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This paper considers the real-time recovery of a fast time series (e.g., updated every T seconds) by using sparsely sampled measurements from two sensors whose sampling intervals are much larger than T (e.g., MT and NT, where M and N are integers). 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 the sensor speeds MT and NT, and by exploiting a model-based fusion of the sparsely collected data. We provide the collaborative sensing design, parametric analysis and optimization of the algorithm. Application to a closed-loop disturbance rejection problem reveals the feasibility to annihilate 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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Proceedings of ASME 2018 Dynamic Systems and Control Conference
DSCC2018ASME
September 30 - October 3, 2018, Atlanta, Georgia, USA
DSCC2018-9088
MODEL-BASED SPARSE INFORMATION RECOVERY BY A COLLABORATIVE
SENSOR MANAGEMENT
Hui Xiao
Dept. of Mechanical Eng.
University of Connecticut
Storrs, Connecticut, 06269
Email: hui.xiao@uconn.edu
Yaakov Bar-Shalom
Dept. of Electrical and Computer Eng.
University of Connecticut
Storrs, Connecticut, 06269
Email: yaakov.bar-shalom@uconn.edu
Xu Chen
Dept. of Mechanical Eng.
University of Connecticut
Storrs, Connecticut, 06269
Email: xchen@uconn.edu
ABSTRACT
This paper considers the real-time recovery of a fast time
series (e.g., updated every T seconds) by using sparsely sam-
pled measurements from two sensors whose sampling intervals
are much larger than T (e.g., MT and NT , where M and N are
integers). 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 the sensor speeds MT and NT , and
by exploiting a model-based fusion of the sparsely collected data.
We provide the collaborative sensing design, parametric analysis
and optimization of the algorithm. Application to a closed-loop
disturbance rejection problem reveals the feasibility to annihilate
fast disturbance signals with the slow and not fully aligned sen-
sor 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.
1 INTRODUCTION
Fast feedback response is key for safe and high-performance
operation of a control system. Whether the application is to mon-
itor thermal conditions in a nuclear power plant, to track ground
and aerial targets for defense purposes, or to maintain material
temperature when additively manufacturing personalized pros-
thetic implants for patients, we build mathematical models, col-
lect measurements, and analyze performances by assuming or
desiring fast sampled measurements (e.g., 20 times the desired
closed-loop bandwidth in a servo problem [5, 11]). However,
many sensors update at intrinsically limited speeds. For instance,
the update rate of a radar scanner is constrained by the rota-
tion rate of the antenna; for imaging-based automation (using,
e.g., sonar or infrared vision), complex elaborations must be per-
formed to extract information from the raw image frames. In the
presence of fast dynamics and disturbances that happen between
the slow sampling instances, the resulting lack of sight constrains
the overall situational awareness of the system, and can lead to
unsafe system operations in a wide range of engineering applica-
tions that can benefit from fast real-time closed-loop operations.
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 ex-
ist to generate dense signals from limited 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 strat-
egy interprets sampling as a projection operator – one that com-
putes a band-limited approximation of the input signal. The aim
here is to approximate the original signal instead of insisting on a
perfect reconstruction [7,13]. Both the first and the second strate-
gies focus on regular, uniform sampling. A third and more recent
strategy involves irregular data collection. Perhaps the fastest
growing in this category is compressed, or compressive sensing
(CS) [1, 2, 4], which advocates randomized sampling and L1op-
timization to approximate the original signal in a transformed
domain — one that allows a sparse, compressive representation
of the data.
From the viewpoint of control design, real-time closed-loop
functionality and causality are key factors when manipulating a
temporal signal flow. This implies the obstacle that the full se-
quence of measured data will not be available when recovering
a particular element in the middle of the experiment. A natural
question is then: what methodology can be used for desparsify-
ing a slowly measured data online, with assurance of causality
and real-time computation? Aligned with the first two discussed
1
strategies of information processing, advanced digital to analog
converter (DAC) and filtering have been proposed for real-time
controls considering inter-sample behaviors. However, the re-
construction is an approximate one, and connecting feedback
with multi-sensory sparse data collection has also not entered the
radar yet. Within the third strategy of sparse signal processing,
CS has been proposed as a nice fit for networked feedback con-
trol when the remotely transmitted measurement data is large and
compressible [8–10]. In these studies, CS is used to store and
process imaging information; the focus is not on sparsity in time
but in the pixel space of the images.
