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Interlaced Direct Adaptive Regulation Scheme Applied to a

Benchmark Problem

Ioan Dor´

e Landau∗,Abraham Castellanos Silva∗,

Luc Dugard∗, Xu Chen†.

Abstract—An adaptive regulation scheme using the Inter-

nal Model Principle and an IIR Youla-Kuˇ

cera controller

parametrization is proposed for the suppression of multiple

unknown and time varying narrow band disturbances. This

development has been motivated on one hand by the need

of simplifying the design of the central controller for direct

adaptive regulation schemes using the Internal Model Principle

and FIR Youla-Kuˇ

cera controller parametrization and on the

other hand by the high computation load in indirect adaptive

regulation schemes based on the shaping of the output sensitivity

functions using band stop ﬁlters. Real time results and compar-

ison with previous approaches used for the EJC International

benchmark on adaptive regulation[10] will be provided.

Index Terms—Adaptive Regulation, Active Vibration Control,

Inertial Actuators, Multiple Narrow Band Disturbances, Youla-

Kuˇ

cera Parametrization, Internal Model Principle

I. INTRODUCTION

Adaptive rejection of unknown and time-varying multiple

narrow band disturbances is an important challenge with

applications in AVC (Active Vibration Control) and ANC

(Active Noise Control).

In [10] the results of an international benchmark on

adaptive regulation of an AVC problem were presented. There

were a number of contributions [1], [2], [6], [5], [14], [9]

and [7] which have been evaluated experimentally. Among

the best results considering performance, robustness and

complexity stand the ones obtained by [1], [6] and [5].

In [1] band stop ﬁlters (BSF) are used to shape the sensitiv-

ity functions using band stop ﬁlters. Although this approach

shows good results and a good robustness, the drawback is

the signiﬁcant computer load related to the solution of a

Bezout equation at each sampling instant. In [6], an efﬁcient

adaptive regulation scheme has been proposed using the ap-

proximate inverse of the plant model and an estimation of the

disturbance model using a representation of the disturbance

model with the help of polynomials with mirror coefﬁcients.

In [5], one uses the Internal Model principle combined with

an adaptive YK-FIR controller parametrization. While this

direct adaptive regulation scheme will require on one hand a

very low computer load on the other hand it requires a careful

design of the central controller (problem dependent). The

problem comes from the fact that the Internal Model Principle

does too much by assuring asymptotically total rejection of

the disturbance while in practice attenuation of narrow band

disturbances by 40 to 60 dB is largely enough.

The novel approach proposed in this paper is based on the

use of an IIR Youla Kucera parametrization. The numerator

of the IIR Youla Kucera ﬁlter will introduce the internal

model of the disturbance and the denominator of the IIR

Youla Kucera ﬁlter will assign in real time additional poles

to the closed loop which will allow to reduce the water

bed effect and to improve robustness. This poles will be

deﬁned by a polynomial Dp(ρz−1)with 0 <ρ<1 where

Dp(z−1)is the denominator of the model of the narrow band

disturbance . This will drastically simplify the design of the

central controller and the additional computation load related

to the estimation in real time of the Dp(z−1)is low. This

approach will be comparatively evaluated with repsect to the

best results obtained previously within the EJC Benchmark

on adaptive regulation [10]

II. THE EXPERIMENTAL SETTING

A photo of the active vibration control experimental set up

used in this study is presented in Fig. 1 along with the basic

actions performed by the system. A detailed description can

be found in [10].

Passive

damper

Inertial

actuator

(partly

visible)

Mechanical

structure

Shaker

(disturbance

generator)

Force

measurement

Control

action

Disturbance

action

Fig. 1. Active vibration control using an inertial actuator (photo).

The system consists of a passive damper, an inertial

actuator, a mechanical structure, a transducer for the residual

force, a controller, a power ampliﬁer and a shaker. The

system input, u(t)is the position of the mobile part (magnet)

of the inertial actuator, the output y(t)is the residual force

measured by a force sensor. The transfer function between

the disturbance force (δ(t)) and the residual force (y(t)) is

called primary path.

