How to graph the closed-loop response of MIMO Control System?

Hi everyone,

I have MIMO transfer function matrix G and controller K given

G = [10.014/(1+15.021*s) -0.5/(1+10.014*s); 0.5/(1+5.007*s) 5.007/(1+20.028*s)]
K= k*eye(2)

How would I plot the step response and bode plot of the closed loop system with a unity feedback gain as attached in the picture? I used the code below but I have doubts if this is as straight forward as this.

CL = feedback(G*K,eye(2))
step(CL)
bode(CL)

Would be very glad to hear your response on this.

Answer 1

[Birdman]

Actually what you did to obtain closed loop forms is wrong. I suggest you to obtain them in a for loop separately as follows:

s=tf('s');
k=1;
G = [10.014/(1+15.021*s) -0.5/(1+10.014*s); 0.5/(1+5.007*s) 5.007/(1+20.028*s)];
for i=1:size(G,1)
    for j=1:size(G,2)
        CL(i,j)=feedback(G(i,j)*k,1);
    end
end
figure(1);
step(CL);
figure(2);
bode(CL);

This gives the correct closed loop forms and plots the correct step responses and Bode plots.

[Mr. NailGuy]

Hi @Birdman thanks for this. How can verify if this is correct in theory? How would I know if I got the correct closed loop transfer function matrix of the form CL =[H11 H12; H21 H22]? In your algorithm, when I increase the k to a value of 10, u2 to y1 becomes unstable already which I don't expect it to be. How can I relate this to the plant characteristic loci? Thanks

[Birdman]

Well, if increasing k value to 10 causes unstability, it can be understood from root locus. See the attached figure. I obtained it by typing

 rlocus(G(1,2))

You see that the critical gain value for that plant is 2. If you give a k value more than 2, then you will see that the closed loop pole goes to the right plane, which means the system will be unstable and as you increase k to 10, you will assign the closed loop pole to a greater value than 0.

[Mr. NailGuy]

Hi Birdman, thanks for the explanation. However, I am considering the closed loop system with the controller K which will affect the overall closed-loop system right? Will increasing k value affect element H12 directly like this? I am just wondering on it. Thanks

[Birdman]

Well, it depends on your code. In my code, since all the closed loop tfs are formed depending on k, it will affect all of them directly.

[Mr. NailGuy]

Thanks Birdman. I am considering G*K with a negative feedback closing the loop and just checking the step response of the closed loop from each input to each output. So I wonder if the code you provided satisfies this, tho I greatly appreciate it :)

[Birdman]

If you explicitly provide gains for the systems, then you need to define k as a matrix as well, as follows:

 k=[1 -1;1 1];

and also a change in for loop,

 CL(i,j)=feedback(G(i,j)*k(i,j),1);

with this, you will provide negative feedback for your unstable plant. Hope this answers your question.

[Mr. NailGuy]

Hi Birdman, my gain K is equals to k*eye(2) where k is a scalar (1,5,10,etc.) So the gain K is always a diagonal matrix. Thanks