IEEE TRANSACTIONS ON TRANSPORTATION ELECTRIFICATION, VOL. 10, NO. 2, JUNE 2024
3991
A Safety Requirements’ Adaptive NMPC Strategy
for Electric Vehicle Stability Control With
Computationally Efficient Optimization
Hanghang Liu , Lin Zhang , Shen Li , Member, IEEE, Rongjie Yu, Guofa Li , and Hong Chen , Fellow, IEEE
Abstract— Handling and stability are vital for vehicle safety,
especially under extreme conditions, such as low friction surfaces,
violent steering, and urgent acceleration/deceleration. Electric
vehicles (EVs) are a promising way to improve the stability
due to their rapid and accurate responses. However, vehicle
states are influenced by highly coupled and nonlinear dynamics.
The safety requirements are different under various conditions.
To solve the above problems, a nonlinear model predictive control
(NMPC)-based strategy is proposed for stability control. First, a
3-D stability space, which considers yaw, lateral, and longitudinal
motions, is proposed to analyze the degree of vehicle stability
under different conditions. Then, an adaptive control strategy
is proposed to meet various safety requirements. Moreover,
to improve the control performance under extreme conditions,
a nonlinear vehicle dynamic model is adopted to predict the
future states. To enable vehicular applications with low-cost hardware, a Pontryagins minimum principle (PMP)-based solving
method is proposed for computationally efficient online optimization. Finally, hardware-in-loop experiments are conducted to
check the effectiveness and superiority of the proposed method.
The experimental results show that the proposed strategy has
a better performance improving vehicle handling and stability
under extreme conditions.
Index Terms— Active safety control, computationally efficient
optimization, electric vehicles (EVs), nonlinear model predictive
control (NMPC), stability spaces.
I. I NTRODUCTION
HE increasing attention given to electric vehicles (EVs)
has encouraged the development of related technologies, e.g., energy consumption, active safety, and autonomous
driving. More than 1.35 million people lose their lives
T
Manuscript received 24 June 2023; revised 8 August 2023;
accepted 3 September 2023. Date of publication 6 September 2023;
date of current version 18 June 2024. This work was supported in part by
the National Natural Science Foundation of China under Grant 62003238,
Grant 52372393, and Grant 62073152; in part by the Dongfeng Technology
Center (Research and Application of Next-Generation Low-Carbon Intelligent
Architecture Technology); and in part by the Shanghai Municipal Science and
Technology Major Project under Grant 2021SHZDZX0100. (Corresponding
author: Lin Zhang.)
Hanghang Liu and Hong Chen are with the College of Electronics and
Information Engineering, Tongji University, Shanghai 201804, China (e-mail:
Hanghang_L@163.com; chenhong2019@tongji.edu.cn).
Lin Zhang is with the School of Automotive Studies, Tongji University,
Shanghai 201804, China (e-mail: zhanglin_jlu@foxmail.com).
Shen Li is with the School of Civil Engineering, Tsinghua University,
Beijing 100084, China (e-mail: sli299@tsinghua.edu.cn).
Rongjie Yu is with the College of Transportation Engineering, Tongji
University, Shanghai 201804, China (e-mail: yurongjie@tongji.edu.cn).
Guofa Li is with the College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing 400044, China (e-mail: liguofa@
cqu.edu.cn).
Digital Object Identifier 10.1109/TTE.2023.3312397
in traffic accidents worldwide each year [1], and safety
issues are receiving increasing attention. Benefiting from
the flexible, rapid, and high-accuracy demand response, distributed drive EVs (DDEVs) have been widely adopted for
advanced active safety control technologies [2]. Based on
the above advantages of DDEVs, many studies have been
conducted to fully exploit their potential in active safety
control.
In [3], an analytical sliding mode control (SMC) method
was proposed to improve handling stability under high friction
road surface. For a low friction surface, a model predictive
control (MPC)-based strategy was proposed, in which both
yaw rate tracking and sideslip suppression were considered
in [4]. The above studies were evaluated by experiments
with real vehicles, and the results showed the superiority of
DDEVs compared with traditional vehicles. However, tire slip
is not considered, which is also important for stability control,
especially under some extreme conditions [5]. To meet the
requirements of handling, stability and anti-skid, a hierarchical
control strategy with double layers was proposed in [6].
In the upper layer, an MPC-based controller was proposed
to generate the desired tire slip ratios, and then a lower layer
PID controller was used to track them. However, the system
constraints cannot be considered in PID controller. To address
this problem, an integrated control strategy was proposed
in [7], where a 7-degree-of-freedom (DoF) vehicle model was
adopted.
In the aforementioned studies, the control requirements
are fixed in advance and remain unchanged throughout the
whole process. As presented in [8], the control requirements
should be adjusted in real-time according to current vehicle
states, and then the potentials can be fully exploited. As a
result, the concept of the stability region is introduced to
evaluate the stability degree of vehicles. The phase plane
(β − β̇, β −γ ) is a common way to describe whether a vehicle
is stable [9], [10], [11]. The phase plane is used to divide the
control regions, safety constraints, and weighting adjustment.
However, inaccurate information (e.g., model simplification
and uncertainties, road friction coefficient errors) will have
a great impact on the stability boundary [12]. Thus, this kind
of offline method cannot provide a reliable real-time stability
state. To address this problem, the envelope method is an
available option, which has been successfully adopted in [13].
A friction limit-based envelope was proposed in [14], where
the yaw rate and sideslip angle limits are used to establish
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an envelope boundary. This envelope can be updated in realtime. However, the friction limit-based envelope is a relatively
rough boundary, which cannot fully reflect the system dynamics in a theoretical way. To solve this problem, a method
based on theoretical analysis was proposed in [15] and [16].
In this method, the yaw rate and lateral speed were chosen
as indictors of the stability boundary. The envelope boundary
was portrayed based on the system state space equation and
Routh stability criterion. However, only the lateral and yaw
motions are considered, and the longitudinal speed is assumed
to be constant. In practice, not all the tire forces can be
used to generate lateral forces, especially under some extreme
conditions (e.g., urgent acceleration and deceleration). The
longitudinal motion must be considered for a more accurate
envelope boundary. As a result, the first target in this study
proposes a more advanced evaluation method to evaluate the
stability degree of vehicles.
Based on the evaluated stability states, some adaptive
control strategies were proposed. In [11], an adaptive SMC
controller was proposed, where the weighting factor can be
adjusted in real-time according to current stability states.
In practice, a vehicle is a complicated nonlinear system.
The stability, handling, and system constraints should be
