# Discrete-Time Sliding Mode Filter with Adaptive Gain
## 1 Introduction  
Sensor signals are important in feedback control of mechatronic systems. However, due to measurement errors and environmental disturbances, sensor signals are usually noisy and uncertain. Various filters are developed for obtaining smooth and reliable signals. Among them, linear filters are perhaps the first option due to their simplicity. They, however, proportionally transfer any noise component into the output. Furthermore, they produce a large phase lag, which may result in instability of feedback controlled systems, in the case of strong noise attenuation.  
In order to avoid drawbacks of linear filters, nonlinear filters have been studied. For instance, median filters [1] are developed for attenuating high-frequency noise. However, it is reported [2] that they require expensive computational costs. As another example, stochastic filters, e.g., Kalman filters [3; 4], are applied in some applications. However, a dynamic model of the signal source, which is not always available, is required. Moreover, their performance is related to the model accuracy.  
The super-twisting algorithm [5; 6] based sliding mode observers [7; 8; 9] have attracted much attention in the last decade. In these observers, finite time convergence can be theoretically realized in continuous-time analysis. However, in discrete-time implementation, the convergence accuracy depends on the sampling period, typically with finite difference [10; 11]. In addition, a dynamic model is also required.  
The sliding mode filter that employs a certain kind of parabolic-shaped sliding surface has been studied [12; 13]. One of the major advantages of the filter is that finite time convergence of the output to the input is achieved in the case of a constant input being provided. Moreover, the filter does notrequire a dynamic model. Some evaluation results [14; 15; 16] of the filter and its applications [17; 18; 19] were reported. However, the filter is prone to overshoot.  
Regarding the drawback of the filter [12; 13], [PERSON] et al. [20] proposed a new model-free parabolic sliding mode filter, which is referred to as PSMF, for effectively removing noise in feedback control systems. It is reported [21] that the noise removing capability of parabolic sliding mode filter (PSMF) is almost the same as that of the second-order Butterworth low-pass filter (2-LPF), but its phase lag is smaller than that of 2-LPF. In addition, PSMF is less prone to overshoot, and it produces smaller phase lag compared with the filter [12; 13]. Moreover, the discrete-time algorithm of PSMF, which is derived by using the backward Euler discretization, does not produce chattering. The effectiveness of PSMF has been experimentally validated [21; 22].  
This paper presents a new model-free discrete-time sliding mode filter for removing noise by balancing the tradeoff between the filtering smoothness and the suppression of delay. The presented filter is an extension of PSMF by including an adaptive gain, of which value is determined by an adaptive method inspired by a first-order adaptive windowing filter [23]. The results of numerical examples and experiments show that the presented filter performs better than PSMF.  
The rest of this paper is organized as follows: Section 2 gives a brief overview of related works; Section 3 presents a new model-free discrete-time sliding mode filter; Sections 4 and 5 validate the effectiveness of the presented filter through numerical examples and experiments, respectively; and Section 6 covers concluding remarks. In addition, the Appendix provides a list of main abbreviations and notations used in the paper.