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Wideband Loop Shaping for Modulation of Energy Transmission in
Nonminimumphase Systems
Tianyu Jiang
Dept. of Mechanical Engineering
University of Connecticut
Storrs, Connecticut, 06269
Email: tianyu.jiang@uconn.edu
Jiong Tang
Dept. of Mechanical Engineering
University of Connecticut
Storrs, Connecticut, 06269
Email: jiong.tang@uconn.edu
Xu Chen
Dept. of Mechanical Engineering
University of Connecticut
Storrs, Connecticut, 06269
Email: xchen@uconn.edu
ABSTRACT
Modulating the closedloop transmission of energy in a wide
frequency band without sacriﬁcing overall system performance is
a fundamental issue in a wide range of applications from preci
sion control, active noise cancellation, to energy guiding. This
paper introduces a loopshaping approach to create such wide
band closedloop behaviors, with a particular focus on systems
with nonminimumphase zeros. Pioneering an integration of the
interpolation theory with a modelbased parameterization of the
closed loop, the work proposes a ﬁlter design that matches the
inverse plant dynamics locally and creates a framework to shape
energy transmission with user deﬁned performance metrics in the
frequency domain. Application to laserbased powder bed fusion
additive manufacturing validates the feasibility to compensate
wideband vibrations and to ﬂexibly control system performance
at other frequencies.
1 INTRODUCTION
Active and ﬂexible shaping of dynamic system responses is
key for modulating inputtooutput energy ﬂows. For example,
adaptive disturbance attenuation minimizes the energy of posi
tion error and achieves precision motion control in nmscale in
a modern hard disk drive (HDD) [1,2]; in modern headphones,
active acoustic noise cancellation reduces acoustic energy trans
mission from the environment to human ears [3]; wave guiding
applications, on the other hand, minimizes loss of energy by re
stricting wave expansion to one dimension or two [4].
As a fundamental element of modulating energy transmis
sion, feedback loop shaping in the frequency domain has at
tracted extensive research attention. Based on the target energy
distribution, relevant control algorithms can be generally classi
ﬁed into two categories: narrow and wideband loop shaping.
In the ﬁrst category, the input energy concentrates at one or sev
eral frequencies. Based on such characteristics, peak ﬁlter [5, 6],
repetitive control [7,8], adaptive notch ﬁlter [9,10], and narrow
band disturbance observer (DOB) [1, 11, 12] generate deep but
narrow notches in the closedloop errorrejection functions at fre
quencies where the input energy (disturbance) dominates. For
wideband loop shaping, the input energy spans over wide peaks
in the spectrum map. A narrow notch cannot provide sufﬁcient
attenuation to such inputs; yet a wide notch tends to cause un
desired ampliﬁcation at other frequencies (the fundamental wa
terbed effect in feedback controls [13]). In view of such chal
lenges, [14] proposed to detune the notch depth and place ﬁxed
zeros according to the performance criterion; [2] utilized an on
line adaptation algorithm to estimate both frequency and notch
width for optimal loop shaping design.
In this paper, we propose a new control scheme for wide
band loop shaping for stable but nonminimumphase systems.
Compared with previous algorithms such as the DOB, this wide
band forward model selective disturbance observer (FMSDOB)
avoids direct inversion of the plant model, thereby offering free
dom and ﬂexibility of loop shaping for nonminimumphase sys
tems. Compared to other loopshaping designs, the proposed al
gorithm inherits beneﬁts of DOB design regarding design intu
ition and strong performance [12,14]. By designing an FIR (ﬁnite
impulse response) ﬁlter that matches both the plant frequency
response and highorder dynamics locally, this scheme creates
wide notches in the closedloop sensitivity function. By integrat
ing a cascaded ﬁltering design into the YoulaKucera (YK) pa
rameterization, the algorithm directly controls energy ampliﬁca
tion at other frequencies. Veriﬁcation is performed by simulation
on a twoaxis galvo scanner in selective laser sintering additive
manufacturing.
