research article control of four-wheel-independent-drive electric … · 2019. 7. 31. · c...

Research Article𝐻

∞Control of Four-Wheel-Independent-Drive Electric Vehicles

with Random Time-Varying Delays

Gang Qin and Jianxiao Zou

School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

Correspondence should be addressed to Gang Qin; [email protected]

Received 12 August 2014; Accepted 15 October 2014

Academic Editor: Jiuwen Cao

Copyright © 2015 G. Qin and J. Zou. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The random time-varying delays would reduce control performance and even deteriorate the EV system. To deal with random time-varying delays and achieve a real-time steady-state response, considering randomness of delay and a rapid response, an𝐻

∞-based

delay-tolerant linear quadratic regulator (LQR) control method based on Taylor series expansion is proposed in this paper. Theresults of cosimulations with Simulink and CarSim demonstrate the effectiveness of the proposed controller through the controlperformance of yaw rate, sideslip angle, and the running track. Moreover, the results of comparison with the other controllerillustrate the strength of explicitly.

1. Introduction

With the rapid development of EV technology, the stabilityof EV is concerned by enterprise and research institution[1]. Due to unique technical advantages such as simplifiedtransmission, regenerative braking system of each wheel,and electronic initiative chassis [2], 4WIDEVs have attractedgreat attention [3].

However, some random time-varying delays of environ-ment interference and network may cause the 4WIDEVsystem to be unstable. On one hand, since the workingenvironment of 4WIDEV changes tremendously, some inter-ferences increase in the conditions of road [4], the variationalparameters ofmotors, the variational parameters of 4WIDEV[5–7], and electromagnetic interference (EMI) [8], whichare random and difficult to measure. On the other hand,4WIDEV controls are also characterized by fast dynamics,whereas the response time and accuracy of controller may beinfluenced under the network-induced delays.

There are a few approaches against network delays [9];for example, an 𝐻

∞controller [10] is proposed to decrease

CAN delays. However, environment interference delays arenot considered in literature [10]. Meanwhile, longitudinalforce and vertical force are ignored in 2-DOFmodel which isproposed in literature [10]. There are numerous approaches

for the appropriate handling based on polytopic inclusions[11]. In literature [12], a method based on Jordan normalform (JNF) is proposed. However, the polytope of JNFis too large to store in real-time system. Then a methodbased on elementwise minimization-maximization (EMM)[13] is used to describe random time-varying delays. ThoughEMM is the same order of magnitude as Taylor seriesexpansion (TA) [14], it is more complex than TA. Therefore,TA is chosen to deal with random time-varying delays inthis paper. However, there is no approach against randomnetwork delays proposed. There are rare research studiesto restrain onboard random time delays of network. Basedon the theoretical research on network-induced delays [9],a practical result of the network-induced delay model wasadopted for the study of automotive system [10]. Even thoughthere are few approaches against CAN network delays, thereis no method which is proposed to deal with network delaysand environment interference delays.

The main work is as follows; firstly, the environmentdelays and network delays are explicitly considered inthe vehicle yawing moment control problem. Consideringthat the random time-varying delays lead to a challengingcontrol problem for the vehicle lateral stability and han-dling, the delays are described via the polytopic technique,which is different from the conventional control strategy.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 245493, 10 pageshttp://dx.doi.org/10.1155/2015/245493

2 Mathematical Problems in Engineering

Drivingintention

recognition

4WIDEVcontroller

Network In-wheelmotors

Transducer system

𝛿 Ti Ti

x, y, 𝜓 𝛾, Vx �̂�F̂zi

Figure 1: Electric control system structure.

Furthermore, an 𝐻∞-based LQR tracking control scheme is

designed tomake the control performance and the robustnessof 4WIDEV against random time-varying delays.

The remaining sections of this paper are organized asfollows. In Section 2, the 4WIDEV control system is studied.In Section 3, the𝐻

∞-based LQR is designed to solve random

time-varying delays with LMI theory. In Section 4, the effec-tiveness of controller is demonstrated via cosimulation withSimulink and CarSim. Section 5 presents some concludingremarks.

