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Multi-band beyond-Nyquist Disturbance Rejection on a Galvanometer Scanner System

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In many laser processing applications, galvanometer scanners are integrated into a larger system that creates external disturbances to the actual laser-material interaction. To reject such disturbance with feedback-based control schemes, the sampling of the output needs to be fast enough to capture all major frequency components of the disturbance. In some applications, however, the sensor’s sampling speed is limited, such that the disturbance is beyond the sampler’s Nyquist frequency. This paper introduces a multi-rate control scheme to fully reject narrow-band beyond-Nyquist disturbances in galvanometer scanner systems. The proposed algorithm consists of a special bandpass filter with tailored frequency response, and a model-based predictor that reconstructs signals from limited sensor data. Verification of this algorithm is conducted by both simulated and experimental results on a commercial galvanometer scanner testbed.
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Multi-band beyond-Nyquist Disturbance Rejection on a Galvanometer
Scanner System
Hui Xiao and Xu Chen
Abstract In many laser processing applications, galvanome-
ter scanners are integrated to a larger system that creates
external disturbances to the actual laser-material interaction. To
reject such disturbance with feedback-based control schemes,
the sampling of the output needs to be fast enough to capture
all major frequency components of the disturbance. In some
applications, however, the sensor’s sampling speed is limited,
such that the disturbance is beyond the sampler’s Nyquist
frequency. This paper introduces a multi-rate control scheme
to fully reject narrow-band beyond-Nyquist disturbances in
galvanometer scanner systems. The proposed algorithm consists
of a special bandpass filter with tailored frequency response,
and a model-based predictor that reconstructs signals from
limited sensor data. Verification of this algorithm is conducted
by both simulated and experimental results on a commercial
galvanometer scanner testbed.
Index Terms disturbance rejection, beyond Nyquist, gal-
vanometer scanner.
I. INTRODUCTION
Galvanometer scanner are one of the widely used scanning
systems for many high-end applications in laser market
including marking, micro-machining, rapid prototyping, cut-
ting/routing and welding. It offers a high-speed, accurate and
flexible way to position the laser beam under a reasonable
cost. The same benefits also make it an attractive tool in some
other application domains, such as optical-resolution pho-
toacoustic microscopy [1] and a galvanometer-laser-scanning
based indoor localization system [2]. Among those applica-
tions, accurate and fast scanning is a crucial requirement for
galvanometer scanner systems, as the direction of the laser
beam is extremely sensitive to the scanner mirror. Therefore,
designing high precision controllers that are robust to noise
and disturbances is of great importance for a galvanometer
scanner system.
In most cases the galvanometer scanners are integrated to
a more complex system. Take the selective laser sintering
(SLS) in additive manufacturing as an example. Usually a
two-dimensional galvanometer scanner is used to position the
laser spot on the working surface [3]. On the one hand, the
increasing demand of printing speed and product quality is
pushing the requirement of a larger bandwidth in controlling
the galvanometer scanners. On the other hand, the frequency
of the disturbance—caused by platform vibration and atmo-
spheric turbulence—is increased by a higher printing speed.
†: corresponding to: 191 Auditorium Road U-3139, Storrs, CT, USA,
06269-3139, Tel.:860-486-3688.
Xu Chen (email: xchen@engr.uconn.edu) and Hui Xiao (email:
hui.xiao@uconn.edu) are with the graduate students and the faculty
in the Department of Mechanical Engineering, University of Connecticut,
respectively.
In order to compensate the system disturbance, an image
sensor can collect the laser spot positions on the working
surface. However, the image sensor’s sampling rate is usually
slow, because it takes time to transfer and process the massive
image data. Consequently, the disturbance frequency could
be near or beyond the Nyquist frequency of the image
sensor, and the controller will be incapable to reject the
disturbance, leading to defective 3D-printed products. The
same problem here could arise in other galvanometer scanner
applications. One can choose a feedback sensor with a higher
sampling rate, but the cost may be prohibitive, and it may
not be technically feasible in certain applications. This paper
provides a control-oriented method to reject beyond-Nyquist
disturbances, enabling controlled system response beyond
conventional performance limits.
Recent research results on disturbances/jitters suppression
on galvanometer scanner systems include adaptive con-
trol algorithms based on recursive least-squares [4], [5] or
frequency-weighted minimum-variance control [6], extended
PID control [7], [8], predictive control [9], linear quadratic
Gaussian control [10], and iterative learning control [11].
