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A Multirate Repetitive Control for Fractional-order Servos in Laser-based Additive Manufacturing

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This paper studies repetitive control (RC) algorithms to advance the quality of repetitive energy deposition in laser-based additive manufacturing (AM). An intrinsic limitation appears in discrete-time RC when the period of the exogenous signal is not an integer multiple of the sampling time. Such a challenge hampers high-performance applications of RC to laser-based AM because periodicity of the exogenous signal has no guarantees to comply with the sampling rate of molten-pool sensors. This paper investigates three RC algorithms to address such fractional-order RC cases. A wide-band RC and a quasi RC apply the nearest integer approximation of the period, yielding overdetermined and partial attenuation of the periodic disturbance. A new multirate RC generates high-gain control signals exactly at the fundamental frequency and its harmonics. Experimentation on a dual-axis galvo scanner in laser-based AM compares the effectiveness of different algorithms and reveals fundamental benefits of the proposed multirate RC.
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A Multirate Repetitive Control for Fractional-order Servos in
Laser-based Additive Manufacturing
Dan Wang and Xu Chen
Abstract This paper studies repetitive control (RC)
algorithms to advance the quality of repetitive energy
deposition in laser-based additive manufacturing (AM). An
intrinsic limitation appears in discrete-time RC when the
period of the exogenous signal is not an integer multiple
of the sampling time. Such a challenge hampers high-
performance applications of RC to laser-based AM because
periodicity of the exogenous signal has no guarantees to
comply with the sampling rate of molten-pool sensors. This
paper investigates three RC algorithms to address such
fractional-order RC cases. A wide-band RC and a quasi
RC apply the nearest integer approximation of the period,
yielding overdetermined and partial attenuation of the
periodic disturbance. A new multirate RC generates high-
gain control signals exactly at the fundamental frequency
and its harmonics. Experimentation on a dual-axis galvo
scanner in laser-based AM compares the eectiveness of
dierent algorithms and reveals fundamental benefits of
the proposed multirate RC.
Index Terms repetitive control, disturbance observer,
additive manufacturing, sampled-data control
I. Introduction
Repetitive control (RC) [1] is designed to track/reject
periodic exogenous signals in applications with repet-
itive tasks. By learning from memories of previous
iterations in the task, RC can drastically enhance cur-
rent control performance in the structured task space
and has been adopted in many fields, such as tracking
controls in magnetic and optical disk drives [2], wafer
scanners [3], and robotic manipulators [4], as well as
regulation controls in wind turbines [5], power con-
verters [6], and unmanned aerial vehicles [7].
This paper studies RC in laser-based additive manu-
facturing (AM) processes, with a focus on the selective
laser sintering (SLS) subcategory. This AM technology
applies laser beams as the energy source to melt and
join powder materials layer by layer [8]. A typical
workpiece is built from many thousands of thin layers.
Within each layer, laser gets reflected by coordinated
mirrors in a beam deflection mechanism (e.g., a dual-
axis galvo scanner) to follow trajectories predefined
by the object geometry (in a “slicing” process). This
process contains highly repetitive patterns [9], [10],
with the mirrors driven by periodic or near-periodic
reference signals. As practical servo control systems
inevitably have limited bandwidths, distinct periodic
errors are induced by dierent part geometries and
The authors are with Department of Mechanical Engineer-
ing, University of Connecticut, Storrs, CT, 06269, USA (email:
dan.wang@uconn.edu;xchen@uconn.edu)
laser path plannings. Therefore, RC algorithms, de-
signed to enhance the tracking/regulation control of
periodic references/disturbances, has the potential to
fundamentally improve the quality of the repetitive
scanning of the laser beam. Indeed, such periodicity has
been validated and leveraged upon to improve control
processes in other laser-based AM technologies [11],
[12].
The internal model principle [13] is the theoretical
basis of RC. In digital RC, an internal model 1/1zN
is implemented in the controller, where zis the complex
indeterminate in the z-transform and Nis the period
of the disturbance/reference. Nequals the sampling
frequency (denoted in this paper as 1/Tsor fs) divided
by the fundamental frequency ( f0) of the periodic sig-
nal. When Nis an integer, the repetitive controller can
generate high-gain control signals at the fundamental
frequency and its harmonics, yielding small gains in
the error-rejection dynamics. When fsis not divisible by
f0, that is, Nis not an integer, RC with the rounded N
cannot target the aimed harmonic frequencies exactly,
resulting in degraded performance. Unfortunately, such
a fundamental ineciency occurs to SLS AM because
frequency of the exogenous reference or disturbance is
closely related to scanning trajectories and is not guar-
anteed to divide the sampling frequencies of molten-
pool sensors or the galvo scanner.