2. WIDEV Control System

2.1. Electric Control System Structure. As is shown in Figure 1,a 4WIDEV structure consists of driving intention recog-nition, 4WIDEV controller, network, in-wheel motors, andtransducer and estimate system.

As is descripted in Figure 1, the transducer system isdesigned to gather accelerate pedal location, braking pedallocation, longitudinal and transverse acceleration, yaw rate,and velocity as inputs. Driving intention recognition deter-mines expected vehicle velocity 𝑉des and steering angle 𝛿.4WIDEV controller picks up expected yaw moment 𝑀

𝑍

and expected vehicle driving moment 𝐹des. Moreover, on thepremise of satisfying proposed𝑀

𝑍and𝐹des, torque allocation

set is in charge of calculating driving moment 𝑇𝑖and steering

angle of four wheels 𝛿𝑓. The control signals are exchanged

using control area network (CAN or FlexRay) [10].

2.2.Model of 4WIDEV. Without any loss of generality, we canget three assumptions [10].

Assumption 1. 4WIDEV is symmetrical and its moving coor-dinate origin is center of gravity (CG).

Assumption 2. The steering wheel angle is equal to frontwheel angle and same characteristics of each wheel.

Assumption 3. Air resistance is ignored.

In this paper, a 4-DOF model of 4WIDEV is used forcontroller design, as is shown in Figure 2. Through the stressanalysis of 4WIDEV and Newton’s second law of motion, wecan get 4-DOF vehicle model as follows [10]:

�̇� = 𝐴𝑐𝑥 + 𝐵

𝜔𝑐𝜔 + 𝐵

𝑢𝑐𝐹Δ𝑥, (1)

where 𝑥 = [𝛽 𝑟 𝜙 ̇𝜙] 𝜔 = 𝛿, 𝐹Δ𝑥

= (𝐹𝑥𝑓𝑟

− 𝐹𝑥𝑓𝑙) + (𝐹

𝑥𝑟𝑟−

𝐹𝑥𝑟𝑙)

𝐴𝑐=

[[[[[[[[[[[[

[

−

𝐼𝑒𝑞

𝐼𝑥𝑥

𝐶0

𝑚𝑢

−1 −

𝐼𝑒𝑞

𝐼𝑥𝑥

𝐶1

𝑚𝑢2

ℎCG (mg ⋅ ℎCG − 𝑘𝜙)𝐼𝑥𝑥𝑢

−

ℎCG𝑏 ̇𝜙

𝐼𝑥𝑥𝑢

−

𝐶𝑓

𝐼𝑧𝑧

−

𝐶2

𝐼𝑧𝑧𝑢

0 0

0 0 0 1

−

ℎCG𝐶0𝐼𝑥𝑥

−

ℎCG𝐶1𝐼𝑥𝑥𝑢

mg ⋅ ℎCG − 𝑘𝜙𝐼𝑥𝑥

−

𝑏 ̇𝜙

𝐼𝑥𝑥

]]]]]]]]]]]]

]

,

𝐵𝜔𝑐= [0

𝑑

2𝐼𝑧𝑧

0 0]

𝑇

,

𝐵𝑢𝑐= [

1

𝑚𝑢

(

2𝐶𝑓𝐼𝑒𝑞

𝐼𝑥𝑥

+ 𝐹𝑥𝑓)

𝑎

𝐼𝑧𝑧

(2𝐶𝑓+ 𝐹

𝑥𝑓) 0

2𝐶𝑓+ ℎCG𝐹𝑥𝑓

𝐼𝑥𝑥

]

