Frequency Domain Analysis for An Adaptive Windowing Parabolic Sliding Mode Filter
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Frequency Domain Analysis for An Adaptive Windowing Parabolic Sliding Mode Filter
Shanhai Jin 1
Xiaodan Wang 1
Yonggao Jin 1
Xiaogang Xiong 0
0 Singapore Institute Of Manufacturing Technology , 71 Nanyang Dr , Singapore 638075 , Singapore
1 School of Engineering, Yanbian University , Yanji 133002 , China
This paper analyses a frequency domain performance of an adaptive windowing parabolic sliding mode filter (AW-PSMF) by using Describing Function method. The analysis results show that AW-PSMF has similar gain characteristics to that of the second-order Butterworth low-pass filter (2-LPF), but AW-PSMF produces flatter gain behaviour than 2-LPF does at cutoff frequency. In addition, AW-PSMF produces smaller phase lag than 2-LPF does.
1 Introduction
In feedback control of mechatronic systems, feedback
signals are usually corrupted by noise. Thus, a filter is
required for removing noise from feedback signals.
Linear filters are commonly applied for reducing noise
because of their simplicity. However, in a linear filter,
any noise component is proportionally transferred into
the output. In addition, a large phase lag, which may
result in the instability of feedback systems, is caused by
strong noise attenuation.
Nonlinear filters have been applied for avoiding
drawbacks of linear filters. For example, median filters [
1
]
are used for removing high-frequency noise, but they are
computationally expensive [
2
]. As another example,
stochastic filters, e.g., Kalman filter [
3
], [
4
], are also
applied in some applications. However, a dynamics
model of the source of the signal, which is usually not
available, is required. In addition, their performance
depends on the model accuracy.
In the last decade, sliding mode observers based on
the super-twisting algorithm [
5
], [
6
] has been attracted
much attention. One advantage of these observers is that
they theoretically realize finite time convergence in
continuous-time analysis. However, typically with finite
difference, the convergence accuracy in discrete-time
implementation depends on the sampling period, as
reported in [
5
]. Moreover, they are prone to overshoot
during the convergence. Furthermore, they also require a
system dynamics model.
The sliding mode filter that employs a parabolic
sliding surface has been studied [
7
], [
8
]. One of major
advantages of this filter is that, in the case of a constant
input is provided, the output converges to the input in
finite time. However, the filter is prone to overshoot. In
addition, the discrete-time implementation of the filter
produces high-frequency chattering due to inappropriate
discretization.
Toward to the drawbacks of the parabolic sliding
mode filter [
7
], [
8
], Jin et al. [
9
] presented a new
parabolic sliding mode filter, which is referred to PSMF,
for effectively removing noise in feedback control
systems. It is reported in [
9
] that PSMF produces smaller
phase lag than linear filters, and it is less prone to
overshoot than the sliding mode filter [
7
], [
8
]. In addition,
the algorithm of PSMF, which is derived by using the
backward Euler differentiation, does not produce
chattering. After that, Jin et al. [
10
] presented an adaptive
windowing parabolic sliding mode filter, which is named
as AW-PSMF, by extending PMSF. It is stated in [
10
]
that AW-PSMF adaptively adjusts its window size for
obtaining the largest window size that optimizes the
trade-off between the filtering smoothness and the delay
suppression. The effectiveness of AW-PSMF has been
experimentally validated in feedback control of a
mechatronic system. In [
10
], however, the frequency
domain performance of AW-PSMF is not evaluated.
This paper presents a frequency domain performance
of AW-PSMF. It is shown that AW-PSMF has similar
gain characteristics to that of the second-order
Butterworth low-pass filter (2-LPF), but AW-PSMF
produces smaller phase lag (maximum 140 degree) than
2-LPF (maximum 180 degree) does.
The rest of this paper is organized as follows. Section
2 provides a brief overview of parabolic sliding mode
filters. Section 3 analysis the performance of AW-PSMF
in frequency domain. Section 4 gives concluding remarks.
2 Parabolic sliding mode filters
In [
9
], Jin et al. presented a parabolic sliding mode filter
(PSMF), of which continuous-time expression is given as
follows:
x1
x2
x2
F (H
1)
2
F (H
2
sgn( (F, u, x1, x2 ))
1)
sgn(x2 ) ,
where
(F,u, x1, x2 )
2F( x1
u)
x2 x2 .
Here, u is the input, x1 and x2 are the outputs, and
F 0 and H
1 are constants. In addition, sgn() is the
set-valued signum function defined as follows:
1 if z 0
sgn(z)
[
1,1
] if z
1
if z
0
0
It should be noted that sgn(z) returns a set instead of a
single value when z 0 .
Figure 1. illustrates the sliding surface and state
trajectories of PSMF in x1-x2 space. It is shown that
PSMF employs a parabolic-shaped sliding surface, and
the state is attracted to the sliding (...truncated)