Integrated Active Suspension and Anti-Lock Braking Control for Four-Wheel-Independent-Drive Electric Vehicles

Abstract

This paper presents an integrated control scheme for enhancing the ride comfort and handling performance of a four-wheel-independent-drive electric vehicle through the coordination of active suspension system (ASS) and anti-lock braking system (ABS). First, a longitudinal-vertical coupled vehicle dynamics model is established by integrating a road input model. Then the coupling mechanisms between longitudinal and vertical vehicle dynamics are analyzed. An ASS-ABS integrated control system is proposed, utilizing an H_∞ controller for ASS to optimize load transfer effect and a neural network sliding mode control for ABS implementation. Finally, the effectiveness of the proposed control scheme is evaluated through comprehensive tests conducted on a hardware-in-loop (HIL) test platform. The HIL test results demonstrate that the proposed control scheme can significantly improve the braking performance and ride comfort compared to conventional ABS control methods.

1 Introduction

Automotive electrification is expanding rapidly worldwide in response to the formidable challenges of greenhouse gas emissions and fossil oil depletion [1, 2]. Four-wheel-independent-drive electric vehicles (FWID EVs) have garnered attention in past years. They utilize four in-wheel motors (IWMs) to achieve direct propulsion [3], and the independent control of each wheel presents enormous potential for enhancing overall vehicle performance [4, 5]. In the meantime, vehicle safety control (VSC) systems, such as active suspension system (ASS) and anti-lock braking system (ABS), are being increasely used in production vehicle models [6]. The coordinated control of these systems holds promise for simultaneously enhancing vehicle handling performance and ride comfort during braking execution [7, 8].

In previous studies, ASS has been often employed to optimize vehicle ride comfort by directly controlling suspension actuation forces. It can also contributes to mitigating the negative impact of increased unsprung mass resulting from the use of in-wheel motors in FWID EVs. Numerous research endeavors have been dedicated to developing effective ASS controllers to optimize sprung mass acceleration based on quarter-vehicle dynamics modeling [9]. Commonly used control methods include explicit model predictive control (MPC) [10], fuzzy logic control [11], ceiling damping control [12], sliding mode control [13], H_∞ control [14] and others. In particular, H_∞ control is favored over other control algorithms due to its robustness to actuator faults, control network delays, and model uncertainties [15]. However, the conventional quarter-vehicle model cannot describe the pitch motion of vehicle. To address this limitation, researchers proposed half/full-vehicle models to account for vertical vibration and pitch motion of vehicle. For example, Sun et al. [16] presented an adaptive back-stepping ASS controller based on a half-vehicle model. Similarly, Youn et al. [17] proposed a linear quadratic optimal level-attitude ASS controller by employing a full-vehicle model. These studies to a large extent mitigated the pitch and level vibrations without considering load transfer during braking execution.

ABS implementation in FWID EVs can be realized either by conventional hydraulic braking or by regenerative braking [18]. Researchers have devoted significant efforts towards efficiently implementing ABS under varying driving conditions, using slide mode control, MPC, and H₂/H_∞ mixed control. Slide mode control is widely used to design wheel slip regulators and to develop nonlinear sliding observers for tire slip ratio and friction force estimation [18, 19]. To improve vehicle stability during braking execution, a sliding mode variable structure controller has been developed to regulate the wheel slip ratio near its optimal value [20]. Other approaches include MPC based on three-point road friction estimation [21] and H₂/H_∞ mixed control for slip ratio search optimization [22]. Generally, the primary control objective of these studies is to regulate the wheel slip ratio to track the reference or to keep it within a specified range. However, the existing methods have constantly overlooked the influence of tire vertical motion on the performance of ABS.

The implementation of ASS and ABS can result in coupled vertical and longitudinal vehicle dynamics. The integrated control of ASS and ABS has the potential to further enhance braking performance. In Ref. [23], Lin et al. simulated the implementations of ABS and ASS using a quarter-car model. They proposed a two-back-stepping controller to achieve independent control of these systems, and optimized the braking performance by controlling tire deflection and normal tire force. However, they failed to consider the impact of ASS control on vehicle vertical motion. Similarly, Lu et al. [24] introduced a fuzzy sliding mode control scheme that coordinates the control of a semi-active suspension system with a braking and steering control system. Nevertheless, there are few studies investigating the coupled effects of ASS and ABS on vehicle motions for FWID EVs [25]. Besides, the rapid changes in braking torque during ABS implementation can possibly induce severe pitch motions of the sprung mass and fluctuation of the load transfer, making it challenging for the controlled wheels to track the optimal slip ratio. Moreover, these pitching motions can significantly compromise vehicle ride comfort.

This study aims to address the aforementioned research gaps by developing an integrated control scheme for ASS and ABS to improve the ride comfort and handling performance of FWID EVs. To achieve this objective, a longitudinal-vertical coupled vehicle dynamics model is first established by integrating a road input model. Then the coupling mechanisms between the longitudinal and vertical vehicle dynamics are analyzed. An H_∞ controller for ASS control and a neural network sliding mode control (NNSMC) for ABS control are respectively proposed. Finally, the effectiveness of the integrated control scheme is verified through comprehensive hardware-in-loop (HIL) tests. The exclusive contributions of this study to the related research can be summarized as follows.

• A comprehensive full-vehicle longitudinal-vertical coupled dynamics model is developed by incorporating the dynamics of IWMs.

• The load transfer resulting from vehicle pitch motion during braking execution is considered as an external disturbance to ASS to improve the robustness of the proposed controller to vehicle longitudinal dynamics.

