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Optimal Dynamic Scheduling of Wireless Networked Control
Systems
Yehan Ma
Washington University in St. Louis
yehan.ma@wustl.edu
Jianlin Guo, Yebin Wang,
Ankush Chakrabarty, Heejin
Ahn, Philip Orlik
Mitsubishi Electric Research Labs
{guo, yebinwang, chakrabarty, hahn,
porlik}@merl.com
Chenyang Lu
Washington University in St. Louis
lu@wustl.edu
ABSTRACT
Wireless networked control system is gaining momentum in in-
dustrial cyber-physical systems, e.g., smart factory. Suering from
limited bandwidth and nondeterministic link quality, a critical chal-
lenge in its deployment is how to optimize the closed-loop control
system performance as well as maintain stability. In order to bridge
the gap between network design and control system performance,
we propose an optimal dynamic scheduling strategy that optimizes
performance of multi-loop control systems by allocating network
resources based on predictions of both link quality and control
performance at run-time. The optimal dynamic scheduling strategy
boils down to solving a nonlinear integer programming problem,
which is further relaxed to a linear programming problem. The
proposed strategy provably renders the closed-loop system mean-
square stable under mild assumptions. Its ecacy is demonstrated
by simulating a four-loop control system over an IEEE 802.15.4
wireless network simulator – TOSSIM. Simulation results show
that the optimal dynamic scheduling can enhance control system
performance and adapt to both constant and variable network back-
ground noises as well as physical disturbance.
CCS CONCEPTS
•Computer systems organization →Sensor networks
;Sen-
sors and actuators;•Networks →Cross-layer protocols.
KEYWORDS
Cyber-physical system, wireless network, multi-loop control sys-
tem, dynamic scheduling, optimization, link quality
ACM Reference Format:
Yehan Ma, Jianlin Guo, Yebin Wang, Ankush Chakrabarty, Heejin Ahn,
Philip Orlik, and Chenyang Lu. 2019. Optimal Dynamic Scheduling of Wire-
less Networked Control Systems. In 10th ACM/IEEE International Confer-
ence on Cyber-Physical Systems (with CPS-IoT Week 2019) (ICCPS ’19), April
This work was done while Y. Ma was an intern with Mitsubishi Electric Research Labs.
Permission to make digital or hard copies of all or part of this work for personal or
classroom use is granted without fee provided that copies are not made or distributed
for prot or commercial advantage and that copies bear this notice and the full citation
on the rst page. Copyrights for components of this work owned by others than ACM
must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
to post on servers or to redistribute to lists, requires prior specic permission and/or a
fee. Request permissions from permissions@acm.org.
ICCPS ’19, April 16–18, 2019, Montreal, QC, Canada
©2019 Association for Computing Machinery.
ACM ISBN 978-1-4503-6285-6/19/04. . . $15.00
https://doi.org/10.1145/3302509.3311040
16–18, 2019, Montreal, QC, Canada. ACM, New York, NY, USA, 10 pages.
https://doi.org/10.1145/3302509.3311040
1 INTRODUCTION
Wireless technology is gaining rapid adoption in industry automa-
tion for lowering deployment and maintenance costs in challenging
industrial environments. Industrial standard organizations such
as ISA100 [
20
], WirelessHART [
14
], and ZigBee [
1
], which are all
based on the IEEE 802.15.4 physical layer [
19
], are strong propo-
nents of wireless network for industrial automation. However, in
wireless networked control systems (WNCSs), the primary use of
wireless network is in monitoring. The status quo is that, it remains
challenging to close the loop at the control-to-actuation side over
wireless network due to multiple reasons.
First, wired networks, such as Ethernet, use twisted pairs and
ber optic links, resulting in high data rate of up to hundreds of
Gbit/s. In contrast, wireless networks, especially low-power and
low-cost industrial wireless networks, have limited throughput. For
instance, IEEE 802.15.4 physical layer supports data rate of up to
250 kbit/s. The control performance of WNCSs largely depends on
how much network resource they are able to obtain. Second, the
physical isolation of wired networks ensure supreme link quality
and resiliency to external environment changes. However, link
qualities of wireless networks are prone to environmental factors
such as obstacles, noises, interferences, extreme weather, as well as
human interference in the form of cyber attacks. Poor link quality
can cause signicant data packet loss, resulting in degradation of the
control performance. Finally, most wireless network designs focus
on network performances, overlooking control performances which
directly determine the prots and the safety of a factory. Therefore,
a practical wireless network design for WNCSs must target the
improvement of the control performance by taking limited network
resource allocation and the impact of link quality into consideration.
In this paper, we bridge the gaps between control performance
and network design by exploring the direct impact of network link
quality and network resource allocation on the physical control.
We design an optimal dynamic scheduling strategy to optimize
the control performance by allocating more network resources to
needy loops and reducing the eects of network on physical control
system, based on run-time predictions of link quality and physical
control performance.
Our major contributions in this paper include:
(1) incorporate link quality prediction of wireless network;
ICCPS ’19, April 16–18, 2019, Montreal, QC, Canada Y. Ma et al.
(2)
provide a tractable method for optimal network scheduling
based on predictions of both link quality and the control
performance;
(3)
establish stability guarantees for the closed-loop system with
optimal scheduling;
(4)
illustrate the ecacy of our strategy on the high-delity
TOSSIM simulation environment in spite of constant and
variable background noises, and physical disturbance.
The rest of the paper is organized as follows. Sec. 2 reviews
related works. Sec. 3 overviews WNCS. Sec. 4 describes wireless link
quality prediction and its simulation results. Sec. 5 formulates the
optimal scheduling problem and its linear programming relaxation.
Sec. 6 details the stability analysis and condition of the proposed
optimal scheduling method. Sec. 7 evaluates the simulation results.
We present our conclusions in Sec. 8.
2 RELATED WORK
The past decade has witnessed sustained interest in exploring
WNCSs and expanding their applications over industry automa-
tion [
26
,
30
], in the views of network design, control system design,
and more recently, network and control co-design.
From a network design perspective, several approaches are pre-
sented to address resource allocation. For example, Huang et al. [
18
]
propose an adaptive time slot allocation scheme for IEEE 802.15.4,
which considers low latency and fairness of packet waiting time;
Zhan et al. [
44
] allocate network resource by adjusting the slot
length adaptively in accordance with the data size of the end device.
