# Transfer function into block diagram and matrix form

I'm reading a book on Control System Design and running into the same issue I had in Kinematic Design. When things are presented in a format without any numerical examples it just appears as gibberish to me and always has (I do have a learning disability which maybe my issue.) So I'm working the problems and trying to take the problem statement of $$H(s) = \frac{(s+2)(s+4)}{(s+1)(s+3)(s+5)}$$ and place it into a block diagram as well as a matrix. The issue I'm running into is that the only examples I'm seeing involved $$H(s)= \frac{b_0s^k+b_1s^{k-1}+b_k}{s^k+a_1s^{k-1}+a_k}$$ which is makes absolutely no sense to me. Any help would be appreciated.

• if you can find a copy of it, have a look at System Dynamics: A Unified Approach by Karnopp and Rosenberg, it does a good job of transfer functions, block diagram and matrix representations. – niels nielsen Oct 11 '20 at 6:07

## 1 Answer

Since $$H(s)$$ describes a transfer function, then:

$$\tag{1}\label{eq1} H(s) = \frac{Y(s)}{X(s)}$$

For some arbitrary input $$X(s)$$ and arbitrary output $$Y(s)$$. It is usually the case that you are given the input $$X(s)$$ and the transfer function $$H(s)$$, and so equation \eqref{eq1} is re-arranged as:

$$Y(s) = X(s) \cdot H(s) \tag{2}\label{eq2}$$

So, the purpose of the transfer function is to describe the behavior (output) of a system to an arbitrary input $$X(s)$$, as described by equation \eqref{eq2}. For example, suppose that $$H(s)=(s+2)$$ and $$X(s)=s$$. Then:

$$Y(s) = X(s) \cdot H(s) = s(s+2) = s^2 + 2s$$

Now, note that:

$$H(s)= \frac{b_0s^k+b_1s^{k-1}+b_k}{s^k+a_1s^{k-1}+a_k}$$

Can be obtained by expanding:

$$H(s) = \frac{(s+2)(s+4)}{(s+1)(s+3)(s+5)}$$

For example, for the numerator:

$$(s+2)(s+4) = s^2 + 6s + 8$$

So:

$$H(s) = \frac{s^2 + 6s + 8}{(s+1)(s+3)(s+5)}$$

For the denominator:

$$(s+1)(s+3)(s+5)=(s+1)(s^2+8s+15)=s^3+9s^2+23s+15$$

So:

$$H(s) = \frac{s^2 + 6s + 8}{s^3+9s^2+23s+15}$$

and place it into a block diagram as well as a matrix.

Referring back to equation \eqref{eq1}, the block diagram of this system is just: Where:

$$H(s) = \frac{s^2 + 6s + 8}{s^3+9s^2+23s+15}$$

However, I am not sure what you mean by placing $$H(s)$$ into a matrix. Perhaps you mean convert $$H(s)$$ into its state space representation?

• They certainly mean statespace – morbo Oct 11 '20 at 11:46
• Yes it would be to convert into state space representation. See it as putting the equation into matrix form. – Barrett Cloud Oct 11 '20 at 20:42
• @DKScidmore search for "Control Canonical form" online. it is a simple method that can be used to transform a transfer function to a state-space representation. – Petrus1904 Oct 13 '20 at 10:36