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Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.

Digital Object Identiﬁer 10.1109/ACCESS.2017.DOI

Packet-Drop Design in URLLC for

Real-Time Wireless Control Systems

BO CHANG1, (Student Member, IEEE), GUODONG ZHAO2, (Senior Member, IEEE), ZHI

CHEN3, (Senior Member, IEEE), LIYING LI4, (Member, IEEE), and MUHAMMAD ALI IMRAN5,

(Senior Member, IEEE),

1National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China (UESTC), Chengdu

611731, China (emails: changb3212@163.com)

2School of Engineering, University of Glasgow, Glasgow, G12 8LT, UK (e-mail: Guodong.Zhao@glasgow.ac.uk)

3National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China (UESTC), Chengdu

611731, China (emails: chenzhi@uestc.edu.cn)

4School of Engineering, University of Glasgow, Glasgow, G12 8QQ, UK (e-mail: Liying.Li@glasgow.ac.uk)

5School of Engineering, University of Glasgow, Glasgow, G12 8QQ, UK (e-mail: muhammad.imran@glasgow.ac.uk)

Corresponding author: Zhi Chen (e-mail: chenzhi@uestc.edu.cn).

This work was supported in part by the National Natural Science Foundation of China under Grant 61631004, by the National Key R&D

Program of China under Grant 2018YFC0807101, by EPSRC Global Challenges Research Fund - the DARE project - EP/P028764/1, and

by the scholarship from China Scholarship Council (CSC) under the Grant CSC 201806070120.

ABSTRACT In real-time wireless control systems, ultra-reliable and low-latency communication (URLL-

C) is critical for the connection between remote controller and its control objective. Since both transmission

delay and packet loss can lead to control performance loss, our goal is to optimize control performance

by jointly considering control and URLLC constraints in this paper. To achieve this goal, we formulate

an optimal problem to minimize control cost by optimizing packet drop and wireless resource allocation.

To solve the problem, we analyze the relationship between communication and control. Then, based on the

relationship, we decompose the original problem into two subproblems: (1) an optimal packet-drop problem

to minimize control cost and (2) an optimal resource allocation problem to minimize communication

packet error. Finally, the corresponding solutions for each subproblem can be obtained. Compared with

the traditional method only considering communication aspect, the proposed packet-drop and resource

allocation method shows remarkable performance gain in terms of control cost.

INDEX TERMS URLLC, real-time wireless control, packet drop, wireless resource allocation.

I. INTRODUCTION

In real-time wireless control systems, ultra-reliable and low-

latency communication (URLLC) is critical for the connec-

tion between remote controller embedded in the base station

(BS) and its control objective (i.e., the plant) [1]. In such

a system, the inevitable transmission delay and packet loss

in URLLC lead to control performance loss. Furthermore,

when the controller handles plenty of plants, guaranteeing

both ultra-reliable and low latency is extremely challenging,

which may lead to signiﬁcant control performance loss. It

is expected that both communication and control aspects

can be jointly considered to maintain good overall system

performance.

Recently, some works have been done on the impact of

communication on control performance [2]–[10]. For exam-

ple, by modeling transmission time delay and packet loss into

control systems, the authors in [3] analyzed the consequence

of the imperfect transmission on the control performance,

where imperfect transmission coefﬁcients are introduced by

communication protocols, e.g., transmission control protocol

(TCP) or user datagram protocol (UDP). In [4]–[9], the au-

thors further investigated the control performance of different

control categories with imperfect communications. In [10],

the authors provided a tutorial and reviewed the existing

advances of wireless network design and optimization for

wireless networked control systems. However, the aforemen-

tioned works are based on the existing wireless networks and

cannot be used in control scenarios with ultra-reliable and

low-latency requirements.

To deal with the issue, URLLC is proposed in the coming

ﬁfth generation (5G) cellular networks to support real-time

wireless control systems [11]. Some research have been done

to maintain the extremely high QoS requirements in URLLC

[12]–[19]. For instance, the authors in [12] discussed re-

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source reservation strategy to maintain extreme high QoS. In

[13], the authors investigated resource allocation schedules in

physical layer for URLLC downlink. In [14], the authors ana-

lyzed resource allocation method when URLLC coexists with

another important scenario, i.e., enhance mobile broadband

(eMBB). In [15], the authors derived the closed-form solution

for resource allocation based on the concept of effective

capacity by considering the queueing delay violation prob-

ability. In [16], the authors considered the joint blocklength

and location optimization to minimize the decoding error

probability while adopting the channel capacity expression

for ﬁnite blocklength. In addition, the authors in [18] devel-

oped a protocol of device-to-device (D2D) communications

in URLLC. However, the packet drop introduced by limited

computing capacity is not discussed in the aforementioned

works, which may lead to signiﬁcant queueing packet drop.

