PROPT Lee-Ramirez Bioreactor - Revision history

Per: /* Plot result */

2012-02-14T09:21:44Z

Plot result

← Older revision		Revision as of 04:21, 14 February 2012
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	[[File:leeBioReactor_02.png]]		[[File:leeBioReactor_02.png]]

			[[Category:PROPT Examples]]

Mbot at 12:09, 9 November 2011

2011-11-09T12:09:12Z

Mbot at 13:06, 8 November 2011

2011-11-08T13:06:12Z

← Older revision		Revision as of 08:06, 8 November 2011
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	<pre>		<pre>
	Problem type appears to be: qpcon		Problem type appears to be: qpcon
	Time for symbolic processing: 0.~~85283~~ seconds		Time for symbolic processing: 0.75819 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 1 ConstrEv 744 ConJacEv 744 Iter 573 MinorIter 3217		FuncEv 1 ConstrEv 744 ConJacEv 744 Iter 573 MinorIter 3217
	CPU time: 2.~~542816~~ sec. Elapsed time: 2.~~538000~~ sec.		CPU time: 2.308815 sec. Elapsed time: 2.305000 sec.

	</pre>		</pre>
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	<pre>		<pre>
	Problem type appears to be: qpcon		Problem type appears to be: qpcon
	Time for symbolic processing: 0.~~86391~~ seconds		Time for symbolic processing: 0.74722 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 1 ConstrEv 548 ConJacEv 548 Iter 486 MinorIter 1855		FuncEv 1 ConstrEv 548 ConJacEv 548 Iter 486 MinorIter 1855
	CPU time: 5.~~709637~~ sec. Elapsed time: 5.~~969000~~ sec.		CPU time: 5.787637 sec. Elapsed time: 5.447000 sec.

	</pre>		</pre>
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	Problem type appears to be: qpcon		Problem type appears to be: qpcon
	Time for symbolic processing: 0.~~84929~~ seconds		Time for symbolic processing: 0.75518 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 1 ConstrEv 508 ConJacEv 508 Iter 443 MinorIter 2428		FuncEv 1 ConstrEv 508 ConJacEv 508 Iter 443 MinorIter 2428
	CPU time: 16.~~520506~~ sec. Elapsed time: 17.~~194000~~ sec.		CPU time: 16.052503 sec. Elapsed time: 16.059000 sec.

	</pre>		</pre>
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	Problem type appears to be: qpcon		Problem type appears to be: qpcon
	Time for symbolic processing: 0.~~83028~~ seconds		Time for symbolic processing: 0.75258 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 1 ConstrEv 689 ConJacEv 689 Iter 616 MinorIter 2438		FuncEv 1 ConstrEv 689 ConJacEv 689 Iter 616 MinorIter 2438
	CPU time: 83.~~694536~~ sec. Elapsed time: 85.~~082000~~ sec.		CPU time: 79.123707 sec. Elapsed time: 79.308000 sec.

	</pre>		</pre>

Mbot at 18:24, 2 November 2011

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	{{Part Of Manual\|title=the PROPT Manual\|link=[[PROPT \|PROPT Manual]]}}		{{Part Of Manual\|title=the PROPT Manual\|link=[[PROPT\|PROPT Manual]]}}

	Dynamic optimization of chemical and biochemical processes using restricted second-order information 2001, Eva Balsa-Canto, Julio R. Banga, Antonio A. Alonso Vassilios S. Vassiliadis		Dynamic optimization of chemical and biochemical processes using restricted second-order information 2001, Eva Balsa-Canto, Julio R. Banga, Antonio A. Alonso Vassilios S. Vassiliadis
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	</pre>		</pre>

			[[File:leeBioReactor_01.png]]

			[[File:leeBioReactor_02.png]]

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Dynamic optimization of chemical and biochemical processes using restricted second-order information 200..."

2011-11-02T15:29:45Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Dynamic optimization of chemical and biochemical processes using restricted second-order information 200..."

New page

{{Part Of Manual|title=the PROPT Manual|link=[[PROPT |PROPT Manual]]}}

Dynamic optimization of chemical and biochemical processes using restricted second-order information 2001, Eva Balsa-Canto, Julio R. Banga, Antonio A. Alonso Vassilios S. Vassiliadis

Case Study II: Lee-Ramirez bioreactor

==Problem description==

This problem considers the optimal control of a fed-batch reactor for induced foreign protein production by recombinant bacteria, as presented by Lee and Ramirez (1994) and considered afterwards by Tholudur and Ramirez (1997) and Carrasco and Banga (1998). The objective is to maximize the profitability of the process using the nutrient and the inducer feeding rates as the control variables. Three different values for the ratio of the cost of inducer to the value of the protein production (Q) were considered.