The remainder of this paper is organized as follows. Section
1
Copyright © 2018 ASME
Proceedings of the ASME 2018
Dynamic Systems and Control Conference
DSCC2018
September 30October 3, 2018, Atlanta, Georgia, USA
DSCC20189089
2 reviews the algorithmic foundation of the proposed method and
its connection to our previous work on narrowband loop shap
ing; Section 3 provides the main results of wideband loop shap
ing design, following which Section 4 presents the supporting
simulation results, and ﬁnally, Section 5 concludes this paper.
Notations: Throughout the paper, the calligraphic Sand
Rdenote, respectively, the set of stable proper rational trans
fer functions, and the set of proper rational transfer functions.
When a linear time invariant (LTI) plant Pis stabilized by an LTI
controller C(in a negative feedback loop), S(,1/(1+PC)) and
T(,PC/(1+PC)) denote, respectively, the sensitivity function1
and the complementary sensitivity function. The calligraphic ℜ
and ℑdenote, respectively, the real and imaginary part of a com
plex number.
2 REVIEW OF RELEVANT LITERATURE
2.1 Review of YoulaKucera parameterization
Theorem 1. (YK Parameterization [15, 16]). If an SISO plant
P=N/D can be stabilized by a negativefeedback controller
C=X/Y, with (N,D)and (X,Y)being coprime factorizations
over S, then any stabilizing feedback controller of P can be pa
rameterized as
Call =X+DQ
Y−NQ :Q∈S,Y(∞)−N(∞)Q(∞)6=0.
Here, (N,D)is called a coprime factorization of P∈Rover
Sif: (i) P=ND−1, (ii) N(∈S)and D(∈S)are coprime trans
fer functions, and (iii) D−1∈R.
A main advantage of YK Parameterization is that it renders
the sensitivity function of the closed loop (i.e., the error rejection
function of the feedback system) to:
˜
S=1
1+PCall =1
1+PC 1−N
YQ,(1)
which is decoupled into the product of the baseline sensitivity
1/(1+PC)and the addon afﬁne module 1−NQ/Y. Because
stability is assured under the controller parameterization, design
ers can now focus on the addon module (which depends afﬁnly
on Q) to achieve desired performance features.
2.2 Forward Model Selective Disturbance Observer
When Pand Care stable, one can factorize them in Theo
rem 1 as N(z) = P(z),D(z) = 1,X(z) = C(z),Y(z) = 1.(These
are special plantcontroller pairs that often arise in mechatronic
systems.) Then
Call =C(z) + Q(z)
1−P(z)Q(z),Q(z)∈S
1The transfer function from the output disturbance to the plant output.
parameterizes all stabilizing controllers for P(z), and the new
sensitivity function is
S=1
1+P(z)C(z)(1−P(z)Q(z)) ,S0(z)(1−P(z)Q(z)).(2)
The factorization above renders the addon module in (1) to
1−P(z)Q(z). This parameterization has made the added module
simple and do not depend on baseline controller C(z). Certainly,
from the viewpoint of implementation, a perfect plant model is
unrealistic in practice. In this sense, instead of P(z), a nominal
model of the plant ˆ
P(z)is used to formulate the YK parameteri
zation, which gives the new stabilizing controller and sensitivity
function
˜
Call =C(z) + Q(z)
1−ˆ
P(z)Q(z),Q(z)∈S
˜
S=1
1+P(z)C(z) + (P(z)−ˆ
P(z))Q(z)(1−ˆ
P(z)Q(z)).(3)
At frequencies where P(ejω) = ˆ
P(ejω), i.e. where the nominal
model is accurate, (3) gives
˜
S=1
1+P(ejω)C(ejω)(1−P(ejω)Q(ejω)),
i.e. the decoupling of sensitivity in (2) remains valid in the fre
quency domain. Loop shaping and energy transmission modu
lation then translate to designing 1 −P(ejω)Q(ejω) = 0, which
gives ˜
S(ejω) = 0. One intuitive design is thus to make Q(z) =
P−1(z), namely, direct inversion of the plant model. However,
when the plant contains unstable zero(s) or is strictly proper, a di
rect inversion will introduce instability or nonproperness to the
closed loop. In our previous work [17, 18], we proposed a point
wise inverse design to overcome these fundamental challenges.
The main result is provided in the following proposition.