considered. To address this problem, MPC is a promising
method and has been widely used in vehicle safety control
for solving multiple objectives and constrained problems [17].
It is widely recognized that too high of a computational cost is
one of the main reasons preventing MPC from being adopted
in many fields. To satisfy the real-time performance, linear
MPC (LMPC) is a common method for vehicular applications [7], [8], [18]. For nonlinear MPC (NMPC) strategies,
some methods can be used to overcome the above disadvantage. One of the methods is to solve the optimization problems
offline and store the control law into the memory [19], [20].
During the run time, the control law will be evaluated by the
lookup table. This method is called as explicit MPC (eMPC).
The main drawbacks of eMPC are that the offline optimization
is very computationally expensive for long prediction horizon
and time-varying objective function, and storing and online
evaluating the lookup tables are challenging for real-time
applications [21]. Another way is to simplify the original
optimization problem and solve it with a commercial toolkit
(FORCEs Pro, etc.) [22]. However, not everyone has access to
these toolkits due to price, copyright, or other reasons. Thus,
a computationally efficiency solution method is also necessary
to satisfy the requirement of real-time online optimization for
vehicular applications.
To address the aforementioned problems, this study proposes a novel stability space, where the longitudinal,
lateral, and yaw motion are taken into account. Then,
the safety requirements are analyzed and combined with
NMPC to design an adaptive control strategy. To improve
the computational efficiency, a Pontryagins minimum principle (PMP)-based solution method is also developed. The
main contributions of this article can be summarized as
follows.
1) To accurately capture the nonlinear characteristics under
extreme conditions, a 3-D stability space that considers
longitudinal, lateral, and yaw motions is proposed based
on the nonlinear vehicle dynamic model. Then, the
stability space is divided into stable, critically stable,
and unstable subspaces according to the different tire
states. The stability space is updated in real-time with
the current vehicle states.
2) Based on the stability space, the stability degree of
the vehicle is evaluated in real-time. Then, safety
requirements under different vehicle states are analyzed.
To capture the nonlinear and coupled relationship and
satisfy the time-varying safety requirements, an adaptive
NMPC strategy is proposed.
3) In the upper layer controller, a PMP-based, indirect solution method is proposed to improve the computational
efficiency of the nonlinear optimization problem. With
the proposed method, the NMPC strategy can be used
on low-cost hardware. In the lower layer controller,
the slack variables are introduced to soften the system constraints, which can improve the computational
efficiency.
The remainder of this article is organized as follows:
In Section II, the 3-D stability space and subspaces division are introduced. In Section III, the design of the upper
layer controller and lower layer controllers is provided.
In Section IV, the proposed PMP-based, indirect solution
method is illustrated, and the effectiveness is checked by
simulations. In Section V, the hardware-in-the-loop (HIL)
experimental results are provided to show the feasibility
of the proposed control method. We conclude our research
in Section VI.
II. S TABILITY S PACE E STIMATION AND S AFETY
R EQUIREMENTS ’ A NALYSIS
In this section, the 3-D stability space is estimated. For high
accuracy under extreme conditions, the longitudinal motion is
considered. To reflect the nonlinear characteristics, nonlinear
vehicle and tire models are adopted. Then, to avoid the shaking
problem between the stable and unstable spaces [23], the
stability space is divided into stable, critically stable, and
unstable subspaces. All the symbols used in this study and
their physical meanings are listed in Table I. In this article,
the vehicle parameters are consistent with a production EV
(DongFeng E70), and details of this vehicle can be found
in [4]. The subscripts 1, 2, 3, and 4 represent the front left
wheel, front right wheel, rear left wheel, and rear right wheel,
respectively.
A. Related Models
A 2-DoF vehicle model considering lateral and yaw motion
is adopted to depict the handling and stability characteristics.
In some literatures, lateral and yaw motions are considered in the same dimension. In fact, the lateral speed of
the vehicle represents the lateral motion along the y-axis,
and the yaw rate represents the rotary motion about the
z-axis. They are essentially different state variables of the
vehicle. In this article, the lateral speed and yaw rate
must be treated as two independent variables. According to
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LIU et al.: SAFETY REQUIREMENTS’ ADAPTIVE NMPC STRATEGY FOR ELECTRIC VEHICLE STABILITY CONTROL
3993
where vr x = Re ω − Vx and vr y = −Vx α are defined as
the longitudinal and lateral slip, respectively. The detailed
calculation process of the sideslip angles of the tires can be
found in [4] and is not given in this study for brevity. The
total slip vector is defined by vr = [vr x , vr y ]T , and the norm
of total slip can be written as follows:
q
∥vr ∥ = vr2x + vr2y .
(3)
TABLE I
V EHICLE PARAMETERS AND M ODEL VARIABLES
The Stribeck function can be written as follows:
s
!
∥vr ∥
g(vr ) = C2 + (C1 − C2 ) exp −
.
C3
(4)
As discussed above, the longitudinal acceleration can also
have a great impact on the tire forces. According to the tire
model in (3), the longitudinal tire forces are mainly influenced
by longitudinal slips. If the longitudinal slips and vertical load
transfers are directly introduced as independent variables, the
stability space will be a 6-D hyperspace. Then, it will be
impossible to estimate the stability space. To avoid the curse of
dimensionality, the longitudinal acceleration A x is chosen to
represent the longitudinal motions and vertical load transfers.
Then, the stability space can be described as a 3-D space with
Vy , γ , and A x .
Based on the above consideration, the original tire model
must be modified. As presented in [25], the decaying effect
of longitudinal forces on lateral forces is approximately
parabolic. Then, a modified tire model with longitudinal tire
forces is introduced as follows:
Fx
(5)
F̃ y = Fy vr x , vr y 1 − ρ
µFz
vehicle dynamics, the vehicle model can be written as
follows:
V̇ y = −γ Vx +
γ̇ =
1
((Fy1 + Fy2 ) cos δ f + Fy3 + Fy4 )
m
1
(L f (Fy1 + Fy2 ) cos δ f − L r (Fy3 + Fy4 )
Iz
d
+ (Fy1 − Fy2 ) sin δ f ).
2
(1a)
(1b)
It should be noted that the longitudinal speed is not considered
in the above 2 DoF models. In practice, the longitudinal
tire forces and longitudinal acceleration are closely related.
Longitudinal acceleration is a critical factor for stability
space estimation and must be considered. Under extreme
conditions, the tire force is highly nonlinear about longitudinal and lateral slips. Lateral and longitudinal tire forces
are also highly coupled. Based on the above considerations,
a nonlinear combined-slip LuGre tire model is adopted.
As presented in [24], the LuGre tire model can be described as
follows:
!
σ0
Fx vr x , vr y = σ0 ∥vr ∥
+ σ1 vr x Fz
(2a)
+ σ2 Re |ω|
µ·g(vr )
!
σ0
