Stability of PID control system for vehicle platoon with input delay and communication delay
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摘要: 针对含输入时延与通信时延的车辆队列PID控制系统，分析了其内部稳定性和队列稳定性，研究了内部稳定的充要条件，求解了完整、精确的时延边界；在内部稳定性分析中，考虑输入时延与通信时延影响下车辆队列PID控制系统为中立型双时延系统的特点，结合Rekasius代换和劳斯表，提出了关于中立算子的系统强稳定充要条件；在此基础上，为了便于PID参数的快速选取，推导了一种形式更为简练的系统强稳定充分条件；在强稳定条件下，基于特征根聚类法求解了系统完整、精确的时延边界；针对具有奇数辆跟随车的车辆队列，推导了无关车辆队列规模的输入时延上界；在队列稳定性分析中，为了保证干扰和误差沿车辆队列向后传播不发散，分析了车间误差传递函数，给出了双时延影响下队列稳定的充分条件。仿真结果表明：在含输入时延与通信时延的分布式PID控制器作用下，车辆队列控制系统可同时保证内部稳定和队列稳定；车间状态误差可在15 s内快速减小并趋近于零；在所有车辆恒速行驶时，车间保持50 m期望安全距离；在领航车以0.5 m·s-2加速和0.8 m·s-2减速时，跟随车的速度和加速度随领航车变化，并在领航车速度稳定时一致；车辆队列在不同行驶工况下，由领航车加、减速引起的车间位置误差小于0.2 m，且沿车辆队列向后传播不发散。Abstract: The internal stability and string stability of the PID control system were analyzed for vehicle platoon with input delay and communication delay, the sufficient and necessary conditions of the internal stability were emphatically studied, and the exhaustive and exact time delay margins were derived. In the internal stability analysis, considering that the PID control system for vehicle platoon is a neutral time delay system with input delay and communication delay, the sufficient and necessary strong stability conditions were proposed by analyzing the stability of the neutral operator via Rekasius substitution and Routh table. In order to facilitate selecting the PID parameters, a sufficient condition with a more concise form was derived. Then, the clustering method of characteristic roots was applied to obtain the exhaustive and exact time delay margins. Considering the vehicle platoon with an odd number of following vehicles, the upper bound of the input delay, which was independent of the scale of the vehicle platoon, was derived. In order to ensure that the interference and error propagated backward along the vehicle platoon without divergence, the error transfer function among the vehicles was analyzed, and the sufficient condition of string stability under the influence of two delays was given. Simulation results show that the internal stability and string stability of vehicle platoon can be guaranteed simultaneously by the distributed PID controllers under communication delay and input delay. The state errors quickly converge to zero within 15 s. When the velocities of the vehicles are constant, an desired safe distance maintains 50 m between the successive vehicles. When the leader vehicle accelerates at 0.5 m·s-2 and decelerates at 0.8 m·s-2, the velocities and accelerations of the following vehicles asymptotically change with those of the leader and are consistent with the leader when the leader's velocity is constant. Under the different driving conditions, the spacing errors caused by the acceleration and deceleration of the leader are less than 0.2 m, and propagate backward along vehicle platoon without divergence. 1 tab, 11 figs, 36 refs.
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图 4 试验2中车辆队列各个子系统的时延边界
Figure 4. Time delay margins of subsystems of vehicle platoon in Example 2
图 5 试验2中车辆队列的时延边界与时延谱域
Figure 5. Time delay margins and spectral domains of vehicle platoon in Example 2
图 6 试验2中点a处车辆队列的位置误差和速度
Figure 6. Spacing errors and velocities of vehicle platoon at point a in Example 2
图 7 试验2中点b处车辆队列的位置误差和速度
Figure 7. Spacing errors and velocities of vehicle platoon at point b in Example 2
图 8 试验2中点c处车辆队列的位置误差和速度
Figure 8. Spacing errors and velocities of vehicle platoon at point c in Example 2
图 9 试验3中车辆队列的状态与位置误差
Figure 9. States and spacing errors of vehicle platoon in Example 3
图 10 试验4中车辆队列的状态与位置误差
Figure 10. States and spacing errors of vehicle platoon in Example 4
图 11 试验5中车辆队列的状态与位置误差
Figure 11. States and spacing errors of vehicle platoon in Example 5
表 1 车辆队列参数和分布式PID控制器参数
Table 1. Parameters of vehicle platoon and distributed PID controller
η/s d0/m kb kf kl kPr kPv kPa kIr kIv kIa kDr kDv kDa 0.79 50 0.794 0.026 1.000 1.300 3.800 1.293 0.907 0.221 0.197 0.213 0.047 0.051 
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