To deal with this problem, the authors in [19] proposed a

queueing policy and a random packet drop policy to maintain

the QoS requirement, where only communication aspect is

taken into account. However, the interaction between control

and URLLC is not obtained, which is very important since

the communication design in URLLC is actually determined

by the control requirement.

In this paper, we consider the design of the uplinks from

sensors to the BS, where we formulate an optimal problem

to minimize control cost by optimizing packet drop and

resource allocation method. To solve the problem, the key is

to decouple the binary packet drop and continuous resource

allocation variables. The main contributions of this paper are

summarized as follows.

•We formulate an optimal problem to minimize the con-

trol cost, where the limited wireless resource and ex-

tremely high QoS requirements are taken into account.

The formulated problem allows us to use optimal packet

drop and wireless resource allocation method to support

real-time wireless control with minimum control cost.

•We analyze the relationship between optimal control

law and communication parameter design, and then

we ﬁnd that the binary packet drop and continuous

bandwidth allocation can be decoupled based on their

contributions to the control and communication.

•We decompose the formulated original problem into

two subproblems based on the variable decouple: (1)

an optimal packet drop problem to minimize control

cost and (2) an optimal resource allocation problem to

minimize communication packet error. By solving the

subproblems, we obtain the packet drop and resource

allocation method.

The rest of this paper is organized as follows. In Section

II, the system model is presented. In Section III, the optimal

problem is formulated. In Section IV, we obtain the resource

allocation method and packet drop method for the formulated

problem. In Section V, simulation results are provided to

show the performance of our method. Finally, Section VI

concludes the paper.

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-()&+(..$+%$,/$00$01

2$)#(+%3

4.")&%3

2$)#(+%5

4.")&%5

2$)#(+%6

4.")&%6

FIGURE 1: Communication model.

II. SYSTEM MODEL

As shown in Fig. 1, we consider a centralized wireless control

system, where a base station (BS) embedded Mremote

controllers conducts the control process for Mplants. In the

control process, each plant has one sensor sampling the state

of the plant. Once the BS receives the sampling signal from

each sensor, the corresponding remote controller embedded

in the BS calculates the control command. Then, the BS

transmits the command to the corresponding the plant to

update its current state. With the control process performing,

the plant state turns to the target state. In this section, the

system model considering both communication latency and

reliability is presented for the performance evaluation in real-

time wireless control systems.

A. COMMUNICATION

In this subsection, we focus on the uplink from the sensors

to the BS, where we assume that only the uplink experiences

time delay and packet loss. We consider orthogonal frequen-

cy division multiple access (OFDMA), where we assume

that each sensor is allocated with independent continuous

bandwidth Bm. In addition, we consider ﬂat fading channel,

where the channel gains for each sensor are approximately

identical and perfectly known for the sensor. Furthermore,

we assume that transmission duration for the m-th sensor is

Tm.

1) Channel Model

We consider that the channel model consists of the small-

scale fading and large-scale attenuation coefﬁcients between

transceiver, which are represented as hmand gmfor the

uplink from the m-th sensor to the BS, respectively. Ac-

cording to [20], the large-scale attenuation coefﬁcient can be

expressed as

gm[dB]=−128.1−37.6 lg(lm),(1)

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where lm≥0.035 km is the distance between transceiver

[20].

The small-scale fading hmfollows Rayleigh distribution

with mean zero and variance σ2

0= 1 [21]. However, since

the end-to-end (E2E) latency is no more than 1ms in URLLC

[19], the transmission time delay from the sensors to the BS

or that from the BS to the plants is less than the channel

coherence time, which means that the small-scale fading hm

is constant during the transmission period of the uplink and

the downlink [22].

2) Channel Capacity

According to [19] and [23], we can obtain the uplink channel

capacity expression for the m-th sensor in URLLC as

Rm=Cm−rVm

TmBm

f−1

Q(εe

m) + log(TmBm)

2TmBm

,(2)

where the ﬁrst term on the right hand of (2) is the achievable

Shannon capacity without transmission error, the second term

is the minus error bits introduced by channel dispersion

Vm, the third term is the approximation of the reminder

terms of order log(TmBm)/(TmBm),f−1

Q(·)is inverse of

Qfunction, and εe

mis the transmission error probability.

Furthermore, we assume that the single-sided noise spectral

density is represented by N0, then according to [19], we have

Shannon capacity Cmand channel dispersion Vmas follows,

respectively,

Cm= log (1 + γm),(3)

and

Vm= (log e)21−1

(1 + γ2

m),(4)

where γmis the received signal-to-noise-ratio (SNR) at the

BS and can be expressed as

γm=|hm|2Bmgmpm

N0Bm

=|hm|2gmpm

N0

,(5)

where Pmis the transmission power spectral density of the

m-th sensor.