The plant transfer function (G=q−dB

A) between the input

of the inertial actuator (u(t)) and the residual force is called

secondary path. The parametric model of the secondary path

can be straightforwardly obtained by system identiﬁcation

techniques. The sampling frequency is fs=800 Hz.

The system itself in the absence of the disturbances fea-

tures a number of low damped vibration modes as well as

low damped complex zeros (anti-resonance). This makes the

design of the controller difﬁcult for rejecting disturbances

close to the location of low damped complex zeros (very

low system gain). The parametric model of the secondary

path has a signiﬁcant order, nA=22 and nB=25.

The frequency range of operation is between 50 and 95

Hz. See [10] for more details on benchmark speciﬁcations

and measurement procedures.

III. PLA NT A ND C ON TRO LL ER D ES CR IP TI ON

Consider the adaptive regulation scheme depicted in Fig. 2

where the IIR YK-parametrized controller is shown. The

linear case with known disturbances characteristics will be

considered subsequently in order to clarify the plant and

controller structure (the adaptive loop is dropped out) and

the control objectives.

The structure of the identiﬁed linear time-invariant

discrete-time model of the plant (the secondary path) used

for controller design is:

G(z−1) = z−dB(z−1)

A(z−1)=z−d−1B∗(z−1)

A(z−1),(1)

with dis equal to the plant integer time delay (number of

sampling periods),

A(z−1) = 1+a1z−1+· · · +anAz−nA; (2)

B(z−1) = b1z−1+· · · +bnBz−nB=z−1B∗(z−1); (3)

B∗(z−1) = b1+· · · +bnBz−nB+1,(4)

where A(z−1),B(z−1),B∗(z−1)are polynomials in the com-

plex variable z−1and nA,nBand nB−1 represent their

orders1. Details on system identiﬁcation of the models con-

sidered in this paper can be found in [13].

Adaptive Part

Central Controller

Fixed Part

Fig. 2. Direct adaptive scheme using an IIR YK-parametrization of the

controller. Dashed box: ﬁxed part, Point-dash box: adaptive part.

1The complex variable z−1will be used for characterizing the system’s

behavior in the frequency domain and the delay operator q−1will be used

for describing the system’s behavior in the time domain.

In this scheme, the central controller is described through

R0(z−1)and S0(z−1), which are polynomials in z−1having

the orders nR0and nS0, respectively, with the following

expressions:

R0=r0

0+r0

1z−1+. . . +r0

nR0z−nR0=R0

0(z−1)·HR0(z−1);

(5)

S0=1+s0

1z−1+. . . +s0

nS0z−nS0=S0

0(z−1)·HS0(z−1),(6)

where HR0and HS0are pre-speciﬁed parts of the controller

(used for example to incorporate the internal model of a

disturbance or to open the loop at certain frequencies).

R0(z−1)and S0(z−1)are minimal degree solutions of

P

0(z−1) = A(z−1)S0(z−1) + z−dB(z−1)R0(z−1),(7)

where P

0(z−1)deﬁnes the nominal closed loop poles related

to the central controller.

With an IIR Youla Kucera parametrization, the controller

polynomials are deﬁned as follows2:

R=AQR0+HS0HR0ABQ(8)

S=AQS0−HS0HR0z−dBBQ(9)

where the optimal Q-ﬁlter has the following structure:

Q(z−1) = BQ(z−1)

AQ(z−1)=

bQ

0+bQ

1z−1+· · · +bQ

nBQz−nBQ

1+aQ

1z1+· · · +aQ

nAQz−nAQ

.(10)

One deﬁne the output sensitivity function:

Syp (z−1) = A(z−1)S(z−1)

P(z−1); (11)

where

P=AS +z−dBR =AQP

0

=AQAS0+z−dBR0(12)

deﬁnes the poles of the closed loop (roots of P(z−1)).