Fy vr x , vr y = σ0 ∥vr ∥
+ σ1 vr y Fz
(2b)
+ σ2 Re |ω|
µ·g(vr )
where ρ = c1 µ + c2 is the coefficient of attenuation, which
is a function of the friction coefficient µ, and c1 and c2 are
parameters. When using this model, the longitudinal slip is
assumed to be zero, and the normal lateral fire force can be
derived by the tire model in (2). Then, the longitudinal tire
force can be derived by Fx = m A x /4. Substituting the normal
lateral fire force and longitudinal tire force into (5), a lateral
tire force under composite conditions can also be derived.
The longitudinal and lateral tire forces under composite
conditions are given in Fig. 1. The modified model has a
satisfied accuracy and consistency with the original LuGre tire
model. When the longitudinal force is zero (i.e., A x = 0), the
lateral force of the modified model equals that of the LuGre
model.
B. Stability Space Estimation
The vehicle model in (1) is a highly nonlinear model.
For nonlinear systems, it is very difficult and complicated to
determine whether the system is stable. In the classical control
theory, the Routh stability criterion is a common method for
stability judgment and has been widely applied. To apply the
Routh method, the vehicle model should be linearized by
Taylor expansion [15], [16]. Combining (1), (2), and (5), the
system is defined as follows:
T
V̇ y 1 = f Vy , γ , δ f .
(6)
γ̇
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IEEE TRANSACTIONS ON TRANSPORTATION ELECTRIFICATION, VOL. 10, NO. 2, JUNE 2024
Fig. 2.
Fig. 1.
Tire forces of LuGre and the modified models. (a) µ = 0.35.
(b) µ = 0.85.
The symbol o represents the subscript of the current equilibrium point. At the current equilibrium point (Vyo , γo ), and
the driver’s front wheel steering angle δ f , the system can be
written as follows:
V̇ y 1 V̇ yo
1V̇ y
=
+
γ̇ o
γ̇
1γ̇
T
≈ f Vyo , γo , δ f o
+
+
∂f
∂f
1Vy +
1γ
∂ Vy (Vyo ,γo )
∂γ (Vyo ,γo )
∂f
1δ f .
∂δ f (Vyo ,γo )
(7)
Ignoring the higher order terms, the linearized system model
can be written as follows:
1V̇ y
1Vy
= Ao
+ Bo 1δ f
(8)
1ṙ
1r
where the matrices are defined as follows:
Ao11 Ao12
Bo1
Ao =
, Bo =
.
Ao21 Ao22
Bo2
(9)
The detailed definitions of the matrices are listed in the
Appendix. Based on the above definitions, the characteristic
polynomial of Jacobian matrix Ao can be written as
λ − Ao11
−Ao12
λ I − Ao =
−Ao21
λ − Ao22
= λ 2 + p1 λ + p0 = 0
(10)
where
p1 = −Ao11 − Ao22 ,
p0 = Ao11 Ao22 − Ao12 Ao21 .
According to the Routh stability criterion, when the roots
Diagram of stability space.
of the characteristic polynomial have a negative real part
(i.e., p1 > 0 and p2 > 0), the system is stable. To response
to the driver’s steering demand, the system should also be
controllable. According to the control theory, when the rank
of matrix P = [Bo , Ao Bo ] equals the dimension of the
system, it is controllable. In this study, when the system is
stable and controllable, the corresponding equilibrium point
is inside the stability space.
Based on the system equations in (1), (2), and (5), the stability space can be derived by checking whether the space points
satisfy the stable and controllable criteria. As shown in Fig. 2,
a 3-D stability space with friction coefficient µ = 0.85 and
speed Vx = 60 km/h is given as an example. Where a and c
represent the oversteering surfaces (stable criterion), and
b and d represent the understeering surfaces (controllable criterion). By the diagram of stability space, when the vehicle is
in acceleration and deceleration conditions, the corresponding
stability space will shrink with the amplitude of acceleration/deceleration. Vehicle acceleration/deceleration is achieved
by the longitudinal tire forces. As shown in Fig. 1, with the
increase in longitudinal tire forces, the lateral tire forces will
undergo a considerable decay. Then, the insufficient lateral
forces will further lead to a contraction of the stability space.
In addition, the load transfer will also impact the stability
space during acceleration/deceleration. For the deceleration,
a part of rear axis load will transfer to the front axis. Then,
the available rear tire forces will be reduced with a rear axis
load transfer. Under this condition, the lateral load capacity
of rear tires will be reduced, and vehicle will easily lose
stability. For the stability space, the oversteering surfaces will
shrink more when the vehicle is in deceleration. A similar
analysis can be adopted for the case of acceleration. The
vehicle easily loses maneuverability in acceleration conditions,
and the understeering surfaces shrink more when the vehicle is
in acceleration.
It should be noted that the stability space and lateral tire
forces are closely related. A diagram of lateral tire forces
under different vertical loads is shown in Fig. 3. The tire force
curve can be divided into three parts: linear area (green area),
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LIU et al.: SAFETY REQUIREMENTS’ ADAPTIVE NMPC STRATEGY FOR ELECTRIC VEHICLE STABILITY CONTROL
3995
TABLE II
S AFETY R EQUIREMENTS ’ A NALYSIS
Fig. 3.
Diagram of lateral tire forces under different vertical loads.
located in a stable subspace. When ξ ∈ [−1, 0], the vehicle
states are located in the critical stable subspace. The value
of ξ increases as the vehicle states move toward the center
of the stability space. When ξ < −1, the vehicle states are
located outside of the stability space. The minimal value of ξ
is limited to no less than −2 for clarity.
C. Safety Requirements’ Analysis
Fig. 4.
Diagram of stability index.
nonlinear area (yellow area), and saturated area (red area).
In the linear area, the lateral force varies linearly with the
tire slip angle. As the slip angle continues to increase, the
lateral force transitions to the nonlinear area, and the slope of
the lateral force curve declines substantially. The black lines
in Fig. 3 are used to show the changes in slope. In the linear
area, the slope can be treated as a constant. In the nonlinear
area, a significant decline of slope can be observed. If the slip
angle increases further, it will enter the saturated area, and the
stability will be lost. Ideally, the lateral forces should be kept
within a linear area to guarantee stability. Under some extreme
conditions, a nonlinear area is allowed to fully explore the
vehicle potentials. The saturated area should be strictly avoided
under any condition. With the above conditions, the stability
space can be divided into a stable subspace (2in ), a critical
stable subspace (2mid ) and an unstable subspace (2out ). In the
stable subspace, the tire force is maintained in the linear area
(green area in Fig. 3), and the stability can be hold in a high
level. In the critical stable subspace, the tire force enters the
nonlinear area (yellow areas in Fig. 3), and the stability is
weakened due to the tire force attenuation. Once the tire force
enters saturated area (red areas in Fig. 3), the stability will
be lost, and the corresponding subspace is defined as unstable
subspace.
As discussed above, the stability space is divided into three
subspaces. To depict the relationship between vehicle states
and stability subspaces, a stability index ξ is introduced.
As shown in Fig. 4, when ξ ∈ [0, 1], the vehicle states are
Based on the stability space introduced in Section II-B, the
safety requirements of different stability subspaces can be analyzed. In this study, the safety requirements can be summarized
