Comparison of two approaches for adaptive feedforward compensation

in active vibration control with mechanical coupling

Ioan Dore Landau, Tudor-Bogdan Airimitoaie and Marouane Alma

Abstract— Adaptive feedforward broadband vibration (ornoise) compensation is currently used when a correlated mea-surement with the disturbance (i.e. an image of the distrubance)is available. However, in most of the systems there is a”positive” feedback coupling between the compensator systemand the measurement of the image of the disturbance. Thismay lead to the instability of the system. In [9] a stableadaption algorithm for IIR compensators operating in thecontext of internal coupling has been proposed and evaluatedexperimentally. In [11] a stable adaptation algorithm for afeedforward compensator having a Youla Kucera structure hasbeen proposed. The objective of this paper is to comparativelyevaluate the two approaches from theoretical and experimentalpoints of view. An active vibration control system featuring aninternal mechanical coupling will serve as a test bench.

Index Terms— active vibration control, adaptive feedforwardcompensation, adaptive control, Youla-Kucera parametrization,parameter estimation

I. INTRODUCTION

Adaptive feedforward broadband vibration (or noise) com-

pensation is currently used in ANC (Active Noise Control)

and AVC (Active Vibration Control) when an image of

the disturbance (i.e. a correlated measurement) is available

([3], [8], [4], [7], [13]). However in many systems there

is a ”positive” feedback coupling between the compensator

system and the measurement of the image of the disturbance

(vibration or noise) ([7], [13], [6]). The positive feedback

may destabilize the system. The system is no more a pure

feedforward compensator. One of the solutions to overcome

this problem ([8]) is to try to compensate the positive

feedback ([8], [5]). However since the compensation can not

be perfect, the potential instability of the system still exists

([12], [2]).

Another approach discussed in the literature is the analysis

in this new context of existing algorithms for adaptive

feedforward compensation developed for the case without

feedback. An attempt is made in [12] where the asymptotic

convergence in a stochastic environment of the so called

”Filtered-U LMS” (FULMS) algorithm is discussed. Further

results on the same direction can be found in [5].

A stability approach for developing appropriate adaptive

algorithms in the context of internal positive feedback is

discussed in [7]. Unfortunately the results are obtained in

the context of very particular assumptions upon the system.

In [9] a stable adaptation algorithm for adapting an IIR

feedforward compensator with matrix adaptation gain has

Ioan Dore Landau, Tudor-Bogdan Airimitoaie andMarouane Alma are with the Control system depart-ment of GIPSA-Lab, St Martin d’Heres, 38402 FRANCEIoan-Dore.Landau@gipsa-lab.grenoble-inp.fr.

been proposed, analyzed and tested experimentally on an

active vibration control system featuring an internal ”pos-

itive” feedback coupling. A different path is considered

in [11]. One starts with a feedforward compensator which

stabilizes the internal ”positive” feedback loop. This will

constitute the central filter for a Youla-Kucera parametrized

compensator. The Youla-Kucera parameters (Q-parameters)

will be adapted for optimizing the feedforward compensator

in order to minimize the residual acceleration (in active

vibration control) or the residual noise (in active noise

control). This approach has been inspired by [13].

The objective of this paper is to evaluate comparatively the

stable algorithms proposed in [9] and [11] from theoretical

and experimental points of view. An active vibration control

system featuring an internal mechanical coupling will serve

as a test bench.

The paper is organized as follows. In Section II the AVC

system which will be used as a test bench will be presented.

The system representation and feedforward compensator

structures are given in section III. The algorithms for adaptive

feedforward compensation will be presented in section IV.

A comparative theoretical analysis is presented in section V.

Section VI will present experimental results obtained on the

active vibration control system with the two approaches.

II. AN ACTIVE VIBRATION CONTROL SYSTEM USING AN

INERTIAL ACTUATOR AS A TEST BENCH

Figures 1 and 2 represent an AVC system using a cor-

related measurement with the disturbance and an inertial

actuator for reducing the residual acceleration. The structure

is representative for a number of situations encountered in

practice.

The system consists of three mobile metallic plates (M1,

M2, M3) connected by springs. The first and the third plates

are also connected by springs to the rigid part of the system

formed by two other metallic plates connected themselves

rigidly. The upper and lower mobile plates (M1 and M3)

are equipped with inertial actuators. The one on the top

serves as disturbance generator (inertial actuator 1 in figure

2), the one at the bottom serves for disturbance compensation

(inertial actuator 2 in figure 2). The system is equipped

with a measure of the residual acceleration (on plate M3)

and a measure of the image of the disturbance made by an

accelerometer posed on plate M1.

