VI International Telecommunications Symposium (ITS2006), September 3-6, 2006, Fortaleza-CE, Brazil
1
FxLMS Algorithm with Variable Step Size and
Variable Leakage Factor for Active Vibration Control
Walter A. Gontijo, Orlando J. Tobias, Rui Seara, and Eduardo M. O. Lopes
Abstract—This paper discusses a novel adaptive algorithm
termed variable step size and variable leakage factor filtered-x
least-mean-square (VSSVLFxLMS) algorithm for applications
in active vibration control (AVC). For such applications, the
leaky filtered-x LMS (LFxLMS) algorithm has been considered
for stability purposes. However, when the standard LFxLMS
algorithm (with fixed leakage) is used, the control performance
is strongly dependent on the selected leakage value. A similar
result is noticed concerning the chosen step-size value. Thus, by
using the proposed VSSVLFxLMS algorithm better step-size
and leakage values are obtained, improving considerably the
control performance. The conjectures of this work are
confirmed by implementing the referred algorithm in an
experimental setup of an AVC system.
Controller
Error
sensor
Primary
source
Secondary
source
Reference
sensor
Aluminium beam
Support
Support
50 mm
50 mm
100 mm
900 mm
Index Terms—FXLMS algorithm, leaky FXLMS algorithm,
active vibration control.
I. INTRODUCTION
In recent years, the area of active vibration control (AVC)
has experienced a considerable development coming from
two important facts. The increasing interest in using that
technique to develop smart structures and, secondly, the
great advance and wide diffusion of the digital signal
processor (DSP) boards. The latter has allowed
implementing more complex and better control algorithms.
In AVC applications (see Fig. 1), the filtered-x
least-mean-square (FxLMS) algorithm is extensively used
[1]-[3]. Such popularity is mainly due to its robustness and
low complexity [1], [2]. However, the direct implementation
of FxLMS algorithm is not always possible. For instance,
the finite-precision arithmetic of the DSPs or an insufficient
spectral content of the input signal can lead the adaptive
algorithm to instability [4], [5]. A widely used solution to
overcome such a problem is the use of the leaky FxLMS
algorithm. It results from introducing a penalizing term into
the standard cost function, giving rise to the leaky FxLMS
(LFxLMS) algorithm [4]-[6].
Fig. 1. Experimental setup of the AVC system.
When the LFxLMS algorithm is applied to an AVC
system, it is noticed that the canceling performance
achieved is strongly dependent on the leakage value used. In
[4] and [7], it is demonstrated that in the presence of
nonlinearities in the adaptation path, there is an optimum
value for the leakage factor to attain a maximum
cancellation. Such an optimum can be theoretically
determined if the system parameters (plant, nonlinearity
model, input signal power, among others) are known. In
practice, these parameters are obviously unknown; thus, the
best leakage factor value must be determined by a trial and
error procedure. In addition, since the weight update
equation is dependent on the product of the step-size
parameter and leakage factor, for a given leakage factor
value there is also a particular step-size value that leads to
the condition of maximum cancellation. Thus, to achieve an
optimal tradeoff between step size and leakage factor for
algorithm performance, the novel variable step size and
variable leakage factor FxLMS (VSSVLFxLMS) algorithm
is proposed in this paper. The advantages of this improved
version of the FxLMS algorithm are the following:
i) There is no need to know the system parameters.
Manuscript received March 31, 2006. This work was supported in part
by the Brazilian National Council for Scientific and Technological
Development (CNPq).
Walter A. Gontijo, Orlando J. Tobias, and Rui Seara are with the
LINSE - Circuits and Signal Processing Laboratory of the Department of
Electrical Engineering at the Federal University of Santa Catarina,
Florianópolis, Santa Catarina, Brazil (e-mails: [email protected],
[email protected]; [email protected]).
Eduardo M. O. Lopes is with PISA – Group for Integrated Research on
Vibrating and Acoustic Systems of the Department of Mechanical
Engineering at the Federal University of Santa Catarina, Florianópolis,
Santa Catarina, Brazil (e-mail: [email protected]).
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ii) Both step size and leakage factor are self-adjustable,
aiming to achieve the maximum vibration cancellation.
iii) Ability to accommodate time-varying conditions.
