Enhancement of Yaw Moment Control for Drivers with Excessive Steering in Emergency Lane Changes

Abstract

When a ground vehicle runs at high speeds, even a slight excess in the wheel steering angle can immediately cause the vehicle to slide sideways and lose control. In this study, we propose an active safety control system designed to address emergency situations where the driver applies excessive steering input and the vehicle speed varies significantly during control. The system combines the direct yaw moment (DYM) method with a steering saturation scheme that prevents excessive driver steering input from adversely influencing the front-wheel steering. Consequently, the control system allows the DYM to focus more on other stabilization tasks and maintain tire/road friction within its workable linear range. The implementation relies on a reference steering angle and a reference vehicle state, derived from a linear vehicle model considering tire/road friction limitations. When the driver’s steering angle and the system state deviate from these reference values, the control system intervenes by applying both the steering saturation scheme and DYM method. This ensures the front-wheel steering angle and system state remain close to the reference values. The control strategy is developed using the polytopic Linear Parameter Varying (LPV) technique and Linear Matrix Inequality (LMI) to account for the changes in vehicle speed. It is further enhanced with an input saturation technique based on a high-gain approach, which improves control utilization and system response during emergency situations. The advantages of the proposed control strategy are demonstrated through simulation results.

1. Introduction

The active safety system in vehicles is a key aspect of vehicle dynamics control. It comprises two primary categories based on their relevance to driving conditions (see [1,2] and references therein). The first category involves auxiliary control of the vehicle’s longitudinal motion to prevent wheels from locking or slipping due to a driver’s excessive braking or traction. Well-known examples of such systems include the Anti-lock Brake System (ABS) and Traction Control System (TCS). The second category is Vehicle Stability Control (VSC), which stabilizes the lateral motion of the vehicle by controlling its body sideslip angle and yaw rate. The development of VSC is motivated by the fact that most drivers lack the ability or experience to manage sudden skids while driving. When a driver makes an emergency lane change in panic, they may overreact and steer too aggressively. This can cause the tire/road friction to saturate, resulting in the vehicle skidding sideways and potentially colliding with other vehicles or crash barriers.

There are three main approaches to stabilizing vehicle lateral motion in literature. The first approach involves using differential torque between the left and right wheels to produce DYM for stabilization. Control design methods for DYM stabilization include sliding mode control [3], model predictive control [4,5], LPV control [6,7], and other techniques (see [1,2,3,4,5,6,7] and references therein). The second approach is to manipulate the steering angle to achieve lateral motion stabilization. Active Front Steering (AFS), Active Rear Steering (ARS), and Four-Wheel Steering (4WS) are examples of this approach (see [8,9,10,11] and references therein). In AFS, for instance, a compensation steering angle is coupled to the driver steering angle through the planetary gear mechanism to adjust the front-wheel steering angle. The third approach combines an active wheel steering scheme with the DYM method (see [12,13,14] and references therein). This integration method adjusts the longitudinal and lateral tire/road frictions simultaneously within the linear friction region, effectively stabilizing the vehicle’s lateral state and providing promising performance. However, directly integrating AFS may lead to excessive steering of the front wheels, particularly when the driver steers aggressively in a moment of panic. In such scenarios, the literature often relies solely on the DYM method and treats the driver’s excessive steering as an external disturbance. As a result, the DYM method has to dedicate most of its capacity to counteracting the instability caused by excessive steering, thereby reducing its effectiveness in managing other stabilization factors concurrently. On the other hand, time-varying vehicle speed is a significant factor of the vehicle’s lateral dynamics. During an emergency intervention, the vehicle speed is usually constantly changing. Although certain controllers have been developed to ensure performance around constant vehicle speeds, they might not be theoretically appropriate for situations where the vehicle speed changes over time.

In the literature, a modeling technique with the polytopic LPV is often used to deal with the situation of time-varying vehicle speed (see [6,7] and references therein). The polytopic LPV control offers the advantage of applying linear control techniques to nonlinear systems. This type of controller design often employs a quadratic Lyapunov function, in which the Lyapunov matrix is constant or parameter-dependent, to ensure some specific performance across an interested polytope. When the parameter varies within the polytope, the design and analysis of the controller can be focused solely on the polytope vertices. This approach guarantees the stabilization of every parameter point within the polytope. The polytopic LPV control design is typically performed with rectangular polytopes. However, when the actual parameter trajectories do not reach certain vertices of the rectangular polytope, such as the vehicle speed parameter in the LPV vehicle control design, some studies suggest reducing the polytope size and using other forms of polytopes like trapezoidal, triangular, or two-vertex polytopes for the design (see [6] and references therein). The LMI technique usually accompanies the polytopic LPV control design to find feedback gains that satisfy design specifications [6,7,15,16]. Reference [7] developed a lateral reference state tracking control using the polytopic LPV-LMI modeling and gain scheduling method. It copes with issues of cornering stiffness uncertainty by using a norm-bounded structured uncertainty approach. However, the reference state provided in the study does not include safety limit ranges. As a result, when the driver’s steering angle exceeds a certain threshold, the corresponding reference state will exceed the safety range. Therefore, their proposed solutions are more suitable for driving circumstances where the driver’s steering angle falls within a normal operation range.

This study proposes an active safety control system to handle emergencies where the driver applies excessive steering input and the vehicle speed changes significantly during control. The system combines the DYM method with a steering saturation scheme to prevent excessive driver steering input from affecting the wheel steering angle. This approach keeps the tire/road friction within the linear operation range during emergencies and enables the DYM to allocate more capacity to other stabilization control tasks. The control system uses a reference state as well as a reference steering angle, obtained from the steady state of a linear vehicle lateral model and from the limitations imposed by tire/road friction saturation, to guide its operation. When the driver’s steering angle and the system state diverge from the established reference values, the control system steps in by implementing both the steering saturation scheme and the DYM method. This intervention maintains the front-wheel steering angle and system state in close alignment with the reference values.

To deal with the problem of varying vehicle speeds, this study employs the polytopic LPV modeling technique. The Lyapunov redesign concept is also utilized to decompose overall uncertainties into structured uncertainties of matching and unmatching conditions (see [17,18,19] and references therein). This simplifies the controller design process. Moreover, the design of the DYM control with input constraint uses numerical solutions to LMIs, with the restriction on the DYM value obtained by considering the limit condition of the friction circle on the combined wheel slip. Once the design is finalized, an input saturation technique utilizing a high-gain approach is employed to enhance control utilization and improve system response during emergency situations [20,21]. Finally, the resulting yaw moment is distributed to the four wheels to generate the wheel control torque. The results from simulation indicate that the enhanced integration strategy achieves not only safety reference state tracking, but also ensures the tire/road friction operates in its linear range and enables the DYM to handle more instability factors.

This article presents two primary contributions. First, it proposes a control system designed to compensate for excessive driver steering by using the reference steering angle. This approach ensures that the front-wheel steering angle remains within a safe range while regulating the yaw moment to minimize deviations between the safety reference state and the vehicle’s actual state. Integrating these controls also keeps tire/road friction within a manageable linear range, allowing the DYM to focus on other stabilization tasks. Second, the study introduces a high-gain input saturation technique aimed at enhancing the efficiency of the control law when faced with input constraints. This enhancement aims to improve the system’s emergency response capabilities. Building on our previous work [22], this paper highlights several key features:

1. Front-wheel steering saturation scheme: Specifically designed for managing crises during emergency lane changes involving driver’s excessive steering. This technique intervenes only in cases of excessive steering, allowing drivers to steer normally under regular conditions.

2. Comprehensive control theory explanation: Provides a detailed explanation of the control techniques, including the following: (I) Analysis and consideration of matching and unmatching uncertainties; (II) Detailed derivations of the constraints on the required yaw moment and wheel torques for the emergency management; and (III) Detailed derivations and proofs of lemmas and theorems, presented in Appendix A.

3. Simulation section for effectiveness demonstration: Focuses on demonstrating and discussing the effectiveness of the control functionality in handling excessive steering during emergency lane changes.

This article is organized as follows: The lateral control model section outlines the derivation of the safety reference states and reference steering angle essential for safe vehicle operation. In the control system configuration and yaw moment control design sections, we propose a strategy to limit front-wheel steering angles and develop an LPV-LMI-based DYM control approach under yaw moment constraints. Enhancements to this approach include the integration of a high-gain input saturation scheme. Finally, through simulations, we validate the efficacy of the control design and give the conclusions of the study.

