01.12.2020  Original Article  Ausgabe 1/2020 Open Access
Parallel Distributed Compensation /H_{∞} Control of Lanekeeping System Based on the TakagiSugeno Fuzzy Model
 Zeitschrift:
 Chinese Journal of Mechanical Engineering > Ausgabe 1/2020
1 Introduction
Although the dramatic increase in the number of vehicles has made the daily life and travel convenient, it has also incurred a variety of safety problems. Statistics show that lane departure is one of the main factors in traffic accidents [
1]. With recent progress in information, communication, and sensor technology, advanced driver assistance systems (ADASs) have increasingly attracted the attention of many scholars. An ADAS mainly consists of an adaptive cruise system (ACS), forward collision warning system (FCWS), lane departure warning system (LDWS), lanekeeping assist system (LKAS), etc [
2]. A lanekeeping system functions on the premise of not interfering with the driver’s normal driving, when the vehicle deviates from the lane’s centerline due to the driver’s negligence, the system takes active action, steering to help the driver avoid any traffic accidents caused by the inadvertent lane departure [
3]. Many scholars have conducted indepth research into LKASs, yielding good results. A videobased vehicle embedded PID autonomous steering control test was proposed in Refs. [
4,
5]. Yaw rate tracking error based PID active frontwheel steering control was adopted to improve the vehicle steering dynamics, and an embedded control structure with two independent control loops was proposed. Test validation was carried out by the prototype vehicle on a testing road.
In Ref. [
6] the videobased lanekeeping and active steering were studied indepth, considering the camera’s sampling frequency and image processor’s lag time, a multirate Kalman filter and stochastic measure timedelay compensated for the time delay of the video signal transmission and processing. A preview and following theorybased driver model was developed, and the sliding mode controller was designed by using the position and direction errors at the preview point as the switching function in Ref. [
7]. A lanekeeping coordinated control method combining crossingroad time and driver maneuvering behavior judgement was proposed, integrating an electric power steering system and lanekeeping system [
8]. In Ref. [
9], a back propagation neuronetwork PID based lanekeeping control system was designed by analyzing the neuronetwork algorithm principle. In Ref. [
10], a new impulse multirate lanekeeping control method based on virtual prediction was proposed. To improve the vehicle keeping performance on a curved road, the integral of the lateral offset error was included in the design of the state feedback controller. The multirate Kalman filter could rapidly recognize the vehicle state and the lane detection instantaneous fault could be compensated for based on the proposed virtual lane predictive method. The linear quadratic regulator of the vehicle steering control switching system was proposed, the control layer of the lateral error and yaw angle was developed, and the PID controllers were subsequently designed [
11]. A simple feedforward control scheme with stability and robustness was discussed, and a driver coordinated stability keeping system was proposed, under stable cruise conditions on a high way and emergency conditions [
12].
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The predictive control model provides a method for danger estimation and decision and optimal control, it is thus used in lanekeeping assisted systems by many researchers [
13,
14]. Many controllers are designed using nonlinear vehicle models or simplified linear models. Although, the nonlinear computation cost is larger, this method satisfies the requirements of a realtime application for a vehicle with low speeds [
15,
16]. In Ref. [
17], the vehicle lateral dynamics and tire sideslip characteristics were analyzed and the lateral dynamics model was studied by using the displacement deviation, yaw angle deviation, and the differential items of both as the state variables. The frontwheel steering angle was used as the control input; the optimized performance index and automatic lane keeping control system constraints were developed and the MPC based steering control strategy was designed. A linear timevariant driverinloop system model was proposed, the vehicle nominal trajectory was predicted using the proposed model and vehicle current states, and the lanekeeping predictive controller was designed [
18]. Two different drivervehicle combined control methods were proposed by the twopoint preview driver model to coordinate the driver to accomplish lane tracking [
19]. The first design therein was based on linear multivariable output regulator theory, weakening the driver control to properly regulate the controller input to guide the vehicle motion. The second design was a model predictive control theory for accomplishing the controller design by considering the overlap between the lane tracking steering assisted system and driver.
Another method of lane departure assisted control is braking control [
20]. Ref. [
21] first proposed the concept of steering via differential braking based on the theory of a lane departure assisted system through differential braking. In Ref. [
22], the proposed assisted control was based on the conventional ABS/ESP hydraulic braking system. The assisted control decision was made using vehicleroad information, the desired yaw angle velocity was calculated according to the vehicleroad deviation at the preview point and the sliding mode controller was adopted for active differential braking control to realize vehicle yaw motion tracking control; thus achieving the goal of lane departure assisted control. In Ref. [