𝑇

,

𝐼𝑒𝑞= 𝐼

𝑥𝑥+ 𝑚ℎ

2

CG,

𝐶0= 2𝐶

𝑓+ 2𝐶

𝑟,

𝐶1= 2𝑎𝐶

𝑓− 2𝑏𝐶

𝑟,

𝐶2= 2𝑎

2

𝐶𝑓+ 2𝑏

2

𝐶𝑟,

(2)

where𝑚 is the mass of 4WIDEV, 𝐼𝑧is the inertia of 4WIDEV,

𝐼𝑧𝑧is yaw inertia, 𝐼

𝑥𝑥is roll inertia, 𝑙

𝑓is the distance from CG

to front axles, 𝑙𝑟is the distances from CG to rear axles, 𝑏 is

the distance from front axle to CG, and 𝑎 is the distance fromrear axle to CG. 𝑑 is track width, ℎCG is height of CG, 𝛿 is thefront wheels steering angle which is inputs, 𝛼

𝑓is front tires

slip angle, 𝛼𝑟is rear tires slip angle, 𝛽 is sideslip angle, 𝜙 is roll

angle and ̇𝜙 derivative of roll angle, 𝐹𝑥𝑓

is longitudinal forceof front tire, 𝐹

𝑟𝑓is rear longitudinal force, 𝐹

𝑦𝑓is lateral force

of front tires,𝐹𝑦𝑟is rear tires lateral force,𝑉 is the CG velocity

[15] of 4WIDEV, 𝑢 is longitudinal velocity, V is lateral velocity,𝑟 is yaw rate, 𝑏

𝜙is roll damping, 𝐶

𝑓is cornering stiffness of

front tire, 𝐶𝑟is cornering stiffness of rear tire, and 𝑘

𝜙is roll

angle stiffness. 𝛿 is the input of 4WIDEV model, and 𝛽, 𝑟, 𝜙,and ̇𝜙 are four states of 4WIDEV model.

2.3. Reference State Responses. Generally, the referencesideslip angle which is mainly concerned with the vehiclestability [16] is set as zero to make 4WIDEV stable [17],whereas the reference yaw rate is defined in terms of vehicleparameters, longitudinal speed, and steering input of thedriver as [18]

𝑟ref =𝑢

𝑎 + 𝑏 + (𝑚𝑢2(𝐶

𝑟𝑏 − 𝐶

𝑓𝑎) /2𝐶

𝑓𝐶𝑟(𝑎 + 𝑏))

𝛿. (3)

Applying (3) to (1), the reference state responses can beexpressed as follows:

𝑦 = 𝑅𝜔, (4)

Mathematical Problems in Engineering 3

Y

YX

O

vy

vx

vcog

a b

Tr

Tr

Tf

l

𝛿

Z

G

𝛽𝜙

𝜙

Fy rl

Fy rl

Fx rl

Fy rr

Fy rr

Fx rr

Fy flFx fl

Fy fr Fx fr

Fz rl Fz rr

Figure 2: Bicycle model of vehicle lateral dynamics.

where

𝑦 =

[

[

[

[

𝛽ref𝑟ref𝜙reḟ

𝜙ref

]

]

]

]

,

𝑅 = [0

𝑢

𝑎 + 𝑏 + (𝑚𝑢2(𝐶

𝑟𝑏− 𝐶

𝑓𝑎) /2𝐶

𝑓𝐶𝑟(𝑎 + 𝑏))

0 0]

𝑇

,

(5)

where 𝑟ref is reference yaw rate, 𝛽ref is reference sideslip angle,𝜙ref is reference roll angle, and ̇𝜙ref is reference derivative ofroll angle.

2.4. Analysis of Random Time-Varying Delays

Assume 1. Random time-varying delays are bounded withinan in-vehicle network and EV’s working environment [19].

