[Luyben] Process Mod.. - Student subdomain for University of Bath

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608 MULTIVARIABLE PROCESSES TABLE 17.3 (continued) CALL IDENT(HIN,N) CALL MADD(HIN,Q,H,N) CALL LEQ2C(H,NN,IA,HIN,M,IB,IJOB,WA,WK,IER) CALL LOADTF(GL,W,N,KL,TAUL,DL) CALL MVMULT(HIN,GL,XTLC,W,N) DO 6 I=l,N 6 DB(I)=20.*ALOGlO(CABS(XTLC(I))) WRITE(6,7)W,(DB(I),I=l,N) 7 FORMAT(lX,F7.4,4F9.3) W=W*DW IF(W.LT.lO.)GO TO 100 RETURN END C SUBROUTINE LOADTF(GL,W,N,KL,TAUL,DL) COMPLEX GL(4),ZL,ZD DIMENSION DL(4),TAUL(4,4) REAL KL(4) ZL(X)=CMPLX(l.,X*W) ZD(X)=CMPLX(COS(WtX),-SIN(W*X)) DO 10 I=l,N GL(I)=KL(I)*ZD(DL(I))*ZL(TAUL(l,I))/ZL(TAUL(2,1))/ + ZL(TAUL(3,I))/ZL(TAUL(4,1)) 10 CONTINUE RETURN END C C MATRIX-VECTOR MULTIPLICATION SUBROUTINE MVMULT(A,B,C,W,N) COMPLEX A(4,4),B(4)$(4) DO 30 I=l,N C(I)=CMPLX(O.,O.) DO 20 J=l,N 20 C(I)=A(I,J,*B(J)+C(I) 30 CONTINUE RETURN END Results W TLC1 TLC2 TCL3 .OlOO -51.187 -18.386 -7.112 .0126 -47.967 -17.091 -3.642 .0158 -45.022 -16.026 .240 .0200 -42.393 -15.209 3.093 .0251 -40.119 -14.645 6.361 .0316 -38.231 -14.331 9.540 ,039s -36.722 -14.266 12.547 .0501 -35.442 -14.440 15.252 .0631 -34.002 -14.781 17.553 .0794 -32.012 -15.072 19.407 .lOOO -29.484 -14.857 20.770 .1259 -26.696 -13.620 21.450 .1585 -23.911 -11.456 20.922 .I995 -21.406 -9.153 17.734 .2512 -19.702 -7.744 7.871 .3162 -19.614 -8.386 15.111 W TLC1 TLC2 TCL3 .3981 -21.309 -11.531 15.413 .5012 -24.020 -16.312 8.943 6310 -27.079 -19.446 9.338 .7943 -29.796 -19.764 6.413 1.0000 -32.102 -21.345 5.627 1.2589 -35.142 -24.907 3.107 1.5849 -37.406 -26.733 ,995 1.9953 -39.624 -28.397 -1.342 2.5119 -41.039 -29.725 -3.389 3.1623 -42.692 -32.305 -5.623 3.9811 -45.035 -34.280 -7.692 5.0119 -46.915 -36.055 -9.727 6.3096 -49.044 -38.224 -11.729 7.9433 -50.828 -40.280 -13.540 10.0000 -52.814 -42.443 -15.796
DESIGN OF CONTROLLERS FOR MULTIVARIABLE PROCESSES 609 Figure 17.3 gives some comparisons of the performance of the multivariable DMC structure with the diagonal structure. Three linear transfer-function models are presented, varying from the 2 x 2 Wood and Berry column to the 4 x 4 sidestream column/stripper complex configuration. The DMC tuning constants used for these three examples are NP = 40 and NC = 15. See Chap. 8, Sec. 8.9. The weighting factors (f) for the changes in the manipulated variables were 10, 0.1, and 0.1 for the three cases. These were chosen so that the maximum magnitudes of the changes in the manipulated variables did not exceed those experienced when the diagonal structure was used. If these weighting factors were made smaller, tighter control could be achieved, but the changes in the manipulated variables would be larger. In most chemical engineering systems, we cannot permit rapid and large swings in manipulated variables. For example, such large swings in steam to the reboiler of a distillation column could flood the column or even blow out trays in the lower section of the tower. Using the multivariable DMC produces some improvement for setpoint disturbances. However, as Fig. 17.3 shows, there are only slight differences between DMC and the diagonal structure (with BLT tuning) for load disturbances. Thus the real economic incentive for the more complex multivariable configuration has yet to be demonstrated by impartial evaluators. 17.7.2 Multivariable IMC In theory, the internal model control methods discussed for SISO systems in Chap. 11 can be extended to multivariable systems (see the paper by Garcia and Morari in IEC Process Design and Development, Vol. 24, 1985, p. 472). In practice, however, this extension is not as straightforward as in DMC. In multivariable DMC, there is a definite design procedure to follow. In multivariable IMC, there are steps in the design procedure that are not quantitative but involve some “art.” The problem is in the selection of the invertible part of the process transfer function matrix. Since there are many possible choices, the design procedure becomes cloudy. The basic idea in multivariable IMC is the same as in single-loop IMC. The ideal controller would be the inverse of the plant transfer function matrix. This would give perfect control. However, the inverse of the plant transfer function matrix is not physically realizable because of deadtimes, higher-order denominators than numerators, and RHP zeros (which would give an openloop unstable controller). So the openloop plant transfer function matrix GM is split up into two parts : an invertible part and a noninvertible part. GAf=G+G- (17.12) Then the multivariable IMC controller is set equal to the invertible part times a “filter” matrix which slows up the closedloop response to give the system more robustness. The filter acts as a tuning parameter (like setting the closedloop time constant in the SISO case in Chap. 8).
