An Integrated Control Framework for Torque Vectoring and Active Suspension System

Abstract

Four-wheel independently driven electric vehicles (FWID-EV) endow a flexible and scalable control framework to improve vehicle performance. This paper integrates the torque vectoring and active suspension system (ASS) to enhance the vehicle’s longitudinal and vertical motion control performance. While the nonlinear characteristic of the tire model leads to a relatively heavier computational burden. To facilitate the controller design and ease the load, a half-vehicle dynamics system is built and simplified to the linear-time-varying (LTV) model. Then a model predictive controller is developed by formulating the objective function by comprehensively considering the safety, energy-saving and comfort requirements. The in-wheel motor efficiency and the power loss of tire slip are treated as optimization indices in this work to reduce energy consumption. Finally, the effectiveness of the proposed controller is verified through the rapid-control-prototype (RCP) test. The results demonstrate the enhancement of the energy-saving as well as comfort on the basis of vehicle stability.

1 Introduction

Electric vehicles are regarded as a promising solution to deal with the increasing emissions and high energy-efficiency requirements [1,2,3]. Nowadays, four wheel independently driven electric vehicles (FWID-EVs) with a modular powertrain layout have attracted lots of attention from academic and industrial researchers [4, 5]. Thanks to the simple mechanical structure and short transmission chain, FWID-EVs can quickly respond to the execution command through the in-wheel motor torque output. Moreover, four independently driven motors provide a considerable enhancement for vehicle stability with a flexible control mode [6,7,8]. However, the frequent acceleration/deceleration behaviors in urban conditions would affect the vehicle’s vertical motion, especially when the torque outputs increase/decrease rapidly. The driver’s comfort cannot be guaranteed. To this end, this work integrates the torque vectoring and active suspension system of FWID-EVs focusing on the straight-ahead driving condition. It aims to improve the overall performance on the premise of ensuring vehicle stability.

The four in-wheel motors could generate different torque outputs according to designed torque vectoring strategies [9]. It brings a novel control scheme for the direct yaw moment system rather than only depending on the braking force [10]. By taking the energy efficiency into consideration, the energy consumption can also be reduced [11, 12]. Comprehensively considering the motor efficiency and tire slip energy consumption of FWID-EVs, Zhang et al. [13] propose a torque distribution method based on discrete adaptive sliding control, which effectively reduces the energy loss of vehicles. For the active suspension system, the vehicle heave, pitch, and roll motions can be adjusted with differential active forces [14,15,16], which is related to ride comfort, road holding stability, and roll prevention.

It is widely accepted that the integration of these subsystems is better to realize holistic control [17, 18]. Many studies focus on longitudinal and lateral integrated control by combining active front wheel steering (AFS) and direct yaw moment control (DYC). An adaptive controller based on a Lyapunov method is employed to coordinate AFS and DYC in Ref. [19]. The adaption law is designed to deal with the uncertain cornering stiffness. A multi-agent system-based control framework is proposed in Ref. [20] to realize the integration of AFS and DYC, in which these subsystems are treated as agents and realize the cooperation by Pareto-optimality theory. Considering the time delay of the vehicle network, a novel H∞ controller combined with a delay-tolerant linear quadratic regulator is designed in Ref. [21] to improve vehicle stability based on AFS and DYC. Furthermore, to handle the tire nonlinear characteristic, some studies present the Takagi-Sugeno (TS) fuzzy and polytope methods to build the vehicle model, based on which the robust controller is developed. Jin et al. [22] propose a robust state-feedback controller to ensure the vehicle handling performance, in which the T-S fuzzy method is used for the nonlinear Brush tire modeling. Zhang et al. [23, 24] adopt the polytope to describe the uncertainties in the model. A robust gain-scheduled controller is designed to enhance the vehicle’s lateral stability. It is worth mentioning that an inappropriate Lyapunov function would lead to being conservative of robust controller when handling the tire’s nonlinear characteristic [16, 25].

In addition to the vehicle longitudinal and lateral motion control, some studies also integrate vertical motion. Zhao et al. [17] develop a hierarchical framework to coordinate AFS, ASS and DYC, thereby improving the overall performance. A trigger mechanism is established to decide the working regions of different subsystems based on the tire reserve forces. Hussein et al. [18] propose a high-order sliding model method to realize a global chassis control. The results show the effectiveness of enhancing the vehicle stability and ride comfort. Qin et al. [26] investigate the couplings between the dynamic vibration-absorbing structures and in-wheel motors. A particle optimization method is adopted to enhance the vertical dynamics performance. Nevertheless, the energy-saving is seldom considered in the integration of these subsystems. The researches in Refs. [1, 27, 28] demonstrate that energy consumption can be reduced through a reasonable torque vectoring strategy, as well as improving the vehicle stability.

