1. Inverse problem (from experimental data to model construction)
Part 2
Parameter fitting for ODEs using fmincon function
Example by Xueyang Feng: Nov 16th
X = FMINCON(FUN,X0,A,B,Aeq,Beq) minimizes FUN subject to the linear
equalities Aeq*X = Beq as well as A*X <= B. (Set A=[] and B=[] if no
inequalities exist.)
4/10/2017

2. d) Parameter fitting using nlinfit
Inverse problem
Unlike forward problems, inverse problems require experimental data, and an iterative solution. Because inverse problems require solving the forward problem numerous time, the ode45 solver will be nested within a nonlinear regression routine called “nlinfit.”
The syntax is: [param, r, J, COVB, mse] = nlinfit(X, y, fun, beta0);
returns the fitted coefficients param, the residuals r, the Jacobian J of function fun, the estimated covariance matrix COVB for the fitted coefficients, and an estimate MSE of the variance of the error term.
X is a matrix of n rows of the independent variable
y is n-by-1 vector of the observed data
fun is a function handle to a separate m-file to a function of this form:
yhat = fun(b,X)
where yhat is an n-by-1 vector of the predicted responses, and b is a vector of the parameter values. beta0 is the initial guesses of the parameters.

3. Example 1: Model fitting ODE equation
Based on experimental data, find y0 and k.
data =xlsread('exp_data.xls'); %read data from excel
yo=100; k=0.6;
beta0(1)=yo;
beta0(2)=k;
x=data(:,1); yobs=data(:,2);
[param,resids,J,COVB,mse] = nlinfit(x,yobs,'forderinv',beta0);
rmse=sqrt(mse); %root mean square error = SS/(n-p)
%R is the correlation matrix for the parameters, sigma is the standard error vector
[R,sigma]=corrcov(COVB);
%confidence intervals for parameters
ci=nlparci(param,resids,J);
%computed Cpredicted by solving ode45 once with the estimated parameters
ypred=forderinv(param,x);
%mean of the residuals
meanr=mean(resids);

4. Function with ode45 figure hold on
h1(1)=plot(x,ypred,'-','linewidth',3); %predicted y values
h1(2)=plot(x,yobs,'square', 'Markerfacecolor', 'r');
legend(h1,'ypred','yobs')
xlabel('time (min)')
ylabel('y')
%residual scatter plot
plot(x, resids, 'square','Markerfacecolor', 'b');
YLine = [0 0];
XLine = [0 max(x)];
plot (XLine, YLine,'R'); %plot a straight red line at zero
ylabel('Observed y - Predicted y')
xlabel('time (min)‘)
Function with ode45
function y = forderinv(param,t)
%first-order reaction equation
tspan=t; %we want y at every t
tspan, param(1)); %param(1) is y(0)
function dy = ff(t, y) %function that computes the dydt
dy(1)= -param(2)*y(1);
end

5. One ODE Raw data File name: exp_data.xls Time y Time y 0 107.2637
Time y
417395
One ODE

6. Three ODES for batch fermentationX P
Yp/s=Yx/s*Yp/x

7. Three ODEs parameter fitting
File name: HW217.xls t X P S
0.05 15
2.7
5.4
8.1 14
20.5
24.1

8. Fitting four parameters
Script
clear all %clear all variables
global y0
data =xlsread('HW217.xls');
%initial conditions
y0=[ ];
%initial guess
Umax=0.1; Ks=10; Ypx=0.11; Yxs=0.45;
beta0(1)=Umax; beta0(2)=Ks; beta0(3)=Ypx; beta0(4)=Yxs;
%Measured data
x=data(:,1);
yobsX=data(:,2);
yobsP=data(:,3);
yobsS=data(:,4);
yobs=[yobsX; yobsP; yobsS];
%nlinfit returns parameters, residuals, Jacobian (sensitivity coefficient matrix),
%covariance matrix, and mean square error. ode45 is solved many times
%iteratively
[param,resids,J,COVB,mse] = nlinfit(x, yobs,'forderinv2', beta0);