23], layered control for the lanekeeping assisted control algorithm was proposed, consisting of upper and lower layer controllers. The upperlayer controller assesses the departure and determines whether the lane departure assisted control should execute. The lowerlayer computes the required yaw moment to track the desired yaw rate and allocates each tire’s braking force to track the desired yaw moment.
Meanwhile, the coordination between the driver and assisted system has attracted the attention of various researchers [
24]. In Ref. [
25], a switch strategy was designed for managing the connection between the driver and assisted system. Under the condition of ensuring that the assisted system does not interfere with the driver, the driver can be assisted to avoid the lane departure occurrence. The prototype vehicle is used to validate the effectiveness of the proposed method. In Ref. [
26], during the design of vehicle lateral control, an active safety system framework of coordinated sharing control was proposed, integrating driver and assisted system. The simplified linear system state was converted to a linear quadratic regulator, solving the problem via optimal control to reduce computing complexity; the control authority decision algorithm implemented the coordinated switching between the driver and control system. In Refs. [
27,
28], the LQR/H∞ lateral controller was designed for a nonlinear vehicle. The driver model of a farnear visual angle was built and the H∞ controller was designed by parallel distribution and LMI, solving the problem of the driver’s involvement in both vehicle control and controller disturbances. A steering override for lanekeeping systems was proposed [
29]. The computeraided engineering software was applied to coordinate the interaction of the driver and lanekeeping assisted system. In Ref. [
30], considering the interaction between the driver and assisted system, a drive program model integrated the drive program, effectively predicting behavior during the controller design process. Tactile sharing control was applied in coordinating the driver and driving assisted system [
31]. The gain regulation control method based on estimating the cooperative states was proposed. When the two intentions are inconsistent, the assisted system action rate was reduced. The lane tracking via switched control was guaranteed to be capable of matching the intentions of the driver and assisted system. The driving simulator was utilized to validate the method’s effectiveness.
The research given above is of great significance for vehicle lateral control but most of them alter the steering angle using control signals to address the vehicle lateral control problem (or by considering the longitudinal velocity as constant). This control strategy is very robust but ignores the influence of the driver and external resistance due to ground friction forces and tire aligning moments during travel; this resistance will cause a target steering angle tracking delay. Therefore, this article builds a vehicledriverroad closedloop system by taking into consideration the change in vehicle longitudinal velocity within a certain range when traveling, treating vertical speed as the fuzzy variables of a TS fuzzy model, with the steering torque as the vehicle’s lateral control signal. A lanekeeping robust controller is designed to obtain the desired torque by using parallel distributed compensation (PDC) and linear matrix inequality (LMI). Considering the influence of external resistance when the vehicle is steered, a feedforward compensation controller is designed to compensate the assist torque for the lanekeeping system. Finally, the simulation experiment and real vehicle experiment are carried out to verify the reliability of the method.
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2 System Modelling
2.1 VehicleRoad Model
Suppose the vehicle can identify a lane via a lane identification system.
\(\psi_{\nu }\) and
\(\psi_{d}\) are actual and desired yaw angle of the vehicle respectively, so the yaw bias is
\(\psi_{L} = \psi_{v}  \psi_{d}\) and the clockwise direction is assumed to be positive.
\(Y_{CG}\) is the distance from the center of mass of the vehicle to the lane centerline and the vehicle center of mass is negative on the left side of the lane. The vehicleroad reference model is shown in Figure
1.
×
The pilot horizontal deviation at the driver near viewpoint is the following:
$$Y_{L} = Y_{CG} + L_{\text{near}} \sin \psi_{L} .$$
(1)
The position relationship between the vehicle and road position is shown in Figure
1. The system state vector is
\({\varvec{x}}_{v} = \left( {\begin{array}{*{20}c} {v_{y} } & r & {Y_{L} } & {\psi_{L} } & {\delta_{d} } & {\dot{\delta }_{d} } \\ \end{array} } \right)^{\text{T}}\), and the system input is
\(u_{v} = T_{d} + T_{c}  T_{r}\), where
\(T_{d}\),
\(T_{c}\), and
\(T_{r}\)are the driver’s torque, assist torque, and external torque acting on the vehicle by the road environment, respectively.
\(w_{v} = \rho\) is the curvature of the road. The state equation of the vehicleroad system is as follows [
14]:
where
\({\varvec{A}}_{v} {\mathbf{ = }}\left( {\begin{array}{*{20}c} {a_{11} } & {a_{12} } & 0 & 0 & {a_{15} } & 0 \\ {a_{21} } & {a_{22} } & 0 & 0 & {a_{25} } & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & {L_{near} } & {v_{x} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ {a_{61} } & {a_{62} } & 0 & 0 & {a_{65} } & {  \frac{{B_{S} }}{{I_{z} }}} \\ \end{array} } \right),\)
\({\varvec{B}}_{v} = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ {\frac{1}{{I_{S} }}} \\ \end{array} } \right),\)
\(a_{11} =  \frac{{2(C_{f} + C_{r} )}}{{mv_{x} }},\)
\(a_{12} = \frac{{2(C_{r} L_{r}  C_{f} L_{f} )}}{{mv_{x} }}  v_{x} ,\)
\(a_{21} = \frac{{2(C_{r} L_{r}  C_{f} L_{f} )}}{{I_{z} v_{x} }},\)
\(a_{22} = \frac{{2(C_{r} l_{r}^{2} + C_{f} l_{f}^{2} )}}{{I_{z} v_{x} }},\)