However, beyond-Nyquist disturbance rejection on a gal-
vanometer systems has not entered the radar yet. If han-
dled incorrectly, the disturbance could cause unobserved
performance loss in the actual scanner system. Our former
theory work [12] provides a mixed-rate feedback solution
for rejecting beyond-Nyquist disturbance by introduction
of a forward-model disturbance observer (FMDOB) and a
model-based multirate intersample data predictor (MMP)
with finite impulse response (FIR) structure. In this paper,
we substantially enhanced this algorithm and integrated it
into a galvanometer scanner control system. Built on top of
the scanner’s baseline controller, a new design of FMDOB is
adopted here to ensure the close-loop stability and separate
multi-band disturbance signal from the output signal. Based
on the frequency distribution of the disturbance, a new MMP
with infinite impulse response (IIR) structure is designed to
reconstruct the disturbance signal into a fast sampled one,
enabling the possibility of exact disturbance rejection at a
higher sampling rate.
The remainder of this paper is organized as follows.
Section II formulates the disturbance rejection problem. The
proposed multi-rate servo scheme for disturbance rejection
beyond the Nyquist frequency is presented in section III. Sec-
tion IV discusses the design of the model-based predictor that
uses the disturbance structure to recover the faster sampled
signals. Section V shows simulation as well as experimental
results on a two degree of freedom (DOB) galvanometer
Fig. 1. Block diagram of a galvanometer control system.
scanner testbed, and finally Section VI concludes this paper.
II. PROB LEM FORMULATION
Consider the case when a galvanometer scanner is used to
position a laser beam. The scanner is placed before an optical
system, so that the scanner mirror can reflects the laser beam
to its desired position. Figure 1 shows the control block
diagram of such system. The main elements here include
the continuous-time plant Pc(s), the discrete-time controller
C(z), and the signal holder H.y0is the angular position of
the scanner mirror, measured by an encoder with a sampling
rate of fs1=1/Ts.yis the actual planar position of the laser
spot. Here, Gstands for the coordinate transformation and
the optical path from y0to y, which is defined by the optical
system’s forward kinematics.
For a more concise description, we classify the system
disturbances into two parts. d0is the disturbance that influ-
ences the angular position of the galvanometer mirror, such
as the vibration of the mirror support, or torque disturbance
of the motor; dis the disturbance caused by the vibration
of the optical system, or the atmospheric turbulence that can
impact the beam path. Suppose both d0and dcontain high
frequency components. In order to reject do, one can choose
a high-speed encoder and design a suitable PID controller,
Hcontroller, or observer-base controller, etc. The focus of
this paper is on the more challenging rejection of d, which
the existing controller does not have direct access.
In this paper, the focused problem is as follows. Suppose
dconsists of narrow-band disturbances that can be approxi-
mated as
d(t) =
m
i=1
λisin(2πfit+φi),(1)
where mdenotes the number of frequency bands. The
frequencies fi’s are assumed known (can be derived or
identified, see, e.g., [13]–[15]), but the amplitude λiand
phase φiare unknown. In order to reject the disturbance d, we
use another sensor (e.g. camera) to capture the beam position
y. This sensor has a much slower sampling rate ( fs<fs1),
which is not fast enough to capture all disturbance frequency
bands (i.e. there exist fisuch that fi>fs/2). Under such a
problem configuration, we design a multi-rate control system
to fully reject the disturbance dat a higher sampling rate of
fs1.
III. MULTI RATE FORWAR D-MODEL DISTURBANCE
OBS ERVER
The proposed multirate control scheme is presented in
Figure 2. The sampled signals are divided into two groups,
Fig. 2. Multi-rate control scheme for beyond-Nyquist disturbance rejection.
each with a different sampling rate: one is fast sampled at
fs1—indicted by the dotted line, and the other one is slow
sampled at fs=fs1/L,LZ+, which is indicted by the
dashed line. To avoid amplifying the system disturbances
d0, the sampling rate fs1needs to be high enough such that
fs1/2>fmax, where fmax is the highest frequency band of
the system disturbances. Suppose the ZOH (zero-order hold)
equivalent of Pc(s)is Pd(z), then P
d(z)4
=Pd(z)C(z)/(1+
Pd(z)C(z)) is the identified close-loop galvanometer scan-
ner model sampled at fs1. Let G1stand for the inverse
coordinate transformation of the optical system. Then one
can observe that a slow sampled disturbance is estimated by
ˆ
dL:[n]after the proposed signal processing in Figure 2. For
this proposed servo scheme, Q(z)and MMP are the two key
elements, which will be discussed in the remainder of this
section and the next section, respectively.