• An integrated ABS-ASS control scheme is proposed to improve braking safety while maintaining ride comfort, particularly in emergency braking scenarios.

The remainder of this paper is organized as follows: Section 2 introduces the longitudinal-vertical coupled vehicle dynamics model combined with the road input model. Section 3 discusses the interactions between ASS and ABS during the braking process, and provides a detailed description about the proposed integrated control scheme. Section 4 verifies the effectiveness of the proposed control scheme through HIL tests. Finally, the key conclusions drawn from this study are summarized in Section 5.

2 Longitudinal-Vertical Coupling Vehicle Dynamics Model

The coupling effect between longitudinal and vertical vehicle motions primarily arises from nonlinear dynamics of tire. Particularly during braking maneuvers, vehicle deceleration induces load transfer between the front and rear axles, influencing both the vertical motion of the vehicle and the interactions between tires and road. Hence, it becomes crucial to accurately model the longitudinal-vertical coupling relationship when implementing longitudinal and vertical vehicle control strategies during braking.

2.1 Full Vehicle Models

A comprehensive 7-Degree-of-Freedom (7-DOF) vehicle model is developed by incorporating the vertical translation and pitch motions of the sprung mass, horizontal translation of the vehicle, and vertical translation and rolling motions of the front and rear wheels, as illustrated in Figure 1.

Figure 1

Vehicle dynamics model

In Figure 1, the suspension force F_i is the normal force generated by the load transfer with the subscripts i=fl, fr, rl, rr representing the left front, right front, left rear, and right rear wheels, respectively; m_bi is the equivalent quarter sprung mass; k_si and c_si are the suspension stiffness and damping coefficients, respectively; m_si is the stator and axle mass of IWM; m_ri is the equivalent overall mass of the IWM rotor and tire; k_b is the bearing stiffness of IWM; k_t is the tire stiffness; T_bi is the braking torque; F_xi is the frictional force transferred from road to tires; u_i is the active suspension actuation force; l_f, l_r, and h_g are the distances from the Centre-of-Cravity (CoG) of the sprung mass to the front axle, rear axle and road surface, respectively; x_s is the longitudinal displacement of the sprung mass; $\varphi$ is the pitch angle of the vehicle body; ω_i is the angular speed of tires.

The dynamics of the sprung mass are given by [26]

$$\begin{gathered} m_{s} \ddot{z}_{s} \,{ = }\,F_{sfr} + F_{sfl} + F_{srr} + F_{srl} , \hfill \\ m_{s} \ddot{x}_{s} = \sum {F_{xi} } ,i = fl,fr,rl,rr, \hfill \\ I_{s} \ddot{\varphi } = - l_{f} \left( {F_{sfr} + F_{sfl} } \right) + l_{r} \left( {F_{srr} + F_{srl} } \right) - m_{s} \ddot{x}_{s} h_{g} /g, \hfill \\ \end{gathered}$$

(1)

where m_s is the sprung mass; z_s is the vertical displacement of the sprung mass; F_si is the vertical suspension force; I_s is the pitch rotational inertia of vehicle. F_si can be calculated by

$$\begin{gathered} F_{sfl} = F_{sfr} = k_{sfl} (z_{sfl} - z_{bfl} ) + c_{sfl} (\dot{z}_{sfl} - \dot{z}_{bfl} ) + u_{fl} , \hfill \\ F_{srl} = F_{srr} = k_{srl} (z_{srl} - z_{brl} ) + c_{srl} (\dot{z}_{srl} - \dot{z}_{brl} ) + u_{rl} . \hfill \\ \end{gathered}$$

(2)

The center of the sprung mass moves forward during the braking process, and the normal force transmitted by the load transfer can be given by [27]

$$\begin{gathered} F_{fr} = F_{fl} = - \ddot{x}_{s} m_{s} h_{g} /2g(l_{r} + l_{f} ), \hfill \\ F_{rr} = F_{rl} = \ddot{x}_{s} m_{s} h_{g} /2g(l_{r} + l_{f} ). \hfill \\ \end{gathered}$$

(3)

From Eq. (3), it can be seen that the longitudinal acceleration affects the vertical suspension force, which represents the coupling effect between the longitudinal and vertical motions.

Based on the Newton's law, the dynamic equation of the quarter vertical vibration model can be given by

$$\left\{ \begin{gathered} z_{bfl} = z_{bfr} = z_{s} - l_{f} \sin (\varphi ), \hfill \\ z_{brl} = z_{brr} = z_{s} + l_{r} \sin (\varphi ), \hfill \\ m_{bfl} = m_{bfr} = m_{s} l_{r} /2(l_{r} + l_{f} ), \hfill \\ m_{brl} = m_{brr} = m_{s} l_{f} /2(l_{r} + l_{f} ), \hfill \\ m_{si} \ddot{z}_{si} + F_{si} + k_{b} (z_{si} - z_{ri} ) = 0, \hfill \\ m_{ri} \ddot{z}_{ri} + k_{t} \left( {z_{ri} - q_{i} } \right) + k_{b} (z_{ri} - z_{si} ) = 0, \hfill \\ F_{zi} = m_{bi} {\kern 1pt} + m_{si} + m_{ri} - k_{t} \left( {z_{ri} - q_{i} } \right), \hfill \\ i = fl,fr,rl,rr, \hfill \\ \end{gathered} \right.$$

(4)

where z_bi, z_si and z_ri represent the vertical displacements of the equivalent quarter sprung mass, stator mass and rotor mass of IWM, with the subscript "i" representing the wheel position; F_zi is the vertical tire force; q_i is the generated road profile. The nonlinear magic formula (MF) model is employed to calculate the tire force due to its high fitting accuracy with test data [28]. A longitudinal single-wheel dynamics model can be given by

$$\left\{ \begin{gathered} F_{xi} = F_{zi} \mu (\lambda_{i} )\sin (C\arctan (B\lambda_{i} - E(B\lambda_{i} - \arctan (B\lambda_{i} )))), \hfill \\ I_{i} \dot{\omega }_{i} = T_{bi} - RF_{xi} , \hfill \\ i = fl,fr,rl,r, \hfill \\ \end{gathered} \right.$$

(5)

where λ_i is the wheel slip ratio; μ is the tire-road adhesion coefficient; B, C, and E are the parameters of the MF model; I_i is the rotational inertia of each wheel; R is the effective wheel radius.