The mathematical formulation, following the modified parameter function set presented by Tholudur and Ramirez (1997) to increase the sensitivity to the controls, is as follows:

Find u1(t) and u2(t) over t in [t0 t_f] to maximize:

<math> J = x_4(t_f)*x_1(t_f) - Q*\int_{t_0}^{t_f} u_2 \mathrm{d}t </math>

subject to:

<math> \frac{dx_1}{dt} = u_1+u_2 </math>
<math> \frac{dx_2}{dt} = g_1.*x_2-(u_1+u_2).*\frac{x_2}{x_1} </math>
<math> \frac{dx_3}{dt} = \frac{u_1}{x_1}.*c_1-(u_1+u_2).*\frac{x_3}{x_1}-g_1.*\frac{x_2}{c_2} </math>
<math> \frac{dx_4}{dt} = g_2.*x_2-(u_1+u_2).*\frac{x_4}{x_1} </math>
<math> \frac{dx_5}{dt} = \frac{u_2*c_3}{x_1}-(u_1+u_2).*\frac{x_5}{x_1} </math>
<math> \frac{dx_6}{dt} = -g_3.*x_6 </math>
<math> \frac{dx_7}{dt} = g_3.*(1-x_7) </math>

<math> t_1 = 14.35+x_3+(\frac{x_3^2}{111.5}) </math>
<math> t_2 = 0.22+x_5 </math>
<math> t_3 = x_6+\frac{0.22}{t_2}*x_7 </math>

<math> g_1 = \frac{x_3}{t_1}*(x_6+x_7*\frac{0.22}{t_2}) </math>
<math> g_2 = 0.233*\frac{x_3}{t_1}*(\frac{0.0005+x_5}{0.022+x_5}) </math>
<math> g_3 = 0.09*\frac{x_5}{0.034+x_5} </math>

<math> c_1 = 100 </math>
<math> c_2 = 0.51 </math>
<math> c_3 = 4 </math>

where the state variables are the reactor volume (x1), the cell density (x2), the nutrient concentration (x3), the foreign protein concentration (x4), the inducer concentration (x5), the inducer shock factor on cell growth rate (x6) and the inducer recovery factor on cell growth rate (x7). The two control variables are the glucose rate (u1) and the inducer feeding rate (u2). Q is the ratio of the cost of inducer to the value of the protein production, and the final time is considered fixed as t_f = 10 h. The model parameters were described by Lee and Ramirez (1994). The initial conditions are:

<math> x(t_0) = [1 \ 0.1 \ 40 \ 0 \ 0 \ 1 \ 0]' </math>

The following constraints on the control variables are considered:

<math> 0 <= u1 <= 1 </math>
<math> 0 <= u2 <= 1 </math>

<source lang="matlab">
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
</source>

==Problem setup==

<source lang="matlab">
toms t

for n=[20 35 55 85]
</source>

<source lang="matlab">
p = tomPhase('p', t, 0, 10, n);
setPhase(p);

tomStates z1 z2 z3s z4 z5 z6 z7

% Declaring u as "states" makes it possible to work with their
% derivatives.
tomStates u1 u2

% Scale z3 by 40
z3 = z3s*40;

% Initial guess
if n == 20
x0 = {icollocate({z1 == 1; z2 == 0.1
z3 == 40; z4 == 0; z5 == 0
z6 == 1; z7 == 0})
icollocate({u1==t/10; u2==t/10})};
else
x0 = {icollocate({z1 == z1opt
z2 == z2opt; z3 == z3opt
z4 == z4opt; z5 == z5opt
z6 == z6opt; z7 == z7opt})
icollocate({u1 == u1opt
u2 == u2opt})};
end

% Box constraints
cbox = {mcollocate({0 <= z1; 0 <= z2
0 <= z3; 0 <= z4; 0 <= z5})
0 <= collocate(u1) <= 1
0 <= collocate(u2) <= 1};

% Boundary constraints
cbnd = initial({z1 == 1; z2 == 0.1
z3 == 40; z4 == 0; z5 == 0
z6 == 1; z7 == 0});

% Various constants and expressions
c1 = 100; c2 = 0.51; c3 = 4.0;
Q = 0;

t1 = 14.35+z3+((z3).^2/111.5);
t2 = 0.22+z5;
t3 = z6+0.22./t2.*z7;

g1 = z3./t1.*(z6+z7*0.22./t2);
g2 = 0.233*z3./t1.*((0.0005+z5)./(0.022+z5));
g3 = 0.09*z5./(0.034+z5);

