Final answer:
To convert the transfer function H(s) = (2s+1)/(s^2+7s+9) to state-space, use controllable canonical form to derive the state matrix (A), input matrix (B), output matrix (C), and feedthrough matrix (D), leading to a state-space representation.
Step-by-step explanation:
The student has asked to convert the transfer function H(s) = (2s+1) / (s^2+7s+9) into a state-space representation. To accomplish this, we start by assigning state variables. A common approach is to use the controllable canonical form. Let's define x1 as the output of the first integrator and x2 as the output of the second integrator.
Step-by-Step Process:
- Write down the differential equations based on the transfer function.
- Express the differential equations in matrix form.
- Identify the state-space matrices: state matrix (A), input matrix (B), output matrix (C), and feedthrough matrix (D).
For the given transfer function, we obtain the following state-space representation:
A = \[ \begin{bmatrix} 0 & 1 \\ -9 & -7 \end{bmatrix} \]
B = \[ \begin{bmatrix} 0 \\ 2 \end{bmatrix} \]
C = \[ \begin{bmatrix} 1 & 0 \end{bmatrix} \]
D = \[ \begin{bmatrix} 1 \end{bmatrix} \]