If one replaces the closed-loop galvanometer plant with
P
d(z), omits the MMP block, and cancels out Gwith G1,
the basic structure of the disturbance compensation algorithm
becomes a forward model disturbance observer in section II-
A of [16] (redrawn in fig 3). For completeness, we just briefly
introduce the key concepts of FMDOB below.
In Figure 3, the system output yd[n]can be derived as
Y(z) = Pd(z)U(z)+(1Pd(z)Q(z))D(z).(2)
The relationship between the command signal u[n]and the
system output yd[n]is thus independent from the feedback
loop. In addition, the feedback loop introduces additional
dynamics between disturbance d[n]and system output yd[n],
which enables the possibility of perfect disturbance rejection.
More specifically, if we design the filter Q(z)such that [16]
1Pd(ejωi)Q(ejωi) = 0,(3)
where ωiis the disturbance frequency in radians per second,
then the last term of equation (2) will be canceled out, result-
ing in full rejection of disturbance. It is worth mentioning
that it is not always feasible to assign a full inversion of
Pd(z)to Q(z), because P1
d(z)may not be a proper transfer
function, or have unstable poles that will lead to instability.
In this paper, we give an extended design of Q(z). Suppose
the disturbance can be approximated by equation (1). The
following theorem provides a point-wise stable inversion of
Pd(z)at multiple frequencies, while maintaining a small gain
to |1Pd(ejω)Q(ejω)|when ω6=ωi.
Fig. 3. Forward-model disturbance observer.
Theorem 1: Let ωi=2πfiTsbe the frequency of a dis-
turbance component, Pd(ejωi)be the frequency response
of the plant Pd(z)at ωi, and assume that |Pd(ejωi)| 6=0,
i=1,2,...,m(otherwise no feedback design can achieve the
disturbance rejection). Let p=2m1,and
Q(z) = Q0(z)(q0+q1z1+· · · +qpzp),(4)
with
q0
.
.
.
qp
=
1 cosω1··· cos pωp
0 sinω1··· sin pωp
.
.
..
.
.....
.
.
.
.
..
.
.....
.
.
1 cosωp··· cos pωp
0 sinωp··· sin pωp
1
RPd(ejω1)
|Pd(ejω1)|2
IPd(ejω1)
|Pd(ejω1)|2
.
.
.
.
.
.
RPd(ejωp)
|Pd(ejωp)|2
IPd(ejωp)
|Pd(ejωp)|2
.
(5)
Here,1
Q0(z) = 1
m
i=11
2
1+2k1,i(1+k2,i)z1+ (1+k2,i)z2
1+k1,i(1+k2,i)z1+k2,iz2,
(6)
where
k1,i=cos(ωi),(7)
k2,i=1tan(πBw,iTs)
1+tan(πBw,iTs).(8)
Then equation (3) holds for each ωi, and the amplification
at ω6=ωiis controlled by choosing Bw,i, which is the 3-dB
disturbance-rejection bandwidth of Q0(z)centered around ωi.
Proof: For each ωi, equation (3) has the solution
Q(ejωi) = 1
Pd(ejωi)=Pd(ejωi)
|Pd(ejωi)|2,
i.e.
RQ(ejωi) = RPd(ejωi)
|Pd(ejωi)|2
IQ(ejωi) = IPd(ejωi)
|Pd(ejωi)|2
,i=1,2,...,m.(9)
Define
Q(z) = q0+q1z1+qpzp(10)
1RPd(ejω0)and IPd(ejω0)are the real part and the imaginary part of
Pd(ejω0), respectively.
such that Pd(ejωi)Q(ejωi) = 0, then by equation (9), we must
have, for i=1,2,...,m,
q0+q1cosωi+· · · +qpcospωi=RPd(ejωi)
|Pd(ejωi)|2,
q1sinωi+· · · +qpsinpωi=IPd(ejωi)
|Pd(ejωi)|2.
