Adaptive neural network sliding mode control for steer-by-wire-based vehicle stability control

Abstract

This study develops a novel vehicle stability control (VSC) scheme using adaptive neural network sliding mode control technique for Steer-by-Wire (SbW) equipped vehicles. The VSC scheme is designed in two stages, i.e., the upper and lower level control stages. An adaptive sliding mode yaw rate controller is first proposed as the upper one to design the compensated steering angle for enabling the actual yaw rate to closely follow the desired one. Then, in the implementation of the yaw control system, the developed steering controller consists of a nominal control and a terminal sliding mode compensator where a radial basis function neural network (RBFNN) is adopted to adaptively learn the uncertainty bound in the Lyapunov sense such that the actual front wheel steering angle can be driven to track the commanded angle in a finite time. The proposed novel stability control scheme possesses the following prominent superiorities over the existing ones: (i) No prior parameter information on the vehicle and tyre dynamics is required in stability control, which greatly reduces the complexity of the stability control structure. (ii) The robust stability control performance against parameter variations and road disturbances is obtained by means of ensuring the good tracking performance of yaw rate and steering angle and the strong robustness with respect to large and nonlinear system uncertainties. Simulation results are demonstrated to verify the superior control performance of the proposed VSC scheme.

Keywords

Finite time convergence radial basis function neural network robustness steer-by-wire

1 Introduction

In the past few decades, research on neural networks (NNs) in the control field has been attracting a considerable attention. The reasons are that: (i) Most industrial control systems exhibit a wide range of unknown nonlinearities and external disturbances; (ii) Artificial NNs have strong ability to approximate arbitrary linear or nonlinear functions to any desired degree of accuracy under certain conditions.

Many studies focus on applying NNs to model unknown functions in dynamical control systems [1 –4]. It has been seen from these methods that the neural network (NN) is directly used to learn the system uncertainty through back-propagation and radial basis function NNs [3, 4], based on which the control design can be formulated. However, due to the approximation error between the system uncertainties and the NN outputs in practical control applications, how to guarantee sufficiently accurate estimations is an open issue. Recently, several NN control schemes have been developed based on the Lyapunov analysis [1 , 5–7]. The main advantage of these control methods is that the adaptive laws for the system uncertainty is derived in the sense of Lyapunov such that the closed-loop system stability can be ensured. Furthermore, it has been seen from [8 –11] that, the NNs are adopted in the NN-based sliding mode (SM) control schemes to adaptively learn the system uncertainty bounds and the outputs of the NNs can adaptively update the SM control gain. Thus, not only the effects of the lumped uncertainty can be removed, but also the asymptotical error convergence can be achieved.

It is well known that the vehicle stability control (VSC) systems, also referred to as yaw stability control systems, enable vehicles to avoid spinning and drifting out. Recently, modern Steer-by-Wire based VSC systems have received considerable attention among the automotive industry in terms of improving handling behaviors and stability. Generally, in VSC systems, vehicle yaw rate and side slip angle play an important role in stabilizing vehicle motion. Although the low-cost gyro sensors can be used to measure the actual yaw rate [12], the side slip angle information is difficult to obtain owing to the high-cost sensors. Thus, there is a growing trend of estimating vehicle side slip angle with the help of the available sensors, such as steering angle sensors, gyro sensors, and lateral acceleration sensors [13 –16]. In [13], a novel linear slip angle observer by using pole placement method was developed to ensure a smaller estimation error and robustness property with respect to disturbances. In [14], an identification method of the tyre cornering stiffness using recursive least-squares algorithm was incorporated into the slip angle observer design to improve the estimation accuracy. In [15], the estimation of side slip angle was obtained by combining the kinematical formula with the state observer based on the measurements of vehicle lateral acceleration, speed and yaw rate. In [16], an observer using an Extended Kalman Filter method was proposed to provide side slip angle information with the robustness property against the variations of the cornering stiffness parameters. However, since the performance of the VSC systems largely depends on the estimation accuracy of the slip angle, whether the tyre slip angle for the VSC systems can be accurately and robustly on-line estimated under the disturbances is still an open issue. In addition, due to the use of many sensors, to implement real-time estimation becomes difficult and complicated in the VSC systems.

On the other hand, in the implementation of the yaw control system of the SbW equipped vehicles, due to the fact that the actual steering angle is generated via the front wheel steering motor, steering control becomes of vital importance to drive the actual steering angle to exactly track the reference angle provided by the yaw control. Note that the modern SbW systems which replace the mechanical steering shaft between the hand-wheel and front wheels with electric motors and sensors involve various types of nonlinearities and disturbances, such as Coulomb friction, tyre self-aligning torque and so on. Thus, in order to ensure the robust steering performance under parameter uncertainties and unpredicted road changes, many control techniques have been developed for SbW systems, for instance, conventional proportional-derivative (PD) control [17 –20], state feedback control using linear quadratic technique [21], PD control with online estimation for tyre disturbances [22], adaptive pole placement control [23]. Nevertheless, using these control schemes, excellent steering performance cannot be guaranteed, especially when road conditions are significantly changing. Recently, due to the advantages of simplicity and robustness against parameter variations and disturbances, SM techniques [24 –26] has been used in SbW control systems using the bound information of uncertain parameters and disturbances, where the commonly used linear sliding surface is adopted [27 –30]. However, how the proper bound of the lumped uncertainty of SbW systems can be obtained to ensure a robust steering performance and avoid control saturation is still a challenging issue. Motivated by the benefits of NN control, we thus apply the NNs to SbW control systems for adaptively estimating the lumped uncertainty bound such that not only the effects of the lumped uncertainty can be removed, but also the control scheme can be easily implemented.

In this paper, we will develop a novel VSC scheme using adaptive sliding mode technique for the SbW equipped vehicles under the system parameter uncertainties and varying road conditions. The developed stability control scheme is composed of two stages, i.e., the upper and lower level control stages, respectively. First, at the upper level controller, an adaptive sliding mode yaw rate controller (ASM-YRC) is proposed to make the actual yaw rate track the desired one. It is worth noting that, since the upper bound of the vehicle parameters and uncertainties are adaptively estimated in Lyapunov sense, the estimations of the tyre cornering stiffness and slip angles in yaw control design are no longer required. Second, in the SbW control systems, the proposed steering control is composed of a nominal control and a terminal sliding mode (TSM) compensator using a radial basis function neural network (RBFNN) for estimating the uncertainty bound. It will be shown from the simulation results that the proposed neural control scheme exhibits excellent steering performance and behaves with strong robustness with respect to parameter variations and external varying road disturbances. Also, it will be seen that due to the adopted RBFNN for adaptively updating the lumped uncertainty bound, not only the prior knowledge of the lumped uncertainty bound is not required resulting in the simplicity of the control design, but also the control gain is greatly reduced compared with the SM-based SbW control systems. As a result, it will be shown from the simulation results in this paper that, the proposed VSC scheme is capable of improving vehicle stability on different road environment and driving conditions and ensuring the good VSC performance in terms of the tracking performance for both vehicle yaw rate and steering angle.

