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This paper considers the real-time recovery of a fast time series by using sparsely sampled measurements from sensors whose sampling speeds are prohibitively slow originally. Specifically, when the fast signal is an autoregressive process, we propose an online information recovery algorithm that reconstructs the dense underlying temporal dynamics fully by systematically modulating two slow sensors, and by exploiting a model-based fusion of the sparsely collected data. We provide the design of collaborative sensing and model-based information recovery algorithm, impacts of parameter choosing and model singularity, and methods to reduce computational complexity and increase prediction robustness. The proposed method is experimentally verified in an optical beam steering platform for additive manufacturing. Application to a closed-loop disturbance rejection problem reveals the feasibility to eliminate fast disturbance signals with the slow and not fully aligned sensor pair in real time, and in particular, the rejection of narrow-band disturbances whose frequencies are much higher than the Nyquist frequencies of the sensors.
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS
A Collaborative Sensing and Model-Based Real-Time
Recovery of Fast Data Flows from Sparse
Measurements
Hui Xiao, Yaakov Bar-Shalom, IEEE Fellow, and Xu Chen,IEEE Member
Abstract—This paper considers the real-time recovery of
a fast time series by using sparsely sampled measurements
from sensors whose sampling speeds are prohibitively slow
originally. Specifically, when the fast signal is an autoregressive
process, we propose an online information recovery algorithm
that reconstructs the dense underlying temporal dynamics
fully by systematically modulating two slow sensors, and by
exploiting a model-based fusion of the sparsely collected data.
We provide the design of collaborative sensing and model-
based information recovery algorithm, impacts of parameter
choosing and model singularity, and methods to reduce com-
putational complexity and increase prediction robustness. The
proposed method is experimentally verified in an optical beam
steering platform for additive manufacturing. Application to a
closed-loop disturbance rejection problem reveals the feasibil-
ity to eliminate fast disturbance signals with the slow and not
fully aligned sensor pair in real time, and in particular, the
rejection of narrow-band disturbances whose frequencies are
much higher than the Nyquist frequencies of the sensors.
Index Terms —multirate signal processing, information re-
covery, disturbance beyond Nyquist frequency
I. Introduction
Fast feedback response is key for safe and high-performance
operation of a control system. Whether the application is to
monitor thermal conditions in a nuclear power plant, to track
ground and aerial targets for defense purposes, or to maintain
material temperature when additively manufacturing personal-
ized prosthetic implants for patients, we build mathematical
models, collect measurements, and analyze the performance
by assuming or desiring fast sampled measurements (e.g., 20
times the desired closed-loop bandwidth in a servo problem
[1]). However, many sensors update at intrinsically limited
speeds. For instance, the update rate of a radar scanner is
constrained by the rotation rate of the antenna; for imaging-
based automation, complex computations must be performed
to extract information from the raw image frames. In the
: corresponding to: Mechanical Engineering Building, 3900 E Stevens
Way NE, Seattle, WA 98195.
At the initial submission of this manuscript, Xu Chen (email:
chx@uw.edu) and Hui Xiao (email: huix27@uw.edu) were with
the faculty and the graduate students at the University of Connecticut,
respectively. Xu Chen and Hui Xiao are now with the faculty and
the graduate students in the Department of Mechanical Engineering,
University of Washington - Seattle, respectively.
Yaakov Bar-Shalom (email: yaakov.bar-shalom@uconn.edu)
is with the faculty in the Department of Electrical and Computer
Engineering, University of Connecticut.
presence of fast dynamics and disturbances that happen be-
tween the slow-rate sampling instances, the resulting lack of
information constrains the overall situational awareness of
the system, and can lead to unsafe system operation in a
wide range of engineering applications that need fast real-time
closed-loop operation. In pursuit of resolving this significant
barrier, this paper aims to provide a new information feedback
mechanism for systematic fast controls under slow information
feedback.
From a signal processing view point, a few strategies exist
to generate dense signals from sparse sensor measurements.
Under the first and perhaps the most commonly adopted
strategy, practitioners typically rely on simple techniques such
as linear interpolation. A second and mathematically more
elegant strategy interprets sampling as a projection operator
– one that computes a band-limited approximation of the
input signal. The concept here is to approximate the original
signal instead of insisting on a perfect reconstruction [2]. Both
the first and the second strategies focus on regular, uniform
sampling. A third strategy is compressed, or compressive
sensing (CS) [3], [4], [5] that involves irregular data collection.
It advocates randomized sampling and L1-norm minimization
to approximate the original signal in a transformed domain —
one that allows for a sparse, compressive representation of the
data. The fourth and still rapidly evolving strategy focuses
on utilizing co-prime sensor arrays for data collection [6],
[7], [8]. Methods in this category root from the theory of
prime numbers, which allows one to statistically estimate the
autocorrelation Rc(t)of a band-limited signal (with a sampling
time of Tseconds) at t=kT for any k, even though the
signal is only sampled sparsely at t=kP
MTand t=kP
NTwith
P
Mand P
Nbeing coprime integers. The fifth and most recent
strategy applies deep neural networks to learn the projection
from sparse to dense signals [9], [10], [11] from a training
data set that contains both sparse and dense signals.