2. Vehicle Lateral Control Model

2.1. Lateral Control Model and Steady State

As shown in Figure 1, consider the following lateral control model for a ground vehicle traveling at speeds $V_0\in [V_{\min },V_{\max }]$ (see [23] and references therein):

$${\dot{x}}_v=A_v(V_0)x_v+B_v(V_0){\delta }_f+B_m{M}_z+d_v, V_0\in [V_{\min },V_{\max }]$$

(1a)

$$a_y=C_v(V_0)x_v+D_v{\delta }_f$$

(1b)

$${\alpha }_f=E_f(V_0)x_v+{\delta }_f, {\alpha }_r=E_r(V_0)x_v$$

(1c)

$$\begin{array}{l}{x}_v={\left[\begin{array}{cc}\beta & \gamma \end{array}\right]}^T\\ A_v(V_0)=\left[\begin{array}{cc}-\frac{C_f+C_r}{m{V}_0}& \frac{-C_f{l}_f+C_r{l}_r}{m{V}_0^2}-1\\ \frac{-C_f{l}_f+C_r{l}_r}{J_z}& -\frac{C_f{l}_f^2+C_r{l}_r^2}{J_z{V}_0}\end{array}\right],\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{B}_v(V_0)=\left[\begin{array}{c}\frac{C_f}{m{V}_0}\\ \frac{C_f{l}_f}{J_z}\end{array}\right],\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{B}_m=\left[\begin{array}{c}0\\ \frac{1}{J_z}\end{array}\right]\\ C_v(V_0)=\left[\begin{array}{cc}-\frac{C_f+C_r}{m}& \frac{-C_f{l}_f+C_r{l}_r}{m{V}_0}\end{array}\right], D_v=\frac{C_f}{m},\\ E_f(V_0)=\left[\begin{array}{cc}-1& -\frac{l_f}{V_0}\end{array}\right], E_r(V_0)=\left[\begin{array}{cc}-1& \frac{l_r}{V_0}\end{array}\right]\end{array}$$

where $\beta$, $\gamma$, and $a_y$ are the side slip angle, yaw rate, and lateral acceleration of the vehicle body measured at the center of gravity (CG), respectively; ${\alpha }_f$ and ${\alpha }_r$ are the wheel slip angles of the front and rear wheels; ${\delta }_f$ is the front-wheel steering angle; $C_f$ and $C_r$ denote the cornering stiffness of the front and rear axles, respectively; $m$ represents the mass of the vehicle; and $J_z$ denotes the inertia about the $z$ axis. $l_f$ and $l_r$ are the distances from the center of gravity (CG) to the front and rear axles, while $l_d$ is the distance from the CG to the wheel side. The yaw moment $M_z$, which is mainly generated by the longitudinal tire/road friction forces $f_{xj}$, $j=1,2,\dots ,4$ of the four wheels, is calculated using

$$M_z=-l_d{f}_{x1}\cos ({\delta }_f)+l_d{f}_{x2}\cos ({\delta }_f)-l_d{f}_{x3}+l_d{f}_{x4}$$

Since the front-wheel steering angle ${\delta }_f$ is small in operation, the yaw moment $M_z$ can approximate to

$$M_z=-l_d{f}_{x1}+l_d{f}_{x2}-l_d{f}_{x3}+l_d{f}_{x4}$$

(2)

The yaw moment $M_z$ is restricted due to the limitation on the tire/road friction. We assume that the parameters above are nominal values, and any errors resulting from model linearization and parameter estimation are represented as the uncertain disturbance $d_v\in {\Re }^2$.

When the vehicle is driving at a constant speed $V_0$ and operating normally, the system matrix $A_v(V_0)$ is asymptotically stable and the matrix pair $(A_v(V_0),B_m)$ is controllable with the augmented matrix $[B_v(V_0)B_m]$ of full column rank. Additionally, it is typical in vehicle system analysis to consider the driver steering angle ${\delta }_d$ as the front-wheel steering angle ${\delta }_f$, i.e., ${\delta }_f={\delta }_d$. Therefore, a reference vehicle lateral model traveling at speed $V_0$ from (1) with ${\delta }_f={\delta }_d$, $M_z=0$ and $d_v=0$, is provided as

$${\dot{x}}_v=A_v(V_0)x_v+B_v(V_0){\delta }_d$$

(3a)

$$a_y=C_v(V_0)x_v+D_v{\delta }_d$$

(3b)

$${\alpha }_f=E_f(V_0)x_v+{\delta }_d, {\alpha }_r=E_r(V_0)x_v$$

(3c)

The steady-state solution of the reference model (3) with constant speed $V_0$ can be obtained as

$${\beta }_{ss}(V_0)=\frac{l_r}{l_f+l_r}\left(1-\frac{m{l}_f}{C_r{l}_r(l_f+l_r)}{V}_0^2\right)\left(\frac{1}{1+K{V}_0^2}\right){\delta }_d$$

(4)

$${\gamma }_{ss}(V_0)=\frac{V_0}{l_f+l_r}\left(\frac{1}{1+K{V}_0^2}\right){\delta }_d$$

(5)

$$a_{y,ss}(V_0)=\frac{V_0^2}{l_f+l_r}\left(\frac{1}{1+K{V}_0^2}\right){\delta }_d$$

(6)

$${\alpha }_{f,ss}(V_0)=\frac{l_{r}m{V}_0^2}{{(l_f+l_r)}^2{C}_f}\left(\frac{1}{1+K{V}_0^2}\right){\delta }_d$$

(7)

$${\alpha }_{r,ss}(V_0)=\frac{l_{f}m{V}_0^2}{{(l_f+l_r)}^2{C}_r}\left(\frac{1}{1+K{V}_0^2}\right){\delta }_d$$

(8)

where

$$K=\frac{m}{{(l_f+l_r)}^2}\left(\frac{C_r{l}_r-C_f{l}_f}{C_f{C}_r}\right)$$

(9)

is the vehicle stability factor.

2.2. Vehicle Safety Reference State and Reference Steering Angle

Figure 2 illustrates the flow diagram depicting the relationship between the steady-state solution (4)–(9), the vehicle speed $V_0$, and the driver steering angle ${\delta }_d$.

As shown in the figure, these steady-state solutions are not independent but coupled. By using the steady-state yaw rate ${\gamma }_{ss}$ as a design parameter, we can reverse the flow in Figure 2 and derive a new relationship as follows:

$${\beta }_{ss}(V_0)=\left(l_r-\frac{m{l}_f}{C_r(l_f+l_r)}{V}_0^2\right)\frac{{\gamma }_{ss}(V_0)}{V_0}$$

(10)

$$a_{y,ss}(V_0)=V_0{\gamma }_{ss}(V_0)$$

(11)

$${\delta }_d(V_0)=(l_f+l_r)\left(1+K{V}_0^2\right)\frac{{\gamma }_{ss}(V_0)}{V_0}$$

(12)

$${\alpha }_{f,ss}(V_0)=\frac{l_{r}m{V}_0}{(l_f+l_r)C_f}{\gamma }_{ss}(V_0)=\frac{l_{r}m}{(l_f+l_r)C_f}{a}_{y,ss}(V_0)$$

(13)

$${\alpha }_{r,ss}(V_0)=\frac{l_{f}m{V}_0}{(l_f+l_r)C_r}{\gamma }_{ss}(V_0)=\frac{l_{f}m}{(l_f+l_r)C_r}{a}_{y,ss}(V_0)$$

(14)

In the active safety design, the steady-state yaw rate ${\gamma }_{ss}$ is considered to be limited as below (see [24,25,26] and references therein):

$$\left|{\gamma }_{ss}(V_0)\right|\le {\tilde{\gamma }}_{\lim }(V_0)\hspace{0.33em}\hspace{0.33em}=0.85\frac{{\mu }_{\max }g}{V_0}$$

(15)

where ${\mu }_{\max }$ is the maximum tire/road friction coefficient and $g$ is the gravity constant. The limitation (15) is used to maintain the lateral acceleration $a_y$ unsaturated. By using the relationship (10)–(14) and the restriction (15), the allowable operating ranges of these values can be obtained as follows:

$$\left|{\beta }_{ss}(V_0)\right|\le {\tilde{\beta }}_{\lim }(V_0)\hspace{0.33em}\hspace{0.33em}=\left|\frac{l_r}{V_0}-\frac{m{l}_f{V}_0}{C_r(l_f+l_r)}\right|{\tilde{\gamma }}_{\lim }(V_0)$$

(16)

$$\left|a_{y,ss}(V_0)\right|\le {\tilde{a}}_{y,\lim }(V_0)=V_0{\tilde{\gamma }}_{\lim }(V_0)=0.85{\mu }_{\max }g$$

(17)

$$\left|{\delta }_d\right|\le {\tilde{\delta }}_{\lim }(V_0)=\left(l_f+l_r\right)\left|\frac{1+K{V}_0^2}{V_0}\right|{\tilde{\gamma }}_{\lim }(V_0)$$

