﻿ 四驱混合动力轿车转弯工况路径跟踪控制
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 同济大学学报(自然科学版)  2019, Vol. 47 Issue (5): 695-703.  DOI: 10.11908/j.issn.0253-374x.2019.05.015 0

### 引用本文

ZHAO Zhiguo, ZHOU Liangjie, WANG Kai. Path Tracking Control of Four-Wheel Drive Hybrid Electric Car in Steering[J]. Journal of Tongji University (Natural Science), 2019, 47(5): 695-703. DOI: 10.11908/j.issn.0253-374x.2019.05.015

### 文章历史

1. 同济大学 汽车学院，上海 201804;
2. 同济大学 新能源汽车工程中心，上海 201804

Path Tracking Control of Four-Wheel Drive Hybrid Electric Car in Steering
ZHAO Zhiguo 1,2, ZHOU Liangjie 1,2, WANG Kai 1,2
1. School of Automotive Studies, Tongji University, Shanghai 201804, China;
2. Clean Energy Automotive Engineering Center, Tongji University, Shanghai 201804, China
Abstract: A path tracking control strategy integrating lateral and longitudinal control under steering conditions is proposed for a four-wheel hybrid vehicle. Based on vehicle dynamics and motion model, a driver preview steering model based on trajectory tracking error is designed. The desired speed is determined by using fuzzy controller, and the distribution of torque is optimized. The model predictive controller of vehicle speed and trajectory tracking is designed, and a driving simulator integrating CarSim and MATLAB/Simulink co-simulation platform is built to simulate the control strategy. The results show that the tracking effect of vehicle path and speed during steering is good, which meets the requirement of path tracking in steering condition.
Key words: hybrid vehicle    path tracking    predictive control    driver model

1 四驱混合动力系统及动力学模型 1.1 四驱混合动力系统结构

 图 1 四驱混合动力轿车动力系统结构 Fig.1 Power system structure of hybrid vehicle
1.2 七自由度车辆动力学模型

 $\left\{ \begin{array}{l} \begin{array}{*{20}{c}} {{{\dot v}_x} = \gamma {v_y} + \frac{1}{m}\left[ {\left( {{F_{x1}} + {F_{x2}}} \right)\cos \delta - \left( {{F_{y1}} + } \right.} \right.}\\ {\left. {\left. {{F_{y2}}} \right)\sin \delta + {F_{x3}} + {F_{x4}} - \frac{1}{2}{C_{\rm{d}}}A\rho v_x^2} \right]} \end{array}\\ {{\dot v}_y} = - \gamma {v_x} + \frac{1}{m}\left[ {\left( {{F_{x1}} + {F_{x2}}} \right)\sin \delta + \left( {{F_{y1}} + } \right.} \right.\\ \;\;\;\;{F_{y2}})\cos \delta + {F_{y3}} + {F_{y4}}]\\ \dot \gamma = \frac{1}{{{I_z}}}\left\{ {\left[ {\left( {{F_{x1}} + {F_{x2}}} \right)\sin \delta + \left( {{F_{y1}} + {F_{y2}}} \right)\cos \delta } \right]{l_{\rm{f}}} - } \right.\\ \;\;\;\;\;\;\left( {{F_{y3}} + {F_{y4}}} \right){l_{\rm{r}}} - \left[ {\left( {{F_{x1}} - {F_{x2}}} \right)\cos \delta - } \right.\\ \;\;\;\;\;\;\left( {{F_{y1}} - {F_{y2}}} \right)\sin \delta ]\frac{{{d_1}}}{2} - \left( {{F_{x3}} - {F_{x4}}} \right)\frac{{{d_2}}}{2} + \\ \;\;\;\;\;\;\left. {{M_{z1}} + {M_{z2}} + {M_{z3}} + {M_{z4}}} \right\} \end{array} \right.$ (1)
 图 2 车辆模型 Fig.2 Model of vehicle

1.3 电池模型

 $\begin{array}{*{20}{c}} {{U_{\rm{o}}}\left( t \right) = {V_{\rm{o}}} + {R_0}I\left( t \right) + {U_{\rm{c}}}\left( t \right)}\\ {I\left( t \right) = \frac{{{U_{\rm{c}}}\left( t \right)}}{{{R_1}}} + {C_1}\frac{{{\rm{d}}{U_{\rm{c}}}\left( t \right)}}{{{\rm{d}}t}}} \end{array}$ (2)

