How to constrain genetic algorithm input u?

My real system (buck converter) can only take an input of 0 to 1 (duty ratio) and I need to constrain the system so the controller action does not go past this. How can I do this?

The code for the project is below, it is steve bruntons genetic algorithm coding. https://youtu.be/S5C_z1nVaSg?t=470

I assume the u variable is the duty ratio but I am not sure.

---------------------------------------------------- First live script with Ga cmd in it-------------------------------------

clear all, close all, clc

dt = 0.000001; % this is 10^-6 
PopSize = 30 % was 500
MaxGenerations = 10; %was 1000
s = tf('s');
G = (1439928003.68621)/(s*s+5333.33333333333*s+95995200.2457475) % got this by doing feedback cmd

options = optimoptions(@ga,'PopulationSize',PopSize,'MaxGenerations',MaxGenerations,'OutputFcn',@myfun);
[x,fval] = ga(@(K)pidtest(G,dt,K),3,-eye(3),zeros(3,1),[],[],[],[],[],options);

-----------------------------------------------in adjacent live script------------------------------------------------

function J = pidtest(G,dt,parms)
s = tf('s');
K = parms(1)+ parms(2)/s + parms(3)*s/(1+0.000001*s)% this is 10^-6 
Loop = series(K,G);
ClosedLoop = feedback(Loop,1);
t = 0:dt:0.05; % this indicates length of time to show
[y,t] = step(ClosedLoop,t);

CTRLtf = K/(1+K*G);
u = lsim(K,1-y,t); % not sure if this is what I should change

Q = [1]; % Q weighting (couldn't get it so it looked at inductor current also) effecfts response of system?
R = 0.005; % R weighting effects controller action
J = dt*sum(Q*(1-y(:)).^2 + R*u(:).^2) % LQR is my cost function 

step(5*ClosedLoop,t)
h = findobj(gcf,'type','line');
set(h,'linewidth',2);
drawnow

---------------------------------------------------------------------------------------------------------------------------------

Thanks in advance, if anymore information is desired please ask away.

The PID controller h

[Walter Roberson]

I am not sure which is the input that needs to be constrained to 0 to 1 ?

[JAMES KEEN]

hi @Walter Roberson The image below shows the PID controller being constrained to a upper and lower bound of 1 to 0. I hope to make it so my PID controller does not exceed these bounds in the above code. If i do not bound the PID controller output the simulation and simulink model show disparities as a real system cannot realisitically have a duty ratio which is negative or above 1 (100%).

in short I need the code to realise that the PID controller output cannot exceed 1 or go below 0.

If the above explanation is not of assitance please say as I can retry since I am not 100% sure myself.

[Walter Roberson]

In your code, which variable needs to be constrained to that range at what step ?

[JAMES KEEN]

@Walter Roberson I will get back to you tomorrow.

To be quite honest I am not sure but I want it so the controller does not put out a signal of over 300 when the output (duty ratio) shouldn't leave the bounds of 0 to 1.

I appreciate the provided interest and will attempt to find an answer to that question.

[Walter Roberson]

I do not have much experience with PID controllers so it takes me a while to figure how the pieces fit together.

The general structure of your program would suggest that you are trying to optimize A + B/s + (C*s)/(s/1000000 + 1) to replace... something... but I am having trouble figuring out what you are trying to replace ? Perhaps you are trying to find A + B/s + (C*s)/(s/1000000 + 1) to modulate (something) to give result about as good as (some other way of calculating something)... but I don't see how yet how it all fits into your diagram ???

[JAMES KEEN]

@Walter Roberson

The image below explains the system and how i want the controller K to be limited to an output of 0 to 1. The numbers that make up K can be any value they want but I don't want the controller K output to be allowed to exceed the bounds of 0 to 1. The reason I want to make sure the controller K does not exceed the constraint of 0 to 1 is because the duty ratio (a percentage) can only go in the range of 0 to 1 without problems occuring.

This second image shows what the code is producing (it produces varying values of P, I, D values to find most optimal controller which are gathered together under the name (K)). I have also shown why I need the PID controller (K) to be within the limits of 0 to 1, it is because the real system cannot handle a duty ratio (output of PID) that is below 0 or above 1.

If there are any other questions please feel free to ask as I am quite new to this sector also.

[Walter Roberson]

Sorry, I have no idea how the output desired is intended to connect to your code. You do have a K in your code but it is

K = parms(1)+ parms(2)/s + parms(3)*s/(1+0.000001*s)% this is 10^-6 

and it is not returned. It is a transfer function, but transfer functions are not any particular value: transfer functions describe a relationship between inputs and outputs. You can talk about the value of a transfer function evaluated at a particular frequency, or the maximum value over a range of frequencies or such things, but not about a transfer function having a definite value.

If the point is just that the transfer function must be between 0 and 1 over a particular range of frequencies, then if we assume that the parameters are non-negative, you get

K = ((P1 + 1000000*P3)*s^2 + (1000000*P1 + P2)*s + 1000000*P2)/(s^2 + 1000000*s)

and for non-negative P1, P2, P3, then the numerator is always positive, and restricting it to the range 0 to 1 can be done by requring that the numerator <= denominator. Solving for equality we get

 ((P1 + 1000000*P3)*s^2 + (1000000*P1 + P2)*s + 1000000*P2) == (s^2 + 1000000*s)
 ((P1 + 1000000*P3)*s^2 + (1000000*P1 + P2)*s + 1000000*P2) - (s^2 + 1000000*s)
 (P1 + 1000000*P3 - 1)*s^2 + (1000000*P1 + P2 - 1000000)*s + 1000000*P2

Really this is <= 0 rather than a == 0 .

Maximize:

 syms P1 P2 P3 s
 eqn = (P1 + 1000000*P3 - 1)*s^2 + (1000000*P1 + P2 - 1000000)*s + 1000000*P2
 scrit = solve(diff(eqn,s), s)
 scurl = subs(diff(eqn,s,s),s,scrit)

which gives

 2*P1 + 2000000*P3 - 2

and that needs to be negative for scrit to be discussing a maxima, and the boundary for that is P3 = (1-P1)/1000000. This pretty much tells you that the s^2 coefficient here needs to be negative for the K to be guaranteed bounded to the proper range for all s.

However... if you are only operating over a particular range of s values then you might be able to achieve the desired limits even if the limit would not hold for a wider range of s values.