6 Control Sytem Analyi and Deign by the Root-Locu Method 6 1 INTRODUCTION The baic characteritic of the tranient repone of a cloed-loop ytem i cloely related to the location of the cloed-loop pole. If the ytem ha a variable loop gain, then the location of the cloed-loop pole depend on the value of the loop gain choen. It i imptant, therefe, that the deigner know how the cloed-loop pole move in the plane a the loop gain i varied. From the deign viewpoint, in ome ytem imple gain adjutment may move the cloed-loop pole to deired location. Then the deign problem may become the election of an appropriate gain value. If the gain adjutment alone doe not yield a deired reult, addition of a compenat to the ytem will become neceary. (Thi ubject i dicued in detail in Section 6 6 through 6 9.) The cloed-loop pole are the root of the characteritic equation. Finding the root of the characteritic equation of degree higher than 3 i labiou and will need computer olution. (MATLAB provide a imple olution to thi problem.) However, jut finding the root of the characteritic equation may be of limited value, becaue a the gain of the open-loop tranfer function varie, the characteritic equation change and the computation mut be repeated. A imple method f finding the root of the characteritic equation ha been developed by W. R. Evan and ued extenively in control engineering. Thi method, called the root-locu method, i one in which the root of the characteritic equation 269
are plotted f all value of a ytem parameter. The root creponding to a particular value of thi parameter can then be located on the reulting graph. Note that the parameter i uually the gain, but any other variable of the open-loop tranfer function may be ued. Unle otherwie tated, we hall aume that the gain of the open-loop tranfer function i the parameter to be varied through all value, from zero to infinity. By uing the root-locu method the deigner can predict the effect on the location of the cloed-loop pole of varying the gain value adding open-loop pole and/ open-loop zero. Therefe, it i deired that the deigner have a good undertanding of the method f generating the root loci of the cloed-loop ytem, both by hand and by ue of a computer oftware program like MATLAB. In deigning a linear control ytem, we find that the root-locu method prove to be quite ueful, ince it indicate the manner in which the open-loop pole and zero hould be modified o that the repone meet ytem perfmance pecification. Thi method i particularly uited to obtaining approximate reult very quickly. Becaue generating the root loci by ue of MATLAB i very imple, one may think ketching the root loci by hand i a wate of time and efft. However, experience in ketching the root loci by hand i invaluable f interpreting computer-generated root loci, a well a f getting a rough idea of the root loci very quickly. Outline of the Chapter. The outline of the chapter i a follow: Section 6 1 ha preented an introduction to the root-locu method. Section 6 2 detail the concept underlying the root-locu method and preent the general procedure f ketching root loci uing illutrative example. Section 6 3 dicue generating root-locu plot with MATLAB. Section 6 4 treat a pecial cae when the cloed-loop ytem ha poitive feedback. Section 6 5 preent general apect of the root-locu approach to the deign of cloed-loop ytem. Section 6 6 dicue the control ytem deign by lead compenation. Section 6 7 treat the lag compenation technique. Section 6 8 deal with the lag lead compenation technique. Finally, Section 6 9 dicue the parallel compenation technique. 6 2 ROOT-LOCUS PLOTS Angle and Magnitude Condition. Conider the negative feedback ytem hown in Figure 6 1. The cloed-loop tranfer function i C() R() = G() 1 + G()H() (6 1) R() + G() C() Figure 6 1 Control ytem. H() 270 Chapter 6 / Control Sytem Analyi and Deign by the Root-Locu Method
The characteritic equation f thi cloed-loop ytem i obtained by etting the denominat of the right-hand ide of Equation (6 1) equal to zero. That i, 1 + G()H() = 0 G()H() =-1 (6 2) Here we aume that G()H() i a ratio of polynomial in. [It i noted that we can extend the analyi to the cae when G()H() involve the tranpt lag e T.] Since G()H() i a complex quantity, Equation (6 2) can be plit into two equation by equating the angle and magnitude of both ide, repectively, to obtain the following: Angle condition: /G()H() =;180 (2k + 1) (k = 0, 1, 2, p ) (6 3) Magnitude condition: G()H() = 1 (6 4) The value of that fulfill both the angle and magnitude condition are the root of the characteritic equation, the cloed-loop pole. A locu of the point in the complex plane atifying the angle condition alone i the