vibration suppression control for a two-inertia system...
TRANSCRIPT
Vibration Suppression Control for a Two-InertiaSystem using Load-Side High-Order State Variables
Obtained by a High-Resolution EncoderShota Yamada*, Hiroshi Fujimoto**
The University of Tokyo5-1-5, Kashiwanoha, Kashiwa, Chiba, 227-8561 Japan
Phone: +81-4-7136-3873*, +81-4-7136-4131**Fax: +81-4-7136-3881*, +81-4-7136-4132**
Email: [email protected]*, [email protected]**
Abstract—For high-precision control of a two-inertia system,information of both the drive side and the load side is usuallyrequired for obtaining high control bandwidth. In order to reducethe implementation cost and space, a novel control method, whichemploys the load side information only, is proposed using ahigh-resolution encoder which is today widely used in variousfields. Simulation and experimental results demonstrate thatthe proposed method is implementable and shows good controlperformance.
I. INTRODUCTION
Developments of position control methods have enabledus to obtain a rapid response, or high control bandwidthfor improving high control performance. However, there isan emerging problem that control system with high controlbandwidth may excite mechanical vibration, which deterio-rates control performance [1]. It is the case especially in amechanism which has a flexible joint between a motor and aload, such as the feeding tables of machine tools, the arms ofindustrial robots and rolling mills. Therefore, there is a strongdemand for vibration suppression control with high controlbandwidth, while maintaining productivity.
A two-inertia system, which has a flexible joint betweentwo rigid bodies is well-known as a model representingcharacteristics of a mechanism having mechanical resonancesin low frequency. In order to deal with the shaft twistingvibration in a two-inertia system, a lot of control methodshave been proposed, for such as the resonance ratio controland state feedback control etc. [2]– [5]. In previous studies,both the drive side and the motor side information are requiredfor vibration suppression control with high control bandwidthfor a two-inertia system [6].
In order to reduce the implementation cost and space, thispaper proposes a novel vibration suppression control methodusing a high-resolution encoder at the load side, which isnow used in various industrial fields. This proposed methodenables vibration suppression control by obtaining phase-leadinformation at the load side, which has a non-collocationproblem. Control performance of the proposed control methodis verified by simulations and experiments.
Fig. 1. Block diagram of a two-inertia system.
II. REMARKABLE CHARACTERISTICS OFA TWO-INERTIA SYSTEM
Fig. 1 shows the block diagram of a two-inertia system. Letangle, angular velocity, torque input, torsional rigidity, inertiamoment, viscous coefficient, input torque be θ, ω, T , K, J ,D, T , respectively. Suffix M denotes the motor side (or driveside), on the other hand, suffix L means the load side. For thesake of simple theoretical studies, the terms DM and DL areneglected and the transfer functions (t.f.) from TM to θM , TM
to θL are given by
θMTM
=1
JMs2s2 + ω2
z
s2 + ω2p
, (1)
θLTM
=1
JMs2ω2z
s2 + ω2p
, (2)
where ωp is the resonance angular frequency and ωz is theanti-resonance angular frequency.
ωp =
√K
(1
JM+
1
JL
)(3)
ωz =
√K
JL(4)
The equations (1), (2) indicate that t.f. θMTM
has anti-resonancewhile t.f. θL
TMdoes not have it.
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0
50
Magnitude [dB
]
100
101
102
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−100
0
Frequency [Hz]
Phase [deg]
θL
TMθM
TM
Fig. 2. Frequency responses of a two-inertia system from drive-side inputtorque to motor-side angle and load-side angle.
Fig. 3. Structure of Conventional Method 1.
Load-side feed back control bandwidth is limited by reso-nance due to phase delay. Fig. 2 shows the bode diagram oft.f. θM
TMand t.f. θL
TM. As we can see in Fig. 2, the phase of t.f.