2.2 Road Input Model

The road grade (RG) and road type (RT) are considered in the road input model. RG represents the vertical road profiles for generating different vertical wheel movements, while RT characterizes the longitudinal road friction.

(1) Vertical road input

The power spectral density can describe the statistical characteristics of RG in the vertical direction. The Harmonic superposition algorithm is used to generate the time-domain road profiles as [29, 30]

$$\begin{gathered} q(t) = \sum\limits_{k = 1}^{M} {\sqrt {2 \cdot G_{q} \left( {f_{{{\text{mid}} - k}} } \right) \cdot \frac{{f_{2} - f_{1} }}{M}} } \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \cdot \sin \left( {2\pi f_{{{\text{mid }} - k}} + \chi_{k} } \right), \hfill \\ \end{gathered}$$

(6)

where f_mid-k is the kth middle frequency, k=1, 2, …, M; G_q (f_mid-K) is the power spectral density at f_mid-K; χ_k is an identifiably distributed phase with a range of [0, 2π]. The upper and lower time-domain frequency boundaries are denoted as f₁ and f₂, respectively. According to ISO-8608 [31], the road profiles of ISO-A, ISO-B and ISO-C are shown in Figure 2.

Figure 2

Road profiles of three typical roads

(2) Longitudinal road input

The longitudinal friction force is transferred to the vehicle through tire-road interactions. The tire-road adhesion coefficient μ and the reference slip ratio λ^* can be given by

$$\left\{ \begin{gathered} \mu (\lambda ) = c_{1} \left( {1 - e^{{ - c_{2} \lambda }} } \right) - c_{3} \lambda , \hfill \\ \lambda ^* = \frac{1}{{c_{2} }}{\text{In}}\frac{{c_{1} c_{2} }}{{c_{3} }}, \hfill \\ \end{gathered} \right.$$

(7)

where c₁, c₂, and c₃ define the road friction conditions (see Table 1) [27]. The rightmost column of Table 1 is the reference slip ratio λ^*, around which the peak adhesion coefficient can be obtained.

Table 1 Road condition constants

3 Design of the ASS and ABS Controllers

The ASS and ABS controllers are respectively developed in the integrated control framework base on the longitudinal-vertical coupling vehicle model.

3.1 Design of the ASS Controller

An ASS controller is developed to achieve the desirable suspension response in the stationary state, such as sprung mass motion, suspension deflection, and tire deflection [32]. A robust H_∞ control scheme is established to derive the external control forces for realizing desired vehicle states.

The vehicle is assumed to operate under pure braking conditions by ignoring vehicle yaw motion. Due to the symmetrical characteristics of the vehicle, the left half-vehicle model is selected as the control object. When $\varphi$ is small with sin($\varphi$)≈$\varphi$, the equivalent half-vehicle model can be expressed as

$$\begin{gathered} \ddot{z}_{bfl} = \ddot{z}_{s} - l_{f} \ddot{\varphi } = 2(F_{sfl} + F_{srl} )/m_{s} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - l_{f} ( - 2l_{f} F_{sfl} + 2l_{r} F_{srl} - m_{s} \ddot{x}_{s} h_{g} /g)/I_{s} , \hfill \\ \ddot{z}_{brl} = \ddot{z}_{s} + l_{r} \ddot{\varphi } = 2(F_{sfl} + F_{srl} )/m_{s} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + l_{r} ( - 2l_{f} F_{sfl} + 2l_{r} F_{srl} - m_{s} \ddot{x}_{s} h_{g} /g)/I_{s} , \hfill \\ \ddot{z}_{sfl} = - (F_{sfl} + k_{b} (z_{sfl} - z_{rfl} ))/m_{sfl} , \hfill \\ \ddot{z}_{srl} = - (F_{srl} + k_{b} (z_{srl} - z_{rrl} ))/m_{srl} , \hfill \\ \ddot{z}_{rfl} = - (k_{t} \left( {z_{rfl} - q_{fl} } \right) + k_{b} (z_{rfl} - z_{sfl} ))/m_{rfl} , \hfill \\ \ddot{z}_{rrl} = - (k_{t} \left( {z_{rrl} - q_{rl} } \right) + k_{b} (z_{rrl} - z_{srl} ))/m_{rrl} . \hfill \\ \end{gathered}$$

(8)

