lqr controller with saturation for closed loop system

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)

[Sam Chak]

Hi @Junhwi,

This is an interesting problem when using the functions from the Control System Toolbox. I need some time to think. By the way, could you plot to show the output of the LQR?

[Junhwi]

I think I plotted the output :)

[Paul]

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?

Answer 1

[Ayush Modi]

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.

[Sam Chak]

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