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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 53 "Dynamics of a Heated Tank
 with PI Temperature Control" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "T
he energy balance for the stirred tank is" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "EB := Diff(T[t],t)=(W*C[P]*(T[i]-T[t])+q)/(rho*V*C[
P]): EB;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$&%\"TG6#%\"tGF*
**,&*(%\"WG\"\"\"&%\"CG6#%\"PGF/,&&F(6#%\"iGF/F'!\"\"F/F/%\"qGF/F/%$rh
oGF8%\"VGF8F0F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "T[t]" "&%\"TG6#%\"tG" }{TEXT -1 79 " is the temperature
 in the tank. The temperature of the exit stream is given by" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Teqn:=Diff(T[0],t)=(T[t]-T[0
]-tau[d]/2*Diff(T[t],t))*2/tau[d]: Teqn;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%%DiffG6$&%\"TG6#\"\"!%\"tG,$*&,(&F(6#F+\"\"\"F'!\"\"*&&%$tauG
6#%\"dGF1-F%6$F/F+F1#F2\"\"#F1F4F2F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 58 "The thermocouple shielding and electronics are modelled by" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Teqn3:=Diff(T[m],t)=(T[0]-T[
m])/tau[m]: Teqn3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$&%\"T
G6#%\"mG%\"tG*&,&&F(6#\"\"!\"\"\"F'!\"\"F1&%$tauGF)F2" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 43 "The energy input to the tank is modelled \+
by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "qeqn:=q=q[s]+K[c]*(T[
r]-T[m])+K[c]/tau[I]*epsilon: qeqn;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/%\"qG,(&F$6#%\"sG\"\"\"*&&%\"KG6#%\"cGF),&&%\"TG6#%\"rGF)&F16#%\"mG!
\"\"F)F)*(F+F)&%$tauG6#%\"IGF7%(epsilonGF)F)" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 5 "where" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eeqn
:=Diff(epsilon,t)=T[r]-T[m]: eeqn;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%%DiffG6$%(epsilonG%\"tG,&&%\"TG6#%\"rG\"\"\"&F+6#%\"mG!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "and" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "qseqn:= q[s]=W*C[P]*(T[r]-T[i,s]): qseqn;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/&%\"qG6#%\"sG*(%\"WG\"\"\"&%\"CG6#%\"PGF*,&
&%\"TG6#%\"rGF*&F16$%\"iGF'!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 31 "The parameters in the model are" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 90 "params :=\{V=4000/rho/C[P],W=500/C[P],T[i,s]=60,T[r]=
80,tau[d]=1,tau[m]=5,tau[I]=2,K[c]=0\};" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'paramsG<*/%\"VG,$*&%$rhoG!\"\"&%\"CG6#%\"PGF+\"%+S/%\"WG,$*$F
,F+\"$+&/&%\"TG6$%\"iG%\"sG\"#g/&F86#%\"rG\"#!)/&%$tauG6#%\"dG\"\"\"/&
FD6#%\"mG\"\"&/&%\"KG6#%\"cG\"\"!/&FD6#%\"IG\"\"#" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 82 "The initial values of the differential variables a
re given as a list of equations." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "Start := [t=0,T[t]=80,T[0]=80,T[m]=80,epsilon=0]: Sta
rt;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'/%\"tG\"\"!/&%\"TG6#F%\"#!)/&
F)6#F&F+/&F)6#%\"mGF+/%(epsilonGF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 163 "Several different approaches for solving this system of equati
ons can now be followed. A difficulty is that the step change in the i
nlet temperature occurs not at " }{XPPEDIT 18 0 "t=0" "/%\"tG\"\"!" }
{TEXT -1 9 ", but at " }{XPPEDIT 18 0 "t=10" "/%\"tG\"#5" }{TEXT -1 
396 ". One possible approach is to integrate the equations for the fir
st 10 minutes, then make the step change and integrate the equations w
ith the new parameters for as long as needed. Maple's own dsolve/numer
ic will accept a procedure for evaluation of the right hand sides of t
he differential equations. We can program the step change in that proc
edure. Here is our code for this particular system." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 461 "deproc := proc(n,t,y,dy)\nlocal q,qs,k,e