The state vector is given by

$$\begin{gathered} {\varvec{x}}(t) = \hfill \\ [\dot{z}_{bfl} {\kern 1pt} {\kern 1pt} {\kern 1pt} \dot{z}_{brl} {\kern 1pt} {\kern 1pt} {\kern 1pt} \dot{z}_{sfl} {\kern 1pt} {\kern 1pt} {\kern 1pt} \dot{z}_{srl} {\kern 1pt} {\kern 1pt} {\kern 1pt} \dot{z}_{rfl} {\kern 1pt} {\kern 1pt} {\kern 1pt} \dot{z}_{rrl} {\kern 1pt} {\kern 1pt} {\kern 1pt} z_{bfl} - z_{sfl} ... \hfill \\ z_{brl} - z_{srl} {\kern 1pt} {\kern 1pt} {\kern 1pt} z_{sfl} - z_{rfl} {\kern 1pt} {\kern 1pt} {\kern 1pt} z_{srl} - z_{rrl} {\kern 1pt} {\kern 1pt} {\kern 1pt} z_{rfl} - q_{fl} {\kern 1pt} {\kern 1pt} {\kern 1pt} z_{rrl} - q{}_{rl}]^{{\text{T}}} . \hfill \\ \end{gathered}$$

(9)

The dynamic equations of the ASS system can be written as

$$\dot{\user2{x}}(t) = {\varvec{A}}{\varvec{x}}(t) + {\varvec{B}}_{w} {\varvec{w}}(t) + {\varvec{B}}_{u} {\varvec{u}}(t).$$

(10)

By substituting Eqs. (9) and (10) into Eq. (8), the state space can be rewritten as

$$\begin{gathered} {\varvec{A}} = \hfill \\ \left[ {\begin{array}{*{20}c} { - \frac{{2c_{sfl} }}{{m_{s} }} - \frac{{2c_{sfl} l_{f}^{2} }}{{I_{s} }}} & { - \frac{{2c_{srl} }}{{m_{s} }} + \frac{{2c_{srl} l_{f} l_{r} }}{{I_{s} }}} & {\frac{{2c_{sfl} }}{{m_{s} }} + \frac{{2c_{sfl} l_{f}^{2} }}{{I_{s} }}} & {\frac{{2c_{srl} }}{{m_{s} }} - \frac{{2c_{srl} l_{f} l_{r} }}{{I_{s} }}} & 0 & 0 & { - \frac{{2k_{sfl} }}{{m_{s} }} - \frac{{2k_{sfl} l_{f}^{2} }}{{I_{s} }}} & { - \frac{{2k_{srl} }}{{m_{s} }} + \frac{{2k_{srl} l_{f} l_{r} }}{{I_{s} }}} & 0 & 0 & 0 & 0 \\ { - \frac{{2c_{sfl} }}{{m_{s} }} + \frac{{2c_{sfl} l_{r} l_{f} }}{{I_{s} }}} & { - \frac{{2c_{srl} }}{{m_{s} }} - \frac{{2c_{srl} l_{r}^{2} }}{{I_{s} }}} & {\frac{{2c_{sfl} }}{{m_{s} }} - \frac{{2c_{sfl} l_{r} l_{f} }}{{I_{s} }}} & {\frac{{2c_{srl} }}{{m_{s} }} + \frac{{2c_{srl} l_{r}^{2} }}{{I_{s} }}} & 0 & 0 & { - \frac{{2k_{sfl} }}{{m_{s} }} + \frac{{2k_{sfl} l_{r} l_{f} }}{{I_{s} }}} & { - \frac{{2k_{srl} }}{{m_{s} }} - \frac{{2k_{srl} l_{r}^{2} }}{{I_{s} }}} & 0 & 0 & 0 & 0 \\ {\frac{{c_{sfl} }}{{m_{sfl} }}} & 0 & { - \frac{{c_{sfl} }}{{m_{sfl} }}} & 0 & 0 & 0 & {\frac{{k_{sfl} }}{{m_{sfl} }}} & 0 & { - \frac{{k_{b} }}{{m_{sfl} }}} & 0 & 0 & 0 \\ 0 & {\frac{{c_{srl} }}{{m_{srl} }}} & 0 & { - \frac{{c_{srl} }}{{m_{srl} }}} & 0 & 0 & 0 & {\frac{{k_{srl} }}{{m_{srl} }}} & 0 & { - \frac{{k_{b} }}{{m_{srl} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{k_{b} }}{{m_{rfl} }}} & 0 & { - \frac{{k_{t} }}{{m_{rfl} }}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{k_{b} }}{{m_{rrl} }}} & 0 & { - \frac{{k_{t} }}{{m_{rrl} }}} \\ 1 & 0 & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right], \hfill \\ \end{gathered}$$

$$\begin{gathered} {\varvec{B}}_{w} = \left[ {\begin{array}{*{20}l} {\frac{{l_{f} m_{s} h_{g} }}{{gI_{s} }}} \hfill & {\frac{{ - l_{r} m_{s} h_{g} }}{{gI_{s} }}} \hfill & {{\mathbf{0}}_{1 \times 8} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {{\mathbf{0}}_{1 \times 8} } \hfill & { - 1} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {{\mathbf{0}}_{1 \times 8} } \hfill & 0 \hfill & { - 1} \hfill \\ \end{array} } \right]^{\text T} , \hfill \\ {\varvec{w}}(t) = \left[ {\begin{array}{*{20}l} {\ddot{x}_{s} } \hfill & {\dot{q}_{fl} } \hfill & {\dot{q}_{rl} } \hfill \\ \end{array} } \right]^{\text T} , \hfill \\ {\varvec{B}}_{u} = \left[ {\begin{array}{*{20}l} {\frac{2}{{m_{s} }} + \frac{{2l_{f}^{2} }}{{I_{s} }}} \hfill & {\frac{2}{{m_{s} }} - \frac{{2l_{r} l_{f} }}{{I_{s} }}} \hfill & { - \frac{1}{{m_{sfl} }}} \hfill & 0 \hfill & {{\mathbf{0}}_{1 \times 8} } \hfill \\ {\frac{2}{{m_{s} }} - \frac{{2l_{r} l_{f} }}{{I_{s} }}} \hfill & {\frac{2}{{m_{s} }} + \frac{{2l_{r}^{2} }}{{I_{s} }}} \hfill & 0 \hfill & { - \frac{1}{{m_{srl} }}} \hfill & {{\mathbf{0}}_{1 \times 8} } \hfill \\ \end{array} } \right]^{\text T} , \hfill \\ {\varvec{u}}(t) = \left[ {\begin{array}{*{20}l} {u_{fl} } \hfill & {u_{rl} } \hfill \\ \end{array} } \right]^{\text T} . \hfill \\ \end{gathered}$$