Several strategies exist to potentially address prob-
lems related to fractional-order periods in RC. High-
order repetitive controllers [14], [15] introduce a series
of delay elements to widen the high-gain region around
the harmonic frequencies. Spatial repetitive controllers
[16], [17] employ a spatial domain to obtain a time-
invariant disturbance period. Adaptive RC schemes
[18], [19] adjust the sampling rate adaptively to get an
integer N. Delay-varying repetitive controllers [20] use
knowledge of the repetitive variable to continuously
adjust the time-varying delay. [6], [21] propose dierent
filters to approximate the fractional order of delay.
Complementing previous RC designs, this paper
aims at generating enhanced control eorts exactly
at desired frequencies in the fractional-order RC. The
main result is the development of a multirate RC
algorithm along with two indirect RC schemes for
fractional-order periods. First, a wide-band RC is
achieved by applying the nearest integer of Nwhile
widening the attenuation width of each frequency
notch in the error-rejection dynamics. In the second
2018 Annual American Control Conference (ACC)
June 27–29, 2018. Wisconsin Center, Milwaukee, USA
978-1-5386-5428-6/$31.00 ©2018 AACC 4831
indirect quasi RC, the fundamental frequency is shifted
to get an integer N, which creates an overdetermined
rejection of the original repetitive errors. The proposed
new multirate RC designs the internal model under a
second sampling frequency f0
ssuch that N=f0
s/f0is an
integer and hence high-gain control signals are created
at the exact frequencies. The remainder of this paper
will discuss theory, implementation guidance, and per-
formance comparison of the proposed algorithms. In
Section II, we review the basis of plug-in RC design and
introduce cross-channel periodic disturbances in laser-
based AM. The three fractional-order RC algorithms
are built in Section III. Theoretical analyses are verified
by an experimental study on a dual-axis galvo scanner
platform in SLS. Section IV provides the experimental
verification of the algorithms. Finally, Section V con-
cludes the paper.
II. Preliminaries
In this section, a plug-in RC design is first reviewed.
Then we introduce the cross-channel periodic distur-
bances in a dual-axis galvo scanner used in SLS.
A. Fundamentals of repetitive control
Consider a discrete-time feedback control system
composed of the plant Pz1and the baseline con-
troller Cz1(Fig. 1 with the dotted box removed).
Cz1can be designed by means of regular servo
algorithms, such as PID, H, and lead-lag compensa-
tion. The signals r(k),e(k),d(k), and yd(k)represent
the reference, tracking error, input disturbance, and
system output, respectively. Throughout this paper, we
assume that 1) the coecients of all transfer functions
are real; 2) both Pz1and Cz1are rational, proper,
linear, and time-invariant; 3) the baseline feedback loop
consisting of Pz1and Cz1is stable.
C(z1)P(z1)
Q(z1)
zmˆ
P1(z1)zm
d(k)
+
r(k)+e(k)
+ud(k)+
yd(k)
+
w(k)
+
+
Plug-in compensator
Fig. 1. Block diagram for a plug-in RC design.
The proposed RC algorithms are based on a plug-
in RC design [3] shown in Fig. 1. Here, the plug-in
compensator utilizes the internal signals e(k)and ud(k)
to generate a compensation signal w(k).mdenotes the
relative degree of Pz1. A nominal plant model ˆ
Pz1
is adopted to approximate the plant dynamics. With
the plug-in compensator, the transfer function of the
overall controller from e(k)to ud(k)is
Call z1=Cz1+zmˆ
P1z1Qz1
1zmQz1.(1)
The Q-filter here is designed such that 1zmQz1
contains the internal model 1 zN:
1zmQz1=1zN
1αNzN,(2)
where α[0,1) is a tuning factor that determines
the attenuation bandwidth of 1 zmQz1. A wider
attenuation bandwidth can be achieved by choosing a
smaller α. However, to get better steady-state perfor-
mance, αis preferred to be close to 1.
