# how to obtain state space diagram and state space model for transfer function

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?

• Consider using LaTeX for formatting. – Epictetus Dec 4 '12 at 9:56
• Yes, please, please learn some LaTeX for posting huge formulas like this. It was very difficult to read as it was. – Simon Hayward Dec 4 '12 at 10:05
• Some MSE users tried to improve your post using TeX (for better readability). Please check whether these edits did not unintentionally change the meaning of your post. – Julian Kuelshammer Dec 4 '12 at 10:10
• Yes, that is the other problem! Cheers Julian. :) – Simon Hayward Dec 4 '12 at 10:37
• yes the edits done are correct what is latex I am hearing it first time the edits are correct – Registered User Dec 4 '12 at 10:53

## 1 Answer

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.