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Proceedings of International Conference on Mechanical, Electrical and Medical Intelligent System 2019

Implementation of Disturbance Observer without Differentiator and Suppression of High Frequency Vibration

Hikaru Satoa, Shinji Wakuib Graduate School of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho,

Koganei-shi, Tokyo 184-8588, Japan a<[email protected]>, b<[email protected]>

Keywords: disturbance observer, differentiator, linear motor, high frequency vibration Abstract. Disturbance observer (abbreviated as DOB) is used to suppress disturbances. This paper proposes 3 path DOB as a new DOB. This DOB does not include a differentiator, and is easier to be implemented to the plant than conventional DOB without differentiator. In addition, an adjustment method of nominal models is proposed. This is because there are many cases that the nominal model is not obtained accurately. It is a simple adjustment method that only comparing two waveforms in the DOB. Using this method, 3 path DOB was implemented on the linear stage and confirmed its effectiveness as a DOB. Furthermore, the effect of excluding the differentiator from DOB is examined. A method of attenuating high frequencies by adding a low pass filter in the 3 path DOB is proposed.

1. Introduction

In the field of control engineering, DOB have been developed by Onishi et al. [1]. DOB can be classified into three types according to its structure. One is the DOB used in Noritsugu's research [2]. This is widely known and has a simple structure. In addition, this includes a differentiator. On the other hand, the DOB used by Onishi et al. has a complex structure and there is no differentiator. In this paper, there are two arguments for DOB without a differentiator.

The first is difficulty to use DOB without differentiator. It has a complex structure compared to one with a differentiator. Therefore, it is considered difficult to implement in the actual industrial field. DOB requires a nominal model and a low-pass filter (abbreviated as LPF). In the study, a known nominal model is set in the DOB, and the time constant of LPF is adjusted appropriately. However, there are many cases that the nominal model is not obtained accurately. For example, it is not easy to measure the mass of the machine after the assembly. In some cases, the actual torque cannot be reproduced on the actual machine, and it is difficult to measure the viscous damping coefficient and the spring constant. There are various studies to measure them [3 4], however, cost is required for using them. In addition, implementation of DOB without differentiator is difficult to determine the polarity of the summing point. This is shown in Chapter 3. In this paper, we propose a DOB without a differentiator with a new structure. The proposed DOB is easier to be implemented and adjusted than the conventional one, and has the same function as a DOB.

The second is attenuation of high-frequency signals as an application of DOB without a differentiator. The research [1], which shows a DOB without a differentiator, does not show the benefits of not having any differentiators. Therefore, the effect of excluding the differentiator is verified in this paper. This prevents high frequency signals from being amplified, and the passband of LPF can be expanded. This improves the performance of DOB. However, the effect of not using differentiators could not be shown. As a modification, the attenuation of high-frequency signals is verified by adding LPF to the proposed DOB without differentiator.


2. Linear Stage

In this paper, a linear stage is used as a plant. Figs. 1 and 2 show the appearance and experimental setup of the linear stage. The movable part of this linear stage is 4 kg, and the dimension is 270 mm wide, 180 mm long, and 15 mm high. the stage is driven by coreless type linear motor and moves in a linear direction along the linear guide. As shown in Fig. 2, the control input from the digital signal processor (DSP) is converted into a current command by the driver, which drives the stage. When the stage is driven, the position is detected by an optical encoder attached to the side of the movable portion. From the above, the plant is the linear stage and driver, and its block diagram is shown in Fig. 3. Table 1 shows the parameters used in this paper, including those shown in the figure. The transfer function from the manipulated variable u to the position x is given by Eq. (1).

2( ) tK

P sMs Ds K

(1)

Fig. 1 Appearance of linear stage Fig. 2 Experimental setup

Fig.3 Block diagram of plant

Table. 1 Parameters of this paper

Symbol Description and unit M Mass of the movable part [kg] D Viscous damping coefficient [N·s/m] K Spring constant [N/m] Kt Thrust constant of the stage with its driver [N/V] CPI PI compensator [-] KP Proportional gain [-] TI Integration time [N/m] r Reference [V] ur Output of PI compensator [V] u Manipulated variable of input signal to the plant [V] v Velocity [m/s] x Position [m]

dN Disturbance of force [N]

DSPDriver

Voltage [V]

Voltage [V]