Meanwhile, model predictive control (MPC) has been increasingly used in the vehicle dynamics control [29, 30]. The MPC does not adopt a constant global optimization target, but repeatedly optimizes online in a rolling method to obtain a global sub-optimal solution. It has strong adaptability to complex and changeable driving environments, and can meet the needs of real-time online optimization control under multi-objective compound constraints. Hence, a model predictive controller is proposed in this paper to integrate the torque vectoring and active suspension system by comprehensively considering safety, energy-saving and comfort. A non-linear model predictive controller is applied to improve the stability of distributed drive electric vehicles under critical driving scenarios [29]. The experiments on the snowy road validate the feasibility. The path-tracking problem is transformed into a standard MPC optimization problem in Ref. [30]. Multi-constraints are also considered in the solver.

The main contributions of this paper are as follows. First, a half-vehicle dynamics model is employed to describe the vehicle’s longitudinal and vertical motions, in which the nonlinear tire dynamics is described by an LTV model. It facilitates the controller design and reduces the computational load. Then, a model predictive controller is developed by optimizing the multi-objectives, including energy-saving, vehicle safety, and comfort. The RCP experiments are conducted to validate the effectiveness of the proposed method.

The remainder of this paper is organized as follows. A half-vehicle system combined with a tire dynamics model is built in Section 2. In Section 3, the MPC controller and the test bench are discussed. The optimization objective functions are given in detail. Section 4 shows the test results of the proposed controller. Finally, Section 5 presents the conclusion.

2 Vehicle System Modeling

This work focuses on vehicle control for the straight-ahead driving condition. Hence, the vehicle's longitudinal motion and vertical motion are investigated in this section. A half-vehicle model is established. Moreover, the nonlinear characteristic of the tire model is fully considered and simplified to facilitate the controller design.

2.1 Wheel Dynamics

Considering the effect of vehicle vertical motion on the straight line driving, the vehicle longitudinal dynamics model can be expressed as:

$$m\dot{V}_{x} + m_{s} V_{z} \dot{\theta } = \sum\nolimits_{i = f,r} {F_{xi} } ,$$

(1)

where $m$ and $m_{s}$ represent the total mass and sprung mass of the vehicle, respectively. $V_{x}$ and $V_{z}$ are the vehicle longitudinal and vertical velocities. $\theta$ is the vehicle pitch angle. $F_{xf}$ and $F_{xr}$ denote the longitudinal forces of the front and rear tires, respectively, which are generated by the torque inputs of the in-wheel motors. Furthermore, the wheel dynamics as shown in Figure 1 is given by:

$$J_{w} \dot{w}_{i} = T_{wi} - r_{e} F_{xi} ,$$

(2)

where $J_{w}$ and $r_{e}$ are the inertia moment and rolling radius of wheel, respectively. $T_{wi}$ is the torque input of the in-wheel motor $i$. $w_{i}$ is the rotational speed of the wheel $i$, $i = f,r$; $f$ represents the front tire, and $r$ represents the rear tire. When the vehicle runs in the linear region, the tire longitudinal force can be represented by:

$$F_{xi} = K_{xi} \lambda_{i} ,$$

(3)

where $K_{xi}$ is the tire longitudinal stiffness, which is related with the vertical load, tire type and tire pressure, etc. $\lambda_{i}$ is the tire longitudinal slip ratio. It can be further expressed as:

$$\lambda_{i} = \frac{{w_{i} r_{e} - V_{x} }}{{V_{x} }}.$$

(4)

Figure 1

Wheel dynamics model

For Eq. (3), when the vehicle is in the nonlinear region, it would be invalid if using the longitudinal tire stiffness $K_{xi}$ obtained by the linear state. A conventional method is to adopt the tire fitting formulas and calculate the tire force in real time [31]. This can guarantee the model’s accuracy. However, the tire force is directly employed in the system. Due to the strong nonlinearity, it would cause a large computational burden for the optimization. Hence, a modified tire longitudinal stiffness $K^{\prime}_{xi}$ is defined in this work to ensure accuracy while reducing the computational burden through the linearization method. The tire longitudinal force $F_{xi}$ is calculated by the Magic formula. Then the modified tire longitudinal stiffness $K^{\prime}_{xi}$ is calculated by Eq. (3) in real-time and presented by a time-varying parameter. The $K^{\prime}_{xi}$ would change according to different driving conditions. By introducing the modified parameter $K^{\prime}_{xi}$ into the Eq. (2), the wheel dynamics can be rewritten as:

$$\dot{\lambda }_{i} = - \frac{{r_{e}^{2} }}{{J_{w} V_{x} }}K^{\prime}_{xi} \lambda_{i} + \frac{{r_{e} }}{{J_{w} V_{x} }}T_{wi} .$$

(5)

The Magic formula [32] fitting the tire longitudinal force is given by:

$$F_{xi} = \mu F_{zi} \sin \left\{ {C_{x} \arctan \left[ \begin{aligned}& B_{x} \lambda_{i} - \hfill \\ &D_{x} \left( {B_{x} \lambda_{i} - B_{x} \arctan \left( {\lambda_{i} } \right)} \right) \hfill \\ \end{aligned} \right]} \right\},$$

