Duality

Given a system on State Space Representation form where $D=0$

We have:

$$\begin{array}{rl}\dot{z}& =A^{T}z+C^{T}u\\ \bar{y}& =B^{T}u\end{array}$$

Which has the following Observability Gramian:

$$W_o^z(t)={\int }_0^t{e}^{A\tau }B{B}^T{e}^{A^T\tau }d\tau$$

Which is the exact same as the Controllability Gramian for the original system:

Link to original

Theorem of duality

Which means we can check the observability of the original system by checking the controllability of the dual system.