Building upon the above knowledge and moving beyond ex-
isting architectures of constrained real-time functionality, this
paper proposes an online computation friendly algorithm to re-
cover a discrete signal d[n]from sparsely sampled measurements.
Specifically, we consider the case when d[n]is an autoregressive
process and when the sparsely sampled measurements are from
two sensors S1and S2with slow sampling periods MT and NT ,
where Mand Nare distinct integers greater than one. By col-
lecting parallel sets of very slowly measured samples from the
linear flow, we show that the dense and fast intersample infor-
mation can be fully recovered in real time, by a unique model-
ing of the signal-sensor pair. This signal-reconstruction method,
which builds the correlation formula between the missing sig-
nal data and collaborative measurements, is made possible by
elaborately designing and re-parameterizing the internal signal
model of d[n]. The result is that we will be able to not only facil-
itate the real-time sparse information processing, but also seam-
lessly integrate the magnified sensing with closed-loop model-
based controls to achieve agile feedback response to structured
disturbances and high-level control inputs.
Notations: LCM(M,N)denotes the least common multiple
of Mand N. If a uniformly sampled sequence d[n]has sampling
period T,t{d[n]}=nT is the timestamp (the time when a data
point is measured) of signal d[n]. The ceiling function dxemaps
a real number xto the smallest following integer.
2 MECHANISMS OF THE PROPOSED COLLABORA-
TIVE SENSING
Let the discrete measurements from S1and S2be denoted as
dM[n]and dN[n], respectively. The following simple and direct
connections hold:
dM[n]d[Mn](1)
dN[n]d[Nn](2)
Here, the sign “” represents that two samples are equal and
aligned in time (i.e., two samples have the same timestamps)1.
In order to better describe the collaborative sampling pro-
cess, we divide d[n]into a list of sequences {d{i}}i=1,2,3,..., where
1We use this notation rather than “=” because data points having an identical
value could have distinct time stamps. For example, a periodic signal x(n) =
x(n+T), but x(n)and x(n+T)are not aligned in time.
FIGURE 1. CONNECTIONS BETWEEN dM[n],dN[n]AND d[n]
WHEN M=2, N=3 AND L=6.
d{i}is called the i-th batch in d[n]. Each batch contains Lcon-
secutive data points in d[n], that is,
d{i}[k]d[iL +k],k=1,2,...,L(3)
where d{i}[k]denotes the k-th data points 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 d{i}[k]dX[n],
then d{i+1}[k]dX[n+k1],where k1=L/X and X denotes M
or N.
Proof. If d{i}[k]dM[n], then combining Eqs. (1) and (3), one
can get d[iL +k]d{i}[k]dM[n]d[Mn], or equivalently,
their time stamps are equal: t{d[iL +k]}/T=iL +k=Mn =
t{d[MN]}.Now for the time stamp of d{i+1}[k]it holds that
t{d{i+1}[k]}/T= (i+1)L+k
=M(n+L/M) = t{dM[n+L/M]}/T(4)
where L/Mis an integer. Thus we have d{i+1}[k]dM[n+
L/M]. Analogously, d{i+1}[k]dN[n+L/M]if d{i}[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 chosen
batch size L=LCM(M,N). This property of repeated connec-
tions 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 algo-
rithm under the following batch configurations.
Definition 1. The batch d{i}[k]used in this paper (see Fig. 1) is
defined based on the following rules:
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., d{i}[L]dM[n1]dN[n2].
With the definition above, a signal batch d{i}has the following
properties:
2
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}.
The above definition of data sets will be used in the following
information recovering algorithm design.