Proposition 2. Let Q(z) = b0+∑m
l=1blz−l,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
,(4)
where m =2n−1. Then
Q(ejωi)P(ejωi) = 1,∀i=1,2,. . .,n.
2
Copyright © 2018 ASME
++
+
+
+

FIGURE 1.PROPOSED FORWARD MODEL SELECTIVE DISTUR
BANCE OBSERVER SCHEME.
Proof. See [18].
By focusing on the local inversion of P(ejω), Proposition
2 relaxes the requirement of a stable plant inversion. Proof of
the closedloop stability and robustness of the proposed scheme
is provided in [18]. Brieﬂy speaking, since the forward model
based controller is branched out of the YK parameterization,
nominal stability follows directly. On the other hand, at frequen
cies where there are large model uncertainties and mismatches,
highperformance control intrinsically has to be sacriﬁced for ro
bustness based on robust control theory. It is not difﬁcult to make
Q(ejω)small at these frequencies to keep the inﬂuence of the
mismatch element (P(z)−ˆ
P(z))Q(z)small in (3).
Fig. 1 shows a realization of the scheme discussed above.
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 (not measurable) 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
disturbance d(k).
Remark 3.We assume that the magnitude of P(ejω)is not zero
at the target frequencies (otherwise, disturbances are directly re
jected by the plant).
Remark 4.By block diagram analysis, if P(ejωi) = ˆ
P(ejωi), one
can show that when r(k) = 0 and P(ejωi)Q(ejωi) = 1, P(z)˜u(k)
approximates −d(k)at ωi(hence canceling the disturbance).
We will use this intuition of disturbance estimation for con
troller tuning.
3 PROPOSED WIDEBAND LOOP SHAPING FOR
NONMINIMUMPHASE SYSTEMS
Compared with singlefrequency excitation, wideband sig
nals (see, e.g., Fig. 2) induce widely spanned spectral peaks.
For such cases, pointwise inversion of the plant frequency re
sponse alone will not generate satisfying loopshaping result.
For Review Only
SUBMITTED TO IEEE TRANSACTIONS ON MAGNECTICS, VOL. XX, NO. XX, XX XXXX 2
0 500 1000 1500 2000 2500 3000 3500
0
2
4
6
Magnitude
0 500 1000 1500 2000 2500 3000 3500
Frequency (Hz)
0
1
2
3
4
Magnitude
(b) wideband energy
(a) energy with narrow spectrum peaks
Fig. 1. Examples of the disturbance spectrums
limitation imposed by the Bode’s Integral Theorem. It states
that in linear feedback control systems, it is impossible to
attenuate disturbances at all frequencies unless for some
special cases that are rarely feasible in motion control ap
plications. The attenuation of disturbances at one frequency
range will inevitably cause ampliﬁcations at other spectral
regions. Although approachs (iii) and (iv) can theoretically
attenuate disturbances at any customerspeciﬁed frequencies
(even above the bandwidth of the servo loop), they focus on the
rejection of narrowband disturbances (see Fig. 1(a)) or single
peak bandlimited disturbances where the “waterbed” ampli
ﬁcation is relatively small. Hence, in these approaches, the
adaptation focuses mainly on ﬁnding the optimal attenuation
frequencies. Yet for multiple wideband disturbance attenua
tion, such undesired ampliﬁcations are much more dangerous,
and should be systematically considered in controller design
to avoid deterioration of the overall performance.
In view of the needs and challenges, an adaptive loop
shaping scheme is proposed in this paper for multiple wide
band disturbance attenuation. By considering adaptations w.r.t
not only the attenuation frequencies but also the attenuation
widths, this scheme automatically tunes for the optimal con
troller parameters that offer better overall performance. Such a
twodegreeoffreedom (2DOF) adaptation allows an optimal
allocation of control efforts over all frequencies. It balances the
preferred disturbance attenuation and undesired ampliﬁcation
with minimum position errors. As an extension to our previous
work [18]–[20], [24]–[26], this paper contributes in three
aspects: (i) the adaptive controller design covers both single
and dualstage HDDs; (ii) wideband disturbances with the
important extension to multiple spectral peaks are addressed;
and (iii) experimental veriﬁcation on a VoiceCoilDriven
Flexible Positioner (VCFP) system is performed.