θMTM
is not delayed more than 180 degree due to the existenceof anti-resonance, while the phase of t.f. θL
TMis delayed more
than 180 degree in a frequency band higher than the resonancefrequency because there is no anti-resonance. The problemthat instability is caused when load-side feed back controlbandwidth is higher than the resonance frequency can beconsidered in the perspective of collocation [7]. A collocatedsystem is a system which has an encoder and an actuator in thesame position while a non-collocated system is one which hasin different positions. For example a two-inertia system havingan encoder at the load side is a non-collocated system. In anon-collocated system it is difficult to retain stability underhigh gain feedback because information with displacement isfedback.
III. CONVENTIONAL RESEARCHES ABOUT TWO-INERTIASYSTEM CONTROL
TABLE I shows the comparison of control methods of atwo-inertia system. If pole placement is possible, “Possible”is written down in “Pole assignment”. Pole assignment enablesus to increase the degree of freedom in designing controlsystem and therefore it becomes easy to obtain good controlperformance. If the control system has no state error, ‘✓’is marked in “Servo”. The state error can be suppressed byapplying integral control to the load-side position measuredby a load-side encoder as far as the system does not lose itsstability.
A. Dual loop control
In the industrial society where there is a strong demand forcost reduction, semi-closed loop control was often applied [7].
Fig. 4. Structure of Conventional Method 2.
Fig. 5. Structure of the proposed method.
In semi-closed loop control it is easy to retain its stability butdifficult to have a precise positioning. Recently, a strongerdemand for a precise positioning has changed the controlmethod from semi-closed loop control to dual loop control,which requires not only the drive-side information but alsothe load-side information.
1) P-PI control: Today in the industrial society, it is com-mon to synthesize a cascade control system, which has a drive-side velocity feedback inner loop with high control bandwidthand a load-side position feedback outer loop. In this paper, thisP-PI control method shown in Fig. 3 is denoted as Conven-tional Method 1. In order to make load-side feedback controlbandwidth higher, collocated drive-side velocity control isrequired. Conventional Method 1 is easy to apply becausethe relationship between gain parameters KP , KωP , KωI andcontrol performances are clear. However, because closed-looppoles cannot be assigned arbitrarily, vibration caused by theplant resonance may appear.
2) Semi-dual loop control: Semi-dual loop control method,proposed by M. Ruderman et al. [8], aims at cost and spacesaving by eliminating a drive-side encoder. Drive-side velocityis estimated by Luenberger state observer and disturbanceobserver. The proposed method in this paper also aims ateliminating the drive-side encoder for the same purpose. Insemi-dual loop control, it is difficult to have good controlperformance because it introduces estimation delay caused byobserver and in addition it can not do pole placement.
B. State feedback control
Vibration suppression control becomes possible by statefeedback [5]. It is common to use state observer because statefeedback control needs all state variables. Observer enablesus to reduce an encoder but induces deterioration of controlperformance due to estimation delay and modeling error.
On the other hand, there is a control method, which doesstate feedback with four state variables (i.e. drive-side position,velocity and load-side position, velocity) obtained by twoencoders at the drive side and the load side. The structure ofthis method is shown in Fig. 4. This method has good control
TABLE ICOMPARISON OF CONTROL METHODS OF A TWO-INERTIA SYSTEM.
Motor-side encoder Load-side encoder Pole assignment ServoP-PI (Conv1) Required Required Impossible ×
Semi-dual loop control [8] Unrequired Required Impossible ✓State feedback by observer [5] Required Unrequired Possible ×
State feedback by encoders(Conv2) Required Required Possible ✓Proposed Unrequired Required Possible ✓
performance because it can place closed-loop poles arbitrarilyand in addition it is free of estimation delay. Let this methodbe Conventional Method 2 as a good comparison target to theproposed method.
The control method which feedbacks both drive-side po-sition and load-side position like Conventional Method 2 isused on the assumption that both-side home positions haveno displacement, but in fact there is a displacement in microor nano scale caused by the mounting error of encoders. Thisdisplacement leads to deterioration or instability. ConventionalMethod 1, which is widely used in the industrial society is freefrom this problem because it uses load-side position and drive-side velocity, not drive-side position. Therefore it is importantin terms of industrial application that control methods do notneed both drive-side position and load-side position.
