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Abstract

Direct yaw-moment control systems have been proven effective in enhancing vehicle stability and handling. The existing direct yaw-moment control designs commonly involve computation of tire side-slip angles, which is susceptible to measurement and estimation errors. The fixed control gain of the conventional sliding mode direct yaw-moment control design cannot adapt to variations and uncertainties in vehicle parameters. As a result, its robustness against parametric variations and uncertainties is limited. To improve the control performance, a novel adaptive sliding mode direct yaw-moment control approach is proposed in this article for electric vehicles with independent motors. The proposed method utilizes a varying control gain to adapt to the variations of front and rear tire side-slip angles. Comparative simulation results show that the proposed scheme outperforms the conventional method with inaccurate tire side-slip angle feedback. With the proposed direct yaw-moment control system on-board, the adverse effects of inaccuracies on tire side-slip angles are suppressed and the vehicle’s robustness against parametric variations and uncertainties is enhanced.

Keywords

Electric vehicles direct yaw-moment control sliding mode control adaptive control vehicle handling

Introduction

The swift development of electric vehicles has significantly facilitated the design and implementation of advanced vehicle chassis control systems. One of such systems, benefiting from the configuration of independent driving motors, is direct yaw-moment control (DYC). The simplest way of achieving DYC is through differential braking; however, the inevitable deceleration may disturb the driver and cause accidents.¹ With the distributed motor configuration, DYC achieves independent control over both driving and braking forces, thereby offering a more effective and flexible way for enhancing vehicle stability than four wheel steering² and braking (or driving) only DYC.¹

Existing DYC designs normally employ the simple 2-degree-of-freedom bicycle model in the controller design process.^3–7 This approach generally requires the knowledge of front and rear tire side-slip angles to obtain the lateral tire forces. To calculate these angles, the following approximations are commonly used^8,9

α_{f} \approx β + \frac{l_{f} r}{v_{x}} - δ

(1)

α_{r} \approx β - \frac{l_{r} r}{v_{x}}

(2)

where $α_{f}$ and $α_{r}$ denote the front and rear tire side-slip angles, respectively; $l_{f}$ and $l_{r}$ represent the distances from the front and rear axles to the vehicle center of gravity (COG), respectively; r stands for the yaw rate; $β$ is the vehicle side-slip angle at the COG; $v_{x}$ denotes the vehicle longitudinal velocity; and $δ$ is the steer angle of the front wheels.

Using equations (1) and (2) to compute the tire side-slip angles requires the vehicle side-slip angle $β$ which cannot be easily measured using inexpensive sensors.^6,10 To deal with this problem, the existing DYC methods inevitably design and employ an observer in the control system to estimate the required $β$ information.¹¹ For example, Yu et al.⁴ adopted an extended Kalman filter to provide the vehicle side-slip angle information, Xiong et al.⁶ employed a Luenberger observer for vehicle side-slip angle estimation based on a non-linear single-track vehicle model, and Geng et al.¹¹ proposed a vehicle side-slip angle fuzzy observer for their DYC algorithm.

The main shortcoming with the existing DYC solutions is that their control performances are dependent on the estimation accuracy of $β$ and the estimation errors, if not properly dealt with, can lead to performance deterioration or even vehicle instability. Note that regardless of the type of vehicle side-slip angle observers, perfect estimation is not practical and some extent of estimation error is always inevitable. Therefore, it is desirable to devise a controller that automatically adapts to the changes of plant parameters, thereby suppressing the effects of estimation errors and maintaining consistent control performance in the presence of parametric variations/uncertainties.

To this end, we propose in this article a novel adaptive sliding mode DYC approach for electric vehicles. The major contribution of this work lies in the design of a varying sliding mode control gain which automatically adapts to the changes in tire side-slip angles. By this means, the adverse effects of inaccuracies on $α_{f}$ and $α_{r}$ are suppressed and the system robustness against parametric variations and uncertainties is enhanced. It should be noted that according to equations (1) and (2), the inaccuracies on $α_{f}$ and $α_{r}$ can possibly result from the inaccuracies on $β$ , $v_{x}$ , r, or $δ$ . In other words, the proposed method not only improves robustness against estimation errors on $β$ but also enhances robustness against measurement or estimation inaccuracies on $v_{x}$ , r, and $δ$ .

Since sliding mode control technique has been widely employed in recent DYC solutions,¹² a conventional sliding mode control design for DYC is introduced for comparisons in this study. The proposed control approach and the conventional control scheme are applied on a simulated electric vehicle equipped with independently driven rear wheels. The control performances of these two methods are compared, with three types of feedback signals (tire side-slip angles) sent to the two controllers: accurate values, inaccurate values with constant estimation errors, and inaccurate values with convergent estimation errors. The simulation results show that the proposed controller is capable of suppressing the adverse effects of estimation errors and, as a result, presents superior control performance to the conventional approach with inaccurate parameter estimations.

The rest of this article is organized as follows. Section “Problem statement” provides a comprehensive problem statement, including the vehicle models employed in this study, the DYC design problem, and a conventional sliding mode DYC approach. Section “Proposed adaptive sliding mode control” elucidates the detailed design process of the proposed adaptive sliding mode DYC approach. Section “Simulation results” presents the comparative simulation results in different conditions. Section “Conclusions and recommendations” concludes the article and proposes the future work.

