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An Inverse-Free Disturbance Observer for Adaptive Narrow-Band Disturbance Rejection With Application to Selective Laser Sintering


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Selective laser sintering (SLS) is an additive manufacturing (AM) process that builds 3-dimensional (3D) parts by scanning 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 urging the applications of customized controls in SLS. In this work, a Youla-Kucera parameterization based forward-model 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 always 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.
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Tianyu Jiang
Dept. of Mechanical Engineering
University of Connecticut
Storrs, Connecticut, 06269
Hui Xiao
Dept. of Mechanical Engineering
University of Connecticut
Storrs, Connecticut, 06269
Xu Chen
Dept. of Mechanical Engineering
University of Connecticut
Storrs, Connecticut, 06269
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 dual-axis 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
Scanner mirrors
Sintered part
F-theta lens
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 beam-steering
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 narrow-band 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
narrow-band vibrations but also wide-band noises. Adaptive
feedforward cancellation [6] composes an estimation of a sinu-
Proceedings of the ASME 2017 Dynamic Systems and Control Conference
October 11-13, 2017, Tysons, Virginia, USA
Selective laser sintering (SLS) is an additive manufactur-
ing (AM) process that builds 3-dimensional (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 Youla-Kucera parameterization based forward-model 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.
Copyright © 2017 ASME
In view of the limitations above, this paper proposes a new
DOB scheme for rejection of narrow-band disturbance. Instead
of an explicit plant-inversion, 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 and
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. Highly-accurate, controlled motion is achieved with a po-
sition detector that enables closed-loop 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
Laser source Galvo scanne r
Power supply
PC server with
dSPACE and
White screen
Servo driver
3 Forward-model 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
narrow-band disturbance in d(k);
Adaptive parameter estimation algorithm: provides online
information of the characteristics of ˆ
The basic structure of the closed-loop 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. Block-diagram analysis gives
D(z)= (P(z)˜
At frequencies where the identified plant model ˆ
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.
Copyright © 2017 ASME
soidal disturbance using a Phase-Locked 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 state-space 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 high-gain 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-
Adaptive Parameter
Add-on Compensator
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=ωji6=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
Q(ejωi) = 1
Q(ejωi) = P(ejωi)
Q(ejωi) = P(ejωi)
,i=1,2,. . ., n.(4)
4.1 Basic solution
The general solution to (3) is provided in the following
Proposition 1. Let
Q(ejω) = (b0+
blel jω),(5)
1 cosω1. . . cosmω1
0 sinω1. . . sinmω1
1 cosωn. . . cosmωn
0 sinωn. . . sinmωn
where m =2n1. Then
Q(ejωi)P(ejωi) = 1,i=1,2,. . ., n
Proof. Let Q(ejω) = b0+m
l=1blel jω, where m(Z)is the or-
der of the filter. Based on (4), we must have, for i=1,2,.. . ,n,
In matrix form, the above is equivalent to
1 cosωi. .. cos mωi
0 sinωi. . . sinmωi
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 ωis. 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
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 lattice-structure [19] band-
pass filter
QBP(ejω) = 11
Copyright © 2017 ASME
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 (7) to (5) gives the improved design
Q(z) = QBP(z)(b0+
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 frequency-domain closed-
loop properties and how to control the relevant performance lim-
Corollary 1. Take any P and Q that are stable and causal. The
magnitude response of 1−P(ejω)Q(ejω)satisfies
ln1−P(ejω)Q(ejω)dω=π nγ
ln|γi|ln |σ+1|!,(9)
where {γi}nγ
i=1(nγ0)is the set of unstable zeros of 1−P(z)Q(z)
i=1,Φif nγ=0), and
Proof. See [20].
Corollary 1 specifies potential fundamental performance
limits similar to all linear designs. For plants whose relative de-
gree is zero, limzP(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), limzP(z) =
0 and σ=0, (9) simplifies to Rπ
i=1ln|γi|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)
ρcosωp. . . 1
ρsinωp. . . 1
10210 3
Magnitude (dB)
Frequency (Hz)
2 3
Magnitude (dB)
Basic solution
With fixed zero near z=1
10210 3
Magnitude (dB)
Frequency (Hz)
10210 3
Magnitude (dB)
Basic solution
With fixed zero near z=1
Frequency (Hz)
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
ρcosωp. . . 1
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. Block-diagram algebra gives the new
sensitivity function
S= (1PQ)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 high-frequency 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.
Copyright © 2017 ASME
Frequency (Hz)
10210 3104
Magnitude (dB)
Basic Q
w/ cascaed IIR enhancement
Frequency (Hz)
Based on the technical discussions in this section, the pro-
posed scheme offers the following benefits for disturbance rejec-
No explicit plant inversion: In a regular DOB, a full stable
inversion ˆ
P1(z)is required, which achieves ˆ
at all frequency regions where disturbance attenuation is de-
sired. However, this is usually infeasible due to instabil-
ity and non-properness. Thus, a chosen stable nominal in-
version ˆ
n(z)is adopted instead of ˆ
P1(z). In the region
where there is large mismatch between ˆ
n(z)and ˆ
(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
P1(z)to the forward model based selective DOB in
Fig. 4. Based on (4), Qcould be designed as a point-wise
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 low-pass 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 broad-band 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)1P(z)Q(z)
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 (1P(z)Q(z))w(k).
