Content uploaded by Hui Xiao

Author content

All content in this area was uploaded by Hui Xiao on Oct 29, 2019

Content may be subject to copyright.

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

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

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.

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

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 k∈KM={M,2M,3M,...,L}.

2) There are L/Ndata points in a batch that are aligned to

dN[n], with index k∈KN={N,2N,3N,...,L}.

3) There are L−L/M−L/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=

{k∈Z+|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. k∈KMor KN, a direct

measurement is available and no data recovery is needed.

However, if k∈KU,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(z−1) = 1+∑m

i=1aiz−i(am6=0) such that A(z−1)d[n] = 0at

the steady state (z−1is the one-step delay operator such that

z−1d[n] = d[n−1]), then the k-th data point in the i-th batch

can be recovered by

bi[k] =

t1

∑

i=0

wk,idM[n1−i] +

t2

∑

j=0

vk,jdN[n2−j],(3)

where t1and t2are ﬁnite integers, n1and n2denote indices

of dMand dNsuch that dM[n1]↔dN[n2]↔bi−1[k](such

relationship is ensured by the third rule of Deﬁnition 1). The

unknown parameters wk,i’s and vk,j’s 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}+k−m; 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(z−1)A(z−1) + z−kWk(z−M) + z−kVk(z−N) = 1,(7)

where

Fk(z−1) = 1+f1z−1+· · · +flz−l,(8)

Wk(z−M) = wk,0+wk,1z−M+·· · +wk,t1z−t1M,(9)

Vk(z−N) = vk,0+vk,1z−N+·· · +vk,t2z−t2N.(10)

Multiplying both sides of (7) with d[n]and dropping the trivial

term Fk(z−1)A(z−1)d[n], we have

d[n] = z−kWk(z−M)d[n] + z−kVk(z−N)d[n],(11)

namely,

d[n] =

t1

∑

i=0

wk,id[n−k−iM] +

t2

∑

j=0

vk,jd[n−k−jN].(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[n−k]↔d[iL]↔bi−1[k].Recall that the indices n1and n2

are chosen such that dM[n1]↔dN[n2]↔bi−1[L]. Thus we

get d[n−k]↔dM[n1]↔dN[n2], or (n−k)T=n1MT =n2NT

based on their time-stamp equivalence. Now the time stamps

of the summation terms in (12) are

t{d[n−k−iM]}= (n−k−iM)T

= (n1−i)MT =t{dM[n1−i]},(13)

t{d[n−k−jN]}= (n−k−jN)T

= (n2−j)NT =t{dN[n2−j]}.(14)

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

Thus we get

d[n−k−iM]↔dM[n1−i],(15)

d[n−k−jN]↔dN[n2−j].(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 z−i’s (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(z−1) = 1+a1z−1+a2z−2. The model A(z−1)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 k∈KUin 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[n1−1]

+vk,0d2[n2] + vk,1d2[n2−1],k=1,5.(17)

When k=1, we have l=max {t1M,t2N}+k−m=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}+k−m=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+t2≥m+nd−2 (20)

where

nd=mint1+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+2≥l+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+2−ndand 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=D−EB−1C†a.(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

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

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 Core™i7-6800K 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(z−1) = 1+a1z−1+a2z−2+a3z−3,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

(D−EB−1C). 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+D−EB−1C†∆xTwhere qkxTis the true value. The

prediction error cased by the measurement noise is thus

e=D−EB−1C†∆xT.(27)

It is well understood that if the matrix D−EB−1Cis

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(z−1) = 1+a1z−1+· ·· +amz−m

deﬁnes a linear function between a data point d[n]and its m

consecutive prior data points: d[n] = −∑m

i=1amd[n−m].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(z−1)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(z−1) = 1+a6z−6, that is,

d[n] = −a6d[n−6]. 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(z−1) = 1+

t

∑

i=1

aiQz−iQ ,(28)

where Q∈Z+,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[n−iQ].(29)

Suppose d[n]↔bi[k]is the missing point in a batch that needs

to be recovered (i.e. k∈KU). If d[n−iQ]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 k∈KU,such that for any i∈Z,mod(k+iQ,L)∈KU

(30)

then a signal with model AQ(z−1)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

p−Qi

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

n∈Z. Because Lis a multiple of M, mod(c,M) = mod(k+

iQ −nL,M) = mod(k+iQ,M). Therefore, for k=1∈KU,

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.