IV. VIBRATION SUPPRESSION CONTROL BY STATEFEEDBACK ONLY USING LOAD-SIDE STATE VARIABLES
A. The reason for unnecessity of drive-side information
Relative order of t.f. θMTM
indicated in (1) is the second orderand therefore relative order of t.f. ωM
TMis the first order. On
the assumption that integral control is applied, relative ordershould be the first order in order not to delay in phase morethan 180 degree. Meanwhile t.f. θL
TMshown in (2) delays in
phase more than 180 degree because relative order of the t.f.is the forth order. Consequently, it is difficult to obtain highcontrol bandwidth at the load side. However, in case that load-side jerk can be obtained, relative order of the t.f. at the loadside becomes the first order, which realizes the control systemincluding integral control with only load-side state variables.
Fig. 6(a) shows the comparison of the resolution of velocityobtained by a backward difference when the resolution of anencoder is set as 1 nm and 100 nm. Though longer samplingtime can improve the resolution obtained by backward differ-ences, it induces more phase delay. Fig. 6(b) shows that thecomparison of phase delay caused by a backward difference bydifferent sampling time. More phase delay deteriorates controlperformance and even induces instability. It is quite difficultto obtain high order state variables because they requiremultiple backward differences. However, fast improvement ofthe resolution of an encoder in these days enables us to obtainhigh order state variables. The proposed method makes thesame condition at the non-collocated load side as that at thecollocated drive side in terms of phase delay. This is enabledby a high-resolution encoder which can obtain load-side jerk.The essential point of the proposed method is to recover thephase delay at the load side.
10−5
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10−3
0
2
4
6
8
10
Sampling time [s]
Ve
locity r
eso
lutio
n [
mm
/s]
Resolution 1 nmResolution 100 nm
(a) Comparison of velocity resolu-tion obtained by a backward differ-ence.
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0
100
Frequency [Hz]
Phase [deg]
Ts=1 us
Ts=100 us
Ts=200 us
(b) Comparison of phase delay causedby a backward difference.
Fig. 6. Trade-off between velocity resolution and phase delay
B. Synthesizing the proposed method
The order of t.f. θLTM
shown in (2) is the forth order, andtherefore the state space realization of θL
TMcan be given by
(5)–(10) with four state variables x = [θL ωL ωL ωL]T.
x = Ax+ bTM , y = cx (5)
A =
0 1 0 00 0 1 00 0 0 10 A42 A43 A44
(6)
A42 = − K
JMJL(DM +DL) (7)
A43 = −(K
JL+
DMDL
JMJL+
K
JM
)(8)
A44 = −(DL
JL+
DM
JM
)(9)
b =[0 0 0 K
JMJL
]T, c =
[1 0 0 0
](10)
Then state feedback control and integral control for the load-side angle θL with a new state variable xI are applied;
TM = −Fx+ xI , (11)
xI =KI
s(θ∗L − θL), (12)
d
dt
[xxI
]=
[A− bF b−KIc 0
] [xxI
]+
[0KI
]θ∗L, (13)
where KI is a gain of integral control and θ∗L is a referenceof the load-side angle. The structure of the proposed methodis shown in Fig. 5. Vibration suppression control is achievedby placing the poles of the new augmented matrix shown in(13) such that vibration can be attenuated.
(a) plant of precise positioningstage.
(b) Model of the plant.
Fig. 7. plant of precise positioning stage and its model.
TABLE IIPARAMETERS OF THE PLANT.
Carriage mass M 7.7 kgTable mass m 5.3 kgTable Inertia J 1.5 ×10−2 kgm2
Viscosity C 24 N/(m/s)Spring constant kθ 1.7×103 Nm/radDecay constant µθ 0.20 Nm/(rad/s)Length L 9.2×10−2 mLength l 8.5×10−2 mThrust coefficient Kt 27 N/A
C. The advantages of the proposed method
TABLE I indicates four advantages of the proposed method;unnecessity of a drive-side encoder, no state error, capabil-ity of pole placement, being free of the problem that thedisplacement of the home position between the drive sideand the load side (discussed in III–B). When AC motor isimplemented as an actuator, though an encoder is required atthe drive side for vector control, a lower-resolution encodercan be implemented to reduce cost compared with an encoderused in the conventional control methods.