(18)

$$\left|{\alpha }_{f,ss}\right|\le {\tilde{\alpha }}_{f,\lim }=\frac{l_{r}m{V}_0}{(l_f+l_r)C_f}{\tilde{\gamma }}_{\lim }(V_0)=\frac{l_{r}m}{(l_f+l_r)C_f}{\tilde{a}}_{y,\lim }(V_0)$$

(19)

$$\left|{\alpha }_{r,ss}\right|\le {\tilde{\alpha }}_{r,\lim }=\frac{l_{f}m{V}_0}{(l_f+l_r)C_r}{\tilde{\gamma }}_{\lim }(V_0)=\frac{l_{f}m}{(l_f+l_r)C_r}{\tilde{a}}_{y,\lim }(V_0)$$

(20)

By using the restriction $({\tilde{\gamma }}_{\lim },{\tilde{\beta }}_{\lim },{\tilde{\delta }}_{\lim })$ above, we can establish the safety reference state $({\beta }_{ref},{\gamma }_{ref})$ and the reference driver steering angle ${\delta }_{d,ref}$ as the following functions of the vehicle speed $V_0$:

$${\beta }_{ref}(V_0)=\left\{\begin{array}{c} {\beta }_{ss}\hspace{0.33em}(V_0)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em},if\hspace{0.33em}\left|{\beta }_{ss}(V_0)\right|\le {\tilde{\beta }}_{\lim }(V_0)\\ {\tilde{\beta }}_{\lim }(V_0)\hspace{0.33em}\hspace{0.33em}sign(\hspace{0.33em}{\beta }_{ss}\hspace{0.33em}(V_0)),if\hspace{0.33em}\left|{\beta }_{ss}(V_0)\right|>{\tilde{\beta }}_{\lim }(V_0)\end{array}\right.$$

(21)

$${\gamma }_{ref}(V_0)=\left\{\begin{array}{c}{\gamma }_{ss}(V_0)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em},if\hspace{0.33em}\left|{\gamma }_{ss}(V_0)\right|\le {\tilde{\gamma }}_{\lim }(V_0)\\ {\tilde{\gamma }}_{\lim }(V_0)sign({\gamma }_{ss}(V_0)),if\hspace{0.33em}\left|{\gamma }_{ss}(V_0)\right|>{\tilde{\gamma }}_{\lim }(V_0)\end{array}\right.$$

(22)

$${\delta }_{d,ref}(V_0,{\delta }_d)=\left\{\begin{array}{c}{\delta }_d\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em},if\hspace{0.33em}\left|{\delta }_d\right|\le {\tilde{\delta }}_{\lim }(V_0)\\ {\tilde{\delta }}_{\lim }(V_0)sign({\delta }_d),if\hspace{0.33em}\left|{\delta }_d\right|>{\tilde{\delta }}_{\lim }(V_0)\end{array}\right.$$

(23)

The steady state solution $({\beta }_{ss},{\gamma }_{ss},{\delta }_d)$ and the restriction parameter $({\tilde{\beta }}_{\lim },{\tilde{\gamma }}_{\lim },{\tilde{\delta }}_{\lim })$ have the relationship

$$\begin{gathered} A_v(V_0)\left[\begin{array}{c}{\beta }_{ss}\\ {\gamma }_{ss}\end{array}\right]+B_v(V_0){\delta }_d=0 \\ {A}_v(V_0)\left[\begin{array}{c}{\tilde{\beta }}_{\lim }sign({\beta }_{ss})\\ {\tilde{\gamma }}_{\lim }sign({\gamma }_{ss})\end{array}\right]+B_v(V_0){\tilde{\delta }}_{\lim }sign({\delta }_d)=0 \end{gathered}$$

Therefore, the reference state $({\beta }_{ref},{\gamma }_{ref})$ and the reference steering angle ${\delta }_{d,ref}$ in (21)–(23) satisfy

$$A_v(V_0)x_{ref}+B_v(V_0){\delta }_{d,ref}=0$$

(24)

where $x_{ref}={[{\beta }_{ref} {\gamma }_{ref}]}^T$.

3. Control System Configuration

Configuration of the control system in this study is depicted in Figure 3. It consists of three modules: the Reference Model Module, Control Law Module, and 4-Wheel Torque Distribution Module. The Reference Model Module provides the reference state $({\beta }_{ref},{\gamma }_{ref})$ and reference steering angle ${\delta }_{d,ref}$, which are used by the Control Law Module to implement the schemes of saturation angle ${\delta }_{sat}$ and enhanced yaw moment $M_z$. These schemes work together to ensure the front-wheel steering angle ${\delta }_f$ and vehicle state $(\beta ,\gamma )$ closely follow their references. Additionally, this collaborative effort maintains tire/road friction within the linear operation range and allows the DYM to focus on other critical stabilization tasks, especially when drivers overreact with excessive steering. Finally, the Torque Distribution Module generates 4-wheel torques $T_j$, $j=1,2,\dots ,4$, to implement the enhanced yaw moment. More details are provided in the subsequent sections.

3.1. Active Steering Saturation Control

In this study, the front-wheel steering angle ${\delta }_f$ is considered to be mechanically connected to the driver steering angle ${\delta }_d$ through the planetary gear mechanism. After appropriate gear matching, the mechanism adjusts the front-wheel steering angle ${\delta }_f$ by

$${\delta }_f={\delta }_d-{\delta }_{sat}$$

(25)

where the saturation angle ${\delta }_{sat}$ is designed to dynamically block excessive driver steering angle ${\delta }_d$ to keep the wheel steering angle ${\delta }_f$ within the safety range $\left|{\delta }_f\right|\le {\tilde{\delta }}_{\lim }$, or specifically, to make the wheel steering angle ${\delta }_f$ behave as

$${\delta }_f\cong {\delta }_{d,ref}(V_0,{\delta }_d)\hspace{0.33em}\hspace{0.33em}$$

(26)

A control scheme for the saturation angle ${\delta }_{sat}$ to have the effect (26) is proposed as

$${\dot{\delta }}_{sat}=-\alpha {\delta }_{sat}+\alpha ({\delta }_d-{\delta }_{d,ref}(V_0,{\delta }_d)\hspace{0.33em}\hspace{0.33em})+{\omega }_d,\hspace{0.33em}\alpha >0$$

(27)

where ${\omega }_d$ is the angular speed of the driver’s steering angle ${\delta }_d$ and a sufficiently large $\alpha >0$ is chosen such that the saturation angle ${\delta }_{sat}$ can be regarded as much faster in comparison with the driver steering angle ${\delta }_d$ and vehicle speed $V_0$. Using (25) provides

$${\dot{\delta }}_f={\dot{\delta }}_d-{\dot{\delta }}_{sat}$$

(28)

By substituting (27) into (28) with ${\omega }_d={\dot{\delta }}_d$, we can arrive at

$${\dot{\delta }}_f=-\alpha {\delta }_f+\alpha {\delta }_{d,ref}(V_0,{\delta }_d)\hspace{0.33em}\hspace{0.33em}$$

(29)

On the fast convergence condition, the wheel steering angle ${\delta }_f$ in (29) can be viewed as

$${\delta }_f={\delta }_{d,ref}(V_0,{\delta }_d)+\upsilon ,\hspace{0.33em}\upsilon \underset{faster}{\to }0$$

(30)

With the effect of the saturation angle (30), the relationship (24) can be rewritten as

$$A_v(V_0)x_{ref}+B_v(V_0){\delta }_f=B_v\upsilon , \upsilon \to 0$$

(31)

3.2. Control System with Active Front Steering Saturation

Let $e_v=x_v-x_{ref}$ be the error between the vehicle state and reference state. The speed-variable-dependent dynamical error equation $e_v$ can be obtained from (1) as

$${\dot{e}}_v={\dot{x}}_v-{\dot{x}}_{ref}=A_v(V_0)e_v+B_m{M}_z+A_v(V_0)x_{ref}+B_v(V_0){\delta }_f+d_v-{\dot{x}}_{ref}, V_0\in [V_{\min },V_{\max }]$$

Substituting (31) into the equation above yields

$${\dot{e}}_v=A_v(V_0)e_v+B_m{M}_z+{\overline{d}}_v(t), V_0\in [V_{\min },V_{\max }]$$

(32)

where

$${\overline{d}}_v(t)=B_v(V_0)\upsilon +d_v-{\dot{x}}_{ref}$$

(33a)

$$\left|\upsilon \right|=\left|{\delta }_f-{\delta }_{d,ref}\right|\to 0$$

(33b)