One can write the output of the system as:

y(t) = A(q−1)S(q−1)

P(q−1)·p(t) = Syp(q−1)·p(t).(13)

IV. DISTURBANCE DESCRIPTION

In the present approach one considers an optimal attenu-

ation of the disturbance taking into account both the zeros

and poles of the disturbance model. It is assumed that the

model of the disturbance is a notch ﬁlter and the disturbance

is represented by:

p(t) = Dp(ρz−1)

Dp(z−1)e(t)(14)

where e(t)is a centered white Gaussian noise sequence and

Dp(z−1) = 1+αz−1+z−2,(15)

2The arguments (z−1)and (q−1)will be omitted in some of the following

equations to make them more compact.

is a polynomial with roots on the unit circle.3In (15), α=

−2cos (2πω1Ts),ω1is the frequency of the disturbance in

Hz, and Tsis the sampling time. Dp(ρz−1)is given by:

Dp(ρz−1) = 1+ραz−1+ρ2z−2,(16)

with 0 <ρ<1.

This model is pertinent for representing narrow band

disturbances.

Using the output sensitivity function for the case of the

IIR Youla-Kucera parametrization, the output of the plant in

the presence of the disturbance can be expressed as (using

the factorization S=S0HS)

y(t) = AS0

P

0

β(t)(17)

where

β(t) = HS

AQ

Dp(ρq−1)

Dp(q−1)e(t)(18)

In order to minimize the effect of the disturbance upon y(t),

one should minimize the variance of β(t). One has two tuning

devices HSand AQ. Minimization of the variance of β(t)is

equivalent of searching HSand AQsuch that β(t)becomes a

white noise [3]. The obvious choices are HS=Dp(which

corresponds to the IMP) and AQ=Dp(ρq−1). Of course

this development can be generalized for the case of multiple

narrow band disturbances.

V. INTERN AL MO DE L PRINCIPLE WITH YK IIR

PARAMETRIZATION

Consider the case when the frequencies of the disturbance

are known, i.e. Dp(z−1)is known and a given central con-

troller R0(z−1)and S0(z−1)is already computed

In order to asymptotically reject the effect of p(t)over y(t),

the polynomial S=S0HSshould incorporate the denominator

Dp(z−1)(Internal Model Principle - [8]), as is shown next:

S(z−1) = S0(z−1)·HS(z−1)

=S0(z−1)·HS0(z−1)·Dp(z−1).(19)

Looking at the eq. (9), is possible to deﬁne a diophantine

equation allowing to compute the optimal Q-IIR ﬁlter which

introduces the model of the disturbance into the controller.

The diophantine equation is

S0Dp+HR0z−dBBQ=AQS0

0,(20)

where the common term HS0(z−1)has been eliminated. Here

Dp(z−1),HR0(z−1),d,B(z−1)and S0

0(z−1)are known, and

BQ(z−1)and S0(z−1)are unknown. In order that eq. (20) be

solvable, AQ(z−1)should be deﬁned. Suppose temporarily

that AQ(z−1)is known and asymptotically stable (a.s.), since

this polynomial will deﬁne additional poles for the closed-

loop (see eq. (12)). Then, eq. (20) has a unique and minimal

degree solution for S0(z−1)and BQ(z−1)with nAQ+nS0

0−1≤

nDp+nHR0+nB+d−1, nS0=nB+d+nHR0−1 and nBQ=

nDp−1.

3Its structure in a mirror symmetric form guarantees that the roots are

always on the unit circle.

In eq. (20), the computed numerator BQ(z−1)intro-

duces zeros in the polynomial S(z−1), through the YK-

parametrization. This allows the rejection of the narrow-band

disturbance.

AQwill be chosen as:

AQ(z−1) = Dp(ρz−1) = 1+ραz−1+ρ2z−2,(21)

where α=−2cos (2πf Ts)and using a constant ρ,0<ρ<1.

