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Chapter 2 Linear System Representation
Xmath Control Design Module 2-18 ni.com
Many of the discretization techniques discussed in the Hold Equivalence
Methods: exponential and firstorder section can be easily reversed to
obtain a continuous equivalent to a discrete system. The
makecontinuous( ) function implements these reverse algorithms
based on the keyword you specify as shown in Example 2-10. Although
makecontinuous( ) accepts an input system in any form, it returns the
continuous-time system as a state-space system.
The forward, backward, and Tustin methods for mapping from the s-plane
to the z-plane can be easily reversed using the equivalencies shown in
Table 2-3.
Discrete-to-continuous algorithms using matrix logarithms (to reverse the
exponential operations involved in doing the z-transform for the impulse
invariant zero-order hold) are available for the
exponential
(step-invariant) transformation and the
ztransform (impulse-invariant)
methods. A limitation of these methods, however, is that they will not return
a meaningful continuous equivalent to a discrete system that has pure
delays (1/z terms), because the logarithm of zero is infinite.
Example 2-10 Verifying a Discretization Using makecontinuous( )
Create a system:
H = 0.5*polynomial([-0.36])/...
polynomial([-1,-1,-0.395+0.06305*jay,
-0.395-0.06305*jay]);
Form the discrete equivalent using the forward approximation:
Hd_f = discretize(H,0.1, {forward});
Table 2-3. Mapping Methods for makecontinuous( )
Method of Approximation Discrete to Continuous
Forward rectangular rule:
Keyword:
forward
Backward rectangular rule:
Keyword:
backward
Tustin’s rule:
Keyword:
tustins
z 1 sdt()+→
z
1
1 sdt()–
---------------------
→
z
1 sdt()+ 2⁄
1 sdt()– 2⁄
----------------------------
→