Given link quality, end-to-end packet delivery ratio (PDR) can be
eectively improved by retransmission [
11
], channel selection [
15
],
routing [
37
], and reachability-aware scheduling [
9
], etc. However,
few are targeting optimizing control performance.
On the control system side, many control designs based on the
physical plant models as well as on network parameters are per-
formed to maintain the performance. To name a few, Sinopoli et
al. [
39
] discuss Kalman ltering with intermittent measurement;
Gao et al. [
12
] investigate robust output tracking control subject to
time delay between controllers and actuators; Ma et al. [
33
] explore
the design freedom of system architectures and propose a smart
actuation architecture; Wang et al. [
25
,
42
,
43
] model packet loss as
a Bernoulli or Markov-type process and establish stochastic stabil-
ity of the resultant WNCS. However, most control designs consider
only application-level network parameters, such as latency and
PDR, instead of lower-level parameters, such as link quality and
signal-to-noise ratio (SNR). However, with only application level
information, it is hard to fully utilize and manage network resource
for control performance.
More recently, network and control co-designs aim to jointly
design the control and network to eliminate the eects of limited
throughput and poor link quality of wireless networks, among
which there are network resource allocation designs tailored for
control performance of WNCSs. Saifullah et al. determine [
7
,
36
]
sampling rates to optimize control performance. Gatsis et al. [
13
]
propose distributed control-aware random network access policies
for each sensor so that all control loops are stabilizable. Lješnjanin
et al. [
29
] allocate network resource by nding optimal node, which
minimize cost function of model predictive control (MPC), in every
Actuators
()uk
()yk
ˆ()yk
Plant
ˆ()uk
Actuators
Plant
Controller Actuators
Sensors Plant
Network
Manager
Figure 1: Architecture of WNCS ( red and blue dashed arrows
indicate actuation and sensing ows, respectively)
network time instant. Ma et al. [
31
,
32
] propose the concept of
holistic control that cojoins network reconguration and physical
control over multi-hop mesh network. However, [
13
,
36
] assume
perfect link quality, and none of [
7
,
29
,
31
] models the eects of link
quality on control performance. Peters et al. [
35
] present co-design
of scheduler and controller by deriving optimal control as well as
determining transmitting control commands in contention access
period (CAP) or contention free period (CFP), or no transmission at
all, targeting IEEE 802.15.4 MAC. However, they assume that PDR is
constant and do not consider retransmission in scheduling, which
is a key factor of improving PDR and control performance [11].
In this paper, we explore the direct impact of network link qual-
ity and network resource allocation on the physical control system
performance, and formulate an optimal dynamic scheduling strat-
egy to optimize the control performance by balancing the number
of transmissions among multiple control loops.
3 OVERVIEW OF WNCS
Fig. 1 shows the architecture of the multi-loop WNCS. The con-
trollers are typically located far from the physical plants. One reason
is that plants operate in environments which may not be conductive
to hardware implementation of control algorithms. Another reason
is one control algorithm may be responsible for multiple plants, and
therefore, a larger centralized unit of computation may be required
to implement such an algorithm.
3.1 Physical plant and controller
We consider
N
control loops that share the same wireless network.
Each control loop is associated with an individual plant. For the
i
th
loop, the corresponding plant is modeled as a nonlinear discrete-
time system of the form:
xi(k+1)=fi(xi(k),ui(k)),(1)
where
k
is the time index,
i∈ {
1
,
2
, . . . , N}
is the loop index,
xi(k) ∈
Rni
is the state vector, and
ui(k) ∈ Rmi
is the actuation vector that
renders the closed-loop system asymptotically stable when there is
no packet loss in network. For simplicity, we state all denitions and
theorems for the case when the equilibrium point is at the origin of
Rni
. There is no loss of generality because any equilibrium point
can be shifted to the origin via a change of variables [27].
At time
k
, a sensor sends measurements
yi(k)
to a controller over
the wireless network. At the controller side, a state observer [
39
]
estimates the states of the plant. Based on the estimated state
ˆ
xi(k)
,
the controller generates the control command
ui(k)
and sends it to
the actuator over the wireless network. The actuator then applies
ˆ
ui(k)
to the plant. If
ui(k)
fails to be delivered by the deadline, the
actuator reuses the control input of last period, ˆ
ui(k−1).
Optimal Dynamic Scheduling of Wireless Networked Control Systems ICCPS ’19, April 16–18, 2019, Montreal, QC, Canada
3.2 Wireless network
3.2.1 Wireless sensor-actuator network (WSAN). Using IEEE 802.15.4-
based network, we schedule sensing and actuation ows of the
control loops. A superframe is a collection of timeslots repeating
in time. For IEEE 802.15.4-based network, in beacon enabled mode,
the superframe is bounded by beacons sent by the coordinator. As
shown in Fig. 2, the beacon frame transmission starts at the begin-
ning of the rst slot of each superframe. The beacons are used to
synchronize the attached devices, to identify the network, and to de-
scribe the structure of the superframes. During the inactive period,
the coordinator and end nodes are able to enter a low-power mode,
such as sleep mode. The active period is composed of contention-
access period (CAP) and contention-free period (CFP). During CAP,
devices compete for media access using the MAC scheme of carrier
sense multiple access/ collision avoidance (CSMA/CA). For applica-
tions with real-time performance requirements, the network man-
ager (NM) dedicates guaranteed time slots (GTSs) during CFP. As
specied by IEEE 802.15.4 MAC protocol [
19
], the NM can allocate
up to 7 slots in CFP. The limitations of IEEE 802.15.4 MAC protocol
was discussed and modied by [
3
,
17
], such that the number of
slots assigned to CAP and CFP becomes a free design parameter.
WirelessHART and ISA100 also support customized number of slots
in CFP. In this paper, we target WNCSs with real-time performance
requirements, thus focus on the scheduling of the CFP, whereas
CAP can be reserved for other uses.
B B
Contention
Access Period Contention
Free Period
Inactive
Period
Beacons
Active Period
t
Figure 2: Structure of the IEEE 802.15.4 superframe
3.2.2 Network manager with beer authority. The NM manages the
network and its devices. In most network architectures, the NM, the
application controller, and the coordinator of WSAN are co-located.
