# Direct Yaw-Moment Control of All-Wheel-Independent-Drive Electric Vehicles with Network-Induced Delays through Parameter-Dependent Fuzzy SMC Approach

Authors: Wanke Cao; Zhiyin Liu; Yuhua Chang; Antoni Szumanowski
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

This paper investigates the robust direct yaw-moment control (DYC) through parameter-dependent fuzzy sliding mode control (SMC) approach for all-wheel-independent-drive electric vehicles (AWID-EVs) subject to network-induced delays. AWID-EVs have obvious advantages in terms of DYC over the traditional centralized-drive vehicles. However it is one of the most principal issues for AWID-EVs to ensure the robustness of DYC. Furthermore, the network-induced delays would also reduce control performance of DYC and even deteriorate the EV system. To ensure robustness of DYC and deal with network-induced delays, a parameter-dependent fuzzy sliding mode control (FSMC) method based on the real-time information of vehicle states and delays is proposed in this paper. The results of cosimulations with Simulink® and CarSim® demonstrate the effectiveness of the proposed controller. Moreover, the results of comparison with a conventional FSMC controller illustrate the strength of explicitly dealing with network-induced delays.


---

## Body

## 1. Introduction

In recent years, all-wheel-independent-drive electric vehicles (AWID-EVs) have attracted increasing research efforts from both the academia and industry [1–3]. Equipped by advanced electric motors with more accurate and quicker torque generations than internal combustion engine (ICE) and hydraulic braking systems, AWID-EVs have obvious advantages in terms of direct yaw-moment control (DYC) through flexible differential driving/braking functions over traditional centralized drive vehicles [1, 3–6]. Plenty of existing studies have focused on the more flexible DYC and its integration control with active steering for AWID-EVs [2, 7–13]. However, considering the presence of model uncertainties, system parameter variations and external disturbances such as road rough or wind gust, it is one of the most principal issues for AWID-EVs to ensure the robustness of DYC.To improve and ensure the robustness of DYC, in the existing lateral dynamics control strategies of AWID-EVs, sliding mode control (SMC) has been widely adopted, which is robust and suitable for nonlinear systems such as vehicles. Goodarzi and Esmailzadeh used a sliding mode controller in low-level control to improve the robustness of vehicle dynamics control system for AWID-EVs [1]. Li et al. employed a sliding mode controller in the main loop to offer enough robustness for an integrated vehicle chassis control system based on DYC, active steering, and active stabilizer [14]. Wang and Longoria designed three sliding mode controllers to ensure control system robustness in a new coordinated-reconfigurable vehicle dynamics control strategy for AWID-EVs [15] and [16, 17] presented a terminal sliding mode control method to improve the robustness of yaw rate tracking control and torque distribution control. But, in all of these aforementioned control strategies based on SMC, the interference from electronic control systems such as networked-induced delays is not considered.However, in a modern AWID-EV, the control signals from controllers and the measurements from sensors are usually exchanged through an in-vehicle communication network, for example, controller area network (CAN) or FlexRay [2]. In other words, a modern AWID-EV is a networked control system (NCS) rather than a conventional centralized control system [2, 7, 9, 10]. Thus, the network-induced delays cannot be ignored. According to these research results in [2, 7, 9, 10], the network-induced delays caused by CAN could reduce the control performance of DYC and even deteriorate the EV system. Some researchers proposed H∞-based linear quadratic regulator (LQR) control method against CAN network delays as in [2, 7, 10]. However, research on SMC-based DYC of AWID-EVs is rare.Furthermore, as a variable structure control technology, SMC is more vulnerable to network-induced delays than continuous control technologies such as LQR or PID [18, 19]. There are numerous approaches to improve SMC. Among these approaches, the state-dependent boundary layer tuning method [20] has been widely used to improve the robustness of SMC. However the conventional state-dependent boundary layer tuning method is not sufficiently effective to deal with network-induced delay.The main work is as follows: firstly, the network- induced delays are explicitly considered in the DYC through SMC control method. Considering that the network-induced delays lead to a challenging problem for the DYC based on SMC, the chattering problem of SMC caused by the delays is analyzed in detail and the delays are determined with a command-first scheme, which is a more accurate method than the existing approaches. Furthermore, a parameter-dependent fuzzy sliding mode control (FSMC) method based on the real-time information of vehicle states and system delays is proposed to ensure the robustness of DYC for AWID-EVs against network-induced delays.The remaining sections of this paper are organized as follows: In Section2, problem formulation is described containing overall structure for DYC of networked AWID-EVs, control-oriented vehicle lateral dynamics model, and reference state model. The negative impact that resulted from NCS on the lateral dynamics model of AWID-EVs is also analyzed in detail in this section. In Section 3, an integrated state-dependent and delay-dependent fuzzy SMC is proposed to improve the robustness of DYC for networked AWID-EVs. In Section 4, the results of cosimulations with Matlab®/Simulink and CarSim are demonstrated. Conclusions are summarized in Section 5.


