A new accurate discretization method for high-frequency component mechatronics systems

Mechatronics 62 (2019) 102250

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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics

A new accurate discretization method for high-frequency component mechatronics systems ✩ Tae-Il Kim a, Ji-Seok Han a, Tae-Ho Oh a, Young-Seok Kim a, Sang-Hoon Lee b, Dong-Il “Dan” Cho a,∗ a b

ASRI/ISRC, Department of Electrical and Computer Engineering, Seoul National University, Seoul, 08826, Republic of Korea RS Automation Co., Ltd., Gyeonggi-do, 17709, Republic of Korea

a r t i c l e

i n f o

Keywords: Discretization Discrete-time differentiator Nyquist frequency Adaptive notch filter Servo system Resonance suppression

a b s t r a c t Modern mechatronics systems are implemented digitally. However, the magnitude and phase errors caused in the discretization process severely restrain the control performance in systems with high-frequency components where only a few sampled data points are available per period. This paper presents a new accurate discretization method for mechatronics systems that provide good performance even when intrinsic frequency components are close to the Nyquist frequency which is one-half of the sampling frequency. The bilinear transform method is most commonly used, but it causes oscillations when the initial state error of the output is not zero or when rapid changes occur in the input signal. Several variations of the bilinear transform method have been proposed to improve these problems, but as a tradeoff, they introduce large magnitude and/or phase errors at high frequencies. In this paper, a more accurate discretization method is developed, which combines a modified bilinear transform method with a new method to compensate for the frequency and damping ratio warping caused by approximate discretization. The proposed method reduces the magnitude and phase errors over the entire frequency range. The proposed method is experimentally evaluated in a mechatronics system with a mechanical resonance frequency that is about 0.6 times the Nyquist frequency.

1. Introduction

is used:

Controllers and filters in modern mechatronics systems are commonly designed in the continuous-time domain and are discretized for digital implementation. However, all discretization methods are inevitably accompanied by magnitude and/or phase errors, and can even introduce unwanted oscillations in the discretized signals. The performance of a discretized system at relatively high frequencies, especially near the Nyquist frequency, is critically dependent upon which discretization method is chosen. In many cases, the fast frequency components of a system can be close to the Nyquist frequencies due to the limitations in sampling time. In this paper, an industrial mechatronics system, where the controller sampling frequency is 8 kHz (hence, 4 kHz Nyquist frequency) and a hardware intrinsic mechanical resonance is near 2.5 kHz, is used to compare the performance of various discretization methods to the method developed here. This example represents a typical servo system used in manufacturing robots. To suppress this resonance, a notch filter

𝐻𝑁𝐹 (𝑠) =

✩ ∗

𝑠2 + 2𝜁𝑁 𝜔𝑛 𝑠 + 𝜔𝑛 2 𝑠2 + 2𝜁𝐷 𝜔𝑛 𝑠 + 𝜔𝑛 2

,

(1)

where 𝜔𝑛 = 5000𝜋(2.5k Hz), 𝜁𝑁 = 0.0707 and 𝜁𝐷 = 0.707. Fig. 1 shows the frequency responses of discretized notch filters using exact discretization, the Euler forward method and the bilinear transform method. When the forward Euler method is used, relatively large magnitude and phase errors occur due to the phase lag near the Nyquist frequency. By using the bilinear transform method with the frequency pre-warping technique, relatively small magnitude and phase errors occur, and the errors become zero at the notch frequency. However, the bilinear transform method may generate an oscillating transient response when the initial state error of the output is not zero [1]. Rapid changes, such as sensor noise or abrupt disturbances in the input signal may also cause this phenomenon, which is inevitable in many real systems. Numerous discretization methods have been proposed to make the high-frequency performance of discretized filters or controllers close to that of the original continuous-time designs. Schneider et al. [2] presented a family of higher-order numerical integration formulas and their

This paper was recommended for publication by Associate Editor Dr. Yayou Li. Corresponding author. E-mail address: [email protected] (D.-I.“. Cho).

https://doi.org/10.1016/j.mechatronics.2019.102250 Received 8 December 2018; Received in revised form 10 June 2019; Accepted 10 July 2019 Available online xxx 0957-4158/© 2019 Published by Elsevier Ltd.

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Mechatronics 62 (2019) 102250

Fig. 1. Frequency responses of discretized notch filters.

corresponding s-to-z mapping functions. Pei and Hsu [3] developed a fractional bilinear transform derived from fractional delay filters. Gupta et al. [4] proposed wideband recursive digital integrators designed by interpolating some of the popular digital integration techniques and digital differentiators obtained by modifying the transfer functions of the integrators. Xin et al. [5] presented a digital differentiator based on a generalized integrator for power converters. Nguyen-Van [6] developed an observer-based sampled-data feedback control where a discrete-time model was used for the observer, and the controller design was derived using a continual discretization method. Pan et al. [7] proposed two discrete differentiators for the active damping of an LCL-type gridconnected inverter: a first-order differentiator based on the backward Euler method and a digital lead compensator, and a second-order differentiator, based on a Tustin method plus digital notch filter. Several discretization methods for the fractional-order differentiator and integrator have been proposed and utilized in fractional-order controller designs [8–15]. Barbosa et al. [10] presented least-squares based methods for obtaining digital rational approximations to fractional-order integrators and differentiators. Romero et al. [12] proposed a discretization method for fractional-order differentiators and integrators using Chebyshev’s polynomials theory to achieve accurate discrete-time approximation. Muresan et al. [13] developed an FPGA based fractional-order controller for a DC motor using a discretization approach that consists of the 9th-order recursive Tustin method. Many discretization methods, such as the generalized integrator based methods [5,16] or the methods for fractional-order systems [8–15], are mainly concerned with improving the differentiation performance by using the differentiators with complex structures. However, the frequency and the damping ratio become warped in these methods because of various approximations of the exact discretization. Moreover, because of the limitations of computational power and numerical precision in real-time embedded systems, simple transformation functions for discretization are commonly used, where numerators and denominators are first-order functions or constants, such as the bilinear transform method. To overcome the discretization error in the limitations of computational power and numerical precision, several discretization methods [1,17,18] were developed that combine the simple transformation methods with compensation methods for the frequency warping caused by approximate discretization. Al-Alaoui [18] proposed a discrete-time integrator obtained by interpolating the trapezoidal and the rectangular integration rules and a discretization method developed using the inverse of the integrator equation.