The ASS performance is typically assessed using the criteria such as ride comfort, suspension working space, and tire-road adhesion. Furthermore, the vertical-longitudinal coupling effect that occurs during braking leads to pitch movement of vehicle, resulting in load transfer and compromising vehicle stability. To evaluate system response, the acceleration of the sprung mass and the angular acceleration of the pitch motion are studied. The former is used to evaluate the ride comfort while the latter reflects the handling stability during braking.

In the meantime, it is essential to ensure that the suspension deflection and the air gap between the stator and rotor of IWM must maintain within a specified range to prevent structural failure. Furthermore, the output force of the ASS actuator is constrained due to limited power supply. Considering these conditions, the regulated output z₁ and the normalized constraint output z₂ are defined as

$$\begin{aligned} {\varvec{z}}_{1} & = [ {\begin{array}{*{20}l} {\ddot{z}_{s} } \hfill & {\ddot{\varphi }} \hfill \\ \end{array} } ]^{{\text{T}}} \\ & ={\varvec{C}}_{x1} {\varvec{x}}(t) + {\varvec{D}}_{w1} {\varvec{w}}(t) + {\varvec{D}}_{u1} {\varvec{u}}(t), \\ {\varvec{z}}_{2} & = \left[ {\frac{{z_{bfl} - z_{sfl} }}{{z_{flm} }}} \right.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{z_{brl} - z_{srl} }}{{z_{rlm} }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{z_{rfl} - z_{sfl} }}{{z_{flam} }} \, ... \\ & \left. {\begin{array}{*{20}c} {} & {} & {} & {\frac{{z_{rrl} - z_{srl} }}{{z_{rlam} }}} & {\frac{{u_{fl} }}{{u_{flm} }}} & {\frac{{u_{fr} }}{{u_{rlm} }}} \\ \end{array} } \right]^{{\text{T}}} \\ & = {\varvec{C}}_{x2} {\varvec{x}}(t) + {\varvec{D}}_{u2} {\varvec{u}}(t), \\ \end{aligned}$$

(11)

where

${\varvec{C}}_{x1} = \left[ {\begin{array}{*{20}c} {\frac{{ - 2c_{sfl} }}{{m_{s} }}} & {\frac{{ - 2c_{srl} }}{{m_{s} }}} & {\frac{{2c_{sfl} }}{{m_{s} }}} & {\frac{{2c_{srl} }}{{m_{s} }}} & {{\mathbf{0}}_{1 \times 2} } & {\frac{{ - 2k_{sfl} }}{{m_{s} }}} & {\frac{{ - 2k_{srl} }}{{m_{s} }}} & {{\mathbf{0}}_{1 \times 4} } \\ {\frac{{2l_{f} c_{sfl} }}{{I_{s} }}} & {\frac{{ - 2l_{r} c_{srl} }}{{I_{s} }}} & {\frac{{ - 2l_{f} c_{sfl} }}{{I_{s} }}} & {\frac{{2l_{r} c_{srl} }}{{I_{s} }}} & {{\mathbf{0}}_{1 \times 2} } & {\frac{{2l_{f} k_{sfl} }}{{I_{s} }}} & {\frac{{ - 2l_{r} k_{srl} }}{{I_{s} }}} & {{\mathbf{0}}_{1 \times 4} } \\ \end{array} } \right]$, ${\varvec{D}}_{w1} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill \\ {\frac{{ - m_{s} h_{g} }}{{gI_{s} }}} \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right],{\varvec{D}}_{u1} = \left[ {\begin{array}{*{20}l} {\frac{2}{{m_{s} }}} \hfill & {\frac{2}{{m_{s} }}} \hfill \\ {\frac{{ - 2l_{f} }}{{I_{s} }}} \hfill & {\frac{{2l_{r} }}{{I_{s} }}} \hfill \\ \end{array} } \right]$,

${\varvec{C}}_{x2} = \left[ {\begin{array}{*{20}c} {{\mathbf{0}}_{1 \times 6} } & {z_{flm}^{ - 1} } & 0 & 0 & 0 & {{\mathbf{0}}_{1 \times 2} } \\ {{\mathbf{0}}_{1 \times 6} } & 0 & {z_{rlm}^{ - 1} } & 0 & 0 & {{\mathbf{0}}_{1 \times 2} } \\ {{\mathbf{0}}_{1 \times 6} } & 0 & 0 & {z_{flam}^{ - 1} } & 0 & {{\mathbf{0}}_{1 \times 2} } \\ {{\mathbf{0}}_{1 \times 6} } & 0 & 0 & 0 & {z_{rlam}^{ - 1} } & {{\mathbf{0}}_{1 \times 2} } \\ {{\mathbf{0}}_{1 \times 6} } & 0 & 0 & 0 & 0 & {{\mathbf{0}}_{1 \times 2} } \\ {{\mathbf{0}}_{1 \times 6} } & 0 & 0 & 0 & 0 & {{\mathbf{0}}_{1 \times 2} } \\ \end{array} } \right]$ ,

${\varvec{D}}_{u2} = \left[ {\begin{array}{*{20}l} {{\mathbf{0}}_{1 \times 4} } \hfill & {u_{flm}^{ - 1} } \hfill & 0 \hfill \\ {{\mathbf{0}}_{1 \times 4} } \hfill & 0 \hfill & {u_{rlm}^{ - 1} } \hfill \\ \end{array} } \right]^{\text T}$, z_ilm is the maximum suspension deflection; z_ilam is the maximum air gap of IWM; u_ilm is the maximum output force of the actuator. The subscript, i=f, r, represent the front and rear wheels, respectively.