Dp(z−1)has its roots over the unit circle (see eq. 15) but as

a consequence of the change of z−1to ρz−1, the roots of

AQ(z−1)are located in the same radial line but inside of

the unit circle, and therefore it is asymptotically stable. In

this approach the constant ρis deﬁned as a function of the

desired attenuation. This is also a parameter for tuning the

robustness, since it has inﬂuence over the water bed effect

occurring on Syp (z−1).

In Fig. 3 the magnitude of the frequency responses of the

output sensitivity function with a single central controller

but for different structures of the YK ﬁlter used for distur-

bance compensation are shown. The ﬁrst case corresponds

to the use of an YK-FIR parametrization (as used in [5])

for implementing the model of the disturbance and it is

depicted using a dotted line. The ampliﬁcations outside of the

frequency of the disturbance are important and could lead to

insufﬁcient robustness (the computed modulus margin - ∆M

- is 0.0961 corresponding to an ampliﬁcation of 20.6 dB)4.

The second case, represented with a dashed line, corresponds

to the use of a BSF ﬁlter approach (as in [1]) for computing

the optimal BQ(z−1)and AQ(z−1). The BSF was computed

using the disturbance frequency, a desired attenuation of -

60 dB and a denominator damping of 0.09 (the ∆Mis

0.4318 corresponding to an ampliﬁcation of 7.3 dB). The

third case, represented with a solid line corresponds to a

YK-FIR parametrization using to ρ-notch type ﬁlter structure

with AQgiven in (21). A constant ρ=0.97 was used for

this case (the numerator structure corresponds to the YK-

FIR case considered earlier). The computed ∆Mis 0.4527

corresponding to a maximum ampliﬁcation of 6.9 dB.

Clearly the ρ-notch type structure for AQcan achieve a

strong reduction of the water bed effect. Only an estimation

of αand a given constant ρare required for directly

implementing the YK-IIR ﬁlter. This type of structure has

been chosen subsequently for the denominator AQ(z−1)in

order to develop an interlaced direct adaptive scheme.

VI. PARAMETER ADAPTATI ON ALGORITHMS

Using the ρtype YK-IIR ﬁlters it is necessary to estimate

ﬁrst the parameters of Dp(z−1). Then one estimates the

parameters of ˆ

BQ(z−1).

A. Estimation of Dp(q−1)

Assume that the signal p(t)contains nnarrow-band com-

ponents. p(t)will then satisfy

n

∏

i=11−2cos(ωi)z−1+z−2p(t) = 0,(22)

4However adding a number of auxiliary resonant poles conveniently

located the water bed effect can be strongly reduced in order ot get a good

modulus margin [5]

0 20 40 60 80 100

−50

−40

−30

−20

−10

0

10

20

Frequency [Hz]

Magnitude [dB]

Output Sensitivity Function for different structures

FIR ∆M=0.0961

BSF ∆M=0.4318

ρ−notch ∆M=0.4527

Fig. 3. Zoom of the frequency response of the output sensitivity function

for different YK-ﬁlters. FIR case: dotted line, BSF case: dashed line and

ρ-notch case: solid line.

where ωi(i=1,...,n)is the frequency of the ith narrow-band

component in p(t). Eq (22) can be equivalently written:

Dp(z−1)p(t+1) = 0.(23)

The disturbance model can be expressed by:

p(t+1) = −

n−1

∑

i=1

αi[p(t+1−i) + p(t+1−2n+i)] − · · ·

· · · − αnp(t+1−n)−p(t+1−2n) = θT

DpφDp(t).

(24)

where the parameter vector is:

θDp= [α1,α2,...,αn]T.(25)

and regressor vector at the time tis:

φDp(t) = φ1,Dp(t),φ2,Dp(t),φn,Dp(t)T,(26)

where

φj,Dp(t) = −p(t+1−j)−p(t+1−2n+j),j=1,...,n−1

(27)

φn,Dp(t) = −p(t+1−n).(28)

Eq. (24) can then be simply represented by

p(t+1) = θT

DpφDp(t)−p(t+1−2n).(29)

One deﬁnes the a priori prediction of p(t+1):

ˆp0(t+1) = ˆ

θT

Dp(t)φDp(t)−p(t+1−2n),(30)

where ˆ

θDp(t)is the predicted parameter vector at time t.