Assume 2. In order to simplify calculation, we suppose thatall delays are equivalent to network delays [20]. An explicitexpression can be demonstrated as follows:

𝜏𝑙 arg 𝑒,𝑗 =

(𝑗 + 2) 𝑙

𝑅 − ∑𝑗−1

𝑖=0(𝑙/𝑐

𝑖)

, (6)

where 𝜏𝑙 arg 𝑒,𝑗 is the upper bound of the delay of the 𝑗th

message, 𝑙 is the maximum frame length, 𝑅 is the rate ofcommunication protocol, and 𝑐

𝑖is the cycle length of the 𝑖th

priority message.From (6), we can get an integral term as follows:

Γ (𝑥) = ∫

𝑥

0

𝑒𝐴𝑐(𝑇𝑠−𝜃)

𝑑𝜃; (7)

using Taylor series expansion in (7), we can get the following:

Γ (𝑥) = −

ℎ

∑

𝑞=1

(−𝑥)𝑞

𝑞!

𝐴𝑞−1

𝑐𝑒𝐴𝑐𝑇𝑠

+ Θℎ

. (8)

With a proper selection of the numberℎ, the high-order termsin the remainder can be relatively small. We can neglect theremainder Θℎ and obtain the ℎ-order approximation as isshown in the following:

Γℎ

(𝑥) = −

ℎ

∑

𝑞=1

(−𝑥)𝑞

𝑞!

𝐴𝑞−1

𝑐𝑒𝐴𝑐𝑇𝑠

; (9)

the random terms can be expressed as a linear combination[11] as follows:

Δ𝑖,𝑘=

ℎ+1

∑

𝑙=1

𝜇𝑖,𝑙(𝑘) Δ

1,𝑙, 𝑖 = 0, 1, . . . , Υ − 1

𝑑𝑦

𝑑𝑥

,

Δ𝑖,𝑘=

ℎ+1

∑

𝑙=1

𝑢𝑖,𝑙(𝑘) Δ

0,𝑙, 𝑖 = Υ,

(10)

where 𝜇𝑖,𝑙(𝑘) is a random time-varying coefficient with

respect to 𝜏𝑘and

ℎ+1

∑

𝑙=1

𝑢𝑖,𝑙(𝑘) = 1, 𝑢

𝑖,𝑙(𝑘) > 0. (11)

2.5. Control Model with Random Time-Varying Delays. Withthe random time-varying delays, (1) is derived as

𝑥𝑘+1

= 𝐴𝑑𝑥𝑘+ 𝐵

𝑤𝑟𝑑𝑤𝑘+ 𝐵

𝑢𝑑𝑢𝑘. (12)

Considering control delay from the EV controller, we canwrite the input of the vehicle at time 𝑡 in the following:

𝑢 (𝑡) = 𝑢𝑘

∀𝑡 ∈ [𝑘𝑇𝑠+ 𝜏

𝑘, (𝑘 + 1) 𝑇

𝑠+ 𝜏

𝑘+1] . (13)

Define the maximum value delay as

𝜏𝑙 arg 𝑒 = (𝑟 + ]) 𝑇𝑠, (14)

where Υ ∈ 𝑍+and ] ∈ 𝑅

[0,1).

Applying (13) to (12), we obtain

𝑥𝑘+1

= 𝐴𝑑𝑥𝑘+ 𝐵

𝑤𝑟𝑑𝑤𝑘+ 𝐵

𝑢𝑑𝑢𝑘+ Δ

0,𝑘(𝑢𝑘−1

− 𝑢𝑘)

+ Δ1,𝑘

(𝑢𝑘−2

− 𝑢𝑘−1

) + ⋅ ⋅ ⋅ + ΔΥ,𝑘

(𝑢𝑘−Υ−1

− 𝑢𝑘−Υ

) ,

(15)


where

Δ𝑖,𝑘=

{{{{{{{{

{{{{{{{{

{

0, 𝜏𝑘−𝑖

− 𝑖𝑇𝑠≤ 0

∫

𝜏𝑘−1−𝑖𝑇𝑠

0

𝑒𝐴(𝑇𝑠−𝜃)

𝑑𝜃 ⋅ 𝐵𝑢, 0 ≤ 𝜏

𝑘−𝑖− 𝑖𝑇

𝑠≤ 𝑇

𝑠

∫

𝑇𝑠

0

𝑒𝐴(𝑇𝑠−𝜃)

𝑑𝜃 ⋅ 𝐵𝑢, 𝑇

𝑠≤ 𝜏

𝑘−𝑖− 𝑖𝑇

𝑠.