Problem statement

In this section, we first present the vehicle models for control system design and simulation validation purposes, respectively. Then, we employ the same notations to introduce the DYC design problem in section “DYC design problem,” as well as a common sliding mode DYC approach in section “Conventional sliding mode control.”

Control model

The typical linear vehicle model, that is, the bicycle model,^3–7,13 is commonly used in the design process of various chassis control systems. This planar vehicle model includes 2 degrees of freedom (lateral and yaw motions), and it depicts the basic characteristics of handling dynamics. This model is mathematically expressed as follows

\sum F_{y} = m ({\overset{\cdot}{v}}_{y} + v_{x} r)

(3)

\sum M_{z} = I_{z} \overset{\cdot}{r}

(4)

where m denotes the vehicle total mass, $I_{z}$ represents the yaw moment of inertia, $v_{y}$ stands for the vehicle lateral velocity, and $\sum F_{y}$ and $\sum M_{z}$ are the total lateral force and total yaw moment applied on the vehicle, respectively.

In the bicycle model, the lateral tire force is assumed to be linearly proportional to the tire side-slip angle. Thus, the force $\sum F_{y}$ and moment $\sum M_{z}$ are written as follows

\sum F_{y} = F_{f} + F_{r} = - C_{f} α_{f} - C_{r} α_{r}

(5)

\sum M_{z} = F_{f} l_{f} - F_{r} l_{r} = - C_{f} α_{f} l_{f} + C_{r} α_{r} l_{r}

(6)

where $F_{f}$ and $F_{r}$ denote the front and rear lateral tire forces, respectively, and $C_{f}$ and $C_{r}$ represent the front and rear tire cornering stiffness, respectively.

During normal driving conditions, the bicycle model is adequate to capture the fundamental characteristics of vehicle dynamics,¹⁴ and its simplicity facilitates the design of chassis control systems. Thus, the bicycle model is employed in this study to devise a direct yaw moment controller.

Simulation model

To produce realistic vehicle responses and verify the control designs, a comprehensive 8-degree-of-freedom simulation model is established. In this model, the vehicle body is composed of two major parts: the sprung rigid body and the unsprung rigid body. To establish the vehicle equations of motion, two Cartesian coordinate systems are attached to the sprung and unsprung rigid bodies. The details of the two-rigid-body configuration and the associated coordinate systems are available in Abe and Manning.¹⁵

Based on the above coordinate systems, one can derive the following 4-degree-of-freedom vehicle equations of motion,^12,15,16 governing the vehicle longitudinal, lateral, roll, and yaw motions

\sum F_{x} = m ({\overset{\cdot}{v}}_{x} - v_{y} r) + m_{s} h_{s} pr

(7)

\sum F_{y} = m ({\overset{\cdot}{v}}_{y} + v_{x} r) - m_{s} h_{s} \overset{\cdot}{p}

(8)

\sum M_{x} = I_{x} \overset{\cdot}{p} - I_{xz} \overset{\cdot}{r} - m_{s} h_{s} ({\overset{\cdot}{v}}_{y} + v_{x} r)

(9)

\sum M_{z} = I_{z} \overset{\cdot}{r} - I_{xz} \overset{\cdot}{p}

(10)

where $m_{s}$ denotes the vehicle sprung mass, $h_{s}$ represents the distance between the roll axis and the COG of the sprung rigid body, $I_{x}$ denotes the roll moment of inertia of the sprung rigid body, $I_{xz}$ represents the product of inertia with respect to the vehicle symmetrical plane, and p stands for the roll rate of the sprung rigid body.

The external forces and moments $\sum F_{x}$ , $\sum F_{y}$ , $\sum M_{x}$ , and $\sum M_{z}$ in the above equations are given by

\sum F_{x} = \sum_{i = 1}^{4} (F_{xi} \cos δ_{i} - F_{yi} \sin δ_{i})

(11)

\sum F_{y} = \sum_{i = 1}^{4} (F_{xi} \sin δ_{i} + F_{yi} \cos δ_{i})

(12)

\sum M_{x} = - K_{ϕ} ϕ + m_{s} g h_{s} \sin ϕ - C_{ϕ} p

(13)

\begin{matrix} \sum M_{z} = \sum_{i = 1}^{4} x_{i} (F_{xi} \sin δ_{i} + F_{yi} \cos δ_{i}) \\ - \sum_{i = 1}^{4} y_{i} (F_{xi} \cos δ_{i} - F_{yi} \sin δ_{i}) \end{matrix}

(14)

where $δ_{i}$ denotes the steer angle of the ith wheel (for rear wheels, $δ_{3} = δ_{4} = 0$ ); $x_{i}$ and $y_{i}$ represent the coordinates of the ith wheel in the horizontal plane; $K_{ϕ}$ and $C_{ϕ}$ are the suspension roll stiffness and suspension roll damping, respectively; and $F_{xi}$ and $F_{yi}$ are the longitudinal and lateral tire forces exerted on the ith wheel. The subscript i takes on values of 1, 2, 3, and 4, which represent the front left, front right, rear right, and rear left wheel, respectively.