For a vibration disturbance that is composed of d(k) =
i=1Cisin(ωik+φi) (Ci6=0, ωi6=0, φiR), the denominator
of D(z)is
Expanding the product gives
A(z1) = 1+a1z1+· · · +anzn+·· · +a1z2n+1+z2n,(13)
where we have mapped the parameters {ωi}n
iin (12) to {ai}n
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 narrow-band signals that contain nunknown
frequency components. Reference [2] has proposed a two-stage
adaption scheme that combines recursive least squares (RLS) and
output error based PAA for identification of θ, where the param-
eter adaption equations are
θ(k1)+ P(k1) (φ(k1)) νo(k)
P(k)= 1
ν(k)=e(k) + α2ne(k2n) + ϕ(k1)Tθc
νo(k)=eo(k)+ α2ne(k2n) + ϕ(k1)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(z1) = 1+c1αz1+
· ·· +cnαnzn+· ·· +c1α2n1z2n+1+α2nz2n,α(0,1); and
ϕ(k1) = [ϕ1(k1),ϕ2(k1),. . ., ϕn(k1)]Tare computed
from: ϕi(k1) = αie(ki) + α2nie(k2n+i);i=1,..., n
1,ϕn(k1) = αne(kn). The regressor vector in (14) is
composed of φi(k1) = w(ki)+ w(k2n+i)αie(ki)
α2nie(k2n+i);i=1,..., n1,and φn(k1) = w(kn)
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 time-varying disturbance, we refer readers
to [22].
Copyright © 2017 ASME
6 Stability and robustness
This section analyzes the closed-loop 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 closed-loop
stability is guaranteed.
Proof. Recall Fig. 4. With some block-diagram algebra, the fol-
lowing transfer function is obtained
Y(z)= 1P(z)Q(z)+Q(z)(P(z)ˆ
P(z)) D(z).(15)
Let P(z) = BP(z)/AP(z),ˆ
P(z) = Bˆ
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 closed-loop characteristic equation becomes
P(z)[(AP(z)AC(z) + BP(z)BC(z)] = 0.
Hence the closed-loop 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 closed-loop stability,
the closed-loop stability of the new system is automatically ob-
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 Hnorm), applying the Small Gain The-
orem [25] yields the following robust-stability condition
where the complementary sensitivity function is given by T=
1S. After substituting (11) in, (16) becomes
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∆ωi0P(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
P(z) = 0.5465z3+0.4598z20.0717z0.0721
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 time-varying narrow-band disturbance series with
step change is considered in both simulation and experi-
ment. The disturbance frequency changes in the pattern
of null500Hz1500Hz500Hz3000Hz500Hznull.
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 steady-state position errors
indeed converge to zero with a maximum value less than 1 ×
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 steady-state 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 2-norm 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 steady-state 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 inverse-free forward model selective dis-
turbance observer scheme is introduced for narrow-band 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
Copyright © 2017 ASME
0 0.5 1 1.5 2 2.5
Time (s)
Estimated Frequency (Hz)
0 0.5 1 1.5 2 2.5
Time (s)
Voltage (V)
Open Loop
Close Loop
500Hz 1500Hz 500Hz 3000Hz 500Hz
0 0.5 1 1.5 2 2.5
Time (s)
Voltage (V)
Open Loop
Close Loop
500Hz 1500Hz 500Hz 3000Hz 500Hz
Period (s) 0.390.49 0.690.79 0.991.09
(Baseline) (500Hz) (1500Hz)
2(×101V2)1.652 1.451 1.373
Period (s) 1.291.39 1.591.69 1.891.99
(500Hz) (3000Hz) (500Hz)
2(×101V2) 1.485 1.521 1.458
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Frequency [Hz]
Power Spectrum Density (PSD) [dB]
significant performance gain for disturbance attenuation. The
proposed scheme avoids the explicit plant-inversion 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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Copyright © 2017 ASME
... Active and flexible shaping of dynamic system responses is central for ubiquitous modern precision systems. For example, modern hard disk drives (HDD) leverage adaptive disturbance attenuation to achieve nm-scale precision control of the read/write heads [1]- [3]; in smart headphones, active noise cancellation reduces acoustic disturbance transmission from the environment to human ears [4]; in selective laser sintering (SLS) additive manufacturing, minimizing position errors of galvo mirrors is key to improve achievable part geometry [5]. In these applications, a certain form of output profile is desired given various band-limited disturbances such as mechanical and acoustic vibrations, yielding the problem of local loop shaping (LLS). ...
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... Note that 1 − P (e jωi )Q(e jωi ) = 0. Under Assumption 2 (P (e jωi ) = 0), (12) becomes ( dP (e jω ) dω 5 Given Q(z) = q 0 + m l=1 q l z −l , the left hand side of (13) is ...
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Closed-loop disturbance rejection without sacrificing overall system performance is a fundamental issue in a wide range of applications from precision motion control, active noise cancellation, to advanced manufacturing. The core of rejecting band-limited disturbances is the shaping of feedback loops to actively and flexibly respond to different disturbance spectra. However, such strong and flexible local loop shaping (LLS) has remained underdeveloped for systems with nonminimum-phase zeros due to challenges to invert the system dynamics. This article proposes an LLS with prescribed performance requirements in systems with nonminimum-phase zeros. Pioneering an integration of the interpolation theory with a model-based parameterization of the closed loop, the proposed solution provides a filter design to match the inverse plant dynamics locally and, as a result, creates a highly effective framework for controlling both narrowband and wideband vibrations. From there, we discuss methods to control the fundamental waterbed limitation, verify the algorithm on a laser beam steering platform in selective laser sintering additive manufacturing, and compare the benefits and tradeoffs over the conventional direct inverse-based loop-shaping method. The results are supported by both simulation and experimentation.
... However, when the plant contains unstable zero(s) or is strictly proper, a direct inversion will introduce instability or non-properness to the closed loop. In our previous work [17,18], we proposed a pointwise inverse design to overcome these fundamental challenges. ...
... , n. Proof. See [18]. ...
... By focusing on the local inversion of P(e jω ), 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. ...
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