discretizing and resampling models -凯发k8网页登录

this example shows how to use the commands for continuous/discrete, discrete/continuous, and discrete/discrete conversions.

related commands

control system toolbox™ offers extensive support for discretization and resampling of linear systems including:

c2d discretizes continuous-time models
d2c compute continuous-time extensions of discrete-time models
d2d resamples discrete-time models.

several algorithms are available to perform these operations, including:

zero-order hold
first-order hold
impulse invariant
tustin
matched poles/zeros.

continuous/discrete conversion

for example, consider the second-order system with delay:

$g(s) = e^{-s} {s - 2 \over s^2 3 s 20}$

to compute its zero-order hold (zoh) discretization with sampling rate of 10 hz, type

g = tf([1 -2],[1 3 20],'inputdelay',1);
ts = 0.1;   % sampling interval
gd = c2d(g,ts)

gd =
 
             0.07462 z - 0.09162
  z^(-10) * ----------------------
            z^2 - 1.571 z   0.7408
 
sample time: 0.1 seconds
discrete-time transfer function.

compare the continuous and discrete step responses:

step(g,'b',gd,'r')
legend('continuous','discretized')

discrete/continuous conversion

conversely, you can use d2c to compute a continuous-time "interpolant" for a given discrete-time system. starting with the discretization gd computed above, convert it back to continuous and compare with the original model g:

gc = d2c(gd);
step(g,'b',gd,'r',gc,'g--')
legend('original','discretized','d2c interpolant')

the two continuous-time responses match perfectly. you may not always obtain a perfect match especially when your sampling interval ts is too large and aliasing occurs during discretization:

ts = 1;  % 10 times larger than previously
hd = c2d(g,ts);
hc = d2c(hd);
step(g,'b',hd,'r',hc,'g--',10)
legend('original','discretized','d2c interpolant')

resampling of discrete-time systems

resampling consists of changing the sampling interval of a discrete-time system. this operation is performed with d2d. for example, consider the 10 hz discretization gd of our original continuous-time model g. you can resample it at 40 hz using:

gr = d2d(gd,0.025)

gr =
 
             0.02343 z - 0.02463
  z^(-40) * ----------------------
            z^2 - 1.916 z   0.9277
 
sample time: 0.025 seconds
discrete-time transfer function.

compare this with a direct discretization at 40 hz:

step(g,'b',gr,'r',c2d(g,0.025),'g--',4)
legend('continuous','resampled from 0.1 to 0.025','discretized with ts=0.025')

notice that both approaches lead to the same answer.

which algorithm and sampling rate to choose?

see the example entitled discretizing a notch filter for more details on how the choice of algorithm and sampling rate affect the discretization accuracy.