There are msuch equation sets, or 2mlinear equations. Since
ωi[0,π),and ωi6=ωjif i6=j, those linear equations are
independent from each other. Then we have 2mlinearly
independent equations and p+1=2munknowns, thus there
exists unique solution of qi’s, which are given by (5).
Q0(z)in (6) is a multi-band bandpass filter that has m
narrow band-pass range centered at ωi. It is produced by
1¯
Q(z), where ¯
Q(z)is composed of mcascaded lattice-
based band-stop filters [17], [18] whose bandwidth is related
to k2,iin equation (8). One can show that Q0(ejωi) = 1 at
each center frequency ωi, then combining (6) and (10), we
have equation (3) for each ωi.
For frequency range ω6=ωi,|Q0(ejωi)|can be made
arbitrarily small by reducing the bandwidth Bw,i. Thus |1
Pd(ejωi)Q(ejωi)|is controlled to be small, avoiding large
noise amplification.
IV. MOD EL-BASED PREDICTO R DESIGN
Recall that in Figure 2, the forward model disturbance
observer is working at a fast sampling rate of fs1, but the
sampling rate of the system output is limited at fs=fs1/L.
Therefore, an algorithm is needed to upsample the feedback
signal. More specifically, we need to design a predictor that
can estimate the intersample data, which is missing due to the
slow sampling rate. In this paper, we introduce a model-based
predictor with an IIR structure. Compared to the former FIR
design [12], it is more robust to noise, and is capable of more
accurate prediction.
Theorem 2: If a multi-band disturbance defined
by equation (1) has mfrequency components
f= [ f1f2... fm],and dL[n] = dc(nLTs)is the
slow sampled disturbance signal with sampling time LTs,
d[n] = dc(nTs)is the fast sampled disturbance signal with
sampling time Ts. Then d[n]can be fully recovered from
the slow sampled disturbance dL[n]by
d[nL] = dL[n],
and for k=1,2,...,L1,
d[nL +k] = yk[n] = wk·ϕd[n]b·ϕy[n].(11)
Here, ykis the k-th recovered intersample signal between
d[nL]and d[(n+1)L],ϕd[n]and ϕy[n]are column vectors
that are defined as
ϕd[n] = [dL[n],dL[n1],··· ,dL[n(2m1)]]T,(12)
ϕy[n] = [yk[n1],yk[n2],··· ,yk[n2m]]T.(13)
wkand bare parameter vectors that are defined as
wk=wk,0,wk,1,··· ,wk,(2m1),(14)
b= [b1,b2,··· ,b2m].(15)
The parameter vector bis composed of the coefficients of
polynomial B(z1) = 1+b1z1+··· +b2mz2m,computed
from
B(z1) =
m
i=1
(12αcos(2πfiLTs)z1+α2z2),(16)
in which αis a constant that is less than 1 but larger than
0. wkis solved by constructing the linear equation
Mk
fk,1
.
.
.
fk,2m(L1)
wk,0
.
.
.
wk,2m1
=
a1
a2
.
.
.
a2m
0
.
.
.
0
+¯
b,(17)
Mk
4
= [ ˜
Mk|ekek+L··· ek+(2m1)L],(18)
where Mkis a square matrix with a dimension of 2mL×2mL,
and ˜
Mkis defined as
˜
Mk
4
=
1 0 . . . 0
a1
.......
.
.
.
.
.......0
a2m
......1
0......a1
.
.
........
.
.
0... 0a2m
2mL×2m(L1).
(19)
[a1,a2,··· ,a2m]in equation (17) are the parameters of the
polynomial A(z1) = 1+a1z1+a2z2+··· +a2mz2m,
which is computed by
A(z1) =
m
i=1
(12cos(2πfiTs)z1+z2).(20)
The column vector ¯
bin the rightmost of equation (17)
contains all zeros, except for the L,2L,··· ,2mL-th entries,
which equals b1,b2,··· ,b2m.