1.4 电机模型

 ${T_{{\rm{r}}1}} = \frac{1}{{1 + {t_1}s}}{T_{{\rm{el}}}}$ (3)
 $T_{\mathrm{r} 2}=\frac{1}{1+t_{2} s} T_{\mathrm{e} 2}$ (4)

1.5 轮胎模型

 ${s_{x{\rm{i}}}} = \left\{ {\begin{array}{*{20}{c}} {\frac{{{\omega _{\rm{i}}}{R_{\rm{d}}}\cos {\alpha _{\rm{i}}} - {v_{\rm{i}}}}}{{{v_{\rm{i}}}}},}&{{v_{\rm{i}}} \ge {\omega _{\rm{i}}}{R_{\rm{d}}}\cos {\alpha _{\rm{i}}}}\\ {\frac{{{\omega _{\rm{i}}}{R_{\rm{d}}}\cos {\alpha _{\rm{i}}} - {v_{\rm{i}}}}}{{{\omega _{\rm{i}}}{R_{\rm{d}}}\cos {\alpha _{\rm{i}}}}},}&{{v_{\rm{i}}} < {\omega _{\rm{i}}}{R_{\rm{d}}}\cos {\alpha _{\rm{i}}}} \end{array}} \right.$ (5)
 ${s_{y{\rm{i}}}} = \left\{ \begin{array}{l} \left( {1 + {s_{x{\rm{i}}}}} \right)\tan {\alpha _{\rm{i}}},\;\;\;\;{v_{\rm{i}}} \ge {\omega _{\rm{i}}}{R_{\rm{d}}}\cos {\alpha _{\rm{i}}}\\ \tan {\alpha _{\rm{i}}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{v_{\rm{i}}} < {\omega _{\rm{i}}}{R_{\rm{d}}}\cos {\alpha _{\rm{i}}} \end{array} \right.$ (6)

2 路径跟踪控制 2.1 控制系统架构

 图 3 路径跟踪控制系统架构 Fig.3 Control system structure of path tracking
2.2 前轮期望转角决策及其滑模变结构跟踪控制 2.2.1 前轮期望转角决策

 图 4 驾驶员预瞄模型 Fig.4 Preview model of driver

 $\left. \begin{array}{l} {{\dot y}_{\rm{e}}} = - {v_y} - {L_{\rm{d}}}\gamma + {v_x}{\phi _{\rm{e}}}\\ {{\dot \phi }_{\rm{e}}} = {v_x}\rho - \gamma \end{array} \right\}$ (7)

2.2.2 前轮转角滑模变结构跟踪控制

 $\dot{x}=f(x)+b u+d$ (8)

 $s=\dot{y}_{\mathrm{e}}+\lambda y_{\mathrm{e}}$ (9)

 $\dot{s}=-\eta g(s)$ (10)
 $g(s)=\left\{\begin{array}{l}{1, s>\Delta} \\ {k s, s \leqslant|\Delta|, k=1 / \Delta} \\ {-1, s<-\Delta}\end{array}\right.$ (11)

 $\dot{s}=\ddot{y}_{\mathrm{e}}+\lambda \dot{y}_{\mathrm{e}}=-\eta g(s)$ (12)

 $\left[ \begin{array}{l}{x_{\mathrm{e}}} \\ {y_{\mathrm{e}}} \\ {\phi_{\mathrm{e}}}\end{array}\right]=\left[ \begin{array}{ccc}{\cos \phi_{C}} & {\sin \phi_{C}} & {0} \\ {-\sin \phi_{C}} & {\cos \phi_{C}} & {0} \\ {0} & {0} & {1}\end{array}\right] \left[ \begin{array}{l}{X_{P}-X_{C}} \\ {Y_{P}-Y_{C}} \\ {\phi_{P}-\phi_{C}}\end{array}\right]$ (13)

 $\dot{\boldsymbol{x}}=\boldsymbol{A}_{\mathrm{v}} x+\boldsymbol{B}_{\mathrm{v}} \delta+\boldsymbol{W}_{\rho}$ (14)