root locu. The root of the characteritic equation (the cloed-loop pole) creponding to a given value of the gain can be determined from the magnitude condition. The detail of applying the angle and magnitude condition to obtain the cloed-loop pole are preented later in thi ection. In many cae, G()H() involve a gain parameter, and the characteritic equation may be written a 1 + A + z 1BA + z 2 B p A + z m B A + p 1 BA + p 2 B p A + p n B = 0 Then the root loci f the ytem are the loci of the cloed-loop pole a the gain i varied from zero to infinity. Note that to begin ketching the root loci of a ytem by the root-locu method we mut know the location of the pole and zero of G()H(). Remember that the angle of the complex quantitie iginating from the open-loop pole and open-loop zero to the tet point are meaured in the counterclockwie direction. F example, if G()H() i given by G()H() = A + z 1 B A + p 1 BA + p 2 BA + p 3 BA + p 4 B Section 6 2 / Root-Locu Plot 271
Tet point A 2 u 2 A 4 B 1 A 3 p 2 A 1 u 4 f 1 u 1 u 4 Tet point u 2 p 2 u 1 Figure 6 2 (a) and (b) Diagram howing angle meaurement from open-loop pole and open-loop zero to tet point. p 4 z 1 (a) p 3 u 3 p 1 0 p 4 z 1 f 1 p 3 (b) u 3 p 1 0 where p 2 and p 3 are complex-conjugate pole, then the angle of G()H() i /G()H() = f 1 - u 1 - u 2 - u 3 - u 4 where f 1, u 1, u 2, u 3, and u 4 are meaured counterclockwie a hown in Figure 6 2(a) and (b). The magnitude of G()H() f thi ytem i G()H() = B 1 A 1 A 2 A 3 A 4 where A 1, A 2, A 3, A 4, and B 1 are the magnitude of the complex quantitie +p 1, +p 2, +p 3, +p 4, and +z 1, repectively, a hown in Figure 6 2(a). Note that, becaue the open-loop complex-conjugate pole and complex-conjugate zero, if any, are alway located ymmetrically about the real axi, the root loci are alway ymmetrical with repect to thi axi. Therefe, we only need to contruct the upper half of the root loci and draw the mirr image of the upper half in the lower-half plane. Illutrative Example. In what follow, two illutrative example f contructing root-locu plot will be preented. Although computer approache to the contruction of the root loci are eaily available, here we hall ue graphical computation, combined with inpection, to determine the root loci upon which the root of the characteritic equation of the cloed-loop ytem mut lie. Such a graphical approach will enhance undertanding of how the cloed-loop pole move in the complex plane a the openloop pole and zero are moved. Although we employ only imple ytem f illutrative purpoe, the procedure f finding the root loci i no me complicated f higherder ytem. Becaue graphical meaurement of angle and magnitude are involved in the analyi, we find it neceary to ue the ame diviion on the abcia a on the dinate axi when ketching the root locu on graph paper. 272 Chapter 6 / Control Sytem Analyi and Deign by the Root-Locu Method
EXAMPLE 6 1 Conider the negative feedback ytem hown in Figure 6 3. (We aume that the value of gain i nonnegative.) F thi ytem, G() = ( + 1)( + 2), H() = 1 Let u ketch the root-locu plot and then determine the value of uch that the damping ratio z of a pair of dominant complex-conjugate cloed-loop pole i 0.5. F the given ytem, the angle condition become The magnitude condition i /G() = n ( + 1)( + 2) =-/ - / + 1 - / + 2 =;180 (2k + 1) (k = 0, 1, 2, p ) G() = 2 ( + 1)( + 2) 2 = 1 A typical procedure f ketching the root-locu plot i a follow: 1. Determine the root loci on the real axi. The firt tep in contructing a root-locu plot i to locate the open-loop pole, =0, = 1, and = 2, in the complex plane. (There are no openloop zero in thi ytem.) The location of the open-loop pole are indicated by croe. (The location of the open-loop zero in thi book will be indicated by mall circle.) Note that the tarting point of the root loci (the point creponding to =0) are open-loop pole. The number of individual root loci f thi ytem i three, which i the ame a the number of open-loop pole. To determine the root loci on the real axi, we elect a tet point,. If the tet point i on the poitive real axi, then / = / + 1 = / + 2 = 0 Thi how that the angle condition cannot be atified. Hence, there i no root locu on the poitive real axi. Next, elect a tet point on the negative real axi between 0 and 1. Then Thu / = 180, / + 1 = / + 2 = 0 - / - / + 1 - / + 2 =-180 and the angle condition i atified. Therefe, the ption of the negative real axi between 0 and 1 fm a ption of the root locu. If a tet point i elected between 1 and 2, then and / = / + 1 = 180, / + 2 = 0 - / - / + 1 - / + 2 =-360 R() + ( + 1) ( + 2) C() Figure 6 3 Control ytem. Section 6 2 / Root-Locu Plot 273