From the viewpoint of control design, real-time closed-loop
functionality and causality are key factors when manipulating
a temporal signal flow. This leads to the difficulty that the
full sequence of measured data will not be available when
recovering a particular element in the middle of the experi-
ment. Then, what methodology can be used for desparsifying
a slowly measured information flow online, with assurance of
causality and real-time realizability? Aligned with the first
two discussed strategies of information processing, advanced
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digital to analog converter (DAC) and filtering have been
proposed for real-time controls considering inter-sample be-
haviors. However, the reconstruction is an approximate one,
and feedback operation based on multi-sensory sparse data
collection has not entered the field yet. Within the third
strategy of sparse signal processing, CS has been proposed as
a nice fit for networked feedback control when the remotely
transmitted measurement data is large and compressible [12],
[13], [14]. In these studies, the focus is not on data recovery
but on data compression. The co-prime methods in the fourth
strategy do not directly reconstruct data in the dense space.
Spectral estimation such as estimating the frequencies of
sinusoids and the Direction-Of-Arrival (DOA) of signal waves
are the current main focuses. The deep learning based methods
in the fifth strategy focus on sparsity in the pixel space of the
images rather than sparsity in the update rate of measurements,
and the recovering performance depends on the quality of the
training data.
Building upon the above knowledge and moving beyond
existing architectures of constrained real-time functionality,
this paper proposes an online computation-friendly algorithm
to recover a discrete signal d[n]with a sampling time of
Tseconds from sparsely sampled measurements and subse-
quently embeds the algorithm in feedback closed-loop con-
trols. Specifically, we consider the case when d[n]is a known
autoregressive process and propose to obtain the sparsely
sampled measurements from two sensors S1and S2with slow
sampling periods MT and N T . The integers Mand Nare
greater than one and M6=N. Our technical contribution is
that we provide the design and modeling of the collaborative
sensor pair, and that we show the dense and fast intersample
information can be fully recovered in real time, by collecting
parallel sets of very sparse sets of measured samples from the
data flow. This signal-reconstruction method is made possible
by elaborately designing and re-parameterizing the internal
signal model [15] of d[n]. Such an approach is completely new
for multi-sensor signal recovery to our best knowledge. It not
only facilitates first-of-its-kind real-time sparse information
processing, but can also seamlessly integrate the magnified
sensing with closed-loop model-based controls to achieve agile
feedback response to structured disturbances and reference
trajectories.
The remainder of this paper is organized as follows. Section
II introduces the mechanisms of the proposed collaborative
sensing and formulates the problem. Section III presents
the model-based information recovery algorithm. Discussions
on how to choose parameters, how to reduce computational
complexity, prediction robustness, and noise mitigation are
presented in section IV. Section V shows a numeric result
in a beam steering system for additive manufacturing, fol-
lowed by an example application to rejecting beyond-Nyquist
disturbance in section VI. Section VII concludes this paper.
A preliminary version of this paper was presented in [16].
This paper is a substantially extended study that includes new
results and the complete analysis. In particular, subsections
IV-C and IV-D and section V are completely new, and section
VI contains brand new validation results.
Notations: LCM(M,N)denotes the least common multiple
of Mand N. If a uniformly sampled sequence x[n]has sam-
pling period T,t{x[n]}=nT is the timestamp (the time when
a data point is measured) of signal x[n]. We use x1[n]x2[n]
to indicate that two sequences are equal and aligned in time
(i.e., x1[n] = x2[n],x1[n]and x2[n]have the same timestamps
for any n)1. The ceiling function dxemaps a real number xto
the smallest following integer. The remainder after division of
aby bis denoted by mod(a,b). The Moore–Penrose inverse
of a matrix Ais denoted as A.
II. Mechanisms of the Proposed Collaborative
Sensing
Let d[n]be a discrete time sequence with sampling time T,
dM[n]and dN[n]be the discrete measurements from two sen-
sors S1and S2with sampling times MT and N T , respectively.
The following direct connections hold:
dX[n]d[Xn],X=Mor N.(1)
In order to better describe the collaborative sampling process,
we divide d[n]into a list of subsequences {bi}i=1,2,3,..., where
biis referred to as the i-th batch in d[n]. Each batch contains
Lconsecutive data points in d[n], that is,
bi[k]d[iL +k],k=1,2,...,L,(2)
where bi[k]denotes the k-th data point in the i-th batch.
As a first result, when the batch size Lis properly set, it
can be shown that if the k-th data point in a batch is equal and
aligned to a data point in dM[n](or dN[n]), then the k-th data
point in the next batch will be equal and aligned to another
data point in dM[n](or dN[n]):
Lemma 1. Let the batch size L =LCM(M,N), if bi[k]dX[n],
then bi+1[k]dX[n+k1],where k1=L/X and X =M or N .
