Sage Journals: Discover world-class research

Abstract

This paper introduces a novel control method to address lateral stability challenge in distributed driving electric vehicles. By considering inherent time delay and unknown sensor measurement error generated from design, manufacturing, and external disturbances, the proposed approach achieves better robustness. Firstly, we establish a two-degree-of-freedom (2-DOF) dynamic model for lateral motion of an electric vehicle driven by in-wheel motors (EV-DIM). Based on this model, a control system considering time delay and unknown sensor measurement error is constructed. The control system incorporates a controller based on a dual domination gains observer, and its global asymptotic stability is rigorously proved by the Lyapunov method. Numerical simulations and co-simulations via CarSim and Simulink verify the method’s effectiveness. Compared to the sliding mode control, the proposed method shows better performance which indicates this approach promising for enhancing lateral stability of an electric vehicle.

Keywords

Electric vehicle lateral motion stability time-delay nonlinear systems unknown sensor measurement error

Introduction

Distributed drive electric vehicles present a novel chassis structure compared to traditional counterparts, significantly simplifying the powertrain system. This advancement facilitates vehicle lightweighting and augments energy efficiency. The incorporation of multi-wheel independent drive permits individual wheel operation, while enabling coordinated control in accordance with overall vehicle requirements. This attribute holds promise for enhancing overall vehicle stability during operation.¹

When an electric vehicle is driven at high speed, sudden disturbances can induce hazardous lateral movements, challenging drivers to respond swiftly in emergencies.² Hence, developing lateral stability control techniques for enhancing the safety of electric vehicles is paramount.

In recent years, researchers have introduced diverse approaches to tackle lateral stability control in vehicles. For instance, Nam et al.³ proposed a method to enhance the stability of electric vehicles by estimating the vehicle’s slip angle using tire lateral force sensors. a vehicle stability controller was formulated leveraging the LuGre tire model to achieve the specified control objectives.^4,5 Zhang et al.⁶ developed an optimal control approach grounded in quadratic optimal control theory to effectively mitigate yaw angle deviation of the center of gravity during cornering maneuvers. Li et al.⁷ have developed a linear time-varying model predictive control (LTV-MPC) method based on a two-degree-of-freedom vehicle model.

In contrast to traditional vehicles, which require additional hydraulic control units to produce yaw moments for stabilizing vehicle motion through differential hydraulic braking, distributed drive electric vehicles can conveniently utilize motors to generate differential driving torques for stability control without compromising vehicle dynamics. Feng et al.⁸ integrated the principles of active disturbance rejection control (ADRC) theory and sliding mode control to develop a novel direct yaw moment control (DYC) controller customized for 4 WID-EV. In order to enhance the longitudinal and lateral stability of vehicles under extreme conditions, Li et al.⁹ introduced a real-time controller for four-wheel independent motor-driven electric vehicles based on NMPC. Jin et al.¹⁰ employed linear parameter-varying techniques to devise a robust gain-scheduled H $\infty$ controller aimed at enhancing the lateral stability of four-wheel independent drive electric vehicles. To improve robustness against estimation errors of disturbances, Hosseini-Pishrobat and Keighobadi¹¹ introduced an extended state observer based on a single-track vehicle model and an established mathematical model of the vehicle to estimate the total disturbance signal. Li et al.¹² devised a robust gain-scheduled output feedback controller based on measurable velocity. This controller accounts for uncertainties such as longitudinal velocity, tire lateral stiffness, and actuator saturation. To address constraints in manually adjusting parameters for the predictive Stanley lateral controller, Abdelmoniem et al.¹³ proposed an integrated fuzzy monitoring controller. This method automatically adapts to changes in vehicle velocity and maneuvers by estimating the prediction horizon length and assigning different weights to each controller.

Sliding mode control, as a classic control method, also enjoys extensive application in vehicle lateral stability control. Tian et al.¹⁴ have developed an approach that integrates sliding mode control with optimal allocation algorithms, utilizing a hierarchical structure to control vehicle lateral stability. Xiao¹⁵ partitioned the original model of the unmanned aerial vehicle into two components to accomplish trajectory planning and ensure flight stability: an attitude control model regulated by a sliding mode controller, and an altitude control model managed by an extended state observer. To represent the reference longitudinal velocity reflecting the actual speed, Chen et al.¹⁶ employed a combination of a neural network observer and a fuzzy logic controller. They devised a novel lateral control system comprising high, medium, and low-level subsystems to ensure the smooth and safe operation of vehicles on roads with low adhesion. Wang et al.¹⁷ utilized sliding mode control with longitudinal and lateral velocities, along with yaw angular velocity, as control variables to determine required longitudinal and lateral forces, and yaw moments. They integrated a cost function with adjustable weight coefficients to consider motor power and tire load. Ma et al.¹⁸ implemented an integral terminal sliding mode (ITSM) controller, considering parameter uncertainties, to swiftly achieve lateral stability, while employing velocity tracking for longitudinal stability.

As neural networks advance, their utilization in enhancing lateral stability control in automotive vehicles is developing. Cao et al.¹⁹ devised a robust steering control system utilizing BVSC to mitigate lateral path tracking deviations, counteract unknown external disturbances, and ensure the lateral stability of autonomous driving vehicles. Huang et al.²⁰ employed the BP-PID algorithm to devise a DDEV multi-model control system (MMCS) based on direct yaw moment control.

In real-world scenarios, disturbances affecting vehicles are often uncertain, and control systems mostly involve nonlinear terms. Hence, taking into account the time-varying nature of vehicle velocity and the uncertainty of external disturbances, Meng et al.²¹ proposed a method more suitable for computer implementation, termed the Sampled-Data Output Feedback Control with Almost Disturbance Decoupling method. To address potential uncontrollability near the origin resulting from nonlinearity, Meng et al.²² developed a homogenous control method based on sampled-data output feedback. Considering the uncertainty in vehicle longitudinal velocity and controller saturation, Li et al.¹² introduced a homogeneous polynomial parameter-correlation method to address the issue. They designed a multi-objective controller to enhance the lateral dynamic stability and maneuverability of distributed drive electric vehicles by achieving the desired external yaw moment.

The aforementioned research method does not tackle practical engineering issues such as system delays and unknown errors in sensor sensitivity. Fortunately, some researchers have explored these issues in theory. Sun et al.^23,24 studied the global output feedback stabilization problem of a class of uncertain nonlinear systems. They utilized a dual control approach to address time-varying measurement errors and delayed states. This control strategy relies on the selection of Lyapunov-Krasovskii functionals and the development of novel state observers, thus eliminating the requirement for output functions and nonlinear information. Shao et al.²⁵ stabilized a nonlinear system by introducing the power integral method. They devised a novel continuous controller to resolve the global stabilization issue of the system.

In this paper, we investigate the lateral motion stability of an EV-DIM considering time delay and unknown sensor measurement error, and propose a vehicle stability control method based on a dual domination gains observer. The contributions of this study are as follows.