% ODEs and path constraints
ceq = collocate({
dot(z1) == u1+u2
dot(z2) == g1.*z2-(u1+u2).*z2./z1
dot(z3) == u1./z1.*c1-(u1+u2).*z3./z1-g1.*z2/c2
dot(z4) == g2.*z2-(u1+u2).*z4./z1
dot(z5) == u2*c3./z1-(u1+u2).*z5./z1
dot(z6) == -g3.*z6
dot(z7) == g3.*(1-z7)});

% Objective
J = -final(z1)*final(z4)+Q*integrate(u2);

spenalty = 0.1/n; % penalty term to yield a smoother u.
objective = J + spenalty*integrate(dot(u1)^2+dot(u2)^2);
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Lee Bio Reactor';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);

% Optimal z, u for starting point
z1opt = subs(z1, solution);
z2opt = subs(z2, solution);
z3opt = subs(z3, solution);
z4opt = subs(z4, solution);
z5opt = subs(z5, solution);
z6opt = subs(z6, solution);
z7opt = subs(z7, solution);
u1opt = subs(u1, solution);
u2opt = subs(u2, solution);
</source>

<pre>
Problem type appears to be: qpcon
Time for symbolic processing: 0.85283 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Lee Bio Reactor f_k -6.158551034261496900
sum(|constr|) 0.000000279053783607
f(x_k) + sum(|constr|) -6.158550755207713200
f(x_0) 0.001000000000000196

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

FuncEv 1 ConstrEv 744 ConJacEv 744 Iter 573 MinorIter 3217
CPU time: 2.542816 sec. Elapsed time: 2.538000 sec.

</pre>

<pre>
Problem type appears to be: qpcon
Time for symbolic processing: 0.86391 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Lee Bio Reactor f_k -6.149281027809802700
sum(|constr|) 0.000002151758382440
f(x_k) + sum(|constr|) -6.149278876051420500
f(x_0) -6.161215441652444700

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

FuncEv 1 ConstrEv 548 ConJacEv 548 Iter 486 MinorIter 1855
CPU time: 5.709637 sec. Elapsed time: 5.969000 sec.

</pre>

<pre>
Problem type appears to be: qpcon
Time for symbolic processing: 0.84929 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Lee Bio Reactor f_k -6.148479046961865500
sum(|constr|) 0.000000948376662244
f(x_k) + sum(|constr|) -6.148478098585203000
f(x_0) -6.151342657272863300

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

FuncEv 1 ConstrEv 508 ConJacEv 508 Iter 443 MinorIter 2428
CPU time: 16.520506 sec. Elapsed time: 17.194000 sec.

</pre>

<pre>
Problem type appears to be: qpcon
Time for symbolic processing: 0.83028 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Lee Bio Reactor f_k -6.149330296208792600
sum(|constr|) 0.000000385398306198
f(x_k) + sum(|constr|) -6.149329910810486400
f(x_0) -6.149452089871609000

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

FuncEv 1 ConstrEv 689 ConJacEv 689 Iter 616 MinorIter 2438
CPU time: 83.694536 sec. Elapsed time: 85.082000 sec.

</pre>

<source lang="matlab">
end

t = subs(collocate(t),solution);
z1 = subs(collocate(z1),solution);
z2 = subs(collocate(z2),solution);
z3 = subs(collocate(z3),solution);
z4 = subs(collocate(z4),solution);
z5 = subs(collocate(z5),solution);
z6 = subs(collocate(z6),solution);
z7 = subs(collocate(z7),solution);
u1 = subs(collocate(u1),solution);
u2 = subs(collocate(u2),solution);
</source>

==Plot result==

<source lang="matlab">
figure(1)
plot(t,z1,'*-',t,z2,'*-',t,z3/10,'*-',t,z4,'*-' ...
,t,z5,'*-',t,z6,'*-',t,z7,'*-');
legend('z1','z2','z3/10','z4','z5','z6','z7');
title('Lee Bio Reactor state variables.');

figure(2)
plot(t,u1,'+-',t,u2,'+-');
legend('u1','u2');
title('Lee Bio Reactor control');

disp('J = ');
disp(subs(J,solution));
</source>

<pre>
J =
-6.1510

</pre>

← Older revision		Revision as of 07:09, 9 November 2011
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	<math> J = x_4(t_f)x_1(t_f) - Q\int_{t_0}^{t_f} u_2 \mathrm{d}t </math>		<math> J = x_4(t_f)x_1(t_f) - Q\int_{t_0}^{t_f} u_2 \mathrm{d}t </math>


	subject to:		subject to:

	<math> \frac{dx_1}{dt} = u_1+u_2 </math>		<math> \frac{dx_1}{dt} = u_1+u_2 </math>