V. SIMULATIONS AND EXPERIMENTS
A. Plant description
Aiming at industrial application, we chose the precisepositioning stage shown in Fig. 7(a) as the plant. This plantcan be modeled as a two-inertia system which consists of thecarriage at the drive side and the table at the load side likeFig. 7(b). It has two linear encoders whose resolution is 1 nmat both sides.
By linear approximation assuming θ ≃ 0, the t.f. X1
F frominput force f to the drive-side position x1 and the t.f. X2
Ffrom input force f to the load-side position x2 are given asfollowing (14)–(16).
X1
F=
b12s2 + b11s+ b10
a4s4 + a3s3 + a2s2 + a1s(14)
X2
F=
b22s2 + b21s+ b20
a4s4 + a3s3 + a2s2 + a1s(15)
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Ma
gn
itu
de
[d
B]
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102
103
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0
Frequency [Hz]
Ph
ase
[d
eg
]
Measurement
Model
Fig. 8. Frequency responses of the plant from input to drive-side position.
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Ma
gn
itu
de
[d
B]
100
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0
Frequency [Hz]P
ha
se
[d
eg
]
Measurement
Model
Fig. 9. Frequency responses of the plant from input to load-side position.
a4 = MmL2 +MJ +mJ
a3 = Mµθ +mµθ + (mL2 + J)C
a2 = Mkθ +mkθ −MmgL−m2gL+ µθC
a1 = (kθ −mgL)C
b22 = mL2 + J −mLl
b12 = mL2 + J
b21 = b11 = µθ
b20 = b10 = kθ −mgL
(16)
Fig. 8, 9 show the frequency responses of t.f. X1
F and t.f.X2
F . Solid lines indicate the measurement results and dashedlines indicate the fitted models based on (14)–(16). Identifiedparameters by fitting are shown in TABLE II. T.f. X1
F doesnot delay in phase more than 180 degree because the anti-resonance frequency is lower than the resonance frequency.However, t.f. X2
F delays in phase more than 180 degree be-cause the resonance frequency is lower than the anti-resonancefrequency. This plant has the same problem as the two-inertiasystem discussed in section II where the control bandwidth ofthe load-side feedback is limited.
The model of the plant can be realized in state spacecanonical form with state variables z = [z1 z2 z3 z4]
T. Thet.f. X2
F of the plant shown in (15) has zeros differing from thet.f. θL
TMshown in (2) of the two-inertia system discussed in
section II. Therefore the vector of the output equation of theplant is c′ = [ b20a4
b21a4
b22a4
0] = [c1 c2 c3 0], which means thatthe plant output (or load-side position) differs from z1.
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103
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0
50
100
Magnitude [dB
]
100
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103
−600
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−200
0
Frequency [Hz]
Phase [deg]
PropConv1Conv2
Fig. 10. Comparison of open-loop characteristics in ideal condition.
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0
20
Magnitude [dB
]
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102
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−90
0
Frequency [Hz]
Phase [deg]
PropConv1Conv2
Fig. 11. Comparison of closed-loop characteristics in ideal condition.
With the model shown in Fig. 7(b), three methods, Conven-tional Method 1, 2 and the proposed method are comparedin simulations. It should be noted that the proposed methodneeds only one encoder while the conventional methods needtwo encoders.
B. Simulations in ideal condition
1) Ideal condition: In this section B, simulation is con-ducted in continuous time. In order to apply the proposedmethod to the plant, x2 is converted to z1, z2, z3 with (17).
Z1
X2=
1
c3s2 + c2s+ c1,
Z2
X2=
s
c3s2 + c2s+ c1,
Z3
X2=
s2
c3s2 + c2s+ c1
(17)
Then z4 is obtained by pseudo differential whose cut offfrequency is 2 kHz. These state variables z1, z2, z3, z4 arefedback in the proposed method. Also in the conventionalmethods, pseudo differential whose cut off frequency is 2 kHzis applied to obtain velocity.