A 2-DOF dynamic model of an EV-DIM for lateral motion is constructed. This model reflects the errors between actual values and ideal values for sideslip angle and yaw velocity that provide the basis for the following controller design.

A lateral stability error tracking control system is constructed which considers the unknown sensor measurement error and time delay. The unknown sensor measurement error is caused by external disturbances, sensor design, manufacture, etc. The time delay arises due to factors such as sensor data processing, system synchronization, gaps in the mechanical system, etc.

A lateral stability controller based on dual domination gains observer is designed which has better robustness. The two domination gains are used to deal with the influences of the unknown sensor measurement error and time delay.

The rest of paper is organized as follows. Section 2 introduces the lateral motion dynamics model of EV-DI M taking into account uncertain disturbances. In Section 3, a controller and observer based on dual control gain are designed. Section 4 conducts numerical simulations of the lateral stability of an EV-DIM. Section 5 carries out co-simulations via CarSim and Simulink and makes comparisons with sliding mode control, thereby validating the effectiveness and superiority of the proposed method.

The lateral motion dynamic model of EV

In order to simplify the lateral stability control method for four-wheel-drive electric wheel vehicles, this paper reduces the four-wheel-drive electric wheel vehicle to a 2-DOF single-track model as shown in Figure 1.

Figure 1.

The two-degree-of-freedom model of an electric vehicle driven by in-wheel motors.

Although this model is similar with the fuel vehicle model, the key distinction lies in the fact that each wheel is individually powered by a hub motor, which is different from the fuel automotive structure. Due to the hub motor’s rapid and precise torque response, it is easy to generate an additional torque $M_{z}$ to regulate the lateral motion of the vehicle. To design an effective controller, the ideal two-wheel vehicle’s lateral dynamics mathematical model is constructed by disregarding roll and pitch motions. The expression for this model is as follows.

\begin{matrix} {\overset{\cdot}{β}}_{d} = \frac{F_{xf} + F_{xr}}{m v_{y}} - γ_{d} \\ {\overset{\cdot}{γ}}_{d} = \frac{L_{f} F_{xf} - L_{r} F_{xr} + M_{zd}}{I_{z}}, \end{matrix}

(1)

where $β_{d}$ is the ideal sideslip angle of vehicle centroid, $γ_{d}$ is the ideal yaw velocity, $F_{xf}$ and $F_{xr}$ are the lateral forces of front and rear wheels respectively, $v_{y}$ is the longitudinal velocity, $m$ and $I_{z}$ are the mass and moment of inertia respectively, $L_{f}$ and $L_{r}$ are the distance between the front axle and centroid and the distance between the rear axle and centroid respectively, $M_{zd}$ is the ideal additional yaw moment.

The lateral forces of the front and rear tires can be calculated using the following linear tire model.²⁶

F_{xf} = C_{f} α_{f}, F_{xr} = C_{r} α_{r},

(2)

where $C_{f}$ and $C_{r}$ are the front and real tire cornering stiffness values respectively, $α_{f}$ and $α_{r}$ are the wheel slip angles which can be calculated as

\begin{matrix} α_{f} = - (β_{d} + \frac{L_{f} γ_{d}}{v_{y}} - δ (t)) \\ α_{r} = - (β_{d} - \frac{L_{r} γ_{d}}{v_{y}}), \end{matrix}

(3)

where $δ (t)$ is the front wheel turning angle. According to equations (2) and (3), the system (1) can be expressed as

\begin{matrix} {\overset{\cdot}{β}}_{d} = - \frac{(C_{f} + C_{r}) β_{d}}{m v_{y}} + \frac{(C_{r} L_{r} - C_{f} L_{f}) γ_{d}}{{mv}_{y}^{2}} - γ_{d} \\ + \frac{C_{f}}{m v_{y}} δ (t) \\ {\overset{\cdot}{γ}}_{d} = \frac{(C_{r} L_{r} - C_{f} L_{f}) β_{d}}{I_{z}} - \frac{(C_{f} L_{f}^{2} + C_{r} L_{r}^{2}) γ_{d}}{I_{z} v_{y}} \\ + \frac{M_{zd}}{I_{z}} + \frac{C_{f} L_{f}}{I_{z}} δ (t) . \end{matrix}

(4)

In practical operating conditions, electric vehicles suffer various external disturbances while driving within their lanes. These disturbances can significantly impact the vehicle’s behavior and performance. Consequently, the actual 2-DOF electric vehicle model can be succinctly expressed as

\begin{matrix} {\overset{\cdot}{β}}_{p} = - \frac{(C_{f} + C_{r}) β_{p}}{m v_{y}} + \frac{(C_{r} L_{r} - C_{f} L_{f}) γ_{p}}{{mv}_{y}^{2}} - γ_{p} + \frac{C_{f}}{m v_{y}} (δ (t) + Δ δ (t)) \\ {\overset{\cdot}{γ}}_{p} = \frac{(C_{r} L_{r} - C_{f} L_{f}) β_{p}}{I_{z}} - \frac{(C_{f} L_{f}^{2} + C_{r} L_{r}^{2}) γ_{p}}{I_{z} v_{y}} + \frac{M_{zd}}{I_{z}} + \frac{C_{f} L_{f}}{I_{z}} (δ (t) + Δ δ (t)), \end{matrix}

(5)

where $β_{p}$ is the actual sideslip angle, $γ_{p}$ is the actual yaw velocity, $Δ δ (t)$ represents the uncertain disturbance arising from factors such as road profile variations, lateral wind, and differences in tire pressure during vehicle operation, $M_{zp}$ is the actual additional yaw moment. For the purpose of simplifying analysis, researchers commonly treat the vehicle speed $v_{y}$ as a constant. In practical scenarios, the velocity of a moving vehicle is not constant. To accurately reflect real-world conditions, this paper takes into account the time-varying nature of the quantity denoted by $v_{y}$ . In the methods of analysis, $v_{y}$ can be expressed with the following form of uncertainty.

ρ_{1} (t) = \frac{1}{v_{y}}, ρ_{2} (t) = \frac{1}{v_{y}^{2}}

(6)

Despite $v_{y}$ varying with time, it still possesses a maximum value. Therefore

0 < ρ_{1} (t) \leq θ_{1}, 0 < ρ_{2} (t) \leq θ_{2}

(7)

where $θ_{1}$ and $θ_{2}$ are the parameters related to speed.

Furthermore, to accurately reflect practical conditions, this study also takes into account the presence of unknown sensor measurement errors and time delay. As a result, we introduce the variable $θ (t)$ and $τ (t)$ to simulate these issue.