(6)

where $\mu$ is the road-tire friction coefficient. $B_{x}$, $C_{x}$ and $D_{x}$ are obtained through the parameter fitting method. $F_{zi}$ is the vertical load of the $i$-wheel, which can be calculated by:

$$F_{zf} = \frac{mg}{2}\left[ {\frac{{l_{r} }}{{l_{f} + l_{r} }} - \frac{h}{{\left( {l_{f} + l_{r} } \right)g}}a_{x} } \right],$$

(7)

$$F_{zr} = \frac{mg}{2}\left[ {\frac{{l_{f} }}{{l_{f} + l_{r} }} + \frac{h}{{\left( {l_{f} + l_{r} } \right)g}}a_{x} } \right],$$

(8)

where $l_{f}$ and $l_{r}$ are the distances from the vehicle centre of gravity (CoG) to the front and rear axles. $h$ and $a_{x}$ are the height of CoG and vehicle longitudinal acceleration, respectively. $g$ is the gravitational acceleration. To show the accuracy of the Magic formula, the fitting tire longitudinal forces under different vertical loads (i.e., 2.1 kN, 6.1 kN and 10.1 kN) are compared with the real test data. The detailed results are shown in Figure 2. During the straight driving condition, the tire slip angle $\alpha$ is 0.

Figure 2

Tire longitudinal force fitting results

From the test results, it can be seen that the fitting tire longitudinal force is closely approaching the test data. Consequently, the tire longitudinal force $F_{xi}$ is obtained through the Magic formula in this work. Here, the tire test data are collected from a high-fidelity vehicle model (CarSim). Then the modified tire longitudinal stiffness $K^{\prime}_{xi}$ is obtained in real-time based on Eq. (3). Eq. (1) can be expressed as:

$$m\dot{V}_{x} + m_{s} V_{z} \dot{\theta } = K^{\prime}_{xf} \lambda_{f} + K^{\prime}_{xr} \lambda_{r} .$$

(9)

2.2 Vehicle Vertical Dynamics

The vehicle would have obvious heave and pitch motions when existing frequent acceleration/deceleration behaviors. To describe the vehicle vertical motion, a 4-degree-of-freedom (4-DoF) half-vehicle active suspension system as shown in Figure 3 is used to establish the vehicle vertical dynamics model. $F_{f}$ and $F_{r}$ are active suspension forces, which are generated by the actuators. The roll motion is ignored due to the straight line driving. With the assumption of a small pitch angle. The heave motion of the sprung mass and unsprung mass can be represented by:

$$m_{s} \left( {\ddot{z}_{o} - V_{x} \dot{\theta }} \right) = \left[ \begin{aligned} &- k_{sf} \left( {z_{sf} - z_{uf} } \right) - b_{sf} \left( {\dot{z}_{sf} - \dot{z}_{uf} } \right){ + }F_{f} \hfill &\\ - k_{sr} \left( {z_{sr} - z_{ur} } \right) - b_{sr} \left( {\dot{z}_{sr} - \dot{z}_{ur} } \right) + F_{r} \hfill \\ \end{aligned} \right],$$

(10)

$$m_{uf} \ddot{z}_{uf} = \left[ \begin{aligned}& k_{sf} \left( {z_{sf} - z_{uf} } \right) + b_{sf} \left( {\dot{z}_{sf} - \dot{z}_{uf} } \right) \hfill \\ &- F_{f} - k_{tf} z_{uf} \hfill \\ \end{aligned} \right],$$

(11)

$$m_{ur} \ddot{z}_{ur} = \left[ \begin{aligned}& k_{sr} \left( {z_{sr} - z_{ur} } \right) + b_{sf} \left( {\dot{z}_{sr} - \dot{z}_{ur} } \right) \hfill \\ &- F_{r} - k_{tr} z_{ur} \hfill \\ \end{aligned} \right].$$

(12)

Figure 3

4-DoF vehicle vertical dynamics model

It should be noted that the road excitation is not considered in this study. $z_{o}$ is the vertical displacement of the vehicle CoG. $k_{sf}$ and $k_{sr}$ are the equivalent stiffness of the front and rear suspensions, respectively. $b_{sf}$ and $b_{sr}$ are the equivalent damping of the front and rear suspensions. $k_{tf}$ and $k_{tr}$ are the equivalent stiffness of the front and rear wheels, respectively. $z_{sf}$ and $z_{sr}$ are the vertical displacements of the front and rear sprung mass. $z_{uf}$ and $z_{ur}$ are the vertical displacements of the front and rear unsprung mass, respectively. $F_{f}$ and $F_{r}$ are the active force inputs of the front and rear suspensions, respectively. $m_{uf}$ and $m_{ur}$ are the unsprung mass of the front and rear wheels, respectively. Furthermore, the vertical displacements of the front and rear sprung mass can be expressed as:

$$z_{sf} = z_{o} - l_{f} \theta ,$$

(13)

$$z_{sr} = z_{o} + l_{r} \theta .$$

(14)

