What is the primary implication of a system being completely controllable in the context of state-space representation?
Question 2
For a linear time-invariant (LTI) system described by $\dot{x} = Ax + Bu$, where $x \in \mathbb{R}^n$ and $u \in \mathbb{R}^m$, what is the structure of the controllability matrix $C$?
Question 3
Which of the following scenarios would most likely lead to an uncontrollable system?
Question 4
In the context of state feedback control, why is controllability a prerequisite for arbitrary pole placement?
Question 5
Consider a system with state matrix $A$ and input matrix $B$. If the rank of the controllability matrix $C = \begin{bmatrix} B & AB & A^2B & \dots & A^{n-1}B \end{bmatrix}$ is less than the number of states $n$, what can be concluded?
