Frequency Domain Analysis for An Adaptive Windowing Parabolic Sliding Mode Filter

MATEC Web of Conferences, Jan 2017

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.

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Frequency Domain Analysis for An Adaptive Windowing Parabolic Sliding Mode Filter

MATEC Web of Conferences 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)


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Shanhai Jin, Xiaodan Wang, Yonggao Jin, Xiaogang Xiong. Frequency Domain Analysis for An Adaptive Windowing Parabolic Sliding Mode Filter, MATEC Web of Conferences, 2017, pp. 14005, 95, DOI: 10.1051/matecconf/20179514005