The pitch motion of the suspension system can be described by

$$I_{y} \ddot{\theta } = - l_{f} \Re_{f} + l_{r} \Re_{r} ,$$

(15)

where $I_{y}$ is the vehicle inertia moment around the y-axis.

$$\Re_{f} { = } - k_{sf} \left( {z_{sf} - z_{uf} } \right) - b_{sf} \left( {\dot{z}_{sf} - \dot{z}_{uf} } \right){ + }F_{f} ,$$

(16)

$$\Re_{r} { = } - k_{sr} \left( {z_{sr} - z_{ur} } \right) - b_{sr} \left( {\dot{z}_{sr} - \dot{z}_{ur} } \right){ + }F_{r} .$$

(17)

Combining Eqs. (1)–(17), the vehicle system model for the straight-ahead driving condition is given by

$${\dot{\varvec{x}}}\left( t \right) = {\varvec{A}}{\varvec{x}}\left( t \right) + {\varvec{B}}{\varvec{u}}(t),$$

(18)

where ${\varvec{x}} = \left[ {V_{x} ,\lambda_{f} ,\lambda_{r} ,\theta ,\dot{\theta },z_{o} ,\dot{z}_{o} ,z_{uf} ,z_{ur} ,\dot{z}_{uf} ,\dot{z}_{ur} } \right]^{{\text{T}}} ,$${\varvec{u}} = \left[ {T_{f} ,T_{r} ,F_{f} ,F_{r} } \right]^{{^{{\text{T}}} }},$

$${\varvec{A}} = \left[ \begin{array} {lllllllllll} \frac{{K^{\prime}_{xf} }}{m} & 0 & \frac{{K^{\prime}_{xr} }}{m} & 0 & - V_{z} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & a_{{{22}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & a_{{{33}}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & a_{54} & a_{55} & a_{{{5}6}} & a_{{{5}7}} & a_{{{58}}} & a_{{{59}}} & a_{{{5},{10}}} & a_{{5,11}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & a_{{7{4}}} & a_{75} & a_{76} & a_{77} & a_{{7{8}}} & a_{{7{9}}} & a_{{7,{10}}} & a_{{7,{11}}}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 &a_{{{10},{4}}} & a_{{{10},5}} & a_{{{10},6}} & a_{{{10},{7}}} & a_{{{10},{8}}} & 0 & a_{{{10},{10}}} & 0 \\ 0 & 0 & 0 & a_{{{11},{4}}} & a_{{{11},5}} & a_{{{11},6}} & a_{{{11},{7}}} & 0 & a_{{{11},{9}}} & 0 & a_{{{11},{11}}} \end{array} \right]_{11 \times 11},$$

$${\varvec{B}} = \left[\begin{array} {lllllllllll} 0 & \frac{{r_{e} }}{{J_{w} * V_{x} }} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\ 0 & 0 & \frac{{r_{e} }}{{J_{w} * V_{x} }} & 0 & 0 & 0 & 0 & 0 & 0 & 0& 0 \\ 0 & 0 &0 & 0 & - \frac{{l_{f} }}{{I_{y} }} & 0 &\frac{1}{{m_{s} }} & 0 & 0 &- \frac{1}{{m_{uf} }} & 0 \\ 0 & 0 & 0 &0 &- \frac{{l_{r} }}{{I_{y} }} & 0 &\frac{1}{{m_{s} }} & 0 & 0 &0 & - \frac{1}{{m_{ur} }} \end{array} \right]^{{\text{T}}}_{4 \times 11},$$