as stability, maneuverability, and anti-skid. According to the
current vehicle states (Vy , γ , A x ), the corresponding stability
subspace can located. The safety requirements’ analysis is
listed in Table II. The deceleration condition is taken as
an example to analyze the safety requirements in different
stability subspaces as follows.
1) If the vehicle states (Vy , γ , A x ) ∈ 2in , it is considered
to be in a stable subspace.
2) If the vehicle states (Vy , γ , A x ) ∈
/ 2in and (Vy , γ , A x ) ∈
2mid , it is considered to be in a critical stable subspace.
3) If the vehicle states (Vy , γ , A x ) ∈
/ 2mid and
(Vy , γ , A x ) ∈ 2out , it is considered to be in an stable
subspace.
When the vehicle is located in 2in , the maneuverability
is mainly considered, and the stability and anti-skid are
also considered. For subspace 2mid , the vehicle tends to be
oversteering, and then the stability and rear tires anti-skiding
should be prioritized. For subspace 2out , stability has the
highest priority over other performance indices. In practice,
the space surfaces are updated in real-time to match the
time-varying vehicle states.
III. S AFETY R EQUIREMENTS ’ A DAPTIVE
C ONTROL S TRATEGY D ESIGN
In this section, a safety requirement adaptive control strategy is proposed for stability control. The diagram of the
proposed control system is shown in Fig. 5. The reference
generation block (green block) is used to generate the desired
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3996
Fig. 5.
IEEE TRANSACTIONS ON TRANSPORTATION ELECTRIFICATION, VOL. 10, NO. 2, JUNE 2024
Control scheme of the NMPC strategy.
V̇ y = − γ Vx +
γ̇ =
1
((Fy1 + Fy2 ) cos δ f + (Fx1 + Fx2 ) sin δ f
m
+ Fy3 + Fy4 )
(11b)
1
L f (Fx1 + Fx2 ) sin δ f + L f (Fy1 + Fy2 ) cos δ f
Iz
d
− L r (Fy3 + Fy4 ) + (Fx2 − Fx1 ) cos δ f
2
d
d
+ (Fx4 − Fx3 ) + (Fy1 − Fy2 ) sin δ f .
2
2
(11c)
For the vehicle wheels, a rotational dynamic model including tire forces and motor torques is adopted. The rotational
motions of vehicle wheels ωi (i = 1, 2, 3, 4) can be written as
follows:
1
ω̇i =
(−Fxi Re + Ti )
(12)
Iw
Fig. 6.
Three-DoF vehicle model.
yaw rate based a second-order vehicle model. The stability
state will be determined in the vehicle state determination
block, and then the stability index will be provided. Based
on the stability states and index, the safety requirements are
analyzed, and the weighting factors and constraint conditions
are adjusted accordingly. Then, an NMPC controller with a
computationally efficient optimization algorithm is proposed in
the upper layer. The desired tire slip ratios will be generated to
realize the control targets. To track the desired tire slip ratios,
an LMPC controller is proposed in the lower layer, and the
generated motor torques are executed by the vehicle.
A. Vehicle and Wheel Dynamic Models
As shown in Fig. 6, a DDEV is adopted in this study. For the
vehicle body, a 3-DoF double track model is adopted. In this
model, the longitudinal, lateral, and yaw motions are taken
into consideration. The equations of vehicle dynamics can be
written as follows:
1
V̇ x = γ Vy + ((Fx1 + Fx2 ) cos δ f − (Fy1 + Fy2 ) sin δ f
m
+ Fx3 + Fx4 )
(11a)
where the total torque Ti = Td + 1Ti is the sum of the
driver’s demand and additional torque, and the longitudinal
tire forces Fxi can be estimated by the tire model in (2). Then,
the tire slip ratios can be derived as κi = (ωi Re − Vxi )/Vxi ,
and Vxi (i = 1, 2, 3, 4) represents the different longitudinal
tire speed. It should be noted that the longitudinal speed of
tires is different when the vehicle is cornering. To improve the
accuracy of the slip ratios, the longitudinal speed of different
tires can be derived as follows:
s
2
2L f
d
Vx1 = Vx − γ L 2f +
cos tan−1
− δf
(13a)
2
d
s
2
d
−1 2L f
2
Vx2 = Vx + γ L f +
cos tan
− δf
(13b)
2
d
s
2
d
−1 2L r
2
Vx3 = Vx − γ L r +
cos tan
(13c)
2
d
s
2
d
2L r
cos tan−1
.
(13d)
Vx4 = Vx + γ L r2 +
2
d
B. Controller Design
1) Upper Layer Controller: In the upper layer controller,
the state vector is defined as x = [x1 , x2 , x3 ]T = [Vx , Vy , γ ]T ,
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LIU et al.: SAFETY REQUIREMENTS’ ADAPTIVE NMPC STRATEGY FOR ELECTRIC VEHICLE STABILITY CONTROL
and the control input is defined as u = [u 1 , u 2 , u 3 , u 4 ]T =
[κ˜1 , κ˜2 , κ˜3 , κ˜4 ]T . The κ̃ represents the desired tire slip ratio,
which is treated as the virtual control input in this system.
Combining the vehicle dynamic model in (11), the state-space
equation can be written as ẋ = f (x, u). To facilitate the
controller design, the continuous time model should be transformed into a discrete model. Let the sample time be Ts ,
combining the Euler method, the discrete model can be
written as x(k + 1) = Ts f (x(k), u(k)) + x(k). Although
there is some simplification in the discretization process, the
main nonlinear characteristics are retained. In the upper layer
controller, the main targets are maneuverability, stability, and
slip suppression. In addition, the variations in the desired slip
ratios should also be considered to avoid torque oscillation.
Then, the objective items at time instant ki (k+1 ≤ ki ≤ k+ N )
can be defined as follows:
L 1 (ki ) = [x3 (ki ) − γref ]2
L 2 (ki ) = [x2 (ki ) − Vy,ref ]
(14b)
L 3 (ki ) = u(ki − 1)2
(14c)
L 4 (ki ) = 1u(ki − 1)
2
(14d)
where N is the prediction horizon of the upper layer controller,
γref is the desired yaw rate, and Vy,ref is the desired lateral
speed. In addition, the system constraints should also be
considered. For the safety constraint, the lateral speed should
be kept within Vy,low < Vy < Vy,up to satisfy the friction
limit [22]. For the control input, the constraint u low < u < u up
should be satisfied. Combining the objective items and constraints, the nonlinear optimization problem of the upper layer
controller can be written as follows:
min L =
k+N
X
variables ρ̄ = [ρ̄ 1 , ρ̄ 2 , ρ̄ 3 , ρ̄ 4 ]T are introduced to allow a
small range limit breakout. The slack variables should be
minimized as much as possible. Then, the objective items of
slack variables in time instant ki (k + 1 ≤ ki ≤ k + n) can be
written as follows:
2
J3 (ki ) = ρ̄(ki − 1) P
(17)
where P = diag[0ρ̄ , 0ρ̄ , . . . , 0ρ̄ ] is the weighting matrix of
slack variables. With the slack variables, the state constraints
can be written as −κup − ρ̄ ≤ κ ≤ κup + ρ̄. Considering the
limited motor torques, the control input should also satisfy
the constraint −1Tup ≤ ũ i ≤ 1Tup . For the slack variable ρ̄,
it should also be constrained within a small range −ρ̄ up ≤
ρ̄ ≤ ρ̄ up to avoid the excessive slip overshooting. Then, the
optimization problem of lower layer controller can be written
as follows:
(14a)
2
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min J =
k+n
X
[0κ J1 (ki ) + 0T J2 (ki ) + 0ρ̄ J3 (ki )]
ki =k+1
s.t. − I¯1Tup ≤ ũ(ki ) ≤ I¯1Tup
− I¯ρ̄ up ≤ ρ̄(ki ) ≤ I¯ρ̄ up
I¯(−κup − ρ̄) ≤ κ(ki ) ≤ I¯(κup + ρ̄)
(18)
where I = [1, 1, 1, 1]T . With the introduction of slack variables, the total control input is redefined as u = [ũ T , ρ̄ T ]T .
By discretizing the rotational dynamic model in (12), the
state-space equation of lower layer controller can be written
as x(k + 1) = Ax(k) + E + Bu(k). Detailed definitions of the
above matrices are given in the Appendix.
C. Safety Requirements’ Adaptive Strategy
[0γ L 1 (ki ) + 0 y L 2 (ki )
ki =k+1
+ 0u L 3 (ki ) + 0du L 4 (ki )]
s.t. Vy,low ≤ x2 (ki ) ≤ Vy,up
u low ≤ u(ki ) ≤ u up .