The a priori prediction error is given by

e0(t+1) = p(t+1)−ˆp0(t+1) = −˜

θT

Dp(t)φDp(t),(31)

where ˜

θDp(t) = ˆ

θDp(t)−θDpis the parameter estimation

error.

The following a posteriori signals are deﬁned:

•the a posteriori prediction of p(t+1):

ˆp(t+1) = ˆ

θT

Dp(t+1)φDp(t)−p(t+1−2n),(32)

•the a posteriori prediction error:

e(t+1) = p(t+1)−ˆp(t+1) = −˜

θT

Dp(t+1)φDp(t).

(33)

Equation (33) has the standard form of an a posteriori

adaptation error which allows to associate the standard pa-

rameter adaptation algorithm (PAA) introduced in [11]

ˆ

θDp(t+1) = ˆ

θDp(t) + F2(t)φDp(t)e0(t+1)

1+φDp(t)TF2(t)φDp(t)(34)

e0(t+1) = p(t+1)−ˆp0(t+1)(35)

ˆp0(t+1) = ˆ

θT

Dp(t)φDp(t) + p(t+1−2n)(36)

F2(t+1)−1=λ1(t)F2(t)−1−λ2(t)φDp(t)φDp(t)T(37)

0<λ1(t)≤1; 0 ≤λ2(t)<2; F2(0)>0

B. Estimation of BQ(z−1)

Consider eqs. (13) and (9). From Fig. 2, the signal w(t+1)

is deﬁned as follows

w(t+1) = A(q−1)y(t+1)−B∗(q−1)u(t−d)

=A(q−1)p(t+1),(38)

then, the output of the closed-loop system can be expressed

as follows

y(t) = ˆ

AQS0−HS0HR0q−dBˆ

BQ

ˆ

AQP

0

w(t).(39)

Following the principles given in [12], it is possible

to develop a direct adaptive algorithm for estimating ˆ

BQ

provided that ˆ

AQis available. Using eq. (39), the a posteriori

error is deﬁned as

ε(t+1) = v1(t+1) + ·· ·

· · · BQ−ˆ

BQ(t+1)wf(t+1)·· ·

−A∗

Q−ˆ

A∗

Q(t+1)ˆuf

Q(t)−A∗

Qε(t)(40)

where

wf(t+1) = HS0HR0q−dB

P

0

w(t+1)(41)

ˆuf

Q(t) = HS0HR0q−dB

P

0

ˆuQ(t)(42)

v1(t+1) = S0HS0ANp

AQP

0

δ(t+1)(43)

(see also Fig. 2). The signal v1(t+1)tends asymptotically

towards zero (an asymptotically stable system excited by a

Dirac pulse) and can be neglected.

The equation for the a posteriori error takes the form

ε(t+1) = 1

AQθT

1−ˆ

θT

1(t+1)φ1(t+1) + ·· ·

· · · +vf

1(t+1) + v2(t+1),(44)

TABLE I

COMPARISON OF ALGORITHMS FOR THE ADAPTATION OF THE

NU MER ATOR PAR AM ETE RS BQ(z−1)

Adaptation Prediction Regressor Positive Observations

error error vector Real Cond.

v(t+1)ε(t+1)Φ1(t)H0(z−1)

ε(t+1)Eq. (44) φ1(t)1

AQ−λ2

2-

ˆ

AQε(t+1)Eq. (44) φ1(t)ˆ

AQ

AQ−λ2

2-

ε(t+1)Eq. (44) φf

1(t)ˆ

AQ

AQ−λ2

2-

ε(t+1)Eq. (44) φf

1(t)˜

AQ(t)