(16)

Defining a new state vector 𝜉𝑘= [𝑥

𝑇

𝑘𝑢𝑇

𝑘−1⋅ ⋅ ⋅ 𝑢

𝑇

𝑘−Υ−1]

𝑇

, from(15), we can get

𝜉𝑘+1

= 𝐴aug𝜉𝑘 + 𝐵𝑤𝑟,aug𝑤𝑘 + 𝐵𝑢,aug𝑢𝑘, (17)

where

𝐴aug =

[

[

[

[

[

[

[

𝐴𝑑Δ0,𝑘

− Δ1,𝑘

⋅ ⋅ ⋅ ΔΥ−1,𝑘

− ΔΥ,𝑘

ΔΥ,𝑘

0 0 ⋅ ⋅ ⋅ 0 0

0 𝐼 ⋅ ⋅ ⋅ 0 0

.

.

.

.

.

. d...

.

.

.

0 0 ⋅ ⋅ ⋅ 𝐼 0

]

]

]

]

]

]

]

,

𝐵𝑤𝑟,aug =

[

[

[

[

[

[

[

𝐵𝑤𝑟𝑑

0

0

.

.

.

0

]

]

]

]

]

]

]

, 𝐵𝑢,aug =

[

[

[

[

[

[

[

𝐵𝑢𝑑− Δ

0,𝑘

𝐼

0

.

.

.

0

]

]

]

]

]

]

]

.

(18)

In (17), 𝜉𝑘is a state vector, 𝑢

𝑘is control variables vector, and

𝑤𝑘is random time-varying delays vector. Thus, a discrete-

time control model with random time-varying delays isproposed.

3. 𝐻∞

Controller Design

With the system in (17), the control objective is to min-imize the tracking error and the control input signals. Aperformance index is formulated as a combination of thetracking error and the control signals. In this paper, we selecta quadratic form of the tracking error and the control signalsas follows:

𝐽 =

∞

∑

𝑖=0

(𝑒𝑇

𝑖𝑄𝑒

𝑖+ 𝑢

𝑇

𝑖𝑅𝑢

𝑖) , (19)

where 𝑄 and 𝑅 are two positive definite weighting matricesto regulate the weight of steering angle correction and directyaw moment. In this paper,𝑄 and 𝑅 are selected as constantsfor controller design.

The 𝐻∞-based LQR tracking controller is obtained by

finding the optimized state-feedback gain 𝐾(𝑢𝑘= 𝐾𝜉

𝑘) to

minimize index 𝐽, which is also equal to the 2-norm of thefollowing constructed signal:

𝑧𝑘= 𝐸𝜉

𝑘+ 𝐹𝑢

𝑘, (20)

where

𝐸 = [0 𝑄

1/2

0 ⋅ ⋅ ⋅ 0

0 0 0 ⋅ ⋅ ⋅ 0

] , 𝐹 = [

0

𝑅1/2] . (21)

Considering𝑤 = 𝛿𝑓is bounded in 𝑙

2space [11], we introduce

an 𝐻∞

performance index 𝜂 such that 𝐽 < 𝜂2 ‖𝑤‖2. Then,the optimization problem instead that an optimal𝐻

∞control

problem for the following system:

𝜉𝑘+1

= 𝐴aug𝜉𝑘 + 𝐵𝑤𝑟,aug𝑤𝑘 + 𝐵𝑢,aug𝑢𝑘,

𝑧𝑘= 𝐸𝜉

𝑘+ 𝐹𝑢

𝑘.

