08 - Feedback ANC

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Luigi Piroddi “Active Noise Control” course notes (January 2012)

8. Feedback active noise control 2

(Adaptive) feedforward or (non-adaptive) feedback ANC? Feedback control systems differ from feedforward systems in the manner in which the control signal is derived. � Feedforward systems rely on some predictive measure of the incoming

disturbance to generate an appropriate “canceling” disturbance. � Feedback systems generate the control signal by processing the error signal,

with the goal of attenuating the residual effects of the disturbance after it has passed (the reference sensor is not required).

� one sensor less

� no acoustic feedback

� we do not need to worry about the low coherence between reference and disturbance

� feedback ANC cannot make any difference between the noise and the useful signal measured by the error microphone: everything is attenuated

A feedforward system should be implemented whenever it is possible to obtain a suitable reference signal, because the performance of an adaptive feedforward system is, in that case, superior to a feedback system.



Feedback ANC is required for applications in which it is not possible or practical to measure or internally generate a coherent reference signal: � spatially incoherent noise generated from turbulence

� noise generated from many sources and propagation paths

� resonant response of an impulsively excited structure, where no coherent reference signal is available

Unlike feedforward systems for which the physical system and controller can be optimized separately, feedback systems must be designed by considering the physical system and controller as a coupled system.

For noise problems: � Adaptive feedforward control has been applied successfully to ducts,

aircraft cabins and motor vehicle interiors and exteriors.

� Feedback control has been applied successfully to ear defenders where it is not easy to sample the incoming signal in advance, making it difficult to generate an appropriate reference signal for a feedforward controller.



Location of sensors and actuators The physical arrangement of control sources and error sensors plays a very important role in determining the effectiveness of an active control system.

Moving the locations of the control sources and sensors affects both system controllability and stability: � For feedforward systems, the physical system arrangement can

be optimized independently of the controller.

� For feedback systems, the physical system arrangement is an important part of the controller design. For duct applications:

� the microphone should be placed at the surface of the duct (zero airflow velocity)

� the loudspeaker should be placed in the centre of the transverse cross-section of the duct to minimize the number of transverse modes



Classical feedback ANC scheme

Noise source

Primary noise

Feedback ANC

Canceling loudspeaker

Error microphone

y(n)

e(n)

duct

d

The sensor output is processed by an amplifier that has: � an overall gain higher than unity

� a 180° phase reversal

The system requires only one sensor (the acoustic secondary-to-reference feedback problem is avoided).

The feedback control system employs a secondary source located in the vicinity of an error sensor (to minimize loop delay).



Problems: � limited broadband noise attenuation � limited frequency range of operation (up to a few hundred Hz) � the presence of a delay from the secondary source to the error sensor

implies that only periodic signals can be completely canceled � possible instability caused by positive feedback at high frequencies

� the control system will oscillate when the combined loop delays are equivalent to a 180° phase shift at a particular frequency and the overall gain is greater than unity

The smaller the distance between the error microphone and the secondary source: � the higher the critical frequency over which the system cannot work, but � the less uniform the sound field generated by the secondary source

(due to physical limitations of conventional loudspeakers)

Therefore, the non-uniform near-field of the loudspeaker determines an upper bound on the frequencies that can be canceled.

To prevent system oscillation, the loop gain must be less than 1 before the critical frequency is reached (gain roll-off).



Improved feedback ANC system implementation The working range of a feedback ANC system can be greatly extended by feeding the loudspeaker’s output into a partially closed volume (with dimensions much smaller than the wavelength of the highest frequency to be controlled) containing the microphone.

Primary noise

Feedback ANC


Error microphone

y(n)

e(n)

duct

In this way, � the near-field of the loudspeaker becomes more uniform

� the acoustic effect of the partially closed volume yields a well-behaved roll-off characteristic

� the microphone is automatically isolated from nearby reflecting surfaces (and, in cases of highly hostile environments, it is placed in a safer zone)



Reported applications of feedback ANC � Duct ANC system comprising two (electrically) independent feedback ANC

systems in cascade (Hong et al., 1987) � each feedback stage provides additional attenuation (although less than the

sum of the individual actions due to the interaction between the two systems)

� Active electronic muffler for automobile exhaust noise control (Taki, 1993) � the noise attenuation performance is approximately equal to that of a passive

muffler, but a great reduction of exhaust back pressure is obtained, providing a 5−10 % increase of engine power