Current [A]Voltage [V] Stage

tKu

Driver

M

1

s

1

DK

s

1 x

Stage


dV Disturbance equivalent to voltage [V] kvel Detection sensitivity of velocity [V·s/m] kpos Detection sensitivity of position [V/m] Mn Nominal model of mass [kg] Dn Nominal model of viscous damping coefficient [N·s/m] Kn Nominal model of spring constant [N/m] Ktn Nominal model of thrust constant [N/V] g Angular frequency of LPF [rad/s]

Tin Time constant of LPF in Inverse DOB [s] Ttp Time constant of LPF in 3 path DOB [s] Tf Time constant of LPF added 3 path DOB [s]

KDOB Feedback gain of DOB [s] d̂N Estimated disturbance of force [N] d̂V Estimated disturbance of voltage [V] v1 Output of summing point in w/o differentiator DOB [N] va Output of LPF [V] vb Output of product of nominal model and LPF [V]

3. DOB without Differentiator

3.1 DOB

DOB estimates the disturbance from the manipulated variable to the plant and the control variable from the sensor. Then, DOB suppresses disturbance by negatively feeding back the estimated disturbance to the manipulated variable. Fig. 4 shows DOB divided into three types. Inverse DOB [5] shown in Fig. 4(a) is the most used form among the three. In this DOB, dimensions of the manipulated variable u and the control variable v are converted into disturbance dimension and these variables are calculated at the summing point for estimation. This mechanism is easy to understand from the structure, therefore inverse DOB is easy to use. In forward DOB [6] shown in Fig. 4 (b) the dimension of the manipulated variable u is converted into that of the control variable v and both variables are compared at the summing point. The estimated disturbance is estimated by converting the output dimension of the summing point into the disturbance dimension. The calculation process is larger than the inverse. These DOBs have differentiators in the process. On the other hand, the w/o differentiator DOB shown in Fig. 4 (c) [7] consists only of the integrator and has no differentiator. It has a complicated structure with two summing points, and the structure of the DOB cannot be clearly explained like DOBs above. In this research, we are studying DOB without differentiator based on this w/o differentiator DOB.

The transfer function from disturbance dN to the estimated disturbance d̂N is the same regardless of the difference in the structure. In addition, we confirmed that each DOB can be represented to one another by the equivalent transformation of the block diagram. Therefore, each DOB is mathematically equal and has the same disturbance estimation capability.

1 1ˆ

1 1 1 1n n

N tn t NM M

d K K u dg s M g s M

(2)


Fig. 4 3 types DOB

3.2 New DOB without Differentiator

As mentioned above, the w/o differentiator DOB has the same performance as other DOBs. However, since the structure is complicated, there are difficulties in implementation. In this study, we propose no differentiator DOB with a different structure. The 2 path DOB is proposed and shown in Fig. 5. This structure is created by dividing the manipulated variable into two parts by applying partial fraction expansion to the product of the nominal model and LPF in the inverse DOB. This makes the structure of the inverse DOB without differentiator. To implement this type of DOB, first implement the inverse DOB. Next, the polarity is maintained, and DOB is converted to the 2 path DOB by the equivalent transformation. This method overcomes the difficulty of implementing the w/o differentiator DOB. This is shown in the simulation assuming the situation of implementation of (a) inverse DOB and (c) w/o differentiator DOB.

Fig. 5 2 path DOB Fig. 6 Simulation block diagram

When implementing DOB on the plant, the polarity of the control variable must be determined

first. The polarity varies depending on the mounting position and the direction of the sensor attached to the plant. Therefore, it is necessary to be able to determine substantially according to the plant. This simulation of polarity determination is performed.

A block diagram of the simulation is shown in Fig. 6. The simulation is performed by changing the DOB part of the figure to Inverse DOB or w/o differentiator DOB of Fig. 4. The plant is the speed control system modeled as an integrator, and PI compensation is implemented. However, in order to reproduce the state where the plant is not accurately modeled, the nominal model value was set to 70% of the true value. In addition, feedback gain of DOB KDOB=0. The simulation is performed by applying a square wave disturbance to the disturbance dN. In order to determine whether the sensor polarity is correct or not, simulations are performed with both polarities. In addition, we changed the parameters of the PI compensator and simulated two patterns. A waveform observed for polarity determination is shown in Fig. 7.

