If you rewrite the equations for the transfer function model to show the locations of the zeros and poles of the dynamic system, you obtain the zero-pole-gain model.

SISO

The following is the equation for the continuous zero-pole-gain SISO model:

G s = k s - Z 1 s - Z 2 . . . s - Z m s - P 1 s - P 1 . . . s - P n

where

  • s is the Laplace variable and continuous time
  • k is the transfer function gain
  • Zi are the zeros
  • Pj are the poles

The following is the equation for the discrete-time zero-pole-gain SISO model:

G z = k z - Z 1 z - Z 2 . . . z - Z m z - P 1 z - P 1 . . . z - P n

where z is discrete time

When s or z equals 0, you can calculate the static gain from the two equations.

s t a t i c   g a i n = - 1 m - n k z 1 z 2 . . . z m P 1 P 2 . . . P n

MISO

The following is the equation for the continuous zero-pole-gain MISO model:

y i = ∑ f = 1 n G i f s u j

where

  • Gij are the transfer functions between the stimulus and the response
  • i is the input number of the system
  • j is the output number of the system

The following is the equation for the discrete-time zero-pole-gain MISO model:

y i = ∑ j = 1 n G i j s u j

The System Identification VIs do not include a VI to estimate zero-pole-gain models directly because you can use the SI Model Conversion VI to convert another model representation to a zero-pole-gain model.