspodig to the value of = 0 are at the poles of G(s)H(s). This ay iclude the poles at ifiity where the uber of zeros is greater tha that of the poles (occurrig i a derived loop gai i soe desig approaches). If p j are the fiite poles ad z i are fiite zeroes of the loop gai G (s)h(s), the characteristic polyoial (7.4) ca be writte as F(s) = + G (s)h(s) = + Π ( s+ zi ) = 0 Π ( s + p ) j = j (7.9)
96 INTRODUCTION TO CONTROL ENGINEERING So, the agitude coditio (7.6) becoes Π ( s+ zi) = Π ( s + p ) (7.0) j = j Now as approaches zero, the right had side of Equatio (7.0) becoes ifiite which is satisfied whe s approaches the poles p j. 2. Poits of the loci correspodig to = The poits o the root loci correspodig to the value of = are at the zeros of G(s)H(s), icludig those at the ifiity. This is evidet fro Equatio (7.0), as approaches ifiity, the right had side approaches zero i value, which requires that s ust approach the zeros z i. Agai, if the deoiator is of order higher tha the uerator ( > ), the s =, akes the left had side zero correspodig to =. Therefore, soe roots are located at s =. I Fig. 7., the root loci origiates at the fiite poles p = 0 ad p 2 = 5 for = 0 ad teriates at the fiite zero at s = 8 ad at s 2 = for =. 3. Nuber of braches of separate root loci The uber of root loci is equal to the uber of fiite poles or zeros of G(s)H(s) whichever is greater. This is apparet, sice the root loci ust start at the poles ad teriate at the zeros of G(s)H(s), the uber of braches of loci is equal to the axiu of the two ubers fiite poles or zeros. I the syste of Equatio (7.2), sice the uber of fiite poles is 2, which is higher tha the fiite zero, which is i this case, the uber of braches of root loci is 2 (see Fig. 7.). 4. Syetry of root loci The root loci are syetric with respect to the real axis, sice the coplex roots occur i coplex cojugate pairs. 5. Root loci o the real axis Root loci are foud o a give sectio of the real axis of the s-plae oly if the total uber of real poles ad real zeros of G(s)H(s) to the right of the sectio is odd for > 0. With referece to the distributio of poles ad zeros of G (s)h(s) as show i Fig. 7.2, if we take the search poit s s, aywhere o the real axis betwee p ad p 2, the agular cotributio of the coplex cojugate poles p 3 ad p 4 at the search poit is 360, (this is also true for ay coplex cojugate zeros, whe preset). The poles ad zeros o the real axis to the right of s s each cotribute 80 with appropriate sig icluded. Therefore, with referece to Fig. 7.2, we ca write the phase coditio (7.7) as ϕ + ϕ 2 [θ + θ 2 + (θ 3 + θ 4 ) + θ 5 + θ 6 ] = (2k + )80 (7.) where ϕ, ϕ 2 are the phase agles cotributed by the zeros z ad z 2 at the search poit s s, ad θ i, to 6 are the phase agles cotributed by the poles p i, to 6. Coputig the agles fro Fig. 7.2, we have; 0 + 0 [80 + 0 + (360 ) + 0 + 0 ] = (2k + )80 or 540 =(2k + )80
ROOT LOCUS TECHNIQUE 97 This ca be satisfied with k = 2. This observatio is true if the search poit is take aywhere o the stretch of the real axis lyig betwee p ad p 2. Hece, the etire stretch of the real axis betwee p ad p 2 will be part of the root locus. I this way, we ca show that the part of the real axis lyig betwee p 5 ad z, as well as betwee p 6 ad z 2 are part of the root locus (vide Fig. 7.2). Fig. 7.2 Root loci o the real axis 6. Asyptotes of root loci For large values of s the root loci are asyptotic to the straight lies with agles give by θ k = ( 2k + ) 80 (7.2) where k = 0,, 2,... ; is the uber of fiite poles of G(s)H(s), ad is the uber of fiite zeros of G(s)H(s). Plottig the root loci are greatly facilitated if oe ca deterie the asyptotes approached by the various braches as s takes o large values. Now fro Equatio (7.9) we ca write or li s Π ( s+ zi) = Π ( s + p ) j = j = = s or = s s ad = s = (2k + )80 s = θ k = ( 2k + ) 80 (7.3) where θ k is the cotributio to phase agle by a fiite pole or zero at a search poit which is far away fro the.
98 INTRODUCTION TO CONTROL ENGINEERING 7. Itersectio of the asyptotes o the real axis The itersectio of the asyptotes lies oly o the real axis of the s-plae. The poits of itersectio of the asyptotes o the real axis is give by σ = Σ real parts of poles of G( s )H( s ) Σ real parts of zeros of G( s )H( s ) (7.4) The ceter of the liear asyptotes, ofte called the asyptote cetroid, is deteried by cosiderig the characteristic equatio (7.9). For large values of s, oly the higher-order ters eed be cosidered, so that the characteristic equatio reduces to + s s = 0 + = 0 This approxiate relatio idicates that the cetroid of ( ) asyptotes is at the origi, s = 0. However, a ore geeral approxiatio is obtaied if we cosider a characteristic equatio of the for + = 0 ( s σ ) with the cetroid at σ. Expadig the deoiator ad retaiig the first two ters of the above expressio, we have + s = 0 (7.5) s ( ) σ s +... Agai fro Equatio (7.9) we ca write + Π ( s+ zi ) ( s + bs +... + b) = + Π ( s + p ) s + as +... + a j = j = + (7.6) s + ( a b ) s +... + R( s)/ N( s) where b = Σ z i ad a = Σ p j j =, R(s) is the residue ad N(s) = Π ( s+ z i ) For large values of s the characteristic equatio ay be writte by cosiderig oly the first two ters of relatio (7.6) + s + ( a b ) s (7.7) Now, equatig the coefficiets of s i relatios (7.7) ad (7.5) we obtai (a b ) = ( ) σ Hece σ = a b, which is equivalet to Equatio (7.4).