When the compensator system is active, the actuator acts

upon the residual acceleration, but also upon the measure-

ment of the image of the disturbance (a positive feedback).

The measured quantity u(t) will be the sum of the correlated

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Fig. 1. An AVC system using a feedforward compensation - photo

disturbance measurement d(t) obtained in the absence of the

feedforward compensation (see figure 3(a)) and of the effect

of the actuator used for compensation.

The corresponding block diagrams in open loop operation

and with the compensator system are shown in Figures

3(a) and 3(b), respectively. In figure 3(b), u(t) denotes the

effective output provided by the measurement device and

which will serve as input to the adaptive feedforward filter

N. The output of this filter denoted by y(t) is applied to the

actuator through an amplifier. The transfer function G (the

secondary path) characterizes the dynamics from the output

of the filter N to the residual acceleration measurement (am-

plifier + actuator + dynamics of the mechanical system). The

transfer function D between d(t) and the measurement of the

residual acceleration (in open loop operation) characterizes

the primary path.

The coupling between the output of the filter and the mea-

surement u(t) through the compensator actuator is denoted

by M. As indicated in figure 3(b) this coupling is a ”positive”

feedback. This unwanted coupling raises problems in practice

(source of instabilities) and makes the analysis of adaptive

(estimation) algorithms more difficult.

The objective is to develop stable recursive algorithms for

online estimation and adaptation of the parameters of the

feedforward filter compensator (which will be denoted N)

such that the measured residual error (acceleration or force

in AVC, noise in ANC) be minimized in the sense of a certain

criterion. This has to be done for broadband disturbances d(t)(or s(t)) with unknown and variable spectral characteristics

M1

M2

M3

SupportMeasurement of the image

of the disturbance u(t)Inertial actuator I

(Disturbance source)

Inertial actuator II

(Compensator)

Positive coupling

(Reverse path) M(z-1)

Secondary path G(z-1)

Residual acceleration X (t)

Primary path D(z-1)

Global primary path :

W(z-1).D(z-1)

Support

N

Parametric Adaptation Algorithm

Power Amplifier

Feedforward compensator

^

^

M1

M2

M3

SupportMeasurement of the image

of the disturbance u(t)Inertial actuator I

(Disturbance source)

Inertial actuator II

(Compensator)

Positive coupling

(Reverse path) M(z-1)

Secondary path G(z-1)

Residual acceleration X (t)

Primary path D(z-1)

Global primary path :

W(z-1).D(z-1)

Support

N

Parametric Adaptation Algorithm

Power Amplifier

Feedforward compensator

^

^

Fig. 2. An AVC system using a feedforward compensation - scheme

W Ds(t)

d(t)

d(t) x(t)

Global primary path

Measurement of the

image of the disturbance

Residual

acceleration

measurement

Primary path

χ (t)

(a)

W D

M

G+

+

+

+

s(t)

u(t)

ν (t)

d(t)

z (t)

x(t)

Global primary path

Positive feedback coupling

Measurement of the

image of the disturbance

Secondary path

Residual

acceleration

measurement

Feedforward filter

PAA

Primary path

Parameter adaptation algorithm

y (t)0

χ (t)

-1

N 0

0

0

(b)

Fig. 3. Feedforward AVC: in open loop (a) and with adaptive feedforwardcompensator (b)

and an unknown primary path model1.

III. BASIC EQUATIONS AND NOTATIONS

The different blocks of the AVC system (Figure 3(b)) will

be described in this section.

The primary path is characterized by the asymptotically

stable transfer operator:

D(q−1) =BD(q

−1)

AD(q−1)(1)

where

BD(q−1) = bD

1 q−1 + ...+bDnBD

q−nBD (2)

AD(q−1) = 1+aD

1 q−1 + ...+aDnAD

q−nAD (3)

The unmeasurable value of the output of the primary path

(when the compensation is active) is denoted x(t).The secondary path is characterized by the asymptotically

stable transfer operator

G(q−1) =BG(q

−1)

AG(q−1)(4)

where

BG(q−1) = bG

1 q−1 + ...+bGnBG

q−nBG (5)

AG(q−1) = 1+aG

1 q−1 + ...+aGnAG

q−nAG (6)

The positive feedback coupling is characterized by the

asymptotically stable transfer operator

M(q−1) =BM(q−1)

AM(q−1)(7)

1Variations of the unknown model W , the transfer function betweenthe disturbance s(t) and d(t), are equivalent to variations of the spectralcharacteristics of s(t).