It can be een that the angle condition i not atified. Therefe, the negative real axi from 1 to 2 i not a part of the root locu. Similarly, if a tet point i located on the negative real axi from 2 to q, the angle condition i atified. Thu, root loci exit on the negative real axi between 0 and 1 and between 2 and q. 2. Determine the aymptote of the root loci. The aymptote of the root loci a approache infinity can be determined a follow: If a tet point i elected very far from the igin, then and the angle condition become lim G() = lim Sq Sq ( + 1)( + 2) = lim Sq 3-3/ =;180 (2k + 1) (k = 0, 1, 2, p ) Angle of aymptote = ;180 (2k + 1) 3 (k = 0, 1, 2, p ) Since the angle repeat itelf a k i varied, the ditinct angle f the aymptote are determined a 60, 60, and 180. Thu, there are three aymptote. The one having the angle of 180 i the negative real axi. Befe we can draw thee aymptote in the complex plane, we mut find the point where they interect the real axi. Since G() = if a tet point i located very far from the igin, then G() may be written a G() = ( + 1)( + 2) 3 + 3 2 + p F large value of, thi lat equation may be approximated by G() ( + 1) 3 (6 5) A root-locu diagram of G() given by Equation (6 5) conit of three traight line. Thi can be een a follow: The equation of the root locu i which can be written a n 3 =;180 (2k + 1) ( + 1) -3/ + 1 =;180 (2k + 1) / + 1 =;60 (2k + 1) 274 Chapter 6 / Control Sytem Analyi and Deign by the Root-Locu Method
By ubtituting =+ into thi lat equation, we obtain / + + 1 =;60 (2k + 1) tan -1 v = 60, -60, 0 + 1 Taking the tangent of both ide of thi lat equation, which can be written a v = 13, -13, 0 + 1 + 1 - v 13 = 0, + 1 + v 13 = 0, v = 0 Thee three equation repreent three traight line, a hown in Figure 6 4.The three traight line hown are the aymptote. They meet at point = 1. Thu, the abcia of the interection of the aymptote and the real axi i obtained by etting the denominat of the right-hand ide of Equation (6 5) equal to zero and olving f. The aymptote are almot part of the root loci in region very far from the igin. 3. Determine the breakaway point. To plot root loci accurately, we mut find the breakaway point, where the root-locu branche iginating from the pole at 0 and 1 break away (a i increaed) from the real axi and move into the complex plane. The breakaway point crepond to a point in the plane where multiple root of the characteritic equation occur. A imple method f finding the breakaway point i available. We hall preent thi method in the following: Let u write the characteritic equation a f() = B() + A() = 0 (6 6) j 3 v = 0 + 1 v = 0 3 1 0 + 1 + v = 0 3 j 3 Figure 6 4 Three aymptote. Section 6 2 / Root-Locu Plot 275
where A() and B() do not contain. Note that f()=0 ha multiple root at point where df() = 0 d Thi can be een a follow: Suppoe that f() ha multiple root of der r, where r 2.Then f() may be written a f() = A - 1 B r A - 2 B p A - n B Now we differentiate thi equation with repect to and evaluate df()/d at = 1. Then we get df() 2 = 0 (6 7) d = 1 Thi mean that multiple root of f() will atify Equation (6 7). From Equation (6 6), we obtain df() = B () + A () = 0 (6 8) d where A () = da(), B () = db() d d The particular value of that will yield multiple root of the characteritic equation i obtained from Equation (6 8) a =- B () A () If we ubtitute thi value of into Equation (6 6), we get f() = B() - B () A () A() = 0 B()A () - B ()A() = 0 (6 9) If Equation (6 9) i olved f, the point where multiple root occur can be obtained. On the other hand, from Equation (6 6) we obtain and d d =- B() A() =-B ()A() - B()A () A 2 () If d/d i et equal to zero, we get the ame equation a Equation (6 9). Therefe, the breakaway point can be imply determined from the root of d d = 0 It hould be noted that not all the olution of Equation (6 9) of d/d=0 crepond to actual breakaway point. If a point at which d/d=0 i on a root locu, it i an actual breakaway break-in point. Stated differently, if at a point at which d/d=0 the value of take a real poitive value, then that point i an actual breakaway break-in point. 276 Chapter 6 / Control Sytem Analyi and Deign by the Root-Locu Method
