A state-space system is controllable when suitable inputs can move the state between arbitrary states in finite time.

For the continuous-time linear system

$$\dot x=Ax+Bu,$$

with $n$ state variables, form the controllability matrix

$$\mathcal C=\begin{bmatrix}B&AB&A^2B&\cdots&A^{n-1}B\end{bmatrix}.$$

The pair $(A,B)$ is controllable exactly when

$$\operatorname{rank}(\mathcal C)=n.$$

Each block $A^kB$ describes directions that input influence can reach after being propagated through the internal dynamics. Full rank means these directions span the entire state space.

If the system is not controllable, some internal modes cannot be moved by any choice of the available input. Those modes may still evolve, but the actuator has no authority over them.

Controllability is therefore a structural question about plant and actuator placement, not about whether a particular controller has been tuned well.