The proposed algorithm has been implemented in the
EZ-Kit LiteTM 21161, Analog Devices’ DSP platform, with
32 bits floating-point arithmetic. Section III shows some
numerical results of the proposed algorithm for performance
assessment.
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VI International Telecommunications Symposium (ITS2006), September 3-6, 2006, Fortaleza-CE, Brazil
II. VSSVLFXLMS ALGORITHM
In this section, the weight updating equation for the
proposed algorithm is derived. To this end, the block
diagram of the AVC experimental setup (Fig. 1), shown in
Fig. 2, is considered. In Fig. 2, variables d (n) and e(n)
denote the primary and error signals, respectively. Vector
x(n) represents the reference signal and yl (n) is the output
signal of the secondary source. The filter impulse response
cascaded with the adaptive filter, termed secondary path, is
given by s [ s0 s1 " sM 1 ]T . The estimate of the secondary
proposing an algorithm in which both the step-size and
leakage parameters are variable. In this way, (3) and (4)
must be modified as follows:
w (n 1)
x(n)
d(n)
wo
+
w(n)
y(n)
6
Q ( n ) w ( n ) P ( n )e( n ) x f ( n ) ,
Q ( n ) 1 P ( n) J ( n ) .
(7)
In next the section, the proposed procedure to adjust the
leakage factor will be derived.
B. Variable Leakage Factor
To obtain a variable leakage factor we use the stochastic
gradient rule. In this way, the leakage factor is adjusted in
accordance with the negative gradient of the quadratic error.
Thus,
J ( n)
yl (n)
s
(6)
and
path s, denoted by sˆ [ sˆ0 sˆ1 " sˆM 1 ]T , is obtained by using
a system identification procedure as, for instance, the one
suggested in [1].
2
J (n 1) U w e 2 ( n)
,
2 w J (n 1)
(8)
where U is a positive constant with 0 U 1 . The second
term on the r.h.s of (8), by considering (6), can be written as
s^
(n)
f
LMS
we2 (n)
wJ (n 1)
e(n)
we2 (n)
w w ( n)
Let us consider the stochastic gradient rule
ww (n)
wJ (n 1)
(1)
where J (n) denotes the cost function, given by [8]
J ( n)
e2 (n) JwT (n)w (n) ,
(2)
and J t 0 is the leakage factor.
Qw (n) Pe(n)xf (n) ,
(4)
w{[1 P(n) J (n 1)]w (n 1) P(n)e(n 1)xf (n 1)}
wJ (n 1)
P(n)w (n 1).
(11)
J ( n)
(3)
where
Q 1 PJ ,
(10)
Finally, from (9), and substituting (10) and (11) into (8),
the expression for adjusting the leakage factor is given by
Now, determining the gradient of (2) and substituting it
into (1), one obtains
w (n 1)
2e(n)xf (n) .
Now, the second term on the r.h.s. of (9) is obtained from
(6), as follows:
A. Weight Update and Leakage Adjust Equations
P
w ( n) ’J ( n ) ,
2
(9)
where
Fig. 2. Block diagram of the AVC system using the FxLMS
algorithm.
w (n 1)
T
ª we 2 (n) º w w (n)
,
«
»
¬« w w (n) ¼» wJ (n 1)
J (n 1) UP(n)e(n)w T (n 1)xf (n) .
(12)
C. Variable Step Size
The step-size value is adjusted by using a similar
approach to (8), resulting in the following expression
adapted from [9]:
and
P( n )
P(n 1) Ue(n)xTf (n)e(n 1)xf (n 1) ,
(13)
M 1
x f ( n)
¦ sˆi x(n i) .
(5)
i 0
Vector xf (n) represents the reference signal filtered by the
secondary path estimate.
In the different approaches of the LFxLMS algorithm
found in the open literature, the parameter J is considered
constant [2], [8]. On the other hand, in our case, we are
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where xf (n) is obtained from (5). Note that by monitoring
the error signal, when the adaptive filter weights have
converged (error signal approaches to zero) the adjustment
process, defined by (12) and (13), is stopped. In practice an
upper and lower limit, for (12) and (13), should be
specified. Such a procedure is to avoid that both parameters
do not become too small or too large, making the adaptive
algorithm very slow or even unstable, respectively.
