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586 MULTIVARIABLE PROCESSES ‘) log w FIGURE 16.9 SISO closcdloop plots. If we plotted just the denominator of the right-hand side of Eq. 16.42, the curve would be the reciprocal of the closedloop servo transfer function and would look something like that shown in Fig. 16.9b. The lower the dip in the curve, the lower the damping coefficient and the less robust the system. B. MATRIX MULTIVARIABLE SYSTEMS. For multivariable systems, the Doyle- Stein criterion for robustness is very similar to the reciprocal plot discussed above. The minimum singular value of the matrix given in Eq. (16.43) is plotted as a function of frequency o. This gives a measure of the robustness of a closedloop multivariable system. Doyle-Stein criterion: minimum singular value of u + &,$ (16.43) The Q matrix is defined as before Quo) = GM,,-, &,,, . The tuning and/or structure of the feedback controller matrix B~ic, is changed until the minimum dip in the curve is something reasonable. Doyle and Stein gave no definite recommendations, but a value of about - 12 dB seems to give good results. The procedure is summarized in the program given in Table 16.4 which calculates the minimum singular value of the B + &$J matrix for the Wood and Berry column with the empirical controller settings. Figure 16.10 gives plots of the singular values as a function of frequency. The lowest dip occurs at 0.23 radians per minute and is about - 10.4 dB.
ANALYSIS OF MULTIYARIABLE SYSTEMS 587 TABLE MA Doyle-Stein criterion for Wood and Berry column REAL KP(4,4),KC(4) DIMENSION RESET(4),TAU(4,4,4),D(4,4),WK(4),WKEIG(12),SVAL(4) COMPLEX GM(4,4),B(4,4)&(4,4),QIN(4,4),YID(4,4) COMPLEX A(4,4),ASTAR(4,4),H(4,4),CEIGEN(4),WA(24),VECT(4,4) C PROCESS TRANSFER FUNCTION DATA DATA KP(1,1),KP(1,2),KP(2,1),KP(2,2)/12.8,-18.9,6.6,-19.4/ DATA TAU(1,1,1),TAU(1,1,2),TAU(1,2,1),TAU(1,2,2)/4~0./ DATA TAU(2,1,1),TAU(2,1,2),TAU(2,2,1),TAU(2,2,2) + /16.7,21.,10.9,14.4/ DO 1 Id,2 DO 1 J=1,2 TAU(B.I,J‘)=O. 1 TAUi4,i,Jj=O. DATA D(l,l),D(1,2),D(2,1),D(2,2)/1.,3.,7.,3./ C EMPIRICAL CONTROLLER SETTINGS DATA KC(l),KC(2)/0.2,-0.04/ DATA RESET(l),RESET(2)/4.44,2.67/ C SET INITIAL FREQUENCY AND NUMBER OF POINTS PER DECADE W=O.Ol DW=(lO.)w(1./20.) N=2 IA=4 M=N IB=4 IJOB=O C MAIN LOOP FOR EACH FREQUENCY WRITE(6,lO) 10 FORMAT(’ FREQUENCY SINGULAR VALUES’) 100 CALL PROCTF(GM,W,N,KP,TAU,D) CALL FEEDBK(ti,W,k,Ik,RESET) CALL MMULT(GM,B,Q,N) CALL IDENT(QIN,N) CALL LEQ2C(Q,N,IA,QIN,M,IB,IJOB,WA,WK,IER) CALL IDENT(YID,N) CALL MADD(YID,QIN,A,N) CALL CONJT(A,ASTAR,N) CALL MMULT(ASTAR,A,H,N) IZ=4 JOBN=O CALL EIGCC(H,N,IA,JOBN,CEIGEN,VECT,IZ,WKEIG,IER) DO 6 I=l,N 6 SVAL(I)=SQRT(REAL(CEIGEN(I))) WRITE(6,7)W, (20.*ALOGlO(SVAL(I)),I=l,N) 7 FORMAT(lX,4F10.5) W=W*DW IF(W.LT.l.) GO TO 100 STOP END FREQUENCY SINGULAR VALUES .OlOOO -.00281 -.02777 .01122 -.00410 -.03454 .01259 -.00573 -.04311 .01413 -.00775 -.05402 .01585 -.01024 -.06790 .01778 -.01323 -.08560
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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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