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

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(z−1) = 1+∑t

i=1a6iz−6i. All signal models other

than AQ(z−1)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(z−1)

of d[n], we ﬁrst show that the downsampled signals dM[n]

and dN[n]also contain internal signal models AM(z−1)and

AN(z−1), then use them in the ﬁlter design for model-based

noise compensation.

Remark 3.If there exists a polynomial A(z−1)such that

A(z−1)d[n] = 0 at the steady state, and dKis downsampled

from dby dK[n] = d[Kn], then there also exists a polynomial

AK(z−1)such that AK(z−1)dK[n] = 0 at the steady state, where

AK(z−1) =

K−1

∏

p=0

A(z−1

Kei2π

Kp).(35)

Proof. Let A(z−1)d[n] = B(z−1)δ[n],where δ[n]is the delta

impulse signal. Then A(z−1)d[n] = 0 when n>nb, where nbis

the order of polynomial B(z−1). Note that d[n]can be viewed

as the impulse response of a system with transfer function

D(z) = B(z−1)/A(z−1). 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

K−1

∑

p=0

D(z1

Ke−i2π

Kp) = 1

K

K−1

∑

p=0

B(z−1

Kei2π

Kp)

A(z−1

Kei2π

Kp)

=B∗(z)

∏K−1

p=0A(z−1

Kei2π

Kp).(36)

Thus the internal model of d[Kn]can be found from the

denominator of DK(z)in (36), which is AK(z−1)in (35).

The proposed noise attenuation is to pre-ﬁlter dK[n](K=M

or N) by

GK(z) = Aα(z−1)−AK(z−1)

Aα(z−1)(37)

and feed the ﬁltered results to the collaborative sensor fusion.

Here, AK(z−1)is the internal model of dK[n] = d[Kn]and

Aα(z−1)is to be designed. Such a GK(z−1)can pass a signal

dK[n]with an internal model AK(z−1)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(z−1)are in the unit circle. To be more speciﬁc, we

calculate

GK(z)dK[n] = Aα(z−1)dK[n]−

AK(z−1)dK[n]

Aα(z−1)=dK[n](38)

at the steady state. Let the polynomial AK(z−1)have an order

of m, then it can be decomposed to the product of mﬁrst-order

polynomials

AK(z−1) =

m

∏

i=1

(z−1−λiejθi),(39)

where {λiejθi}i=1,...,mare the roots of AK(z−1). 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(z−1)as

Aα(z−1) =

m

∏

i=1

(z−1−αλiejθi),(40)

where α∈(0,1)and is close to 1. Figure 5 shows the differ-

ence of GK(ejω)under different αselections. Here A(z−1)

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).

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

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(z−1)d[n] = 0 at the steady state [15], where

A(z−1) =

n

∏

i=1

(1−2cos(2πfiT)z−1+z−2)(41)

and Tis the sampling time of d[n]. Substituting in

the frequency values yields the polynomial model of

d[n]:A(z−1) = 1−1.9619z−1+4.1664z−2−4.9797z−3+

6.2201z−4−4.9797z−5+4.1664z−6−1.9619z−7+z−8.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

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

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.

References

[1] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital control of

dynamic systems, vol. 3. Addison-wesley Menlo Park, CA, 1998.

[2] M. Unser, “Sampling-50 years after shannon,” Proc. IEEE, vol. 88, no. 4,

pp. 569–587, 2000.

[3] D. L. Donoho, “Compressed sensing,” IEEE Transactions on information

theory, vol. 52, no. 4, pp. 1289–1306, 2006.

[4] E. J. Candès and M. B. Wakin, “An introduction to compressive

sampling,” IEEE signal processing magazine, vol. 25, no. 2, pp. 21–

30, 2008.