The remainder of the paper is organized as follows. Sec
tion II presents the proposed adaptive control structure for
both single and dualstage HDDs. Section III discusses the
performanceoriented Qﬁlter design with tunable passband
widths. In Section IV, the 2DOF parameter adaptation algo
rithms are formulated. Stability analysis is given in Section
V, followed by the experimental results on a VCFP system in
Section VI. Finally, Section VII concludes the paper.
II. PROPOSED ADAPTATION STRUCTURE WITH MODEL
INVERSION
A. General structure
Throughout the paper, we focus on the trackfollowing prob
lem of HDD systems in the presence of the above discussed
wideband disturbances. Figure 2 shows the proposed adaptive
control structure in singlestage HDDs. Without the addon
compensator, it reduces to a basic feedback loop where the
system P(z1)is stabilized by a controller C(z1), which
achieves a baseline sensitivity function whose magnitude re
sponse is similar to the one shown in Fig. 3. Such a baseline
servo design is typical in practice, and can commonly achieve
a bandwidth of around 1000Hz for singlestage HDDs and
around 2000Hz for dualstage HDDs. Above the bandwidth,
disturbances are ampliﬁed due to the “waterbed” effect. There
fore, new customized compensator is desired for enhanced
disturbance attenuation at high frequencies.
d(k)
+
✏✏
r=0
+//e(k)//C(z1)+//u(k)
+////P(z1)y(k)
//
+✏✏
//zm//ˆ
d(k)
P1
n(z1)
+
oo
oon(k)
+
oo
c(k)
OO
Q(z1)oo
✏✏
F(z1)
dF(k)
✏✏
paramters
KS
Adapt module
addon compensator
OO
Fig. 2. The structure of proposed control scheme for singlestage HDDs
10
1
10
2
10
3
10
4
−50
−40
−30
−20
−10
0
10
Magnitude (dB)
Frequency (Hz)
Fig. 3. Typical magnitude response of the baseline sensitivity function
Within the addon compensator in Fig. 2, zmsep
arates1the delay out of system P(z1)such that
P(z1),zmPm(z1)and Pn(z1)represents the nominal
model of Pm(z1), i.e., Pn(z1)⇡Pm(z1)at least at fre
quencies where disturbance attenuation is desired. The objec
tive of the addon compensator is to generate a cancellation
signal c(k)which can effectively cancel out the inﬂuence of
1For discretetime systems, we always have m1and the separation
of zmassures a realizable inverse of Pmz1and its nominal model
described afterwards.
Page 2 of 28
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FIGURE 2.EXAMPLE OF WIDEBAND DISTURBANCE SPEC
TRUM [2].
102103104
30
20
10
0
10
20
30
Magnitude (dB)
Plant response inversion
Basic Q via (4)
Bode Diagram
Frequency (Hz)
1=2000Hz
2=4000Hz
3=7000Hz
FIGURE 3.COMPARISON OF THE INVERSE MAGNITUDE RE
SPONSE OF A PLANT AND FREQUENCY RESPONSE OF Q FILTER
GENERATED BY (4).
This can be seen in Fig. 3, which compares the inverse mag
nitude response of a plant and corresponding Q(ejω)designed
from (4). Note the different slope rates of these two lines at
the target frequencies (2000Hz, 4000Hz, 7000Hz). As the fre
quency deviates from these points, the response of addon mod
ule 1−P(ejω)Q(ejω)can easily fall out of desired performance
threshold. In this section, we propose an augmented Q ﬁlter de
sign so that both 1 −P(ejω)Q(ejω)and its ﬁrstorder derivative
are zero at the center of each frequency band. The result is that
the frequency response of Q ﬁlter matches the inverse response of
the plant not only at but also around the target frequencies, which
introduces wide loopshaping notches in the frequency response
of the addon module 1−P(ejω)Q(ejω).