F the preent example, the characteritic equation G()+1=0 i given by By etting d/d=0, we obtain Since the breakaway point mut lie on a root locu between 0 and 1, it i clear that = 0.4226 crepond to the actual breakaway point. Point = 1.5774 i not on the root locu. Hence, thi point i not an actual breakaway break-in point. In fact, evaluation of the value of creponding to = 0.4226 and = 1.5774 yield 4. Determine the point where the root loci cro the imaginary axi. Thee point can be found by ue of Routh tability criterion a follow: Since the characteritic equation f the preent ytem i the Routh array become The value of that make the 1 term in the firt column equal zero i =6. The croing point on the imaginary axi can then be found by olving the auxiliary equation obtained from the 2 row; that i, 3 2 + = 3 2 + 6 = 0 which yield =;j12 The frequencie at the croing point on the imaginary axi are thu v =;12. The gain value creponding to the croing point i =6. An alternative approach i to let = in the characteritic equation, equate both the real part and the imaginary part to zero, and then olve f v and. F the preent ytem, the characteritic equation, with =,i ( + 1)( + 2) + 1 = 0 =-A 3 + 3 2 + 2B d d =-A32 + 6 + 2B = 0 =-0.4226, =-1.5774 = 0.3849, f =-0.4226 =-0.3849, f =-1.5774 3 + 3 2 + 2 + = 0 3 2 1 0 1 3 6-3 () 3 + 3() 2 + 2() + = 0 A - 3v 2 B + ja2v - v 3 B = 0 Equating both the real and imaginary part of thi lat equation to zero, repectively, we obtain - 3v 2 = 0, 2v - v 3 = 0 Section 6 2 / Root-Locu Plot 277 2
+ 1 + 2 j1 2 u 1 u 3 u 2 1 0 Figure 6 5 Contruction of root locu. j1 from which v =;12, = 6 v = 0, = 0 Thu, root loci cro the imaginary axi at v =;12, and the value of at the croing point i 6. Alo, a root-locu branch on the real axi touche the imaginary axi at v=0. The value of i zero at thi point. 5. Chooe a tet point in the broad neighbhood of the axi and the igin, a hown in Figure 6 5, and apply the angle condition. If a tet point i on the root loci, then the um of the three angle, u 1 +u 2 +u 3, mut be 180. If the tet point doe not atify the angle condition, elect another tet point until it atifie the condition. (The um of the angle at the tet point will indicate the direction in which the tet point hould be moved.) Continue thi proce and locate a ufficient number of point atifying the angle condition. 6. Draw the root loci, baed on the infmation obtained in the fegoing tep, a hown in Figure 6 6. j2 ` = 6 = 6 = 1.0383 60 j1 = 1.0383 3 2 1 0 1 j1 j2 Figure 6 6 Root-locu plot. ` 278 Chapter 6 / Control Sytem Analyi and Deign by the Root-Locu Method
7. Determine a pair of dominant complex-conjugate cloed-loop pole uch that the damping ratio z i 0.5. Cloed-loop pole with z=0.5 lie on line paing through the igin and making the angle ;co -1 z =;co -1 0.5 =;60 with the negative real axi. From Figure 6 6, uch cloedloop pole having z=0.5 are obtained a follow: 1 =-0.3337 + j0.5780, 2 =-0.3337 - j0.5780 The value of that yield uch pole i found from the magnitude condition a follow: = ( + 1)( + 2) =-0.3337 + j0.5780 = 1.0383 Uing thi value of, the third pole i found at = 2.3326. Note that, from tep 4, it can be een that f =6 the dominant cloed-loop pole lie on the imaginary axi at =;j12. With thi value of, the ytem will exhibit utained ocillation. F >6, the dominant cloed-loop pole lie in the right-half plane, reulting in an untable ytem. Finally, note that, if neceary, the root loci can be eaily graduated in term of by ue of the magnitude condition. We imply pick out a point on a root locu, meaure the magnitude of the three complex quantitie, +1, and +2, and multiply thee magnitude; the product i equal to the gain value at that point, + 1 + 2 = Graduation of the root loci can be done eaily by ue of MATLAB. (See Section 6 3.) EXAMPLE 6 2 In thi example, we hall ketch the root-locu plot of a ytem with complex-conjugate openloop pole. Conider the negative feedback ytem hown in Figure 6 7. F thi ytem, G() = ( + 2) 2 + 2 + 3, H() = 1 where 0. It i een that G() ha a pair of complex-conjugate pole at =-1 + j12, =-1 - j12 A typical procedure f ketching the root-locu plot i a follow: 1. Determine the root loci on the real axi. F any tet point on the real axi, the um of the angular contribution of the complex-conjugate pole i 360, a hown in Figure 6 8. Thu the net effect of the complex-conjugate pole i zero on the real axi.the location of the root locu on the real axi i determined from the open-loop zero on the negative real axi.a imple tet reveal that a ection of the negative real axi, that between 2 and q, i a part of the root locu. It i noted that, ince thi locu lie between two zero (at = 2 and = q), it i actually a part of two root loci, each of which tart from one of the two complex-conjugate pole. In other wd, two root loci break in the part of the negative real axi between 2 and q. R() + ( + 2) 2 + 2 + 3 C() Figure 6 7 Control ytem. Section 6 2 / Root-Locu Plot 279
