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digital notch filters are more popular choice for tackling actuator resonance
3.2.3 Cancelling Sensor Noise using Notch Filter
It is shown in the previous section how notch filters can be used to eliminate
unwanted resonant oscillations of the actuator. Notch filter can also be used
to eliminate sensor noise whose energy is concentrated in a narrow band of
frequencies. When different signals are measured in a control system, the
process of measurement often contribute to noise entering into the system.
These measurement noises are usually random in nature containing wide band
of frequencies. However, in many practical systems, the noise from a sensor
either can be sinusoidal or has its energy concentrated in a narrow band of
frequencies. Such noise has severe detrimental effects on the performance of
the closed loop. A method useful for elimination of the effects of narrow band
sensor noise is explained next.
As an illustrative example, let us assume that the noise contaminating the
sensor output has a peak at f n kHz. Such noise can be contributed by many
practical issues, for example, the switching noise of a motor driver. The open-
loop transfer function L(s), error rejection (or sensitivity) transfer function
S(s), complementary sensitivity transfer function T (s), and shock transfer
function S h (s) are identical to those defined in the previous sections, given the
plant model G p (s), controller G c (s)andfilter F(s).
Figures 3.15-3.19 show the responses of the plant model shown in Fig-
ure 3.2 controlled by the same lag-lead compensator as before but with an
additional notch filter whose center frequency is 2.88 kHz and is derived from
equation (3.17) by letting ω n1 = ω n2 =2π × 2.88 rad/sec, and setting the
values of ζ n1 to a small positive number (0.02 → 0.2) and ζ n2 close to 1. The
controller is designed to obtain a crossover frequency f v equal to 1000 Hz or
6284 rad/s.
The bode plots of the sensitivity transfer function and the complementary
transfer function for this design are shown in Figure 3.17 (dashed line). The
complementary transfer function shows a notch at the designed frequency, i.e.,
2.88 kHz when the designed notch filter for eliminating sensor noise is included.
These figures also show the bode plots when the notch filter cancelling sensor
noise is not included (solid line). It is evident from the comparison between the
two that the inclusion of the notch filter offers approximately 30 dB additional
attenuation at the designed frequency of 2.88 kHz, i.e., the frequency of sensor
noise. However, the output becomes more oscillatory for a step command in
the reference input as illustrated by the closed loop step response shown in
Figure 3.19.
Since the notch filter is used in this case for attenuating the sensing noise,
the filter can be chosen such that its ferquency response is deeper and narrower
than that of the notch filter used to attenuate actuator resonance.
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