9. Script
rmse=sqrt(mse); %root mean square error = SS/(n-p)
n=size(x); nn=n(1);
%confidence intervals for parameters
ci=nlparci(param,resids,J);
%computed Cpredicted by solving ode45 once with the estimated parameters
ypred=forderinv2(param,x);
ypredX=ypred(1:nn);
ypredP=ypred(nn+1:2*nn);
ypredS=ypred(2*nn+1:3*nn);
%mean of the residuals
meanr=mean(resids);

10. Script
figure
hold on
plot(x, ypredX, x, ypredP, x, ypredS); %predicted y values
plot(x, yobsX, 'r+', x, yobsP, 'ro', x, yobsS, 'rx');
xlabel('time (min)')
ylabel('y')
%residual scatter plot
x3=[x; x; x];
plot(x3, resids, 'square', 'Markerfacecolor', 'b');
YLine = [0 0];
XLine = [0 max(x)];
plot (XLine, YLine,'R'); %plot a straight red line at zero
ylabel('Observed y - Predicted y')
xlabel('time (min)‘)

11. Function and sub-function
function y = forderinv2(param,t)
%first-order reaction equation
global y0;
tspan=t; %we want y at every t
%param(1) is y(0)
function dy = ff(t,y) %function that computes the dydt
dy(1)= param(1)*y(3)/(param(2)+y(3))*y(1); %biomass
dy(2)= param(1)*param(3)*y(3)/(param(2)+y(3))*y(1); %product
dy(3)= -1/param(4)*dy(1)-1/(param(3)*param(4))*dy(2); %substrate
dy=dy';
end
% after the ode45, rearrange the n-by-3 y matrix into a 3n-by-1 matrix
% and send that back to nlinfit.
y1=y(:,1); y2=y(:,2); y3=y(:,3);
y=[y1; y2; y3];

12. 3. Simulink Basic elements Blocks
Blocks: generate, modify, combine, and display signals
Typical blocks
Continuous:  Linear, continuous-time system elements (integrators, transfer functions, state-space models, etc.)
Discrete:  Linear, discrete-time system elements (integrators, transfer functions, state-space models, etc.)
Functions & Tables:  User-defined functions and tables for interpolating function values
Math:  Mathematical operators (sum, gain, dot product, etc.)
Nonlinear:  Nonlinear operators (coulomb/viscous friction, switches, relays, etc.)
Signals:  Blocks for controlling/monitoring signals
Sinks:  Used to output or display signals (displays, scopes, graphs, etc.)
Sources: Used to generate various signals (step, ramp, sinusoidal, etc.)
Lines
transfer signals from one block to another

13. The graphic interfaces
3. Simulink
Simulink
Type “simulink” in the command window
The graphic interfaces
Matlab

14. Tutorial example
x = sin (t)
simout
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15. Simulink example 2 Creep compliance of a wheat protein film
Using formaldehyde cross-linker
Creep compliance of a wheat protein film (determination of retardation time and free dashpot viscosity in the Jefferys model)
Where J is the strain, J1 is the retarded compliance (Pa-1); λret=µ1/G1 is retardation time (s); µ0 is the free dashpot viscosity (Pa s); t is the time.
The recovery of the compliance is following the equation (t >t1):
Where t1 is the time the stress was released.

16. Creep compliance of a wheat protein film
4. Simulink examples
Creep compliance of a wheat protein film
Simulink model
Parameters: J1 = a = 0.38 Mpa-1; λret = b = s; µ0 = c = Mpa s

17. Creep compliance of a wheat protein film
4. Simulink examples
Creep compliance of a wheat protein film
Simulation result

18. 4. Simulink example Fermentation system Agitation Regular fermenter
Ethanol fermenter
Heater
Heater
Products
Ethanol
Biomass
Biomass
Aeration

19. Ethanol production kinetics
4. Simulink examples
Ethanol production kinetics
Growth kinetics
Substance consumption
Product production
Where: η is the toxic power, Pmax is the maximum product concentration at which the growth is completely inhibited

20. 4. Simulink examples a. Using blocks to construct the model
Using S-function to construct the model (you may also writing the programs for simulink function)

21. Special help session November 28 (Monday, 6~9pm) I will be in Sever 201 computer Lab to answer your modeling questions.
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22. Introduction to Optimization
The maximizing or minimizing of a given function subject to some type of constraints.
Make your processes as effective as possible
Good afternoon all. This talk is about addressing challenges in the simulation of lithium-ion battery models.