(c) w/o differentiator(a) Inverse (b) Forward

Ms

1

sM n

1

11 sg

sM n

u

Nd

ˆNd

tK

tnK

v

tK

11

1

sg

u

Nd

ˆNd

tnK ngM

v

Ms

1

ngM

tK

11

1

sg

u

Nd

ˆNd

tnK

v

Ms

1

11 sg

sM n

1v

tK

11

1

sg

uNd

ˆNd

tnK

v

Ms

1

1

1 1g s

ngM

tKr Nd

v u

DOB

1

Msru

velk

pK

sTI

1

1

tnKDOBK


Fig. 7 Simulations of polarity determination

When the polarity is determined, the output signal of the summing point comparing the control

variable from the sensor and the manipulated variable to the plant is observed. In inverse DOB in Fig. 4 (a), this signal is the estimated disturbance itself. Polarity can easily be determined by judging whether the estimated disturbance is similar to the disturbance or not.

Fig. 7 shows a square waveform when the polarity is correct regardless of the PI compensator parameters. On the other hand, if it is wrong, the inverse response shown in the dashed line occurs. This will be explained using mathematical expressions. If the polarity is correct, the transfer function from the applied disturbance d to the estimated disturbance d̂ is Eq. (3), and if it is incorrect, it is (4). In contrast to Eq. (3), in (4) the s2 term of the numerator is negative and there is an unstable zero. In short, the inverse response occurs when it is wrong. Therefore, it is easy to determine the polarity error from the waveform observation of the estimated disturbance in inverse DOB.

2

2

1ˆ1 1

I n P I tn P tn

I P I t P t

T M s K T K s K Kd d

g s T Ms K T K s K K

(3)

2

2

1ˆ1 1

I n P I tn P tn

I P I t P t

T M s K T K s K Kd d

g s T Ms K T K s K K

(4)

KP=0.1, TI=0.1KP=1, TI=1

Fals

eC

orre

ct0

-0.4

0.8

1.2

0.4

0

-0.4

0.8

1.2

0.4

Inve

rse

DO

B̂

Fals

eC

orre

ct

Time [s]5 100

Time [s]

0

-1

2

3

1

0

-1

2

3

1

w/o

dif

fere

ntia

tor

DO

B

5 100

Est

imat

ed d

istu

rban

ce d

[V

]v 1

[V

]


Next, w/o differentiator DOB shown in Fig. 4 (c) is considered. The output signal the summing point v1 is generated by comparison of the control variable from the sensor and the manipulated variable to the plant. In the case of inverse DOB, the signal used to determine the polarity is estimated disturbance itself. On the other hand, meaning of signal v1 is obscure in the case of w/o differentiator DOB. With KDOB=0, the v1 is expressed by Eq. (5) using the target value r and disturbance d.

2

1 2

2

tn P I tn P n t P I n t P

I t P I t P

tn P I n I tn P

I t P I t P

K MK T s K MK gM K K T s gM K Kv r

MT s K K T s K K

K K T gM T s K Kd

MT s K K T s K K

(5)

When r=0 and d = d0/s, the engineering meaning of the equation is not clear compared to inverse DOB. Also, depending on the PI compensator parameters, the s1 term becomes negative. From the simulation result of w/o differentiator DOB in Fig. 7, the shape of v1 is almost a square wave. Comparing the two waveforms under the condition of Kp=1, TI=1(left side of Fig. 7), the transient response is different. Since the difference is slight, it is difficult to judge the polarity in the actual machine. Next, when the two waveforms under the condition of Kp=0.1, TI=0.1(right side of Fig. 7), the inverse response occurs when the polarity is correct, and does not occur when the polarity is incorrect. It seems that the sensor polarity can be determined by an inverse response. In this way, implementation is difficult because the appearance of the inverse response depends on the parameter of the PI compensator.

From the above, the w/o differentiator DOB in Fig. 4 (c) is difficult to implement. On the other hand, the proposed 2 path DOB can be easily installed with the implementation of inverse DOB and equivalent transformation of its structure.

3.3 Implementation for Linear Stage

Since the plant of this research is a linear stage, the position of the stage is detected by an optical encoder. Therefore, the disturbance observer must assume a controlled object with a second order lag system. For this reason, the 3 path DOB without differentiator in Fig. 8 (b) is used. This is obtained by the equivalent transformation from inverse DOB which also assume a second order lag system. The product of the nominal model and LPF of the inverse DOB in Fig. 8 (a) is divided into three manipulated variables by applying partial fraction expansion.