\(a_{15} = \frac{{2C_{f} }}{{mR_{s} }}\),
\(a_{25} = \frac{{2C_{f} l_{f} }}{{I_{z} R_{s} }}\),
\(a_{61} = \frac{{2C_{f} \eta_{t} }}{{J_{s} R_{s} v_{x} }}\),
\(a_{62} = \frac{{2C_{f} l_{f} \eta_{t} }}{{J_{s} R_{s} v_{x} }},\)
\(a_{65} =  \frac{{2C_{f} \eta_{t} }}{{J_{s} R_{s}^{2} }},\) where
\(C_{f} ,\;C_{r}\) are front and rear cornering stiffness, respectively,
\(l_{f}\),
\(l_{r}\) are the distance from the CG to the front and rear axle, respectively,
\(\delta_{d}\) is the steering angle of the front wheels,
\(m\) is the total mass,
\(I_{z}\) is the vehicle yaw moment of inertia,
\(J_{s}\) is the equivalent moment of inertia of the steering system,
\(R_{s}\) is the ratio between the front steering wheel to the steering wheel,
\(\eta_{t}\) is tire drag, and
\(B_{s}\) is the steering system equivalent damping coefficient.
$$\begin{aligned} {\dot{\varvec{x}}}_{v} ={\varvec{A}}_{v} {\varvec{x}}_{v}+ {\varvec{B}}_{v} {\varvec{u}}_{v} +{\varvec{H}}_{v} {\varvec{w}}_{v} , \hfill \\ {\varvec{y}}_{v}= {\varvec{C}}_{v} {\varvec{x}}_{v} , \hfill \\ \end{aligned}$$
(2)
$${\varvec{H}}_{v} = \left( {\begin{array}{*{20}c} 0 & 0 & {  v_{x} } & {  L_{near} v_{x} } & 0 & 0 \\ \end{array} } \right)^{\text{T}} ,$$
2.2 Steering System Model
The electric power steering (EPS) system is used not only as a device assisting steering mechanism but also as the actuator of lane departure alert system (LDAS). The steering system consists of a torque and angle sensor, motor and its drive circuit, ECU, and mechanical steering mechanism (see Figure
2).
×
The torque of the steering column is
where
\(\theta_{c}\) is the steering wheel angle,
\(x_{r}\) is the rack displacement,
\(r_{p}\) is the pitch radius of the pinion,
\(J_{c}\),
\(B_{c}\) and
\(K_{c}\) are the inertia moment, damping, and steering column stiffness, respectively, and
\(T_{d}\) is the driver torque.
$$T_{d} = J_{c} \ddot{\theta } + B_{c} \dot{\theta } + K_{c} \theta_{c}  K_{c} \frac{{x_{r} }}{{r_{p} }},$$
(3)
The torque of the motor’s output shaft is
$$T_{m} = J_{m} \ddot{\theta } + B_{m} \dot{\theta } + K_{r} \theta_{m}  NK_{m} \frac{{x_{r} }}{{r_{p} }}.$$
(4)
And the relationship between the motor torque and current is
where
\(\theta_{m}\) is the motor rotation angle,
\(N\) is the transmission ratio of the motor reducer,
\(J_{m}\),
\(B_{m}\) and
\(K_{m}\) are the inertia moment, damping, and motor stiffness, respectively,
\(T_{m}\) is the motor output torque,
\(K_{t}\) is the motor torque constant, and
\(I_{m}\) is the motor armature current.
$$T_{m} = K_{t} I_{m} ,$$
(5)
The force on the rack is [
15]:
where
\(F_{r}\) is the external force acting on the rack, and
\(m_{r}\) is the equivalent mass,
\(B_{r}\) is the equivalent damping, and
\(K_{r}\) is the equivalent stiffness of the gear rack. The actual steering wheel torque,
\(T_{d}\) is measured by a sensor approximating
\(T_{se} = K_{c} \left( {\theta_{c}  \theta_{m} /N} \right)\),
\(T_{a} = NT_{m}\) is motor torque on the steering column, and
\(T_{r} = F_{r} r_{p}\) is the external resistance effect on the tire when the vehicle is running.
$$\begin{aligned} m_{r} \ddot{x}_{r} + B_{r} \dot{x}_{r} + K_{r} x_{r}= \hfill \\ NK_{m} \frac{{(\theta_{m}  N\frac{{x_{r} }}{{r_{p} }})}}{{r_{p} }} + \frac{{K_{c} (\theta c  \frac{{x_{r} }}{{r_{p} }})}}{{r_{p} }}  F_{r} , \hfill \\ \end{aligned}$$
(6)
2.3 Driver Model
Physiologists and psychologists found that the driver’s visual information for steering control includes the far and near areas. Far area information is used to estimate the distance road curvature and near area information is used to compensate position error [
16]. As shown in Figure
1, the near visual point is generally on the midline 5‒10 m in front of the vehicle. The angle between the vertical axis and the line between the near visual point and the vehicle mass center is the near visual angle,
\(\theta_{near}\); this angle reflects the deviation degree of the vehicle. The far visual point is the tangent point on the edge of the inside lane as the vehicle travels around a corner. The ligature between the far visual point and vehicle center is a line tangent to the inside lane edge, and the angle between the tangent and the vehicle longitudinal axis is a far visual angle,
\(\theta_{far}\); this angle is predominantly used to predict the curvature of the road ahead and compensate for the steering wheel angle [
17].
The near and far vision points are
$$\left\{ \begin{aligned} \theta_{near} \approx \frac{{Y_{L} }}{{L_{near} }} + \psi_{L} , \hfill \\ \theta_{far} \approx \frac{{L_{far} }}{{R_{v} }} = L_{far} \frac{r}{{v_{x} }}. \hfill \\ \end{aligned} \right.$$
(7)
The driver model, shown in Figure
3, mainly includes advance steering gain based on visual compensation control,
where
\(T_{L}\) and
\(T_{I}\) are lead and delay time constants, respectively.
\(G_{L} = e^{  \tau _q}\) is the equivalent information delay for the driver’s eye and other sensory organs to the central nervous system and
\(\tau_{q}\) is the driver’s reaction time. Considering the neuromuscular system of the driver’s arm, the firstorder transfer function
\(G_{NM} = 1/\left( {T_{N} s + 1} \right)\) is introduced, where
\(T_{N}\) is the neuromuscular delay time constant [
18]. The driver’s perceptive control of muscle movement consists of perceptual
\(G_{k1} = K_{D} s/\left( {\left( {s + 1} \right)/T_{1} } \right)\) and action
\(G_{k2} = K_{G} \left( {T_{k1} s + 1} \right)/\left( {T_{k2} s + 1} \right)\), where
\(T_{1}\) is the time constant for
\(G_{k1}\), and
\(T_{k1}\) and
\(T_{k2}\) are time constants for
\(G_{k2}\). The parameters of the driver model are presented in Table
1.
$$\begin{aligned} G_{a} = K_{a} , \hfill \\ G_{c} = {{K_{c} (T_{L} s + 1)} \mathord{\left/ {\vphantom {{K_{c} (T_{L} s + 1)} {(T_{I} s + 1)}}} \right. \kern0pt} {(T_{I} s + 1)}} \hfill \\ \end{aligned}$$
(8)
Table 1
Driver model parameters
Parameters

