# Penicillin Plant

Fed-batch Fermentor Control: Dynamic Optimization of Batch Processes II. Role of Measurements in Handling Uncertainty 2001, B. Srinivasan, D. Bonvin, E. Visser, S. Palanki

Illustrative example: Nominal Optimization of a Fed-Batch Fermentor for Penicillin Production.

## Problem description

This particular example was featured in the work of B. Srinivasan et al. 2001. The optimal trajectories for the problem was provided in the work.

In this problem, the objective is to maximize the concentration of penicillin, P, produced in a fed-batch bioreactor, given a finite amount of time.

```Reactions: S -> X, S -> P
Conditions: Fed-batch, isothermal.
Objective: Maximize the concentration of product P at a given final time.
Manipulated variable: Feed rate of S.
Constraints: Input bounds; upper limit on the biomass concentration,
which is motivated by oxygen-transfer limitation typically occurring at
large biomass concentration.``` subject to:     Programmer: Wee Kiat Lim (Nanyang Technological University)

```% Copyright (c) 2009-2009 by Tomlab Optimization Inc.
```

## Problem setup

Penalty for variations in u

```penalty_constant = 0.001;

% Various constants
miu_m = 0.02;  Km   = 0.05;  Ki = 5;
Yx    = 0.5;   Yp   = 1.2;   v = 0.004;
Sin   = 200;   umin = 0;     umax = 1;
Xmin  = 0;     Xmax = 3.7;   Smin = 0;

% no. of collocation points to use
narr = [30 100];
for n=narr
toms t1
toms tcut
p1 = tomPhase('p1', t1, 0, tcut, n);

setPhase(p1);
tomStates X1 S1 P1 V1 %Vs %Scaling is disabled here
tomControls u1

% Initial guess
if n == narr(1)
x01 = {tcut == 75
icollocate({X1 == 1+2.7*t1/tcut; S1 == 0.5;
P1 == 0.6*t1/tcut; V1 == 150})
collocate(u1 == 0.03+0.06*t1/tcut)};
else
x01 = {tcut == tcutg
icollocate({X1 == Xg1; S1 == Sg1; P1 == Pg1; V1 == Vg1})
collocate(u1 == ug1)};
end

% Box constraints
cbox1 = {75 <= tcut <= 85
0 <= icollocate(X1) <= Xmax
Smin <= icollocate(S1) <= 100
0 <= icollocate(P1) <= 5
1 <= icollocate(V1) <= 300
umin <= collocate(u1) <= umax};

% Boundary constraints
cbnd1 = initial({X1 == 1; S1 == 0.5;
P1 == 0; V1 == 150});

miu1 = (miu_m*S1)/(Km + S1 + S1^2/Ki);

% ODEs and path constraints
temp11 = miu1*X1;
temp21 = u1/V1;
temp31 = v*X1;
ceq1 = collocate ({
dot(X1) == temp11 - u1/V1*X1
dot(S1) == -temp11/Yx - temp31/Yp + temp21*(Sin - S1)
dot(P1) == temp31 - temp21*P1
dot(V1) == u1});

if n == narr(1)
% No objective in first phase
objective = 0;
else
% Variation penalty
objective = penalty_constant*integrate(dot(u1)^2);
end

toms t2
p2 = tomPhase('p2', t2, tcut, 150-tcut, n);

setPhase(p2);
tomStates X2 S2 P2 V2 %Vs %Scaling is disabled here
tomControls u2

% Initial guess
if n == narr(1)
x02 = {icollocate({X2 == Xmax; S2 == 0; P2 == 0.6+t2/150;
V2 == 150}); collocate(u2 == 0.01)};
else
x02 = {icollocate({X2 == Xg2; S2 == Sg2; P2 == Pg2; V2 == Vg2})
collocate(u2 == ug2)};
end

% Box constraints
umax2 = 0.03;
cbox2 = {0 <= icollocate(X2) <= Xmax
Smin <= icollocate(S2) <= 100
0 <= icollocate(P2) <= 5
1 <= icollocate(V2) <= 300
umin <= collocate(u2) <= umax2
initial(S2) <= 0.2};

miu2 = (miu_m*S2)/(Km + S2 + S2^2/Ki);