## 2. Problem Formulation

### 2.1. Overall Structure for DYC of Networked AWID-EVs

According to vehicle dynamics, the main working principle of the DYC of AWID-EVs is to keep the vehicle lateral motion state variables such as the yaw rate and the slip angle tracking the reference states by using the external yaw moment [21], which is directly generated by active longitude tire forces distribution of all wheels. As shown in Figure 1, a typical overall structure for DYC of networked 4-wheel-independent drive vehicles mainly consists of AWID-EV controller, controller area network (CAN), 4 motor controllers, motion state sensors, and BMS. The overall control system is integrated by CAN.Figure 1
Overall structure for DYC of networked AWID-EVs.The DYC function is implemented by AWID-EV controller, which is usually designed as a hierarchical controller including reference state model, motion controller unit, torque distribution unit, and estimation and processing unit as in Figure1. The reference state model is used to solve the reference states such as the reference sideslip angle βres and the reference yaw rate γres according to vehicle speed V and the front wheel steering angle δ from the driver. The reference states indicate the desired motion state by the driver. The motion controller unit is used to calculate the external yaw-moment ΔMz to keep the sideslip angle β and the yaw rate γ tracking the reference states. The torque distribution unit is used to solve the longitude forces Fxfl, Fxfr, Fxrl, and Fxrr for 4 motors according to the ΔMz. The estimation and processing unit is used to measure or estimate states such as V,β,γ.In this study, the motion controller unit, which influences the system robustness against network-induced delays, will be studied, whereas the reference state model, torque distribution unit, and estimation and processing unit are simplified.


### 2.2. Control-Oriented Vehicle Lateral Dynamics Model

As shown in Figure3, a two-degree-of-freedom (2-DOF) vehicle model, which has been widely studied as the control-oriented vehicle lateral dynamics model in various researches on DYC of vehicles [2, 21], is used in the paper, where CG is the center of gravity; m is the vehicle mass; Iz is the vehicle yaw inertia; Mz is the yaw moment; Fxf and Fxr are the longitude tire forces of front and rear wheels, respectively; Fyf and Fyr are the lateral tire forces of front and rear wheels, respectively; αf and αr are the slip angle of front and rear wheels, respectively.With the 2-DOF vehicle model, the state-space formulation of control-oriented vehicle lateral dynamics model for DYC of AWID-EVs is expressed as follows [10]:(1)x˙=Ax+Bu+Eδ,where (2)x=βγT,u=ΔMz,A=-2cf+crmV-2cflf-crlrmV2-1-2cflf-crlrIz-2cflf2+crlr2IzV,B=01Iz,E=2cfmV2cflfIz.c f and cr are the cornering stiffness of the front and rear tires, respectively.


### 2.3. Reference State Model

In vehicle lateral motion control, the desired sideslip angle is generally selected to be zero to ensure vehicle stability, while the desired yaw rate is usually defined to ensure good handling performance [10]. A widespread expression of the desired yaw rate is described in [2, 10]. Therefore, the reference state model can be written as follows: (3)r=Rδ,where(4)r=βresγres,R=0Vlf+lr+mV2crlr-cflf/2cfcrlf+lrT.


### 2.4. The Impact Resulted from NCS on Vehicle Lateral Dynamics Model

Firstly, without loss of generality as in [22], it is possible to make following assumption. In the NCS of DYC for AWID-EVs shown in Figure 2, the sensor node periodically samples the vehicle states with fixed period Ts, the controller node and the actuator node operate in event-driven mode which means a task will be immediately implemented once a message arrives via CAN, and the task implementation time in each node is ignored. With such assumption and without considering network-induced delays, the NCS of DYC for AWID-EVs runs like an ideal centralized control system with fixed sampling period Ts. The control-oriented discrete-time model of the vehicle lateral dynamics along with reference state model can be written as [2] (5)xk+1=Adxk+Bduk+Edδk,rk=Rdδk,where(6)Ad=eATs,Rd=R,Bd=∫0TseATs-θdθ·B,Ed=∫0TseATs-θdθ·Ewith xk, δk, uk, and rk indicating the state vector, steering angle vector, control input vector, and the reference state vector at time kTs, respectively.Figure 2
General structure of the NCS for DYC of AWID-EVs.Figure 3
2-DOF model of control-oriented vehicle lateral dynamics.Secondly, considering the network-induced delays and the same assumption mentioned above, the control inputu will be delayed by CAN as shown in Figure 4.Figure 4
The network-induced delays.Thus, the control inputu of the vehicle model at time t can be expressed as follows [22]:(7)ut=uk,∀t∈kTs+τk,k+1Ts+τk+1.If the delay τk is expressed as(8)τk=Υ+vTs,where Υ∈Z+ and v∈R[0,1).Then, the control-oriented discrete-time model of the vehicle lateral dynamics with network-induced delays can be expressed as follows [2]:(9)xk+1=Adxk+Edδk+Bduk+Δ0,kuk-1-uk+Δ1,kuk-2-uk-1+⋯+ΔΥ,kuk-Υ-1-uk-Υ=Adxk+Edδk+Bduk+∑i=0ΥΔi,kuk-i-1-uk-i,where the coefficient of each disturbance element induced by network-induced delays is expressed as follows [2]:(10)Δi,k=0,τk-i-i·Ts≤0∫0τk-i-i·TseATs-θdθ·B,0≤τk-i-i·Ts≤Ts∫0TseATs-θdθ·B,Ts≤τk-i-i·Ts.For analyzing, expression (9) is rewritten as follows:(11)xk+1=Adxk+Bduk+Edδk+fk,where the disturbance item(12)fk=∑i=0ΥΔi,kuk-i-1-uk-i.f k is the function of the network-induced delay τ and the input u.Thus, the control-oriented model of vehicle lateral dynamics for networked AWID-EVs is described as a discrete-time model with the disturbance elements caused by network-induced delays.