Proof. If bi[k]dM[n], then combining (1) and (2), one can
get d[iL+k]bi[k]dM[n]d[Mn], hence their time stamps
are equal: t{d[iL +k]}/T=iL +k=Mn =t{d[MN]}/T.
Then it can be shown that the time stamps of bi+1[k]and
dM[n+L/M]are equal: t{bi+1[k]}/T= (i+1)L+k=M(n+
L/M) = t{dM[n+L/M]}/T, where L/Mis an integer. In
addition, bi+1[k] = d[iL +k+L] = d[M(n+L/M)] = dM[n+
L/M].Thus we have bi+1[k]dM[n+L/M]. Analogously,
bi+1[k]dN[n+L/N]if bi[k]dN[n].
Lemma 1 suggests that the connections between dM[n],
dN[n]and d[n]are repeated over batches (see Fig. 1), if
the batch size Lis chosen as LCM(M,N). This property of
repeated connections makes it possible to design a procedure
to recover one batch of signal points, then use the procedure
repetitively to recover other batches. With this in mind, we
design our recovering algorithm under the following batch
configurations.
Definition 1. The batch bi[k]used in this paper (An example
is shown in Fig. 1) is defined based on the following rules:
1We use this notation rather than “=” because data points having an
identical value could have distinct time stamps. For example, a periodic
signal can have identical values x[n] = x[n+T], but x[n]and x[n+T]are
not aligned in time.
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Fig. 1. Connections between dM[n],dN[n]and d[n]when M=2,N=3
and L=6.
1) The first data points in d[n],dM[n]and dN[n]are aligned
in time, i.e., d[0]dM[0]dN[0].
2) The batch size L=LCM(M,N).
3) The last data point in a batch is aligned to both dM[n]
and dN[n], i.e., bi[L]dM[n1]dN[n2].
With the definition above, a signal batch bihas the following
properties:
1) There are L/Mdata points in a batch that are aligned to
dM[n], with index kKM={M,2M,3M,...,L}.
2) There are L/Ndata points in a batch that are aligned to
dN[n], with index kKN={N,2N,3N,...,L}.
3) There are LL/ML/N+1 data points in
a batch that are not aligned to either dM[n]
or dN[n]. This index set is denoted as KU=
{kZ+|k<L,mod(k,M)6=0,mod(k,N)6=0}.
For example, for M=3 and N=2, the batch size L=6,
KM={3,6},KN={2,4,6}and KU={1,5}. In the case with
M=8 and N=7, we have KM={8,16,24,32,40,48,56},
KN={7,14,21,28,35,42,49,56}and 42 unmeasured data
points exist in KU.
III. Proposed Model-based Information
Recovery with Collaborative Sensing
If the time index of the fast underlying signal bi[k]is aligned
to any of the sensor measurements, i.e. kKMor KN, a direct
measurement is available and no data recovery is needed.
However, if kKU,bi[k]is lost in the sampling process. The
following theorem shows that if d[n]satisfies an internal signal
model, the lost information can be recovered by combining
historical measurements form S1and S2.
Theorem 1. Let dM[n], dN[n], d [n], and bi[k]be defined as
described in the previous section. If there exists a polynomial
A(z1) = 1+m
i=1aizi(am6=0) such that A(z1)d[n] = 0at
the steady state (z1is the one-step delay operator such that
z1d[n] = d[n1]), then the k-th data point in the i-th batch
can be recovered by
bi[k] =
t1
i=0
wk,idM[n1i] +
t2
j=0
vk,jdN[n2j],(3)
where t1and t2are finite integers, n1and n2denote indices
of dMand dNsuch that dM[n1]dN[n2]bi1[k](such
relationship is ensured by the third rule of Definition 1). The
unknown parameters wk,i’s and vk,js come from the solution
to the following system of linear equations
Mk
fk,1
.
.
.
fk,l
wk,0
.
.
.
wk,t1
vk,0
.
.
.
vk,t2
=
a1
a2
.
.
.
am
0
0
.
.
.
0
.(4)
Here, l =max {t1M,t2N}+km; Mkis a matrix of dimension
(l+m)×(l+t1+t2+2), and is defined as
Mk= [ ˜
Mkekek+M.. . ek+t1Mekek+N. . . ek+t2N],(5)
where
˜
Mk=
1··· 0
a1
....
.
.
.
.
....0
am
...1
0...a1
.
.
.....
.
.
0··· am
(l+m)×l
,(6)
and eiis the elemental column vector whose entries are all
zeros except for the i-th entry, which equals 1.
