Consider the plant shown in the figure,plant. Find a state-space model for the plant.
I know, I have to use the transfer function to get the state-state model. here I have a confusion about the transfer function.is it $\frac{1}{s(s^{2}+1.4as +a^{2})}$ or $\frac{s}{s^{2}+1.4as +a^{2}}$?
Your plant has multiple inputs, denoted with $U(s)$ and $D(s)$, and one output, denoted with $Y(s)$. Writing the blockdiagram out into one equation for $Y(s)$ yields
$$ Y(s) = \frac{1}{s(s^{2}+1.4as+a^{2})} U(s) + \frac{1}{s} D(s). $$
So for the state space model you could for example use $\begin{bmatrix}u & d\end{bmatrix}^\top$ as the input vector, with $U(s)$ and $D(s)$ equal to the Laplace transform of $u$ and $d$ respectively. Though, it might also be possible that $d$ would represent a disturbance, in which case both $u$ and $d$ would have their own input matrix.