If the system has a v ariable loop gain then the location of the closedloop poles depends on the value of the loop gain chosen It is important therefore that the designer know how the closedloop poles move in the splane as the loop gain is varied Fr ID: 27082 Download Pdf

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If the system has a v ariable loop gain then the location of the closedloop poles depends on the value of the loop gain chosen It is important therefore that the designer know how the closedloop poles move in the splane as the loop gain is varied Fr

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Root Locus Analysis

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Introduction The basic characteristic of the transient response of a clos ed-loop system is closely related to the location of the closed-loop poles. If the system has a v ariable loop gain, then the location of the closed-loop poles depends on the value of the loop gain chosen. It is important therefore, that the designer know how the closed-loop poles move in the s-plane as the loop gain is varied. From the designer viewpoint, in some systems simple gain adjustments may move the closed-loop poles to the desired locations. Then th e design

problem may become the selection of an appropriate gain value. In this chapter w e discuss the design problems that involve the selection of a particular parameter value (usua lly the open-loop gain value) such that the transient response characteristics are satisfact ory. If the gain adjustment alone does not yield a desired result, addition of a compensator to the s ystem will become necessary. The closed-loop poles are the roots of the characteristic eq uation. Finding the roots of the characteristic equation of degree higher than three is l aborious and will need a computer solution.

However, just ﬁnding the roots of the characteris tic equation may be of limited value, becauseas the gain of theopen-loop transfer functio n varies, thecharacteristic equation changes and the computations must be repeated. A simple method for ﬁnding the roots of the characteristic eq uation has been developed by W.R. Evans and used extensively in control engineering. T he method, called the root-locus method , is one in which the roots of the characteristic equation are plotted for all values of a system parameter. The roots corresponding to a particular v alue of this parameter

can then be located on the resulting graph. Note that the parameter is usually the gain, but any other variable of the open-loop transfer function may be used. By using the root-locus method the designer can predict the e ﬀects of the location of the closed-loop poles when varying the gain value or adding open -loop poles and/or open-loop zeros. Therefore, it is desired that the designer have a good understanding of the method for sketching the root loci of the closed-loop system. Root-locus method Thebasicideabehindtheroot-locus methodisthat thevalue s of that make transferfunction

around the loop equal 1 must satisfy the characteristic equation of the system. The locus of the roots of the characteristic equation of the c losed-loop system as the gain varies from zero to inﬁnity gives the method its name. Su ch a plot clearly shows the contributions of each open-loop pole or zero to the location s of the closed-loop poles. The root-locus method enables us to ﬁnd the closed-loop poles fr om the open-loop poles and zeros with the gain as parameter. Root-locus plots Consider the closed-loop system shown in Figure 1, where is the open-loop gain, the open- loop

transfer function is: ) = kG The closed-loop transfer function is: kG 1+ kG (1)

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R(s) C(s) k G(s) H(s) Figure 1: Closed-loop control system The characteristic equation for this closed-loop system is obtained by setting the denom inator of the right side of equation (1) equal to zero. That is 1+ kG ) = 0 or kG ) = 1 (2) kG ) is a ratio of polynomials in . Since kG ) is a complex quantity, equation (2) can be split in two equations by equating the angles and magni tudes of both sides, to obtain: kG kG ) = 1+ or: Angle condition: kG ) = 180 (2 +1) , k = 0 ,... (3) Magnitude

condition: kG = 1 (4) The values of that fulﬁll the angle and magnitude conditions are the roots of the charac- teristic equation, or the closed-loop poles. A plot of point s of the complex plane satisfying the angle condition alone is the root-locus. The roots of the cha racteristic equation (the closed- loop poles) corresponding to a given value of the gain can be d etermined from the magnitude condition. Note that to begin sketching the root locus of a system we must know the location of the poles and zeros of ). The angles of the complex quantities originating from the open-loop

poles and open-loop zeros to the test point are measured in the counterclockwise direction. Example. If, for example kG ) is given by: kG ) = )( )( )( where and are complex-conjugate poles, then the angle of kG ) is: kG ) = where , , , , are measured counterclockwise as shown in Figure 2.

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Test point -p -p -p -z -p Figure 2: Angle measurement from a test point to the open-loop poles and zeros The magnitude of ) for this system is: kB where ,A ,A ,A ,A are the magnitudes of complex quantities and , respectively, as shown in Figure 2.

Notethatbecausetheopen-loopcomplex-conjugatepoles,i fany, arelocatedsymmetrically about the real axis, the root loci are always symmetrical wit h respect to this axis. Root locus plot of a second order system Before presenting a method for constructing such plots in de tail, a root-locus plot of a simple second-order system is illustrated. Consider the sy stem shown in Figure 3, where the time constant = 1. The open-loop transfer function ), where ) = 1 in this k R(s) C(s) s(Ts+1) Figure 3: Control system system, is: ) = +1) The closed-loop transfer function is then The characteristic equation

is: = 0 We wish to ﬁnd the locus of the roots of this system as is varied from zero to inﬁnity.