3 MODEL-BASED INFORMATION RECOVERY WITH
COLLABORATIVE SENSING
Intuitively, if the time index of the fast underlying sig-
nal d{i}[k]is aligned to any of the sensor measurements, i.e.
kKMorKN, a direct measurement is available and no data re-
covery is needed. However, if kKU,d{i}[k]is lost in the sam-
pling 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 d {i}[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
d{i}[k] =
t1
i=0
wk,idM[n1i] +
t2
j=0
vk,idN[n2j](5)
where t1and t2are finite integers, n1and n2denote indices of dM
and dNsuch that dM[n1]dN[n2]d{i1}[L](such relation-
ship is ensured by the third rule of Definition 1). The unknown
parameters wk,i’s and vk,is come from the solution to the follow-
ing system of linear equations
Mk
fk,1
.
.
.
fk,l
wk,0
.
.
.
wk,t1
vk,0
.
.
.
vk,t2
=
a1
a2
.
.
.
am
0
0
.
.
.
0
(6)
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](7)
where
˜
Mk=
1·· · 0
a1
....
.
.
.
.
....0
am
...1
0...a1
.
.
.....
.
.
0·· · am
(8)
and eiis the elemental column vector whose entries are all zeros
except for the i-th entry, which equals 1.
Proof. To see first (5), we construct
Fk(z1)A(z1) + zkWk(zM) + zkVk(zN) = 1 (9)
where
Fk(z1) = 1+f1z1+· · · +flzl(10)
Wk(zM) = wk,0+wk,1zM+·· · +wk,t1zt1M(11)
Vk(zN) = vk,0+vk,1zN+·· · +vk,t2zt2N(12)
Multiplying both sides of (9) 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](13)
namely,
d[n] =
t1
i=0
wk,id[nkiM] +
t2
j=0
vk,id[nkjN](14)
Let d[n]be the k-th data point of the k-th batch, i.e. d[n]d(i)[k],
then based on the batch definition (Eq. (3)), we have d[nk]
d[iL]d{i1}[L].Recall that the indices n1and n2are chosen
such that dM[n1]dN[n2]d{i1}[L]. Thus we get d[nk]
dM[n1]dN[n2], or (nk)T=n1MT =n2N T based on their
time-stamp equivalence. Now the time stamps of the summation
terms in (14) are
t{d[nkiM]}= (nkiM)T
= (n1i)MT =t{dM[n1i]}(15)
t{d[nkjN]}= (nkjN)T
= (n2j)NT =t{dN[n2j]}(16)
3
Thus we get
d[nkiM]dM[n1i](17)
d[nkjN]dN[n2j](18)
In other words, (5) will be satisfied as long as (14), or its equiva-
lent from (9) is satisfied.
Now consider solving (9). Expanding the equation and col-
lecting 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 from as (6).
Example 1. Consider an illustrative example with M=3, N=
2 and A(z1) = 1+a1z1+a2z2.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 Eq. (5). Here we choose t1=
t2=1 (there are more discussions about choosing t1and t2in the
following section), then the recovering equations become:
d{i}[k] = wk,0d3[n1] + wk,1d3[n11]
+vk,0d2[n2] + vk,1d2[n21],k=1,5 (19)
Following the procedures in Theorem 1, parameters w1,0,w1,1,
v1,0,v1,1are obtained from the solution of
1 0 1 0 1 0
a11 0000
a2a10001
0a20100
f1,1
f1,2
w1,0
w1,1
v1,0
v1,1
=
a1
a2
0
0
(20)
and parameters w5,0,w5,1,v5,0,v5,1are 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 0000
0000a2a10001
00000a20100
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
(21)
4 DISCUSSION
4.1 Choosing t1and t2
In Theorem 1, (t1+1)data points from dM[n]and (t2+1)
data points from dN[n]are used in the recovery equation (5). In
fact, the number of data points used in the recovery process is
flexible, as we discuss next.
Corollary 1. A necessary condition for the system of equations
(6) to have a solution is
t1+t2m+nd2 (22)
where
nd=mint1+1
L/M,t2+1
L/N (23)
Proof. Recall that a solvable system of linear equations must not
be overdetermined, so an obvious necessary condition for (6) to
have solutions is
l+t1+t2+2l+m(24)
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 Mk
are identical, yielding redundant variable pairs in (6) (say there
are ndnumber of them). Then, the number of independent vari-
ables becomes l+t1+t2+2ndand the necessary condition
(24) reduces to (22).