 $\begin{array}{l} {\mathit{\boldsymbol{A}}_{\rm{v}}} = \\ \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}&0&0&0\\ {{A_{21}}}&{{A_{22}}}&0&0&0\\ 0&0&1&0&0\\ { - \left( {{A_{11}} + {L_{\rm{d}}}{A_{21}}} \right)}&{ - \left( {{A_{12}} + {L_{\rm{d}}}{A_{22}} + {v_x}} \right)}&0&0&0\\ 0&{ - 1}&0&0&0 \end{array}} \right]; \end{array}$
 ${A_{11}} = \frac{{ - {C_{\rm{f}}} - {C_{\rm{r}}}}}{{m{v_x}}},{A_{12}} = \frac{{b{C_{\rm{r}}} - a{C_{\rm{f}}}}}{{m{v_x}}} - {v_x};$
 ${A_{21}} = \frac{{b{C_{\rm{r}}} - a{C_{\rm{f}}}}}{{{I_z}{v_x}}},{A_{22}} = - \frac{{{a^2}{C_{\rm{f}}} + {b^2}{C_{\rm{r}}}}}{{{I_z}{v_x}}};$
 $\boldsymbol{B}_{\mathrm{v}}=\left[ \begin{array}{lll}{B_{1}} & {B_{2}} & {-\left(B_{1}+L_{\mathrm{d}} B_{2}\right)} & {0} & {0}\end{array}\right]^{\mathrm{T}};$
 $B_{1}=-\frac{C_{\mathrm{f}}}{m}, B_{2}=\frac{a C_{\mathrm{f}}}{I_{z}};$
 $\boldsymbol{W}=\left[ \begin{array}{lllll}{0} & {0} & {0} & {v_{x}^{2}} & {v_{x}}\end{array}\right]^{\mathrm{T}}$

 $\begin{array}{l} {\delta _{\rm{d}}} = u = - \frac{1}{{{B_1} + {L_{\rm{d}}}{B_2}}}\left[ {\left( {{A_{11}} + {L_{\rm{d}}}{A_{21}} + \lambda } \right){v_y} + } \right.\\ \;\;\;\;\;\;\;\left( {{v_x}\rho - \gamma } \right){v_x} - \lambda {v_x}{\phi _{\rm{e}}} - \eta g(s) + \\ \;\;\;\;\;\;\;\left( {{A_{12}} + {L_{\rm{d}}}{A_{22}} + \lambda {L_{\rm{d}}}} \right)\gamma ] \end{array}$ (15)
2.3 期望车速模糊决策

 图 5 模糊推理规则曲面 Fig.5 Rule surface of fuzzy control
3 速度跟踪MPC控制器设计

MPC控制方法鲁棒性强，约束处理方便，其在车辆运动控制中已广泛应用[10-11].采用离散形式对车辆纵向运动学特征进行描述，取车辆纵向车速和加速度为系统状态变量，即$\boldsymbol{x}(k)=[v(k), a(k)]^{\mathrm{T}}$，假设在很小的采样时间内，车辆的运动速度变化不大，车辆运动的状态空间表达为

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{x}}(k + 1) = {\mathit{\boldsymbol{A}}_{\rm{m}}}\mathit{\boldsymbol{x}}(k) + {\mathit{\boldsymbol{B}}_{\rm{m}}}\mathit{\boldsymbol{u}}(k)}\\ {{\mathit{\boldsymbol{A}}_{\rm{m}}} = \left[ {\begin{array}{*{20}{c}} 1&{{T_{\rm{s}}}}\\ 0&{1 - \frac{{{T_{\rm{s}}}}}{t}} \end{array}} \right],{\mathit{\boldsymbol{B}}_{\rm{m}}} = \left[ {\begin{array}{*{20}{c}} 0\\ {\frac{{{T_{\rm{s}}}}}{t}} \end{array}} \right]} \end{array}$ (16)

 $y(k) = \mathit{\boldsymbol{Cx}}(k) - z(k)$ (17)