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I WILLIAM WIlllAM 1. LUYBEN PROCESS
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The Series Bailey and OUii: Biochem
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PROCESS MODELING, SIMULATION, AND C
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ABOUT THE AUTHOR William L. Luyben
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CONTENTS Preface Xxi 1 1.1 1.2 1.3
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CONTENTS .. . xlu 5.7 Batch Reactor
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CONTENTS xv 10.3 10.4 Performance S
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comwrs xvii 151.3 Transpose of a Ma
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CONTENTS Xix 20.4 Sampled-Data Cont
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. Xxii PREFACE Another significant
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PROCESS MODELING, SIMULATION, AND C
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2 PROCESS MODELING, SIMULATION. AND
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4 PROCESS MODELING, SIMULATION, AND
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6 PROCESS MODELING. SIMULATION, AND
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8 PROCESS MODELING, SIMULATION, AND
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10 PROCESS MODELING, SIMULATION, AN
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12 PROCESS MODELING, SIMULATION, AN
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CHAPTER 2 FUNDAMENTALS 2.1 INTRODUC
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FUNDAMENTALS 17 big, complex system
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FUNDAMENTALS 19 z=o I z + dz z=L FI
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FUNDAMENTALS 21 The minus sign come
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FUNDAMENTALS 23 Molar flow of A lea
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FUNDAMENTALS 25 For liquids the PP
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FUNDAMENTALS 27 Flow of energy (ent
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FUNDAMENTALS 29 The units of this f
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FUNDAMENTALS 31 FIGURE 2.9 Laminar
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FUNDAMENTALS 33 Then Eq. (2.48) bec
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FUNDAMENTALS 35 Wenzel in Chemical
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FUNDAMENTALS .37 This exponential t
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FUNDAMENTALS 39 a direction 0 = 45
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EXAMPLES OF MATHEMATICAL MODELS OF
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EXAMPLES OF MA~EMATICAL MODELS OF C
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EXAMPLES OF MATHEMATICAL MODELS OE
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88 COMPUTER SIMULATION Our discussi
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90 COMPUTER SIMULATION lation packa
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92 COMPUTER SIMULATION For this rea
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94 COMPUTER SIMULATION TABLE 4.1 Ex
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96 COMPUTER SIMULATION Clearly, the
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98 COMPUTER SIMULATION TABLE 4.2 Ex
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I()() COMPUTER SIMULATION Settingfl
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