Value


Steering gain
K
_{a}

18

Visual compensation
K
_{c}

10

Visual compensation lead time constant
T
_{L}(s)

2.4

Visual compensation delay time
T
_{I} (s)

0.2

Perceived link gain
K
_{D}

1.1

Action link gain
K
_{G}

− 0.63

Driver reaction time
τ
_{q} (s)

0.12

Neuromuscular hysteresis constant
T
_{N} (s)

0.11

Perceived link constant
T
_{1} (s)

0.046

Action link constant
T
_{k1} (s)

1.99

Action link constant
T
_{k2} (s)

0.013

×
Simplifying the model shown in Figure
3, we obtain
where
$$T_{d} = K_{d1} \theta_{near} + K_{d2} \theta_{far} + K_{d3} \delta_{d} ,$$
(9)
$$\begin{aligned} K_{d1} =  G_{c} G_{L} G_{NM} ,K_{d2} = G_{a} G_{L} G_{NM} \theta_{far} , \hfill \\ K_{d3} =  (G_{k1} G_{k2}  G_{k1} )G_{NM} \delta_{d} \hfill \\ \end{aligned}$$
3 Design of LaneKeeping System
In this paper, a hierarchical lane keeping system is designed, consisting of a decision layer, control layer, and execution layer. The specific structure is shown in Figure
4.
(1)
The decision layer mainly includes the driver status recognition system and lane recognition system based on machine vision. The principle of the driver’s state recognition system is shown in Figure
5. When the steering light turns on, it is regarded as active steering by the driver and the driver dominates the vehicle control. The default normal vehicle driving condition occurs when light is on and the lateral deviation, Y
_{cg}, relative to the lane centerline does not exceed a certain threshold (Y
_{cg}<Y
_{th}). If the lateral deviation exceeds a threshold and the driver is not in the operating state, the controller dominants vehicle driving to achieve rectification control. If the driver does not have enough time to turn on the light, i.e., in an emergency, the driver torque will exceed the threshold (T
_{d}>T
_{th}), regarded as a driver emergency operation. To avoid interference with the controller, the driver should dominate in the vehicle driving; while the driver torque does not exceed the threshold, the controller dominates [
19].
×
(2)
The execution layer mainly consists of the composite controller and control allocation module (see Figure
6). The composite controller consists of a H∞ controller based on the TS model and a feedforward compensation controller. The H∞ controller is designed to obtain the expected assist torque based on the direction deviation and state information of the vehicle, obtained by the lane detection system and the vehicle sensors in realtime. The feedforward compensation controller is used to compensate for the additional torque and reduce the influence of external resistance on the desired target tracking. The control module mainly receives the vehicle’s dominant signal, which is transmitted by the driver’s state recognition system to send the driver torque signal and total assist torque signal to the motor controller.
×
(3)
The execution layer is mainly composed of an EPS motor and its controller; it is used to receive all assist torque signals by the execution layer and subsequently operate the motor to realize active steering for deviation correction, ensuring that the vehicle is always near the centerline of the lane.
×
3.1 H∞ Controller Based on TS Model
3.1.1 DriverVehicleRoad System Based on TS Fuzzy Model
The nonlinear system can be described by the following TS model:
then
where
\(z\left( t \right) = \left( {z_{1} \left( t \right),\;z_{2} \left( t \right), \cdot \cdot \cdot ,\;z_{p} \left( t \right)} \right)\) are fuzzy variables of the drivervehicleroad closedloop system. The system fuzzy variable is
\(v_{x} \left( t \right) = 55,\;70, \cdots ,\;115\;{\text{km/h}}\) and, in this paper, the fuzzy ruler is 5 and the fuzzy membership function is triangular.
\(u_{d} \left( t \right)\) is the input to the system,
\(x_{d} \left( t \right)\) is the state matrix of the system, and
