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AN INVERSEFREE DISTURBANCE OBSERVER FOR ADAPTIVE NARROWBAND
DISTURBANCE REJECTION WITH APPLICATION TO SELECTIVE LASER
SINTERING
Tianyu Jiang
Dept. of Mechanical Engineering
University of Connecticut
Storrs, Connecticut, 06269
Email: tianyu.jiang@uconn.edu
Hui Xiao
Dept. of Mechanical Engineering
University of Connecticut
Storrs, Connecticut, 06269
Email: hui.xiao@uconn.edu
Xu Chen
Dept. of Mechanical Engineering
University of Connecticut
Storrs, Connecticut, 06269
Email: xchen@uconn.edu
ABSTRACT
1 Introduction
Selective laser sintering is a powder based additive manu
facturing process for directly fabricating 3D objects. Over the
last two decades, it has evolved from a novelty to a multibillion
dollar industry [1]. A central step in SLS is to direct a high
energy laser beam at high speed on a bed of powder materials.
This is achieved by using a dualaxis galvo mirror system and
an optical lens that focuses the beam onto the powder surface
(Fig. 1). The motion control here must be performed at very
high precision, because the planary motion of the energy beam
deﬁnes the achievable geometric feature of the 3D printed ob
ject. However, the long focusing length creates a magnifying
Laser
Scanner mirrors
Sintered part
Ftheta lens
Figure 1. SCHEMATIC ILLUSTRATION OF SLS.
effect of the angular position error of the galvo, and any relative
motions between the powder bed and the beam’s optical path.
These error sources are difﬁcult, or infeasible, to eliminate com
pletely by better mechanical designs. For instance, the interfer
ence between the galvo mirrors is intrinsic to the beamsteering
mechanism; and vibrations in a complex mechanical system is
unavoidable [2–4].
This paper focuses on control methods that reject the afore
mentioned vibrations, or more generally, disturbances that can
be decomposed to narrowband components. The problem has
attracted great research attention, and many control design al
gorithms have been proposed in the related literature. Adap
tive Noise Cancellation (ANC) [5] is a very popular feedfor
ward compensation scheme that has been used to reject not only
narrowband vibrations but also wideband noises. Adaptive
feedforward cancellation [6] composes an estimation of a sinu
Proceedings of the ASME 2017 Dynamic Systems and Control Conference
DSCC2017
October 1113, 2017, Tysons, Virginia, USA
DSCC20175184
Selective laser sintering (SLS) is an additive manufactur
ing (AM) process that builds 3dimensional (3D) parts by scan
ning a laser beam over powder materials in a layerwise fashion.
Due to its capability of processing a broad range of materials,
the rapidly developing SLS has attracted wide research attention.
The increasing demands on part quality and repeatability are urg
ing the applications of customized controls in SLS. In this work,
a YoulaKucera parameterization based forwardmodel selective
disturbance observer (FMSDOB) is proposed for ﬂexible servo
control with application to SLS. The proposed method employs
the advantages of a conventional disturbance observer but avoids
the need of an explicit inversion of the plant, which is not al
ways feasible in practice. Advanced ﬁlter designs are proposed
to control the waterbed effect. In addition, parameter adaptation
algorithm is constructed to identify the disturbance frequencies
online. Simulation and experimentation are conducted on a galvo
scanner in SLS system.
1
Copyright © 2017 ASME
In view of the limitations above, this paper proposes a new
DOB scheme for rejection of narrowband disturbance. Instead
of an explicit plantinversion, a forward plant model is employed
and a dynamic ﬁlter is designed to achieve a pointwise inver
sion and perfect disturbance rejection at only the needed fre
quencies. This construction leads to what we shall refer to as a
froward model selective disturbance observer (FMSDOB). Also
discussed is a parameter adaptation method that is able to identify
the disturbance frequencies.
Notations:P(z)and P(ejω)denote respectively, a transfer
function and its frequency response. The calligraphic ℜand ℑ
denote, respectively, the real part and the imagine part of a com
plex number. S(,1/(1+PC)) and T(,PC/(1+PC)) denote
the sensitivity function1and complementary sensitivity function
where a linear time invariant (LTI) plant Pis stabilized by an LTI
controller Cin a feedback loop.