B. CONTROL

In this subsection, we provide the control model for each

plant mwith communication time delay and reliability. As

shown in Fig. 2, the control process is conducted as follow-

ing. First, a sensor takes the sample of the corresponding

plant’s current state and transmits it to the BS. Then, the

controller in the BS estimates the state by Kalman Filter

[3], calculates the control command, and sends it to the

plant. Finally, the plant state updates by the received control

command. Based on the above control process, the linear

differential equation of the m-th plant can be expressed as

[3]

dxm(t) = Axm(t)dt +Bum(t)dt +dnm(t),(6)

where xm(t)is the plant state, um(t)is the control input, and

nm(t)is the disturbance caused by additive white gaussian

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)##*+$,- .$/0$"&1!/(#%

2#03(#&,3"(%3//#%&$(&(4#&56

7/$"(&0 6#"'3%&0

839"/!"- :;/!"-

FIGURE 2: Control model.

noise (AWGN) with zero mean and variance R. In addition,

we assume that each plant mhas the same Aand B, which

represent the physical system parameter matrices (more de-

tails can be obtained in [24]).

We assume that sm,n represents the sample period at time

index n, which consists of the wireless transmission time

delay Tm,n and an idle period ¯sm,n. Their relationship can

be expressed as

sm,n = ¯sm,n +Tm,n,(7)

where n= 1,2,··· , N represents the sampling time index

in the control process. Then, the discrete time control model

with time delay dm,n can be obtained as [3]

xm,n+1=Ωm,nxm,n+Φm,n

0un+Φm,n

1um,n−1+nm,n,(8)

where Ωm,n =eAsm,n ,Φm,n

0=R¯sm,n

0eAtdt·B, and

Φm,n

1=Rsm,n

¯sm,n eAtdt·B.

Assuming ξm,n = (xT

m,n uT

m,n−1)Tis the generalized

state, then the control function in (8) can be rewritten as

ξm,n+1 =Ωm,dξm,n +Φm,d um,n +¯nm,n,(9)

where ¯nm,n = (nT

m,n 0)Tand Φm,d = Φm,n

0

I!.

We assume Ωm,n =Ωm. Then, we have Ωm,d =

ΩmΦm,n

1

0 0 !.

Considering the packet loss, we have the close-loop system

in (9) can be rewritten as (10) on the top of next page.

In the above discussion, we have obtained the wireless

control model 1where both communication time delay and

packet loss have been taken into account. In the following

of this paper, we will formulate the optimal problem and

1According to [27], to maintain the stability of the wireless control

system, the following assumption should be satisﬁed: The packet loss

probability in URLLC and the control system parameters satisfy ρ(1 −

εth)(Ωm,d +Φm,d L)⊗(Ωm,d +Φm,dL)+εthΩm,d ⊗Ωm,d , where

ρ(·)is the spectral radius, Lis the control command feedback and will be

discussed in Appendix A, ⊗is the Kronecker product, and εth is the upper

bound of the packet loss probability.

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ξm,n+1 =Ωm,dξm,n +Φm,d um,n +¯nm,n,if βm,n = 1,and αm,n = 1,

Ωm,dξm,n +¯nm,n,if βm,n = 0,or αm,n = 0,(10)

where we use αm,n to indicate if a packet should be discarded or not (αm,n = 0 is to discard the packet while αm,n = 1 is

to keep the packer for transmission) and use βm,n to indicate if a packet is successfully transmitted or not (βm,n = 0 is failed

transmission while αm,n = 1 is successful transmission).

propose corresponding method to obtain packet drop and

resource allocation.

III. PROBLEM FORMULATION

Our goal is to ﬁnd the optimal packet drop method and

wireless resource allocation method by minimizing control

cost. To achieve this goal, we ﬁrst provide the objective func-

tion, i.e., the control cost. Then, we obtain the packet drop

constraint in wireless communications. Finally, we formulate

the optimal problem.

A. OBJECTIVE

The quadratic control cost is one the most important criteri-

ons to evaluate the control performance [25][26], which are

composed by the sum of the deviations of the plant state from

its desired setpoint and the magnitude of the control input.

Then, the quadratic control cost can be expressed as [10]

Jm,N =E[ξT

m,N Wξm,N

+

N−1

X

n=0

(ξT

m,nWξm,n +uT

m,nUum,n )],(11)

where Wand Uare the weight of the state and that of

the control input, respectively, and they can be adjusted

according to the emphasis of the control system. The control

variables in (11) can be obtained by Appendix A.

From (10), we can obtain that the generalized plant state

ξm,n is a function of transmission time delay and packet

loss, i.e., ξm,n(Tm,n , αm,n). In addition, um,n is also a

function of Tm,n and αm,n. Furthermore, the total number

of packet loss is determined by constraints on the packet loss

probability. Thereby, the control cost Jm,N is a function of

communication parameters.

B. CONSTRAINT

In Section II.A, we have introduced transmission error prob-

ability εe

m. However, when we consider the packets from M

sensors, the queueing delay violation probability cannot be

ignored [18], which results in some part of the packet loss at

the BS to maintain the extreme high QoS in URLLC.