The rotational motions of the four wheels provide the other 4 degrees of freedom. Assuming that all four wheels are identical, the equation of motion for the ith wheel is as follows

J \frac{d ω_{i}}{dt} = T_{i} - F_{xi} R

(15)

where J represents the mass moment of inertia of each wheel assembly, $ω_{i}$ denotes the angular velocity of the ith wheel, $T_{i}$ stands for the motor torque applied on the ith wheel (for front wheels, $T_{1} = T_{2} = 0$ ), $F_{xi}$ is the ith longitudinal tire force, and R is the tire radius.

The tire forces involved in the preceding equations, $F_{xi}$ and $F_{yi}$ , are obtained by the well-known Magic Formula tire model.¹⁷ This tire model is widely employed in the literature and has been proven adequately accurate to predict the actual tire dynamics.¹⁸ The Magic Formula tire model is mathematically expressed as follows¹⁷

y (x) = D \sin (C \arctan (Bx - E (Bx - \arctan Bx)))

(16)

with

Y (X) = y (x) + S_{V}

(17)

x = X + S_{H}

(18)

where X represents the input variable (wheel slip ratio or tangent of tire side-slip angle); $Y (X)$ denotes the output variable $F_{xi}$ or $F_{yi}$ ; $S_{H}$ and $S_{V}$ are the horizontal shift and vertical shift respectively; and B, C, D, and E are the stiffness factor, shape factor, peak value, and curvature factor, respectively. The detailed expressions for B, C, D, E, $S_{H}$ , and $S_{V}$ are available in Pacejka et al.¹⁷

DYC design problem

With a DYC system on-board, a corrective yaw moment can be generated to tune the dynamics of the electric vehicle, thereby enhancing its stability and handling. Accordingly, the yaw dynamics ( i.e. equations (4) and (6)) in the bicycle model is modified to

- C_{f} α_{f} l_{f} + C_{r} α_{r} l_{r} + Δ M_{z} = I_{z} \overset{\cdot}{r}

(19)

where the new term $Δ M_{z}$ represents the corrective yaw moment generated by the DYC system. Note that with a DYC system, $Δ M_{z}$ is generated through tuning the driving/braking torques produced by the electric motors, as opposed to by altering the side-slip angles $α_{f}$ and $α_{r}$ .⁶

A typical DYC system is schematically shown in Figure 1. The controller receives measured and estimated vehicle states from sensors and observers, respectively, and calculates the corrective yaw moment $Δ M_{z}$ by means of some certain type of control algorithm. This yaw moment $Δ M_{z}$ is then converted to motor torque commands and distributed to independent motors. Since the vehicle yaw dynamics is directly related to the corrective yaw moment $Δ M_{z}$ by equation (19), one major task in the DYC process is to come up with an appropriate $Δ M_{z}$ to improve vehicle stability and handling.

Figure 1.

Schematic of a typical DYC system.

In section “Conventional sliding mode control,” we present a common sliding mode DYC approach for illustrative and comparative purposes.

Conventional sliding mode control

In this study, we refer to the commonly used sliding mode DYC approach as the conventional method, and its details are available in previous studies.^7,19–21 The design process of this method is generalized in this section.

In the conventional method, the switching function s of the sliding mode controller is normally defined as the error between the actual yaw rate and reference yaw rate

s = r - r^{*}

(20)

where $r^{*}$ represents the reference yaw rate, given by

r^{*} = \frac{v_{x} δ}{l (1 + {Kv}_{x}^{2})}

(21)

where l represents the wheel base. The term K denotes the stability factor and takes the following form

K = \frac{m}{l^{2}} (\frac{l_{r}}{C_{f}} - \frac{l_{f}}{C_{r}})

(22)

Note that equation (21) is indeed the steady-state yaw rate response derived from the bicycle model,^3,9,22,23 which has been widely employed as the reference yaw rate in vehicle stability and handling control.

The control objective of this sliding mode controller is to drive the scalar s to zero and thereafter maintain it at zero. As a result, the actual yaw rate r converges to the reference yaw rate $r^{*}$ . To achieve this control objective, the following sliding condition should be satisfied²⁴

\frac{1}{2} \frac{d}{d t} s^{2} = s \overset{\cdot}{s} \leq - η | s |

(23)

where $η$ is a strictly positive constant.

In order for the above sliding condition to be met, in the conventional method, the following $Δ M_{z}$ is proposed as the control input

Δ M_{z} = {\hat{C}}_{f} {\hat{α}}_{f} l_{f} - {\hat{C}}_{r} {\hat{α}}_{r} l_{r} + I_{z} {\overset{\cdot}{r}}^{*} - k sgn (s)

(24)

where ${\hat{C}}_{f}$ and ${\hat{C}}_{r}$ denote the estimated values of the front and rear cornering stiffness, respectively; ${\hat{α}}_{f}$ and ${\hat{α}}_{r}$ represent the estimated values of the front and rear tire side-slip angles, respectively; and k stands for a positive control gain. As previously mentioned, ${\hat{α}}_{f}$ and ${\hat{α}}_{r}$ are obtained using equations (1) and (2).