Al-Alaoui [18] also presented a compensation method for frequency warping caused by the approximation of the exact discretization. Yoon et al. [1] modified the bilinear transform method to move the pole location from 𝑧 = −1 to the inside of the unit circle, and thus this method does not produce the oscillating transient response in the steady state. Kim et al. [17] presented a frequency mapping method to compensate for the frequency warping phenomenon that occurs when the method by Yoon et al. [1] is utilized. These modified bilinear transform methods can attenuate the oscillating transient response that may occur in the bilinear transform method. However, these methods still have a relatively large magnitude and/or phase errors near the Nyquist frequency. In this paper, we report a new accurate discretization method for control systems with intrinsic frequency components that are close to the Nyquist frequency. A modified bilinear transform method is developed to attenuate the oscillation caused by the initial state error in the output signal and rapid changes in the input signal. Furthermore, a compensation method for the warping of the frequency and damping ratio are developed. As an example, an adaptive notch filter (ANF) for industrial servo systems is discretized using the proposed method. We simulated the ANF using MATLAB to show the frequency estimation performance. The ANF is also implemented to a servo controller and applied to a printed circuit board (PCB) inspection machine with a mechanical resonance frequency that is about 0.6 times the Nyquist frequency. Experiments were performed to show that the proposed method significantly improves the performance of the controller. 2. Discretization methods In this section, the problems of conventional discretization methods are analyzed using discrete-time differentiators, because the s-to-z transform equations of discretization methods are the transfer functions of discrete-time differentiators. To ameliorate the problems of the previous methods, a new discretization method is proposed. The frequency responses of the discrete-time differentiators are shown in Fig. 2, where the sampling frequencies are 8 kHz. In Fig. 2, the method by Al-Alaoui [18] and the proposed method show smaller magnitude errors and the bilinear transform method shows the smallest phase error. Fig. 3 shows the differentiation results of the bilinear transform method, the method by Al-Alaoui [18], the methods by Yoon et al. [1] and Kim et al. [17], and the proposed method in several sampling frequencies, where the input signal is 1 Hz sine wave. Note that the initial values of the differentiators are zero and that the outputs have initial state errors. In

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Mechatronics 62 (2019) 102250

Fig. 2. Frequency responses of discrete-time differentiators.

Fig. 3, the method by Al-Alaoui [18] and the proposed method show better differentiation results than the other methods. 2.1. Problems in the conventional discretization methods 2.1.1. Bilinear transform method The bilinear transform method is commonly used due to its simplicity and because its phase response is the same as that of exact discretization. The bilinear transform approximates the Laplace operator s to the ztransform in the discrete-time domain as: ( ) 2 𝑧−1 𝑠 ≃ 𝐻 𝑏 (𝑧 ) = (2) , 𝑇 𝑧+1 where T is sampling time. Since the bilinear transform is a first-order approximation of the exact mapping of the s-plane to the z-plane, the frequency-warping phenomenon occurs near the Nyquist frequency. To compensate for this phenomenon, the frequency pre-warping method is utilized: ( ) 𝑇 2 𝜔𝑏 = tan 𝜔𝑑 , (3) 𝑇 2 where 𝜔d is the given frequency specification for the design, and 𝜔b is the pre-warped frequency. However, the transient response characteristics of this method can cause unwanted oscillations in the discretized signals. As shown in Fig. 3(a), the oscillating transient responses are not attenuated by the initial state error of the output [1]. Rapid changes in the input signal, such as sensor noise or abrupt disturbances, may also cause an oscillating transient response. In these cases, the oscillations are not attenuated because Eq. (2) has one pole at 𝑧 = −1, which is on the unit circle [1]. Several variations of bilinear transform that move this pole inside the unit circle have been proposed to attenuate the oscillations.

Eq. (5) has a pole at 𝑧 = −(1 − 𝑎)∕(1 + 𝑎), which is in the unit circle, and thus the oscillation caused by the initial-state error of the output or by rapid changes in the input is attenuated as shown in Fig. 3(b). This transformation maps the imaginary axis of the s-plane to a circle on the z-plane of which the real and imaginary parts are given by ( )2 ( 2 ) 𝑎 −1 4 + 𝜔𝐴 𝑇 Re(𝑧) = ( , (6) )2 𝜔𝐴 𝑇 (𝑎 + 1)2 + 4 Im(𝑧) = (

𝜔𝐴 𝑇

4𝜔𝐴 𝑇 , )2 (𝑎 + 1)2 + 4

(7)

respectively, where 𝜔A is the pre-warped frequency. Al-Alaoui [18] presented a mapping between the continuous-time and discrete-time frequency variables: ( ) ( ) 4𝜔𝐴 𝑇 Im(𝑧) 1 1 𝜔𝑑 = tan−1 = tan−1 (8) ( )2 ( ) . 𝑇 Re(𝑧) 𝑇 𝑎2 − 1 4 + 𝜔𝐴 𝑇 However, by using Eq. (8), the magnitude and phase errors are relatively large near the Nyquist frequency, especially when a is close to 1. This problem occurs because Eq. (8) approximates the damped frequency to the natural frequency. Fig. 4 shows the mapping of the left half of the s-plane to the blue circle on the z-plane using Eq. (5). The damping ratio of the mapped point in the z-plane is larger than that of the original point on the s-plane. This effect gets more pronounced as a increases, especially in the relatively high-frequency range. Because of this warping of the damping ratio, the difference between the damped frequency and the natural frequency becomes large, and thus the approximation in Eq. (8) introduces the unwanted magnitude and phase errors near the Nyquist frequency.

(4)

2.1.3. Methods by Yoon et al. [1] and Kim et al. [17] To ameliorate the oscillating transient response of the bilinear transform method, Yoon et al. [1] proposed a modified bilinear transform in which the pole location is moved from 𝑧 = −1 to the inside of the unit circle: ( ) 2 𝑧−1 𝑠 ≃ 𝐻𝑌 = , (9) 𝑇 𝑧 + 𝛼𝑌

where a is a design parameter, and 0 < a < 1. By approximating the ideal continuous-time integrator 1/s to HAI (z), an s-to-z transformation is derived: 2(𝑧 − 1) 1 𝑠≃ = 𝐻 𝐴 (𝑧 ) = . (5) 𝐻𝐴𝐼 (𝑧) 𝑇 [(1 − 𝑎) + (1 + 𝑎)𝑧]

where 𝛼 Y is a design parameter, and 0 < 𝛼 Y < 1. Kim et al. [17] proposed a compensation method for frequency warping of this modified bilinear transform: ( ) √ −𝐴𝐵𝐶 + 1 + 𝐴2 𝐶 − 𝐵 2 𝐶 1 𝜔𝑑 = cos−1 , (10) 𝑇 𝐴2 𝐶 + 1

2.1.2. Method by Al-Alaoui [18] The Al-Alaoui integrator [18] is obtained by interpolating the trapezoidal and the rectangular integration rules: 𝑇𝑧 𝑇 𝑧+1 + (1 − 𝑎) 𝑧−1 2 𝑧−1 𝑇 {(1 − 𝑎) + (1 + 𝑎)𝑧} = , 2(𝑧 − 1)

𝐻𝐴𝑙 (𝑧) = 𝑎

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Mechatronics 62 (2019) 102250

Fig. 3. Outputs of the discretized differentiators for 1 Hz sine input.

where 𝐴 = 2𝛼𝑌 , 𝐵 = 𝛼𝑌 2 + 1, 𝐶 = Re(𝜔𝐾 , )2 𝑇 2 ∕(4(𝛼 + 1)2 ), and 𝜔K is the pre-warped frequency. However, when these methods are applied, the amplitude errors remain in the steady state, which is shown in Fig. 3(c) and (d). As shown in Fig. 2, the methods by Yoon et al. [1] and Kim et al.