$a_{{{22}}} = - K^{\prime}_{xf} * r_{e} * r_{e} /\left( {J_{w} * V_{x} } \right),a_{{{33}}} { = } - K^{\prime}_{xr} * r_{e} * r_{e} /\left( {J_{w} * V_{x} } \right),$$a_{54} = \left( { - k_{sf} *l_{f} *l_{f} - k_{sr} *l_{r} *l_{r} } \right)/I_{y}$, $a_{56} = \left( {k_{sf} *l_{f} - k_{sr} *l_{r} } \right)/I_{y}$, $a_{55} = \left( { - b_{sf} *l_{f} *l_{f} - b_{sr} *l_{r} *l_{r} } \right)/I_{y}$$a_{57} = \left( {b_{sf} *l_{f} - b_{sr} *l_{r} } \right)/I_{y}$, $a_{{5{8}}} = - k_{sf} *l_{f} /I_{y}$, $a_{{5{9}}} = k_{sr} *l_{r} /I_{y}$, $a_{5,10} = - b_{sf} *l_{f} /I_{y}$, $a_{{5,{11}}} = b_{sr} *l_{r} /I_{y}$, $a_{74} = \left( {k_{sf} *l_{f} - k_{sr} *l_{r} } \right)/m_{s}$, $a_{{7{5}}} = \left( {b_{sf} *l_{f} - b_{sr} *l_{r} + m_{s} * V_{x} } \right)/m_{s}$,$a_{{{7}6}} = \left( { - k_{sf} - k_{sr} } \right)/m_{s}$, $a_{{{77}}} = \left( { - b_{sf} - b_{sr} } \right)/m_{s}$, $a_{{{78}}} = k_{sf} /m_{s}$,$a_{{{79}}} = k_{sr} /m_{s}$, $a_{{{7},{10}}} = b_{sf} /m_{s}$,$a_{{7,11}} = b_{sr} /m_{s}$, $a_{{{10},{4}}} = - k_{sf} * l_{f} /m_{uf}$, $a_{{10,5}} = - b_{sf} * l_{f} /m_{uf}$, $a_{{10,6}} = k_{sf} /m_{uf}$, $a_{{10,7}} = b_{sf} /m_{uf}$,$a_{{10,8}} = \left( { - k_{tf} - k_{sf} } \right)/m_{uf}$, $a_{{10,10}} = - b_{sf} /m_{uf}$, $a_{{11,4}} = k_{sr} * l_{r} /m_{ur}$, $a_{{11,5}} = b_{sr} * l_{r} /m_{ur}$, $a_{{11,6}} = k_{sr} /m_{ur}$, $a_{{11,7}} = b_{sr} /m_{ur}$, $a_{{11,9}} = \left( { - k_{tr} - k_{sr} } \right)/m_{ur}$, $a_{{11,11}} = - b_{sr} /m_{ur}$.

It could be found that the model (18) is a continuous-time nonlinear system. The parameter matrices ${\varvec{A}}$ and ${\varvec{B}}$ exist time-varying state variables, including longitudinal velocity $V_{x}$ and modified tire longitudinal stiffness $K^{\prime}_{xi}$. Here, to facilitate the model predictive control (MPC) design and reduce the computational burden. A linear-time-varying (LTV) discrete model is established. The state variables in the parameter matrices are treated as a constant during the sampling time, which would update at different sampling time. The sampling time is given by $\Gamma$. Then based on the Eular method, the continuous system (18) at time step $k$ can be discretized as:

$${\varvec{x}}\left( {k + 1} \right) = {\varvec{A}}_{c} {\varvec{x}}\left( k \right) + {\varvec{B}}_{c} {\varvec{u}}(k),$$

(19)

where ${\mathbf{x}}\left( k \right)$ and ${\mathbf{u}}\left( k \right)$ represent system states and inputs at time step $k$, respectively.

$$\left\{ \begin{aligned} &{\varvec{A}}_{c} = e^{{{\varvec{A}}\Gamma }} , \, \hfill \\ &{\varvec{B}}_{c} = \int\limits_{k\Gamma }^{(k + 1)\Gamma } {e^{{{\varvec{A}}\left[ {\left( {k + 1} \right)\Gamma - t} \right]}} } {\varvec{B}}{\text{d}}t. \hfill \\ \end{aligned} \right.$$

(20)

The handling of model linearization and discretization simplifies the system and facilitates the controller design [33]. Meanwhile, the tire characteristic can also be guaranteed.

3 Optimal Control Design

The schematic integrated control framework for torque vectoring and active suspension system is shown in Figure 4. It includes the wheel dynamics model, nonlinear tire model, 4-DoF vehicle vertical dynamics model, reference model and MPC optimal controller. The MPC controller is proposed to realize the integrated control of vehicle longitudinal and vertical motions. A combination of multi-performance indices, including energy saving, vehicle safety, and comfort, are considered during the controller design. The relaxation factors are introduced to dynamically modulate the weight coefficients of different control objectives. Furthermore, to better verify the controller control effect, the rapid-control-prototype test bench is also described.

Figure 4

Schematic integrated control framework for torque vectoring and active suspension system

3.1 Multi-objectives Function

The control objective of energy saving can be represented as follows:

$$J_{1} = J_{1}^{1} { + }J_{1}^{2} { + }J_{1}^{3} ,$$

(21)

where

$$J_{{1}}^{{1}} { = }\sum\limits_{t = 1}^{{N_{p} - 1}} {\left\| {T_{f} (k + t\left| k \right.) - T_{f,ref} } \right\|_{{Q_{1} }}^{2} } ,$$

(22)

$$J_{{1}}^{{2}} { = }\sum\limits_{t = 1}^{{N_{p} - 1}} {\left\| {F_{x,i} \lambda_{i} V_{x} } \right\|_{{Q_{2} }}^{2} } ,$$

(23)

$$J_{{1}}^{{3}} { = }\sum\limits_{t = 1}^{{N_{p} - 1}} {\left( {\left\| {{\varvec{T}}_{i} \left( {k + t\left| k \right.} \right)} \right\|_{{R_{1} }}^{2} + \left\| {{\varvec{F}}_{i} \left( {k + t\left| k \right.} \right)} \right\|_{{R_{2} }}^{2} } \right)} ,$$