The rest of the paper is organized as follows. In Section 2 and 3, the modelling of vehicle dynamics is formulated, and the proposed VSC scheme as well as stability analysis is presented. In Section 4, the plant model of SbW systems with uncertain dynamics is discussed. In Section 5, the proposed steering control scheme in which the RBFNN estimator is used in the compensator to compensate the effects of the lumped uncertainty is described. Simulation results are demonstrated to validate the excellent performance of the proposed VSC scheme in Section 6. Section 7 concludes the paper and gives future work.

2 Modelling of vehicle dynamics

In this section, for design simplicity, vehicle dynamics in the horizontal plane is represented by a linear bicycle model using the state variables as vehicle body slip angle β and yaw rate γ, as shown in Fig. 1. In Fig. 1, the two front-wheels and the two rear-wheels of the vehicle are represented by a central front-wheel (CFW) and a central rear-wheel (CRW), respectively, V_CG is the vehicle velocity at the centre of gravity (CG), δ_f is the steering angle of the CFW, V_x and V_y are the longitudinal and lateral components of the CG velocity, V_f and V_r are the velocities of the CFW and CRW, l_f and l_r are the distances of the CFW and the CRW from the CG of the vehicle, $F_{f}^{y}$ and $F_{r}^{y}$ are the lateral forces of the CFW and the CRW, respectively, α_f and α_r are the tire slip angles of the CFW and the CRW, respectively, β is the vehicle body slip angle at CG, and γ is the vehicle yaw rate.

The dynamical equations about the lateral and yaw motions are written as follows [18, 22]: ${MV}_{x} (\dot{β} + γ) = F_{f}^{y} cos δ_{f} + F_{r}^{y}$ (1) $I_{z} \dot{γ} = l_{f} F_{f}^{y} cos δ_{f} - l_{r} F_{r}^{y}$ (2) where M is the vehicle mass and I_z is the moment of inertia of vehicle with respect to its yawaxis.

At a small slip angle, such as 4 degrees or less, the tyre lateral forces $F_{f}^{y}$ and $F_{r}^{y}$ are linearly related to the tire slip angles α_f and α_r, which can be modelled as follows [18, 36]: $F_{f}^{y} = - C_{f} . α_{f}$ (3) $F_{r}^{y} = - C_{r} . α_{r}$ (4) where C_f and C_r are the cornering stiffness coefficients of the front tyre and rear tyre, respectively, which are parameters closely related to the tire-road friction. Assuming that the vehicle body slip angle is close to zero, we can approximately calculate α_f and α_r [14, 18]: $α_{f} = β + \frac{γ . l_{f}}{V_{x}} - δ_{f}$ (5) $α_{f} = β - \frac{γ . l_{r}}{V_{x}}$ (6)

Taking the small angle approximations for (1) and (2) (i.e., cos(δ_f) ≈ 1), we obtain the following linearized dynamical equations related to the lateral and yaw motions: $\begin{matrix} {MV}_{x} \dot{β} + \frac{1}{V} ({MV}_{x}^{2} + C_{f} l_{f} - C_{r} l_{r}) γ \\ + (C_{f} + C_{r}) β - C_{f} δ_{f} = 0 \end{matrix}$ (7) $\begin{matrix} I_{z} \dot{γ} + \frac{1}{V_{x}} (C_{f} l_{f}^{2} + C_{r} l_{r}^{2}) γ - (C_{r} l_{r} - C_{f} l_{f}) β \\ - C_{f} l_{f} δ_{f} = 0 \end{matrix}$ (8)

Rewritting (7) and (8) into state space form, we obtain $\dot{y} = Ay + b δ_{f}$ (9) where $y = {[β γ]}^{T}$ $A = [\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}] = [\begin{matrix} \frac{- C_{f} - C_{r}}{{MV}_{x}} & - 1 + \frac{C_{r} l_{r} - C_{f} l_{f}}{{MV}_{x}} \\ \frac{C_{r} l_{r} - C_{f} l_{f}}{I_{z}} & \frac{- C_{f} l_{f}^{2} - C_{r} l_{r}^{2}}{I_{z} V_{x}} \end{matrix}]$ $b = [\begin{matrix} b_{11} \\ b_{21} \end{matrix}] = [\begin{matrix} \frac{C_{f}}{{MV}_{x}} \\ \frac{C_{f} l_{f}}{I_{z}} \end{matrix}]$

Thus, based on the state-space equation in (9), we obtain the simplified yaw dynamic equation given as follows: $\dot{γ} = a_{21} β + a_{22} γ + b_{21} δ_{f}$ (10)

Re-arranging (10), we have $d_{1} \dot{γ} = d_{2} + d_{3} γ + δ_{f}$ (11) where $d_{1} = \frac{1}{b_{21}}$ , $d_{2} = \frac{a_{21}}{b_{21}} β$ , and $d_{3} = \frac{a_{22}}{b_{21}}$ . Let the upper bounds of a₂₁, a₂₂ and β be ${\bar{a}}_{21}$ , ${\bar{a}}_{22}$ and $\bar{β}$ , respectively, and the lower bound of b₂₁ be b_21,0. The following bound conditions can always be satisfied: $| d_{1} | \leq {\bar{d}}_{1}$ (12) $| d_{2} | \leq {\bar{d}}_{2}$ (13) $| d_{3} | \leq {\bar{d}}_{3}$ (14) where ${\bar{d}}_{1} = \frac{1}{b_{21, 0}}$ , ${\bar{d}}_{2} = \frac{{\bar{a}}_{21}}{b_{21, 0}} \bar{β}$ and ${\bar{d}}_{3} = \frac{{\bar{a}}_{22}}{b_{21, 0}}$ are all positive constants.

3 Vehicle yaw stability control

In this paper, the target of vehicle stability control is to improve the vehicle handling responses in terms of steady and transient response characteristics, i.e., the actual vehicle yaw rate γ is able to exactly track the reference yaw rate for an active steering vehicle. Figure 2 shows the overall vehicle stability control diagram, where the augmented front wheel steering angle input $δ_{f}^{*}$ is defined as follows [13]: $δ_{f}^{*} = δ_{cmd} + Δ δ$ (15) where δ_cmd is the hand-wheel commanded steering angle and Δδ is the compensated steering angle provided by the yaw rate controller.