At the harmonic frequencies (ωk=k2πf0Ts,kZ+,
the set of positive integers), the magnitude responses
of 1 zmQ(z1) are zero because 1 ejωkN=1
ejk2πf0Ts/(f0Ts)=1ejk2π=0. Hence, |Call(z1)| → ∞
and Gdyd(z1)=P(z1)[1zmQ(z1)]
1+P(z1)C(z1)=0 when z=ejωk.
At the intermediate frequencies, Q(ejω)0, and |1
zmQ(z1)|z=ejω1 when αis close to 1; thus Call(z1)
C(z1), and the original loop shape is maintained.
During implementation, a zero-phase lowpass fil-
ter q0z1is incorporated into Qz1for robustness
against plant uncertainties at high-frequency regions:
Qz1=1αNz(Nm)
1αNzNq0z1,(3)
and one such example is
q0z1=1+z1n0(1+z)n0
4n0,(4)
where n0Zis the number of the added zero pairs at
the Nyquist frequency. The cut-ofrequency of Qz1
can be additionally adjusted by adding extra zero-
phase pairs [3].
B. Cross-channel disturbances in laser-based AM
A dual-axis galvo scanner platform in Fig. 2 is a key
component in laser-based AM for laser path planning.
Typically, it is composed of two sets of mirrors, motors,
and control systems, that is, an X channel and a Y
channel. With the collaborative rotation of the two
mirrors, the input laser beam is reflected to generate
a predefined scanning trajectory at high speed with
high precision. The rotation angles of the mirrors are
measured by encoders mounted coaxially with the
motor shaft in the scanner enclosure.
The collaborative control of the two channels intro-
duces periodic disturbances. The mechanical motion of
one rotating mirror can transmit to the other mirror. In
addition, high currents in the ground lines of the two
servo drivers can cause the channels to crosstalk [22].
When actuating one channel with a simple harmonic
signal at f0, the FFT of the non-actuated channel output
4832
Power supply
Monitor
Servo driver x
Servo driver y
ADC
ADC
DAC
DAC
Laser source
Galvo scanne r
White scr een
Server with
dSPACE and
Matlab
Fig. 2. Schematic of the hardware platform.
is observed to contain a frequency spike at f0caused
by mechanical vibrations and frequency spikes at 2n f0
(nZ+) due to crosstalk. Numerically, if the Y channel
is driven with a sine wave at 600 Hz and the X channel
has no input, frequency spikes at 1200Hz, 2400Hz,
3600Hz, etc arise at the FFT of the X output when the
crosstalk is unaccounted for.
III. Proposed fractional-order RC algorithms
We propose three RC algorithms to tackle a non-
integer Nin the internal model. Two indirect schemes, a
wide-band RC and a quasi RC, are first explored. Then
we develop the analyses and applications of the main
new multirate RC.
A. Wide-band and quasi repetitive control
The wide-band and quasi RC algorithms are variants
of the plug-in compensator in (1) and Fig. 1. Both
of them are implemented at the baseline sampling
rate of Ts. In the wide-band RC, Nis rounded to
the nearest integer. To insert high control gains at the
desired harmonic frequencies, the proposed wide-band
RC increases the attenuation width of each frequency
notch in 1 zmQz1, as shown in Fig. 3. With α
decreasing from 0.99 to 0.8, the magnitude response
of 1 zmQz1decreases from 2.8dB to 14.2dB at
1200Hz, yielding increased control eorts at the aimed
frequencies. Yet, as the attenuation width is increased
further, the amplification of intermediate frequencies
due to the waterbed eect also increases [23]. Ad-
ditional design considerations are therefore necessary
for achieving better performance, such as the gain
scheduling and the use of a damped notch filter [24].
In the proposed quasi RC, Nis designed to be
an integer by shifting the fundamental frequency to
the least common multiple (LCM) between itself and
the sampling frequency, that is, N=fs/LCMfs,f0.
The resulting new fictitious fundamental frequency
f0
sis fs/N. For example, with Ts=1/16ms and f0=
1200Hz, N=16000/LCM (16000,1200)=40, and f0
s=
16000/40 =400Hz; high-gain control signals are gen-
erated at {400iHz}(iZ+). The desired harmonics
at {1200iHz}(iZ+)are thus covered in the quasi
RC with extra servo enhancement at frequencies
{400iHz}(i,kZ+andi,3k), as shown in the dotted
line in Fig. 4.
-20
-10
0
Magnitude (dB)
1-z-mQ(z -1)
102103
Frequency (Hz)
-80
-60
-40
-20
0
Magnitude (dB)
Q(z-1 )
=0.8
=0.95
=0.99
Fig. 3. Example magnitude responses of 1 zmQz1and Qz1
in wide-band RC.