[17] have the magnitude error of 20log10 ((1 + 𝛼𝑌 )∕2) dB in relatively low-frequency range. This magnitude error causes amplitude errors as shown in Fig. 3(c) and (d). The amplitude errors increase when 𝛼 Y becomes smaller. The magnitude and phase errors become even larger near

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Mechatronics 62 (2019) 102250

( ) ( ) = Re 𝑧𝑑 + 𝑗 Im 𝑧𝑑 ,

( ) 1 + 𝛼𝑝 𝑠𝑝 = 𝐻𝑝 𝑧𝑑 = 𝑇

(

(13)

) 𝑧𝑑 − 1 , 𝑧 𝑑 + 𝛼𝑝

(14)

| | 𝜔 𝑝 = |𝑠 𝑝 | | |

√ √( ( ) )2 ( )2 1 + 𝛼𝑝 √ √ Re 𝑧𝑑 − 1 + Im 𝑧𝑑 √ = ( ( ) )2 ( )2 , 𝑇 Re 𝑧 + 𝛼 + Im 𝑧 𝑑

Fig. 4. Mapping of the s-plane to the z-plane using the method by Al-Alaoui [18].

𝜁𝑝 = −

𝜔𝑝 (

(

= − cos tan

2.2. Proposed discretization method The proposed method consists of two parts: an s-to-z mapping method and a compensation method for the frequency and the damping ratio warping. Note that the responses of the differentiator discretized using the proposed method shown in Figs. 2 and 3(e) can only show the effects of the s-to-z mapping method, because the proposed compensation method does not change the pole frequency or damping ratio of the ideal differentiator placed at the origin on the s-plane. We present an s-to-z mapping method that removes the constant magnitude error in the method by Yoon et al. [1] by modifying the constant coefficient as follows: ( ) 1 + 𝛼𝑝 𝑧 − 1 𝑠 ≃ 𝐻 𝑝 (𝑧 ) = , (11) 𝑇 𝑧 + 𝛼𝑝 where 𝛼 p is a design parameter. As shown in Section 2.1.3, the method by Yoon et al. [1] has the amplitude error of (1 + 𝛼𝑌 )∕2 in relatively low-frequency range. By replacing the constant term, 2/T, in Eq. (9) with (1 + 𝛼𝑌 )∕𝑇 , Eq. (11) is derived, and this amplitude error can be removed. The design parameter, 𝛼 p , is similar to 𝛼 Y , where 0 < 𝛼 p < 1, and 𝛼 p determines the attenuation speed of the oscillation caused by the initial state error of the output or rapid changes in the input signal. When 𝛼 p becomes small, the oscillation attenuates rapidly as shown in Fig. 3(e), and the magnitude error decreases as shown in Fig. 2. However, in this case the phase error increases, as a tradeoff. The transformation, Eq. (11), becomes equivalent to the method by Al-Alaoui [18], Eq. (5), by selecting 𝛼𝑝 = (1 − 𝑎)∕(1 + 𝑎). In Figs. 1–3, the method by Al-Alaoui [18] with 𝑎 = 0.2 and the proposed method with 𝛼𝑝 = 0.67 show similar responses because 𝛼 p is close to (1 − 𝑎)∕(1 + 𝑎) = 2∕3. The presented s-to-z mapping method has a relatively small magnitude error near the Nyquist frequency as shown in Fig. 2. However, a small magnitude error still remains, and it has a relatively large phase error compared to the bilinear transform. To reduce the effects of the magnitude and the phase errors, new mapping methods are proposed that pre-warp the frequencies and the damping ratios of the poles and the zeros. For the given frequency, 𝜔d , and damping ratio, 𝜁 d , as the specifications for a filter or a controller design, the proposed mapping methods are: ( ) √ 𝑠𝑑 = −𝜁𝑑 + 1 − 𝜁𝑑 2 𝑗 𝜔𝑑 , (12) ( ) 𝑧𝑑 = exp 𝑠𝑑 𝑇

{ (√ ) (√ )} 1 − 𝜁𝑑 2 ⋅ 𝜔𝑑 𝑇 + 𝑗 sin 1 − 𝜁𝑑 2 ⋅ 𝜔 𝑑 𝑇 = exp −𝜁𝑑 𝜔𝑑 𝑇 ⋅ cos (

)

(15)

𝑑

( ) Re 𝑠𝑝

−1

the Nyquist frequency because of the warping of the damping ratio. This problem occurs because the method by Kim et al. [17] approximates the damped frequency to the natural frequency.

𝑝

( ( ) ) ( ) )) Im 𝑧𝑑 Im 𝑧𝑑 −1 − tan , ( ) ( ) Re 𝑧𝑑 − 1 Re 𝑧𝑑 + 𝛼𝑝

(16)

where 𝜔p is the pre-warped frequency, and 𝜁 p is the pre-warped damping ratio. Fig. 5 shows the proposed pre-warping process using Eqs. (12)– (16). The left half of the s-plane is mapped to the blue circle on the zplane using the exact discretization, while the red circle on the z-plane is mapped to the left half of the s-plane for pre-warping using Eq. (14). Note that sp can be placed at the right half, even when sd is on the left half of the s-plane. In this case, 𝜁 p is a negative value. However, in the discretized filter or controller sp is mapped to the point inside the unit circle on the z-plane by the warping effect of the proposed s-to-z transformation. This pre-warping method compensates for the difference in frequency responses between the exact discretization method and the proposed method at the design point, sd . The inverse of this pre-warping process can be utilized in several applications. For the given frequency, 𝜔p , and damping ratio, 𝜁 p , the inverse mapping is:

𝑠𝑝 =

( ) √ −𝜁𝑝 + 1 − 𝜁𝑝 2 𝑗 𝜔𝑝 ,

𝑧𝑑 = 𝐻𝑝

−1

( ) ( ) 𝑠 𝑝 + 𝛼𝑝 + 1 ∕ 𝛼 𝑝 𝑇 ( ) 𝑠 𝑝 = − 𝛼𝑝 , ( ) 𝑠 𝑝 − 𝛼𝑝 + 1 ∕ 𝑇

( )) Im 𝑧𝑑 ( ) ln ||𝑧𝑑 || + 𝑗 tan−1 ( ) Re 𝑧𝑑 √ (√ )2 |⎞ ⎛|| √ √{ ( ) ( )}2 | ⎜| √ + 1 − 𝜁𝑝 2 ⋅ 𝜔𝑝 ||⎟ √ − 𝜁𝑝 𝜔 𝑝 + 𝛼𝑝 + 1 ∕ 𝛼𝑝 𝑇 | 1 ⎜| √ |⎟ = ln |𝛼 √ (√ )2 ||⎟ 𝑇 ⎜⎜|| 𝑝 √ ( ) }2 √ { |⎟ 2 |⎟ − 𝜁𝑝 𝜔 𝑝 − 𝛼𝑝 + 1 ∕ 𝑇 + 1 − 𝜁𝑝 ⋅ 𝜔 𝑝 ⎜|| |⎠ ⎝| | √ √ ⎛ ⎛ ⎛ ⎞ 2 2 1 − 𝜁𝑝 ⋅ 𝜔 𝑝 1 − 𝜁𝑝 ⋅ 𝜔 𝑝 ⎜ ⎟ ⎜ 1⎜ + 𝑗 ⎜tan−1 ⎜ ( ) ( ) ⎟ − tan−1 ⎜ ( ) 𝑇⎜ 𝛼 ∕ 𝛼 − 𝜁 𝜔 + + 1 𝑇 − 𝜁 𝜔 𝑝 𝑝 ⎜ 𝑝 𝑝 ⎟ ⎜ 𝑝 𝑝 − 𝛼𝑝 + 1 ∕ 𝑇 ⎝ ⎝ ⎠ ⎝

(17)

(18)

(

𝑠𝑑 =

1 𝑇

𝜔𝑑 = ||𝑠𝑑 ||, 𝜁𝑑 = −

⎞⎞ ⎟⎟ ⎟⎟, (19) ⎟⎟ ⎠⎠

(20)

( ) Re 𝑠𝑑 𝜔𝑑

.

(21)

For the given transfer function, Gd (s), the final pre-warped transfer function, Gp (s), can be obtained by multiplying a constant to match the lowfrequency magnitude:

(

𝐺𝑝 (𝑠) =

) ( ) 𝐺𝑑 𝑠′ || ⋅ 𝐺𝑝′ (𝑠), | 𝐺′ 𝑝 (𝑠′ ) ||𝑠′ =0

(22)

where 𝐺𝑝′ (𝑠) is the pre-warped transfer function using the pre-warped frequencies and damping ratios. By utilizing the proposed mapping method, the warping phenomena of the frequency and the damping

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Mechatronics 62 (2019) 102250

Fig. 5. Pre-warping of frequency and damping ratio in the proposed method.

ratio can be ameliorated which are especially severe in the relatively high-frequency range. We examine the stability of s-to-z mapping method, Eq. (11), for an initial value problem, where the model equation and the initial value are: 𝑌̇ (𝑡) = 𝜆𝑌 (𝑡) + 𝑔 (𝑡),

(23)

𝑌 (0 ) = 𝑌 0 ,

(24)

respectively. Let 𝑌̇ 𝜀 (𝑡) be the solution with the perturbed initial data 𝑌0 + 𝜀: 𝑌̇ 𝜀 (𝑡) = 𝜆𝑌𝜀 (𝑡) + 𝑔 (𝑡),

(25)

𝑌𝜀 (0) = 𝑌0 + 𝜀,

(26)

and define Zɛ (t) as follows: ( ) 𝑍𝜀 (𝑡) = 𝑌𝜀 (𝑡) − 𝑌 (𝑡) ∕𝜀.

(27)

Then the equation, the initial condition, and the solution for Zɛ (t) become: 𝑍̇ 𝜀 (𝑡) = 𝜆𝑍𝜀 (𝑡),

(28)

3. Numerical example: discretization of ANF In this section, a discretization procedure of ANF is presented using the proposed method as a numerical example. The ANF is briefly reviewed and discretized. Simulations using MATLAB and the experiments utilizing an industrial servo system were carried out to understand the performance of the discretized ANF. 3.1. Review of ANF The ANF was originally proposed by Regalia [19] as a discrete infinite impulse response filter, and it was transformed and analyzed in the continuous-time domain by Bodson and Douglas [20]; Hsu et al. [21]; Mojiri and Bakhshai [22]; and Bahn et al. [23]. The ANF identifies the frequency of a sinusoidal signal with unknown frequency online and filters it out [17]. Recently, the ANF is applied to industrial servo systems to suppress resonance automatically [1,17,23,24]. Note that the ANF proposed by Bahn et al. [23] was used to develop a resonance suppression method as a discretization example in this paper. The frequency estimator of the ANF consists of a second-order adaptive resonator and an estimation law that calculates the first derivative of the estimated frequency: 𝑥̈ + 2𝜁 𝜔̂ 𝑥̇ + 𝜔̂ 2 𝑥 = 2𝜁 𝜔̂ 2 𝑢,

𝑍 𝜀 (0 ) = 1 , 𝑍𝜀 (𝑡) = exp (𝜆𝑡),

(29) 𝜔̂̇ = −𝛾

𝑍̄ 𝜀 [𝑘] =

(31)

( ) 1 + 𝛼𝑝 ∕ 𝑇 − 𝛼 𝑝 𝜆 𝑍̄ 𝜀 [𝑘 − 1] ( ) 1 + 𝛼𝑝 ∕ 𝑇 − 𝜆

= 𝐶 𝑍̄ 𝜀 [𝑘 − 1].

(√ ) √ (𝜔̂ 𝑥)2 − |𝑥𝑥̈ | ,

(35)

(30)

respectively. When 𝜆 has a negative real part, Zɛ (t) will go to zero as t → ∞, and thus the effect of perturbation decays. Using Eq. (11), Eq. (28) is discretized as follows: 1 + 𝛼𝑝 ( ( ) ) 𝑍̄ 𝜀 [𝑘] − 𝑍̄ 𝜀 [𝑘 − 1] = 𝜆 𝑍̄ 𝜀 [𝑘] + 𝛼𝑝 𝑍̄ 𝜀 [𝑘 − 1] , 𝑇

(34)

where x, 𝜁 , 𝜔̂ , u, and 𝛾 are the filter state, the damping ratio of the adaptive resonator, the estimated frequency, the input signal, and the estimation gain, respectively. By integrating 𝜔̂̇ , the estimated frequency is calculated: 𝜔̂ (𝑡) = 𝜔̂ 0 +

𝑡

∫0

𝜔̂̇ (𝜏)𝑑𝜏,

(36)

where 𝜔̂ 0 is the initial value of the estimated frequency.