Proof. To see first (3), we construct
Fk(z1)A(z1) + zkWk(zM) + zkVk(zN) = 1,(7)
where
Fk(z1) = 1+f1z1+· · · +flzl,(8)
Wk(zM) = wk,0+wk,1zM+·· · +wk,t1zt1M,(9)
Vk(zN) = vk,0+vk,1zN+·· · +vk,t2zt2N.(10)
Multiplying both sides of (7) with d[n]and dropping the trivial
term Fk(z1)A(z1)d[n], we have
d[n] = zkWk(zM)d[n] + zkVk(zN)d[n],(11)
namely,
d[n] =
t1
i=0
wk,id[nkiM] +
t2
j=0
vk,jd[nkjN].(12)
Let d[n]be the k-th data point of the i-th batch, i.e. d[n]
bi[k], then based on the batch definition (see (2)), we have
d[nk]d[iL]bi1[k].Recall that the indices n1and n2
are chosen such that dM[n1]dN[n2]bi1[L]. Thus we
get d[nk]dM[n1]dN[n2], or (nk)T=n1MT =n2NT
based on their time-stamp equivalence. Now the time stamps
of the summation terms in (12) are
t{d[nkiM]}= (nkiM)T
= (n1i)MT =t{dM[n1i]},(13)
t{d[nkjN]}= (nkjN)T
= (n2j)NT =t{dN[n2j]}.(14)
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Thus we get
d[nkiM]dM[n1i],(15)
d[nkjN]dN[n2j].(16)
In other words, (3) will be satisfied as long as (12), or its
equivalent from (7) is satisfied.
Now consider solving (7). Expanding the equation and
collecting the coefficients of zis (i=1,2,...,l+m), one can
get (l+m)linear equations with (l+t1+t2+2)unknowns,
which can be written in the matrix form as (4).
Example 1. Consider an illustrative example with M=3, N=
2 and A(z1) = 1+a1z1+a2z2. The model A(z1)has an
order of m=2. Based on Definition 1, the batch size is chosen
as L=LCM(3,2) = 6, then KU={1,5}.In the recovering
process, data points with index kKUin batches of d[n]will
be recovered from (3). Here we choose t1=t2=1 (there are
more discussions about choosing t1and t2in the following
section), then the recovering equations become:
bi[k] = wk,0d3[n1] + wk,1d3[n11]
+vk,0d2[n2] + vk,1d2[n21],k=1,5.(17)
When k=1, we have l=max {t1M,t2N}+km=2. Then
parameters w1,0,w1,1,v1,0,v1,1can be obtained from the
solution of
"1 0 1 0 1 0
a11 0000
a2a10 0 0 1
0a20 1 0 0 #
f1,1
f1,2
w1,0
w1,1
v1,0
v1,1
=a1
a2
0
0.(18)
When k=5 and l=max {t1M,t2N}+km=6, parameters
w5,0,w5,1,v5,0,v5,1can be obtained from the solution of
1 0 0 0 0 0 0 0 0 0
a11 0 0 0 0 0 0 0 0
a2a11 0 0 0 0 0 0 0
0a2a11 0 0 0 0 0 0
0 0 a2a11 0 1 0 1 0
000a2a11 0 0 0 0
0 0 0 0 a2a10 0 0 1
00000a20 1 0 0
f5,1
f5,2
f5,3
f5,4
f5,5
f5,6
w5,0
w5,1
v5,0
v5,1
=
a1
a2
0
0
0
0
0
0
.(19)
IV. Discussion
A. Minimal Historical Data Used in Recovery
In Theorem 1, (t1+1)data points from dM[n]and (t2+1)
data points from dN[n]are used in the recovery equation (3). In
fact, the number of historical data points used in the recovery
process is flexible, as we discuss next.
Corollary 1. A necessary condition for the system of equations
(4) to have a solution is
t1+t2m+nd2 (20)
where
nd=mint1+1
L/M,t2+1
L/N.(21)
Proof. Recall that a solvable system of linear equations must
not be overdetermined, so an obvious necessary condition for
(4) to have solutions is
l+t1+t2+2l+m(22)
In addition, when iM =jN holds for some i[0,t1]and
j[0,t2], the corresponding columns ek+iM and ek+jN in
matrix Mkare identical (see, e.g., the 7th and 9th columns
of Mkin (19)), yielding redundant pairs of variables in (4)
(say there are ndnumber of them). Then, the number of
independent variables becomes l+t1+t2+2ndand the
necessary condition (22) reduces to (20).
To more quantitatively define nd, we recall that a signal
batch could provide at most L/Mmeasurements from sensor
S1and L/Nmeasurements from sensor S2, hence the number
of prior batches used in the recovery that contain measure-
ments from S1(denoted as nd,M)or S2(denoted as nd,N)are
nd,M=t1+1
L/M,nd,N=t2+1
L/N.(23)
It can be seen from Definition 1 that the condition iM =jN
holds only once in a single batch, then the number of redun-
dant variable pairs ndis the number of prior batches where
measurements from both sensors are involved in the recovery
process. That is, nd=min{nd,M,nd,N}.
B. Method to Reduce Computation Complexity
Taking the pseudoinverse inverse of M
kgives a particular
solution of (4):
fk
qk=M
ka
0,(24)
where fk=fk,1,·· · ,fk,lT,qk=
wk,0,·· · ,wk,t1,vk,0,· · · ,vk,t2T,and a=[a1,· · · ,am]T. It is
worth noting that the computing time of taking pseudoinverse
is sensitive with the matrix size. We discuss next an reduced-
order procedure to solve (4) that will drastically reduce the
computation load for real-time applications.