(15)
2) Lower Layer Controller: In the lower layer controller,
the system vector is defined as x = [x 1 , x 2 , x 3 , x 4 ]T =
[ω1 , ω2 , ω3 , ω4 ]T , and the control input is defined as
ũ = [ũ 1 , ũ 2 , ũ 3 , ũ 4 ]T = [1T1 , 1T2 , 1T3 , 1T4 ]T . The main
targets are to track the desired slip ratios and suppress energy
consumption by adjusting motor torques. Then, the objective
items in time instant ki (k + 1 ≤ ki ≤ k + n) can be written as
follows:
2
J1 (ki ) = Re V˜x x̄(ki ) − I¯ − u ref Q
2
J2 (ki ) = ũ(ki − 1) R
(16a)
(16b)
where n is the prediction horizon of the lower layer controller,
and Q = diag[0κ , 0κ , . . . , 0κ ] and R = diag[0T , 0T , . . . , 0T ]
are the weighting matrices. In addition, the tire slip ratios
should be kept within the safety range to prevent tires form
skidding. However, the slip ratios may exceed the limit under
some extreme conditions. Once the limits are broken, the optimization problem may be unsolvable. To avoid this problem
and improve the computational efficiency, a group of slack
The stability spaces under different conditions and safety
requirements are discussed in Section II. The control targets
should be adjusted dynamically to guarantee stability and
maneuverability. As presented in our previous study [8],
an adaptive weighting factor strategy is adopted to balance the
requirements between stability and maneuverability. Combining the stability index introduced in Section II-B, the adaptive
strategy can be written as follows:
(
(
S1 ξ 2 + S2 , ξ ≤ 0
S3 − S4 0 y , ξ ≤ 0
0y =
0γ =
S2 ,
ξ > 0,
S3 − S2 ,
ξ > 0.
(19)
When the stability index ξ > 0, the vehicle is inside the stable
subspace. The main focus is yaw rate tracking to improve
maneuverability. When the stability index decreases to ξ < 0,
the vehicle is inside the critical stable subspace or unstable
subspace. Then, more attention should be focused on stability
control to prevent the vehicle from being unstable.
As presented in Section II, the steering characteristics are different when accelerating or decelerating. The
vehicle tends to be oversteering/understeering when decelerating/accelerating. Then, the tire slip limits should also be
adjusted dynamically. A relaxation function is introduced as
A x −1
A x −1
ζ = ν 1 + eτ 1+ µg
+ ν 1 + eτ 1− µg
(20)
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where τ is the sharpness of the function, and ν is large number
for the constraints. With the relaxation function, the tire slip
limits can be written as follows:
(
(
κup − ζ, A x ≥ 0
κup + ζ, A x ≥ 0
κ̃ up, f =
κ̃ up,r =
κup + ζ, A x < 0,
κup − ζ, A x < 0.
(21)
Then, the control input constraints in (15) can be expressed
as u up = [κ̃ up, f , κ̃ up, f , κ̃ up,r , κ̃ up,r ]T , u low = −u up . It should
be noted that by the application of the proposed adaptive
strategy in the upper layer NMPC controller, the safety
requirements can be reflected in the desired slip ratios. Then,
the LMPC controller in the lower layer is only used to track
the desired slip ratios and does not need to consider safety
requirements.
D. Reference Generation
In this article, a second-order reference model is adopted
for the reference signal generation. According to the vehicle
dynamic theory, the transfer function G(s) from the front
wheel steering angle δ f to the desired yaw rate γref can be
written as
1 + τγ s
.
(22)
G(s) = K γ 1
2
s + ω2ζn s + 1
ω2
The calibration of relaxation function is the same as in our
previous study [26]. Due to the limited space and the fact
that the calibration process is not the focus of this article,
the detailed calibration process is not given in this article.
Then, the objective item of lateral speed can be reformulated as L ′2 (ki ) = L 2 (ki ) + ζ̃ (ki ). With the above objective
item, the optimization problem in (15) can be rewritten as
follows:
min L =
IV. C OMPUTATIONALLY E FFICIENT
O PTIMIZATION A LGORITHM
An MPC-based hierarchical control strategy is proposed in
this article. The proposed control strategy can be realized
by solving the optimization problems. In the upper layer
controller, the optimal control problem is transformed into a
nonlinear optimization problem. To improve the computational
efficiency, a PMP-based method is adopted for the upper
layer controller. In the lower layer controller, a linear model
is adopted. Then, the optimization problem is transformed
into a quadratic programming (QP) problem and solved
in real-time.
A. PMP-Based Method for Upper Layer Controller
The state constraint Vy,low ≤ x2 (ki ) ≤ Vy,up is very important for vehicle stability and must be guaranteed. To reduce
the complexity of the nonlinear optimization problem in (15),
the relaxation function of the state constraint is set as
follows:
x (k )
x (k )
τ̃ 1+ V2 i −1
τ̃ 1− V2y,upi −1
y,low
ζ̃ (ki ) = ν̃ 1 + e
+ ν̃ 1 + e
.
(24)
0γ L 1 (ki ) + 0 y L ′2 (ki )
ki =k+1
+ 0u L 3 (ki ) + 0du L 4 (ki )
s.t. u low ≤ u(ki ) ≤ u up .
(25)
Then, combining the state-space equation in (11) and the
objective function in (25), the Hamilton function is defined
as follows:
H (x(ki ), u(ki )) = 01 L 1 (ki ) + 02 L ′2 (ki ) + 03 L 3 (ki )
+ 04 L 4 (ki ) + λ (ki )T F̃(ki )
(26)
where λ (ki ) = [λ1 (ki ), λ2 (ki ), λ3 (ki )]T are the Lagrangian
multipliers. The definition of F̃(ki ) can be found in the
Appendix. Based on the theory of PMP, the necessary conditions of optimal control can be written as follows:
λ (ki ) = λ (ki + 1) +
n
The detailed definitions of the parameters in (22) are given
in the Appendix. Consider the limited friction ability of road
surface, the original yaw rate reference γref should be bounded
within a range as |γref | ≤ γup , where γup = |µg/Vx |. With the
friction limit, the final reference signal of yaw rate can be
written as
(
G(s)δ f ,
if |G(s)δ f | ≤ γup
γref =
(23)
sgn(δ f )γup , else.
k+N
X
∂H
Ts .
∂ x ki +1
(27)
The terminal conditions of a fixed terminal time with
the free terminal states problem can be written as
follows:
λ (k + N + 1) = 0.
(28)
To satisfy the conditions of optimal control, the Hamilton
function must be minimized by the optimal control input
u ∗ (ki ) at every time instant as follows:
H (u ∗ (ki ), λ ∗ (ki )) ≤ H (u(ki ), λ ∗ (ki )).
(29)
It should be noted that the Hamilton function in (26) is a
highly nonlinear multivariable function and does not have
an explicit solution. To solve this problem, the original
Hamilton function is approximated by a first-order Taylor
expansion. Then, the Hamilton function can be rewritten as
follows:
H̃ (x(ki ), u(ki )) = p1 u(ki )T u(ki ) + ( p2 + p3 )u(ki )
+ g(x(ki ), u(ki − 1))
(30)
where
p1 = 03 + 04 , p2 = −204 u(ki − 1)T
∂ F̃ ∂ F̃ ∂ F̃ ∂ F̃
,
,
,
.
p3 = λ (ki )T
∂u 1 ∂u 2 ∂u 3 ∂u 4 ki −1
(31)
The detailed definition of constant part g(x(ki ), u(ki − 1)) is
given in the Appendix. Let ∂ H̃ /∂u = 0, and the expression
of the extreme value point is
p T + p3T
∂ H̃
= 0 ⇒ Ps = 2
.
∂u
−2 p1
(32)
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LIU et al.: SAFETY REQUIREMENTS’ ADAPTIVE NMPC STRATEGY FOR ELECTRIC VEHICLE STABILITY CONTROL
3999
Based on the theory of calculus, the minimal value of the
Hamilton function H̃ can be found at the extreme point Ps .
Then, the explicit solution of the unconstrained problem can
be written as follows:
u ∗j (ki ) =
∂ F̃
λ (ki )T ∂u
j
ki −1
− 204 u j (ki − 1)
−2 p1
.
(33)
Considering the constraints of the control inputs, the final
control input of the upper layer controller can be derived as
follows:


if u ∗j (ki ) ≤ u low
u low ,
∗
∗
(34)
u j (ki ) = u j (ki ), if u low < u ∗j (ki ) < u up


∗
u up ,
if u j (ki ) ≥ u up .
With the proposed solving method, the original optimization
problem is transformed into an optimal initial problem as
follows:
λ ∗ (k) = arg min
λ (k)∈R3
∥λ (k + N + 1)∥.
Fig. 7. Solving times of different algorithms. (a) Solving time of IPOPT.
(b) Solving time of PMP. (c) Average solving time with different prediction
horizons.
(35)
The above optimal initial problem can be solved using numerical methods, and more details can be found in our previous
study [26].
B. QP-Based Method for the Lower Layer Controller
In the lower layer controller, a linear model is adopted to
predict the future states of wheel rotation. Then, combining the
state-space equation in (12) and objective function in (18), the
optimization problem can be formulated as a QP problem as
follows:
1
min J = Ū (k)T G Ū (k) + c T Ū (k)
2
s.t. Au Ū (k) ≤ b
(36)
where U (k) = [ū(k)T , ū(k + 1)T , . . . , ū(k + n)T ]T represents
the independent variables of the lower layer controller. The
detailed definitions of the matrices are listed in the Appendix.
By solving the QP problem in (36), the optimal control input
can be derived. Then, the first group of torque sequences will
be executed by the motors.
C. Simulation Verification
To verify the computational efficient optimization method,
simulations with MATLAB and CarSim are conducted, and the
test case is set as double-lane change (DLC) manipulation.
A laptop computer with an Intel1 Core2 i7-9750H CPU
at 2.60 GHz is adopted to run the simulations. To compare
the real-time and closed-loop performance, an inner point
optimization (IPOPT) nonlinear programming algorithm [27]
is adopted.
The real-time performance comparisons of different solving
methods are given in Fig. 7. As can be seen in Fig. 7(a) and (b),
the proposed PMP-based solving method shows a much
higher computational efficiency than IPOPT. With the same
1 Registered trademark.
2 Trademarked.
Fig. 8. Simulation results of DLC test. (a) Yaw rate tracking of IPOPT.
(b) Yaw rate tracking of PMP. (c) Yaw rate tracking without control. (d) Lateral
speed.
prediction horizon, the computation time can be reduced by
more than ten times with the proposed PMP-based method.
The average computation times are given in Fig. 7(c), with
the prediction horizon, the computation time of PMP-based
method increases linearly, while IPOPT shows an exponential
growth.
In addition to real-time performance, the closed-loop performance should also be verified. The simulation results of
a DLC manipulation are given in Fig. 8. It can be seen
that both the handling performance and stability can be
improved significantly with two solving methods. In contrast,
the vehicle cannot finish the DLC test and lose its stability when the controller is removed. Then, the closed-loop
performance of the proposed PMP-based method can be
verified.
V. C ONTROL S TRATEGY E VALUATION
In this section, the proposed control strategy is evaluated by
a group of HIL experiments with a human driver. As shown
in Fig. 9(a), a full vision simulator is used to provide the
driving scene, a 6-DOF motion platform is used to simulate
the vehicle motions (yaw, pitch, and roll motions) and a human
driver manipulates the vehicle in the cockpit. In Fig. 9(b),
three industrial personal computers (IPCs) are adopted. IPC1
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IEEE TRANSACTIONS ON TRANSPORTATION ELECTRIFICATION, VOL. 10, NO. 2, JUNE 2024
Fig. 9. Full vision driving simulator with an HIL system. (a) Full vision
driving simulator with a 6-DOF motion platform. (b) HIL system.
Fig. 10. Experimental results of DLC with deceleration test. (a) Longitudinal
speed. (b) Longitudinal acceleration.
TABLE III
C ONTROLLER PARAMETERS AND S CENARIO S ETTING
TABLE IV
P ERFORMANCE E VALUATION I NDICES
is used to provide a high-accuracy vehicle dynamic model,
IPC2 is used to run SCANeR and provide driving scenarios,
and the real-time target machine IPC3 acts as a prototype
controller. All the signals are transmitted by a controller area
network (CAN) bus. To highlight the superiorities of the
proposed method, two controllers are evaluated. The controller
designed in Section III-B with the safety requirements’ adaptive strategy is called controller A, while the controller without
safety requirements’ adaptive strategy is called controller B.
All the controller parameters and experimental settings are
given in Tables I and III. In addition, several performance
indices are introduced for the HIL evaluation. As shown
in Table IV, the indices Pγ , PVy , PT , and Pκ represent
the handling ability, stability, energy consumption, and antiskid performance, respectively. Due to the limited space,
the specific meanings of performance indices are not given
in this article, and more details can be found in previous
studies [4], [8].
A. DLC With Deceleration
To verify the effectiveness of the proposed control strategy, a DLC maneuver with deceleration is adopted. In this
Fig. 11. Experimental results of DLC with deceleration test. (a) Yaw rate
tracking with adaptive strategy. (b) Yaw rate tracking without adaptive strategy.
(c) Yaw rate tracking without control. (d) Lateral speed results.
Fig. 12. Tire slip ratios of DLC with deceleration test. (a) Tire slip ratios
with adaptive strategy. (b) Tire slip ratios without adaptive strategy.
experiment, the human driver is supposed to track a standard
DLC trajectory while decelerating the vehicle. As can be
seen in Fig. 10(a) and (b), the initial speed is approximately 83 km/h. At about t = 4.5 s, the vehicle starts to
decelerates, and the deceleration is approximately −1.5 m/s2 .
During the deceleration, the vertical load of rear axis will
be reduced, and further resulting in a reduction in available tire force. Then, the vehicle tends to be oversteering,
and the stability will be considerably weakened. As shown
in Fig. 11, the yaw rate tracking and lateral speed results
are given. When the controller is off, the vehicle cannot
track the reference of the yaw rate, and the stability is
lost at approximately t = 6 s. For controller B, the stability
and maneuverability are improved substantially. However,
the oversteering characteristic can be obviously observed at
approximately t = 6.5 s. For controller A, the safety requirements are adjusted dynamically, and the yaw rate can be
properly tracked with a smaller lateral speed.
The tire slip ratios of DLC with deceleration are provided
in Fig. 12. With the assistance of controllers, the slip ratios
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4001
Fig. 13. Addition motor torques of DLC with deceleration test. (a) Addition
motor torques with adaptive strategy. (b) Addition motor torques without
adaptive strategy.
Fig. 15.
Stability index of DLC with deceleration test.
Fig. 16. Experimental results of DLC with acceleration test. (a) Longitudinal
speed. (b) Longitudinal acceleration.
Fig. 14.
Web assessment of DLC with deceleration test.
can be suppressed within a small range. Due to the vertical
load reduction rear axis, the rear slip ratios of controller B
exceed the limits at approximately t = 7 s. In contrast, with
the assistance of the safety requirements’ adaptive strategy, the
slip ratios of controller A can be strictly constrained within
the limits. In addition, the final additional motor torques are
given in Fig. 13.
Based on the aforementioned performance indices, a web
assessment is given in Fig. 14. With the help of adaptive strategy, controller A has a much better performance in yaw rate
tracking and anti-skid ability. However, the energy consumption and stability indices of controller A are slightly higher
than controller B. For the stability indices, due to the longer
operation time, the integration of lateral speed will make a
lager terminal value. It should be noted that the maximum
of lateral speed of controller A is much smaller than that of
controller B. Then, stability ability of controller A is better
than controller B. The energy consumption of controller A
is slightly higher than controller B. It can be explained that
the cornering manipulation time of controllers A and B is
about 6 and 4 s, respectively. Then, a higher energy consumption of controller A can be observed.
The stability index ξ results are given in Fig. 15. When
the controller is off, due to the overshooting of the yaw rate
and the excessive lateral speed at approximately t = 5.5 s,