(32)

The bar on the variable denotes that it is discretized using the proposed method. Assuming that 𝜆 has a negative real part, C is bounded as follows: ( ) | 1 + 𝛼𝑝 ∕ 𝑇 − 𝛼 𝑝 𝜆 | | | |𝐶 | = | ( | ) | 1 + 𝛼 ∕𝑇 − 𝜆 | 𝑝 | | {( ) } | 1 + 𝛼𝑝 ∕𝑇 − 𝛼𝑝 Re(𝜆) − 𝑗 𝛼𝑝 Im(𝜆) | | | = | {( (33) | < 1, ) } | | + 𝛼 𝑇 − Re 𝜆) 𝑗 Im 𝜆) 1 ∕ − ( ( 𝑝 | | which shows that for any step size T, the solution is absolutely stable.

3.2. Discretization of ANF using the proposed method The adaptive resonator equation, Eq. (34) is discretized using Eq. (11) as: 𝑥̄ [𝑘] =

−2𝜁𝑝 𝜔̄̂ [𝑘]𝑓1 [𝑘] − 𝑓2 [𝑘] + 2𝜁𝑑 𝜔̄̂ [𝑘]2 𝑢[𝑘] , ( )2 1+𝛼 1+𝛼 ̄ ̄̂ [𝑘]2 𝜔 ̂ 𝜔 + 2 𝜁 𝑘 + [ ] 𝑝 𝑇 𝑇

(37)

where 𝑓1 [𝑘] = −𝛼𝑝 𝑥̄̇ [𝑘 − 1] −

1 + 𝛼𝑝 𝑇

𝑥̄ [𝑘 − 1],

(38)

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Fig. 6. Frequency responses of discretized adaptiveresonator equations.

𝑓2 [𝑘] = −2𝛼𝑝 𝑥̄̈ [𝑘 − 1] − 𝛼𝑝 2 𝑥̄̈ [𝑘 − 2] ( ) 1 + 𝛼𝑝 2 + (−2𝑥̄ [𝑘 − 1] + 𝑥̄ [𝑘 − 2]), 𝑇

(39)

and 𝜁 p is the pre-warped damping ratio. 𝑥̇ [𝑘] and 𝑥̈ [𝑘] can be calculated as: 𝑥̄̇ [𝑘] = −𝛼𝑝 𝑥̄̇ [𝑘 − 1] + = 𝑓 1 [𝑘 ] +

1 + 𝛼𝑝 𝑇

1 + 𝛼𝑝 𝑇

(𝑥̄ [𝑘] − 𝑥̄ [𝑘 − 1])

𝑥̄ [𝑘],

𝑥̄̈ [𝑘] = −2𝛼𝑝 𝑥̄̈ [𝑘 − 1] − 𝛼𝑝 2 𝑥̄̈ [𝑘 − 2] + ( = 𝑓 2 [𝑘 ] +

1 + 𝛼𝑝 𝑇

)2

(40)

(

1 + 𝛼𝑝 𝑇

)2 (−2𝑥̄ [𝑘 − 1] + 𝑥̄ [𝑘 − 2])

𝑥̄ [𝑘].

(41)

Fig. 6 shows the frequency responses of the discretized adaptive resonator equations, where the estimated frequency is 2500 Hz. The proposed method shows the relatively small magnitude and phase errors when compared to other modified bilinear transform methods because this method compensates for both frequency and damping ratio warping phenomena. The estimation law, Eq. (35), becomes: ) (√ ( )2 √ 𝜔̄̂ [𝑘]𝑥̄ [𝑘] − ||𝑥̄ [𝑘]𝑥̄̈ [𝑘]|| . (42) 𝜔̄̂̇ [𝑘] = −𝛾 Eq. (36) is discretized using Eqs. (11) and (42) as: ( ) 𝑇 𝜔̄̂̇ [𝑘] + 𝛼𝑝 𝜔̄̂̇ [𝑘 − 1] 1 + 𝛼𝑝 ) { (√ ( )2 √ 𝑇 𝜔̄̂ [𝑘]𝑥̄ [𝑘] − ||𝑥̄ [𝑘]𝑥̄̈ [𝑘]|| −𝛾 = 𝜔̄̂ [𝑘 − 1] + 1 + 𝛼𝑝 } ̄ ̇ + 𝛼𝑝 𝜔̂ [𝑘 − 1] .

𝜔̄̂ [𝑘] = 𝜔̄̂ [𝑘 − 1] +

( 1+𝛾

)−1 { 𝑇 |𝑥̄ [𝑘]| ⋅ 𝜔̄̂ [𝑘 − 1] 1 + 𝛼𝑝

+

( √ )} 𝑇 𝛾 ||𝑥̄ [𝑘]𝑥̄̈ [𝑘]|| + 𝛼𝑝 𝜔̄̂̇ [𝑘 − 1] . 1 + 𝛼𝑝

(44)

Note that the ANF outputs the warped frequency, and thus the actual frequency of the vibration can be achieved using the inverse of frequency pre-warping method as follows: √ ( ) 1 − 𝜁̂𝑝2 ⋅ 𝜔̂ 𝑝 = Im 𝑠̂𝑝 ( ( )) ( ) ( ) 1 + 𝛼𝑝 cos 𝜔̂ 𝑑 𝑇 + 𝑗 sin 𝜔̂ 𝑑 𝑇 − 1 = Im ( ) ( ) 𝑇 cos 𝜔̂ 𝑑 𝑇 + 𝑗 sin 𝜔̂ 𝑑 𝑇 + 𝛼𝑝 ( ) ( ) 1 + 𝛼𝑝 𝛼𝑝 + 1 sin 𝜔̂ 𝑑 𝑇 = , ( ) 𝑇 2𝛼𝑝 cos 𝜔̂ 𝑑 𝑇 + 𝛼𝑝 2 + 1

𝜔̂ =

(43)

Since 𝜔̂ [𝑘] is the estimated frequency, it is assumed that 𝜔̂ [𝑘] is nonnegative, and thus Eq. (43) becomes: 𝜔̄̂ [𝑘] =

Fig. 7. Frequency estimation results of case 1, where the inputs are 800 Hz sine waves with white noise.

⎛ −2𝛼𝑝 𝐾2 𝜔̂ 2 + 1 −1 ⎜ 𝜔̂ 𝑑 = cos ⎜ 𝑇 ⎜ ⎝

√( ( ) ) ⎞ 4𝛼𝑝 2 𝐾1 − 𝛼𝑝 2 + 1 𝐾2 𝜔̂ 2 + 1 ⎟ ⎟, 4𝛼𝑝 𝐾1 𝜔̂ 2 ⎟ ⎠

(45)

(46)

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Fig. 8. Frequency estimation results of case 2, where the inputs are 2500 Hz sine waves with white noise.

Fig. 11. Frequency estimation results of case 5, where the inputs are 2500 Hz sine waves with white noise and a large pulse at time 0.

Fig. 9. Frequency estimation results of case 3, where the inputs are 3000 Hz sine waves with white noise.