u 1 j 2 Figure 6 8 Determination of the root locu on the real axi. Tet 2 1 0 point u 2 j 2 Since there are two open-loop pole and one zero, there i one aymptote, which coincide with the negative real axi. 2. Determine the angle of departure from the complex-conjugate open-loop pole. The preence of a pair of complex-conjugate open-loop pole require the determination of the angle of departure from thee pole. nowledge of thi angle i imptant, ince the root locu near a complex pole yield infmation a to whether the locu iginating from the complex pole migrate toward the real axi extend toward the aymptote. Referring to Figure 6 9, if we chooe a tet point and move it in the very vicinity of the complex open-loop pole at = p 1, we find that the um of the angular contribution from the pole at =p 2 and zero at = z 1 to the tet point can be conidered remaining the ame. If the tet point i to be on the root locu, then the um of f1 œ, u 1, and -u2 œ mut be ;180 (2k + 1), where k=0,1,2,p. Thu, in the example, f œ 1 - Au 1 + u œ 2B =;180 (2k + 1) u 1 = 180 - u œ 2 + f œ 1 = 180 - u 2 + f 1 The angle of departure i then u 1 = 180 - u 2 + f 1 = 180-90 + 55 = 145 u 1 p 1 z 1 f9 1 f 1 0 Figure 6 9 Determination of the angle of departure. p 2 u9 2 u 2 280 Chapter 6 / Control Sytem Analyi and Deign by the Root-Locu Method
Since the root locu i ymmetric about the real axi, the angle of departure from the pole at = p 2 i 145. 3. Determine the break-in point. A break-in point exit where a pair of root-locu branche coalece a i increaed. F thi problem, the break-in point can be found a follow: Since we have which give =- 2 + 2 + 3 + 2 d d =-(2 + 2)( + 2) - A2 + 2 + 3B = 0 ( + 2) 2 2 + 4 + 1 = 0 =-3.7320 =-0.2680 Notice that point = 3.7320 i on the root locu. Hence thi point i an actual break-in point. (Note that at point = 3.7320 the creponding gain value i =5.4641.) Since point = 0.2680 i not on the root locu, it cannot be a break-in point. (F point = 0.2680, the creponding gain value i = 1.4641.) 4. Sketch a root-locu plot, baed on the infmation obtained in the fegoing tep. To determine accurate root loci, everal point mut be found by trial and err between the breakin point and the complex open-loop pole. (To facilitate ketching the root-locu plot, we hould find the direction in which the tet point hould be moved by mentally umming up the change on the angle of the pole and zero.) Figure 6 10 how a complete root-locu plot f the ytem conidered. z = 0.7 line 145 j2 j1 4 3 2 1 0 1 j1 j2 Figure 6 10 Root-locu plot. Section 6 2 / Root-Locu Plot 281
The value of the gain at any point on root locu can be found by applying the magnitude condition by ue of MATLAB (ee Section 6 3). F example, the value of at which the complex-conjugate cloed-loop pole have the damping ratio z=0.7 can be found by locating the root, a hown in Figure 6 10, and computing the value of a follow: Or ue MATLAB to find the value of. (See Section 6 4.) It i noted that in thi ytem the root locu in the complex plane i a part of a circle. Such a circular root locu will not occur in mot ytem. Circular root loci may occur in ytem that involve two pole and one zero, two pole and two zero, one pole and two zero. Even in uch ytem, whether circular root loci occur depend on the location of pole and zero involved. To how the occurrence of a circular root locu in the preent ytem, we need to derive the equation f the root locu. F the preent ytem, the angle condition i If =+ i ubtituted into thi lat equation, we obtain which can be written a Taking tangent of both ide of thi lat equation uing the relationhip we obtain which can be implified to A + 1 - j12ba + 1 + j12b = 2 2 + 2 v tan -1 a + 2 b - tan-1 a v - 12 + 1 b - tan-1 a v + 12 b =;180 (2k + 1) + 1 tan -1 a v - 12 + 1 b + tan-1 a v + 12 + 1 b = v tan-1 a b ; 180 (2k + 1) + 2 tan x ; tan y tan (x ; y) = 1 < tan x tan y tan c tan -1 a v - 12 + 1 b + tan-1 a v + 12 + 1 b d = tan v - 12 + 1 + v + 12 + 1 1 - a v - 12 + 12 bav + 1 + 1 b 2v( + 1) ( + 1) 2 - Av 2-2B = 1 < = v + 2 vc( + 2) 2 + v 2-3D = 0 =-1.67 + j1.70 c v tan-1 a b ; 180 (2k + 1)d + 2 v + 2 ; 0 = 1.34 / + 2 - / + 1 - j12 - / + 1 + j12 =;180 (2k + 1) / + 2 + - / + 1 + - j12 - / + 1 + + j12 =;180 (2k + 1) v + 2 * 0 (6 10) Thi lat equation i equivalent to v = 0 ( + 2) 2 + v 2 = A 13B 2 282 Chapter 6 / Control Sytem Analyi and Deign by the Root-Locu Method
