How to use MATLAB's lqi.m function?

Hello,

I'm trying to figure out how to use MATLAB's lqi.m function, but the documentation of the function is somewhat vague and unfortunately does not run through an exmaple. I have the following system that I would like to implement:

, , ,

As for the weighting matrices Q, and R, I will end up choosing and modifying them based on results. They're important, but they aren't the issue I'm experiencing. I would like to try and use the lqi function on the above system to track the output to be 1. How can I do this? I know I need to use the lqi.m function but I don't know how to use it. How do I specify the input r? Or how do I design for this at all? Can someone give me an in-depth example of this?

Thank you for your patience and time

edit: I forgot to mention the system is in discrete time with a step interval of Ts = 1. Apologies.

Answer 1

[Sam Chak]

Hi @Craig Chambers

I'm unfamiliar with you system. So, I tested with the LQR first.

It seems that a properly scaled step input can make the control system track it.

A   = [1.1 2.0;
       0.0 0.95];
B   = [0;
       0.0787];
C   = [-1 1];
D   = 0;
K   = lqr(A, B, eye(2), 1)  % control gain matrix

K = 1×2

28.8132 52.2105

sys = ss(A-B*K, B, C, D);
N   = 1/dcgain(sys)         % scale at the reference input

N = -4.3461

sys = ss(A-B*K, B*N, C, D)  % closed-loop system with scaled input

sys =
 
  A = 
           x1      x2
   x1     1.1       2
   x2  -2.268  -3.159
 
  B = 
           u1
   x1       0
   x2  -0.342
 
  C = 
       x1  x2
   y1  -1   1
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

step(sys), grid on

[Craig Chambers]

Hello Sam, thanks! This is an interesting approach using the DC gain of the system. Not the solution I had in mind but this is also something else to consider. Truth be told I was only looking at LQI since other posts pointed to it.

I forgot to mention the system is discrete with time step interval of 1 but I don’t think that changes the approach as much. I’ll make an edit. Thanks.