� Noise attenuation in transport aircraft cabin (Legrain and Goulain, 1988) � a loudspeaker is mounted at one end of the cabin and microphones are located

at passenger ear level along the fuselage

� Broadband noise control in an active headset (Carme,1988; Veit, 1988; Wheeler and Smeatham, 1992) � the confined space minimizes phase shift and acoustic noise travel time between

the headset’s loudspeaker and a closely spaced miniature microphone; more than 10 dB attenuation is achieved at the ear drum below 3 kHz



Analysis of a general feedback ANC system

+

d(n)

S(z) −

W(z) e(n) y(n)

H(z)

d(n) = primary noise (at the error location)

e(n) = residual noise measured by the error sensor

y(n) = secondary control signal

W(z) = controller transfer function

S(z) = secondary path transfer function

Under steady state conditions:

E(z) = 1

1+S(z)W(z) D(z)

The error goes to 0 as the magnitude of the loop gain |S(ejω)W(ejω)| approaches infinity. Therefore, significant noise reduction can be achieved by designing the controller to have a large gain over the frequency band of interest.

H(z) = closed loop transfer function from the primary noise to the error signal



In the ideal case S(z) = 1, arbitrary noise reduction can be obtained by employing a constant controller W(z) = µ with large gain.

In practice, the frequency response of S(z) is never perfectly flat and free of phase shift: � The response of the secondary source introduces a considerable phase shift.

� The physical path from the secondary source to the error sensor introduces some delay, due to propagation time.

Using a constant controller, as the phase shift in the secondary path approaches 180°, the control system may become unstable (if the loop gain is greater than unity at the corresponding frequency) → see, e.g., Bode stability criterion

Define

L(ejω) = S(ejω)W(ejω) = ML(ω)ejφL(ω)

where

� ML(ω) = |L(ejω)|

� φL(ω) = ∠L(ejω)



Then:

|1 + S(ejω)W(ejω)|2 = 1 + ML(ω)2 + 2ML(ω)cos(φL(ω))

The design of a feedback ANC involves finding a W(z) such that ML(ω) is maximized while −180° < φL(ω) < 180°, ensuring at the same time the following two constraints: � causality of W(z)

� stability of H(z) → open-loop gain |ML(ω)| < 1 at 180° phase shift

Remarks: � If not compensated for stability at high frequencies, the ANC system will

diverge into uncontrolled oscillation initiated by low level noise.

� Additional filters may be introduced into W(z) to compensate for the phase shift of S(ejω), thereby increasing the ANC bandwidth.

� High level of noise attenuation is obtained for periodic or very low-frequency noise.

� The gain-bandwidth limitation may be reduced by using cascaded feedback stages.



The waterbed effect Bode (1945) showed that the integral of the log sensitivity function H(z) (which is linearly proportional to the disturbance attenuation) in dB, calculated over the whole frequency range, must be zero. → constraint on H(z)

This performance limitation of feedback controllers implies that

� we can obtain good attenuation (small |H(ejωT)|) only in a limited bandwidth (where the disturbance has significant energy), but

� we must allow for small enhancements elsewhere (typically, where the disturbance has little energy)

This disturbance enhancement can be made arbitrarily small for minimum-phase plants, but not for non-minimum-phase ones.

This is known as the waterbed effect.

Notice that this limitation is not shared by feedforward control systems (using a time-advanced reference signal), that can achieve broadband attenuation of a disturbance without any out-of-band enhancements.



Closed loop rejection of harmonic disturbances for narrowband ANC The z-transform of e(n) is given by:

E(z) = 1

1+L(z) D(z)

where

L(z) = S(z)W(z)

+

d(n)

S(z) −

W(z) e(n) y(n)

H(z)

Since

1

1+L(ejω) ≅

1|L(ejω)|

ω ≤ ωc

1 ω > ωc

the control system must be designed so that: � closed loop stability is guaranteed

� ωc >> ω (otherwise no attenuation can be achieved)

� |L(jω )| ≥ A (level of attenuation)



Since the z-transform of the disturbance signal d(n) = cos(ωnTS) is equal to

D(z) = z2 − cos(ωTS) z

z2 − 2cos(ωTS) z + 1 ,

if the controller is stabilizing and includes a factor z2 − 2cos(ωTS) z + 1 at the denominator (internal model principle), perfect rejection of the disturbance is obtained at steady state. In fact, defining:

W(z) = NW(z)DW(z) =

NW(z)(z2−2cos(ωTS)z+1)DW(z)

and S(z) = NS(z)DS(z)

one obtains:

E(z) = 1

1+L(z) D(z) = 1

1 + NW(z)NS(z)

(z2−2cos(ωTS)z+1)DW(z)DS(z)

z2 − cos(ωTS) z

z2 − 2cos(ωTS) z + 1 =

= (z2−cos(ωTS)z)DW(z)DS(z)NW(z)NS(z)+DW(z)DS(z)



Since the poles of function E(z) are all in |z| < 1 (stabilizing regulator), e(n) → 0 in view of the final value theorem.

