Fminsearch curve fitting does not fit properly

Hi,

i am trying to fit a simple nonlinear pendulum model to measured data by adjusting the damping constant (parameter estimation).

my pendulum function:

function simY = PendelOde2(b,varargin)
u =1;
m  = 0.5035;     % [kg]      mass 
l  =  0.1315;   % [m]       length of pendulum 
g  = 9.81;      % [m/s²]    gravitational acceleration
expTime = 0:1e-2:20.02;
tic
ODE_Sol    = ode45(@(tt,x) myNonlinearPendulum(tt,x,u,m,g,l,b),[0 200.2],[deg2rad(13.5253),deg2rad(0)]);
simY       =   deval(ODE_Sol, expTime);
simY = simY(1,:);
toc
    function [dx,y] = myNonlinearPendulum(t,x,u,m,g,l,b,varargin)

    % Output equation.
    y = x(1); % Angular position.

    % State equations.
    dx = [x(2);                             ... % Angular position
      -(g/l)*sin(x(1))-b/(m*l^2)*x(2)   ... % Angular velocity
     ];
    end
end

how i am trying to fit (using fminsearch):

clear;
clc; 
load('Pendel1_11_eigen.mat')
x = Pendel1_11_eigen.X.Data;
y = Pendel1_11_eigen.Y.Data;

m  = 0.5035;     % [kg]      mass 
l  =  0.1315;   % [m]       length of pendulum 
g  = 9.81;      % [m/s²]    gravitational acceleration

b0  = 3.1e-4;    % [Nm s]    damping constant (to estimate)


[bmin, Smin]  = fminsearch(@(b) norm(PendelOde2(b) - y), b0)

y1 = rad2deg(PendelOde2(bmin))
plot(x,y)
hold on
plot(x,y1)
grid

This code is running, but the solution of the optimization does not fit to the data (see plot below). At this point i am a little confused why fminsearch thinks it has found a optimal solution and what i should change to make this work.

Any help is highly appreciated. Thanks in advance.

Answer 1

[Matt J]

Plotting your function in an interval around bmin shows that it is a minimum of the function you've provided. So, fminsearch did its job correctly.

The only thing to assume is either that this is a local minimum (you need a better initial guess), or else it is, indeed, the best fit possible given the model you're using.

[bmin, Smin, ef]  = fminsearch(@(b) norm(PendelOde2(b) - y), b0);

fun=@(b) norm(PendelOde2(b) - y);
fun=@(b)arrayfun(fun,b);
t=linspace(-1,1,21);
Ft=fun(bmin+t*b0);
plot(t,Ft)

[Jonas Gentner]

Thanks for the reply. The initial guess i am making here with b0 is already a really good fit that i found manually. It is way better than what fminsearch is returning as an optimum. It is hard to find a better initial guess and it shows that the model is capable of fitting to this data.

What i found out in my despair, is that fminsearch is fitting perfectly when i change the length of pendulum parameter from 0.1315 to 0.135. I guess i need to check the pendulum length of my test setup again.