Proposition 5. Let ωi∈(0,π),i=1,2, . .. , n be center frequen
cies of the target loopshaping regions, (N,D)be a coprime fac
torization of P such that P =ND−1, where N =∑p
u=0cuzuand
D=zq+∑q−1
v=0avzv. Design
Q(z) = b0+
m
∑
l=1
blz−l,(5)
3
Copyright © 2018 ASME
with m =4n−1and
b0
.
.
.
bm
=
1 cosω1cos2ω1. .. cosmω1
0 sinω1sin2ω1. .. sinmω1
.
.
..
.
.
1 cosωncos2ωn. .. cosmωn
0 sinωnsin2ωn. .. sinmωn
0 cosω12cos 2ω1.. . mcos mω1
0 sinω12sin 2ω1.. . msin mω1
.
.
..
.
.
0 cosωn2cos 2ωn.. . mcos mωn
0 sinωn2sin 2ωn.. . msin mωn
−1
ℜP(ejω1)
P(ejω1)2
ℑP(ejω1)
P(ejω1)2
.
.
.
ℜP(ejωn)
P(ejωn)2
ℑP(ejωn)
P(ejωn)2
−ℜH(ejω1)
ℑH(ejω1)
.
.
.
−ℜH(ejωn)
ℑH(ejωn)
,
(6)
where
H(ejω) = (qejqω+∑q−1
v=1vavejvω)N(ejω)−(∑p
u=1ucuejuω)D(ejω)
N2(ejω),
then
(1−P(ejωi)Q(ejωi) = 0
d
dω(1−P(ejω)Q(ejω)) ω=ωi=0.(7)
Proof. We ﬁrst derive the conditions for the ﬁrst equation in (7).
When P(ejωi)6=0, this can be rewritten as
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.(8)
Given Q(z) = b0+∑m
l=1blz−l, (8) becomes
b0+
m
∑
l=1
blcoslωi=ℜP(ejωi)
P(ejωi)2
m
∑
l=1
blsinlωi=ℑP(ejωi)
P(ejωi)2.
In matrix form, these conditions are shown in the upper section
of the right hand side of (6).
Now we prove the conditions of the second equation in (7),
which is equivalent to
(dP(ejω)
dωQ(ejω)+ dQ(ejω)
dωP(ejω))ω=ωi=0.
When 1−P(ejωi)Q(ejωi) = 0 and P(ejωi)6=0, this becomes
(dP(ejω)
dω
1
P(ejω)+dQ(ejω)
dωP(ejω)) ω=ωi=0,
i.e.
dQ(ejω)
dωω=ωi=−1
P2(ejω)
dP(ejω)
dωω=ωi=d
dω
1
P(ejω)ω=ωi.
(9)
Given again Q(z) = b0+∑m
l=1blz−l, the left hand side of (9) is
dQ(ejω)
dωω=ωi=−
m
∑
l=1
bllsinlωi−jm
∑
l=1
bllcoslωi.(10)
Given the coprime factorization of P=ND−1, the right hand side
of (9) becomes
d
dω
1
P(ejω)ω=ωi
=d
dz
1
P(z)
dz
dωz=ejωi
=d
dz
D(z)
N(z)
dz
dωz=ejωi
=j
d
dzD(z)N(z)−d
dzN(z)D(z)
N2(z)zz=ejωi
=j(qzq+∑q−1
v=1vavzv)N(z)−(∑p
u=1ucuzu)D(z)
N2(z)z=ejωi
=jH(ejωi).(11)
Matching the real and imaginary parts of (10) and (11) for i=
1,2,. . ., ngives the lower section of (6).
There are 4nlinear independent equations in (6), the mini
mum order for Qis m=4n−1.
Corollary 6. If (7) is true, then
d
dω1−P(ejω)Q(ejω)ω=ωi=0.
Proof. Assume that 1 −P(ejω)Q(ejω) = A(ω)ejθ(ω), where
A(ω)and θ(ω)are the magnitude and frequency responses, re
spectively. Then
d
dω(1−P(ejω)Q(ejω))= dA(ω)
dωejθ(ω)+jA(ω)dθ(ω)
dωejθ(ω)=0.