With this control input, the derivative of the switching function, $\overset{\cdot}{s}$ , can be expressed as follows

\begin{matrix} \overset{\cdot}{s} = \overset{\cdot}{r} - {\overset{\cdot}{r}}^{*} \\ = \frac{1}{I_{z}} (M_{z} + Δ M_{z}) - {\overset{\cdot}{r}}^{*} \\ = \frac{1}{I_{z}} (- C_{f} α_{f} l_{f} + C_{r} α_{r} l_{r} + {\hat{C}}_{f} {\hat{α}}_{f} l_{f} - {\hat{C}}_{r} {\hat{α}}_{r} l_{r} \\ + I_{z} {\overset{\cdot}{r}}^{*} - k sgn (s)) - {\overset{\cdot}{r}}^{*} \\ = \frac{1}{I_{z}} (({\hat{C}}_{f} {\hat{α}}_{f} - C_{f} α_{f}) l_{f} - ({\hat{C}}_{r} {\hat{α}}_{r} - C_{r} α_{r}) l_{r} \\ - k sgn (s)) \end{matrix}

(25)

Note that when the switching function s is positive, the sliding condition (i.e. inequality equation (23)) reduces to

\overset{\cdot}{s} \leq - η

(26)

Substituting equation (25) in inequality equation (26) yields

k \geq I_{z} η + ({\hat{C}}_{f} {\hat{α}}_{f} - C_{f} α_{f}) l_{f} - ({\hat{C}}_{r} {\hat{α}}_{r} - C_{r} α_{r}) l_{r}

(27)

On the contrary, when the switching function s is negative, the sliding condition becomes

\overset{\cdot}{s} \geq η

(28)

which leads to

k \geq I_{z} η - ({\hat{C}}_{f} {\hat{α}}_{f} - C_{f} α_{f}) l_{f} + ({\hat{C}}_{r} {\hat{α}}_{r} - C_{r} α_{r}) l_{r}

(29)

Inequalities in equations (27) and (29) imply that to satisfy the sliding condition, the control gain k must lie in the following range

k \geq I_{z} η + | ({\hat{C}}_{f} {\hat{α}}_{f} - C_{f} α_{f}) l_{f} - ({\hat{C}}_{r} {\hat{α}}_{r} - C_{r} α_{r}) l_{r} |

(30)

Since the estimation errors are practically bounded, we might as well assume that

| ({\hat{C}}_{f} {\hat{α}}_{f} - C_{f} α_{f}) l_{f} - ({\hat{C}}_{r} {\hat{α}}_{r} - C_{r} α_{r}) l_{r} | \leq F

(31)

where F represents the upper bound of $| ({\hat{C}}_{f} {\hat{α}}_{f} - C_{f} α_{f}) l_{f} - ({\hat{C}}_{r} {\hat{α}}_{r} - C_{r} α_{r}) l_{r} |$ . By setting $k = I_{z} η + F$ , the sliding condition is satisfied and in turn, the system is stable and its trajectories approach the sliding surface $s = 0$ in finite time.

In practice, in order to avoid chattering, the “ $sgn$ ” term in the control input (equation (24)) is replaced by the saturation function “ $sat$ .” Thus, the final control input is rewritten as follows

Δ M_{z} = {\hat{C}}_{f} {\hat{α}}_{f} l_{f} - {\hat{C}}_{r} {\hat{α}}_{r} l_{r} + I_{z} {\overset{\cdot}{r}}^{*} - k sat (\frac{s}{ϕ_{1}})

(32)

where $ϕ_{1}$ denotes the boundary layer thickness.

Proposed adaptive sliding mode control

Employing a constant control gain k in the control input (e.g. equation (32)) is not desirable for systems with variations and uncertainties in the plant parameters. This fixed control gain k can be insufficient or excessive to produce satisfactory control performance, in the presence of parametric variations and uncertainties.

To overcome the above shortcoming, we propose an adaptive sliding mode controller that employs a varying control gain to adapt to the parametric variations and uncertainties. The learning nature of this adaptive controller improves its control performance as adaptation goes on. As a result, compared to the conventional method, the adaptive control scheme achieves superior overall control performance. The design process of the proposed adaptive sliding mode controller is introduced in the following text.

Based on the control input of the conventional method (equation (24)), the control input of the adaptive controller is proposed as follows

\begin{matrix} Δ M_{z} = {\hat{C}}_{f} {\hat{α}}_{f} l_{f} - {\hat{C}}_{r} {\hat{α}}_{r} l_{r} + I_{z} {\overset{\cdot}{r}}^{*} \\ - a (| {\hat{α}}_{f} | l_{f} k_{1} + | {\hat{α}}_{r} | l_{r} k_{2}) sgn (s) \end{matrix}

(33)

where $a > 1$ is a positive design parameter used to adjust the magnitude of the control gain, and $k_{1}$ and $k_{2}$ are two positive control parameters updated according to the following rules

{\overset{\cdot}{k}}_{1} = | s | | {\hat{α}}_{f} | l_{f}

(34)

{\overset{\cdot}{k}}_{2} = | s | | {\hat{α}}_{r} | l_{r}

(35)

Equations (33)–(35) imply that unlike the conventional sliding mode controller, the control gain $a (| {\hat{α}}_{f} | l_{f} k_{1} + | {\hat{α}}_{r} | l_{r} k_{2})$ is no longer a constant, and it changes with the estimated tire side-slip angles as well as the yaw rate error s. The adaptive nature of the variables $k_{1}$ and $k_{2}$ provides this controller with superior robustness against estimation errors on vehicle states ( ${\hat{α}}_{f}$ and ${\hat{α}}_{r}$ ).