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where

BM(q−1) = bM1 q−1 + ...+bM

nBMq−nBM = q−1B∗

M(q−1) (8)

AM(q−1) = 1+aM1 q−1 + ...+aM

nAMq−nAM (9)

The optimal IIR (infinite impulse response) feedforward

compensator (unknown) is defined by

N(q−1) =R(q−1)

S(q−1)(10)

where

R(q−1) = r0 + r1q−1 + ...+ rnRq−nR (11)

S(q−1) = 1+ s1q−1 + ...+ snSq−nS = 1+q−1S∗(q−1) (12)

Using the Youla-Kucera parametrisation (Q-

parametrisation), the filter polynomials R(q−1) and S(q−1)of the optimal feedforward compensator (subsequently

denoted YK feedforward compensator) get the form

R(q−1) = R0(q−1)−AM(q−1)Q(q−1) (13)

S(q−1) = S0(q−1)−BM(q−1)Q(q−1) (14)

where AM(q−1) and BM(q−1) denote respectively the de-

nominator and numerator of the positive feedback coupling

path. S0(q−1) and R0(q

−1) are the polynomials of the central

(stabilizing) filter.

In [11], the optimal polynomial Q(q−1) is considered to

be of the form

Q(q−1) = q0 +q1q−1 + ...+qnQq−nQ (15)

The estimated optimal feedforward filter is denoted by

N(q−1) or N(θ ,q−1) when it is a linear filter with constant

coefficients or N(t,q−1) during estimation (adaptation) of its

parameters.

For the estimated IIR feedforward compensator, the esti-

mated R and S polynomials are denoted by R(q−1), S(q−1)or R(θ ,q−1), S(θ ,q−1). For the estimated YK feedforward

compensator, the estimated Q polynomial is denoted by

Q(q−1) or Q(θ ,q−1) when it is a linear filter with constant

coefficients or Q(t,q−1) during estimation (adaptation).The input of the feedforward filter is denoted by u(t) and

it corresponds to the measurement provided by the primarytransducer (force or acceleration transducer in AVC or amicrophone in ANC). In the absence of the compensationloop (open loop operation) u(t) = d(t). The ”a posteriori”output of the feedforward filter (which is the control signalapplied to the secondary path) is denoted by y(t + 1) =y(t +1|θ(t +1)). The ”a priori” output of the estimated IIRfeedforward filter is given by:

y0(t +1) = y(t +1|θ(t))

= −S∗(t,q−1)y(t)+ R(t,q−1)u(t +1)

= θ T (t)φ(t) = [θ TS (t), θ T

R (t)]

[φy(t)φu(t)

](16)

where

θ T (t) = [s1(t)...snS(t), r0(t)...rnR

(t)] = [θ TS (t), θ T

R (t)] (17)

φ T (t) = [−y(t)...− y(t −nS +1, u(t +1), u(t)...u(t −nR +1)]

= [φ Ty (t),φ T

u (t)] (18)

and y(t), y(t − 1) ... are the ”a posteriori” outputs of the

feedforward filter generated by:

y(t +1) = y(t +1|θ(t +1)) = θ T (t +1)φ(t) (19)

while u(t +1), u(t)... are the measurements provided by the

primary transducer2.

For the YK adaptive feedforward filter, the ”a priori” output

of the feedforward compensator is given by:

y0(t +1) = y(t +1|θ(t))

= −S∗0y(t)+R0u(t +1)

+Q(t,q−1)[B∗M y(t)−AM u(t +1)]

= −S∗0y(t)+R0u(t +1)+ θ T (t)φ(t) (20)

where

θ T = [q0, q1, q2, . . . , qnQ] (21)

φ T (t) = [α(t +1),α(t), . . . ,α(t −nQ +1)] (22)

α(t +1) = BM y(t +1)−AM u(t +1) (23)

The unmeasurable ”a priori” output of the secondary path

will be denoted z0(t +1).

z0(t +1) = z(t +1|θ(t)) =B∗

G(q−1)

AG(q−1)y(t) (24)

The ”a posteriori” unmeasurable value of the output of the

secondary path is denoted by:

z(t +1) = z(t +1|θ(t +1)) (25)

The measured primary signal (called also reference) satisfies

the following equation:

u(t +1) = d(t +1)+B∗

M(q−1)

AM(q−1)y(t) (26)

The measured residual error satisfies the following equation:

χ0(t +1) = χ(t +1|θ(t)) = x(t +1)+ z0(t +1) (27)

The ”a priori” adaptation error used in the algorithms will

have the form

ν0(t +1) =−χ0(t +1) =−x(t +1)− z0(t +1) (28)

while the ”a posteriori” adaptation error will be given by

ν(t +1) = ν(t +1|θ(t +1)) =−x(t +1)− z(t +1) (29)

When using an estimated filter N with constant parameters:

y0(t) = y(t), z0(t) = z(t), χ0(t) = χ(t) and ν0(t) = ν(t).

