Transfer function into block diagram and matrix form

Question

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.

Answer 1

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?