Observability

Remarks

Consider a linear system $x' = A x + B u$, $y = C x$. The output difference of two initial states does not depend on $u$. The system is observable if and only if, for any $T > 0$, the output $y$ on $[0, T]$ determines $x(0)$. Observability of the linear system is equivalent to $\rank \begin{pmatrix} C \\ CA \\ \vdots \\ CA^{n-1} \end{pmatrix} = n$. Example: $x_1' = x_2$, $x_2' = 0$, $y = x_1$ is observable, since $y(t) = x_1(0) + t \, x_2(0)$ gives $x_1(0) = y(0)$ and $x_2(0) = y'(0)$. Non-example: the same system with output $y = x_2$ is not observable, since $(0, 0)$ and $(1, 0)$ give the same output $y \equiv 0$.