The proposed adaptive sliding mode controller must guarantee the stability of the control system. We may employ the following Lyapunov function candidate to evaluate the stability of the system

V = \frac{1}{2} I_{z} s^{2} + \frac{1}{2} (k_{1} - k_{1 \infty})^{2} + \frac{1}{2} (k_{2} - k_{2 \infty})^{2}

(36)

where $k_{1 \infty}$ and $k_{2 \infty}$ represent the values to which the control gains $k_{1}$ and $k_{2}$ will converge as time approaches infinity, and they take the following forms

k_{1 \infty} = | \frac{C_{f} α_{f} - {\hat{C}}_{f} {\hat{α}}_{f}}{{\hat{α}}_{f}} |

(37)

k_{2 \infty} = | \frac{C_{r} α_{r} - {\hat{C}}_{r} {\hat{α}}_{r}}{{\hat{α}}_{r}} |

(38)

Combining equations (33)–(38), the derivative of this Lyapunov function candidate can be derived as follows

\begin{matrix} \overset{\cdot}{V} = I_{z} s \overset{\cdot}{s} + (k_{1} - k_{1 \infty}) {\overset{\cdot}{k}}_{1} + (k_{2} - k_{2 \infty}) {\overset{\cdot}{k}}_{2} \\ = I_{z} s (\frac{1}{I_{z}} (- C_{f} α_{f} l_{f} + C_{r} α_{r} l_{r} + {\hat{C}}_{f} {\hat{α}}_{f} l_{f} - {\hat{C}}_{r} {\hat{α}}_{r} l_{r} + I_{z} {\overset{\cdot}{r}}^{*} - a (| {\hat{α}}_{f} | l_{f} k_{1} + | {\hat{α}}_{r} | l_{r} k_{2}) sgn (s) - {\overset{\cdot}{r}}^{*}) \\ + (k_{1} - | \frac{C_{f} α_{f} - {\hat{C}}_{f} {\hat{α}}_{f}}{{\hat{α}}_{f}} |) | s | | {\hat{α}}_{f} | l_{f} \\ + (k_{2} - | \frac{C_{r} α_{r} - {\hat{C}}_{r} {\hat{α}}_{r}}{{\hat{α}}_{r}} |) | s | | {\hat{α}}_{r} | l_{r} \\ = s (- C_{f} α_{f} l_{f} + C_{r} α_{r} l_{r} + {\hat{C}}_{f} {\hat{α}}_{f} l_{f} - {\hat{C}}_{r} {\hat{α}}_{r} l_{r} - a (| {\hat{α}}_{f} | l_{f} k_{1} + | {\hat{α}}_{r} | l_{r} k_{2}) sgn (s)) \\ - | s | | C_{f} α_{f} - {\hat{C}}_{f} {\hat{α}}_{f} | l_{f} + k_{1} | s | | {\hat{α}}_{f} | l_{f} \\ - | s | | C_{r} α_{r} - {\hat{C}}_{r} {\hat{α}}_{r} | l_{r} + k_{2} | s | | {\hat{α}}_{r} | l_{r} \\ = s (- (C_{f} α_{f} - {\hat{C}}_{f} {\hat{α}}_{f}) l_{f} + (C_{r} α_{r} - {\hat{C}}_{r} {\hat{α}}_{r}) l_{r}) \\ - | s | | C_{f} α_{f} - {\hat{C}}_{f} {\hat{α}}_{f} | l_{f} - | s | | C_{r} α_{r} - {\hat{C}}_{r} {\hat{α}}_{r} | l_{r} \\ + (1 - a) k_{1} | s | | {\hat{α}}_{f} | l_{f} + (1 - a) k_{2} | s | | {\hat{α}}_{r} | l_{r} \\ ⩽ | s | (| C_{f} α_{f} - {\hat{C}}_{f} {\hat{α}}_{f} | l_{f} + | C_{r} α_{r} - {\hat{C}}_{r} {\hat{α}}_{r} | l_{r}) \\ - | s | | C_{f} α_{f} - {\hat{C}}_{f} {\hat{α}}_{f} | l_{f} - | s | | C_{r} α_{r} - {\hat{C}}_{r} {\hat{α}}_{r} | l_{r} \\ + (1 - a) k_{1} | s | | {\hat{α}}_{f} | l_{f} + (1 - a) k_{2} | s | | {\hat{α}}_{r} | l_{r} \\ = (1 - a) k_{1} | s | | {\hat{α}}_{f} | l_{f} + (1 - a) k_{2} | s | | {\hat{α}}_{r} | l_{r} \\ < 0 \end{matrix}

(39)

Inequality in equation (39) implies that the derivative $\overset{\cdot}{V}$ is negative definite. Since the Lyapunov function candidate is obviously positive definite, the proposed control system is asymptotically stable according to Lyapunov’s stability theorem.