	<math> \frac{dx_2}{dt} = g_1.x_2-(u_1+u_2).\frac{x_2}{x_1} </math>		<math> \frac{dx_2}{dt} = g_1.x_2-(u_1+u_2).\frac{x_2}{x_1} </math>

	<math> \frac{dx_3}{dt} = \frac{u_1}{x_1}.c_1-(u_1+u_2).\frac{x_3}{x_1}-g_1.*\frac{x_2}{c_2} </math>		<math> \frac{dx_3}{dt} = \frac{u_1}{x_1}.c_1-(u_1+u_2).\frac{x_3}{x_1}-g_1.*\frac{x_2}{c_2} </math>

	<math> \frac{dx_4}{dt} = g_2.x_2-(u_1+u_2).\frac{x_4}{x_1} </math>		<math> \frac{dx_4}{dt} = g_2.x_2-(u_1+u_2).\frac{x_4}{x_1} </math>

	<math> \frac{dx_5}{dt} = \frac{u_2c_3}{x_1}-(u_1+u_2).\frac{x_5}{x_1} </math>		<math> \frac{dx_5}{dt} = \frac{u_2c_3}{x_1}-(u_1+u_2).\frac{x_5}{x_1} </math>

	<math> \frac{dx_6}{dt} = -g_3.*x_6 </math>		<math> \frac{dx_6}{dt} = -g_3.*x_6 </math>

	<math> \frac{dx_7}{dt} = g_3.*(1-x_7) </math>		<math> \frac{dx_7}{dt} = g_3.*(1-x_7) </math>


	<math> t_1 = 14.35+x_3+(\frac{x_3^2}{111.5}) </math>		<math> t_1 = 14.35+x_3+(\frac{x_3^2}{111.5}) </math>

	<math> t_2 = 0.22+x_5 </math>		<math> t_2 = 0.22+x_5 </math>

	<math> t_3 = x_6+\frac{0.22}{t_2}*x_7 </math>		<math> t_3 = x_6+\frac{0.22}{t_2}*x_7 </math>


	<math> g_1 = \frac{x_3}{t_1}(x_6+x_7\frac{0.22}{t_2}) </math>		<math> g_1 = \frac{x_3}{t_1}(x_6+x_7\frac{0.22}{t_2}) </math>

	<math> g_2 = 0.233\frac{x_3}{t_1}(\frac{0.0005+x_5}{0.022+x_5}) </math>		<math> g_2 = 0.233\frac{x_3}{t_1}(\frac{0.0005+x_5}{0.022+x_5}) </math>

	<math> g_3 = 0.09*\frac{x_5}{0.034+x_5} </math>		<math> g_3 = 0.09*\frac{x_5}{0.034+x_5} </math>


	<math> c_1 = 100 </math>		<math> c_1 = 100 </math>

	<math> c_2 = 0.51 </math>		<math> c_2 = 0.51 </math>

	<math> c_3 = 4 </math>		<math> c_3 = 4 </math>


	where the state variables are the reactor volume (x1), the cell density (x2), the nutrient concentration (x3), the foreign protein concentration (x4), the inducer concentration (x5), the inducer shock factor on cell growth rate (x6) and the inducer recovery factor on cell growth rate (x7). The two control variables are the glucose rate (u1) and the inducer feeding rate (u2). Q is the ratio of the cost of inducer to the value of the protein production, and the final time is considered fixed as t_f = 10 h. The model parameters were described by Lee and Ramirez (1994). The initial conditions are:		where the state variables are the reactor volume (x1), the cell density (x2), the nutrient concentration (x3), the foreign protein concentration (x4), the inducer concentration (x5), the inducer shock factor on cell growth rate (x6) and the inducer recovery factor on cell growth rate (x7). The two control variables are the glucose rate (u1) and the inducer feeding rate (u2). Q is the ratio of the cost of inducer to the value of the protein production, and the final time is considered fixed as t_f = 10 h. The model parameters were described by Lee and Ramirez (1994). The initial conditions are:

	<math> x(t_0) = [1 \ 0.1 \ 40 \ 0 \ 0 \ 1 \ 0]' </math>		<math> x(t_0) = [1 \ 0.1 \ 40 \ 0 \ 0 \ 1 \ 0]' </math>


	The following constraints on the control variables are considered:		The following constraints on the control variables are considered:

	<math> 0 <= u1 <= 1 </math>		<math> 0 <= u1 <= 1 </math>

	<math> 0 <= u2 <= 1 </math>		<math> 0 <= u2 <= 1 </math>


	<source lang="matlab">		<source lang="matlab">