The system of linear equations (4) can be rewritten into
the following form, where Mkis segmented into four smaller
matrices with dimensions defined below.
Em×lDm×(t1+t2+2)
Bl×lCl×(t1+t2+2) fk
qk=a
0.(25)
Then the following reduced-order solution can be obtained by
expanding the above the matrix equation:
qk=DEB1Ca.(26)
Instead of directly computing the pseudoinverse of the
large matrix Mk, the reduced-order method saves computation
cost by reducing the matrix dimension by lin height and
width before taking the pseudoinverse. Furthermore, efficient
algorithms exist for inverting the upper triangular matrix B
[17]. Figure 2 shows the changes of the computing cost as
kincreases when computing the prediction parameters in a
batch. The test results shows that the proposed method reduces
the computation costs to a significantly lower level under
different configurations; furthermore, the computation cost
remains largely invariant when kincreases.
Remark 1.The computational complexity of the prediction
step (i.e., equation (3)) is O(t1+t2+2),that is, linear with
respect to the total number of historical data used in the
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0 10 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
1.2 10-3
Fig. 2. The average time for computing the system solution using the
direct method and the reduced-order method. The signal model has an
order of 6 and M=8,N=7,t1=t2=3. The tests were performed on a
computer (with Intel Corei7-6800K CPU and 64GB memory) running
MATLAB 2017b.
Ori gnal mo del:
Re-param eteri zed
model: Recovered
data po int
Fig. 3. Illustration of model re-parameterizing for sparse information
recovery. Here, M=3,N=2,A(z1) = 1+a1z1+a2z2+a3z3,t1=
t2=1.
prediction. The computational complexity of calculating the
prediction parameters (i.e., equation (26)) depends mainly
on the complexity of inverting the m×(t1+t2+2)matrix
(DEB1C). The prediction parameters can be computed off-
line and only need to be recalculated when the signal model
changes.
C. Prediction Robustness and Model Singularity
Let xm= [dM[n1],...,dN[n2],. . . ]Tbe the measurement vec-
tor and xbe the measurement noise vector. Then the predic-
tion equation (3) can be formulated as d[n] = qk(xT
m+xT) =
qkxT+DEB1CxTwhere qkxTis the true value. The
prediction error cased by the measurement noise is thus
e=DEB1CxT.(27)
It is well understood that if the matrix DEB1Cis
ill-conditioned (i.e., close to becoming singular), then its
inversion will contain larger numbers, and consequently, the
measurement noise will be amplified.
In this section, we discuss the extreme case and provide
solutions to mitigate the model singularity. To see this, we first
note that the signal model A(z1) = 1+a1z1+· ·· +amzm
defines a linear function between a data point d[n]and its m
consecutive prior data points: d[n] = m
i=1amd[nm].An
essential part of our proposed recovery algorithm is to re-
parameterize the signal model and build a new connection
between d[n]and prior data points available from sparse
sensor measurements (recall Eqns. (3), (12), (15) and (16)).
Figure 3 illustrates the model re-parameterizing process when
recovering the first point in a batch. If the signal model
A(z1)contains scarce connections between d[n]and its prior
Fig. 4. An example when a missing data point is unrelated to any
measurements.
data points, the new connection would be difficult or even
impossible. For example, consider A(z1) = 1+a6z6, that is,
d[n] = a6d[n6]. As shown in Figure 4, when M=3,N=2,
or M=4, N=3, there exist missing data points in a batch
that are unrelated to any other measurements. Thus it becomes
impossible to find the re-parameterized model that can recover
the missing data point.
More generally, consider a signal model:
AQ(z1) = 1+
t
i=1
aiQziQ ,(28)
where QZ+,and at least one aiQ is nonzero. Then d[n]are
only connected to its iQ-th prior data points:
d[n] =
t
i=1
aiQd[niQ].(29)
Suppose d[n]bi[k]is the missing point in a batch that needs
to be recovered (i.e. kKU). If d[niQ]is not picked up by
the sensors for all i=1,2,3,·· · , then d[n]is impossible to
recover. Mathematically, given the batch configuration with
M,Nand L=LCM(M,N), if Qsatisfies the condition:
Exist kKU,such that for any iZ,mod(k+iQ,L)KU
(30)
then a signal with model AQ(z1)will not be fully recoverable.
Remark 2.If both (Q,M)and (Q,N)are not coprime pairs2,
then condition (30) is satisfied.