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VI International Telecommunications Symposium (ITS2006), September 3-6, 2006, Fortaleza-CE, Brazil
1.0
Firstly, experimental results illustrating the behavior of
the leaky FxLMS algorithm (with fixed leakage factor) are
presented, considering several leakage values. The aim of
this experiment is to support the development of the
variable-leakage factor algorithm. The experimental results
are obtained by measuring the steady-state error value of the
LFxLMS algorithm as a function of the leakage factor value
(see Fig. 3). Note from this figure that the behavior of the
steady-state error is a convex function, evidencing the
existence of an optimum leakage value. The process to
obtain such an optimum is manual, being cumbersome and
time-consuming. Thus, the possibility of implementing an
automatic procedure is of great interest from a practical
point of view.
0.8
Signal from the error sensor
III. EXPERIMENTAL RESULTS
3
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
0
1000
2000
3000
4000
5000
6000
Iterations
(a)
0.26
Leaky FXLMS
-20
Error signal spectrum
0.24
Steady-state MSE
Control off
Control on
0.22
0.20
Optimum leakage value
0.18
0.16
0.05
-40
-60
-80
-100
0.10
0.15
0.20
0.25
0.30
0.35
-120
Leakage factor
0
50
100
200
250
300
350
400
450
Frequency [Hz]
Fig. 3. Variation of the steady-state error as a function of the
leakage factor value.
(b)
-5
Zoom around 300 Hz
The VSSVLFxLMS algorithm has been implemented in
the EZ-Kit LiteTM 21161 platform from Analog Devices. In
Figs. 4 and 5, the obtained experimental results are
illustrated using the active vibration control setup given in
Fig. 1. These figures show the evolution of the error signal
as well as the power spectrum of the steady-state error
signal with and without control, for two frequencies of the
perturbation signal (80 Hz and 300 Hz).
In Fig. 4, the system is disturbed by a sinusoidal signal of
300 Hz. Fig. 4(a) shows the evolution of the error signal
coming from the error sensor (see Fig. 1), illustrating the
algorithm convergence. In Fig. 4(b), the spectrum of the
error signal after algorithm convergence is plotted. A detail
of Fig. 4(b) around the frequency of 300 Hz is shown in
Fig. 4(c). From these figures, we can observe the controlling
effect of the proposed algorithm. The attenuation obtained
for the 300 Hz case is 11 dB. In Fig. 5, the obtained results
for a disturbance signal of 80 Hz are shown following the
same presentation pattern as in Fig. 4. Here, the same
observations drawn from Fig. 4 are also verified for Fig. 5,
except that now the obtained attenuation is 7 dB. We
attribute such a difference to the mechanical characteristics
of the system under control. Further studies are being
carried out aiming to improve the attenuation characteristics
of the proposed algorithm.
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150
Control off
Control on
-10
-15
-20
-25
270
280
290
300
310
320
330
340
Frequency [Hz]
(c)
Fig. 4. Experimental results for 300 Hz. (a) Error signal. (b) Error signal
spectrum with and without control. (c) Detailed plot around 300 Hz.
IV. CONCLUDING REMARKS
In this work, the variable step size and variable leakage
factor FxLMS algorithm has been derived and tested in a
real AVC application. The proposed algorithm exhibits a
good performance at the expense of a small computational
580
VI International Telecommunications Symposium (ITS2006), September 3-6, 2006, Fortaleza-CE, Brazil
complexity increase. Since different procedures for step size
and leakage factor adjustment can be performed, further
studies for improving the behavior of the VSSVLFxLMS
algorithm are currently underway.
REFERENCES
[1]
[2]
[3]
0.8
[4]
Signal from the error sensor
0.6
0.4
[5]
0.2
[6]
0
-0.2
[7]
-0.4
-0.6
-0.8
[8]
0
1000
2000
3000
4000
5000
6000
[9]
Iterations
(a)
Control off
Control on
Error signal spectrum
-20
-40
-60
-80
-100
-120
0
50
100
150
200
Frequency [Hz]
(b)
Control off
Control on
Zoom around 80 Hz
-8
-10
-12
-14
-16
-18
50
60
70
80
90
100
Frequency [Hz]
(c)
Fig. 5. Experimental results for 80 Hz. (a) Error signal. (b) Error signal
spectrum with and without control. (c) Detailed plot around 80 Hz.
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