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

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570 MULTIVARIABLE PROCESSES TABLE 16.2 (continued) C SUBROUTINE PROCTF(GM,W,N,KP,TAU,D) DIMENSION KP(4,4),TAU(4,4,4),D(4,4) COMPLEX GM(4,4),ZL,ZD REAL KP ZL(X)=CMPLX(l.,X*W) ZD(X)=CMPLX(COS(W*X),-SIN(W*X)) DO 10 I=l,N DO 10 J=l,N 10 GM(I,J)=KP(I,J)*ZD(D(I,J))*ZL(TAU(l,I,J))/ZL(TAU(2,I,J)) + /ZL(TAU(3,I,J))/ZL(TAU(4,I,J)) RETURN END Results W WQl) WQl) RE(Q2) IWQ2) .OlOOO -10.93861 -73.93397 -.69893 -10.97331 .01259 -10.77362 -57.94015 -.70154 -8.69403 .01585 -10.52296 -45.07131 -.70426 -6.87520 .01995 -10.15061 -34.67506 -.70560 -5.41860 .02512 -9.61544 -26.25261 -.70209 -4.24563 .03162 -8.88105 -19.43858 -.68745 -3.29489 .03981 -7.93375 -13.98019 -.65334 -2.52193 .05012 -6.80296 -9.70591 -.59328 -1.89904 .06310 -5.56933 -6.48106 -.50896 -1.41125 .07943 -4.34690 -4.16391 -.41292 -1.04679 .10000 -3.24441 -2.58527 -.32303 -.78823 .12589 -2.33036 -1.56020 -.25321 -.61188 .15849 -1.62209 -.91593 -.20913 -.49410 .19953 -1.09771 -.51329 -.19052 -.41728 .25119 -.71786 -.25031 -.19364 -.37372 .31623 -.19787 -.35982 -.45773 -.06077 .39811 -.19106 -.33411 -.29072 .05551 .50119 -.19649 -.28201 -.16390 .10739 .63096 -.20761 -.20640 -.06256 .10977 .79433 -.19927 -.11268 .00200 .07250 1 .ooooo -.14416 -.03580 .00868 .03019 1.25893 -.10863 -.03576 .02118 .04213 1.58489 -.10893 .01474 .04596 -.01127 1.99526 -.05624 .03026 -.00732 -.01991 2.51189 .06183 -.02432 -.01433 Note that these eigenvalues are neither the openloop eigenvalues nor the closedloop eigenvalues of the system! They are eigenvalues of a completely different matrix, not the 4 or the 4cL matrices. Characteristic loci plots for the Wood and Berry column are shown in Fig. 16.3. They show that the empirical controllers settings give a stable closedloop system, but the ZN settings do not since the q1 eigenvalue goes through the (- JO) point. A brief justification for the characteristic loci method (thanks to C. C. Yu) is sketched below. For a more rigorous treatment see McFarland and Belletrutti (Automatica 1973, Vol. 8, p. 455). We assume an openloop stable system so the closedloop characteristic equation has no poles in the right half of the s plane.
ANALYSIS OF MULTNARIABLE SYSTEMS 571 Wood and Berry 2 2 - 4, 42 I I I I I - 2 2 - 2 I Gains = 0.20 Resets = 4.44 -0.04 2.61 Wood and Berry 2 - 2 - - - 2 Gains = 0.96 - 0 . 1 9 Resets = 3.25 9.20 I FIGURE 16.3 The closedloop characteristic equation for a multivariable closedloop process is Det Cl + GM,,, &I = Det U + &I = 0 Let us canonically transform the Q matrix into a diagonal matrix 4. which has as its diagonal elements the eigenvzlues of $I$ Q = kY’llQW’-’ (16.11)
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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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Page 107 and 108: 88 COMPUTER SIMULATION Our discussi
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ANALYSIS OF MULTIVARIABLE SYSTEMS 5
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ANALYSIS OF MULTIYARIABLE SYSTEMS 5
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FIGURE 16.5 Multivariable INA (- 1,
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ANALYSIS OF MULTIYARIABLE SYSTEMS 5
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DESIGN OF CONTROLLERS FOR MULTIVARI
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DESIGN Ok CONTROLLERS FOR MULTIVARI
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