## 2.1. Overall Structure for DYC of Networked AWID-EVs

According to vehicle dynamics, the main working principle of the DYC of AWID-EVs is to keep the vehicle lateral motion state variables such as the yaw rate and the slip angle tracking the reference states by using the external yaw moment [21], which is directly generated by active longitude tire forces distribution of all wheels. As shown in Figure 1, a typical overall structure for DYC of networked 4-wheel-independent drive vehicles mainly consists of AWID-EV controller, controller area network (CAN), 4 motor controllers, motion state sensors, and BMS. The overall control system is integrated by CAN.Figure 1
Overall structure for DYC of networked AWID-EVs.The DYC function is implemented by AWID-EV controller, which is usually designed as a hierarchical controller including reference state model, motion controller unit, torque distribution unit, and estimation and processing unit as in Figure1. The reference state model is used to solve the reference states such as the reference sideslip angle βres and the reference yaw rate γres according to vehicle speed V and the front wheel steering angle δ from the driver. The reference states indicate the desired motion state by the driver. The motion controller unit is used to calculate the external yaw-moment ΔMz to keep the sideslip angle β and the yaw rate γ tracking the reference states. The torque distribution unit is used to solve the longitude forces Fxfl, Fxfr, Fxrl, and Fxrr for 4 motors according to the ΔMz. The estimation and processing unit is used to measure or estimate states such as V,β,γ.In this study, the motion controller unit, which influences the system robustness against network-induced delays, will be studied, whereas the reference state model, torque distribution unit, and estimation and processing unit are simplified.


## 2.2. Control-Oriented Vehicle Lateral Dynamics Model

As shown in Figure3, a two-degree-of-freedom (2-DOF) vehicle model, which has been widely studied as the control-oriented vehicle lateral dynamics model in various researches on DYC of vehicles [2, 21], is used in the paper, where CG is the center of gravity; m is the vehicle mass; Iz is the vehicle yaw inertia; Mz is the yaw moment; Fxf and Fxr are the longitude tire forces of front and rear wheels, respectively; Fyf and Fyr are the lateral tire forces of front and rear wheels, respectively; αf and αr are the slip angle of front and rear wheels, respectively.With the 2-DOF vehicle model, the state-space formulation of control-oriented vehicle lateral dynamics model for DYC of AWID-EVs is expressed as follows [10]:(1)x˙=Ax+Bu+Eδ,where (2)x=βγT,u=ΔMz,A=-2cf+crmV-2cflf-crlrmV2-1-2cflf-crlrIz-2cflf2+crlr2IzV,B=01Iz,E=2cfmV2cflfIz.c f and cr are the cornering stiffness of the front and rear tires, respectively.


## 2.3. Reference State Model

In vehicle lateral motion control, the desired sideslip angle is generally selected to be zero to ensure vehicle stability, while the desired yaw rate is usually defined to ensure good handling performance [10]. A widespread expression of the desired yaw rate is described in [2, 10]. Therefore, the reference state model can be written as follows: (3)r=Rδ,where(4)r=βresγres,R=0Vlf+lr+mV2crlr-cflf/2cfcrlf+lrT.


## 2.4. The Impact Resulted from NCS on Vehicle Lateral Dynamics Model

Firstly, without loss of generality as in [22], it is possible to make following assumption. In the NCS of DYC for AWID-EVs shown in Figure 2, the sensor node periodically samples the vehicle states with fixed period Ts, the controller node and the actuator node operate in event-driven mode which means a task will be immediately implemented once a message arrives via CAN, and the task implementation time in each node is ignored. With such assumption and without considering network-induced delays, the NCS of DYC for AWID-EVs runs like an ideal centralized control system with fixed sampling period Ts. The control-oriented discrete-time model of the vehicle lateral dynamics along with reference state model can be written as [2] (5)xk+1=Adxk+Bduk+Edδk,rk=Rdδk,where(6)Ad=eATs,Rd=R,Bd=∫0TseATs-θdθ·B,Ed=∫0TseATs-θdθ·Ewith xk, δk, uk, and rk indicating the state vector, steering angle vector, control input vector, and the reference state vector at time kTs, respectively.Figure 2
General structure of the NCS for DYC of AWID-EVs.Figure 3
2-DOF model of control-oriented vehicle lateral dynamics.Secondly, considering the network-induced delays and the same assumption mentioned above, the control inputu will be delayed by CAN as shown in Figure 4.Figure 4
The network-induced delays.Thus, the control inputu of the vehicle model at time t can be expressed as follows [22]:(7)ut=uk,∀t∈kTs+τk,k+1Ts+τk+1.If the delay τk is expressed as(8)τk=Υ+vTs,where Υ∈Z+ and v∈R[0,1).Then, the control-oriented discrete-time model of the vehicle lateral dynamics with network-induced delays can be expressed as follows [2]:(9)xk+1=Adxk+Edδk+Bduk+Δ0,kuk-1-uk+Δ1,kuk-2-uk-1+⋯+ΔΥ,kuk-Υ-1-uk-Υ=Adxk+Edδk+Bduk+∑i=0ΥΔi,kuk-i-1-uk-i,where the coefficient of each disturbance element induced by network-induced delays is expressed as follows [2]:(10)Δi,k=0,τk-i-i·Ts≤0∫0τk-i-i·TseATs-θdθ·B,0≤τk-i-i·Ts≤Ts∫0TseATs-θdθ·B,Ts≤τk-i-i·Ts.For analyzing, expression (9) is rewritten as follows:(11)xk+1=Adxk+Bduk+Edδk+fk,where the disturbance item(12)fk=∑i=0ΥΔi,kuk-i-1-uk-i.f k is the function of the network-induced delay τ and the input u.Thus, the control-oriented model of vehicle lateral dynamics for networked AWID-EVs is described as a discrete-time model with the disturbance elements caused by network-induced delays.