We assume that εq

mrepresents the queueing delay violation

probability at the BS. Considering both queueing delay vio-

lation probability and transmission error probability, we have

the total packet loss probability of the m-th sensor as

εm=εe

m+εq

m≤εth,(12)

!""#$%&'(&()#& *

*#+%,$&-

*#+%,$&-./

000

000

1m n

i

!m n

i

m n

Q

!"m n

Q

m n

j

!"m n

j

!"#$%&

!"#$%&'(

)))

)))

))) )))

FIGURE 3: Queueing at the BS.

where εth is the upper bound of the total packet loss proba-

bility. From (2), we have

εe

m=fQTmBmCm−λ+ log(TmBm)/2

(log e)√TmBm,(13)

where λ=TmBmRmis payload information for each

sensor, and fQ(·)is the Q function. Next, we discuss the the

queueing delay violation probability εq

min details.

As shown in Fig. 3, each E2E communication pair, i.e.,

sensor-BS-plant, has the corresponding buffer at the BS,

where imrepresents the packets uploaded to the BS from the

m-th sensor and Qmrepresents the queue length for the m-th

plant. Furthermore, we assume that jmrepresents the packets

departed from the m-th queue. Then, according to [19], we

have the following Lemma 1.

Lemma 1: The queueing delay violation probability εq

m

can be expressed as

εq

m= exp{−φmEB

m(φm)Dq

max},(14)

where φmis the QoS exponent for the m-th plant, Dq

max

is the queueing delay bound, and EB

m(φm)is the effective

bandwidth and can be expressed as

EB

m(φm) = lim

N→∞

1

NTu,m φm

ln

(E

"exp

φm

N

X

n=1

im,n

!#).

(15)

Proof 1: The details of the proof for Lemma 1 can be

obtained in [19].

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C. OPTIMAL PROBLEM

In this subsection, we formulate the optimal problem, which

is described in Problem 0, i.e., P0, and can be expressed as

P0: min

αm,n, Bm,n Tm,n

Jsum =

M

X

m=1

Jm,N (16a)

s.t.

αm,n ∈ {0,1},(16b)

εe

m+εq

m≤εth,(16c)

Tm,n ≤Tth,(16d)

Bm,n ≤Bth,(16e)

where the objective of this optimal problem is to minimize

the total control cost Jsum constrained by communication

parameters, Bth is the upper bound of the allocated band-

width for each sensor at any time index, Tth is the upper

bound of the communication time delay. In addition, εth is

the upper bound of the packet loss probability in URLLC,

which is in the region that guarantees the convergence of the

control systems [1]. Furthermore, αm,n is used to indicate

if a packet should be discarded or not. Then, we obtain

that the successful packet transmission probability can be

expressed as Pr{βm,n = 1|αm,n = 1}= 1 −εe

mand the

failed packet transmission probability can be expressed as

Pr{βm,n = 0|αm,n = 1}=εe

m.

In (16), the constraint in (16e) maintains the URLLC

QoS requirements to guarantee successful transmission when

wireless transmission is triggered by control process. Con-

straints in (16b) and (16c) are related to both communication

and control. On the one hand, (16c) maintains the total packet

loss probability to guarantee the URLLC QoS requirements

for successful transmission, which determines packet drop

strategy αm. On the other hand, the total control cost of M

plants is determined by how to arrange αmin (16b), where

αmis related to communication packet loss probability. Fur-

thermore, the constraint on transmission time delay in (16d)

is also related to communication performance and control

performance. Thus, P0in (16) is extremely challenging to

be solved under the constraints in (16b), (16c), and (16d). In

the next section, we will discuss the solution for the problem

P0in details.

IV. PACKET DROP AND WIRELESS RESOURCE

ALLOCATION METHOD

In this section, we ﬁrst analyze the relationship between

communication and control. Based on this, we decompose

(16) into two subproblems: (1) an optimal wireless resource

allocation problem to minimize transmission error proba-

bility and (2) an optimal packet drop problem to minimize

control cost. Then, the solution for the two subproblems can

be obtained.

A. RELATIONSHIP BETWEEN CONTROL AND

COMMUNICATION

As shown in Fig. 2, the linear feedback control law is used in

this paper (more details about this law can be obtained in [2]).

Then, the optimal expression for the control cost in (11) can

be rewritten as (17) on the top of next page [2]. Then, we can

obtain the relationship between control and communication

by the following Theorem 1.

Theorem 1: Once the communication time delay and

packet loss probability is determined, the optimal control cost

in (17) is related with αm,n from communication aspect.

Thus, the minimization of the objective function in (16a)

is relatively independent with communication constraints on

communication time delay and packet loss probability in

(16).

