# Active Noise Cancellation for Ventilation Ducts Using a Pair of Loudspeakers by Sampled-Dataℋ∞ Optimization

Authors: Yasuhide Kobayashi; Hisaya Fujioka
Journal: Advances in Acoustics and Vibration
 (2008)

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## Abstract

We deal with the design of active noise control systems for ventilation ducts with a pair of loud-speakers based on sampled-dataℋ∞ optimization, in order to improve our previous study with a single loudspeaker. The resultant controller requires lower computational complexity for the implementation,  compared to the standard time-varying adaptive controllers, since it is time-invariant and the order is comparable. Moreover, the effect of Swinbanks' source is automatically recovered by the ℋ∞ optimization. The benefit of the proposed method is also confirmed from experimental results for a real house, where the achieved noise attenuation level is about twice as much as that of the single loudspeaker case.


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## Body

## 1. Introduction

Active noise cancellation (ANC) is to attenuate noise
by the same sound but opposite phase. This technique has been practically used
in air conditioning systems in large-scale buildings, aircraft cabins, and so
on [1]. However, there
are many fields such as ventilation ducts for small buildings where ANC is desired
but has not been applied yet because the conventional adaptive algorithms for
ANC require expensive implementation.In the previous study [2], by assuming that
temperature variation is small as in recent energy-efficient houses so
that time-invariant controller can achieve sufficient performance, two types of
time-invariant low-order controllers are compared: one is based on the
conventional adaptive filter (the filtered-U recursive LMS algorithm) but the
coefficients are fixed by the stationary values. Another is obtained by a
robust control method (sampled-data ℋ∞ control). It has been shown that robust
control design has advantage to implement inexpensive ANC system for
ventilation ducts, because it gives a systematical way to design a controller
which is robust against not only plant dynamics variation but also aliasing
components of noise. However, further improvement of the sound attenuation is
desired.The method originally proposed by Swinbanks [3] is well known as an
effective one for the improvement of the system performance, where an
additional loudspeaker is attached to cancel out the upstream sound generated
by a control source [4]. The method has been examined in detail under adaptive
control setup [5].
However, the effect of Swinbanks' source under robust control setup has not
been studied. Moreover, experimental results applied to actual ventilation
systems installed in houses have not been reported.In this paper, we examine robust control design of
active noise control systems with a pair of loudspeakers in order to improve
the system performance. By regarding the loudspeakers as two independent
sources, a single-input-multiple-output (SIMO) controller is also designed to
be compared with  Swinbanks' source. The validity of robust control design
is shown by experimental results using a ventilation duct installed in a real
house.It would be
worth mentioning that there have been many studies that deal with the design
problem for duct active noise control systems not only in the adaptive control
framework [1, 6] but also in the robust
control framework [7–10]
under the simple duct setup, while the comparison of the adaptive and robust
control with a real ventilation system has been firstly reported in  [2]. Furthermore, robust control
system with Swinbanks' source has not been studied yet.