## 3. Controller Design

### 3.1. Sliding Mode Controller Design for DYC of 4WID-EV

Firstly, without considering network-induce delays, a general SMC for the discrete-time model (5) is designed. According to the typical design methodology of an general SMC [18], the sliding mode surface can be defined as (13)sx=cTx-r=cTe.e denotes the tracing error of motion states and cT=[c1c2] is the weight coefficient of elements of e.A reach law has been widely used [15, 18], which is written as(14)sk+1-sk=-qTssk-εTssgnsk,ε>0,q≥0,1-qTs>0,where sk denotes s(xk) at the time Tk.With the reach law (14), the control law can be solved as follows:(15)uk=-cTBd-1cTAdxk+cTEdδk-cTrk+1-cTxk-rk+qTscTxk-rk+εTssgncTxk-rk.However, according to the research in [18], for a discrete-time system, the state trajectory hardly occurs on the sliding mode surface (13) but zigzags around the sliding mode surface cause a quasi-sliding mode with a quasi-sliding mode band (QSMB) as in Figure 5. The QSMB is expressed as follows [18]:(16)x∣sx<εTs1-qTs.Figure 5
Trajectories of quasi-sliding mode.In order to avoid the chattering phenomenon in the QSMB, a boundary layer technique [1, 14] is usually adopted by defining a saturation function sat(sk) instead of sgnsk in (14) and (15)(17)satsk=1,ifsk>wblksk,ifsk≤wbl,k=1wbl-1,ifsk>-wbl,where wbl is the boundary layer width.Thus, the reach law (14) and control law (15) can be rewritten as (18)sk+1-sk=-qTssk-εTssatsk,(19)uk=-cTBd-1cTAdxk+cTEdδk-cTrk+1-cTxk-rk+qTscTxk-rk+εTssatcTxk-rk.According to the expression (16), it is necessary to tune the boundary layer width wbl based on the dynamics of sx in the controller design stage. When the control law (19) is used for the discrete-time system (5), the sx dynamics can be expressed as [18]:(20)sk+1=sk-qTssk-εTssatsk.Thesx dynamics can be solved by (5) and (12).However, when the control law (19) is used for the DYC of networked AWID-EVs, according to formulas (11), (13), (14), and (19), the sx dynamics will be changed as follows:(21)sk+1=sk-qTssk-εTssatsk-cTfk.Comparing (21) with (20), the disturbance item cTf(k), which is caused by network-induced delays, will impose a new uncertainty item on the sx dynamics and will result in adverse impact on the robustness of DYC.


### 3.2. Integrated State-Dependent and Delay-Dependent Sliding Mode Controller Design

A state-dependent boundary layer tuning method [20], which can tune the boundary layer width actively according to the dynamic state such as s(x) instead of a fixed boundary layer, has been widely used to improve the robustness of SMC in the real-time system applications [20].However, according to (21), the network-induced delays cause the uncertainty of the sx dynamics. Consequently, it is reasonable to create an integrated state-dependent and delay-dependent method to tune the boundary layer width for SMC dynamically (see Figure 6).Figure 6
Fuzzy sliding mode controller for DYC of AWID-EV.However, according to expressions (10), (12), and (21), the mathematic function between the sx and the network-induced delay τ is complicated nonlinear. It is too difficult to be solved online in general in-vehicle ECUs. Thus, a fuzzy logic unit is designed to deal with the complicated nonlinear mathematic problem as in Figure 6.


### 3.3. Fuzzy Logic Unit Design for the Integrated State-Dependent and Delay-Dependent Method

As shown in Figure6, the fuzzy logic unit is used to solve the boundary layer width wbl according to the state norm s and the network-induced delay τ. Thus the fuzzy logic unit has two input values and one unique output value, which are, respectively, defined as follows.Input 1 (22) S ∈ N B , N S , Z E , P S , P BInput 2 (23) τ ∈ N B , N S , Z E , P S , P BOutput (24) W b l ∈ N B , N S , Z E , P S , P B , P B 1 + , P B 2 + , P B 3 + , P B 4 +And the fuzzy linguistic variable values are defined in Table1. As shown in Figure 7, the triangular membership functions are used for the fuzzification of the two input variables and the one output variable. The scaling factors SF1 and SF2 (see Figure 6), which are tuned at the design stage by trial and extensive simulations performed in this study, are used to map the actual values of the input and output variables to their fuzzified values [23]. The rule base for the proposed fuzzy logic unit is described in Table 2.Table 1
Fuzzy linguistic variable value terms.
Name Description (relative value) NB Negative big NS Negative small ZE Zero PS Positive small PB Positive big PB1+ Positive big + level 1 PB2+ Positive big + level 2 PB3+ Positive big + level 3 PB4+ Positive big + level 4Table 2
Rule base of the fuzzy logic unit.
| S | τ W b l NB NB NB NS NB NS ZE NB ZE PS NB PS PB NB PB NB PS PS NS PS PB ZE PS PB1+ PS PS PB2+ PB PS PB3+ NB NS NS NS NS ZE ZE NS PS PS NS PB PB NS PB1+ NB PB PB NS PB PB1+ ZE PB PB2+ PS PB PB3+ PB PB PBPB4+ NB ZE ZE NS ZE PS ZE ZE PB PS ZE PB1+ PB ZE PB1+Figure 7
Membership functions for three variables.The variable domains such as the inputs|s|∈[0,0.5]and τ∈[0,20] and the output wbl∈[0.6,1.4] are selected based on the simulation results with a high-fidelity full-vehicle model in CarSim and Matlab in this study.The fuzzy unit employs the Mamdani Fuzzy Inference System (FIS), which is described by the following schema:(25)IFSisA,τisB,THENWblisC.IFSisA′,τisB′,THENWblisC′,where A, A′, B, B′, C, and C′ are fuzzy values defined as the input and output variables, respectively. Thecentre of area method is used in the defuzzification to solve Wbl.The rule base in detail between the inputs and the outputs is shown in Figure8. Thus, once the state norm s and the network-induced delay τ are known, the boundary layer width can be calculated by the fuzzy unit, and these rules could be easily implemented into the microprocessor of vehicle controller with a look-up mode. According to definition (13), the state norm |s| is known, and the unknown network-induced delay τ will be discussed in the following section.Figure 8
The rule relations between the inputs and the outputs for the proposed controller.