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To give an idea of what a root-locus plot looks like for this sy stem, we shall ﬁrst obtain the roots of the characteristic equation analytically in te rms of and then vary from zero to inﬁnity. It should be noted that this is not the proper way t o construct root locus. The graphical work can be simpliﬁed greatly by applying the gene ral rules to be presented in next section. (Obviously, if an analytical solution for the char acteristic roots can

be found easily, there is no need for the root-locus method). The roots of the characteristic equation are: kj, s kj The roots are real for 4 and complex for k > 4. The locus of the roots corresponding to all positive values of is plotted in Figure 4. The gain is a parameter on the root j2 j1 -j1 -j2 -1 k=0 k=0 k=1 k=4 k=1 k=4 k=1/4 -1 s+1 Figure 4: Root-locus diagram. Left - Root locus. Right - Angl es measured from a test point locus. The motion of the poles with increasing is shown by arrows. Once we draw such a plot we can immediately determine the value of that will yield a root, or a

closed-loop pole, at a desired point. The closed-loop poles correspondi ng to = 0 are the same as the poles of ). As the value of increases from zero to 1 4, the closed-loop poles move toward the point ( 0). For values of between zero and 1 4, all the closed-loop poles are on the real axis. This corresponds to an overdamped syste m, and the impulse response is non-oscillatory. At = 1 4, the two real poles are real and equal. This corresponds to a critically damped system. As increases from 1 4, the closed-loop poles break away from the real axis, becoming complex, and since the real part of

the cl osed-loop poles is constant for k > 4, the closed-loop poles move along the line 4. Hence, for k > 4, the system becomes underdamped. For a given value of , one of the conjugate closed-loop poles moves toward 4+ We shall show that any point on the root locus satisﬁes the ang le condition. The angle condition given by equation (3) is: +1) +1 = 180 (2 +1) , k = 0 ,..

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Consider point P on the root locus shown in Figure 4 (- right). The complex quantities and + 1 have angles and respectively, and magnitudes and + 1 . Note that all angles are considered positive if

they are measured in th e counterclockwise direction. The sum of angles and is clearly 180 If point P is located on the real axis between -1 and 0, then = 180 and = 0 Thus, we see that any point on the root locus satisﬁes the angl e condition. We also see that if the point P is not located on the root locus, then the sum of and is not equal to 180 (2 +1). Thus, points that are not on the root locus do not satisfy the angle condition, and therefore, they cannot be closed-loop poles for any valu e of If the closed-loop poles are speciﬁed on the root locus, then the corresponding

value of is determined from the magnitude condition, equation (4). I f, for example the closed-loop poles selected are 2, then the corresponding value of is found from: +1) 2+ = 1 or +1) 2+ 17 From the root-locus diagram of Figure 4 we see the eﬀects of cha nges in the value of on the transient response behavior of the second-order system . An increase in the value of will decrease the damping ratio , resulting in an increase in the overshoot of the response. A increase in the value of will also result in increases in the damped and undamped natu ral frequencies. From the root-locus

plot, it is clear that the closed-loop po les are always in the left-half of the s-plane; thus, no matter how much is increased, the system remains always stable. Root locus procedure Although computer approaches are available to the construc tion of theroot locus herewe shall use graphical computation combined with inspection, to det ermine the root locus upon which the roots of the characteristic equation of the closed-loop system must lie. For complicated systems with many poles and zeros, constructing a root locus plot may seem complicated, but actually it is not diﬃcult if the

rules for constructing the r oot loci are applied. By locating particular points and asymptotes, and by computing angles o f departure from complex poles and angles of arrival at complex zeros we can construct the ge neral form of the root loci without diﬃculty. As a matter of fact, the full advantage of t he root locus method can be realized in case of higher-order systems for which other met hods of ﬁnding the closed-loop poles are extremely laborious. In the followings some rules for constructing a root locus ar e given. 1. Write the characteristic equation so that the

parameter o f interest appears as a mul- tiplier 1+ kP ) = 0 2. Factor ) in terms of poles and zeros 1+ =1 =1 = 0

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3. Locate the open-loop poles and zeros of ) in the s-plane with symbols: - poles, - zeros. 4. Determine the number of separate loci SL SL , when = number of ﬁnite poles, = number of ﬁnite zeros. 5. Locate the segments of the real axis that are root loci: (a) Locus begins at a pole and ends at a zero (or inﬁnity) (b) Locus lies to the left of an odd number of poles and zeros 6. The root loci are symmetrical with respect to the horizont al real

axis. 7. Theloci proceedto thezeros at inﬁnityalong asymptotes c entered at andwithangles +1 180 , q = 0 ,... 1) 8. By utilizing the Routh-Hurwitz criterion, determine the point at which the locus crosses the immaginary axis (if it does so). 9. Determine the breakaway point on the real axis (if any) (a) Set ), (from the characteristic equation 1+ kP ) = 0) (b) Obtain dp /ds = 0 (c) Determine roots of (b) or use graphical method to ﬁnd maxi mum of ). 10. Determine the angle of locus departurefrom complex pole s and the angle of locus arrival at complex zeros, using the phase

criterion ) = 180 (2 +1) , at s or z 11. Determine the root locations that satisfy the phase crit erion ) = 180 (2 +1) at a root location s 12. Determine the parameter value at a speciﬁc root =1 =1 Illustrative examples Example 1. Consider a closed-loop system as the one shown in Figure 1, wh ere ) = 1, and the open-loop transfer function: ) = +1)( +2) Let us sketch the root-locus plot and determine the value of so that the damping factor of a pair of dominant complex-conjugate closed-loop poles i s 0 5. By following the procedure presented in the previous sectio n we obtain:

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1. The characteristic equation is written as: 1+ kG ) = 0 or 1+ +1)( +2) = 0 2. The open-loop transfer function has no zeros, thus = 0, and three poles, = 0, 1, 2, thus = 3. 3. Locate the open-loop poles of the open-loop transfer func tion with symbol x, as shown in Figure 5. 4. Locus has = 3 branches. 5. Locate the segments of the real axis that are root loci: On the negative real axis, the locus exists between = 0 and 1, and from 2 to . As the locus begins at pole and ends at a zero (or inﬁnity), be tween 0 and -1 we must fund a breakaway point. 6. The root locus is symmetrical

with respect to the horizont al real axis. 7. The locus proceed to the zeros at inﬁnity along asymptotes centered at and with angles +1 180 , q = 0 or = 60 180 300 Thus, there are three asymptotes. The one having the angle 18 is the negative real axis. The asymptotes are shown in Figure 5. 8. The point at which the locus crosses the immaginary axis ca n be determined using the Routh-Hurwitz criterion. The characteristic equation is: +3 +2 = 0 and the Routh array: : 1 2 : 3 : (6 The crossing points on the imaginary axis that makes the term in the ﬁrst column equal zero is obtained

from (6 3 = 0, or = 6. Replacing this value in the characteristic equation we obtain: +3 +2 +6 = 0

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or +3)( +2) = 0 The crossing points on the imaginary axis are the imaginary r oots of the characteristic equation when = 6: 9. The breakaway point on the real axis will be determined fro m the solution of ) = 0 where ) = /G ), or ) = +1)( +2). dp ds (3 +6 +2) = 0 yields 4226 , s 5774 Since the breakaway point must lie between 0 and -1, 4226 corresponds to the actual breakaway point. Based on the information obtained in the foregoing steps, th e root locus obtained is shown in

Figure 5. -1 -2 k=6 1.4j -1.4j 60 s=-0.42 Figure 5: Root-locus diagram In order to determine a pair of complex conjugate closed-loo p poles such that the damping ratio is 0.5 we have to remember some properties of the complex pole s of the second order system: Remember. The complex poles of a general second-order transfer functi on with the char- acteristic equation: +2 = 0

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-s=zw z- w= Figure 6: Location of complex poles are j They are represented in Figure 6. From the ﬁgure, the damping factor results as follows: cos or arccos Closed-loop poles with = 0 5 lie on the

lines passing through the origin and making the angles arccos arccos 5 = 60 with the negative real axis, as shown in Figure 7. From Figure 7, such closed-loop poles having = 0 5 are obtained as follows: -1 -2 60 k=1.03 k=1.03 Figure 7: Root-locus diagram

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3337 5780 The value of that yields such poles is found from the magnitude condition +1)( +2) = 1 0383 Using this value of , the third pole is found at 3326. Constructing the root locus when a variable parameter does not appear as a multiplying factor. Insome cases the variable parameter may not appear as a multiplying factor

of ). In such cases it may be possible to rewrite the characteris tic equation such that the variable appears as a multiplying factor of ). This example illustrates how to proceed in this case. R(s) C(s) H(s) G(s) Figure 8: Closed-loop system Consider a closed-loop system shown in Figure 8, where ) = 20 +1)( +4) and ) = 1+ ks The open-loop transfer function is then: ) = 20(1+ ks +1)( +4) Notice that the adjustable variable does not appear as a multiplying factor. The character- istic equation of this system is: 1+ ) = 1+ 20(1+ ks +1)( +4) = 0 or +5 +4 +20+20 ks = 0 By deﬁning 20 and

dividing by the sum of terms that do not contain , we get: 1+ Ks +5 +4 +20 = 0 We shall now sketch the root locus of the system given from the new characteristic equation. There are: one zero at = 0 and three poles: 2, 5. Thus the number of open-loop zeros and poles are = 1, = 3, respectively. The root locus will have = 3 10

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branches. On the real axis the region which starts at pole -5 t o the origin (open-loop zero) belongs to the ﬁnal root locus. The other two branches start o n the imaginary axis, at the poles 2 and approach the asymptotes for increasing The

intersection of the two ( = 3 1 = 2) asymptotes with the real axis can be found from: 2+ The angles of the asymptotes are: 180 (2 +1) , q = 0 or = 90 270 With this information available the root locus may be sketch ed and is presented in Figure 9. j2 -j2 -5 0 -2.5 Figure 9: Root locus 11

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