The simplest regulator structure that contains the cancelling factor is

W(z) = k z (z − b)

z2 − 2cos(ωT) z + 1

which is called notch filter.

The real zero b of the transfer function is generally introduced to reduce the phase loss determined by the poles on the unit circle.

Choosing b = cos(ωT) centers the zero at the same frequency of the poles.



-50

0

50

100

150

200

Mag

nitu

de (

dB)

10-2

10-1

100

101

102

103

-90

-45

0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Regulator transfer function



-200

-150

-100

-50

0

Mag

nitu

de (

dB)

10-2

10-1

100

101

102

103

-90

-45

0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Closed loop sensitivity transfer function

(S(z) = 1)



Problems: � The regulator is at the stability limit.

� As the relative degree of the system’s transfer function increases it gets more difficult to ensure stability (check the asymptotes of the root locus).

� Stability is even harder to obtain if the controlled system adds resonances (and anti-resonances) at frequencies near to those of the filter.

� The precise positioning of the notch is critical.

The root locus is typically employed in the design process.



Singularities of the loop transfer function on the unit circle The control of resonant systems (e.g., vibration control of mechanical structures or acoustic control of enclosures) involves loop transfer functions with multiple singularities on the unit circle or near to it: � resonances and anti-resonances due to the vibrational/acoustic normal

modes of the controlled system

� unit circle poles introduced by the controller for harmonic disturbance rejection

If the low damped poles and zeros of the loop transfer function are alternated on the unit circle, it can be shown (e.g., with the root locus approach) that the stabilization of the system is always possible, even in the presence of significant system perturbations (robust stability).

On the contrary, if such property does not hold, stabilization is very critical or impossible.

The correct placement of the controller singularities on the unit circle is therefore crucial.



Case 1: L(z) with one couple of poles on (or near to) the unit circle

-1 0 1

-1

-0.5

0

0.5

1

rel. degree of L(z) = 0

Real Axis

Imag

inar

y A

xis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Imag

inar

y A

xis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Imag

inar

y A

xis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Imag

inar

y A

xis

OK (always) OK

(but the gain must be small)



Case 2: L(z) with two couples of poles on (or near to) the unit circle

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Imag

inar

y A

xis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Imag

inar

y A

xis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Imag

inar

y A

xis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Imag

inar

y A

xis

-1 0 1

-1

-0.5

0

0.5

1


Real Axis

Imag

inar

y A

xis

-1 0 1

-1

-0.5

0

0.5

1


Real AxisIm

agin

ary

Axi

s

OK (stabilizable)

not OK (not stabilizable)



Adaptive feedforward ANC vs. non-adaptive feedback ANC Control algorithm

Advantages Disadvantages

adaptive feedforward ANC

� error signal driven to 0 � large stability bounds � precise modeling not required

� transient suppression is difficult

� coherent reference signal required

non-adaptive feedback ANC

� no reference microphone � no acoustic feedback � active damping provides

transient signal suppression � relatively simple control

algorithm

� stability not guaranteed � non-selective attenuation � small delay requirement

constrains control setting � modeling uncertainty

reduces robustness � limited cancellation over

limited bandwidth



Single-channel adaptive ANC using a feedback-generated reference A simple idea to recover some of the results found for adaptive feedforward ANC consists in using feedback to estimate the primary noise, in order to use this estimate as a mock reference signal for the ANC filter.

This technique can be viewed as an adaptive feedforward system that synthesizes its reference signal based only on the adaptive filter output and the error signal.

+

d(n)

S(z) −

e(n) y(n)

S(z)

+

+ x(n) = d(n)

W(z)

D(z) = E(z) + S(z)Y(z)

X(z) = D(z) = E(z) + S(z)Y(z)

Y(z) = W(z)

1−S(z)W(z) E(z)

This scheme is very similar to the Internal Model Control (IMC) scheme.



Interpretation of the IMC -like feedback ANC as a feedforward ANC

E(z) = 1

1+S(z)W(z)

1−S(z)W(z)

D(z) = 1−S(z)W(z)

1+[S(z)−S(z)]W(z)D(z)

Under ideal conditions where S(z) = S(z) (and therefore x(n) = d(n)) we obtain:

E(z) = [1 − S(z)W(z)]D(z)

and the feedback control scheme can be seen to correspond to an adaptive feedforward ANC system.