The designed state feedback controller ${\varvec{u}}(t) \,= \,\user2{{\varvec K}x}(t)$ holds under the following assumptions.

• (1) Without external perturbations, the closed-loop system described in Eqs. (10) and (11) is asymptotically stable.

• (2) The performance $\left\| {{\varvec{z}}_{{1}} (t)} \right\|_{\infty } \le \gamma \left\| {w(t)} \right\|_{\infty }$ is minimized subject to Eqs. (10) and (11), where $\gamma$ is the anti-disturbance level of the H_∞ controller.

• (3) The time-domain constraint $\vert{{\varvec{z}}_{{2}} (t)} \vert \le 1$|z₂(t)|≤1 must be satisfied.

The parameters of the robust H_∞ controller can be obtained by solving the following theorem.

Theorem

If there exist positive scalars ρ and γ and a positive definite symmetric matrix X that make the inequalities described in Eqs. (12) and (13) hold at any time instant under LMIs, the system is globally asymptotic stable. In Eq. (13), ρ²=w_maxγ², and w_max is the upper perturbation energy of w(t). K=WX⁻¹ can be used to derive the state feedback gain K, and the detailed proof is given in Ref. [33],

$$\left[ {\begin{array}{*{20}c} {\left( {{\varvec{AX}} + {\varvec{B}}_{u} {\varvec{W}}} \right)^{{\text{T}}} + \left( {{\varvec{AX}} + {\varvec{B}}_{u} {\varvec{W}}} \right)} & {{\varvec{B}}_{w} } & {\left( {{\varvec{C}}_{x1} {\varvec{X}} + {\varvec{D}}_{u1} {\varvec{W}}} \right)^{{\text{T}}} } \\ {{\varvec{B}}_{w}^{{\text{T}}} } & { - \gamma^{2} {\varvec{I}}} & {{\varvec{D}}_{w1}^{{\text{T}}} } \\ {{\varvec{C}}_{x1} {\varvec{X}} + {\varvec{D}}_{u1} {\varvec{W}}} & {{\varvec{D}}_{w1} } & { - {\varvec{I}}} \\ \end{array} } \right] < 0,$$

(12)

$$\left[ {\begin{array}{*{20}c} { - {\varvec{X}}} & {{\varvec{C}}_{x2} {\varvec{X}} + {\varvec{D}}_{u2} {\varvec{W}}} \\ {({\varvec{C}}_{x2} {\varvec{X}} + {\varvec{D}}_{u2} {\varvec{W}})^{{\text{T}}} } & { - \frac{1}{\rho }{\varvec{I}}} \\ \end{array} } \right] < 0.$$

(13)

3.2 Design of the ABS Controller

The neural network sliding mode control (NNSMC) has gained widespread recognition due to its capability to accommodate system constraints in many applications [34, 35]. In this study, NNSMC is employed to track the optimal slip ratio λ^* during the braking process. The wheel slip ratio is defined as

$$\lambda_{i} = \frac{{v_{s} - \omega_{i} R}}{{v_{s} }},$$

(14)

where v_s is the longitudinal vehicle velocity.

Define the sliding surface as

$$\begin{gathered} s = e = \lambda^{*} - \lambda_{i} , \hfill \\ \dot{s} = - \dot{\lambda }_{i} . \hfill \\ \end{gathered}$$

(15)

The differentiation of Eq. (14) is obtained as

$$\begin{gathered} \dot{\lambda }_{i} = \left( {1 - \frac{{\omega_{i} R}}{{v_{s} }}} \right)^{\prime } = - \left( {\frac{{\dot{\omega }_{i} R}}{{v_{s} }} - \frac{{\dot{v}_{s} \omega_{i} R}}{{v_{s}^{2} }}} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{\dot{v}_{s} \omega_{i} R - v_{s} \dot{\omega }_{i} R}}{{\dot{v}_{s}^{2} }}. \hfill \\ \end{gathered}$$

(16)

The single-wheel dynamics model can be given by

$$\dot{\omega }_{i} = \frac{{T_{b} }}{{I_{i} }} - \frac{{RF_{xi} }}{{I_{i} }}.$$

(17)

By substituting Eq. (17) into Eq. (16), we get

$$\begin{gathered} \dot{\lambda }_{i} = \frac{{\dot{v}_{s} \omega_{i} R}}{{v_{s}^{2} }} + \frac{{R^{2} }}{{v_{s} I_{s} }}F_{xi} - \frac{{T_{bi} R}}{{v_{s} I_{s} }} + d \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = GF_{xi} + PT_{bi} + D + d, \hfill \\ \end{gathered}$$

(18)

where $G = \frac{{\omega_{i} R}}{{m_{s} v_{s}^{2} }}, \, P = - \frac{R}{{v_{s} I_{s} }}, \, D = \frac{{\dot{v}_{s} R}}{{v_{s}^{2} }}$, and d is the disturbance.