2u(t +1) is available before adaptation of parameters starts at t +1

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IV. THE ALGORITHMS

The algorithms for adaptive feedforward compensation

presented in [9] and [11] have been developed under the

following hypotheses:

1) The signal d(t) is bounded (which is equivalently to

say that s(t) is bounded and W (q−1) in figure 3 is

asymptotically stable).

2) The primary path model D(z−1) is unknown and con-

stant.

3) The effect of the measurement noise upon the mea-

surement of the residual acceleration is neglected (de-

terministic context).

Note that the resulting algorithms are afterwards analyzed in

the stochastic context (presence of the measurement noise)

The specific additional hypotheses for the IIR compensator

are:

• (Perfect matching condition for IIR compensator). There

exists a filter N(q−1) of finite dimension such that3:

N

(1−NM)G =−D (30)

and the characteristic polynomial of the ”internal” feed-

back loop:

P(z−1) = AM(z−1)S(z−1)−BM(z−1)R(z−1) (31)

is a Hurwitz polynomial.

The specific additional hypotheses for the Youla-Kucera

(YK) compensator are:

• It exists a central feedforward compensator N0(R0,S0)

which stabilizes the inner positive feedback loop formed

by N and M and the characteristic polynomial of the

closed loop:

P(z−1) = AM(z−1)S0(z−1)−BM(z−1)R0(z

−1) (32)

is a Hurwitz polynomial.

• (Perfect matching condition for YK compensator) It

exists a value of the Q parameters such that

G ·AM(R0 −AMQ)

AMS0 −BMR0=−D (33)

The parameter adaptation algorithm has the same structure

for the two adaptive compensators, however the signification

of the parameter vector and of the regressor vector are totally

different. The parameter adaptation algorithm has the form:

θ(t +1) = θ(t)+F(t)Φ(t)ν(t +1) ; (34)

ν(t +1) =ν0(t +1)

1+ΦT (t)F(t)Φ(t); (35)

F(t +1) =1

λ1(t)

F(t)−

F(t)Φ(t)ΦT (t)F(t)λ1(t)λ2(t)

+ΦT (t)F(t)Φ(t)

(36)

1 ≥ λ1(t)> 0;0 ≤ λ2(t)< 2;F(0)> 0 (37)

Φ(t) = φ f (t) (38)

3In many cases, the argument q−1 or z−1 will be dropped.

φ f (t) = L(q−1)φ(t) (39)

where λ1(t) and λ2(t) allow to obtain various profiles for

the matrix adaptation gain F(t) (see section VI and [10]).

By taking λ2(t) ≡ 0 and λ1(t) ≡ 1 one gets a constant

adaptation gain matrix (and choosing F = γI, γ > 0 one gets

a scalar adaptation gain). L(q−1) = BLAL, is an asymptotically

stable filter. ν0(t + 1) is the measured residual error (given

in eq. 28).

For the IIR adaptive feedforward compensator θ(t) and

φ(t) are given by:

θ T = [s1...snS, r0...rnR

] = [θ TS , θ

TR ] (40)

φ T (t) = [−y(t)...− y(t −nS +1, u(t +1), u(t)...u(t −nR +1)]

= [φ Ty (t),φ T

u (t)] (41)

For the YK adaptive feedforward compensator θ(t) and φ(t)are given by:

θ T = [q0, q1, q2, . . . , qnQ] (42)

φ T (t) = [α(t +1),α(t), . . . ,α(t −nQ +1)] (43)

α(t +1) = B∗M y(t)−AM u(t +1) (44)

Three choices for the filter L will be considered, leading

to three different algorithms.

