ZPK transfer functions in Matlab

Consider the transfer function: \[\begin{aligned} H(s) &= \frac{2 s + 1}{s^2 + 7 s + 12} \\\ &= 2 \frac{s+1/2}{(s+3)(s+4)}. \end{aligned}\]

In the second equality, we have factored the polynomials and expressed them in terms of poles \(p_i\) and zeros \(z_i\) with terms \((s - p_i)\) and \((s - z_i)\). Note the gain factor \(2\) that emerges in this form.

Both forms are useful. In the former, two polynomials in \(s\) define the transfer function; in the latter, a list of zeros, poles, and a gain constant define the transfer function.

In Matlab, there are two corresponding manners of defining a transfer function. We demonstrate the first, already familiar, method using the tf command, which takes polynomial coefficients, as follows.

H_tf = tf([2,1],[1,7,12])

H_tf =
 
     2 s + 1
  --------------
  s^2 + 7 s + 12
 
Continuous-time transfer function.

Alternatively, we can define the transfer function model with the zpk command using the zeros, poles, and gain constant.

H_zpk = zpk([-1/2],[-3,-4],2)

H_zpk =
 
   2 (s+0.5)
  -----------
  (s+3) (s+4)
 
Continuous-time zero/pole/gain model.

This zpk model will work with all the usual functions tf models do. However, if you’d like to convert zpk to tf, simply use tf as follows.

tf(H_zpk)

ans =
 
     2 s + 1
  --------------
  s^2 + 7 s + 12
 
Continuous-time transfer function.

Alternatively, we can convert a tf model to a zpk model.

zpk(H_tf)

ans =
 
   2 (s+0.5)
  -----------
  (s+4) (s+3)
 
Continuous-time zero/pole/gain model.