Based on Theorem 1,αm,n is the connection between

communication and control. Then, P0can be divided into

two subproblems. The ﬁrst subproblem is to optimize the

communication time delay and packet loss probability by

wireless resource allocation. Once they are obtained, the

second subproblem is to minimize control cost by packet

drop design.

B. PACKET DROP AND RESOURCE ALLOCATION

Compared with transmission time delay in URLLC, packet

loss introduces more control performance loss, i.e., larger

control cost. This is because packet loss can be treated as

larger time delay than the required time delay in URLLC.

Then, minimizing packet loss probability is critical for better

control performance. Thus, our goal is to minimize packet

loss probability by optimizing transmission time delay and

bandwidth allocation in the ﬁrst subproblem. By the ob-

tained transmission time delay and packet loss probability,

the second subproblem is to design the packet drop αm,n to

minimize control cost.

1) Resource Allocation

Since the queueing delay violation probability is independent

with wireless communications, the ﬁrst sub-problem is to

minimize transmission error probability by optimal wireless

resource allocation, which can be expressed as P1, i.e.,

P1: min

Tm,n,Bm,n

εe

m(18a)

s.t.

εe

m+εq

m≤εth,(18b)

Tm,n ≤Tth,(18c)

Bm,n ≤Bth,(18d)

By minimizing error probability, the transmission require-

ment for the control process can be guaranteed. To obtain

the optimal resource allocation for P1, we assume that the

resource block consists of time resource and bandwidth re-

source, i.e., Tm,n ×Bm,n. Then, to solve the problem (18),

we need the following property about εe

m,n.

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J∗

m,N=ξT

m,0Sm,0ξm,0+Tr(Sm,0Pm,0)+

N−1

X

n=0

(Tr((ΩT

m,dSm,n+1 Ωm,d +W−Sm,n)Eαm,n [Pm,n|n])+Tr(Sm,n+1Rm,n )),

(17)

where the parameters can be obtained by Appendix A. Observing the expression in (17), the optimal control cost J∗

m,N is

related with αm,n for given transmission time delay.

Property 1: The function εe

m,n(Tm,n ×Bm,n )is convex

in Tm,n ×Bm,n.

Proof: See Appendix B.

By Property 1, we can apply the exact linear search

method to ﬁnd the optimal (Bm,nTm,n )∗to minimize εe

m,n

[28].

From the above discussion, we can obtain the optimal

resource block allocation to minimize transmission error

probability. To reduce control cost, the transmission time

delay is desired to be short enough. Then, the time delay

Tm,n in resource block can be calculated by

Tm,n =(Bm,nTm,n )∗

Bth

.(19)

By the obtained minimum transmission error probability

εe

mand time delay Tm,n in (19), we discuss the packet drop

design in the following.

2) Packet Drop

The second subproblem minimizing the overall control cost

can be expressed as P2, i.e.,

P2: min

αm,n

J∗

sum =

M

X

m=1

J∗

m,N (20a)

s.t.

αm,n ∈ {0,1},(20b)

1−PN−1

n=0 αm,n

N≤εe

m+εq

m,(20c)

where (20b) is the overall communication packet loss prob-

ability for each E2E (i.e., sensor-BS-plant pair). To deal

with P2in (20), we assume that Mplants have the same

control parameters. In addition, we assume that the weight

on the plant state Wis much larger than that on the control

input U. This assumption holds in this paper, since the plant

state is more important than the control input in mission-

critical real-time wireless control systems [1]. In addition,

we have proved that J∗

m,N increases strictly with the overall

communication packet loss probability in [1].

Since control process is a sequential process, packet drop

strategy leads to different control cost. In addition, it is

extremely difﬁcult to predict the plant state since the state

update in (6) has disturbance term. Then, it is challenging to

obtain global optimal packet drop method to minimize total

control cost. Instead, we propose a suboptimal packet drop

method, where we obtain the point-wise minimum control

cost by the suboptimal packet drop.

In the proposed method, we assume that Em,n =

xT

m,nWxm,n represents the instantaneous control cost of the

m-th plant at time index n. When packet drop occurs at time

index n, the BS will drop the packet that contributes to mini-

mum plant state, which leads to point-wise minimum control

cost. The detailed method is summarized in Algorithm 1.

Algorithm 1 The proposed suboptimal packet drop method.

Input: εe

m,εq

m,W,A,B,C,xm,0, and Tm,n.

1: Set 4m= 1, where m= 1,2,··· , M

2: Set αm,n = 1, where m= 1,2,··· , M , and n=

0,1,2,··· , N −1

3: while n≤Ndo

4: Calculate Em,n =ξT

m,nWξm,n , where m=

1,2,··· , M

5: [Emin, mmin , nmin] = min{Em,n },

6: while PM

m=1 EB

m(φm)≥PM

m=1 Rmdo

7: if 4mmin > εe

mmin +εq

mmin then

8: m0=mmin,

9: n0=nmin,

10: αm0,n0= 0,

11: {Em,n}={Em,n } \ Emin,

12: {m}={m} \ mmin,

13: {n}={n} \ nmin,

14: else

15: m0=mmin,

16: n0=nmin,

17: αm0,n0= 1,

18: {Em,n}={Em,n } \ Emin,

19: {m}={m} \ mmin,

20: {n}={n} \ nmin,

21: end if

22: end while

23: n=n+ 1

24: end while

Output: Packet drop method αm,n.