The NM communicates with controllers and the coordinator via a
reliable wired network with ignorable packet drop and latency.
We propose a NM that utilizes the information of predicted link
quality and the knowledge of predicted control performance from
the controller to obtain optimal scheduling. As a result, the NM
dynamically schedules the data ows of the WNCS based on its
knowledge of both the wireless network and the physical plants
at run-time. Then the NM noties the coordinator of the updated
schedule, and the coordinator broadcasts the updated schedule in
the beacon at the beginning of the next superframe. In this way, eld
nodes that receive the beacon update their schedules accordingly.
Remark 3.1. In a multi-loop WNCS, the NM allocates the network
resource based on the predicted link quality and control performance
of each loop. When the scheduled number of transmissions of loop
i
, denoted by
ηi
, is assigned to be zero, the actuation event of loop
i
is not triggered. By determining
ηi
as 0or
Z+
, actuation events of
control loops are skipped or triggered by the NM. Thus, the network
resource allocation is a special kind of event-triggered control.
3.3 Recover from beacon packet loss
In a star network, for upstream (sensing) ows, if a beacon message
is received by an upstream node, the node wakes up and sends
sensing ow at the assigned time slots indicated by the beacon.
However, if the beacon message, which contains the updated sched-
ule generated by the NM, is lost, dynamic scheduling may cause
collisions between ows. For instance, if a sensor fails to receive the
updated schedule, it will not be able to update its newly assigned
time slots, and will keep transmitting sensing ows to the controller
at previous assigned time slots, which may be assigned to other
ows according to the updated schedule. Therefore, for simplicity,
we propose to reserve xed time slots for sensing ows and only
dynamically schedule actuation ows.
For downstream (actuation) ows, if a beacon message is re-
ceived by a downstream node, the node shall wake up and listen
at the assigned time slots indicated by the beacon. We propose a
packet loss recovery strategy to improve the resiliency of beacon
packet loss. If no beacon packet has been received by a node, the
node wakes up and keeps listening for the whole superframe. This
strategy results in longer listening time and higher energy expen-
diture of wireless nodes if and only if the beacon message is lost.
Besides, the longer listening time will not cause any collision.
4 LINK QUALITY
We adopt a general metric – packet reception ratio (PRR) – to
represent the link quality since maximization of the successfully
transmitted packets is the basic objective to most networks [
4
].
The NM dynamically generates schedules for the WSAN based on
predicted PRRs of all links. Besides, physical layer characteristics
such as received signal strength indicator (RSSI), SNR, and link
layer characteristics such as link quality indicator and expected
transmission count also indicate the quality of wireless link [4].
4.1 Link quality prediction
Holt’s additive trend prediction method [
16
,
40
] is employed to
predict PRR of next mtransmissions,
S(k)=αPRR(k)+(1−α)S(k−1)+T(k−1)
T(k)=γS(k) − S(k−1)+(1−γ)T(k−1)
d
PRR(k+m|k)=S(k)+mT (k)
(2)
where
PRR(k)
is the current measured PRR of a specic link,
S(k)
denotes an estimate of the current level of the series,
T(k)
repre-
sents an estimate of current trend (slope),
m
is a positive integer
representing the steps ahead,
d
PRR(k+m|k)
is the predicted PRR
m
transmissions ahead,
α
and
γ
(0
<α,γ<
1) are the level and slope
smoothing parameter, respectively.
4.2 Results of link quality prediction
In our study, wireless traces from 4 links of the WSAN testbed at
Washington University [
37
] have been collected, which contain
the connectivity and RSSI data [
24
]. In addition, we use controlled
background noise strength to simulate various network conditions.
Both the RSSI and controlled noise strength are fed into a high-
delity wireless simulator – TOSSIM [
22
,
23
]. Fig. 3 shows PRRs
(91,000 packets for each data point) of four links under controlled
noise levels. The PRRs vary among links under the same noise levels
since the RSSIs are dierent. The PRR under the lowest noise level
ICCPS ’19, April 16–18, 2019, Montreal, QC, Canada Y. Ma et al.
(
−
84 dBm) is the highest. Under the same noise levels, links with
higher RSSIs (link1>link 2=link4>l ink3) yield higher PRRs.
-84 -82 -80 -78 -77 -76 -75
Noise (dBm)
0.5
0.6
0.7
0.8
0.9
PRR
link1 (RSSI:-64 dBm)
link2 (RSSI:-66 dBm)
link3 (RSSI:-67 dBm)
link4 (RSSI:-66 dBm)
Figure 3: PRRs under various noise levels
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Sequence No.
0
0.2
0.4
0.6
0.8
1
PRR
Noise:-84 dBm
Noise:-75 dBm
Figure 4: Sliding-window PRRs of link 3
Fig. 4 shows the sliding-window PRRs of link 3 under noise levels
of
−
84 dBm and
−
75 dBm, respectively. The horizontal axis is the
number of packets transmitted via link 3. The window size is 15 in
this case study. 1-step PRR prediction results are shown in Fig. 5.
We use link 3 under noise level of
−
75 dBm as an example, and we
choose
α=
0
.
9
,γ=
0
.
1in
(2)
. We can see that PRR prediction (red
dashed line) matches well with measured PRR (blue solid line). The
mean absolute error (MAE) of the PRR predictions is shown in Fig. 6.
The prediction error increases as the prediction step size increases.
1-step prediction error is less than 4%, and 5-step prediction error is
less than 10%. Note that as the noise level increases from
−
84 dBm
to
−
75 dBm, the prediction error increases. This indicates that the
noise level aects the prediction accuracy. However, we achieve
more than 90% of prediction accuracy for all simulated scenarios.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Sequence Number
0
0.2
0.4
0.6
0.8
1
PRR
PRR
PRR prediction
Figure 5: 1-step PRR prediction under noise −
75
dBm (link 3)
-84 -82 -80 -78 -77 -76 -75
Noise (dBm)
0
0.02
0.04
0.06
0.08
0.1
MAE of PRR Prediction
1-step prediction
2-step prediction
3-step prediction
4-step prediction
5-step prediction
Figure 6: PRR prediction errors of link 3 under various noise
5 OPTIMAL SCHEDULING
In this section, we propose an optimal dynamic scheduling strategy
that optimizes control performance by allocating limited network
resources based on predictions of both link quality and control
performance at run-time. We formulate the optimal scheduling
strategy as a nonlinear integer program, which is relaxed into a
linear programming (LP) problem. Finally, we present a heuristic
algorithm of sorting control loops by the descending order of their
costs in each superframe for shortening the latency of needy loops.