## 2. Problem Setup

Figure1 and Table 1 show the block diagram and
instalments of  the experimental apparatus which are the same in [2] except that SPK3 and the
corresponding D/A channel are attached so that Swinbanks' source is
composed of the pair of loudspeakers, SPK2 and SPK3.Table 1
Experimental instruments.
Ventilation fanKaneka corp. SV-200U (250m3/h,
energy-recovery ventilation)Loudspeaker (SPK1)FOSTEX FW208N woofer speaker with wooden box
enclosureLoudspeaker (SPK2 & SPK3)FOSTEX FW108N woofer speaker with PVC pipe
enclosureMicrophonesElectlet condenser typeSound-level meterRION NL-20Power amplifierTOSHIBA TA8213KHigh-pass filterNF ELECTRONIC INSTRUMENTS FV-664 (2 ch, 80 Hz,
24 dB/oct)Low-pass filter500 Hz 4th-order ButterworthPCDell Dimension 2200 (RT-Linux 3.2, kernel
2.4.22)A/D, D/ACONTEC AD12-16(PCI), DA12-4(PCI) (12 bit,±5 V, 10μs)Figure 1
Experimental apparatus.In addition, for simplicity of robust control design
in this paper, SPK1 is used as a noise source in the modeling stage of robust
control design to examine frequency response of the plant model, while in
[2] actual fan noise
is used.Figure2 shows the configuration of the ventilation
system installed in a two-stored real house which is also the same as in
[2]. The grilles are
attached to the ceiling of each floor, and the ANC system is connected between
fresh-air grilles and the ventilation fan.Ventilation system configuration.
(a)
1st floor(b)
2nd floorIn this paper, we examine the following cases to derive
the control sources SPK2 and SPK3.Case (a)—a single loudspeaker—by settingv(t)=0,
only SPK2 is used to generate control sound.Case (b)—swinbanks' source [5]—by setting(1)v(t)=−u(t−τ),τ=dc0,SPK3 is driven to cancel out the
upstream sound generated by SPK2, where d is the distance between SPK2 and SPK3, and c0 is the sound speed.Case (c)—an array of two loudspeakers—by settingv(t) free to u(t), SPK2 and SPK3 are driven as independent sources.Our primal
motivation is to improve the performance by using a pair of loudspeakers (cases (b) and (c)) instead of a single loudspeaker (case (a)) in the robust control
framework. In Swinbanks' source, the upstream loudspeaker is driven by
a downstream one in an ad  hoc way by (1). Hence our second motivation is to
further improve the performance by giving more freedom to the upstream
loudspeaker in case (c), although it has been reported in adaptive control
framework that the performance is not improved if we replace the delay in case (b) by an adaptive filter to enlarge the design freedom [5].In the experiments for case (b) below, (1) is
approximately implemented as a real-time module of RTLinux that updates the
signal v(t) at every 0.1 millisecond which is considered to be short enough to
avoid aliasing effect. In addition, by letting d=0.34 m from Figure 1 and c0=344 m/s from normal temperature environment, we use τ=1 millisecond which exactly corresponds to 10 times of
the period of the real-time module mentioned above. Moreover, the cutoff
frequency of HPF is determined by considering the frequency range of Swinbanks' source given as [f0,5f0], where f0:=d/12c0 [5]. The sampling period of controller is 1 millisecond throughout this paper.


## 3. Robust Control Design

The design procedure for case (a) is the same as in
[10]. The detail of the
design procedure for case (c) is omitted but it is done by simply replacing the
signal [uv]T to u to apply the design process of case (a).


## 3.1. Modeling

The plant models for Figure1 are determined by frequency
response experiment. The system from [wu]T to [zy]T is considered as the plant transfer function G(s) for cases (a) and (b) as
(2)G(s):=[Gzw(s)Gzu(s)Gyw(s)Gyu(s)],
while for case (c) G(s) is defined by(3)G(s):=[Gzw(s)Gzu(s)Gzv(s)Gyw(s)Gyu(s)Gyv(s)],where Gab(s) means the transfer function from the signal b to the signal a.Figure3 shows the frequency response of G(s) and corresponding nominal plant obtained by
subspace-based method where the order is 85. In the figures for Gzu(s) and Gyu(s),
two frequency response results are shown in blue and yellow curves where the former corresponds to cases (a) and (c), while the latter corresponds to case (b). On the other hand, the frequency responses for Gzw(s) and Gyw(s) shown in blue curves are commonly used to
determine nominal plant for all the
cases.Frequency response of plant.
(a)
Gzw(b)
Gzu(c)
Gzv(d)
Gyw(e)
Gyu(f)
GyvIn experimental results of case (a), the phase lag becomes larger in the order ofGyw, Gzu, Gyu,
and Gzw,
of which order coincides with that of the distance from corresponding
microphone to speaker. In case (b), remarkable change on Gyu(s) is observed compared with case (a): the gain
is smaller in the whole frequency range, and the phase lag becomes larger,
which can be considered as the result that the distance for sound traveling
from the control source to reference microphone becomes larger. This implies
that the separation of control input and measured output is improved so that
the better performance is expected [11]. Although such remarkable change is not observed in Gzu(s),
the gain is slightly larger in the middle frequency range of Swinbanks'
source, which is the nature of the source reported in [5].In addition, in order to guarantee the closed-loop
system stability against the modeling error of the nominal plant, additive
uncertainty model is introduced for feedback-path transfer function. That is,
for cases (a) and (b),Gyu(s) is considered as(4)Gyu(s)=G¯yu(s)+W(s)δ(s),where G¯yu(s) is the nominal plant for Gyu(s), δ(s) is normalized modeling error whose ℋ∞ norm is less than or equal to 1, and W(s) is a weighting function which is determined to
cover the modeling error as shown in Figure 4.
Similarly, for case (c) we define
(5)[Gyu(s)Gyv(s)]=[G¯yu(s)G¯yv(s)]+W(s)Δ(s),where Δ(s) is 1×2 normalized modeling error. Note that the common
weighting function is used for all cases as(6)W(s)=0.015(s2+2ζ1ω1s+ω12)ω22ω32ω12(s2+2ζ2ω2s+ω22)(s2+2ζ3ω3s+ω32),where ω1=200, ζ1=0.9, ω2=650, ζ2=0.7, ω3=2200, ζ3=0.6.Additive uncertainty and weight.
(a)(b)


