Parameter estimation in kinetic study of reforming reaction involving ODEs

Hi!

I am trying to estimate 13 unknown parameters involved in 4 ordinary differential equations (ODEs) by fitting it to experimental data. There are six variables (4 dependent variables and 2 independent variables). Eight set of data points are available. Equations are as follows.

1. dYB/dt = -(r1) - (r2)
2. dYH2/dt = (8*r1) + (12*r2) + (r3) - (r4)
3. dYCO/dt = (4*r1) - (r3) + (r4)
4. dYCO2/dt = (4*r2) + (r3) - (r4)

Where r1 = k10*e^(-E1/(RT))*(PB^a), r2 = k20*e^(-E2/(RT))*(PB^b), r3 = k30*e^(-E3/(RT))*(PCO^c), r4 = k40*e^(-E4/(RT))*(PCO2^d)*(PH2^e).

I tried to estimate parameters using lsqcurvefit and ode15s, but there are errors which I am not able to rectify.

Thank you in advance.

[Star Strider]

‘... there are errors which I am not able to rectify.’

And those are ...?

[Are Mjaavatten]

I am afraid this problem is quite a bit more challenging than you imagine. You should read the Matlab tutorials on differential equation solving and curve fitting.

You should always test and debug the parts before combing them in a large system:

First, you should make Difeq a separate file and have P as an extra input. Next you must debug it: You refer to V(5,:), but V should be a 1 by 4 vector. V(5) seems to be time, so maybe you should use t instead? Make sure that the results of Difeq(0,V0,P0) look reasonable.

Next try to solve the ODE for P = P0:

P = P0;
rhs = @(t,V) Difeq(0, V, P);
[t,s] = ode45(rhs,[0;[Time]],V0);

It may be wise to first try with a much smaller time interval, though.

Once this works, you may in principle be ready to try to fit the output. However, trying to fit 13 parameters with values that vary by a factor of 10^11 is not likely to succeed. It is a good idea to normalise your parameters, for instance by dividing them with your initial guess P0. Even so, fitting all 13 at once is going to be quite difficult. I suggest you try to find good guesses for most parameters and fit only one or two at first. Then increase the number of unknowns by one parameter at a time. Still, 13 seems a bit much.

Good luck! (You will need it.)

[Ghansyam Vadodaria]

Star Strider, I received following errors while executing commands in the attached file.

Index in position 1 exceeds array bounds (must not exceed 4).

Error in kinetic_model_3/DifEq (line 57)

r1 = ((P(1)*exp((-P(5)/R)*((1./V(5,:))-(1/Tm)))).*((V(1,:).^P(9))));

Error in odearguments (line 90)

f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.

Error in ode15s (line 150)

odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Error in kinetic_model_3 (line 40)

[T,Sv] = ode15s(@DifEq, t, V0);

Error in lsqcurvefit (line 213)

initVals.F = feval(funfcn_x_xdata{3},xCurrent,XDATA,varargin{:});

Error in KM_3 (line 31)

[P,RESNORM,RESIDUAL,EXITFLAG] = lsqcurvefit(@kinetic_model_3,P0,Time,Ydata)

[Torsten]

V has 4 elements, namely [YH, YH2,YCO, YCO2].

What are you trying to do when you refer to V(5,:) ?

[Ghansyam Vadodaria]

@ Torsten, In my system I have four dependent variables (YB, YH2, YCO, YCO2) and two independent variables (Time and Temperature) V(5,:) is temperature.