Fig. 8 Second order lag DOB before and after equivalent transformation

(b) 3 path DOB(a) Inverse DOB

KDsMs

Kt

2

2

1

1inT s

Vd

V̂d av

bv

2

2

1

1n n n

tnin

M s D s K

KT s

u xKDsMs

Kt

2

2

1

1tpT s

Vd

x

ˆVd

u

2n

tn tp

M

K T

2

2

1

1

n ntp tp

n n

tp

D KT T

M M

T s

2

1

ntp

n

tp

DT

M

T s

av

bv


The implementation of the 3 path DOB requires following four steps.

Step 1: Design of Inverse DOB

Step 2: Equivalent transformation to 3path DOB

Step 3: Adjustment of nominal model Kn of spring constant K from va and vb

Step 4: Adjustment of nominal model Dn of viscous damping coefficient D from va and vb

Fig. 9 shows the block diagram of the entire experimental system.

Fig. 9 Block diagram of the entire experimental system.

3.3.1 Design of Inverse DOB

As described above, the inverse DOB is first designed in the implementation procedure of 3 path DOB. The time constant of LPF is determined from previous studies. Next, the polarity is determined on the actual machine. The reference r=0, and step disturbance is intentionally applied as the disturbance dV in Fig. 8 (a). The waveform of the estimated disturbance d̂V is compared with the waveform of the applied disturbance. Fig. 10 shows the experimental results. From this figure, the inverse response does not occur. Therefore, the polarity is correct. In addition, the estimated disturbance has the same rising direction with the input disturbance. Therefore, the polarity of the feedback is negative so that the applied disturbance is canceled by the estimated disturbance.

Fig. 10 Polarity determination

2tK

Ms Ds K

2

1

1tpT s

Vd

x

ˆVd

2n

tn tp

M

K T

2

2

1

1

n ntp tp

n n

tp

D KT T

M M

T s

2

1

ntp

n

tp

DT

M

T s

av

bv

r u

x

( )PIC s

DOBK

posk

ru

0.1

0.2

0.3

-0.10 10 20

Est

imat

ed d

istu

rban

ce [

V]

Time [s]

0

Estimated disturbance

Disturbance


3.3.2 Equivalent Transformation to 3 path DOB from Inverse DOB

The polarity of the summing point is determined. Once inverse DOB is implemented, it as converted to 3 path DOB while maintaining these polarities. Fig. 9 shows that the manipulated variable is divided into three by partial fraction expansion of the product of the inverse nominal model and LPF.

3.3.3 Adjustment of Nominal Model Kn of Spring Constant K from va and vb

DOB requires a nominal model. There are cases where the nominal model is not accurately obtained in the industrial field. We propose an adjustment method that assumes that situation. The parameters to be adjusted are the nominal model Kn and Dn of spring constant K and viscous damping coefficient D. Mass Mn and thrust constant Ktn are already known. The mass and thrust constant should be described in the specifications, and the mass can be obtained from actual measurements. From the above, similar conditions can be assumed outside of this study.

The two signals va and vb that are compared to generates the estimated disturbance are observed during the adjustment. The steady state value of the waveform is the easiest to observe. This value can be confirmed mathematically and shown in below. The transfer functions expressing va and vb using reference r and disturbance dV are respectively obtained by the block diagram shown in Fig. 9. The results of applying the final value theorem to both transfer functions are Eqs. (6) and (7). Here, reference r and disturbance dV are step functions with the amplitude r0 and dV0 (r=r0/s, dV=dV0/s).

0 0 0 0t PP

a V Vt P t P t

K KK K Kv r d r d

K K K K K (6)

0 0 00

t P n pos

tn nb V pos

t P tn

K K K k

K Kv r d k r

K K K (7)

In those equations, the sensitivity of position sensor kpos is considered as 1 since it is used to convert the dimensions of the signal. In addition, disturbance dV is not considered. The results are Eqs. (8) and (9).

at

Kv r

K (8)

0n

btn

Kv r

K (9)

The difference between right sides of two equations is whether the values of the spring constant and the torque constant are derived from the plant or the nominal model. Therefore, when the nominal model matches the value of the plant, steady state values of va and vb match. Ktn is known, so the values of Kt and Ktn are sufficiently equal. In this situation, if the gain containing Kn is adjusted and the steady state values of va and vb match, Kn will be close to K.