## 3.2. Controller Design

According to the preparation above, sampled-dataℋ∞ control synthesis [12] is applied to the following
digital controller design problem: find a discrete-time controller Kd(z) which maximizes positive scalar α so that the following conditions hold: (i)
the closed-loop system of Figure5 is
internally stable;(ii)
there exists positive scalard such that ℒ2 induced norm of the closed-loop system is less
than 1, where S is the sampler with sampling period h=1 millisecond, H is the zeroth order hold, and Wp(s) is a bandpass filter given by(7)Wp(s)=(ss+ωp1)2(ωp2s+ωp2)2,ωp1=2π×80,ωp2=2π×400.Note that the closed-loop system
gain is robustly minimized by maximizing α to improve control performance to attenuate
fan noise.Figure 5
Robust performance problem with scalings.The design results are as follows. The maximalα=4.64 was achieved for d=1.07 for case (a), and the maximal α=5.87 was achieved for d=1.56 for case (b), which implies that the
closed-loop performance will be improved by Swinbanks' source. Furthermore, α was further improved by case (c) as α=6.10 for d=1.23,
because of the less conservative design. The order of Kd(z) is 93.


## 4. Comparison of Controllers

In this section, both adaptive and robust controllers
are examined, where each adaptive controller for cases (a) and (b) is
determined as a fixed 100th IIR filter by using the filtered-U RLMS method
(see [2] for design
process in detail), while the controller for case (c) is determined as a
single-input-double-output fixed 100th IIR filter by using the multiple-channel
filtered-U RLMS method [6].Figure6 shows adaptive and robust
controllers for cases (a) and (b). It can be seen that for the adaptive
controllers, the peak gains at about 60, 90, and 100 Hz become smaller when Swinbanks' source is used, while the similar phenomenon has been reported in
[5]. The effect of Swinbanks' source is also observed for the robust controllers shown as the flat
gain characteristic within the frequency range from 80 to 400 Hz, while for the
case with a single loudspeaker, relatively large peak at about 180 Hz is
appeared.Effect of Swinbanks' source.
(a)
Adaptive control(b)
Robust controlFigure7 shows adaptive and robust controllers for cases (b) and (c). It can be seen for the robust controller of case (c) that Swinbanks' source characteristic is automatically obtained, since (i) the gain
characteristics of both channels of the controller are similar and (ii) the
phase difference is around 180 deg. Note  that
this fact is firstly reported in this paper. Furthermore, by examining the result
in detail, advantages of robust control design with case (c) are observed.
Firstly, from the gain characteristic, the controller for case (c) has
relatively large peaks at about 120 and 170 Hz compared with case (b), which
suggests that the flat gain characteristic of  Swinbanks' source is not
essential for performance improvement for actual ventilation system. Secondary,
in the frequency range around 60∼400,
the gain from y to v is slightly smaller than from y to u,
which can be interpreted as the result of robust control design to compensate
the attenuation of sound due to propagation. In contrast, it might be difficult
to give such a reasonable interpretation for the adaptive controller of case (c).Effect of SIMO design.
(a)
Adaptive control(b)
Robust control