### 3.4. Determination of the Network-Induced Time Delay

To implement the proposed method, the network-induced delayτ should be determined. According to the assumption above (see Figures 2 and 4), the network-induced delay τk, which consists of the delays in both the forward and feedback links in the kth cycle, can be expressed as follows:(26)τk=dk+σk.Generally, the feedback link delayσk in a NCS is known, which can be measured by using a “time sampling technology” with the time tag within each received message sent by sensor node. However, the current delay dk in the forward link cannot be measured. In this paper, a “delay estimating technology,” which is based onNetwork Calculus Theory[24], is introduced to estimate the forward link delay dk with an explicit expression as follows [24]:(27)τlarge,j=j+2lR-∑i=0j-1l/ci,where τlarge,j is the upper bound of delay of a message with thejth priority sent by CAN; l indicates the maximum data frame length; R is the baud rate of the CAN; ci is the cycle length of the message with the ith priority.According to the research result in [24], the networked-induced delay of the message with the highest priority sent by CAN network can be accurately estimated through the expression (27). Therefore, this paper uses a “command-first scheme,” in which the command message in the forward link is sent with the highest priority to ensure the accuracy of dk as follows:(28)dk=τlarge,j=j+2lR-∑i=0j-1l/ci,settingj=0.Thus, the network-induced delayτk can be precisely determined by the following formula:(29)τk=σk+j+2lR-∑i=0j-1l/ci,settingj=0.


## 3.1. Sliding Mode Controller Design for DYC of 4WID-EV

Firstly, without considering network-induce delays, a general SMC for the discrete-time model (5) is designed. According to the typical design methodology of an general SMC [18], the sliding mode surface can be defined as (13)sx=cTx-r=cTe.e denotes the tracing error of motion states and cT=[c1c2] is the weight coefficient of elements of e.A reach law has been widely used [15, 18], which is written as(14)sk+1-sk=-qTssk-εTssgnsk,ε>0,q≥0,1-qTs>0,where sk denotes s(xk) at the time Tk.With the reach law (14), the control law can be solved as follows:(15)uk=-cTBd-1cTAdxk+cTEdδk-cTrk+1-cTxk-rk+qTscTxk-rk+εTssgncTxk-rk.However, according to the research in [18], for a discrete-time system, the state trajectory hardly occurs on the sliding mode surface (13) but zigzags around the sliding mode surface cause a quasi-sliding mode with a quasi-sliding mode band (QSMB) as in Figure 5. The QSMB is expressed as follows [18]:(16)x∣sx<εTs1-qTs.Figure 5
Trajectories of quasi-sliding mode.In order to avoid the chattering phenomenon in the QSMB, a boundary layer technique [1, 14] is usually adopted by defining a saturation function sat(sk) instead of sgnsk in (14) and (15)(17)satsk=1,ifsk>wblksk,ifsk≤wbl,k=1wbl-1,ifsk>-wbl,where wbl is the boundary layer width.Thus, the reach law (14) and control law (15) can be rewritten as (18)sk+1-sk=-qTssk-εTssatsk,(19)uk=-cTBd-1cTAdxk+cTEdδk-cTrk+1-cTxk-rk+qTscTxk-rk+εTssatcTxk-rk.According to the expression (16), it is necessary to tune the boundary layer width wbl based on the dynamics of sx in the controller design stage. When the control law (19) is used for the discrete-time system (5), the sx dynamics can be expressed as [18]:(20)sk+1=sk-qTssk-εTssatsk.Thesx dynamics can be solved by (5) and (12).However, when the control law (19) is used for the DYC of networked AWID-EVs, according to formulas (11), (13), (14), and (19), the sx dynamics will be changed as follows:(21)sk+1=sk-qTssk-εTssatsk-cTfk.Comparing (21) with (20), the disturbance item cTf(k), which is caused by network-induced delays, will impose a new uncertainty item on the sx dynamics and will result in adverse impact on the robustness of DYC.


## 3.2. Integrated State-Dependent and Delay-Dependent Sliding Mode Controller Design

A state-dependent boundary layer tuning method [20], which can tune the boundary layer width actively according to the dynamic state such as s(x) instead of a fixed boundary layer, has been widely used to improve the robustness of SMC in the real-time system applications [20].However, according to (21), the network-induced delays cause the uncertainty of the sx dynamics. Consequently, it is reasonable to create an integrated state-dependent and delay-dependent method to tune the boundary layer width for SMC dynamically (see Figure 6).Figure 6
Fuzzy sliding mode controller for DYC of AWID-EV.However, according to expressions (10), (12), and (21), the mathematic function between the sx and the network-induced delay τ is complicated nonlinear. It is too difficult to be solved online in general in-vehicle ECUs. Thus, a fuzzy logic unit is designed to deal with the complicated nonlinear mathematic problem as in Figure 6.