By substituting Eq. (18) into Eq. (15), we get

$$T_{bi} = - \frac{G}{P}\hat{F}_{xi} + \frac{{\eta {\text{sign}}(s) + D}}{P},$$

(19)

where $\hat{F}_{xi}$ is the estimated F_xi; η is the coefficient of the NNSMC controller.

The Lyapunov asymptotical stability condition can be rewritten as

$$\begin{gathered} \dot{L} = s \cdot \dot{s} + \tau \tilde{\omega }_{i}^{{\text{T}}} \dot{\tilde{\omega }}_{i} \hfill \\ = s\left[ { - G\tilde{\omega }^{{\text{T}}} h(x) - G\varepsilon - D - d - \eta {\text {sign}}(s)} \right] - \tau \tilde{\omega }^{{\text{T}}} \dot{\hat{\omega }} \hfill \\ = - \tilde{\omega }^{{\text{T}}} \left[ {sGh(x) + \tau \dot{\hat{w}}} \right] - s\left[ {G\varepsilon + D + d + \eta {\text {sign}}(s)} \right], \hfill \\ \end{gathered}$$

(20)

where $\dot{\hat{\omega }} = - \frac{G}{\tau }s \cdot h(x)$, and τ and ε are the estimated error and the reaching law.

Let $\left| \eta \right| \ge \left| {G\varepsilon + D + d} \right|$, so that $\dot{L} < 0$. The stability of the ABS controller is proven.

The PID controller is selected as a comparison to highlight the efficacy of the proposed contrller, which is given by

$$T_{bi} = K_{P} \left( {\lambda - \lambda^{*} } \right) + K_{I} \int_{0}^{t} {\left( {\lambda - \lambda^{*} } \right)} {\text{d}}t + K_{D} \frac{{{\text{d}}\left( {\lambda - \lambda^{*} } \right)}}{{{\text{d}}t}},$$

(21)

where K_P, K_I and K_D are the adjustable parameters of the PID controller.

Figure 3 presents a control block diagram illustrating the flowchart of the proposed integrated control scheme. To trigger ABS and ASS control, a hard brake maneuver is performed by the test vehicle on a straight road. In the developed longitudinal-vertical coupling model, the vertical tire motion is influenced by RG, while the longitudinal tire motion is affected by RT. During the braking process, the coupling effect between vertical and longitudinal vehicle dynamics primarily manifests in the sprung and unsprung masses. In the case of the sprung mass, the braking-induced load transfer induces pitching movement that impacts vehicle ride comfort. As for the unsprung mass, the coupling effect mainly occurs in the tires, where the vertical dynamic load significantly influences the longitudinal friction. The proposed integrated control scheme has a specific workflow, which is explained as follows: The ASS controller utilizes the vibration responses of components as inputs and generates an output force based on the synthesized control law. This ASS actuator output force effectively mitigates the load transfer caused by rapidly changing braking forces when ABS is activated, ensuring that the ABS controller can promptly track the peak tire-road adhesion and thus reduce the braking distance.

Figure 3

Schematic of the proposed integrated control scheme

4 Hardware-in-Loop (HIL) Verifications

To thoroughly evaluate the effectiveness of the proposed control scheme, comprehensive Hardware-in-Loop (HIL) tests were conducted based on a dedicated HIL platform. Virtual real-time vehicle models were employed in combination with a real electronic control unit (ECU) to assess the performance and reliability of the developed control strategy. Figure 4 illustrates the configuration and testing principle of the HIL platform.

Figure 4

Block diagram of the HIL tests

Figure 5 depicts the HIL setup, which consists of two host computers, a real-time personal controller (RTPC), an OpenECU controller, a CANape, and a DC power source. LABCAR serves as a real-time vehicle simulator to provide a virtual environment for examing the efficacy of the control scheme. The resource configuration of LABCAR is performed using the LABCAR IP software on the host computer PC1, and the generated C code is downloaded to RTPC, operating with a cycle time of less than 0.1 ms. The ABS and ASS controllers in MATLAB/Simulink calculate the braking torque and the suspension force. The MATLAB/Simulink code is then compiled into executable codes and downloaded to the OpenECU controller for real-time implementation. The OpenECU controller is connected to LABCAR via the CAN bus, and CANape is utilized for signal collection and calibration.

Figure 5

The HIL setup

The vehicle specifications are listed in Table 2, and the magnetic ride control active suspension is used as the actuator [36].

Table 2 Specifications of the test vehicle

The ISO-B and dry asphalt roads are used in the HIL test. Prior to the execution of braking, the initial vehicle speed is set as 60 km/h. ABS deactivation occurs when the vehicle speed drops below 5 km/h. The LMI toolbox utilizes the solver-MINCX to determine the ASS actuation force. Through the LMI algorithm, the control gain matrix K for the active suspension controller is obtained, ensuring a guaranteed H_∞ performance index of γ=78 and ρ=0.17, which is given by

$$\begin{gathered} {\varvec{K}} = \hfill \\ \left[ {\begin{array}{*{20}r} \hfill {272} & \hfill {{182}} & \hfill {{1789}} & \hfill {2819} & \hfill {2649} & \hfill {{2639}...} \\ \hfill {{\kern 1pt} 4714} & \hfill {1154} & \hfill {{3}{\text{.96}} \times {10}^{{5}} } & \hfill {{2}{\text{.53}} \times {10}^{{5}} } & \hfill {{3}{\text{.49}} \times {10}^{{5}} } & \hfill {{2}{\text{.05}} \times {10}^{{ - 4}} ;} \\ \hfill {{318}} & \hfill {238} & \hfill {1174} & \hfill {{1745}} & \hfill {{1494}} & \hfill {1505...} \\ \hfill {{2454}} & \hfill {4822} & \hfill {{8}{\text{.23}} \times {10}^{{5}} } & \hfill {{3}{\text{.28}} \times {10}^{{5}} } & \hfill {{3}{\text{.78}} \times {10}^{{5}} } & \hfill {{2}{\text{.45}} \times {10}^{{ - 4}} } \\ \end{array} } \right]. \hfill \\ \end{gathered}$$

Figure 6 illustrates the active suspension actuation forces under two scenarios. In both cases, the actuation forces remain below the threshold of u_flm. Additionally, the control force applied under the PID and ASS scheme is lower compared to the integrated NNSMC and ASS scheme, indicating that the latter necessitates higher force to counteract the adverse impact of braking on vertical vehicle motion.