23. Example 1 (fminsearch) c) Introduction to Optimization
To find the minimum of a multidimensional function in [-1.2, 1]:
The plot of the function can be generated by the following code:
[x, y]=meshgrid ([-2:0.1:2]);
z=100*(y-x.^2).^2+(1-x).^2;
mesh(x, y, z);
grid on
clear
clc
[x, y]=meshgrid ([-2:0.1:2]);
z=100*(y-x.^2).^2+(1-x).^2;
mesh(x, y, z);
grid on
[-1.2,1]);
x=a(1)
y=a(2)
z=100*(y-x.^2).^2+(1-x).^2
The function of ex2c is written as follow:
function f=ex2c(x)
f=100*(x(2)-x(1)^2)^2+(1-x(1))^2;
The function of ex2cMinimum is written as follow:
function ex2cMinimum
[-1.2,1]);

24. Example 2 Using fmin for curve fitting function sumsqe= two_var(aa)
%===========================================
% Michaelis & Menten Model Fit.
% File name "SimpleMMplot.m".
clear
clf
global S V;
% experiemntal data
S=[ ];
V=[ ];
v0=[1; 1];
a=fminsearch('two_var',v0); %least square errors
Km=a(1);
Vmax=a(2);
Vmodel=Vmax*S./(Km+S);
plot(S,V,'ro', S, Vmodel)
xlabel('Concentrations')
ylabel('rate')
legend('Data','Predict')
Example 2
Using fmin for curve fitting
function sumsqe= two_var(aa)
global S V;
Km =aa(1); Vmax =aa(2); Vmodel=Vmax*S./(Km+S);
err=V-Vmodel; err2=err.*err; sumsqe=sum(err2);
return

25. Find values of x that minimize
Example 3
Find values of x that minimize
starting at the point x = [10; 10; 10] and subject to the constraints
X = FMINCON(FUN, X0, A, B, Aeq, Beq, LB, UB) minimizes FUN subject to the linear equalities Aeq*X = Beq as well as A*X <= B. (Set A=[] and B=[] if no inequalities exist.)

26. Use fmincon for optimization
FMINCON attempts to solve problems of the form:
min F(X) subject to: A*X <= B, Aeq*X = Beq (linear constraints)
C(X) <= 0, Ceq(X) = 0 (nonlinear constraints)
LB <= X <= UB (bounds)
%file name example1
clc;close all;clear all;
int_guess=[10;10;10];% INITIAL GUESS FOR U'S
A=[ ;1 2 2]; % equality constraints
B=[0 72];
int_guess, A, B) % CALL FOR OPTIMIZER
%file name eg_1.m
function [err]=eg_1(u)
x1=u(1);
x2=u(2);
x3=u(3);
err=-x1*x2*x3;

27. Example 4 (dynamic optimization)
Consider the batch reactor with following reaction
A B C
Find the temperature, at which the product B is maximum
Mathematical Representation of the system is as :

28. T=335.3244 FVAL = -0.6058 clc;close all;clear all; warning off
int_guess=300;% INITIAL GUESS FOR U'S
LB=298; % LOWER BOUND OF U
UB=398;% UPPER BOUND ON U
% CALL FOR OPTIMIZER PROBABLY NOT TO CHANGE BE USER
[err]=reactor_problem(u);
load data_for_system
plot(t,Y(:,1),'k')
hold on
plot(t,Y(:,2),'m')
legend('opt c_A','opt c_B') u T=
FVAL =

