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.

  • $\begingroup$ 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. $\endgroup$ – niels nielsen Oct 11 '20 at 6:07

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:

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


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

For the denominator:



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

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:

enter image description here


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

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?

  • 1
    $\begingroup$ They certainly mean statespace $\endgroup$ – morbo Oct 11 '20 at 11:46
  • 1
    $\begingroup$ Yes it would be to convert into state space representation. See it as putting the equation into matrix form. $\endgroup$ – Barrett Cloud Oct 11 '20 at 20:42
  • $\begingroup$ @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. $\endgroup$ – Petrus1904 Oct 13 '20 at 10:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.