psilon;\nglobal params,T;\nT[tank]:=y[1];\nT[0]:=y[2];\nT[m]:=y[3];\ne
psilon:=y[4];\nif t < 10 then T[i]:=60 else T[i]:=40 fi;\nqs:=W*C[P]*(
T[r]-T[i,s]); \nq:=qs+K[c]*(T[r]-T[m])+K[c]/tau[I]*epsilon;\ndy:=vecto
r(n):\ndy[1]:=(W*C[P]*(T[i]-T[tank])+q)/(rho*V*C[P]):\ndy[2]:=(T[tank]
-T[0]-tau[d]/2*dy[1])*2/tau[d]:\ndy[3]:=(T[0]-T[m])/tau[m]:\ndy[4]:=T[
r]-T[m]:\nfor k from 1 to n do dy[k] := subs(params,dy[k]); od;\nRETUR
N(eval(dy));\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 443 "The argume
nts (required by Maple) are the number of equations, the independent v
ariable, the vector of dependent variables and a vector of right hand \+
sides (computed by the procedure). Note where we have programmed the s
tep change in the inlet temperature and that set of model parameters i
s declared to be a global variable. This facilitates changing the para
meters without changing the procdure, as we will have to do for later \+
calculations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 391 "We solve the problem using dsolve/numeri
c using the following command. Note that by using the optional argumen
ts we can tell dsolve/numeric to use the proceure above for the right \+
hand sides. In this case the first three arguments serve no role excep
t that dsolve/numeric expects them to be there. The last argument tell
s dsolve/numeric to return an array with the results tabulated as show
n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "result:=dsolve(\{diff
(x(t),t)=0,diff(y(t),t)=0,diff(w(t),t)=0,diff(z(t),t)=0\}, \{w(t),x(t)
,y(t),z(t)\},type=numeric, method=rkf45, initial=vector([80,80,80,0]),
start=0,procedure=deproc,value=array([0,seq(i,i=9..20)]));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'resultG-%'MATRIXG6#7$7#-%'VECTORG6#7'%\"t
G-&%1dsolve/numeric_FG6#\"\"\"6#F.-&F16#\"\"#F4-&F16#\"\"$F4-&F16#\"\"
%F47#-F&6#7/7'\"\"!\"#!)FGFGFF7'\"\"*FGFGFGFF7'\"#5$\"+;0++!)!\")$\"+%
[*****zFN$FGFFFF7'\"#6$\"+'R\"okxFN$\"+R%pU'zFN$\"+J+l-!)FN$!+P\\\"y@#
!#67'\"#7$\"+!ye\"evFN$\"+WtufxFN$\"+CA)p(zFN$\"+i#oU&\\Ffn7'\"#8$\"+r
G^#Q(FN$\"+?%psb(FN$\"+vBJ=zFN$\"+:n!3\\&!#57'\"#9$\"+)Gx3C(FN$\"+Nj;#
Q(FN$\"+zE*f$yFN$\"+1Z5i<!\"*7'\"#:$\"+!QJi8(FN$\"+m=kSsFN$\"+Py$*RxFN
$\"+p%*)[(QFfp7'\"#;$\"+xE!42(FN$\"+yW2OrFN$\"+0#f!RwFN$\"+$\\/$zpFfp7
'\"#<$\"+$3[i/(FN$\"+Rx$32(FN$\"+rn;TvFN$\"+8KO36FN7'\"#=$\"+=vViqFN$
\"+z#)GYqFN$\"+oC*HX(FN$\"+DbM7;FN7'\"#>$\"+\\_K=rFN$\"+%)**eiqFN$\"+[
X=!Q(FN$\"+cPC(>#FN7'\"#?$\"+uDS6sFN$\"+*p!f=rFN$\"+[^FFtFN$\"+82LXGFN
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "To extract the table, perhaps
 for plotting purposes we use the op command in the following form" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "op([1,3,2,2],result);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7/7'\"\"!\"#!)F)F)F(7'\"\"
*F)F)F)F(7'\"#5$\"+;0++!)!\")$\"+%[*****zF0$F)F(F(7'\"#6$\"+'R\"okxF0$
\"+R%pU'zF0$\"+J+l-!)F0$!+P\\\"y@#!#67'\"#7$\"+!ye\"evF0$\"+WtufxF0$\"
+CA)p(zF0$\"+i#oU&\\F>7'\"#8$\"+rG^#Q(F0$\"+?%psb(F0$\"+vBJ=zF0$\"+:n!
3\\&!#57'\"#9$\"+)Gx3C(F0$\"+Nj;#Q(F0$\"+zE*f$yF0$\"+1Z5i<!\"*7'\"#:$
\"+!QJi8(F0$\"+m=kSsF0$\"+Py$*RxF0$\"+p%*)[(QFhn7'\"#;$\"+xE!42(F0$\"+
yW2OrF0$\"+0#f!RwF0$\"+$\\/$zpFhn7'\"#<$\"+$3[i/(F0$\"+Rx$32(F0$\"+rn;
TvF0$\"+8KO36F07'\"#=$\"+=vViqF0$\"+z#)GYqF0$\"+oC*HX(F0$\"+DbM7;F07'
\"#>$\"+\\_K=rF0$\"+%)**eiqF0$\"+[X=!Q(F0$\"+cPC(>#F07'\"#?$\"+uDS6sF0
$\"+*p!f=rF0$\"+[^FFtF0$\"+82LXGF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 100 "The first column is the time, the next three are the tank, the
rmocouple, and measured temperatures. " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 24 "A. Open Loop Perf
ormance" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Open loop performance \+
is simulated by setting " }{XPPEDIT 18 0 "K[c]=0" "/&%\"KG6#%\"cG\"\"!