Proof: We construct
Fk(z1)A(z1) + zkWk(zL)B(zL) = 1,(21)
where A(z1)is defined by equation (20),
Fk(z1) = 1+fk,1z1+· · · +fk,2m(L1)z2m(L1),(22)
Wk(zL) = wk,0+wk,1zL+·· · +wk,2m1z(2m1)L,(23)
B(zL) = b1zL+b2z2L+· · · +b2mz2mL.(24)
The coefficients of B(zL)are the same as those in B(z1)
(computed by equation (16)). Based on internal model prin-
ciple, for multiple narrow-band disturbances, an important
property of A(z1)is that, at the steady state, we have
A(z1)d[n] = 0.Combining this with equation (21) yields
that, at steady state,
1zkWk(zL) + B(zL)d[n] = 0,
The corresponding difference equations are,
d[n] = wk,0d[nk] + · ·· +wk,2m1d[nk(2m1)L]
b1d[nL]b2d[n2L]− · ·· − b2md[n2mL].(25)
Changing variables and replacing nwith nL +k, we have
d[nL +k] = wk,0d[nL] + · · · +wk,2m1d[((n(2m1))L]
b1d[(n1)L+k]− · ·· − b2md[(n2m)L+k].(26)
Recall that dL[n] = d[nL]and yk[n] = d[nL+k]. Then equation
(26) can be written in the form of equation (11).
Now consider solving (21). expanding the equation and
collecting the coefficients of zis (i=1,2,...,2mL),one can
get 2mL linear equations with 2mL unknowns. Thus there
exists a unique solution that can be written in the matrix
form (17).
For a predictor with an IIR structure, the intersample signals
yk[n]are computed not only from weighted sum of d[n], but
also by using the historical prediction values. This prediction
algorithm can be characterized by the transfer function from
d[n]to yk[n], which is derived from 11:
W
k(z1) = wk,0+wk,1z1+· ·· +wk,(2m1)z(2m1)
1+b1z1+· · · +b2mz2m,(27)
where the denominator comes from equation (16), and the
numerators are solved by equation (17). Compared to the
case with an FIR design, the IIR structure W
k(z1)has a
new design freedom in the denominator. The benefits of this
new freedom is elaborated by an example below.
In equation (16), αis a constant that determines the
weighting of input signals and historical prediction signals
when combining them together to make a prediction. The
closer αis to 1, the heavier the weighting of historical
prediction signals will be. Figure 4 shows the bode plot
of W
k(z1)when choosing different α. In the plot, the
disturbances/signals consist of three frequency bands, f1,
f2and f3, where f1is below Nyquist and the other two
are beyond-Nyquist. The result shows that the frequency
response of the predictors are the same at f=fi,i=1,2,3,
but are dramatically different at other frequency regions.
Generally speaking, a greater αwill lead to a much smaller
magnitude response below 0dB; but an FIR predictor (α=0)
usually has a magnitude response above 0dB.In other words,
an IIR predictor with a larger αis more robust to input noise
compared to the one with smaller α; an FIR predictor is
sensitive to input noise (see Figure 5). In practice, in order
to reduce the undesired influence of measurement noise and
increase the prediction accuracy, αshould be chosen closer
to 1, e.g, starting with α=0.9. The value can be further
increased when dealing with noisy applications.
V. SIMULATED AND EXPERIMENTAL RESULTS
In this section, we present the simulated and experimental
results of beyond-Nyquist disturbance rejection on a gal-
vanometer scanner. We use the 6215H galvanometer scanner
100 200 300 400 500 600 700
-30
-20
-10
0
10
20
Magnitude (db)
= 0.5
= 0.8
= 0.95
FIR predictor
100 200 300 400 500 600 700
Frequency (Hz)
0
200
400
600
800
Phase (deg)
f1 = 187.5Hz
f3 = 1125Hz
Aliasing band
Nyquist frequency
= 625Hz
f2 = 812.5Hz
Aliasing band
Fig. 4. Bode plot of the IIR and FIR predictor. Suppose the sensor sampling
rate is limited at T s =0.8ms, and the disturbance has three bands at f1=
187.5Hz,f2=812.5Hz and f3=1125Hz.Two of them is beyond-Nyquist
frequency ( fN=625Hz)and are marked as red dashed lines.