## 3.3. Fuzzy Logic Unit Design for the Integrated State-Dependent and Delay-Dependent Method

As shown in Figure6, the fuzzy logic unit is used to solve the boundary layer width wbl according to the state norm s and the network-induced delay τ. Thus the fuzzy logic unit has two input values and one unique output value, which are, respectively, defined as follows.Input 1 (22) S ∈ N B , N S , Z E , P S , P BInput 2 (23) τ ∈ N B , N S , Z E , P S , P BOutput (24) W b l ∈ N B , N S , Z E , P S , P B , P B 1 + , P B 2 + , P B 3 + , P B 4 +And the fuzzy linguistic variable values are defined in Table1. As shown in Figure 7, the triangular membership functions are used for the fuzzification of the two input variables and the one output variable. The scaling factors SF1 and SF2 (see Figure 6), which are tuned at the design stage by trial and extensive simulations performed in this study, are used to map the actual values of the input and output variables to their fuzzified values [23]. The rule base for the proposed fuzzy logic unit is described in Table 2.Table 1
Fuzzy linguistic variable value terms.
Name Description (relative value) NB Negative big NS Negative small ZE Zero PS Positive small PB Positive big PB1+ Positive big + level 1 PB2+ Positive big + level 2 PB3+ Positive big + level 3 PB4+ Positive big + level 4Table 2
Rule base of the fuzzy logic unit.
| S | τ W b l NB NB NB NS NB NS ZE NB ZE PS NB PS PB NB PB NB PS PS NS PS PB ZE PS PB1+ PS PS PB2+ PB PS PB3+ NB NS NS NS NS ZE ZE NS PS PS NS PB PB NS PB1+ NB PB PB NS PB PB1+ ZE PB PB2+ PS PB PB3+ PB PB PBPB4+ NB ZE ZE NS ZE PS ZE ZE PB PS ZE PB1+ PB ZE PB1+Figure 7
Membership functions for three variables.The variable domains such as the inputs|s|∈[0,0.5]and τ∈[0,20] and the output wbl∈[0.6,1.4] are selected based on the simulation results with a high-fidelity full-vehicle model in CarSim and Matlab in this study.The fuzzy unit employs the Mamdani Fuzzy Inference System (FIS), which is described by the following schema:(25)IFSisA,τisB,THENWblisC.IFSisA′,τisB′,THENWblisC′,where A, A′, B, B′, C, and C′ are fuzzy values defined as the input and output variables, respectively. Thecentre of area method is used in the defuzzification to solve Wbl.The rule base in detail between the inputs and the outputs is shown in Figure8. Thus, once the state norm s and the network-induced delay τ are known, the boundary layer width can be calculated by the fuzzy unit, and these rules could be easily implemented into the microprocessor of vehicle controller with a look-up mode. According to definition (13), the state norm |s| is known, and the unknown network-induced delay τ will be discussed in the following section.Figure 8
The rule relations between the inputs and the outputs for the proposed controller.


## 3.4. Determination of the Network-Induced Time Delay

To implement the proposed method, the network-induced delayτ should be determined. According to the assumption above (see Figures 2 and 4), the network-induced delay τk, which consists of the delays in both the forward and feedback links in the kth cycle, can be expressed as follows:(26)τk=dk+σk.Generally, the feedback link delayσk in a NCS is known, which can be measured by using a “time sampling technology” with the time tag within each received message sent by sensor node. However, the current delay dk in the forward link cannot be measured. In this paper, a “delay estimating technology,” which is based onNetwork Calculus Theory[24], is introduced to estimate the forward link delay dk with an explicit expression as follows [24]:(27)τlarge,j=j+2lR-∑i=0j-1l/ci,where τlarge,j is the upper bound of delay of a message with thejth priority sent by CAN; l indicates the maximum data frame length; R is the baud rate of the CAN; ci is the cycle length of the message with the ith priority.According to the research result in [24], the networked-induced delay of the message with the highest priority sent by CAN network can be accurately estimated through the expression (27). Therefore, this paper uses a “command-first scheme,” in which the command message in the forward link is sent with the highest priority to ensure the accuracy of dk as follows:(28)dk=τlarge,j=j+2lR-∑i=0j-1l/ci,settingj=0.Thus, the network-induced delayτk can be precisely determined by the following formula:(29)τk=σk+j+2lR-∑i=0j-1l/ci,settingj=0.


## 4. Simulation Results

To study the effectiveness of the proposed controller, the cosimulations are carried out in Matlab/Simulink with a full-vehicle model constructed by CarSim (see Figure9). The vehicle parameters used in the simulations are based on a prototyped 4WID-EV, and main parameters are listed in Table 3.Table 3
Main vehicle parameters.
Parameter Description Quantity M Vehicle mass 1350 kg I Z Yaw moment of inertia 1975 kg⋅m2 l f Front semiwheelbase 1.085 m l r Rear semiwheelbase 1.386 m c f Cornering stiffness of front tires 58000 N/rad c r Cornering stiffness of rear tires 60000 N/radFigure 9
The cosimulation model with Simulink and CarSim.The proposed controller is used in motion controller unit (see Figure1) and a simple torque distribution strategy, which distributes the direct yaw-moment equally to the driving or braking torques of 4 wheels, is used in torque distribution unit (see Figure 1). For comparison, a conventional state-dependent fuzzy SMC controller (the conventional FSMC) without considering network-induced delays is also designed. And the designed rule base of the conventional FSMC is shown in Figure 10.Figure 10
The rule relations between the input and the output for conventional FSMC.The reaching law parameters of SMC are chosen asε=27.5 and q=0. According to the results in [22], the upper bound of network-induced delays in a practical vehicle control system is about as high as 1.7Ts. The sampling period of the closed-loop system is adopted as Ts=10ms. Thus, CAN-induced delays in simulations are assumed to change randomly in time range [0, 1.7Ts] (see Figure 11).Figure 11
The network-induced delays in control loop over CAN.Two different steering maneuvers, which are commonly used in vehicle tests, are considered: a ramp steering maneuver and a double lane-changing maneuver. The ramp steering maneuver is often adopted in theJ-turn test and the double lane-changing maneuver is usually used in extreme cases, for example, high-speed overtaking or obstacle avoidance.In each driving maneuver, simulations are carried out in two stages. The first stage is under the ideal network condition to evaluate the effectiveness of controllers without considering network-induced delays. The second stage is to verify the robustness of the proposed controller with network-induced delays.


