Page 75 - Mechatronics with Experiments

CLOSED LOOP CONTROL 61

Control computer
Clock
u (t) u(t)
1
u(kT)
Control algorithm D/A Post filter Actuator Process

A/D Pre filter Sensors
y(kT)
y (t) y (t) y(t)
1
2
FIGURE 2.11: Filtering and bandwidth issues in a digital closed loop control system.

ideal filter would have frequency response characteristics similar to the ideal reconstruction
filter. It would pass identically all the frequency components up to 1∕2 of the sampling
frequency, and cancel the rest. However, it is not practical to use such filters in control and
data acqusition systems.
Generally, a second order or higher order filter is used as a noise filter. Typically the
filter transfer function for a second-order filter is

w 2 n
G (s) =
F 2 2
s + 2 w s + w n
n n
If a higher order filter is desired, multiple second-order filters can be cascaded in series. The
exact parameters of the noise filter ( , w ) are selected based on the class of the filter. The
n n
Butterworth, ITAE, and Bessel filter are popular filter parameters used for that purpose.
Similarly, since the output of the D/A converter is a sequence of step changes, it may
be useful to smooth the control signal before it is applied to the amplification stage. The
same type of noise filters can be used as a post-filtering device to reduce the high frequency
content of the control signal.
The pre-filters and post-filters add time delay into the closed loop system due to
their finite bandwidth. In order to use the pre- and post-filters for noise cancellation and
smoothing purposes without significantly affecting the closed loop system bandwidth, the
following general guidelines should be followed. We have four frequencies of interest:

1. closed loop system bandwidth, w ,
cls
2. sampling frequency, w ,
s
3. pre- and post-filter bandwidth, w ,
filter
4. the maximum frequency content of the signal presented to the sampling and hold
circuit of A/D converter, w signal .

In order to make sure that the pre- and post-filtering does not affect the closed loop system
bandwidth, the filters must have about 10 times or more higher bandwidth than the closed
loop system bandwidth. The filter bandwidth is also a good estimate of the highest frequency
content allowed to enter the sampling circuit.

w filter ≈ w signal ≈ 5 to 10 ∗ w cls