\(y_{d} \left( t \right) = R^{q \times l}\) is the output of the system.
\(A_{di} \in R^{n \times n}\),
\(B_{di} \in R^{n \times l}\),
\(H_{di} \in R^{n \times l}\) and
\(C_{di} \in R^{n \times q}\) are matrices corresponding to the section’s
ith subsystem of the fuzzy system [
20].
$$R_{p}^{i} :{\text{ if}}\quad z_{1} \left( t \right)\;{\text{is}}\;M_{i1} \;{\text{and}}\;z_{p} \left( t \right)z_{p} (t)\;{\text{is}}\;M_{ip} ,$$
$${\dot{\varvec{x}}}_{d} \left( {\varvec{t}} \right)={\varvec{A}}_{di} {\varvec{x}}_{d} \left( {\varvec{t}} \right)+{\varvec{B}}_{di} {\varvec{u}}_{d} \left( {\varvec{t}} \right)+{\varvec{H}}_{di} {\varvec{w}}_{d} \left( {\varvec{t}} \right),$$
$${\varvec{y}}_{d} \left( {\varvec{t}} \right)={\varvec{C}}_{di} {\varvec{x}}_{d} \left( {\varvec{t}} \right),\;i = 1,2, \cdots ,r,$$
(10)
The state parameters of the drivervehicleroad closedloop system are the parameters in the vehicleroad system model, including the driver’s state parameters. The input of the system is
\(u_{d} = T_{a}  T_{r}\), not including driver torque. The state equation of the system model can be expressed as:
where
$${\dot{\varvec{x}}}_{d} = \left[ \begin{aligned} \dot{x}_{v} \hfill \\ \dot{T}_{d} \hfill \\ \end{aligned} \right] = {\varvec{A}}_{dr} \left[ \begin{aligned} x_{v} \hfill \\ T_{d} \hfill \\ \end{aligned} \right] + {\varvec{B}}_{dr} u_{d} + {\varvec{H}}_{dr} w_{d} ,$$
(11)
$$\left[ \begin{aligned} \dot{x}_{v} \hfill \\ \dot{T}_{d} \hfill \\ \end{aligned} \right] = \left[ \begin{aligned} \begin{array}{*{20}c} {A_{v} } & {B_{v} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {A_{dr1} } & 0 \\ \end{array} \hfill \\ \end{aligned} \right]\left[ \begin{aligned} x_{v} \hfill \\ T_{d} \hfill \\ \end{aligned} \right] + \left[ \begin{aligned} B_{v} \hfill \\ 0 \hfill \\ \end{aligned} \right]u_{d} + \left[ \begin{aligned} H_{v} \hfill \\ 0 \hfill \\ \end{aligned} \right]w_{d} ,$$
(12)
$$A_{dr1} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{K_{d1} }}{{L_{near} }} + \frac{{K_{d2} L_{far} a_{21} }}{{v_{x} }}} & {2K_{d1} + a_{22} } & {\frac{{K_{d1} v_{x} }}{{L_{near} }}} & 0 & {a_{25} } \\ \end{array} } & {K_{d3} } \\ \end{array} } \right],$$
X
_{v} is the state parameter of the vehicleroad system, and
A
_{v} and
B
_{v} are state matrices of the vehicleroad system. The fuzzy variables, state parameters, and input to the drivervehicleroad closedloop system are given and the TS fuzzy model is as follows:
where
\(\omega_{di} (z(t)) = \prod\limits_{j = 1}^{p} {M_{ij} (z(t))} = M_{ij} (v_{x} (t))\) is the validity degree of the
ith rule and the value is the product of the membership functions associated with section
i. Furthermore,
$${\dot{\varvec{x}}}_{d} (t) = \frac{{\sum\limits_{r = 1}^{i} {\omega_{di} (z(t))} ({\varvec{A}}_{di} {\varvec{x}}_{d} (t) + {\varvec{B}}_{di} u_{d} (t) + {\varvec{H}}_{di} \omega_{d} (t))}}{{\sum\limits_{r = 1}^{i} {\omega_{di} (z(t))} }},\quad {\varvec{y}}_{d} (t) = \frac{{\sum\limits_{r = 1}^{i} {\omega_{di} (z(t))} {\varvec{C}}_{di} {\varvec{x}}_{d} (t)}}{{\sum\limits_{r = 1}^{i} {\omega_{di} (z(t))} }},$$
$$h_{i} (z(t)) = \frac{{\omega_{di} (z(t))}}{{\sum\limits_{r = 1}^{i} {\omega_{di} (z(t))} }}.$$
The TS fuzzy model of the drivervehicleroad closedloop system is
$$\begin{aligned} \dot{x}_{d} (t) = \sum\limits_{i = 1}^{r} {h_{i} (z(t)){\varvec{A}}_{di} x_{d} (t) + {\varvec{B}}_{di} u_{d} (t) + {\varvec{H}}_{di} w_{d} (t} ), \hfill \\ y_{d} (t) = \sum\limits_{i = 1}^{r} {h_{i} (z(t)){\varvec{C}}_{di} x_{d} (t)}. \hfill \\ \end{aligned}$$
(13)
3.1.2 Design of the H_{∞} Controller Based on the TS Model
In this paper, the output of the controller based on the TS fuzzy model is the desired assist torque for lanekeeping. Local state feedback controllers for 5 subsystems of the drivervehicleroad TS fuzzy model from Eqs. (
12) and (
13) are designed via parallel distributed compensation (PDC).
where