ROOT LOCUS TECHNIQUE 99 8. Agles of departure fro coplex poles ad agles of arrival at coplex zeros The agle of departure of the root loci fro a coplex pole or the agle of arrival at a coplex zero of G(s)H(s) ca be deteried by cosiderig a search poit s s very close to the pole, or zero that satisfy the phase coditio of relatio (7.7) For illustratio, let us cosider the loop gai G(s)H(s) = s( s+ 4)( s+ 2+ j4)( s+ 2 j4 ). There are four fiite poles located at p = 0, p 2 = 4, p 3 = 2 j4 ad p 4 = 2 + j4, which are show i Fig. 7.3. We take a search poit s = s s very close to the coplex pole p 3 = 2 + j4 ad apply the coditios of phase agles as follows: s s (s s + 4) (s s + 2 + j4) (s s + 2 j4) = (2k + )80 or θ θ 2 θ 3 θ 4 = (2k + ) 80 Usig the values fro the graph, we have, 35 45 θ 3 90 = (2k + )80 Therefore, θ 3 = (2k + )80 270 = 540 270 = 270, for k = 2 6 3 4 2 s s p 3 Agle of departure fro coplex pole p 3 Iage axis 0 2 2 p 2 p 4 p 4 4 6 6 5 4 3 2 0 2 Real axis Fig. 7.3 Deteriatio of agle of departure fro coplex poles 9. Itersectio of the root loci with the iagiary axis i the s-doai The value of at the poit of itersectio of the root loci with the iagiary axis s = jω ay be deteried by usig the Routh-Hurwitz test. 0. Breakaway poits The breakaway poits o the root loci are poits at which ultiple-order roots lie (poit B i Fig. 7.). We have oted that the locus starts at the poles with = 0 ad teriates at the fiite zeroes or at with =. We have also foud i Fig. 7. that the part of the real axis betwee the pole s = 0 ad the pole s = 5 is lyig o the root loci. So the two braches of loci start at the poles with = 0 ad approach each other as is icreased util the two braches eet at poit B i Fig. 7..
200 INTRODUCTION TO CONTROL ENGINEERING Ay further icrease of will cause the roots breakaway fro the real axis. Therefore, so far as the real axis loci is cocered, as we ove fro oe pole o the real axis towards the other pole, the value of gradually icreases ad reaches a axiu value ad the decreases to zero at the other pole (see Fig. 7. ad 7.4). Siilarly, i case a portio of real axis lyig betwee two fiite zeros is a part of root locus, two braches of loci will breaki o the real axis ad ove towards the fiite zeros as approaches ifiity. It is apparet, that startig fro oe fiite zero lyig o the real axis correspodig to ifiite value of ad ovig towards the other fiite zero o the real axis where is agai ifiite, the value of will attai a iiu value i betwee the two real axis zeros (see Fig. 7.5). For coputig the breakaway or breaki poits, we ca rearrage the characteristic equatio to isolate the ultiplyig factor such that it ca be i the for = F (s), where F (s) does ot cotai. The breakaway ad breaki poits are the deteried fro the roots of the equatio obtaied by settig d = 0. ds.4.2 =.2 ax 0.8 Value of 0.4 ats= 5,=0 breakaway poit s = 3.0 ats=0,=0 0 5 4 3. 2 0 Variatio of s o the real axis i s-plae ss ( + 5) Fig. 7.4 The breakaway poit of = = F s + 8 (s) 50 Variatio of 40 30 20 5.6 0 is iiu at the break i poit 0 zeroats= 8 zeroats= 5 8 7 6.23 5 Variatio of s o the real axis ( s + 4s + 20) Fig. 7.5 The breaki poit of = ( s + 5)( s + 8 ) = F (s) 2
ROOT LOCUS TECHNIQUE 20. Values of o the root loci The value of at ay poit s o the root loci is deteried fro the followig equatio = G( s) H( s) Product of legths of vectors fro poles of G( s)h( s) to s = Product of legths of vectors draw fro zeros of G( s)h( s) to s (7.8) With referece to Fig. 7.6, we draw the costat dapig ratio lie θ = cos (0.707) ( s + 5) which eets the root locus of G (s) = ( s+ 2+ j4)( s+ 2 j4) at s = 5 + j5 ad s 2 = 5 j5. The value of at the closed loop poles s is foud as (see Fig. 7.6) : = L p. L p2 /L z = (3.6)(9.5)/5 = 6.0 6 s 4 Lp p 2 Iag axis of s 0 2 z Lz Lp 2 4 p 2 s 2 6 2 0 8 6 4 2 0 Real axis of s Fig. 7.6. Coputatio of at a specified root-locatio s o the locus diagra 7.4 STEP BY STEP PROCEDURE TO DRAW THE ROOT LOCUS DIAGRAM I order to fid the roots of the characteristic equatio graphically o the plae of the root, we shall preset a step by step procedure icorporatig the properties listed above by cosiderig soe illustrative exaples. Exaple 7. Cosider a cotrol syste with loop gai L(s) give by ( s + 6) L(s) = G(s)H(s) = s( s+ 2)( s+ 3) We are iterested i deteriig the locus of roots for positive values of i the rage 0.