Similar to [22], in terms of the desired vehicle model response, the desired vehicle yaw rate can be obtained from the given steering angle, vehicle speed and vehicle parameters as follows: $γ_{ref} = \frac{ω_{ref}}{s + ω_{ref}} γ_{ss} δ_{cmd}$ (16a) where ω_ref is the cutoff frequency of the desired model filter and γ_ss is given by $γ_{ss} = \frac{2 V_{x} C_{f} C_{r} (l_{f} + l_{r})}{2 C_{f} C_{r} {(l_{f} + l_{r})}^{2} - {MV}_{x}^{2} (l_{f} C_{f} - l_{r} C_{r})}$ (16b)

The tracking error between the actual yaw rate γ and the desired yaw rate γ_ref is defined as $z = γ - γ_{ref}$ (17)

A sliding variable is then defined as $s_{1} = z$ (18)

Thus, the time derivative of (18) is expressed as

$\begin{matrix} {\dot{s}}_{1} & = & \dot{γ} - {\dot{γ}}_{ref} = a_{21} β + a_{22} γ + b_{21} δ_{f} - {\dot{γ}}_{ref} \\ = & a_{21} β + a_{22} γ + b_{21} (δ_{cmd} + Δ δ) - {\dot{γ}}_{ref} \\ = & a_{21} β + a_{22} γ + b_{21} δ_{cmd} + b_{21} Δ δ - {\dot{γ}}_{ref} \end{matrix}$ (19)

Re-arranging (19), we obtain $\frac{1}{b_{21}} {\dot{s}}_{1} = \frac{a_{21}}{b_{21}} β + \frac{a_{22}}{b_{21}} γ + δ_{cmd} + Δ δ - \frac{1}{b_{21}} {\dot{γ}}_{ref}$ (20)

Following (11), (20) becomes $d_{1} {\dot{s}}_{1} = d_{2} + d_{3} γ + δ_{cmd} + Δ δ - d_{1} {\dot{γ}}_{ref}$ (21)

In this paper, the designed ASM-YRC is proposed as follows: $Δ δ = - sign (s_{1}) [{\hat{\bar{d}}}_{2} + {\hat{\bar{d}}}_{3} | γ | + | δ_{cmd} | + {\hat{\bar{d}}}_{1} | {\dot{γ}}_{ref} |]$ (22) where sign (s₁) is the sign function with $sign (s_{1}) = {\begin{matrix} 1 & for s_{1} > 0 \\ 0 & for s_{1} = 0 \\ - 1 & for s_{1} < 0 \end{matrix}$ (23) and the adaptive laws are derived as $\overset{\cdot}{\hat{{\bar{d}}_{1}}} = η_{1} | s_{1} | | {\dot{γ}}_{ref} |$ (24) $\overset{\cdot}{\hat{{\bar{d}}_{2}}} = η_{2} | s_{1} |$ (25) $\overset{\cdot}{\hat{{\bar{d}}_{3}}} = η_{3} | s_{1} | | γ |$ (26) where η_i, (i = 1, 2, 3) are positive adaptation gains.

For the case that the upper bounds of d₁, d₂, and d₃ in (12–14) are unknown, the design of the ASM-YRC and the analysis of the error convergence are given in the following theorem:

Theorem 3.1. Consider the yaw motion model in (10), if the yaw rate control input in (22) with the adaptive laws in (24–26) is used, the yaw rate tracking error z defined in (17) will converge to zero asymptotically.

Proof. Consider a Lyapunov function $V = \frac{d_{1} s_{1} s_{1}^{2}}{2} + \frac{η_{1}^{- 1}}{2} {\tilde{\bar{d}}}_{1}^{2} + \frac{η_{2}^{- 1}}{2} {\tilde{\bar{d}}}_{2}^{2} + \frac{η_{3}^{- 1}}{2} {\tilde{\bar{d}}}_{3}^{2}$ (27) where ${\tilde{\bar{d}}}_{1} = {\hat{\bar{d}}}_{1} - {\bar{d}}_{1}$ , ${\tilde{\bar{d}}}_{2} = {\hat{\bar{d}}}_{2} - {\bar{d}}_{2}$ , ${\tilde{\bar{d}}}_{3} = {\hat{\bar{d}}}_{3} - {\bar{d}}_{3}$ . The time derivative of the Lyapunov function is obtained as $\begin{matrix} \dot{V} & = & d_{1} s_{1} {\dot{s}}_{1} + η_{1}^{- 1} {\tilde{\bar{d}}}_{1} \overset{\cdot}{\hat{{\bar{d}}_{1}}} + η_{2}^{- 1} {\tilde{\bar{d}}}_{2} \overset{\cdot}{\hat{{\bar{d}}_{2}}} + η_{3}^{- 1} {\tilde{\bar{d}}}_{3} \overset{\cdot}{\hat{{\bar{d}}_{3}}} \\ = & d_{1} s_{1} (\dot{γ} - {\dot{γ}}_{ref}) + η_{1}^{- 1} {\tilde{\bar{d}}}_{1} \overset{\cdot}{\hat{{\bar{d}}_{1}}} + η_{2}^{- 1} {\tilde{\bar{d}}}_{2} \overset{\cdot}{\hat{{\bar{d}}_{2}}} + η_{3}^{- 1} {\tilde{\bar{d}}}_{3} \overset{\cdot}{\hat{{\bar{d}}_{3}}} \\ = & d_{1} s_{1} \dot{γ} - d_{1} s_{1} {\dot{γ}}_{ref} + η_{1}^{- 1} {\tilde{\bar{d}}}_{1} \overset{\cdot}{\hat{{\bar{d}}_{1}}} + η_{2}^{- 1} {\tilde{\bar{d}}}_{2} \overset{\cdot}{\hat{{\bar{d}}_{2}}} \\ + η_{3}^{- 1} {\tilde{\bar{d}}}_{3} \overset{\cdot}{\hat{{\bar{d}}_{3}}} \\ = & d_{2} s_{1} + d_{3} s_{1} γ + s_{1} δ_{cmd} + s_{1} Δ δ - d_{1} s_{1} {\dot{γ}}_{ref} \\ + η_{1}^{- 1} {\tilde{\bar{d}}}_{1} \overset{\cdot}{\hat{{\bar{d}}_{1}}} + η_{2}^{- 1} {\tilde{\bar{d}}}_{2} \overset{\cdot}{\hat{{\bar{d}}_{2}}} + η_{3}^{- 1} {\tilde{\bar{d}}}_{3} \overset{\cdot}{\hat{{\bar{d}}_{3}}} \\ = & d_{2} s_{1} + d_{3} s_{1} γ + s_{1} δ_{cmd} \\ - | s_{1} | [{\hat{\bar{d}}}_{2} + {\hat{\bar{d}}}_{3} | γ | + | δ_{cmd} | + {\hat{\bar{d}}}_{1} | {\dot{γ}}_{ref} |] \\ - d_{1} s_{1} {\dot{γ}}_{ref} + η_{1}^{- 1} {\tilde{\bar{d}}}_{1} \overset{\cdot}{\hat{{\bar{d}}_{1}}} + η_{2}^{- 1} {\tilde{\bar{d}}}_{2} \overset{\cdot}{\hat{{\bar{d}}_{2}}} \\ + η_{3}^{- 1} {\tilde{\bar{d}}}_{3} \overset{\cdot}{\hat{{\bar{d}}_{3}}} \\ = & d_{2} s_{1} + d_{3} s_{1} γ + s_{1} δ_{cmd} - | s_{1} | {\hat{\bar{d}}}_{2} - | s_{1} | {\hat{\bar{d}}}_{3} | γ | \\ - | s_{1} | | δ_{cmd} | - | s_{1} | {\hat{\bar{d}}}_{1} | {\dot{γ}}_{ref} | - d_{1} s_{1} {\dot{γ}}_{ref} \\ + η_{1}^{- 1} ({\hat{\bar{d}}}_{1} - {\bar{d}}_{1}) η_{1} | s_{1} | | {\dot{γ}}_{ref} | \\ + η_{2}^{- 1} ({\hat{\bar{d}}}_{2} - {\bar{d}}_{2}) η_{2} | s_{1} | \\ + η_{3}^{- 1} ({\hat{\bar{d}}}_{3} - {\bar{d}}_{3}) η_{3} | s_{1} | | γ | \\ = & d_{2} s_{1} + d_{3} s_{1} γ + s_{1} δ_{cmd} - | s_{1} | | δ_{cmd} | - d_{1} s_{1} {\dot{γ}}_{ref} \\ - {\bar{d}}_{1} | s_{1} | | {\dot{γ}}_{ref} | - {\bar{d}}_{2} | s_{1} | - {\bar{d}}_{3} | s_{1} | | γ | \\ \leq & - ({\bar{d}}_{1} - | d_{1} |) | s_{1} | | {\dot{γ}}_{ref} | - ({\bar{d}}_{2} - | d_{2} |) | s_{1} | \\ - ({\bar{d}}_{3} - | d_{3} |) | s_{1} | | γ | \\ \leq & - ({\bar{d}}_{1} - | d_{1} |) | s_{1} | | {\dot{γ}}_{ref} | < 0 for | s_{1} | \neq 0 \end{matrix}$ (28)

