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

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Mohamed Elasmar 2023-11-9

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评论： Matt J 2023-11-13

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

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.

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Matt J 2023-11-9

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Yes, there is polyfit.

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Mohamed Elasmar 2023-11-13

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:

Torsten 2023-11-13

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 2023-11-13

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,