How to change the reference ?

we have this system by its state representation :

the formulation of the quardatic command is described by the following equations:

exemple :

we would like to apply the quardatic command so that the output y(k) = x1(k) follows a certain reference.

For reference = 5 , this is the code :

clear, clc
Ac=[0 1;0 0]; Bc=[0;1]; Cc=[1 0]; Dc=0;
[A,B,C,D]=c2dm(Ac,Bc,Cc,Dc,0.1)   %Converting to discrete mode
N=600;
ref=5;       %The reference
Q=[1 2;2 1] ; R=0.1;
P=Q;
x_ref=[ref;0];
for k=N:-1:0
    F=R+B'*P*B;               %Equation 1
    K=inv(F)*B'*P*A;          %Equation 2
    M=P-P*B*inv(F)*B'*P;      %Equation 3
    P=A'*M*A+Q;               %Equation 4
end
u=0 ; x=[0;0];
for i=1:N
    x=A*x+B*u;
    u=-K*(x-x_ref);           %Equation 5
    y(i)=C*x;
end
plot(1:N,ref*ones(1,N),'r',1:N,y)

I get the following figure:

And now i would like to use the same code by modifying the reference signal, so as to obtain nearly these 2 curves:

y= red curve , reference= black curve

The goal is that the signal y (i) follows a non-fixed reference:

ref = 8 in [0.200]

ref = 5 in [200,400]

ref = 12 in [400,600]

This is the reference to follow :

if anyone can help me, i will be very grateful

[Image Analyst]

I'm not tackling that until you do two things: (1) add comments, and (2) use descriptive variable names. To me it just looks like an impenetrable alphabet soup of a program. And I'm not sure why you can't just make the changes since (evidently) you understand the original code far better than us. We're willing to help but you have to make it easy for us to help you.

[Alan Robertson]

I hope it's better now.

I apologize because I do not master it as it should be, and that is why I ask your help. thank you for your understanding

Answer 1

[Raj]

I don't think this is right:

ref = 8 in [0.200]

ref = 5 in [200,400]

ref = 12 in [400,600]

You cannot have 2 references at same time instant. So I am assuming you meant

ref = 8 in [0.200]

ref = 5 in [201,400]

ref = 12 in [401,600]

Now coming to the solution, the easiest way is to simply shift the reference inside 'for' loop like this:

clear, clc
Ac=[0 1;0 0]; Bc=[0;1]; Cc=[1 0]; Dc=0;
[A,B,C,D]=c2dm(Ac,Bc,Cc,Dc,0.1)   %Converting to discrete mode
N=600;
Q=[1 2;2 1] ; R=0.1;
P=Q;
for k=N:-1:0
    F=R+B'*P*B;               %Equation 1
    K=inv(F)*B'*P*A;          %Equation 2
    M=P-P*B*inv(F)*B'*P;      %Equation 3
    P=A'*M*A+Q;               %Equation 4
end
u=0 ; x=[0;0];
for i=1:N
        if i<=200
            ref=8; %The reference
        elseif i>200 && i<=400
            ref=5;
        else
            ref=12;
        end
    x_ref=[ref;0];
    x=A*x+B*u;
    u=-K*(x-x_ref);
    y(i)=C*x;
end
  plot([0 200 201 400 401 600],[8 8 5 5 13 13],'r',1:N,y)

[Alan Robertson]

Thanks a lot for your answer ! Indeed I made a mistake, I wanted to say: ref = 8 in [0.200] ref = 5 in [201,400] ref = 12 in [401,600] thanks again