4
Copyright © 2018 ASME
2000 3000 4000 5000 6000 7000 8000 9000
5
0
5
10
15
Magnitude (dB)
Plant response inversion
Q via (4): narrowband design
Q via (6): wideband design
Bode Diagram
Frequency (Hz)
1=2000Hz
2=4000Hz
3=7000Hz
FIGURE 4.COMPARISON OF MAGNITUDE RESPONSE FOR DIF
FERENT Q FILTER DESIGNS.
Note that A(ωi) = 0. The above equation then gives
d
dω(1−P(ejω)Q(ejω)) ω=ωi=dA(ω)
dωejθ(ω)ω=ωi=0,
which is equivalent to
dA(ω)
dω=d
dω1−P(ejω)Q(ejω)ω=ωi=0.
Consider again the case in Fig. 3. The frequency response
of the proposed Q(ejω)from (6) is added, which is shown in Fig.
4. Compared with the basic solution (4), the magnitude response
of the ﬁlter that incorporates higherorder plant dynamics is seen
to match the inverse magnitude response of P(z)within a large
band around the target frequencies.
The magnitude response of 1 −PQ in Fig. 5 further illus
trates the beneﬁt of (6) for wideband loop shaping. It can be
seen that the new ﬁlter design gives a wider notch shape at all
three target frequencies. A zoomin view of the magnitude re
sponse of 1 −PQ shows that the proposed wideband design in
deed achieves zero derivative (which is guaranteed by Corollary
6) at the target frequencies, creating lower magnitudes at the fre
quency regions centered around these points.
For implementation, one can calculate H(ejω)based on the
analytic transfer function, or directly calculate the derivative of
the inverse frequency response on the right hand side of (9) by
using measured frequency responses of P(ejω).
The proposed FIR ﬁlter achieves desired energy transmis
sion modulation at ωi. Yet, because there is no constraint on
the overall magnitude, this basic solution tends to induce unde
sired ampliﬁcation when ω6=ωi, especially at frequencies far
away from the target frequency. Such waterbed effect is partic
ularly severe and dangerous in wideband loopshaping design.
Meanwhile, at frequencies where there are large model uncer
tainties and mismatches, highperformance control intrinsically
has to be sacriﬁced for robustness based on robust control the
ory. Thus, the proposed implementation form is to incorporate
special bandpass characteristics to maintain the magnitude of
1000 2000 3000 4000 5000 6000 7000 8000
100
50
0
50
Magnitude (dB)
Q via (4): narrowband design
Q via (6): wideband design
(a). magnitude response of 1PQ
1000 2000 3000 4000 5000 6000 7000 8000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Magnitude (abs)
(b). zoomin view of 1PQ
Frequency (Hz)
FIGURE 5.MAGNITUDE RESPONSE OF 1PQ FOR DIFFERENT
Q DESIGNS.
Q(ejω)small when ω6=ωi. More speciﬁcally, we propose the
following latticestructure [19] bandpass ﬁlter
QBP(z) = 1−1
2n
n
∏
i=1
(1+k2,i)(1+2k1,iz−1+z−2)
1+k1,i(1+k2,i)z−1+k2,iz−2,(12)
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 (12) to (5) gives
Q(z) = QBP(z)(b0+
m
∑
l=1
blz−l),(13)
which not only maintains the desired wideband loop shape, but
also blocks noises in d(k)outside the target frequency ranges.
4 SIMULATION VERIFICATION
The proposed algorithm is veriﬁed by simulation on a two
axis galvo scanner in selective laser sintering additive manufac
turing. The identiﬁed plant transfer function is
ˆ
P(z) = 0.0282z2+0.1504z+0.1146
z4−1.3190z3+0.929z2−0.6073z−0.0035,(14)
where the sampling time Ts=0.025ms. A ﬁgure that shows the
frequency response of (14) compared with the measured response
is provided in [18]. Note that this is a nonminimumphase sys
tem with an unstable zero at z=−4.419. A vendorintegrated
baseline controller is already embedded in the plant. We thus set
C(z) = 1 in Fig. 1. The magnitude response of the baseline sensi
tivity function is provided in Fig. 6. The system has a bandwidth
around 1000Hz.