V. SIMULATION RESULTS

In this section, we provide simulation results to demonstrate

the performance of our analysis in this paper. In communica-

tion sub-systems, we assume that the payload information is

100 bits. The maximum time delay of URLLC is 1ms and

the maximum packet loss εis 10−5. The control parameters

are as follows: A= 2 14

0 1 !,B= 0

1!,C= 1 0

0 1!,

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Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS

1 2 3 4 5 6 7 8 9 10

Total available bandwidth (Hz) 106

4.5

5

5.5

6

6.5

7

7.5

8

Total control cost

107

Proposed method

Random method

Channel gain based method

Exhaustive search method

FIGURE 4: Total control cost with different available band-

width Bth.

150 200 250 300 350 400 450 500

Total number of plants, M

0

1

2

3

4

5

6

7

8

Total control cost

107

Proposed method

Random method

Channel gain based method

Exhaustive search method

FIGURE 5: Total control cost with different number of

sensor-BS-plant pairs M.

P0= 0.01I,W=I,U=I,Rn=I, and Rn0= 0.01I.

Furthermore, we assume that the initial state is (100,100).

Each curve is obtained by 10000 Monte Carlo trails if there

is no extra declaration. Moreover, the random packet drop

method and channel gain based packet drop method are

considered as comparison. In addition, the exhaustive search

method solving the formulated problem is considered to

justify the beneﬁts of the proposed algorithm.

Fig. 4 demonstrates the total control cost when the avail-

able bandwidth Bth is different, where the queueing delay

bound is 0.1ms. From the ﬁgure, all the curves decrease

monotonously with Bth. This is reasonable since larger Bth

can guarantee less packet loss, which maintains the timely

control input for control systems to reduce the control cost.

0 100 200 300 400 500 600 700 800

Arriving rate (packet/s)

1

2

3

4

5

6

7

8

Total control cost

107

Proposed method

Random method

Channel gain based method

Exhaustive search method

FIGURE 6: Total control cost with different arriving rate of

sampling packet.

1 2 3 4 5 6 7 8 9 10

Queuing delay constraint (s) 10-7

1

2

3

4

5

6

7

8

Total control cost

108

Proposed method

Random method

Channel gain based method

Exhaustive search method

FIGURE 7: Total control cost with different constraints on

queueing delay.

In addition, the decreasing rate of all the curves is smooth and

low when Bth is more than 2×106Hz, which is because that

Bth is saturated. Furthermore, the total control cost is similar

for both random method and channel gain based method,

since they have equal contribution to the control cost. On

the one hand, all the three curves are similar when Bth is

small. This is reasonable since small Bth leads to large packet

error probability, which results in that the control system is

not very sensitive to the packet drop method. From the ﬁg-

ure, compared with random method and channel gain based

method, the proposed method decreases the total control cost

by almost 40% when Bth ≥2×106Hz, which indicates

that the proposed method in this paper has large advantage

compared with only considering the communication aspect.

In addition, compared with exhaustive search method, the

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Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS

control cost of the proposed method is raised by at most

6.5%, which indicates that the solution of the proposed

method is close to the global optimal solution.

Fig. 5 shows the total control cost when the number of

sensor-plant pairs Mis different, where the total available

bandwidth is Bth = 2 ×106Hz, the queueing delay bound is

0.1ms. From the ﬁgure, all the curves increase monotonously

with the number of sensor-plant pairs M. This is reasonable

since the supported number of sensor-plant pairs is ﬁxed with

given Bth, which further leads to large control cost when

Mincreases. In addition, the curves of the random method

and the channel gain based method are similar, which is

because that both of them have the same effect on the control

performance. Furthermore, the advantage of the proposed

method is approximative when the number Mis very large,

i.e., M≥300. This is reasonable since the dropped packets

to minimize control cost has minor effect on the total cost

when Mis too large compared with the traditional methods.

In addition, the control cost of the proposed method is is

similar to the global optimal solution obtained by exhaustive

search method.