Thus, expression (28) indicates that the sliding variable s₁ will asymptotically converge to zero and then, on the sliding mode surface, the actual yaw rate γ will track the desired yaw rate γ_ref [34].

As seen from Theorem 3.1 and the stability proof, the following facts have been noted: First, the adaptive laws in (24–26) are designed in the sense of Lyapunov, which indicates that ${\hat{\bar{d}}}_{i} (i = 1, 2, 3)$ , the estimates of ${\bar{d}}_{i} (i = 1, 2, 3)$ , do not converge to their exact values; However, they are adjusted as the specific sliding mode control parameters, in the sense that the sliding variable s₁ can converge to zero asymptotically. Then the yaw rate tracking error will converge to zero. Second, it is seen from the adaptive laws in (24–26) that, no prior information about the vehicle parameters, such as vehicle body slip angle β and the cornering stiffness coefficients of the front tyre and rear tyre C_f and C_r, is required for adaptively estimating ${\bar{d}}_{i} (i = 1, 2, 3)$ in (12)–(14) in Lyapunov sense. This is one of significant merits of the proposed ASM-YRC for practical application.

In order to eliminate the effects of the uncertainties, the sign function is involved in the yaw rate control signal in (22), which ensures that not only the influences of uncertain system dynamics can be removed but also the asymptotic convergence of the closed-loop system is ensured. However, the chatters lead to a wide frequency bandwidth of the controlled system and may excite undesired high frequency response in the closed-loop system. In addition, the augmented front wheel steering angle with the help of the yaw rate control in (15) will serve as the commanded signal for the steering angle control of the SbW systems in the next section, and thus the continuous signal should be desired. Using the boundary layer control technique in [27, 31], we can modify the yaw rate control law in (22) as follows: $Δ δ = - sat (s_{1}) [{\hat{\bar{d}}}_{2} + {\hat{\bar{d}}}_{3} | γ | + | δ_{cmd} | + {\hat{\bar{d}}}_{1} | {\dot{γ}}_{ref} |]$ (29) where $sat (s_{1}) = {\begin{matrix} \frac{s_{1}}{δ_{1}} & for | s_{1} | < δ_{1} \\ sign (s_{1}) & for | s_{1} | \geq δ_{1} \end{matrix}$ (30) with the constant δ₁ is positive.

Expression (29) is called the boundary layer adaptive sliding mode yaw rate controller (BL-ASMYRC). As shown in [27], the zero-error convergence will be sacrificed as the sign function is replaced by the sigmoid function. However, through the proper selection of the value of the positive constant δ₁, the tracking error can be small enough to meet the tracking precision requirement in practice.

4 Modelling of SbW systems

In the previous section, the compensated steering angle Δδ has been designed to ensure that the actual yaw rate is able to closely follow the reference yaw rate. However, in this paper, the actual steering angle is generated by the steering mechanism of the SbW systems including the front wheel steering motor and the rack and pinion gear drive train. Thus, the control objective of the steering system is to drive the actual front wheel steering angle δ_f to well track the reference signal given by the augmented steering angle $δ_{f}^{*}$ in (15), with a strong robustness with regard to the uncertain steering parameters and varying road conditions. The integral control scheme of the proposed stability control system including the proposed yaw motion control and steering control for the SbW equipped vehicles is summarized in Fig. 3.

A standard SbW system is divided into two parts: the hand-wheel and front wheel parts, respectively. For the hand-wheel part, a motor termed as hand-wheel feedback motor (HFM), is coupled to the hand-wheel column for providing a driver with force feedback. In terms of the front wheel part, a motor called front-wheel steering motor (FSM) is installed on the pinion side to steer the two front wheels closely following the hand-wheel reference command through the retained rack and pinion gearbox. Figure 4 clearly shows the main components of the SbW system, addressing that the mechanical shaft is replaced by two electric DC motors.