2 Hardware description
The developed algorithm in this paper was veriﬁed via simu
lation and experimentation on a galvo scanner, which is a central
element in SLS for laser beam delivery. Each galvo axis consists
of a mini actuation motor with an optical mirror mounted on the
shaft. Highlyaccurate, controlled motion is achieved with a po
sition detector that enables closedloop servo drivers to control
the scanner motors collaboratively.
Fig. 2 shows the frequency response of one axis of the galvo,
from the voltage input to the voltage output denoting the motor’s
position sensed by the encoder. A picture of the physical system
is shown in Fig. 3.
1the transfer function from the output disturbance to the plant output
Figure 2. FREQUENCY RESPONSE OF THE GALVO SCANNER’S
ONEAXIS DYNAMICS.
Laser source Galvo scanne r
Power supply
Monitor
PC server with
dSPACE and
Matlab
White screen
Servo driver
Figure 3. PICTURE OF THE PHYSICAL GALVO SCANNER SYSTEM.
3 Forwardmodel selective disturbance observer
Fig. 4 shows the proposed control structure. We have the
following relevant signals and transfer functions:
P(z)and ˆ
P(z): the plant and its identiﬁed model;
C(z): a baseline controller designed to provide a robustly
stable closed loop;
d(k)and ˆ
d(k): the actual (unmeasurable) disturbance and its
online estimate;
˜u(k)and u(k): the control command with and without the
compensation signal;
y(k): measured residual error;
c(k): the compensation signal that asymptotically rejects the
narrowband disturbance in d(k);
Adaptive parameter estimation algorithm: provides online
information of the characteristics of ˆ
d(k).
The basic structure of the closedloop employs the under
lying principle of internal model control [9, 10, 13]. To better
explain the mechanism of this disturbance rejection, consider the
input to Q(z)in Fig. 4. Blockdiagram analysis gives
ˆ
D(z)= (P(z)˜
U(z)+D(z))−ˆ
P(z)˜
U(z).(1)
At frequencies where the identiﬁed plant model ˆ
P(ejω)=P(ejω),
the right hand side (RHS) of (1) becomes D(z), i.e. d(k)in time
domain. This disturbance estimation is then processed by the
cancellation ﬁlter Q(z), the design of which constitutes the main
result of this study.
2
Copyright © 2017 ASME
soidal disturbance using a PhaseLocked Loop (PLL). In feed
back control, many more algorithms have been developed. Ref
erences [7, 8] used repetitive Control (RC) and its adaptive ver
sions. References [9, 10] developed statespace design based
on the Internal Model Principle (IMP). References [11–14] ap
plied Youla parameterization with adaptive ﬁnite impulse re
sponse (FIR) ﬁlters. Peak ﬁlters [15,16] and disturbance observer
(DOB) [17, 18] are also popular methods in this category.
Among the above algorithms, as a ﬂexible and powerful add
on control approach, DOB has been applied to broad control ap
plications. The central concept of a DOB is that, if the plant
dynamics can be properly inverted, then an equivalent input dis
turbance can be extracted and feed back to enhance servo per
formance. However, a direct inverse of plant dynamics is usu
ally infeasible for plants with unstable zeros and not realizable
for strictly proper plants. Another general challenge in observer
based highgain feedback control is the waterbed effect raised
from Bode’s Integral Theorem, which guarantees that for most
practical systems, attenuation of disturbances at some frequen
cies will inevitably cause error ampliﬁcation at some other fre
quencies.
++

+
Adaptive Parameter
Estimation
r= 0
++
ˆ()Pz
()Pz
()Cz
Addon Compensator
()uk
()uk
()ck
()dk
ˆ()dk
()yk
()Qz
Figure 4. PROPOSED FORWARD MODEL DISTURBANCE OB
SERVER SCHEME.
4 Main results
If ˆ
P(z)=P(z), one can obtain that
Y(z) = P(z)U(z) + (1−P(z)Q(z))D(z),(2)
where the relationship between u(k)and the output remains in
tact compared with that of the baseline system; at the same time,
additional dynamics is introduced between d(k)and y(k).
