Design of a control system
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1 SE3 Prof. Davide Manca Poliecnico di Milano Dynamics and Conrol of Chemical Processes Soluion o Lab #3 Design of a conrol sysem Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 1
2 Sysem represenaion A 1 A F i LC h 1 r 1 h F o Daa: F i A A r m s 30 m 50 m 1. s m I.C.: h 6.6m F o changes linearly wih he level: F o 1.43h Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3
3 Tasks 1. Deerminaion of he parameers of he proporional and he proporionalinegral conrol sysems by means of he Cohen-Coon mehod assuming here is a sep disurbance on he inle flowrae, such ha i doubles. The ask of he conrol sysem is o keep he seady sae, i.e. o keep consan he level in he second ank. Assume ha he sysem is in seady sae when he sep disurbance occurs.. Evaluae he sysem dynamics in closed loop. Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 3
4 Conrol philosophy Conrolled variable: Level of he second ank Manipulaed variable: Valve opening = = second ank oule flowrae Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 4
5 Design of a conrol sysem wih Cohen-Coon mehod A. Assign a sep disurbance on he variable ha could change during he plans operaions (inle flowrae). Find he dynamics of he conrolled variable (liquid level in he second ank) in open-loop, i.e. wihou he conrol sysem. B. Find he characerisic parameers of he sysem: A. gain of he sysem ; d B. ime delay: ; K C. Characerisic ime o reach he new seady sae:. C. Find he conrol sysem parameers according o specific formulas. Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 5
6 A- Evaluaion of he open-loop dynamics A 1 A F i h 1 r1 h LC Fo Dynamics of he sysem A A 1 dh d 1 dh d F i h 1 1 h h r h 1 r 1 F o Open-loop oule flow rae law: Fo 1.43h Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 6
7 Evaluaion of he seady sae condiions 0 0 F h i s s 1 1 h h r s 1 s h r 1 F s s o s s 3 Fo Fi 9.4 m s s s s h1 r1 Fi h 17.9 m Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 7
8 MATLAB implemenaion Main Fi_s = 9.4 y0 = [ ] Span = [ ]; [,h] = ode45(@(,y)sisdif(,y,a1,a,r1,fi_s),span,y0,opions); Sisdif funcion dy = Sisdif(,y,A1,A,r1,Fi_s) h1 = y(1); h = y(); Fi = Fi_s*; dy(1) = (Fi - (h1-h)/r1) / A1; dy() = ((h1-h)/r1-(1.43*h))/a; Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 8 8
9 Open-loop dynamics Alezza liquido [m] Liquid level [m] Serbaoio Tank 1 1 Serbaoio Tank Tempo Time [s] [s] 35.7 m 13.1 m Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 9
10 Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 10 B - Conroller parameers d c d K K d c d d I d d K K Proporional conrol Proporional-inegral conrol
11 Characerisic parameers of he sysem h old = seady sae condiion before he disurbance new = seady sae condiion afer he disurbance h s, new h s, old Gain of he sysem: K h s, new s, old F i h new old F i d ime Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 11
12 Liquid level [m] Sysem dynamics for a sep disurbance 14 Alezza liquido [m] Parameri del Sysem sisema parameers: K K 0.696s m = s = D 8.8s D Time Tempo [s] [s] Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 1
13 C1- Proporional conroller model The general form of he model of he proporional conrol is: c s c c K y y s F F K h h o s c SP c Where is he bias, i.e. he value of he manipulaed variable when he conrolled variable is a he se-poin. In his case a valve on he inle flowrae is manipulaed and he ne resul is he modificaion of he second ank oule flowrae. Assuming a linear relaionship beween he valve opening degree and he oule flowrae, he conroller model can be re-wrien as: s The model of he sysem becomes: dh h1 h 1 A1 Fi d r1 dh h1 h s s A Fo Kc h h d r 1 Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 13
14 MATLAB implemenaion Main Fi_s = 9.4 y0 = [ ] Span = [ ]; [,h] = ode45(@(,y)sisdif(,y,a1,a,r1,fi_s),span,y0,opions); Sisdif funcion dy = Sisdif(,y,A1,A,r1,Fi_s) Fi = Fi_s*; %Sep h_s = 6.6; %Se poin Fo_s= 1.43*h_s; %Oule flow dy(1) = (Fi - (h1-h)/r1) / A1; dy() = ((h1-h)/r1-(fo_s + Kc*(h-h_s)))/A; Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE
15 Proporional conrol 14 Alezza liquido [m] Liquid level [m] Open-loop P conrol Tempo Time [s] [s] Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 15