### 4.1.J-Turn Steering Maneuver

In this case, the vehicle runs at a low speed of 40 km/h on a slippery road with a low road friction (μ = 0.4). During the J-turn maneuver, the steering wheel angle first increases from 0 deg to 18 deg in 0.5 s, which is used to simulate a sharp turn. Then it decreases to 0 deg again in 4 s.Figure12 shows the results of cosimulations in the J-turn steering maneuver under the ideal network condition. It is obvious in Figures 12(a), 12(b), and 12(c) that the vehicle yaw rate can precisely track the desired reference by the driver, the vehicle sideslip angle can be restrained in a narrow scope around 0.014 rad, and the lateral acceleration can be restrained in a narrow scope around 0.013 g. For the two controllers, the control performance is satisfactory. The results demonstrate the effectiveness of the conventional FSMC and the proposed controller.Figure 12
Control performance inJ-turn maneuver under the ideal network condition. (a) Vehicle yaw rate. (b) Vehicle sideslip angle. (c) Vehicle lateral acceleration.
(a) (b) (c)Figure13 shows the results of cosimulations in the J-turn steering maneuver with the network-induced delays caused by CAN. It is obvious in Figures 13(a), 13(b), and 13(c) that, with the conventional FSMC, the vehicle motion cannot track the desired yaw rate in the transient phase but can keep tracking the desired yaw rate in the steady phase, and the yaw rate overshoot is about 10.3% in the transient phase, while, for the proposed controller, the adverse impact of delays can be eliminated and the control performance is still satisfactory. The yaw rate overshoot is about 3.4%. Therefore the results of comparison explicitly illustrate the strength of the proposed controller dealing with network-induced delays.Figure 13
Control performance inJ-turn maneuver with the network-induced delays caused by CAN. (a) Vehicle yaw rate. (b) Vehicle sideslip angle. (c) Vehicle lateral acceleration.
(a) (b) (c)Figure14 shows the torque response of 4 motors in the J-turn steering maneuver with the network-induced delays caused by CAN. It is obvious that, with the conventional FSMC, the chattering phenomenon of each motor is severe in the transient phase, which would reduce control performance of DYC and even deteriorate the EV system, while, for the proposed controller, the performance of the torque response of each motor is satisfactory.Figure 14
Motor torques of 4 wheels with two different controllers for DYC of 4WID-EV in doubleJ-turn maneuver with the network-induced delays caused by CAN.Figures15 and 16 show the dynamic boundary layers of two controllers in the J-turn steering maneuver with the network-induced delays caused by CAN.Figure 15
Tuning the boundary layer width process by the conventional FSMC in doubleJ-turn steering maneuver with the network-induced delays caused by CAN.Figure 16
Tuning the boundary layer width process by the proposed controller in doubleJ-turn steering maneuver with the network-induced delays caused by CAN.The results show that the conventional FSMC can tune the boundary layer width according to the vehicle states but not the network-induced delays. The proposed controller can tune the boundary layer width according to both the vehicle states and the network-induced delays.


### 4.2. Double Lane-Changing Steering Maneuver

In this case, the vehicle runs at a high speed of 100 km/h on a road with a high road friction (μ = 0.85).The following cosimulation process is quite similar to that for theJ-turn steering maneuver. Under different network conditions, the results are shown in Figures 17, 18, 19, 20, and 21. Under the ideal network condition, the actual vehicle yaw rate can track the desired reference very well, and the vehicle sideslip angle and the vehicle lateral acceleration can also be kept regulated for both controllers. When with the network-induced delays, the proposed control can still keep the vehicle yaw rate tracking the desired reference very well, whereas the conventional FSMC results in significant oscillations due to the effect of the network-induced delays. Furthermore, a similar observation can be found in the vehicle sideslip angle and the vehicle lateral acceleration. The torque response of each motor is also shown as in Figure 19. The results show that network-induced delays have a significant impact on the stability of the closed-loop control system and can obviously reduce the robustness of the conventional FSMC.Figure 17
Control performance in double lane-changing maneuver under the ideal network condition. (a) Vehicle yaw rate. (b) Vehicle sideslip angle. (c) Vehicle lateral acceleration.
(a) (b) (c)Figure 18
Control performance in double lane-changing maneuver with the network-induced delays caused by CAN. (a) Vehicle yaw rate. (b) Vehicle sideslip angle. (c) Vehicle lateral acceleration.
(a) (b) (c)Figure 19
Motor torques of 4 wheels with two different controllers for DYC of 4WID-EV in double lane-changing maneuver with the network-induced delays caused by CAN.Figure 20
Tuning the boundary layer width process by the conventional FSMC in double lane-changing steering maneuver with the network-induced delays caused by CAN.Figure 21
Tuning the boundary layer width process by the proposed controller in double lane-changing steering maneuver with the network-induced delays caused by CAN.The tuning boundary layer width processes of two controllers in double lane-changing maneuver are shown in Figures20 and 21. The results also show that the conventional FSMC only can tune the boundary layer width according to the vehicle states but not the network-induced delays, while the proposed controller can tune the boundary layer width actively according to both the vehicle states and the system delays, which makes the control system more robust.