In [27], we have presented a complete mathematical model for the SbW system. To proceed with the control design, we will give the plant model of the SbW system as follows: $J_{eq} {\ddot{δ}}_{f} + B_{eq} {\ddot{δ}}_{f} + F_{s} sign ({\dot{δ}}_{f}) + τ_{e} = k_{r} u$ (31) where F_s is the Coulomb friction constant, δ_f is the front wheel steering angle, k_r is the combined steering ratio, u is the FSM control torque, sign(.) denotes the standard signum function, J_eq and B_eq are the total inertia and damping coefficient of the SbW system model, respectively, which are defined as $J_{eq} = J_{fw} + {(k_{r})}^{2} J_{sm}$ (32a) $B_{eq} = B_{fw} + {(k_{r})}^{2} B_{sm}$ (32b) where J_fw and J_sm are the moments of inertia of the front wheels and FSM, respectively, and B_fw and B_sm are the viscous friction of the front wheels and FSM.

The Equation (31) can also be written as: $a {\ddot{δ}}_{f} + b {\dot{δ}}_{f} + F_{sa} sign ({\dot{δ}}_{f}) + τ_{ea} = u$ (33) where $a = \frac{J_{eq}}{k_{r}}$ (34) $b = \frac{B_{eq}}{k_{r}}$ (35) $F_{sa} = \frac{F_{s}}{k_{r}}$ (36) $τ_{ea} = \frac{τ_{e}}{k_{r}}$ (37)

In this paper, we consider the parametric uncertainty as follows: $| a - a_{0} | \leq \bar{a}$ (38) $| b - b_{0} | \leq \bar{b}$ (39) $| F_{sa} - F_{sa 0} | \leq {\bar{F}}_{sa}$ (40) $| τ_{ea} - τ_{ea 0} | \leq {\bar{τ}}_{ea}$ (41) where a₀, b₀, F_sa0 represent the nominal model parameters, and τ_ea0 is the predefined nominal tyre disturbance on the specific wet asphalt road. $\bar{a}$ , $\bar{b}$ , ${\bar{F}}_{sa}$ , and ${\bar{τ}}_{ea}$ are the bounds of the uncertain parameters and the external disturbance.

Then, we rewrite (33) as $a_{0} {\ddot{δ}}_{f} + b_{0} {\dot{δ}}_{f} + F_{sa 0} sign ({\dot{δ}}_{f}) + τ_{ea 0} = u + ρ$ (42) where ρ denotes the lumped uncertainty as

$\begin{matrix} ρ & = & (a_{0} - a) {\ddot{δ}}_{f} + (b_{0} - b) {\dot{δ}}_{f} \\ + (F_{sa 0} - F_{sa}) sign ({\dot{δ}}_{f}) + (τ_{ea 0} - τ_{ea}) \end{matrix}$ (43)

Thus, the nominal system is given by $a_{0} {\ddot{δ}}_{f} + b_{0} {\dot{δ}}_{f} + F_{sa 0} sign ({\dot{δ}}_{f}) + τ_{ea 0} = u_{1}$ (44) where u₁ is the nominal control input.

It should be noted from [30] that the terms a, b, F_sa and τ_ea are all proved to be bounded and if the closed-loop control signal u is selected to be upper bounded by the following polynomial function: $| u | < λ_{0} + λ_{1} | δ_{f} | + λ_{2} | {\dot{δ}}_{f} |$ (45) where λ₀, λ₁, and λ₂ are positive constants, the lumped uncertainty in (43) can also be upper bounded [30]: $| ρ | < \bar{ρ}$ (46) with $\bar{ρ} = c_{0} + c_{1} | δ_{f} | + c_{2} | {\dot{δ}}_{f} |$ (47) where c₀, c₁, and c₂ are designed positive constants.

Remark 4.1. It can be seen that the lumped uncertainty in (43) is composed of the parameter uncertainties in the SbW components and the estimation error of the tire self-aligning torque (τ_ea0 - τ_ea). The detail of the modelling of the trye disturbance torque τ_ea was developed in [30]. When the road condition frequently changes, the estimation error (τ_ea0 - τ_ea) may be very large and greatly affects the steering performance of the SbW systems.

Remark 4.2. It is worth noting that the proper selection of the c_i (i = 1, 2, 3) in (47) largely depends on the determination of the bounds of the uncertain parameters and the external disturbance $\bar{a}$ , $\bar{b}$ , ${\bar{F}}_{sa}$ , and ${\bar{τ}}_{ea}$ in (38)–(40). However, it is difficult to assign the appropriate values to c_i (i = 1, 2, 3) in real applications, since too large bounds may lead to control saturation and small ones may not ensure the closed-loop system stability.

Our control objective for the steering system is to design a robust steering controller using RBFNN for relaxing the constraints on the lumped uncertainty bound and terminal sliding mode for the finite-time convergence, such that the front wheels can track the hand-wheel commands fast and accurately in the presence of the model uncertainties and disturbances.

5 Neural network sliding mode control of SbW systems

In this section, the robust control design consists of a nominal controller and a RBFNN-based TSM compensator. We will first present a nominal control law for stabilizing the nominal model in (44). Then, we design an TSM compensator using RBFNN for adaptively adjusting the lumped uncertainty bound such that finite-time error convergence of the closed-loop system and strong robustness are ensured.

5.1 Nominal control design

First, the tracking error between the front wheel steering angle δ_f and the desired reference angle θ_hr is defined as follows: $ɛ_{θ} = δ_{f} - \frac{θ_{h}}{N_{hr}} = δ_{f} - θ_{hr}$ (48) where N_hr denotes the ratio between the hand-wheel angle θ_h and the front wheel steering angle δ_f.

Given the system model in (42), we obtain the error dynamics of the closed-loop SbW system as follows:

$\begin{matrix} \underset{θ}{\ddot{ɛ}} & = & - \frac{b_{0}}{a_{0}} {\dot{ɛ}}_{θ} + \frac{u - F_{sa 0} sign ({\dot{δ}}_{f}) - τ_{ea 0}}{a_{0}} + ρ^{'} \\ - \frac{b_{0}}{a_{0}} {\dot{θ}}_{hr} \end{matrix}$ (49) where ρ′ is the new bounded lumped uncertainty in the closed-loop error dynamics and defined as: $ρ^{'} = \frac{ρ}{a_{0}} - \underset{hr}{\ddot{θ}}$ (50)

Thus, we have $| ρ^{'} | \leq \bar{ρ^{'}} = \frac{\bar{ρ}}{a_{0}} + {\bar{\ddot{θ}}}_{hr}$ (51)

where $\bar{ρ^{'}}$ is the upper bound of new lumped uncertainty in (50) and ${\bar{\ddot{θ}}}_{hr}$ is the upper bound of the second derivative of θ_hr.

