Energy Efficiency Oriented Robust Model Predictive Stability Control for Autonomous Electric Vehicles

Abstract

The four-wheel independent steering and drive autonomous vehicle is a typical over-actuated system. The complexity of controlling it is increasing with the number of actuators. Since the model-based approach can solve the constrained multiple output problem, it is mostly utilized in the existing works. However, they usually investigate a single objective optimization, while employing simplified prediction models to relieve computational burdens. In this case, the robustness of the controller will inevitably suffer from model mismatch, which makes it hard to fulfill the various demands of autonomous driving. This work proposes a multi-objective control framework, which optimizes stability and energy efficiency simultaneously. Furthermore, robust model predictive control is introduced to address the model mismatch. Compared with the state-of-the-art, the effectiveness of the proposed approach has been validated by hardware-in-the-loop tests. Under the double lane change Maneuver, the longitudinal speed is maintained 1.7% higher. The vehicle stability is enhanced, while the motor energy loss and tire slip energy are reduced by 23.3% and 8.3%, respectively.

1 Introduction

With the development of automotive intelligence and electrification, the requirements for autonomous electric vehicles (AEVs) are expanded from safety to energy efficiency. Four-wheel independent steering and drive (4WIS-4WID) technologies are the current hotspots. Their prominent advantage is flexible control, which can further enhance stability when reasonably allocating torque vectoring and steering [1]. However, improper allocation may result in tire wear and additional energy consumption, thus reducing stability and energy efficiency.

In existing works, MPC has shown remarkable superiority in solving AEV stability control problems with multiple constraints [2]. However, it is always affected by discrepancies between actual vehicles and models, which always appear in practice and impact vehicle safety [3]. To address this issue, Tube-based RMPC is proposed to satisfy the original constraints under the worst conditions and shows outstanding robustness [4]. Besides stability, EVs also have a critical issue of energy efficiency. Reducing motor energy consumption and tire energy dissipation are the main approaches [5]. The UniTire model is widely applied to calculate tire slip power because of its accuracy and simplicity. To optimize energy efficiency while enhancing stability simultaneously, hierarchical control is a common approach [6]. The stability is ensured in the upper layer, and the lower layer controls actuators while reducing energy consumption. However, objectives are not optimized together in these works, causing suboptimal results.

Reviewing the state-of-the-art, the impacts of excessive tire slippage on stability and energy consumption are rarely considered while designing a path tracking strategy. In addition, most hierarchical approaches construct single-objective optimization problems in each layer, which may lead to underperformance. To reduce the computational burden, simplified models are typically used to predict vehicle states, inevitably leading to model mismatches that affect the robustness of the MPC controller. To address the problems above, we propose an RMPC-based multi-objective path tracking strategy in this work. The original contributions are: 1) An integrated path tracking framework is proposed, which realizes steering and torque vectoring while simultaneously considering stabilization, motor power loss, and tire slip power. 2) To ensure performance despite model deviations, an RMPC-based controller is designed to address the tire stiffness error of the model.

The rest of this paper is organized as follows. In the second section, the models built for multi-objective control are introduced. The RMPC-based integrated path tracking framework is proposed in the third section. Additionally, the performance of the proposed strategy is verified in the fourth section by hardware-in-the-loop (HIL) tests. Finally, conclusions are presented in the fifth section.

Fig. 1.

4DOF dynamics model.

$$\begin{aligned} m({\dot{V}_x - V_x\beta r}) + {F_a} & = \sum {F_x},\end{aligned}$$

(1a)

$$\begin{aligned} m{V_x}(\dot{\beta }+ r) + {m_s}{h_s}\ddot{\phi }& = \sum {F_y},\end{aligned}$$

(1b)

$$\begin{aligned} {I_z}\dot{r} - {I_{xz}}\ddot{\phi }& = \sum {M_z},\end{aligned}$$

(1c)

$$\begin{aligned} {I_x}\ddot{\phi }- {I_{xz}}\dot{r} & = \sum {M_x}. \end{aligned}$$

(1d)

2 System Modeling

The 4-degree-of-freedom (4DOF) vehicle dynamic model is shown in Fig. 1. Its lateral, longitudinal, yaw, and roll dynamics equations are given as (1), where $m_{s}$ represent the sprung mass, respectively, $h_{s}$ represents the height of the sprung mass, $I_{z}$, $I_{x}$ and $I_{xz}$ represent the moment and product of inertia, respectively. $\beta$ is the sideslip angle, r is the yaw rate, $\phi$ is the roll angle. $F_{a}$ donates the air resistance, $\Sigma M_{z}$, and $\Sigma M_{x}$ donate the combined moments, respectively.