## 4.1.J-Turn Steering Maneuver

In this case, the vehicle runs at a low speed of 40 km/h on a slippery road with a low road friction (μ = 0.4). During the J-turn maneuver, the steering wheel angle first increases from 0 deg to 18 deg in 0.5 s, which is used to simulate a sharp turn. Then it decreases to 0 deg again in 4 s.Figure12 shows the results of cosimulations in the J-turn steering maneuver under the ideal network condition. It is obvious in Figures 12(a), 12(b), and 12(c) that the vehicle yaw rate can precisely track the desired reference by the driver, the vehicle sideslip angle can be restrained in a narrow scope around 0.014 rad, and the lateral acceleration can be restrained in a narrow scope around 0.013 g. For the two controllers, the control performance is satisfactory. The results demonstrate the effectiveness of the conventional FSMC and the proposed controller.Figure 12
Control performance inJ-turn maneuver under the ideal network condition. (a) Vehicle yaw rate. (b) Vehicle sideslip angle. (c) Vehicle lateral acceleration.
(a) (b) (c)Figure13 shows the results of cosimulations in the J-turn steering maneuver with the network-induced delays caused by CAN. It is obvious in Figures 13(a), 13(b), and 13(c) that, with the conventional FSMC, the vehicle motion cannot track the desired yaw rate in the transient phase but can keep tracking the desired yaw rate in the steady phase, and the yaw rate overshoot is about 10.3% in the transient phase, while, for the proposed controller, the adverse impact of delays can be eliminated and the control performance is still satisfactory. The yaw rate overshoot is about 3.4%. Therefore the results of comparison explicitly illustrate the strength of the proposed controller dealing with network-induced delays.Figure 13
Control performance inJ-turn maneuver with the network-induced delays caused by CAN. (a) Vehicle yaw rate. (b) Vehicle sideslip angle. (c) Vehicle lateral acceleration.
(a) (b) (c)Figure14 shows the torque response of 4 motors in the J-turn steering maneuver with the network-induced delays caused by CAN. It is obvious that, with the conventional FSMC, the chattering phenomenon of each motor is severe in the transient phase, which would reduce control performance of DYC and even deteriorate the EV system, while, for the proposed controller, the performance of the torque response of each motor is satisfactory.Figure 14
Motor torques of 4 wheels with two different controllers for DYC of 4WID-EV in doubleJ-turn maneuver with the network-induced delays caused by CAN.Figures15 and 16 show the dynamic boundary layers of two controllers in the J-turn steering maneuver with the network-induced delays caused by CAN.Figure 15
Tuning the boundary layer width process by the conventional FSMC in doubleJ-turn steering maneuver with the network-induced delays caused by CAN.Figure 16
Tuning the boundary layer width process by the proposed controller in doubleJ-turn steering maneuver with the network-induced delays caused by CAN.The results show that the conventional FSMC can tune the boundary layer width according to the vehicle states but not the network-induced delays. The proposed controller can tune the boundary layer width according to both the vehicle states and the network-induced delays.


## 4.2. Double Lane-Changing Steering Maneuver

In this case, the vehicle runs at a high speed of 100 km/h on a road with a high road friction (μ = 0.85).The following cosimulation process is quite similar to that for theJ-turn steering maneuver. Under different network conditions, the results are shown in Figures 17, 18, 19, 20, and 21. Under the ideal network condition, the actual vehicle yaw rate can track the desired reference very well, and the vehicle sideslip angle and the vehicle lateral acceleration can also be kept regulated for both controllers. When with the network-induced delays, the proposed control can still keep the vehicle yaw rate tracking the desired reference very well, whereas the conventional FSMC results in significant oscillations due to the effect of the network-induced delays. Furthermore, a similar observation can be found in the vehicle sideslip angle and the vehicle lateral acceleration. The torque response of each motor is also shown as in Figure 19. The results show that network-induced delays have a significant impact on the stability of the closed-loop control system and can obviously reduce the robustness of the conventional FSMC.Figure 17
Control performance in double lane-changing maneuver under the ideal network condition. (a) Vehicle yaw rate. (b) Vehicle sideslip angle. (c) Vehicle lateral acceleration.
(a) (b) (c)Figure 18
Control performance in double lane-changing maneuver with the network-induced delays caused by CAN. (a) Vehicle yaw rate. (b) Vehicle sideslip angle. (c) Vehicle lateral acceleration.
(a) (b) (c)Figure 19
Motor torques of 4 wheels with two different controllers for DYC of 4WID-EV in double lane-changing maneuver with the network-induced delays caused by CAN.Figure 20
Tuning the boundary layer width process by the conventional FSMC in double lane-changing steering maneuver with the network-induced delays caused by CAN.Figure 21
Tuning the boundary layer width process by the proposed controller in double lane-changing steering maneuver with the network-induced delays caused by CAN.The tuning boundary layer width processes of two controllers in double lane-changing maneuver are shown in Figures20 and 21. The results also show that the conventional FSMC only can tune the boundary layer width according to the vehicle states but not the network-induced delays, while the proposed controller can tune the boundary layer width actively according to both the vehicle states and the system delays, which makes the control system more robust.


## 5. Conclusions

This paper proposed an integrated state-dependent and delay-dependent fuzzy sliding mode control method to improve the robustness of DYC of AWID-EVs subject to network-induced delays. The SMC, which can effectively deal with model uncertainties, system parameter variations, and external disturbances, has been widely used to improve the robustness of DYC of AWID-EVs instead of common continuous control technologies such as LQR and PID. However, on the other hand, the SMC, which is a variable structure control, is more vulnerable to the system delays from electronic control systems. Meanwhile, in modern AWID-EVs, the networked control system based on in-vehicle networks such as CAN would inevitably impose network-induced delays on the vehicle control system. In order to improve the robustness of DYC of AWID-EVs, this paper first analyzed the adverse impact resulted from NCS on DYC based on SMC in detail. Then an integrated state-dependent and delay-dependent fuzzy SMC method is proposed to improve the robustness of DYC for AWID-EVs.The results of comparison in both typical steering maneuver cases show that the proposed controller can effectively improve the robustness of DYC for AWID-EVs subject to network-induced delays. Moreover, the proposed controller also inherits the robustness of SMC in terms of dealing with model uncertainties, system parameter variations, and external disturbances.



---

*Source: 5170492-2017-01-22.xml*