5.1 Multi-loop control system modeling
5.1.1 Simplifications and assumptions. We use
s
to represent the
schedule of next superframe. The number of transmission (
η
) is at
the center of the tradeo between reliability and network resources,
i.e., more transmissions lead to a higher packet delivery ratio (PDR)
at a cost of network resources [
31
]. Denote
ηi
the number of trans-
mission of
loop i
in schedule
s
. For example,
ηi=
2indicates that
loop
i
is assigned 2transmission slots. Our scheduling problem is
to determine and balance
ηi
among control loops by predicting link
quality and physical system performance.
We focus on the actuation (downstream) packet scheduling
problem. This is because the state observer provides robust and
theoretically sound protection against loss of sensing informa-
tion [
28
,
38
,
39
], and the WNCSs are more sensitive to packet loss
on the actuation side of the wireless network [
23
]. We refer readers
who are interested in sensing packet scheduling problem to [10].
In Secs. 5.2, 5.3, and 6, we focus on modeling packet loss and
schedule the actuation packets for the control loops in the ascending
order of the loop number in each superframe. For ease of analysis,
we assume strict periodicity of actuation packets. This restriction
is lifted in our simulation to allow realistic packet timing.
To simplify the problem, we assume all loops have the same
sampling period. A potential method for relaxing this assumption
is to use sampled-data control techniques discussed in, for example,
[
6
], i.e., rewriting systems with dierent periods in the slowest time
frame (least common multiple of all sampling periods).
5.1.2 Packet delivery modeling. Let a binary variables
ϕi(k)
denote
end-to-end packet reception
ϕi(k)=
1
or loss
ϕi(k)=
0
. PDR of
actuation packets for loop
i
under schedule
s
is denoted as
µϕi(s)=
Pϕi(k)=
1
. Note that
µϕi(s)
depends on PRR of the link and the
number of transmissions in schedule
s
. Given link failure ratio of
link i (loop i) as βi=1−P RRi, we have PDR
µϕi(s)=1−βηi
i.(3)
Here, PDR is a function of link quality and schedule.
5.2 Optimal scheduling formulation
At time
k
, controller determines control
u(k)
based on state
x(k)
and system model
(1)
. Network manager ought to come up with
a schedule
s(k)
based on
x(k),u(k)
, PRR, and system model
(1)
.
In fact, optimal scheduling solves for
s(k)
based on the predicted
state
x(k+
1
)
which implicitly depends on schedule
s(k)
through
PDR. Specically, state
ˆ
xi(k+
1
)
for loop
i
can be inferred from
xi(k),ui(k), and ϕias follows
(1) packet of loop i at t=karrives (closed loop):
ˆ
uc
i(k)=ui(k),xi(k+1)=ˆ
xc
i(k+1)=fixi(k),ˆ
uc
i(k),(4)
(2) packet ui(k)is lost, and ˆ
ui(k−1)is actuated (open loop):
ˆ
uo
i(k)=ˆ
ui(k−1),xi(k+1)=ˆ
xo
i(k+1)=fixi(k),ˆ
uo
i(k),(5)
Optimal Dynamic Scheduling of Wireless Networked Control Systems ICCPS ’19, April 16–18, 2019, Montreal, QC, Canada
For illustration purpose, we dene a quadratic cost function of
loop ias follows:
Jixi(k)=xT
i(k)Wixi(k),(6)
where
Wi≻
0is a positive denite matrix. Dene the overall cost
function as follows:
Jx(k)=
N
Õ
i=1Jixi(k)=xT(k)Wx (k),(7)
where x(k)=x1(k)x2(k). . . xN(k)T, and
W=blkdiaд(W1,W2, .. ., WN).blkdiaдis the block-diagonalize op-
erator that constructs a diagonal matrix from input matrices. We
will see in Sec. 6 that this objective function provides some benets
in terms of the guarantee of mean-square stability for LTI systems.
Given schedule s(k), the expectation of Jixi(k+1)is:
EJixi(k+1)=µϕi(s)Jiˆ
xc
i(k+1)+1−µϕi(s)Jiˆ
xo
i(k+1),
(8)
where Eis the expectation operator. Substituting (3) into (8) gives
EJixi(k+1)=Jiˆ
xc
i(k+1)+Jiˆ
xo
i(k+1)−Jiˆ
xc
i(k+1)βηi
i.
(9)
The optimal scheduling problem is formulated as:
minimize
ηi
EJx(k+1)(10a)
subject to
N
Õ
i=1
ηi≤L(10b)
ηi∈ {0,1, . . . , L},∀i∈ {1,2, ..., N},(10c)
where
L
is the total number of slots assigned for all actuation ows
in each superframe. The constraint
(10b)
indicates the requirement
of schedulability. The constraint
(10c)
means that the transmission
number should be a non-negative integer. Problem
(10)
is an inte-
ger programming problem. Furthermore, the objective function is
nonlinear in
η
as can be seen from
(9)
. It is well-known that this
class of problems is NP-hard [21].
5.3 Run-time optimal scheduling
Since we are targeting a scheduling problem that must be solved
for every superframe, its tractability is of vital importance.
5.3.1 Binary linear programming. We propose a transformation of
variables to recast Problem
(10)
into a binary linear programming
(BLP) problem. The resultant BLP problem is equivalent to Problem
(10)
by introducing the binary variable
˜
Ti j ∈ {
0
,
1
}
that ags the
magnitude of
ηi
, which implies the change of decision space from
{0,1, . . . , L}Nto {0,1}N(L+1):
˜
Ti j =(1,ηi=j,j∈ {0,1, . . . , L}
0,otherwise. (11)
We can represent EJixi(k+1)in (9) as
EJixi(k+1)=
L
Õ
j=0Jiˆ
xc
i(k+1)+Jiˆ
xo
i(k+1)− Jiˆ
xc
i(k+1)βj
i
| {z }
qi j
˜
Ti j .
