Topic: Control Systems
When converting the following system into the standard state space form $\ddot{x}+3\dot{x}-9x=3~sin(wt)$ the resultant A matrix is
- A. $\begin{pmatrix}0&1\\ -9&-3\end{pmatrix}$
- B. $\begin{pmatrix}0&-1\\ 9&-3\end{pmatrix}$
- C. $\begin{pmatrix}0&1\\ 9&-3\end{pmatrix}$
- D. $\begin{pmatrix}0&-1\\ -9&3\end{pmatrix}$
Answer: C. $\begin{pmatrix}0&1\\ 9&-3\end{pmatrix}$
Explanation: Let the state variables be $x_1 = x$ and $x_2 = \dot{x}$.
Then, $\dot{x}_1 = x_2$.
From the differential equation: $\ddot{x} = 9x - 3\dot{x} + 3\sin(wt)$.
Substituting the state variables: $\dot{x}_2 = 9x_1 - 3x_2 + 3\sin(wt)$.
In matrix form, $\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 9 & -3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 0 \\ 3 \end{bmatrix} \sin(wt)$.
The system matrix $A$ is $\begin{bmatrix} 0 & 1 \\ 9 & -3 \end{bmatrix}$.