\(K_{i}\) is the state feedback controller of the section of the
ith local linear model. Considering the disturbance of the nonlinear system, the
\(H_{\infty }\) control method should minimize the influence of disturbance on the system. Therefore, for a given scalar, the performance index should satisfy:
$$u_{di} (t) =  K_{i} x_{d} (t),\quad i = 1,2, \ldots, 5,$$
(14)
$$\left\ {G(s)} \right\_{\infty } = \frac{{\left\ y \right\_{2} }}{{\left\ \rho \right\_{2} }} < \gamma .$$
(15)
The smaller the norm of the performance index, the smaller the disturbance to the system.
The necessary conditions for the design of the traditional
\(\text{H}_{\infty }\) state feedback controller is the existence of the positive definite matrix
\(\varvec{X}\) and matrix
\(\varvec{M}\), which satisfy the following formula [
15]:
where
X
_{v} is the state parameters matrix of the reference vehicleroad system model, established in the second section of the present paper, and
A
_{v} and
B
_{v} are the state matrixes of the reference vehicleroad system model. The feedback controller as the reference vehicle road system can be obtained using LMIs. Thus, the traditional
\(\text{H}_{\infty }\) controller for the feedback matrix is as follows:
$$\left[ {\begin{array}{*{20}c} {\varvec{A}_{v} \varvec{X} + \varvec{B}_{v} \varvec{X} + \left( {\varvec{A}_{v} \varvec{X} + \varvec{B}_{v} \varvec{M}} \right)^\text{T} } & {\varvec{H}_{V} } & {\varvec{C}_{V} \varvec{X}^\text{T} } \\ {\varvec{H}^\text{T} } & {  \varvec{I}} & 0 \\ {\varvec{C}_{v} \varvec{X}} & 0 & {  \gamma^{2} \varvec{I}} \\ \end{array} } \right] \le 0,$$
(16)
$$K_{v} = \left( {\begin{array}{*{20}c} {  0.0658} & {0.2357} & {  1.1625} & {2.8865} & {0.0575} & {0.0004} \\ \end{array} } \right).$$
(17)
When designing the local state controller, the feedback gain
\(K_{i}\) and minimum disturbance rejection
\(\gamma\) can be obtained by using the LMI constraint and the linear objective function convex optimization problem, obtaining the optimal solution. The feedback gain and the minimum disturbance rejection are obtained. The necessary condition to satisfy the following formula is the existence of the positive definite matrix [
21]:
where
\({\varvec{Q}} = {\varvec{W}}^{\text{T}} {\varvec{W}},\)
\({\varvec{W}} > {\mathbf{0}}\) indicates a positive definite matrix and
\(M_{i}\) is the fuzzy membership function curve of fuzzy rule
i. The controller parameters of each subsystem can be solved via the LMI toolbox in Matlab. The output of each local subsystem controller is:
$$\begin{array}{*{20}c} {\hbox{min} :\gamma^{2} } \\ {X,M_{1} ,M_{2}, \ldots, M_{r} }, \\ {X > 0}, \\ \end{array} \left[ {\begin{array}{*{20}c} {\varvec{X}\varvec{A}_{di}^\text{T} + \varvec{A}_{id} \varvec{X}  \left( {\varvec{B}_{di} \varvec{M}_{i} + \varvec{M}_{i}^\text{T} \varvec{B}_{di}^\text{T} } \right)} & {\varvec{XC}_{di}^\text{T} } & {\varvec{M}_{i} } \\ {\varvec{C}_{di} \varvec{X}} & {  \varvec{Q}} & 0 \\ {\varvec{M}_{i}^\text{T} } & 0 & {  \gamma^{2} \varvec{I}} \\ \end{array} } \right] \le 0$$
$$u_{di} (t) =  K_{i} x_{d} (t) =  \varvec{M}_{i} \varvec{X}^{  1} \varvec{x}_{d} (t).$$
(18)
The output of the controller based on the TS fuzzy model is
$$u_{d} (t) =  \sum\limits_{i = 1}^{5} {h_{i} (z(t)} )\varvec{K}_{i} \varvec{x}_{d} (t).$$
(19)
The local controller feedback gain is established in the global performance index under the condition of control; when considering the global system control, to ensure the stability of each local subsystem under feedback when considering the global system control, gain subsystems
\({\varvec{K}}_{i} = \left( {k_{1} ,k_{2} ,k_{3} ,k_{4} ,k_{5} ,k_{6} } \right)\) are considered such that global system stability can be guaranteed. The partial parameters of five local subsystem controllers are presented in Table
2.
Table 2
Parameters of H∞ controller of local subsystems
Controller parameters