Thee two equation are the equation f the root loci f the preent ytem. Notice that the firt equation, v=0, i the equation f the real axi. The real axi from = 2 to = q crepond to a root locu f 0. The remaining part of the real axi crepond to a root locu when i negative. (In the preent ytem, i nonnegative.) (Note that < 0 crepond to the poitive-feedback cae.) The econd equation f the root locu i an equation of a circle with center at = 2, v=0 and the radiu equal to 13. That part of the circle to the left of the complex-conjugate pole crepond to a root locu f 0. The remaining part of the circle crepond to a root locu when i negative. It i imptant to note that eaily interpretable equation f the root locu can be derived f imple ytem only. F complicated ytem having many pole and zero, any attempt to derive equation f the root loci i dicouraged. Such derived equation are very complicated and their configuration in the complex plane i difficult to viualize. General Rule f Contructing Root Loci. F a complicated ytem with many open-loop pole and zero, contructing a root-locu plot may eem complicated, but actually it i not difficult if the rule f contructing the root loci are applied. By locating particular point and aymptote and by computing angle of departure from complex pole and angle of arrival at complex zero, we can contruct the general fm of the root loci without difficulty. We hall now ummarize the general rule and procedure f contructing the root loci of the negative feedback control ytem hown in Figure 6 11. Firt, obtain the characteritic equation 1 + G()H() = 0 Then rearrange thi equation o that the parameter of interet appear a the multiplying fact in the fm 1 + A + z 1BA + z 2 B p A + z m B A + p 1 BA + p 2 B p A + p n B (6 11) In the preent dicuion, we aume that the parameter of interet i the gain, where >0. (If <0, which crepond to the poitive-feedback cae, the angle condition mut be modified. See Section 6 4.) Note, however, that the method i till applicable to ytem with parameter of interet other than gain. (See Section 6 6.) 1. Locate the pole and zero of G()H() on the plane.the root-locu branche tart from open-loop pole and terminate at zero (finite zero zero at infinity). From the facted fm of the open-loop tranfer function, locate the open-loop pole and zero in the plane. CNote that the open-loop zero are the zero of G()H(), while the cloed-loop zero conit of the zero of G() and the pole of H().D = 0 R() + G() C() Figure 6 11 Control ytem. H() Section 6 2 / Root-Locu Plot 283
Note that the root loci are ymmetrical about the real axi of the plane, becaue the complex pole and complex zero occur only in conjugate pair. A root-locu plot will have jut a many branche a there are root of the characteritic equation. Since the number of open-loop pole generally exceed that of zero, the number of branche equal that of pole. If the number of cloed-loop pole i the ame a the number of open-loop pole, then the number of individual root-locu branche terminating at finite open-loop zero i equal to the number m of the open-loop zero. The remaining n-m branche terminate at infinity (n-m implicit zero at infinity) along aymptote. If we include pole and zero at infinity, the number of open-loop pole i equal to that of open-loop zero. Hence we can alway tate that the root loci tart at the pole of G()H() and end at the zero of G()H(), a increae from zero to infinity, where the pole and zero include both thoe in the finite plane and thoe at infinity. 2. Determine the root loci on the real axi. Root loci on the real axi are determined by open-loop pole and zero lying on it. The complex-conjugate pole and complexconjugate zero of the open-loop tranfer function have no effect on the location of the root loci on the real axi becaue the angle contribution of a pair of complex-conjugate pole complex-conjugate zero i 360 on the real axi. Each ption of the root locu on the real axi extend over a range from a pole zero to another pole zero. In contructing the root loci on the real axi, chooe a tet point on it. If the total number of real pole and real zero to the right of thi tet point i odd, then thi point lie on a root locu. If the open-loop pole and open-loop zero are imple pole and imple zero, then the root locu and it complement fm alternate egment along the real axi. 3. Determine the aymptote of root loci. If the tet point i located far from the igin, then the angle of each complex quantity may be conidered the ame. One open-loop zero and one open-loop pole then cancel the effect of the other. Therefe, the root loci f very large value of mut be aymptotic to traight line whoe angle (lope) are given by Angle of aymptote = ;180 (2k + 1) n - m (k = 0, 1, 2, p ) where n = m = number of finite pole of G()H() number of finite zero of G()H() Here, k=0 crepond to the aymptote with the mallet angle with the real axi. Although k aume an infinite number of value, a k i increaed the angle repeat itelf, and the number of ditinct aymptote i n-m. All the aymptote interect at a point on the real axi. The point at which they do o i obtained a follow: If both the numerat and denominat of the open-loop tranfer function are expanded, the reult i G()H() = Cm + Az 1 + z 2 + p + z m B m - 1 + p + z 1 z 2 p zm D n + Ap 1 + p 2 + p + p n B n - 1 + p + p 1 p 2 p pn 284 Chapter 6 / Control Sytem Analyi and Deign by the Root-Locu Method