100101102103104
Frequency (Hz)
-80
-60
-40
-20
0
Magnitude (dB)
Q(z-1 )
Multirate RC
Quasi RC
-30
-20
-10
0
Magnitude (dB)
1-z-mQ(z -1)
Fig. 4. Example magnitude responses of 1 zmQ(z1) and Q(z1)
in quasi and multirate RCs.
The stability conditions for the wide-band and quasi
RCs are the same as those for the plug-in RC design in
[24], that is, ˆ
P1z1,Qz1,and Cz1belong to the
set of stable, proper, and rational transfer functions.
The wide-band and quasi RCs are indirect solutions,
only partially attenuating the disturbances. For appli-
cations that require stronger servo enhancement, we
propose more fundamental structural changes in the
controller design, as shall be discussed next.
B. Multirate repetitive disturbance attenuation
The proposed multirate RC directly addresses the
fractional-order period by introducing a second sam-
pling frequency f0
s. Let f0
sequal the greatest common
divisor (GCD) of the sampling and the fundamental
periodic frequencies, namely, f0
s=GCDfs,f0. Then
N=f0
s/f0is an integer.
The proposed multirate RC (Fig. 5) is configured by
inserting the upsampling and downsampling blocks
into Fig. 1 before and after the all-stabilizing controller.
The solid and dashed lines represent the slow and fast
signals sampled by Tsand T0
s=1/f0
s, respectively. T0
s
and Tsare related by T0
s=Ts/F(F>1 and FZ+).
The transfer functions inside the Call z1block are
implemented at T0
s.
Based on multirate signal processing (see the Ap-
pendix), the frequency response of the open-loop trans-
fer function from ydto the summing junction before
Pz1is
4833
Cdh(z1)FP(z1)F
Q(z1)
zmˆ
P1(z1)zm
d(k)
+
r(k)+e
+
udh(k)ud
+
yd
ydh(k)
+
w(k)
+
+
Call(z1)
Fig. 5. Block diagram of multirate RC.
˜
CejTs=1
F
F1
X
k=0
Call ejT0
s2πk
F.(5)
Thus, when the reference r(k) is zero (i.e., in regula-
tion problems), block diagram manipulations in Fig. 5
give the Fourier transform of the plant output yd(k):
YdejTs=PejTsDejTs
1+PejTs˜
CejTs(6)
=PejTsDejTs
1+1
FPejTsPF1
k=0Call ejT0
s2πk
F.
For example, let Ts=1/16ms and f0=1200 Hz. The
multirate RC gives T0
s=1/GCD(16000,1200)=1/48 ms.
The plug-in compensator is designed under the sam-
pling time of T0
ssuch that small gains of 1 zmQz1
are generated at the harmonic frequencies of 0=2πn×
1200rad/s (nZ+), as shown in the solid line of Fig. 4.
For disturbances at 0, since the repetitive controller
gives Call ej0T0
s→ ∞, we have the summation form
of Call in (5) also goes to infinity. Thus, YdejTsin
(6) converges to zero, yielding yd(kTs)=0 at 0.
When there is (stable and bounded) model uncer-
tainty z1such that ˆ
Pz1=Pz11+ ∆ z1,
standard robust-stability analysis gives that the closed-
loop system of the multirate RC algorithm is stable if
and only if both of the following hold:
1) Nominal stability condition is satisfied, that is, the
closed loop is stable when z1=0.
2) Robust stability requirement is met by ap-
plying the small gain theorem [25]: for any
,ejTsTejTs<1, where Tz1=
P(z1)C(z1)
1+P(z1)C(z1)is the complementary sensitivity
function.
The second condition translates to ejTs<
1/TejTs. Thus, 1/TejTsspecifies the upper
bound of the plant uncertainty at all frequencies. Fig.
6 shows the magnitude responses of 1/Tz1from
the example in Section IV. We can see that compared
with the baseline PID control, the introduction of the
multirate RC extensively preserves and increases the
robust stability bounds, especially at high frequen-
cies. With the multirate RC (solid line), the minimal
1/TejTsis 2.9 dB at 1203 Hz, which requires the
magnitude response of the uncertainty not to be greater
than 71.6% of the magnitude response of the plant at
this frequency. Without the lowpass filter q0z1in
multirate RC (dash-dot line), the minimal 1/TejTs
decreases to 10.6 dB (29.5%) at 4801 Hz.