If the uncertainty in (50) is not considered, (49) can be simplified into the error dynamics of the nominal system in (44) as: $\underset{θ}{\ddot{ɛ}} = - \frac{b_{0}}{a_{0}} {\dot{ɛ}}_{θ} + \frac{u_{1} - F_{sa 0} sign ({\dot{δ}}_{f}) - τ_{ea 0}}{a_{0}} - \frac{b_{0}}{a_{0}} {\dot{θ}}_{hr}$ (52)

In order to stabilize the error dynamics in (52), the nominal control signal u₁ is chosen as:

$\begin{matrix} u_{1} & = & F_{sa 0} sign ({\dot{δ}}_{f}) + τ_{ea 0} + a_{0} (k_{1} ɛ_{θ} + k_{2} {\dot{ɛ}}_{θ}) \\ + b_{0} {\dot{θ}}_{hr} \end{matrix}$ (53) where k₁ and k₂ are the designed feedback gains.

5.2 Control design for uncertain SbW system with unknown uncertainty bound using RBFNN estimator

It is noted in [27], [30] that the bound information of system uncertainty and disturbances is assumed to be known in the SM controller design. However, in practical applications, the upper bound of ρ′ in (51) is normally unknown and difficult to measure. Thus, we adopt RBFNN to adaptively learn the uncertainty bound and the output of the RBFNN is then used as the parameter of the controller.

Consider the SbW system with uncertainties in (42). The closed-loop control input u consists of two components: $u = u_{1} + u_{0}$ (54) where u₁ is given in (53) and u₀ is the compensator for dealing with the lumped uncertainty, which will be described later.

By substituting (53) and (54) into (49), the closed-loop error dynamics can be written as: $\underset{θ}{\ddot{ɛ}} = - \frac{b_{0}}{a_{0}} {\dot{ɛ}}_{θ} + k_{1} ɛ_{θ} + k_{2} {\dot{ɛ}}_{θ} + \frac{u_{0}}{a_{0}} + ρ^{'}$ (55)

Generally, in order to use the TSM technique for the compensator design, a nonsingular terminal sliding surface is designed as $s = ɛ_{θ} + \frac{1}{λ} {\dot{ɛ}}_{θ}^{(p / q)}$ (56) where λ > 0, p and q are two positive odd integers satisfying 1 < p/q < 2.

The estimate of the lumped uncertainty bound $\bar{ρ^{'}}$ is given by $\hat{\bar{ρ^{'}}} (X, \hat{ω}) = {\hat{ω}}^{T} φ (X)$ (57) where $X = {[\begin{matrix} δ_{f} {\dot{δ}}_{f} \end{matrix}]}^{T}$ is the input of the RBFNN, $\hat{ω}$ is the weight vector of the RBFNN and the vector φ (X) ∈ R^N is Guassian type of function defined as $φ_{i} (X) = \exp (- \frac{{∥ X - g_{i} ∥}^{2}}{σ_{i}^{2}}) i = 1, 2, \dots, N$ (58) where g_i and σ_i are the centre and width of number N neurons, respectively, and both of them are predetermined.

For the use of RBFNN in the compensator design, we have the following assumptions:

A1: Given an arbitrary small positive constant ξ₁ and continuous function $\bar{ρ^{'}}$ , the optimal weights of RBFNN ω^* satisfy $| ω^{*^{T}} φ (X) - \bar{ρ^{'}} | = | ξ (X) | < ξ_{1}$ (59)

A2: The lumped uncertainty bound in (51) satisfy the following inequality $\bar{ρ^{'}} - | ρ^{'} | > ξ_{2} > ξ_{1}$ (60) where ξ₂ is a positive constant.

It is worth noting that the two assumptions A1 and A2 are tenable. Assumption A1 indicates the estimation ability of RBFNN, while A2 shows that the range of $\bar{ρ^{'}}$ can be flexibly assigned.

Then, for the design of the robust controller using RBFNN estimator, the adaption of the weights, and the analysis of the error convergence, we have the following theorem. The control diagram of the proposed RBFNN-based SbW control system is shown in Fig. 5.

Theorem 5.1.Consider the error dynamics in ( 4 9) with Assumptions A1 and A2. If the control inputuis designed as $u = u_{1} + u_{0}$ (61) where u₁ is the nominal control in (53) and u₀ is the SMC designed as follows:

$\begin{matrix} u_{0} & = & - a_{0} sign (s) [| k_{1} | | ɛ_{θ} | + | - \frac{b_{0}}{a_{0}} + k_{2} | | {\dot{ɛ}}_{θ} |] \\ - a_{0} λ \frac{q}{p} {({\dot{ɛ}}_{θ})}^{(2 - \frac{p}{q})} - a_{0} sign (s) \hat{\bar{ρ^{'}}} \end{matrix}$ (62) where $\hat{\bar{ρ^{'}}}$ is estimated by using (57) and the weights are adjusted by using the following adaption law: $\overset{\cdot}{\hat{ω}} = η \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | φ (X)$ (63) with adaption gain η > 0, then the output tracking error converges to zero in finite time.

Proof. Defining a Lyapunov function candidate $V = \frac{1}{2} s^{2} + \frac{1}{2} η^{- 1} {\tilde{ω}}^{T} \tilde{ω}$ (64) where $\tilde{ω} = ω^{*} - \hat{ω}, \overset{\cdot}{\tilde{ω}} = - \overset{\cdot}{\hat{ω}}$ .