(22)

With the state-feedback control, the closed-loop system is

𝜉𝑘+1

= (𝐴aug + 𝐵𝑢,aug𝐾) 𝜉𝑘 + 𝐵𝑤𝑟,aug𝑤𝑘,

𝑧𝑘= (𝐸 + 𝐹𝐾) 𝜉

𝑘,

(23)

and inequality is as follows:

‖𝑧‖2< 𝜂 ‖𝑤‖

2. (24)

From (24), we can get an optimized control gain 𝐾, and thefollowing is achievable:

[

[

[

[

[

[

[

−Ω 0 (𝐴aug,𝑖 + 𝐵𝑢,aug,𝑖𝐾)𝑀 𝐵𝑤𝑟,aug

∗ −𝐼 (𝐸 + 𝐹𝐾)𝑀 0

∗ ∗ Ω −𝑀 −𝑀𝑇

0

∗ ∗ ∗ −𝜂2

𝐼

]

]

]

]

]

]

]

< 0. (25)

For 𝑄 and 𝑅, when 𝜂 is small, the controlled output is small,and vice versa. Therefore, the controller design can be finallyexpressed as

𝑚

Ω,𝑀,𝑌,𝜂

𝜂2

s.t.[

[

[

[

−Ω 0 𝐴aug,𝑖𝑀+ 𝐵𝑢,aug,𝑖𝑌 𝐵𝑤𝑟,aug∗ −𝐼 𝐸𝑀 + 𝐹𝑌 0

∗ ∗ Ω −𝑀 −𝑀𝑇

0

∗ ∗ ∗ −𝜂2

𝐼

]

]

]

]

< 0

∀𝑖 = 1, 2, . . . (ℎ + 1)Υ+1

.

(26)

Equation (26) is a typical minimization problem of alinear objective function with constraints of LMIs and canbe solved with the LMI Toolbox in MATLAB. Therefore, theminimization under constraints is solved, and a controlleragainst random time-varying delays is proposed. Then thecontroller is obtained by 𝐾 = 𝑌𝑀−1, where 𝐾 is a fixed gainmatrix which can be calculated offline. The PID controllerrequires the integral and derivative of the error signal.

4. Simulation and Interpretation of Results

The vehicle model parameter values are listed in Table 1.As is shown in Figure 3, take a PID controller for an

example for comparison. Sample time Ts is 10ms which is thesample period of the closed-loop system. In the simulations,choose the longitudinal vehicle velocity as 80 km/h and thetire-road friction coeffient as 0.85 in all the maneuvers.


Steering angle

C

Controller clock Side slip angle

Yaw rate

ax

ay

FL v

FR v

RL v

RR v

Vx

Side slip angle

Yaw rate

ax

ay

FL v

FR v

RL v

RR v

Vx

4WD EV controller

u1

u2

u3

u4

Random time-varying delay

Input1

Input2

Input3

Input4

Output1

Output2

Output3

Output4

Network In-wheel motor

FL torque acquired

FR torque acquiredOutput torque

RL torque acquired

RR torque acquired

CarSim S-functionVehicle code: i i

Figure 3: Simulation diagram constructed by CarSim.

Table 1: Main simulation parameters of vehicle.

Vehicle quality𝑚 1296 (unit)Distance of centroid and front axle 𝑙

𝑓1.25

Distance of centroid and rear axle 𝑙𝑟

1.32Distance between two front wheels 𝐵

𝑓1.405

Distance between two rear wheels 𝐵𝑟

1.399Vertical height of vehicle centroid to ground ℎ

𝜀0.415

Tire radius 𝑟 0.1651Rotational inertia of vehicle spin on 𝑍 axle 𝐼

𝑧1750

0 500 1000 1500 2000 25000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Message

Dela

y tim

e (m

s)

Figure 4: Random time-varying delays series.

4.1. Straight Maneuver. Random time-varying delays areshown in Figure 4. The max delay is 5ms, and the min delayis 0ms. It means the delay time of each message.

As is shown in Figure 5, reference velocity of 4WIDEV is80 km/h. Under the condition without delays, the responsetime of the PID controller and the 𝐻

∞controller is about 1

second.The overshoot of system is zero. Both controllers havegood control performance.