(24)

where ${\varvec{T}}_{i} = \left[ {T_{f} \,\, T_{r} } \right]^{{\text{T}}},$ ${\varvec{F}}_{i} = \left[ {F_{f} \,\, F_{r} } \right]^{{\text{T}}},$ $Q_{1} \in \Re^{1 \times 1}$, $Q_{2} \in \Re^{2 \times 2}$, $R_{1} \in \Re^{2 \times 2}$, and $R_{2} \in \Re^{2 \times 2}$ are the positive diagonal matrices, which represent the weight parameters. $N_{p}$ is the predictive horizon. In this work, the predictive horizon is equal to the control horizon. $T_{f,ref}$ is the reference torque inputs of the front in-wheel motor. The optimization objective (21) is a combination of different energy consumption. The cost function (22) considers energy saving through improving the motor efficiency, based on optimizing the torque inputs. To reduce energy consumption, the selection principle of the reference torque inputs is to provide a high-efficiency zone for in-wheel motors. The efficiency map for the in-wheel motor is given in Figure 5, in which the test data is collected from a Protean PD-18 in-wheel motor. A wheel speed-motor efficiency mapping table is designed to obtain the reference torque inputs $T_{f,ref}$ by matching the wheel speed. Under some specific driving conditions, the torque inputs of the in-wheel motors may deviate from the high-efficiency zone. In this case, the energy consumption would be reduced if matching the high-efficiency zone for some in-wheel motors first. Simultaneously when rear tires approach the saturation state, a small lateral force could cause the vehicle sideslip. Considering the longitudinal stability control, the in-wheel motors of the front axle would satisfy the high-efficiency state in priority. The cost function (23) is the power loss of the longitudinal slip for the front and rear tires. It can be further represented by:

$$J_{{1}}^{{2}} { = }\sum\limits_{t = 1}^{{N_{p} - 1}} {\left\| {K^{\prime}_{xi} \lambda_{i}^{2} \left( {k + t\left| k \right.} \right)V_{x} \left( {k + t\left| k \right.} \right)} \right\|_{{Q_{2} }}^{2} } .$$

(25)

Figure 5

Efficiency map of in-wheel motors

For Eq. (24), it denotes the energy consumption caused by the actuators. The control objective of the vehicle safety can be expressed by

$$J_{2} = J_{2}^{1} { + }J_{2}^{2} + J_{2}^{3} ,$$

(26)

where

$$J_{2}^{1} { = }\sum\limits_{t = 1}^{{N_{p} - 1}} {\left\| {{\varvec{\lambda}}_{i} \left( {k + t\left| k \right.} \right)} \right\|_{{{\varvec{W}}_{1} }}^{2} } ,$$

(27)

$$J_{2}^{2} { = }\sum\limits_{t = 1}^{{N_{p} - 1}} {\left\| {{\varvec{z}}_{si} \left( {k + t\left| k \right.} \right) - {\varvec{z}}_{ui} \left( {k + t\left| k \right.} \right)} \right\|_{{{\varvec{W}}_{2} }}^{2} } ,$$

(28)

$$J_{2}^{{3}} { = }\sum\limits_{t = 1}^{{N_{p} - 1}} {\left\| {V_{x} \left( {t + k\left| k \right.} \right) - V_{x,ref} } \right\|_{{{\varvec{W}}_{3} }}^{2} } ,$$

(29)

where ${\varvec{\lambda}}_{i} = \left[ {\lambda_{f} \, \lambda_{r} } \right]^{{\text{T}}}$, ${\varvec{z}}_{si} = \left[ {z_{sf} \, z_{sr} } \right]^{{\text{T}}}$, ${\varvec{z}}_{ui} = \left[ {z_{uf} \, z_{ur} } \right]^{{\text{T}}}$. ${\varvec{W}}_{1} \in \Re^{2 \times 2}$, ${\varvec{W}}_{2} \in \Re^{2 \times 2}$ and ${\varvec{W}}_{{3}} \in \Re^{{{1} \times {1}}}$ are the positive diagonal matrices. The cost function (27) aims to minimize the tire slip ratio and improve the vehicle safety. In addition, as designed in the cost function (22), the front in-wheel motors are endowed with the high priority to match the efficiency zone, thereby guaranteeing the vehicle longitudinal stability. Furthermore, to enhance the vehicle vertical safety, the cost function (28) is presented by a soft constraint to depress the suspension deflections, which should also satisfy the hard constraint [34] shown as follows:

$$\left| {{\varvec{z}}_{si} - {\varvec{z}}_{ui} } \right| \le {\varvec{\rho}}z_{mzx} ,$$