100101102103104
Frequency (Hz)
-10
0
10
20
30
Magnitude (dB)
Multirate RC with no lowpass filter q0 in Q
Multirate RC
Baseline control
Fig. 6. Magnitude responses of 1/Tz1, which specify the upper
bounds of the plant uncertainties to keep robustness.
IV. experimental verification in a dual-axis galvo
scanner platform
This section provides implementation guidance and
performance comparison of the proposed algorithms
for rejecting fractional-order periodic disturbances in
the dual-axis galvo scanner platform in Fig. 2.
To attenuate the crosstalk-induced disturbances with
fractional-order periodicity in the X channel, the wide-
band, quasi, and multirate RCs in Section III are imple-
mented on top of the baseline X-channel controller.
The identified plant model with the sampling time
T0
s=1/48ms is
P0z1=0.061z2+0.103z+0.061
z51.485z4+1.032z30.433z20.057z0.061 .(7)
The plant model under the sampling time Ts=
1/16ms is:
Pz1=0.061z4+0.737z3+0.351z2+0.034z+0.0001
z5+0.144z40.773z30.359z20.034z0.0001 .(8)
Fig. 7 shows the frequency responses of the measured
and identified P0z1of the X channel in (7).
-100
-50
0
50
Magnitude (dB)
measured system
identified system
102103104
Frequency (Hz)
-200
0
200
Phase (deg)
Fig. 7. Bode plot of P0z1sampled at T0
s.
The plant models already contain a factory built-in
PID-type controller. The baseline feedback loop (Fig.
1 without the plug-in compensator) is thus designed
under the sampling time of Ts=1/16ms by applying
Pz1in (8) and by letting Cz1=1. Such a design
provides a 4400Hz bandwidth in Tz1. Throughout
this section, n0in the low-pass filter (4) is chosen to be
3, based on the robust stability condition.
4834
The control schemes are implemented on a dSPACE
DS1104 processor board. Without loss of general-
ity, the Y channel is run with a harmonic signal
Asin2πf0t+φwith A=8 V, f0=600 Hz, and φ=0. The
periodic disturbance emerges on the X-channel output,
with frequency spikes at 1200Hz, 2400Hz, 3600 Hz, etc
(the top plot in Fig. 8). The baseline PID-type controller
is unable to reduce the crosstalk between the X and Y
channels (the dotted line in Fig. 9).
0 1000 2000 3000 4000 5000 6000 7000 8000
0
2
4
Spectrum amplitude
10-3
Baseline control 3 = 0.013159
0 1000 2000 3000 4000 5000 6000 7000 8000
0
2
4
10-3
Wide-band RC 3 = 0.0074306
0 1000 2000 3000 4000 5000 6000 7000 8000
0
2
4
Spectrum amplitude
10-3
Quasi RC 3 = 0.0070742
0 1000 2000 3000 4000 5000 6000 7000 8000
Frequency(Hz)
0
2
4
10-3
Multirate RC 3 = 0.0044765
44%
46%
66%
Fig. 8. FFT of yd(t)with the sampling of Ts.
The wide-band and quasi RC algorithms are both
implemented at the sampling time of Ts=1/16ms.
The relative degree of Pz1in (8) equals 1 in these
configurations, that is, m=1. In the wide-band RC, N=
round1
Tsf0=round40
3=13. Under this configuration,
the frequency spikes the plug-in compensator targets
are at 1
TsN=1230.77Hz and its integer multiples. A
wider attenuation width is needed in 1 zmQz1to
include 1200Hz, 2400 Hz, 3600 Hz, etc. To achieve this
goal, αis set as 0.8, as shown in Fig. 3.
In the quasi RC, the fictitious fundamental frequency
is configured at 400Hz such that N=16000
400 =40. Thus,
the plug-in compensator generates high-gain control
signals at 400Hz and its integer multiples, covering the
target frequencies at {1200iHz}(iZ+).αis designed
to be 0.999 to achieve good steady-state performance.
The time-domain result of the quasi RC is similar to
the wide-band RC result shown in Fig. 9 and is thus
omitted here. The outputs in the time domain show
that the wide-band and quasi RCs work better than
the baseline control. The frequency-domain analysis in
Fig. 8 shows the same result, and the first two fre-
quency spikes are remarkably reduced compared with
the baseline control. Due to the decreased control eort
at high frequencies, the frequency spikes at 4800Hz and
Fig. 9. yd(t)of baseline control, wide-band RC, and multirate RC.