Fig. 12. Frequency estimation results of case 6, where the inputs are 3000 Hz sine waves with white noise and a large pulse at time 0.

where 𝜔̂ 𝑑 is the actual frequency, and K1 and K2 are: 𝑇2 𝐾1 = ( )4 , 1 + 𝛼𝑝 𝐾2 =

( ) 1 + 𝛼𝑝 2 𝑇 2 ( )4 , 1 + 𝛼𝑝

(47)

(48)

respectively. 3.3. Simulations

Fig. 10. Frequency estimation results of case 4, where the inputs are 800 Hz sine waves with white noise and a large pulse at time 0.

To evaluate the performance of the discretization methods, ANFs are implemented using MATLAB. In the simulations four ANFs were compared that are discretized using the bilinear transform with the frequency warping method, the method by Al-Alaoui [18], the methods by Yoon et al. [1] and Kim et al. [17], and the method proposed here, respectively. Note that the ANFs output the warped frequencies, and thus the actual frequencies of the vibrations can be achieved using the inverse of frequency pre-warping method. In the bilinear transform method, the

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Fig. 13. Experimental setup of the VRS.

Fig. 15. Results of ANF that uses the bilinear transform method with the frequency pre-warping method to suppress the resonance near 800 Hz. Fig. 14. Frequency response of the horizontal axis of the VRS.

inverse of Eq (3) is: ( ) 2 𝑇 𝜔𝑑 = tan−1 𝜔𝑏 . 𝑇 2

Table 1 Common simulation condition for each case.

(49)

In the method by Al-Alaoui [18], the actual frequency is achieved using Eq (8). In the methods by Yoon et al. [1] and Kim et al. [17], Eq. (10) is utilized. The inverse of frequency pre-warping for the proposed method is derived as Eq. (46). Six cases were simulated varying inputs of the ANFs to estimate the frequencies of the inputs. The inputs are ( ) 𝑢𝑚 [𝑘] = 𝐴 cos 𝜔𝑚 𝑘𝑇 + 𝑣[𝑘] + 𝑝𝑚 [𝑘], (50) where m is the case number, v is white noise, and { 𝑃𝑚 𝑘 = 1, 𝑝 𝑚 [𝑘 ] = 0 otherwise.

(51)

The parameters utilized in the simulation are presented in Tables 1 and 2. In cases 1–3, three frequencies, 800 Hz, 2500 Hz and 3000 Hz, were tested, respectively. In cases 4–6, a large pulse was added to the same inputs of cases 1–3. Note that the initial values of the estimated frequencies were 3000 Hz, and the upper and lower limits of the estimated frequencies were 3500 Hz and 100 Hz, respectively. In Table 2, steadystate error results of each case are shown. Every response in cases 1–3 reach steady state in 30 ms as shown in Figs. 7–9. The bilinear transform has the smallest steady-state errors in these cases, where the steady-state errors are less than 0.1%. However, the ANFs discretized using the method by Al-Alaoui [7] and the methods

Parameter

Value

T: sampling time A: amplitude of sinusoidal Standard deviation of white noise,v 𝛾: estimation gain a: design parameter of the method by Al-Alaoui [18] 𝛼 Y : design parameter of the method by Yoon et al. [1] 𝛼 p : design parameter of the proposed method

125 μs 10 0.5 600 0.2 0.67 0.67

by Yoon et al. [1] and Kim et al. [17] showed relatively large steadystate errors of the estimated frequencies in these cases. The steady-state errors of the method by Al-Alaoui [7] are 0.3%, 6.4% and 11.5% in cases 1–3, respectively. The steady-state errors of the methods by Yoon et al. [1] and Kim et al. [17] are 12.8%, 1.3% and −5.4% in cases 1–3, respectively. The method by Al-Alaoui [7], Yoon et al. [1], and Kim et al. [17] approximate the damped frequency to the natural frequency. The damping ratio of the mapped point in the z-plane is larger than that of the original point on the s-plane. Because of this warping of the damping ratio, the difference between the damped frequency and the natural frequency becomes large, and thus these approximations introduce the relatively large steady-state error. Moreover, the methods by Yoon et al. [1] and Kim et al. [17] have the magnitude error caused by the constant term, 2/T, in Eq. (10) that increases the frequency-estimation error. The proposed method has relatively small steady-state errors, where the steady-state errors are 0.4%, 2.2% and 0.1% in cases 1–3, respectively.

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Table 2 Parameters of the input and the steady-state error result of each case in simulations. Case #, m

Frequency, 𝜔m (Hz)

Amplitude of pulse, Pm

1 2 3 4 5 6

800 2500 3000 800 2500 3000

0 0 0 1000 1000 1000

Steady-state error (%) Bilinear transform + pre-warping

Al-Alaoui [18]

Yoon et al. [1] and Kim et al. [17]

Proposed method

0.0 0.0 0.1 Failed 0.0 0.1

−0.3 −6.4 −11.5 −0.3 −6.4 −11.5

12.8 1.3 −5.4 12.8 1.3 −5.4

0.4 −2.2 −0.1 0.4 −2.2 −0.1

Fig. 16. Results of ANF that uses the method by Al-Alaoui [18] to suppress the resonance near 800 Hz.

Fig. 17. Results of ANF that uses the methods by Yoon et al. [1], Kim et al. [17] to suppress the resonance near 800 Hz.

The results of cases 4–6 are shown in Figs. 10–12. In case 4, the ANF discretized using the bilinear transform fails to estimate the frequency of the input and becomes unstable. The results of the bilinear transform method have about 50 ms and 40 ms longer settling time than the others in case 5 and 6, respectively. However, the transient responses of the modified bilinear methods including the proposed method end in 15 ms. When rapid changes occur in the input signal due to the large pulse, the performance of the ANF discretized using the bilinear transform is deteriorated because this impact decays slower in the bilinear transform method than the modified bilinear transform methods. The steady-state responses of cases 4–6 are similar to those of cases 1–3, respectively, except the result of the bilinear transform in case 4. In this case, the ANF discretized using the bilinear transform become unstable and the estimated frequency reaches the lower limit at 22 ms.

3.4. Experiments and results 3.4.1. Experimental environments The proposed method is evaluated by the experiments using a verifyand-rework station (VRS) shown in Fig. 13, which was also utilized by Bahn et al. [23]. This machine is an automated visual inspection machine used to correct PCB pattern errors. This machine has a moving stage that includes two axes of servo systems: horizontal and vertical axes. Note that the horizontal axis is utilized in the experiments. This servo system includes an RS Automation (Pyeongtaek-si, Korea) CSD7 800 W servo drive and an 800 W servo motor. The servo system uses a proportional-integral-derivative (PID) controller, which has a global system gain parameter related to the system bandwidth. The global gain adjusts the individual gains of the PID controller. The open-loop frequency characteristics of the machine are obtained by

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Table 3 Parameters utilized in the experiments. Parameter

Value

T: sampling time 𝛾: estimation gain a: design parameter of the method by Al-Alaoui [18] 𝛼 Y : design parameter of the method by Yoon et al. [1] 𝛼 p : design parameter of the proposed method

125 μs 200 0.2 0.67 0.67

Fig. 18. Results of ANF that uses the proposed method to suppress the resonance near 800 Hz.