Figure 6

Actuation forces of the active suspension

The results of the HIL tests depicting the dynamic response of the sprung mass under road excitation are presented in Figure 7.

Figure 7

Comparison of vehicle acceleration response: (a) Acceleration of the sprung mass, (b) Pitch acceleration of the vehicle body, (c) Longitudinal acceleration of the sprung mass

Figure 7 shows that the integrated NNSMC and ASS control scheme results in reduced vertical and pitch accelerations compared to the stand-alone PID and the PID with ASS controller. This indicates that the proposed control scheme surpasses traditional PID control in terms of ride comfort and handling stability. Notably, the sprung acceleration under road excitation is considerably optimized, indicating improved vehicle ride comfort, while the braking performance benefits from smoother vertical vehicle body movement. Additionally, the integrated control scheme demonstrates reduction in both the braking distance and the braking time.

The dynamics of the wheel under different control strategies are illustrated in Figure 8.

Figure 8

Comparison of the dynamic wheel loads and wheel slip ratios: (a) Front wheel vertical load, (b) Front-wheel frictional force, (c) Rear wheel vertical load, (d) Rear-wheel frictional force

Figures 8(a), (c) demonstrate the effective suppression of vertical load fluctuations on the front and rear wheels through the implementation of the NNSMC and ASS controller. This can be attributed to the active regulation of the vehicle body's pitching motion and the elimination of load transfer achieved by the proposed ASS controller. In Figures 8(b), (d), the introduction of ASS, in comparison to the PID controller, leads to reduced fluctuation in the longitudinal frictional force. Moreover, the integrated NNSMC and ASS exhibit more effective control of the frictional force compared to the combination of PID and ASS controllers, resulting in further suppression of the fluctuation of the longitudinal frictional force.

Figure 9 presents a comprehensive comparison of the ABS control target, i. e., the slip ratio.

Figure 9

Comparison of the wheel slip ratios: (a) Front wheel slip ratio, (b) Rear wheel slip ratio

Figures 9(a), (b) demonstrate the effectiveness of the integrated ABS and ASS control scheme through the HIL test on a single road surface. The smooth vertical load facilitates faster and smoother tracking of the optimal slip ratio by the ABS controller. The slip ratio remains stable around the optimal value. Furthermore, the proposed NNSMC control strategy for ABS outperforms the PID control in terms of optimal slip ratio tracking accuracy.

To further assess the effectiveness of the proposed integrated control scheme, statistical comparison results are presented in Table 3. The parameters evaluated include braking time (t_s) and braking distance (d_s). In order to accurately measure the error in slip ratio, the signal-to-noise ratio (SNR) is introduced as an evaluation metric. The slip ratio of the front wheel is considered as the target, which is given by

$${\text{SNR}} = \frac{{\sqrt {\frac{1}{t}\int_{0}^{t} {\left( {\lambda_{fl} (t) - \lambda ^*} \right)}^{2} {\text{d}}t} }}{\lambda ^*}.$$

(22)

Table 3 Statistical results between the comparison and proposed controller

Additionally, the root mean square error (RMS) is utilized to quantify the optimization effect. For a sequence that contains n elements, the RMS value-x_rms is given by

$$x_{rms} = \frac{\left\| x \right\|}{{\sqrt n }} = \sqrt {\frac{1}{n}\sum\limits_{j = 1}^{n} {x_{j}^{2} } } ,\quad j = 1, \, \ldots , \, n.$$

(23)

In Table 3, it can be seen that the proposed integrated controller leads to an improvement of 16.4% in $\ddot{z}_{s}$. Additionally, the braking duration and distance are reduced by 15.3% and 14.9%, respectively. The evaluation indexes for both vertical and longitudinal performance surpass those of the integrated PID and ASS controller. In summary, the proposed integrated ASS and ABS controllers enhance vehicle braking performance while ensuring ride comfort.

5 Conclusions

This paper presents an integrated active suspension system (ASS) and anti-lock braking system (ABS) control scheme for four-wheel-independent-drive electric vehicles. To capture the longitudinal and vertical coupling effect of the vehicle, a comprehensive longitudinal-vertical coupled vehicle dynamics model is established by integrating a road input model. Furthermore, an H_∞ controller is designed for ASS, while a neural network sliding mode control (NNSMC) is developed for ABS. The effectiveness of the proposed scheme is thoroughly examined using a Hardware-in-Loop (HIL) platform. The results of the HIL tests demonstrate significant improvements achieved by the proposed control scheme. Specifically, the acceleration of the sprung mass and the pitch acceleration of the vehicle body are effectively reduced, leading to enhanced vehicle ride comfort. Moreover, the braking time and distance are reduced by 15.3% and 14.9%, respectively. These findings validate the efficacy of the proposed integrated control scheme in optimizing both braking performance and vehicle ride comfort.