28.3 Adaptive Notch Filter

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The task of eliminating or suppressing undesirable

narrowband frequencies can be efficiently accomplished

using a notch filter (also known as a

narrowband-stop filter), if the frequencies are

known. The filter highly attenuates a particular

frequency component and leaves the rest of the

spectrum relatively unaffected. An ideal notch

filter has a unity gain at all frequencies except in

the so-called null frequency band, where the gain is

zero. A single-notch filter is effective in removing

single-frequency or narrowband interference; a

multiple-notch filter is useful for the removal of

multiple narrowbands, which is necessary in

applications requiring the cancellation of harmonics.

Digital notch filters are widely used to

retrieve sinusoids from noisy signals, eliminate

sinusoidal disturbances, and track and enhance

time-varying narrowband signals with wideband

noise. They have found extensive use in the areas

of radar, signal processing, communications,

biomedical engineering, and control and instrumentation

systems.

To create a null band in the frequency response of a digital filter at a normalized frequency, b0; a pair of

complex-conjugate zeros can be introduced on the unit circle at phase angles ^b0, respectively. The zeros

are defined as

z1;2 ¼ e^jb0 ¼ cos b0 ^ j sin b0 ð28:8Þ

where the normalized null frequency, b0, is defined as

b0 ¼ 2p

f0

fs ð28:9Þ

FIGURE 28.12 Examples for applying six constraints to a rigid body: (a) three sets of twin reactive forces;

(b) 3 – 2 –1 reactive forces.

FIGURE 28.11 Coupling of a triangular plane structure

to a tetrahedron space structure with six members.

28-10 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Note that fs is the sampling frequency in Hz (or rad/sec) and f0 is the notch frequency in Hz (or rad/sec).

This yields a finite impulse response (FIR) filter given by the following transfer function:

HðzÞ ¼ 1 2 2 cos b0z21 þ z22 ð28:10Þ

A FIR notch filter has a relatively large notch bandwidth, which means that the frequency components in

the neighborhood of the desired null frequency are also severely attenuated as a consequence. The

frequency response can be improved by introducing

a pair of complex-conjugate poles. The poles

are placed inside the circle with a radius of a at

phase angles ^b0. The poles are defined as

p1;2 ¼ a e^jb0 ¼ aðcos b0 ^ j sin b0Þ ð28:11Þ

where a # 1 for filter stability, and ð1 2 aÞ is the

distance between the poles and the zeros.

The poles introduce a resonance in the vicinity

of the null frequency, thus reducing the bandwidth

of the notch. The transfer function of the filter is

given by

HðzÞ ¼ ðz 2 z1Þðz 2 z2Þ

ðz 2 p1Þðz 2 p2Þ ð28:12Þ

Substituting the expression for zi and pi; and

dividing throughout by z2; the resulting filter has

FIGURE 28.13 Examples of support with (a) one reactive force; (b) two reactive forces; and (c) three reactive forces.

FIGURE 28.14 Kinematical vs. semikinematical

design: (a) ideal condition — point contact; (b) line

contact; (c) area contact.

Vibration Suppression and Monitoring in Precision Motion Systems 28-11

© 2005 by Taylor & Francis Group, LLC

the following transfer function:

HðzÞ ¼

a0 þ a1z21 þ a2z22

1 þ b1z21 þ b2z22 ð28:13Þ

HðzÞ ¼

1 2 2 cos b0z21 þ z22

1 2 2a cos b0z21 þ a2z22 ð28:14Þ

Digitally, the filtered signal, y, is thus obtained from the raw signal, u, via the recursive formula in the

discrete time domain as follows:

yðnÞ ¼ a0uðnÞ þ a1uðn 2 1Þ þ a2uðn 2 2Þ 2 b1yðn 2 1Þ 2 b2yðn 2 2Þ ð28:15Þ

where the coefficients ai and bi are the same as those in Equation 28.13 because z21 corresponds to the

time-shift (delay through sampling period) operator.

