Hw 5 ex 2

Hint on how to solve ex 2: assume that the system is observable, and try an argument by contradiction. If the controller makes the system unstable, then the corresponding matrix ${\tilde {A}}=A-BK$ must have an eigenvalue with positive real part, to which corresponds a certain eigenvector $v$.

One can rewrite the Algebraic Riccati equation using ${\tilde {A}}$, where you should note the changes of signs:

$P{\tilde {A}}+{\tilde {A}}^{T}P+PBQ_{u}^{-1}B^{T}P+Qx=0$

Pre-post multiplying by the unstable eigenvector (as if you were evaluating a quadratic form), you will see that the only case in which the corresponding form can be zero is only if P=0 and $v^{*}Q_{v}v$ is zero. Which contradicts the initial assumption.

--Elisa