(12)
According to the linearity of mathematical expectation, the ex-
pectation of the overall cost function is equal to the sum of the
expectations of the cost function of each loop, that is
EJx(k+1)=
N
Õ
i=1
EJixi(k).(13)
By dening
˜
T=˜
T10 ˜
T11 ... ˜
T1L˜
T20 ˜
T21 ... ˜
T2L... ˜
TN L T,
we can see that the objective function is a linear function of ˜
T,
EJx(k+1)=Q˜
T,(14)
where
Q=q10 q11 ... q1L, ..., qN0... qN L
. Problem
(10)
is reduced to a binary linear programming problem as follows
minimize
˜
Ti j
Q˜
T(15a)
subject to
N
Õ
i=1
L
Õ
j=0
j˜
Ti j ≤L(15b)
L
Õ
j=0
˜
Ti j =1(15c)
˜
Ti j ∈ {0,1},∀i∈ {1,2, ..., N},∀j∈ {0,1,2, ..., L}
(15d)
Note that we rewrite the constraint
(10b)
as
(15b)
. In order to ensure
each loop
i
has unique
ηi
, we impose constraints
(15c)
-
(15d)
. The
transmission numbers can be recovered from ˜
Tusing
ηi=0 1 2 ... L˜
Ti0˜
Ti1˜
Ti2... ˜
TiL T.(16)
There are many integer linear programming solvers such as Gurobi,
CPLEX, and MATLAB.
5.3.2 Linear programming relaxation. By relaxing the binary con-
straint
(15d)
to
˜
Ti j ∈ [
0
,
1
],
we have a typical LP problem, which
can be solved eciently using linprog in MATLAB or other LP
solvers. We then convert the resultant relaxed solution to integral
form by rounding
ηi
of
(16)
. The complexity of LP is
O(m3
ln (m)D)
[
2
],
where
m
is the space dimension, i.e.
N(L+
1
)
,
D
denotes the bit
length of the input data. When we set
N=
4and
W=I4
, among
57,600 results, 99
.
98% of cases yield the optimal solutions (found
by brute-force search in the feasible set). As shown in Fig. 7, the
advantage of LP relaxation in computational complexity appears
when Nincreases.
5 10 15 20 25
N (N=L)
100
105
1010
1015
Complexity
Brute force
LP relaxation
Figure 7: Complexity of optimal scheduling problem
Remark 5.1. The resultant
ηi
might be infeasible (
ÍN
i=1ηi>L
)
due to relaxation and rounding. Since there is a diminishing return
in PDR improvement as
ηi
increases [
31
], we propose a heuristic
ICCPS ’19, April 16–18, 2019, Montreal, QC, Canada Y. Ma et al.
method to achieve a feasible solution by iteratively reducing the largest
element max
1≤i≤Nηiby one, until ÍN
i=1ηi≤L.
5.4 Heuristics of sorting loops in superframes
In previous sections, we assume that we schedule the actuation
packet of each loop in the ascending order of the loop number. In
this section, we provide an algorithm of determining the order of
loops in each superframe given the solution
ηi
of optimal scheduling
problem
(15)
. As shown in Alg. 1, we propose to sort the actuation
packet of each loop in the descending order of their costs (
Costi
),
i.e., the loops with larger costs will be scheduled earlier so that
the actuation packets of those loops will obtain shorter latency. In
addition, we spread the retransmissions of same loop to shorten
the latency of other loops.
Algorithm 1: Algorithm of sorting loops in each superframe
input :
Transmission numbers returned by optimal scheduling:
ηi
,
predicted costs: Costi=wixi(k),wi,i∈ {1,2, . . . , N}
are customized weight of each loop, number of slots for
actuation packets: L.
output : The schedule of actuation packets in next superframe:
Schedule
Schedule ←zeros(L);Slot ←1;
for all ido
ηi_left ←ηi;
//ηi_left represents unscheduled transmissions;
Cost_matrix ←1 2 3 . . . N
Cost1Cost2. . . CostN;
Sorted_loop_number ←sort loops (rst row of Cost_matrix) in
descending order of Costi(second row of Cost_matrix);
while slot ≤Land ÍN
i=1ηi>0do
for iin Sorted_loop_number do
if ηi_left >0then
Schedule(Slot) ←i;Slot ←Slot +1;
ηi_left ←ηi_left −1;
return Schedule;
6 STABILITY ANALYSIS
The aforementioned optimal scheduling strategy can improve the
control performance of the multi-loop WNCS without loss of sta-
bility. In this section, we provide a condition of stability in the
mean-square sense. According to [
5
], a discrete-time stochastic
system is mean-square stable (MSS) if for any initial state x(0),
lim sup
k→∞
E∥x(k)xT(k)∥=0.
A closed-loop system is MSS if there exists a stochastic Lyapunov
function V(x), such that
(1) V(0)=0and V(x)>0,∀x,0;
(2) ∥x∥ → ∞ ⇒ V(x)→∞;
(3) EV(x)decreases along system trajectories. That is,
EVx(k+1)−EVx(k)≤0.(17)
Next we show that our optimal dynamic scheduling strategy
can ensure mean-square stability of the closed-loop system under
mild assumption: the existence of any xed schedule such that
the resultant system is MSS. A xed schedule can be a typical
periodic schedule or any static schedule that are calculated oine.
We rst need to determine whether there is a xed schedule that
makes the closed-loop system MSS. Here, we provide a condition
to check whether systems resulted from a xed schedule are MSS
for discrete-time LTI (DT-LTI) systems as an example.
6.1 MSS check of LTI system with xed schedule
Consider a multi-loop DT-LTI system, where system dynamics of
the loop iare given by
xi(k+1)=Aixi(k)+Biui(k),ui(k)=Kixi(k),(18)
where
xi(k) ∈ Rni
is the state vector, and
ui(k) ∈ Rmi
is the control
input. Assume that the state feedback gain
Ki
renders the closed-
loop subsystem (loop i) asymptotically stable in ideal network.