 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat X}}}_p}\left( {k + p\left| k \right.} \right) = {{\mathit{\boldsymbol{\bar A}}}_{\rm{m}}}\mathit{\boldsymbol{x}}\left( k \right) + {{\mathit{\boldsymbol{\bar B}}}_{\rm{m}}}\mathit{\boldsymbol{U}}\left( {k + m} \right) + \mathit{\boldsymbol{\bar H}}e\left( k \right)}\\ {{{\mathit{\boldsymbol{\hat Y}}}_p}\left( {k + p\left| k \right.} \right) = \mathit{\boldsymbol{\bar Cx}}\left( k \right) + \mathit{\boldsymbol{\bar DU}}\left( {k + m} \right) + }\\ {\mathit{\boldsymbol{\bar F}}e\left( k \right) - \mathit{\boldsymbol{\bar Z}}} \end{array}$ (18)

 $y_{\mathrm{r}}(k+i)=\varphi^{i} y(k)$ (19)

 $\begin{array}{*{20}{c}} {J = \sum\limits_{i = 1}^p {{{\left[ {{{\mathit{\boldsymbol{\widehat y}}}_p}(k + i|k) - {y_{\rm{r}}}(k + i|k)} \right]}^{\rm{T}}}} \mathit{\boldsymbol{Q}} \cdot }\\ {\left[ {{{\mathit{\boldsymbol{\widehat y}}}_p}(k + i|k) - {y_{\rm{r}}}(k + i|k)} \right] + }\\ {\sum\limits_{i = 0}^{m - 1} u {{(k + i)}^{\rm{T}}}\mathit{\boldsymbol{R}}u(k + i)} \end{array}$ (20)

 $\left\{\begin{array}{l}{v_{\min } \leqslant v(k) \leqslant v_{\max }} \\ {a_{\min } \leqslant a(k) \leqslant a_{\max }} \\ {u_{\min } \leqslant u(k) \leqslant u_{\max }}\end{array}\right.$ (21)

 $m{a_x} = \frac{{{i_{\rm{g}}}}}{{{R_{\rm{r}}}}}{T_{\rm{d}}} - {F_x} - \frac{1}{2}{C_{\rm{d}}}A\rho v_x^2 - mg\sin \theta$ (22)

 $m a_{x}=-\frac{1}{R_{\mathrm{r}}} T_{\mathrm{b}}-F_{x}-\frac{1}{2} C_{\mathrm{d}} A_{\rho} v^{2}-m g \sin \theta$ (23)

4 转弯工况下的转矩优化分配策略 4.1 转矩的分配关系及约束条件

 $\left\{\begin{array}{l}{F_{x}=F_{x 1}+F_{x 2}+F_{x 3}+F_{x 4}} \\ {M_{\mathrm{a}}=\frac{l_{\mathrm{w}}}{2}\left(-F_{x 1}+F_{x 2}-F_{x 3}+F_{x 4}\right)}\end{array}\right.$ (24)

 $\left\{ \begin{array}{l} {F_{x1}} = - {r^{ - 1}}{T_{{\rm{hfl}}}}\\ {F_{x2}} = - {r^{ - 1}}{T_{{\rm{hfr}}}}\\ {F_{x3}} = {r^{ - 1}}\left( {{T_{{\rm{mrl}}}} - {T_{{\rm{hrl}}}}} \right)\\ {F_{x4}} = {r^{ - 1}}\left( {{T_{{\rm{mrr}}}} - {T_{{\rm{hrr}}}}} \right) \end{array} \right.$ (25)

 $\mathit{\boldsymbol{v}} = \mathit{\boldsymbol{Bu}}$ (26)

 $\boldsymbol{B}=\left[ \begin{array}{ccccc}{\frac{1}{r}} & {\frac{1}{r}} & {-\frac{1}{r}} & {-\frac{1}{r}} & {-\frac{1}{r}} & {-\frac{1}{r}} \\ {-\frac{l_{\mathrm{w}}}{2 r}} & {\frac{l_{\mathrm{w}}}{2 r}} & {\frac{l_{\mathrm{w}}}{2 r}} & {-\frac{l_{\mathrm{w}}}{2 r}} & {\frac{l_{\mathrm{w}}}{2 r}} & {-\frac{l_{\mathrm{w}}}{2 r}}\end{array}\right]$ (27)