" }{TEXT -1 161 ", which is what was done for the above calculations. \+
We need to integrate for a longer time, however, so we repeat the abov
e command but integrate to 60 minutes." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 215 "result:=dsolve(\{diff(x(t),t)=0,diff(y(t),t)=0,diff(
w(t),t)=0,diff(z(t),t)=0\}, \{w(t),x(t),y(t),z(t)\},type=numeric, meth
od=rkf45, initial=vector([80,80,80,0]),start=0,procedure=deproc,value=
array([0,seq(i,i=9..60)]));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "We
 extract the results table" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "Ttable :=op([1,3,2,2],result): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 32 "and plot the three temperatures." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 96 "plot(\{seq([seq([Ttable[i,1],Ttable[i,k]],i=1..rowdim(Ttable))
],k=2..4)\},color=[red,blue,black]);" }}{PARA 13 "" 1 "" {INLPLOT "6%-
%'CURVESG6$7W7$\"\"!$\"#!)F(7$$\"\"*F(F)7$$\"#5F($\"1+++;0++!)!#97$$\"
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\\lF\\\\lF\\\\lF(" 2 431 231 231 2 0 1 0 2 9 0 4 2 1.000000 45.000000 
45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 
1 0 0 0 349 175 0 0 0 0 0 0 }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 26 "B. \+
Closed loop performance" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "This r
equires a change in " }{XPPEDIT 18 0 "K[c]" "&%\"KG6#%\"cG" }{TEXT -1 
36 " to 50 (it was 0 in the first case)." }}}{EXCHG {PARA 0 "> " 0 "" 
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**>=;J?)F37$Fgcn$\"1+++o`\\q%)F37$F\\dn$\"1+++zwP@()F37$Fadn$\"1+++#oa
c%*)F37$Ffdn$\"1+++F@>M\"*F37$F[en$\"1+++b5?z#*F37$F`en$\"1+++0&eXP*F3
7$Feen$\"1+++OY1;%*F37$Fjen$\"1+++7bi,%*F37$F_fn$\"1,++GPNJ$*F37$Fdfn$
\"1+++*4#e2#*F37$Fifn$\"1+++j#*yM!*F37$F^gn$\"1+++o$R%>))F3-Fcgn6&Fegn
F(F(F(" 2 430 200 200 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 
10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 1 0 0 0 0 
8700 0 0 0 0 0 0 }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 23 "D. Proportiona
l Control" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Proportional only co
ntrol is simulated by setting the term " }{XPPEDIT 18 0 "K[c]/tau[I]" 
"*&&%\"KG6#%\"cG\"\"\"&%$tauG6#%\"IG!\"\"" }{TEXT -1 39 " = 0. We can \+
accomplish this by giving " }{XPPEDIT 18 0 "tau[I]" "&%$tauG6#%\"IG" }
{TEXT -1 33 " a very high value as shown here." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 95 "params :=\{V=4000/rho/C[P],W=500/C[P],T[i,s]=6
0,T[r]=80,tau[d]=1,tau[m]=5,tau[I]=1e99,K[c]=500\};" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'paramsG<*/%\"VG,$*&%$rhoG!\"\"&%\"CG6#%\"PGF+\"%+S
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\"dG\"\"\"/&FD6#%\"mG\"\"&/&FD6#%\"IG$FG\"#**/&%\"KG6#%\"cGF5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "We carry out the integrations for \+
this situation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "result:
=dsolve(\{diff(x(t),t)=0,diff(y(t),t)=0,diff(w(t),t)=0,diff(z(t),t)=0
\}, \{w(t),x(t),y(t),z(t)\},type=numeric, method=rkf45, initial=vector
([80,80,80,0]),start=0,procedure=deproc,\nvalue=array([0,seq(i,i=9..60
)])):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The output table is extr
acted from the result:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "T
table :=op([1,3,2,2],result): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 
"and the three temperatures plotted as a function of time." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot(\{seq([seq([Ttable[i,1],Ttable
[i,k]],i=1..rowdim(Ttable))],k=2..4)\},color=[red,blue,black]);" }}
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