Differentiating V with respect to time, we have

$\begin{matrix} \dot{V} & = & s \dot{s} - η^{- 1} {\tilde{ω}}^{T} \overset{\cdot}{\hat{ω}} \\ = & s [{\dot{ɛ}}_{θ} - \frac{1}{λ} \frac{p}{q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} \underset{θ}{ɛ}] - η^{- 1} {\tilde{ω}}^{T} \overset{\cdot}{\hat{ω}} \\ = & s {{\dot{ɛ}}_{θ} + \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} \\ [k_{1} ɛ_{θ} + (k_{2} - \frac{b_{0}}{a_{0}}) {\dot{ɛ}}_{θ} + \frac{u_{0}}{a_{0}} + ρ^{'}]} \\ - η^{- 1} {\tilde{ω}}^{T} \overset{\cdot}{\hat{ω}} \\ = & \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} ρ^{'} s \\ - \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | \hat{\bar{ρ^{'}}} - η^{- 1} {\tilde{ω}}^{T} \overset{\cdot}{\hat{ω}} \\ = & \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} (ρ^{'} s - \hat{\bar{ρ^{'}}} | s | + \bar{ρ^{'}} s - \bar{ρ^{'}} s) \\ - η^{- 1} {\tilde{ω}}^{T} \overset{\cdot}{\hat{ω}} \\ \leq & - \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} (\bar{ρ^{'}} | s | - | ρ^{'} | | s |) \\ + \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} (- \hat{\bar{ρ^{'}}} | s | + \bar{ρ^{'}} | s |) - η^{- 1} {\tilde{ω}}^{T} \overset{\cdot}{\hat{ω}} \\ = & - \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | (\bar{ρ^{'}} - | ρ^{'} |) - η^{- 1} {\tilde{ω}}^{T} \overset{\cdot}{\hat{ω}} \\ + \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | \\ [- {\overset{\cdot}{\hat{ω}}}^{T} φ (X) + ω^{*^{T}} φ (x) - ξ (X)] \\ = & - \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | (\bar{ρ^{'}} - | ρ^{'} |) \\ - \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | ξ (X) \\ + \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | {\tilde{ω}}^{T} φ (X) - η^{- 1} {\tilde{ω}}^{T} \overset{\cdot}{\hat{ω}} \\ = & - \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | (\bar{ρ^{'}} - | ρ^{'} |) \\ - \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | ξ (X) \\ + \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | {\tilde{ω}}^{T} φ (X) \\ - η^{- 1} {\tilde{ω}}^{T} η \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | φ (X) \\ \leq & - \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | ξ_{2} + \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | | ξ (X) | \\ \leq & - \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | (ξ_{2} - ξ_{1}) \\ < & - \frac{p}{λ q} {({\dot{ɛ}}_{θ})}^{(\frac{p}{q} - 1)} | s | ξ_{3} for | s | \neq 0, | {\dot{ɛ}}_{θ} | \neq 0 \end{matrix}$ (65) where the positive constant ξ₃ always satisfies: $ξ_{2} - ξ_{1} > ξ_{3}$ (66)

In equality (65) means that s converges to zero in finite time according to Lyapunov stability theory [34, 35]. Then, for $| {\dot{ɛ}}_{θ} | = 0$ , the control input u in (61) can be rewritten as: $u = F_{sa 0} sign ({\dot{δ}}_{f}) + τ_{ea 0} + b_{0} {\dot{θ}}_{hr} - a_{0} sign (s) \hat{\bar{ρ^{'}}}$ (67)

Using (67) in (49), we have ${\ddot{ɛ}}_{θ} = ρ^{'} - \frac{b_{0}}{a_{0}} {\dot{θ}}_{hr} + \frac{- a_{0} sign (s) \hat{\bar{ρ^{'}}} + b_{0} {\dot{θ}}_{hr}}{a_{0}}$ (68)

For s > 0, (68) can be expressed as ${\ddot{ɛ}}_{θ} = - \hat{\bar{ρ^{'}}} + ρ^{'} \leq - (\hat{\bar{ρ^{'}}} - | ρ^{'} |) \leq - ξ_{4}$ (69) where $\hat{\bar{ρ^{'}}}$ is chosen to be a sufficiently large positive constant and thus ξ₄ can always be founded. Similarly, for s < 0, we obtain ${\ddot{ɛ}}_{θ} \geq ξ_{4}$ .

Hence, there exists to be a vicinity of ${\dot{ɛ}}_{θ} = 0$ so that for an arbitrarily small ɛ > 0, we can obtain $Ω (ɛ) = {ɛ_{θ} \in R^{2} | - ɛ < {\dot{ɛ}}_{θ} < ɛ}$ and ${\ddot{ɛ}}_{θ} \leq - ξ_{4}$ for s > 0, ${\ddot{ɛ}}_{θ} \geq ξ_{4}$ for s < 0. Therefore, by LaSalle’s theorem [30], every trajectory starting in Ω (ɛ) approaches s = 0 in finite time.

Therefore, it is easily seen that the TSM s = 0 can be reached from anywhere in finite time, such that the output tracking error can converge to zero in finite time [34], which means that the front wheel steering angle will closely track the reference hand-wheel angle in finite time.

Here completes the proof.

Remark 5.1. As can be seen from Theorem (3.1), the following facts have been noted: (i) The weights of the RBFNN are adjusted in Lyapunov sense, which indicates that the weights do not converge to their optimal values; Instead, they are adjusted to ensure the finite-time convergence of both the terminal sliding variable s and the output tracking error ɛ_θ. (ii) The proposed control does not need the prior bound knowledge of the system lumped uncertainty, which successfully overcomes the difficulty in obtaining the appropriate bound information in the controller design.

Remark 5.2. In order to eliminate the chatters in the control input caused by the use of the signum function sign (s), we have the following boundary layer-based robust control [30]:

$\begin{matrix} u_{0} & = & - a_{0} sat (s) [| k_{1} | | ɛ_{θ} | + | - \frac{b_{0}}{a_{0}} + k_{2} | | {\dot{ɛ}}_{θ} |] \\ - a_{0} λ \frac{q}{p} {({\dot{ɛ}}_{θ})}^{(2 - \frac{p}{q})} - a_{0} sat (s) \hat{\bar{ρ^{'}}} \end{matrix}$ (70) where sat (s) has the same definition as in (30) and the corresponding boundary layer constant is δ₂.

6 Simulation results

6.1 Simulation results of the proposed ASM-YRC

In this sub-section, in order to test the performance of the proposed ASM-YRC, the steering controller is not included, which indicates that the actual front wheel steering angle applying to the vehicle model is assumed to be equal to the desired steering angle input in (15). The simulation environment is assumed as follows:

The desired hand-wheel commanded angle is generated by a periodic sinusoidal signal as δ_cmd = 0.4 sin(0.7 πt), which represents the input to perform a standard slalom manoeuvre;

The vehicle velocity is set as V_x = 36 km/h;

The road surface changes from snowy road to dry asphalt road at 8.0 second.

Nominal parameters of vehicle dynamics and SbW system are listed in Table 1. The cutoff frequency of the desired vehicle model filter ω_ref in (16a) is chosen as 20 rad/s and the adaptation gains for the proposed ASM-YRC in (24)–(26) are selected as η₁ = 0.022, η₂ = 0.05, andη₃ = 0.25.

Figure 7 shows the control performance of the proposed ASM-YRC. It is seen that the actual yaw rate can well track the ideal yaw rate of the vehicle for two different road conditions. Even though the road surface changes from snowy to dry conditions at 8 seconds, the vehicle yaw tracking can still be maintained well due to the robustness of the proposed ASM-YRC. Moreover, as seen in Fig. 7(d), the estimated bounds ${\bar{d}}_{i} (i = 1, 2, 3)$ are online estimated for the purpose of ensuring the closed-loop error convergence, such that the information of the vehicle dynamics is indeed no longer required in the yaw control design.

