Chapter 2 Linear System Representation
© National Instruments Corporation 2-15 Xmath Control Design Module
Pole-Zero Matching: polezero
The pole-zero matching method of discretizing a continuous system
follows from the relation between the continuous s and discrete z frequency
domains:
where T is the sampling interval to be used for the discrete system.
Continuous-time poles and finite zeros are mapped to the z-plane using this
relation. Zeros at infinity are mapped into z = 0, where they do not affect
the frequency response.
After the poles and zeros have been mapped, the algorithm tries to
make sure the system gains are equivalent at some critical frequency.
If the systems have no poles or zeros at DC(s=0,z=1), the d iscrete-time
gain is selected such that the system gains match at DC. Alternatively,
if the systems have no poles or zeros at the Nyquist frequency
(s=p*j/T,z=–1), the gains are equalized at that frequency. In the
event that neither of these gains can be matched, no gain is chosen.
Z-Transform: ztransform
This method is a direct Z-transform of the continuous-time transfer
function, which corresponds to the Z-transform of the impulse response of
the system. If ztransform is used, you will match the impulse responses
of the continuous and discrete systems. The responses may differ slightly
due to round off error.
Hold Equivalence Methods: exponential and firstorder
The discretization methods for exponential and firstorder both rely
on the approximation that the discrete-time response can be represented as
a hold on the sampled values of the continuous-time response.
Backward rectangular rule:
Keyword: backward
Tustin’s rule:
Keyword: tustins
Table 2-2. Mapping Methods for discretize( ) (Continued)
Method of Approximation Continuous to Discrete
sz1–
zdt
-----------
→
s2z1–()
dt z 1+()
---------------------
→
ze
sT
=