Fig. 19. Results of ANF that uses the bilinear transform method with the frequency pre-warping method to suppress the resonance near 2600 Hz.

adding white noise to the system, which is shown in Fig. 14. There are three main resonance points, near 150 Hz, 800 Hz, and 2600 Hz. Several notch filters are required to suppress the resonances of the system. When the system gain is in the range of 50 Hz to 200 Hz, resonances near 150 Hz and 800 Hz occur. However, an additional resonance near 2600 Hz occurs when the system gain is larger than 200 Hz. In the experiments, the ANF is activated when the level of the resonances becomes large [17]. When the resonance is suppressed by the ANF, the ANF is deactivated, and a fixed-frequency notch filter is applied, the notch frequency is the same as the estimated frequency of the ANF [17]. The four ANFs that were compared in the MATLAB simulation were implemented, and their resonance-suppression performances were evaluated. The parameters used in the experiments are shown in Table 3. Note that the initial value, the upper limit and the lower limit of the estimated frequency are 3000 Hz, 3500 Hz and 100 Hz, respectively. In the experiments, the estimated frequencies and electrical current commands were measured. Two experiments were performed; the system gain parameter was set to 150 Hz and 220 Hz, respectively in each experiment.

3.4.2. Experimental results In the first experiment, the system gain parameter was set to 150 Hz, and the resonances near 150 Hz and 800 Hz occurred. Every ANF in the experiment successfully suppressed the resonance near 150 Hz which has high damping. To show the resonance-suppression performance near 800 Hz more clearly, a fixed-frequency notch filters was set at 150 Hz. The results of the ANFs to suppress the resonance near 800 Hz are shown in Figs. 15-18. The resonance near 800 Hz was suppressed by the ANFs discretized using the method by Al-Alaoui [7] and the developed method, and the final values of the estimated frequencies are 807 Hz and 798 Hz, respectively. However, the ANFs discretized using the bilinear transform method and the methods by Yoon et al. [1] and Kim et al. [17] failed to suppress the resonance near 800 Hz. These results are similar to the results of the simulation case 4. Using the bilinear transform method, the effect of rapid changes, such as sensor noise or abrupt disturbances in the input signal were not sufficiently attenuated to estimate the resonance frequency successfully. In the methods by Yoon et al. [1] and Kim et al. [17], the frequency estimation error that was also shown in the simulation results was responsible for the failure of the resonance suppression.

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Table 4 Experimental result of each method. Case #

Resonance frequency, (Hz)

1 2

800 2600

Resonance suppression result (Final value of the estimated frequency) Bilinear transform + pre-warping

Al-Alaoui [18]

Yoon et al. [1] &Kim et al. [17]

Proposed method

Failed (-) Suppressed (2534 Hz)

Suppressed (807 Hz) Failed (2427 Hz)

Failed (-) Failed (2398 Hz)

Suppressed (798 Hz) Suppressed (2563 Hz)

Fig. 20. Results of ANF that uses the method by Al-Alaoui [18] to suppress the resonance near 2600 Hz.

In the second experiment, the system gain parameter was set to 220 Hz and a new resonance near 2600 Hz occurred. Note that two fixed-frequency notch filters were set in the second experiment, of which notch frequencies are at 150 Hz and 800 Hz, respectively. The results of the ANFs to suppress the resonance near 2600 Hz are shown in Figs. 19–22. The resonance near 2600 Hz was successfully suppressed by the ANFs discretized using the bilinear transform and the developed method, and the final values of the estimated frequencies are 2534 Hz and 2563 Hz, respectively. In the simulations, the amplitude of the sinusoidal in the input signals was constant. However, in the experiments, when the estimated frequency becomes close to the resonance frequency, the amplitude decreases as the notch filter attenuates the vibration. Fig. 6(b) shows that the resonators discretized using the method by Al-Alaoui [7] and the methods by Yoon et al. [1] and Kim et al. [17] have about 17 dB smaller magnitude than that of the exact discretization at the estimated frequency, because the damping-ratios become warped near the Nyquist frequency. Because of small magnitudes at estimated frequency, the convergence speeds become too slow when the input amplitude decreases, and thus the estimated frequencies fail

Fig. 21. Results of ANF that uses the methods by Yoon et al. [1], Kim et al. [17] to suppress the resonance near 2600 Hz.

to converge before the ANFs become deactivated. These phenomena cause the estimation errors in the experiments to become larger than those in the simulations. However, the bilinear transform method and the proposed method have relatively large magnitudes at the estimated frequency, and the estimated frequencies converge to the resonance frequency. The experimental results are summarized in Table 4, where the green cells are the cases that the ANFs successfully suppressed the resonances, and the red cells are the cases that the ANFs failed to suppress the resonances. Only the ANF discretized using the developed method successfully suppressed the resonances in both experiments. The proposed method can attenuate the effect of the rapid change in the input with sufficiently fast speed and has relatively small magnitude and phase error near Nyquist frequency, which enabled the ANF discretized using the proposed method to suppress both resonances. These results show that the developed method improves the frequency estimation performance of the ANF in the entire frequency range and enables the ANF to suppress the resonance.

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Acknowledgement This work was supported by the World Class 300 Project(R&D)(S2563339) of the Ministry of SMEs and Startups (Korea) and Brain Korea 21 Plus Project funded by National Research Foundation of Korea. References

Fig. 22. Results of ANF that uses the proposed method to suppress the resonance near 2600 Hz.