Proof. Because (Q,M)is not coprime, they have a common
divisor (denoted as p) that is greater than 1. For any integers
i,j,
M j Qi =pM j
pQi
p6=1 (31)
or equivalently,
mod(1+iQ,M)6=0.(32)
Let c=mod(k+iQ,L), then we have c=k+iQ nL, where
nZ. Because Lis a multiple of M, mod(c,M) = mod(k+
iQ nL,M) = mod(k+iQ,M). Therefore, for k=1KU,
mod(mod(k+iQ,L),M) = mod(k+iQ,M)6=0.(33)
Similarly, since (Q,N)is not coprime,
mod(mod(k+iQ,L),N) = mod(k+iQ,N)6=0.(34)
2(A,B)is a coprime pair if and only if their greatest common divisor
is 1.
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Combining (33) and (34), we conclude that mod(k+iQ,L)
KUfor any integer iand for k=1, so condition (30) is satisfied.
In practice, the discussed model singularities are uncommon
and can be overcome by designing the sensor pair. For
example, given M=3 and N=2, the smallest Qsuch that
both (Q,M)and (Q,N)are not coprime pairs is 6, then (28)
becomes AQ(z1) = 1+t
i=1a6iz6i. All signal models other
than AQ(z1)will be sufficient for the proposed recovering
algorithm with M=3 and N=2. Selecting M=5 and N=2,
on the other hand, provides a feasible solution.
D. Signal Pre-filtering
To improve the recovery accuracy in presence of measure-
ment noise, we propose in this subsection a filter design
to pass through signals dM[n]and dN[n]without modifying
their amplitude and phase, while rejecting noise signals in a
board frequency range. Given the internal signal model A(z1)
of d[n], we first show that the downsampled signals dM[n]
and dN[n]also contain internal signal models AM(z1)and
AN(z1), then use them in the filter design for model-based
noise compensation.
Remark 3.If there exists a polynomial A(z1)such that
A(z1)d[n] = 0 at the steady state, and dKis downsampled
from dby dK[n] = d[Kn], then there also exists a polynomial
AK(z1)such that AK(z1)dK[n] = 0 at the steady state, where
AK(z1) =
K1
p=0
A(z1
Kei2π
Kp).(35)
Proof. Let A(z1)d[n] = B(z1)δ[n],where δ[n]is the delta
impulse signal. Then A(z1)d[n] = 0 when n>nb, where nbis
the order of polynomial B(z1). Note that d[n]can be viewed
as the impulse response of a system with transfer function
D(z) = B(z1)/A(z1). Thus we conclude that internal model
of d[n]can be found from the denominator of its transfer
function D(z). Now consider the transfer function DK(z)of
d[Kn]
DK(z) = 1
K
K1
p=0
D(z1
Kei2π
Kp) = 1
K
K1
p=0
B(z1
Kei2π
Kp)
A(z1
Kei2π
Kp)
=B(z)
K1
p=0A(z1
Kei2π
Kp).(36)
Thus the internal model of d[Kn]can be found from the
denominator of DK(z)in (36), which is AK(z1)in (35).
The proposed noise attenuation is to pre-filter dK[n](K=M
or N) by
GK(z) = Aα(z1)AK(z1)
Aα(z1)(37)
and feed the filtered results to the collaborative sensor fusion.
Here, AK(z1)is the internal model of dK[n] = d[Kn]and
Aα(z1)is to be designed. Such a GK(z1)can pass a signal
dK[n]with an internal model AK(z1)as long as the roots
0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1
-25
-20
-15
-10
-5
0
5
Fig. 5. Magnitude of G(ejω)when choosing different values of αin
(40).
0 20 40 60 80 100 120 140 160 180
Steps
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Average noise in
500 Monte Carlo runs
Unfiltered noise
Filtered noise =0.8
Filtered noise =0.95
Fig. 6. Measurement noise before and after filtering by G(z)when
choosing a different α.
of Aα(z1)are in the unit circle. To be more specific, we
calculate
GK(z)dK[n] = Aα(z1)dK[n]
AK(z1)dK[n]
Aα(z1)=dK[n](38)
at the steady state. Let the polynomial AK(z1)have an order
of m, then it can be decomposed to the product of mfirst-order
polynomials
AK(z1) =
m
i=1
(z1λiejθi),(39)
where {λiejθi}i=1,...,mare the roots of AK(z1). For the purpose
of reducing the measurement noise energy, it is important to
maintain a small value to GK(ejω)(i.e., the magnitude of
the frequency response of GK(z)) at the noise frequencies. To
that end, we design Aα(z1)as
Aα(z1) =
m
i=1
(z1αλiejθi),(40)
where α(0,1)and is close to 1. Figure 5 shows the differ-
ence of GK(ejω)under different αselections. Here A(z1)
has 6 roots located at {e±jπ/5,e±j2π/5,0.9e±j3π/5}. It is
shown that GK(ejω)is maintained under 0dB at frequencies
other than the desired signal frequencies (0.2πand 0.4π),
so the noise energy can be reduced. When αis closer to 1,
GK(ejω)will be smaller in a wide frequency range, which
yields a better noise reduction rate. As a trade-off, GK(z)with
αcloser to 1 has a slower converging speed (see Fig. 6).
