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Wide-Band Loop Shaping for Modulation of Energy Transmission in Nonminimum-Phase Systems

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Modulating the closed-loop transmission of energy in a wide frequency band without sacrificing overall system performance is a fundamental issue in a wide range of applications from precision control, active noise cancellation, to energy guiding. This paper introduces a loop-shaping approach to create such wideband closed-loop behaviors, with a particular focus on systems with nonminimum-phase zeros. Pioneering an integration of the interpolation theory with a model-based 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 laser-based powder bed fusion additive manufacturing validates the feasibility to compensate wide-band vibrations and to flexibly control system performance at other frequencies.
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Wide-band Loop Shaping for Modulation of Energy Transmission in
Nonminimum-phase 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 closed-loop 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 loop-shaping approach to create such wide-
band closed-loop behaviors, with a particular focus on systems
with nonminimum-phase zeros. Pioneering an integration of the
interpolation theory with a model-based 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 laser-based powder bed fusion
additive manufacturing validates the feasibility to compensate
wide-band vibrations and to flexibly control system performance
at other frequencies.
1 INTRODUCTION
Active and flexible shaping of dynamic system responses is
key for modulating input-to-output energy flows. For example,
adaptive disturbance attenuation minimizes the energy of posi-
tion error and achieves precision motion control in nm-scale 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 wide-band 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 closed-loop error-rejection functions at fre-
quencies where the input energy (disturbance) dominates. For
wide-band 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 nonminimum-phase 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 nonminimum-phase sys-
tems. Compared to other loop-shaping 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 high-order dynamics locally, this scheme creates
wide notches in the closed-loop sensitivity function. By integrat-
ing a cascaded filtering design into the Youla-Kucera (YK) pa-
rameterization, the algorithm directly controls energy amplifica-
tion at other frequencies. Verification is performed by simulation
on a two-axis 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 30-October 3, 2018, Atlanta, Georgia, USA
DSCC2018-9089
2 reviews the algorithmic foundation of the proposed method and
its connection to our previous work on narrow-band loop shap-
ing; Section 3 provides the main results of wide-band 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
and denote, respectively, the real and imaginary part of a com-
plex number.
2 REVIEW OF RELEVANT LITERATURE
2.1 Review of Youla-Kucera parameterization
Theorem 1. (YK Parameterization [15, 16]). If an SISO plant
P=N/D can be stabilized by a negative-feedback 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
YNQ :QS,Y()N()Q()6=0.
Here, (N,D)is called a coprime factorization of PRover
Sif: (i) P=ND1, (ii) N(S)and D(S)are coprime trans-
fer functions, and (iii) D1R.
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 1N
YQ,(1)
which is decoupled into the product of the baseline sensitivity
1/(1+PC)and the add-on affine module 1NQ/Y. Because
stability is assured under the controller parameterization, design-
ers can now focus on the add-on 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 plant-controller pairs that often arise in mechatronic
systems.) Then
Call =C(z) + Q(z)
1P(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)(1P(z)Q(z)) ,S0(z)(1P(z)Q(z)).(2)
The factorization above renders the add-on module in (1) to
1P(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)
1ˆ
P(z)Q(z),Q(z)S
˜
S=1
1+P(z)C(z) + (P(z)ˆ
P(z))Q(z)(1ˆ
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)(1P(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) =
P1(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 non-properness 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=1blzl,with
b0
.
.
.
bm
=
1 cosω1. . . cosmω1
0 sinω1. . . sinmω1
.
.
..
.
.
.
.
..
.
.
1 cosωn. . . cosmωn
0 sinωn. . . sinmωn
1
P(ejω1)
|P(ejω1)|2
P(ejω1)
|P(ejω1)|2
.
.
.
.
.
.
P(ejωn)
|P(ejωn)|2
P(ejωn)
|P(ejωn)|2
,(4)
where m =2n1. 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 closed-loop 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,
high-performance 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 WIDE-BAND LOOP SHAPING FOR
NONMINIMUM-PHASE SYSTEMS
Compared with single-frequency excitation, wide-band 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 loop-shaping 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) wide-band 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 customer-specified frequencies
(even above the bandwidth of the servo loop), they focus on the
rejection of narrow-band disturbances (see Fig. 1(a)) or single-
peak band-limited 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 wide-band 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
two-degree-of-freedom (2-DOF) 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 dual-stage HDDs; (ii) wide-band disturbances with the
important extension to multiple spectral peaks are addressed;
and (iii) experimental verification on a Voice-Coil-Driven
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 dual-stage HDDs. Section III discusses the
performance-oriented Qfilter design with tunable passband
widths. In Section IV, the 2-DOF 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 track-following prob-
lem of HDD systems in the presence of the above discussed
wide-band disturbances. Figure 2 shows the proposed adaptive
control structure in single-stage HDDs. Without the add-on
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 single-stage HDDs and
around 2000Hz for dual-stage 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
add-on compensator
OO
Fig. 2. The structure of proposed control scheme for single-stage 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 add-on 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 add-on compensator is to generate a cancellation
signal c(k)which can effectively cancel out the influence of
1For discrete-time systems, we always have m1and the separation
of zmassures a realizable inverse of Pmz1and its nominal model
described afterwards.