Observe the structure of the afﬁne Qparameterization
1−P(z)Q(z)in (2). Let ωi∈(0,π),i=1,2, .. .,nbe the distur
bance frequencies, with ωi6=ωj∀i6=j, we consider a pointwise
inverse of Psuch that
Q(ejωi)P(ejωi) = 1.(3)
Under the assumption that P(ejωi)6=0, the equation has the
solution
Q(ejωi) = 1
P(ejωi)=P(ejωi)
P(ejωi)2,
i.e.
ℑQ(ejωi) = −ℑP(ejωi)
P(ejωi)2
ℜQ(ejωi) = ℜP(ejωi)
P(ejωi)2
,i=1,2,. . ., n.(4)
4.1 Basic solution
The general solution to (3) is provided in the following
proposition.
Proposition 1. Let
Q(ejω) = (b0+
2n−1
∑
l=1
ble−l jω),(5)
with
b0
.
.
.
bm
=
1 cosω1. . . cosmω1
0 sinω1. . . sinmω1
.
.
..
.
.
.
.
..
.
.
1 cosωn. . . cosmωn
0 sinωn. . . sinmωn
−1
ℜP(ejω1)
P(ejω1)2
ℑP(ejω1)
P(ejω1)2
.
.
.
.
.
.
ℜP(ejωn)
P(ejωn)2
ℑP(ejωn)
P(ejωn)2
,(6)
where m =2n−1. Then
Q(ejωi)P(ejωi) = 1,∀i=1,2,. . ., n
Proof. Let Q(ejω) = b0+∑m
l=1ble−l jω, where m(∈Z)is the or
der of the ﬁlter. Based on (4), we must have, for i=1,2,.. . ,n,
b0+
m
∑
l=1
blcoslωi=ℜP(ejωi)
P(ejωi)2
m
∑
l=1
blsinlωi=ℑP(ejωi)
P(ejωi)2.
In matrix form, the above is equivalent to
1 cosωi. .. cos mωi
0 sinωi. . . sinmωi
b0
.
.
.
bm
=
ℜP(ejωi)
P(ejωi)2
ℑP(ejωi)
P(ejωi)2
.
There are nsuch equation sets, or 2nlinear equations. When
ωi∈(0,π), the rows of the matrix on the left side are all linearly
independent for different values of ωi’s. We thus have 2nlinearly
independent equations and m+1 unknowns. The minimum order
is m+1=2n. Under this case, the solutions of bi’s are given by
(6).
Proposition 1 provides an FIR ﬁlter design that achieves the
desired disturbance rejection at ωi. However, because there is no
constraint on the overall magnitude, this basic solution tends to
induce disturbance ampliﬁcation when ω6=ωi, especially at fre
quencies far away from the target frequency. The proposed im
plementation form is to incorporate special bandpass character
istics to maintain the magnitude of Qsmall when ω6=ωi. More
speciﬁcally, consider the following latticestructure [19] band
pass ﬁlter
QBP(ejω) = 1−1
2n
n
∏
i=1
(1+k2,i)(1+2k1,ie−jω+e−j2ω)
1+k1,i(1+k2,i)e−jω+k2,ie−j2ω,(7)
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Copyright © 2017 ASME
where k1,i=−cosωiand k2,i= [1−tan(Bw,i/2)]/[1+
tan(Bw,i/2)],Bw,i(in radian) is the 3dB bandwidth of QBP(z)
centered around ωi. It can be shown that QBP(ejωi) = 1,∀i=
1,2,. . ., n. Applying (7) to (5) gives the improved design
Q(z) = QBP(z)(b0+
2n−1
∑
l=1
blz−l),(8)
which not only maintains the disturbance rejection properties, but
also blocks noises in d(k)outside the target frequency ranges. To
elucidate this fact, we discuss next the frequencydomain closed
loop properties and how to control the relevant performance lim
itations.
Corollary 1. Take any P and Q that are stable and causal. The
magnitude response of 1−P(ejω)Q(ejω)satisﬁes
Zπ
0
ln1−P(ejω)Q(ejω)dω=π nγ
∑
i=1
lnγi−ln σ+1!,(9)
where {γi}nγ
i=1(nγ≥0)is the set of unstable zeros of 1−P(z)Q(z)
({γi}nγ
i=1,Φif nγ=0), and
σ=lim
z→∞P(z)Q(z)/(1−P(z)Q(z)).
