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SIMULATION OF CONTROL SYSTEMS
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FIRST ORDER SYSTEH WITH TRANSPORT LAG LO REM 15 DIH S(11) 20 DEF FNDY(Y) = -Y+KC-KC*X 30 KC=8.399999 r(o Y = 0 50 T = 0 bU DT=.02 70 FOR I = L TO LZ 80 S(I) = 0 90 NEXT I 100 PRINT "T","Y","X" LLO X=S(LL) L20 lZL=FNDY(Y)*DT I,30 K2=FNDY(Y+.S*KZ)*DT 140 K3=FNDY(Y+.S*KZ)*DT l&O K't=FNDY(Y+K3)*DT Lb0 Y=Y+(KL+Z*KZ+Z*K3+K I)/b 170 T=T+DT ";T,Y,X *' 1tSO PRINT USING "#.### la90 K-LO 200 FOR I = L TO-20 2LiJ S(K+L)=S(K) 220 K=K-L 230 REXT I 2r(o S(L)=Y 250 IF T > la.001 THEN END 2b0 GOT0 LLLl 270 END
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FIGURE 34-9 BASIC computer program for control of a first-order system with transport lag (Example 34.3).
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(34.44) (34.45) (34.46)
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2 = -j 2 = -9 From Fig. 34.10, we may write
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+ m - y]
In the time domain, this equation becomes
j = (lh)[u(t)
(34.47)
where U(S) = l/s has been written as u(t) (a unit step) in the time domain. Taking the derivative of both sides of Eq. (34.47) gives y = (l/T)[c (f) + rit - j] (34.48)
where use has been made of the fact that the derivative of a unit-step function is a unit-impulse function (see Chap. 4). Combining Eqs. (34.44), (34.45), and (34.46) with Eqs. (34.47) and (34.48) gives
TABLE 34.3
Computer output for control of first-order system with transport lag (Example 34.3)
T 0.020 0.0110 o.ol o 0.080 0.L00 o.lJ20 0.140 O.llbO o.lll lo 0.200 0.220 0.2r10 0.260 0.280 0.300 0.320 0.340 0.3bO 0.380 o.r100 o.lrzo 0.11110 O. ibO O. fBO 0.500 0.520 0.540 O.SbO 0.580 O-b00 0.620 0.6110 0.660 O.b80 0.700 0.720 0.7YO 0.7bO 0.780 0.800 0.820 0.840 0.8bO 0.880 0.700 0.920 0.990 0.9bO 0.980 1.000 Il.020 Y 0.166 0.327 0.489 O.b ib 0.799 0.950 ll.097 L.242 L.38 i Il.523 L.b59 1.765 L.B lL 1.890 ll.sLL L.907 la.877 L.821r L.7 +8 1.649 la.530 Il.390 Il.235 1.07ll 0.9OL 0.732 0.5b7 0.410 0.2brl 0.135 0.024 -.Ob4 -.L28 -.I65 -.L73 -.lJ53 -.LOb -.03ll O.Ob7 0.188 0.329 O.ltBI, O.b52 0.827 Il.0011 L.ll77 L.3 +8 Il.505 l.br17 L.7b=l It.869 X 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 O.Lbb 0.329 0.489 O.b ib 0.799 0.950 Il.077 Il.242 1.38 4 lt.523 lo.659 L.7b5 L.B iL 1.890 L.7Lz la.907 I.877 It.824 L.7 +8 2.649 It.530 1.390 Il.235 Il.071 0.7OL 0.732 0.5b7 o.rllto 0.2b t o.ll35 0.024 -.Obr( -.L28 -.Lb5 -.ll73 -.lJ53 -.LOb -.031 O.Ob7 O.ll88
DIGITAL COMPUTER SIMULATION OF CONTROL SYSTEMS
FIGURE 34-10
PID control of a first-order process (Example 34.4).
t = -j, = -(l/~)[u(t) + m - y] Z = -ji = -(lh){S(t) + rh - (lh)[u(t) + m - y]}
(34.49) (34.50)
Substituting the expressions fore, .G, and 2 from Eqs. (34.44), (34.49), and (34.50) into Eq. (34.43) gives after simplification rir = c[-Tdd(t) + A + By + Am] where A = Td -,r B=y t-~~ 71 (34.51)
The right side of Eq. (34.51) contains the forcing term -Cr&(t). If Eq. (34.51) were integrated, this term would contribute a constant value of -Crdr. The reason for this is that the integration of a unit impulse is a unit step, thus r I0 G(t)& = u(t) = 1
We may now write Eq. (34.51) in the form with riz = C(A+By+Am)
m(o) = -cT d = -Kc7,f(T + Kcrd) The differential equations to be solved by the Runge-Kutta method now can be summarized j = (l/r)(l + m - y) rh = CA+CBy+CAm with Y(O) = 0 m(o) = -(K&)/(T + & d) (34.52) (34.53)
COMPUTERSINPROCESSCONTROL
Solving Eqs. (34.52) and (34.53) with the initial conditions given will produce a response for the control system of Fig. 34.10. The procedure for programming Eqs. (34.52) and (34.53) by use of the Runge-Kutta method is straightforward and will not be done here.
SIMULATION
SOFTWARE
In the first part of this chapter, we have seen how one can write a digital computer program for the solution of a control problem. Even for the simple examples
presented, there is considerable work in writing and debugging the program. A number of software programs have been written to solve dynamic problems, including process control problems. One of the earliest was CSMP developed by IBM. More recent programs include ACSL, TUTSIM, Simnon, and CC. The sources of these simulation programs are given in the list of references at the end of the chapter. Some of these programs provide blocks that simulate the basic transfer functions of process control such as integrator, first-order, second-order, lead-lag, and transport lag.
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