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I am trying to design a LQR controller with Integral action (Linear-Quadratic-Integral control) for a below plant. The plant has 4 states, 2 inputs and 4 output. Is it possible to track 4 reference using LQI controller?

$$ A = \left(\begin{matrix}-1.340&0.672&-12.9669&9.775\\-2.070&-3.275&1.707&0\\4.405&0.2345&-4.3911&0\\0&1&0.0713&0\end{matrix}\right) $$

$$ B = \left(\begin{matrix}0&-3.0234\\18.624&24.110\\14.073&-7.060\\0&0\end{matrix}\right) $$

$$ C= \left(\begin{matrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}\right) $$ Click here for Controller structure (source- https://bit.ly/2LUpU7X)

The system is Controllable and is open loop stable.

Below is the script which I tried to solve in MATLAB:

A_aug = [A, zeros(4,4); -C, zeros(4,4)];
B_aug = [B; zeros(4,2)];

Q = eye(8);
R = eye(2);

F_aug = -lqr(A_aug,B_aug,Q,R);

k = F_aug(:,1:n);
k_i = F_aug(:,end);

But I was getting this error.

Cannot compute the stabilizing Riccati solution S for the LQR design. This could be because:
* R is singular,
* [Q N;N' R] needs to be positive definite,
* The E matrix in the state equation is singular.

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For linear quadratic integral (LQI) control to work, the augmented system has to be stabilizible. This limitation can be also be found in the Matlab documentation of the LQI function.

A linear time invariant (LTI) system is stabilizible, if all its uncontrollable modes are stable. You can first check with the ctrb function if there are uncontrollable modes:

A = [-1.34, 0.672, -12.9669, 9.775; 
     -2.07, -3.275, 1.707, 0;
     4.405, 0.2345, -4.3911, 0; 
     0, 1, 0.0713, 0];
B = [0, -3.0234; 
     18.624, 24.11; 
     14.073, -7.06; 
     0, 0];
C = eye(4);

Aa = [A, zeros(4, 4); -C, zeros(4, 4)];
Ba = [B; zeros(4, 2)];
Ca = [C, zeros(4, 4)];

rank(ctrb(Aa, Ba))

This script gives the output

ans =

     6

so your augmented system has two uncontrollable modes. You can also use the ctrbf function to get a Kalman decomposition, which seperates controllable and uncontrollable portions: it finds a similarity transform $T$ such that

$$ \bar{A}_a = T A_a T^T = \begin{bmatrix} A_{a,uc} & 0 \\ A_{a,21} & A_{a,c} \end{bmatrix} $$

where $A_{a,c}$ is the controllable and $A_{a,uc}$ the uncontrollable portion of your augmented system matrix $A_a$. In code:

[Aa_bar, Ba_bar, Ca_bar, T, k] = ctrbf(Aa, Ba, Ca);
n_uc = size(Aa, 1) - sum(k); % Number of uncontrollable modes is 8 - 6 = 2
Aa_uc = Aa_bar(1:n_uc, 1:n_uc)

which outputs

Aa_uc =

   1.0e-16 *

   0.330254728851448   0.215513706491097
   0.433511198605747   0.089609131250558

So $A_{a, uc}$ is (practically, up to numerics) a zero matrix, so the uncontrollable modes are not stable because $A_{a, uc}$ is not a Hurwitz matrix.

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    $\begingroup$ Thanks a lot for your answer. Instead of tracking 4 reference, now I track only 2. It is working all fine. My augumented plant now is stabilizible. $\endgroup$ – Sarish Patil Jul 17 '19 at 7:48

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