Proof. See [20].
Corollary 1 speciﬁes potential fundamental performance
limits similar to all linear designs. For plants whose relative de
gree is zero, limz→∞P(z)6=0. It is then possible that σ>0 and
the integral on the RHS of (9) is less than zero. However, for
strictly proper plants (the more common case), limz→∞P(z) =
0 and σ=0, (9) simpliﬁes to Rπ
0ln1−P(ejω)Q(ejω)dω=
π∑nγ
i=1lnγi≥0. Then it is inevitable that there exist frequencies
where 1−P(ejω)Q(ejω)>1. In other words, some disturbances
are ampliﬁed in (2).
Although the overall area integral is constrained in (9), by
proper structural design in Q(z), the waterbed effect can be con
trolled based on the disturbance spectrum, performance goals,
and robustness of the system in different regions. This is the pri
mary reason of the bandpass design in (8). Further enhancement
can be made, as we will now discuss in the next subsections.
4.2 Modulation of zeros
To avoid ampliﬁcation of noise, one can add ﬁxed zeros to
(8) to constrain the magnitude at desired frequencies. Consider
adding ﬁxed zeros ρe±jωp,ωp∈(0,π)to (8). Two additional
equations need to be added to (6)
"11
ρcosωp. . . 1
ρmcosmωp
01
ρsinωp. . . 1
ρmsinmωp#
b0
.
.
.
bm
=0
0,(10)
10210 3
30
20
10
0
Magnitude (dB)
1P(z)Q(z)
Frequency (Hz)
2 3
150
100
50
Magnitude (dB)
Basic solution
With fixed zero near z=1
Q(z)
10210 3
30
20
10
0
Magnitude (dB)
1P(z)Q(z)
Frequency (Hz)
10210 3
150
100
50
Magnitude (dB)
Basic solution
With fixed zero near z=1
Q(z)
Frequency (Hz)
Figure 5. EFFECT OF A FIXED ZERO AT LOW FREQUENCY REGION.
and the minimum order becomes m+1=2n+2. When ωp=0
or π, the second equation in (10) can be removed, we just need to
add
h11
ρcosωp. . . 1
ρmcosmωpi
b0
.
.
.
bm
=0,
and the minimum order becomes m+1=2n+1.
Fig. 5 shows the effect of placing a zero near z=1 (zero
DC gain). Recall Fig. 4. Blockdiagram algebra gives the new
sensitivity function
S= (1−PQ)S0,(11)
where S0=1/(1+PC)is the baseline sensitivity function. The
induced small gain of Q(ejω)at low frequency successfully re
duces 1−P(ejω)Q(ejω)in the highlighted region and hence, re
duces the magnitude response of the sensitivity function in (11).
Similarly, introducing a ﬁxed zero near z=−1 provides en
hanced small gain for Q(ejω)in the highfrequency region. This
method is especially effective when the noise frequency in d(k)
is available, in which case, combined ﬁxed zeros near the noise
frequency could be introduced into Q(z)by (10).
4.3 Cascaded IIR ﬁlters
In addition to the lattice bandpass ﬁlter described in (7),
other IIR ﬁlters can be further cascaded to (8) to enhance the
frequency response. This method was discussed in [21]. Fig.
6 presents the comparison results of the magnitude response of
Q(ejω)after introducing the IIR design described in [21]. It can
be seen that the response remains unchanged at ωi, but is attenu
ated at all other frequencies.
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Copyright © 2017 ASME
Frequency (Hz)
10210 3104
150
100
50
0
Magnitude (dB)
Basic Q
w/ cascaed IIR enhancement
Q(z)
Frequency (Hz)
Figure 6. CASCADED IIR ENHANCEMENT IN Q(z).