16 C- Proporional-Inegral conroller model The general form of he model of he proporional-inegral conrol is: K c c K y y y y d c s c SP 0 SP I K c F F K h h h h d s s s o c 0 I Assuming a linear relaionship beween he valve opening degree and he oule flowrae, he conroller model can be re-wrien as: The model of he sysem becomes: dh h1 h 1 A1 Fi d r1 1 dh h h s s Kc s A Fo Kc h h h h d d r 0 1 I Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 16
17 Inegral evaluaion PROBLEM: he inegral should be evaluaed a he curren ime-sep of he ODE sysem inegraor. The definie inegral can be numerically evaluaed by means of he rapezoidal rule. For example: s I h h d h h 0 h 0 s h The error is oo high!! Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 17
18 Inegral evaluaion I is possible o reduce he numerical error by increasing he number of rapezoids used o fi he inegral: Number of seps (a priori assigned) N s s I h h d h n 1 h n h 0 n1 h Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 18
19 Behaviour of a general inegraor The inegraor calls he funcion of he differenial sysem is wrien (Sisdif in he following) a non-equispaced ime seps; The inegraor can re-call he Sisdif a former ime seps han he las one; No all he evaluaions of he Sisdif conribue o he problem soluion. Some of hem can be used, for insance, for he evaluaion of he Jacobian marix. Furhermore, if he error overcomes he olerance, he inegraion seps fails and i is no used for he soluion esimaion. In his case, he inegraion sep is reduced. Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 19
20 Behaviour of a general inegraor h : successful ime sep : failed sep decrease of he inegraion sep Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 0
21 Soluion procedure 1. The inegral is solved for equispaced inervals (ex. ), i.e. he inegraion horizon 0 is broken ino equispaced sub-inervals: 1 s h Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 1
22 Soluion procedure. The differenial sysem is solved in each one of hese inervals, evaluaing he inegral a each sep made by he inegraor as sum of rapezoids. As reference value o compue he inegral, he value of he inegral in he firs sure former sep is used. Assume o sar from ime and o inegrae up o. The inegral I 1 up o reach has been formerly compued. The inegraor will proceed sep-by-sep 1. In he inermediae seps, he inegral is evaluaed as: s I h h d I h h h s wih 1 Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3
23 Soluion procedure Firs inegraion sep h 1 h I I h h h s Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 3
24 MATLAB implemenaion Main I d I h h h s Sisdif s I d I h h h 0 d Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 4
25 MATLAB implemenaion (Main) Variable iniializaion: global Fi_s r1 A1 A h_s Kc aui inegral Old fold Span = [ ]; y0 = [ ]; A1 = 30., A = 50.; r1 = 1.; Fi_s = 9.4; h_s = 6.6; % Iniial condiions % sqm % s/sqm % cum/s % Se poin Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 5
26 MATLAB implemenaion (Main) Sysem dynamics evaluaion: opions = odese('reltol',1e-8,'abstol',1e-10,'oupufcn',@prino); [,y] = ode113(@exe31p,span,y0,opions); h11 = y(:,1); h = y(:,); variabmeasured = h(lengh(h)); % Oupu o saionary (ake he las elemen of he vecor of he ank level ) offse = abs(variabmeasured - h_s); % Offse Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 6
27 MATLAB implemenaion (Sisdif) funcion dy = EXE31P(,y) global Fi_s r1 A1 A h_s Kc aui inegral Old fold h11 = y(:,1); h = y(:,);... Fi = Fi_s*; h_s = 6.6; err = h_s - h; inegraltempo = inegral + (fold + err) * ( - Old) /.; Fou = Fo_s + Kc * err + Kc / aui * inegraltempo;... dy(1) = (Fi - (h1-h)/r1) / A1; dy() = ((h1-h)/r1-fou)/a; Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 7
28 MATLAB implemenaion (Funcion) funcion saus = Prino(,y,flag) global inegral Old fold h_s if srcmp(flag, 'ini') elseif srcmp(flag, 'done') else h = y(); er = h_s - h; fnew = er; New = ; inegral = inegral + (fold + fnew) * (New - Old) /.; fold = fnew; Old = New; end Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 8 8
29 Liquid level [m] Proporional-Inegral conroller Alezza liquido [m] Open-loop P conrol PI conrol Time Tempo [s] [s] Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 9
30 Liquid level [m] Proporional-Inegral conroller Se poin P conrol PI conrol Time [s] Se poin change Davide Manca Dynamics and Conrol of Chemical Processes Maser Degree in ChemEng Poliecnico di Milano SE3 30
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