0 10 20 30 40 50 60 70 80 90 100
Step
-1.5
-1
-0.5
0
0.5
1
Prediction error
IIR preidictor
FIR predictor
(a)
0 10 20 30 40 50 60 70 80 90 100
Step
-1
-0.5
0
0.5
1
Prediction error
IIR preidictor
FIR predictor
(b)
Fig. 5. Prediction error of IIR (α=0.95) and FIR (α=0) predictor, under
the same configuration as Figure 4, and the regressor vectors ϕdand ϕyare
initialed as zero vectors. (a) the input has a random noise with maximum
amplitude of 0.05. The IIR predictor shows better robustness to noise. (b) if
the input is noise-free, both the IIR and FIR predictors can made an accurate
prediction in steady state.
from Cambridge Technology Inc (Figure 6). It is a two-
axis scanner. Each axis has ±20 degrees of scan angles and
8µrad of repeatability. It comes with a driver board that has a
built-in motor driver circuit and pre-tuned PID-type control
algorithms, also with a position sensor to capture the mirror
angles. The closed-loop galvanometer scanner model P
d(z)is
identified by system identification techniques [19] at a high
sampling rate of 10kHz, and the result is
P
d(z) = 0.551z1+0.356z20.0496z3+0.00673z4
10.164z1+0.0278z20.00166z3+0.000152z4.(28)
Suppose the optical system has high-frequency disturbances.
We use Ethernet cameras to detect the output beam positions
of the optical system. Because of the computation budget in
image processing, the minimum sampling time of the vision
sensor is limited at 0.3ms. For narrow-band disturbances,
one can use the regular disturbance observer [20] to achieve
perfect disturbance rejection. However, If the disturbance
has frequency bands greater than fN=1667Hz (the Nyquist
frequency of the vision sensor), the actual plant output will
be significantly amplified [12].
Fig. 6. The galvanometer scanner testbed.
In order to show the effectiveness of the proposed al-
gorithm, we first build the same control scheme as in
Figure 2 in MATLAB. The disturbance in the simulation
has three frequency components at 0.8fN,1.6fNand 2.3fN,
respectively. The the fast and slow sampling times are fs1=
10kHz and fs=10/3kHz. Figure 7 shows the system output
sampled at 10 kHz. The doted and solid lines are the system
outputs when proposed disturbance observer is turned off
and on, respectively. The results indicate that the proposed
algorithm has the ability of full disturbance rejection, as
the fast sampled outputs y[n]converge to zero when the
disturbance observer is turned on.
0 0.5 1 1.5
Time (sec)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Normalized output
compensation off
compensation on
Fig. 7. Plant output sampled at 10kHz.
Figure 8 shows the time-domain disturbance reconstruc-
tion results by the IIR predictor with α=0.95. The dashed
line represents the real-time disturbance signal. The distur-
bance samples are marked with ×, and the reconstructed
disturbance samples is marked with #. In the simulation, we
include a white noise with the maximum amplitude of 0.03
to the input of the IIR predictor. The predictor successfully
recovered the intersample data from the noisy and slow
sampled signal.
In the experiment on the galvanometer scanner, we feed
a 2kHz beyond-Nyquist disturbance into the system. Figure
9 shows 10kHz sampled time-domain system outputs. After
0.3085 0.309 0.3095 0.31 0.3105 0.311 0.3115
Time (sec)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
normalized output
Real-time disturbance signal
Ts sampled disturbance
Reconstructed disturbance
Fig. 8. Disturbance reconstruction results.
Fig. 9. Galvanometer scanner outputs sampled at 10kHz.
the disturbance compensation loop was enabled at t=2.5
seconds, the outputs drop dramatically, yielding a 90%
disturbance amplitude rejection (not 100% because there
is some saturation nonlinearity and wide band noise). The
same disturbance rejection results can also be shown in the
frequency domain (Figure 10). When the compensation loop
was disabled, the outputs has a large spike at 2kHz, which
disappeared after turning on the proposed algorithm.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
0.05
0.1
Normalized amplitude
compensation on
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency (Hz)
0
0.05
0.1
Normalized amplitude
compensation off
Fig. 10. FFT of Galvanometer scanner outputs sampled at 10kHz.
VI. CONCLUSION
In this paper, the problem of beyond-Nyquist disturbance
rejection on the galvanometer scanner is addressed. Based on
the frequency information about the disturbance, we designed
the multi-rate model-based predictor that can accurately
recover the intersample information from a slow sampled dis-
turbance. Combined with a forward model disturbance struc-
ture, the MMP enables the possibility of rejecting beyond-
Nyquist narrow-band disturbances. Both the simulated and
experimental results show the effectiveness of the proposed
algorithm in disturbance rejection.