To more quantitatively define nd, we recall that a signal
batch could provide at most L/Nmeasurements from sensor S1,
hence the number of prior batches containing measurements from
S1that are used in the recovery process is
nd,M=t1+1
L/M(25)
Similarly, for sensor S2,
nd,N=t2+1
L/N(26)
It can be seen from Definition 1 that the condition iM =jN holds
only once in a single batch, hence the number of redundant vari-
able pairs ndis the number of prior batches where measurements
from both sensors are involved in the recovery process, which is
the minimum value between nd,Mand nd,N.
Note that although the dimension of Mkvaries when recov-
ering different data points in a batch, the necessary condition (22)
only needs to be checked once because kis not involved in (22).
This can be understood by realizing that the recovered data points
in a batch are calculated using the same set of prior sensor mea-
surements.
4.2 Solution to the System of Linear Equations, and a
Method to Reduce Computation
If t1and t2are chosen based on (22), solutions are guaran-
teed to exist for the system of equations defined in Eq. (6). One
4
particular solution is given by
fk
qk=M
ka
0(27)
where M
kis the Moore-Penrose inverse or the pseudoinverse of
Mk,fk=fk,1,·· · ,fk,lT,qk=wk,0,· ·· ,wk,t1,vk,0,· ·· ,vk,t2T,
and a=[a1,·· · ,am]T. The solution to (27) has the minimum
Euclidean norm among all possible solutions. However, comput-
ing the pseudoinverse is time-consuming for a large Mk(whose
dimension grows quickly as kincreases (see Fig. 2)). We discuss
next an reduced-order procedure to solve (6) that will drastically
reduce the computation load for real-time applications.
The system of linear equations (6) can be rewritten into the
following form, where Mkis segmented into four smaller matri-
ces with dimensions defined below.
Am×lDm×(t1+t2+2)
Bl×lCl×(t1+t2+2) fk
qk=a
0(28)
Then the system solution is given by
qk=DAB1Ca(29)
To see this, unfold the matrix equation (28) as
A f k+Dqk=a(30)
B f k+Cqk=0 (31)
Notice that Bis an invertible upper triangular matrix. Thus fk
can be solved from (31) as
fk=B1Cqk(32)
Inserting (32) into (30) yields (29) .
Instead of directly computing the pseudoinverse of the large
matrix Mk, the reduced-order method reduces the matrix dimen-
sion by lin height and width before taking the pseudoinverse, and
efficient algorithms exist for the inversion of the upper triangular
matrix B[12], This allows for a significant reduction of compu-
tation cost in configurations with large parameters (M,N,t1,t2).
Figure 2 shows the changes of the computing cost as kincreases
when computing the prediction parameters in a batch2. 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.
2The tests were done on a same computer running MATLAB 2017b; func-
tions were called multiple times, the average time used in the computation was
recorded.
0 10 20 30 40 50 60
index k in a batch
0
0.2
0.4
0.6
0.8
1
1.2
time / seconds
10-3
direct method
reduced-order method
FIGURE 2. THE TIME FOR COMPUTING THE SYSTEM SOLU-
TION USING DIRECT METHOD AND OPTIMIZED METHOD.
5 EXAMPLE APPLICATION: BEYOND-NYQUIST DIS-
TURBANCES 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. Our preliminary study [14, 15] have reported by sim-
ulation and experimentation that a well-designed classic high-
gain control could amplify instead of attenuate the actual distur-
bance 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 case when a micro-servo motor is controlled
by a discrete PID controller with a feedback loop. The contin-
uous transfer function of the motor from the input applied volt-
age to the output speed (rad/sec) is P
c(s) = 360000/(s2+660s+
36000). The controller has the transfer function
C(z) = kP+kITs
1
z1+kD
1
Ts
z1
z(33)
with kP=0.9628, kITs=0.0640, kD/Ts=1.55 and sampling
time Ts=0.9 msec. The closed loop achieves a rise time of
0.0036 sec and a settling time of 0.018 sec. Suppose the plant
is subject to narrow-band disturbances with high frequencies at
516Hz, 783Hz and 1150Hz, which are close to or beyond the
Nyquist frequency (i.e. 555Hz) of the sensor. Many control al-
gorithms exist for rejecting such narrow-band disturbances. For
example, [3,16] provides narrow-band disturbance observers that
can achieve infinite high-gain control at selected disturbance fre-
quency ranges. The focused application will construct a multirate
control system combining the narrow-band disturbance observer
and our proposed algorithm of information recovery with collab-
orative sensors.