To apply the stability analysis in [
34
], we model the closed-
loop system dynamics over actuation networks with schedule
s
as
a discrete-time stochastic system. According to [
34
], the closed-
loop system dynamics of loop
i
are equivalent to the following
augmented system
zi(k+1)=˜
Asi (s,k)zi(k),(19)
where
˜
Asi (s,k)=
AiBi0
0 1 −ϕi(s)ϕi(s)
KiAi0KiBi
,zi(k)=
xi(k)
ˆ
ui(k)
ui(k)
.(20)
Similar to (4) and (5), zi(k+1)can be determined as follows
(1) packet at t=karrives (ϕi(k)=1):
ˆ
ui(k)=ui(k),zi(k+1)=ˆ
zc
i(k+1)=˜
Ac
si zi(k),(21)
(2) packet at t=kis lost, and ˆ
ui(k−1)is adopted (ϕi(k)=0):
ˆ
ui(k)=ˆ
ui(k−1),zi(k+1)=ˆ
zo
i(k+1)=˜
Ao
si zi(k),(22)
where
˜
Ac
si =
AiBi0
0 0 1
KiAi0KiBi
,˜
Ao
si =
AiBi0
0 1 0
KiAi0KiBi
.
Analogously, the multi-loop control system can be rewritten as
z(k+1)=˜
A(s,k)z(k)(23)
where ˜
A(s,k)=blkdiaд˜
As1(s,k),˜
As2(s,k), . . . , ˜
As N (s,k),
z(k)=z1(k)z2(k). . . zN(k)T.
In order to prove stability prop-
erties of the closed-loop system, besides assumptions in Sec. 5.1.1,
we make the following assumption.
Assumption 6.1. Sequences
{ϕi(k),k∈N},∀i∈ {
1
,
2
, ..., N}
,
are i.i.d.
Note that this assumption is lifted in evaluation section to allow
much more realistic radio propagation and noise models in TOSSIM
[22]. Under Assumption 6.1, we can rewrite ˜
A(s,k)in (23) as
˜
A(s,k)=˜
A0+
N
Õ
i=1
˜
Aipi(k),(24)
where
pi(k)
are i.i.d. random variables with
Epi(k)=
0, variance
Varpi(k)=σ2
pi, and Epi(k)pj(k)=0,∀i,j∈ {1,2, . . . , N},
˜
A0=blkdiaд(˜
A01,˜
A02, . . . , ˜
A0N),˜
A1=blkdiaд(Aϕ1,0, . . . , 0),
˜
A2=blkdiaд(0,Aϕ2,0, . . . , 0)
,
...
,
˜
AN=blkdiaд(0,0, . . . , AϕN)
,
Optimal Dynamic Scheduling of Wireless Networked Control Systems ICCPS ’19, April 16–18, 2019, Montreal, QC, Canada
˜
A0i=
AiBi0
01−µϕi(s)Iµϕi(s)I
KiAi0KiBi
,
Aϕi=
0 0 0
0µϕi(s)I−µϕi(s)I
0 0 0
,σ2
pi(s)=1
µϕi(s)−1.
Here, we let
pi(k)=
1
−ϕi(k)
µϕi
be binary random variable that takes 1
or 1
−1
µϕi
with
Ppi(k)=
1
=
1
−µϕi
and
Ppi(k)=
1
−1
µϕi=µϕi
.
From Assumption 6.1, we have that
pi(k)
is i.i.d with
Epi(k)=
0
and Varpi(k)=σ2
pi.
For the discrete-time stochastic system
(24)
, the following lemma
gives a condition to check whether the system is MSS.
Lemma 6.1. [
5
](p131) The system
(19)
is MSS if and only if there
exists a positive denite matrix Psatisfying
˜
AT
0P˜
A0−P+
N
Õ
i=1
σ2
pi˜
AT
iP˜
Ai<0.(25)
Remark 6.2. If control loops are independent,where the states of
one loop do not interact with those of other loops, each loop
i
can derive
its own positive denite matrix (denoted as
Pi
) separately as single
control loop in Lemma 6.1. We have P=blkdiaд(P1,P2, ..., PN).
6.2 Stability condition of optimal scheduling
Given the existence of a xed schedule which renders the closed-
loop system MSS, we can establish that the closed-loop system
resulted from the optimal schedule is also MSS.
Proposition 6.3. If there exists a xed schedule
sf
such that
the resultant closed-loop system is MSS, and
J(x)
is a stochastic
Lyapunov function with
sf
, then the closed-loop system with the
optimal schedule s∗derived by solving (10) is also MSS.
x∗(k)
x′(k+1)
x∗(k+1)
sf
s∗
Figure 8: Diagram of stability proof
Proof.
As shown in Fig. 8, we apply both the stabilizing xed
schedule
sf
and the optimal schedule
s∗(k)
to any state
x∗(k)
, and
then get x′(k+1)and x∗(k+1), respectively.
Since
J(x)
is a stochastic Lyapunov function of the closed-loop
system resulted from a xed schedule
sf
,
J(x)
satises
J(x)>
0
,∀x,
0,
J(x) → ∞
as
∥x∥ → ∞
, and
EJ(x)
decreases along
trajectories of the system, according to (17). Therefore,
EJx′(k+1)≤EJx∗(k).(26)
Because the schedule
s∗
minimizes the objective function
EJx(k+
1)in the optimization problem (10), we have
EJx∗(k+1)≤EJx′(k+1).(27)
Combining (26) and (27), we derive
EJx∗(k+1)≤EJx∗(k).(28)
For the optimally scheduled system (i.e.
s=s∗
),
EJ(x)
decreases
along trajectories of the system, and satises
J(x)>
0
,∀x,
0,
and
J(x) → ∞
as
∥x∥ → ∞
. Therefore,
J(x)
is also a stochastic
Lyapunov function of the optimally scheduled system.
Remark 6.4. For DT-LTI system
(23)
, for
P≻
0satisfying Lemma
6.1 with
sf
, we can interpret the function
J(x)=xTPx
as a stochas-
tic Lyapunov function with
sf
([
5
] p132), and thus
J(x)
is also a
Lyapunov function of the optimally scheduled system. This is why we
choose a quadratic objective function in (7).
Remark 6.5. Although we set
J(x)
as a quadratic function to
analyze MSS for DT-LTI systems, Proposition 6.3 holds for other forms
of
J(x)
. That is, if there is a stochastic Lyapunov function
V(x)
for nonlinear systems with a xed schedule [
8
], then
V(x)
is also a
stochastic Lyapunov function for the closed-loop system rendered by
the optimal schedule that minimizes EV(x)in (10).