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V. Verification in Optical Beam Steering
Fast and accurate optical beam steering is crucial in many
applications such as selective laser sintering (SLS) [18], op-
tical coherence tomography (OCT) [19], and confocal mi-
croscopy [20]. For example, following a repetitive trajectory,
the galvo scanner used in SLS directs a high-energy laser
beam onto the surface of a powder bed to form a cross-
section layer of a part. The laser-material interaction here
happens in a short time scale (the laser beam moves at
meters per second on the powder bed), and the scanning
accuracy relates directly to the part quality [21]. In practice,
however, various disturbance sources exist in the optical path,
including, for example, mechanical vibrations of the platform
and thermal fluctuations due to smoke and circulation of the
inert gas in the process chamber. In order to compensate
such disturbances, it is important to have external sensors
(e.g. cameras) to measure the beam position on the target
surface. However, the sampling speed of such external sensors
is usually low and incapable of capturing the fast dynamics of
the physical process. In this section, we show that the proposed
information recovery technique can reconstruct a fast-sampled
beam position measurement from such sparse measurements.
Specifically, the dual-axis 6215H galvo scanner from Cam-
bridge Technology Inc was used. The recovered measurement
d[n]is sampled at 1kHz, and the two slow measurements dM[n]
and dN[n]are sampled at 1/8kHz and 1/7kHz, respectively.
In the experiment, the input to one axis of the galvo scanner
contains four frequency components at 120Hz, 167Hz, 240Hz,
and 300Hz. Such frequency components are much higher
than the Nyquist sampling frequencies of slow measurements,
that is, 62.5Hz for dM[n]and 71.4Hz for dN[n]. Hence the
discretized measurements will lose the original shape of the
trajectory (see the upper plot of Fig. 7) due to aliasing in the
sampling process.
Based on the internal signal model of a narrow-band signal
d[n]with nfrequency components fi,i=1,...,n,we have
A(z1)d[n] = 0 at the steady state [15], where
A(z1) =
n
i=1
(12cos(2πfiT)z1+z2)(41)
and Tis the sampling time of d[n]. Substituting in
the frequency values yields the polynomial model of
d[n]:A(z1) = 11.9619z1+4.1664z24.9797z3+
6.2201z44.9797z5+4.1664z61.9619z7+z8.In the
signal pre-filtering design, the internal model of dM[n]and
dN[n]can be obtained from (35) by replacing Kwith Mand
N. Then prefilters for dM[n]and dN[n]are designed as in Eqns.
(37, 40) with α=0.95. Figure 7 shows the slow measurements
dM[n]and dN[n]as well as the reconstructed signal d[n]. As
shown in the bottom plot in Fig. 7, the recovered data in d[n]
matches with its true value. The recover accuracy is further
confirmed in the frequency domain (see Fig. 8). The result
reveals the algorithm’s capability of recovering information
beyond the Nyquist frequency of sensors in real time. This
enables an approach to reject beyond-Nyquist disturbances in
a control system, as we show in the following section.
2 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.1
-20
-10
0
10
20
Beam position (mm)
dc(t)
dM[n]
dN[n]
2 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.1
Time (seconds)
-20
-10
0
10
20
Beam position (mm)
dc(t)
d[n]
Fig. 7. Collaborative sensor measurements and the recovered signal for
optical beam scanning in additive manufacturing.
0 50 100 150 200 250 300 350 400 450 500
frequency (Hz)
0
5
10
amplitude (mm)
true signal
estimated signal
Fig. 8. Frequency analysis of the recovered signal and the true signal.
VI. Application: Beyond-Nyquist Disturbances
Rejection
An immediate result of slowly sampled data in a feedback
system is that the controlled process will not be able to reject
fast disturbances, or more specifically, signals beyond the
Nyquist frequency of the feedback sensor. Our preliminary
study [22] has reported by simulation and experimentation that
a well-designed classic high-gain control could amplify instead
of attenuating the actual disturbance when its main spectral
components are near or beyond the Nyquist frequency of the
sensor. However, with the proposed model-based information
recovery technique, rejecting beyond Nyquist disturbances
using classic high-gain control becomes possible, as we shall
see from an example below.
Consider the optical beam steering example in section V,
where the beam path is subject to narrow-band disturbances
with spectral components beyond the Nyquist frequency of
the slow sensor. To compensate such disturbances, the focused
application will construct a multirate control system combin-
ing the narrow-band disturbance observer and the proposed
collaborative information recovery.
Figure 9 shows the combined control scheme for beyond-
Nyquist disturbance compensation. The control scheme in the
dash-dot box represent the original beam steering control loop.
The beam position y(t)on the target surface is measured
by a pair of collaborative sensors. In Fig. 9, discrete signals
with sampling times MT ,NT and Tare denoted by sparse
dashed lines, dense dashed lines and dotted lines, respectively;
continuous signals are denoted by solid lines. Components
of the disturbance rejection mechanism include ˆ
P
d(z), the
identified discrete model of the baseline beam steering control
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Fig. 9. Multi-rate control scheme for disturbance rejection using narrow-
band disturbance observer and the proposed information recovery algo-
rithm with collaborative sensing.