Equalizing va and vb signals has engineering implications. There are va and vb in both the inverse DOB and the 3 path DOB. In inverse DOB, va is the manipulated variable before being input to the plant and is not affected by disturbance. On the other hand, vb is a signal matches the dimension of the manipulated variable by applying the inverse nominal model to the control variable obtained by position sensor which is influenced by the disturbance. The engineering meaning of the summing point of inverse DOB is the estimation of disturbance by comparing the signal vb affected by the disturbance with the signal va which is not. This meaning is the same for 3 path DOB equivalently


transformed while maintaining the summing point. Therefore, equalizing va and vb in the absence of disturbance means not causing a response to the estimated disturbance waveform.

The adjustment to match steady state values of va and vb is performed on the actual machine. Fig. 11 shows va and vb after adjustment with KDOB=0. However, the response is slow as a linear motor driven positioning stage. This is because the gain of the PI compensator is decreased so that the output of the PI compensator is stable. Here, the adjustment of the DOB is focused on.

Steady state values of both signals in the figure match. Therefore, Kn is adjusted appropriately.

Fig. 11 Adjustment of Kn

3.3.4 Adjustment of Nominal Model Dn of Viscous Damping Coefficient D from va and vb

Adjustment of the nominal model Kn of spring constant K could be performed at a steady state value. The remaining parameter to be adjusted is the nominal model Dn of viscous damping coefficient D. Since Dn is not included in Eqs. (6), (7), (8), and (9), it cannot be adjusted from steady state values. Therefore, the transient response is observed. Extreme value is given as a transient response that can be easily used for adjustment. The waveform in Fig. 9 also has extreme values. This value can be confirmed mathematically. The transfer function from stage position x to vb is obtained by the simplified block diagram shown in Fig. 12. The time response of vb is obtained by applying the inverse Laplace transform to the transfer function. In this calculation, position x is a step function with the amplitude x0 (x=x0/s). The time at which the extreme value occurs is calculated from the differential of the time response. Substituting the obtained time into the time response vb gives Eq. (10). This equation shows the maximum or the minimum value.

2

2

12 20

21

ntp

n

n ntp tp

n n

DT

MD K

T TM Mn n n n

b tp tp tptn tp n n n

M x K D Kv T T T e

K T M M M

(10)

Since Dn is included in this equation, it is clear that the extreme value of vb is decided by Dn. Therefore, it is possible to adjust Dn by matching the extreme value of vb with that of va.

Dn is adjusted on the actual machine to make the extreme values of va and vb equal. Fig. 13 shows the waveforms of va and vb after the adjustment of Kn and Dn.

From the figure, extreme values of both waveforms match, and Dn can be adjusted appropriately.

0

0.2

0.3

0.4

-0.10 10 20 30

v a ,

vb

[V]

Time [s]

0.1

vb

va


Fig. 12 Block diagram for calculation Fig. 13 Adjustment of Dn

3.4 Verification

The adjustment of the nominal model was completed. Subsequently, the effect of DOB is confirmed by feeding back the estimated disturbance d̂V. The feedback gain KDOB is set to 0.6. The time responses of position x with and without DOB are compared to verify the performance in Fig. 14. In this figure, the vicinity of the target value is enlarged. From the figure, the rise is the same for both with and without DOB, and the positioning is not disturbed. When the DOB is applied, the stage reaches the reference faster than when DOB is not applied. The overshoot is suppressed as well as the drift after the settling. In addition, the settling of the positioning becomes faster.

Fig. 14 Experimental results of positioning before and after using DOB.

4 Addition of LPF to 3 path DOB

The reduction of the high frequency signal including noise was expected as the effect that 3 path DOB does not have a differentiator. However, this effect was not obtained with our linear stage. Therefore, LPF is added to 3 path DOB. LPF is added to one of the three path that has no gain. This addition is expected to reduce high frequency vibration. On the other hand, the addition of LPF causes a change in the waveform vb. This results in the degradation in disturbance estimation accuracy because DOB compares the two signals va and vb. Use of LPF whose DC gain is 1 prevents the steady state value from being affected. Therefore, this is an effective method to improve the accuracy of after positioning settling.

Fig. 15 shows a block diagram of the experimental system including DOB with LPF. There is a block that applies a sine wave to the output of the position sensor. The sine wave is applied in the DSP as high frequency vibration by this block. Suppression of this vibration is the goal of this section.