The bandwidth and the Q-factor of the notch filter are, respectively

BW ¼

2

ffiffi

2 p ð1 2 a2Þ

½16 2 2að1 þ aÞ2􀀉1=2 ð28:16Þ

Q ¼ w0 ½16 2 2að1 þ aÞ2􀀉1=2

2

ffiffi

2 p ð1 2 a2Þ ð28:17Þ

The filter transfer function, HðzÞ; has its zeros on the unit circle. This implies a zero transmission gain at

the normalized null frequency, b0. It is interesting to note that the filter structure, Equation 28.14, allows

independent tuning of the null frequency and the 3-dB attenuation bandwidth by adjusting b0 and a,

respectively. The performance of the notch filter depends on the choice of the constant, a, which controls

the bandwidth, BW, according to Equation 28.16. The bandwidth, which is a function of the distance of

the poles and zeros ð1 2 aÞ; narrows when a approaches unity. Clearly, when a is close to 1, the

corresponding transfer function behaves virtually like an ideal notch filter.

Complete narrowband disturbance suppression requires an exact adjustment of the filter parameters

to align the notches with the resonant frequencies. If the true frequency of the narrowband interference

that is to be rejected is stable and known a priori, a notch filter with fixed null frequency and fixed

bandwidth can be used. However, if no information is available a priori, or when the resonant frequencies

drift with time, the fixed notch may not coincide exactly with the desired null frequency, particularly if

the bandwidth is too narrow (i.e., a < 1). In this case, a tunable or adaptive notch filter is highly

recommended. In Ahlstrom and Tompkins (1985) and Glover (1987), it is proposed to adapt the null

bandwidth of the filter to accommodate the drift in frequency. In Bertran and Montoro (1998), it is

suggested that an active compensator be used to suppress the vibration signals. Kwan and Martin (1989)

adapt the null frequency, b0, while keeping the pole radii, a, constant. In other words, the parameters ai

and bi of Equation 28.13 are adjusted such that the notch will center at the unwanted frequency while

retaining the null bandwidth of the notch filter.

28.3.1 Fast Fourier Transform

The discrete Fourier transform (DFT) is a tool that links the discrete-time domain to the discretefrequency

domain (see Chapter 10, Chapter 21, and Appendix 2A). It is a popular off-line approach,

widely used to obtain the information about the frequency distribution required for the filter design.

However, the direct computation of the DFT is prohibitively expensive in terms of required computation

effort. Fortunately, the fast Fourier transform (FFT) is mathematically equivalent to the DFT, but it is a

more efficient alternative for implementation purposes (with a computational speed that is exponentially

faster) and can be used when the number of samples, n, is a power of two (which is not a serious

constraint). For vibration signals where the concerned frequencies drift with time, the FFT can be

continuously applied to the latest n samples to update the signal spectrum. Based on the updated

spectrum, the filter characteristics can be continuously adjusted for notch alignment. The block diagram

28-12 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

of the adaptive notch filter that has been developed in the present work, with its adjusting mechanism, is

shown in Figure 28.15.

28.3.2 Simulation and Experiments

A simulation study is carried out to explore the application of the adaptive notch filter in suppressing

undesirable frequency transmission in the control system for a precision positioning system that uses

permanent magnet linear motors (PMLM). In the simulation, a sinusoidal trajectory profile is to be

closely followed and an undesirable vibration signal is simulated that drifts from a frequency of 500 Hz in

the first cycle to a frequency of 1 to 5 Hz in the second cycle of the trajectory. Figure 28.16 shows

Adaptive

Notch

Filter

Adjusting

mechanism

based on F F T

Control signal

PMLM

x

Controller

e

xd, xd, xd

. ..

FIGURE 28.15 Block diagram of the adaptive notch filter with adjusting mechanism. (Source: Tan, K.K., Tang, K.Z.,

de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 28.16 Simulation results without a notch filter: (a) error (mm); (b) desired trajectory (mm); (c) control

signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press.

With permission.)

Vibration Suppression and Monitoring in Precision Motion Systems 28-13

© 2005 by Taylor & Francis Group, LLC

FIGURE 28.17 Simulation results using a fixednotch filter: (a) error (mm); (b) desired trajectory (mm); (c) control

signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press.

With permission.)

