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:

$P\tilde{A}+{\tilde{A}}^{T}P+PB{Q}_u^{-1}{B}^{T}P+Qx=0$

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