29. Maximize B ODE calculation function [err]=reactor_problem(u)
A=[1 0]; %initial condition
1], A,[],u); %tspan=1
err= - Y(end,2); %make the maximum B
save data_for_system t Y %just for plotting
ODE calculation
function [YPRIME]=reactor_ODE(t,x,u)
YPRIME=zeros(2,1);
T=u;
k1=4000*exp(-2500/T);
k2=620000*exp(-5000/T);
YPRIME(1)=-k1*(x(1))^2;
YPRIME(2)=(k1*x(1)^2)-k2*x(2);

30. Using MATLAB for solving more complicated dynamic models (Optional)
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31. For example, solve complicated dynamic model in the bioprocess
Many biological systems are dynamic and heterogeneous! Therefore, structured and segregated models have to be used.

32. Definitions • Ordinary Differential Equation (ODE): The dependent
variable is a function of only one independent variable.
• Partial Differential Equation (PDE): The dependent
variable is a function of more than one dependent variable.
Boundary-value problems are those where space conditions are not known; Initial-value problems are those dynamic condition are not known.

33. Boundary-value problems
• Dirichlet boundary conditions are those where a fixed value of a variable is known at a particular location.
• Neumann boundary conditions are those where a derivative is known at a particular location.
Finite-Difference Methods are often used to solve boundary-value problems. In these techniques, finite differences are substituted for the derivatives in the original equation, transforming a linear differential equation into a set of simultaneous algebraic equations.

34. Final equation

35. Aqueous Phase Marine pollution by oil compounds
Sediment
particle
Oil
Aqueous Phase
Sediment particle
Bacteria
Oil transport and degradation in the porous sediment
4/10/2017

36. How to solve PDEs using the finite difference method?
The change of Contaminant concentrations in the porous environment
a: one injection unit
Find the time and space dependent change of Ci,j

37. Sample codes for PDE problem.
function ydot=rhs2d2(t, y)
global dx D ii jj Km Vmax
for j=1:jj
for i=1:ii
vv=vv+1;
yy(i,j) = y(vv);
end
for j=2:jj-1
for i=2:ii-1
R=-Vmax*yy(i,j)/(Km+yy(i,j));
Dffu1=(yy(i+1,j)-2*yy(i,j)+yy(i-1,j))/(dx*dx);
Dffu2=(1/(dx*(i-1)))*(yy(i+1,j)-yy(i-1,j))/2/dx;
Dffu3=(yy(i,j+1)-2*yy(i,j)+yy(i,j-1))/(dx*dx);
dot(i,j) = D*(Dffu1+Dffu2+Dffu3)+R;
bb = ii*(j-1)+i;
ydot(bb) = dot(i,j);
ydot=ydot';
Sample codes for PDE problem.
You can change PDE problem to many mini ODE problem, then you solve them at the same time.
4/10/2017

38. Solving Reaction Engineering Problems with MATLAB By Lian He Solve PDE dynamic equations and nonlinear equilibrium equations

39. Example 1 Porous Catalyst
The spherical catalysts are exposed to reactants at the initial time point. Given the information about the diffusion coefficient De, reaction rate, etc., we want to know the reactant concentration profile. (We assume that the reactant concentration in the catalyst is only a function of time and distance to the center for simplicity. ) R

40. Solution PDE Dimensionless form Initial condition u(0, x)=0
Dimensionless form
Initial condition
u(0, x)=0
Boundary conditions
u(t , 1)=Ssurf
Thiele Module

41. sol = pdepe(m, @pdefun, @icfun, @bcfun, xmesh, tspan, options)
MATLAB tool: pdepe
Syntax
sol @bcfun, xmesh, tspan, options)
function u0=icfun(x)
u0=0;
end
function [c, f, s] = pdefun(x, t, u, DuDx)
c=1;
f=φ^(-2)*DuDx;
s=-u/(1+u/beta);
end
function [pl, ql, pr, qr]=pdebound(xl, ul, xr, ur, t)
pr=ur-1;
qr=0;
pl=0;
ql=1;
end