6.2 Simulation results of the proposed steering control

In this sub-section, the excellent performance of the proposed steering control will be evaluated compared with a PD control [18] and a conventional boundary layer sliding mode (BL-SM) control [27], respectively.

The proposed control parameters are given in Table 2. The adopted RBFNN as shown in Fig. 6 has six nodes and six weights, where the corresponding widths, centres, and initial values of the weights of the Gaussian functions are selected as follows: $\begin{matrix} σ_{i}^{2} & = & 0.36 \\ g_{i} & = & [\begin{matrix} 0 & 0 & 0 & - 1 & 1 & 1 \\ - 1 & 0 & - 1 & - 1 & 0 & 1 \end{matrix}] \\ \hat{ω} (0) & = & {[0.5 0.5 0.5 0.5 0.5 0.5 0.5]}^{T} \end{matrix}$

In order to evaluate the robustness of the proposed neural control scheme, the simulation environment is assumed as follows:

Three different road conditions (wet asphalt, icy, and dry asphalt road) are set in the first 10 seconds, the middle 10 seconds, and the last 10 seconds, respectively.

The vehicle velocity is set as V_x = 36 km/h.

Driver’s input torque is a sinusoidal signal for standard slalom maneuver (Note that the hand-wheel model can be found in [27]).

The Runge-Kutta method with the sampling interval ΔT = 0.01 s is adopted to solve the closed-loop differential equations numerically in this simulation.

Figure 8 shows the steering performance, tracking error, and control torque, respectively, using the following conventional PD control and linear H_∞ control laws, respectively [18]: $u_{PD} = k_{ps} ɛ_{θ} (t) + k_{ds} {\dot{ɛ}}_{θ} (t)$ (71) where the proportional and derivative control gains k_ps and k_ds are properly chosen as k_ps = -4.5, k_ds = -1.4.

It is clearly seen that, the steering performance of the PD control method has significantly deteriorated, after the road surface changes from icy to dry asphalt road condition (t > 25s). The reason is that under the large variations of road conditions, the pre-determined control parameters are not optimal any longer and thus the strong robustness against system uncertainties and large variations of road conditions may not be ensured.

(Figure 9a–c) shows the control performance using the following conventional BL-SM controller:

$\begin{matrix} u_{BLSM} & = & \frac{1}{150} sat (s) [12 * (10 + 12 | {\dot{ɛ}}_{θ} |) \\ + 28 | {\dot{δ}}_{f} | + 3.2 + {\bar{τ}}_{e}] \end{matrix}$ (72) where the sliding variable $s = {\dot{ɛ}}_{θ} + 12 ɛ_{θ}$ , sat (s) is the saturation function with the same form as (46) where the boundary layer constant is 0.8, ${\bar{τ}}_{e}$ is the upper bound of tyre self-aligning torque and can be found in [27].

It is seen that the BL-SM equipped SbW control system behaves much better steering performance and robustness than the PD control, especially on the dry asphalt road during the last 15 seconds. This is because the BL-SM control using the bound information of the system uncertainties and disturbances is able to eliminate the effects of the uncertain system dynamics. However, the superior tracking performance cannot be achieved due to the following two reasons: (i) The linear sliding surface of the BL-SM control leads to the asymptotic convergence of the tracking error such that the finite time convergence cannot be achieved. (ii) Due to the use of the preset boundary layer constant for removing chattering, large steady error exists in the period of wet and icy road such that the overall steering performance have been degraded.

Figure 10(a)–(c) shows the simulation results of the steering performance, tracking error, and control torque, respectively, with the proposed robust neural control. It is shown that the steering performance has been improved significantly, as compared with the ones of the PD control, the linear H_∞ control, and the BL-SM control. The main reasons are threefold: (i) Because the system lumped uncertainty in the proposed control is much smaller than the one in the BL-SM control, good steering performance can be guaranteed by using a small control effort. (ii) The lumped uncertainty bound is adaptively estimated in the Lyapunov sense, based on the RBFNN output such that the control gain is reduced and adaptively updated to eliminate the effects of uncertain dynamics. (iii) The NTSM surface in (26) possesses faster and higher precision tracking characteristics and stronger robustness against uncertainties and disturbances, compared with the conventional BL-SM control systems. Thus, the proposed robust control scheme using RBFNN estimator possess better control performance and stronger robustness with regard to parameter uncertainties and varying road disturbances.

Remark 6.1. It should be noted that due to the perfect learning ability of the RBFNN for the system lumped uncertainty bound in the Lyapunov sense, the proposed neural control technique can also be applied to other automotive applications to estimate the lumped uncertainty bound and ensure the excellent tracking performance, such as braking systems, throttle systems and so on.

6.3 Simulation results of the proposed VSC scheme

In this sub-section, the whole vehicle stability control algorithm including the proposed ASM-YRC and steering controller is verified. Note that all of the control parameters are kept the same as the ones in Subsection 6.1 and 6.2. The simulation environment is assumed as follows:

The driver’s input torque is a periodic sinusoidal signal as τ_h = 1.6 sin(1.9t) and the simulation time is 16 second;

The vehicle velocity is set as V_x = 36 km/h;

SbW system parameters and the road surface are chosen as same as Case 2 in 6.2.

As seen from Figure 11 that, the proposed VSC scheme with the BRFNN-based SbW control system behaves with the both excellent yaw tracking performance and steering angle tracking performance and robustness against road condition change, which indicates that the proposed VSC can ensure a robust vehicle stability performance by rejecting the road disturbance and serve as a good candidate in SbW-based VSC schemes.

7 Conclusion

In this paper, we have proposed a novel VSC scheme using robust adaptive neural sliding mode technique for the SbW equipped vehicles. It has been shown that the proposed control scheme is capable of well tracking the desired vehicle motion trajectory under parameter uncertainties and road disturbances, in terms of yaw rate tracking and steering angle tracking. Since the bound information of all the uncertain parameters and disturbances in vehicle stability control is adaptively estimated using adaptive sliding mode and RBFNN techniques, not only the control gain is reduced, but also the ease of the control design is achieved due to no requirement to estimate the parameter uncertainties and disturbances. The further work is to design a fuzzy neural network-based sliding mode control scheme for SbW systems with the considerations of different vehicle speeds.

Footnotes

Acknowledgments

This research was supported by the National Nature Science Foundation of China under Grant 61503113 and the Cultivation Project of Applicable Technological Achievements of Hefei University of Technology under Grant JZ2016YYTY0032

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