[5] M. Salman Asif and J. Romberg, “Sparse recovery of streaming signals

using L1-Homotopy,” arXiv preprint arXiv:1306.3331, 2013.

[6] A. Koochakzadeh and P. Pal, “Performance of uniform and sparse non-

uniform samplers In presence of modeling errors: a Cramér-Rao bound

based study,” IEEE Transactions on Signal Processing, vol. 65, no. 6,

pp. 1607–1621, 2017.

[7] X.-G. Xia, “On estimation of multiple frequencies in undersampled

complex valued waveforms,” IEEE Transactions on Signal Processing,

vol. 47, no. 12, pp. 3417–3419, 1999.

[8] X.-G. Xia and K. Liu, “A generalized chinese remainder theorem for

residue sets with errors and its application in frequency determination

from multiple sensors with low sampling rates,” IEEE Signal Processing

Letters, vol. 12, no. 11, pp. 768–771, 2005.

[9] A. Mousavi, A. B. Patel, and R. G. Baraniuk, “A deep learning approach

to structured signal recovery,” in 2015 53rd Annual Allerton Conference

on Communication, Control, and Computing (Allerton), pp. 1336–1343.

IEEE, 2015.

[10] H. Gupta, K. H. Jin, H. Q. Nguyen, M. T. McCann, and M. Unser, “Cnn-

based projected gradient descent for consistent ct image reconstruction,”

IEEE transactions on medical imaging, vol. 37, no. 6, pp. 1440–1453,

2018.

[11] A. Mousavi, G. Dasarathy, and R. G. Baraniuk, “A data-driven and

distributed approach to sparse signal representation and recovery,” in

International Conference on Learning Representations, 2019.

[12] Y. Mostoﬁ, “Compressive cooperative sensing and mapping in mobile

networks,” IEEE Transactions on Mobile Computing, vol. 10, no. 12,

pp. 1769–1784, 2011.

[13] M. Nagahara and D. E. Quevedo, “Sparse representations for packetized

predictive networked control,” IFAC Proceedings Volumes, vol. 44, no. 1,

pp. 84–89, 2011.

[14] M. Nagahara, T. Matsuda, and K. Hayashi, “Compressive sampling

for remote control systems,” IEICE Transactions on Fundamentals of

Electronics, Communications and Computer Sciences, vol. 95, no. 4,

pp. 713–722, 2012.

[15] C. E. Garcia and M. Morari, “Internal model control. A unifying review

and some new results,” Industrial & Engineering Chemistry Process

Design and Development, vol. 21, no. 2, pp. 308–323, Apr. 1982.

[16] H. Xiao, B.-S. Yaakov, and X. Chen, “Model-based sparse information

recovery by a collaborative sensor management,” in ASME Dynamic

Systems and Control Conference, Oct. 2018.

[17] L. N. Trefethen and D. Bau III, Numerical linear algebra, vol. 50. Siam,

1997.

[18] I. Gibson, D. W. Rosen, B. Stucker et al.,Additive manufacturing

technologies, vol. 238. Springer, 2010.

[19] W. Drexler, M. Liu, A. Kumar, T. Kamali, A. Unterhuber, and R. A.

Leitgeb, “Optical coherence tomography today: speed, contrast, and

multimodality,” Journal of biomedical optics, vol. 19, no. 7, p. 071412,

2014.

[20] C. Sheppard, D. Shotton, and C. Sheppard, Confocal Laser Scanning

Microscopy. Microscopy Handbook. New York: BIOS Scientiﬁc

Publishers Ltd, 1997.

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

[21] H. Xiao, T. Jiang, and X. Chen, “Rejecting fast narrow-band disturbances

with slow sensor feedback for quality beam steering in selective laser

sintering,” Mechatronics, vol. 56, pp. 166–174, 2018.

[22] D. Wang and X. Chen, “A spectral analysis and its implications of feed-

back regulation beyond Nyquist frequency,” IEEE/ASME Transactions

on Mechatronics, vol. 23, no. 2, pp. 916–926, Apr. 2018.

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).