LQR stands for linear quadratic regulator, where: linear refers to the linear dynamics of the system (which can both be invariant or variant in time); quadratic refers to the cost function which is an integral of a quadratic form, which the LQR minimizes; regulator refers to the goal of the control input to bring the system to zero.
Since your system is not linear you can't directly use LQR for that system. You would either have to resort to linearizing your model around an equilibrium point, or use another technique like some model predictive control (MPC). Linearization will in general only stabilize the system locally around the equilibrium point, while nonlinear MPC might be none convex and therefore potentially really hard to solve.
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.
Best Answer
If $a = 0$ and $b \neq 0$ you can take
$$ u = -\frac{k}{b}x \tag{1} $$
with $k > 0$, so your dynamics $\dot{x} = b \, x$ will be
$$ \dot{x} = -k \, x $$
Because you can choose $k$ freely (only needs to be positive), you can realize any stable linear 1st order dynamic with $(1)$. So yes, you can use LQR in that case.