0 0.5 1 1.5 2 2.5 3
Time (seconds)
-15
-10
-5
0
5
10
15
Beam position (mm)
Disturbance compensation
loop truned on
Fig. 10. The beam position on the scanning surface sampled at 1kHz.
loop, and Q(z), the disturbance compensating filter. With the
disturbance frequency information known, one can design
Q(z)with the procedure provided in [21]. Our model-based
information recovery (the MR block in Fig. 9) is applied
to recover a fast disturbance estimate ˆ
d[n]using sparsely
sampled disturbance estimates ˆ
dM[n]and ˆ
dN[n]. Filters GM(z)
and GN(z)are adopted here for robustness based on algorithms
in subsection IV-D.
To show the effectiveness of the disturbance compensation
loop, we feed a periodic scanning trajectory to the input of
galvo scanner and let the beam path subject to beyond-Nyquist
disturbances with frequencies at 120Hz, 167Hz, 240Hz, and
300Hz. This is the same frequency distribution as in section
V. Thus the same information recovery design can be used.
Figure 10 shows the system output yd[n]sampled at 1kHz. The
disturbance compensation loop is turned on at t=1 second.
The results show that the beyond-Nyquist disturbances are
rejected significantly at a sampling rate seven times faster
than the maximum sampling speed of sensors. The overall
(including the transient response) 3σvalue of the trajectory
tracking error (i.e., the difference between the commanded and
actual value of the beam position on the scanning surface) is
reduced from 3.04 to 0.18, resulting in a 94% disturbance
reduction rate.
VII. Conclusion
In this paper, the problem of reconstructing a fast autore-
gressive signal from two channels of sparsely sampled collab-
orative sensor measurements is addressed. Based on temporal
sensor management and a model-based filtering using sensors
of different sampling speeds, the proposed algorithm can
recover, in real time, highly dense information that is not mea-
sured by the slow sensors. This algorithm was implemented
and validated in a disturbance compensation architecture to
enable full rejection of beyond-Nyquist disturbances in optical
beam steering. Future work includes model estimation, pa-
rameter adaptation, and extended accommodation of stochastic
noises.
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Hui Xiao was born in Yunnan, China. He re-
ceived the B.S. degree from the Department
of Mechanical Engineering, Tsinghua University,
Beijing, China, in 2011.
He is a Ph.D. student in the Department of
Mechanical Engineering, University of Connecti-
cut since 2015. He worked as a teaching assistant
and also a research assistant in the Department
of Mechanical Engineering. Starting fall 2019,
he is a Ph.D. student in the Department of Me-
chanical Engineering, University of Washington.
His research focuses on advanced control, sensor fusion and vision-based
robot control.
Yaakov Bar-Shalom (F’84) was born in
Romania. He received the B.S. (1963) and M.S.
(1967) degrees both in Electrical Engineering
from Technion in Haifa, Israel. Then he received
the Ph.D. degree in Electrical Engineering,
Princeton University in 1970.
He is currently the Board of Trustees
Distinguished Professor of Electrical and
Computer Engineering and the Marianne E.
Klewin Professor in Engineering in University
of Connecticut, Storrs, Connecticut, USA. His
research interests include Bayesian Estimation Theory with application
to remote sensing and information theoretic methods for combination
of data from multiple sources.
Prof. Bar-Shalom is recipient of IEEE Control Systems Society
Distinguished Member Award (1987); UConn AAUP Award for
Excellence in Research (1988); J. Mignona Data Fusion Award from
the DoD JDL Data Fusion Group (2002); IEEE Dennis J. Picard
Medal for Radar Technologies and Applications (2008); Connecticut
Medal of Technology (2012). He has been elected Fellow of IEEE for
contributions to the theory of stochastic systems and of multitarget
tracking.
Xu Chen (M’09) received the B.S. degree
in Mechanical Engineering from Tsinghua
University in 2008; and M.S and Ph.D. degrees
in Mechanical Engineering from University
of California, Berkeley in 2010 and 2013,
respectively.
From 2013 to 2014, he was a Lecturer at
UC Berkeley. In 2014, he joined the faculty
of the Department of Mechanical Engineering,
University of Connecticut, with secondary
appointments at the Institute of Materials
Science and the UTC Institute for Advanced Systems Engineering. In
summer 2019, he joined the faculty of the Department of Mechanical
Engineering, University of Washington. His research interests include
adaptive and learning control, information fusion, robust and
optimization-based control, with applications to additive and advanced
manufacturing, mechatronics, and agile robots.
Prof. Chen is the recipient of the U.S. National Science Foundation
(NSF) CAREER Award (2018), the Young Investigator Award from
ISCIE / ASME International Symposium on Flexible Automation, and
the UTC Institute for Advanced Systems Engineering Breakthrough
Award (2016).
ResearchGate has not been able to resolve any citations for this publication.
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