Based on the technical discussions in this section, the pro
posed scheme offers the following beneﬁts for disturbance rejec
tion:
No explicit plant inversion: In a regular DOB, a full stable
inversion ˆ
P−1(z)is required, which achieves ˆ
P−1(z)P(z)≈1
at all frequency regions where disturbance attenuation is de
sired. However, this is usually infeasible due to instabil
ity and nonproperness. Thus, a chosen stable nominal in
version ˆ
P−1
n(z)is adopted instead of ˆ
P−1(z). In the region
where there is large mismatch between ˆ
P−1
n(z)and ˆ
P−1(z)
(which is very likely when ˆ
P(z)contains multiple unstable
zeros), the Qﬁlter must be designed to have small gain and
cannot successfully compensate disturbance at that region.
On the contrary, there is no need to assign an exact full in
version ˆ
P−1(z)to the forward model based selective DOB in
Fig. 4. Based on (4), Qcould be designed as a pointwise
stable inversion, which avoids the constraint of “fail region”
induced by the mismatch discussed above.
Design intuition, performance, and extensions: A regular
DOB structure with lowpass Qdesign focuses more on the
magnitude and does not explicitly consider the phase com
pensation that can signiﬁcantly deteriorate the achievable
performance at high frequencies. While in (3), both the
magnitude and the phase compensation are taken into con
sideration. On the other hand, compared with other narrow
band algorithms, the proposed scheme inherits the advan
tages of conventional DOB of intuitive design and good per
formance [22, 23]. It also shares the beneﬁts of capability of
easy adaptation and extension to rejecting broadband dis
turbances [24].
5 Parameter adaptation
Knowledge of the disturbance frequencies is needed to gen
erate the parameters in the ﬁlters discussed above. This section
studies the adaptive formulation when such information is not a
priori available.
Recall (1) and (11), the plant output
y(k) = S(z)d(k)≈1−P(z)Q(z)
1+P(z)C(z)ˆ
d(k).
Let w(k),ˆ
d(k)/(1+P(z)C(z)), the parameter adaptation
algorithm (PAA) scheme can then be designed directly based on
minimizing (1−P(z)Q(z))w(k).
For a vibration disturbance that is composed of d(k) =
∑n
i=1Cisin(ωik+φi) (Ci6=0, ωi6=0, φi∈R), the denominator
of D(z)is
A(z−1),
n
∏
i=1
(1−2cos(ωi)z−1+z−2).(12)
Expanding the product gives
A(z−1) = 1+a1z−1+· · · +anz−n+·· · +a1z−2n+1+z−2n,(13)
where we have mapped the parameters {ωi}n
iin (12) to {ai}n
i.
Noticing that the coefﬁcient vector of (13) has a mirror symmet
ric structure, we can deﬁne θ,a1a2.. . anT, such that there
are only nparameters to be identiﬁed. This is the minimum pos
sible number for narrowband signals that contain nunknown
frequency components. Reference [2] has proposed a twostage
adaption scheme that combines recursive least squares (RLS) and
output error based PAA for identiﬁcation of θ, where the param
eter adaption equations are
ˆ
θ(k)=ˆ
θ(k−1)+ P(k−1) (−φ(k−1)) νo(k)
1+φ(k−1)TP(k−1)φ(k−1)(14)
P(k)= 1
λ(k)P(k−1)−P(k−1)φ(k−1)φT(k−1)P(k−1)
λ(k)+φT(k−1)P(k−1)φ(k−1),
with
e(k)=φ(k−1)Tˆ
θ(k)+w(k)+w(k−2n)−α2ne(k−2n)
ν(k)=e(k) + α2ne(k−2n) + ϕ(k−1)Tθc
eo(k)=φ(k−1)Tˆ
θ(k−1)+w(k)+w(k−2n)−α2ne(k−2n)
νo(k)=eo(k)+ α2ne(k−2n) + ϕ(k−1)Tθc
where e(k)and ν(k)are respectively, the a posteriori estimation
and adaptation errors; eo(k)and νo(k)are respectively, the a pri
ori estimation and adaptation errors; θc,[c1,c2, . .. ,cn]Tis the
coefﬁcient vector of a ﬁxed compensator C(z−1) = 1+c1αz−1+
· ·· +cnαnz−n+· ·· +c1α2n−1z−2n+1+α2nz−2n,α∈(0,1); and
ϕ(k−1) = [ϕ1(k−1),ϕ2(k−1),. . ., ϕn(k−1)]Tare computed
from: ϕi(k−1) = αie(k−i) + α2n−ie(k−2n+i);i=1,..., n−
1,ϕn(k−1) = αne(k−n). The regressor vector in (14) is
composed of φi(k−1) = w(k−i)+ w(k−2n+i)−αie(k−i)−
α2n−ie(k−2n+i);i=1,..., n−1,and φn(k−1) = w(k−n)−
αne(k−n).