7 EVALUATION
This section shows a systematic case study of the proposed sched-
uling strategy. On the physical plant side, we use four 3-state non-
linear double water-tank systems that share the same wireless
network. On the network side, we collect IEEE.802.15.4 traces using
TOSSIM, and then empirically evaluate our strategy under con-
stant and variable network background noise levels as well as pulse
physical disturbance.
7.1 Simulation settings
7.1.1 Physical control system. Consider four independent 3-state
nonlinear double water-tank systems, each of which is modeled as
follows [3, 24]:
Û
L1=1
ρA1(αu−√ρд
ρR1pL1)
Û
L2=1
ρA2(√ρд
ρR1pL1−√ρд
ρR2pL2)
Û
LR=1
ρAR(√ρд
ρR2pL2−αu)
(29)
where
L1
,
L2
,
LR
are the liquid levels of the upper tank, lower tank
and the basin, respectively;
A1
,
A2
,
AR
are the cross-sectional areas
of the tanks; and
R1
,
R2
are the resistance parameters of pipes of
upper and lower tanks. We discretize the continuous-time model
(29) using the Euler method with sampling period of ∆t, and have
the discrete-time model
L1(k+1)
L2(k+1)
LR(k+1)
=
1−∆t√ρд
ρ2R1A1√L10 0
∆t√ρд
ρ2R1A2√L11−∆t√ρд
ρ2R2A2√L20
0∆t√ρд
ρ2R2AR√L21
L1(k)
L2(k)
LR(k)
+
α∆t
ρA1
0
−α∆t
ρA2
u.
Table 1: System parameters
PLANT1 PLANT2
par value par value par value par value
A10.01 R10.0006 A10.12 R10.0006
A20.006 R20.0008 A20.007 R20.0008
AR1α10 AR1α10
ICCPS ’19, April 16–18, 2019, Montreal, QC, Canada Y. Ma et al.
Figure 9: Optimal scheduling under constant noise −76 dBm (the upper plot is 0−6s and lower plot is 6−12 s)
There are two types of plants, denoted by PLANT1 and PLANT2,
that have dierent system parameters, shown in Table. 1. Systems
1 and 3 are PLANT1, and systems 2 and 4 are PLANT2.
For the four systems, we design state feedback controllers that
enable reference tracking. To evaluate the tracking performance,
we choose the mean absolute error (MAE) metric:
MAE =1
n+1
n
Õ
k=0|x(k) − xre f (k)|,(30)
where nis the number of samples, and xre f is the reference state.
7.1.2 Wireless network. We simulate the IEEE 802.15.4 beacon-
enabled wireless network. Since we propose to use xed scheduling
for sensing ows in Sec. 5.1.1, in our simulation, we focus on sched-
uling actuation ows by assuming sensors having wired connection
to controllers. Each superframe has ve slots and the slot duration is
8.3 ms. The rst slot is assigned for a beacon message. The following
four CFP slots are assigned for actuation ows of the four control
loops. Given
W
as the identity matrix in the objective function
J(x)
in
(7)
, we solve the relaxed linear optimization problem described
in Sec. 5.3.2 using MATLAB/linprog solver. In simulation, we col-
lect wireless traces from 4 links (8 nodes) of the WSAN testbed at
Washington University. As described in Sec. 4.2, we get packet loss
traces using the RSSI and set controlled noise strength as inputs
of the TOSSIM simulator. For simplicity, we use single channel in
evaluation. Note that the supported number of control loops can be
scaled up by simultaneously accessing up to 16 channels of IEEE
802.15.4 PHY. [41]
7.2 Simulation results
We rst run the WNCS simulations underdierent levels of constant
network background noise. We then evaluate the performance of
our optimal scheduling strategy under variable background noises
to show its adaptability and optimality, comparing with the pe-
riodic scheduling mechanism. In addition, we also evaluate the
performance of our strategy for pulse physical disturbance.
7.2.1 Constant background noise. We run the WNCS simulations
of optimal (OPT) scheduling under several background noise levels.
Our baseline is the WNCS that adopts a static periodic schedule as
shown in Fig. 10, in which GTS slots are uniformly scheduled to
the four control loops. Under noise level of
−
76 dBm, the optimal
schedule is shown in Fig. 9, and the ratios of slot allocation for each
control loop in dierent time intervals are shown in Fig. 11. Since
the sizes of tanks of PLANT1 are smaller than those of PLANT2
as shown in Table.1, PLANT1 (loops 1 and 3) is more sensitive to
packet loss and performs worse than PLANT2 (loops 2 and 4) during
transient responses (rst 4s). During the rst 4s, the NM scheduled
Figure 10: Periodic scheduling under noise −76 dBm
0s-12s 0s-4s 4s-6s 9s-11s
Time Intervals
0
0.2
0.4
0.6
Ratio of Slot Allocation
Loop 1
Loop 2
Loop 3
Loop 4
Figure 11: Slot allocation in various time intervals
0
10
20
Loop1
L1(m)
OPT scheduling
Periodic scheduling
0
10
20
Loop2
L1(m)
0
10
20
Loop3
L1(m)
0 2 4 6 8 10 12
t(s)
0
10
20
Loop4
L1(m)
Figure 12: Response curves under noise level of −76 dBm
most of slots to loop 1 (24
.
2%) and loop 3 (39
.
6%) and much less
slots to loop 2 (17
.
2%) and loop 4 (19
.
0%). More slots are scheduled
to loop 3 than loop 1 since loop 3 has worse link quality as shown
in Fig. 3. Fig. 12 shows the responses of the upper tanks of the
four loops. The OPT scheduling signicantly improves the control
performance of loops 1 and 3 and maintains similar performance
of loops 2 and 4, compared with the periodic scheduling.
In addition, to show the adaptability of our OPT scheduling with
respect to physical disturbance, we add pulse physical disturbance
to loops 1 and 3 at
t=
4s, and to loops 2 and 4 at
t=
9s. As
shown in Figs. 9 and 11, during
t=
4to 6s, most of the slots are
assigned to loop 1 (32.8%) and loop 3 (58.4%), and only a few slots
are assigned to loop 2 (5
.
2%) and loop 4 (3
.
