How to make a multivariable goal optimization

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 with the first variable
      trans_quad = trans_func(tau(2)); %Transfer function 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 two 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°

Answer 1

[Stephan]

Hi,

fgoalattain calculates a result:

f(x) <= goal

Better use fsolve - this will try to calculate:

f(x)=0

Check this:

[x, fval] = phase_delta_90;
fprintf('tau(1) = %.12f\ntau(2) = %.12f\nPhase delta = %2.8f degree',x(1),x(2),fval)

function [x, fval] = phase_delta_90
    goal = 90;
    tau0 = [2E-5, 1E-3];        %Initial Value for tau(1) and tau(2)
    x = fsolve(@calculate_tau,tau0);
    fval = check_result(x);

function tau_result = calculate_tau(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 with the first variable
    trans_quad = trans_func(tau(2)); %Transfer function with the second variable
    tau_result = rad2deg(phase(trans_zero)-phase(trans_quad))-goal; %Phase difference minus goal 
end

function ap_phase = check_result(x)
    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(x(1)); %Transfer function with the first variable
    trans_quad = trans_func(x(2)); %Transfer function with the second variable
    ap_phase = rad2deg(phase(trans_zero)-phase(trans_quad)); %Phase difference after optimization
end
end

[Joe]

Thanks for your answer,but although your code does solve the problem I proposed, it is somehow limited, as far as I know fsolve is not an optimizer instead it solves the system of equations but doesnt do the same job as an optimizer, for instance what happens if I dont have one but many goals, can fsolve be used? Is there an optimizer that instead of solving f(x)<goal or f(x)=goal, it solves lower_goal<f(x)<upper_goal?

[Stephan]

You're right - I misused fsolve here a bit to achieve the goal. I think to solving the problem you have posed, this solver simply fits better than fgoalattain to achieve the goal.

If you want to work with bounds, another solver may be appropriate.

For the given question, the goal was a phase difference of 90 ° as well as possible. This is easier to solve with fsolve than with fgoalattain. That's why I chose it and hope it helped you solve your problem.

Is there an optimizer that instead of solving f(x)<goal or f(x)=goal, it solves lower_goal<f(x)<upper_goal?

Consider:

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 = [90.1 -89.9];                  %Optimization goal
weight = [1 1];                 %Weight
[x,fval] = fgoalattain(@objecfun,tau0,goal,weight,[],[],[],[],lb,ub)

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 with the first variable
    trans_quad = trans_func(tau(2)); %Transfer function with the second variable
    ap_phase(1) = rad2deg(phase(trans_zero)-phase(trans_quad)); %Phase difference
    ap_phase(2) = -(rad2deg(phase(trans_zero)-phase(trans_quad))); %Phase difference
end

What happens here? I used a second Goal with the negative value of your desired Goal with a Little toerance to both sides. This way i told fgoalattain to do exactly what you want:

89.9 <= Goal <= 90.1

If you see the results, it works properly:

x =

     1.0e-03 *

     0.115220205761474   0.993932451929828

fval =

    89.999999999233822 -89.999999999233822

Perhaps you like this solution more ;-)

[Joe]

That is exactly what I was looking for, thank you, just one other thing the lower goal you selected is 85 not -85 right? you only put -85 to invert the inequality sign?

[Stephan]

you only put -85 to invert the inequality sign?

Exactly

[Joe]

Thanks Stephan that was really helpful.