0 1 2 3 4 5 6Time (s)

Velocity with H∞ controllerVelocity with PID controllerReference velocity

Velo

city

(km

/h)

0

20

40

60

80

100

Figure 5: Velocity of 4WIDEVwithout delays in straight maneuver.

0 1 2 3 4 5 6Time (s)


Velo

city

(km

/h)

0

20

40

60

80

100

Figure 6: Velocity of 4WIDEV with random time-varying delays instraight maneuver.


0 2 4 6 8 10 12−0.12

−0.10

−0.08

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.10

Yaw

rate

(rad

/s)

Time (s)


(a) Without delay

0

0 2 4 6 8 10 12

−0.10

−0.12

−0.08

−0.04

−0.02

0.02

0.04

0.06

0.08

0.10

Yaw

rate

(rad

/s)

Time (s)


(b) With random delays

Figure 7: Vehicle yaw rate response in curve steering maneuver.

Velocity with H∞ controllerVelocity with PID controller

0 10 20 30 40 50 600

10

20

30

40

50

60

70

80

90

100

110

Longitudinal distance (m)

Late

ral d

istan

ce (m

)

(a) Without delay


0 10 20 30 40 50 600

10

20

30

40

50

60

70

80

90

100

110


Late

ral d

istan

ce (m

)

(b) With random time-varying delays

Figure 8: Vehicle CG trajectory in curve steering maneuver.

As is shown in Figure 6, reference velocity of 4WIDEVis 80 km/h. Under the condition with random time-varyingdelays, the PID controller significantly oscillates, whereasthe proposed controller demonstrates good robustness. Theresponse time of the proposed controller is about 2 seconds.

Figure 7 shows the simulation results of vehicle yawrate response in the curve steering maneuver. Without timedelays, both the PID controller and the proposed controllergive satisfactory results, and the proposed controller per-forms better without a steady-state tracking error. However,with the random time-varying delays, the PID controlleryields oscillations in the transient process and error in steadystate, whereas the proposed controller can still track thedesired yaw rate, as well as what it does under the idealnetwork condition.

In addition, the effect of the delays is hence significantenough to influence the vehicle global trajectory with the PIDcontroller. Comparatively, for the proposed controller, thetrajectory is just as the driver’s expectation, which is displayedin Figure 8.

4.2. Curve Steering Maneuver. As is shown in Figure 9, refer-ence velocity of 4WIDEV is 80 km/h. Under the conditionwithout delays, the response time of the PID controllerand the 𝐻

∞controller is about 1.5 seconds. The maximum

overshoot of system is less than 2%. Both controllers havegood control performance.

As is shown in Figure 10, reference velocity of 4WIDEVis 80 km/h. Under the condition with random time-varyingdelays, the PID controller significantly oscillates, whereas


0 1 2 3 4 5 6Time (s)


Velo

city

(km

/h)

0

20

40

60

80

100

Figure 9: Velocity of 4WIDEV without delays in curve steeringmaneuver.

0 1 2 3 4 5 6Time (s)


Velo

city

(km

/h)

0

20

40

60

80

100

Figure 10: Velocity of 4WIDEV with random time-varying delaysin curve steering maneuver.

the proposed controller demonstrates good robustness. Theresponse time of the proposed controller is about 2 seconds.

Figure 11 shows the simulation results of vehicle yawrate response in the curve steering maneuver. Without timedelays, both the PID controller and the proposed controllergive satisfactory results, and the proposed controller per-forms better without a steady-state tracking error. However,with the random time-varying delays, the PID controlleryields oscillations in the transient process and error in steadystate, whereas the proposed controller can still track thedesired yaw rate, as well as what it does under the idealnetwork condition.

