Using lsqnonlin for deconvolution

The system I'm trying to solve is as follows:

Ip(t) is the output and ui(t) is the input. The purpose of the exercise is to determine the values of the ui with known values of Ip and k. I have experimental values of the input and the output, and want to create a script that penalizes the difference between consecutive input values and penalizes the difference between the estimated input and the actual input. I've attached the data, and what I've written so far is below. The problem I'm having is that the originally estimated value for the input never changes. It just stays zero.

filename='assignment_5_data.xlsx';
File=readtable(filename);
basal_insulin=File{:,2}/60;
plasma_insulin=File{:,3};
b=zeros(1,89);
A=lsqnonlin(@insulin,b);

function F=insulin(x)
filename='assignment_5_data.xlsx';
File=readtable(filename);
basal_insulin=File{:,2}/60;
plasma_insulin=File{:,3};
x=zeros(89,1);
k=0.0107;
qi2=(plasma_insulin-5.1873)/9.8834;
dqi2dt=[0];

for i=1:1:(length(qi2)-1)
    dqi2dt(end+1)=(qi2(i+1)-qi2(i))/10;
end
qi1=(dqi2dt+k*qi2)/k;

dqi1dt=[0];
for j=1:1:(length(qi1)-1)
    dqi1dt(end+1)=(qi1(j+1)-qi1(j))/10;
end
error=0;
for g=1:length(x)
    error=error+(basal_insulin(g)-x(g))^2;
end

smooth=0;
for h=1:(length(x)-1)
    smooth=smooth+abs(x(h+1)-x(h));
end
F=[error smooth];
plot(x)
end

Answer 1

[Torsten]

lsqnonlin tries to adjust x, and you reset it to 0:

function F=insulin(x)

...

x=zeros(89,1);

This destroys the complete fitting procedure.

Further, lsqnonlin expects you to return a vector of length 89 the elements of which are the errors over the time intervals. You return only 2 single values.