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## Q. 7.EX.6

Flexible Disk Drive in State-Variable Form
Find the state-variable form of the differential equations for Example 2.4, where the output is $θ_2.$

## Verified Solution

Define the state vector to be

$\pmb x = \left[\begin{matrix} θ_1 & \dot{θ}_1 & θ_2 & \dot{θ}_2 \end{matrix} \right] ^T$ .

Then solve Eqs. (2.17) and (2.18) for $\ddot{θ}_1$ and $\ddot{θ}_2$ so that the state-variable form is more apparent. The resulting matrices are

$\pmb F = \left[\begin{matrix}0 & 1 & 0 & 0 \\ -\frac{k}{I_1} & -\frac{b}{I_1} & \frac{k}{I_1} & \frac{b}{I_1} \\0 & 0 & 0 & 1\\ \frac{k}{I_2} & \frac{b}{I_2} & -\frac{k}{I_2} & -\frac{b}{I_2} \end{matrix} \right] , \ \ \ \pmb G = \left[\begin{matrix} 0 \\ \frac{1}{I_1} \\ 0 \\ 0 \end{matrix} \right] , \\ \pmb H = \left[\begin{matrix} 0 & 0 & 1 & 0 \end{matrix} \right] , \ \ \ J =0 .$