8.1. a) For step response, M input is u ( t) Taking inverse Laplace transform. as α 0. Ideal response, K c. = Kc Mτ D + For ramp response, 8-1

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1 8. a For ep repone, inpu i u, U Y a U α α Y a α α Taking invere Laplae ranform a α e e / α / α A α 0 a δ 0 e / α a δ deal repone, α d Y i Gi U i δ Hene a α 0 a i For ramp repone, inpu i u, U Soluion anual for Proe nami and Conrol, nd ediion, Coprigh 004 b ale E. Seborg, Thoma F. Edgar and unan A. ellihamp. 8-

2 8- U Y a α α Y a α α α α α α α Taking invere Laplae ranform [ ] [ ] / / α α α a e e A α 0 a deal repone, Y i i Hene i a a 0 α b ma be diffiul o obain an aurae eimae of he derivaive for ue in he ideal ranfer funion. Ye. The ideal ranfer funion amplifie he noie in he meauremen b aking i derivaive. The approximae ranfer funion redue hi amplifiaion b filering he meauremen. 8. a E P

3 b α α or α Subiuing, α α α α α α Then, α α f 3,, α 0. hen, Hene G

4 a From Eq. 8-4, he parallel form of he P onroller i : G i From Eq. 8-5, for α 0, he erie form of he P onroller i: [ ] G a Comparing G a wih G i b Sine for all,, herefore, and For 4, 0 min, min 4.8, min, min.67 d Conidering onl fir-order effe, a non-zero α will dampen all repone, making hem lower.

5 8.4 Noe ha par a, d, and e require maerial from Chaper 9 o work. a Sem air-o-open valve : v i poiive. Sem air-o-loe valve : v i negaive. b Sem : Flowrae oo high need o loe valve dereae onroller oupu revere aing Sem : Flow rae oo high need o loe valve inreae onroller oupu dire aing. Sem : i poiive Sem : i negaive d v p m Sem : Sem : and v mu have ame ign e An negaive gain mu have a ounerpar ha "anel" i effe. Thu, he rule: # of negaive gain o have negaive feedbak 0, or 4. # of negaive gain o have poiive feedbak or a From Eq. 8- and 8-, [ ] p p p m The liquid-level ranmier haraerii i m T h where h i he liquid level T > 0 i he gain of he dire aing ranmier. 8-5

6 The onrol-valve haraerii i q v p 3 where q i he manipulaed flow rae v i he gain of he onrol valve. From Eq.,, and 3 [ p p] [ h ] q q V p v q q h T V For inflow manipulaion onfiguraion, q> q when p > T h. Hene v > 0 p T hen for "air-o-open" valve v >0, >0 : and for "air-o-loe" valve v <0, <0 : revere aing onroller dire aing onroller For ouflow manipulaion onfiguraion, v <0 hen for "air-o-open" valve, <0 : dire aing onroller and for "air-o-loe" valve, >0 : revere aing onroller b See para above 8.6 For P onrol p p e e * d * 0 p e e * d * 0 Sine e p m and m 8-6

7 Then e - p d * 0 niial repone Slope of earl repone 6-3. min - 5 min 8.7 a To inlude a proe noie filer wihin a P onroller, i would be plaed in he feedbak pah b f The TF beween onroller oupu P and feedbak ignal Y m would be 8-7

8 P Y m f Negaive ign ome from omparaor For Y m P f A B C f The C f erm give rie o an exponenial. To ee he deail of he repone, we need o obain B - f and A b parial fraion expanion. The repone, hown for a negaive hange in Y m, would be Slope / - "deal" P Filered P - - f / ime f Noe ha a 0, he wo repone beome he ame. d f he meaured level ignal i quie noi, hen hee hange migh ill be large enough o aue he onroller oupu o jump around even afer filering. One wa o make he digial filer more effeive i o filer he proe oupu a a higher ampling rae e.g., 0. e while implemening he onroller algorihm a he lower rae e.g., e. A well-deigned digial ompuer em will do hi, hu eliminaing he need for analog oninuou filering. 8-8

9 a From inpeion of Eq. 8-6, he derivaive kik r b Proporional kik r e e e 3. e k- e k- 0 e k e k e k r p p k r r r p p k r i r p p i k, i,, To eliminae derivaive kik, replae e k e k- in Eq. 8-6 b k - k-. k k k k3 r r r p k p k-

10 8.9 a The digial veloi P algorihm i obained b eing / 0 in Eq. 8-8 a p k e k e k- [ ] p k p k [ ] k k The digial veloi P algorihm i obained b eing / 0 in Eq. 8-8 a p k [e k e k- e k e k- e k- ] [ - k k- - k k- k- ] n boh ae, p k doe no depend on p. b For boh hee algorihm p k 0 if k- k- k. Hene ead ae i reahed wih a value of ha i independen of he value of. Ue of hee algorihm i inadviable if offe i a onern. f he inegral mode i preen, hen p k onain he erm p k. Thu, a ead ae when p k 0 and k- k- k, k offe problem i eliminaed. p p and he 8.0 P a E α α α α α α α 8-0

11 Cro- mulipling α P α α E α d p d dp d e α de d d e α d P b E α Cro-mulipling E α P α d p d dp d e de d d e d We need o hooe parameer in order o imulae: e.g.,, 3, 0. 5, α 0., B uing Simulink-ATLAB 0 Sep Repone Parallel P wih a derivaive filer Serie P wih a derivaive filer p' Time Figure S8.0. Sep repone for boh parallel and erie P onroller wih derivaive filer. 8-

12 8. P E a E P dp de e d d d e d b Wih he derivaive mode aive, an impule repone will our a 0. Aferward, for a uni ep hange in e, he repone will be a ramp wih lope / and inerep / for > 0. mpule a 0 p ' lope 8-