How to optimise a specific equation in a model in order to fit the final curve of the model?

Hello,

I am developing a model and trying to validate this model with some experimental data. Two degree polynomial equations are used in this model as follows:

where

where

The validation figure based on the aforementioned equations is as follows:

I would like to optimise the constants of ,or even use higher degree polynomial equations in order to get the best fitting to the experimental data. Is there a code or a function in MATLAB that can simultaneously optimise several constants or pameters?

Thank you in advance.

Answer 1

[Matt J]

Yes, there is polyfit.

[Mohamed Elasmar]

Sorry for ambiguity, but the blue solid line is the final curve of a model, which depends on several parameters and equations. Hence, it is a resultant of integrating many equations together. I do not want to fit the experimental data through one solid polynomial curve because this will produce one general fitting equation and will lead to neglect the whole model. The purpose is to keep all other equations in the model and only optimise one or both of the aforementioned equations so that the final curve of the model fits most of experimental data.

Hopefully, you understood the question correctly and thank you.

[Matt J]

Hopefully, you understood the question correctly...

Nope. The t1 and t2 data sets do not have any unknown parameters in common. Therefore, they can each be fit separately, with polyfit:

c1_b1_a1=polyfit(T,t1,2);
c2_b2_a2=polyfit(T,t2,2);

If you are solving a different problem, which does not have this separable structure, you should post and describe that problem.

[Mohamed Elasmar]

It is an electrochemical model and is composed of different votlage components. For instance, the blue line represents the total voltage as . Each voltage component is calculated through specific equations. The voltage component of is calculated as follows:

, where

.

When I substitute by and and calculate , I get the blue line shown in the figure. Hence, the purpose is to fit to the yellow experimental data by keeping the equations of and without any changes, whereas manipulating the constants from to .

[Matt J]

You could use lsqcurvefit if you have the Optimization Toolbox.

[Mohamed Elasmar]

Thank you very much Matt J. I really appreciate your help. I have reviewed the cases of lsqcurvefit, but I did not know how to apply any of them to my model. Hence, I will put here a summary of my model and would be grateful if you guide me to the proper lsqcurvefit case.

The yellow points are imported from an Excel sheet as follows:

i = xlsread('Table_1.xlsx', '110 °C, 15 bar', 'F3:F452'); %% mA/cm^2

V_total = xlsread('Table_1.xlsx', '110 °C, 15 bar', 'G3:G452'); %% V

As the blue line comes from defining many parameters and solving several equations, it can be summarised as follows:

i = 0:1.388:800; %% (mA/cm^2)

T = 383; %% Kelvin

where , , A and B are calculated by equations whose parameters cannot be optimised. When I simulated the model at the constant values given in my previous comment, the blue curve produced is as shown in the figure:

So my question is how to optimise the constants from to in order to fit the blue curve to the yellow points.

[Torsten]

fun = @(p)V_x+V_z+A*log(i./(p(1)+p(2)*T+p(3)*T.^2))+B*log(i./(p(4)+p(5)*T+p(6)*T.^2))-V_total; %V_total form your Excel file
p0 = [1;1;1;1;1;1];   % Should be improved to reflect your measurements
p = lsqnonlin(fun,p0)

I just see that T is a constant (383) and does not vary with i. In this case, the fitting as suggested doesn't make sense. Alternatively, you can use two different constants in the denominator and fit them:

fun = @(p)V_x+V_z+A*log(i/p(1))+B*log(i/p(2))-V_total; %V_total form your Excel file
p0 = [1;1];   % Should be improved to reflect your measurements
p = lsqnonlin(fun,p0)

[Mohamed Elasmar]

lsqnonlin either returns complex solutions or the same initial input values of p0 !!

[Torsten]

i mustn't start with 0 because you take the log of i/p(1) resp. i/p(2).

[Matt J]

You must bound the parameters so that the log() operations are not given negative numbers.

p = lsqnonlin(fun,p0,[0,0])

[Mohamed Elasmar]

I have followed your steps, @Matt J and @Torsten, but lsqnonlin does not show a reasonable fitting. The code is as follows:

i_total = xlsread(...); %% x_axis of the experimental data

V_total = xlsread(...); %% y-axis of the experimental data

V_x = ...;

v_z = ...;

T = 333;

i = 0.0001:1.388:800; %% Starting from 0.0001 instead of 0 to avoid NaN output

%% Section of Optimization

fun = @(p)V_x + V_z + A * log(i ./ (p(1) + (p(2) * T) + (p(3) * T.^2))) + B * log(i ./ (p(4) + (p(5) * T) + (p(6) * T.^2))) - V_total;

p0 = [29.5;-0.2056;0.000355;13.5;-0.0908;0.000152]; %% Initial values that I want to optimise

p = lsqnonlin(fun,p0,[0,0,0,0,0,0]);

%% Section of defining V_total before and after the optimization

V_before = V_x + V_z + A * log(i ./ (p0(1) + (p0(2) * T) + (p0(3) * T.^2))) ...

+ B * log(i ./ (p0(4) + (p0(5) * T) + (p0(6) * T.^2)));

V_after = V_x + V_z + A * log(i ./ (p(1) + (p(2) * T) + (p(3) * T.^2))) ...

+ B * log(i ./ (p(4) + (p(5) * T) + (p(6) * T.^2)));

%% Plotting the results

plot(i_total,V_total,'o','Color','#EDB120');

hold on

plot(i,V_before,'Color','#0072BD');

hold on

plot(i,V_after,'--','Color','#0072BD');

hold off

lgd = legend('Experimental data','before fitting','after fitting');

This code gives the following figure:

[Matt J]

You are still using the parametrization p(1) + (p(2) * T) + (p(3) * T.^2. This doesn't make any sense, as Torsten mentioned to you, when T is a scalar independent of i. Perhaps T is meant to vary with i, and that is why the model is not accurately fitting the data.

[Torsten]

Further, it was intentionally that I chose p(1) and p(2) not equal to 0 in my code suggestion (you divide by them !).

[Matt J]

I just see that T is a constant (383) and does not vary with i. In this case, the fitting as suggested doesn't make sense. Alternatively, you can use two different constants in the denominator and fit them:

In fact, you don't need more than one parameter, p1. The model can be rewritten,