Besides, we see from equation (37) that

\begin{matrix} k_{1 \infty} = | \frac{C_{f} α_{f} - {\hat{C}}_{f} {\hat{α}}_{f}}{{\hat{α}}_{f}} | \\ = | \frac{(C_{f} - {\hat{C}}_{f}) α_{f} + {\hat{C}}_{f} (α_{f} - {\hat{α}}_{f})}{{\hat{α}}_{f}} | \\ \leq \frac{1}{| {\hat{α}}_{f} |} (| C_{f} - {\hat{C}}_{f} | | α_{f} | + | α_{f} - {\hat{α}}_{f} | | {\hat{C}}_{f} |) \end{matrix}

(40)

Similarly, equation (38) leads to

k_{2 \infty} \leq \frac{1}{| {\hat{α}}_{r} |} (| C_{r} - {\hat{C}}_{r} | | α_{r} | + | α_{r} - {\hat{α}}_{r} | | {\hat{C}}_{r} |)

(41)

Inequalities in equations (40) and (41) indicate that the boundaries of $k_{1 \infty}$ and $k_{2 \infty}$ depend on the absolute values of the estimation errors. Generally speaking, small estimation errors will lead to small $k_{1 \infty}$ and $k_{2 \infty}$ and in turn small $k_{1}$ and $k_{2}$ . Note that these two inequalities guarantee the boundedness of the control parameters $k_{1}$ and $k_{2}$ .

Similar to the conventional sliding mode control, the “ $sgn$ ” term in equation (33) is replaced by “ $sat$ ” to avoid chattering. Thus, the final control input becomes

\begin{array}{l} Δ M_{z} = {\hat{C}}_{f} {\hat{α}}_{f} l_{f} - {\hat{C}}_{r} {\hat{α}}_{r} l_{r} + I_{z} {\overset{\cdot}{r}}^{*} \\ - a (| {\hat{α}}_{f} | l_{f} k_{1} + | {\hat{α}}_{r} | l_{r} k_{2}) sat (\frac{s}{ϕ_{2}}) \end{array}

(42)

where $ϕ_{2}$ denotes the boundary layer thickness.

In this study, an electric vehicle equipped with independently driven rear wheels is considered, and the structure of the proposed control system for this vehicle is illustrated in Figure 2. The entire controller is composed of two controller units: the sliding mode controller unit and the vehicle velocity controller unit. The former one receives measured parameters from on-board sensors as well as estimated vehicle states from two observers, and then it calculates the control input $Δ M_{z}$ as described by equation (42). This corrective yaw moment is converted to the torque difference $Δ T$ between the two motors according to the following relationship

Δ T = \frac{Δ M_{z}}{d_{r}} R

(43)

where $d_{r}$ denotes the rear track width. The latter unit (a proportional–integral–derivative (PID)-type controller) receives the demanded velocity from the driver as well as the actual velocity from the observer and then generates the base torque $T_{base}$ to achieve the required velocity. The outputs from these two controller units are combined to form the motor torque commands, $T_{L}$ and $T_{R}$ .

Figure 2.

Schematic of the proposed control system.

Simulation results

In this section, we present the simulation results for both the conventional control scheme and the proposed adaptive control method. The simulation studies comprise three sections: case studies with accurate vehicle states (no estimation errors), case studies with constant vehicle state estimation errors, and case studies with convergent vehicle state estimation errors. In each section, the yaw rate responses and vehicle paths resulting from both control methods are plotted.

The common J-turn maneuver (i.e. a short straight maneuver followed by a constant turning maneuver) is employed in all case studies. The steering input (the steer angle of the front wheels) for this maneuver is shown in Figure 3. As mentioned in section “Conventional sliding mode control,” the steady-state yaw rate response (equation (21)) is employed as the reference yaw rate $r^{*}$ in both control methods. Besides, the vehicle longitudinal velocity $v_{x}$ is chosen as 60 km/h and is kept constant in the simulation.

Figure 3.

Steer angle input for the J-turn maneuver.

The control performances of these two methods are compared and analyzed, with and without parameter estimation errors. The effectiveness of the proposed control design, and its superiority to the conventional method in the presence of parameter estimation errors, will be clearly shown in the following results.

Results without estimation errors

We first present the simulation results under the ideal situation, in which we assume that the vehicle states required by the controllers, $α_{f}$ and $α_{r}$ , are accurately estimated ( i.e. no estimation errors). Figure 4 shows the yaw rate responses of the simulated vehicle, and we see that when the vehicle states are accurately estimated, the yaw rate responses produced by the two control methods track the ideal curve very well. Analogous results can be observed in the vehicle path plots in Figure 5, where the trajectories resulting from both control methods follow the ideal vehicle path closely.

Figure 4.

Yaw rate response comparison without parameter estimation errors.

Figure 5.

Vehicle path comparison without parameter estimation errors.

Results with constant estimation errors

It is well known that in practice, some vehicle states (e.g. $α_{f}$ and $α_{r}$ ) required by the controller cannot be easily measured and need to be estimated instead. Therefore, the controller must be able to deal with reasonable extent of estimation errors and maintain consistent control performance. In this section, accurate values of $α_{f}$ and $α_{r}$ are no longer fed to the controllers; instead, we intentionally send values with constant errors to the control systems and investigate how these controllers perform.