Based on equations (4) and (5), one can obtain

\begin{matrix} {\overset{\cdot}{e}}_{β} = - \frac{(C_{f} + C_{r}) e_{β}}{m v_{y}} + \frac{(C_{r} L_{r} - C_{f} L_{f}) e_{γ}}{{mv}_{y}^{2}} - e_{γ} \\ + \frac{C_{f}}{m v_{y}} Δ δ (t - τ (t)) \\ {\overset{\cdot}{e}}_{γ} = \frac{(C_{r} L_{r} - C_{f} L_{f}) e_{β}}{I_{z}} - \frac{(C_{f} L_{f}^{2} + C_{r} L_{r}^{2}) e_{γ}}{I_{z} v_{y}} \\ + \frac{Δ M_{zd}}{I_{z}} + \frac{C_{f} L_{f}}{I_{z}} Δ δ (t - τ (t)), \end{matrix}

(8)

where ${\overset{\cdot}{e}}_{β} = {\overset{\cdot}{β}}_{p} - {\overset{\cdot}{β}}_{d}, {\overset{\cdot}{e}}_{γ} = {\overset{\cdot}{γ}}_{p} - {\overset{\cdot}{γ}}_{d}, e_{β} = β_{p} - β_{d}, e_{γ} = γ_{p} - γ_{d}$ . $Δ M_{z} = M_{zp} - M_{zd}$ is extra necessary moment to keep the EV stability.

Let $x (t) = {(x_{1} (t), x_{2} (t))}^{T} \in R^{2}$ denote the system’s state variable, $u (t) \in R$ , $y (t) \in R$ , and $d (t) \in R$ represent the system input, output, and uncertain disturbance element, respectively. Define

x_{1} (t) = e_{β}

(9)

x_{2} (t) = \frac{(C_{r} L_{r} - C_{f} L_{f}) ρ_{2} (t) - m}{m} e_{γ}

(10)

u (t) = \frac{(C_{r} L_{r} - C_{f} L_{f}) ρ_{2} (t) - m}{m I_{z}} Δ M_{z}

(11)

d (t) = Δ δ (t - τ (t)),

(12)

system (8) can be rewritten as

\begin{matrix} {\overset{\cdot}{x}}_{1} (t) = x_{2} (t) + ϕ_{1} (t, x (t)) + g_{1} d (t - τ (t)) \\ {\overset{\cdot}{x}}_{2} (t) = u (t) + ϕ_{2} (t, x (t)) + g_{2} d (t - τ (t)) \\ y (t) = θ (t) x_{1} (t), \end{matrix}

(13)

where

g_{1} (t) = \frac{C_{f}}{m} ρ_{1} (t),

(14)

g_{2} (t) = \frac{(C_{r} L_{r} - C_{f} L_{f}) ρ_{2} (t) - m}{m} \frac{C_{f} L_{f}}{I_{z}},

(15)

ϕ_{1} (t, x (t)) = - \frac{(C_{r} + C_{f})}{m} ρ_{1} (t) x_{1} (t),

(16)

\begin{matrix} ϕ_{2} (t, x (t)) = \frac{[(C_{r} L_{r} - C_{f} L_{f}) ρ_{2} (t) - m]}{m I_{z}} \\ (C_{r} L_{r} - C_{f} L_{f}) x_{1} (t) + [- \frac{C_{r} L_{r}^{2} + C_{f} L_{f}^{2}}{I_{z}} ρ_{1} (t) + \frac{(C_{r} L_{r} - C_{f} L_{f}) {\overset{\cdot}{ρ}}_{2} (t)}{(C_{r} L_{r} - C_{f} L_{f}) ρ_{2} (t) - m}] x_{2} (t) . \end{matrix}

(17)

Lateral motion stability control method based on dual domination gains observer (DDGO)

The aim of this paper is to develop a lateral stability controller for electric vehicles, which tackles challenges associated with unknown sensor measurement error and system time delays. We describe this achievement by Theorem 1.

Theorem 1. Under Assumptions 1 and 2, There exists an output feedback controller that can be employed to achieve globally asymptotic stabilization for system (13).

Assumption 1. There is a known positive parameter $\bar{θ}$ satisfying $| 1 - θ (t) | \leq \bar{θ} < θ^{*} < 1$ with $θ^{*}$ being an allowable sensitivity error of $\bar{θ}$ .

Assumption 2. For $i = 1, \dots, n,$ there is a constant $c \geq 0$ such that

| f_{i} (\cdot) | \leq c \sum_{j = 1}^{i} | x_{j} | + c \sum_{j = 1}^{i} | x_{j} (t - τ_{j} (t)) | .

(18)

Proof. Next, we will prove Theorem 1 using the Lyapunov method. First, we introduce two lemmas that will assist in proving Theorem 1.

Lemma 1. ²⁷ For given positive real numbers $m, n$ and function $a (x, y)$ , there holds

\begin{matrix} | a (x, y) x^{m} y^{n} | \leq c (x, y) {| x |}^{m + n} \\ + \frac{n}{m + n} {(\frac{m}{(m + n) c (x, y)})}^{m / n} \\ \cdot {| a (x, y) |}^{(m + n) / n} {| y |}^{m + n}, \end{matrix}

(19)

where $c (x, y) > 0$ , for any $x \in R$ , $y \in R$ .

Lemma 2. ²⁸ Let $p \in [1, \infty)$ . For any $x_{i} \in R, i = 1, \dots, n$ ,

{(| x_{1} | + \dots + | x_{n} |)}^{p} \leq n^{p - 1} ({| x_{1} |}^{p} + \dots + {| x_{n} |}^{p}) .

(20)

Design of linear observer

In system (13), $x_{1}$ is obtained by a sensor, and $x_{2}$ is unmeasurable. Therefore, it is imperative to construct an observer. A linear observer is designed as²³

\begin{matrix} {\overset{\cdot}{\hat{x}}}_{1} = {\hat{x}}_{2} - K_{1} a_{1} {\hat{x}}_{1} \\ {\overset{\cdot}{\hat{x}}}_{2} = u - K_{1}^{2} a_{2} {\hat{x}}_{1}, \end{matrix}

(21)

where $K_{1} \geq 1$ is the first domination gain. And the positive constants $a_{1}$ and $a_{2}$ are the coefficients of a Hurwitz polynomial. Define the estimation error as

ε_{i} = \frac{x_{i} - {\hat{x}}_{i}}{K_{1}^{i - 1}}, i = 1, 2 .

(22)

Then

\overset{\cdot}{ε} = K_{1} A_{a} ε + K_{1} a x_{1} + Ψ,

(23)

where $ε = [ε_{1} ε_{2}]^{T}$ , $Ψ = [ϕ_{1} \frac{ϕ_{2}}{K_{1}}]^{T}$ , $a = [a_{1} a_{2}]^{T}$ . And $A_{a} = [\begin{matrix} - a_{1} & 1 \\ - a_{2} & 0 \end{matrix}]$ which is a Hurwitz matrix. This matrix satisfies

A_{a}^{T} P_{e} + P_{e} A_{a} = - I,

(24)

where $I$ is the identity matrix of $R^{2 \times 2}$ , $P_{e} \in R^{2 \times 2}$ .