Vehicle speed
v
_{x} (km/h)



55

70

85

100

115


k
_{1}

− 0.0331

− 0.0675

− 0.0760

− 0.1123

− 0.1235

k
_{2}

0.1668

0.2355

0.3025

0.3265

0.3465

k
_{3}

− 1.0814

− 1.1629

− 1.2369

− 1.3265

− 1.4235

k
_{4}

2.6905

2.9868

3.2957

3.3495

3.6905

k
_{5}

0.0457

0.0584

0.0628

0.0826

0.1623

k
_{6}

0.0002

0.0004

0.0007

0.0015

0.0025

3.2 Feedforward Compensation Controller
A large number of experiments show that the vehicle frontwheel cannot track the desired target in a lanekeeping active steering process. The external resistance during steering is not only related to the speed of the vehicle but also the road adhesion coefficient and the size of the frontwheel angle. A higher tireroad adhesion coefficient corresponds to a higher frontwheel angle, lower speed, and greater resistance on the frontwheel. Therefore, it is necessary to ensure tracking precision for lanekeeping to provide the corresponding additional assist torque, compensating for the effects of external resistance.
This paper aims to improve the lanekeeping performance by considering the road adhesion coefficient, the frontwheel angle, and the vehicle speed variation. The fuzzy rules are as follows. When the road adhesion coefficient and frontwheel angle are large, the additional compensation torque should be increased; when the speed is large, the external compensation torque should be reduced. For calculation convenience, the inputs to the fuzzy controller are the road adhesion coefficient
\(\mu\), frontwheel angle
\(\delta_{d}\), and vehicle speed
\(v_{x}\), the output of the fuzzy controller is the compensating torque proportion
\(\lambda \left( {T_{co} = \lambda T_{a} } \right)\). The domain variation range of
\(\mu\) is
\(U_{u} = \left[ {0.4,\;0.85} \right]\) and its fuzzy subset is
\(\left\{\text{S, M, L} \right\}\), i.e., the small, medium, and large states of the road adhesion coefficient. The domain variation range of
\(v_{x}\) is
\(U_{vx} = \left[ {15,\;35} \right]\) and its fuzzy subset is
\(\left\{ \text{S, M, L} \right\}\). The domain variation range of
\(\delta_{d}\) is
\(U_{\delta d} = \left[ {0,25} \right]\) and its fuzzy subset is
\(\left\{ \text{S, M, L} \right\}\). The domain variation range of
\(\lambda\) is
\(U_{\lambda } = \left[ {0.1,\;0.5} \right]\) and its fuzzy subset is
\(\left\{ \text{VS, S, M, L, VL} \right\}\), these are the smaller compensation, small compensation, medium compensation, high compensation and higher compensation [
22], respectively. The input and output are all triangular membership functions and the fuzzy inference rules are presented in Table
3.
Table 3
Fuzzy inference rules
Road adhesion coefficient
μ

Front wheel angle
δ
_{d}

Vehicle speed
v
_{x}



S

M

L


S

S
M
L

S
M
L

VS
S
L

VS
S
VL

M

S
M
L

S
M
L

VS
M
L

VS
S
VL

L

S
M
L

S
L
VL

VS
M
VL

VS
S
L

4 Simulation Experiment
The EPS model, driver model, and lanekeeping controller module were built in Matlab/Simulink. The Carsim simulation vehicle was a Cclass/sedan. The parameters for the EPS model and vehicle model are given in Table
4.
Table 4
Parameters of vehicle and EPS structure
Parameter

Value


Total mass (kg)

1296

Vehicle yaw moment of inertia
I
_{z} (kg·m
^{2})

1750

Distance form CG to front axle
L
_{f} (m)

1.01

Distance from CG to rear axle
L
_{r} (m)

1.56

Front cornering stiffness
C
_{f} (N/rad)

35000

Rear cornering stiffness
C
_{r} (N/rad)

42000

Inertia moment of steering column
J
_{c} (kg·m
^{2})

0.046

Damping of steering column
B
_{c} (N·m·s/rad)

0.361

Stiffness of steering column
K
_{c} (N·m/rad)

115

Equivalent mass of gear rack
m
_{r} (kg)

32

Equivalent damping of gear rack
B
_{r} (N·s/m)

3620

Equivalent stiffness of gear rack
K
_{r} (N/m)

46000

Inertia moment of motor
J
_{m} (kg·m
^{2})

0.00045

Damping of motor
B
_{m} (N·m·s/rad)

0.0033

Transmission ratio of motor reducer
n

22

Gear ratio
i
_{sw}

16.5

Straight road simulation conditions: The vehicle ran in a straight line at a fixed speed and the steering wheel angle was fixed at 10° due to the driver misoperation at 4.5 s, while the steering wheel torque did not exceed the threshold value.
The lane keeping assist system turned on when the lateral deviation reached a certain threshold and the EPS then received a lanekeeping assist torque signal sent by a controller; the control motor subsequently steered to correct deviation and direction. Vehicle lateral deviation and assist torques under speeds of 55 km/h, 70 km/h, and 85 km/h are compared, as shown in Figure
7.
×
Curve road simulation conditions: the vehicle trajectory is shown in Figure
8, the curvature radius of the first, second, and third curve is 155 m, 150 m, and 125 m, respectively.
×
First simulation conditions (low road adhesion coefficient): the road adhesion coefficient was 0.5. The variation in speed was 55‒115 km/h, as shown in Figure
9. Figure
10 Compares the vehicle lateral position deviation, vehicle yaw angle, and yaw rate, under the control of the H
_{∞} controller, the H
_{∞} controller based on the TS model, and the composite controller (feedforward control and H
_{∞} control based on TS).
×
×
Second simulation conditions (high road adhesion coefficient): the road adhesion coefficient was 0.85. Figure
11 Compares the vehicle lateral position deviation, vehicle yaw angle and yaw rate under the control of the H
_{∞} controller, H
_{∞} controller based on the TS model, and the composite controller (feedforward control and H
_{∞} control based on TS).
×
It is shown in Figure
7 that the driver state recognition system in the decision layer of the lanekeeping system can identify the status of the driver; when the driver does not operate the vehicle or misoperates the vehicle, the system can switch vehicle control to ensure that the direction of the vehicle can be immediately corrected and the vehicle travels in the correct lane under the control of the assistance system, guaranteeing traffic safety.
(1)
It is shown in Figures
10, 11 that the H
_{∞} control method based on the TS drivervehicleroad model can ensure that the vehicle travels in the centerline of the lane at different speeds, and both the lateral deviation and yaw angle are within the safe range. This control method can ensure the stale lanekeeping, and, among the three methods, the vehicle has better stability under composite control.
(2)
It is shown in Figures
10, 11 that the H
_{∞} control method based on the TS model can ensure that both the vehicle lateral deviation and yaw angle are within a safe range on roads with both a high and low adhesion coefficient. The vehicle driving deviation and performance index of the three methods, with a high adhesion coefficient, are compared and listed in Table
5. As Table
5 shows, compared with
\(\text{H}_{\infty }\) control, the vehicle lateral deviation is reduced by 39.81% and the performance index is increased by 28.08% under the
\(\text{H}_{\infty }\) control based TS model. Vehicle lateral deviation is reduced by 51.52%, and the performance index is increased by 42.44% under composite control (
\(\text{H}_{\infty }\) control based TS model and feedforward control). The results show that the composite control method has smaller lateral deviation and a smaller performance index, reflecting the higher lanekeeping accuracy.
Table 5
Results of lateral deviation and performance index
Evaluation