## 5. Control Experiments

In this section, robust controllers designed so far
are compared by control experiments.Figure8 shows time response of error microphone signal z,
where the first 12.5 seconds are without control and the following 12.5 seconds are
with control. The smaller sampling period (0.5 millisecond) is used for measurement to
observe intersample behavior within the sampling period of the controller. It
can be seen that cases (b) and (c) show better performance than case (a), while
in [5] it has been
reported by experimental result that the performances of cases (a) and (b) are
similar. This might be caused since directional microphones are used in [5] but not used in this paper.Time response ofz.
(a)(b)(c)Figure9 shows the FFT analysis result of Figure 8. It can
be seen that the amplitude of z is reduced within 80 to 400 Hz.FFT analysis result ofz.
(a)(b)(c)It should be noted that the main frequency component
of noise occurs around 100 Hz whose noise shape differs very much from the open-loop
frequency response ofGzw shown in Figure 3. Therefore, it is
expected that the system performance will be improved by setting the weight
function Wp(s) to consider the noise shape.Table2 shows sound pressure level measured below each
grille. It can be seen that the attenuation level of case (b) is about twice
the level of case (a), which shows the availability of robust control design
for Swinbanks' source. On the other hand, the advantage of case (c) could not
be shown (i.e., the attenuation level is similar to case (b)). It is not what we
expect from the design results with larger α. The reason of this phenomenon is currently
under study. One may pose a conjecture that it is from the modeling error. We
have considered the modeling error from the control input to the measured
output, however, other modeling errors are not reflected in the design.
Especially the error of the transfer function from v to z could greatly affect the experimental result
of case (c). Indeed, the gain characteristic in Figure 3 shows that there is a
relatively large difference between the experimental result and the nominal
plant compared with the other transfer functions. Hence we are working on the
improvement of the modeling for utilizing the potential advantage of the design
setup in case (c), by accounting the modeling error and/or by setting the
nominal plant order higher in control design. We could report the results in
the future.Table 2
Sound pressure level at each grille.
GrilleSound pressure level (LAeq,10s) [dB]Without controlWith controlCase (a)Case (b)Case (c)No. 134.233.0 (−1.2)32.5 (−1.7)32.1 (−2.1)No. 240.438.8 (−1.6)37.4 (−3.0)37.3 (−3.1)No. 331.930.5 (−1.4)28.9 (−3.0)29.2 (−2.7)No. 441.239.1 (−2.1)37.1 (−4.1)37.2 (−4.0)


## 6. Conclusions

In this paper, we have examined a robust control design
of active noise control systems with a pair of loudspeakers,
and the validity of the design method has been shown experimentally
by using a ventilation system installed in a real house. The
results are summarized as follows.(i)
Firstly, by using Swinbanks' source as a
pair of loudspeakers, it has been confirmed that the controller
designed by robust control method has flat gain characteristic
which has been reported in adaptive control literature as an advantage
of using Swinbanks' source.(ii)
Secondly, by considering the pair of loudspeakers as two
independent actuators, it has been shown that the controller designed
by robust control method is more acceptable than the one
designed by adaptive control method, since the similar effect
of Swinbanks' source was automatically obtained by the former design
but  not by the latter one.(iii)
Finally, by comparing the resultant noise attenuation levels of
robust controllers with both a single loudspeaker and a
pair of loudspeakers, the improvement has been shown by the latter
case (i.e., the attenuation level was up to 4 dB while it was up
to 2 dB for the former case).Therefore, we conclude together with the result of
[2] that the robust
control design is useful to implement inexpensive active
noise control systems
with a pair of loudspeakers for ventilation ducts.



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*Source: 253948-2008-04-20.xml*