2

2

1

1

n ntp tp

n n

tp

D KT T

M M

T s

x

2n

tn tp

M

K T

2

1

ntp

n

tp

DT

M

T s

bv

0

0.2

0.3

0.4

-0.10 10 20 30

Time [s]

0.1

0 10 20 30

v a ,

vb

[V]

vb

va

w/ DOB

Commandw/o DOB

10

11

12

80 10 20 30

Pos

ition

[m

m]

Time [s]

9


Fig. 15 block diagram of experimental system with 3 path DOB with LPF The high frequency vibration is assumed to occur in the plant. Therefore, the stage position with

the high frequency signal is observed. Here, the position sensor output is observed for ease of verification.

Positioning signals are compared before and after the addition of LPF. Fig. 16 shows the experimental results. This figure shows the entire positioning waveform. From this figure, the effect of the high frequency signal cannot be seen. The amplitude of the applied high frequency signal is about 1/5 of the amplitude of the reference r. However, as the linear stage is represented by a second order lag, the plant has the characteristic of attenuating high frequency signals. For this reason, the effect of the high frequency signal cannot be observed from the graph where the whole can be observed like this. Therefore, Fig. 17 shows an enlarged waveform after the settling. In this figure, the signal amplitude using DOB with LPF is smaller than that without LPF. Therefore, it was confirmed that the addition of LPF suppressed high frequency vibration.

Fig. 16 Entire positioning waveform Fig. 17 Enlarged waveform after settling

2tK

Ms Ds K

21

1tpT s

Vd

x

ˆVd

2

n

tn tp

M

K T

2

2

1

1

n ntp tp

n n

tp

D KT T

M M

T s

2

1

ntp

n

tp

DT

M

T s

av

bv

r u

x

DOBK

posk

pK

sTI

1

1

1fT s

ru

0 10 20 308

9

10

11x 10

-3

Time [s]

Posi

tion

[m

]

x10-2

0.8Pos

ition

[m

]

1.1

1.0

w/o LPF

w/ LPF

Time [s]

Reference

0.910 20 300 20 22 24 26 28 30

0.01

0.01

0.01

0.01

0.01

Time [s]

Posi

tion

[m

]

Time [s]

w/o LPF

w/ LPF

x10-2

0.995

1.000

1.005

Pos

ition

[m

]

Reference


5 Conclusion

The 3 path DOB was proposed as a DOB without a differentiator and showed two features. One was that the implementation method was easier than the conventional w/o differentiator DOB and showed by simulation. The second was the adjustment method. It was shown that parameters of the ambiguous nominal model are adjusted by simply comparing the two waveforms va and vb. Using this method, 3 path DOB was implemented on the linear stage and confirmed its effectiveness as a DOB. In addition, LPF was added to the 3 path DOB, and suppression of high frequency signals was confirmed.

References

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[2] T. Noritsugu, and M. Takaiwa, " Robust Control of a Pneumatic Servo System Using Disturbance Observer", Transactions of the Society of Instrument and Control Engineers, Vol. 29, No. 1, pp. 86-93, 1993.

[3] J. Otsuka, I. Aoki, and T. Ishikawa, “A Study on Nonlinear Spring Behavior of Rolling Elements (1st Report) -Two Simple Measuring Methods-”, Journal of the Japan Society for Precision Engineering. Supplement. Contributed papers, Vol. 66, No. 6, pp. 944-949, 2000.

[4] Y. Toyozawa, N. Sonoda, H. Harada, and H. Kashiwagi, “Identification of a Machine Model Including Nonlinear Friction -Study of Practical On-line Identification Method for Motor Inertia and Friction-”, Journal of the Japan Society for Precision Engineering. Supplement. Contributed papers, Vol. 77, No. 7, pp. 688-693, 2011.

[5] M. Iwasaki, Y. Kitoh and N. Matsui, " Disturbance Observer-Based Nonlinear Friction Compensation in Servo Drive System", The transactions of the Institute of Electrical Engineers of Japan. D, A publication of Industry Applications Society, Vol. 117-D, No. 4, pp. 456-462, 1996.

[6] Y. Hamada, H. Otsuki, S. Saito, and Y. hata, “Application of Disturbance Observer for Head Positioning Control System of Disk Drives,” Transactions of the Society of Instrument and Control Engineers, Vol. 30, No. 7, pp. 828-835, 1994.

[7] T. Murakami and K. Onishi, "Dynamics Identification Method of Multi-Degrees-of-Freedom Robot Based on Disturbance Observer", Journal of the Robotics Society of Japan, Vol. 11, No. 1, pp. 131~139, 1993.

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