5% constant estimation errors

First, we feed $1.05 α_{r}$ (+5% error), $0.95 α_{f}$ (−5% error), $1.05 α_{f}$ (+5% error), and $0.95 α_{r}$ (−5% error) to both controllers in respective case studies, where $α_{f}$ and $α_{r}$ represent the accurate side-slip angles. These errors are intentionally held constant throughout the entire simulation time. We find out that the results in cases $1.05 α_{r}$ and $0.95 α_{f}$ are fairly similar, so are the results in cases $1.05 α_{f}$ and $0.95 α_{r}$ . This is intuitive because the control inputs in both methods (i.e. equations (32) and (42)) include the terms “ ${\hat{C}}_{f} {\hat{α}}_{f} l_{f} - {\hat{C}}_{r} {\hat{α}}_{r} l_{r}$ ,” which makes a +5% (−5%) error in $α_{r}$ equivalent to a −5% (+5%) error in $α_{f}$ . For the purpose of conciseness, we only present the simulation results for the $1.05 α_{r}$ and $1.05 α_{f}$ cases in Figures 6 –9.

Figure 6.

Yaw rate response comparison with +5% constant estimation error in $α_{r}$ .

Figure 7.

Vehicle path comparison with +5% constant estimation error in $α_{r}$ .

Figure 8.

Yaw rate response comparison with +5% constant estimation error in $α_{f}$ .

Figure 9.

Vehicle path comparison with +5% constant estimation error in $α_{f}$ .

As shown in the $1.05 α_{r}$ case (Figures 6 and 7), the proposed control method and the conventional control method perform very similarly, and their responses track the reference yaw rate and desirable vehicle path closely. However, in the $1.05 α_{f}$ case (Figures 8 and 9), the proposed control method provides superior performance to the conventional one. Figure 8 shows that the conventional controller produces a yaw rate response with significant steady-state error. As for the proposed controller, its yaw rate response deviates slightly from the ideal curve roughly between 1.5 and 1.9 s, and then quickly converges to the reference value. The superiority of the proposed control method is also obvious in Figure 9, where we observe that the trajectory produced by the proposed controller tracks the ideal vehicle path closely, while the resulting trajectory of the conventional controller deviates outwards.

Due to the existence of the estimation error, the conventional controller is no longer able to generate the desired control command to track the reference yaw rate, which in turn results in significant yaw rate error and vehicle path deviation. However, thanks to the adaptive nature, the proposed controller changes its control gain automatically to compensate for the estimation error. As the adaptation goes on, the actual yaw rate response is driven toward the reference yaw rate. Also, the vehicle path closely follows the ideal one.

10% constant estimation errors

We now enlarge the extent of estimation errors to ±10% and investigate the control performances under severer parameter inaccuracies. Likewise, the results in cases $1.1 α_{r}$ and $0.9 α_{f}$ are similar, so are the results in cases $1.1 α_{f}$ and $0.9 α_{r}$ . For the purpose of conciseness, in this section we only show the results for cases $1.1 α_{r}$ and $1.1 α_{f}$ .

As shown in the $1.1 α_{r}$ case (Figures 10 and 11), the proposed control method presents apparent advantages over the conventional control scheme. In terms of the yaw rate responses, we observe in Figure 10 that the conventional controller gives rise to significant deviations for two relatively long periods during the simulation time. In comparison, the proposed controller only produces a very small deviation, and it quickly drives the deviated yaw rate back to the desired value. The superiority of the proposed control method is also shown in Figure 11, where we see that the proposed controller keeps the vehicle close to the ideal path, while the conventional controller deviates the vehicle inwards.

Figure 10.

Yaw rate response comparison with +10% constant estimation error in $α_{r}$ .

Figure 11.

Vehicle path comparison with +10% constant estimation error in $α_{r}$ .

In the $1.1 α_{f}$ case (Figures 12 and 13), the conventional controller cannot provide good yaw rate response and generates a significant steady-state error. On the contrary, the response of the proposed controller only deviates from the desired curve between 1.3 and 1.7 s, after which it closely follows the reference value. As for the vehicle path, it is shown in Figure 13 that the trajectory produced by the proposed controller closely follows the ideal path; however, the conventional controller deviates the vehicle outwards.

Figure 12.

Yaw rate response comparison with +10% constant estimation error in $α_{f}$ .

Figure 13.

Vehicle path comparison with +10% constant estimation error in $α_{f}$ .

As mentioned previously, the estimation errors deprive the conventional controller of its ability to generate the desired control command and in turn dramatically jeopardizes the control performance. On the contrary, the proposed adaptive sliding mode controller adapts to the changes of vehicle states and guarantees the convergence of yaw rate to its reference value in the presence of state estimation errors.

Results with convergent estimation errors

In control systems where the plant states required by the controllers cannot be directly measured, appropriate observers should be designed to estimate the required states. A properly designed observer should be able to guarantee asymptotic convergence of the estimated states; in other words, the estimation error E should satisfy

E = E_{0} e^{- ct}

(44)

where $E_{0}$ denotes the estimation error at time $t = 0$ , and c is a positive constant dependent on the design of the certain observer.

In this section, we show the simulation results with convergent estimation errors. Namely, the estimated vehicle states ${\hat{α}}_{f}$ and ${\hat{α}}_{r}$ converge asymptotically to their actual values, as described by equation (44). In the simulation, $E_{0}$ is chosen as $20 %$ of the actual value and c is set to $1$ , with the other simulation conditions remaining the same. The resulting control performances are plotted in Figures 14 –17.