FIGURE 28.18 Simulation results using an adaptive notch filter: (a) error (mm); (b) desired trajectory (mm);

(c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001,

IOS Press. With permission.)

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© 2005 by Taylor & Francis Group, LLC

the tracking performance of the precision machine

without a notch filter. Figure 28.17 shows the

performance when a fixed notch filter is used, and

Figure 28.18 shows the performance with an

adaptive notch filter. It is clearly evident that a

time-invariant narrowband vibration signal can be

effectively eliminated using just a fixed notch filter.

However, when the vibration frequencies drift, an

adaptive notch filter is able to detect the drift and

align the notch to remove the undesirable

frequencies, with only a short transient period.

The notch filter is subsequently implemented in

the control system of a linear drive tubular linear

motor (LD3810) equipped with a Renishaw optical

encoder having an effective resolution of 1 mm.

The hardware setup for this experimental study is

shown in Figure 28.19 whereas Figure 28.20 shows

the linear motor in more detail.

The components of the linear motor consist of

the thrust rod, the thrust block, the motor cable,

and the optical encoder. The thrust rod is made of

a thin-walled stainless steel tube housing highenergy

permanent magnets. To enable the smooth

translation of the thrust block along the length of

the thrust rod, the thrust block is made from an

aluminum housing that contains cylindrical coils

arranged in a three-phase star pattern. An

electromagnetic field is produced by energizing

these coils. The interactions between the permanent magnetic field of the thrust rod and the changing

magnetic field of the thrust block provide the induced force for the translation of the block. Usually, a

sinusoidal or trapezoidal motor commutation is utilized to smoothen the translation of the block. The

popular proportional – integral – derivative (PID) control is used in this experimental study. The dSPACE

DS1102 (dSPACE User’s Guide, 1996) digital signal processing (DSP) board is used as the data acquisition

and control card. It is a single-board system, which is specifically designed for the development of highspeed

multivariable digital controllers and real-time simulations in various fields. The DS1102 is based

on the Texas Instruments TMS320C31 third-generation floating-point DSP, which builds the main

processing unit, providing fast instruction cycle time for numeric intensive algorithms. It contains 128K

words memory that is fast enough to allow zero wait-state operation. Besides these, the DS1102 DSP

board supports a total memory space of 16M 32-bit words, including program, data, and I/O space. All

off-chip memory and I/O can be accessed by the host, even while the host is running, thus allowing easy

system setup and monitoring. The TMS320C31 is object-code compatible to the TMS320C30. The DSP

is fully supplemented by a set of on-board peripherals, frequently used in digital control systems. Analogto-

digital and digital-to-analog converters, a DSP-microcontroller-based digital-input/output (I/O)

subsystem, and incremental sensor interfaces make the DS1102 an ideal single-board solution for a broad

range of digital control tasks.

The DS1102 DSP board is well supported by popular software design and simulation tools,

including MATLAB and SIMULINK, which offer a rich set of standard and modular design

functions for both classical and modern control algorithms. The SIMULINK model developed for

the system (Figure 28.15), with the notch filter, is shown in Figure 28.21. This model is then

downloaded to the DS1102 DSP board for real-time implementation using one of the options

available from the pull-down menu.

FIGURE 28.20 Experimental platform (LD 3810) —

linear drive tubular motor with optical encoder

attached. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S.,

Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS

Press. With permission.)

FIGURE 28.19 Hardware setup for the experimental

study of the notch filter. (Source: Tan, K.K., Tang, K.Z.,

de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy

Syst., 2001, IOS Press. With permission.)

Vibration Suppression and Monitoring in Precision Motion Systems 28-15

© 2005 by Taylor & Francis Group, LLC

FIGURE 28.21 SIMULINK model created for the system with the notch filter incorporated. (Source: Tan, K.K.,

Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 28.22 Experimental results without a notch filter: (a) error (mm); (b) desired trajectory (mm); (c) control

signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press.

With permission.)

28-16 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

It is now in order to present the experimental results utilizing the notch filter in the system.

Figure 28.22 shows the performance of the PMLM when no filter is used. Figure 28.23 shows the

improvement in the control performance when the notch filter is incorporated into the control

system.

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