If the yaw moment $M_z$ is used alone, i.e., ${\delta }_{sat}=0$ and ${\delta }_f={\delta }_d$, then the disturbance $\upsilon$ in (33) becomes $\upsilon ={\delta }_d-{\delta }_{d,ref}$ with

$$\left|\upsilon \right|=\left|{\delta }_d-{\delta }_{d,ref}\right|=\left\{\begin{array}{cc}0& ,if\hspace{0.33em}\left|{\delta }_d\right|\le {\tilde{\delta }}_{\lim }\\ \left|{\delta }_d\right|-{\tilde{\delta }}_{\lim }& ,if\hspace{0.33em}\left|{\delta }_d\right|>{\tilde{\delta }}_{\lim }\end{array}\right.$$

(34)

Under the condition (34), when the driver steering angle ${\delta }_d$ surpasses the threshold, i.e., $\left|{\delta }_d\right|>{\tilde{\delta }}_{\lim }$, the related disturbance magnitude $\left|\upsilon \right|$ begins to grow in proportion to the difference $\left|{\delta }_d\right|-{\tilde{\delta }}_{\lim }$, and this can lead to tire/road friction saturating if the driver steering angle ${\delta }_d$ becomes large enough. In this case of excessive front-wheel steering, the stabilizing yaw moment $M_z$ in (32) operates in the tire/road friction saturation zone with its capacity almost used in suppressing the instability effect of the difference $\left|{\delta }_d\right|-{\tilde{\delta }}_{\lim }$. In contrast, if the auxiliary saturation angle ${\delta }_{sat}$ is used to limit the wheel steering angle ${\delta }_f$ in the presence of $\left|{\delta }_d\right|>{\tilde{\delta }}_{\lim }$, then the resulting disturbance $\upsilon$ in (33) can be effectively reduced and the tire/road friction can remain within the linear region. This strategy also frees up more capacity of the yaw moment $M_z$ to carry out the reference state tracking and mitigate other potential instability effects as shown in (32) and (33).

It is possible to represent the total uncertain disturbance ${\overline{d}}_v(t)$ in (32) by using the full-column-rank property of the augmented matrix $\hspace{0.33em}[B_v(V_0)B_m]$ where $B_v(V_0)$ and $B_m$ are the column vectors related to the front-wheel steering angle ${\delta }_f$ and yaw moment $M_z$, respectively. Based on the concept of Lyapunov redesign and the structure of the system input matrix $B_m$, the uncertain disturbance ${\overline{d}}_v(t)$ can be decomposed into two parts [17,18,19], namely, a matching condition part $B_m\xi (t)$ and an unmatching condition part $B_v(V_0)\varsigma (t)$ described as follows:

$$B_v(V_0)\upsilon +d_v-{\dot{x}}_{ref}={\overline{d}}_v(t)=\underset{\mathrm{unmatching} \mathrm{condition} \mathrm{part}}{\underbrace{B_v(V_0)\varsigma (t)}}+\underset{\mathrm{matching} \mathrm{condition} \mathrm{part}}{\underbrace{B_m\xi (t)}}$$

(35)

The unmatching condition part $B_v(V_0)\hspace{0.33em}\varsigma$ on the right side of (35) represents a disturbance in the direction of $B_v(V_0)\hspace{0.33em}$ with an uncertain magnitude $\varsigma$. This magnitude is primarily caused by the term $\upsilon \hspace{0.33em}$ on the left side of (35), which arises from the wheel steering angle as described in (33) or (34), and is expected to be on the order of ${10}^{-2}$ radians (the same order with the wheel steering angle ${\delta }_f$). To mitigate this, the auxiliary saturation angle ${\delta }_{sat}$ is used to limit the steering angle ${\delta }_f$, thereby reducing the magnitude $\upsilon$ in (33) and effectively suppressing the disturbance $B_v(V_0)\varsigma (t)$ in (35). Similarly, the matching condition part $B_m\xi$ in (35) represents a disturbance in the direction of $B_m$ with an uncertain magnitude $\xi$. This magnitude can be managed by the yaw moment $M_z$ and is expected to be on the order of ${10}^3$ N-m (the same order with the yaw moment $M_z$). The significant difference in scales between the magnitudes of $\varsigma$ and $\xi$ in (35) presents challenges when managing the bounds on disturbance uncertainty. To address this issue, this study considers the following constraint

$${\left(\varsigma /{\rho }_{\varsigma }\right)}^2+{\left(\xi /{\rho }_{\xi }\right)}^2\le 1\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}$$

(36)

to normalize the combined effect of the uncertainties $(\varsigma ,\xi )$, where ${\rho }_{\varsigma }>0$ and ${\rho }_{\xi }>0$, assumed to be given, are on the orders of ${10}^{-2}$ and ${10}^3$, respectively. By using the representation (35) and the restriction (36), the error dynamical Equation (32) can be rewritten as

$${\dot{e}}_v=A_v(V_0)e_v+B_m{M}_z+B_w(V_0)w, V_0\in [V_{\min },V_{\max }]$$

(37a)

$$w^{T}Hw\le 1, H=diag({\rho }_{\varsigma }^{-2},{\rho }_{\xi }^{-2})$$

(37b)

where $B_w(V_0)=[B_v(V_0)B_m]$ and $w={[\varsigma \xi ]}^T$. Unlike the previous study in Reference [7] that utilized structured uncertainty to address variations in the cornering stiffness, our approach employs an uncertainty reconfiguration that deals with overall disturbances through the type of structured uncertainty (35). This approach reduces the complexity of the control design.

4. Yaw Moment Control Design

4.1. Yaw Moment Limit and Wheel Torque Distribution

During emergency lane changes, vehicles under safety control often engage in combined cornering, traction, and braking maneuvers, which result in significant coupling between lateral and longitudinal wheel slips and friction forces. By considering combined slip, we can effectively account for these coupling effects and ensure that the resulting friction forces remain within the friction circle (see [27,28,29] and references therein). In this section, our focus is on addressing the combined slip effect and specifically on generating a constrained yaw moment as outlined in (32) and (2), while adhering to restricted combined wheel slip. This restriction is crucial as it prevents the generation of excessive longitudinal friction forces, which could otherwise reduce the necessary cornering forces during emergency lane changes and potentially compromise stability and safety. Let $S={[s_L,s_S]}^T$ represent the combined wheel slip and $‖S‖=\sqrt{s_L^2+s_S^2}$ stand for its magnitude, where $s_L$ and $s_S$ denote the longitudinal and lateral wheel slips, respectively. When operating in the linear region of tire/road friction, the traction/brake force of each wheel can be expressed as

$$\frac{T_j}{r_{ej}}=f_{xj}=f_{zj}{k}_j{s}_{Lj},\hspace{0.33em}\hspace{0.33em}j=1,2,\dots ,4$$

(38)

where $T_j$ and $r_{ej}$ are the wheel torque and effective radius, respectively; $k_j$ and $s_{Lj}$ are the slope of the tire/road friction coefficient and longitudinal wheel slip in that order; and notation $f_{zj}$ is the normal load of each wheel estimated by (see [29] and references therein)

$$f_{z1}=\frac{1}{2}\frac{l_r}{l_f+l_r}mg-\frac{1}{2}\frac{h}{l_f+l_r}m{a}_x-\frac{h}{2l_d}\frac{k_{f\varphi }}{k_{f\varphi }+k_{r\varphi }}m{a}_y$$

(39a)

$$f_{z2}=\frac{1}{2}\frac{l_r}{l_f+l_r}mg-\frac{1}{2}\frac{h}{l_f+l_r}m{a}_x+\frac{h}{2l_d}\frac{k_{f\varphi }}{k_{f\varphi }+k_{r\varphi }}m{a}_y$$

(39b)

$$f_{z3}=\frac{1}{2}\frac{l_f}{l_f+l_r}mg+\frac{1}{2}\frac{h}{l_f+l_r}m{a}_x-\frac{h}{2l_d}\frac{k_{r\varphi }}{k_{f\varphi }+k_{r\varphi }}m{a}_y$$

(39c)

$$f_{z4}=\frac{1}{2}\frac{l_f}{l_f+l_r}mg+\frac{1}{2}\frac{h}{l_f+l_r}m{a}_x+\frac{h}{2l_d}\frac{k_{r\varphi }}{k_{f\varphi }+k_{r\varphi }}m{a}_y$$

(39d)

In (39), notations $a_x$ and $a_y$ are the longitudinal and lateral accelerations and $k_{f\varphi }$ and $k_{r\varphi }$ are the front and the rear roll stiffness, respectively. This study assumes, in emergency situations, the proposed wheel torque control engages all wheels with the same amount of longitudinal wheel slip, i.e., $\left|s_L\right|=\left|s_{Lj}\right|$, $j=1,2,\dots ,4$, and the tire/road friction characteristics and the effective radius are the same for the four wheels, i.e., $k=k_j$, $r_e=r_{ej}$, $j=1,2,\dots ,4$. After the stabilizing yaw moment $M_z$ is calculated, the four wheels are assigned with control torques as

$$T_1=-sign(M_z)r_e{f}_{z1}k\left|s_L\right|, T_2=sign(M_z)r_e{f}_{z2}k\left|s_L\right|$$