The tire model utilized in this work is linear. Its rotational dynamics equation is derived as:

$$\begin{aligned} {J_w}{\dot{\omega }_i} = {T_{m,i}} - {T_{r,i}} - {T_{b,i}} - {R_w}{F_{x,i}}, \end{aligned}$$

(2)

where $J_{w}$ and $R_{w}$ donate the wheel rotational inertia and effective radius, respectively. $\omega _{i}$ represents the rotational speed. $T_{m,i}$ and $T_{b,i}$ are the driving and braking torques, $T_{r,i}$ is the rolling resistance torque, and $F_{x,i}$ is the longitudinal driving force, where $i=\{fl, fr, rl, rr\}$ represents the wheel location.

A UniTire semi-empirical model is employed to accurately calculate the tire slip power, which calculates slip force and velocity by analyzing forces between slip regions and adhesion in the tire contact patch as follows:

$$\begin{aligned} {V_{sj,i}} = - \frac{{{s_{j,i}}}}{{1 - {s_{j,i}}}}\cdot {V_{j,i}} \cdot \cos {\alpha _i},{F_{sj,i}} = {F_{s,i}} \cdot \frac{{{s_{j,i}}}}{{\sqrt{s_{j,i}^2 + s_{j,i}^2} }}, \end{aligned}$$

(3)

where $F_{s,i}$ is the resultant slip force, $j=\{x, y\}$ is the direction of force.

The power of tire slip $P_{s,t}$ is given as follows:

$$\begin{aligned} {P_{s,t}} = \sum \limits _{i = fl,fr,rl,rr} {F_{sx,i}}{V_{sx,i}} + {F_{sy,i}}{V_{sy,i}}. \end{aligned}$$

(4)

The power loss generated by the in-wheel motor $P_{ml,t}$ can be fitted by a cubic polynomial:

$$\begin{aligned} {P_{ml,t}} = \sum \limits _{i = fl,fr,rl,rr} {a(n_i){T_{m,i}}^3 + b(n_i){T_{m,i}}^2 + c(n_i){T_{m,i}} + d(n_i)}, \end{aligned}$$

(5)

where a, b, c, and d are fitted coefficients determined by motor speed $n_i$.

Applying the Taylor expansion and zero-order hold (ZOH) methods to linearize and discretize the dynamics equation as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} x(k + 1)& = {\bar{A}_k}x(k) + {\bar{B}_{k1}}u(k) + {\bar{B}_{k2}}w(k) + d_{t1}\\ y(k)& = {C_t}x(k) + {D_t}u(k) + d_{t2} \end{array}\right. }, \end{aligned}$$

(6)

where the state vector $x = [{V_x},Y,\beta ,r,\varphi ,\phi ,\dot{\phi },{s_{x,fl}},{s_{x,fr}},$ ${s_{x,rl}},{s_{x,rr}}]^{T}$, the control vector $u = [{T_{m,fl}},{T_{m,fr}},{T_{m,rl}},$ ${T_{m,rr}},{\delta _{f,l}},{\delta _{f,r}},{\delta _{r,l}},{\delta _{r,r}}]^{T}$, the output vector $y= [{V_x},Y,\beta ,\varphi ,r,{P_{ml,t}}$ $,{P_{s,t}}]^{T}$, and $w = {[ {{C_{x,f}},{C_{x,r}},{C_{\alpha ,f}},{C_{\alpha ,r}}}]}^{T}$ represents the bounded disturbance.

Fig. 2.

Framework of the multi-objective integrated path tracking controller.

3 Controller Design

As shown in Fig. 2, a multi-objective controller framework is proposed. To handle multiple objectives in a single layer, an integrated tube-based RMPC is designed, which includes contractive constraints and multi-objective cost functions. The motion states constraints are set to ensure vehicle stability. However, under aggressive driving scenarios, changes in cornering or longitudinal stiffness will cause bounded disturbances to the system.

Equation (6) can be rewritten as a nominal system without disturbance in the actual system. The state variable is donated by z and satisfies $z(0)=x(0)$, while the control variable is represented by v and the output by $\eta =y$. Therefore, the state error between the actual system and the nominal system is calculated as $e(k) = x(k) - z(k)$. The RMPC introduces a linear state feedback for e(k) to compute the control $u(k) = v(k) + Ke(k)$, where the feedback gain K is designed to satisfy $|eig({\bar{A}_{t}}-{\bar{B}_{t1}}K)<1|$, thus ${A_K} = {\bar{A}_{t}}-{\bar{B}_{t1}}K$ is Hurwitz.