AQ−λ2

2

Local

Convergence

where

vf

1(t+1) = 1

AQ

v1(t+1)→0,since AQis a.s.(45)

v2(t+1) = 1

AQA∗

Q−ˆ

A∗

Q(t+1)−ˆuf

Q(t)→0,(46)

θ1=hbQ

0,· · · ,bQ

2n−1iT

(47)

ˆ

θ1(t+1) = hˆ

bQ

0(t+1),· · · ,ˆ

bQ

2n−1(t+1)iT

(48)

φ1(t+1) = wf(t+1),·· · ,wf(t+2−2n)T(49)

where nis the number of narrow-band disturbances. The

signal v2(t+1)goes to zero asymptotically since it can be

shown that ˆ

A∗

Q(t)→AQ. Eq. (44) has the standard form of

an adaptation error equation [11], and the following PAA is

proposed:

ˆ

θ1(t+1) = ˆ

θ1(t) + F1(t)Φ1(t)ν(t+1)(50)

ν(t+1) = ν0(t+1)

1+ΦT

1(t)F1(t)Φ1(t)(51)

w1(t+1) = S0

P

0

w(t+1)(52)

F1(t+1)−1=λ1(t)F1(t)−1−λ2(t)Φ1(t)ΦT

1(t)(53)

0<λ1(t)≤1; 0 ≤λ2(t)<2; F1(0)>0 (54)

For the case ν(t+1) = ε(t+1)one has ν0(t+1) = ε0(t+

1)where the a priori prediction error is given by

ε0(t+1) = w1(t+1)−ˆ

θT

1(t)Φ1(t)(55)

For the case where ν(t+1) = AQε(t+1)one has:

ν0(t+1) = ε0(t+1) +

nAQ

∑

i=1

ˆaiε(t+1−i)(56)

Since in the equation of the a posteriori error (44) there is a

term 1/AQ, according to [11] there will be a sufﬁcient posi-

tive real condition to be satisﬁed. There are several possible

choices for the regressor vector Φ1(t)and the ﬁltering of

the adaptation error in order to satisfy this condition. Table I

gives the various options and the corresponding sufﬁcient

positive real condition. The various options have the objective

of relaxing the basic positive real condition 1/AQ. It is the

last option which has been used for the experiments which

will be presented next. A stability analysis can be found in

[4].

TABLE II

REA L-TI ME R ESU LTS FO R THE YK-IIR ALGORIHTM - S IMPLE STE P

TES T.

Level Case (Hz) GA (dB) DA (dB) MA (dB@Hz) TD %

50 34.5 40.3 9.3@62.5 92.2

55 33.1 45.4 8.2@50.0 100

60 33.3 45.6 6.8@125.0 100

1 65 31.8 45.4 9.1@56.3 100

GA≥30 70 29.9 45.6 8.1@131.3 100

DA≥40 75 30.3 47.9 8.6@70.3 100

MA≤6 80 29.5 48.6 7.7@6.3 100

85 29.5 43.6 6.3@117.2 100

90 29.1 43.7 7.5@117.2 100

95 27.1 39.0 6.8@375.0 100

50-70 38.2 40.9-43.9 10.3@64.1 100

2 55-75 35.9 46.1-47.2 11.9@60.9 100

GA≥30 60-80 37.8 45.6-45.9 7.9@70.3 100

DA≥40 65-85 35.2 42.9-42.9 7.9@212.5 100

MA≤7 70-90 36.1 43.7-44.9 10.0@115.6 100

75-95 35.0 44.9-40.0 9.9@128.1 100

3 50-65-80 40.1 38.3-39.7-43.7 8.9@125.0 100

GA≥30 55-70-85 40.1 45.2-45.1-42.7 7.8@78.1 100

DA≥40 60-75-90 38.7 45.2-42.2-43.3 10.8@78.1 100

MA≤9 65-80-95 38.8 43.9-41.7-40.5 10.2@85.9 80.9

VII. APP LI CATI ON TO THE EJC BENCHMARK

The Pole Placement with sensitivity function shaping [13]

is used to calculate the central controller. All the stable poles

of the system are included in P

0(z−1). Also 6 real poles are

added for robustness reasons. Four band stop ﬁlters have been

introduced in HR0(z−1)in order to shape Sup(z−1)outside

the operation zone. The loop is opened at 0 fsand 0.5fs.