Construct a positive definite matrix

V_{1} = ε^{T} P_{e} ε + \sum_{i = 1}^{2} \frac{c}{1 - σ_{i}} \int_{t - τ_{i} (t)}^{t} \frac{x_{i}^{2} (s)}{K_{1}^{2 i - 2}} d s,

(25)

Under Assumption 2, the derivative of $V_{1}$ along (23) is

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - K_{1} ∥ ε ∥^{2} + 2 K_{1} ε^{T} P_{1} a x_{1} + 2 ε^{T} P_{1} Ψ \\ + \sum_{i = 1}^{2} \frac{{cx}_{i}^{2}}{(1 - σ_{i}) K_{1}^{2 i - 2}} - \sum_{i = 1}^{2} \frac{{cx}_{i}^{2} (t - τ_{i} (t))}{K_{1}^{2 i - 2}}, \end{matrix}

(26)

where $1 - {\overset{\cdot}{τ}}_{i} (t) \geq 1 - σ_{i}$ .

By Lemma 1, we deduce the following inequality from (26).

\begin{array}{l} {\dot{V}}_{1} \leq - \frac{K_{1}}{2} ∥ ε ∥^{2} + (6 c ∥ P_{e} ∥ + 5 c ∥ P_{e} ∥^{2}) ∥ ε ∥^{2} \\ + 2 K_{1} ∥ P_{e} ∥^{2} \cdot ∥ a ∥^{2} x_{1}^{2} \\ + \sum_{i = 1}^{2} (\frac{c (3 - i) ∥ P_{e} ∥}{2 K_{1}^{2 i - 2}} + \frac{c}{(1 - σ_{i}) K_{1}^{2 i - 2}}) x_{i}^{2} \\ ≜ - \frac{K_{1}}{2} ∥ ε ∥^{2} + l_{1} ∥ ε ∥^{2} + K_{1} l_{2} x_{1}^{2} + \sum_{i = 1}^{2} \frac{ℓ_{i}}{K_{1}^{2 i - 2}} x_{i}^{2}, \end{array}

(27)

where

$l_{1} = 6 c ∥ P_{e} ∥ + 5 c ∥ P_{e} ∥^{2} \geq 0$ , $l_{2} = 2 ∥ P_{e} ∥^{2} \cdot ∥ a ∥^{2} > 0$ , $ℓ_{i} = (c / 2) (3 - i) ∥ P_{e} ∥ + c / (1 - σ_{i}) \geq 0, i = 1, 2$ are independent of the domination gain $K_{1}$ .

Design of output feedback control law

According to equations (13) and (21), there exists

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} + ϕ_{1} (t, x (t)) + g_{1} d (t - τ (t)) \\ {\overset{\cdot}{\hat{x}}}_{2} = u - K_{1}^{2} a_{2} {\hat{x}}_{1} . \end{matrix}

(28)

We denote $f_{1} = ϕ_{1} (t, x (t)) + g_{1} d (t - τ (t))$ . Because $ε_{1} = x_{1} - {\hat{x}}_{1}$ , there is

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} + f_{1} \\ {\overset{\cdot}{\hat{x}}}_{2} = u + K_{1}^{2} a_{2} (ε_{1} - x_{1}) . \end{matrix}

(29)

The coordinate transformation for system (29) is

z_{1} = x_{1}, z_{2} = \frac{{\hat{x}}_{2}}{K_{1} K_{2}}, v = \frac{u}{{(K_{1} K_{2})}^{2}},

(30)

where $K_{2} \geq 1$ is the second domination gain. Subsequently, the augmented system (29) is reformulated as

\begin{matrix} {\overset{\cdot}{z}}_{1} & = K_{1} K_{2} ξ_{2} + K_{1} ε_{2} + f_{1} \\ {\overset{\cdot}{z}}_{2} & = K_{1} K_{2} v + \frac{K_{1} a_{2}}{K_{2}} (ε_{1} - z_{1}) . \end{matrix}

(31)

Therefore, we design an output feedback controller for system (30) as

υ = - b_{2} y - b_{1} z_{2},

(32)

where the positive constants $b_{1}$ and $b_{2}$ are the coefficients of a Hurwitz polynomial. Thus, system (29) can be reformulated as

\begin{matrix} \overset{\cdot}{z} = K_{1} K_{2} A_{b} z + K_{1} K_{2} N b_{2} (1 - θ (t)) z_{1} \\ + K_{1} D_{2} ε_{2} + K_{1} D_{1} (ε_{1} - z_{1}) + Φ, \end{matrix}

(33)

where $A_{b} = [\begin{matrix} 0 & 1 \\ - b_{2} & - b_{1} \end{matrix}]$ , $N = [0 1]^{T}$ , $D_{2} = [1 0]^{T}$ , $D_{1} = [0 a_{2}]^{T}$ , $Φ = [f_{1} 0]^{T}$ .

Similar to the information provided above, $A_{b}$ is a Hurwitz matrix. Therefore, there exists a positive definite matrix $P_{z}$ which satisfies $A_{b}^{T} P_{z} + P_{z} A_{b} = - I$ . We define the following function

V_{2} = z^{T} P_{2} z + \frac{c}{1 - σ_{1}} \int_{t - τ_{1} (t)}^{t} x_{1}^{2} (s) ds .

(34)

With Assumption 2, the derivative of $V_{2}$ along (33) is

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq 2 z^{T} P_{2} (K_{1} K_{2} A_{b} z + K_{1} K_{2} N b_{n} (1 - θ (t)) z_{1} \\ + K_{1} D_{2} ε_{2} + K_{1} D_{1} (ε_{1} - z_{1}) + Φ) \\ + \frac{{cx}_{1}^{2}}{1 - σ_{1}} - {cx}_{1}^{2} (t - τ_{1} (t)) . \end{matrix}

(35)

By Lemma 1, we deduce the following inequality from (35)

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - K_{1} K_{2} (1 - 2 b_{n} | 1 - θ (t) | \cdot ∥ P_{z} ∥) ∥ z ∥^{2} \\ + K_{1} (2 ∥ P_{z} ∥ \cdot ∥ a ∥ (1 + 4 ∥ P_{z} ∥ \cdot ∥ a ∥) + 8 ∥ P_{z} ∥^{2}) ∥ z ∥^{2} \\ + K_{2} (2 c ∥ P_{z} ∥) ∥ z ∥^{2} \\ + (c ∥ P_{z} ∥^{2} + \frac{c}{1 - σ_{1}}) ∥ z ∥^{2} + \frac{K_{1}}{4} ∥ ε ∥^{2} \\ \leq - K_{1} K_{2} (1 - 2 b_{n} | 1 - θ (t) | \cdot ∥ P_{z} ∥) ∥ z ∥^{2} \\ + K_{1} {\bar{ℓ}}_{1} ∥ z ∥^{2} + K_{2} {\bar{ℓ}}_{2} ∥ z ∥^{2} + \frac{K_{1}}{4} ∥ ε ∥^{2}, \end{matrix}

(36)

where ${\bar{ℓ}}_{1} = 2 ∥ P_{z} ∥ \cdot ∥ a ∥ (1 + 4 ∥ P_{z} ∥ \cdot ∥ a ∥) + 8 ∥ P_{z} ∥^{2} > 0$ , ${\bar{ℓ}}_{2} = 2 c ∥ P_{z} ∥ + c ∥ P_{z} ∥^{2} + c / (1 - σ_{1}) \geq 0$ are independent of the domination gain $K_{1}$ and $K_{2}$ .

