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Once the roots are available, the response of the system to any forcing function can be obtained by the usual procedures of partial fractions and inversion given in Chap. 3.
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Table 15.1A gives a BASIC computer program for finding the roots of a polynomial equation by the Lin-Bairstow method [see Hovannessian and Pipes (1969)]. To use this program, arrange the polynomial equation in the form a,s + u,-l&s -l + u,-*r2 + . . . + a, = 0 Before running the program, a DATA statement is used to list the order (n) and the coefficients of the polynomial equation as follows: DATA n, a,,, a,,-~, . . ., a, An example of the use of the root-finding program is shown in Table 15.2A; the example involves finding the roots of Eq. (15.5) for the case of K = 26.5.
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TABLE 15.1A
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BASIC program for finding roots of a polynomial equation
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LO 20 30 110 50 REM ROOTS OF POLYNOMIAL EQUATION REM USING LIN-BAIRSTOW METHOD REH AN*S*+N + A(N-L)*S**(N-1) + A(N-Z)*S**(N-2) + . ..+ A0 = II REM DATA N, AN, A(N-II), . . ..A0 REM REFERENCE: DIGITAL COHP HETH IN ENGRG, HOVANNESSIAN, S.A. AND L. A. PIPES LOO DIM A(LO),B(LO),C(LO),D(10) LLO READ N 120 PRINT "DEGREE"N 130 PRINT "COEFFICIENT" I,'+0 FOR I = N TO 0 STEP -L l&O READ A(I) Iah0 PRINT A(I) ; I170 NEXT I LAO PRINT 190 PRINT 200 LET R=A(L)/A (2) 220 LET S=A(O)/A (2) 220 LET B(N)=A(N 1 230 LET C(N)=0 2110 LET D(N)=0 250 LET B(N-lt)=A (N-L);R*B(N) 260 LET C(N-L)=-B(N) 270 LET D(N-L)=O 280 FOR I=2 TO N-Z 290 LET B(N-I)=A(N-I)-R*B(N-I+L)-S*B(N-I+Z) 300 LET C(N-I)=-B(N-I+l,)-R*C(N-I+L)-ScC(N-1+2) 310 LET D(N-I)=-B(N-1+2)-S*D(N-1+2)-R*D(N-I+Z)
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ROOTLOCUS
TABLE 15.1A (Continued)
BASIC program for finding roots of a polynomial equation
320 330 340 350 3b0 370 380 390 rlO0 '410 '420 930 440 1150 rlb0 470 480 990 500 510 520 530 5110 550 5bO 570 580 590 bO0 bL0 b20 630 NEXT I LET Rl=A(la)-R*B(Z)-S*B(3) LET Slt=A(O)-S*B(i ) LET T=-B(Z)-R*C(i )-S*C(3) LET U=-B(3)-S*D(3)-R*D(Z) LET V=-S*C(Z) LET W=-B(Z)-S*D(2) LET R2=(-RL*W+SL*U)/(T*W-U*V) LET SZ=(-T*SL+V*RL)/(T*W-lJ*V) LET S=S+S2 LET R=R+R2 IF ABS(R2)<.00001 TEEN 'l50 GOT0 220 LET G=R*R-rl*S IF G<O TEEN '490 PRINT "ROOTS";-R/Z;"+OR-";SQR(G)/Z GOT0 500 PRINT "ROOTS";-R/Z;"+OR-'t;SQR(-G)/Z;"J" LET N=N-2 PRINT IF N=O THEN b30 FOR I = N TO 0 STEP-L LET A(I)=B(I+Z) NEXT I IF N>2 THEN 200 IF NC2 THEN bL0 LET R=A(N-L)/A(N) LET S=A(N-2)/A(N) GOT0 '450 PRINT "ROOT",-A(#-L)/A(N) DATA 3 L,b,LLzsS END j,fl-;&pic,cd*5
TABLE 15.2A
Use of BASIC program of Table 15.1A for finding roots of Eq. (15.5): 8 + t$ + 11s + (K + 6) = 0 with K = 29;.5
RUN DEGREE 3 COEFFICIENT L b ZL 32.5 ROOTS-.'t53'#395 ROOT +OR- 2.485065 J I -5.093Li L
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SYSTEMS
PROBLEMS
Draw the root-locus diagram for the system shown in Fig. P15.1 where GC = K,(l + 0.5s + l/s).
FIGURE P15-1 15.2. Draw the root-locus diagram for the system shown in Fig. P13.4 for (a) 71 =
0.4 mitt and (b) ~1 = 0.2 min. (The proportional controller is replaced by a PI controller.) Determine the controller gain that just causes the system to become unstable. The values of parameters of the system am:
K, = valve constant 0.070 cfm/psi K, = transducer constant 6.74(in. pen travel)/@ R2 = 0.55 ft level/&n
q = time constant of tank 1 = 2.0 min 92 = time constant of tank 2 = 0.5 min
of tank level)
The controller gain K, has the units of pounds per square inch per inch of pen travel. 15.3. Sketch the root-locus diagram for the system shown in Fig. P14.2. If the system is unstable at higher values of K,, find the roots on the imaginary axis and the corresponding value of K,. 15.4. Sketch the root loci for the following equations:
@) l+ (s + 1)(2s + 1) = O
(b) 1 + s(s + 1)(2s + 1) = 0 Cc) 1 + s(s + 1)(2s + 1) = 0
(4 1+ (4 1 + K(1.5~ + 1) o s(s + 1)(2s + 1) =
K(4s + 1)
K(0.5~ + 1) o s(s + 1)(2s + 1) =
On your sketch you should locate quantitatively all poles, zeros, and asymptotes. In addition show the parameter that is being varied along the locus and the direction in which the loci travel as this parameter is increased.
FIGURE PlS-5
15.5. For the control system shown in Fig. P15.5. Casel:rb=$ case2:Q)=$ (a) Sketch the root-locus diagram in each case. (b) If the system can go unstable, find the value of Kc that just causes instability. (c) Using Theorem 3 (Chap. 14) of the Routh test, find the locations (if any) at which the loci cross into the unstable region. 15.6. Draw the root-locus diagram for the control system shown in Fig. P15.6. (a) Determine the value of Kc needed to obtain a root of the characteristic equation of the closed-loop response which has an imaginary part 0.75. (b) Using the value of K, found in part (a), determine all the other roots of the characteristic equation from the mot-locus diagram. (c) If a unit impulse is introduced into the set point, determine the response of the system, C(t).
FlGURE PM-6
15.7. Plot the root-locus diagram for the system shown in Fig. P15.7. We may consider this system to consist of a process having negligible lag, an underdamped, second-order measuring element, and a PD controller. This system may approximate the control of flow rate, in which case the block labeled Kp would represent a valve having no dynamic lag. The feedback element would represent a flow measuring device, such as a mercury manometer placed across an orifice plate. Mercury manometers are known to have underdamped, second-order dynamics. Plot me diagram for 32) = ll3.
K,(1+7gs)-
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