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FIGURE 2.EXAMPLE OF WIDE-BAND 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 add-on mod-
ule 1P(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 first-order 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 loop-shaping notches in the frequency response
of the add-on module 1P(ejω)Q(ejω).
Proposition 5. Let ωi(0,π),i=1,2, . .. , n be center frequen-
cies of the target loop-shaping regions, (N,D)be a coprime fac-
torization of P such that P =ND1, where N =p
u=0cuzuand
D=zq+q1
v=0avzv. Design
Q(z) = b0+
m
l=1
blzl,(5)
3
Copyright © 2018 ASME
with m =4n1and
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
P(ejω1)
|P(ejω1)|2
P(ejω1)
|P(ejω1)|2
.
.
.
P(ejωn)
|P(ejωn)|2
P(ejωn)
|P(ejωn)|2
H(ejω1)
H(ejω1)
.
.
.
H(ejωn)
H(ejωn)
,
(6)
where
H(ejω) = (qejqω+q1
v=1vavejvω)N(ejω)(p
u=1ucuejuω)D(ejω)
N2(ejω),
then
(1P(ejωi)Q(ejωi) = 0
d
dω(1P(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.
Q(ejωi) = P(ejωi)
|P(ejωi)|2
Q(ejωi) = P(ejωi)
|P(ejωi)|2
,i=1,2,. . ., n.(8)
Given Q(z) = b0+m
l=1blzl, (8) becomes
b0+
m
l=1
blcoslωi=P(ejωi)
|P(ejωi)|2
m
l=1
blsinlωi=P(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 1P(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)
dω|ω=ωi=1
P2(ejω)
dP(ejω)
dω|ω=ωi=d
dω
1
P(ejω)|ω=ωi.
(9)
Given again Q(z) = b0+m
l=1blzl, the left hand side of (9) is
dQ(ejω)
dω|ω=ωi=
m
l=1
bllsinlωijm
l=1
bllcoslωi.(10)
Given the coprime factorization of P=ND1, the right hand side
of (9) becomes
d
dω
1
P(ejω)|ω=ωi
=d
dz
1
P(z)
dz
dω|z=ejωi
=d
dz
D(z)
N(z)
dz
dω|z=ejωi
=j
d
dzD(z)N(z)d
dzN(z)D(z)
N2(z)z|z=ejωi
=j(qzq+q1
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=4n1.
Corollary 6. If (7) is true, then
d
dω1P(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(1P(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): narrow-band design
Q via (6): wide-band 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(1P(ejω)Q(ejω)) |ω=ωi=dA(ω)
dωejθ(ω)|ω=ωi=0,
which is equivalent to
dA(ω)
dω=d
dω1P(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 higher-order 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 wide-band loop shaping. It can be
seen that the new filter design gives a wider notch shape at all
three target frequencies. A zoom-in view of the magnitude re-
sponse of 1 PQ shows that the proposed wide-band 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 wide-band loop-shaping design.
Meanwhile, at frequencies where there are large model uncer-
tainties and mismatches, high-performance 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): narrow-band design
Q via (6): wide-band design
(a). magnitude response of 1-PQ
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). zoom-in view of 1-PQ
Frequency (Hz)
FIGURE 5.MAGNITUDE RESPONSE OF 1-PQ FOR DIFFERENT
Q DESIGNS.
Q(ejω)small when ω6=ωi. More specifically, we propose the
following lattice-structure [19] bandpass filter
QBP(z) = 11
2n
n
i=1
(1+k2,i)(1+2k1,iz1+z2)
1+k1,i(1+k2,i)z1+k2,iz2,(12)
where k1,i=cosωiand k2,i= [1tan(Bw,i/2)]/[1+
tan(Bw,i/2)],Bw,i(in radian) is the 3-dB 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
blzl),(13)
which not only maintains the desired wide-band 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
z41.3190z3+0.929z20.6073z0.0035,(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 nonminimum-phase sys-
tem with an unstable zero at z=4.419. A vendor-integrated
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 wide-band disturbance used in
the simulation2. The vibrations contain three major wide-band
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
Wide-band disturbance
FIGURE 7.WIDE-BAND 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 lattice-structure 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 wide-band 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 nonminimum-phase 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)
Wide-band Q
via (6)
Wide-band Q with bandpass filter
1-PQ
Frequency (Hz)
101102103104
-300
-200
-100
0
100
Magnitude (dB)
Q
Frequency (Hz)
FIGURE 8.MAGNITUDE RESPONSES OF THE WIDE-BAND Q
FILTER AND 1PQ.
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 WIDE-BAND Q COMPENSATOR.
in selective laser sintering shows significant performance gains
for attenuating multiple wide-band energy transmission. The re-
sult is achieved by utilizing the first-order derivative of plant dy-
namics response for the loop-shaping 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, higher-order interpolation and
potential trade-offs will be investigated. Briefly, these higher-
order interpolation conditions can be translated to the core matrix
equation analogously as the first-order 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 WIDE-BAND 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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