For good convergence, αshould be initialized with a relative
small value and then increased gradually to a value that is close
to 1. The forgetting factor λ(k)determines how much informa
tion before the time instance kis used for adaption and impacts
the speed of convergence. It needs to be tuned based on the char
acteristics of w(k). For more discussion on the application of the
PAA for identifying timevarying disturbance, we refer readers
to [22].
5
Copyright © 2017 ASME
6 Stability and robustness
This section analyzes the closedloop stability and robust
stability of the proposed control scheme. The effectiveness of
the proposed scheme when there is estimation error of the distur
bance frequency is also discussed.
Proposition 2. Given that P(z)and Q(z)are stable and the
identiﬁed model ˆ
P(z)is exact, i.e. ˆ
P(z) = P(z), the closedloop
stability is guaranteed.
Proof. Recall Fig. 4. With some blockdiagram algebra, the fol
lowing transfer function is obtained
Y(z)= 1−P(z)Q(z)+Q(z)(P(z)−ˆ
P(z))
1+P(z)C(z)+Q(z)(P(z)−ˆ
P(z)) D(z).(15)
Let P(z) = BP(z)/AP(z),ˆ
P(z) = Bˆ
P(z)/Aˆ
P(z),C(z) =
BC(z)/AC(z), and Q(z) = BQ(z)/AQ(z)be the coprime polyno
mial factorization of P(z),ˆ
P(z),C(z)and Q(z). When ˆ
P(z) =
P(z), the closedloop characteristic equation becomes
AQ(z)Aˆ
P(z)[(AP(z)AC(z) + BP(z)BC(z)] = 0.
Hence the closedloop poles are composed of the baseline closed
loop poles and the poles of Q(z)and ˆ
P(z). As the baseline feed
back loop, Q(z), and ˆ
P(z)(=P(z)) are all stable, the new closed
loop is thus stable.
Proposition 2 opens up the design space for Qﬁlter. In
deed, for any stable Q, given the baseline closedloop stability,
the closedloop stability of the new system is automatically ob
tained.
For the robust stability, when the plant is perturbed to be
˜
P(z) = P(z)(1+∆(z)) (the uncertainty ∆(z)is assumed to be sta
ble and has a bounded H∞norm), applying the Small Gain The
orem [25] yields the following robuststability condition
k∆(z)T(z)k∞<1,(16)
where the complementary sensitivity function is given by T=
1−S. After substituting (11) in, (16) becomes
∆(z)P(z)C(z)+Q(z)
1+P(z)C(z)
∞
<1.
Consider the case when there is mismatch between the iden
tiﬁed disturbance frequency and actual disturbance frequency,
which is denoted as ∆ωi. Recall the shape of the frequency re
sponse of 1 −P(z)Q(z)in Fig. 5. The DOB can still reject the
disturbance when ∆ωiis smaller than half of the bandwidth of the
notch, which is adjustable by k2,iin (7). The underlying reason is
that the frequency response of plant P(ejω)is continuous and sat
isﬁes lim∆ωi→0P(ejωi)−P(ej(ωi+∆ωi))=0. If the mismatch is
small enough, then the frequency response of Q(ejω)ω=ωi+∆ωi,
which is the inverse response of P(ejω)ω=ωi+∆ωi, is still close
enough to the inverse of P(ejω)ω=ωi. Thus, the proposed scheme
remains effective.
7 Simulation and experiment
Let
P(z) = 0.5465z3+0.4598z2−0.0717z−0.0721
z4−0.5221z3−0.6208z2+0.0730z+0.0721
in Fig. 4. P(z)here is the nominal model of the plant described in
Section 2 with sampling time Ts=0.1ms. Since there is already
a baseline controller in P(z), we set C(z) = 1.