(40a)

$$T_3=-sign(M_z)r_e{f}_{z3}k\left|s_L\right|, T_4=sign(M_z)r_e{f}_{z4}k\left|s_L\right|$$

(40b)

In accordance with (38) and (40), the longitudinal friction forces of the four wheels become

$$f_{x1}=-sign(M_z)f_{z1}k\left|s_L\right|, f_{x2}=sign(M_z)f_{z2}k\left|s_L\right|$$

(41a)

$$f_{x3}=-sign(M_z)f_{z3}k\left|s_L\right|, f_{x4}=sign(M_z)f_{z4}k\left|s_L\right|$$

(41b)

and therefore the yaw moment (2) develops into

$$M_z=-f_{x1}{l}_d+f_{x2}{l}_d-f_{x3}{l}_d+f_{x4}{l}_d=sign(M_z)l_{d}mgk\left|s_L\right|$$

(42)

By using $M_z=sign(M_z)\left|M_z\right|$, the equation above can be rearranged as

$$\left|M_z\right|=l_{d}mgk\left|s_L\right|$$

(43)

In view of (43), the yaw moment magnitude $\left|M_z\right|$ reaches its maximum allowable amount ${\tilde{M}}_{z,allow}$ when the longitudinal slip magnitude $\left|s_L\right|$ also achieves its allowable limit ${\tilde{s}}_{L,allow}$. Thus, the constraint on the yaw moment (43) can be considered as

$$\left|M_z\right|\le {\tilde{M}}_{z,allow}=l_{d}mgk{\tilde{s}}_{L,allow}$$

(44)

To calculate ${\tilde{s}}_{L,allow}$ in (44), we adopt the concept of the friction circle restriction [27,28,29]:

$${\tilde{s}}_{L,allow}\le \sqrt{{\tilde{s}}_{\mathrm{Re}s,allow}^2-{\tilde{s}}_{S,allow}^2}$$

(45)

where ${\tilde{s}}_{Res,allow}$ represents the allowable combined wheel slip magnitude $‖S‖$ without saturating the tire/road friction and ${\tilde{s}}_{S,allow}$ is the allowable lateral wheel slip. The selection of ${\tilde{s}}_{Res,allow}$ depends on the road conditions; for example, ${\tilde{s}}_{Res,allow}\approx 0.1$ can be used for the dry asphalt road condition and ${\tilde{s}}_{Res,allow}\approx 0.08$ for the wet asphalt road condition. We estimate the allowable lateral slip ${\tilde{s}}_{S,allow}$ in (45) in accordance with the constraint on the steady-state lateral slip (19)–(20), and its selection is proposed with

$${\tilde{s}}_{S,allow}=Max(\tan ({\tilde{\alpha }}_{f,\lim }),\tan ({\tilde{\alpha }}_{r,\lim }))$$

(46)

By calculating (45) with (46) and ${\tilde{s}}_{Res,allow}$ specified by the road condition, the maximum allowable longitudinal slip ${\tilde{s}}_{L,allow}$ and therefore the yaw moment constraint ${\tilde{M}}_{z,allow}$ in (44) are established.

4.2. Control System Modeling with Polytopic LPV

We use the polytopic LPV modeling technique to design the control input for the speed-varying error system (37). In the $(q_1,q_2)$ parameter space shown in Figure 4, we consider the following affine matrices

$${\tilde{A}}_v(q_1,q_2)=\left[\begin{array}{cc}0& -1\\ \frac{-C_f{l}_f+C_r{l}_r}{J_z}& 0\end{array}\right]+q_1\left[\begin{array}{cc}-\frac{C_f+C_r}{m}& 0\\ 0& -\frac{C_f{l}_f^2+C_r{l}_r^2}{J_z}\end{array}\right]+q_2\left[\begin{array}{cc}0& \frac{-C_f{l}_f+C_r{l}_r}{m}\\ 0& 0\end{array}\right]$$

(47a)

$${\tilde{B}}_v(q_1,q_2)=\left[\begin{array}{c}0\\ \frac{C_f{l}_f}{J_z}\end{array}\right]+q_1\left[\begin{array}{c}\frac{C_f}{m}\\ 0\end{array}\right]$$

(47b)

$${\tilde{B}}_w(q_1,q_2)=[{\tilde{B}}_v(q_1,q_2)\hspace{0.33em}{B}_m]=\left[\begin{array}{cc}0& 0\\ \frac{C_f{l}_f}{J_z}& \frac{1}{J_z}\end{array}\right]+q_1\left[\begin{array}{cc}\frac{C_f}{m}& 0\\ 0& 0\end{array}\right]$$

(47c)

and a curve segment

$$C_V:\hspace{0.33em}{q}_2=q_1^2,\hspace{0.33em}{q}_1\in \left[V_{\max }^{-1},V_{\min }^{-1}\right]$$

(48)

It can be seen that the affine matrices (47) on the curve segment (48) indeed are the matrices $A_v(V_0)$, $B_v(V_0)$ and $B_w(V_0)$ in (1) and (37), i.e.,

$$A_v(V_0)={\tilde{A}}_v(q_1,q_1^2), B_v(V_0)={\tilde{B}}_v(q_1,q_1^2), B_w(V_0)={\tilde{B}}_w(q_1,q_1^2), q_1=V_0^{-1}\in \left[V_{\max }^{-1},V_{\min }^{-1}\right]$$

(49)

A triangular polytope $\Theta$ framed by the lines $L_1$, $L_2$, and $L_3$ as depicted in Figure 4 is chosen to contain the curve segment $C_V$ (see [6] and reference therein), where $L_1$ is the line passing through the two points $a\left(V_{\max }^{-1},V_{\max }^{-2}\right)$ and $b\left(V_{\min }^{-1},V_{\min }^{-2}\right)$, while $L_2$ and $L_3$ are the tangent lines to the curve segment $C_V$ at the points $a\left(V_{\max }^{-1},V_{\max }^{-2}\right)$ and $b\left(V_{\min }^{-1},V_{\min }^{-2}\right)$, respectively. The intersection of lines $L_2$ and $L_3$ is the point $c\left((V_{\max }^{-1}+V_{\min }^{-1})/2,V_{\max }^{-1}{V}_{\min }^{-1}\right)$. Thus, the triangular polytope $\Theta$ containing the curve segment $C_V$ has its vertices $a\left(V_{\max }^{-1},V_{\max }^{-2}\right)$, $b\left(V_{\min }^{-1},V_{\min }^{-2}\right)$, and $c\left((V_{\max }^{-1}+V_{\min }^{-1})/2,V_{\max }^{-1}{V}_{\min }^{-1}\right)$. Based on the concept of polytopic LPV, there exists a speed-parameter-dependent coefficient ${\eta }_i(V_0)\ge 0$, $i=1,2,3$ such that ${\displaystyle {\sum }_{i=1}^3{\eta }_i}(V_0)=1$ and the following convex combinations hold:

$$\begin{gathered} A_v(V_0)={\displaystyle {\sum }_{i=1}^3{\eta }_i(V_0)A_{vi}}, B_v(V_0)={\displaystyle {\sum }_{i=1}^3{\eta }_i(V_0)B_{vi}}, \hspace{0.33em}\hspace{0.33em}{B}_w(V_0)=\hspace{0.33em}{\displaystyle {\sum }_{i=1}^3{\eta }_i\hspace{0.33em}(V_0)B_{wi}} \\ , \forall V_0\in \left[V_{\min },V_{\max }\right] \end{gathered}$$

(50)

where

$$\begin{gathered} A_{v1}={\tilde{A}}_v(V_{\max }^{-1},V_{\max }^{-2}), A_{v2}={\tilde{A}}_v(V_{\min }^{-1},V_{\min }^{-2}), A_{v3}={\tilde{A}}_v((V_{\max }^{-1}+V_{\min }^{-1})/2,V_{\max }^{-1}{V}_{\min }^{-1}), \\ {B}_{v1}={\tilde{B}}_v(V_{\max }^{-1},V_{\max }^{-2}), B_{v2}={\tilde{B}}_v(V_{\min }^{-1},V_{\min }^{-2}), B_{v3}={\tilde{B}}_v((V_{\max }^{-1}+V_{\min }^{-1})/2,V_{\max }^{-1}{V}_{\min }^{-1}) \\ \hspace{0.33em}{B}_{w1}=[B_{v1}{B}_m], B_{w2}=[B_{v2}{B}_m], B_{w3}=[B_{v3}{B}_m] \end{gathered}$$

With the polytopic LPV properties (50), we proceed to design the LPV-LMI state feedback gain for the system (37) under the input constraint (44), and then strengthen the restricted control law after completion.