Fig. 6 demonstrates the total control cost when the arriving

rate of the sampling packets is different, where the total

available bandwidth is Bth = 2 ×106Hz, the total number

of sensor-plant pairs is M= 250, and the queueing delay

bound is 0.1ms. From the ﬁgure, all the curves increase

monotonously with the arriving rate until 500 packets/s. In

addition, after 500 packets/sof the arriving rate, the curves

of the total control cost are approximative horizontal. This

can be explained by the following two aspects. On the one

hand, when the arriving rate is less than 500 packets/s, larger

arriving rate means smaller sampling period hk, which leads

to smaller dk/hk. Then, the control cost increases as the ar-

riving rate increasing before 500 packets/s[1]. On the other

hand, when the arriving rate is larger than 500 packets/s, the

number of arriving packets tends to saturated, which leads to

a balance state and the curves of the control cost have little

changes. Furthermore, from the ﬁgure, we can obtain that

the proposed method decreases the control cost by at most

60% compared with only considering the communication

aspect. In addition, the control cost of the proposed method

is approximated to the global optimal solution obtained by

exhaustive search method.

Fig. 7 demonstrates the total control cost when the queue-

ing time delay constraint is different, where the total available

bandwidth is Bth = 2 ×106Hz. From the ﬁgure, all the

curves decrease monotonously with the queueing time delay

constraint. This is reasonable since larger queueing time

delay constraint allows more packets in the queue, which

leads to less packet drop probability and larger transmission

successful probability. Then, the control cost can be reduced.

However, the control cost changes smoothly when the queue-

ing time delay constraint is larger than 5×10−7s. This

is because the allowed number of arriving packets tends to

saturated, and a balance state is maintained. Then, the curves

of the control cost have little changes. Furthermore, from

the ﬁgure, we can obtain that the proposed decreases the

control cost by at most 62% compared with only considering

the communication aspect. In addition, the performance gap

between the proposed method and the exhaustive search

method is minor.

VI. CONCLUSIONS

In this paper, we proposed a packet drop and wireless re-

source allocation method in URLLC for real-time wireless

control systems. To obtain good control performance, we

formulated an optimal problem to minimize the control cost

with communication constraints. To solve the problem, we

discussed the relationship between control and communica-

tion. Based on that, we decomposed the original problem

into two relatively independent sub-problems. By solving

the two subproblems, we obtained the transmission time

allocation, bandwidth allocation and packet drop method.

The proposed approach established a theoretic foundation

for the URLLC enabled real-time wireless control system

performance analysis and algorithm design.

.

APPENDIX A

This appendix provides the detailed calculation of the param-

eters in (17).

According to [2], Skis calculated by

Sk=ΩT

dSk+1Ωd+W−ΩT

dSk+1Φd(ΦT

dSk+1Φd

+U)−1ΦT

dSk+1Ωd.(21)

The generalized state can be estimated by a modiﬁed Kalman

ﬁlter, which can be obtained as follows.

•Step 1: prior generalized state estimation. The prior

estimation for the generalized state can be expressed as

ˆ

ξm,n+1|n=Ωm,d ˆ

ξm,n|n+Φm,dum,n ,(22)

where ˆ

ξm,n|nis the generalized state estimation based

on the current generalized state, and ˆ

ξm,n+1|nis the

generalized state estimation at time n+ 1 based on the

last generalized state at n.

•Step 2: prior error variance estimation. The prior esti-

mation for the error variance can be expressed as

Pm,n+1|n=Ωm,dPm,n|nΩT

m,d +Rn,(23)

where Pm,n|n=E[(ξm,n −ˆ

ξm,n)(ξm,n −ˆ

ξm,n)T]is

the estimation error variance, and Pm,n+1|nis the prior

estimation error variance at time k+ 1.

•Step 3: optimal generalized state estimation. The opti-

mal generalized state estimation is the generalized state

estimation based on ˆ

ξm,n+1|n, and can be expressed as

ˆ

ξm,n+1|n+1 =ˆ

ξm,n+1|n+αm,nKm,n+1 (ym,n+1

−Cm,d ˆ

ξm,n+1|n),

(24)

8VOLUME 4, 2016

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Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS

where Km,n+1 will be discussed in the following Step

4.

•Step 4: optimal control gain estimation. The optimal

control gain estimation Km,n+1 can be expressed as

Km,n+1 =Pm,n+1|nCT

m,d

(Cm,dPm,n+1|nCT

m,d +Rn0)−1.(25)

•Step 5: optimal error variance estimation. The optimal

error variance estimation is the error variance estimation

based on Pm,n+1|n, which can be calculated by

Pm,n+1|n+1 =Pm,n+1|n

−αm,nKm,n+1 Cm,dPm,n+1|n.(26)

Finally, substituting the above parameters into (11), we can

obtain (17). Furthermore, to minimize the control cost in

(11), the control input needs to satisfy the following expres-

sion

um,n =−ΦT

m,dSm,n+1 Φm,d +U−1ΦT

m,dSm,n+1

Ωm,d ˆ

ξm,n|n=−Lm,n ˆ

ξm,n|n.

(27)

APPENDIX B

This appendix provides the detailed proof for Property 1.