If a tet point i located very far from the igin, then by dividing the denominat by the numerat, it i poible to write G()H() a G()H() = n - m + CAp 1 + p 2 + p + p n B - Az 1 + z 2 + p + z m BD n - m - 1 + p G()H() = (6 12) c + Ap 1 + p 2 + p + p n B - Az 1 + z 2 + p + z m B n - m d n - m The abcia of the interection of the aymptote and the real axi i then obtained by etting the denominat of the right-hand ide of Equation (6 12) equal to zero and olving f, =- Ap 1 + p 2 + p + p n B - Az 1 + z 2 + p + z m B (6 13) n - m [Example 6 1 how why Equation (6 13) give the interection.] Once thi interection i determined, the aymptote can be readily drawn in the complex plane. It i imptant to note that the aymptote how the behavi of the root loci f 1. A root-locu branch may lie on one ide of the creponding aymptote may cro the creponding aymptote from one ide to the other ide. 4. Find the breakaway and break-in point. Becaue of the conjugate ymmetry of the root loci, the breakaway point and break-in point either lie on the real axi occur in complex-conjugate pair. If a root locu lie between two adjacent open-loop pole on the real axi, then there exit at leat one breakaway point between the two pole. Similarly, if the root locu lie between two adjacent zero (one zero may be located at q) on the real axi, then there alway exit at leat one break-in point between the two zero. If the root locu lie between an open-loop pole and a zero (finite infinite) on the real axi, then there may exit no breakaway break-in point there may exit both breakaway and break-in point. Suppoe that the characteritic equation i given by The breakaway point and break-in point crepond to multiple root of the characteritic equation. Hence, a dicued in Example 6 1, the breakaway and break-in point can be determined from the root of d d B() + A() = 0 =-B ()A() - B()A () A 2 () (6 14) where the prime indicate differentiation with repect to. It i imptant to note that the breakaway point and break-in point mut be the root of Equation (6 14), but not all root of Equation (6 14) are breakaway break-in point. If a real root of Equation (6 14) lie on the root-locu ption of the real axi, then it i an actual breakaway break-in point. If a real root of Equation (6 14) i not on the root-locu ption of the real axi, then thi root crepond to neither a breakaway point n a break-in point. Section 6 2 / Root-Locu Plot 285 = 0
If two root = 1 and = 1 of Equation (6 14) are a complex-conjugate pair and if it i not certain whether they are on root loci, then it i neceary to check the creponding value. If the value of creponding to a root = 1 of d d = 0 i poitive, point = 1 i an actual breakaway break-in point. (Since i aumed to be nonnegative, if the value of thu obtained i negative, a complex quantity, then point = 1 i neither a breakaway n a break-in point.) 5. Determine the angle of departure (angle of arrival) of the root locu from a complex pole (at a complex zero). To ketch the root loci with reaonable accuracy, we mut find the direction of the root loci near the complex pole and zero. If a tet point i choen and moved in the very vicinity of a complex pole ( complex zero), the um of the angular contribution from all other pole and zero can be conidered to remain the ame. Therefe, the angle of departure ( angle of arrival) of the root locu from a complex pole ( at a complex zero) can be found by ubtracting from 180 the um of all the angle of vect from all other pole and zero to the complex pole ( complex zero) in quetion, with appropriate ign included. Angle of departure from a complex pole=180 (um of the angle of vect to a complex pole in quetion from other pole) ± (um of the angle of vect to a complex pole in quetion from zero) Angle of arrival at a complex zero=180 (um of the angle of vect to a complex zero in quetion from other zero) ± (um of the angle of vect to a complex zero in quetion from pole) The angle of departure i hown in Figure 6 12. 6. Find the point where the root loci may cro the imaginary axi. The point where the root loci interect the axi can be found eaily by (a) ue of Routh tability criterion (b) letting = in the characteritic equation, equating both the real part and the imaginary part to zero, and olving f v and.the value of v thu found give the frequencie at which root loci cro the imaginary axi. The value creponding to each croing frequency give the gain at the croing point. 7. Taking a erie of tet point in the broad neighbhood of the igin of the plane, ketch the root loci. Determine the root loci in the broad neighbhood of the axi and the igin. The mot imptant part of the root loci i on neither the real axi n the aymptote but i in the broad neighbhood of the axi and the igin.the hape Angle of departure f u 1 Figure 6 12 Contruction of the root locu. [Angle of departure =180 - (u 1 +u 2 )+f.] u 2 0 286 Chapter 6 / Control Sytem Analyi and Deign by the Root-Locu Method