HS0(z−1) = 1

A value of ρ=0.97 has been used for all the levels and

all the test. This value provides a good compromise between

performance and robustness. The value is not very critical.

A. Real-time results

Table II summarizes the real-time results for the Simple

Step Test. The performance objective are shown in column

1. DA is the disturbance attenuation, GA is the global

attenuation, MA is the maximum ampliﬁcation. TD indicates

the percentage of fulﬁlment of the transient duration (2s).

B. Performance Comparison

The results which have been presented has to be evaluated

comparatively with the the most relevant schemes presented

for the EJC benchmark [10]. This comparison will be done

on a global basis using the procedure presented in [10]. The

results presented above will be compared with those of [1],

[6] and [5].

The following two global evaluation criteria are considered

for comparison

•Benchmark Satisfaction Index (BSI) for steady state

performance, known also as Tuning capabilities. This

criterion uses the results from the Simple Step Test

in order to show how ”good” is the performance of

a speciﬁed scheme, by measuring the fulﬁlment of the

speciﬁcations (column Level in Table II) and assigning

a percentage.

•Complexity evaluation is done in terms of measurement

of the Task Execution Time5. The value of the criterion

is obtained from the average task execution time (TET)

measured in the xPC-Target environment from MAT-

LAB. Low values correspond to less complexity of the

control scheme.

In Fig. 4 the comparison of the BSI for the steady state

performance is presented for the four approaches mentioned.

As shown, the adaptive scheme proposed in this paper

(named YK-IIR) achieves the highest performance in real-

time for the ﬁrst level (BSI1-RT), a very good performance

for the second level (BSI2-RT) and the second best (only

behind [5]) for the third level (BSI3-RT).

100,00

98,69

93,30

99,07

86,63

81,11

80,87

89,37

100,00

98,38

97,29

99,84

86,65

88,51

89,56

87,38

99,78

99,44

99,13

100,00

92,52

90,64

97,56

96,39

0,00 20,00 40,00 60,00 80,00 100,00

YK-IIR

%

Benchmark Satisfaction Index For

Steady State Performance (Tuning)

BSI1-Sim

BSI1-RT

BSI2-Sim

BSI2-RT

BSI3-Sim

BSI3-RT

[5]

[1]

[6]

BSI1-Sim

BSI1-RT

BSI2-Sim

BSI2-RT

BSI3-RT

BSI3-Sim

BSI1-Sim

BSI1-RT

BSI2-Sim

BSI2-RT

BSI3-RT

BSI3-Sim

BSI1-Sim

BSI1-RT

BSI2-Sim

BSI2-RT

BSI3-RT

BSI3-Sim

Fig. 4. Benchmark Satisfaction Index (BSI) comparison for four approaches

in the three levels. RT = Real Time, Sim = Simulation.

The complexity, the YK-IIR approach is higher compared

to the one obtained with the YK-FIR approach [5] (which is

the lowest). However the complexity of the YK-IIR approach

is comparable with that [6] and still signiﬁcantly smaller than

the complexity of [1]

5In fact the difference between the task execution time in closed loop and

in open loop is considered in the criterion.

VIII. CONCLUDING REMARKS

The results on this paper suggest that with an adaptive

IIR Youla-Kuˇ

cera scheme one can drastically simplify the

design of the central controller with respect to the adaptive

FIR Youla-Kuˇ

cera for the same level of performance and

with an acceptable increase of the complexity in terms of

task execution time.

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