According to Lemma 2, we select $p = 2$ to obtain

\sum_{i = 1}^{2} \frac{ℓ_{i}}{K_{1}^{2 i - 2}} x_{i}^{2} \leq 2 \sum_{i = 1}^{2} ℓ_{i} ∥ ε ∥^{2} + 2 \sum_{i = 1}^{2} ℓ_{i} K_{2}^{2 i - 2} ∥ z ∥^{2} .

(37)

Then (26) can be rewritten as

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - \frac{K_{1}}{2} ∥ ε ∥^{2} + (l_{1} + 2 \sum_{i = 1}^{2} ℓ_{i}) ∥ ε ∥^{2} \\ + (K_{1} l_{2} + 2 \sum_{i = 1}^{2} ℓ_{i} K_{2}^{2 i - 2}) ∥ z ∥^{2} . \end{matrix}

(38)

Next, we define the upper limit for the allowable sensor measurement error as

\bar{θ} < θ^{*} = \frac{1}{2 b_{n} ∥ P_{zb} ∥} .

(39)

According to Lemma 1, it is easy to obtain

1 - 2 b_{n} | 1 - θ (t) | \cdot ∥ P_{z} ∥ \geq 1 - 2 b_{n} ∥ P_{z} ∥ \bar{θ} \overset{Δ}{=} ρ,

(40)

where $0 < ρ < 1$ . Then according to (36) and (38), there exists

\begin{matrix} \overset{\cdot}{V} \leq - K_{1} K_{2} (ρ - \frac{{\bar{ℓ}}_{1} + l_{2}}{K_{2}} \\ - \frac{{\bar{ℓ}}_{2} + 2 \sum_{i = 1}^{n} ℓ_{i} K_{2}^{2 i - 3}}{K_{1}}) ∥ z ∥^{2} \\ - K_{1} (\frac{1}{4} - \frac{l_{1} + 2 \sum_{i = 1}^{n} ℓ_{i}}{K_{1}}) ∥ ε ∥^{2} . \end{matrix}

(41)

We choose the two domination gains $K 1$ and $K 2$ as

K_{1} \geq {1, \frac{16 ({\tilde{ℓ}}_{2} + 2 \sum_{i = 1}^{2} ℓ_{i} K_{2}^{2 i - 3})}{ρ}, \frac{16 (l_{1} + 2 \sum_{i = 1}^{2} ℓ_{i})}{4 - ρ}},

(42)

K_{2} \geq max {1, \frac{8 ({\bar{ℓ}}_{1} + l_{2})}{7 ρ}} .

(43)

Then (41) can be rewritten as

\overset{\cdot}{V} \leq - \frac{ρ}{16} ∥ ε ∥^{2} - \frac{ρ}{16} ∥ z ∥^{2} .

(44)

It is obvious that equation (44) is negative definite. Therefore, we have proven that by choosing appropriate domination gains $K_{1}$ and $K_{2}$ , the derivative of $V$ is negative definite. Next, we continue to analyze the global stability of the closed-loop system.

The following notations will be used in the analysis. For a real vector $x = [x_{1}, \dots, x_{n}]^{T} \in R^{n}$ , the norm $∥ x ∥$ is defined by $∥ x ∥ = \sqrt{\sum_{i = 1}^{n} x_{i}^{2}}$ , and $∥ x_{t} ∥_{c} =_{- δ_{M} \leq Ξ \leq 0} ∥ x (t + Ξ) ∥$ , $t \geq 0$ , where $δ_{M}$ is a known positive constant. A continuous function $ν : R^{+} \to R^{+}$ satisfying $ν (0) = 0$ is called a $K_{\infty}$ function if it is strictly increasing and ${}_{s \to + \infty}{ν (s)} = + \infty$ .

Based on the existence and continuity of solutions, the closed-loop system state composed of $Z (t) \overset{Δ}{=} [ε (t), z (t)]^{T}$ can be defined on the time interval $[- δ_{M}, t_{m})$ , where $t_{m} > 0$ may be a finite constant or $+ \infty$ . The analysis is divided into three parts.

(i) The bound of $Z (t)$ on $[- δ_{M}, t_{m})$ . From the definition of $V$ , it can be deduced that

V \geq ε^{T} P_{e} ε + z^{T} P_{z} z \geq λ_{m} ∥ Z ∥^{2} \overset{Δ}{=} ν_{1} (∥ Z ∥),

(45)

where $λ_{m} = {λ (P_{e}), λ (P_{z})}, and ν_{1} is a K_{\infty}$ function. And according to $τ_{i} (t) \leq d_{i} \leq d_{M}$ , we can get

\begin{matrix} c \sum_{i = 1}^{2} \int_{t - τ_{i} (t)}^{t} \frac{x_{i}^{2} (s)}{(1 - σ_{i}) K_{1}^{2 i - 2}} d s + \frac{c}{1 - σ_{1}} \int_{t - τ_{1} (t)}^{t} x_{1}^{2} (s) d s \\ \leq \sum_{i = 1}^{2} \frac{2 c}{1 - σ_{i}} \int_{t - τ_{i} (t)}^{t} (ε_{i}^{2} (s) + K_{2}^{2 i - 2} z_{i}^{2} (s)) d s \\ + \frac{2 c}{1 - σ_{1}} \int_{t - τ_{1} (t)}^{t} (ε_{1}^{2} (s) + z_{1}^{2} (s)) d s \\ \leq \sum_{i = 1}^{2} \frac{2 {cK}_{2}^{2 i - 2}}{1 - σ_{i}} \int_{- τ_{i} (t)}^{0} ∥ Z (t + Ξ) ∥^{2} d Ξ \\ + \frac{2 c}{1 - σ_{1}} \int_{- τ_{1} (t)}^{0} ∥ Z (t + Ξ) ∥^{2} d Ξ \\ \leq {(\sum_{i = 1}^{2} \frac{2 {cK}_{2}^{2 i - 2} δ_{M}}{1 - σ_{i}} + \frac{2 c δ_{M}}{1 - σ_{1}})}_{- δ_{M} \leq Ξ \leq 0} ∥ Z (t + Ξ) ∥^{2} \\ \leq φ {(\sup_{- δ_{M} \leq Ξ \leq 0} ∥ Z (t + Ξ) ∥)}^{2}, \end{matrix}

(46)

where $φ = \sum_{i = 1}^{2} 2 {cK}_{2}^{2 i - 2} δ_{M} / (1 - σ_{i}) + 2 c δ_{M} / (1 - σ_{1})$ . Then it can be deduced that

\begin{matrix} V \leq λ (P_{e}) ∥ ε ∥^{2} + λ (P_{z}) ∥ z ∥^{2} \\ + φ {(\sup_{- δ_{M} \leq Ξ \leq 0} ∥ Z (t + Ξ) ∥)}^{2} \\ \leq λ_{M} ∥ Z ∥^{2} + φ {(\sup_{- δ_{M} \leq Ξ \leq 0} ∥ Z (t + Ξ) ∥)}^{2} \\ \leq (λ_{M} + φ) {(\sup_{- δ_{M} \leq Ξ \leq 0} ∥ Z (t + Ξ) ∥)}^{2} \overset{Δ}{=} ν_{2} (∥ Z_{t} ∥_{C}), \end{matrix}

(47)

where $λ_{M} = \max {λ_{\max} (P_{e}), λ_{\max} (P_{z})}, and ν_{2} (\cdot) is a K_{\infty}$ function. Combining equations (45) and (47), we can obtain the following relationship,

ν_{1} (∥ Z (t) ∥) \leq V (Z_{t}) \leq ν_{2} (∥ Z_{t} ∥_{C}) .