Algtorithm I L = G

Algorithm II L = G

Algorithm III

L =AM

PG (45)

where

P = AM S− BMR (46)

V. COMPARATIVE ANALYSIS OF THE ALGORITHMS

A. The deterministic case - perfect matching

For both adaptive feedforward filters, the equation for the

a posteriori adaptation error has the form:

ν(t +1) = H(q−1)[θ − θ(t +1)]T Φ(t) (47)

where:

H(q−1) =AM(q−1)G(q−1)

P(q−1)L(q−1),Φ = φ f (48)

One has the following result valid for both schemes:

Lemma 5.1: Assuming that eq. (47) represents the evolu-

tion of the a posteriori adaption error and that the parameter

adaption algorithm (34) through (38) is used one has:

limt→∞

ν(t +1) = 0 (49)

limt→∞

[ν0(t +1)2]

1+Φ(t)T F(t)Φ(t)= 0 (50)

||Φ(t)|| is bounded (51)

limt→∞

ν0(t +1) = 0 (52)

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for any initial conditions θ(0),ν(0) provided that:

H ′(z−1) = H(z−1)−λ2

2,max

t[λ2(t)]≤ λ2 < 2 (53)

is a strictly positive real transfer function.

Proof of this result can be found in [9] for the IIR adaptive

feedforward filter and in [11] for the YK adaptive feedfor-

ward filter.

It is interesting to remark that for Algorithm III, the pos-

itive real condition always holds provided that the estimates

of AM , P and G are close to the true values.

B. The stochastic case - perfect matching

The effect of the measurement noise on the residual ac-

celeration measurement cannot be neglected. The algorithms

for IIR and YK adaptive feedforward have been analyzed

in this context in [9] and [11] respectively. Under the

same positive real condition (53), the parameter estimations

will be unbiased (provided that the measurement noise is

independent of d(t)).Lets focus now on the differences between the IIR adaptive

compensator and the YK adaptive compensator.

Remark 1: When using the IIR adaptive compensator, pro-

vided that the strictly positive real condition is satisfied, the

poles of the internal ”positive” loop will be asymptotically

stable but they can be very close to the unit circle.

Remark 2: The central YK controller allows to assign the

desired closed loop poles. The first consequence is that one

can impose a minimum damping on the closed loop poles.

Remark 3: Since the system as well as the initial char-

acteristics of the disturbance can be identified, a model

based design of an initial feedforward compensator can be

done. Unfortunately this information can not be easily used

to efficiently initialize the parameters of the IIR adaptive

compensator. There are two reasons for this: a) The model

based design (like H∞) will lead in general to a controller

with a larger number of parameters than the number of

parameters used in the adaptive IIR filter, b) Initialization

of the parameters will require to operate from the beginning

with very low adaptation gain and this will reduce the

adaptation capabilities of the algorithm.

Remark 4: Using an YK adaptive compensator it is easier to

satisfy the positive real condition since we can assign P (for

example close to AM).

VI. EXPERIMENTAL RESULTS - COMPARISON OF THE

TWO APPROACHES

The active vibration control system has been described in

section II.

A. System identification

The structure of the linear discrete time models for the

different paths have been given in section III. These models

have been identified and details can be found in [9], [11].

The frequency characteristics of the identified models are

given figure 4.

0 50 100 150 200 250 300 350 400−80

−60

−40

−20

0

20

40

Frequency (Hz)

Am

plitu

de (

dB

)

Primary, Secondary and Reverse Paths Models

Secondary path

Reverse path

Primary path

Fig. 4. Frequency characteristics of the primary, secondary and reversepaths (identified models)

B. Broadband disturbance rejection

For the YK adaptive feedforward compensator two difer-

ent central compensators have been used. One central com-

pensator has been designed using a pole placement method

adapted for the case of positive feedback systems (denoted

subsequently PP). The end result was a controller of orders

nR0= 14 and nS0

= 16. The other central compensator (H∞) is

a reduced order model based H∞ compensator with nR0= 19

and nS0= 20 designed and tested in [1]4.

An adaptation algorithm with decreasing matrix adapta-

tion gain (λ1(t) = 1, λ2(t) = 1) combined with a constant

trace adaptation gain has been used in all the experiments.

Time domain results obtained in open loop and with the

IIR adaptive feedforward compensation (algorithm III) on

the AVC system are shown in figure 5. The IIR adaptive

compensator has 20 parameters. Figure 6 shows the power

0 50 100 150 200−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Plant output using broadband disturbance and Algo. III after 50 sec

Time [sec]

Resid

ual a

ccele

ratio

n [V

]

Fig. 5. Real time results obtained with IIR adaptive compensator (Algo-rithm III )using matrix adaptation gain (20 parameters)

spectral densities of the residual acceleration measured on

the AVC in open loop (without compensator) and using the

IIR adaptive feedforward compensator for the case of 20

parameters.

