System design of heterogeneous vehicle platoon based on multi-delay proportional-retarded controller
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摘要: 针对异构车辆队列，为有效利用时滞提升系统响应速度等控制性能，引入多时滞设计了一种比例-时滞(PR)控制器；使用隐函数定理及柯西-黎曼方程对车辆队列各子系统的特征方程进行了分析，提出了子系统的最右侧极点配置方法；确定了最右侧极点的可配置范围，并给出了关于控制器增益和时滞的设计指导规则；为了逐次增强各跟随车所对应子系统的稳定性，合理分离了各子系统的极点，提出了整个车辆队列的最右侧极点配置方法，并给出了弦稳定的充分条件。仿真结果表明：设计的多时滞PR控制器及所提出的最右侧极点配置方法，可精确地将异构车辆队列的极点配置在期望位置，并能同时保证车辆队列的内部稳定性与弦稳定性，且最右侧极点越小，系统响应速度越快；相较于单时滞PR控制器，该方法灵活地分离了各子系统的最右侧极点，并将系统调节时间减少了2.45%；与传统的比例-微分控制器(PD)相比，PR控制器在车辆乘坐舒适度和燃油效率方面有显著提升，改进幅度达到2~4个数量级；相同频率扰动下，PR与PD控制器作用下的加速度和控制输入的振幅比均始终小于1/3，当扰动频率为9Hz时，振幅比降至0.031。可见，所设计的多时滞PR控制器在响应速度、极点分离与抑制干扰等方面均有显著优势。Abstract: To effectively utilize delays to enhance system response speed and other control performances, a proportional-retarded (PR) controller was designed for the heterogeneous vehicle platoon by introducing multi-delays. The characteristic equations of each subsystem of the vehicle platoon were analyzed by applying the implicit function theorem and Cauchy-Riemann equation, and the right-most pole assignment method was proposed for these subsystems. The available assignment ranges of the right-most poles were determined, and the guidelines for designing the controller gains and delays were given. To gradually increase the stability of the subsystems that corresponded to the follower vehicles, the poles of subsystems were separated reasonably, and a right-most pole assignment method was proposed for the entire vehicle platoon. Moreover, sufficient conditions for string stability were derived. Simulation results indicate that the designed multi-delay PR controller and the proposed right-most pole assignment method can accurately assign the pole of the heterogeneous vehicle platoon at the desired location, and both the internal stability and string stability of the vehicles are ensured. Moreover, a smaller right-most pole indicates a faster system response speed. Compared to a single-delay PR controller, the right-most poles of subsystems are flexibly separated via this method, the system setting time reduces by 2.45%. Compared with traditional proportional-derivative (PD) controllers, the PR controller shows significant improvements in ride comfort and fuel economy, with enhancement ranging from 2-4 orders of magnitude. Under the disturbances of the same frequency, the amplitude ratios of the acceleration and control input for both the PR and PD controllers remain consistently below 1/3, dropping to 0.031 when the frequency of the disturbances is 9 Hz. It can be seen that the designed multi-delay PR controller has significant advantages in response speed, pole separation, and disturbance suppression.
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图 2 试验1中第1组参数下车辆队列的极点分布与误差
Figure 2. Pole distribution and errors of vehicle platoon for first group parameters in experiment 1
图 3 试验1中第2组参数下车辆队列的极点分布与误差
Figure 3. Pole distribution and errors of vehicle platoon for second group parameters in experiment 1
图 4 试验2中单时滞PR控制器作用下车辆队列的极点分布与误差
Figure 4. Pole distribution and errors of vehicle platoon with single-delay PR controller in experiment 2
图 6 试验4中PR控制器作用下车辆队列的状态及控制输入
Figure 6. States and control inputs of vehicle platoon with PR controller in experiment 4
图 7 试验4中PD控制器作用下车辆队列的状态及控制输入
Figure 7. States and control inputs of vehicle platoon with PD controller in experiment 4
图 8 试验4中PR与PD控制器作用下车辆队列的加速度与控制输入频谱
Figure 8. Frequency spectrums of accelerations and control inputs of vehicle platoons with PR controller and PD controller in experiment 4
表 1 kpi、kri和τi关于γi的变化趋势
Table 1. Variation tendencies of kpi, kri and τi with respect to γi
参量 变化趋势 左边界 右边界 γi 减小 -1/(3Ti) 0 kpi 增大 0 +∞ kri 增大 0 +∞ τi 减小 0 +∞ 
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表 2 PR控制器参数
Table 2. Parameters of PR controller
试验组别 控制器参数 跟随车1 跟随车2 跟随车3 跟随车4 跟随车5 第1组 kpi 0.023 8 0.039 5 0.060 0 0.086 1 0.117 7 kri 0.018 7 0.031 4 0.048 3 0.070 1 0.967 0 τi/s 4.012 0 3.008 0 2.366 0 1.921 0 1.603 0 第2组 kpi 0.148 8 0.216 5 0.296 9 0.392 9 0.498 9 kri 0.127 1 0.188 7 0.261 8 0.349 7 0.447 4 τi/s 1.269 0 0.984 0 0.800 0 0.667 0 0.574 0
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表 3 单时滞PR控制器参数
Table 3. Parameters of single-delay PR controller
控制器参数 跟随车1 跟随车2 跟随车3 跟随车4 跟随车5 kpi 0.276 0.403 0.594 0.900 1.452 kri 0.233 0.348 0.524 0.813 1.344
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表 4 PD控制器参数
Table 4. Parameters of PD controller
控制器参数 跟随车1 跟随车2 跟随车3 跟随车4 跟随车5 kpi 0.080 0.082 0.081 0.080 0.078 kdi 0.356 0.396 0.439 0.474 0.510
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表 5 加速度与控制输入的振幅比
Table 5. Amplitude ratios of acceleration and control input
频率/Hz 1 3 5 7 9 振幅比 加速度绝对值/(m·s-2) 0.312 0.143 0.066 0.034 0.031 控制输入绝对值 0.320 0.143 0.066 0.034 0.031
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表 6 异构车辆队列的乘坐舒适度与燃油经济性
Table 6. Ride comforts and fuel economies of heterogeneous vehicle platoon
跟随车 乘坐舒适度 燃油经济性 PR控制器 PD控制器 PR控制器 PD控制器 1 0.003 3 58.358 0 0.331 8 70.169 4 2 0.021 7 77.909 5 2.030 3 96.475 8 3 0.046 6 106.663 0 2.096 2 128.560 0 4 0.096 2 144.659 0 4.528 8 177.310 0 5 0.204 9 195.254 0 12.511 8 244.370 0
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