Except for galvanometer scanner systems, the proposed
control scheme can also be implemented in many other
system where beyond-Nyquist disturbances exist, but the
sensor sampling rate is limited, such as in vision servo
applications, chemical systems, etc.
REFERENCES
[1] Y. Yuan, S. Yang, and D. Xing, “Optical-resolution photoacoustic mi-
croscopy based on two-dimensional scanning galvanometer,Applied
Physics Letters, vol. 100, no. 2, p. 023702, 2012.
[2] J. Kokert, F. Höflinger, and L. M. Reindl, “Indoor localization system
based on galvanometer-laser-scanning for numerous mobile tags (ga-
locate),” in Indoor Positioning and Indoor Navigation (IPIN), 2012
International Conference on. IEEE, 2012, pp. 1–7.
[3] I. Gibson, D. W. Rosen, B. Stucker et al.,Additive manufacturing
technologies. Springer, 2010, vol. 238.
[4] P. K. Orzechowski, N. Y. Chen, J. S. Gibson, and T.-C. Tsao, “Optimal
suppression of laser beam jitter by high-order rls adaptive control,
IEEE Transactions on Control Systems Technology, vol. 16, no. 2, pp.
255–267, 2008.
[5] P. K. Orzechowski, S. Gibson, T.-C. Tsao, D. Herrick, M. Mahajan,
and B. Wen, “Adaptive suppression of optical jitter with a new liquid
crystal beam steering device,” in Defense and Security Symposium.
International Society for Optics and Photonics, 2007, pp. 65 690V–
65 690V.
[6] N. O. Perez-Arancibia, J. S. Gibson, and T.-C. Tsao, “Frequency-
weighted minimum-variance adaptive control of laser beam jitter,”
IEEE/ASME Transactions On Mechatronics, vol. 14, no. 3, pp. 337–
348, 2009.
[7] C. A. Mnerie, S. Preitl, and V.-F. Duma, “Performance enhancement of
galvanometer scanners using extended control structures,” in Applied
Computational Intelligence and Informatics (SACI), 2013 IEEE 8th
International Symposium on. IEEE, 2013, pp. 127–130.
[8] C. Mnerie, S. Preitl, and V.-F. Duma, “Galvanometer-based scanners:
Mathematical model and alternative control structures for improved
dynamics and immunity to disturbances,” International Journal of
Structural Stability and Dynamics, p. 1740006, 2017.
[9] C. A. Mnerie, S. Preitl, and V.-F. Duma, “Classical pid versus predic-
tive control solutions for a galvanometer-based scanner,” in Applied
Computational Intelligence and Informatics (SACI), 2015 IEEE 10th
Jubilee International Symposium on. IEEE, 2015, pp. 349–353.
[10] P. K. Orzechowski, S. Gibson, and T.-C. Tsao, “Disturbance rejection
by optimal feedback control in a laser beam steering system,” in ASME
2004 International Mechanical Engineering Congress and Exposition.
American Society of Mechanical Engineers, 2004, pp. 55–62.
[11] H. W. Yoo, S. Ito, and G. Schitter, “High speed laser scanning
microscopy by iterative learning control of a galvanometer scanner,”
Control Engineering Practice, vol. 50, pp. 12–21, 2016.
[12] X. Chen and H. Xiao, “Multirate forward-model disturbance observer
for feedback regulation beyond nyquist frequency,” Systems & Control
Letters, vol. 94, pp. 181–188, 2016.
[13] D. Li, S. L. Shah, and T. Chen, “System identification issues in
multirate systems,” in Electrical and Computer Engineering, 1999
IEEE Canadian Conference on, vol. 3. IEEE, 1999, pp. 1576–1578.
[14] W. Yan, C. Du, and C. K. Pang, “A general multirate approach for
direct closed-loop identification to the nyquist frequency and beyond,
Automatica, vol. 53, pp. 164–170, 2015.
[15] C. K. Pang, W. Yan, and C. Du, “Multirate identification of mechanical
resonances beyond the nyquist frequency in high-performance mecha-
tronic systems,” IEEE Transactions on Systems, Man, and Cybernetics:
Systems, vol. 46, no. 4, pp. 573–581, 2016.
[16] X. Chen and H. Xiao, “Multirate forward-model disturbance observer
for feedback regulation beyond nyquist frequency,” in American Con-
trol Conference (ACC), 2016. IEEE, 2016, pp. 839–844.