Considering the implementation, controllers in ideal condi-tion simulations are synthesized such that they have enoughstability margin. Firstly, Conventional Method 1 is synthesizedsuch that the control bandwidth of velocity control loop is 80Hz and that of position control loop is 2.8 Hz. In this case thephase margin is 69 degree. Then though Conventional Method
0 0.1 0.2 0.3 0.40
0.02
0.04
0.06
0.08
0.1
0.12
Time [s]
X2 [
mm
]
RefPropConv1Conv2
Fig. 12. Comparison of the step responses in ideal condition.
TABLE IIICOMPARISON OF PHASE MARGIN AND BANDWIDTH IN IDEAL
SIMULATIONS.
Phase margin[deg]
Bandwidth[Hz]
2% settling time ofreference the response [ms]
Conv1 69 2.8 240Conv2 69 9.2 67
Prop 69 9.2 67
2 and the proposed method can place poles arbitrarily, thepoles of the closed loop are placed in quintuple roots way suchthat the phase margins are 69 degree. These two methods havethe same poles because they have the same control structure:state feedback and integral control for the load-side position.
2) Comparison of the control bandwidth and the step re-sponses: Simulation results are shown in Fig. 10–12. Fig. 10,11 indicate that three methods’ phase margins are 69 degreeand the control bandwidth is 2.8 Hz in Conventional Method 1,9.2 Hz in Conventional Method 2 and the proposed method.TABLE III shows that the comparison of phase margin andbandwidth in ideal condition. Fig. 12 demonstrates that theproposed method can suppress the vibration and the fastresponse can be obtained by the proposed method.
C. Simulations in realistic condition
1) Realistic condition: In simulations of this section andexperiments, considering applying not only to this precisepositioning stage plant but also to a general two-inertia systemdiscussed in section II, x2 is converted to z1 by (18) not by(17) and then z2, z3, z4 are obtained by multiple backward dif-ferences. This conversion enables us to apply the synthesizingmethod proposed in section IV–B to the plant.
Z1
X2=
1
c3s2 + c2s+ c1(18)
Velocity, acceleration and jerk are obtained by multiple 5 kHzbackward differences with the high-resolution encoder whoseresolution is 1nm. Controllers are discretized by samplingfrequency 5 kHz. The Butterworth low pass filter (LPF:F (n, fc)) is designed, where n denotes the order of the LPFand fc denotes the cut off frequency. One LPF (F (2, 2k)) is
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Ma
gn
itu
de
[d
B]
100
101
102
103
−600
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0
Frequency [Hz]
Ph
ase
[d
eg
]
PropConv1Conv2
Fig. 13. Comparison of open-loop characteristics.
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0
20
Ma
gn
itu
de
[d
B]
100
101
102
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−90
0
Frequency [Hz]
Ph
ase
[d
eg
]
PropConv1Conv2
Fig. 14. Comparison of closed-loop characteristics.
applied to velocity, two LPFs (F 2(2, 2k)) to acceleration, andthree LPFs (F 3(2, 2k)) to jerk.
Conventional Method 2 and the proposed method are syn-thesized such that their poles are placed in the same places asthose of the two methods synthesized in section V–B–2). Thenthe phase margins of Conventional Method 2 and the proposedmethod become 56 degree and 45 degree respectively. Thisdifference is due to the influence of LPFs. In order to obtainhigh-order state variables, the proposed method needs moreLPFs than the conventional methods. Conventional Method 1is synthesized such that its phase margin becomes 45 degreeas same as that of the proposed method. Then the controlbandwidth of velocity control loop becomes 80 Hz and thatof position control loop becomes 5.2 Hz.
2) Comparison of the control bandwidth and the step re-sponses: Simulation results are shown in Fig. 13–15. Fig. 13,14 indicate three methods’ phase margins and that the controlbandwidth is 5.2 Hz in Conventional Method 1, 9.2 Hz inConventional Method 2 and the proposed method.
The step responses are compared in Fig. 15. TABLE IVshows that the comparison of phase margin and bandwidthin realistic simulations. The proposed method can suppressthe vibration and the fast response is obtained thanks to thehigh control bandwidth compared with Conventional Method1.
TABLE IVCOMPARISON OF PHASE MARGIN AND BANDWIDTH IN REALISTIC
SIMULATIONS.