S(z)

+

d(n)

S(z) −

e(n) y(n) W(z) x(n)

LMS x’(n)



Filter adaptation with the FxLMS algorithm The adaptive filter W(z) can be adapted using the FxLMS algorithm, employing

another instance of S(z) to compensate for the secondary path.

x(n) = d(n) =

= e(n) + ∑m=0

M−1

s^m(n)y(n−m)

y(n) = ∑l=0

L−1

wl(n)x(n−l)

wl(n+1) = wl(n) + µ x’(n−l)e(n)

x’(n) = ∑m=0

M−1

s^m(n)x(n−m)

+

d(n)

S(z) −

e(n) y(n)

S(z)

+

+ d(n)

W(z) x(n)

S(z)

LMS x’(n)



Example: active control of road noise in cars (Elliott and Sutton, 1996) A comparison is made between adaptive feedforward ANC and adaptive feedback ANC, developed on the basis of the IMC method.

Performance variations with respect to the system’s delay are studied.

This delay is due to: � the physical propagation time from loudspeaker to microphone

� the processing time of the digital controller

� the delays through the anti-aliasing and reconstruction filters

The feedforward control system operates with reference signals derived from six accelerometers.

The controller used six FIR filters with 128 coefficients operating at a sample rate of 1 kHz.

A single FIR filter having 128 coefficients is used to implement the W(z) filter in the feedback controller at a sample rate of 1 kHz.



The following figures report: � the spectrum of the measured pressure inside a small car driven at

60 km/h over a rough road surface (solid curve)

� the residual spectrum predicted after using a feedforward/feedback control system assuming the plant response to be a pure delay (1 ms delay = dashed line, 5 ms delay dotted line).

The attenuations achieved are relatively insensitive to delays in this range.

The main factor limiting the performance of the feedforward controller is the fact that the multiple coherence between the reference signals and the disturbance signal is less than unity.

The feedback control system does not use any reference signals (and therefore is not limited by low coherence problems) and the disturbance is being cancelled by a filtered version of the disturbance itself.



Feedforward control system



Feedback control system



Remarks: � The predictability of the disturbance over a time scale equal to the delay of

the plant is expected to affect the feedback control system performance.

� In fact, the performance is significantly degraded by the larger delay. � For the 1 ms delay, the spectrum of the residual error is flat, indicating that

it is almost white.

� With the 5 ms delay, the residual error has a more colored spectrum, but the sharp peaks in the original disturbance, which correspond to more predictable components in the primary pressure signal, have been largely attenuated.

� The overall attenuation falls to less than 1 dB for a 5 ms delay.

� For very short plant delays, the feedback controller can achieve a higher attenuation at the error microphone than the feedforward controller, although the error microphone would have to be so close to the loudspeaker to achieve these small delays that the acoustic effect of control would not be very widespread.



� In any case, the variation of attenuation with delay is much dependent on the statistical properties of the disturbance. � If the disturbance were a pure tone, for example, both feedforward and feedback control

systems could, in principle, give infinite attenuations at a single microphone, regardless of the plant delay.

� If, on the other hand, the disturbance were white noise, the feedback system would be unable to achieve any attenuation if there were a finite delay in the plant.

� The residual error below ~20 Hz after control in the feedback case is somewhat higher than before. This increase is due to amplification of the disturbance by the control system and may overload the loudspeaker used as the secondary source.



Waveform synthesis using a feedback-generated reference

Noise source

Primary noise


Error microphone

e(n)

duct

PLL

y(n) S(z)

+ +

d(n)

W(z)

Synchronization pulse

The regenerated reference signal x(n) = d(n) is used to synthesize a low frequency component locked at the fundamental driving frequency of the primary noise source, which is then used as an input to the waveform synthesizer W(z).



Multiple-channel adaptive feedback ANC: K×1 system (1 error sensor)

+

d(n)

S1(z) − e(n) y1(n)

x(n) K×1

Feedback ANC SK(z)

... −

Reference signal

estimator

yK(n)



e(n) = d(n) − ∑k=1

K

sk(n)*y k(n)

where � sk(n), k = 1, 2, …, K, are the impulse responses of the secondary paths Sk(z)

� yk(n), k = 1, 2, …, K, are the secondary signals of the adaptive filters Wk(z)

x(n) = d(n) = e(n) + ∑k=1

K

s^k(n)*yk(n)

The FxLMS algorithm is used to minimize the error signal e(n) by adjusting the weight vector for each adaptive filter Wk(z):

wk(n+1) = wk(n) + µ xk’(n)e(n)

where

� xk’(n) = s^k(n)*x(n)



Example of a 2×1 adaptive feedback ANC system

+

LMS

LMS

S1(z)

S2(z)

W2(z)

W1(z) y1(n)

y2(n)

x1’(n)

x2’(n)

x(n)

S1(z)

S2(z)

e(n) e(n)

+

+



Multiple-channel adaptive feedback ANC: K×M system (multiple error sensors) If a single error signal does not capture all the desired spatial and frequency components, the general K×M scheme can be adopted (M error sensors and K secondary sources).