of the root loci in thi imptant region in the plane mut be obtained with reaonable accuracy. (If accurate hape of the root loci i needed, MATLAB may be ued rather than hand calculation of the exact hape of the root loci.) 8. Determine cloed-loop pole. A particular point on each root-locu branch will be a cloed-loop pole if the value of at that point atifie the magnitude condition. Converely, the magnitude condition enable u to determine the value of the gain at any pecific root location on the locu. (If neceary, the root loci may be graduated in term of. The root loci are continuou with.) The value of creponding to any point on a root locu can be obtained uing the magnitude condition, product of length between point to pole = product of length between point to zero Thi value can be evaluated either graphically analytically. (MATLAB can be ued f graduating the root loci with. See Section 6 3.) If the gain of the open-loop tranfer function i given in the problem, then by applying the magnitude condition, we can find the crect location of the cloed-loop pole f a given on each branch of the root loci by a trial-and-err approach by ue of MATLAB, which will be preented in Section 6 3. Comment on the Root-Locu Plot. It i noted that the characteritic equation of the negative feedback control ytem whoe open-loop tranfer function i G()H() = Am + b 1 m - 1 + p + b m B n + a 1 n - 1 + p (n m) + a n i an nth-degree algebraic equation in. If the der of the numerat of G()H() i lower than that of the denominat by two me (which mean that there are two me zero at infinity), then the coefficient a 1 i the negative um of the root of the equation and i independent of. In uch a cae, if ome of the root move on the locu toward the left a i increaed, then the other root mut move toward the right a i increaed. Thi infmation i helpful in finding the general hape of the root loci. It i alo noted that a light change in the pole zero configuration may caue ignificant change in the root-locu configuration. Figure 6 13 demontrate the fact that a light change in the location of a zero pole will make the root-locu configuration look quite different. Figure 6 13 Root-locu plot. Section 6 2 / Root-Locu Plot 287
Cancellation of Pole of G() with Zero of H(). It i imptant to note that if the denominat of G() and the numerat of H() involve common fact, then the creponding open-loop pole and zero will cancel each other, reducing the degree of the characteritic equation by one me. F example, conider the ytem hown in Figure 6 14(a). (Thi ytem ha velocity feedback.) By modifying the block diagram of Figure 6 14(a) to that hown in Figure 6 14(b), it i clearly een that G() and H() have a common fact +1. The cloed-loop tranfer function C()/R() i C() R() = The characteritic equation i Becaue of the cancellation of the term (+1) appearing in G() and H(), however, we have ( + 1) 1 + G()H() = 1 + ( + 1)( + 2) The reduced characteritic equation i ( + 1)( + 2) + ( + 1) C( + 2) + D( + 1) = 0 = ( + 2) + ( + 2) ( + 2) + = 0 The root-locu plot of G()H() doe not how all the root of the characteritic equation, only the root of the reduced equation. To obtain the complete et of cloed-loop pole, we mut add the canceled pole of G()H() to thoe cloed-loop pole obtained from the root-locu plot of G()H(). The imptant thing to remember i that the canceled pole of G()H() i a cloed-loop pole of the ytem, a een from Figure 6 14(c). R() + + ( + 1) ( + 2) 1 C() (a) Figure 6 14 (a) Control ytem with velocity feedback; (b) and (c) modified block diagram. R() + G() ( + 1) ( + 2) + 1 H() (b) C() R() + ( + 2) (c) 1 + 1 C() 288 Chapter 6 / Control Sytem Analyi and Deign by the Root-Locu Method
Typical Pole Zero Configuration and Creponding Root Loci. In ummarizing, we how everal open-loop pole zero configuration and their creponding root loci in Table 6 1. The pattern of the root loci depend only on the relative eparation of the open-loop pole and zero. If the number of open-loop pole exceed the number of finite zero by three me, there i a value of the gain beyond which root loci enter the right-half plane, and thu the ytem can become untable.a table ytem mut have all it cloed-loop pole in the left-half plane. Table 6 1 Open-Loop Pole Zero Configuration and the Creponding Root Loci Section 6 2 / Root-Locu Plot 289