Creating and Manipulating Models

Zero-Order Hold

Zero-order hold (ZOH) devices convert sampled signals to continuous-time signals for analyzing sampled continuous-time systems. The zero-order-hold discretization of a continuous-time LTI model is depicted in the following block diagram.

The ZOH device generates a continuous input signal u(t) by holding each sample value u[k] constant over one sample period.

The signal is then fed to the continuous system , and the resulting output is sampled every seconds to produce .

Conversely, given a discrete system , the d2c conversion produces a continuous system whose ZOH discretization coincides with . This inverse operation has the following limitations:

• d2c cannot operate on LTI models with poles at when the ZOH is used.
• Negative real poles in the domain are mapped to pairs of complex poles in the domain. As a result, the d2c conversion of a discrete system with negative real poles produces a continuous system with higher order.

The next example illustrates the behavior of d2c with real negative poles. Consider the following discrete-time ZPK model.

hd = zpk([],-0.5,1,0.1)
Zero/pole/gain:
   1
-------
(z+0.5)
 
Sampling time: 0.1

Use d2c to convert this model to continuous-time

hc = d2c(hd)     

and you get a second-order model.

Zero/pole/gain:
   4.621 (s+149.3)
---------------------
(s^2 + 13.86s + 1035)

Discretize the model again

c2d(hc,0.1)      

and you get back the original discrete-time system (up to canceling the pole/zero pair at z=-0.5):

Zero/pole/gain:
 (z+0.5)
---------
(z+0.5)^2
 
Sampling time: 0.1

Continuous/Discrete Conversions of LTI Models

First-Order Hold