A timevarying narrowband disturbance series with
step change is considered in both simulation and experi
ment. The disturbance frequency changes in the pattern
of null→500Hz→1500Hz→500Hz→3000Hz→500Hz→null.
The ﬁrst disturbance is injected at 0.5s and the duration time for
each frequency is 0.3s. The amplitude of the disturbance is 0.1V.
Fig. 7 persents the time series of the identiﬁed frequencies
using PAA with αvarying from 0.865 to 0.99 and λvarying from
0.93 to 0.99. The disturbance frequencies can be seen to have
been identiﬁed rapidly and correctly.
Fig. 8 shows the simulation result of the time trace of the
residual position errors. Based on the identiﬁed frequencies, the
proposed algorithm using a Q(z)ﬁlter in (8), with the bandwidth
of the lattice bandpass ﬁlter set as 50Hz, is seen to provide rapid
and strong vibration compensation. Inspection of data at 0.79s,
1.09s, and 1.69s indicates that the steadystate position errors
indeed converge to zero with a maximum value less than 1 ×
10−4V.
The experimental result of rejecting the same vibrations dis
cussed above is shown in Fig. 9. Comparing the data at 0.45s,
0.75s, 1.05s and 1.65s, we observe that the steadystate errors
with the compensation scheme have been reduced to be at the
same or smaller magnitude as the baseline case where no distur
bance is present. The observations are further veriﬁed by calcu
lating the the 2norm of the position errors before the injection
of any disturbance and at the last 0.1s of each period with a dis
turbance injected (denoted as kerk2
2). The results are shown in
Table 1.
The steadystate error spectra further reveals the effective
ness of the compensation scheme. Fig. 10 shows that when the
system is subjected to a 1500Hz vibration, the proposed scheme
reduces the spectral peak from around −18dB to −101dB, indi
cating a disturbance attenuation of about 83dB.
8 Conclusion
In this paper, an inversefree forward model selective dis
turbance observer scheme is introduced for narrowband distur
bance rejection. This algorithm is constructed by designing a
pointwise model inversion ﬁlter. Simulation and experimenta
tion on a galvo scanner platform in additive manufacturing show
6
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0 0.5 1 1.5 2 2.5
Time (s)
0
1000
2000
3000
Estimated Frequency (Hz)
Figure 7. IDENTIFIED FREQUENCIES USING PAA
0 0.5 1 1.5 2 2.5
Time (s)
0.25
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0.25
Voltage (V)
Open Loop
Close Loop
500Hz 1500Hz 500Hz 3000Hz 500Hz
Figure 8. TEST1: SIMULATION RESULTS OF REJECTING NARROW
BAND DISTURBANCE WITH SINGLE STEP CHANGING FREQUENCY.
0 0.5 1 1.5 2 2.5
Time (s)
0.25
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0.25
Voltage (V)
Open Loop
Close Loop
500Hz 1500Hz 500Hz 3000Hz 500Hz
Figure 9. TEST1: EXPERIMENTAL RESULTS OF REJECTING NAR
ROW BAND DISTURBANCE WITH SINGLE STEP CHANGING FRE
QUENCY.
Table 1. 2NORM OF BASELINE NOISE AND RESIDUAL ERRORS.
Period (s) 0.39→0.49 0.69→0.79 0.99→1.09
(Baseline) (500Hz) (1500Hz)
kerk2
2(×10−1V2)1.652 1.451 1.373
Period (s) 1.29→1.39 1.59→1.69 1.89→1.99
(500Hz) (3000Hz) (500Hz)
kerk2
2(×10−1V2) 1.485 1.521 1.458
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]
120
100
80
60
40
20
Power Spectrum Density (PSD) [dB]
PSDOL
PSDCL
Figure 10. EXPERIMENTAL RESULTS OF ERROR SPECTRA OF RE
JECTING A 1500HZ VIBRATION.
signiﬁcant performance gain for disturbance attenuation. The
proposed scheme avoids the explicit plantinversion in a conven
tional DOB, hence is particularly useful for plants with unstable
zeros or when a stable plant inversion is prohibitively expensive
over the full frequency range.
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