After recieving a great solution to an earlier question, I'm posting another one, because the book "Linear Systems Theory" by Joao Hespanha is very vague on the subject and I'm unable to find the right answer on google. Thanks in advance for the help.
The question:
Consider a continuous-time system given by:
$\dot{x}(t) = Ax(t) + Bu(t) + Gv(t), \qquad y(t)=Cx(t)+w(t), \qquad z(t)=Hx(t)$
Where $v$ and $w$ are uncorrelatted zero mean white noise signals and with A to be chosen and
$\ B = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 2 \\ \end{bmatrix} $ , $\ G =\begin{bmatrix} 0 \\ 2 \\ 0 \\ 1 \\ \end{bmatrix} , C = \begin{bmatrix} 2 & 3 & 0 & 0 \\ \end{bmatrix} , H = \begin{bmatrix} 2 & 1 & 0 & 0 \\ \end{bmatrix} $
For which of the following $A$ matrices do there exist weighting matrices $Q$ and $R$ such that the corresponding LQG controller results in an asymptotically stable closed loop system?
1) $A = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -3 \\ \end{bmatrix} $
2) $A = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -3 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix} $
3) $A = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{bmatrix} $
3) $A = \begin{bmatrix} -3 & 0 & 0 & 0 \\ 0 & -4 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & -2 \\ \end{bmatrix} $
4) None of the above
I think I solved it:
A necessary condition for LQR is that the pair $(A,B)$ must be stabilizable. To check this you can use the Hautus test for stabilizability, which state for the continuous time case:
$\ Rank\quad [A-\lambda I \quad B] = n \qquad \forall \quad \lambda \geq 0 $
in which $\lambda$ represents the eigenvalues and $n$ is the dimension of the $A$ matrix.
For example the $\lambda$'s of matrix 1) are $-3 ,-2, -1$ and $ 0$, the only $\lambda$ that meets the requirements is $0$. Substituting $0$ in the Hautus test yields a rank of $3$ which is unequal to $n$ (which is $4$).
Repeating this procedure for the remaining matrices shows that only matrix 2) meets the requirements of having rank $4$ (with corresponding eigenvalue $0$).