MATLAB - How to define a multivariable goal optimization - matlab

Question

Im trying to understand multivariable goal optimization, I will need to optimize complex functions, but to start, I need to optimize the following function:
function ap_phase = objecfun(tau)
f = 1000; %Frequency
w = 2*pi*f; %Angular Frequency
trans_func = #(taux) (1-1i*w*taux)./(1+1i*w*taux); %Transfer function
trans_zero = trans_func(tau(1)); %Transfer function evaluated with the first variable
trans_quad = trans_func(tau(2)); %Transfer function evaluated with the second variable
ap_phase = rad2deg(phase(trans_zero)-phase(trans_quad)); %Phase difference
end
The function objecfun takes one vector of length 2 as an input, computes 2 transfer functions, then substracts the phase of the transfer functions.
My goal is that the phase should be around 90°
The script im using to make the optimization is the following
tau0 = [2E-5, 1E-3]; %Initial Value for tau(1) and tau(2)
lb = [1E-7, 1E-7]; %Lower bound for tau(1) and tau(2)
ub = [1E-2, 1E-2]; %Upper bound for tau(1) and tau(2)
goal = 90; %Optimization goal
weight = 1; %Weight
[x,fval] = fgoalattain(#objecfun,tau0,goal,weight,[],[],[],[],lb,ub)
The optimizer converges but im getting a wrong answer, im getting
x =
0.0100 0.0000
fval =
-178.1044
That's wrong, fval should be near 90°
What am I doing wrong?

Answer 1

I think you need to replace your objective function and goal value so that it fit the problem formulation. You can use as an objective function say the L2 norm of the difference between your function output and your desired angle, and set the goal as some tolerance.
I checked also with "fmincon":
new_goal = 1e-4;
objectfun = #(x) norm(objecfun(x) - goal);
options = optimoptions('fgoalattain');
options.PlotFcns = 'optimplotfval';
[tau_star,fval] = fgoalattain(objectfun,tau0,new_goal,weight,[],[],[],[],lb,ub,[],options);
options = optimoptions('fmincon');
options.PlotFcns = 'optimplotfval';
[tau_star2,fval,exitflag,output] = fmincon(objectfun,tau0,[],[],[],[],lb,ub,[], options);
fgoalattain_solution_phase_diff = objecfun(tau_star)
fmincon_solution_phase_diff = objecfun(tau_star2)
And got:
fgoalattain_solution_phase_diff =
90.0000
fmincon_solution_phase_diff =
90.0006
Note: you can also omit the rad2deg in your function and use as the desired angle its value in [rad].