Content uploaded by Xu Chen

Author content

All content in this area was uploaded by Xu Chen on Oct 25, 2019

Content may be subject to copyright.

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,

H∞controller, 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,L∈Z+, 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 G−1stand 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 G−1,

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)+(1−Pd(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]

1−Pd(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 P−1

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 |1−Pd(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=2m−1,and

Q(z) = Q0(z)(q0+q1z−1+· · · +qpz−p),(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)z−1+ (1+k2,i)z−2

1+k1,i(1+k2,i)z−1+k2,iz−2,

(6)

where

k1,i=−cos(ωi),(7)

k2,i=1−tan(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+q1z−1+qpz−p(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,...,L−1,

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[n−1],··· ,dL[n−(2m−1)]]T,(12)

ϕy[n] = [yk[n−1],yk[n−2],··· ,yk[n−2m]]T.(13)

wkand bare parameter vectors that are deﬁned as

wk=wk,0,wk,1,··· ,wk,(2m−1),(14)

b= [b1,b2,··· ,b2m].(15)

The parameter vector bis composed of the coefﬁcients of

polynomial B(z−1) = 1+b1z−1+··· +b2mz−2m,computed

from

B(z−1) =

m

∏

i=1

(1−2αcos(2πfiLTs)z−1+α2z−2),(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(L−1)

wk,0

.

.

.

wk,2m−1

=−

a1

a2

.

.

.

a2m

0

.

.

.

0

+¯

b,(17)

Mk

4

= [ ˜

Mk|ekek+L··· ek+(2m−1)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(L−1).

(19)

[a1,a2,··· ,a2m]in equation (17) are the parameters of the

polynomial A(z−1) = 1+a1z−1+a2z−2+··· +a2mz−2m,

which is computed by

A(z−1) =

m

∏

i=1

(1−2cos(2πfiTs)z−1+z−2).(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(z−1)A(z−1) + z−kWk(z−L)−B∗(z−L) = 1,(21)

where A(z−1)is deﬁned by equation (20),

Fk(z−1) = 1+fk,1z−1+· · · +fk,2m(L−1)z−2m(L−1),(22)

Wk(z−L) = wk,0+wk,1z−L+·· · +wk,2m−1z−(2m−1)L,(23)

B∗(z−L) = b1z−L+b2z−2L+· · · +b2mz−2mL.(24)

The coefﬁcients of B∗(z−L)are the same as those in B(z−1)

(computed by equation (16)). Based on internal model prin-

ciple, for multiple narrow-band disturbances, an important

property of A(z−1)is that, at the steady state, we have

A(z−1)d[n] = 0.Combining this with equation (21) yields

that, at steady state,

1−z−kWk(z−L) + B∗(z−L)d[n] = 0,

The corresponding difference equations are,

d[n] = wk,0d[n−k] + · ·· +wk,2m−1d[n−k−(2m−1)L]

−b1d[n−L]−b2d[n−2L]− · ·· − b2md[n−2mL].(25)

Changing variables and replacing nwith nL +k, we have

d[nL +k] = wk,0d[nL] + · · · +wk,2m−1d[((n−(2m−1))L]

−b1d[(n−1)L+k]− · ·· − b2md[(n−2m)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 z−i’s (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(z−1) = wk,0+wk,1z−1+· ·· +wk,(2m−1)z−(2m−1)

1+b1z−1+· · · +b2mz−2m,(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(z−1)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(z−1)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) = 0.551z−1+0.356z−2−0.0496z−3+0.00673z−4

1−0.164z−1+0.0278z−2−0.00166z−3+0.000152z−4.(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.

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Time (sec)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Mirror angle (deg)

compensation turned on

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.