(48)

For any $ϵ > 0$ , noting ${}_{s \to + \infty}{ν_{1} (s)} = + \infty$ , one can always find $β = β (ϵ)$ with $β > ϵ > 0$ such that $ν_{2} (ϵ) \leq ν_{1} (β)$ . From (44), it follows that $V (Z_{t}) \leq V (Z_{0}), \forall t \in [0, t_{m})$ . Then by (48), if $∥ Z_{0} ∥_{C} < ϵ$ , we have

\begin{matrix} ν_{1} (∥ Z (t) ∥) \leq V (Z_{t}) \leq V (Z_{0}) \leq ν_{2} (∥ Z_{0} ∥_{C}) \\ < ν_{2} (ϵ) \leq ν_{1} (β), \forall t \in [0, t_{m}) \end{matrix} .

(49)

Based on the aforementioned inequalities, $∥ Z_{0} ∥_{C} < ϵ$ shows $∥ Z (t) ∥ < β, \forall t \in [- δ_{M}, t_{m})$ .

(ii) The proof of $t_{m} = + \infty$ can be achieved by contradiction. If $t_{m}$ is finite, it would indicate a finite escape time, which would necessitate that at least one element of the state vector $Z (t)$ becomes unbounded as $t$ approaches positive infinity. However, the solution’s continuity ensures that $Z (t)$ is bounded at $t = t_{m}$ , given the bounded nature of $Z (t)$ on the interval $[- δ_{M}, t_{m})$ . This circumstance clearly presents a contradiction. Hence, it must be that $t_{m} = + \infty$ , and the state of the closed-loop system is uniformly bounded over the interval $[- δ_{M}, + \infty)$ .

(iii) $\lim_{t \to + \infty} Z (t) = 0$ . Equations (44) and (48) show that the function $V (Z_{t})$ is monotonically decreasing with respect to $t$ and retains a nonnegative value throughout. Thus, the limit ${}_{t \to + \infty}{V (Z_{t})}$ is finite and

\begin{matrix} \int_{0}^{+ \infty} ∥ Z (s) ∥^{2} d s = \int_{0}^{+ \infty} (ε^{2} (s) + z^{2} (s)) d s \\ \leq - \frac{16}{ρ} \int_{0}^{+ \infty} \overset{\cdot}{V} (Z_{s}) d s \\ = \frac{16}{ρ} (V (Z_{0}) - \lim_{t \to + \infty} V (Z_{t})) < + \infty . \end{matrix}

(50)

On the other hand, the boundedness of $Z (t)$ on $[- δ M, + \infty)$ ensures that $\overset{\cdot}{Z} (t)$ is uniformly bounded over the same interval. Therefore, $Z (t)$ is uniformly continuous for $t$ over the interval $[- δ M, + \infty)$ , and $Z^{2} (t)$ is as well. Using Lemma A.6 proposed by Krstic et al.²⁹ One obtains $\lim_{t \to + \infty} Z^{2} (t) = 0$ , which in turn indicates that $\lim_{t \to + \infty} z_{i} (t) = 0$ and $\lim_{t \to + \infty} ϵ_{i} (t) = 0$ for $i = 1, . . ., n$ .

According to equation (30), it is evident that ${\hat{x}}_{2} (t)$ is bounded over the interval $[- δ M, + \infty)$ and $\lim_{t \to + \infty} {\hat{x}}_{2} (t) = 0$ . Then $x_{2} (t) = {\hat{x}}_{2} (t) + K_{1} ϵ_{2} (t)$ implies $x_{2} (t)$ is bounded on the interval $[- δ_{M}, + \infty)$ and $\lim_{t \to + \infty} x_{2} (t) = 0$ . According to equation $x_{2} (t) = {\hat{x}}_{2} (t) + K_{1} ϵ_{2} (t)$ , it is inferred that $x_{2} (t)$ is bounded over the interval $[- δ_{M}, + \infty)$ and $\lim_{t \to + \infty} x_{2} (t) = 0$ . Additionally, $x_{1} (t) = z_{1} (t)$ shows that $x_{1} (t)$ is also bounded on the interval $[- δ_{M}, + \infty)$ and $\lim_{t \to + \infty} x_{1} (t) = 0$ .

Therefore, the closed-loop system composed of (13), (29), (30), and (32) is uniformly asymptotically stable.

Numerical simulation

To assess the control effectiveness of the designed controller for vehicle stability, we employed a sliding mode controller for comparison. Numerical simulations were conducted using a MATLAB/Simulink numerical solver model. Simulations were carried out at vehicle speeds of 60, 80, and 120 km/h.

Table 1 provides the vehicle parameters utilized in this numerical simulation, while Table 2 provides the controller parameters.

Table 1.

Vehicle parameters.

Vehicle parameters	Nominal value
Vehicle mass $m$ /kg	1270
Distance between the front axle and centroid $L_{f}$ /mm	1015
Distance between the rear axle and centroid $L_{r}$ /mm	1895
Cornering stiffness of front wheels $C_{f}$ /N/rad	31,308
Cornering stiffness of rear wheels $C_{r}$ /N/rad	50,000
Rotational inertia $I_{z}$ /kg $m^{2}$	1536.7

Table 2.

Parameters of two controllers in numerical simulation.

DDGO parameter	Nominal value	SMC parameter	Nominal value
$a_{1}$	1.2	$c$	1
$a_{2}$	0.6	$ϵ$	0.50
$b 1$	1.19
$b 2$	1.21
$K_{1}$	25
$K_{2}$	3

Numerical simulation at 60 km/h

At a speed of 60 km/h, the yaw velocity using the designed DDGO controller and the sliding mode controller is depicted in Figure 2. From the figure, it is evident that under the DDGO controller, the maximum oscillation value is approximately 0.27 rad/s. After 1.45 s, the oscillation value at the peak is approximately 0.22 rad/s. And it is stabilized to zero after 3.356 s. In contrast, when employing the sliding mode controller, the maximum oscillation value of the lateral angular velocity reaches as high as 0.44 rad/s. After 1.627 s, the oscillation value at the peak remains at 0.253 rad/s. Subsequently, it is stabilized to zero after 3.593 s.