4.3. LPV-LMI Yaw Moment Control Design with Input Constraint

The state feedback control for the system (37) is proposed as

$$M_z=Ke, K=Y{Q}^{-1}$$

(51)

where the matrices $Y$ and $Q$ satisfy the following LMI conditions

$$Q>0$$

(52a)

$$\left[\begin{array}{ccc}Q{A}_{vi}^T+A_{vi}Q+({\alpha }_c+{\mu }_c)Q+Y^T{B}_m^T+B_{m}Y& B_{vi}& B_m\\ B_{vi}^T& -{\alpha }_c{\rho }_{\varsigma }^{-2}& 0\\ B_m^T& 0& -{\alpha }_c{\rho }_{\xi }^{-2}\end{array}\right]<0,\hspace{0.33em}i=1,2,3$$

(52b)

$$\left[\begin{array}{cc}-Q& Y^T\\ Y& -({\tilde{M}}_{z,allow}/g_c{)}^{2}I\end{array}\right]<0$$

(52c)

$$\left[\begin{array}{cc}-Q& Q\\ Q& -{\gamma }_c^{2}I\end{array}\right]<0$$

(52d)

In (52), the matrices $A_{vi}$, $B_{vi}$, $\hspace{0.33em}{B}_{wi}$, $\hspace{0.33em}i=1,2,3$, and ${\tilde{M}}_{z,allow}$ are defined in (50) and (44), respectively; the parameters ${\alpha }_c>0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\mu }_c>0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\gamma }_c>0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{g}_c>1$ are adjustable via trial and error. To reduce the computation load in emergency situations, the feedback gain $K$ in (51) is chosen as constant. For the application of on-line gain scheduling methods, interested readers are referred to the literature (see [6] and references therein).

Substituting the feedback control (51) into the system (37) yields the closed-loop error system

$$\dot{e}=(A_v(V_0)+B_{m}K)e+B_w(V_0)w, V_0\in [V_{\min },V_{\max }]$$

(53a)

$$w^{T}Hw\le 1, H=diag({\rho }_{\varsigma }^{-2},{\rho }_{\xi }^{-2})$$

(53b)

Let $V(e)=e^{T}Pe$, $P=Q^{-1}>0$ be the Lyapunov function for the system (53). The Lyapunov derivative along the trajectories of (53) can be derived as

$$\begin{array}{c}\dot{V}(e)={\dot{e}}^{T}Pe+e^{T}P\dot{e}\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=e^T(A_v^T(V_0)P+P{A}_v(V_0)+K^T{B}_m^{T}P+P{B}_{m}K)e+e^{T}P{B}_w(V_0)w+w^T{B}_w^T(V_0)Pe,\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\forall V_0\in \left[V_{\min },V_{\max }\right]\hfill \end{array}$$

(54)

As shown in Figure 5, we define the stability regions: $L_V(1)=\left\{\left.e\in {\Re }^2\right|e^{T}Pe\le 1\right\}$, ${\Omega }_{ball}({\gamma }_c)=\left\{\left.e\in {\Re }^2\right|e^{T}e\le {\gamma }_c^2\right\}$ and $L_V(g_c)=\left\{\left.e\in {\Re }^2\right|e^{T}Pe\le g_c^2,\hspace{0.33em}{g}_c>1\right\}$ for analysis in Lemma 1 and Theorem 1. By using the Lyapunov derivative (54) and the LMIs (52), we have Lemma 1 below:

Lemma 1.

Consider the state feedback system (53) with the control input (51) and input constraint (44). The Lyapunov derivative (54) can be estimated as

$$\mathbf{\left(}\mathsf{I}\mathbf{\right)} \dot{V}(e)\le -{\mu }_c{e}^{T}Pe=-{\mu }_{c}V(e), \mathrm{if} e^{T}Pe\ge 1, \forall V_0\in \left[V_{\min },V_{\max }\right]$$

(55a)

Furthermore,

$$\mathbf{(}\mathsf{II}\mathbf{)} \underset{e\in L_V(g_c)}{Max}{e}^T{K}^{T}Ke<{\tilde{M}}_{z,allow}^2$$

(55b)

$$\mathbf{(}\mathsf{III}\mathbf{)} \underset{e\in L_V(1)}{Max}{e}^{T}e<{\gamma }_c^2$$

(55c)

Accordingly,

(a) 

The system (53) has an invariant set $L_V(g_c)$ where the control input (51) satisfies the constraint (44).

(b) 

The system (53) has a region of ultimate boundedness $L_V(1)$.

(c) 

The region ${\Omega }_{ball}({\gamma }_c)$ contains the region $L_V(1)$.

Proof. 

See Appendix A. □

Remark 1.

Lemma 1 shows that if the state $e$ starts in the set of region $L_V(g_c)$, then it will stay within the region $L_V(g_c)$, and after a finite time, the state $e$ enters the ultimate boundedness region $L_V(1)$ [19]. The design parameter ${\mu }_c>0$ is used for estimating the convergence rate of the state $e$ starting from the region $L_V(g_c)\backslash L_V(1)$.

Remark 2.

During the control processwith initial condition $e(0)\in L_V(g_c)$ , the control input satisfies the constraint (44), i.e.,

$$e(0)\in L_V(g_c)\Rightarrow e(t)\in L_V(g_c),\forall t\ge 0\Rightarrow \left|M_z(t)\right|=‖Ke(t)‖\le {\tilde{M}}_{z,allow},\forall t\ge 0$$

Remark 3.

Because the region ${\Omega }_{ball}({\gamma }_c)$ contains the set $L_V(1)$, it is recommended to minimize the parameter $\hspace{0.33em}{\gamma }_c>0$ to reduce the ultimate boundedness region $L_V(1)$ when solving LMIs (52). Further, it is beneficial to maximize the parameter$\hspace{0.33em}{g}_c>1$ to expand the stabilization region $L_V(g_c)$.

4.4. Enhanced Yaw Moment Control

In emergency situations, it is crucial to prioritize achieving the shortest stabilization time as a design objective for the control system. It is known in optimal control that the Bang–Bang control is the optimal solution with the shortest time, as it maximizes capacity with limited input. This study adopts a reinforcement scheme with input constraint (see [20,21] and references therein). Like the Bang–Bang control, the following scheme enhances the utilization of the control (51) while working within the input constraint (44). The enhanced scheme is provided as

$$M_z=Sa{t}_M(Ke-{\gamma }_H{B}_m^{T}Pe), {\gamma }_H>0$$

(56a)

$$Sa{t}_M(u)=\left\{\begin{array}{ll}u,& \left|u\right|<{\tilde{M}}_{z,allow}\\ {\tilde{M}}_{z,allow}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}sign(u),& \left|u\right|\ge {\tilde{M}}_{z,allow}\end{array}\right.,$$

(56b)

where ${\gamma }_H>0$ is chosen as sufficiently large. The enhanced scheme (56) can improve the stabilization property of the original control (51) in the region $L_V(g_c)$ as outlined below. With the enhanced scheme (56), the system (37) turns into

$$\dot{e}=A_v(V_0)e+B_{m}Sa{t}_M(Ke-{\gamma }_H{B}_m^{T}Pe)+B_w(V_0)w, {\gamma }_H>0, V_0\in [V_{\min },V_{\max }]$$

(57a)

$$w^{T}Hw\le 1, H=diag({\rho }_{\varsigma }^{-2}{\rho }_{\xi }^{-2})$$

(57b)

Due to the fact of Item (II) in Lemma 1, there exits ${\tilde{\gamma }}_H(e)\ge 0$ depending on the state $e$ such that

$$\forall e\in L_V(g_c)\Rightarrow Sa{t}_M(Ke-{\gamma }_H{B}_m^{T}Pe)=Ke-{\tilde{\gamma }}_H(e)B_m^{T}Pe, {\tilde{\gamma }}_H(e)\ge 0$$

(58)

Choose the Lyapunov function $V(e)=e^{T}Pe$, $P=Q^{-1}>0$ in Lemma 1 for the state feedback system (57). The Lyapunov derivative along the trajectories of (57) is obtained as

$$\begin{array}{c}\dot{V}(e)={\dot{e}}^{T}Pe+e^{T}P\dot{e}\hfill \\ \hfill =e^T(A_v^T(V_0)P+P{A}_v(V_0))e+Sa{t}_M{(Ke-{\gamma }_H{B}_m^{T}Pe)}^T{B}_m^{T}Pe+e^{T}P{B}_{m}Sa{t}_M(Ke-{\gamma }_H{B}_m^{T}Pe)+e^{T}P{B}_w(V_0)w+w^T{B}_w^T(V_0)Pe,\\ \hfill \forall V_0\in \left[V_{\min },V_{\max }\right]\hfill \end{array}$$

(59)

By using (58), the Lyapunov derivative (59) has the property

$$\begin{array}{r}\forall e\in L_V(g_c)\Rightarrow \dot{V}(e)=e^T(A_v^T(V_0)P+P{A}_v(V_0)+K^T{B}_m^{T}P+P{B}_{m}K)e+e^{T}P{B}_w(V_0)w+w^T{B}_w^T(V_0)Pe\\ -2{\tilde{\gamma }}_H(e)e^{T}P{B}_m{B}_m^{T}Pe, \forall V_0\in \left[V_{\min },V_{\max }\right]\hspace{1em}\hspace{1em}\hspace{1em}\end{array}$$

(60)

Theorem 1.