6000Hz remain largely unchanged.
In the multirate RC, the plug-in compensator is
designed at T0
s=1/48ms with F=Ts/T0
s=3. The relative
degree of P0z1in (7) is 3 (m=3). αis chosen to
be 0.999 to reduce the waterbed eect. In this case,
N=1
T0
sf0
=48000
1200 =40. In the solid line of Fig. 4, small
gains of 1zmQz1are exactly generated at 1200 Hz
and its integer multiples. The increased control eorts
at high frequencies yield a further-attenuated output in
the time domain (Fig. 9) and in the frequency domain
(the bottom plot of Fig. 8). Besides the attenuated
low-frequency spikes, the frequency spikes at 3600Hz,
4800Hz, and 6000 Hz are obviously reduced.
The experimental results of the four control systems
are compared in Fig. 8. As a performance metric, for
each control system, the 3σvalue of the time-domain
result is provided, where σdenotes the standard devi-
ation. Compared with the baseline control, the appli-
cation of the multirate, wide-band, and quasi RC algo-
rithms decreases the 3σvalues, thereby achieving the
desired disturbance-attenuation eect. In more details,
the wide-band and quasi RCs have similar performance
gains of 44% and 46%, respectively. The multirate RC
outperforms the other two by reaching a 66% decrease
of the 3σvalue.
V. Conclusion
In this paper, three repetitive control (RC) algorithms
are proposed to overcome the intrinsic limitation of
the internal model 1/1zNwhen the sampling fre-
quency is not divisible by the fundamental disturbance
frequency. The proposed wide-band RC scheme widens
the attenuation widths to include the desired distur-
bance frequencies. The quasi RC method creates an
integer Nby selecting the least common multiple of the
sampling and fundamental disturbance frequencies as
the fictitious fundamental frequency. The multirate RC
applies a second divisible sampling frequency, that is,
the greatest common divisor of the baseline sampling
frequency and the fundamental frequency. Experimen-
tal results verify the eectiveness of all three algo-
4835
rithms, and the multirate RC algorithm outperforms
the wide-band and quasi ones by enhancing control
eorts at high frequencies.
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Appendix
Proof of (5)
The downsampling operation in Fig. 5 gives
Ud(ejTs)=1
F
F1
X
k=0
Udh(ej(T0
s2πk
F)).(9)
From the upsampling operation, the Fourier trans-
forms of ydand ydh are related by
Yd(ejTs)=Ydh(ejT0
s).(10)
Let r(k)=0. Note that
˜
C(ejTs)=Ud(ejTs)
Yd(ejTs),Call(ejT0
s)=Udh(ejT0
s)
Ydh(ejT0
s),(11)
and thus
F1
X
k=0
Call(ej(T0
s2πk
F))=
F1
X
k=0
Udh(ej(T0
s2πk
F))
Ydh(ej(T0
s2πk
F))
.(12)
The denominators of the terms on the right side of
(12) satisfy
Ydh(ej(T0
s2πk
F))=Ydh(ej(2πk
Ts)T0
s)
=Yd(ej(2πk
Ts)Ts)=Yd(ejTs),(13)
where the second equal sign is due to the upsampling
in (10). Substituting (13) and (9) into (12) gives
F1
X
k=0
Call(ej(T0
s2πk
F))=PF1
k=0Udh(ej(T0
s2πk
F))
Yd(ejTs)
=FUd(ejTs)
Yd(ejTs).(14)
Hence, according to (11),
F1
X
k=0
Call(ej(T0
s2πk
F))=F˜
C(ejTs),(15)
which is equivalent to (5).
4836
... [18] uses a correction factor to correct the deviated poles of the fractional-order repetitive controller. [19] introduces two wide-band and quasi RCs together with a multirate RC in a plug-in configuration; the full closed-loop properties, however, are not investigated. ...
... Note that the transfer functions inside the C all (z) block are all implemented at T s . Base on multirate signal processing [19], the frequency response of the open-loop transfer function from y d to the summing junction before P(z) isC(e jΩT s ) = 1 F ∑ F−1 k=0 C all (e j(ΩT s − 2πk F ) ). Thus, when the reference r(k) is zero (i.e., in regulation problems), the Fourier transform of the plant output y d (k) is ...
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