(30)

where ${\varvec{\rho}} = \left[ {1\,\,{1}} \right]^{{\text{T}}}$, $z_{\max }$ is the permitted maximum value of the suspension deflection. As for the cost function (29), it represents the speed control requirement of drivers. When the vehicle speed is lower than the drivers’ intention, drivers would have an acceleration behavior and increase the torque inputs. Conversely, drives have a deceleration behavior. Therefore, the weight parameter $W_{{3}}$ has a bigger value to follow the reference speed and satisfy the drivers’ demand. The control objective of the comfort is expressed as follows:

$$J_{{3}} = J_{3}^{1} { + }J_{3}^{2} ,$$

(31)

where

$$J_{{3}}^{1} { = }\sum\limits_{t = 1}^{{N_{p} - 1}} {\left\| {\ddot{z}_{o} \left( {t + k\left| k \right.} \right)} \right\|_{{P_{1} }}^{2} } { = }\sum\limits_{t = 1}^{{N_{p} - 1}} {\left\| {{\varvec{C}}x\left( {t + k\left| k \right.} \right)} \right\|_{{P_{1} }}^{2} } ,$$

(32)

where ${\varvec{C}} = \left[ {{0 \quad 0 \quad 0 \quad }a_{{7{4}}} \quad a_{75} \quad a_{76} \quad a_{77} \quad a_{{7{8}}} \quad a_{{7{9}}} \quad a_{{7,{10}}} \quad a_{{7,{11}}} \, } \right]$,

$$J_{{3}}^{2} { = }\sum\limits_{t = 1}^{{N_{p} - 1}} {\left( {\left\| {\theta \left( {t + k\left| k \right.} \right)} \right\|_{{P_{2} }}^{2} + \left\| {z_{o} \left( {t + k\left| k \right.} \right)} \right\|_{{P_{3} }}^{2} } \right)} .$$

(33)

Here, the vertical displacement and acceleration of vehicle CoG, and pitch angle are employed to quantify and describe the ride comfort. The cost functions (32) and (33) aim to suppress the vehicle heave motion and vertical motion, thereby improving the drivers’ comfort. Combining Eqs. (21)–(33), the global optimization objective function is expressed as follows:

$$J = \vartheta_{{1}} J_{1} { + }\vartheta_{{2}} \left( {J_{2} + J_{3} } \right),$$

(34)

where $\vartheta_{1}$ and $\vartheta_{2}$ are relaxation factors to balance the performance indices between energy-saving, ride comfort and safety control. The vehicle safety should be put in the first place during some extreme conditions. In this work, the slip ratio $\lambda_{i}$ is selected to evaluate the vehicle longitudinal stability, which also satisfies Eq. (35):

$$F_{xi} = K^{\prime}_{xi} {{\varvec{\lambda}}}_{i} \le \mu F_{zi} .$$

(35)

To guarantee the vehicle stability when the available tire force is small, the relaxation factors is calculated by:

$$\left\{ \begin{gathered} \vartheta_{2} = \max \left| {\frac{{K_{xi} {{\varvec{\lambda}}}_{i} }}{{\mu F_{zi} }}} \right|{, } \hfill \\ \vartheta_{1} = 1 - \vartheta_{2} . \hfill \\ \end{gathered} \right.$$

(36)

3.2 Rapid-Control-Prototype (RCP) Test

The optimal problem is represented by:

$$\mathop {\min }\limits_{u} J.$$

(37)

Subject to $u_{\min } \le u \le u_{\max }$ and hard constraint (30).

The rapid-control-prototype (RCP) test bench as shown in Figure 6 is built to validate the effectiveness of the proposed control method. It includes a real-time simulation system based on NI-PXI, a high-speed solution system based on dSPACE, and a high-fidelity vehicle system based on CarSim. In the RCP system established in this paper, a high-fidelity vehicle dynamic modeling business software (CarSim) is embedded into a real-time simulator (NI-PXI system). The optimal control solver is downloaded to the dSPACE-1401(900 MHz, 16 MByte) and running in real time. Based on the vehicle state feedback from the CarSim, the dSPACE can calculate the optimal inputs, which would be sent to the NI-PXI system and executed by the vehicle. The CAN bus is adopted to realize the data transmission. To avoid potential system disturbances, the first sequence of the optimal inputs is employed in the system. In addition, the updating states in the LTV model (18) would also transmit to the controller at each sampling time.

Figure 6

RCP test bench

4 Test Results

This section validates the effectiveness of the proposed controller through the RCP test. The Economic Commission for Europe (ECE) is chosen as the test condition. To show the control performance visually, a part of the velocity profile for the ECE elementary urban cycle is adopted to test the stability and energy-saving with the proposed torque vectoring strategy, while the ECE extra-urban driving cycle with frequent acceleration behavior is to test the ride comfort with the proposed active suspension system. The vehicle parameters used in this work are given in Table 1. The RCP has been conducted through the GRAMPC optimization solver [35], which is adapted to the nonlinear MPC and can be employed in the dSAPCE through code generation technology. In addition, the LQR controller is set as a comparison test in this work. The road-tire friction coefficient is set by 0.5. The control horizon is 3 and the prediction horizon is 5.