Consider the state feedback system (57). The Lapunov derivative (60) can be estimated as

$$\begin{gathered} \mathbf{\left(}\mathsf{I}\mathbf{\right)} \dot{V}(e)\le -{\mu }_{c}V(e)-2{\tilde{\gamma }}_H(e)e^{T}P{B}_m{B}_m^{T}Pe\le -{\mu }_{c}V(e), \forall V_0\in \left[V_{\min },V_{\max }\right], \mathrm{if} \\ 1\le e^{T}Pe\le g_c \end{gathered}$$

(61a)

Moreover,

$$\mathbf{\left(}\mathsf{II}\mathbf{\right)} \left|M_z\right|=\left|Sa{t}_M(Ke-{\gamma }_H{B}_m^{T}Pe)\right|\le {\tilde{M}}_{z,allow}, \forall e\in {\Re }^2$$

$$\mathbf{(}\mathsf{III}\mathbf{)} \underset{e\in L_V(1)}{Max}{e}^{T}e<{\gamma }_c^2$$

(61b)

Accordingly,

(a) 

The system (57) has an invariant set  $L_V(g_c)$ where the control input (56) satisfies the constraint (44).

(b) 

The system (57) has a region of ultimate boundedness  $L_V(1)$.

(c) 

The region  ${\Omega }_{ball}({\gamma }_c)$ contains the region $L_V(1)$.

Proof. 

Item (I): The result from (A8) in the Appendix A provides the following estimate

$$1\le e^{T}Pe\Rightarrow e^T(A_v^T(V_0)P+P{A}_v(V_0)+K^T{B}_m^{T}P+P{B}_{m}K)e+e^{T}P{B}_w(V_0)w+w^T{B}_w^T(V_0)Pe\le -{\mu }_c{e}^{T}Pe, \forall V_0\in \left[V_{\min },V_{\max }\right]$$

(62)

Thus, by combining the properties in (60) and (62), the Lyapunov derivative (59) can be estimated as

$$1\le e^{T}Pe\le g_c\Rightarrow \dot{V}(e)\le -{\mu }_c{e}^{T}Pe-2{\tilde{\gamma }}_H(e)e^{T}P{B}_m{B}_m^{T}Pe, {\tilde{\gamma }}_H(e)\ge 0, \forall V_0\in \left[V_{\min },V_{\max }\right]$$

Item (II): The result can be immediately obtained by the definition in (56).

Item (III): Same as in the proof of Lemma 1.

The property provided in Item (I) and Item (II) leads to the results of Item (a) and Item (b). The result of Item (c) comes from the property in Item (III). □

In emergency situations, the enhanced control (56) can make the system response faster. By calculating the enhanced control (56), we obtain the longitudinal slip magnitude $\left|s_L\right|$ in (43), and proceed to calculate the control torque of each wheel using (40). Finally, when integrated with the saturation angle scheme (25) and (27), the enhanced active safety system is completed.

5. Simulations

The control scheme is evaluated using a nonlinear vehicle model (D-Class Sedan) within the CarSim-Simulink co-simulation environment where the wet asphalt road (${\mu }_{\max }=0.8$) is used for the test. The vehicle parameters provided by CarSim are listed in

Table 1. Vehicle parameters.

$m=1530 \mathrm{kg}$	$J_z=2315.3\hspace{0.33em}\hspace{0.33em}{\mathrm{kg}\text{-}\mathrm{m}}^2$	$l_f=1.110 \mathrm{m}$
$l_r=1.67 \mathrm{m}$	$l_d=0.775 \mathrm{m}$	$r_e=0.325 \mathrm{m}$
$C_f=11.6130\hspace{0.33em}\times \hspace{0.33em}{10}^4 \mathrm{Nt}/\mathrm{rad}$	$C_r=8.3900\times {10}^4 \mathrm{Nt}/\mathrm{rad}$	${\mu }_{\max }=0.8$
${\tilde{a}}_{y,\lim }=0.68 \mathrm{g} (6.6708 {\mathrm{m}/\mathrm{s}}^2)$	${\tilde{\alpha }}_{f,\lim }=0.0528 \left(\mathrm{rad}\right)$	${\tilde{\alpha }}_{r,\lim }=0.0486 \left(\mathrm{rad}\right)$

. In the scenario, assume the driver executes an emergency lane change on the wet asphalt road while the vehicle is traveling in an initial speed $V_0=120 \mathrm{km}/\mathrm{h}$. The overreacting response of the driver steering angle (in degrees) is provided as follows:

$${\delta }_d(t)=\left\{\begin{array}{c}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}0\le t<0.375\\ \min \left\{3.75, 5\sin (2(t-0.375))\right\},if\hspace{0.33em}0.375\le t<0.375+\pi \\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}0,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}if\hspace{0.33em}0.375+\pi \le t\end{array}\right.$$

(63)

Without any control assistance, i.e., ${\delta }_{sat}=0$, ${\delta }_f={\delta }_d$, and $M_z=0$, Figure 6a depicts the history of the driver steering angle ${\delta }_d$ and the related steering angle limit ${\tilde{\delta }}_{\lim }$ which is updated with the vehicle speed $V_0$ (in black) as shown in Figure 6b. The responses of the four wheel speeds ${\omega }_j$ are also shown in Figure 6b. Figure 6c,d displays the subsequent responses of the vehicle state $(\beta ,\gamma )$. It is noticeable that the vehicle state $(\beta ,\gamma )$ rapidly deviates from their reference $({\beta }_{ref},{\gamma }_{ref})$. Figure 6e,f illustrates the history of the vehicle acceleration $(a_x,a_y)$ and the combined slip magnitude $‖S_j‖$ of the four wheels. At around 0.8 s, the lateral acceleration $a_y$ begins to saturate with $a_y\approx {\tilde{a}}_{y,\lim }=0.85\times 0.8 \mathrm{g}=0.68 \mathrm{g}$, and the combined slips of the two front wheels go beyond their linear region, i.e., $‖S_j‖\ge 0.08,j=1,2$, resulting in the vehicle skidding and losing maneuverability from about 0.8 s.

The performance of the vehicle control system under (I) the integration control (25), (27), and (51); (II) the enhanced integration control (25), (27), and (56); and (III) the acting-alone enhanced yaw moment control (56) is demonstrated and compared in simulations. The related parameters for the saturation angle scheme (27), the yaw moment control (51), (56), LMIs (52), and the wheel control torque (40) are chosen below:

$$\begin{array}{l}\alpha =30, {\tilde{s}}_{Res,allow}=0.08, {\tilde{s}}_{S,allow}=0.0528, {\tilde{s}}_{L,allow}=0.0601, k=14, {\tilde{M}}_{z,allow}=9781.2, {\alpha }_c=7,\\ {\gamma }_c=0.3, {\mu }_c=0.2, {\rho }_{\varsigma }=0.044, {\rho }_{\xi }=5868.73, g_c=1.5, V_{\max }=34 \mathrm{m}/\mathrm{s}, V_{\min }=20 \mathrm{m}/\mathrm{s}\\ Q=\left[\begin{array}{cc}0.08152& 0.00082\\ 0.00082& 0.08535\end{array}\right], Y=\left[\begin{array}{cc}-797.97698& -1832.24857\end{array}\right], P=\left[\begin{array}{cc}12.26665& -0.11762\\ -0.11762& 11.71726\end{array}\right],\\ K=\left[\begin{array}{cc}-9572.98975& -21375.07610\end{array}\right], {\gamma }_H{B}_m^{T}P=[-508.0318 50607.9673], {\gamma }_H=1\times {10}^7\end{array}$$

All the cases use the same parameter values as provided above.