Fig. 7 shows spectrum of the wideband disturbance used in
the simulation2. The vibrations contain three major wideband
5
Copyright © 2018 ASME
101102103104105
60
40
20
0
Magnitude (dB)
Bode Diagram
Frequency (rad/s)
FIGURE 6.MAGNITUDE RESPONSE OF BASELINE SENSITIVITY
FUNCTION.
0 1000 2000 3000 4000 5000 6000 7000
Frequency (HZ)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Amplitude
Wideband disturbance
FIGURE 7.WIDEBAND DISTURBANCE IN SIMULATION.
peaks centered around 100Hz, 900Hz, and 2500Hz.
The baseline sensitivity function already attenuates the low
frequency signal. Thus, in the Q ﬁlter design, we only focus on
the higher bands (900Hz and 2500Hz). Fig. 8 shows magnitude
responses of the Q ﬁlter and 1 −PQ. As expected, two wide
attenuation notches are located at the target frequencies in the
upper plot. To mitigate the large ampliﬁcation in high frequency
region due to waterbed effect, the red dashed line shows the result
of applying the latticestructure bandpass ﬁlter in (12). The cor
responding performance in time and frequency domain is shown
in Fig. 9 and Fig. 10, respectively. It can be seen that the low
frequency band is rejected in both plots thanks to the baseline
sensitivity function (recall Fig. 6). However, as the input energy
frequency increases, baseline sensitivity function is not powerful
enough anymore. On the other hand, the proposed scheme is able
to effectively attenuate the large spectral peaks.
5 CONCLUSION AND FUTURE WORK
In this paper, a wideband loop shaping scheme for modu
lating energy transmission in a feedback system is introduced.
This algorithm is constructed by designing a pointwise model
inversion ﬁlter. The proposed scheme avoids explicit plant inver
sions and is particularly useful for nonminimumphase systems
or when a stable plant inversion is prohibitively expensive over
the full frequency range. Simulation on a galvo scanner platform
2The disturbance is a scaled analogy of actual disturbance in high precision
motion system [2].
101102103104
300
200
100
0
100
Magnitude (dB)
Wideband Q
via (6)
Wideband Q with bandpass filter
1PQ
Frequency (Hz)
101102103104
300
200
100
0
100
Magnitude (dB)
Q
Frequency (Hz)
FIGURE 8.MAGNITUDE RESPONSES OF THE WIDEBAND Q
FILTER AND 1−PQ.
0 0.5 1 1.5
1.5
1
0.5
0
0.5
1
1.5
Voltage (V)
w/o compensation: 3 =0.90379
0 0.5 1 1.5
Time (s)
1.5
1
0.5
0
0.5
1
1.5
Voltage (V)
w/ compensation, 3 =0.38075
FIGURE 9.TIME SERIES COMPARISON WITH AND WITHOUT
THE PROPOSED WIDEBAND Q COMPENSATOR.
in selective laser sintering shows signiﬁcant performance gains
for attenuating multiple wideband energy transmission. The re
sult is achieved by utilizing the ﬁrstorder derivative of plant dy
namics response for the loopshaping ﬁlter design. Experimenta
tion on actual hardware is underway. As a future work, while the
achieved selective model interpolation was observed to be effec
tive for mechatronic applications, higherorder interpolation and
potential tradeoffs will be investigated. Brieﬂy, these higher
order interpolation conditions can be translated to the core matrix
equation analogously as the ﬁrstorder derivative condition, and
we expect the tools and knowledge from this paper will apply
directly to extensional cases.
6
Copyright © 2018 ASME
0 1000 2000 5000 6000 7000
3000 4000
0
0.005
0.01
0.015
0.02
0.025
Amplitude
w/o compensation
0 1000 2000 3000 4000 5000 6000 7000
Frequency (HZ)
0
0.005
0.01
0.015
0.02
0.025
Amplitude
w/ compensation
FIGURE 10.SPECTRUM COMPARISON WITH AND WITHOUT
THE PROPOSED WIDEBAND Q COMPENSATOR.
ACKNOWLEDGEMENT
T. J., J. T., and X. C. are grateful to support from the Depart
ment of Mechanical Engineering at the University of Connecti
cut. T. J. would like to acknowledge the partial ﬁnancial support
from the GE Fellowship for innovation.
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