6%) since they are in
steady states. During
t=
9to 11 s, most of the slots in OPT schedule
are scheduled to loop 2 (43
.
2%) and loop 4 (38
.
1%). This result shows
that our OPT scheduling can adjust to physical disturbance.
We run simulations of three scheduling strategies: (1) combining
OPT scheduling and sorting with identical weights (OPT sched-
uling + Sorting), (2) OPT scheduling, and (3) periodic scheduling,
for 50 times. Fig. 14 shows the boxplots of MAEs of each schedul-
ing strategy under dierent noise levels. The control performance
Optimal Dynamic Scheduling of Wireless Networked Control Systems ICCPS ’19, April 16–18, 2019, Montreal, QC, Canada
Figure 13: Optimal scheduling under variable noise level
-84 -82 -80 -78 -77 -76 -75
Noise (dBm)
2
4
6
8
10
12
14
MAE
OPT scheduling + sorting
OPT scheduling
periodic scheduling
Figure 14: MAE under constant background noise levels
0
0.5
1
PRR1
Noise:-75 dBmNoise:-78 dBm
0
0.5
1
PRR2
Noise:-75 dBmNoise:-78 dBm
0
0.5
1
PRR3
Noise:-84 dBmNoise:-75 dBm
0 2 4 6 8 10 12
t(s)
0
0.5
1
PRR4
Noise:-84 dBmNoise:-75 dBm
PRR
PRR prediction
Figure 15: Run-time link quality variation
degrades as the background noise increases. The OPT scheduling
outperforms the periodic scheduling for all background noise levels.
The advantage of the OPT scheduling becomes more apparent as
the link quality degrades. This is because the OPT schedule adjusts
transmissions based on link quality and control performance and
thus is more robust to noise. The sorting algorithm can further
improve the control performance by considering the latency.
7.2.2 Variable background noise. In this section, we evaluate our
OPT scheduling under variable background noise to show its adapt-
ability and optimality when network conditions change. Variable
background noise patterns are shown in Fig. 15. In the rst 5s, the
noise levels of links 1 and 2 are
−
78 dBm, and those of links 3 and
4 are
−
75 dBm. Therefore the PRRs of links 3 and 4 are lower than
links 1 and 2. The PRR of link 3 is the worst as shown in Fig. 3. The
background noise changes at
t=
5
s
. The noise strengths of links
1 and 2 increase to
−
75 dBm, and that of links 3 and 4 decrease to
−84 dBm. The PRR of link 2 becomes the worst in this case.
Under the noise levels shown in Fig. 15, the OPT schedule is
shown in Fig. 13, and the ratios of slot allocation are shown in
Fig. 16. The NM schedules more slots to loop 3 (52
.
3%) than other
loops during the rst 5s because loop 3 has the worst network
condition. The NM in the variable noise levels schedules more slots
to loop 4 than in the constant noise case during the rst 5s since link
4 has the worse network condition than links 1 and 2. More slots
are scheduled to loop 2 (36
.
1%) during the last 7s (5s to 12 s) since
0s-12s 0s-5s 5s-12s 4s-6s 9s-11s
Time Intervals
0
0.2
0.4
0.6
Ratio of Slot Allocation
Loop 1
Loop 2
Loop 3
Loop 4
Figure 16: Slot allocation under variable noise level
0
10
20
Loop1
L1(m)
OPT scheduling
Periodic scheduling
0
10
20
Loop2
L1(m)
0
10
20
Loop3
L1(m)
0 2 4 6 8 10 12
t(s)
0
10
20
Loop4
L1(m)
Figure 17: Response curves under variable noise level
link 2 has the worst network condition. Due to physical disturbance
at 4s to loops 1 and 3, and at 9s to loops 2 and 4, many slots from
4s to 6s are assigned to loops 1 (22
.
4%) and 3 (54
.
2%), and many
slots from 9s to 11 s are assigned to loops 2 (54
.
7%) and 4 (30
.
2%).
The response curves of OPT and periodic scheduling are shown
in Fig. 17. The control performance using the OPT scheduling is
improved for loops 1 and 3 compared with the periodic scheduling,
and remains similar for loops 2 and 4. Therefore, we can conclude
that our OPT scheduling can adapt to both physical disturbance
and varying network condition at the same time.
Statistical results of control performance under variable noise
levels are shown in Fig. 18. In terms of the total MAEs of four
control loops (rst group of the boxplots), the OPT scheduling
outperforms the periodic scheduling, the OPT scheduling combined
with sorting is better than only the OPT scheduling. The OPT
scheduling optimizes the total cost function of all control loops by
allocating more network resources to needy loops and links at run-
time. When we look into the performance of individual control loop,
compared with the periodic scheduling, the control performance of
loop 3 is signicantly improved by the OPT scheduling since loop
3 is allocated more network resource by the OPT scheduling. The
performance of loops 2 and 4 downgrades a little since they have
relatively low MAEs and therefore less allocated network resource.
Note that the extent of improvement in loop 3 is much larger than
the downgrade in loops 2 and 4. The results show that the OPT
scheduling can balance the network resource allocation according
to link quality and control performance among multiple loops.
ICCPS ’19, April 16–18, 2019, Montreal, QC, Canada Y. Ma et al.
4 loops Loop1 Loop2 Loop3 Loop4
0
2
4
6
8
MAE
OPT scheduling + Sorting
OPT scheduling
Periodic scheduling
Figure 18: MAE under variable noise level
8 CONCLUSIONS
In order to bridge the gap between wireless network design and
physical control system performance, we propose an optimal dy-
namic scheduling strategy that optimizes control performance of
multi-loop systems by allocating limited network resources based
on predictions of both link quality and control performance at run-
time. We formulate our optimal scheduling problem as a nonlinear
integer programming problem, and then relax it to a linear pro-
gramming problem for computational eciency. Also, we provide
a stability condition for the wireless networked control system
that adopts the optimal scheduling. A systematic evaluation is per-
formed based on four nonlinear double water-tank systems over a
realistic IEEE 802.15.4 wireless network. Simulation results show
that our optimal scheduling has signicantly enhanced the adapt-
ability of the system under both constant and variable background
noise as well as physical disturbance.
ACKNOWLEDGMENTS
The authors would like to thank Dr. Arvind Raghunathan at MERL
for helpful discussions.
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