Table 1 Vehicle parameters

4.1 ECE Elementary Urban Cycle

The simulation results under the ECE elementary urban cycle are shown in Figure 7. Figure 7(a) shows the torque inputs of the in-wheel motors. Figure 7(b) is the velocity tracking performance with the different strategies. It can be seen that the vehicle can track the reference velocities closely with the proposed control strategy, while there exists obvious overshoots and undershoots with the LQR controller. The maximum tracking errors with the two strategies are 0.05 m/s and 1 m/s, respectively. This is because the tire longitudinal stiffness is a constant value when calculating the feedback control law for the LQR. However, the accurate time-varying tire longitudinal stiffness is obtained with the proposed controller in real time and used to calculate the optimized torque inputs.

Figure 7

The simulation results under the ECE elementary urban cycle: (a) Torque inputs of in-wheel motors, (b) Vehicle velocity, (c) Motor efficiency with the proposed controller, (d) Motor efficiency with LQR controller, (e) Front tire slip ratio, (f) Rear tire slip ratio

The efficiency maps of the front in-wheel motor with different strategies are given in Figure 7(c) and (d). It is clear that with the proposed control strategy, the in-wheel motor can work in a relatively high-efficiency zone, which basically maintains a level of 85%–95%. The motor efficiency of the LQR controller keeps low in some regions, which is between 40% and 55%. This proves that the proposed torque vectoring strategy can improve energy efficiency. It could be noted that the motor efficiency reduces when the torque input of the front wheel reaches − 300 N·m (i.e., 25–32 s) with the proposed controller. This can be attributed to the velocity tracking. As shown in Figure 7(b), the reference velocity drops quickly during 25–32 s. Hence, the torque inputs are reduced to track the reference velocities, which causes a lower motor efficiency zone.

Figure 7(e) and (f) are the longitudinal slip ratio of the front and rear tires. The tire slip ratio has a smaller value with the proposed controller compared to the LQR. This means that the vehicle can run in a more stable state with less power loss of the longitudinal slip. Some fluctuations can be seen in the tire slip ratio. This is because the MPC controller is a real-time optimization solver. External disturbances, such as time delays in the data transmission, would have an effect on the optimization results, thereby affecting the system states. According to these comparative test results, it can be concluded that the proposed torque vectoring can effectively guarantee the velocity tracking performance while ensuring vehicle stability and energy efficiency.

4.2 ECE Extra-Urban Driving Cycle

The ECE extra-urban driving cycle with frequent acceleration behavior is employed to value ride comfort with different strategies. Figure 8 shows the test results. As shown in Figure 8(a), it can be found that the vehicle velocity is close to the reference value with the proposed controller. Figure 8(b) represents the active force inputs of the front and rear suspensions, respectively. The vertical displacements of vehicle CoG with different controllers are given in Figure 8(c). The maximum vertical displacements are 0.028 and 0.005 m, respectively. This demonstrates that the vehicle has a smaller displacement with the proposed strategy, thereby improving the driver’s comfort. Meanwhile, it can be seen that when a vehicle has a positive displacement, the active suspension generates a negative force to reduce the vertical deformation.

Figure 8

The simulation results under the ECE extra-urban driving cycle with frequent acceleration behavior: (a) Vehicle velocity, (b) Active suspension force inputs, (c) Vertical displacements, (d) Vehicle pitch angle, (e) Vehicle vertical acceleration

Figure 8(d) illustrates the vehicle pitch angle. The maximum pitch angle with the proposed controller can be reduced by 78% compared to the LQR controller. The differential active suspension forces as shown in Figure 8(b) significantly depress the pitch angle. It can relieve the driver’s discomfort when suffering frequent acceleration/deceleration behavior. The vertical acceleration of vehicle CoG is shown in Figure 8(e). It is clear that the vehicle has a better performance to improve the vehicle’s vertical motion. These prove that the proposed control strategy is effective for the vehicle vertical dynamics control.

5 Conclusions

• (1) This paper applies the LTV-MPC controller to integrate the vehicle torque vectoring and active suspension system for the straight-ahead driving condition. The vehicle dynamics model considers the nonlinear characteristic of the tire. Then the fitting magic formula validated by the experimental data is used to obtain the tire longitudinal stiffness in real-time, based on which updating the system model parameters. To satisfy the tracking performance of the system states while guaranteeing vehicle safety, energy-saving and comfort, a combination of the optimization functions is formulated. Rapid control prototype tests are conducted to verify the proposed control strategy. The control effect of the LQR controller is compared with that of the proposed control strategy. The results show that compared with the LQR control strategy, the proposed control strategy can improve motor efficiency by nearly 73%.

• (2) Two types of ECE test results show that the proposed controller can effectively guarantee the velocity tracking performance while ensuring vehicle stability and improving energy efficiency and comfort. In the future, the vehicle steering behavior is also expected to be combined in the system.