42. Results

43. function [pl, ql, pr, qr]=bcfun(xl, ul, xr, ur, t) pr=ur-1; qr=0;
Three Functions
Main script
function [pl, ql, pr, qr]=bcfun(xl, ul, xr, ur, t)
pr=ur-1;
qr=0;
pl=0;
ql=1;
end
clc;
clear all;
x=0:0.02:1; %radius range
t=0:0.04:8; %timespan
%Show the concentration profile as a function of time
for k=1:length(t)
plot(x,u(k,:),'linewidth',2);
xlabel('distance from the center');
ylabel('substrate concentration');
pause(.2)
end
figure(2)
%3D plot
mesh(x,t,u);
xlabel('x');
ylabel('t');
zlabel('u');
axis([ ]);
function u0=icfun(x)
u0=0;
end
function[c,f,s]=pdefun(x,t,u,DuDx)
De=3*10^-4;
Ss=1*10^-4;
Km=0.1*Ss;
R=0.001;
umax=0.5;
Mt=R*sqrt(umax/Km/De);
beta=Km/Ss;
c=1;
f=Mt^(-2)*DuDx;
s=-u/(1+u/beta);
end

44. Example 2

45. Solution # Reaction Molar Extent 1 C+2H2O=CO2+2H2 X1 2 C+CO2=2CO X2 3
C+2H2=CH4 X3
Species
nj0 nj
H2O 1
1-2X1
CO2
X1-X2 CO
2X2
CH4 X3 H2
2(X1-X3)
sum
1+X1+X2-X3

46. You can use MATLAB tool-fsolve to solve the problem
Nonlinear equations
You can use MATLAB tool-fsolve to solve the problem

47. Main script clear all; clc; %initial guess of molar extent
%set the intial values
T=600;
%creat different vectors for recording temperature, molar extent and molar
%fraction
t=zeros(101,1); X=zeros(101,3); yH2O=zeros(101,1); yCO2=zeros(101,1);
yCO=zeros(101,1); yCH4=zeros(101,1); yH2=zeros(101,1);
for i=1:101
%find the solution
[x,fval] =
%record the results
t(i)=T; X(i,:)=x;
%caculate the molar fraction of each species
yH2O(i)=(1-2*x(1))/(1+x(1)+x(2)-x(3)); yCO2(i)=(x(1)-x(2))/(1+x(1)+x(2)-x(3));
yCO(i)=2*x(2)/(1+x(1)+x(2)-x(3)); yCH4(i)=x(3)/(1+x(1)+x(2)-x(3));
yH2(i)=2*(x(1)-x(3))/(1+x(1)+x(2)-x(3));
%change values for the next loops
T=T+10;
end
%plotting
subplot(2,2,[1 3]);
plot(t,X(:,1),'o',t,X(:,2),'*',t,X(:,3),'.');
legend('C+H_2O=>CO_2+2H_2','C+CO_2=>2CO','C+2H_2=>CH_4');
xlabel('Temperature (K)');
ylabel('molar extent'); axis([ ]);
subplot(2,2,[2 4]);
plot(t,yH2O,'o',t,yCO2,'-o',t,yCO,'.',t,yCH4,'*',t,yH2,'^');
legend('H_2O','CO_2','CO','CH_4','H_2');
ylabel('molar fraction');
axis([ ]);
Main script

48. Fsolve function to solve nonlinear equation
function F = nlfunc(x,T)
%coefficient values
K1=9.73*10^(-12)*exp(-10835/T+36.36);
K2=9.92*10^(-22)*exp(-20740/T+69.60);
K3=8.00*10^8*exp(8973/T-30.11);
%three column vectors
F=zeros(1,3);
%functions
F(1)=4*(x(1)-x(3))^2*(x(1)-x(2))-K1*(1-2*x(1))^2*(1+x(1)+x(2)-x(3));
F(2)=4*x(2)^2-K2*(x(1)-x(2))*(1+x(1)+x(2)-x(3));
F(3)=x(3)*(1+x(1)+x(2)-x(3))-4*K3*(x(1)-x(3))^2;
end

49. Results