5.1. Case I: The Integrated Control

In this case, the vehicle performance under the integration control, (25), (27), and (51), is simulated. Figure 7a depicts the responses of the vehicle speed $V_0$ (in black) and wheel speeds ${\omega }_j$; Figure 7b shows the responses of the restricted front-wheel steering angle ${\delta }_f$ (in black) and the driver steering angle ${\delta }_d$ (in brown), along with the saturation angle ${\delta }_{sat}$ (in blue) and the steering angle limit ${\tilde{\delta }}_{\lim }$ (in gray). We can observe that when the vehicle speed $V_0$ changes, the steering angle limit ${\tilde{\delta }}_{\lim }$ is correspondingly updated. The saturation angle ${\delta }_{sat}$, which mainly acts in the periods of [0.5, 1.8] and [2.1, 3.5] in seconds, can effectively make the wheel steering angle ${\delta }_f$ behave as ${\delta }_f\cong {\delta }_{d,ref}$. Figure 7c demonstrates that the yaw moment control $M_z$ mostly takes place in the intervals around 0.5 s, 2 s, and 3.5 s; during these intervals, the rapid changes in the driver steering angle ${\delta }_d$ can be seen in Figure 7b. It can be observed that the two control schemes ${\delta }_{sat}$ and $M_z$ carry out different tasks, that is to say, the saturation angle ${\delta }_{sat}$ serves to prevent the wheel steering angle ${\delta }_f$ from developing the unmatching disturbance $(\upsilon ,\varsigma )$ that could have an instability effect on the vehicle in the periods of [0.5, 1.8] and [2.1, 3.5] in seconds, while the yaw moment $M_z$ mainly performs the reference-following operation in the intervals around 0.5 s, 2 s, and 3.5 s. Figure 7d shows the history of the distributed wheel control torques $T_j$. Figure 7e,f demonstrates that the vehicle state $(\beta ,\gamma )$ remains near their reference $({\beta }_{ref},{\gamma }_{ref})$, meaning the integration control helps to maintain the safety reference following the situation of a driver’s excessive steering. The resulting vehicle acceleration $(a_x,a_y)$ in Figure 7g shows the lateral acceleration $a_y$ operates within the safety range $\left|a_y\right|\le {\tilde{a}}_{y,\lim }=0.68 \mathrm{g}$ during the emergency lane change. Figure 7h depicts the responses of the combined wheel slip magnitudes $‖S_j‖=\sqrt{s_{Lj}^2+s_{Sj}^2}$ of the four wheels, and Figure 7i–l shows the histories of the combined wheel slips $S_j={\left[s_{Lj},s_{Sj}\right]}^T$ of the four wheels. We can see that all the four combined wheel slips $S_j$ lie in the linear region with $‖S_j‖\le {\tilde{s}}_{Res}\approx 0.08$ (wet asphalt).

5.2. Case II: The Enhanced Integration Control

We evaluate the effectiveness of the enhanced integration control (25), (27), and (56) with simulation. The results are presented in Figure 8a–l. These figures are arranged in the same order as Case I to facilitate a direct comparison with the results of Case I. We first observe that the vehicle speed $V_0$ in Figure 8a becomes lower after the control process, and yaw moment $M_z$ and wheel torques $T_j$ appear larger during the control process in comparison to those of Case I. In Figure 8e,f, the performance of the side slip angle $\beta$ appears similar to that of Case I, but the yaw rate $\gamma$ shows an improved response to the reference state ${\gamma }_{ref}$, meaning the enhanced integration control meets the urgent lane change demand with a faster yaw rate response. We see in Figure 8g the corresponding lateral acceleration $a_y$ still operates within the safety range $\left|a_y\right|\le {\tilde{a}}_{y,\lim }=0.68 \mathrm{g}$. Figure 8h–l depicts the history of the combined wheel slip magnitudes $‖S_j‖$ and the combined wheel slips $S_j={\left[s_{Lj},s_{Sj}\right]}^T$ of the four wheels. We observe that all the combined wheel slips still fall within the linear region, i.e., $‖S_j‖\le {\tilde{s}}_{Res}\approx 0.08$. When compared with the longitudinal slips $s_{Lj}$ of Case I, the enhanced control performs with larger longitudinal slips $s_{Lj}$ in three intervals around 0.5, 2, and 3.5 s, and therefore gives rise to a noticeable variation in longitudinal acceleration $a_x$ as shown in Figure 8g. However, the enhanced control still exhibits levels of lateral acceleration $a_y$ similar to that of Case I. As demonstrated in the simulation, the enhanced integration control (56) prioritizes safety while driving by improving the agility of yaw rate response to its reference. Noticeably, it still operates within the linear region of tire/road friction and allows the DYM to effectively handle other disturbances like $(d_v,{\dot{x}}_{ref})$ described in (35).

5.3. Case III: The Acting-Alone Enhanced Yaw Moment Control

By applying the same parameter values as in Case I and Case II, we demonstrate in this case the performance of the vehicle under the acting-alone enhanced yaw moment control (56) (i.e., ${\delta }_{sat}=0$ and ${\delta }_f={\delta }_d$), which is the traditional DYM approach to managing the instability effect of excessive driver steering. Figure 9a,b depicts the responses of the vehicle speed $V_0$, wheel speeds ${\omega }_j$, the front-wheel steering angle ${\delta }_f$, and the steering angle limit ${\tilde{\delta }}_{\lim }$. Compared with the results of Case I and Case II, the vehicle speed $V_0$ is the lowest after the control process. Figure 9c,d shows the responses of the yaw moment $M_z$ and the distributed wheel control torques $T_j$, respectively. In comparison to the results of Case II, the acting-alone yaw moment control appears to generate considerable yaw moment and wheel torques in the periods of [0.5, 1.8] and [2.1, 3.5] in seconds to suppress the instability effect caused by the excessive driver steering $\left|{\delta }_f\right|>{\tilde{\delta }}_{\lim }$ as shown in Figure 9b. Although the vehicle state $(\beta ,\gamma )$ shown in Figure 9e,f remains close to the reference state $({\beta }_{ref},{\gamma }_{ref})$, there is some obvious overshoot and deviation in the response of yaw rate $\gamma$. Figure 9g shows the resulting acceleration $(a_x,a_y)$ with the lateral acceleration $a_y$ close to the safety limit ${\tilde{a}}_{y,\lim }=0.68 \mathrm{g}$ in the periods of [1, 1.7] and [2.8, 3.5] in seconds, and Figure 9h–l displays the responses of the combined wheel slip magnitudes $‖S_j‖$ and the wheel slips $S_j={\left[s_{Lj},s_{Sj}\right]}^T$ of the four wheels in that order. We observe that the two front wheels experience significant coexistence of longitudinal and lateral wheel slips (i.e., $‖S_j‖$, $j=1,2$) in the periods of [1, 1.7] and [2.8, 3.5] in seconds. During these periods, the friction forces of the two front wheels are operating within the friction saturation region, and therefore the control system has a limited capacity to address additional stabilization challenges. The deviation of the yaw rate from its safety reference in Figure 9f and the two front wheels operating in the friction saturation region in Figure 9h are the drawbacks when the acting-alone yaw moment control is employed to counter the driver’s excessive steering.

In Case III, we specifically evaluated the effectiveness of using DYM alone during emergency lane changes. This serves as a baseline for comparison with our proposed control system. As expected, the standalone DYM effectively regulates the vehicle state close to the reference state, as shown in Figure 9e,f. However, it struggles to maintain cornering within the tire/road friction saturation zone due to excessive driver input, as depicted in Figure 9h–l. Our proposed control system, demonstrated in Cases I and II, shows significant improvements. By incorporating the reference steering angle and a high-gain input saturation technique, we ensured that the tire/road friction operated within the feasible linear range and achieved a faster response during emergencies. This highlights the effectiveness of our approach in alleviating the burden on DYM and enhancing vehicle agility during emergencies involving excessive driver steering.

6. Conclusions

An active safety control system is proposed in this study to respond to emergency situations where the driver steering angle exceeds the safety range and the vehicle speed undergoes noticeable changes during control. A reference state and a reference steering angle obtained from a linear vehicle model with a tire/road friction saturation limitation are used by the control system. In case of any discrepancy between the intended reference values and that of the driver steering angle and vehicle state, the system intervenes by implementing the saturation angle scheme and DYM method. To accommodate varying speeds, polytopic LPV modeling is employed, and uncertainties are addressed through Lyapunov redesign, categorizing them into structured matching and non-matching uncertainties. The design of DYM control utilizes numerical solutions to LMIs, incorporating constraints on DYM values based on the friction circle’s limit condition for combined wheel slip. Before completing the design, we apply an input saturation technique using a high-gain approach to improve the utilization of the control and enhance the system response in emergency situations. The resulting enhanced yaw moment is distributed to the wheels to generate wheel control torque. The proposed control system integrates the enhanced DYM with a steering saturation scheme, which prevents the driver’s overreacting steering from destabilizing the vehicle. As a result, the control system maintains tire/road friction within the feasible linear operating range, enabling the DYM to allocate more resources to other stabilization tasks. Simulation examples demonstrate the effectiveness of the enhanced integration control.