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How do we obtain a state-space realization and a block diagram of a given transfer function?

Consider the transfer function

$\frac{C(s)}{R(s)}=\frac{5s}{3s^{2}+3s+1}$

Steps for solution are

$\frac{C(s)}{R(s)}=\frac{5s}{3s^{2}+3s+1}\frac{Q(s)}{Q(s)}\\$

$C(s)=5(s^{-1})Q(s)\\$ $\Rightarrow R(s)=(3+3s^{-1}+s^{-2})Q(s)$ $\Rightarrow R(s)=3Q(s)+3(s^{-1})Q(s)+s^{-2}Q(s)$ $\Rightarrow 3Q(s)=R(s)-3(s^{-1})Q(s)-s^{-2}Q(s)$

$\Rightarrow Q(s) = \frac13R(s)-s^{-1}Q(s)-\frac13s^{-2}Q(s)$

$\Rightarrow Q(s) = \frac13R(s)-Q(s)\left[s^{-1}+\frac13s^{-2}\right]$
the graph which is plotted in the book is of last equation of above solution.

I do not know how to post the graph here on Stack Exchange, but what I want to understand is:

How is the graph of this equation plotted?

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    yes the edits done are correct what is latex I am hearing it $f$irst time the edits are correct2012-12-04

1 Answers 1

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Only some parts of your question make sense. In particular, given the transfer function

$ H(s) = \frac{5s}{3s^2+3s+1} $

you invert the Laplace transforms to obtain the ode

$3y''(t) + 3y'(t) + y = 5u'(t).$

Define $X_1 = y$, $X_2 = y'$, $U_1 = u$, $U_2 = u'$. The state equations may therefore be written as $ \begin{aligned} X_1' &= X_2\\ X_2' &= -X_2 - \frac{1}{3}X_1 + \frac{5}{3}U_2\\ \end{aligned} $ and thus $ \begin{bmatrix} X_1\\ X_2\\ \end{bmatrix}' = \begin{bmatrix} 0 & 1\\ -\frac{1}{3} & -1 \end{bmatrix} \begin{bmatrix} X_1\\ X_2\\ \end{bmatrix} + \begin{bmatrix} 0 & 0\\ 0 & \frac{5}{3} \end{bmatrix} \begin{bmatrix} U_1\\ U_2\\ \end{bmatrix}, $ corresponding to the partial realization $ A = \begin{bmatrix} 0 & 1\\ -\frac{1}{3} & -1 \end{bmatrix} ,\ B = \begin{bmatrix} 0 & 0\\ 0 & \frac{5}{3} \end{bmatrix}. $ This is the only interpretation I can think of for "state space model". The only thing I can imagine is meant by "state variable diagram" (other than a block diagram which there is no "plotting" involved in making) is the $X_1$ vs. $X_2$ contour plot which you get by plotting the vector field of the RHS minus the controller parametrically. I have no idea what the calculation you show is trying to do and without any reference to where you got it from have no idea where to start with that.