Figure 2.

The yaw velocity with DDGO controller and sliding mode controller at 60 km/h of numerical simulation.

As depicted in Figure 3, with the DDGO controller, the maximum oscillation value of the side slip angle is 0.0032 rad/s. After 1.576 s, the oscillation value at the peak is 0.0026 rad/s. Followed by rapid attenuating, it reaches zero after 3.351 s. In contrast, when employing the sliding mode controller, the maximum oscillation value of the side slip angle is about 0.0138 rad/s. After 1.2 s, the oscillation value at the peak is about 0.0077 rad/s. Subsequently, it is stabilized to zero after 5.096 s.

Figure 3.

The sideslip angle with the DDGO controller and sliding mode controller at 60 km/h of numerical simulation.

Numerical simulation at 80 km/h

Figure 4 illustrates the velocity with the DDGO controller at 80 km/h, where the maximum oscillation value is 0.26 rad/s. After 1.363 s, the yaw velocity stabilizes near zero. In contrast, the sliding mode controller exhibits a higher maximum oscillation value of approximately 0.205 rad/s. It takes 4.1 s for the oscillations to converge to zero. As shown in Figure 5, under the DDGO controller, the maximum oscillation value of the side slip angle is only 0.0027 rad. Subsequently, the slip angle rapidly approaches zero after 3 s. The sliding mode controller, on the other hand, reaches an oscillation peak of 0.0188 rad. After several oscillation peaks, the side slip angle also tends toward zero after 4 s.

Figure 4.

The yaw velocity with the DDGO controller and sliding mode controller at 80 km/h of numerical simulation.

Figure 5.

The sideslip angle with the DDGO controller and sliding mode controller at 80 km/h of numerical simulation.

Numerical simulation at 120 km/h

Figure 6 illustrates the yaw velocity with the DDGO controller at 120 km/h, where the maximum oscillation value is 0.18 rad/s. After 4.012 s, the yaw velocity is stabilized to zero. In contrast, the sliding mode controller exhibits a higher maximum oscillation value of approximately 0.413 rad/s. It takes 4.1 s for the oscillations to converge to zero. As shown in Figure 7, under the DDGO controller, the maximum oscillation value of the side slip angle is only 0.0042 rad. Subsequently, the side slip angle rapidly approaches zero after 2.8 s. The sliding mode controller, on the other hand, reaches an oscillation peak of 0.067 rad. After several oscillation peaks, the lateral yaw angle also tends toward zero after 4.5 s.

Figure 6.

The yaw velocity with DDGO controller and sliding mode controller at 120 km/h of numerical simulation.

Figure 7.

The sideslip angle with the DDGO controller and sliding mode controller at 120 km/h of numerical simulation.

Simulation results indicate that the DDGO controller demonstrates superior control performance in terms of yaw velocity and sideslip angle compared with the sliding mode controller. And the DDGO controller exhibits robustness, rendering it more suitable for lateral stability control system.

Co-simulation analysis

In the previous section, we conducted simulations and comparisons of control laws based on 2-DOF model by Matlab software. The results affirm the viability of the proposed method. However, it is important to note that an automobile is complex comprised of numerous components. For further verifying the effectiveness of the proposed method in engineering, we will employ CarSim, a specialized software for automotive dynamics simulation, in conjunction with Simulink within the Matlab environment, to conduct comprehensive analysis of the system’s performance in this section. The simulated vehicle’s speeds are set at 80 and 120 km/h, respectively, while the road’s surface exhibits a coefficient of friction of 0.85.

Analysis of double lane-change test

This test is typically used to assess the stability and handling of a vehicle during rapid, consecutive lane changes. In the simulation, the vehicle is required to quickly switch from one lane to another and then back to the original lane at a certain speed. The objective of this test is to observe the vehicle’s response to such abrupt maneuvers, including body roll, yaw response, and the time required to regain stability. The specific input parameters of the double lane-change of CarSim software are shown in Figure 8.

Figure 8.

Double lane change input settings in CarSim.

In Figures 9 to 14, it is evident that the DDGO controller shows markedly superior attenuation effects compared to the SMC method. The fluctuation of control parameters is comparatively smoother, indicating better resistance to disturbance.

Figure 9.

The yaw velocity with the DDGO controller and sliding mode controller at 60 km/h of co-simulation for double lane-change.

Figure 10.

The sideslip angle with the DDGO controller and sliding mode controller at 60 km/h of co-simulation for double lane-change.

Figure 11.

The yaw velocity with the DDGO controller and sliding mode controller at 80 km/h of co-simulation for double lane-change.

Figure 12.

The sideslip angle with the DDGO controller and sliding mode controller at 80 km/h of co-simulation for double lane-change.

Figure 13.

The yaw velocity with DDGO controller and sliding mode controller at 120 km/h of co-simulation for double lane-change.

Figure 14.

The sideslip angle with the DDGO controller and sliding mode controller at 120 km/h of co-simulation for double lane-change.

Analysis of steering wheel impulse input test

This test simulates a driver’s quick and sudden steering action in an emergency, such as avoiding an obstacle. In the simulation setup, the vehicle travels at a specified speed, and a sudden angular displacement is input to the steering wheel. The test focuses on the vehicle’s response speed, stability, and ability to return to a stable driving condition after the sudden steering input. The specific parameters of the impulse input in CarSim software are detailed in Figure 15.

Figure 15.

Steering wheel impulse input settings in CarSim.

This set of simulations replicates angular step conditions. According to Figures 16 to 21, in terms of magnitude and fluctuation indices, the designed DDGO controller also exhibits effective control performance compared to the SMC method.

Figure 16.

The yaw velocity with the DDGO controller and sliding mode controller at 60 km/h of co-simulation for steering wheel impulse input.

Figure 17.

The sideslip angle with the DDGO controller and sliding mode controller at 60 km/h of co-simulation for steering wheel impulse input.

Figure 18.

The yaw velocity with the DDGO controller and sliding mode controller at 80 km/h of co-simulation for steering wheel impulse input.

Figure 19.

The sideslip angle with the DDGO controller and sliding mode controller at 80 km/h of co-simulation for steering wheel impulse input.

Figure 20.

The yaw velocity with DDGO controller and sliding mode controller at 120 km/h of co-simulation for steering wheel impulse input.

Figure 21.

The sideslip angle with the DDGO controller and sliding mode controller at 120 km/h of co-simulation for steering wheel impulse input.

In conclusion, the two controllers effectively enhance the lateral stability of the vehicle. The DDGO controller demonstrates superior performance compared to the sliding mode controller, as it effectively addresses the issues related to system time delay and unknown sensor measurement error.