Phase margin[deg]
Bandwidth[Hz]
2% settling time ofthe reference response [ms]
Conv1 45 5.2 390Conv2 56 9.2 68
Prop 45 9.2 67
0 0.1 0.2 0.3 0.4 0.50
0.02
0.04
0.06
0.08
0.1
0.12
Time [s]
X2 [
mm
]
RefPropConv1Conv2
Fig. 15. Step responses in realistic simulations.
D. Experiments
1) Comparison of the step responses: The conditions ofthe experiments are the same as those in realistic simulations.Experimental results of the step response are shown in Fig. 16.Experimental results are close to the simulation results shownin Fig. 15. It is presumed that the disappearance of the vibra-tion of Conventional Method 1 seen in the simulation is dueto non-linear friction.
2) Comparison of the input-disturbance responses: Theinput-disturbance responses are compared in Fig. 17. Impulse-like input disturbance which is constant at -10 N from 0.020 sto 0.030 s was applied. The response of Conventional Method1 has larger maximum amplitude and slow settling. Conven-tional Method 2 and the proposed method have the comparableperformance. TABLE V shows that the comparison of thesettling time and the maximum amplitude of the disturbanceresponse.
3) Problem of the proposed method: A difficulty of theproposed method lies in obtaining high-order state variables.Fig. 18 shows the simulation and experimental results of z4with LPFs in the step response of the proposed method shownin Fig. 15, 16. Because z4 is obtained by the third orderdifference, it is considered to be equivalent to jerk if plantis the two-inertia system shown in Fig. 1. Fig. 18 shows thatz4, which is equivalent to jerk is badly subject to the influenceof noise.
In the simulations and experiments, LPFs are applied toimprove the resolution of high-order state variables. As aresult, the phase margin of the proposed method has becomesmaller than that of Conventional Method 2. In order to obtainbetter control performance, novel methods to obtain highorder state variables precisely with less delay by applying thenovel encoder system [9] and utilizing the efficient polynomial
TABLE VCOMPARISON OF THE 2% SETTLING TIME AND THE MAXIMUM
AMPLITUDE OF THE DISTURBANCE RESPONSE IN EXPERIMENTS.
2% settling time ofthe reference response [ms]
Max amplitude ofthe disturbance response [um]
Conv1 130 20Conv2 74 6.1
Prop 66 5.1
0 0.1 0.2 0.3 0.40
0.02
0.04
0.06
0.08
0.1
0.12
Time [s]
X2 [m
m]
RefPropConv1Conv2
Fig. 16. Step responses in the experiment.
approximation [10], [11] etc. will be studied in the future.
VI. CONCLUSION
Though it was considered that both the drive-side and theload-side information were required for vibration suppressioncontrol with high control bandwidth, this paper proposed anovel vibration suppression control method only using load-side state variables by utilizing a high-resolution encoder,which is now widely used in various industrial fields.
The proposed method can remove the drive-side encoderand therefore it is advantageous in terms of cost reductionof an encoder and space saving. The proposed method hasseveral merits: unnecessity of a drive-side encoder, no esti-mation delay, no steady error, capability of pole placement,being free of the problem that the displacement of the homepositions between the drive side and the load side. Since theproposed method’s target is a two-inertia system, it has variousapplications such as industrial robots and machine tools etc.
The comparable performance of the proposed method isdemonstrated by the simulations in ideal condition. It shouldbe noted that the proposed method needs only one encoderwhile the conventional methods need two encoders.
The control performance of the proposed method is lim-ited by LPFs, which are applied to obtain high-order statevariables. Therefore, novel methods to obtain high order statevariables precisely with less delay by applying the novelencoder system [9] and utilizing the efficient polynomialapproximation [10], [11] etc. will be studied in the future.
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0 0.05 0.1 0.15 0.2−30
−25
−20
−15
−10
−5
0
5
X2 [um
]
Time [s]
0 0.05 0.1 0.15 0.2−12
−10
−8
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−4
−2
0
2
Dis
turb
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]
PropConv1Conv2Disturbance
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−0.01
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z4
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