Time domain results obtained with the YK adaptive

feedforward compensators on the active vibration control

system when using nQ = 31 (32 parameters) are shown in

7. The upper part of figure 7 shows the results obtained

when using H∞ central compensator. The lower part of 7

shows the results obtained when using the pole placement

PP central compensator. The adaptation transient is similar

to that obtained for IIR adaptive feedforward compensator.

The spectral density of the residual acceleration in open loop

and, when using the H∞ central compensator and with the

4The initial H∞ controller had (nRH∞= 70 and nSH∞

= 70)

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0 50 100 150 200 250 300 350 400−80

−70

−60

−50

−40

−30

−20

−10

Frequency [Hz]

PS

D e

stim

ate

[d

B]

Power Spectral Density Estimates

Open loopAdaptive feedforward (Algo. II): −15.689dBAdaptive feedforward (Algo. III): −16.2312dB

Fig. 6. Power spectral densities of the residual acceleration in open loopand with IIR adaptive feedforward compensation (experimental)

adapted H∞ Q-parametrized feedforward compensator are

shown in figure 8 for nQ = 19 (20 parameters) and nQ =31 (32 parameters). The spectral densities of the residual

acceleration in open loop, with the pole placement (PP)

central controller and with the adapted pole placement Q-

parametrized controller are shown in figure 9. The influence

of the number of adjustable parameters upon the performance

of the system is summarized in Table I (algorithm III).

VII. CONCLUSIONS

The paper has presented a comparison of two stable adap-

tive feedforward compensation schemes applicable to active

vibration control systems with mechanical coupling (IIR and

YK parametrized compenstors). Comparative experimental

results indicate close performances of the two approaches.

However from a practical point of view the YK adaptive

feedforward compensator seems more interesting in terms

of initialization, assignment of the closed loop poles and

satisfaction of the positive real condition for stability and

convergence.

The results are certainly applicable to active noise control

systems with acoustic coupling.

0 50 100 150 200−0.6−0.4−0.2

00.20.40.6

Time [sec]

Re

sid

ua

l a

cce

lera

tio

n [

V]

Plant output using broadband disturbance and Q32 (PP) param. after 50 seconds

0 50 100 150 200−0.6−0.4−0.2

00.20.40.6

Plant output using broadband disturbance and Q32 (H∞

) param. after 50 seconds

Fig. 7. Real time results obtained with YK adaptive compensator(Algorithm III )using matrix adaptation gain

REFERENCES

[1] M. Alma, J.J. Martinez, I.D. Landau, and G. Buche. Design andtuning of reduced order h-infinity feedforward compensators for activevibration control. IEEE Transactions on Control Systems Technology,to be printed.

[2] M.R. Bai and H.H.Lin. Comparison of active noise control structuresin the presence of acoustical feedback by using the hinf synthesistechnique. J. of Sound and Vibration, 206:453–471, 1997.

0 50 100 150 200 250 300 350 400−80

−70

−60

−50

−40

−30

−20

−10

Frequency [Hz]

PS

D E

stim

ate

[d

B]

Power Spectral Density Estimate "Disturbance = PRBS"

Open loop

H∞

: −14.7026dB

Q(H∞

) 20 pars: −16.7633dB

Q(H∞

) 32 pars: −16.5215dB

Fig. 8. Experimental spectral densities of the residual accelerations (H∞)

0 50 100 150 200 250 300 350 400−80

−70

−60

−50

−40

−30

−20

−10

Frequency [Hz]

PS

D E

stim

ate

[d

B]

Power Spectral Density Estimate "Disturbance = PRBS"

Open loop

PP: −4.6136dB

Q(PP) 20 pars: −12.9291dB

Q(PP) 32 pars: −15.1631dB

Fig. 9. Experimental spectral densities of the residual accelerations (PP)

[3] S.J. Elliott and P.A. Nelson. Active noise control. Noise / News

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[11] I.D. Landau, T.B. Airimitoaie, and M. Alma. A youla-kuceraparametrized adaptive feedforward compensator for active vibrationcontrol. Proceedings of the IFAC WC 2011, Milano, Italy, 2011.

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Number of parameters 20 32 40

Global attenuation-IIR (db) 16.2316.4916.89

Global attenuation-YK/H∞ (db)16.7616.5216.03

Global attenuation-YK/PP (db)12.9215.1615.55

TABLE I

INFLUENCE OF THE NUMBER OF PARAMETERS UPON THE GLOBAL

ATTENUATION

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