Figure 14.

Yaw rate response comparison with convergent estimation error in $α_{r}$ .

Figure 15.

Vehicle path comparison with convergent estimation error in $α_{r}$ .

Figure 16.

Yaw rate response comparison with convergent estimation error in $α_{f}$ .

Figure 17.

Vehicle path comparison with convergent estimation error in $α_{f}$ .

We see in Figures 14 and 15 that when the estimated rear side-slip angle ${\hat{α}}_{r}$ converges asymptotically to its actual value, both the proposed and conventional controllers perform well in terms of the yaw rate and vehicle path responses. However, when the estimated front side-slip angle ${\hat{α}}_{f}$ possesses a convergent estimation error, the proposed control method significantly outperforms the conventional one. As shown in Figure 16, the yaw rate response produced by the proposed controller presents only a slight error roughly between 1.5 and 2.0 s, after which the yaw rate curve is driven back to its reference value. The conventional controller, on the contrary, generates an obvious steady-state error in the yaw rate response. Besides, we observe in Figure 17 that the proposed scheme achieves close vehicle path tracking, while the conventional one deviates the car outwards.

To clarify the causes that lead to the performance difference between the two control methods, we plot in Figure 18 the control gains of these two controllers with convergent estimation error in $α_{f}$ , namely, the value of k in control law equation (32) and that of $a (| {\hat{α}}_{f} | l_{f} k_{1} + | {\hat{α}}_{r} | l_{r} k_{2})$ in control law equation (42). Obviously the gain k in control law equation (32) remains constant throughout the simulation time, as a result, the controller’s capability of dealing with parametric variations and uncertainties is limited. In comparison, the magnitude of the proposed control gain adapts to the variations in ${\hat{α}}_{f}$ and ${\hat{α}}_{r}$ , according to the rules specified by equations (34) and (35). We see in Figure 18 that the initial values of the two gains are the same, but as the simulation goes by, the proposed control gain augments itself to compensate for the estimation error and significantly improves the control performance.

Figure 18.

Control gain comparison with convergent estimation error in $α_{f}$ .

It is important to note that the proposed control gain $a (| {\hat{α}}_{f} | l_{f} k_{1} + | {\hat{α}}_{r} | l_{r} k_{2})$ is bounded and converges to a certain value when ${\overset{\cdot}{k}}_{1} \to 0$ and ${\overset{\cdot}{k}}_{2} \to 0$ (i.e. $k_{1} \to k_{1 \infty}$ and $k_{2} \to k_{2 \infty}$ ), as follows

\begin{matrix} a (| {\hat{α}}_{f} | l_{f} k_{1} + | {\hat{α}}_{r} | l_{r} k_{2}) \to \\ a (l_{f} | C_{f} α_{f} - {\hat{C}}_{f} {\hat{α}}_{f} | + l_{r} | C_{r} α_{r} - {\hat{C}}_{r} {\hat{α}}_{r} |) \end{matrix}

(45)

Besides, the proposed control gain will be reset to zero as soon as the vehicle completes cornering and returns to straight motion (i.e. when $| {\hat{α}}_{f} |$ and $| {\hat{α}}_{r} |$ drop to zero).

Conclusion and recommendations

The existing distributed-motor-type DYC designs commonly involve the computation of tire side-slip angles, which is susceptible to measurement and estimation errors. In this article, we first introduce the conventional sliding mode control scheme with a fixed control gain and then propose a novel adaptive sliding mode DYC approach that utilizes a varying control gain to adapt to the variations of tire side-slip angles. With the proposed control method, the adverse effects of inaccuracies on tire side-slip angles are suppressed, which in turn enhances the system robustness against parametric variations and uncertainties.

The conventional control scheme and the proposed control method are compared under various simulation conditions. In an ideal situation where the tire side-slip angles are accurately estimated, both methods produce good yaw rate and vehicle path responses. However, in practical situations where the tire side-slip angle information contains estimation errors (different levels of constant errors or convergent errors), the control performance of the conventional control scheme deteriorates significantly. In comparison, the proposed control method maintains consistent control performance with inaccurate tire side-slip angles and outperforms the conventional control scheme in terms of tracking the reference yaw rate and reference vehicle path.

The proposed DYC method can provide a corrective yaw moment to effectively enhance the stability and handling of electric vehicles. The focus of our following research will be on the integration of DYC and other advanced intelligent safety systems, to further enhance the safety and dynamic performance of electric vehicles and pave the way for future autonomous driving.

Footnotes

Handling Editor: Zhixiong Li

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 51805055), the National Key Research and Development Project (Grant Nos 2018YFB0106102 and 2018YFB0106104), and the Specialized Research Fund for Chongqing Key Industries Common Key Technology Innovation (Grant No. CSTC2015ZDCY-ZTZX60009).

ORCID iD

Chunyun Fu

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A novel adaptive sliding mode control approach for electric vehicle direct yaw-moment control

Abstract

Keywords

Introduction

Problem statement

Control model

Simulation model

DYC design problem

Conventional sliding mode control

Proposed adaptive sliding mode control

Simulation results

Results without estimation errors

Results with constant estimation errors

5% constant estimation errors

10% constant estimation errors

Results with convergent estimation errors

Conclusion and recommendations

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References