H
_{∞} control

H
_{∞} control based on TS model

composite control


Deviation peak (m)

0.332

0.218

0.183

Deviation square root

0.0338

0.02496

0.02115


G(
s)
_{∞}

41.42

29.79

23.84

5 Real Vehicle Experiment
To evaluate the reliability of the LKAS more accurately, a real vehicle test was conducted. The specific process is as follows.
(1)
When the vehicle is driving on the road, the vehicle camera mounted on the rearview mirror obtains a forward road image, which is sent to the digital signal processor (DSP) for lane detection and identification to determine lane information, i.e., lane shape, position, and vehicle information, direction, etc.
(2)
The vehicle position and direction deviation related to lane line are obtained through the transformation of the actual road environment and image coordinates.
(3)
DSP sends the information in the form of CAN to the CompactRIO device interface system.
(4)
Via speed, steering wheel angle, and torque sensors, the system obtains realtime vehicle speed,
\(v_x\), steering wheel torque measurement,
\(T_{se}\), and motor rotation signal,
\(\theta_m\), and sends these signals to the CompactRIO interface system.
(5)
The CompactRIO interface system calculates the assist torque that the vehicle requires, according to the location information and the vehicle state information, which is received from the vehicle sensors.
(6)
The system sends the expected assist torque signal and the feedforward compensation control signal to the independently researched and developed motor angle controller. Then, the motor controller drive tracks the desired angle and performs active steering correction, finally achieving lanekeeping. The specific test flow is shown in the Figure
12.
×
Experimental equipment included the experiment vehicle, steering wheel angle/torque sensor, gyroscope, VBOX (with GPS), CompactRIO, digital camera, image processor, motor controller and a CAN interface to connect the controllers, etc. The test equipment was installed in the experiment vehicle. Except for the accelerator pedal, controlled by the driver, the vehicle steering wheel was controlled by the controller and the system used a lane detection system installed in the vehicle to conduct automatic driving.
During the experiment, the road surface was processed and the high and low road adhesion coefficients were obtained. The speed ranged from 30 km/h to 65 km/h. The results are shown in Figure
13, comparing the lateral deviation under H
_{∞} control based on the TS model and composite control with feedforward compensation. The results of the real vehicle and simulation experiment with a high road adhesion coefficient are compared, as shown in Table
6. The experimental results show that the proposed control method ensures that the vehicle tracks the lane and achieves lanekeeping. Furthermore, they also show that the lanekeeping effect improves with a low adhesion coefficient condition, vehicle lateral deviation is smaller under composite control, and lanekeeping performance improves.
Table 6
Results of simulation experiment and real vehicle experiment
Evaluation

Simulation

Real vehicle



Compensation

Compensation


No

Yes

No

Yes


Deviation peak (m)

0.218

0.183

0.2901

0.2714

Deviation square root

0.02496

0.02115

0.0249

0.0212


G(
s)
_{∞}

29.79

23.84

51.57

47.32

×
6 Conclusions
(1)
The drivervehicleroad closedloop system based on the TS model with the fuzzy variable of the longitudinal velocity is established. The lanekeeping controller was designed by the twopoint preview driver model. The experiment results show that the designed controller can maintain the vehicle in proximity to the lane centerline and the system has good robustness and stability.
(2)
The driver state identification module can identify the driver status well and ensure that the LKAS can control the vehicle direction when the vehicle deviates from the correct route and the lateral deviation reaches a certain threshold in the driver unconscious or incorrect operation condition. Therefore, the vehicle is always traveling near the centerline, and traffic safety is guaranteed.
(3)
The
\(\text{H}_{\infty }\) controller based on the TS model is designed for the desired assist torque and considers external resistance affecting the wheel when steering. The feedforward compensation controller is designed to provide additional assist torque to compensate for external resistance. The simulation results show that the precision of lanekeeping is higher under composite control.
(4)
The lane recognition system, based on machine vision, and lanekeeping control system are installed on the real vehicle. The image processor, decision controller, and motor controller communicate via CAN. Multiple real vehicle tests were conducted to demonstrate that the vehicle experiment platform was built to achieve the lanekeeping function and to guarantee the stability of the vehicle during the process.
Acknowledgements
The authors would like to thank Dr. Dongkui Tan and Dr. Zhenya Wei for their guidance and assistance in the vehicle tests.
Competing Interests
The authors declare that they have no competing interests.
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