Since the disturbance $w \in \mathbb {W}$ is bounded, it is proven to exist a robust positively invariant set $A_K \Gamma \oplus \mathbb {W} \subset \Gamma$, where $\oplus$ is Minkowski sum. Therefore, contractive constraints of the state and control for the nominal system are given as $\mathbb {Z} \in \mathbb {X} \ominus \Gamma ,\mathbb {V} \in \mathbb {U} \ominus K \Gamma$, where $\ominus$ is Pontryagin difference, $\mathbb {X}$ and $\mathbb {U}$ are the polyhedron of the state and control constraints. The cost function of the multi-objective optimization problem is given as:

$$\begin{aligned} {J_{MPC}} = \sum \limits _{i = 1}^{{N_p}} {\left\| {\eta (i) - {\eta _{zref}}(i)} \right\| _w^2} + \left\| {z\left( {{N_p}} \right) - {z_{ref}}\left( {{N_p}} \right) } \right\| _P^2 + \sum \limits _{i = 0}^{{N_c} - 1} {\varDelta v(i)_R^2} \end{aligned}$$

(7)

where, $N_{p}$ and $N_{c}$ donate the prediction and control horizon. w, P,and R represent weight matrices. Therefore, the nominal MPC problem is designed as:

$$\begin{aligned} \min _{\varDelta u} & J_{M P C},\end{aligned}$$

(8a)

$$\begin{aligned} \text{ s.t. } z(k + 1) & = {A_t}(k)z(k) + {B_{t1}}(k)v(k) + d_{t1} ,\end{aligned}$$

(8b)

$$\begin{aligned} |\varDelta v| & \le \varDelta v_{\max } ,\end{aligned}$$

(8c)

$$\begin{aligned} z_i & \in \mathbb {Z}, \quad \forall i \in \left[ 0, N_p-1\right] ,\end{aligned}$$

(8d)

$$\begin{aligned} v_i & \in \mathbb {V}, \quad \forall i \in \left[ 0, N_p-1\right] ,\end{aligned}$$

(8e)

$$\begin{aligned} z_{N_p} & \in \mathbb {Z}_f , \end{aligned}$$

(8f)

where $\mathbb {Z}_{f}$ is set to ensure asymptotic stability and recursive feasibility. A quadratic programming problem (QP) is built, while the active set method is utilized to generate the real control input $u^* = {v}^* + K\left( x-z^*\right)$.

4 HIL Test Results

Considering driving safety under high-speed conditions and the reliability of the steer-by-wire system (SBW), HIL tests are employed to evaluate the performance of the proposed strategy. A double lane change (DLC) case is designed. The friction coefficient of the road is 0.85, the target speed is 100 km/h. Optimization objectives include vehicle stability (VS), tire slip energy (TSE), and motor energy loss (MEL). The state-of-the-art approaches are compared to show the advantages of the proposed method: 1) Hierarchical MPC (HMPC), a hierarchical control strategy introduced in [7]; 2) Single-layer MPC (SMPC), a strategy that employs a single layer instead of a hierarchical structure; 3) Single-layer robust MPC (SRMPC), the strategy proposed in this work.

Fig. 3.

Comparison of torques and steering angles under DLC case. (a)–(d) each wheel torque; (e)–(h) each wheel steering angle.

Fig. 4.

Comparison of reference path tracking errors under DLC case.

Fig. 5.

Comparison of the results under DLC case. (a) longitudinal speed; (b) side slip angle; (c) yaw rate; (d) sideslip angle rate - sideslip angle phase diagram; (e)–(f) energy consumption; (g)–(h) tire stiffness changes.

Fig. 3–5 shows the inputs and outputs. Under aggressive maneuvers, the HMPC tends to input higher torques and track the reference path better. However, its speed-holding ability and energy efficiency are underperformed. In contrast, the single-layer strategies input smaller torques due to the optimization of multiple objectives and reduce more energy consumption. Additionally, the approach proposed in this work shows the best performance. Compared with the other strategies, it maintains longitudinal speed 2.8% and 1.7% higher, reduces 43.3% and 23.3% motor energy loss, 12.4% and 8.3% tire slip energy dissipation. Meanwhile, as shown in Fig. 5(b)–(d), the vehicle applying the proposed method has the best stability. This is achieved by effectively addressing the model mismatch shown in Fig. 5(g)–(h).

5 Conclusion

In this work, an RMPC-based integrated multi-objective path tracking framework is devised and validated in HIL. Experiment results demonstrate that the proposed strategy can effectively enhance vehicle stability while improving energy efficiency. Compared with the state-of-the-art, the longitudinal speed is maintained 1.7% higher, while the motor energy loss and tire slip energy are reduced by 23.3% and 8.3% under the DLC case, respectively. HIL test shows that the designed RMPC-based controller can address the model mismatch well and further enhance vehicle performance. In the future, the proposed strategy will be verified under more complex maneuvers.