[17] P. A. Regalia, S. K. Mitra, and P. Vaidyanathan, “The digital all-pass
filter: A versatile signal processing building block,Proceedings of
the IEEE, vol. 76, no. 1, pp. 19–37, 1988.
[18] P. A. Regalia, “An improved lattice-based adaptive iir notch filter,”
IEEE transactions on signal processing, vol. 39, no. 9, pp. 2124–2128,
1991.
[19] L. Ljung, System identification. Wiley Online Library, 1999.
[20] X. Chen and M. Tomizuka, “A minimum parameter adaptive approach
for rejecting multiple narrow-band disturbances with application to
hard disk drives,IEEE Transactions on Control Systems Technology,
vol. 20, no. 2, pp. 408–415, 2012.
... Many control algorithms exist for rejecting such narrow-band disturbances. For example, [3,16] provides narrow-band disturbance observers that can achieve infinite high-gain control at selected disturbance frequency ranges. The focused application will construct a multirate control system combining the narrow-band disturbance observer and our proposed algorithm of information recovery with collaborative sensors. ...
... Components of the disturbance rejection mechanism in-cludeP d (z), the identified discrete model of the continuous plant P s (s), and Q(z), the disturbance compensating filter. With the disturbance frequency information known, one can design Q(z) with the procedure provided in [16]. Our model-based recover technique (i.e. ...
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Full-text available
This paper considers the real-time recovery of a fast time series (e.g., updated every T seconds) by using sparsely sampled measurements from two sensors whose sampling intervals are much larger than T (e.g., MT and NT, where M and N are integers). Specifically, when the fast signal is an autoregressive process, we propose an online information recovery algorithm that reconstructs the dense underlying temporal dynamics fully, by systematically modulating the sensor speeds MT and NT, and by exploiting a model-based fusion of the sparsely collected data. We provide the collaborative sensing design, parametric analysis and optimization of the algorithm. Application to a closed-loop disturbance rejection problem reveals the feasibility to annihilate fast disturbance signals with the slow and not fully aligned sensor pair in real time, and in particular, the rejection of narrow-band disturbances whose frequencies are much higher than the Nyquist frequencies of the sensors.
... To reduce the tracking error for galvanometer trajectories, Zaeh et al. combined an adaptive inverse control with dynamic behaviors of structures, thus proposing a photo adaption inverse control method, whose tracking error was 90% lower than that of the reference controller [10]. Xiao et al. developed a scheme based on a multi-rate control to resist external disturbance caused by laser-material interaction, thereby inhibiting narrow-band non-Nyquist disturbance [11]. Considering the requirements for high accuracy and high dynamics of GSs, Qin et al. proposed a high-accuracy position control method based on an active disturbance rejection control (ADRC): this overcame the problem of the high-frequency noise incurred by the traditional speed-calculating method based on a differentiator and ensured a high-accuracy position control of the system [12]. ...
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Laser processing plays an important role in industrial manufacturing, in which a galvanometer scanner (GS) functions as the core component of the laser processing equipment. With the development of the digital system, the GS based on the digital system finds a broader range of potential application. In order to address the slow step-responses of a GS with disturbance and parameter perturbation, the mathematical model of the motor of the GS is derived and a discrete-time sliding-mode variable-structure controller (DSVC) based on a decoupled disturbance compensator (DDC) (DSVC+DDC) is designed. The step-response performance of a GS is the key factor affecting the quality evaluation of laser processing. Experiments are conducted on the step responses of the motor of the GS on a digital experimental platform. The experiment results show that when guaranteeing a steady-state error within 20 urad and an overshoot of less than 5%, the rise time for step-responses in 1% and 10% of the whole stroke is 1 and 2 ms, decreasing by 23% and 58% compared with the reference performance index, which indicates the effectiveness of the proposed method. The proposed approach can not only compensate for the external disturbance online and improve the step-response speed of the GS, but also relax the traditionally assumed limit of the upper bound of external disturbance to the limit of the change rate of external disturbance, which reduces the difficulty of a practical application. This method has great significance for further applications in high precision machining.
... Section 5 shows simulation as well as experimental results on a dual-axis galvo scanner system, and finally, Section 6 concludes this paper. A preliminary version of the paper is presented in [27] . This paper is a substantially extended study that includes the full mathematical proofs and new theoretical and experimental results. ...
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