In addition, the effect of the delays is hence significantenough to influence the vehicle global trajectory with thePID controller. Comparatively, for the proposed controller,

0 2 4 6 8 10 12−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Time (s)


Yaw

rate

(rad

/s)

(a) Without delay

0 2 4 6 8 10 12−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Time (s)


Yaw

rate

(rad

/s)


Figure 11: Vehicle yaw rate response in curve steering maneuver.

the trajectory is just as the driver’s expectation, which isdisplayed in Figure 12.

4.3. Lane-Changing Maneuver. Figure 13 shows the simula-tion results of vehicle speed response in single lane-changingmaneuver. Under the condition without delays, the responsetime of the PID controller and the proposed controller isabout 1.5 seconds. The maximum overshoot of system is lessthan 3.5%. Both controllers have good control performance.There are some drops in the response curve, which is becauseof deceleration of the vehicle when it turns a corner.

As is shown in Figure 14, reference velocity of 4WIDEVis 80 km/h. Under the condition with random time-varyingdelays, the PID controller significantly oscillates, whereas theproposed controller demonstrates good robustness.

Figure 15 shows the simulation results of the vehicleyaw rate under lane-changing maneuver. Under the ideal



0 10 20 30 40 50 600

10

20

30

40

50

60

70

80

90

100

110


Late

ral d

istan

ce (m

)

(a) Without delay


0 10 20 30 40 50 600

10

20

30

40

50

60

70

80

90

100

110


Late

ral d

istan

ce (m

)


Figure 12: Vehicle CG trajectory in curve steering maneuver.

0 1 2 3 4 5 6Time (s)


Velo

city

(km

/h)

0

20

40

60

80

100

Figure 13: Velocity of 4WIDEV without delays in lane-changingmaneuver.

network condition without delays, both the PID controllerand the proposed controller can track the desired yaw ratewell. Nevertheless, with the random time-varying delays,there are significant oscillations in the vehicle yaw rate withthe PID controller, and the oscillations still exist when thesteering wheel angle returns to zero in the last 1 s, whichindicates that the vehicle system is almost at the criticality ofinstability, while, for the proposed controller, there is almostno negative influence on the tracking performance evenwhenthe random network delays are introduced.

In addition, the effect of the delays is hence significantenough to influence the vehicle global trajectory with the PIDcontroller. Comparatively, for the proposed controller, thetrajectory is just as the driver’s expectation, which is displayedin Figure 16.

0 1 2 3 4 5 6Time (s)


Velo

city

(km

/h)

0

20

40

60

80

100

Figure 14: Velocity of 4WIDEV with delays in lane-changingmaneuver.

5. Conclusion

In this paper, a yawingmoment control method of 4WIDEVsto restrain random time-varying delays is proposed. An𝐻

∞-

based LQR tracking controller is introduced and adoptedin the control system against random time-varying delayseffectively. Meanwhile, the original system with the terms isinduced by random time-varying delays and then describesthe randomness as polynomial. Two simulation maneuversare carried out on an EV model constructed by CarSim toverify the performance of the proposed controller. Simulationresults show that the𝐻

∞controller not only achieves a good

control effect under the network conditionwithout delays butalso guarantees enough robustness and performance whenthere are network-induced random time-varying delays in


−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0 2 4 6 8 10 12

Time (s)


Yaw

rate

(rad

/s)

(a) Without delay

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0 2 4 6 8 10 12

Time (s)


Yaw

rate

(rad

/s)


Figure 15: Vehicle yaw rate response in lane-changing maneuver.

0 10 20 30 40 50 60 70 80 90 100

4

3

2

1

0

−1

−2

−3

−4


Lateral distance (m)

Long

itudi

nal d

istan

ce (m

)

(a) Without delay

0 10 20 30 40 50 60 70 80 90 100

4

3

2

1

0

−1

−2

−3

−4


Lateral distance (m)

Long

itudi

nal d

istan

ce (m

)


Figure 16: Vehicle CG trajectory in lane-changing maneuver.

the closed-control loop as well. Comparisons with a PIDcontroller without delays further evidenced the effectivenessof the proposed controller.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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