We assume x=Bm,n Tm,n,G1= log(1 + |hm|2gmpm

N0),

and G2= (log e), then εe

m,n can be rewritten as

εe

m,n =fQ(xG1−λ+ log(x)/2

G2√x).(28)

Let

f1(x) = xG1−λ+ log(x)/2

G2√x.(29)

Taking derivative with respect to x, we can obtain

∂f1(x)

∂x =G2

2x1

2·G1x+ (1 + λ)−log(x)

2(30)

Then, the second derivative with respect to xcan be ex-

pressed as

∂2f1(x)

∂x2=G2[G1(ln 4)x−(1 + λ)(ln 4) + ln x−2]

8x3

2(ln 2) .

(31)

Since Bm,n is with MHz order of magnitude and Tm,n is

with ms order of magnitude, we can obtain [G1(ln 4)x−(1+

λ)(ln 4)] >0and (ln x−2) >0. Then, we have that ∂2f1(x)

∂x2

is more than zero, i.e.,

∂2f1(x)

∂x2>0.(32)

Thus, f1(x)is convex in x. Furthermore, since fQ(·)is

convex. Thus, εe

m,n(x)is convex, i.e., εe

m,n is convex in

Bm,n ×Tm,n.

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BO CHANG received his Bachelor and Master

degrees from University of Electronic Science and

Technology of China (UESTC), Chengdu, China,

where he is currently pursuing a Ph.D. degree at

the National Key Laboratory of Science and Tech-

nology on Communications. From May 2019 to

May 2020, he is a visiting student at University of

Glasgow. His research interests include cognitive

radio, wireless localization, ultra-reliable and low-

latency communications, and communication and

control co-design for Industrial Internet-of-Things (IIoT).

GUODONG (PHILIP) ZHAO (SM’16) received

his Ph.D. Degree from Beihang University, Bei-

jing, China, in 2011 and his B.E. degree from

Xidian University, Xi’an, China, in 2005. In 2011-

2018, he worked as an associate professor at U-

niversity of Electronic Science and Technology of

China (UESTC) in China. In 2012-2013, he visited

the Hong Kong University of Science and Tech-

nology, Hong Kong. In 2016, he visited Lehigh

University, USA. In 2018, he joined University

of Glasgow in UK as a lecturer (assistant professor). His current research

interests are within the areas of wireless communications and control. He has

authored 50+ papers in IEEE journals and conferences. He received the Best

Paper Award from IEEE Global Telecommunication Conference (GLOBE-

COM) in 2012 and the Best Ph.D. Thesis Award from Beihang University

in 2012. He has served as a TPC for many international conferences, e.g.,

ICC and VTC. He also served a reviewer for many IEEE Transactions, e.g.,

the IEEE Transactions on Signal Processing and IEEE Journal on Selected

Areas in Communications.

ZHI CHEN (SM’16) received the B. Eng, M.

Eng., and Ph.D. degrees in electrical engineering

from the University of Electronic Science and

Technology of China (UESTC), in 1997, 2000,

and 2006, respectively. In 2006, he joined the Na-

tional Key Laboratory of Science and Technology

on Communications, UESTC, where he has been

a Professor since 2013. He was a Visiting Scholar

with the University of California at Riverside,

Riverside, from 2010 to 2011. His current research

interests include 5G mobile communications, tactile internet, and Terahertz

communication. He has served as a reviewer for various international jour-

nals and conferences, including IEEE Transactions on Vehicular Technology

and IEEE Transactions on Signal Processing.

LIYING LI received her B.E. and Ph.D. Degrees

from University of Electronic Science and Tech-

nology of China (UESTC) in 2005 and 2011, both

in Electrical Engineering. She visited Georgia In-

stitute of Technology, GA, USA, in 2008-2010

and Lehigh University, PA, USA, in 2016. Since

2011, she has been with the School of Automa-

tion Engineering, University of Electronic Science

and Technology of China (UESTC), where she

is currently an Associate Professor. Her research

interests are within the areas of wireless communications, smart grid, and

data science.

MUHAMMAD ALI IMRAN (M’03, SM’12) Fel-

low IET, Senior Member IEEE, Senior Fellow

HEA is a Professor of Wireless Communication

Systems with research interests in self organised

networks, wireless networked control systems and

the wireless sensor systems. He heads the Com-

munications, Sensing and Imaging CSI research

group at University of Glasgow. He is an Afﬁliate

Professor at the University of Oklahoma, USA

and a visiting Professor at 5G Innovation Centre,

University of Surrey, UK. He has over 20 years of combined academic

and industry experience with several leading roles in multi-million pounds

funded projects. He has ﬁled 15 patents; has authored/co-authored over 400

journal and conference publications; was editor of 3 books and author of

more than 20 book chapters; has successfully supervised over 40 postgrad-

uate students at Doctoral level. He has been a consultant to international

projects and local companies in the area of self-organised networks.

10 VOLUME 4, 2016