% ODEs and path constraints
temp12 = miu2*X2;
temp22 = u2/V2;
temp32 = v*X2;
ceq2 = collocate ({
dot(X2) == temp12 - u2/V2*X2
dot(S2) == -temp12/Yx - temp32/Yp + temp22*(Sin - S2)
dot(P2) == temp32 - temp22*P2
dot(V2) == u2});

initial(S2) == final(p1,S1)
initial(P2) == final(p1,P1)
initial(V2) == final(p1,V1)};

if n == narr(1)
% Objective (Negative sign is added to 'maximize' P)
objective = -final(P2);
type = 'lpcon';
solver = 'snopt';
else
objective = objective-final(P2)+penalty_constant*integrate(dot(u2)^2);
type = 'con';
solver = 'snopt';
end

% Solve the problem
options = struct;
options.name = 'Penicillin Plant';
Prob = sym2prob(type, objective, {cbox1, cbnd1, ceq1, cbox2, ceq2, links}, {x01, x02}, options);
Result = tomRun(solver, Prob, 1);
solution = getSolution(Result);

t1 = subs(collocate(p1,t1),solution);
t2 = subs(collocate(p2,t2),solution);
ug1 = subs(collocate(p1,u1),solution);
Xg1 = subs(collocate(p1,X1),solution);
Sg1 = subs(collocate(p1,S1),solution);
Pg1 = subs(collocate(p1,P1),solution);
Vg1 = subs(collocate(p1,V1),solution);
ug2 = subs(collocate(p2,u2),solution);
Xg2 = subs(collocate(p2,X2),solution);
Sg2 = subs(collocate(p2,S2),solution);
Pg2 = subs(collocate(p2,P2),solution);
Vg2 = subs(collocate(p2,V2),solution);
tcutg = solution.tcut;
np=length(t1) + length(t2);
end
```
```===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Penicillin Plant               f_k      -1.682687588966889100
sum(|constr|)      0.000001056758526911
f(x_k) + sum(|constr|)     -1.682686532208362200
f(x_0)     -1.600000000000001400

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv  126 ConJacEv  126 Iter   68 MinorIter 3734
CPU time: 2.625000 sec. Elapsed time: 2.641000 sec.
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Penicillin Plant               f_k      -1.682693617368655700
sum(|constr|)      0.000003716816491676
f(x_k) + sum(|constr|)     -1.682689900552164100
f(x_0)     -1.157798971400167100

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv  102 GradEv  100 ConstrEv  100 ConJacEv  100 Iter   60 MinorIter 9857
CPU time: 51.703125 sec. Elapsed time: 53.094000 sec.
```

## Plot result

Optimal states and control trajectories

```t = [t1;t2];
uopt = [ug1;ug2];
Xopt = [Xg1;Xg2];
Sopt = [Sg1;Sg2];
Popt = [Pg1;Pg2];
Vopt = [Vg1;Vg2];

Pfinal=subs(final(P2),solution);

% Plots of the trajectories
figure(1)
subplot(3,1,1);
plot(t,Popt,'*-');
title(['Final Penicillin concentration is ',num2str(Pfinal),' g/L.'])
ylabel('Penicillin Conc')
xlabel('Time (hrs)')
subplot(3,1,2);
plot(t,Xopt,'*-');
ylabel('Cell Mass Conc')
xlabel('Time (hrs)')
subplot(3,1,3);
plot(t,Sopt,'*-');
ylabel('Substrate Conc')
xlabel('Time (hrs)')

figure(2)
subplot(2,1,1);
plot(t,Vopt,'*-');
title(['Final Penicillin concentration is ',num2str(Pfinal),' g/L.'])
ylabel('Volume of medium')
xlabel('Time (hrs)')
subplot(2,1,2);
plot(t,uopt,'*-');
ylabel('Feed flowrate')
xlabel('Time (hrs)')

fprintf('\n')
fprintf('Optimization completed... \n')
fprintf('Final Penicillin concentration is %5.4f g/L.\n',Pfinal)
```
```Optimization completed...
Final Penicillin concentration is 1.6827 g/L.
```  