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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.
REFERENCES
[1] T.-B. Airimitoaie, A. Castellanos Silva, and I. D. Landau. Indirect
adaptive regulation strategy for the rejection of time varying narrow-
band disturbances applied to a benchmark problem. European Journal
of Control, 19(4), 2013.
[2] S. Aranovskiy and L. B. Freidovich. Adaptive compensation of
disturbance formed as sums of sinusoidal signals with application to
an active vibration control testbench. European Journal of Control,
19(4), 2013.
[3] Karl Johan Astrom and B. Wittenmark. Computer Controlled Systems,
Theory and Design. Prentice-Hall, Englewood Cliffs, N.J., 1984.
[4] A. Castellanos-Silva. Feedback adaptive compensation for active
vibration control in the presence of plant parameter uncertainties.
Ph.D. Thesis, Grenoble Institute of Technology, Grenoble, France,
2014.
[5] A. Castellanos Silva, I. D. Landau, and T.-B. Airimitoaie. Direct
adaptive rejection of unknown time-varying narrow band disturbances
applied to a benchmark problem. European Journal of Control, 19(4),
2013.
[6] X. Chen and M. Tomizuka. Adaptive model inversion for time-varying
vibration rejection on an active suspension benchmark. European
Journal of Control, 19(4), 2013.
[7] R.A. de Callafon and H. Fang. Adaptive regulation via weigthted
robust estimation and automatic controller tuning. European Journal
of Control, 19(4), 2013.
[8] B.A. Francis and W.M. Wonham. The internal model principle of
control theory. Automatica, 12(5):457 – 465, 1976.
[9] A. Karimi and Z. Emedji. H-inf gain-scheduled controller design
for rejection of time-varying narrow-band disturbances applied to a
benchmark problem. European Journal of Control, 19(4), 2013.
[10] I. D. Landau, A. Castellanos Silva, T.-B. Airimitoaie, G. Buche,
and N. Mathieu. Benchmark on adaptive regulation - rejection of
unknown/time-varying multiple narrow band disturbances. European
Journal of Control, 19(4), 2013.
[11] I. D. Landau, R. Lozano, M. M’Saad, and A. Karimi. Adaptive control.
Springer, London, 2nd edition, 2011.
[12] I.D. Landau, A. Constantinescu, and D. Rey. Adaptive narrow band
disturbance rejection applied to an active suspension - an internal
model principle approach. Automatica, 41(4):563–574, 2005.
[13] I.D. Landau and G. Zito. Digital Control Systems - Design, Identifi-
cation and Implementation. Springer, London, 2005.
[14] Z. Wu and F. Ben Amara. Youla parametrized adaptive regulation
against sinusoidal exogenous inputs applied to a benchmark problem.
European Journal of Control, 19(4), 2013.