the vehicle enters the unstable subspace (ξ < −1). Then, the
vehicle will enter an uncontrollable spin. Based on the above
analysis, the effectiveness of the proposed stability space and
index can be proven. For controller B, the vehicle will still
be inside the unstable subspace for a while (t = 5–8 s).
In contrast, with the assistance of the safety requirements’
adaptive strategy, the control targets can be adjusted in a timely
manner once the vehicle enters the critical stable subspace.
Then, the vehicle can be controlled away from the unstable
subspace by controller A during the whole DLC maneuver.
B. DLC With Acceleration
To further verify the effectiveness of the proposed control
strategy, a DLC maneuver with acceleration is also adopted.
In this experiment, the human driver is supposed to track a
standard DLC trajectory while accelerating the vehicle. The
experimental speed and acceleration are provided in Fig. 16.
The initial speed of vehicle is approximately 30 km/h, and the
vehicle starts to accelerate at approximately t = 3 s with an
acceleration of 1.5 m/s2 .
In contrast from the deceleration experiment, the vertical
load of the front axis is reduced during the acceleration,
further resulting in a reduction in available front tire forces.
In this case, the vehicle tends to be understeering, and the
maneuverability will be considerably weakened. As shown
in Fig. 17, the yaw rate tracking and lateral speed results are
given. When the controller is off, an obvious understeering
characteristic can be observed at approximately t = 8 s
and t = 10 s. It can be seen that the maneuverability is
weakened by insufficient front tire forces. With the assistance
of controllers A and B, the understeering characteristic and
yaw rate tracking accuracy can be considerably improved.
In addition, with the assistance of controllers, the lateral speed
can also be constrained within a smaller range to improve
stability.
The tire slip ratios of DLC with acceleration are shown
in Fig. 18. For controller B, the slip ratios can be suppressed
within a smaller range. However, the slip ratio limit will still
be exceeded. In contrast, the slip ratios of controller A can
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Fig. 20.
Web assessment of DLC with acceleration test.
Fig. 21.
Stability index of DLC with acceleration test.
Fig. 17. Experimental results of DLC with acceleration test. (a) Yaw rate
tracking with adaptive strategy. (b) Yaw rate tracking without adaptive strategy.
(c) Yaw rate tracking without control. (d) Lateral speed results.
Fig. 18. Tire slip ratios of DLC with acceleration test. (a) Tire slip ratios
with adaptive strategy. (b) Tire slip ratios without adaptive strategy.
Fig. 19. Addition motor torques of DLC with acceleration test. (a) Addition
motor torques with adaptive strategy. (b) Addition motor torques without
adaptive strategy.
be strictly constrained within the limits. The additional motor
torques are given in Fig. 19.
The web assessment of acceleration test is given in Fig. 20.
The operation times of controllers A and B are very close
to each other. Besides, the maximum of yaw rate tracking
error and lateral speed of controller A is also smaller than
that of controller B. Then, it can be concluded that a much
better overall performance in all the performance indices
can be achieved with the help of the proposed adaptive
strategy.
The stability index ξ of the acceleration experiment is given
in Fig. 21. When the controller is off, the vehicle tends to
be understeering. Due to the large tracking error of yaw
rate and excessive lateral speed, the stability index cannot
be kept within the stable subspace. During the whole DLC
test, the vehicle will enter unstable subspace several times.
For controller B, with the assistance of the controller, the
understeering characteristic can be improved substantially.
However, due to the lack of safety requirements’ adapt ability,
the tire slip ratios and lateral speed cannot be properly
constrained. Similar to the case of the off controller, the
vehicle will also enter the unstable subspace due to the
excessive slip ratios and lateral speed. For controller A, with
the assistance of the safety requirements’ adaptive strategy,
both the slip ratios and lateral speed can be strictly constrained
within the set range. Then, the vehicle can be kept away
from the unstable subspace during the whole experiment.
Compared with the fixed requirements controllers, the proposed adaptive strategy has a better performance in stability
improvement.
VI. C ONCLUSION
In this article, a safety requirements’ adaptive NMPC
strategy is proposed to improve the stability of DDEVs
under extreme conditions. To reflect the stability characteristics, a coupled LuGre tire model is adopted. Combining
the Routh stability criterion and nonlinear 2-DoF vehicle
dynamic model, the stability space is derived. Then, the
stability space is divided into stable/critical stable and unstable
subspaces based on the different vehicle states. The different
safety requirements of subspaces are analyzed. To meet the
requirements of various subspaces, a safety requirements’
adaptive NMPC strategy is proposed as an upper layer controller. In the upper layer controller, the desired slip ratios
are generated. To track the desired slip ratios, an LMPC
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LIU et al.: SAFETY REQUIREMENTS’ ADAPTIVE NMPC STRATEGY FOR ELECTRIC VEHICLE STABILITY CONTROL
controller is proposed to generate the additional motor torques.
To satisfy the requirement of computationally efficient online
optimization for vehicular applications, an indirect solution
algorithm is proposed based on PMP. With the proposed
algorithm, the nonlinear optimization problem of the upper
layer controller is transformed into an optimal initial value
problem and can be optimized in real-time. For the lower layer
controller, the optimization problem is transformed into a QP
problem. The effectiveness and superiority of the proposed
strategy are evaluated by HIL experiments with a human
driver.
The robustness of the proposed strategy is not considered
in this article. In our future research, more attention will be
given to the robustness studies. In addition, the automated
parameter calibration will also be incorporated in our future
research.
A PPENDIX
Ao11 = −
Cα2 + Cα4
Cα1 + Cα3
+
m(Vx − γo ls ) m(Vx + γo ls )
Ao12 = −Vx
l f Vx + Vyo ls Cα1 + −lr Vx + Vyo ls Cα3
1
−
m
+
Ao21 =
1
Iz
Ao22 =
1
Iz
(Vx + γo ls )2
−l f Cα1 + Cα3lr
−l f Cα2 + Cα4lr
+
Vx − γo ls
Vx + γo ls
− l f Vx + Vyo ls l f Cα1 + −lr Vx + Vyo ls lr Cα3
+
B01 =
(Vx − γo ls )2
!
l f Vx − Vyo ls Cα2 + −lr Vx − Vyo ls Cα4
(Vx − γo ls )2
!
− l f Vx −Vyo ls l f Cα2 + −lr Vx −Vyo ls lr Cα4
Cα1 + Cα2
,
m
(Vx + γo ls )2
(Cα1 + Cα2 )l f + (Fy1 − Fy2 )ls
B02 =
Iz
where ls = d/2, and C̃ αi = ∂ Fy /∂α, i = 1, 2, 3, 4 represents
the cornering stiffness at current operating point
s
2 C f Cr (L − K Vx2 )
ωn =
Vx
m Iz
m(C f L 2f + Cr L r2 ) + (C f + Cr )Iz
p
2 Lm Iz C f Cr (L − K Vx2 )
mVx L f
m(C f L f − Cr L r )
Vx
Kγ =
, K =
, τγ =
2
L + K Vx
2Cr L
2C f Cr L
ζ =
where C f and Cr are the linearized cornering stiffness of the
front and rear axes, respectively, and L = L f + L r is defined
as the distance from the front axle to the rear axle
1
F̃ 1 (ki ) = x2 (ki )x3 (ki ) + ((Fx1 (x(ki ), u 1 (ki ))
m
+ Fx2 (x(ki ), u 2 (ki ))) cos δ f −(Fy1 ((x(ki ), u 1 (ki ))
+ Fy2 (x(ki ), u 2 (ki ))) sin δ f
+ Fx3 (x(ki ), u 3 (ki )) + Fx4 (x(ki ), u 4 (ki )))
4003
1
((Fy1 (x(ki ), u 1 (ki ))
m
+ Fy2 (x(ki ), u 2 (ki ))) cos δ f + (Fx1 (x(ki ), u 1 (ki ))
+ Fx2 (x(ki ), u 2 (ki ))) sin δ f + Fy3 (x(ki ), u 3 (ki ))
+ Fy4 (x(ki ), u 4 (ki )))
F̃ 2 (ki ) = −x1 (ki )x3 (ki ) +
1
(L f (Fx1 (x(ki ), u 1 (ki ))+ Fx2 (x(ki ), u 2 (ki )))sin δ f
Iz
+ L f (Fy1 (x(ki ), u 1 (ki )) + Fy2 (x(ki ), u 2 (ki )))
× cos δ f − L r (Fy3 (x(ki ), u 3 (ki ))
d
+ Fy4 (x(ki ), u 4 (ki ))) + (Fx2 (x(ki ), u 2 (ki ))
2
− Fx1 (x(ki ), u 1 (ki ))) cos δ f
d
+ (Fx4 (x(ki ), u 4 (ki ))
2
d
− Fx3 (x(ki ), u 3 (ki ))) + (Fy1 (x(ki ), u 1 (ki ))
2
− Fy2 (x(ki ), u 2 (ki ))) sin δ f )
g(x(ki ), u(ki − 1))
= 01 (x3 (ki ) − γref )2
+ 02 ((x2 (ki ) − Vy,ref )2 + ζ̃ (ki ))
+ 04 u(ki − 1)T u(ki − 1) + λ (ki )T F̃(ki )
∂ F̃ ∂ F̃ ∂ F̃ ∂ F̃
− λ (ki )T
,
,
,
u(ki − 1)
∂u 1 ∂u 2 ∂u 3 ∂u 4 ki −1
Ts
A = I, B = I zeros(4, 4)
Iω
Ts
E =
T̃ d − Re F̃ x
Iω
F̃ 3 (ki ) =
I is the unit matrix with the corresponding dimension
T̃ d = [Td Td Td Td ]T , F̃ x = [Fx1 Fx2 Fx3 Fx4 ]T