4. Conclusion A new discretization method is developed for the mechatronics systems with high-frequency components to reduce the magnitude and phase errors. The proposed method combines the modified bilinear transform method with the compensation method for the frequency and damping ratio warping. As an example, an ANF is discretized using the presented method. In the simulations, the ANF discretized utilizing the proposed method successfully estimated the vibration frequency near the Nyquist frequency with a small steady-state error, even when a large pulse is added to the input signal at the beginning of the simulation. The discretized ANF was also applied to an industrial servo system with a mechanical resonance frequency that is about 0.6 times the Nyquist frequency. The experimental results showed that the proposed method improved the frequency estimation performance, and thus the ANF successfully suppresses the resonance over the entire frequency range. Conflict of Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[1] Yoon J-M, Bahn W, Kim T-I, Han J-S, Lee S-H, Cho D-iD. Discrete derivative method for adaptive notch filter-based frequency estimators. Int J Control Autom Syst 2017;15(2):668–79. [2] Schneider AM, Kaneshige JT, Groutage FD. Higher order s-to-z mapping functions and their application in digitizing continuous-time filters. Proc IEEE 1991;79(11):1661–74. [3] Pei S-C, Hsu H-J. Fractional bilinear transform for analog-to-digital conversion. IEEE Trans Signal Process 2008;56(5):2122–7. [4] Gupta M, Jain M, Kumar B. Novel class of stable wideband recursive digital integrators and differentiators. IET Signal Proc 2010;4(5):560–6. [5] Xin Z, Wang X, Loh PC, Blaabjerg F. Realization of digital differentiator using generalized integrator for power converters. IEEE Trans Power Electron 2015;30(12):6520–3. [6] Nguyen-Van T. An observer based Sampled-data control for class of scalar nonlinear systems using continualized discretization method. Int J Control Autom Syst 2018;16(2):709–16. [7] Pan D, Ruan X, Wang X. Direct realization of digital differentiators in discrete domain for active damping of LCL-type grid-connected inverter. IEEE Trans Power Electron 2017. [8] Chen YQ, Moore KL. Discretization schemes for fractional-order differentiators and integrators. IEEE Trans Circ Syst I 2002;49(3):363–7. [9] Vinagre BM, Chen YQ, Petráš I. Two direct tustin discretization methods for fractional-order differentiator/integrator. J Franklin Inst 2003;340(5):349–62. [10] Barbosa RS, Machado JT, Silva MF. Time domain design of fractional differintegrators using least-squares. Signal Processing 2006;86(10):2567–81. [11] Krishna B. Studies on fractional order differentiators and integrators: a survey. Signal Processing 2011;91(3):386–426. [12] Romero M, de Madrid A, Manoso C, Vinagre B. IIR approximations to the fractional differentiator/integrator using Chebyshev polynomials theory. ISA Trans 2013;52(4):461–8. [13] Muresan CI, Folea S, Mois G, Dulf EH. Development and implementation of an FPGA based fractional order controller for a DC motor. Mechatronics 2013;23(7): 798–804. [14] Shah P, Agashe S. Review of fractional PID controller. Mechatronics 2016;38: 29–41. [15] Barsainya R, Rawat TK, Kumar M. Design of minimum multiplier fractional order differentiator based on lattice wave digital filter. ISA Trans 2017;66:404–13. [16] Xin Z, Loh PC, Wang X, Blaabjerg F, Tang Y. Highly accurate derivatives forLCL– Filtered grid converter with capacitor voltage active damping. IEEE Trans Power Electron 2016;31(5):3612–25. [17] Kim T-I, Han J-S, Park J-H, Oh T-H, Lee J-h, Kim S-O, Lee S-S, Lee S-H, Cho D-ID. Application of discrete derivative method with a new frequency mapping technique for adaptive-notch-filter based vibration control in industrial servo systems. In: Control Technology and Applications (CCTA), 2017 IEEE Conference on. IEEE; 2017. p. 414–19. [18] Al-Alaoui MA. Al-Alaoui operator and the new transformation polynomials for discretization of analogue systems. Electric Eng 2008;90(6):455–67. [19] Regalia PA. An improved lattice-based adaptive IIR notch filter. IEEE Trans Signal Process 1991;39(9):2124–8. [20] Bodson M, Douglas SC. Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency. Automatica 1997;33(12):2213–21. [21] Hsu L, Ortega R, Damm G. A globally convergent frequency estimator. IEEE Trans Automat Contr 1999;44(4):698–713. [22] Mojiri M, Bakhshai AR. An adaptive notch filter for frequency estimation of a periodic signal. IEEE Trans Automat Contr 2004;49(2):314–18. [23] Bahn W, Kim T-I, Lee S-H. Resonant frequency estimation for adaptive notch filters in industrial servo systems. Mechatronics 2017;41:45–57. [24] Bahn W, Yoon J-M, Hahm N-K, Lee S-S, Lee S-H, Cho D-I. Application of adaptive notch filter for resonance suppression in industrial servo systems. In: Control, Automation and Systems (ICCAS), 2014 14th International Conference on. IEEE; 2014. p. 976–9.

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Tae-Il Kim received the B.S. and M.S. degrees from the Department of Electrical Engineering and Computer Science, Seoul National University, Seoul, Korea, in 2010 and 2012, respectively. He was a research engineer in Samsung Techwin from 2012 to 2015. He is currently working toward the Ph. D. degree in the Department of Electrical and Computer Engineering at Seoul National University. His fields of interest are applications of nonlinear control theory for servo drive and sensors for motion control applications.

Young-Seok Kim received the B.S. from the Department of Electronic and Electrical Engineering, Sungkyunkwan University, Suwon, Korea, in 2018. He is currently working toward the Ph. D. in the Department of Electrical and Computer Engineering at Seoul National University. His fields of interest are applications of nonlinear control theory for servo drive and sensors for motion control applications.

Ji-Seok Han received the B.S. degree from the School of Electrical and Electronics Engineering, Chung-Ang University, Seoul, Korea, in 2015. He is currently working toward the Ph. D. in the Department of Electrical and Computer Engineering at Seoul National University. His fields of interest are applications of nonlinear control theory for servo drive and sensors for motion control applications.

Sang-Hoon Lee received the B.S. and M.S. degrees from the Department of Control and Instrumentation Engineering, Seoul National University, Seoul, Korea, in 1991 and 1993, respectively, and the Ph.D. degree from the School of Electrical Engineering, Seoul National University, in 1997. He is currently a Research and Development Center Manager with RS Automation Company, Ltd., Pyeongtaek, Korea. His current research interests include nonlinear control theory and its application to electric machines and factory automation.

Tae-Ho Oh received the B.S. from the Department of Electrical Engineering and Computer Science, Seoul National University, Seoul, Korea, in 2017. He is currently working toward the Ph. D. in the Department of Electrical and Computer Engineering at Seoul National University. His fields of interest are applications of nonlinear control theory for servo drive and sensors for motion control applications.

Dong-Il “Dan” Cho received the B.S.M.E. degree from Carnegie-Mellon University (1980), Pittsburg, PA, and the M.S and Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge (1984 and 1988, respectively). From 1987 to 1993, he was an Assistant Professor in the Department of Mechanical and Aerospace Engineering at Princeton University, Princeton, NJ. Since 1993, he has been a Professor in the Department of Electrical and Computer Engineering at Seoul National University, Seoul, Korea. He has served on the editorial board of many international journals. Currently, he is Senior Editor for IEEE Journal of MEMS and senior Editor for Mechatronics. He was the President of ICROS and BOG Member of IEEE CSS, and is currently Vice President of IFAC, Chair of the Technical Board of IFAC, and AdCom Member of IEEE EDS. He is an elected Senior Member of National Academy of Engineering of Korea.