K×M adaptive filters

Reference signal

estimator

FxLMS

K y(n)

yK(n)

y1(n)

eM(n)

e1(n)

... ...

Noise source

Enclosure

e(n)

M

M

x(n)



This system includes: � M×K secondary paths Smk(z) from the kth secondary source to the mth error sensor

� K×M adaptive filters Wkm(z)

The synthesized reference signals are expressed as:

xm(n) = em(n) + ∑k=1

K

s^mk(n)*yk(n), m = 1, 2, …, M

where

� s^mk(n) is the impulse response of the secondary path estimate Smk(z)

� yk(n) is the kth secondary signal, computed as ∑m=1

M

wkm(n)*xm(n)

� wkm(n) is the impulse response of the adaptive filter Wkm(z)

The filter weights are adapted using the (J×K×M) FxLMS algorithm, with J = M.



Example of a 2×2 multiple-channel adaptive feedback ANC scheme

+

S11(z)

e1(n)

e2(n)

y1(n)

FxLMS

S21(z)

S22(z)

S12(z)

y2(n)

W11(z)

W21(z)

e1(n) x1(n)

x2(n)

+

+

+

+

+ +

+

+ e2(n) +

W12(z)

W22(z)



Hybrid ANC systems

Noise source

Primary noise


Error microphone

y(n) e(n)

duct

+ +

Reference microphone

x(n) Feedback

ANC Feedforward

ANC

� The feedforward ANC system uses two sensors, the reference sensor to measure the

primary disturbance, and the error sensor to monitor the ANC system performance.

� The adaptive feedback ANC uses only the error sensor and cancels only the predictable part of the primary noise

� The feedforward ANC attenuates the primary noise correlated with x(n), while the feedback ANC cancels the narrowband components of the primary noise not observed by the reference sensor.



Hybrid ANC with FIR feedforward ANC

+

LMS S(z)

P(z)

x’(n)

A(z) y(n) S(z)

x(n)

LMS S(z)

C(z)

d(n) d’ (n)

S(z)

e(n)

e(n)

e(n)

e(n)

+ +

−

+

+

W(z)

d(n)



The combined controller has two reference inputs: � x(n) from the reference sensor

� d(n), the estimated primary signal

Filtered versions of these signals are used to adapt the coefficients of two FIR filters (A(z) and C(z)).

Then, the secondary signal is generated as:

y(n) = a(n)Tx(n) + c(n)Td(n)

where

� a(n) = [ a0(n) a1(n) … aL−1(n) ]T is the weight vector of A(z) at time n

� x(n) = [ x(n) x(n−1) … x(n−L+1) ]T

� c(n) = [ c0(n) c1(n) … cL−1(n) ]T is the weight vector of C(z) at time n

� d(n) = [ d(n) d(n−1) … d(n−L+1) ]T



The weights are updated with the FxLMS algorithm:

� a(n+1) = a(n) + µ x’ (n)e(n)

� c(n+1) = c(n) + µ d’ (n)e(n)

where x’ (n) and d’ (n) are the filtered reference signal vectors:

� x’ (n) = s^(n)*x(n)

� d’ (n) = s^(n)* d(n)

Simulation results show that the hybrid ANC system can achieve: � 15 dB additional attenuation with respect to the purely feedforward FxLMS scheme

� 3 dB additional attenuation with respect to the feedback ANC algorithm

A hybrid scheme can also be realized where the feedforward part uses IIR filters adapted with the Filtered-u Recursive LMS algorithm.



Comparison of hybrid and non-hybrid ANC structures Feedforward

ANC Adaptive feedback ANC

Hybrid ANC (FIR)

Hybrid ANC (IIR)

Filter order moderate high low low Spectral capability

broadband and narrowband

narrowband only



Plant noise not canceled good cancellation

good cancellation

good cancellation

Noise field coherence

coherent only coherent or incoherent

coherent or incoherent

coherent or incoherent

Hybrid ANC allows to use a lower-order filter to achieve the same performance.

Feedback ANC cancels only the predictable part of the primary noise, whereas hybrid ANC achieves also broadband noise cancellation.

Feedforward ANC works well if the primary signal is highly correlated with the reference signal: this condition may not be met if the reference sensor picks up plant noise or if the primary noise and the reference sensors are not coherent.

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08 - Feedback ANC

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