ū(k)
E
 ū(k + 1|k) 


AE + E




Ū (k) = 

 Ē = 
..
..




.
.
n−1
A E, · · · + E
ū(k + n − 1|k)




A
lB
0
··· 0
 A2 
 AB
B
· · · 0




Ā =  . , B̄ =  .
.
.. 
..
..
 .. 
 ..
.
.
An
An−2 B · · · B
An−1 B

V̄
0 ··· 0
0
V̄
··· 0


Ṽ = 

..
0
.
0
0
0
0 · · · V̄


1
0
0
0 
 Vx1


1


 0
0
0 


V
x2
V̄ = 

1
 0
0
0 


Vx3



1 
0
0
0
Vx4
T
T
G = 2 B̄ P B̄ + 2R, c = 4 x̄(k) Ā T + Ē T P̄ − P̃ B̄
P̄ = Ṽ T Re P Re Ṽ , P̃ = I¯ + κ̄ P Re Ṽ

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4004
IEEE TRANSACTIONS ON TRANSPORTATION ELECTRIFICATION, VOL. 10, NO. 2, JUNE 2024
I , D̄ = diag[D, D, . . . , D]

I8n Ū (k)


−I8n Ū (k)

Au = 
 Re Ṽ B̄ Ū (k) − D̄Ū (k) 
−Re Ṽ B̄ Ū (k) − D̄Ū (k)


Umax

−Umin


b=
 κ̄ up + I¯4n×1 − Re Ṽ Ā x̄(k) + Ē .
κ̄ up − I¯4n×1 + Re Ṽ Ā x̄(k) + Ē
D= 0

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Hanghang Liu received the B.S. degree in electric engineering from the Henan University of
Technology, Zhengzhou, China, in 2018, and the
M.S. degree in control theory and control engineering from Jilin University, Changchun, China,
in 2021. He is currently pursuing the Ph.D. degree
in control science and engineering with Tongji University, Shanghai, China.
His research interests include model predictive
control and autonomous driving in the ground
vehicle field.
Lin Zhang received the Ph.D. degree in automotive engineering from Jilin University, Changchun,
China, in 2019.
He is currently an Associate Professor with the
School of Automotive Studies, Tongji University,
Shanghai, China. His research mainly include vehicle control in terms of safety, energy-saving, and
intelligence, including vehicle dynamics and control,
HEV control, and trajectory planning.
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LIU et al.: SAFETY REQUIREMENTS’ ADAPTIVE NMPC STRATEGY FOR ELECTRIC VEHICLE STABILITY CONTROL
Shen Li (Member, IEEE) received the B.E. degree
from Jilin University, Changchun, China, in 2012,
and the Ph.D. degree from the University of
Wisconsin Madison, Madison, WI, USA, in 2019.
He is currently a Research Associate with the
School of Civil Engineering, Tsinghua University,
Beijing, China. His research interests include intelligent transportation systems (ITSs), architecture
design of connected automated vehicle highway
(CAVH) systems, cooperative control method of
connected vehicles, autonomous driving safety, traffic data mining based on cellular data, and traffic operations and management.
Rongjie Yu received the B.Sc. degree from Tongji
University, Shanghai, China, in 2010, and the M.Sc.
and Ph.D. degrees in traffic engineering from the
University of Central Florida, Orlando, FL, USA,
in 2012 and 2013, respectively.
He is currently a Professor with the College
of Transportation Engineering, Tongji University.
His research interests include traffic safety, human
behavior, and safety evaluation of connected and
autonomous vehicles.
4005
Guofa Li received the Ph.D. degree in mechanical engineering from Tsinghua University, Beijing,
China, in 2016.
He is currently a Professor with the College of
Mechanical and Vehicle Engineering, Chongqing
University, Chongqing, China. He has published
more than 70 articles in his research areas. His
research interests include environment perception,
driver behavior analysis, and human-like decisionmaking based on artificial intelligence technologies
in autonomous vehicles and intelligent transportation
systems.
Dr. Li is a recipient of the Young Elite Scientists Sponsorship Program in
China, and he receives the Best Paper Awards from the China Association
for Science and Technology (CAST) and the Automotive Innovation Journal.
In addition, he serves as an Associate Editor for IEEE S ENSORS J OURNAL,
as well as a lead Guest Editor for IEEE Intelligent Transportation Systems
Magazine and Automotive Innovation.
Hong Chen (Fellow, IEEE) received the B.S. and
M.S. degrees in process control from Zhejiang
University, Hangzhou, China, in 1983 and 1986,
respectively, and the Ph.D. degree in system dynamics and control engineering from the University of
Stuttgart, Stuttgart, Germany, in 1997.
In 1986, she joined the Jilin University of Technology, Changchun, China. From 1993 to 1997,
she was a Wissenschaftlicher Mitarbeiter with the
Institut fuer Systemdynamik und Regelungstechnik,
University of Stuttgart. Since 1999, she has been
a Professor with Jilin University, and hereafter a Tang Aoqing Professor.
From 2015 to 2019, she served as the Director of the State Key Laboratory
of Automotive Simulation and Control. In 2019, she joined Tongji University
as a Distinguished Professor. Her current research interests include model
predictive control, nonlinear control, and applications in mechatronic systems
focusing on automotive systems.
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