lqr controller with saturation for closed loop system

Question

Junhwi 2023-12-6

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评论： Sam Chak 2024-1-9

Hi, I want to add saturation to regulate the output of the lqr controller. I wonder how to add saturation and wanna check if the input to the plant is really regulated in between +-1. I also tried simulink but it shows not desired results.

I'll really appreciate if you can help. thanks!

the code for lqr is below:

% LQR

%%%%%%%%%%

s = tf('s');

Gs = 4000/(s*(s+10)*(s+20));

[A,B,C,D]=tf2ss(Gs.Numerator{1}, Gs.Denominator{1});

st = 0.06;

Gd=c2d(Gs,st);

[Ad,Bd,Cdl,Ddl]=tf2ss(Gd.Numerator{1}, Gd.Denominator{1});

Q = blkdiag(0.1,0.1,100);

R =0.001;

[K,P]=dlqr(Ad,Bd,Q*st,R/st);

sys12 = ss(Ad-Bd*K,Bd,Cdl,Ddl,st);

N12=1/dcgain(sys12);

sysfin = ss(Ad-Bd*K,Bd*N12,Cdl,Ddl,st);

figure;

step(sysfin)

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Junhwi 2023-12-12

I think I plotted the output :)

Paul 2023-12-13

Hi Junhwi,

Can this statement be clarified: "I wonder how to add saturation and wanna check if the input to the plant is really regulated in between +-1."

I think I understand the second part, which is that for your code above you want to check if the input the plant is between -1 and +1 for a unit step input. Is that correct? And if the plant input is not between those bounds, you want to add a saturation function on the output of the controller to ensure the input to the plant stays within those bounds?

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Answer 1

Ayush Modi 2024-1-9

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Hi,

As per my understanding, you are trying to saturate the output of "lqr controller" and ensure that the input to the "plant" is regulated between -1 and +1. You can manually implement saturation in your code.

Here is an example to demonstrate how you can accomplish this:

u = max(min(u, 1), -1);

Alternatively, you can achieve this by using "Saturation" block in simulink.

Please refer to the following MathWorks documentation for more information:

"Saturation" - https://www.mathworks.com/help/simulink/slref/saturation.html

I hope this resolves the issue you were facing.

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Sam Chak 2024-1-9

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Hi @Ayush Modi

To enhance the value of your response, could you illustrate how to apply the saturation function to the 3rd-order system mentioned by the OP?

s   = tf('s');
Gp  = 4000/(s*(s + 10)*(s + 20))
Gp =
 
          4000
  --------------------
  s^3 + 30 s^2 + 200 s
 
Continuous-time transfer function.
[num, den] = tfdata(Gp, 'v');
A   = [zeros(2, 1), eye(2); 0 -den(3) -den(2)];
B   = [zeros(2, 1); num(4)];
C   = [1 0 0];
D   =  0;
K   = lqr(A, B, eye(size(A)), 1)    % LQR control gains
K = 1×3
    1.0000    1.6830    0.9929