Figure 3 shows the proposed control scheme for beyond
Nyquist disturbance compensation with the collaborative sensing
5
FIGURE 3. MULTI-RATE CONTROL SCHEME FOR DISTUR-
BANCE REJECTION USING NARROW-BAND DISTURBANCE
OBSERVER AND COLLABORATIVE SENSING.
mechanism. Besides the original sensor sampled at 0.9 msec, a
second slow sensor with sampling time 1.2 msec is added to form
the collaborative sensor pair: M=4,N=3 and T=0.3 msec.
In Fig. 3, discrete signals with sampling times M T ,NT and T
are denoted by sparse dashed lines, dense dashed lines and dot-
ted lines, respectively. Continuous signals are denoted by solid
lines. Components of the disturbance rejection mechanism in-
clude ˆ
P
d(z), the identified discrete model of the continuous plant
P
s(s), and Q(z), the disturbance compensating filter. With the
disturbance frequency information known, one can design Q(z)
with the procedure provided in [16]. Our model-based recover
technique (i.e. The MR block in Fig. 3) is applied to recover
a fast disturbance estimate ˆ
d[n]using slow sampled disturbance
estimates ˆ
dM[n]and ˆ
dN[n].
In the recovering process, there are L=LCM(4,3) = 12
points in a signal batch; points with index k=1,2,5,7,10,11
are recovered by (5). Based on the internal signal model [6]
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, where
A(z1) =
n
i=1
(12cos(2πfiTs)z1+z2)(34)
Substituting the parameter values yields the model of the distur-
bance estimate ˆ
d[n]:
A(z1) = 10.1882z1+1.7362z20.1386z3(35)
+1.7362z40.1882z5+z6
Then by Theorem 1, one can get the parameters wk,i’s and vk,is
in the recovering equation (5) from the solution of system of lin-
ear equations (6) for each k. Here, we chose t1=t2=3,which
satisfies the necessary condition (22).
To show the effectiveness of the disturbance compensation
loop, we gave zero reference inputs to the system and turned
on the compensation loop at t=0.3 msec. Figure 5 shows the
system output yd[n]sampled at 0.3 msec. The results show that
1.212 1.214 1.216 1.218 1.22 1.222 1.224 1.226
Time / sec
-1
-0.5
0
0.5
FIGURE 4. SIGNAL RECOVERY RESULTS.
0 0.5 1 1.5 2 2.5 3
Time / sec
-1.5
-1
-0.5
0
0.5
1
1.5
System output
Disturbance conpensation loop turned on
due
to the slow measuring speeds of sensors
FIGURE 5. SYSTEM OUTPUT SAMPLED AT 3.33KHZ.
the beyond-Nyquist disturbances were fully rejected at a sam-
pling rate three times faster than the maximum sampling speed
of sensors. Figure 4 shows the recovered fast sampled distur-
bance ˆ
d[n](black solid line) as well as the slow disturbance mea-
surements ˆ
dM[n](blue dashed line with asterisk marks) and ˆ
dN[n]
(red dash-dotted line with circle marks). The sparsely sampled
disturbances were accurately recovered based on its frequency
information using our purposed algorithm.
6 CONCLUSION
In this paper, the problem of reconstructing the fast discrete
signal d[n]form the sparsely sampled collaborative sensor mea-
surements dM[n]and dN[n]is addressed. Based on a collabora-
tive sensing design and a model-based filtering using sensors of
different sampling speeds, the proposed online algorithm can re-
cover the highly dense information that is not measured by the
slow sensors. This algorithm was implemented and validated in
a disturbance compensation architecture to enable fully rejection
of beyond-Nyquist disturbances.
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