 $\max \left( {{T_{{\rm{m}}i\_{\rm{min}}}}, - {T_{{\rm{road}}\_i}}} \right) \le {T_{{\rm{m}}i}} \le 0$ (28)
 $\begin{array}{l} \max \left( {{T_{{\rm{h}}i\_min}}, - {T_{{\rm{road}}\_i}}} \right) - \\ \;\;\;\;\;\;\;\max \left( {{T_{{\rm{m}}i\_\min }}, - {T_{{\rm{road}}\_i}}} \right) \le {T_{{\rm{h}}i}} \le 0 \end{array}$ (29)

4.2 基于加权最小二乘法的转矩分配问题

 $\min J = \left\| {{\mathit{\boldsymbol{W}}_{\rm{v}}}\left( {\mathit{\boldsymbol{Bu}} - {\mathit{\boldsymbol{v}}_{\rm{d}}}} \right)} \right\|_2^2 + \zeta \left\| {{\mathit{\boldsymbol{W}}_{\rm{u}}}\left( {\mathit{\boldsymbol{u}} - {\mathit{\boldsymbol{u}}_{\rm{d}}}} \right)} \right\|_2^2$ (30)

 $\begin{array}{*{20}{c}} {J = \left\| {{\mathit{\boldsymbol{W}}_{\rm{v}}}\left( {\mathit{\boldsymbol{Bu}} - {\mathit{\boldsymbol{v}}_{\rm{d}}}} \right)} \right\|_2^2 + \zeta \left\| {{\mathit{\boldsymbol{W}}_{\rm{u}}}\left( {\mathit{\boldsymbol{u}} - {\mathit{\boldsymbol{u}}_{\rm{d}}}} \right)} \right\|_2^2 = }\\ {\left\| {\underbrace {\left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{W}}_{\rm{v}}}\mathit{\boldsymbol{B}}}\\ {{\zeta ^{\frac{1}{2}}}{\mathit{\boldsymbol{W}}_{\rm{u}}}} \end{array}} \right)}_\mathit{\boldsymbol{A}}\mathit{\boldsymbol{u}} - \underbrace {\left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{W}}_{\rm{v}}}\mathit{\boldsymbol{v}}}\\ {{\zeta ^{\frac{1}{2}}}{\mathit{\boldsymbol{W}}_{\rm{u}}}{\mathit{\boldsymbol{u}}_{\rm{d}}}} \end{array}} \right)}_\mathit{\boldsymbol{b}}} \right\|_2^2} \end{array}$ (31)

 $\min \left\| {\mathit{\boldsymbol{Au}} - \mathit{\boldsymbol{b}}} \right\|_2^2$ (32)

4.3 WLS控制分配问题的求解

 $\mathop {\min }\limits_x \frac{1}{2}{\mathit{\boldsymbol{x}}^{\rm{T}}}\mathit{\boldsymbol{Hx}} + {\mathit{\boldsymbol{c}}^{\rm{T}}}\mathit{\boldsymbol{x}}$ (33)
 $\begin{array}{*{20}{c}} {{\rm{s}}.{\rm{t}}.\;\;\;{\mathit{\boldsymbol{c}}_i}\left( \mathit{\boldsymbol{x}} \right) = 0,i = 1,2, \cdots ,n}\\ {{g_j}\left( \mathit{\boldsymbol{x}} \right) \ge 0,j = 1,2, \cdots ,m} \end{array}$

 图 6 积极集法流程图 Fig.6 Flowchart of active set method

5.2 仿真结果

 图 8 路径跟踪仿真结果 Fig.8 Result of path tracking simulation
6 驾驶模拟器台架试验

 图 9 驾驶模拟器 Fig.9 Driving simulator

 图 10 路径跟踪控制驾驶模拟器试验结果 Fig.10 Control strategy validation by simulator

 图 11 换挡路径跟踪控制试验结果 Fig.11 Control strategy validation of shifting
7 结论

(1) 基于驾驶员模型设计了滑模变结构转向控制器实现横向路径跟踪，并通过仿真试验证明了其具有良好的跟踪效果.

(2) 根据道路曲率和路径跟踪误差设计了期望车速的模糊控制器，并采用MPC算法实现车速跟踪控制.所提出的基于MPC算法的车辆纵向运动控制，在保证控制目标精确跟踪的同时，通过约束条件兼顾了车辆乘坐舒适性.