Remark 1. The proposed method is based on a simplified 2-DOF linear dynamic model. This linear model assumes that the tire maintains a linear relationship within a specific range of sideslip angles, thus limiting the lateral acceleration within a certain range. This means that under extreme operating conditions such as high-speed driving or sharp turns, the accuracy of the proposed method will be affected. On the other hand, the model does not consider the nonlinear characteristics of the suspension and steering systems and their impacts on vehicle dynamic performance which also influence the stability control effect of the vehicle during actual driving.

Conclusion

In this paper, we propose a control method based on the dual control gain observer for vehicle lateral stability control. Firstly, we built a two-degree-of-freedom dynamic model considering unknown external disturbance for the lateral stability control. Based on the dynamic model, the lateral stability control system is constructed which considers unknown sensor measurement error and system time delay. A dual control gain observer and an output feedback controller are designed. And the designed controller is proven to globally asymptotically stabilize the system. Furthermore, we conduct simulations by software. The results demonstrate that the lateral stability control method based on dual domination gains observer significantly improves performance indicators such as yaw velocity and sideslip angle compared with the sliding mode controller.

Footnotes

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: The work is partially supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ21E050002 and National Natural Science Foundation of China under Grant No. 62173208.

ORCID iD

Ze-Kun Zhang

Data availability statement

Data can be requested and accessed from the authors.

References

Asiabar

Kazemi

A direct yaw moment controller for a four in-wheel motor drive electric vehicle using adaptive sliding mode control. Proc IMechE, Part K: J Multi-Body Dynamics 2019; 233(3): 549–567.

Meng

Sun

Shu

, et al. Lateral motion stability control of electric vehicle via sampled-data state feedback by almost disturbance decoupling. Int J Control 2019; 92(4): 734–744.

Nam

Fujimoto

Hori

Lateral stability control of in-wheel-motor-driven electric vehicles based on sideslip angle estimation using lateral tire force sensors. IEEE Trans Veh Technol 2012; 61(5): 1972–1985.

Wang

Liu

, et al. Coordinated longitudinal and lateral vehicle stability control based on the combined-slip tire model in the MPC framework. Mech Syst Signal Process 2021; 161: 107947.

Hashemi

Jalali

Khajepour

, et al. Vehicle stability control: model predictive approach and combined-slip effect. IEEE/ASME Trans Mech 2020; 25(6): 2789–2800.

Zhang

Wang

, et al. Research on automobile four-wheel steering control system based on yaw angular velocity and centroid cornering angle. Meas Control 2022; 55(1–2): 49–61.

Wang

Cui

, et al. Yaw and lateral stability control based on predicted trend of stable state of the vehicle. Veh Syst Dyn 2023; 61(1): 111–127.

Feng

Wang

Coordinated control of active front steering and active disturbance rejection sliding mode-based DYC for 4WID-EV. Meas Control 2020; 53(9–10): 1870–1882.

Wang

Cai

, et al. NMPC-based controller for vehicle longitudinal and lateral stability enhancement under extreme driving conditions. ISA Trans 2023; 135: 509–523.

10.

Jin

Yin

Chen

Gain-scheduled robust control for lateral stability of four-wheel-independent-drive electric vehicles via linear parameter-varying technique. Mechatronics 2015; 30: 286–296.

11.

Hosseini-Pishrobat

Keighobadi

Extended state observer-based robust non-linear integral dynamic surface control for triaxial mems gyroscope. Robotica 2019; 37(3): 481–501.

12.

Zhao

, et al. Robust gain-scheduling static output-feedback h

\infty

control of vehicle lateral stability with heuristic approach. Inf Sci 2021; 546: 220–233.

13.

Abdelmoniem

Ali

Taher

, et al. Fuzzy predictive stanley lateral controller with adaptive prediction horizon. Meas Control 2023; 56(9–10): 1510–1522.

14.

Tian

Cao

Wang

, et al. Four wheel independent drive electric vehicle lateral stability control strategy. IEEE/CAA J Autom Sin 2020; 7(6): 1542–1554.

15.

Xiao

Trajectory planning of quadrotor using sliding mode control with extended state observer. Meas Control 2020; 53(7–8): 1300–1308.

16.

Chen

Cheng

Gan

, et al. Lateral stability control of a distributed electric vehicle using a new sliding mode controler. Int J Automot Technol 2023; 24(4): 1089–1100.

17.

Wang

Zhang

, et al. Vehicle stability enhancement through hierarchical control for a four-wheel-independently-actuated electric vehicle. Energies 2017; 10(7): 947.

18.

Chen

Zhu

, et al. Lateral stability integrated with energy efficiency control for electric vehicles. Mech Syst Signal Process 2019; 127: 1–15.

19.

Cao

Tian

, et al. Gain-scheduling LPV synthesis H

\infty

robust lateral motion control for path following of autonomous vehicle via coordination of steering and braking. Veh Syst Dyn 2023; 61(4): 968–991.

20.

Huang

Yuan

Shi

, et al. A BP-PID controller-based multi-model control system for lateral stability of distributed drive electric vehicle. J Franklin Inst 2019; 356(13): 7290–7311.

21.

Meng

Qian

Wang

Lateral motion stability control via sampled-data output feedback of a high-speed electric vehicle driven by four in-wheel motors. J Dyn Syst Meas Control 2018; 140(1): 011002.

22.

Meng

Sun

Chen

CC.

A homogeneous domination control method based on sampled-data output feedback for lateral stability of an intelligent electric vehicle. Asian J Control 2023; 25(3): 1851–1865.

23.

Sun

Zhang

Meng

, et al. Feedback stabilisation of time-delay nonlinear systems with continuous time-varying output function. Int J Syst Sci 2019; 50(2): 244–255.

24.

Chen

Qian

Sun

, et al. Global output feedback stabilization of a class of nonlinear systems with unknown measurement sensitivity. IEEE Trans Autom Control 2017; 63(7): 2212–2217.

25.

Shao

Sun

Xie

, et al. Output feedback stabilization for power-integrator systems with unknown measurement sensitivity. Sci China Inf Sci 2021; 64(9): 199201.

26.

Wang

Zhang

Wang

, et al. Robust lateral motion control of four-wheel independently actuated electric vehicles with tire force saturation consideration. J Franklin Inst 2015; 352(2): 645–668.

27.

Sun

Xue

Zhang

A new approach to finite-time adaptive stabilization of high-order uncertain nonlinear system. Automatica 2015; 58: 60–66.

28.

Hardy

Littlewood

Polya

Inequalities. Cambridge: Cambridge University Press, 1952.

29.

Krstic

Kokotovic

Kanellakopoulos

Nonlinear and adaptive control design. New York: John Wiley & Sons, Inc., 1995.

Distributed driving electric vehicle lateral stability control method considering time delay

Abstract

Keywords

Introduction

The lateral motion dynamic model of EV

Design of linear observer

Design of output feedback control law

Numerical simulation

Numerical simulation at 60 km/h

Numerical simulation at 80 km/h

Numerical simulation at 120 km/h

Co-simulation analysis

Analysis of double lane-change test

Analysis of steering wheel impulse input test

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

Data availability statement

References